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Ferroelectric gate control of Rashba–Dresselhaus spin–orbit coupling in ferromagnetic semiconductor (Zn, Co)O Cite as: Appl. Phys. Lett. 119, 012403 (2021); doi: 10.1063/5.0047430 Submitted: 13 February 2021 .Accepted: 23 June 2021 . Published Online: 7 July 2021 Maoxiang Fu,1Jiahui Liu,1 Qiang Cao,2 Zhen Zhang,1Guolei Liu,1,a) Shishou Kang,1 Yanxue Chen,1 Shishen Yan,1,2Liangmo Mei,1and Zhen-Dong Sun1,3,a) AFFILIATIONS 1School of Physics, Shandong University, Jinan 250100, China 2Spintronics Institute, University of Jinan, Jinan 250022, China 3School of Physics and Electrical Engineering, Kashi University, Kashgar 844006, China a)Authors to whom correspondence should be addressed: liu-guolei@sdu.edu.cn and zdsun@sdu.edu.cn ABSTRACT In this paper, we demonstrate the ferroelectric gate control of Rashba–Dresselhaus spin–orbit coupling (R–D SOC) in a hybrid heterostructure consisting of a ferromagnetic semiconductor channel (Zn, Co)O(0001) and a ferroelectric substrate PMN-PT(111). The R–D SOC causes a transverse spin current via the charge-spin conversion, which results in unbalanced transverse spin and charge accumulations due to the spin-polarized band in the ferromagnetic (Zn, Co)O channel. By the reversal of gated ferroelectric polarization, we observed 55%modulation of the R–D SOC correlated Hall resistivity to the magnetization correlated anomalous Hall resistivity and 70% modulation of thelow-field magnetoresistance at 50 K. Our experimental results pave a way toward semiconductor-based spintronic-integrated circuits with anultralow power consumption in ferromagnetic semiconductors. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0047430 The electric field control of ferromagnetism and spin phenomena has been intensively pursued in an information technique, since it offers a promising method for ultra-low power spin manipulation. 1–3 The electric field control of magnetic properties has been demon-strated in several classes of materials such as ferromagnetic semicon-ductors (In, Mn)As, 4(Ga, Mn)As,5–13(In, Fe)Sb,14(Ti, Co)O 2,15 ultrathin ferromagnetic metals,16–20and complex oxides.3,21,22By applying the gate voltage on the ferromagnetic semiconductor chan- nel, the accumulated (or depleted) carriers enhance (or suppress) the carrier density leading to the modulation of magnetization and theCurie temperature, and this type of electric field control of ferromag- netism can be attributed to the carrier-mediated ferromagne- tism. 4,6,14,15On the other hand, on device concept of the spin field effect transistor (spin-FET),23,24the electric field controlled Rashba25 and Dresselhaus26(R–D) spin–orbit coupling (SOC) is an effective and essential way to generate and manipulate a spin-polarized current in nanostructures without an external magnetic field. The Rashba SOC is due to the structure inversion asymmetry, and the Dresselhaus SOC is due to the bulk inversion asymmetry. The electric field con- trolled Rashba as well as Dresselhaus SOC has been demonstrated andextensively studied in non-magnetic semiconductor heterostructuresin the past decades. 24However, the electric field controlled R–D SOC has not yet been realized experimentally in the materials of the ferro- magnetic semiconductor. In this paper, we utilize the gated ferroelectric polarization to control the R–D SOC in the hybrid heterostructure (Zn, Co)O(0001)/ PMN-PT(111). The ferromagnetic semiconductor (Zn, Co)O films are n-type conductivity, and a space charge region is formed by applying t h eg a t ev o l t a g eo nt h ef e r r o e l e c t r i cs u b s t r a t eP M N - P T .T h ep o l eo f ferroelectric polarization induces the built-in electric field inside the(Zn, Co)O channel and also causes the variation of charge density, as shown in Fig. 1(a) . The Rashba spin–orbit coupling is ascribed to the structure inversion asymmetry of the (Zn, Co)O/PMN-PT hetero-strocture and the time inversion asymmetry of ferromagnetism in (Zn, Co)O. The Dresselhaus spin–orbit coupling is ascribed to the bulk inversion asymmetry of the wurtzite structure ZnO. The Hamiltonian by using the k/C1pmethod can be written as H R¼aRðrxky/C0rykxÞ andHD¼c½bkz/C0ðk2 xþk2 yÞ/C138ðrxky/C0rykxÞ,27,28where pxðyÞandrxðyÞ are the components of the electronic momentum operator and the spin Pauli matrices, respectively, and aRand bDare Rashba and Dresselhaus parameters, respectively. Both of the Rashba andDresselhaus SOCs are coexisted. Figure 1(b) shows the diagram of Appl. Phys. Lett. 119, 012403 (2021); doi: 10.1063/5.0047430 119, 012403-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplspin-split of R–D SOC in the k-space without magnetization.24,27It is noted that there is a lack of spin-momentum locking due to the pre- sentation of ferromagnetic exchange coupling in ferromagnetic (Zn, Co)O, though it coexists with the R–D exchange coupling. On the reversal of gated ferroelectric polarization, the R–D SOC results in two aspects: (1) the modulation of the spin-polarized band structure in fer- romagnetic (Zn, Co)O, which relates to a modulated spin-polarized current and (2) a transverse spin accumulation and an unbalanced transverse charge accumulation due to the charge-spin conversion, which corresponds to the Rashba–Edelstein effect in ferromagnetic(Zn, Co)O. In this paper, we reported the ferroelectric gate controlled R–D SOC in the hybrid heterostructure (Zn, Co)O/PMN-PT through the measurements of the Anomalous Hall effect (AHE) and longitudi- nal magnetoresistance, where AHE is a magnetic response of itinerant band carriers caused by asymmetric carrier scattering in the presence of SOC. 29 The high quality (Zn, Co)O thin films in a thickness of 50–100 nm were epitaxially grown on ferroelectric substrates PMN- PT(111) with a 3 nm ZnO buffer layer by using radio frequency oxygen plasma-assisted molecular beam epitaxy. A smooth and high quality interface is very important to eliminate a residual space charge for the efficient carrier transmitting across the interface. The (Zn, Co)O thin film is doped with a high Co concentration 45% to achieve giant magnetization and strong AHE with the high Curie temperature. For Hall measurements, introducing tiny dose of donor dopants Ga of0.2% in atoms increases the conductivity of the (Zn, Co)O film, which helps to enhance the output Hall voltage. The growth temperature is 400/C14C under the oxygen partial pressure 3 /C210/C05Pa. The growth of (Zn, Co)O film is monitored by real time reflected high energy elec- tron deflection, and its chemical states are carried out by in situ x-ray photoelectron spectroscopy (XPS). The crystal structure is character- ized by high resolution x-ray diffraction (HRXRD). Magnetization is measured by a quantum designed superconducting quantum interfer- ence device (SQUID). Detailed growth and characterization refers to our previous works.30Hall effect is measured in the geometry of a Van der Pauw method in the size of 5 /C25m m2. The sheet resistivity of (Zn, Co)O thin films can be chemically tuned by introducing Ga donor dopants with the carrier density in the range of /C241018–1019cm/C03for different purposes of the Hall effect and magne- toresistance measurements.30Four Au electric contacts are deposited through mask shades by magnetron sputtering for the Hall and MR measurements, where (Zn, Co)O is not only a magnetic semiconduc- tor channel but also the top electric conducting layer. Figure 1(a) shows the schematic cross section of the hybrid heter- ostructure (Zn, Co)O(0001)/PMN-PT(111). By applying the gate volt- age between the ferroelectric substrates PMN-PT, the pole of ferroelectric polarization causes a space charge region, which induces a build-in electric field inside the (Zn, Co)O channel and the variation of charge density. The induced electric field results in the R–D SOC in the (Zn, Co)O layer.24,27When the direction of ferroelectric FIG. 1. (a) Cross section of the induced electric field and the carrier variation inside the (Zn, Co)O channel and related ferroelectric polarization in the PM N-PT substrate. (b) Schematic diagram of spin splitting of Rashba and Dresselhaus SOC in the k-space without magnetization. (c) Sheet resistance of (Zn, Co)O thin film s at 300 K as a function of the gate voltage Vgate, where a resistance platform refers to electron accumulation and depletion states. (d) Time duration measurements of a sheet resistance by applying the period pulse gating voltages ( þ300,/C0300 V). Gating voltage lasts 30 s at a 10 min interval.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012403 (2021); doi: 10.1063/5.0047430 119, 012403-2 Published under an exclusive license by AIP Publishingpolarization points upward, it attracts more electrons from the electric circuit and forms an electron accumulation state in the (Zn, Co)O channel, vice versa, the downward ferroelectric polarization forms an electron depletion state in the (Zn, Co)O channel. Figure 1(c) shows the sheet resistance of the (Zn, Co)O channel by applying the gatevoltage, where the resistance platforms correspond to a high resistivity state (HRS) and a low resistivity state (LRS) as shown in Fig. 1(a) .H R S refers to the electron depletion state with carrier density 4.7/C210 18cm/C03and LRS to the electron accumulation state with carrier density 1.8 /C21018cm/C03. The modulation ratio of HRS to LRS is HR LR¼533%, and the modulation ratio of carrier density is 261%. In this paper, without loss of generality, we study the electron accumula- tion and depletion states by applying the remanent polarization P r,where P rþrefers to upward polarization and Pr-to downward polari- zation. Figure 1(d) shows the duration measurements of resistance by applying the period gating voltages þ300 V /C00/C0/C0 300 V /C00 /C0þ300 V, where the gate voltage lasts 30 s at a 10 min interval. It indicates that by the reversal of ferroelectric polarization, the transitionbetween HRS (or electron depletion state) and LRS (or electron accu- mulation state) is reversible and repeatable. It is necessary to exclude from the magnetostriction effect and carrier induced magnetization by the reversal of ferroelectric polariza-tion. Figure 2(a) shows the HRXRD h–2hscans of the (Zn, Co)O channel in the growth direction at electron accumulation (P rþ)a n d depletion (P r-) states. The unchanged lattice constant indicates the same piezoelectric strain at accumulation and depletion states. FIG. 2. (a) High resolution x-ray diffraction h–2hscans for (Zn, Co)O films at the electron accumulation state (blue line, P rþ) and the depletion state (red line, P r/C0). The inset shows theh–2hscans by the reversal of ferroelectric polarization 10 times. (b) XPS of Co 2p 1/2and 2p 3/2peaks and their satellites at electron accumulation and depletion states. The mag- netic hysteresis loops of (Zn, Co)O films at electron accumulation (blue solid lines) and depletion (red solid lines) states at (c) 300, (d) 150, (e) 50, (f) 20, (g) 10, and (h) 5 K.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012403 (2021); doi: 10.1063/5.0047430 119, 012403-3 Published under an exclusive license by AIP PublishingFigure 2(b) shows the XPS measurements for Co 2p 1/2,2 p 3/2photo- emission peaks and their satellites at accumulation and depletion states, which indicates that the chemical states of cobalt dopants are not affected by the carrier variation. For further measurements, wechecked the magnetization of the (Zn, Co)O film at accumulation anddepletion states by using SQUID. Figures 2(c)–2(h) show the tempera- ture dependent magnetic hysteresis loops at accumulation and deple- tion states, which indicates that magnetization has nearly no influenceon the variation of carrier density except that there is /C243% change of the superparamagnetic background at 5 K. It is known that theferromagnetism of (Zn, Co)O is attributed to the percolation of bound magnetic polarons (BMPs). 31Our previous work of angle resolved photoemission spectroscopy shows that the impurity states of Co dop- ants in the case of diluted Co concentration are deep below the Fermilevel, and the impurity states disperse close to the Fermi level when theCo concentration increases up to 40%. 30The character of deep impu- rity states explains why magnetization is not affected by the carrier variation. In other side, because of the inhomogeneous distribution ofBMP, (Zn, Co)O coexists multiple magnetic phases: the ferromagneticregion with a long-rang percolation of BMP and superparamagnetic FIG. 3. Anomalous Hall resistivity qAHE yx as a function of the magnetic field for the 50 nm (Zn, Co)O film doped with 0.2% of Ga at as-grown, electron accumulation, and deple- tion states at (a)50, (b)150, and (c)300 K. The applied current is 1 mA. (d) Diagram of Hall measurements in the geometry of the van der Pauw method in a device size of 5/C25m m2. (e) Temperature dependent resistivity qxxat as-grown, electron accumulation, and depletion states. The inset is the plot of ln qxxvsT/C01=4. (f) Temperature depen- dent carrier density n at as-grown, electron accumulation, and depletion states.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012403 (2021); doi: 10.1063/5.0047430 119, 012403-4 Published under an exclusive license by AIP Publishingclusters with short-rang BMPs,32where the accumulation charges enhance the enhancement of a superparamagnetic phase at low tem- perature 5 K. Figures 3(a)–3(c) show the evolution of anomalous Hall resistiv- ityqAHE yxas a function of magnetic field at as-grown, accumulation and depletion states at 300, 150, and 50 K, where the ordinary Hall resistiv- ity has been subtracted linearly from the raw Hall data. A large signal ofqAHE yxis achieved due to the giant magnetization of (Zn, Co)O films with a high Co concentration (45%). In Fig. 3(c) ,t h em a g n i t u d e so f qAHE yxat 50 K are 2.0, 2.6, and 3.5 lXcm at accumulation, as-grown, and depletion states, respectively, which indicates that qAHE yxis ferro- electric tunable. As expected, qAHE yxhas two origins: the spontaneous magnetization and R–D SOC: qAHE yx¼RsMþqSOC yx. At a fixed ferro- electric polarization, we find out that qAHE yx remains constant ontemperature in the range of 50–300 K, and it also remains constant on the carrier density in the range of 1.8–6.0 /C21019cm/C03,a ss h o w ni n Fig. 3 . Therefore, the magnitude of qAHE yxdepends on magnetization and gated ferroelectric polarization, and it has no influence on thepure carrier variation. 30We also checked the temperature dependent resistivity qxxand the linear fitting of ln qxx/T/C01 4,33which indicates the Mott variable range hopping at as-grown, accumulation, and depletion states, as shown in Fig. 3 . At accumulation and depletion states, we have excluded of the possible magnetic origins of magneto-striction and carrier induced magnetization. The contribution of R sM is constant for the fixed magnetization, while the contribution of qSOC yx is gated controlled. For a simple estimation, we assume that the R–D SOC is symmetric at accumulation and depletion states, then thevariation of q SOC yxbetween accumulation and depletion states is: FIG. 4. (a) Diagram of MR measurements in a device size of 5 /C25m m2. (b) Plots of temperature dependent longitudinal resistivity qxxfor a 100 nm (Zn, Co)O film, the inset shows ln qxxas a function of T/C01=2.l nqxxis linearly depended on T/C01=2at low temperature, and the solid lines show the fitting curve. Plots of MR at electron accumulation and depletion states at (c) 300, (d) 150, (e) 50, (f) 20, (g) 10, and (h) 5 K. The applied charge current is 100 nA at 5 and 10 K, 1 lA at 20 K, and 10 lA at 50–300 K. Solid lines in (e) and (f) are fitting the MR curve by using Eq. (1)with the fitting parameters in Table S2 of the supplementary material .Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012403 (2021); doi: 10.1063/5.0047430 119, 012403-5 Published under an exclusive license by AIP PublishingDqSOC yx¼½qAHE yxðdepletion Þ/C0qAHE yxðaccumulation Þ/C138 ¼ 1:5lXcm, and the contribution of RSM¼½qAHE yxðdepletion ÞþqAHE yxðaccumulation Þ/C138=2 ¼2:75lXcm. Then we can estimate the modulation of AHE by the gated ferroelectric polarizationDqSOC yx RSM¼55%. The R–D SOC exerts an efficient transverse magnetic field, which results in a spin–orbit torque on the magnetization. However, we do not observe the spin–orbit tor- que in our experiments. We carried out the magnetoresistance (MR) measurements in the (Zn, Co)O film to study the spin-dependent scattering under gatedferroelectric polarization. MR is defined as MRðHÞ¼½ q xxðHÞ /C0qxxð0Þ/C138=qxxð0Þ,w h e r e qxxðHÞis the resistivity at magnetic field Hperpendicular to the (Zn, Co)O film. Figure 4(a) shows the sche- matic diagram of MR measurements in a device size of 5 /C25c m2. Figure 4(b) shows the temperature dependent qxx(0T) and qxx(1.5T) at electron accumulation and depletion states. The linear fitting of lnqxxdepending on T/C01=2at low temperature indicates Efros variable range hopping (VRH)34at electron accumulation and depletion states. Figure 4(c)–4(h) show the low field MR- Hcurves for the accumulation and depletion states at 300, 150, 50, 20, 10, and 5 K. The MR– Hcurves show clear hysteresis characteristics, in which the two peak positionsagree with the coercivity of the (Zn, Co)O layer. This finding indicates that MR has the same magnetic origins as magnetization in the (Zn, Co)O layer. Concerning to the magnitude modulation of MR by agated ferroelectric polarization, we estimate the variation DMR at tem- perature 50 K by applying the magnetic field 2 T for the accumulation and depletion states, DMR MR min¼70%. For a qualitative interpretation, the hysteretic MRis attributed to spin dependent scattering according to the phenomenological model of spin-dependent Efros VRH,34where the resistivity qxxcan be written as qxx¼q0 1þP2hcoshiexphTESi T/C18/C19 1 2 ; (1) where hTESioriginates from sum of the effective Coulomb interaction and the effective exchange coupling interaction, Pis the carrier spin polarization ratio, q0is a resistance prefacter, hcoshi¼m2with m stands for the reduced magnetization of whole system, and his the angle between the occupied state and the final vacant state. To avoid the influence of a high-field magnetoresistance, we use qxx(1.5T) as a saturated magnetization state ( hcoshi¼1) and qxx(0T) as a hcoshi ¼0 state to fit qxxandhTESiat accumulation and depletion states. Figures 4(e) and4(f)show the fitting curves at 50 and 20 K by using Eq.(1), where the fitting matches well with experimental measure- ments. The fitting parameters are shown in Table S2 of the supple- mentary material . According to the phenomenological model, we may evaluate the spin polarization, which is 21% at the accumulation state and 28% at the depletion state. The larger spin polarization at thedepletion state indicates the larger equivalent spin splitting due to the exchange coupling and R–D SOC, which agrees to a larger q AHE yxat the depletion state. The results of MR measurements provide anotherexperimental evidence of ferroelectric controlled R–D SOC in (Zn,Co)O. We also check that the ferroelectric controlled MR in (Zn, Co)O has no dependence on the applied current and the external mag- netic field in contrast with the unidirectional magnetoresistance in theRashba system, 35–37where the measurements are shown in Fig. S4 of thesupplementary material .In conclusion, we have epitaxially grown the hybrid heterostruc- ture (Zn, Co)O(0001)/PMN-PT(111) by MBE. We observed the ferro-electric gate controlled AHE in the (Zn, Co)O layer and q AHE yx¼RsMþqSOC yx. It also shows that qAHE yxis not influenced by the variation of temperature and the carrier density at fixed ferroelectric polarization. The modulation change isDqSOC yx RSM¼55% between the accumulation and depletion states. MR measurements provide another experimental evidence for ferroelectric controlled R–D SOC in (Zn,Co)O. The calculated spin polarization is 21% at the accumulationstate and 28% at the depletion state, respectively. Our experimentalresults pave a way toward semiconductor spintronic-integrated circuitswith ultralow power consumption. See the supplementary material for crystal and magnetization of (Zn, Co)O, MR for the Ga 0.002(Zn, Co)O film, and detailed R–T fitting. This research was partially supported by the Natural Science Foundation of Shandong Province Nos. ZR2019MA023 andZR2020ZD28, the National Natural Science Foundation of ChinaNos. 12074216 and 11974145, 111 Project B13029, and the StateKey Project of Fundamental Research of China under Grant No.2015CB921402. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1J. F. I. Zutic and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). 2C. Song, B. Cui, F. Li, X. Zhou, and F. Pan, Prog. Mater. Sci. 87, 33 (2017). 3F. Matsukura, Y. Tokura, and H. Ohno, Nat. Nanotechnol. 10, 209 (2015). 4D. C. H. Ohno, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature 408, 944 (2000). 5F. M. D. Chiba and H. Ohno, Appl. Phys. Lett. 89, 162505 (2006). 6I. Stolichnov, S. W. Riester, H. J. Trodahl, N. Setter, A. W. Rushforth, K. W. Edmonds, R. P. Campion, C. T. Foxon, B. L. Gallagher, and T. Jungwirth, Nat. Mater. 7, 464 (2008). 7S. W. E. R. Riester, I. Stolichnov, H. J. Trodahl, N. Setter, A. W. 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5.0054874.pdf

The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Understanding carbon dioxide capture on metal–organic frameworks from first-principles theory: The case of MIL-53(X), with X =Fe3+, Al3+, and Cu2+ Cite as: J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 Submitted: 22 April 2021 •Accepted: 18 June 2021 • Published Online: 9 July 2021 Giane B. Damas,1,2,a) Luciano T. Costa,3 Rajeev Ahuja,1 and C. Moyses Araujo1,4,a) AFFILIATIONS 1Materials Theory Division, Department of Physics and Astronomy, Uppsala University, 75120 Uppsala, Sweden 2Department of Physics, Chemistry and Biology, Linköping University, 58330 Linköping, Sweden 3MolMod-CS- Department of Physical-Chemistry, Campus Valonguinho, Institute of Chemistry, Fluminense Federal University, Niterói, Rio de Janeiro, Brazil 4Department of Engineering and Physics, Karlstad University, 65188 Karlstad, Sweden a)Authors to whom correspondence should be addressed: giane.benvinda.damas@liu.se and moyses.araujo@physics.uu.se ABSTRACT Metal–organic frameworks (MOFs) constitute a class of three-dimensional porous materials that have shown applicability for carbon dioxide capture at low pressures, which is particularly advantageous in dealing with the well-known environmental problem related to the carbon dioxide emissions into the atmosphere. In this work, the effect of changing the metallic center in the inorganic counterpart of MIL-53 (X), where X=Fe3+, Al3+, and Cu2+, has been assessed over the ability of the porous material to adsorb carbon dioxide by means of first-principles theory. In general, the non-spin polarized computational method has led to adsorption energies in fair agreement with the experimental outcomes, where the carbon dioxide stabilizes at the pore center through long-range interactions via oxygen atoms with the axial hydroxyl groups in the inorganic counterpart. However, spin-polarization effects in connection with the Hubbard corrections, on Fe 3 dand Cu 3 d states, were needed to properly describe the metal orbital occupancy in the open-shell systems (Fe- and Cu-based MOFs). This methodology gave rise to a coherent high-spin configuration, with five unpaired electrons, for Fe atoms leading to a better agreement with the experimental results. Within the GGA +U level of theory, the binding energy for the Cu-based MOF is found to be E b=−35.85 kJ/mol, which is within the desirable values for gas capture applications. Moreover, it has been verified that the adsorption energetics is dominated by the gas–framework and internal weak interactions. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0054874 I. INTRODUCTION The World Meteorological Organization (WMO) has pointed out an expected average temperature of 1.5○C higher than the pre- industrial levels in less than 35 years, as the most abundant green- house gas, carbon dioxide, has reached an increase in 3.3 ppm (0.83%) in one year of analysis, which corresponds to an overall increase of about 145% compared to the pre-industrial levels.1In this context, great efforts are necessary from different sectors of our society for a further change in the current scenario.2,3The carboncapture and storage (CCS) program4–6has different technologies to partially deal with the carbon emissions, finding applications in several industrial installations that include thermodynamic power plants and steel production. In the post-combustion approach, car- bon dioxide is captured from the gas stream due to its affinity to amine-based solutions.7–9In general, these compounds present kinetically favored reactivity with carbon dioxide, as well as low solubility of hydrocarbon compounds that are quite interesting.10 Nonetheless, the low selectivity in the presence of sulfur dioxide and the high energy necessary for solvent regeneration represent a J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 155, 024701-1 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp major problem for capture-related applications.11Thus, the devel- opment and synthesis of new chemical absorbers that can address these problems without losing the capacity for gas adsorption are highly desired.6 Metal–organic frameworks (MOFs) constitute a class of three- dimensional porous materials formed by interconnecting inorganic and organic counterparts, which has found large applicability in this field.12–16In particular, the MIL-53 frameworks comprise an inorganic region formed by the metallic centers connecting oxy- gen atoms from hydroxyl groups (axial positions) or benzene dicar- boxylate (BDC) ligands (equatorial positions) in an octahedral con- figuration.17–19In recent years, the applicability of this series has been widely evaluated by means of experimental14,18,20–25and the- oretical methods17,26for gas adsorption, including carbon diox- ide,14,18,20–23,26,27methane,18,23,25,26and hydrogen sulfide.24 In general, the presence of open metal sites with appropri- ate geometry and pore size in metal–organic frameworks is directly associated with high adsorption capacity and selectivity.12,28Addi- tionally, the material should present high heat or enthalpy of adsorption for good performance at low pressures.28This important macroscopic quantity is directly associated with the gas–framework interaction strength, which is expected to be strong enough to main- tain the latter inside the pore through weak interactions at the end of the process and also provide a suitable post-processing based cat- alytic reaction where the activated carbon dioxide can be converted in raw materials.29On the other hand, the ideal condition of process reversibility is maintained with intermediate values for this quantity. It is important to emphasize that further developments in the field are still made necessary in order to turn these mate- rials competitive in an industrial point of view. In this context, different strategies have been proposed to improve the perfor- mance of metal–organic frameworks for gas capture applications. In analogy with the amine-based solvents that are traditionally employed in post-combustion methods, the functionalization by amine groups has been widely considered to increase the storage capacity and selectivity by improving the interaction strength with carbon dioxide.13,27,28,30For instance, Hu et al.13have evaluated the effects of anchoring alkylamine groups in unsaturated Cr3+cen- ters of MOF-101 at room temperature conditions. In their series, the diethylenetriamine-functionalized MOF exhibits the best CO 2 uptake (3.5 mmol g−1) even with a significant reduction in the sur- face area. In another work, 2-aminoterephthalic acid has been tested as an organic linker in an amino-functionalized Cu-based MOF to increase the gas uptake to 5.85 mmol g−1.15Methacrylamides have also been employed to enhance the carbon dioxide capture in MOFs.31 Furthermore, pore functionalization by other chemical groups, including methyl, hydroxyl, and carboxyl groups, has been reported.12,27,28In this sense, Torrisi et al.27have shown that embed- ding carboxyl and hydroxyl groups into MIL-53(Al3+) is particu- larly advantageous for gas capture applications in comparison to amine functionalities. Nonetheless, anchoring chemical groups in metal–organic frameworks is not always straightforward in a practi- cal point of view since the synthesis conditions, given by high pres- sures and temperatures, do not favor the anchoring process of sev- eral chemical functionalities.32To overcome this issue, Yan et al.32 have initially synthesized the template with active amine groups that were further substituted by the desired acetic acid and trimesoylchloride groups. Although still containing amine groups in the struc- ture, the extra adsorption sites promote an increase of about ∼20% in gas uptake by the resulting MOFs when compared to the initial amine-functionalized material.32 Traces of water can also affect the adsorption capacity and selectivity for carbon dioxide capture in a gas mixture.33–36 Huang et al. have found that strong interactions between water molecules and the framework lead to enhanced water adsorption that could be beneficial or not for gas capture.24In another work, Siegelman et al. have found an improvement in efficiency by an amine-functionalized Mg-based MOF due to hydrogen-bonding interactions between water molecules and carbamate nitrogen atoms, which favor carbon dioxide binding.37However, it is more common that trace amounts of water can exert a negative impact on the adsorption capacity.33For instance, Liu et al.33have verified a decrease in carbon dioxide adsorption from 3.74 to 2.69 mol/kg in a Ni-based MOF with water traces besides the negative effect on the CO 2/N2selectivity. This work aims at understanding the influence of the metal- lic center from MIL-53 (X), where X =Fe3+, Al3+, or Cu2+, on the carbon dioxide capture. Such analysis is performed on a ther- modynamic point of view by means of first-principles calculations based on density functional theory (DFT). The outcomes suggest that the organic counterpart of the metal–organic framework also participates in the adsorption energetics as the oxygen atoms in this region and hydroxyl groups interact in a different extent in each material. In general, the non-spin polarized results have shown consistency to describe the energetics for these systems, but the open shell configuration exhibited by Fe- and Cu-based MOFs is better described with inclusion of such effects. Additionally, the Hubbard corrections have led to a consistent description of the atomic magnetic moment for the metallic center in these systems, a property that has been found to affect severely the adsorption energetics. II. COMPUTATIONAL METHODS The applicability of metal–organic frameworks to gas capture at low pressures has direct association with certain macroscopic properties, such as adsorption capacity and heat/enthalpy of adsorption.12,28The latter has a thermodynamic definition that requires the inclusion of thermal corrections to the total energies for its full assessment, i.e., zero point energies and thermal effects acting over the internal energy of each system, but the main contributing term is the total energy itself.17Hence, variations in this property prior and posterior to adsorption give a reliable estimation of the enthalpy of reaction/adsorption that is crucial to evaluate how good the material is for gas capture. From a microscopic standpoint, such variations are generally dictated by the interaction strength between the metal–organic framework and the guest molecule (carbon dioxide), namely, the binding energy (E b). Therefore, the heat/enthalpy of adsorption has been assessed by calculating the binding energy (E b) of the solid-state system within the framework of the density functional theory (DFT) as imple- mented in the Vienna Ab-Initio Simulation Package (VASP).38Fur- ther details on the computational methodology for modeling the metal–organic framework are given in Subsections II A and II B. J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 155, 024701-2 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A. Bulk structures To model the metal–organic frameworks here under consid- eration, the crystallographic data of MIL-53 (Fe3+) reported by Millange et al have been used .39The original structure has been modified to include hidden hydrogen atoms at the inorganic region, more specifically at the oxygen atoms that are displaced in axial posi- tions related to the metallic center.17Furthermore, it was necessary to remove the water molecules lying inside the pores in order to activate the material for gas adsorption (Fig. 1).40 The Perdew–Burke–Ernzerhof (PBE) functional41was the functional of choice to treat the exchange-correlation potential in the initial solid-state calculations, as it has provided good agree- ment with experimental data obtained for the evaluation of similar thermodynamic properties in previous reports.42–45In the current study, the ionic relaxations were carried out until the total ener- gies reached the convergence criterion of 1.0 ×10−3eV. Dispersion effects and weak interactions were taken into account by including the D3-Grimme corrections46in all steps. The plane wave-basis set was defined with a cutoff energy of 800 eV after convergence tests in the sampled region (400–1000 eV). The Brillouin zone was sam- pled by a 2 ×2×4 Monkhorst–Pack k-point mesh. Spin-polarization effects were further considered for the frameworks with an open shell configuration [MIL-53 (X), where X =Fe3+and Cu2+]. Alterna- tively, the lattice parameters have been fully relaxed for these systems as displayed in Table S1 of the supplementary material. The electronic structure has been attained by calculating the density of states (DOS) and its projected components (pDOS) onFe, Al, Cu, C, O, and H atoms. As the semi-local generalized gradi- ent approximation (GGA) functional fails to describe the bandgap of semiconducting materials, Hubbard corrections have been applied on Fe 3 dand Cu 3 dstates through spin-polarized static calculations within the tetrahedron method with Blöchl corrections. The partial occupancies for each orbital have also been determined by using the Gaussian smearing for visual analysis of the orbital hybridization. Here, the assessment of the electronic structure is basically intended to complement the material description. B. Structure model The gas capture process has been evaluated by expanding the initial bulk structure into a 2 ×2 supercell aiming to avoid adsor- bate interactions with their respective images in the periodic system. Initially, it has been assumed that the pore structures do not vary in a significant way upon gas uptake by allowing partial relaxation of the system, i.e., the ionic positions. This is an oversimplifica- tion that is expected to describe the gas adsorption process in a proper way. Nonetheless, we have also considered eventual changes in the crystal lattice by enabling full relaxation of the system. These results are briefly discussed in this publication. In the former case, the partial relaxations were performed within the Γ-point with a plane-wave cutoff energy of 550 eV. A tighter energy convergence criterion has been applied for electronic/ionic steps (1.0 ×10−5/1.0 ×10−4eV) in order to guarantee that the global minimum on the potential energy surface (PES) has been reached. The final atomic forces over the metal–organic frameworks are found to be less than FIG. 1. Bulk structure of MIL-53 (Fe3+) after ionic position relaxation at the PBE/800 eV level of theory including spin-polarization effects. In detail, it is possible to observe the narrow pore structure of this material. The red, gray, and white spheres correspond to oxygen, carbon, and hydrogen atoms, respectively, whereas the golden sphere is representative of Fe, Al, or Cu. Code: VASP. J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 155, 024701-3 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 0.01 and 0.03 eV/Å for the pore structure and the guest molecule, respectively. The GGA +U relaxations have been carried out using U=7 eV and J =1 eV as Hubbard parameters on Fe 3 dand Cu 3 d states, while employing the same energy convergence criteria for the electronic/ionic steps. The binding energy E bof the adsorbed species to the frame- work has been calculated by subtracting the total energy prior and posterior to the adsorption, i.e., Eb=EMOF−gas−(EMOF+Egas), (1) where E MOF-gas is the total energy of the gas–framework system and the last terms correspond to the individual total energies before adsorption. Zero-point energies and thermal corrections are not considered in this definition. Effects of spin-polarization and Hub- bard corrections (GGA +U) on Fe 3 dand Cu 3 dstates over the quantity expressed by Eq. (1) have also been evaluated for the Fe- and Cu-based metal–organic frameworks. III. RESULTS AND DISCUSSION A. Bulk structures In order to evaluate how the metallic center affects the ability of a non-functionalized metal–organic framework to capture carbon dioxide, we have considered the bulk structure from MIL-53 (Fe3+) with a diamond-shape pore, also called the narrow pore form. This material crystallizes in a P2 1/c space group with unit cell dimensions given by a =19.32 Å, b =15.04 Å, c=6.84 Å, and β=96.3○. Initially, lattice parameters have been constrained during the relaxation step in order to maintain the spatial group symmetry of the crystalline structure. Table S1 shows that the optimization of lattice parameters leads to a slight decrease of ∼1–2 Å of the blattice parameter and variation of <2○in the lattice angles for these systems that are not expected to affect the adsorption thermodynamics upon expansion into the 2 ×2 supercell. In MIL-53 (Fe3+), the inorganic counterpart formed by the iron metallic center is linked to the benzene dicarboxylate (BDC) ligands via oxygen atoms that are located in equatorial positions. In the axial positions, the hydroxyl groups form a region that can interact with the guest molecule as the hydrogen atoms are pointed out vertically to the pore center, while not presenting any steric hindrance. In this sense, the vertical distance between hydrogen atoms from different inorganic counterparts has been calculated as d H–H=5.20–5.40 Å, with iron atoms from adjacent parts being distanced by 19.32 Å. Replacing the metallic center by aluminum or copper in MIL-53 (Al) and MIL-53 (Cu) does not promote variations in the pore width (∼19.3 Å), but its size is diminished for the aluminum case (d H–H =5.11 Å). Additionally, there is a shortage in the Al–O chemical bond of about ∼0.2 Å in comparison to Fe–O or Cu–O, which might be resultant from a stronger interaction between the metallic center and the oxygen connecting the organic counterpart. At this point, it is necessary to emphasize that MIL-53 (Al3+) does not present the same crystal structure as the iron-based material; thus, it is an approximate model for this study.17 B. Electronic structure This analysis has been primarily considered to validate the density functional theory methodology, but also to establish thestructural parameters that are optimized during the relaxation pro- cess, i.e., the ionic positions are always considered but the lattice parameters are usually kept fixed throughout the relaxation. MIL-53 (Fe3+) has shown photoactivity in the visible light region with an experimental optical gap of 2.64 eV, which corre- sponds to an absorption edge at λ=470 nm.47Furthermore, the authors point out that the maximum absorption at λ=220 nm is due to the ligand to metal charge transfer, O (II) →Fe (II).47 Figure 2 and Fig. S1 depict the density of states of MIL-53 (X), where X=Fe3+, Al3+, and Cu2+, as obtained using the Gaussian smear- ing and tetrahedron method with Blöchl corrections, respectively. The latter choice is justified by the semiconducting nature of these materials, which requires such a methodology for an appropriate description of their intrinsic bandgaps, whereas the Gaussian smearing facilitates the plot visualization. For Fe- and Cu-based metal–organic frameworks, the GGA +U methodology has been used to deal with the self-interaction prob- lem from the GGA approximation to density functional theory that often leads to an underestimated bandgap.48–50In this sense, these calculations were performed in a static mode after spin-polarized ionic relaxation within the PBE level of theory. These open-shell systems present an octahedral dorbital splitting with the electron occupancy in Cu d-orbitals expressed as (t3↑↓ 2ge2↑,1↓ g), whereas the magnetic moment ( μ) for Fe3+(4.5μb/atom) suggests a high-spin state with electron occupancy given by (t3↑ 2ge2↑ g). An interesting point is that μfor oxygen atoms is slightly increased at the hydroxyl groups in the Cu-based MOF (∼0.3 μb/atom) in comparison with the Fe-based system (<0.2μb/atom), which is not verified for the oxygen atoms connected to the BDC ligands. Although the Cu-based MOF does not exhibit a significant change in the atomic μfor the metallic centers regarding the level of theory, the iron-based material does have a significant variation in this property (1.0–4.0 μb) within the PBE level, which would lead to Fe3+ions displaying different electronic configurations along the symmetric crystal environment. Such inaccuracy to describe the Fe-based MOF electronic structure could affect the thermodynamic properties if this effect is not propagated upon addition of the carbon dioxide molecule in the further steps. For MIL-53 (Fe3+), the optimum value for the Hubbard param- eter on Fe 3 dstates was estimated to be U =7 eV and J =1 eV to give a theoretical bandgap (E g=2.20 eV) that shows fair agree- ment with the experimental report.47As displayed in Fig. 2(a), this system has the valence band maximum (VBM) mainly com- posed of O 2 porbitals connecting to the metallic center, which persists until −2.6 eV. In the valence band, the spin-up contri- butions from Fe 3 dstates have a rising contribution from ∼−0.3 to−0.7 eV, but the conduction band minimum (CBM) is basi- cally determined by the position of the spin-down contributions from these atoms. Fingerprints from C–H and O–H bonds can be easily identified in the H 1 splot [Fig. 2(a)- bottom] with four clear peaks in the interval from −1.8 to −5.7 eV that match well with C 2 pstates and O 2 pstates that are present in the same interval. In Fig. S1(b), the calculated bandgap for MIL-53 (Al3+) is Eg=3.23 eV, which also shows good agreement with the experi- mental value reported by Guo et al. (Eexp=3.56 eV).51In the same work, the authors have found that this material has an absorption J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 155, 024701-4 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. Density of states obtained for the metal–organic frameworks under investigation at the level of theory: PBE/800 eV for MIL-53 (Al3+) and GGA +U/800 eV, with U=7 eV on Fe 3 dor Cu 3 dstates of MIL-53 (X =Fe3+, Cu2+). Code: VASP /Gaussian smearing method with σ=0.1. edge at λ=348 nm; therefore, it would not exhibit activity in the visible region for eventual photocatalytic purposes. Here, it is veri- fied that the valence band is formed mainly of O 2 pand C 2 pstates from the top ( −0.3 eV) until −2.3 eV. Al 3 pstates do not participate in the VBM or CBM composition, which are prominent just in the range of −2.1 to −7.1 eV with very low density of states ( <2.6 den- sity of states/eV). In the VBM, a very low contribution from Al 3 s states (0.11 DOS/eV) can be seen at −2.9 eV. This is the reason for the large bandgap shown by this material, since the CBM is formed by the overlapping of 2 pstates from carbon and oxygen atoms that lie much higher in energy than the unoccupied dstates from Fe and Cu atoms. Here, the C–H and O–H bonds are verified upon orbital overlapping from −1.74 to −10.0 eV with four major peaks that are shifted in about +0.25 eV compared to the Fe-based MOF. As displayed in Fig. 2(c), MIL-53 (Cu2+) has a similar den- sity of states profile shown by the iron-based system with Cu 3 d unoccupied states lying much lower in energy compared to Fe 3 d states. Therefore, the overlapping with unoccupied states from oxy- gen atoms is promoted initially at +0.23 eV (1.4 eV lower than Fe-based MOF) to significantly reduce the bandgap. On the other hand, carbon unoccupied orbitals will just appear with a higher intensity at about +3.0 eV. This material has a calculated bandgap of E g=0.83 eV using U =7 eV and J =1 eV for Cu 3 dstates [see Fig. S1(c)]. C. Gas capture Figure 3 displays the adsorption sites (labeled by different num- bers) here under consideration for CO 2capture. At site (1), the interaction takes place via hydrogen atoms from hydroxyl groupsthat are connected to the metallic center in the axial positions. At site (2), the interaction occurs with the ligand carbon and hydrogen atoms through the oxygen atom. At site (3), the molecule is expected to move freely inside the pore to interact via carbon or oxygen atoms. Horizontal interactions with hydrogen from the BDC ligand have been considered at site (4). Finally, at site (5), the guest molecule has been placed to interact with both inorganic (via hydroxyl groups) and organic (via carbon) counterparts. Table I contains the gas–framework binding energies (E b) calculated via Eq. (1) for all configurations (in kJ/mol). Inclusion of spin-polarization has been considered at the third and fourth columns, in which the latter column is estimated within the GGA +U level of the- ory. The experimental values are available in the last column for comparison. Mahdipoor et al. have previously determined the absolute heat of adsorption for MIL-53 (Fe3+) in 58.7 kJ/mol by experi- mental methods.52Table I (second column) indicates that E blies between −47.60 and −73.42 kJ/mol for this material, which gives an overestimation of ∼25% for the most favorable configuration (site 1). In this system, the final configuration shows a small angular shift for a better (CO 2)O⋅ ⋅ ⋅H(MOF) interaction at ∼1.97 Å. Such an interaction does not alter the O–H bond ( ∼0.98 Å) from the hydroxyl groups or the C =O bond (1.18 Å), a typical behavior for weak van der Waals interactions. Thus, one should not expect any changes in the electronic structure of this system since there is no orbital hybridization between O 2 p(CO 2) and H 1 sorbitals (-OH group). Here, the comparison between experiment/calculation methods is given with the absolute values for heat of adsorption. J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 155, 024701-5 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. Initial configurations for CO 2adsorption inside the MOF structures under investigation: the interaction takes place vertically in 1, 2, and 5, whereas in 3 (in detail) and 4, the carbon dioxide molecule has been placed in a horizontal position. The red, gray, and white spheres correspond to oxygen, carbon, and hydrogen atoms, respectively, whereas the golden sphere is representative of Fe, Al or Cu. It is interesting to note that applying spin-polarization effects, within the DFT/GGA theory level (third column), does not improve the theory–experiment agreement, instead leading to an even more significant overestimation in terms of absolute values(Eb=−189.58 to −245.43 kJ/mol). In order to investigate the under- lying reasons for such a discrepancy, the atomic magnetic moment (μ) at the metallic site has been evaluated for each case (see Table S2). The supercell prior to adsorption has ∼1–3 unpaired electrons at the TABLE I. Binding energies (E b) for several possible configurations upon carbon dioxide adsorption within the GGA level without spin-polarized effects (second column, ISPIN =1), as well as with its inclusion (third column, ISPIN-2). GGA +U values correspond to spin-polarized calculations with U =7 eV and J =1 eV on Fe 3 dor Cu 3 dstates. Note that the comparison between the heat of adsorption and E b, which is based on the total energy variation prior and posterior to the adsorption, is held using the absolute values. In the last column, the absolute values of heat of adsorption are taken from the literature. The boldfaces denote the most favorable Eb for each case. Binding energies (Eb, kJ/mol) GGA GGA +UHeat of adsorption Configuration ISPIN-1 ISPIN-2 ISPIN-2 (kJ/mol) MIL-53 (Fe3 +) 1 −73.42 −242.04 −47.13 58.752 2 −69.57 −198.47 −38.19 3 −48.34 −189.58 −21.26 4 −47.60 −215.50 −29.11 5 −51.55 −245.43 −17.71 MIL-53 (Al3+) 1 −36.19 ⋅ ⋅ ⋅ 2 −35.61 ⋅ ⋅ ⋅ 3 −36.73 ⋅ ⋅ ⋅ 35.018 4 −34.17 ⋅ ⋅ ⋅ 5 −19.39 ⋅ ⋅ ⋅ MIL-53 (Cu2+) 1 −39.80 −42.87 −35.85 2 −28.34 −39.90 −32.19 3 −33.23 −46.97 −33.66 n/a. 4 −30.38 −47.36 −30.97 5 −29.71 −37.49 −34.18 J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 155, 024701-6 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. Final configurations for CO 2 adsorption inside MIL-53 (Fe3+) after ionic relaxation. In (a), the most favor- able configuration at site 1 is shown, whereas the other systems are rep- resented in (b). Note: the position of the hydrogen (from -OH groups) slightly varies for each case. The red, gray, and white spheres correspond to oxygen, carbon, and hydrogen atoms, respec- tively, whereas the golden sphere is rep- resentative of Fe. metallic center ( μ=1.0–2.7 μB), but the introduction of the guest molecule promotes oscillations in the electron occupancy across the framework for all sites. As a result, the total magnetization does not remain constant in the series, even with multiple reoptimizations being carried out after the convergence is reached. These data clearly indicate the lack of consistency in the GGA level of theory to describe the open shell configuration of these frameworks, i.e., to find the correct minimum energy configuration in the potential energy surface, since the gas capture does not involve the metallic center in a direct way to justify the change in its electronic structure. On the other hand, μremains constant upon addition of car- bon dioxide within the GGA +U approximation as displayed in the fourth column in Table S2. In this case, the use of Hubbard cor- rections has returned E b=−47.13 kJ/mol, which has an agreement of 80.3% with the experimental reported value.52The total magne- tization determined for this material (mag =80.00 μB) establishes a coherent high spin configuration with five unpaired electrons for each Fe atom. Changes in the U parameter (U =6 and 8 eV) have been tested for better tuning of E b, but no significant improvement has been observed ( <1 kJ/mol) for site (1). The absolute heat of adsorption measured by Bourrelly et al.18 (35 kJ/mol) is indicative of a much weaker gas–framework inter- action in MIL-53 (Al3+) in comparison with the Fe-based MOF (58.7 kJ/mol).52This property has been properly described by our calculations, where the most favorable configuration (3) over- estimates the experimental value by only 1.73 kJ/mol ( <5%, Eb=−36.73 kJ/mol). For matters of comparison, Ramsahye et al. have determined E b=−41 kJ/mol for MIL-53 (Al3+) within the PW91 level of theory/double numerical basis set with polarizationfunctions applied on hydrogen atoms, using a different model for MIL-53 (Al3+) that is based on its crystallographic data.17Thus, the present study shows better agreement, and Fig. S2 (a) and (b) illus- trate the final structures after ionic relaxation with the experimental result than that found by previous works.17,27 The Cu-based MOF has site (1) as the most favorable config- uration within the PBE level of theory, with E b=−39.80 kJ/mol in the non-spin polarized case. The final geometry can be visualized in Figs. S3(a) and S3(b), where the guest molecule is also rotated in rela- tion to site (1) in a similar position to that verified for the Fe-based MOF. Here, it is remarkable that the inclusion of spin-polarization effects provides a smoother trend with E b=−37.49 to −47.36 kJ/mol in comparison with the Fe-based MOF. This can be associated with the lower number of unpaired electrons in the Cu-based MOF that approximates the solution to the non-spin polarized case, but it still provides a different chemical trend for this framework series. In the GGA +U case, E b=−35.85 kJ/mol is 7.0 kJ/mol lower than that predicted by the PBE method, suggesting a very simi- lar heat of adsorption for this material in comparison with MIL-53 (Al3+). The final magnetization in the supercell (mag =32.00) is coherent with the presence of one unpaired electron in each metallic center (∼0.9μB/atom) and a slight magnetization on O atoms from the metal–organic framework ( ∼0.1–0.3 μB). Hence, the presence of an unpaired electron confirms the Cu d9electronic configuration arising from the Cu2+oxidation state in these systems. Therefore, this methodology is consistent to determine the electronic config- uration for these materials prior and posterior to the adsorption, thus providing total energies that can be compared with each other. The method itself determines the correct magnetization in the first J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 155, 024701-7 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5. The main H ⋅ ⋅ ⋅O interactions in MIL-53 (X), where X =Fe3+, Al3+, or Cu2+before the gas adsorption. The red, gray, and white spheres correspond to oxygen, carbon, and hydrogen atoms, respectively, whereas the golden sphere is representative of Fe, Al, or Cu. Level of theory: PBE/non-spin polarized. electronic convergence that remains constant in the further steps until the ionic relaxation is finished. This is not verified for the PBE method, where the magnetization tuning along with the ionic relax- ation process could lead to different electronic structure descriptions for some systems. Albeit these frameworks have crystal structures that only dif- fer by the metallic center, it is noticeable that the Fe-based MOF has a binding energy (E b) for carbon dioxide capture that is much higher than the other frameworks (see Table I). Such disparity is not explained by the gas–framework interaction strength since the inter- action distance remains unaltered (1.97–2.04 Å) regardless of the material. Moreover, the van der Waals nature of these interactions points out the inactivity of the metallic center in the gas adsorp- tion. Thus, we should account for other weak interactions inside the framework that could play a significant role in the adsorption energetics. Subsection III D will address this point in detail. D. Structural analysis: The non-spin polarized case It has been discussed in Subsection III C that the inactivity of the metallic center to affect the gas–framework interactions should lead to more similar values for E b. In order to address this ques- tion, the main gas–framework and framework–framework interac- tions are investigated in this subsection. Figures 5 and 6 highlight these interactions prior andposterior to the gas adsorption, respec- tively, where O1–O3 correspond to oxygen atoms from the organic counterpart. This analysis has been performed using the non-spin polarized case for the same interaction site [site (1) for Fe has thesame final configuration as site (3) for Al and Cu] as the GGA +U methodology has not been employed for structural relaxation of the Al-based MOF. All relevant data are reported in Table II. It is noticeable that the vertical distance between hydrogen atoms from different inorganic counterparts d(H1 ⋅ ⋅ ⋅H2) increases in the order Fe <Al<Cu as the H–O bond in the hydroxyl group is bent toward the organic counterpart. Such geometrical distortion is an overall effect of the electrostatic attraction between hydrogen (H1) and the surrounding oxygen atoms connected to that counterpart. In MIL-53 (Cu2+), the attraction is more evident due to the shorter H1 ⋅ ⋅ ⋅O1 and H1 ⋅ ⋅ ⋅O2 distances (2.55 and 2.62 Å), while the (O1 ⋅ ⋅ ⋅H1⋅ ⋅ ⋅O2) angle is about 26○higher than that in the other MOFs. Nonetheless, these interactions are weakened upon gas adsorption, as indicated by the stretching of H1 ⋅ ⋅ ⋅O1 and H1⋅ ⋅ ⋅O2 distances to up to 2.98 Å (an increase of ∼0.4 Å) in MIL- 53 (Cu2+), which is a more significant variation than that observed for the other frameworks. Furthermore, the decrease of 56.4○in the a(O1 ⋅ ⋅ ⋅H1⋅ ⋅ ⋅O2) angle for this framework upon adsorption indicates a weakened interaction between the gas and the organic region. Other interactions inside the framework after adsorption are quite constant regardless of the metallic center. For instance, d(H1 ⋅ ⋅ ⋅O4) and d(H2 ⋅ ⋅ ⋅O5) distances lie in the range 1.97–2.05 Å for all systems, whereas the interaction between the oxygen atom from the organic counterpart and the carbon atom, i.e., d(C1⋅ ⋅ ⋅O6), is about 2.81–2.90 Å. Thus, these interactions are not dictating the differences seen in the gas adsorption energetics of these frameworks. FIG. 6. The main interactions after gas adsorption at site (1) inside the MIL-53 (Fe3+). Note that the interactions may vary according to each system. The red, gray, and white spheres correspond to oxygen, carbon, and hydrogen atoms, respectively, whereas the golden sphere is representative of Fe, Al, and Cu. Level of theory: PBE/non-spin polarized. J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 155, 024701-8 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. Structural parameters given by atomic distances ( d, in Å) and angle ( a, in○) of the main contributions to the gas adsorption energetics in MIL-53 (X), with X =Fe3+, Al3+, or Cu2+. In the second column, the oxygen atoms (O1–O6) are located either at the organic counterpart (organic) or the gas molecule (CO 2). Level of theory: PBE/non-spin polarized. MIL-53 (Fe3+) MIL-53 (Al3+) MIL-53 (Cu2+) Atomic distance (Å)/Angle(○) Oxygen Initial +CO 2 Δd Initial +CO 2 Δd Initial +CO 2 Δd d(O1 ⋅ ⋅ ⋅H1) Organic 2.87 2.82 −0.05 2.90 2.85 −0.05 2.55 2.69 0.14 d(O2 ⋅ ⋅ ⋅H1) Organic 2.64 2.64 0.00 2.71 2.69 −0.02 2.62 2.98 0.36 d(O3 ⋅ ⋅ ⋅H1) Organic 2.71 2.95 0.14 2.68 2.89 0.21 3.12 2.97 −0.15 d(H1 ⋅ ⋅ ⋅H2) ⋅ ⋅ ⋅ 5.16 5.44 0.28 5.22 5.39 0.17 5.50 5.76 0.26 d(X–X) ⋅ ⋅ ⋅ 7.50 8.46 1.16 7.52 8.42 0.90 7.52 8.47 0.95 d(H1 ⋅ ⋅ ⋅O4) CO 2 ⋅ ⋅ ⋅ 1.97 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.98 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2.04 ⋅ ⋅ ⋅ d(H2 ⋅ ⋅ ⋅O5) CO 2 ⋅ ⋅ ⋅ 1.97 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1.98 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2.05 ⋅ ⋅ ⋅ d(C1⋅ ⋅ ⋅O6) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2.88 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2.90 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2.81 ⋅ ⋅ ⋅ a(O1⋅ ⋅ ⋅H1⋅ ⋅ ⋅O2) Organic 114.6 115.1 0.5 112.0 113.2 1.2 138.9 82.5 −56.4 Eb(kJ/mol) −73.42 −36.73 −39.18 The process of gas accommodation inside the framework pore promotes an increase in the distance between metallic centers in about d(X–X) =0.9–1.2 Å. As this distance is decreased in the adja- cent unit cells by ∼0.5 Å, it is suggested that the presence of the guest molecule slightly opens the pore, whereas the flexibility of this model is clarified. Thus, the wider space opened between dif- ferent inorganic regions in MIL-53 (Cu2+) could slightly decrease the interaction strength with the guest molecule, i.e., the adsorp- tion enthalpy or heat of adsorption, as the hydrogen atoms are the main sites contributing to the adsorption. However, the constant gas–framework interactions suggest that internal interactions inside the framework involving the hydroxyl group and organic counter- part have significant contributions to the binding energy of these materials. IV. CONCLUSIONS In the current study, we have investigated the effect of vary- ing the metallic center in the inorganic counterpart of MIL-53 (X), where X =Fe3+, Al3+, Cu2+, on the carbon dioxide adsorp- tion by using first-principles methods. The relevance of applying spin-polarization and Hubbard corrections (GGA +U method) to describe the electronic structure and gas adsorption energetics has been investigated. The Hubbard parameters for Cu- and Fe-based MOFs have been initially estimated through electronic structure assessment, in which the values of U =7 eV and J =1 eV are found to be appropriate to treat the Fe dand Cu dstates. Within this theory level, MIL-53 (Cu2+) has a calculated bandgap of 0.83 eV, whereas MIL-53 (Fe3+) and MIL-53 (Al3+) display bandgaps of 2.20 and 3.23 eV, respectively, in fair agreement with experimental reports. In fact, the proper description of the metal orbital occupancy in the open shell systems is achieved using the spin-polarized GGA +U calculations. The atomic magnetic moment on the metallic center is, in this context, an important parameter to be tracked throughout the adsorption process, as it should remain constant prior andpos- terior to the adsorption. Here, MIL-53(Fe3+) is found to stabilize on the high-spin configuration with five unpaired electrons per atomand with a CO 2binding energy of −47.13 kJ/mol in good agreement with the experimental finding for heat of adsorption. It should be pointed out that our thermodynamics assessment includes only the total energy contribution for the reaction enthalpy, i.e., temperature- dependent contributions to the internal energy, zero-point energy, and pV term are not included. Therefore, the agreement with the experimental outcome could be further improved if such contri- butions are included and specific thermodynamics conditions are properly simulated. However, it lays beyond the scope of the cur- rent study. In the case of the Cu-based MOF, we have obtained a CO 2binding energy of −35.85 kJ/mol. The latter is similar to the one obtained for the Al-based MOF, viz., −36.73 kJ/mol. These results indicate that Cu-based MIL-53 is a promising framework for CO 2capture applications. Concerning the structure, the CO 2guest molecule is stabilized within the MOF pore center through weak interactions with the hydroxyl groups of the inorganic counterpart, which shows the relevance of the coordinating molecule on the metal site. These results provide insights for future design of suitable MOF compounds for CO 2capture and storage. SUPPLEMENTARY MATERIAL See the supplementary material for density of states obtained for the metal–organic frameworks under investigation using the tetrahedron method with Blöchl corrections and final configura- tion for gas adsorption inside MIL-53(Al3+) and MIL-53(Cu2+) after ionic relaxation. The supplementary material is available free of charge on the ACS Publications website. ACKNOWLEDGMENTS This research project received financial support from the Swedish Research Council (VR) and STandUP for Energy collab- oration, with computational resources provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC Cen- ter for High Performance Computing and National Supercom- puter Centre (NSC). G.B.D. acknowledges CAPES (Coordenação de Aperfeiçoamento de Pessoal de Ensino Superior) for financial J. Chem. Phys. 155, 024701 (2021); doi: 10.1063/5.0054874 155, 024701-9 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp support of her Ph.D. studies. L.T.C. acknowledges support from CAPES/Print/UFF Grant No. 8881.310460/2018-01 and CAPES- STINT Grant No. 88887.465528/2019-00 and the CNPq Fellowship. The authors declare no conflicts of interest. DATA AVAILABILITY The data that support the findings of this study are available within the article. 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5.0048612.pdf

APL Materials ARTICLE scitation.org/journal/apm Engineering the spin conversion in graphene monolayer epitaxial structures Cite as: APL Mater. 9, 061113 (2021); doi: 10.1063/5.0048612 Submitted: 24 February 2021 •Accepted: 3 June 2021 • Published Online: 23 June 2021 Alberto Anadón,1,a) Adrián Gudín,1 Rubén Guerrero,1Iciar Arnay,1Alejandra Guedeja-Marron,1,2 Pilar Jiménez-Cavero,3,4 Jose Manuel Díez Toledano,1,5Fernando Ajejas,1,b)María Varela,6 Sebastien Petit-Watelot,7 Irene Lucas,3,4Luis Morellón,3,4 Pedro Antonio Algarabel,3,4 Manuel Ricardo Ibarra,3,4,8 Rodolfo Miranda,1,5,9Julio Camarero,1,5,9Juan Carlos Rojas-Sánchez,7 and Paolo Perna1,c) AFFILIATIONS 1IMDEA Nanociencia, C/Faraday 9, 28049 Madrid, Spain 2Departamento de Física de Materiales and Instituto Pluridisciplinar, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain 3Instituto de Nanociencia y Materiales de Aragón, Universidad de Zaragoza and Consejo Superior de Investigaciones Científicas, 50018 Zaragoza, Spain 4Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain 5Departamento de Física de la Materia Condensada and Departamento de Física Aplicada and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, 28049 Madrid, Spain 6Departamento de Física de Materiales and Instituto Pluridisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain 7Université de Lorraine, CNRS, IJL, Nancy, France 8Laboratorio de Microscopías Avanzadas, Universidad de Zaragoza, 50018 Zaragoza, Spain 9IFIMAC, Universidad Autónoma de Madrid, 28049 Madrid, Spain Note: This paper is part of the Special Topic on Emerging Materials for Spin–Charge Interconversion. a)Author to whom correspondence should be addressed: alberto.anadon@univ-lorraine.fr b)Current address: Unité Mixte de Physique, CNRS, Thales, Univ. Paris-Sud, Université Paris-Saclay, Palaiseau, France. c)Electronic mail: paolo.perna@imdea.org ABSTRACT Spin Hall and Rashba–Edelstein effects, which are spin-to-charge conversion phenomena due to spin–orbit coupling (SOC), are attracting increasing interest as pathways to manage rapidly and at low consumption cost the storage and processing of a large amount of data in spintronic devices as well as more efficient energy harvesting by spin-caloritronics devices. Materials with large SOC, such as heavy metals (HMs), are traditionally employed to get large spin-to-charge conversion. More recently, the use of graphene (gr) in proximity with large SOC layers has been proposed as an efficient and tunable spin transport channel. Here, we explore the role of a graphene monolayer between Co and a HM and its interfacial spin transport properties by means of thermo-spin measurements. The gr/HM (Pt and Ta) stacks have been prepared on epitaxial Ir(111)/Co(111) structures grown on sapphire crystals, in which the spin detector (i.e., top HM) and the spin injector (i.e., Co) are all grown in situ under controlled conditions and present clean and sharp interfaces. We find that a gr monolayer retains the spin current injected into the HM from the bottom Co layer. This has been observed by detecting a net reduction in the sum of the spin Seebeck and interfacial contributions due to the presence of gr and independent from the spin Hall angle sign of the HM used. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0048612 APL Mater. 9, 061113 (2021); doi: 10.1063/5.0048612 9, 061113-1 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm Spin–charge current interconversion based on spin–orbit cou- pling is an essential operation in present spintronics applications.1–6 Systems showing these properties are promising candidates for the realization, for instance, of a new generation of nonvolatile magnetic random access memories or efficient energy harvesting devices,7–9 among other examples. The most widespread systems providing large spin Hall conversion efficiency toward these applications are based on heavy metals, e.g., Pt, Ta, or W, because of their strong spin–orbit coupling (SOC). Recently, two-dimensional (2D) materials, such as Rashba interfaces,10,11topological insulator surfaces,12–14and transition metal dichalcogenides,15–24have been proposed to obtain efficient spin–charge current interconversion25and their wide range of func- tional properties. Some can present large SOC,15,17,19,26while oth- ers such as gr can exhibit micrometer spin diffusion lengths and long spin lifetimes.27In addition, the properties of gr can be tuned by proximity with other materials, such as ferromagnets (FMs),28,29 heavy metals,30or even other 2D materials.17 In this regard, it has been observed recently that the gr/Pt inter- face presents a very high spin-to-charge output voltage at room temperature (RT) in lateral spin valve devices using exfoliated gr and electrodes grown ex situ by electron beam lithography.31,32The enhanced spin–charge signal was due to the combination of current shunting suppression, highly resistive platinum, and efficient spin injection into gr. However, in contrast, it has also been observed that gr can significantly reduce the spin pumping voltage33,34or even generate a spin pumping voltage by itself without the necessity of a HM due to interfacial spin–orbit interactions.35,36These discre- pancies, together with the low intrinsic SOC of gr, point toward the relevance of the quality of the interfaces in determining the overall spin transport properties. Here, we study the interface between the gr monolayer and a HM and its effect on spin-to-charge current conversion in epi- taxial systems in which the spin detector (i.e., top HM), the gr layer, and the spin injector (i.e., Co) are all grown in situ under controlled conditions and with clean and sharp interfaces. All the samples have an (111)Ir 10 nm buffer layer and a 1.6 nm-thick Co layer on top of it. Then, we have two different types of stacks on top of the Ir/Co: gr/HM and HM. The role of gr in determining the overall spin-to-charge current conversion has been disentan- gled by means of thermo-spin experiments, as shown in Fig. 1. In these experiments, which are done in the so-called longitudinal spin Seebeck effect (SSE) configuration,8,37the SSE and the anomalous Nernst effect (ANE)38coexist in this geometry. In order to sepa- rate both contributions, we first use an Ir/Co/Ir control sample to obtain the ANE in the Co layer. We subtract this contribution in all the other heterostructures in order to obtain the overall spin–charge current contribution. We demonstrate that the spin–charge conver- sion in a Co/gr/HM system is not enhanced compared to the refer- ence Co/HM and independent from the spin Hall angle sign of the HM used as spin detectors, i.e., Pt or Ta. This experimental find- ing highlights the importance of gr to engineer the spin conversion and for the development of spin-caloritronics and spin-orbitronics devices. The samples incorporating gr (i.e., gr/HM) and the ones without gr (i.e., HM) were all fabricated in situ on epitaxial Ir(111)/Co(111) grown on sapphire crystals under controlled conditions, that is, they present similar structural quality and clean FIG. 1. Schematic of thermo-spin measurements in graphene metal hybrid het- erostructures. When a thermal gradient is applied in an Ir/Co/Pt structure in the z direction as well as a magnetic field in the y direction, a spin current ( Js) is gener- ated in the z direction and we will observe two different thermo-spin contributions, the anomalous Nernst effect ( EANE) and the spin Seebeck effect ( ESSE). When a graphene monolayer is introduced, we will need to consider not only the effect of graphene itself but also the additional contributions of the two new interfaces in the system ( Egr), which may induce the inverse Rashba–Edelstein effect as well as spin memory loss, a partial loss of spin current coherence. interfaces. We followed the methodology described in Refs. 28 and 39. In brief, we first deposited a 10 nm-thick epitaxial Ir(111) on Al2O3(0001) single crystal substrates by DC sputtering at 670 K with a partial Ar pressure of 8 ⋅10−3mbar and low deposition rate (of 0.3 Å/s). Subsequently, in the case of the gr-based heterostructures, the monolayer gr was prepared by chemical vapor deposition by ethylene dissociation at 1025 K at a partial pressure of 5.5 ⋅10−6 mbar. The samples were then cooled down to RT and Co was deposited by molecular beam epitaxy, and then, the Co intercalation below gr was promoted by thermal annealing at 550 K. This proce- dure produces the formation of a homogeneous Co layer with high structural order and sharp interfaces.28,39The Co layer is monitored in every step of the growth by x-ray photoemission spectroscopy to assure that it is not oxidized. In the case of samples without gr, we deposited a 1.6 nm-thick Co layer by DC sputtering at RT on top of the Ir(111) buffer. Finally, in all samples, a 5 nm capping layer of Pt or Ta was DC sputtered at RT. To prove the structural quality of the samples, we resorted to x-ray diffraction (XRD) and high resolution scanning transmis- sion electron measurements (STEMs) at RT. The XRD measure- ments were performed using a commercial Rigaku SmartLab SE multipurpose diffractometer with a monochromatic Cu K αsource (λ=1.54 Å). STEM observations were carried out in a JEOL ARM200cF at 200 kV and RT. The microscope is equipped with a CEOS spherical aberration corrector and a Gatan Quantum electron energy-loss spectrometer.28Specimens were prepared by mechanical polishing and Ar ion milling. Figure 2(a) shows a θ–2θdiffraction pattern recorded in a Al 2O3//Ir/Co/gr/Ta heterostructure. Besides the Al 2O3[0006] and Al 2O3[00012] crystallographic reflections from the substrate, maximum intensity appears at 2 θ=40.6○and 87.9○, which corre- sponds with Ir[111] and Ir[222] reflections, respectively. The forma- tion of thickness fringes around the Ir[111] and Ir[222] reflections confirms the low roughness of the interfaces. In the inset, the ωscan (rocking curve) around the Ir[222] reflection shows a sharp pro- file. The curve was fitted using a pseudo-voigt function obtaining a full width at half maximum (FWHM) of 0.27○, which proves a low degree of mosaicity in the deposited films. Figure 2(b) shows φscans APL Mater. 9, 061113 (2021); doi: 10.1063/5.0048612 9, 061113-2 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm FIG. 2. Structural and microscopic char- acterization of epitaxial Ir/Co/gr/HM het- erostructures. (a) X-ray θ–2θdiffraction pattern recorded in an Al 2O3//Ir/Co/gr heterostructure. In the inset, a θ–2θ scan recorded around the Ir(111) reflec- tion is shown. (b) φscan plots of the Al2O3[20–210] and Ir[002] reflections. (c) and (d) Scanning transmission elec- tron microscopy characterization of a Ir[111]/Co/gr sample grown on a SrTiO 3 substrate (with tCo=1 nm and tIr =10 nm), capped with a Ta oxide thick layer in order to protect the gr sur- face. Atomic resolution high-angle annu- lar dark-field images of the STO/Ir and Ir/Co interfaces, respectively. The scale bars represent a length of 2 nm. around the Al 2O3[202⋅110]and Ir[002] reflections. The rotation scan around the Ir[002] reflection shows a sixfold symmetry instead of the expected threefold symmetry. This is related with the presence of two equivalent twin-boundary domains rotated by 180○.40Similar curves, including ω- andφ-scans, are obtained for samples with- out gr (not shown). From Figs. 2(a) and 2(b), the following epi- taxial relations are obtained: out-of-plane [0001]Al2O3∣∣[111]Ir and two in-plane configurations, (1) [01–10]Al2O3∣∣[1–10]Ir (−90○and 30○) and (2) [01–10]Al2O3∣∣[1–10]Ir (30○and 90○). The positions of the Ir[111] and Ir[002] reflections indicate an incommensurate growth of iridium with a bulk-like afcc lattice parameter within the experimental error. This is explained by the large mismatch ( ∼13%) between Al 2O3[0001] (0.4785 nm) and Ir[111]. The STEM observations confirm the quality of the stacks. Figures 2(c) and 2(d) display the atomic resolution STEM high- angle dark-field image of an Ir/Co/gr/Ta heterostructure, showing a high crystalline quality and sharp and coherent interface. No major hints of chemical interdiffusion or disorder are visible. These results, along with x-ray diffraction, suggest that the Co layer is epitaxial and the Co layer on the Ir buffer is fully strained and coherent. Thermo-spin measurements were performed in an Oxford spectrostat NMR40 continuous flow He cryostat with a thermoelec- tric measurement system.41–43Experimentally, the sample is put in place between two ceramic AlN plates, which are electrically insu- lating but have high thermal conductivity. They are attached using thermal paste to a large Cu piece that acts as a cold feet and is in direct contact with the cryostat. A resistive heater on the upper AlN piece provides the thermal gradient by application of an elec- tric current in the order of several milliamperes. The temperature difference between the upper and lower plate is measured by two T-type thermocouples near to the sample in order to obtain accu- rate temperature values. The samples were contacted electrically with thin Al wires with a diameter of 25 μm using commercial thermal silver paste. The voltage was measured using a Keithley2182A nanovoltmeter. The sketch of the measurement geometry is shown in Fig. 1: the thermal gradient is applied in the z direction, while a magnetic field is swept in the y direction. A thermo-spin voltage is then measured in the x direction. It is worth recalling that in systems containing metallic FMs, the thermo-spin voltage has three main contributions: (1) the anoma- lous Nernst effect (ANE), i.e., the thermal counterpart of the anoma- lous Hall effect, which has a similar physical origin;8,38,43(2) the spin Seebeck effect, which comprises the generation of a spin cur- rent from incoherent thermal excitation and its conversion on an electric voltage by means of the inverse spin Hall effect (ISHE); and (3) the interfacial spin–orbit contribution, arising from the Rashba interfacial spin–orbit field, which can give rise to a wide range of phenomena, from spin memory loss to spin current generation.6,10,44 We first identified the ANE signal contribution of the Co layer, which is proportional to the Co magnetization. We acquired the thermo-spin voltage in a symmetric epitaxial Ir(10)/Co(1.6)/Ir(5) stack [panel (b)] as a function of the in-plane applied magnetic field and compared it to the sample magnetization along the y direc- tion normalized by the saturation value. The identical behavior of both magnitudes is shown in Fig. 3(b), as expected from the ANE phenomenological relation EANE=QS(μ0M×∇T), (1) with QS,μ0,M, and∇T being the Nernst coefficient, the vacuum magnetic permeability, the thermal gradient, and the magnetization of the FM, respectively. Since Ir has a much smaller spin Hall than other heavy metals, such as Pt,45this symmetric stack can be hence used as a reference to check the size of the anomalous Nernst effect of the Co layer in the asymmetric stacks, which will be subtracted from the overall voltage measured. Note that in Figs. 3(a) and 3(b), we can observe a very APL Mater. 9, 061113 (2021); doi: 10.1063/5.0048612 9, 061113-3 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm FIG. 3. Thermo-spin voltage in epitaxial hybrid gr/HM and HM stacks. (a) Thermo- spin voltage in Ir/Co/Ir, Ir/Co/Pt, and Ir/Co/gr/Pt. The observed thermo-spin voltage in the sample with the Ir capping layer (yellow) is significantly smaller than the that with Pt, and it is mainly due to the anomalous Nernst effect in Co. A reduction in the voltage at saturation field is observed when gr is introduced into the stack. (b) Close-up view on the anomalous Nerst effect voltage in the Ir/Co/Ir sample and its magnetization measured by vibrating sample magnetometry. (c) Angular dependence on the thermo-spin voltage in the Ir/Co/Pt sample. The angle θrep- resents the relative angle between the measured voltage (x direction) and the applied magnetic field (xy plane). small voltage in the Co/Ir sample mainly due to the electrical screen- ing by the buffer layer of Ir because of its low resistivity, about three times lower than Pt in this range of thickness.46–48This is specially the case for epitaxial Ir,49which leads to smaller values of the spin Hall angle when comparing to polycrystalline metals (see Ref. 48). The second contribution to the measured voltage is the SSE generated by the inverse spin Hall effect,50,51which has a similar geo- metric behavior, since the spin current lies in the z direction as it is induced by an out-of-plane thermal gradient, JC=θSHρ A(2e ̵h)JS×σ, (2) where JCandJsare the charge and spin currents in the HM, respec- tively,θSHandρare the spin Hall angle and the electrical resistivity of the HM, respectively, Arepresents the contact area between the FM and the HM, eis the elementary charge, and σis the mean spin polarization direction of the electrons in the FM close to the interface with the HM. It is important to note here that JS∝∇Tandσ∝M in the FM at saturation. We have thus performed thermo-spin measurements in the Pt and gr/Pt samples. This is shown in Fig. 3(a) where we observe that the introduction of gr reduces the total observed thermo-spin voltage in the Ir/Co/Pt system by about 40%. As can be seen in Eqs. (1) and (2), the SSE and ANE voltages follow a cross product relation between the thermal gradient and the magnetization; therefore, when magnetization rotates in the xy plane and the x component of the thermo-spin electric field is measured, we will observe a sinusoidal relation, as shown in Fig. 3(c), where the angleθrepresents the relative angle between the measured voltage and the applied magnetic field. At this point, it is important to notice that (i) the dependence of the thermo-spin voltage with an external magnetic field is simi- lar for both effects and (ii) the comparison of thermal gradients in Co in these stacks is reliable. For the latter, we routinely checked that the total thermal conductivity of the system, i.e., the sample(mainly substrate) with its holder, is maintained unchanged in all experiments and all samples. In fact, the main contributions to the thermal resistance of the system come from the substrate and sample holder because their total thermal resistance is orders of magnitude larger than that of the thin film stack. Consequently, the heat current that flows through Co, which has the same thickness in all samples, is similar in all the cases. This implies that the inclusion of gr or dif- ferent metallic detecting layers does not modify the (perpendicular) thermal gradient in Co and that the corresponding spin current is kept reasonably unchanged for all samples. As remarked before, the ANE signal of the Co layer taken from the measurements of the symmetric Ir/Co/Ir system is subtracted from the voltages acquired in the asymmetric stacks with the Pt detecting layer with and without gr. We carefully considered the resistivities and thickness of the films in the system. This is shown in the supplementary material. Thus, the ANE (Vcontr ANE) contribution to the voltage in the xdirection for a multilayer system can be estimated for each sample as42,51 Vcontr ANE=(r 1+r)VANE, (3) where VANEis the anomalous Nernst effect voltage of a single metal- lic FM layer with the same thickness subjected to the same thermal gradient and r=ρHM ρFMtFM tHM, withρHMandρFMrepresenting the HM and Co resistivities and tHMand tFMrepresenting their thickness, respectively. The resulting voltage dependences on the applied magnetic field after subtraction of the ANE contribution are shown in Fig. 4. Here, the voltage signals are normalized by the sample resistance to rule out the possibility of a shunting effect in the gr monolayer in the inverse spin Hall signal. We also assume for this calculation that FIG. 4. Interfacial contribution to the thermo-spin voltage. (a) Thermo-spin volt- age after subtraction of the anomalous Nernst effect component. This value is divided by the sample resistance in order to reduce artifacts and compare the val- ues adequately. L x=0.8 mm represents the lateral dimension of the sample in the x direction and ∇T=(Thot−Tcold)/Lz, where Lz=0.4 mm is the sample thickness, including the substrate. The absolute saturation voltage observed in the Pt sample (blue open circles) is reduced by 60% when comparing with the gr/Pt sample (green triangles). This is also the case for the absolute voltage in the Ta sample (wine squares) compared to gr/Ta (red filled circles) where the observed reduction is 11%. APL Mater. 9, 061113 (2021); doi: 10.1063/5.0048612 9, 061113-4 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm Ms is the same in all the samples. To break down the contribution of the gr monolayer, in addition to the gr/Pt- and Pt-based stacks, we have considered a second set of samples capped by a 5 nm-thick Ta layer with a naturally oxidized surface (i.e., gr/Ta- and Ta-based stacks). As clearly shown in Fig. 4, the voltage dependence with the external magnetic field has an opposite sign when comparing Ta and Pt samples, as expected from their different signs of the spin Hall angle. Although the signal reductions in the two types of samples are of different magnitudes, that is, 60% in Pt-based samples and 11% in the Ta-based samples, our experimental finding suggests a uni- versal behavior regardless of the detecting layer. Here, the reduction percentage is calculated by subtracting the voltage at μ0H=0.7 T as ∣(VHM−Vgr/HM)/Vgr/HM⋅100∣. There are different mechanisms that may be behind the ori- gin of this observation. In this experiment, gr may support a non- negligible SOC, induced by the adjacent metals through electronic hybridization. This, in turn, produces a significant Rashba-type Dzyaloshinskii–Moriya interaction (DMI).28,29,40On this basis, we envisage three different mechanisms responsible for the reduction of the measured thermo-spin signals. (i) The introduction of gr could produce a shunting of the ISHE current, reducing the effective spin- to-charge conversion in the HM. This artifact is avoided normalizing the thermo-spin voltages by the sample resistance, as shown in Fig. 4. (ii) A Rashba interface, such as the Co/gr in our system, can induce spin–charge conversion by the so-called inverse Rashba–Edelstein effect (IREE). This would be translated in a voltage contribution of similar sign and magnitude for both systems. As we observe, this sce- nario cannot explain our findings unless the hybridization of gr due to the HM changes substantially the effective IREE of the interface. The IREE has already been observed in YIG/gr by spin pumping,35,36 and after normalization by sample resistance, its magnitude is signif- icantly smaller than the ISHE in Pt, although it could be different in the case of Co/gr. (iii) The gr interfaces are characterized by the pres- ence of an interfacial spin–orbit coupling field that can affect the spin coherence,6,46,52–54depolarizing the spin current traveling across it and thus reducing the total observed signal. This effect, referred to as spin memory loss (SML), may happen in both Co/gr28and gr/HM46 interfaces. The fact that the reduction is smaller in the case of Ta could be explained by its smaller SML when compared to Pt inter- faces.6,55Another plausible scenario could also arise considering a combination of the IREE effect and SML. In summary, we may have a different enhancement or attenuation depending on the nature of the HM. In addition, even though saturation magnetization can play an important role in ANE measurements,42,56this interpretation still holds even if the value of the saturation magnetization (Ms) is sig- nificantly different in both systems. As shown in the supplementary material, we obtain a higher average Ms in the gr samples, sug- gesting that the thermo-spin voltage suppression by graphene could be even larger than the estimation that we provide in Fig. 4. Fur- ther experiments including the direct injection of spin current are necessary in order to discern between both contributions since while spin Hall and inverse spin Hall are reciprocal effects, this is not necessarily the case of the Rashba–Edelstein effect and its inverse counterpart. Summarizing, we have fabricated high quality epitaxial hybrid metallic/monolayer graphene stacks with coherent, roughness-free interfaces as confirmed by x-ray diffraction and atomically resolved scanning transmission electron microscopy experiments. We haveexplored the spin–charge conversion by means of thermo-spin mea- surements in which we have carefully disentangled the anoma- lous Nernst effect from the spin Seebeck and interfacial contribu- tions. Furthermore, we estimated the interfacial contribution when a graphene monolayer is inserted. Although in other experiments the gr/Pt system has been shown to increase the spin Hall effect efficiency, we demonstrate that, for thermally induced spin cur- rents in the longitudinal spin Seebeck configuration, the presence of graphene reduces the overall spin–charge conversion regardless of the heavy metal (Ta or Pt with different spin Hall angle signs) layer used. We disregard any possible effect of the introduction of graphene in the thermal gradient in Co due to the insignificant change that the thermal resistance of graphene introduces in the system compared to the total thermal resistance of the sample and sample holder. We ascribe the reduction in the thermo-spin volt- age mainly to the combination of spin memory loss and the inverse Rashba–Edelstein effect. See the supplementary material for more information on the anomalous Nernst effect contribution in thermo-spin measurements and the saturation magnetization in ultra-thin cobalt films. We thank V. P. Amin, S. Sangiao, A. Fert, and F. Casanova for valuable discussions. This research was supported by the Regional Government of Madrid through Project No. P2018/NMT-4321 (NANOMAGCOST-CM) and the Spanish Minis- try of Economy and Competitiveness (MINECO) through Project Nos. RTI2018-097895-B-C42, RTI2018-097895-B-C43 (FUN-SOC), PGC2018-098613-B-C21 (SpOrQuMat), PGC2018-098265-B-C31, and PCI2019-111867-2 (FLAG ERA 3 grant SOgraphMEM). J.M.D.T. and A.G. acknowledge support from MINECO and CM through Grant Nos. BES-2017-080617 and PEJD-2017-PREIND- 4690, respectively. I.A. acknowledges financial support from the Regional Government of Madrid through Contract No. PEJD- 2019-POST/IND-15343. IMDEA Nanoscience is supported by the “Severo Ochoa” Program for Centres of Excellence in R & D, MINECO (Grant No. SEV-2016-0686). A.A., S.P.-W., and J.-C.R.-S. acknowledge support from Toptronic ANR through Project No. ANR-19-CE24-0016-01. P.J.-C., I.L., L.M., P.A.A., and M.R.I. acknowledge support from Project No. MAT2017-82970-C2-R. Electron microscopy observations were carried out at the Cen- tro Nacional de Microscopía Electrónica at the Universidad Com- plutense de Madrid. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011). 2X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V. O. Lorenz, and J. Q. Xiao, Nat. Commun. 5, 3042 (2014). 3K. Jhuria, J. Hohlfeld, A. Pattabi, E. Martin, A. Y. A. Córdova, X. Shi, R. L. Conte, S. 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APL Mater. 9, 061113 (2021); doi: 10.1063/5.0048612 9, 061113-6 © Author(s) 2021

5.0057392.pdf

Characteristics of Thin Film Organic Photovoltaic Solar Cells Jagriti Dewan1Dand Mani Kant Yadav2E 1Pt. J.L.N. Government College, Faridabad ,QGLD 2J.C. Bose University of Science and Technology, Faridabad ,QGLD Djagritidewan22@gmail.com E&RUUHVSRQGLQJDXWKRU yadavmanikant64@gmail.com Abstract. The field of Organic Photovoltaics is gaining widespread popularity owing to the cost-effectiveness, efficiency, and stability of Thin Film Organic Photovoltaic Solar cells (OSC’s) for application especially in the relevance of the needs of the poor and remote areas of our vast country. Thin Film OSC aims at reducing the thickness of the active layer too few nanometres. This can help in enhancing the light- trapping ability of the OSC’s and thereby reducing the number of optical losses. A major proble m in OSC’s is the poor mobility and recombination of photogenerated charge carriers. The key role is to create appropriate designs of the device architecture with feeble losses. This paper aims at the device and material study of the OSC’s which could enha ncethe Power Conversion Efficiency (PCE) of the Thin Film OSC’s. This research area is in tune with the present International R&D trends to develop flexible and cost-effective solar cells Keywords —Organic Solar cells, Photovoltaics, Thin film, Power Conversion Efficiency, nanometer, recombination. INTRODUCTION Organic Photovoltaic Solar cells have gained widespread popularity over the years. Recent researches have shown a marked improvement in the Power Conversion Efficiency (PCE) which has made this an extremely popular R&D activity in the Solar cells field. The Solar Cell which was developed at Bell Laboratories in 1954 is a type of Photovoltaic device that converts optical en ergy to electrical energy[1]. Though it seems to be an extremely simple concept, successful Photovoltaic (PV) devices for solar energy production will require the optimization of many crucial factors involving material electron donor properties, electrode configuration, substrate mechanics, light trapping schemes, and fabrication methods. This paper covers a study of varioustechniques of material development an d light trapping methodologies to develop efficient OSC’s. Thin-Film Organic Solar Cells Silicon (Si) based solar cell s are designed primarily owing to its non-toxic nature and availability. Si however has an indirect bandgap. This leads to high optical lo sses due to reflection and hence the light absorption capacity is poor. To meet the requirement of light-harvest ing, the thickness of the Si active layer should be no less than 100 μm. The thickness of silicon solar cells is accompanied by efficient crystallization and purification methods which makes Si-based solar cells expensive. To solve these problems, pr oper device architecture and light trapping methods such as periodic gratings, photon ic crystals, plasmonic structures Si nanowire arrays (SiNWs), and Si nanocone arrays(SiNCs) have been proposed and investigated widely.Though efficiencies of these thin-film organic devices have not reached their inorganic counterpart’s dynamic methodologies will help achieve an optimum PCE. The field of OPV began with the use of small organic Advanced Materials and Radiation Physics (AMRP-2020) AIP Conf. Proc. 2352, 020040-1–020040-4; https://doi.org/10.1063/5.0057392 Published by AIP Publishing. 978-0-7354-4105-7/$30.00020040-1molecules (pigments) and further developed with semico nducting polymers. Also, a variety of small-molecular- weight electron-acceptor materials are available easily . A major advantage of these small-molecule materials compared with large-molecule polymers is that vacuum sublimation can be used to form well-controlled amorphous or polycrystalline thin films on flexible polymeric substrates. As a result, they can be used to fabricate complex multilayer devices, and there is no need to make the molec ules soluble. Moreover, they also are very easy to purify [2]. Preparation Techniques The most common among various techniques employed ar e the dry thermal evaporation of organic constituents. For small molecules, evaporation is the best choice. Therm al evaporation involves high vacuum conditions. In addition to it, contaminants such as oxygen and water are eliminated. The mean free path of the molecules in such an ultra-vacuum condition is greatly enhanced as co mpared to the distance from the evaporating source to the sample. This method allows good thickness and dopant control, eliminates parasitic coating on the walls of the chamber, and allows the fabrication of complex multilayer devices [3]. In addition to it, there are numerous techniques one of wh ich is Spincoating. This technique has indisputably the most important for the development of solar cells. In spite of the complexity of film formation, it helps in producing a homogenous film over a large area. The typi cal form of spin coating involves the application of a liquid to a substrate followed by an acceleration of the substrate to a chosen rotational speed [4]. The acceleration results in the ejection of most of the liquid an d what is left is a thin film of liquid on the substrate. The film thickness d obtained can be expressed as [4] d = k ὠª where ὠis the angular speed and k and a are constants related to the properties of the liquid (solvent), solute, and substrate. Formation of (xcitons Inorganic solids, the intermolecular overlap of electron ic wavefunctions is very weak which results in making the energy bands very narrow, and thus they can be ap proximated as molecular orbitals [5]. This results in the formation of two energy levels termed as Highest Occu pied Molecular orbital (HOM O) which is analogous to the valence band and Lowest unoccupied Molecular Orbital (LUMO) which is analogous to the conduction band in the case of inorganic materials. The energy bandga p is the difference betwee n the two energy levels. When the solar radiation is incident on the thin fi lm OSC’s absorption of energy greater than or e qual to the bandgap results in the formation of an excited electron (e) in the LUMO and a corresponding hole (h) in the HOMO. Due to the attractive Coulomb potential, the excite d electron and hole get drawn closer towards each other and become bound in a hydrogenic electronic state called an exciton. An exciton is neutral in charge and is capable of moving throughout the material. The formatio n of excitons is unfavorable in OSCs because one needs to generate free electrons and holes to be co llected at the opposite electrodes to generate current. For the successful operation of an OSC, the excitons must be diss ociated into free charge ca rriers with the aid of an energy greater than their binding energy [6]. Types of Solar Cells The structure and design of the OSC’s have been improved over the ye ars and there are mainly four kinds of Organic cells (1) single layer (2) bilayer (3) bu lk heterojunction (4) hybrid OSC. The first organic solar cells were based on single th ermally evaporated molecular organic layers sandwiched between two metal electrodes of different work functions [4]. The top layer (anode) is kept transparent for the absorption of light and made up of a thin film of organic materials. On absorbing a photon of energy equal to or greater than the bandgap an exciton is created and the on ly external energy available to dissociate excitons and draw the free charge carriers to opposite electrodes is due to the electric field established by the difference in the work functions, Φanode and Φcathode , of the anode and the cathode, respectively. The efficiencies achieved in single layer OSCs is low because of the insufficient tr ansport of free charge carriers to the electrodes. The poor efficiency of single layer OSCs introduced the idea of a bilayer OSCs which consist of two layers of organic material. The first layer is of donor material and the second layer is of an acceptor material and both have different ionization potential and electr on affinities. In this design, exc iton dissociation and charge carrier 020040-2collection are far more efficient than in a single layer OSC. However, the exciton diffusion length is very less and thus the free charge carriers so generated ar e limited in number [5]. Electrode 1 Electrode 1 Organic materialElectron donor Electron acceptor Electrode 2 Electrode 2 (a) (b) ),*85( 6WUXFWXUHRI DVLQJOHOD\HUDQG EPXOWLOD\HURUJDQLF VRODUFHOOV The essence of the bulk heterojunction (BHJ) is to intimat ely mix the donor and acceptor components in a bulk volume so that each donor-acceptor interface is within a distance less than the exciton diffusion length of each absorbing site.[7] As a result, this stru cture enables charge carrier generation everywhere within the active layer, which increases the photon to electron conversion efficiency dramatically.[6-7] The structure and mechanism of a hybrid solar cell are sim ilar to that of a BHJ with the only difference that the organic acceptor is replaced by an inorganic material. This is done to enhance the PCE of solar cells by utilizing both the type of materials. A combination of silicon nanowires (SiNWs) and poly (3,4-ethylene dioxythiophene) poly(styrene sulfonate) (PEDOT: PSS) have produced the best power conversion efficiency of 8.40% in hybrid OSCs to-date [8]. Light Trapping Techniques A major problem in organic solar cells is the poor m obility and recombination of the photogenerated charge carriers. Optical losses result in the loss of a significant portion of the incoming radiation and hence proper light trapping techniques must be incorporated to achieve the desired PCE. Light trapping ability is the capacity of the OSCs to absorb the maximum amount of solar radiations incident on it with feeble optical losses. This can be done by using num erous refractive structures [9], random structures [10-11], random scatterers [12], aperture s [13] and micro lenses [14-16]. These structures can be integrated with proper architecture to avoid recombination losses.For example, Metal nanoparticles placed above the solar cells can scatter most amount of incident light to the substrate and increase the in-coupling efficiency. However, owing to the low refractive index of organic materials a high coupling efficiency is difficult to ach ieve. These nanoparticles can act as optical antennas for that matter and store energy in the form of lo calized surface plasmon resonance (LSPR) [17]. There are other methods also employed su ch as the use of a diffraction grating that couples reflected light into waveguide modes of the solar cell [18]. The structural pr operties of the grating influence the performance of the solar cell.Light trapping elements can be induced by directly st ructuring the substrate of organic solar cells [19-23]. Substrates that have wrinkles or folds were found to have an improved photocurrent as compared to solar cells on a flat substrate [23]. SUMMARY AND FUTURE OUTLOOK The field of OSCs is a promising domain in future res earch. The limited charge ca rrier transport in organic semiconductors requires a thin layer of the material. We need to design more techniques and novel designs to achieve a greater amount of PCE and with minimal losses. In this paper, we reviewed the various material and architectural designs and some of the light-trapping ways . Given the fact that the theoretical calculations have a remarkable effect, however, the experimental realization is an important tool. For this reason, this field of organic solar cells will be an active f ield of research in the coming years. 020040-3REFERENCES [1] "April 25, 1954: Bell Labs Demonstrates the Firs t Practical Silicon Solar Cell". APS News (American Physical Society) 18 (4). April 2009. [2] Organic solar cell research at Stanford University. [3] Organic solar cells: An overview Harald Hoppea and Niya zi Serdar Sariciftci Linz Institute for Organic Solar Cells (LIOS).[4] J-M. Nunzi: Organic photovoltaic materials and devices. C. R. Physique 3, 523 (2002)[5] L.A.A. Pettersson, L.S. Roman, and O. Ingana¨s: Modeling photocurrent action spectra of photovoltaic devices based on organic thin films. J. Appl. Phys. 86, 487 (1999).[6] P. Schilinsky, C. Waldauf, and C.J. Brabec: Reco mbination and loss analysis in polythiophene based bulk heterojunction photodetectors. Appl. Phys. Lett. 81, 3885 (2002).[7] J-M. Nunzi: Organic photovoltaic materials and devices. C. R. Physique 3, 523 (2002).[9] S. Esiner, et al. Adv. Energy Mater. 3 (2013) 1013.[10] C. Cho, et al. Sol. Energy Mater. Sol. Cells 115 (2013) 36. [11] D.H. Wang, et al. Org. Electron. 11 (2010) 285. [12] Z. Hu, J. Zhang, Y. Zhao, J. Appl. Phys. 111 (2012) 104516. [44] P. Peumans, V. Bulovic´, S.R. Forrest, Appl. Phys. Lett. 76 (2000) 2650. [13] K. Tvingstedt, et al. Opt. Express 16 (2008) 21608.[14] S.D. Zilio, et al. Microelectron. Eng. 86 (2009) 1150. [15] J.D. Myers, et al. Energy Environ. Sci. 5 (2012) 6900. [16] ] V.E. Ferry, et al. Appl. Phys. Lett. 95 (2009) 183503.[17] Light trapping in thin-film organic solar cells Zheng Tang, Wolfgang Tress and Olle Ingana¨ Biomolecular and Organic Electronics, IFM, and Center of Organic Electronics, Linko¨ping University, SE-581 83 Linko¨ping, Sweden.[18] Z. Tang, et al. Adv. Energy Mate C. Cocoyer, et al. Appl. Phys. Lett. 88 (2006) 133108. [19] C. Cocoyer, et al. Thin Solid Films 511 (2006) 517. [20] L. Mu¨ller-Meskamp, et al. Adv. Mater. 24 (2012) 906.[21] J.B. Kim, et al. Nat. Photonics 6 (2012) 327. [22] Y. Yang, et al. ACS Nano 6 (2012) 2877. [23] D. Ko, et al. J. Mater. Chem. 21 (2011) 16293.r. 2 (2012) 1467. 020040-4

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APL Photonics ARTICLE scitation.org/journal/app On-demand light wave manipulation enabled by single-layer dielectric metasurfaces Cite as: APL Photon. 6, 086106 (2021); doi: 10.1063/5.0057275 Submitted: 19 May 2021 •Accepted: 20 July 2021 • Published Online: 6 August 2021 Xuyue Guo, Bingjie Li, Xinhao Fan, Jinzhan Zhong, Shuxia Qi, Peng Li,a) Sheng Liu, Bingyan Wei, and Jianlin Zhaoa) AFFILIATIONS Key Laboratory of Light-field Manipulation and Information Acquisition, Ministry of Industry and Information Technology, and Shaanxi Key Laboratory of Optical Information Technology, School of Physical Science and Technology, Northwestern Polytechnical University, Xi’an 710129, China a)Authors to whom correspondence should be addressed: pengli@nwpu.edu.cn and jlzhao@nwpu.edu.cn ABSTRACT Dielectric metasurfaces have been widely developed as ultra-compact photonic elements based on which prominent miniaturized devices of general interest, such as spectrometers, achromatic lens, and polarization cameras, have been implemented. With metasurface applications taking off, realizing versatile manipulation of light waves is becoming crucial. Here, by detailedly analyzing the light wave modulation prin- ciples raising from an individual meta-atom, we discuss the minimalist design strategy of dielectric metasurfaces for multi-dimensionally manipulating light waves, including parameter and spatial dimensions. As proof-of-concepts, those on-demand manipulations in different dimensions and their application potentials are exemplified by metasurfaces composed of polycrystalline silicon rectangle nanopillars. This framework provides basic guidelines for the flexible design of functionalized metasurfaces and the expansion of their applications as well as implementation approaches of more abundant light wave manipulations and applications using hybrid structures. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0057275 I. INTRODUCTION The available functionalities of optical elements come from the effective manipulation of the light wave’s fundamental parame- ters, e.g., amplitude, phase, and polarization. Therefore, structuring materials with capabilities to manipulate light waves has been a long- concerned issue and attracted significant interest. To overcome the physical limitations imposed by conventional natural materials and traditional optical devices, the emerging metamaterials1,2exhibit unprecedented properties and lead to various novel optical effects.3–9 However, challenging problems, e.g., high losses and costly fabrica- tion associated with bulky structures, especially hinder them from practical applications. Until recently, the advent of metasurfaces10–17 that are characterized as reduced dimensionality of metamaterials makes the breakthrough to dramatically reduce the fabrication com- plexity and increase the design flexibility,18–27providing an elegant solution to those problems aroused in metamaterial-based optical devices. In the past decade, metasurfaces have been extensively stud- ied for engineering the fundamental parameters of light waves.28–35A considerable amount of metasurfaces have been developed with impressive applications in realms of holographic imaging,36,37polar- ization conversion,38,39functional devices,16,17multiplexing,26,27and nonlinear optics.40–42Nowadays, particular initiatives have been taken to enable multifunctional metasurfaces, which are based on the multi-dimensional manipulation of light waves. Recent progress has made some achievements, for instance, the introduction of unique structures (few-layer,43diatomic,44and folding45) and com- positional materials (liquid crystal,46phase change material,47and two-dimensional material48) provides additional degree of free- doms (DoFs) for manipulating light waves with metasurfaces in both parameter and space dimensions. In contrast, the use of a simpler structure to achieve multi-dimensional light wave control has greater advantages in practical applications and device fabri- cations. Although some relevant studies have been reported,20,21 the characteristics of multi-dimensional light wave control that can be realized by a minimalist structure have not been sys- tematically analyzed, and the relationship between the manipula- tion of parameter dimension with spatial DoFs has not been well discussed. APL Photon. 6, 086106 (2021); doi: 10.1063/5.0057275 6, 086106-1 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app Here, by detailedly analyzing the structural birefringence of an individual meta-atom in single-layer dielectric metasurfaces, the light wave modulation principle for different parameter dimen- sions and spatial DoFs is discussed, based on which the minimalist design strategy of dielectric metasurfaces for modulation require- ment of multiple dimensions and DoFs is demonstrated. Accord- ing to diverse control principles, we design metasurfaces composed of polycrystalline silicon rectangle nanopillars and then demon- strate multifunctional applications of such minimalist metasur- faces, including phase-only holography, complex-amplitude holog- raphy, 3D holographic scene, axial modulation of light field, and polarization-encrypted holography. Meanwhile, the applicable prin- ciples of manipulating light waves in broadband and 3D space are analyzed. II. THEORY To construct the modulation principles for different parame- ter dimensions, we first investigate the light wave modulation effect of an individual meta-atom in a single-layer dielectric metasur- face. Figure 1 presents the schematic illustration of the wavefront modulation mechanism of the single-layer dielectric metasurface. According to the effective medium theory,49the meta-atom is an effective anisotropic structure that supports large refractive index contrast between orthogonal polarizations of light. Therefore, the complex transmission property of such a birefringent meta-atom can be expressed as J=R(−θ)⎡⎢⎢⎢⎢⎢⎣Toeiφo0 0 Toeiφe⎤⎥⎥⎥⎥⎥⎦R(θ), (1) where R(θ) is the rotation matrix, and the middle matrix accounts for the transmission amplitudes ( To,Te) and phases ( φo,φe) alongthe ordinary and extraordinary axes, respectively, as shown in Fig. 1(a). Assuming that two orthogonal polarizations have uniform transmission amplitude, i.e., To=Te=T, one can further simplify the Jones matrix according to the incident polarization. It is well known that the light–matter interaction is generally described as the response of two kinds of polarization states, that is, the linear polarization (LP) and circular polarization (CP). Thus, we take these two typical polarizations as examples, and the corresponding Jones matrices in the CP basis [ EREL]Tand LP basis [ EHEV]Tsubse- quently can be written as (the subscript R/L denotes the right/left CP state, and H/V denotes the horizontal/vertical LP state, respectively) J(CP)=Teiφ0⎡⎢⎢⎢⎢⎢⎣cos(δ/2) i sin(δ/2)e−i2θ i sin(δ/2)ei2θcos(δ/2)⎤⎥⎥⎥⎥⎥⎦, (2) J(LP)=T⎡⎢⎢⎢⎢⎢⎣eiφ1eiφ2 eiφ2eiφ3⎤⎥⎥⎥⎥⎥⎦,⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩eiφ1=cos2θeiφo+sin2eiφe, eiφ2=cosθsinθeiφe−cosθsinθeiφo, eiφ3=sin2θeiφo+cos2θeiφe, (3) where δ=(φo−φe) and φ0=(φo+φe)/2 depict the phase retar- dation and propagation phase based on ordinary and extraordinary components, respectively. The above Jones matrices cannot be directly connected with the modulable parameter dimensions. Therefore, to address legible modulation principles, we further consider the whole output fields, which are the composition of different polarizations, as schemati- cally shown in Figs. 1(b) and 1(c). For the incidence of the ∣R⟩state, the output field naturally consists of two components with orthog- onal polarizations, namely, the co-polarized and cross-polarized components; thus, the output vector field is expressed as FIG. 1. Wavefront modulation mechanism of the single-layer dielectric metasurface. (a) Schematic illustration of the dielectric metasurface. Inset: transmission property of a meta-atom. (b) and (c) Transmission properties of a meta-atom corresponding to two kinds of bases. (d) Conversion of polarization states on the Poincaré sphere. APL Photon. 6, 086106 (2021); doi: 10.1063/5.0057275 6, 086106-2 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app Ecp out=Teiφocos(δ/2)∣R⟩+iTeiφosin(δ/2)ei2θ∣L⟩. (4) As for the incidence of linear polarized light field with ∣H⟩or ∣V⟩state, the output field consequently presents a [eiφ1eiφ2]Tor [eiφ2eiφ3]Tstate. Here, we consider a superposition state of the ∣H⟩ and∣V⟩states, namely, the ∣D⟩state, as a generalized model, and thus, the output vector field can be further expressed as Elp out=Teiφ1∣H⟩+Teiφ2∣D⟩+Teiφ3∣V⟩. (5) The modulation process introduced by the conversion of polar- ization states on the Poincaré sphere is shown in Fig. 1(d). Clearly, the above equations provide intuitionally controllable parameter dimensions, including amplitude, phase, and polarization. Accord- ing to Eq. (1), meta-atoms can be regarded as waveplates with arbitrary phase retardation ( δ) achieved by structuring the bire- fringence. Meanwhile, this phase retardation results in that the two components with orthogonal CPs have complementary inten- sities of T2cos2(δ/2) and T2sin2(δ/2), as shown in Eq. (4). In this principle, some polarization transformers50and ultrathin energy tailorable splitters16,51have been designed. Obviously, this mod- ulation on amplitude or polarization only refers to single DoF control. To showcase the modulation capabilities of different param- eter dimensions and spatial DoFs, we categorize the correspond- ing controllable wavefront into different cases, which are shown in Table I. It is worth noting that, for the CP basis, this inherent intensity relationship disables the independent control of the wave- front amplitudes corresponding to these two components; therefore, the wavefront modulation has been focused on the cross-polarized component, and given that the co-polarized component is a back- ground noise.52For pure phase modulation, as Eq. (4) shows, these two components have a communal propagation phase exp(i φ0), and the cross-polarized component experiences an abrupt phase change of ±2θ, i.e., the well-known geometric phase.53These two types of phases are determined by the geometric size and azimuthal angle of the meta-atom, respectively, which can be directly modu- lated by the φ0(case 1) and θ(case 2), corresponding to two DoF modulations. The phase modulation has been widely utilized for two- dimensional holographic imaging and reproducing special phase pattern. While by contrast, the manipulation capability with respect to three DoFs greatly improves the performance of metasurfaces in integrated multifunctional optical devices. In scalar optics, thecomplete information of a light field requires both amplitude and phase, namely, complex amplitude. Here, from the complex ampli- tude distribution of the cross-polarized component in the CP basis, i.e.,Tsin(δ/2)exp[i( φ0+2θ)], one can recognize that the amplitude and phase are determined by T,δand φ0, 2θ, respectively (case 3 and case 4). Thus, both the amplitude and phase can be completely and independently controlled, and benefiting from this, the com- plex amplitude modulation has advantages in 3D space imaging over amplitude- or phase-only modulation schemes.52It is important to point out that these two methods have an unavoidable directly trans- mitted component, which especially affects the axial modulation. Therefore, a complex amplitude modulation method with extra axial DoF control is introduced here (case 5). The above discussions focus on the scalar field, while the pos- sibility in simultaneous control of polarization and phase provides huge prospect to develop polarization-dependent optical devices and introduces extra polarization channels to increase the DoFs. For this reason, the optical responses to each component should be taken into account. For instance, in the case of CP basis, the com- bined effect of propagation phase and opposite geometric phases endows independent modulation phases φ0±2θonto two orthog- onal bases54,55(case 6). While for the case of LP basis, as Eq. (3) shows, one can obtain φ1=φoand φ3=φewhen the meta-atoms are arranged without rotation ( θ=0), that is, two independent phase patterns can be implemented on two orthogonal linear polarization states (case 7). Then, taking rotation into consideration, as shown in Eq. (5), three phase patterns, φ1,φ2, and φ3, which are dependent on the geometric parameters φo,φe, and θ, can be implemented on three linear polarization states56(case 8). For modulation with more parameter dimensions, amplitude, phase, and polarization response are inevitably associated with each other, when adjusting the geometric parameters of individual meta- atom, that is, the number of controllable parameters is limited to two, as shown in Table I. To break this limitation, two orthogonal polarization bases whose amplitude and phase can be precisely and independently modulated are primarily required. For the CP basis, this expectation cannot be achieved due to their correlated ampli- tudes, while for the LP basis, the background noise arising from phase-only modulation leads to the inaccuracy of superposition state. Therefore, a capable implementation is using hybrid structures based on exploring the inherent relationship between meta-atoms and associating each parameter dimension with a certain structural parameter of meta-atom. TABLE I. Categorized modulation capabilities of a single meta-atom in a single-layer dielectric metasurface. PD: parameter dimension, SDoF: spatial degree of freedom. Master variable Controllable wavefront PD ×SDoF Case 1 φ0 Eout=exp(i φ0) 1 ×2 Case 2 θ Eout=exp(±i2θ) 1 ×2 Case 3 δ,θ Eout=sin(δ/2)exp(i2 θ) 2 ×3 Case 4 T,δ,θ Eout=Tsin(δ/2)exp(i2 θ) 2 ×3 Case 5 T,φ0,θ Eout=Texp[i( φ0+2θ)] 2 ×3, 1×1 Case 6 φ0,θ Eout=exp[i( φ0+2θ)]∣L⟩+exp[i( φ0−2θ)]∣R⟩ 2×2 Case 7 φo,φe Eout=exp(i φo)∣H⟩+exp(i φe)∣V⟩ 2×2 Case 8 φo,φe,θ Eout=exp(i φ1)∣H⟩+exp(i φ2)∣D⟩+exp(i φ3)∣V⟩ 2×2 APL Photon. 6, 086106 (2021); doi: 10.1063/5.0057275 6, 086106-3 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app As a proof-of-concept, we design and fabricate metasurfaces corresponding to each case. Here, we choose the poly-Si meta-atoms on a fused silica substrate, which have rectangular cross sections with square lattice arrangement, to design and fabricate metasur- faces by using COMS compatible processes (details can be found in the Appendix). The geometric size (height H, length L, width W, and period P) of the meta-atom is variable for different cases, depend- ing on master variables. For simplicity and generality, computer- generated holograms (CGHs)57are chosen to implement most of the following experiments, which are succinct to testify the capability of manipulating the light wave. All experiments were performed at the wavelength of visible light band. III. EXPERIMENT AND RESULTS To testify the performance of pure phase modulation, that is, cases 1 and 2, two-dimensional holographic imaging is implemented experimentally. In practice, phase-only CGHs are generated by use of the typical Gerchberg–Saxton (GS) algorithm.58Crucially, thepure phase modulation based on propagation phase and geometric phase have different modulation precisions and distinct require- ments for the selection of meta-atoms. In the case of propagation phase modulation, the phase-only CGH needs to be discretized, and higher nanopillars are required to ensure sufficient phase modula- tion depth. In contrast, geometric phase modulation only requires a single geometry and has higher modulation accuracy and efficiency via rotating nanopillars. Figures 2(b) and 2(d) show the scanning electron microscope (SEM) images of fabricated metasurfaces cor- responding to cases 1 and 2, where the metasurfaces both have 1000 ×1000 meta-atoms, but different heights (case 1: 610 nm and case 2: 350 nm) and periods (case 1: 450 nm and case 2: 300 nm). Figure 2(a) shows two target images (grayscale and binary images, respectively) and experimentally reconstructed results in case 1 at a wavelength of 633 nm. It can be seen that these target images are reconstructed with high performance. As a contrast, the binary image is also reconstructed by means of case 2 under the same experimental condition, and the corresponding result is shown in the fourth column of Fig. 2(c). Clearly, the reconstructed image FIG. 2. Holographic imaging based on the phase-only modulations of metasurfaces. (a) Target images and experimental results of case 1 at a wavelength of 633 nm. (b) and (d) SEM images of fabricated metasurfaces corresponding to cases 1 and 2, respectively. Scale bars are 1 μm. (c) Experimental results of case 2 at the wavelengths of 473, 488, 532, 633, and 670 nm, respectively. (e) Simulated transmittance and sinusoidal term spectra of the selected meta-atom in case 2. The geometric parameters areL=174 nm and W=104 nm. APL Photon. 6, 086106 (2021); doi: 10.1063/5.0057275 6, 086106-4 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app in this case exhibits a higher fidelity, which results from the high accuracy of geometric phase modulation. In addition, the charac- teristic of independence of wavelength enables us to operate geo- metric phase modulation in a broad bandwidth. Figures 2(c) and 2(e) show the reconstructed results and response spectrum of the selected meta-atom at multiple wavelengths. As shown, although the geometric phase modulation has lower transmittance at short wavelengths, it still exhibits good broadband characteristics as the experimental results present clear reconstructed holographic images at all operating wavelengths. Further testifications of complex amplitude modulation are shown in Figs. 3 and 4, where complex amplitude hologram and 3D holographic scenes are demonstrated. Figures 3(a) and 3(b) show the simulated transmittance, propagation phase, and sinusoidal term of selected meta-atoms in case 3. The complex amplitude holo- grams are calculated by the Fourier transform of the target image. Figures 3(d) and 3(e) show the fabricated metasurface ( H=610 nm, P=450 nm) and reconstructed result of case 3. The experiment is carried out with the setup shown in Fig. 3(c) at a wavelength of 670 nm, and the image reconstructed on the screen is photoed by a camera. Compared with these two previous phase-only modulation methods [Figs. 2(a) and 2(c)], the complex amplitude modulation method intuitively improves the imaging quality and reduces the background noise since both the amplitude and phase are faithfully reproduced. Figure 4(a) shows the combined amplitudes Tsin(δ/2) and propagation phases of selected meta-atoms in case 4. In case 3, the precondition that transmittances of these meta-atoms areconstant limits the geometry selectivity. While in case 4, the con- trol of parameter Tdoes not directly affect the final modulation effect but supplies a greater tolerance to the selection of the meta- atom in case 4. Consequently, under the same height of the meta- atom, the operating wavelength is reduced to 633 nm. In order to fully demonstrate the advantages of complex amplitude modu- lation, a 3D holographic scene, which consists of letters “N,” “P,” and “U” localized at three lateral planes, is performed. Figures 4(b) and 4(c) illustrate the operation principle and experimental setup. For the calculation of CGH, each letter image at certain diffrac- tion distances is back-propagated to the metasurface plane by the beam-propagation method. Figure 4(d) shows the simulated and experimentally observed results at three lateral planes, respectively. In this experiment, we introduce an optical microscopy setup with the cross-polarized analyzer in order to avoid the influence of the co-polarized component. It is noteworthy that the experimentally reconstructed images have almost identical profiles with simulated ones, which powerfully demonstrates the great capability of the com- plex amplitude modulation of the metasurface for reconstructing target images in 3D space. The full control of the amplitude and phase significantly improves the quality and capability of the reconstructed image. Nevertheless, in the above two cases, the unavoidable co-polarized component arising from the incomplete spin conversion, i.e., the non-zero Tcos(δ/2), leads to a drawback that prevents the above methods from axial modulation without polarization filtering. How- ever, eliminating the co-polarized component is difficult to imple- ment in some special situations, such as focusing. Therefore, FIG. 3. Holographic imaging of the metasurface with complex amplitude modulation. (a) and (b) Simulated transmission amplitudes, propagation phases, and sinusoidal term of these selected meta-atoms in case 3. (c) Schematic illustration of the experimental setup. HWP: half-wave plate and QWP: quarter-wave plate. (d) SEM image of the fabricated metasurface corresponding to case 3. The scale bar is 1 μm. (e) Reconstructed results of case 3 at a wavelength of 670 nm. APL Photon. 6, 086106 (2021); doi: 10.1063/5.0057275 6, 086106-5 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app FIG. 4. 3D imaging of the metasurface with complex amplitude modulation. (a) Simulated amplitudes and propagation phase of the selected meta-atoms in case 4. (b) Schematic illustration of the 3D holographic scene. (c) Schematic illustration of the experimental setup. Inset: SEM image of the fabricated metasurface. The scale bar is 1μm. (d) Simulated and experiment results at a wavelength of 633 nm. in case 5, the phase retardation δis fixed as π, making sure that the incident spin polarization is totally transformed into the orthogo- nal one; hence, the amplitude and phase modulations are dependent onTand φ0+2θ, respectively, as shown in Fig. 5(a). Obviously, the amplitude is only dependent on the transmittance of meta-atom [Fig. 5(b)], but the phase term is related to both the propagation and geometric phases. Unfortunately, the transmittance and prop- agation phase are jointly related to the geometry of meta-atoms. To break this relationship, an opposite rotation angle φ0/2 should be added onto θ, i.e., θ’=θ−φ0/2, and then the amplitude and phase are independently and completely controllable. As an example, an axially structured light field with sinc- functional intensity distribution (calculated by the spatial spectrum optimization method based on the Durnin ring59,60) is demonstrated to assess this axial tailoring capability. Figure 5(d) shows the mea- sured intensity distribution (normalized) in the y–zplane, which is observed through the setup shown in Fig. 5(c). The microscope sys- tem is localized on a linear translation stage with a scanning interval of 10 μm. The simulated and measured on-axis intensity distribu- tions (normalized) are displayed in Fig. 5(e). As shown, this method can sustain the construction of axial light field. The above discussions all refer to the wavefront manipula- tion of scalar light field, namely, the cross-polarized component. Inaddition to the enhancement of multiplexing capability, numerous intriguing phenomena related to vector fields, such as their con- struction, enhanced longitudinally polarized component, and super- resolution focusing, are based on the combined modulation of two spin states.61,62To address polarization-dependent light field modu- lation, more parameters should be taken into account. As is known, the geometric phase is always accompanied by a “twin field,” which originates from the phase accumulation of opposite CP state tran- sition. Therefore, by combining the propagation phase, two CPs can obtain independent phase modulation of φ0±2θ, as shown in Fig. 6(a). However, in this case, the inherent amplitude correlation disables the independent amplitude modulation of two CPs. Thus two CPs are commonly considered to have unitary amplitude, i.e., sin(δ/2)=1. Here, a holographic reconstruction of two complemen- tary images [Figs. 6(c) and 6(d)] is employed to showcase the poten- tial in polarization-encrypted application. Figures 6(e)–6(g) display the experimental results under the incidences of light fields with dif- ferent polarizations. As shown, when a linear polarized light field illuminates, the metasurface outputs a uniform spot without pattern, but the polarization-dependent patterns show up for the CP incident light fields. In comparison, LP-based methods provide more optional channels. Thanks to the “structural birefringence” of meta-atoms, APL Photon. 6, 086106 (2021); doi: 10.1063/5.0057275 6, 086106-6 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app FIG. 5. Longitudinal modulation enabled by the metasurface. (a) Schematic of the modulation effect in case 5. (b) Transmittances of 17 selected meta-atoms. (c) Schematic illustration of the experimental setup. Inset: SEM image of the metasurface. The scale bar is 100 μm. (d) Measured intensity distribution in the y–zplane. (e) Simulated and measured on-axis intensity distributions. arbitrary manipulation can be implemented on a certain polarized component modulated along ordinary or extraordinary axis theo- retically, as described in Eq. (1). Here, the transmittance of each meta-atom is set to be unitary. In Eq. (3), when θ=0, one obtains φ1=φoand φ3=φe, namely, two independent phase modulationscan be implemented on two orthogonal linear polarization chan- nels. While taking rotation into account, three independent phase modulations can be implemented on three linear polarization chan- nels. The experimental setup and SEM images of two LP-based metasurfaces without and with rotation are shown in Figs 7(a)–7(c). FIG. 6. Polarization-dependent holographic imaging of the metasurface based on the combined modulations of two CPs. (a) Schematic of the combined modulation of two CPs. (b) SEM image of the metasurface. The scale bar is 1 μm. (c) and (d) Target images encoded on two CPs; reconstructed results for the incidence of a (e) linearly polarized, (f) right-handed CP, and (g) left-handed CP light field. APL Photon. 6, 086106 (2021); doi: 10.1063/5.0057275 6, 086106-7 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app FIG. 7. Polarization-encrypted imaging of the metasurface. (a) Schematic illustration of the experimental setup. (b) and (c) SEM images of the metasurface corresponding to cases 7 and 8. Scale bars are 0.5 μm. (d) and (e) Experimental results of polarization-encrypted imaging based on modulation mechanisms of cases 7 and 8. The red and blue arrows depict the incident and detected polarization directions, respectively. For the metasurface without rotation, four letters are encoded into the horizontal and vertical polarizations by phases φ1and φ3, respectively. The experimental results are depicted in Fig. 7(d). Under the illumination of a diagonal polarized light field, patterns encoded in two polarization channels are simultaneously recon- structed. By rotating the polarization analyzer, the reconstructed pattern is switched from “AB” to “CD.” For the second metasurface, an additional polarization channel is available due to the introduc- tion of geometry rotation. Three letter patterns are encoded into the horizontal, vertical, and diagonal linear polarizations by phases φ1, φ2, and φ3, respectively. As shown in Fig. 7(e), for the incidence of a diagonal polarized light field, three patterns in three polariza- tion channels are simultaneously reconstructed without the ana- lyzer. While for the cases of H- or V-polarization incidence, the pat- tern in the orthogonal polarization channel disappears, respectively. Furthermore, individual polarization channels can be switched by changing the incident and analyzed polarization directions. There- into, the “XYZ” pattern, namely, diagonal polarization channel, can be obtained with the orthogonal polarizer and analyzer. IV. DISCUSSIONS The on-demand modulation principles of single-layer dielectric metasurfaces for multiple dimension control have been theoretically and experimentally exhibited, but more details merit discussions. Notably, arbitrary modulation can be implemented through manip- ulating the “structural birefringence” of meta-atoms and various applications can be realized according to the above principle. How- ever, limited by the properties of natural materials, the modulation depth and width are restricted to meet some particular applications, which also results in confined operating wavelengths and repre- sents a daunting exploratory and computational problem. Therefore, invariant parameters and varying thicknesses are used in different cases to obtain an enough modulation range.Second, the systematic strategy for on-demand light wave manipulation demonstrated here avoids unnecessary complexity in both the design process and experimental operation, which presents the full potential of single-layer dielectric metasurfaces, and leads to a series of applications. The pure phase modulation can be realized in two ways, among which the geometric phase modulation has been widely used in device design due to its convenience, high precision, and broad bandwidth. By contrast, the complex amplitude mod- ulation has an advantage of information density over phase-only hologram, which leads to holographic images with higher quality, higher fidelity, and the reconstruction in 3D space. Moreover, for applications involving holographic data encryption or storage, the complex amplitude hologram can greatly increase the storage capac- ity. Furthermore, the axial modulation method in case 5 enriches the functionalities of complex amplitude modulation and provides an additional DoF in 3D light wave manipulation, as well as an approach for constructing tightly focusing fields with longitudi- nally oscillating polarization. In addition, the polarization-encrypted holography exhibited in cases 6–8 effectively enlarges the design space of polarization-dependent devices, and further applications, such as polarization-multiplexing and information encryption, can be expected. It is no doubt that such a strategy provides a basic guide- line for the flexible design of optical metasurfaces and an effective way for the expansion of their applications. Finally, besides the functionality, the modulation efficiency is another concerned issue. Notably, the pure phase modulation meth- ods have advantages of efficiency because of the excellent encoding techniques. On the other hand, the modulation efficiency is closely related to the amplitude coefficient of the Fourier transform CGHs, i.e.,Tsin(δ/2)≠1. In our experiment, taking the absorption of mate- rial into account, the diffraction efficiency in case 2 is 62.5% (with polarization conversion efficiency exceeding 95%), while in cases 3 and 4, it is about 10%. For axial modulation, the optimized spa- tial spectrum can significantly enhance the generation efficiency to about 20%, but it is still strongly dependent on the pre-established APL Photon. 6, 086106 (2021); doi: 10.1063/5.0057275 6, 086106-8 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app distribution. In polarization-dependent modulations, the response of the meta-atoms and the quality of fabrication are the main factors affecting the efficiency; here, the diffraction efficiency of cases 6–8 is all about 50%. As a whole, the demonstrated design principles and devices can be characterized as low loss. V. CONCLUSION In summary, we have systematically discussed multi- dimensional light wave manipulation via single-layer dielectric metasurfaces. To showcase such a strategy, the “structural birefringence” of meta-atoms on different polarization bases is considered, and the modulation capabilities from single to multiple parameter dimensions are categorized. Based on the proposed mechanism, complete manipulation of the wavefront amplitude, phase, and polarization state has been achieved, and the poly-Si meta-atoms and holographic method are employed to experimen- tally demonstrate how various functionalities are achieved. The results show that single-layer dielectric metasurfaces exhibit strong modulation capability in various light wave manipulation, and the design principle is simple but has powerful extension for the flexible design of optical metasurfaces. This work offers a systematic and generalizable method toward manipulating light waves at will with meta-devices, and provides a possible approach for achieving more abundant manipulation and applications through hybrid structures. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos. 91850118, 11774289, 11634010, 61675168, 12074313, and 11804277), the National Key Research and Development Program of China (Grant No. 2017YFA0303800), the Natural Science Basic Research Program of Shaanxi (Grant No. 2020JM-104), the Fundamental Research Funds for the Central Universities (Grant Nos. 3102019JC008 and 310201911cx022), and the Innovation Foundation for Doctor Dissertation of Northwest- ern Polytechnical University (Grant Nos. CX202046, CX202047, and CX202048). We thank the Zhiwei Song of National Center for Nanoscience and Technology for supplying the materials as well as the Analytical and Testing Center of Northwestern Polytechnical University. APPENDIX: METHOD The metasurfaces were fabricated based on the process of deposition, patterning, lift off, and etching. At first, a 350 nm (610 nm)-thick poly-Si film was deposited on a 500 μm-thick fused silica substrate by inductively coupled plasma enhanced chemical vapor deposition (ICPECVD), and then a 100 nm-thick hydrogen silsesquioxane electron beam spin-on resist (HSQ, XR-1541) was spin-coated onto the poly-Si film and baked on a hot plate at 100○C for 2 min. Next, the desired structures were imprinted by using stan- dard electron beam lithography (EBL, Nanobeam Limited, NB5) and subsequently developed in NMD-3 solution (concentration 2.38%) for 2 min. 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5.0058124.pdf

J. Chem. Phys. 155, 034108 (2021); https://doi.org/10.1063/5.0058124 155, 034108 © 2021 Author(s).First-principles correction scheme for linear-response time-dependent density functional theory calculations of core electronic states Cite as: J. Chem. Phys. 155, 034108 (2021); https://doi.org/10.1063/5.0058124 Submitted: 27 May 2021 . Accepted: 30 June 2021 . Published Online: 15 July 2021 Augustin Bussy , and Jürg Hutter ARTICLES YOU MAY BE INTERESTED IN An improved Slater’s transition state approximation The Journal of Chemical Physics 155, 034101 (2021); https://doi.org/10.1063/5.0059934 Simplified tuning of long-range corrected density functionals for use in symmetry-adapted perturbation theory The Journal of Chemical Physics 155, 034103 (2021); https://doi.org/10.1063/5.0059364 Size-consistent explicitly correlated triple excitation correction The Journal of Chemical Physics 155, 034107 (2021); https://doi.org/10.1063/5.0057426The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp First-principles correction scheme for linear-response time-dependent density functional theory calculations of core electronic states Cite as: J. Chem. Phys. 155, 034108 (2021); doi: 10.1063/5.0058124 Submitted: 27 May 2021 •Accepted: 30 June 2021 • Published Online: 15 July 2021 Augustin Bussya) and Jürg Hutterb) AFFILIATIONS Department of Chemistry, University of Zurich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland a)Current address: Department of Chemistry, University of Zurich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Author to whom correspondence should be addressed: augustin.bussy@chem.uzh.ch b)Electronic mail: hutter@chem.uzh.ch ABSTRACT Linear-response time-dependent density functional theory (LR-TDDFT) for core level spectroscopy using standard local functionals suffers from self-interaction error and a lack of orbital relaxation upon creation of the core hole. As a result, LR-TDDFT calculated x-ray absorption near edge structure spectra needed to be shifted along the energy axis to match experimental data. We propose a correction scheme based on many-body perturbation theory to calculate the shift from first-principles. The ionization potential of the core donor state is first computed and then substituted for the corresponding Kohn–Sham orbital energy, thus emulating Koopmans’s condition. Both self-interaction error and orbital relaxation are taken into account. The method exploits the localized nature of core states for efficiency and integrates seamlessly in our previous implementation of core level LR-TDDFT, yielding corrected spectra in a single calculation. We benchmark the correction scheme on molecules at the K- and L-edges as well as for core binding energies and report accuracies comparable to higher order methods. We also demonstrate applicability in large and extended systems and discuss efficient approximations. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0058124 I. INTRODUCTION X-ray absorption spectroscopy (XAS) is a major characteriza- tion tool used in many fields of natural science. The technique pro- vides a local and element specific probe, yielding insights into the geometric and electronic structure of matter. In particular, the x- ray absorption near edge structure (XANES) region of the spectrum holds information about the chemical state (coordination number, oxidation state, and so on) of the probed atom. As technology pro- gresses and quality light sources become more accessible, XANES is used increasingly often. Following this trend, many theoretical approaches have been developed to help interpret experiments. One of the most widespread computational methods for the simulation of XANES is time-dependent density functional theory (TDDFT). It offers a favorable trade-off between cost and accuracy and is relatively easy to use due to its mostly “black-box” nature.In its standard formulation,1,2TDDFT is best suited for UV–vis spectroscopy, where electronic transitions from valence to low lying unoccupied bound states take place. Much effort has been made in adapting the theory to core state spectroscopy. Most notably, core–valence separation3(CVS) has been used with both flavors of TDDFT, real-time4(RT-TDDFT) and linear-response5–7(LR- TDDFT), allowing for the calculation of excitations from core states at an affordable cost. Other approaches rely on iterative solvers, which can directly target high energy core transitions,8,9thus bypass- ing the CVS. The different implementations yield consistent results, and the choice of the exchange–correlation functional typically has a larger impact on accuracy.10 While TDDFT generated XANES spectra are known to have accurate relative feature intensities and spacing, they usually have to be translated along the energy axis to match experiments. The main reasons behind this shortcoming of TDDFT lie in the lack of orbital J. Chem. Phys. 155, 034108 (2021); doi: 10.1063/5.0058124 155, 034108-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp relaxation upon the creation of a core hole and self-interaction errors11(SIEs). There are multiple ways of dealing with these issues. The simplest is to apply a system dependent empirical shift,12–14 although this limits TDDFT to a purely descriptive role. Similarly, an atom specific shift can be calibrated over multiple systems to be later applied to similar calculations.15,16A common approach to computing the shift from first-principles is by performing a sep- arateΔSCF10,17calculation. By taking the difference between the converged ground and first excited states’ total energies, SIE mostly cancels out and orbital relaxation is taken into account. However, such calculations may be hard to set up and converge. Other tech- niques have been proposed to tackle SIE in TDDFT, for example, by basing the TDDFT calculation on a SIE corrected ground state cal- culation,11,18fitting the fraction of Hartree–Fock exchange in hybrid functionals,6or developing core TDDFT specific functionals19–21 (usually relying on empirically fitted parameters). For calculations involving excitations from heavy atoms, scalar relativistic effects do also play a role. This is usually taken into account at the ground state level using the ZORA22or DKH23approach. In this work, we propose a fully ab initio correction scheme for core LR-TDDFT spectroscopy based on many-body perturba- tion theory. The method exploits the localized nature of core states for efficiency and seamlessly integrates into our previous implemen- tation of core LR-TDDFT24in the CP2K software package.25The method accounts for orbital relaxation and self-interaction error while yielding the core ionization potential as a side-product. We discuss the method’s implementation and demonstrate its applica- bility to molecules at the K- and L-edges and as a means of calculat- ing core binding energies. We also discuss efficient approximations and applications in extended systems. II. THEORY The LR-TDDFT equations are built on the solutions of the time-independent Kohn–Sham (KS) equations. Expanded in a basis of atom-centered functions, the KS orbitals read ϕiσ(r)=∑ pc0 piσφp(r), (1) where the basis elements {φp(r)}, also referred to as atomic orbitals (AOs), are typically non-orthogonal Gaussian type orbitals (GTOs) with Spq=⟨φp∣φq⟩. (2) Note that throughout this work, indices p,q,. . ., always refer to AOs; i,j,. . ., refer to occupied MOs; a,b,. . ., refer to virtual MOs; and σ,τ,. . .refer to spin. In the Sternheimer26,27approach to LR-TDDFT, vertical excita- tion energies are obtained by solving the non-Hermitian eigenvalue equation24 ω⎛ ⎜ ⎝−G 0 0G⎞ ⎟ ⎠⎛ ⎜ ⎝c+ c−⎞ ⎟ ⎠=⎛ ⎜ ⎝A+B−D B −E B−E A +B−D⎞ ⎟ ⎠⎛ ⎜ ⎝c+ c−⎞ ⎟ ⎠, (3) where the c±eigenvectors are the coefficients for the basis expansion of the LR orbitals and ωis the corresponding excitation energy. Thematrix Gis related to the basis set overlap, Gpiσ,q jτ=Spqδijδστ, (4) and the matrix Ais based on the ground state KS matrix Fσ, Apiσ,q jτ=(Fσ pq−εiσSpq)δijδστ, (5) whereεiσis the KS eigenvalue and B,D, and Eare the Hartree exchange–correlation kernel and on- and off-diagonal Hartree–Fock (HF) exchange kernel matrices, Bpiσ,q jτ=∑ rsQσ pr(riσ∣fHxc σ,τ∣sjτ)Qτ qs, (6) Dpiσ,q jτ=cHFδστ∑ rsQσ pr(rs∣iσjτ)Qτ qs, (7) Epiσ,q jτ=cHFδστ∑ rsQσ pr(rjτ∣siσ)Qτ qs, (8) where Qσis a projector on the unoccupied unperturbed space, CHF is the fraction of HF exchange, and fHxc σ,τis defined as fHxc σ,τ(r,r′)=1 ∣r−r′∣+δ2Exc δnσ(r)δnτ(r′)∣ n0(9) within the adiabatic approximation.28 The eigenvalue equation can be greatly simplified by setting c+=0 while retaining the eigenvalue accuracy.29This reduces the matrix dimensions by a factor of 2 and allows ignoring the off- diagonal exact exchange kernel matrix E. This is known as the Tamm–Dancoff approximation30(TDA) and is consistently applied throughout this work. When dealing with core-spectroscopy, specific approximations can be made for efficiency. First, core and valence excitations can be effectively decoupled within the CVS, allowing the neglect of the latter.3Because core MOs are systematically involved in the repul- sion integrals (ERIs) to be evaluated, additional screening can take place31and the related cost is lowered. The dimension of the eigen- value problem in Eq. (3) is reduced as well, bringing down the cost of diagonalization. Moreover, core states of interests can be treated serially within the sudden approximation,7,32further reduc- ing matrix dimensions to that of the ground state KS matrix. In this context, all the four-center ERIs needed for the kernel matrices have either the form (pI∣qJ)or(pq∣IJ), whereϕIandϕJare core donor MOs. In our recent implementation24of core level LR-TDDFT in the CP2K software package,25we introduced a local resolution of the identity (RI) scheme that further reduces the cost of ERIs, (pI∣qJ)≈∑ μ,ν(pI∣μ) (μ∣ν)−1(ν∣qJ) (10) and (pq∣IJ)≈∑ μ,ν(pq∣μ) (μ∣ν)−1(ν∣IJ), (11) J. Chem. Phys. 155, 034108 (2021); doi: 10.1063/5.0058124 155, 034108-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where the RI basis {χμ(r)}only consists of Gaussian functions cen- tered on the excited atom. Such a choice of basis is only possible due to the localized nature of core states. In the case of K-edge spec- troscopy, only the 1s core MO is considered. For the L 2,3-edge, the core MOs span all three 2p states. LR-TDDFT yields excitation energies as corrections to ground state orbital energy differences,43i.e., ω=εa−εI+Δxc, (12) whereεais the orbital energy of a virtual receiving MO and εIis that of the core donor MO. Under Koopmans’s condition, εaand εIare identified as the electron affinity (EA) and the negative ion- ization potential (IP), respectively. However, mostly because of the self-interaction error, DFT orbital energies are far from the actual EAs and IPs and the condition does not hold.44Nonetheless, and especially since ∣εI∣≫∣εa∣, absolute LR-TDDFT core excitation ener- gies are expected to be greatly improved if −εIwere to be substituted by an accurate value of the core IP in Eq. (5). This leads to a rigid shift of the spectrum, while the LR-TDDFT relative energies and intensi- ties are preserved. A similar approach was developed by Verma and Bartlett,45where an IP corrected exchange–correlation potential was designed and successfully applied. In the current work, the focus is on the correction of a single core IP using many-body perturbation theory, regardless of the chosen exchange–correlation functional. Based on a Hartree–Fock ground state calculation, ionization potentials can be computed using second-order electron propagator theory46,47by solving the following equation: IPI=−εI−1 2∑ ajk∣⟨Ia∣∣jk⟩∣2 −IPI+εa−εj−εk−1 2∑ abj∣⟨Ij∣∣ab⟩∣2 −IPI+εj−εa−εb, (13) where indices aand brefer to virtual orbitals and jandkrefer to occupied HF spin-orbitals. Note that the antisymmetrized four- center integrals ⟨Ia∣∣jk⟩and ⟨Ij∣∣ab⟩systematically involve the spin-orbital for which the IP is computed. Equation (13) can be formally adapted to a DFT reference by constructing the general- ized Fock matrix based on the KS orbitals. The occupied and virtual orbitals are then separately rotated such that they become canonical with respect to the generalized Fock matrix. These pseudocanon- ical orbitals and corresponding new eigenvalues are then used in Eq. (13). This is known as the GW2X method.48Alternatively, we also propose the GW2X∗method, in which KS orbitals are directly used, IPGW2X∗ I=−fII−1 2∑ ajk∣⟨Ia∣∣jk⟩∣2 −IPI+faa−fjj−fkk −1 2∑ abj∣⟨Ij∣∣ab⟩∣2 −IPI+fjj−faa−fbb, (14) where fIIis the diagonal element of the generalized Fock matrix corresponding to the ϕIKS orbital. This approach has an effi- ciency advantage over the original GW2X as potentially expensive orbital rotations are avoided. Its implementation is also simpler. The second-order electron propagator method accounts for relax- ation effects upon creation of the core hole as well as electron correlation.49Moreover, it is free of self-interaction error since thegeneralized Fock matrix, which is built with 100% exact exchange, is used. Finally, this is a fully ab initio scheme that does not rely on any empirical parameter. Both GW2X and GW2X∗methods were implemented in CP2K, where they can be employed as a correction scheme for core level LR-TDDFT and/or a way of computing core ionization potentials for x-ray photoelectron spectroscopy (XPS). For each excited core stateϕIin the system, the RI three-center ERIs (pq∣ν)and(μ∣rI)are first computed. It is followed by contraction steps from AOs to MOs such that the anti-symmetrized ERIs needed in Eq. (13) or Eq. (14) can be constructed. The electron propagator equation is then solved using a Newton–Raphson scheme, and the resulting IP is substituted into Eq. (5). From there, the normal LR-TDDFT problem is solved, reusing the precomputed ERIs (pq∣ν)and(μ∣rI). Note that the size of the RI basis {χμ(r)}is independent of the system size. Hence, storing (pq∣μ)scales as 𝒪(n2)in memory at worst. Moreover, con- tracting (pq∣μ)to, e.g., (ab∣μ)has the same computational scaling as a normal matrix–matrix multiplication, namely, 𝒪(n3). Since MOs are not localized, storing a fully contracted tensor (ab∣Ij)would scale cubically in memory. This can be avoided by contracting (ab∣μ)and (ν∣Ij)first and leaving the RI contraction as the last step. The final contraction can then be done in batches, and the (ab∣Ij)integrals are never fully stored. All integral storage and contraction are done using the sparse matrix and tensor library DBCSR.60 III. BENCHMARKS AND RESULTS The implementation of the GW2X method for core states is tested on a wide range of systems, functionals, and basis sets. Bench- marks cover GW2X as a correction to LR-TDDFT for K- and L-edge spectroscopy and as a mean of calculating ionization potentials. We investigate basis set convergence and compare the method to litera- ture benchmarks for molecules. We also discuss the impact of vari- ous approximations on accuracy and apply the method to extended systems in periodic boundary conditions. A. GW2X as correction to LR-TDDFT To assess the applicability of the GW2X method as a cor- rection to core LR-TDDFT, we calculated the first K-edge exci- tation energy of 25 molecules with first and second row atoms and compared to experimental data. The benchmark set includes 16 distinct excitations from molecular C1s levels, 10 from N1s, 13 from O1s, and 4 from F1s. The structures were optimized at the def2-TZVP61/B3LYP62level. For this benchmark, we used the core-specific aug-pcX basis set63and four common hybrid func- tionals with increasing fraction of Hartree–Fock exchange [B3LYP (20%), PBE0 (25%),64PBEh (45%),65and BHHLYP (50%)].66In Fig. 1, the mean absolute deviation (MAD) of LR-TDDFT and LR- TDDFT +GW2X first excitation energies with respect to experiments is displayed for different functional and basis set combinations. The GW2X correction systematically improves the LR-TDDFT results and brings down the very disparate MADs of the different func- tionals to a similar level. This can be explained by the fact that a higher HFX fraction leads to a reduction in the self-interaction error, whereas the GW2X correction is free of it altogether. It can also be observed that increasing the basis set quality from double to quadruple zeta quality does not significantly change the results, J. Chem. Phys. 155, 034108 (2021); doi: 10.1063/5.0058124 155, 034108-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1. Mean absolute deviation of the first K-edge excitation energy calculated using pure LR-TDDFT (thin dark bars) and GW2X corrected LR-TDDFT (wide clear bars) with respect to experimental values. The benchmark set is made of 25 first and second row atom molecules and covers 43 distinct excitations. suggesting rapid convergence. Finally, the GW2X corrected PBEh(α=0.45) functional performs the best overall with MADs of 1.32, 1.38, and 1.37 eV for the aug-pcX-1, aug-pcX-2, and aug- pcX-3 basis sets, respectively. A detailed table containing the cal- culated and reference energies for each molecule is available in the supplementary material. In Table I, LR-TDDFT with GW2X correction is compared to other XANES K-edge calculation methods reported in the literature. A smaller selection of nine molecules (12 excitations) was made such as to maximize overlap with the other studies. The aug-pcX-2 basis (triple zeta quality) was chosen for similar reasons, and the PBEh (α=0.45) functional was selected because it performed best in the previous benchmark. Note that the reported MADs can only be used as the semi-quantitative measure of method quality because of thesmall sample size, patchy data, and varying basis sets. GW2X cor- rected LR-TDDFT performs on par with B3LYP- ΔSCF and equation of motion CCSD while surpassing CIS(D). SRC2-TDDFT performs better, but the core-specific range separated SRC2 functional has four parameters, which were specifically fitted on those molecules.21 Finally, the square gradient minimization (SGM) method developed by Hait and Head-Gordon73combined with ROKS and the SCAN74 functional also yields better results. However, both LR-TDDFT and GW2X correction schemes can be applied without prior knowledge of a target system, whereas SGM requires careful preparation of an initial guess. This “black-box” aspect of LR-TDDFT +GW2X makes it an interesting method for high-throughput studies. GW2X can be used to correct XANES LR-TDDFT at the L- edge, where the ionization potential of the three 2p states is com- puted and spin–orbit coupling is included at the TDDFT level. It is, however, necessary to compare calculated and experimental spectra to reliably extract the first excitation energies at the L 2and L 3peaks, making large scale benchmark studies difficult. Instead, we focus on a few molecules present in similar studies and compare results in Table II. Geometries were optimized at the def2-TZVP/B3LYP level and the PBEh( α=0.45) functional with the aug-pcX-2 basis chosen for the LR-TDDFT +GW2X calculations. All three methods perform similarly well, both at the L 2and L 3peaks and with errors com- parable to those observed at the K-edge. Note that on average, the GW2X correction induces a blue shift of 2.4 eV compared to bare LR-TDDFT, again systematically improving the results. The amount of spin–orbit splitting is well captured by all three methods. B. GW2X for core IP calculations GW2X can also be used as a stand-alone method to compute ionization potentials. In this case, all TDDFT related operations can be ignored for efficiency. Alternatively, the IPs can be obtained as a side-product of a corrected LR-TDDFT calculation. To assess the quality of GW2X in this context, the CORE65 benchmark set67 was used. It contains 65 distinct core states over 32 molecules with TABLE I. Comparison of GW2X corrected LR-TDDFT for XANES K-edge spectroscopy with similar methods from the literature. The difference between the calculated and experimental first allowed excitation is reported (in eV). The bold atom in the chemical formula is the one from which the excitation takes place. B3LYP- ΔSCF33SCAN-ROKS34SRC2-TDDFT21CIS(D)35EOM-CCSD36PBEh-GW2X u6-311(2 +,2+)G∗∗aug-cc-pCVTZ 6-311(2 +,2+)G∗∗aug-cc-pCVQZ aug-cc-pCVTZaaug-pcX-2 Expt. CH4 +0.5 0.0 ⋅ ⋅ ⋅ + 0.3 ⋅ ⋅ ⋅ + 0.9 288.037 C2H4 ⋅ ⋅ ⋅ 0.0 +0.6 ⋅ ⋅ ⋅ + 1.8 +1.2 284.738 CO −0.8 −0.4 −0.7 +2.5 −0.4 +2.5 287.439 CH2O −0.1 ⋅ ⋅ ⋅ 0.0 +2.8 ⋅ ⋅ ⋅ + 1.7 286.040 NH3 ⋅ ⋅ ⋅ − 0.5 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 0.1 +0.9 400.837 NNO −0.7 −0.1 ⋅ ⋅ ⋅ + 2.4 ⋅ ⋅ ⋅ + 0.2 401.041 NNO −0.8 −0.2 ⋅ ⋅ ⋅ + 2.8 ⋅ ⋅ ⋅ + 0.4 404.641 CO −0.6 −0.3 0.0 −0.5 +0.4 −0.7 534.239 H2O ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 0.4 −0.1 534.037 CH 2O −0.4 ⋅ ⋅ ⋅ 0.0 +1.0 ⋅ ⋅ ⋅ − 1.1 530.840 HF ⋅ ⋅ ⋅ − 0.3 −0.5 ⋅ ⋅ ⋅ + 0.4 −0.8 687.442 F2 ⋅ ⋅ ⋅ − 0.2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − 0.7 682.242 MAD 0.6 0.2 0.3 1.8 0.6 0.9 aWith additional Rydberg functions. J. Chem. Phys. 155, 034108 (2021); doi: 10.1063/5.0058124 155, 034108-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. Comparison of GW2X corrected LR-TDDFT for XANES L-edge spectroscopy with similar methods from the literature. The difference between the calculated and experimental first excitation at the L 2- and L 3-edge is reported (in eV). All methods include relativistic treatments for the spin–orbit coupling. The bold atom in the chemical formula is the one from which the excitation takes place. L3 L2 SCAN-ROKS34EOM-CCSD50PBEh-GW2X SCAN-ROKS34EOM-CCSD50PBEh-GW2X aug-cc-pCVTZ uC-6-311(2 +,+)G∗∗aug-pcX-2 Expt. aug-cc-pCVTZ uC-6-311(2 +,+)G∗∗aug-pcX-2 Expt. SiH4 +0.4 −0.3 +1.2 102.651+0.4 −0.3 +1.1 103.251 SiCl4 +0.3 ⋅ ⋅ ⋅ − 0.4 104.252+0.3 ⋅ ⋅ ⋅ − 0.3 104.852 PH3 +0.2 ⋅ ⋅ ⋅ + 0.4 131.953+0.1 ⋅ ⋅ ⋅ + 0.5 132.853 PF3 0.0 ⋅ ⋅ ⋅ + 0.2 134.954+0.1 ⋅ ⋅ ⋅ + 0.5 135.654 H2S +0.3 −0.2 +0.3 164.455+0.3 −0.1 +0.2 165.655 OCS +0.1 +0.1 +0.1 164.356+0.2 −0.1 +0.2 165.556 SO2 ⋅ ⋅ ⋅ + 0.1 0.0 164.457⋅ ⋅ ⋅ + 0.3 +0.2 165.657 HCl +0.1 ⋅ ⋅ ⋅ + 0.4 200.958+0.2 ⋅ ⋅ ⋅ + 0.6 202.458 Cl2 0.0 ⋅ ⋅ ⋅ − 0.6 198.259+0.1 ⋅ ⋅ ⋅ − 0.5 199.859 MAD 0.2 0.2 0.4 0.2 0.2 0.4 first and second row atoms, covering 30 C1s, 21 O1s, 11 N1s, and 3 F1s. Core ionization potentials were calculated with the B3LYP, PBE0, PBEh( α=0.45), and BHHLYP functionals and the aug-pcX basis sets. Mean average deviations from experiments are displayed in Fig. 2, both for absolute and relative energies. Note that relative IPs are defined as the difference with respect to a reference, which was taken to be methane for C1s, ammonia for N1s, water for O1s, and methyl fluoride for F1s. As observed for K-edge excitation ener- gies, the basis set quality does not significantly affect the MAD from experiments, suggesting fast basis set convergence. All functionals perform similarly, with a slight edge for the PBE flavored hybrids. The error on the absolute IPs stands at 1.35 ±0.16 eV, which is very close to the errors observed at the K-edge. Relative IPs are remark- ably well reproduced, which means that core binding energies from different systems can be reliably compared. Detailed values can be found in the supplementary material. FIG. 2. Mean absolute deviation of the core 1s ionization potentials calculated with GW2X over the CORE65 benchmark set with respect to experiment. The thin dark bars represent the absolute IPs, and the wide clear bars represent the relative IPs.In Table III, core 1s ionization potentials calculated using GW2X and similar methods from the literature are displayed. The choice of molecules is such that the overlap with other works is maximized. As discussed earlier, missing data and different basis sets only allow for a semi-quantitative comparison. It is notewor- thy that B3LYP- ΔSCF performs the best, despite being the con- ceptually simplest method. The higher order methods’ equation- of-motion CCSD, PHEh( α=0.45)-G0W0, and transition operator electron propagator (TOEP2) yield comparable results with each other and are relatively close to B3LYP- ΔSCF. The second-order electron propagator method (EP2) and GW2X perform remarkably similarly. This is to be expected since GW2X is the DFT version of EP2, but this nonetheless underlines the validity of the generaliza- tion scheme. Despite exhibiting lower accuracy than the other meth- ods, our implementation of GW2X still improves upon Koopmans’s theorem75by an order of magnitude68while only scaling cubically. Note that the GW2X core IPs increasingly diverge from experiments with atomic number. This behavior is, however, not observed for the GW2X LR-TDDFT corrected K-edge energies of Table I, which suggests that some form of error cancellation takes place. C. Approximations for increased efficiency Various approximations can be employed to speed up GW2X calculations. The auxiliary density matrix method76under its puri- fied (ADMM1) and non-purified flavors (ADMM2) allow for very efficient ground state hybrid functional calculations. We previously showed in Ref. 24 that such an approximate calculation can serve as a base to a LR-TDDFT perturbative treatment with only minor loss of accuracy. In this work, we further propose the ADMM method as a mean to accelerate the construction of the generalized Fock matrix, which is a necessary step of the GW2X method. In Table IV, we investigate the impact of the ADMM approximation on K-edge cor- rections and core IP calculations compared to the full Hartree–Fock exchange treatment. Additionally, we report the results obtained using the GW2X∗method (as described in Sec. II) and the non-core- specific aug-pcseg77/aug-admm78basis sets. There is practically no J. Chem. Phys. 155, 034108 (2021); doi: 10.1063/5.0058124 155, 034108-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III. Comparison of GW2X 1s core ionization potentials with similar methods from the literature. The difference between the calculated and experimental IP is reported (in eV). The bold atom in the chemical formula is the one being ionized. B3LYP- ΔSCF33PBEh-G0W067EOM-CCSD36EP268TOEP268PBEh-GW2X u6-311G∗∗cc-pVQZ aug-cc-pVTZ cc-pVTZ cc-pVTZ aug-pcX-2 Expt. CH4 +0.17 −0.44 ⋅ ⋅ ⋅ + 0.75 +0.31 +0.09 290.8469 C2H4 ⋅ ⋅ ⋅ − 0.28 +0.39 +0.99 +1.37 +0.18 290.8269 CO +0.51 −0.65 +0.45 +1.74 +1.10 +0.91 296.2369 CH2O +0.31 −0.11 ⋅ ⋅ ⋅ + 1.59 +0.93 +0.76 294.3870 NH3 −0.01 −0.61 +0.60 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − 1.04 405.5270 NNO +0.21 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 0.29 +0.44 −1.04 408.6671 NNO +0.24 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + 0.75 +1.76 −0.57 412.5771 CO +0.33 −0.92 +1.33 −0.93 −0.24 −2.17 542.1072 H2O −0.41 −1.30 +0.75 −2.02 −0.42 −2.48 539.9071 CH 2O −0.19 −1.12 ⋅ ⋅ ⋅ − 1.93 −0.83 −2.27 539.3370 HF −0.45 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − 3.44 −0.98 −3.55 694.1871 F2 −0.38 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − 2.42 −1.28 −3.02 696.6971 MAD 0.29 0.68 0.70 1.53 0.88 1.51 difference in accuracy between the two ADMM schemes, making the more efficient non-purified version the default choice. Compared to full HFX, ADMM introduces an additional error of 0.2–0.3 eV for GW2X K-edge correction and 0.4–0.5 eV for core IPs. The error is increased when ADMM is used together with the more approximate GW2X∗scheme, especially for IP calculations. It is noteworthy that the use of the general purpose aug-pcseg-2 and corresponding aug- admm-2 basis sets do not greatly change the results, suggesting that large core-specific basis sets are not strictly necessary in this con- text. This is particularly useful for making large scale calculations more affordable. Finally, the errors induced by the ADMM and/or GW2X∗approximations at the K-edge are rather small compared to the scale of a near-edge x-ray absorption spectrum, which typically spans 15–20 eV. D. GW2X in extended systems There is, in principle, no issue with using GW2X as a correc- tion scheme in periodic boundary conditions (PBCs). Moreover, the high levels of symmetry found in crystal structures can be exploited for efficiency. Since multiple atoms are equivalent, it is only neces- sary to correct the core KS eigenvalue for one of those, leading to a TABLE IV. Mean average deviation (in eV) of the GW2X method with respect to experiments, using various approximations. The same benchmark sets as for Figs. 1 and 2 were used for K-edge and core ionization potentials, respectively. The PBEh(α=0.45) functional and the aug-pcX-2/aug-admm-2 and aug-pcseg-2/aug- admm-2 basis set combinations were used. The results within parentheses refer to the latter. K-edge IP GW2X GW2X∗GW2X GW2X∗ Full HFX 1.2 (1.2) 1.3 (1.2) 1.2 (1.0) 1.6 (1.6) ADMM1 1.4 (1.3) 1.6 (1.4) 1.6 (1.4) 2.2 (1.8) ADMM2 1.5 (1.3) 1.6 (1.5) 1.7 (1.4) 2.2 (1.9)𝒪(n3)scaling with system size. Note that disordered systems such as liquids are limited to a 𝒪(n4)scaling, as each atom may have a slightly different environment. GW2X being related to second- order Møller–Plesset perturbation theory,82it suffers from the same limitations. In particular, it works best for large gap molecular sys- tems, whereas semiconductors require large supercells for converged results.83 We applied the GW2X correction schemes on three molecular crystals, namely, solid ammonia, ice 1h, and solid argon. The respec- tive unit cells contain 128, 288, and 32 atoms, and the structures were first relaxed at the DZVP-MOLOPT-SR-GTH84/PBE85+D386level of theory. LR-TDDFT +GW2X calculations were then performed using the PBEh(α=0.45) hybrid functional with the truncated Coulomb potential87(truncation radii of 5, 6, and 5 Å, i.e., just under half of the cell parameter). The corrected LR-TDDFT spectra are shown in Fig. 3. The first two absorption peaks of crystalline NH 3are well aligned to the experimental spectrum, thanks to the GW2X correc- tion that amounts to a blue shift of 3.7 eV. The relative intensity of the first peak is too small, but this issue lies with LR-TDDFT rather than the correction scheme. The triple zeta aug-pceseg-2/aug- admm-2 basis sets were used to describe a single excited nitrogen atom, while all other atoms relied on GTH pseudopotentials88–90 and DZVP-MOLOPT-SR-GTH/FIT376basis sets. The hexagonal ice spectrum was calculated using the same basis sets, where only one excited oxygen atom was described at the all-electron level as well. The GW2X correction to the spectrum is a blue shift of 2.8 eV, which also aligns the first few peaks remarkably well. The relative intensi- ties are, however, quite poor, and the reasons will be discussed later. The solid argon simulated spectrum is well aligned (GW2X shift of 3.5 eV) and enjoys good relative intensities. Using the higher qual- ity quadruple zeta aug-pcseg-3 was found to be necessary, and the added diffuse functions of the augmented set were found to be espe- cially crucial. Only one atom was described at the all-electron level and all others using DZVP-MOLOPT-SR-GTH basis sets. However, since there is no available auxiliary FIT3 basis set for argon, full HFX was employed instead of the ADMM approximation. Note that using J. Chem. Phys. 155, 034108 (2021); doi: 10.1063/5.0058124 155, 034108-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. GW2X corrected LR-TDDFT spectra of (a) the solid ammonia N K-edge, (b) the ice 1h O K-edge, and (c) the solid argon Ar L 2,3-edge in periodic boundary conditions. The PBEh(α=0.45) functional and a mix of all electron basis sets and pseudopotentials were used. The GW2X corrections amount to blue shifts of 3.7, 2.8, and 3.5 eV for ammonia, ice, and argon, respectively. A Lorentzian broadening of fwhm 0.5 eV was applied, and calculated intensities were uniformly scaled to match experiments. Experimental data come from Refs. 79–81. quadruple zeta basis sets for ammonia and ice did not improve the results. Most of the efficiency aimed approximations discussed in Sec. III C were used in our extended system calculations. General purpose aug-pcseg basis sets were used throughout as was the non- purified ADMM scheme for solid NH 3and ice 1h. Moreover, only one atom per system was described at the all-electron level, while all others relied on pseudopotentials, essentially freezing their core and reducing the number of MOs to include in Eq. (13). Only the original GW2X scheme was preferred over the simplified GW2X∗version as the systems are too small for notable efficiency gains. The spectra obtained with the latter method are available in the supplementary material. Note that a truncated Coulomb potential was used for exact exchange integrals in order to avoid nonphysical self-exchange.87 The same operator was used for ground state HFX, LR-TDDFT exact exchange kernel, and thus also GW2X. The use of such a short range exchange operator also reduces the cost and scaling of the previ- ously mentioned integrals. We did not observe a dependence of the GW2X results on truncation radius, provided that it is large enough (5–6 Å). To illustrate the overall affordability of the method, all cal- culations were run on a 24 CPU core system and execution times reported in Fig. 4. The ground state SCF calculation dominates over- all, mostly because of the ERIs evaluation for the HFX fraction of the functional. This is so even with the initial SCF guess coming from a converged PBE calculation and the ADMM approximation for ammonia and ice. The efficiency of the GW2X correction is empha- sized by the fact that it takes at most 60% of the post-SCF effort in ice, and is almost negligible for the other two smaller systems. The most expensive calculation, solid argon, ran with a wall time of just under 40 min. As previously mentioned, while the spectral features of the hexagonal ice spectrum are well aligned with experimental data, the relative intensities are unsatisfactory. Based on TP-DFT91and CPP-DFT92,93simulations, Zhovtobriukh et al.94have suggested that available ice 1h XANES spectra are not fully representative of the pristine crystal structure. Instead, they show that significant disor- der around the lattice positions must be introduced to reproduce experiments. Using the GenIce95code, we generated a 384-atom dis- ordered model based on hexagonal ice, with Gaussian noise appliedto the position of all water molecules (the –add_noise parameter of GenIce was set to 1.0). The simulated spectrum, displayed in Fig. 5, improves upon that obtained with the pristine 1h structure. In par- ticular, the initially very sharp first peak is broadened and decreases in relative intensity. In periodic systems, GW2X cannot be directly used to calcu- late absolute binding energies. Because the potential reference can be arbitrarily defined, whereas it goes to zero far away in the non- periodic case, KS eigenvalues may be uniformly shifted. Thus, the solution of Eq. (13) for the IP will be shifted by some unknown amount in PBCs. However, once plugged into Eq. (12) for the LR- TDDFT correction, the shifts cancel out. A possible way to calculate core IPs in PBCs using GW2X would be to use slab models and ref- erence the KS eigenvalues against the vacuum level, as it is done in theGW community.96–98 FIG. 4. Execution times of the extended system calculations of Fig. 3. All cal- culations were carried out on an Intel S2600WT2R machine with 24 CPUs and 256 GB of memory. Different colors represent the time spent treating the ground state SCF, the LR-TDDFT equations, and the GW2X correction. Note that the time spent evaluating the AOs’ RI integrals shared by GW2X and LR-TDDFT is credited to the latter. J. Chem. Phys. 155, 034108 (2021); doi: 10.1063/5.0058124 155, 034108-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5. Hexagonal ice XANES spectrum calculated based on a perturbed crystal structure obtained with the GenIce code. The model contains 384 atoms, 8 of which (all oxygen atoms) were randomly selected to be excited. The choice of functional and basis sets is the same as for Fig. 3. The GW2X shift amounts to 3.4±0.1 eV. IV. CONCLUSION We have implemented a correction scheme to address the lack of orbital relaxation and the self-interaction error that afflicts the prediction of absolute excitation energies in LR-TDDFT for core level spectroscopy. The methods are based on the DFT generaliza- tion of the second-order electron propagator theory (GW2X). It allows for the accurate calculation of ionization potentials, which are then substituted into the LR-TDDFT equations, replacing the KS eigenvalues. The implementation exploits the local nature of core states for efficiency, scaling cubically with system size and integrat- ing seamlessly in our existing core level LR-TDDFT implementation. Benchmarks at the K- and L-edges show that the GW2X cor- rection scheme systematically improves LR-TDDFT results for four common hybrid functionals, namely, B3LYP, PBE0, PBEh( α=0.45), and BHHLYP, reaching accuracies comparable to higher level meth- ods, such as the equation of motion CCSD. Moreover, the method can be used to calculate core ionization potentials directly with sim- ilar errors to those observed at the K-edge. Rapid basis set conver- gence was observed in both LR-TDDFT correction and core binding energy calculation, and the use of core-specific basis sets does not seem essential. The method is applicable with periodic boundary conditions for large gap molecular systems, as demonstrated for solid ammo- nia, ice 1h, and crystalline argon. Such calculations are accessible with modest computer resources and are made feasible due to effi- cient approximations. In particular, mixing the ADMM scheme and a hybrid all-electron/pseudopotential description has proven effec- tive. Core ionization potentials are, however, not easily accessible in such systems and would require additional calculations. This work focuses on standard GW2X, and there is room for further exploration. The initial GW2X paper48also describes the GW(2) approach, which could prove more efficient since the exchange contributions are ignored. Alternatively, the accuracy could potentially be improved by using spin scaling approaches, as done in SCS- and SOS-MP2.99,100SUPPLEMENTARY MATERIAL The supplementary material contains tables with the detailed values of the K-edge and ionization potential benchmarks as well as related figures augmented with pure Hartree–Fock results. It also contains simulated XANES spectra of solid ammonia, ice 1h, and solid argon corrected with the approximate GW2X∗method. ACKNOWLEDGMENTS This work was supported by the MARVEL National Centre for Competency in Research funded by the Swiss National Science Foundation (Grant Agreement ID 51NF40-182892). 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5.0051882.pdf

Li-ion intercalation enhanced ferromagnetism in van der Waals Fe 3GeTe 2bilayer Cite as: Appl. Phys. Lett. 119, 012405 (2021); doi: 10.1063/5.0051882 Submitted: 28 March 2021 .Accepted: 27 June 2021 . Published Online: 8 July 2021 Xiaokun Huang,1,2,a) Jinlin Xu,1Renfen Zeng,1Qinglang Jiang,1XinNie,1Chao Chen,1,2Xiangping Jiang,1,2,a) and Jun-Ming Liu3 AFFILIATIONS 1School of Materials Science and Engineering, Jingdezhen Ceramic Institute, Jingdezhen 333001, China 2National Engineering Research Center for Domestic and Building Ceramics, Jingdezhen 333001, China 3Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China a)Authors to whom correspondence should be addressed: hxk_16@126.com and jiangxp64@163.com ABSTRACT Recently, the issue of ferromagnetism enhancement in two-dimensional (2D) van der Waals (vdW) layered magnetic systems has been highly concerned. It is believed that ion intercalation in vdW layered ferromagnets, targeting either enhanced interlayer spin exchanges or intralayer ones, can be an efficient scheme. In this work, by means of the first-principles calculations, we investigate the Li-ion intercalation between the two monolayers of the ferromagnetic (FM) vdW Fe 3GeTe 2(FGT) bilayer and its impact on the ferromagnetism. It is revealed that the Li- ion intercalation provides hopping carriers between the two interfacial Te sublayers, beneficial for the enhancement of the interlayer FM cou-pling at a relatively low intercalation level. On the other hand, the Li-ion intercalation lifted Fermi level promotes the electron transfer fromthe minority spin channel to the majority one for the Fe-3 dbands, favoring the stronger intralayer FM coupling. However, the over- intercalation generated carriers may fill up the majority spin channel, reversely leading to the reduced interlayer FM coupling. Consequently, an optimized intercalation level is expected in terms of ferromagnetism enhancement. This work not only helps to explain the recent experi-mental finding on the gate-controlled Li-ion intercalation in vdW FGT few-layers but also suggests a general scheme for ferromagnetismenhancement in 2D vdW layered ferromagnets using the ion intercalation approach. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0051882 In recent years, owing to the experimental discovery of atomically thin ferromagnetic (FM) vdW materials, such as Cr 2Ge2Te6,C r I 3,a n d Fe3GeTe 2(FGT),1–6ferromagnetism has been widely accepted as a new branch of intrinsic collective properties of 2D vdW materials.7,8 This exciting breakthrough immediately gained extensive attention due to its significant importance to next-generation spintronic devices and stimulated intense works to extend knowledge and applications of2D vdW ferromagnets. 9–15Nevertheless, the reduced dimensionality, due to the weak interlayer vdW interactions, has confined the spin- ordering temperature (Curie point Tc) of a 2D vdW ferromagnet, which is somehow far below room temperature, thus severely hinder-ing practical applications. In this regard, promoting the ferromagne-tism by lifting the T cand enhancing the magnetization has been one of the most critical issues in the field of 2D vdW ferromagnets. Considering the common layered structural feature of 2D vdW materials and evaluating those proposed schemes for magnetismenhancement, ion intercalation has emerged as a very convenient and powerful tool to modulate the properties of 2D vdW systems. 16–21Onone hand, ion intercalation can provide the highest possible doping level and modulate the electronic structure of the internal region in the host without destructing structures of the monolayers.17–19On the other hand, intercalated ions can act like bridges to link the mono-layers separated by vdW gaps, effectively enhancing interlayer interac- tions. 20,21Thus, it is believed that ion intercalation may be a promising scheme for ferromagnetism enhancement of 2D vdW ferromagnets.In fact, a notable experiment tuned the T cvalue of a FGT trilayer from 100 to 300 K using the gate-controlled Li-ion intercalation, and it was proposed that the electron doping may be responsible for the increase inTc.6To study the electron doping effect, a recent theoretical work doped electrons to a monolayer FGT by using the background electronapproximate method. 22It was found that the intralayer magnetic frus- tration can be significantly reduced in a specific range of the electron doping level, which helps to explain the lifted Tcin the Li-intercalated FGT few-layers. Nevertheless, the underlying physics can be more complicated since the electron doping effect in the Li-intercalated FGT few-layers Appl. Phys. Lett. 119, 012405 (2021); doi: 10.1063/5.0051882 119, 012405-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldepends not only on the doping level but also on the positive back- ground charges of the Li ions in the vdW gaps, which have not been explored yet. More importantly, the Li-ion intercalation modulates not only the intralayer spin exchanges but also the interlayer ones. In fact,it would be highly appreciated if the intralayer and interlayer FM cou-plings can be both enhanced ferromagnetically, while the opposite cases including the competing exchanges would be unfavored. However, the impact of the Li-ion intercalation on the interlayer FMcoupling as well as electronic structures of FGT has so far received lit-tle attention. In this work, these issues will be addressed theoretically by studying FM couplings and electronic structures of the FGT bilayer before and after Li-ion intercalation with the first-principlescalculations. We performed first-principles calculations implemented in the Vienna ab initio simulation package (VASP) code. 23–25The general- ized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof parameterization is adopted for exchange and correlationfunctional. 26We adopted the effective Umethod to take into account correlation effects of Fe-3 delectrons,27and we checked the Udepen- dence of the magnetic properties of the FGT (see Figs. S1–S3). A small U¼0.6 eV that appropriately describes the interlayer FM coupling of the FGT is used in this work if not specifically mentioned. More com-putational details are given in the supplementary material . FGT is a vdW layered metallic ferromagnet. 28–32The stacking sequence of the five atomic sublayers in the FGT monolayer can bewritten as Te-Fe I-(Fe IIGe)-Fe I-Te, in which two inequivalent sites of Fe are denoted as Fe Iand Fe II.A ss h o w ni n Figs. 1(a) and1(b),e a c h elemental species in a sublayer occupies one of the three inequivalent in-plane sites, which are denoted as site-A, site-B, and site-C. Weinvestigated the Li-intercalated FGT bilayer as a representative case ofthe FGT few-layers. To study the spatial distribution of the intercalatedLi ions, we started with intercalating one Li ion into each in-plane unit cell of the FGT bilayer, the bilayer model with this intercalation level is denoted as Li 1-FGT 2. To find the best intercalation site for the Li ion, we considered various trial starting positions of the Li ion (see Fig. S4) and fully relaxed the bilayer model to reach the energy minima. Itturned out that the most energetically favorable site for the Li ion is the center of the hollow site (site-A) surrounded by six interfacial Te atoms [see Fig. 1(c) ]. If the Li ion occupies site-B or -C, the energy of the bilayer model will be about 0.5 eV(/in-plane unit cell) higher than occupying site-A. We performed ab initio molecular dynamic (AIMD) simulation at 300 K to evaluate the stability of the Li 1-FGT 2bilayer model. The simulation result indicates that the Li ions at site-A are thermally stable (see Fig. S5). If we further intercalate Li ions into the Li1-FGT 2bilayer model, the interlayer FM spin configuration would be damaged. We designed the bilayer models for Li x-FGT 2(x¼5/4, 4/3, and 2), and their calculated ground states are interlayer AFM states (see Fig. S6 and Table S1), suggesting that over-intercalation isdetrimental to the interlayer FM coupling. Thus, the intercalation level xhigher than 1 will not be further discussed in this paper. It was reported that the electron doping induced by Li-ion inter- calation to the host material is up to the order of 10 14cm/C02per layer,6 which is equivalent to about one electron in every seven in-plane unit cells of the FGT. So we further studied another four different intercala- tion levels at the same order of magnitude, they are one Li ion in every three, four and nine in-plane unit cells, and two Li ions in every three in-plane unit cells, which are denoted as x¼1/3, 1/4, 1/9, and 2/3, respectively. To check whether the Li ions tend to disperse or gather, we studied two different distribution configurations of the same inter- calation level x¼1/3. As shown in Fig. S7, theffiffi ffi 3p /C2ffiffi ffi 3p supercell with one Li ion represents the dispersed distribution, and the 3 /C23 supercell with three neighboring Li ions represents the gathered distri- bution, respectively. The calculated energy of the dispersed distribu- tion is about 8 meV(/in-plane unit-cell) lower than that of the gathered one, indicating that the Li ions tend to disperse. So wedesigned the bilayer models with dispersed Li-ion distributions for x¼1/3, 1/4, 1/9, and 2/3, which areffiffi ffi 3p /C2ffiffi ffi 3p ,2/C22, and 3 /C23s u p e r - cells with one Li ion, and affiffi ffi 3p /C2ffiffi ffi 3p supercell with two Li ions, respectively. Their fully relaxed structures are shown in Figs. 1(d) and S8. Their calculated structural parameters (see Fig. S9) indicate that the Li-ion intercalation slightly expands the interlayer spacing of the bilayer. Then we investigated the Li-ion intercalation modulated FM cou- plings. We started with the interlayer FM coupling. The interlayer magnetic coupling energy can be defined as DE z¼(EAFM-EFM)/u, where E AFMand E FMare the energies of the Li x-FGT 2bilayer model with interlayer AFM and FM spin configurations between the twomonolayers, respectively; uis the number of in-plane unit cells in the supercell. Larger positive DE zindicates the stronger interlayer FM cou- pling. The calculated DEzof the Li x-FGT 2bilayer models are shown in Fig. 2(a) . All the calculated bilayer models are FM states. When the x is relatively low ( x/C201/3), the DEzincreases with the x,i n d i c a t i n gt h e Li-ion intercalation enhanced interlayer FM coupling. As the xfurther increases, the DEzreversely decreases, indicating the over-intercalation reduced interlayer FM coupling. We double checked the DEzwith changing the Hubbard Uparameter [see Fig. 2(b) ], verifying that the non-monotonic change of the DEzaccording to the xis reliable. To check whether the increase in the interlayer distance [see Fig. 2(c) ]i s FIG. 1. (a) and (b) Side and top views of the pristine FGT bilayer, respectively. (c) Side and top views of the Li 1-FGT 2. (d) Top views of the Li 2/3-FGT 2and Li 1/3-FGT 2. The vacuum spacer is not displayed in (a) and (c). To show the Li ions, uppermonolayers of the bilayers are not displayed in the top views.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012405 (2021); doi: 10.1063/5.0051882 119, 012405-2 Published under an exclusive license by AIP Publishingthe cause of the non-monotonic behavior of the DEz,w er e m o v e dL i ions from the Li x-FGT 2bilayers, maintained positions of the other atoms, and calculated the DEzof the remaining FGT bilayers. The DEz [Fig. 2(a) ] slightly decreases as the xincreases, indicating that the enlarged interlayer distance has minor influence on the interlayer FMcoupling. Thus, the dominant cause should be the electron doping bythe Li-ion intercalation. The non-monotonic behavior of the DE zsuggests the competi- tion between at least two competing contributions of the electron dop- ing to the interlayer spin exchanges. The interlayer FM coupling of the FGT bilayer is established by the indirect spin exchanges involving notonly the Fe–Te intralayer exchange channels but also the Te–Te inter- layer exchange channels. The mechanism of such multi-intermediate indirect spin exchanges, 33,34which have the same essence with the conventional double-exchange,35–37is that carriers hop between the magnetic sites with spins of incomplete d-shells pointing to the same direction to reduce the energy of the system, thus establishing the FM order. In this regard, the Li-ion intercalation induced electron dopingmodulates two crucial factors for the interlayer indirect spinexchanges: one is the interlayer hopping carrier and the other is the spin configuration of the Fe site. First, we checked the spatial distribution of the doping electrons. We calculated the Li-ion intercalation induced differential electron density distribution of the Li x-FGT 2bilayer model. The result of the Li1-FGT 2as a representative case is shown in Fig. 3(a) . Apparently, doping electrons are screened by the two interfacial Te sublayers and accumulate in the vdW gap. The other four intercalation levels (x¼2/3, 1/3, 1/4, and 1/9) exhibit almost the same doping electron dis- tributions (see Fig. S10), indicating that the intercalation level has negli-gible influence on spatial distribution of the doping electrons. We also checked the distributions of the doping electrons by evaluating average changes of electron numbers in different parts of one Fe 3GeTe 2for- mula cell (defined as one in-plane unit cell of one monolayer) accord-ing to the x, by using the bader charge analysis [see Fig. 3(b) ]. Dopingelectrons per Fe 3GeTe 2formula cell linearly increase with the x,a n d they are characterized mainly by the interfacial Te site. These doping electrons offer interlayer hopping carriers for the indirect spin exchanges between the two FGT monolayers, which is beneficial for the enhancement of the interlayer FM coupling and explains the increase in the DEzat a relatively low intercalation level. Second, we checked the spin-projected differential electron den- sity distribution to study the Li-ion intercalation modulated spin con- figurations of the Fe sites. The result of the Li 1-FGT 2as a representative case is shown in Fig. 3(c) . Interestingly, although the Fe sites have not got doping electrons [ Fig. 3(b) ], the Li-ion intercalation causes obvious difference between their majority spin (spin up) and minority spin (spin down) channels, indicating the spin redistribution between the two spin channels. The other four intercalation levels exhibit almost the same features of spin redistributions (see Fig. S11). Thus, the average occupation number of the spin up and spin downchannels of the 3 dorbitals of the Fe site increases and decreases line- arly with the x, respectively, leading to the enhanced magnetic moment [see Fig. 3(d) ]. It should be noticed that the spin up channel has already more than four electrons before intercalation, while the spin redistribution further reduces the spin up holes. As the incom- plete d-shells are crucial to the interlayer indirect FM exchanges, a rel- atively high-level intercalation generated over-doping carriers may fill up the spin up channel, thus leading to the reduced interlayer FM cou- pling. Hence, the Li-ion intercalation induced interlayer hopping car- riers and spin redistribution of the Fe-3 dbands are the two competing factors responsible for the non-monotonic change of the DE z. Next, we propose a scenario with the schematic diagrams shown inFigs. 4(a)–4(d) to explain the spin redistribution mechanism and demonstrate that the spin redistribution contributes to the enhanced intralayer FM coupling. Figure 4(a) indicates the density of states (DOSs) for the Fe-3 dbands and the 5 pbands of the interfacial Te sub- layer in each monolayer of the pristine FGT bilayer. The doping elec- trons from the Li ions to the interfacial Te sublayers lift the Fermi level o ft h em e t a l l i cs y s t e m ,w h i l et h eF es i t e sh a v en od o p i n ge l e c t r o n st o accordingly fill their unoccupied 3 dstates above the original Fermi level [ Fig. 4(b) ]. To catch up with the lifted Fermi level, the spin down channel of the Fe-3 dbands has to donate electrons to the spin up channel [ Fig. 4(c) ] and then shifts upward. Consequently, the exchange splitting between two spin channels is enhanced [ Fig. 4(d) ]. To confirm our scenario, we checked the difference between projected DOS (PDOS) of the pristine FGT bilayer and the Li x-FGT 2bilayer in our calculations. To make a clear contrast, we show the results of the Li1-FGT 2that has the highest xamong the five bilayer models. Due to the spatial inversion symmetry of the bilayer model, electronic struc- tures of its two monolayers are identical, so we just plot the PDOS of one monolayer. The Li-ion intercalation causes the PDOS of the inter- facial Te site to shift to the left, indicating the electron doping lifted Fermi level [ Fig. 4(e) ]. The Li-ion intercalation also causes the spin up channel of the Fe-3 dbands to shift to the left, while the shift of the spin down channel is not obvious [ Fig. 4(f) ]. The calculated centers of the spin up and spin down channels of the Fe-3 dbands for the Li 1- FGT 2bilayer model are 109 and 5 meV smaller than that for the pris- tine FGT bilayer, respectively, confirming the substantially enhanced exchange splitting by the Li-ion intercalation. According to the Stoner model, larger exchange splitting corre- sponds to the stronger exchange interaction. As the spin redistribution FIG. 2. (a) Calculated DEzaccording to the intercalation level x. Red solid dots and blue open circles represent the calculated DEzof the Li x-FGT 2bilayers and the FGT bilayers with Li ions being removed from the Li x-FGT 2bilayers, respectively. Red and blue lines are interpolations. (b) Calculated DEzwithU¼0.2, 0,6, and 1.0 eV, respectively. (c) Calculated interlayer distance between the two interfacial Te sublayers.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012405 (2021); doi: 10.1063/5.0051882 119, 012405-3 Published under an exclusive license by AIP Publishingoccurs inside each monolayer, the resultant enhancement of the exchange splitting corresponds to the stronger intralayer FM coupling.It should be noted that the spin redistribution is one of the mechanisms that enhances the intralayer FM coupling, while other previously pro- posed mechanisms, such as the suppression of magnetic frustration bythe electron doping and the additional double-exchange channelsoffered by intercalants, 22,38may also work in Li-intercalated FGT few- layers. However, we did not quantitatively estimate the enhancement ofthe intralayer FM coupling, since the method to fit the intralayer FM coupling to the energy differences between different in-plane magnetic configurations is not suitable here due to the significantly reduced mag-netic moments by forced reorientation of the Fe spins. 22 FIG. 3. (a) Integrals of doping electron density distributions in the ab plane for the Li 1-FGT 2.Dq¼qLi1-FGT2 –qFGT2 . (b) Average changes of electron numbers ( Dn) in different parts of one formula cell according to the x. The labels “Fe 3GeTe 2,” “interfacial Te,” and “Fe 3GeTe” represent a formula cell, the interfacial Te site, and the rest part of one for- mula cell, respectively. (c) Integrals of spin-projected differential electron density distribution in the ab plane for the Li 1-FGT 2. (d) Average occupation number of two spin chan- nels of 3 dorbitals of one Fe site. The inset shows the average change of magnetic moment derived from 3 dorbitals of one Fe site. FIG. 4. (a)–(d) Schematic diagrams for the mechanism of the spin redistribution of the Fe-3 dbands. FL refers to the Fermi level. (e) and (f) PDOS of the interfacial Te site and 3dbands of three Fe sites in one formula cell for the Li 1-FGT 2(x¼1) and pristine FGT ( x¼0) bilayers. Fermi levels are set as 0 eV. The centers of the majority and minority spin channels of the Fe-3 dbands are defined as the first moments of the PDOS.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012405 (2021); doi: 10.1063/5.0051882 119, 012405-4 Published under an exclusive license by AIP PublishingFinally, we compare our results with the experiment findings.6 We have revealed the two mechanisms for the enhanced FM couplings in the Li-intercalated FGT bilayer, which may also be effective forthicker FGT few-layers. One is offering interlayer hopping carriers, a n dt h eo t h e ri si n d u c i n gi n t r a l a y e rs p i nr e d i s t r i b u t i o n .A tar e l a t i v e l y low intercalation level, the interlayer and intralayer FM couplings are both enhanced, explaining the lifted T cof the FGT few-layers. At a rel- atively high intercalation level, over-doping electrons lead to seriously reduced interlayer FM coupling. Consequently, an optimized interca- lation level is expected, which can explain the overall trend in theexperiment that the T cfirst increases and then decreases as the gate- controlled Li-concentration increases. We also suggest that these two mechanisms are not only effective for the Li-intercalated FGT but also may be universally applicable to the other ion-intercalated vdW metal- lic ferromagnet, in which the intercalant offers doping electrons whilethe magnetic sublayers are sandwiched between two nonmagnetic sub- layers of the ferromagnet. Furthermore, two issues should be noted regarding this work. On one hand, our results are unable to explain the complex pattern of the T cas a function of the gate voltage, which may be related to the complicated diffusion process of the Li-ion inter-calation. On the other hand, this work lacks quantitative estimations of the enhanced spin exchanges, thus the Li-ion intercalation induced increase in T cremains unclear. Future investigations regarding these problems would be worthwhile. In summary, we theoretically investigated the Li-ion intercalation in the FGT bilayer and its impact on the ferromagnetism enhancement by using the first-principles calculations. The Li-ion intercalation offers interlayer hopping carriers and induces intralayer spin redistribution, favoring the stronger interlayer and intralayer FM couplings, respectively. On the other hand, an optimized intercalation level is expected becausethe over-intercalation damages the interlayer FM coupling. This work not only helps to explain the enhanced FM couplings in Li-intercalated FGT few-layers but also suggests that ion intercalation may be a general method to enhance the ferromagnetism in 2D vdW ferromagnets. See the supplementary material for computational details, the U dependence of magnetic properties of the FGT, the search for themost energetically favorable site for the Li-ion, AIMD simulation of the Li 1-FGT 2bilayer model, calculated structures and energies of the Lix-FGT 2(x¼5/4, 4/3, and 2) bilayer models, two different Li-ion dis- tribution configurations designed for the x¼1/3, calculated structures of the Li x-FGT 2(x¼1/4 and 1/9) bilayer models, calculated structural parameters of the Li x-FGT 2(x¼0, 1/9, 1/4, 1/3, 2/3, and 1) bilayer models, and calculated differential electron density distributions of theLi x-FGT 2(x¼2/3, 1/3, 1/4, and 1/9) bilayer models. This work was financially supported by grants from the National Natural Science Foundation of China (Nos. 11947092, 52062018, and 51762024) and the Natural Science Foundation of Jiangxi Province (Nos. 20192BAB212002 and 20192BAB206008). Part of the numerical calculations were carried out in the High Performance Computing Center (HPCC) of Nanjing University.We thank Professor Weiyi Zhang for helpful discussions. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material .REFERENCES 1C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature 546,2 6 5 (2017). 2Z. Wang, T. Zhang, M. Ding, B. Dong, Y. Li, M. Chen, X. Li, J. Huang, H. Wang, X. Zhao, Y. Li, D. Li, C. Jia, L. Sun, H. Guo, Y. Ye, D. Sun, Y. Chen, T. Yang, J.Zhang, S. Ono, Z. Han, and Z. Zhang, Nat. Nanotechnol. 13, 554 (2018). 3B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Nature 546, 270 (2017). 4S. Jiang, J. Shan, and K. F. Mak, Nat. Mater. 17, 406 (2018). 5Z. 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5.0054647.pdf

J. Chem. Phys. 155, 034110 (2021); https://doi.org/10.1063/5.0054647 155, 034110 © 2021 Author(s).Scaling up electronic structure calculations on quantum computers: The frozen natural orbital based method of increments Cite as: J. Chem. Phys. 155, 034110 (2021); https://doi.org/10.1063/5.0054647 Submitted: 20 April 2021 . Accepted: 18 June 2021 . Published Online: 16 July 2021 Prakash Verma , Lee Huntington , Marc P. Coons , Yukio Kawashima , Takeshi Yamazaki , and Arman Zaribafiyan ARTICLES YOU MAY BE INTERESTED IN Size-consistent explicitly correlated triple excitation correction The Journal of Chemical Physics 155, 034107 (2021); https://doi.org/10.1063/5.0057426 Nuclear–electronic orbital methods: Foundations and prospects The Journal of Chemical Physics 155, 030901 (2021); https://doi.org/10.1063/5.0053576 An improved Slater’s transition state approximation The Journal of Chemical Physics 155, 034101 (2021); https://doi.org/10.1063/5.0059934The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Scaling up electronic structure calculations on quantum computers: The frozen natural orbital based method of increments Cite as: J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 Submitted: 20 April 2021 •Accepted: 18 June 2021 • Published Online: 16 July 2021 Prakash Verma,1 Lee Huntington,1 Marc P. Coons,2 Yukio Kawashima,1 Takeshi Yamazaki,1,a) and Arman Zaribafiyan1 AFFILIATIONS 11QB Information Technologies (1QBit), 200-1285 W Pender St., Vancouver, British Columbia V6E 4B1, Canada 2Dow, Core R & D, Chemical Science, 1776 Building, Midland, Michigan 48674, USA a)Author to whom correspondence should be addressed: takeshi.yamazaki@1qbit.com ABSTRACT The method of increments and frozen natural orbital (MI-FNO) framework is introduced to help expedite the application of noisy, intermediate-scale quantum (NISQ) devices for quantum chemistry simulations. The MI-FNO framework provides a systematic reduction of the occupied and virtual orbital spaces for quantum chemistry simulations. The correlation energies of the resulting increments from the MI-FNO reduction can then be solved by various algorithms, including quantum algorithms such as the phase estimation algorithm and the variational quantum eigensolver (VQE). The unitary coupled-cluster singles and doubles VQE framework is used to obtain correlation ener- gies for the case of small molecules (i.e., BeH 2, CH 4, NH 3, H 2O, and HF) using the cc-pVDZ basis set. The quantum resource requirements are estimated for a constrained geometry complex catalyst that is utilized in industrial settings for the polymerization of α-olefins. We show that the MI-FNO approach provides a significant reduction in the quantum bit (qubit) requirements relative to the full system simulations. We propose that the MI-FNO framework can create scalable examples of quantum chemistry problems that are appropriate for assessing the progress of NISQ devices. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054647 I. INTRODUCTION Accurate characterization of electron interactions is vital for the computational design of molecules and requires finding exact solutions of the electronic Schrödinger equation. Solving the Schrödinger equation exactly, in a given one-particle basis, on clas- sical computers is a computationally demanding task because the dimension of the Hilbert space of quantum systems increases expo- nentially with system size and the complexity of finding exact solu- tions scales factorially with the number of orbitals and electrons. Thus, on classical hardware, obtaining solutions of Schrödinger equation is only possible for small systems.1 In recent years, there has been increasing interest in quan- tum computation, a new computing paradigm initially conjectured as an efficient framework for simulating quantum mechanical sys- tems.2,3In the decade since this conjecture was put forward, therehas been tremendous theoretical progress toward realizing the con- cept of using a quantum computer for quantum simulations.4Early implementations of quantum algorithms aimed at applications in computational chemistry5were deployed on quantum computers in order to evaluate molecular energies.6–10There has also been accelerated progress in hardware development. For example, IBM,11 Google,12Intel,13Rigetti,14and Quantum Circuits, Inc.15have all developed quantum computing platforms based on superconduct- ing qubits, while IonQ16and Honeywell17have developed platforms based on ion traps. Google’s achievement on a benchmarking mile- stone commonly referred to as “quantum supremacy”18demon- strates a transition of the quantum computing field away from a purely theoretical concept. Despite the progress of hardware devel- opment, current quantum devices are error-prone and have lim- ited computing capacity, hence the introduction of the term noisy, intermediate-scale quantum (NISQ)19to describe them. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The limitations of NISQ hardware have driven significant progress toward the development of algorithms that seek to shorten the timescale for the successful application of quantum comput- ers for solving quantum chemistry problems. Some of these devel- opments include quantum–classical hybrid algorithms for varia- tional optimization6,20–22and wavefunction Ansätze that produce low-depth circuits for efficient quantum simulation.9,8,24–28Research has also focused on incorporating problem decomposition (PD) techniques developed for applications of classical quantum chem- istry9,7,29–35into quantum algorithms to further improve the effi- ciency of simulations on NISQ devices. The advantage of PD tech- niques is the ability to decompose the full electronic structure prob- lem of a molecule into a set of smaller sub-problems that can be solved more efficiently. Problem decomposition approaches also provide a good approximation to the results of calculations per- formed on the corresponding full system. Such approaches have a long history in the literature, originating from early investigations of local electron correlation by Sinano ˘glu,36Nesbet,37and Ahlrichs and Kutzelnigg38during the 1960s. Several comprehensive reviews of PD techniques used in quantum chemistry applications are available in Refs. 39–42. Problem decomposition techniques reduce the effective pro- blem size of a molecular system and create opportunities to char- acterize near-term devices by enabling hardware experiments for larger systems that would otherwise be inaccessible during the NISQ era. Following a similar strategy as some of the authors’ previ- ous work,32we seek an efficient methodology for performing scal- able quantum chemistry simulations on near-term devices based on PD techniques. Specifically, we explore a strategy that combines the method of increments (MI)43–45with the frozen natural orbital (FNO) approach46–51to achieve an MI-FNO framework that enables a twofold reduction of the occupied and virtual molecular orbital (MO) spaces, respectively. The systematic truncation of the occu- pied MO space also becomes essential, in particular, when larger molecular systems that have a considerable number of electrons are targeted. We achieve a reduction in the occupied MO space by adapting a recently proposed incremental full configuration interac- tion (iFCI) approach52–54that is based on the method of increments and provides a polynomial scaling approximation to full configu- ration interaction (FCI). By decomposing the problem into n-body sub-problems (or “increments”), it has been shown that accurate correlation energies can be recovered at low values of nin a highly parallelizable computation.52–54The success of other incremental approaches has also been demonstrated elsewhere for traditional quantum chemistry applications.55–78Furthermore, several recent investigations9,29–35,79have focused on reducing the complexity of chemical systems on quantum computers by utilizing active spaces (i.e., ignoring certain occupied and virtual space orbitals) or truncat- ing the virtual orbital space (i.e., removing higher eigenvalue cano- nical virtual orbitals or systematically reducing the virtual space based on the FNO approach). A similar strategy to ours is deployed by Fielder et al. ,71where an incremental scheme is combined with local pair natural orbitals (LPNOs)80–86to achieve highly efficient and accurate reaction ener- gies for large molecular systems. However, to our knowledge, an approach for systematically and simultaneously reducing the occu- pied and virtual MO spaces has not yet been utilized for applica- tions of quantum chemistry on NISQ devices. Here, we demonstratehow the method of increments can be used to reduce the occupied space of a molecular system, and FNOs can be employed to trun- cate the virtual space. While there are other approaches to reduc- ing the virtual space, such as the LPNOs just mentioned, the FNO procedure employed herein is conceptually simple, easy to imple- ment, and has been shown to be effective in truncating the virtual space of a system48,49,87and has also been extended to the treatment of excited states.88–91Moreover, the effectiveness and usefulness of the FNO procedure for large-scale, coupled-cluster calculations has been demonstrated in recent work.92 As a first step, we validate the accuracy of the MI-FNO approach and demonstrate its ability to reduce both the occupied and virtual spaces while maintaining a reasonable level of accuracy by examining the small molecules BeH 2, CH 4, NH 3, H 2O, and HF using a moderate-sized cc-pVDZ basis set.93The ability to solve the electron correlation problem on classical (conventional) computers depends on the size of the computational space of the molecule. The molecular computational space can be defined in terms of the num- ber of electrons or occupied orbitals and the total number of MOs. For quantum devices, the corresponding computational space can be represented in terms of the number of qubits and the number of one- and two-qubit gates. In order to map the electronic struc- ture problem onto a quantum device, the Fock space representation of the wavefunction is used and the wavefunction is evolved using quantum gates. The number of qubits is equal to the number of molecular spin orbitals, while the complexity of the wavefunction can be represented by the number of one- and two-qubit gates. To demonstrate the efficacy of our MI-FNO approach on larger molecules, we provide a qubit count estimation for an industrially relevantα-olefin polymerization catalyst, a constrained geometry complex (CGC) catalyst,94using the cc-pVDZ and cc-pVTZ basis sets.93 This paper is organized as follows: In Sec. II, a review of the MI-FNO approach within a variational quantum eigensolver (VQE) framework is provided as an example of the quantum solvers suitable for NISQ devices. In Sec. III, the computational details are described and a schematic illustration is provided of our MI-FNO approach for large-scale quantum chemistry simulations on quantum hardware. In Sec. IV, we present the resulting molecular energies obtained using the MI-FNO approach and discuss its applicability for use on near-term devices. Section V provides a summary of results and possibilities for future work. II. THEORY The MI-FNO approach provides a framework for dividing the occupied space of a molecule using MI while the corresponding vir- tual space of each increment is compacted separately using the FNO procedure. In this section, we provide a brief overview of the ingre- dients that make up the framework, namely, MI to decompose the full occupied space into one-, two-, and three-body increments (see Fig. 1) and the use of FNOs to design a tailored virtual space for each increment. The electron correlation problem can then be solved in a reduced computational space using a given quantum algorithm or a conventional quantum chemistry approach when needed. The VQE, coupled with the unitary coupled-cluster singles and dou- bles (UCCSD) wavefunction Ansatz , is explored as an example of a possible quantum approach for NISQ devices. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1. Conceptual and schematic illustration of the MI-FNO framework for scaling up the size of molecules for quantum chemistry simulations on quantum hardware. The full molecular computational space is defined in terms of occupied orbitals (i, j, k, . . .) and virtual orbitals (a, b, c, . . .). The method of increments decomposes the occupied space into smaller one-, two-, and three-body increments, which consists of one, two, and three occupied orbitals, respectively. The virtual space of a given n-body increment can be reduced by using the FNO procedure. A. Conventional problem decomposition techniques 1. Reducing the occupied space: The method of increments The MI approach relies on the many-body or n-body Bethe–Goldstone expansion95of the electron correlation energy of a molecule. It was first introduced in quantum chemistry by Nes- bet.43–45The electron correlation energy ( Ec) is defined as the differ- ence between the exact ( Eexact) and the Hartree–Fock (mean-field) energy ( EHF). Using the many-body expansion, the electron correla- tion energy can be expressed in terms of n-body increments ( ϵi,ϵij, ϵijk, andϵijkl) as Ec=Eexact−EHF =∑ iϵi+∑ i>jϵij+∑ i>j>kϵijk+∑ i>j>k>lϵijkl+⋅⋅⋅ . (1) Then-body increments ϵi,ϵij,ϵijk, andϵijklare, respectively, the one-, two-, three-, and four-body increments defined as ϵi=Ec(i), (2) ϵij=Ec(ij)−ϵi−ϵj, (3) ϵijk=Ec(ijk)−ϵij−ϵik−ϵjk−ϵi−ϵj−ϵk, (4) ϵijkl=Ec(ijkl)−ϵijk−ϵijl−ϵjkl−⋅⋅⋅ (5) ⋮.Here, Ec(i)denotes the correlation energy of the increment i, Ec(ij)denotes the correlation energy of the increment i,j, and so on. These increments [the indices ( i,j,k,. . .)] appearing in the expansion of Eq. (1) can be orbitals, atoms, molecules, or fragments.52,55–64,67–70,72–78It is important to note that the n-body Bethe–Goldstone expansion of the electron correlation energy is not orbital-invariant. However, the lack of orbital invariance becomes much less significant as the expansion converges with n. The one-body increments include one doubly occupied orbital, two-body increments include two distinct doubly occupied orbitals, and so on. Depending on the nature of the correlation problem and/or the available computational resources, any suitable algo- rithm can be chosen to predict the correlation energies, whether geared toward quantum computing or classical architectures. The present framework can work with several quantum algorithms, such as VQE6and phase estimation algorithm (PEA).5,96Note that the MI framework requires size-extensive electronic structure schemes, such as the CC59,67,69or FCI approaches.52–54,72,74,77,78 2. Reducing the virtual space: Frozen natural orbitals The method of increments is a technique that provides an efficient and accurate approach for computing electronic correla- tion energies. One can view the MI approach as a framework for reducing the full occupied space of a molecule into a much smaller space. However, further reduction of the computational space will be required when we target applications on NISQ computers. In the present study, we incorporate the FNO approach46–51into our framework to further reduce the problem size by truncating the vir- tual orbital space. In recent work, this approach has been applied to J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp reduce the computational cost of quantum chemistry calculations in quantum computing.34 The natural orbitals are obtained by diagonalizing the one- particle reduced density matrix (RDM). In the case of “frozen” natural orbitals,97only the virtual–virtual block of the one-particle RDM is diagonalized. The occupied orbitals are just the original canonical Hartree–Fock occupied orbitals while a new transforma- tion and ranking is obtained for the virtual space such that the reference energy is invariant to the transformation. The correla- tion energy is also invariant to the transformation if the method employed is invariant under rotations of the virtual orbitals. Frozen natural orbitals are considered transformed and ranked virtual MOs that can be obtained at any arbitrary level of an ab initio theory. In this work, we use the MBPT(1) (many-body perturbation theory) wavefunction, which constitutes the first-order correction to the Hartree–Fock wavefunction. The one-particle virtual–virtual block of the MBPT(2) density matrix is diagonalized to obtain natural orbitals as eigenvectors and corresponding occupation numbers as eigenvalues. These eigenvalues can be used to truncate the virtual space, while the eigenvectors are employed to transform the virtual space. By choosing a certain threshold or population percentage cri- terion, a certain number of virtual orbitals can be kept, and the rest are ignored. The correlation energy is calculated only in the truncated vir- tual space, and then, the correction term ΔEMBPT(2)=EMO MBPT(2) −EFNO MBPT(2)is added to the correlation energy to recover the full correlation energy. The correction term ΔEMBPT(2)is the MBPT(2) correlation energy in the full molecular orbital space minus the MBPT(2) correlation energy in the truncated FNO space. In the spin-orbital basis, the virtual–virtual ( Dab) block of the one-particle MBPT(2) density matrix49is expressed as D(2) ab=1 2∑ cij⟨cb∥ij⟩⟨ij∥ca⟩ ϵcb ijϵca ij, (6) where the quantity ϵab ijin the denominator is defined as ϵab ij=fii+fjj−faa−fbb, in which fis the Fock matrix. Note that ⟨cb∥ij⟩=⟨cb∣ij⟩−⟨cb∣ji⟩is an antisymmetric two-electron integral. The indices i,jrepresent occupied spin orbitals, while a,b, and c represent virtual spin orbitals. B. A quantum approach to electron correlation 1. The variational quantum eigensolver algorithm We consider the VQE algorithm6as an example of the quan- tum solvers suitable for near-term applications on NISQ devices. The VQE algorithm was originally introduced, within the context of quantum chemistry, as a hybrid quantum–classical algorithm for solving the molecular electronic Schrödinger equation. According to the variational principle, for a (normalized) parameterization of the wavefunction ∣Ψ(⃗θ)⟩, if one minimizes the expectation value of the Hamiltonian operator ̂H, E=⟨̂H⟩=min ⃗θ⟨Ψ(⃗θ)∣̂H∣Ψ(⃗θ)⟩⩾Eexact, (7) an upper bound to the exact ground-state energy is obtained.We wish to estimate values of the parameters {θ1,θ2,. . .,θp} (i.e., the elements of the vector ⃗θ) that minimize the expectation value according to Eq. (7). The VQE algorithm requires a Hamilto- nian operator in qubit form (i.e., written in terms of Pauli operators). Furthermore, a unitary parametric Ansatz for the wavefunction in the qubit basis is required for the state preparation. Once the initial state has been prepared (i.e., an appropriate set of initial parame- ters has been used), an expectation value measurement is performed using quantum hardware or an appropriate simulation tool. Subse- quently, the current value of the expectation value is fed to a classical optimizer in order to estimate a new set of variational parameters. This provides a new wavefunction, and the procedure is repeated until an optimized wavefunction and expectation value have been obtained. The VQE algorithm constitutes a reduced circuit-depth hybrid quantum–classical methodology for solving the molecular electronic Schrödinger equation, as it minimizes the use of quantum hard- ware resources. In the second-quantization picture, the molecular electronic Hamiltonian takes the form ̂H=∑ p,qhp qˆa† paq+1 2∑ p,q,r,shpq rsˆa† pˆa† qˆasˆar, (8) in which p,q,r, and s label general spin-orbitals, and a† pandapare, respectively, creation and annihilation operators associated with the orbital p. The one- and two-electron integrals, hp qandhpq rs, are hp q=⟨p∣̂h∣q⟩=∫φ∗ p(x)⎛ ⎝−1 2∇2−N ∑ μ=1Zμ ∣r−Rμ∣⎞ ⎠φq(x)dx (9) and hpq rs=⟨pq∣rs⟩=∫φ∗ p(x1)φ∗ q(x2)1 r12φr(x1)φs(x2)dx1dx2, (10) respectively, in which Zμand Rμare the charge and position of nucleusμ, respectively, and r12=∣r2−r1∣is the inter-electronic dis- tance. The molecular Hamiltonian can be transformed into the qubit basis by using the Jordan–Wigner transformation,98 ̂H=∑ phpσα p+∑ pqhpqσα pσβ q+∑ pqrhpqrσα pσβ qσγ r+⋅⋅⋅ , (11) or another available transformation technique (e.g., Bravyi–Kitaev99 and Bravyi–Kitaev Superfast100). Here, p,q,r,. . .label qubits, and σα p, whereα∈x,y,z, is a Pauli matrix acting on qubit p. 2. The unitary coupled-cluster Ansatz While there are several strategies for deriving a parametric Ansatz for the wavefunction (e.g., hardware efficient,9Qubit Cou- pled Cluster (QCC),27and iQCC28), we consider the UCC Ansatz in this work. The choice of Ansatz is important for the convergence of the classical optimization and has a marked effect on the circuit depth. The latter issue is beyond the scope of the present study, but we plan to return to it in future work. Let us assume that the Hartree–Fock equations have been solved to obtain a zeroth-order, J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp single-determinantal, mean-field wavefunction ∣Φ0⟩and the one- and two-electron integrals in the spin-orbital basis. The UCC101–104 Ansatz for the correlated wavefunction can then be written as ∣Ψ(⃗θ)⟩=eˆT−ˆT† ∣Φ0⟩, (12) in which the cluster operator is defined as ̂T=̂T1+̂T2+⋅⋅⋅ (13) =∑ i,aθa iˆa† aˆai+1 2∑ i,j,a,bθab ijˆa† aˆa† bˆajˆai+⋅⋅⋅ . (14) The UCC Ansatz is usually truncated up to double excita- tions [i.e., including only ̂T1and̂T2in Eq. (14)], thus defining UCCSD. In analogy with the Hamiltonian in Eq. (11), the Ansatz of Eq. (12) can be transformed into the qubit basis. Due to the non- commuting nature of the operators used in the UCCSD Ansatz , the Suzuki–Trotter decomposition is used to decompose the exponen- tial of the cluster operator as a product of unitary operators acting on the reference wavefunction (obtained from a classical Hartree–Fock calculation) and is subsequently transformed into a qubit represen- tation. This Trotterized UCCSD Ansatz is then used for the state preparation step of the VQE algorithm discussed above to find an approximate expectation value of the molecular electronic Hamil- tonian, thus providing an estimate of the ground-state energy of a given molecule. III. COMPUTATIONAL DETAILS We perform UCCSD calculations using the incremental expansion approach and FNO-based virtual space truncation (MI-FNO-UCCSD). In order to understand the convergence behav- ior of MI-FNO-UCCSD energies as the size of the computational space grows, we also perform the MI-FNO calculation using con- ventional CCSD.105,106The calculations are performed on the exper- imental molecular geometries of BeH 2, CH 4, NH 3, H 2O, and HF obtained from the NIST Computational Chemistry Comparison and Benchmark Database.107The cc-pVDZ basis set93is used for all of the calculations. In the incremental expansion approach, we consider the many- body expansion series including up to two-body terms for BeH 2, as the expansion including up to three-body terms is equivalent to solv- ing the full problem (i.e., BeH 2has three doubly occupied orbitals). The expansion up to two-body terms for BeH 2includes three one- body increments and three two-body increments—in total, six incre- ments. For the rest of the molecules, which have five occupied orbitals, we examine the expansions up to three- and four-body increments. The resulting total numbers of increments are 25 and 30, respectively, for the expansions. For the virtual orbitals of each incre- ment, we examine the effect of the size of the virtual space by adding one virtual orbital (which has a higher FNO occupancy) at a time to the computational space of the increments. The implementation of the MI( n)-FNO approach is numerically validated by comparing the total energies computed with the MI( n)-FNO-CCSD approach with full system CCSD energies. The results are provided in Appendix A. For each molecular system except BeH 2, the electronic struc- ture problems for each of the increments, with a truncated virtualspace up to five virtual orbitals, is solved by using VQE with the UCCSD Ansatz , leading to, at most, a 16-qubit problem. For BeH 2, the expansion including two-body terms along with seven virtual orbitals is considered, which leads to, at most, an 18-qubit problem. An MBPT(2) FNO correction is added to the correlation energies obtained using a truncated virtual space in order to account for the missing correlation energies. The resulting correlation energies for each increment are used to reconstruct the correlation energy of the entire molecule by following the expansion scheme described in Sec. II A 1. We refer to the present approach as MI( n)-FNO-UCCSD [or MI( n)-FNO-CCSD if the classical CCSD approach is used to obtain the correlation energy], where nindicates the expansion up ton-body increments. To obtain an estimate of the number of qubits for a molecule relevant to industry, we consider a CGC catalyst uti- lized in the polymerization of α-olefins.94The configuration of the catalyst is obtained from a crystal structure of the CGC catalyst, and the cc-pVDZ and the cc-pVTZ basis sets are utilized.93 All of the quantum simulations reported are performed using the OpenFermion108and ProjectQ109software packages and the OpenFermion-ProjectQ108interface. The molecular integrals and Hartree–Fock solutions are generated using PySCF.110Incremental decomposition of the occupied orbitals and the corresponding gen- eration of the scalable FNO-transformed virtual space are achieved using the development version of QEMIST Cloud , the Quantum- Enabled Molecular ab Initio Simulation Toolkit .111The Open- Fermion program package is employed to map second-quantized quantities (e.g., Hamiltonian and UCCSD Ansatz ) to the qubit basis. The qubit representation of the truncated molecular Hamiltonian is obtained using the Jordan–Wigner transformation98implemented in OpenFermion. The VQE simulations, using the UCCSD Ansatz , are performed using ProjectQ and OpenFermion-ProjectQ. The OpenFermion-ProjectQ interface is then employed to convert the qubit form of the UCCSD Ansatz into a Trotterized time evolu- tion operator, which can easily be expressed in terms of elementary universal quantum gate operations. The ProjectQ program package, which is an ideal (noiseless) state vector simulator, is used to simu- late the UCCSD circuits and to evaluate the expectation value of the qubit Hamiltonian in the UCCSD state (i.e., using the exact repre- sentation of the state vector). ProjectQ is also employed to perform the gate counts using its resource estimation utility. The classical optimization steps of VQE are performed using the COBYLA algo- rithm112with a convergence tolerance of 10−5. The MBPT(1) ampli- tudes are used as an initial guess of the parameters for the UCCSD trial wavefunction. The conventional CCSD energy of the full prob- lem is also calculated using PySCF and used as a reference energy. In performing the conventional CCSD calculation, we use a tolerance of 10−7hartree. IV. RESULTS AND DISCUSSION A. The quantum computational efficiency of the MI-FNO approach Noisy, intermediate-scale quantum hardware is limited not only in the number of qubits it has but also in the number of gate operations. Therefore, it is important to understand the amount of quantum resources needed to achieve the desired accuracy in elec- tronic structure calculations.31In this section, we discuss to what extent the MI( n)-FNO approach can reduce the quantum resources J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp compared to full UCCSD simulation. The number of one- and two- qubit gates we report should be considered as an upper bound of the gate counts, as the actual number of gate counts can vary depending on the level of circuit optimization. The MI-FNO approach provides a framework for decomposing complex quantum systems into smaller increments that can easily be simulated or computed on NISQ devices. One can apply not only the UCC wavefunction on quantum devices but also the con- ventional coupled-cluster approach on classical machines to obtain electron correlation energies in the truncated computational space. As discussed below, the energy profiles of MI-FNO-UCCSD and MI-FNO-CCSD closely follow each other. This is quite encourag- ing, as we can now estimate the accuracy of the UCCSD approach for systems requiring an increasing number of qubits by using the accuracy of the CCSD method. Our goal is to approximate the CCSD energy of the full system by using the MI-FNO-CCSD or MI-FNO-UCCSD approach. We are interested in knowing the size of virtual space needed to approximate the CCSD energy in the full computational space to within 1 kcal/mol accuracy. The idea is not to provide a magic number for virtual orbitals that will be needed to obtain chemical accuracy but simply to explore the possibility of designing scalable examples of quantum chemistry problems that are appropriate for measuring the progress of NISQ devices. A further goal is to answer the question of to what extent quantum computa- tional resources can be reduced with such an aggressive truncation of the virtual space. Table I gives a summary of the quantum resources required to achieve chemically accurate energies. The convergence behavior of the energies of MI-FNO-UCCSD and MI-FNO-CCSD with respect to CCSD with an incremental increase in FNO-transformed virtual orbitals can be found in Figs. 2 and 3. A more detailed discussion can be found in Sec. IV B. The number of qubits required for the MI-FNO approach to achieve chemical accuracy with respect to the parent CCSD values are 32, 32, 34, and 26 for CH 4, NH 3, H 2O, and HF, respectively, which is a significant reduction, as the number of qubits necessary to perform a direct simulation of the full system is 68, 58, 48, and 38 for CH 4, NH 3, H 2O, and HF, respectively. A more detailed breakdown of the quantum resource estima- tions for MI(2)-FNO-UCCSD with an incremental increase in the virtual space for BeH 2is shown in Appendix B in Fig. 8, while Figs. 9–12 illustrate the resources required to perform MI(3)-FNO- UCCSD calculations on the HF, H 2O, NH 3, and CH 4molecules, respectively. As the main focus of this study is to design aframework that reduces qubit counts in a manner such that system- atically improvable results can be achieved with increasing quantum computational resources, detailed discussion on the reduction of other quantum resources, such as gate counts and measurement, we leave for future work. In brief, we find that our MI( n)-FNO-UCCSD approach considerably reduces the number of gate operation by 74%–98% from those required to perform full UCCSD simulations without PD. The resulting number of gate operations remains very large for NISQ hardware; however, our MI( n)-FNO approach is gen- eral and can be combined with any other Ansätze that may provide shallower circuits than UCCSD, such as the “hardware-efficient” Ansatz9and QCC methods.27,28,113 Due to the larger quantum computational requirements, per- forming VQE calculations based on the full molecular space UCCSD Ansatz are nearly impossible on an existing quantum device. For smaller molecules such as BeH 2, HF, H 2O, NH 3, and CH 4, 38–68 qubits are needed for the cc-pVDZ basis. Detailed information regarding the number of qubits, one-qubit gates, and two-qubit gates for these molecules using the cc-pVDZ basis can be found in Table IV in Appendix B. By limiting the many-body expansion of electron correlation energies to three-body increments, one can reduce any occupied space to a maximum of three doubly occupied orbitals. Assuming that the full virtual space is used to perform MI(3)-UCCSD calcu- lations, the qubit requirements for HF, H 2O, NH 3, and CH 4are reduced by four qubits. Without employing any aggressive strategies for the virtual space truncation, demonstrating the applicability of NISQ devices for chemistry applications is difficult. One can use an active space approach in which a selected number of Hartree–Fock virtual orbitals are included in the computational space to perform correlation energy calculations. Alternatively, one can also rank the virtual orbitals using FNO occupancies and select the most impor- tant virtual orbitals to obtain much improved correlation-energy calculation results in a truncated FNO space (i.e., compared to a truncated canonical Hartree–Fock MO space). It is well-known that the wavefunction-based approaches require larger basis sets to effec- tively capture electron correlation, and the corresponding virtual space of the larger basis is often sparse. Frozen natural orbitals can be used as a tool to recognize the sparsity and to help compress the virtual space. Often, up to a 50%–60% reduction in the virtual space is possible with a loss of only 1% in the correlation energy.48,49,87 The effectiveness of the FNO approach in compressing the vir- tual space can be demonstrated by plotting the cumulative FNO TABLE I. Quantum resources required to obtain chemically accurate energies. The number before the slash represents the quantum resources needed when using the MI( n)-FNO approach, and the number after the slash represents the quantum resources required for full UCCSD simulation without PD. The percentages given in parentheses represent the extent of reduction that the MI( n)-FNO approach achieved. No. of qubits No. of one-qubit gates No. of two-qubit gates BeH 2 18/48 (63%) 4 180/73 230 (94%) 6 944/302 160 (98%) CH 4 32/68 (53%) 143 214/1 726 498 (92%) 384 592/9 482 448 (96%) NH 3 32/58 (45%) 103 830/731 602 (86%) 275 520/3 373 264 (92%) H2O 34/48 (29%) 62 934/241 138 (74%) 182 592/929 648 (80%) HF 26/38 (32%) 36 870/205 498 (82%) 83 328/660 032 (87%) J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. Energy deviation (ΔE=EMI-FNO−ECCSD, in kcal/mol) of the MI(3)-FNO approaches [MI(3)-FNO- UCCSD and MI(3)-FNO-CCSD] with respect to reference energy is plotted as a function of monotonically increasing virtual space size. The energy obtained with CCSD using full MO space is used as a reference energy. Plots are obtained for BeH 2. The area shaded in gray indicates where the results are within chemical accuracy from the reference energy. occupancy percentage as a function of the number of virtual orbitals. A more detailed discussion is presented in Appendix A along with the assembled plots (see Fig. 6). B. The accuracy of the MI( n)-FNO-UCCSD approach To investigate the accuracy of MI( n)-FNO-UCCSD energies, the energies of BeH 2, CH 4, NH 3, H 2O, and HF are obtained using MI-FNO-UCCSD with various truncated virtual spaces. Although the number of virtual orbitals for each increment is the same, it is worth noting that unique FNO transformations are generated for each increment. Figure 2 shows how the total energy of BeH 2, using MI(2)- FNO-UCCSD, behaves as a function of the number of virtual orbitals. The plot shows the difference between the MI(2)-FNO- UCCSD and parent CCSD energies. The area filled in gray shows the region where deviations are within chemical accuracy. We see that the MI(2)-FNO-UCCSD values approach the reference CCSD energy as the number of virtual orbitals increases. When the num- ber of virtual orbitals is seven, the difference from the reference energy becomes as small as 0.000 860 hartree or 0.54 kcal/mol, show- ing that chemical accuracy has been reached. For this calculation, there are three 16-qubit, one-body increments (one occupied and seven virtual orbitals) and three 18-qubit, two-body increments (two occupied and seven virtual orbitals) (Table V in Appendix B). When the number of virtual orbitals is seven, we are able to discard 14 virtual orbitals. As the original problem requires 48 qubits without PD, this is a large reduction in quantum resources. We believe that the present MI-FNO framework can help accelerate the practical application of NISQ hardware in quantum chemistry simulations. We do not run VQE calculations beyond 16 and 18 qubits because storing the exact state vector of the system in a classicaldevice becomes challenging. To estimate the convergence behav- ior of the energy as a function of the virtual space beyond 18 qubits, we first explore whether the MI( n)-FNO-CCSD approach can be used to extrapolate the MI( n)-FNO-UCCSD energies for the BeH 2molecule. Varying the number of virtual orbitals from one to seven, we confirm that the convergence behavior of the MI( n)-FNO- UCCSD and MI( n)-FNO-CCSD approaches closely resemble each other. We then extend the MI( n)-FNO-CCSD calculations to the maximum number of virtual orbitals (21 in the present setup) to gain an understanding of the convergence of the MI(2)-FNO-UCCSD energies. Based on this extrapolation, we find that MI(2)-FNO- UCCSD can provide chemically accurate results when the number of virtual orbitals is larger than seven. Figure 3 contains similar information for HF, H 2O, NH 3, and CH 4. The MI-FNO-UCCSD energy profile for our test molecule closely follows that of MI-FNO-CCSD. As UCCSD and CCSD are fundamentally different theories, an exact equivalence between the results obtained from applying UCCSD and CCSD should not be expected. This is because the antisymmetric cluster operator in UCCSD also includes de-excitation operators and is solved by mini- mizing an energy functional variationally, while the CCSD Ansatz includes only excitation operators and involves the solution of a set of non-linear, projected residual equations. However, for the weakly correlated hydrides considered in this work, we would expect UCCSD to give very similar results to CCSD near the equilibrium geometry. The difference between the MI-FNO-UCCSD and MI-FNO- CCSD energies is just a fraction of kcal/mol. When the virtual space is smaller than the occupied space (i.e., ⩽3), the behavior of the energy convergence is not very smooth and not monotonically decreasing. Large errors (reaching a maximum of 12.8 kcal/mol for CH 4, 9.7 kcal/mol for NH 3, 9.2 kcal/mol for BeH 2, 4.3 kcal/mol for J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. Energy deviation ( ΔE=EMI-FNO−ECCSD, in kcal/mol) of the MI(3)-FNO approaches [MI(3)-FNO-UCCSD and MI(3)-FNO-CCSD] with respect to reference energy is plotted as a function of monotonically increasing virtual space size. The energy obtained with CCSD using full MO space is used as a reference energy. Plots are obtained for HF, H 2O, NH 3, and CH 4. The area shaded in gray indicates where the results are within chemical accuracy from the reference energy. H2O, and 4 kcal/mol for HF) are also associated with the smaller virtual space. When the virtual space become larger than the occu- pied space, the energy profile of the MI-FNO approach systemat- ically converges to within chemical accuracy to the CCSD energy of the full molecular space. When all of the virtual space is taken into consideration, the MI-FNO-CCSD results show excellent con- vergence with respect to CCSD, while for NH 3, H 2O, and HF, errors of approximately 1 kcal/mol are observed. The degree of the error may be mitigated by including higher-order body increments, so we examine the inclusion of four-body terms in the calculation [i.e., MI(4)-FNO-CCSD], as shown in Appendix A in Figs. 7(a)–7(d). In all cases consid- ered, the inclusion of the four-body terms in the MI(4)-FNO- CCSD calculation improves the energy convergence toward the reference energy, and the energy at the point where the num- ber of virtual orbitals is equal to the full virtual space differsfrom the reference energy by only around 5 ×10−5hartree or less. The reason why we observe highly accurate results with the MI(4)-FNO-CCSD approach for these molecules may be because they are all ten-electron systems and the contribution from the core electrons is usually very small when the basis sets have no core- polarization functions, such as in the cc-pVDZ basis used here. We plan to investigate the impact of higher-order body increments (e.g., three-body, four-body, and five-body) on the convergence behavior of the MI-FNO approach by targeting larger molecular systems. We also speculate that the error in the MI-FNO approach is due to the fact that the occupied orbitals are not spatially local- ized, and therefore, the decomposition of the occupied space into a smaller space, based on the method of increments, causes the residual error. We observe that using Foster–Boys localization114 improves the energy convergence of MI(3)-FNO-CCSD toward the J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. Total energy values (hartrees) and the difference from the conventional CCSD values using the MI( n)-CCSD approach. The differences are shown in parentheses. The calculated results for the many-body expansion truncated up to n=2-, 3-, and 4-body increments are listed. CCSD MI(2)-CCSD MI(3)-CCSD MI(4)-CCSD BeH 2−15.835 746 −15.835 806 ( −0.000 060) −15.835 746 (0.000 000) ⋅⋅⋅ ⋅⋅⋅ CH 4−40.385 951 −40.392 778 ( −0.006 827) −40.385 350 (0.000 601) −40.385 952 ( −0.000 001) NH 3−56.400 579 −56.412 272 ( −0.011 693) −56.399 440 (0.001 140) −56.400 581 ( −0.000 002) H2O−76.240 099 −76.254 995 ( −0.014 896) −76.238 622 (0.001 478) −76.240 102 ( −0.000 003) HF −100.228 154 −100.242 731 ( −0.014 577) −100.226 824 (0.001 331) −100.228 157 ( −0.000 003) reference energy. We plan to discuss this improvement in more detail in a future publication. The accuracy of approximating full molecular space corre- lation energies using MI and conventional quantum chemistry approaches, such as FCI and CC, has been explored by various research groups.55–78To validate the implementation of the MI( n) approach for our test molecules, the total energies of BeH 2, CH 4, NH 3, H 2O, and HF are calculated using the MI(2)-CCSD, MI(3)- CCSD, and MI(4)-CCSD methods with no virtual space truncation. A comparison of the total energies using MI and the full virtual space with conventional CCSD energies is presented in Appendix A in Table II. As discussed in the same section, an accuracy of 1 kcal/mol is achieved for these molecules using the MI(3) expansion. To vali- date the MI approach, in conjunction with a virtual space truncation based on the FNO approach (MI-FNO-CCSD), the conventional cri- terion of 99% occupancy for the virtual space truncation is used. The total energies calculated using MI-FNO-CCSD and their differ- ence from the reference CCSD total energies are listed in Table III (Appendix A 2). As discussed in that section, chemical accuracy is achieved for the total energies of the test molecules using MI-FNO- CCSD and a 99% FNO population percentage criterion. By plot- ting the cumulative FNO occupancy percentage as a function of the number of virtual orbitals (see Fig. 6), the FNO procedure is quite effective at exploiting the sparsity in the virtual space. The FNO pro- cedure produces a virtual space that is compact and becomes even more effective at compressing the size of this space as the size of the basis is increased.C. Future outlook of the MI-FNO framework In order to use VQE to test continuously evolving NISQ devices in chemistry applications, one should employ a framework such as MI-FNO, which provides systematically improvable results with increasing quantum computational resources. To demonstrate that the MI-FNO approach is an effective framework for systematically reducing quantum resources for applications of VQE, we provide estimates for the qubit count for an industrially relevant catalyst molecule. We provide these qubit counts for the CGC(1) catalyst from the study by Arriola et al.94(see Fig. 4) as an early indication of the effi- cacy of the MI( n)-FNO approach. The crystal structure of CGC(1) is obtained from Ref. 94, and the cc-pVDZ and the cc-pVTZ basis sets are used for the quantum resource estimations. In Fig. 5, we summarize the qubit count estimations when the FNO virtual space truncation is applied on its own and when the MI(3)-FNO approach is applied. To estimate the number of qubits for the FNO virtual space truncation, we add the number of FNO virtual orbitals Nvafter trun- cation to the number of occupied orbitals Noccin the system. The number of virtual orbitals Nvin the FNO approach is determined based on a given percentage of FNO occupancy for the full molecular system. Hence, the total qubit count is obtained using the expres- sion 2(Nv+Nocc). In the qubit count estimation for the MI(3)-FNO approach, the maximum number of occupied orbitals is three, cor- responding to the three-body increments in the MI(3) expansion. TABLE III. Total energy values (hartrees) and the difference from the conventional CCSD values using the MI( n)-FNO-CCSD approach. The differences are shown in parentheses. We employ an FNO population of 99% to determine the size of the virtual space. CCSD MI-CCSD MI-FNO-CCSD BeH 2MI(2)−15.835 746 −15.835 806 (−0.000 060 )−15.836 066 (−0.000 320 ) CH 4MI(3)−40.385 951 −40.385 350 (0.000 601 )−40.385 716 (0.000 235 ) CH 4MI(4)−40.385 951 −40.385 952 (−0.000 001 )−40.386 196 (−0.000 245 ) NH 3MI(3)−56.400 579 −56.399 440 (0.001 140 )−56.399 468 (0.001 111 ) NH 3MI(4)−56.400 579 −56.400 581 (−0.000 002 )−56.400 693 (−0.000 114 ) H2O MI(3) −76.240 099 −76.238 622 (0.001 478 )−76.238 514 (0.001 585 ) H2O MI(4) −76.240 099 −76.240 102 (−0.000 003 )−76.239 986 (0.000 113 ) HF MI(3) −100.228 154 −100.226 824 (0.001 331 )−100.226 893 (0.001 261 ) HF MI(4) −100.228 154 −100.228 157 (−0.000 003 )−100.227 998 (0.000 156 ) J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. Molecular structure of a CGC catalyst from Ref. 94. Hence, the total qubit count in Fig. 5 is obtained using the expression 2(Nrv+3), where Nrvis obtained using our FNO procedure with the MBPT2 density that has only three occupied orbitals as active orbitals. This is a rough estimate of the number of qubits needed for the MI(3)-FNO approach as we do not scan all the increments. The number reported in Fig. 5 for the MI-FNO approach corresponds to the increment that has three occupied orbitals: HOMO, HOMO-1, and HOMO-2. The MI(3) approach, without truncation of the virtual space, reduces the qubit requirements by 172 for a CGC catalyst. Fur- ther reduction in qubit counts is achieved by employing an FNO population percentage (i.e., 99%) to truncate the virtual space. The MI(3)-FNO approach significantly reduces the qubit requirements.The UCCSD with the full molecular space calculation needs 956 and 2208 qubits, respectively, for the cc-pVDZ and cc-pVTZ basis sets, whereas FNO-UCCSD with a 99% FNO population percentage needs 768 and 1298 qubits, respectively, for the basis sets. This approach leads to a virtual space reduction of 25% and 45%, respectively, for the cc-pVDZ and cc-pVTZ basis sets (see Figs. 13 and 14 and the rest of Appendix C for more details). Given the finding in Appendix A 2 (see Table III) that the MI-FNO approach with a 99% FNO occupancy can produce very accurate results for our test molecules, we speculate that it is possible to obtain results of similar quality for larger molecules. The MI(3)-FNO-UCCSD approach with a 99% FNO population percentage needs 126 and 184 qubits, respectively, for the cc-pVDZ and cc-pVTZ basis sets. This approach leads to a virtual space reduction of 85% and 90%, respectively, for the basis sets (see Figs. 15 and 16 and the rest of Appendix C for more details). The number of qubits is drasti- cally decreased by truncating the virtual space with a smaller FNO population percentage. In addition, for a smaller FNO percentage, the number of qubits required does cause an appreciable change for both basis sets. A comparison of the qubit count between the FNO and MI(3)-FNO approaches demonstrates that the MI(3)-FNO approach has a much smaller qubit requirement than the FNO approach. We believe that the MI-FNO approach is quite beneficial for calculating electron correlation energies for larger molecular systems, where one must deal with many occupied orbitals. It is needless to mention that the number of increments in the MI-FNO approach will become larger for a system with a large occupied space. For example, a CGC catalyst has about 117 569 increments in the three-body expansion. Furthermore, an effective screening procedure, such as the distance-, energy-, and domain-based approaches,59,72can be implemented to reduce the number of incre- ments while maintaining the chemical accuracy of the calcula- tion. We note that many applications of the incremental scheme FIG. 5. Qubit count estimates for a CGC catalyst. The molecule has 89 occupied orbitals. It has 389 and 1015 virtual orbitals for the cc-pVDZ and cc-pVTZ basis sets, respectively. The y axis rep- resents the number of qubits needed to perform the methods, while the x axis represents the FNO% that is used to truncate the virtual space. The meth- ods considered in the plots are FNO and MI(3)-FNO. In FNO, only the virtual space is truncated, while in MI(3)-FNO, both the occupied and virtual spaces are truncated. Numbers on the right (1298, 768, 184, and 126) represent the number of qubits needed when the virtual space is truncated using a 99% FNO population percentage. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp involve systems that are much larger than those considered in Secs. IV A and IV B. The expansion is often made over certain sets of orbitals instead of making the expansion over the occupied orbitals used in this work (e.g., an expansion is made over domains containing a given number of occupied orbitals, as in Refs. 59, 65, and 66). Such an approach can help improve the convergence behavior in an n-body expansion. It would be interesting to explore what the minimum quantum resource requirements would be for obtaining chemically accurate total energies for these large molecular systems after implementing a highly parallel framework. V. CONCLUSION Quantum computing is an alternative computational paradigm with the potential to accelerate the materials innovation process, thereby reducing the time to new discoveries. In the era of NISQ devices, VQE has emerged as a promising algorithm for characteriz- ing the usefulness of NISQ devices for quantum chemistry applica- tions. However, NISQ devices have to overcome many challenges before they become useful for chemical applications. For exam- ple, some of the main bottlenecks are the design of scalable phys- ical quantum states with long coherence times and fast gate oper- ations with low error rates. Efforts have been made to mitigate issues arising from large quantum resource requirements, coher- ence and run times of quantum circuits, noisy gate operations, measurements of energy, and the classical optimization of Ansatz parameters. In the present study, we have focused on reducing the problem size as a strategy for utilizing current and near-term NISQ devices for the simulation of molecular systems. We believe a reduction in qubit count to be essential in helping to advance the timeline for applications of quantum computing in materials science. At the same time, it could provide opportunities for further characteri- zation of the usefulness of NISQ devices for quantum chemistry simulations by allowing hardware experiments to be conducted on smaller, yet more realistic, chemistry problems. We have described a novel framework for the systematic reduction of both the occupied and virtual spaces of molecular systems. Our MI-FNO approach dis- tributes the occupied orbitals among n-body increments based on the many-body expansion of the correlation energy in terms of the occupied orbital space, while a scalable framework for the virtual orbital space is created by using the FNO approach. As a demonstration of the applicability of the MI-FNO approach, we used VQE in combination with the UCCSD Ansatz . We examined its accuracy and feasibility by studying small molecules, namely, BeH 2, CH 4, NH 3, H 2O, and HF, in a cc-pVDZ basis set. We observed that the MI-FNO approach can achieve chem- ical accuracy by significantly reducing both the number of qubits and the number of gate operations, which suggests that it can be used to build a scalable quantum chemistry simulation platform that effec- tively utilizes quantum hardware. Furthermore, as an early demon- stration of the efficacy of this approach for larger molecules, we have presented qubit count estimations for a titanium-metal-based CGC catalyst that has relevance in the large-scale polymerization of α-olefin. We found that by employing a modest truncation of the virtual space using a 99% FNO occupancy, a significant reduction in the qubit requirements can be achieved.ACKNOWLEDGMENTS This work was supported as part of a joint development agree- ment between Dow and 1QBit. We are grateful to Alejandro Garza and Peter Margl from Dow for technical discussions and guidance regarding industrial chemistry use cases and applications and to Paul M. Zimmerman at the University of Michigan for technical discus- sions. The authors thank Marko Bucyk at 1QBit for reviewing and editing the manuscript. APPENDIX A: NUMERICAL VALIDATION OF THE MI( n)-FNO IMPLEMENTATION 1. Accuracy of the method of increments without truncation of the virtual space In the MI-FNO approach, two sources of approximation are used. The correlation energy is approximated by using the MBE, and the virtual space is truncated using the FNO procedure. To val- idate the MI( n)-FNO approach, we first examine the accuracy of the energy calculations using the method of increments, with no vir- tual space truncation. The total energies of BeH 2, CH 4, NH 3, H 2O, and HF are calculated using the MI(2)-CCSD, MI(3)-CCSD, and MI(4)-CCSD methods. The total energies computed using MI and the full virtual space are then compared against conventional full molecular space CCSD energies. The total energies using the MI methods and the compari- son of the energies with the CCSD energies are listed in Table II. We achieve chemical accuracy (0.0015 hartree or 1.0 kcal/mol) with respect to the conventional CCSD value for the six-electron system BeH 2using the MI(2) approach. The MI(3) value agrees with the parent CCSD value because they are equivalent for this six-electron system. Hence, we find that MI(2) is sufficiently accurate for per- forming calculations on BeH 2. For the other four systems, each of which contains ten electrons, we observe a relatively large error using the MI(2) expansion. The CH 4molecule exhibits the smallest error,−0.006 827 hartree (4.28 kcal/mol), which is over four times larger than the target value of 1.0 kcal/mol needed for chemical accuracy. In contrast, with the MI(3) approach, we achieve chemi- cal accuracy in the total energies for all molecules considered. The largest error was observed for the total energy of H 2O, which is 0.001 478 hartree (0.93 kcal/mol) larger than the CCSD value. The error becomes less than 5.0 ×10−6hartree when using MI(4) for these ten-electron systems. The accuracy of the MI(2) expansion is not sufficient for achieving chemical accuracy for the ten-electron systems considered in this work. The number of occupied orbitals in CCSD calculations is reduced by decomposing the original problem into subproblems (increments) using the MI expansion. For BeH 2, the MI(2) calcu- lation includes only two occupied orbitals, while three occupied orbitals are used in the CCSD calculation. For the other ten-electron systems, the MI(3) and MI(4) calculations include three and four occupied orbitals, respectively, while full CCSD calculation includes five occupied orbitals. In this study, we use relatively small-sized molecules; thus, the reduction of occupied orbitals is small. How- ever, as we show later in this section, if we apply MI methods to larger-sized systems, we achieve a large reduction in the number of occupied orbitals. The MI method has the potential to recover accurate total energies while reducing computational costs. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-11 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 2. Accuracy of the method of increments with truncation of the virtual space To investigate the accuracy of molecular energy calculations using the MI approach, in conjunction with virtual space trunca- tion based on the FNO approach (MI-FNO-CCSD), we choose the criterion of virtual orbital selection using a population percentage of 99%. The total energies calculated using MI-FNO-CCSD and their difference from the reference CCSD total energies are listed in Table III. Chemical accuracy is achieved for the total ener- gies calculated with all the MI-FNO-CCSD approaches. The MI(3)-FNO-CCSD calculation performed on the H 2O molecule exhibits the largest error of 0.001 478 hartree (0.99 kcal/mol). The FNO approach reduces the number of virtual orbitals for each sub- problem in the MI expansion. For BeH 2, using the MI(2) expansion, the FNO method with the threshold of 99% occupancy discards 17, 5, and 7 virtual orbitals from the three one-body increments and five, seven, and six virtual orbitals from the three two-body increments, while the full problem of BeH 2has 21 virtual orbitals. Therefore, the FNO method with a 99% threshold is able to discard at least fivevirtual orbitals. For the other ten-electron systems with the MI(3) expansion, the FNO approach discards at least seven, five, three, and two virtual orbitals for the CH 4, NH 3, H 2O, and HF molecules, respectively. Again, we consider smaller systems in this work, so the reduction may not appear significant. However, we observe that the FNO virtual space truncation becomes more efficient as the virtual space becomes larger. Figure 6 shows the cumulative FNO occupancy percentage as a function of the number of virtual orbitals for the molecules we examine. The values on the horizontal axis represent the ratio of the number of virtual orbitals, calculated as “the number of virtual orbitals that are used in the calculation” divided by “the total number of virtual orbitals of the system.” The dotted line shows the FNO occupancy of 99%. The plots are obtained by running FNO-CCSD calculations, not by using the MI-FNO-CCSD approach. As shown, the larger basis set reaches the 99% FNO occupancy faster than the smaller basis sets. This means that the FNO truncation discards more virtual orbitals as the virtual space becomes larger. Therefore, if we were to apply MI-FNO-CCSD to larger-sized systems or employ larger basis sets, we would achieve not only a considerable reduction in the number of occupied orbitals FIG. 6. Cumulative FNO occupancy as a function of the number of virtual orbitals plotted for BeH 2, CH 4, NH 3, H 2O, and HF. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-12 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7. Energy deviation ( ΔE=EMI-FNO −ECCSD, in hartree) of the MI( n)-FNO approaches [MI(3)-FNO-CCSD and MI(4)-FNO-CCSD] with respect to the reference energy is plotted as a function of monotonically increasing virtual space size. The energy obtained with CCSD using the full MO space is used as the reference energy. Plots are obtained for CH 4, NH 3, H 2O, and HF. The area shaded in orange indicates where the results are within chemical accuracy to the reference energy. but also a significant reduction in the number of virtual orbitals. We find that the MI-FNO-CCSD method accurately recovers the total molecular energies while reducing the computational cost, in comparison with the CCSD approach. The many-body expansion of the correlation energy can be truncated to three-body or four-body terms. To show the impactann-body has on energy, a comparative study of energy con- vergence behavior is shown in Fig. 7. Deviation of the energy of MI(3)-FNO-CCSD and MI(4)-FNO-CCSD with respect to the energy of CCSD is plotted against the monotonically increasing vir- tual space. The four-body expansion recovers full CCSD energy well for the ten-electron systems. FIG. 8. Quantum resources required to prepare the quantum state of the subsys- tem of BeH 2as a function of the number of virtual orbitals. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-13 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp APPENDIX B: DETAILED QUANTUM RESOURCE ESTIMATION Figures 8–12 show the quantum resources required to prepare the quantum states of the subsystems for each of the moleculesBeH 2, HF, H 2O, NH 3, and CH 4as a function of the num- ber of virtual orbitals. The MI( n)-FNO approach (where nis restricted up to three-body terms) produces many increments, and the quantum resources that are required for each vary FIG. 9. Quantum resources required to prepare the quantum state of the subsys- tem of HF as a function of the number of virtual orbitals. FIG. 10. Quantum resources required to prepare the quantum state of the subsys- tem of H 2O as a function of the number of virtual orbitals. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-14 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 11. Quantum resources required to prepare the quantum state of the subsys- tem of NH 3as a function of the number of virtual orbitals. FIG. 12. Quantum resources required to prepare the quantum state of the subsys- tem of CH 4as a function of the number of virtual orbitals. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-15 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV. Computational space for CCSD in terms of the number of electrons and molecular orbitals and quantum computational space of UCCSD in terms of the number of qubits, one-qubit gates, and two-qubit gates. CCSD UCCSD Molecules Electrons Mol. orbitals Qubits One-qubit gates Two-qubit gates BeH 2 6 24 48 7.32 ×1043.02×105 HF 10 19 38 2.05 ×1056.58×105 H2O 10 24 48 2.41 ×1059.30×105 NH 3 10 29 58 7.32 ×1053.37×106 CH 4 10 34 68 1.73 ×1069.48×106 TABLE V. Number of qubits required to obtain chemical accuracy using the MI( n)-FNO-UCCSD approach. The number of qubits is estimated based on the energies obtained with the corresponding MI( n)-FNO-CCSD approach for CH 4, NH 3, H2O, and HF. The numbers in parentheses indicate the number of increments the MI( n) approach generates. UCCSD MI(2)-FNO-UCCSD MI(3)-FNO-UCCSD MI(4)-FNO-UCCSD BeH 2 48 18 (6) ⋅⋅⋅ ⋅⋅⋅ CH 4 68 ⋅⋅⋅ 32 (25) 36 (30) NH 3 58 ⋅⋅⋅ 32 (25) 32 (30) H2O 48 ⋅⋅⋅ 34 (25) 28 (30) HF 38 ⋅⋅⋅ 26 (25) 24 (30) depending on the increment size. The largest numbers of qubits and one- and two-qubit gate counts are plotted for the increment that has the largest quantum resource requirements.APPENDIX C: EFFECTIVENESS OF FNO IN COMPACTING VIRTUAL SPACE OF A CGC CATALYST Figures 13–16 show the cumulative FNO occupancy percentage as a function of the number of virtual orbitals for a CGC catalyst. FIG. 13. Cumulative FNO occupancy as a function of the number of virtual orbitals for a CGC catalyst for FNO-CCSD/cc- pVDZ. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-16 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 14. Cumulative FNO occupancy as a function of the number of virtual orbitals for a CGC catalyst for FNO-CCSD/cc- pVTZ. FIG. 15. Cumulative FNO occupancy as a function of the number of vir- tual orbitals for a CGC catalyst for MI(3)-FNO-CCSD/cc-pVDZ. J. Chem. Phys. 155, 034110 (2021); doi: 10.1063/5.0054647 155, 034110-17 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 16. Cumulative FNO occupancy as a function of the number of vir- tual orbitals for a CGC catalyst for MI(3)-FNO-CCSD/cc-pVTZ. The values on the horizontal axis represent the ratio of the num- ber of virtual orbitals, calculated as “the number of virtual orbitals that are used in the calculation” divided by “the total number of vir- tual orbitals of the system,” while the vertical axis represents the cumulative FNO occupancy percentage. The dashed-dotted hori- zontal line shows the FNO occupancy of 99%. The plots are obtained by running FNO or MI-FNO calculations with either a cc-pVDZ or cc-pVTZ basis. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1M. Head-Gordon and E. Artacho, Phys. Today 61(4), 58 (2008). 2Y. Manin, Computable and Noncomputable (Sovetskoye Radio, Moscow, 1980), pp. 13–15 (in Russian). 3R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982). 4S. Lloyd, Science 273, 1073 (1996). 5A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Science 309, 1704 (2005). 6A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, Nat. Commun. 5, 4213 (2014). 7C. Hempel, C. Maier, J. Romero, J. McClean, T. Monz, H. Shen, P. Jurcevic, B. P. Lanyon, P. Love, R. Babbush, A. Aspuru-Guzik, R. Blatt, and C. F. Roos, Phys. Rev. X 8, 031022 (2018). 8P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. 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5.0054731.pdf

The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Exploration of interlacing and avoided crossings in a manifold of potential energy curves by a unitary group adapted state specific multi-reference perturbation theory (UGA-SSMRPT) Cite as: J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 Submitted: 21 April 2021 •Accepted: 14 June 2021 • Published Online: 1 July 2021 Dibyajyoti Chakravarti,1,a) Koustav Hazra,1Riya Kayal,1Sudip Sasmal,2,b) and Debashis Mukherjee3,c) AFFILIATIONS 1School of Chemical Sciences, Indian Association for the Cultivation of Science, Kolkata, India 2Physikalisch-Chemisches Institut, Universität Heidelberg, Heidelberg, Germany 3Centre for Quantum Engineering, Research, and Education (CQuERE), TCG-CREST, Kolkata, India a)Electronic mail: rana.chakravarti@gmail.com b)Electronic mail: sudip.sasmal@pci.uni-heidelberg.de c)Author to whom correspondence should be addressed: pcdemu@gmail.com ABSTRACT The Unitary Group Adapted State-Specific Multi-Reference Perturbation Theory (UGA-SSMRPT2) developed by Mukherjee et al. [J. Comput. Chem. 36, 670 (2015)] has successfully realized the goal of studying bond dissociation in a numerically stable, spin-preserving, and size-consistent manner. We explore and analyze here the efficacy of the UGA-SSMRPT2 theory in the description of the avoided crossings and interlacings between a manifold of potential energy curves for states belonging to the same space-spin symmetry. Three different aspects of UGA-SSMRPT2 have been studied: (a) We introduce and develop the most rigorous version of UGA-SSMRPT2 that emerges from the rigorous version of UGA-SSMRCC utilizing a linearly independent virtual manifold; we call this the “projection” version of UGA-SSMRPT2 (UGA-SSMRPT2 scheme P). We compare and contrast this approach with our earlier formulation that used extra sufficiency conditions via amplitude equations (UGA-SSMRPT2 scheme A). (b) We present the results for a variety of electronic states of a set of molecules, which display the striking accuracy of both the two versions of UGA-SSMRPT2 with respect to three different situations involving weakly avoided crossings, moderate/strongly avoided crossings, and interlacing in a manifold of potential energy curves (PECs) of the same symmetry. Accu- racy of our results has been benchmarked against IC-MRCISD +Q. (c) For weakly avoided crossing between states displaying differently charged sectors around the crossing region, the insufficient inclusion of state-specific orbital relaxation and the absence of dynamic correla- tion induced by orbital relaxation in the first order wavefunction for a second order perturbative theory lead to an artifact of double crossing between the pair of PECs. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054731 I. INTRODUCTION Multireference (MR) electron correlation theories have been a rapidly emerging area of research in recent years1–3for the proper description of excited state behavior and dissociation profile of chemically relevant molecules. Such theories by design are formu- lated in a given N-electron sector of the Fock space, referred toas the Hilbert space, and are thus structurally quite different from the earlier developed multireference correlation theories that used valence universal wave operators.4–8For most of the MR meth- ods, a generic strategy is to divide the electron correlation into non-dynamical, which is a strong correlation resulting from near- degeneracy of certain configurations, and dynamical correlation, which results from the configurations that are excited with respect J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to the near-degenerate configurations (called the active space). The non-dynamical or static correlation, being the stronger, needs a more accurate description that is introduced commonly using a full configurational interaction within an active space (called the com- plete active space, CAS) comprising not only the quasi-degenerate configurations but also the other configurations that are required to make the active space complete. These latter configurations might be high-lying in energy but are necessary to maintain the size extensiv- ity of the resultant multiconfigurational wavefunction. The dynami- cal correlation energy contribution is usually an order of magnitude smaller but is essential to provide the chemical accuracy required for an appropriate theoretical description of a spectroscopic or chemical phenomenon. It can be computed using either a non-perturbative or a perturbative approach, and it is necessary to properly theorize its contribution with certain desirable properties such as (a) size exten- sivity,9(b) size consistency,10and (c) invariance with respect to the orbital basis used. Dynamical correlation effects are usually incorporated using various strategies: MR variants of CI theory,11coupled cluster (CC) theories12–20including the unitary versions thereof,21,22perturbation theories (PT),23–35and other allied approaches.36–45The general problem with Multireference Configuration Interaction is that despite using a complete active space, the inclusion of virtual functions that are excited with respect to the CAS functions up to a given rank leads, in general, to size-inextensive energies, although empirical corrections to alleviate the severity of this error have been in use.46Most of the CC and PT formulations involve construction of an effective Hamiltonian (H eff) in the active space, which contains the effect of both kinds of correlation folded into it via a wave operator ( Ω). The diagonalization of this H effin the active space gives us the energies of the states of interest. More often than not, the use of a complete active space to generate H eff, although seemingly elegant in structure, is seriously affected by numerical instability due to the notorious and ubiquitous “intruder state” problem.47We do not want to go into the details of the intruder problem here, as the practitioners of multirefer- ence correlation theories are well aware of its origin and deleterious effects. Several approaches have been proposed to ameliorate the intruder problem, which differ considerably among themselves both in their approach and in detail. They can be broadly clas- sified into three categories, which differ in their treatment of the dynamical correlation: (i) an effective Hamiltonian in a suitably chosen incomplete active space,48–50(ii) an intermediate Hamilto- nian approach,51and (iii) state-selective (targeting specific states of interest) approach.15–17 The incomplete model space approach involves the selection of an appropriate subspace of the CAS in defining effective Hamil- tonians such that the active space functions that strongly interact with the intruding virtual functions are pushed into the virtual man- ifold and then diagonalizing the H effin this truncated space. It is very difficult to maintain size extensivity for the corresponding energies evaluated by this approach due to the incompleteness of the model space, but a rigorously size-extensive approach was pro- posed by Mukherjee, which involves abandoning the intermediate normalization condition (IN) for the eigenfunctions.48–50 The intermediate Hamiltonian formulation is an ingenious manipulation where the effective Hamiltonian is not truncated froma complete model space, but it is divided into two configurational subspaces, one containing the CSFs dominating the states of inter- est (the so-called main model space) and the other space contains the CSFs that dominate the states that are intruder prone (the inter- mediate space). A common shift in the zeroth order energies for the model functions is given to the Bloch-like equations for the states in the intermediate space, which numerically removes their divergence, and additionally, a condition for vanishing the coupling between the intermediate and the desired states is imposed. This manipulation leads to numerical stability both for the states dominated by the main model space and for the spurious states dominated by the interme- diate space, despite diagonalizing in the complete model space. The only limitation of this approach is the lack of size extensivity of the energies computed, unless explicit constraints are imposed to ensure size extensivity.49,52,53 An extreme approach to remove the intruder problem is to use the effective Hamiltonian approach but solve for cluster amplitudes for one specific state of interest at a time. This has been the most promising approach among the three, but a formulation that sat- isfies all the desirable attributes (a–c as mentioned earlier) for the evaluated energy remains a challenge. Looked at from a very accurate inclusion of both dynamical and non-dynamical correlation, the state-specific version of MRCC developed by Mukherjee et al. ,3,15,54using a decontracted wave operator ansatz of the Jeziorski–Monkhorst (J–M) type,12has been shown to have both the desirable properties of size extensivity and intruder-free solutions but lacks orbital invariance in the energy and other properties computed. On the other hand, an internally con- tracted formulation (IC-MRCC)55–57suffers no orbital invariance problem, but it is at the expense of a huge redundancy problem in the virtual manifold, which has to be removed via a computationally expensive extraction of an orthogonal excitation space. On the other end of the spectrum, a PT approach to dynamic correlation would have been much cheaper and hence more attrac- tive for applications to larger chemically relevant systems and has garnered a lot of attention in the recent past. The error in the abso- lute energy computed by a PT would be far less relevant in com- parison to the correct explanation of trends in the studied chemical phenomenon. The earliest developments were based on perturbative approximations to MRCISD, and two of the most popular of these are the complete active space second order perturbation theory and the multireference moller plesset second order perturbation theory methods developed by Andersson et al.23and Hirao,24respectively. Both of them are second order SS theories and utilize a contracted unperturbed function. CASPT2 suffers from substantial errors in size extensivity and consistency, along with a notorious problem of numerical instability due to the necessity of a larger CAS in the the- ory than is chemically required to describe the studied phenomenon. The numerical instability that plagues the contracted State-Specific Multi-Reference Perturbation Theory (SSMRPTs) is mainly due to lack of coefficient relaxation in the perturbed wavefunction and has been well discussed in a review by Malrieu et al.58Often, ad-hoc “level shifts”59,60have been used in CASPT2 to avoid the instability, but this really throws no light on the reason and nature of it. We may additionally note that, in contrast, the shift parameter introduced in the intermediate Hamiltonian formalism is physically motivated and does not affect the final energies for the states of interest that are dominated by the main model space. The original J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp MRMP2 theory too is size inextensive and not orbital invariant, although an improved version has been later reported by van Dam et al.61The NEVPT2 developed by Angeli et al.27eliminates the need of extracting the linearly independent excitation manifold by an ingenious definition of the excitation operators. It has both SS and intermediate Hamiltonian versions. Several choices of unper- turbed Hamiltonian have been used, with some of them being able to minimize the intruder state problem. The fully general version of the theory is both size extensive and orbital invariant but computationally expensive. The generalized van Vleck second order perturbation theory developed by Hoffmann26is based on a unitary wave operator and is an intermediate Hamiltonian based approach. It is not rigorously size-extensive, but the numerical errors have been found to be small in molecular applications. A recent devel- opment by Liu and Hoffmann31is the SDSPT2 theory, which is also an intermediate Hamiltonian based method, and it too is not strictly size-extensive. The UGA-SSMRPT2 theory developed by Mukherjee et al. is a perturbative approximant of the UGA-SSMRCC theory developed by the group, and it preserves all its desirable qualities: (i) ener- gies are strictly size extensive for each state targeted by the theory and (ii) the solution of the target state is numerically stable, pro- vided that it is energetically well-separated from the virtual space. In practice, targeting a high-lying state within the effective Hamil- tonian space may be numerically unstable; however, if its solution can be reached by some regularization methods, then the energy will still be strictly size-extensive.33,54The UGA-SSMRPT2 stands out from the other MRPTs discussed briefly earlier, owing to the fact that it is systematically improvable order by order upto a fully developed non-perturbative theory, i.e., UGA-SSMRCC. The only other MRPT that has a corresponding non-perturbative analog is the GVVPT2 theory, which is a perturbative version of a unitary MRCC theory.62Neither the UGA-SSMRPT2 nor the UGA-SSMRCC theo- ries are orbital invariant owing to the use of a Jeziorski–Monkhorst (J–M) like12decontracted wave operator. The original spin-orbital-based formulations of both SSM- RCC15and SSMRPT263required the use of suitable sufficiency con- ditions for determining the cluster amplitudes. Although in a spinor- bital based formalism, the virtual functions generated from a given CSF in the active space are all linearly independent, the need for sufficiency conditions arises because a given virtual function can be reached from more than one model functions. This implies that any virtual CSF can be generated from multiple model space CSFs via the various model-function dependent cluster operators. The suffi- ciency conditions provide the number of equations equal to that of all these cluster amplitudes. In spin-adapted SSMRCC54and SSM- RPT33formulations, a virtual CSF is not uniquely specified by the orbital occupancies alone because of the so-called spin-degeneracy problem.64This imposes an additional linear dependence on a set of virtual CSFs in the UGA-based state specific many-body the- ories using the J–M ansatz as is the case in the UGA-SSMRCC theory and UGA-SSMRPT generated therefrom. One may imagine that this requires use of additional sufficiency conditions, but this is not necessarily the case. There are actually two different ways to resolve this problem: (i) To generate suitable amplitude equations, we may posit suitable additional sufficiency conditions , which is the most straight-forward solution. The resultant “amplitude” equations will henceforth be denoted as scheme A. A similar idea was firstsuggested by us in a previous spin-adapted MRPT formulation as well.30(ii) One can, however, completely bypass the use of this extra sufficiency condition by choosing only those virtual CSFs for projections, generated from a given model CSF, which are linearly independent as demanded by the spin degeneracy. The UGA based formalisms allows one to naturally choose only those CSFs. This lat- ter choice leads to another, inequivalent, formalism, which one may call “projection” equations (to be henceforth called scheme P). Both of these developments will be discussed in Sec. II of our paper, first for the UGA-SSMRCC and then for the UGA-SSMRPT2 generated therefrom. We should mention here a recent work by Giner et al. ,65 which formulated a spin-adapted Jeziorski–Monkhorst (J–M) based MRPT2 theory, called by them as JMMRPT2. Being a J–M based ansatz for its wave operator, the theory is also not orbital invari- ant. Size extensivity in JMMRPT2 has been imposed on the the- ory by using a clever manipulation of the denominator, which amounts to the use of a specific unperturbed Hamiltonian, H(0), which is then used to solve for the cluster amplitudes. However, this novelty is probably limited to the perturbative level only and does not seem to be easily generalizable to a non-perturbative theory. One might wonder about the efficacy of an SS-MR theory in either a perturbative or CC approach to describe potential energy curves (PECs) of a manifold of states displaying avoided crossings when the theory is employed for each of the states separately. In fact, this demands more flexibility and accuracy on a theory to sense the presence of neighboring states involved in the various modes of avoided crossing. In this paper, we will explore the effectiveness of our UGA-SSMRPT2 theory using both of its versions (schemes A and P) in treating a manifold of states of the same space-spin sym- metry, particularly highlighting regions of single or multiple avoided crossings, the latter to be called “interlacing.” Such avoided cross- ings demand reproduction of the rather sensitive change in gradient in that region in the dissociation profiles of each state. It is not immediately obvious that the manifold of states that are targeted using separate state-specific computations would “sense” each other in terms of the intricate gradient changes with a change in geome- try and would exhibit their interlacing with each other. Moreover, the positions of these avoided crossings are sensitive to the inclu- sion of dynamical correlation and the state-specific relaxation of active orbitals attendant on such dynamical correlation. We will highlight the importance of state-specific orbital relaxation, espe- cially for describing the weakly avoided crossings involving dramatic differences in the relative weightage of ionic and covalent character of the states concerned. In a perturbative computation of second- order energy, the first-order perturbed function contains insufficient orbital relaxation effects, which can be crucial when the relaxation is strong. Since it is natural to use a common set of orbitals to describe PECs of close-lying states, the interplay of state-specific orbital relaxation and dynamical correlation in a perturbative the- ory such as UGA-SSMRPT2 becomes crucial to ascertain. In this paper, the formulation of scheme P is presented in some detail since it has never been discussed previously. We will also compare the use of operator equations (scheme A) involving sufficiency conditions against the proper projection scheme (scheme P). It will be shown that the errors with respect to MRCI methods is comparable in both the schemes, thus suggesting the efficacy of our extra sufficiency J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp conditions that significantly reduce the computational time of our calculations. Our paper is organized as follows: Sec. II traces the rigorous genesis of the UGA-SSMRPT2 suggested in this paper from the rigorous UGA-SSMRCC theory. The underlying theoretical issues distinguishing the amplitude and projection schemes and the cor- responding working equations are discussed next. The aspects of state-specific orbital relaxation, in particular, for situations involv- ing weakly avoided crossings, are also introduced and discussed here. Section III contains the molecular applications and relevant discussions on various PECs of a selection of manifold of states of prototypical molecules. The molecules chosen by us are such that three different aspects of PEC, viz., the medium and strongly avoided crossing, the weakly avoided crossing, and the interlacing of var- ious PECs in a given symmetry manifold, can be demonstrated. Section IV summarizes the highlights of our findings and presents our future outlook. II. THEORY A. Description of a rigorous UGA-SSMRCC and its approximant, UGA-SSMRPT2, for second order energy As mentioned in Sec. I, the earlier UGA-SSMRPT2 theory was developed by Mukherjee et al. manifestly in the amplitude form.33,66 It has been shown to work very well to describe complex bond disso- ciation profiles of N 2, B2, C2, O 2, etc. The low-lying excited states of these molecules with different space-spin symmetry have also been computed successfully. The theory is an intrinsically low-scaling and a rigorously size-extensive, intruder-free, state-specific multirefer- ence theory. We summarize the main tenets of the theory that help us to arrive at the final working equations. Unlike many of the multireference perturbation theories described in Sec. I, UGA-SSMRPT2 is truly a second order approx- imant originating from a general non-perturbative correlation the- ory. It is derived from the UGA-SSMRCC54theory, preserving all of its desirable qualities, such as size-extensivity and intruder-free nature, with the additional advantage of a low computational cost. The essential motivation in generating the UGA-SSMRPT2 from UGA-SSMRCC is to find an unperturbed Hamiltonian H(0)that cap- tures much of the important physics at the lowest order. In the UGA- SSMRPT2 theory, multi-partitioning of H(0)is the natural choice, i.e., H(0)varies for each model function ϕμin the CAS chosen. The idea of multi-partitioning of the unperturbed Hamiltonian was first introduced by Malrieu et al.67–69 The parent UGA-SSMRCC introduced a wave operator Ωof the Jeziorski–Monkhorst12(J–M) type but posited a normal-ordered exponential cluster ansatz, essentially to exclude the cumbersome contractions between the cluster operators, Ω=∑ μΩμ=∑ μ{eTμ}∣ϕμ⟩⟨ϕμ∣. (1) The curly bracket in Eq. (1) indicates normal ordering with respect to a suitable closed-shell vacuum. For a detailed discussion on this ansatz and its advantages over other spin-free theories, we refer the readers to recent literature from our group.54,70,71We have recently derived a rigorous and systematically improvable UGA-SSMRCCtheory, which is the closest spin-free analog to the spinorbital SSM- RCC theory derived by us earlier.15This derivation and a compre- hensive discussion would be presented in a soon to be published article.72 For any many-body operator A μ, we can decompose it as a sum of a “closed” operator Acl μ, which when acting on a CSF ϕμ, spanning the active space, leads only to transition within the CAS functions, and an “excitation” operator Aex μ, which leads to virtual functions (orthogonal to the CAS functions) when acting on ϕμ. The working equation for determining the cluster amplitudes Tμis of the form Gex μ∣ϕμ⟩=0∀μ. (2) The coefficients {cν∀ν}are to be obtained from the equations of the form ∑ νGcl ν∣ϕν⟩cν=E∣ϕμ⟩cμ∀μ. (3) We present below the operator form of the rigorous CC work- ing equation72for the cluster amplitudes {T μ}, without its detailed derivation, to indicate the genesis of the rigorous UGA-SSMRPT2 following from it, (4) whereθμis defined as71,72 (5) The first order perturbed operator equation will follow from the linearized terms of Eq. (4), (6) It is pertinent to mention here that an earlier UGA-SSMRCC was developed by Maitra et al.54whose working equations were an approximation to Eq. (4), where the operator θμdid not appear. Interestingly enough, it turns out that the first order perturbed equation, either in amplitude or projection form, would still not involve the operator θμsince it starts contributing only from second order onward. This aspect of the theoretical content of the working equations for UGA-SSMRPT2 was not noted before. The final working equation for UGA-SSMRPT2 cluster ampli- tudes is thus identical to the one derived in our earlier paper by Sen et al.33It is obtained by partitioning H as H(0)+V and collecting terms up to first order of perturbation in Eq. (6) as follows: (7) J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp B. Emergence of amplitude equations (scheme A) in UGA-SSMRPT2 As mentioned toward the end of Sec. I, to avoid the compu- tationally involved procedure of extracting the linearly indepen- dent virtual space, one may posit a sufficiency condition called “amplitude” equations. If we use this sufficiency to solve for the clus- ter amplitudes, we can afford to use a set of overcomplete projections onto a linearly dependent basis { χl′ μ}; writing Gex μ=∑ l′gl′ μ{El′ μ} (8) and using the following as sufficiency conditions, we get a set of cluster amplitudes {tl′ μ}, gl′ μ=0,∀l′,μ, (9) where l′denotes all possible changes in orbital occupancy from a particularϕμ. It should be noted here that all the residue blocks ( g), which correspond to a specific orbital excitation operator, have to be clubbed together using the appropriate transformation factors. This essentially means that all direct and exchange spectator scatter- ing blocks that are proportional to a common lower body G-block are multiplied by the reduction factor and added into that common lower body residue. For a detailed diagrammatic discussion of these residue block transfers, we refer to our earlier paper.33 The relaxed second order energy is obtained by diagonalizing the second order (H eff)μν, which is constructed using the converged cluster amplitudes, (Heff)[2] μν=⟨ϕμ∣Hν∣ϕν⟩=Hμν+∑ lHμltl(1) μΓμν, (10) where Γis the Graphical Unitary Group Adapted (GUGA) transition density matrix element correlating the CSFs ϕμandϕν. There is also a possibility to compute the unrelaxed second order energy, which circumvents the diagonalization step, and it is obtained by taking an expectation value involving the unrelaxed coefficients {c0 μ}only, E[2] unrelaxed=∑ μνc[0] μ(Heff)[2] μνc[0] ν. (11) The comparison between these two energies has been extensively studied by Sen et al. ,33which focused on computing PECs for states belonging to different space/spin symmetries, and as a result, there were no regions of avoided crossings and interlacings between the PECs. The general conclusion was that the absolute value of the unrelaxed energy is higher than the relaxed value by the order of a few millihartrees. The PEC features and the parallelity with experimental or CI curves were reported to be similar in both the approaches. However, this would not be true when the first order wavefunction is dramatically revised under the effect of dynamical correlation, as is the case in our current study of avoided crossings and interlacing. Thus, we would present only the relaxed energies in Sec. III. Moreover, even if the relaxation of the zeroth order wave- function seems less influential in terms of the trends in the PECs, it would gain importance when calculating properties other thanenergy. We thank a reviewer for kindly emphasizing this aspect to us. This facet of the theory would be studied extensively in a future work. C. Projection scheme (scheme P) to solve for cluster amplitudes When Eq. (7) is projected onto the set of virtual functions {χl μ}, which are generated by singles and doubles excitation upon the CAS functions {ϕμ}, we get (12) The solution of cluster amplitudes using Eq. (12) requires the extrac- tion of the linearly independent virtual functions manifold in order to avoid singularity in the solution of the linear simultaneous equa- tions. In the ensuing discussion, we shall denote the inactive hole orbitals by i,j,..., the inactive particle orbitals by a,b,..., and the active orbitals by u,v,...;usand udwould correspond to singly occupied and doubly occupied active orbital indices, respectively. The higher body operators, which are obviously proportional to a particular lower body T, are excluded from this extraction proce- dure at the very beginning itself by inspection, e.g., direct spectator active orbital scattered operators, such as Eau iu, or doubly occupied active orbitals in exchange spectator mode, such as Euda i u d. The redun- dancy inχl μspace has its roots in the fact that the action of multiple cluster operators on a given ϕμcould possibly give rise to the same virtual function due to the spin-degeneracy of χl, e.g., the action of{Ea i}and{Eusa i u s}on a model function ∣...jius⟩would lead to the excited functions having same orbital occupancy ∣...jaus⟩. The aforementioned virtual functions could have been distinguishable in a spin-orbital based CC method as the action of {Eua iu}would result in a spin flip of the electron inactive orbital u resulting in a triplet excitation from i to a, but there is obviously no way to distinguish them in a spatial orbital basis. We note here that the T2operators containing exchange “spectator” scatterings of singly occupied active orbitals are two body cluster operators resulting in a single occupancy change when they act upon any CSF, and their inclusion is necessary to cover the complete spin-space of the virtual manifold under a given operator truncation scheme in a “spin-free” formulation. We may call them “pseudo” two body cluster opera- tors. These along with the other T 1’s contribute to orbital relaxation in the perturbed wavefunction via Thouless’ theorem.73The rest of the T 2’s contribute to the dynamical electron correlation energy of the system. As we have emphasized earlier, there generally exists a redun- dancy in the virtual manifold of Eq. (12) arising from the fact that the action of n different operators (say) inducing the same change of orbital occupancy on a particular CSF may give rise to a set of m virtual functions, where m <n. To get a non-singular metric in the set of projection equations, we form the overlap matrix consisting of all such operators for a fixed CSF and try to extract from it the lin- early independent combinations of such operators using a singular- value decomposition (SVD). As cluster operators that change the J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp occupancy of different inactive orbitals can never lead to the same virtual function, we define classes of primitive operators categorized according to the number and type of inactive indices involved in the excitation and go for a SVD classwise for each CSF, ϕμ. Since the overlap matrix formed in this case is a Hermitian one, SVD is equivalent to orthogonal transformation of the overlap matrix, and we will only be using the eigenvectors of such an orthogonal- ization procedure, which will be the combining coefficients (to be denoted as XPl) for the orthogonal combination of the primitive operators. The number of non-zero diagonal elements left after such orthogonalization would be the number of such orthogonal combi- nations ( l), which again will be equal to the number of equations at hand, thus removing redundancy in the virtual space. The scheme of extracting the linearly independent/orthogonal combinations is as follows. A typical element of the overlap matrix (S PQ) would look like SPQ=⟨ϕμ∣{E† P}{EQ}∣ϕμ⟩. (13) Note here that Pand Qbelong to the same primitive class, and we have suppressed the index μfor brevity. SPQis expressed in terms of spin-free reduced density matrices for a given CSF. The SVD transformation in matrix notation is given by X† lPSPQXQl=(Sll)d, (14) where the XPl’s are the orthogonal combining coefficients we are interested in, and the primitive operators ( E) are related to the orthogonal operators ( ϵ) as {ϵl}=∑ P{EP}XPl (15) with ∣χl⟩={ϵl}∣ϕμ⟩. (16) Projecting onto the linearly independent manifold ⟨χl∣, Eq. (12) takes the form ∑ PRX† lPSPRGR=0∀l, (17) where the GRis the primitive residue used in the amplitude equation of (7). Equation (17) will be our projection equation. These two inequivalent approaches to solve for the excitation amplitudes is independent of the choice of H(0)and is also applicable to any order of perturbation right up to the full non-perturbative UGA-SSMRCC theory. D. Choice of H(0) The choice of H(0), i.e., the unperturbed Hamiltonian, has to be such that it does not connect the model space and the virtual manifold. One might imagine that a generalized Fock operator in the natural orbital basis and defined with respect to the unperturbed contracted function Ψ0could be an acceptable choice. However, this is not suitable for a perturbation theory involving an uncontracted treatment of the CAS functions {ϕμ}. A much better description of H(0)would be multi-partitioning67it, where we define the H(0)tobe dependent on the particular CSF it acts on. This choice would include the effective one-body Fock potential of the doubly and singly occupied active orbitals in a given ϕμ. Since the occupancy of active orbitals differs in each reference function ϕμ, we will define the unperturbed Hamiltonian H(0)as ˜fq μp=fq cp+∑ ud∈μ(2Vqud pud−Vudq pud)+∑ us∈μVqus pus, where fcdenotes the core Fock operator, which is common for all ϕμs; p, q are general orbital indices but should belong to the same “class,” i.e., inactive holes ( i,j,...), inactive particles ( a,b,...), or active orbitals ( u,v,...). This effective one-body reference dependent Fock operator as H(0)in the UGA-SSMRPT2 theory was successfully applied by Sen et al. , and for a comprehensive discussion on this choice, we refer the readers to earlier papers from our group.33,66 III. RESULTS AND DISCUSSION We divide our applications into three categories. In Subsec- tions III A–III C, we study (a) the moderate to strong avoided crossings for the potential energy manifolds of LiH, Be 2, and the asymmetric H 2S+; (b) the interlacing behavior in the different space-spin symmetry potential energy manifolds of BC; and (c) very weakly avoided crossing in the PECs of BeH2+, BeF2+, LiF, and BN. As emphasized in Sec. I, the PEC for each state belonging to a particular space-spin symmetry manifold have been computed separately, since in a state-specific MR theory, each root is the target solution of its own specific effective Hamiltonian. It is not obvious to what extent the second order energy of the states in a given symmetry manifold will sense each other’s presence, which would lead to the various gradient changes and interlacing of the PECs between these states. Only if the formalism is rich enough to faithfully approximate each specific root, it is reasonable to infer that each PEC would display interlacing and avoided crossing to an extent demanded by the accuracy of the state-specific method. This is why a study of the behavior of manifolds of states of a given symmetry belonging to the categories (a)–(c) is of paramount importance. In Subsections III A–III C, we will show that both the versions of UGA-SSMRPT2 (schemes A and P) faithfully describe the inter- lacing and avoided crossing features in the PECs for all the systems studied except in LiF and BN, where we will demonstrate the failure to describe a weak ionic-covalent avoided crossing by a state-specific second order perturbative theory and also will try to provide a ratio- nale behind this observation. We will also present a comparative study of the two possible avenues to solve for cluster amplitudes and validate the efficacy of the sufficiency conditions we had imposed in our earlier works.33,66GAMESS-US74version 14 has been used to generate the CASSCF one- and two-particle MO integrals as well as the GUGA one-particle transition density matrix. MOLPRO-1975 has been used to generate the IC-MRCISD +Q energies in all the sys- tems, except BC whose data were kindly provided to us by Mavridis and Tzeli.76,77The UGA-SSMRPT2 code developed by Sen et al.33for scheme A was considerably modified via the exclusive usage of Basic J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Linear Algebra Subprograms wherever applicable in order to utilize its intrinsic openMP thread parallelization. In addition to that, the entire scheme P was developed and implemented as discussed in this paper for the first time. Validation of our sufficiency condition: As discussed earlier, the rigorous method to solve for the cluster amplitudes would be via the projections of the residue equations onto the orthogonal virtual CSFs (χl μ) (scheme P) according to Eq. (17). This could be bypassed if we choose instead a sufficiency condition such that all individual amplitudes of the excitation part of the operator G, Gl′ μ, are set to zero (scheme A), viz., Eq. (7). Scheme A has been extensively used by Sen et al. in their applications66and has been shown to be very promising. In what follows, we validate the relative performance of the computationally cheaper scheme A against the more involved but formally more rigorous scheme P using MRCISD +Q as the benchmark method. We find from Table I that these two schemes have a very similar performance with respect to the computed potential energy curves. For the sake of clarity in the figures and to avoid cluttering of data points in them, hereafter we will present the UGA-SSMRPT2 scheme A numbers in the PECs to demonstrate the proper avoided crossing and interlacing behavior predicted by our theory. Validation of the PEC features: In this subsection, we com- pare the features of avoided crossing given by our theory with the MRCISD+Q numbers. The distance at which avoided crossing occurs is reported in Table II, and the energy gap between the two states at the avoided crossing region is tabulated in Table III. The dissociation energy computed by us is compared with experimental data wherever available in Table IV. We see from Tables II–IV that the features of interlacing and avoided crossing are well reproduced by our state-specific theory. In Subsections III A–III C, we also show the actual PECs of each system studied by our theory. TABLE I. Relative performance of the two inequivalent schemes A and P in UGA- SSMRPT2 shown via the statistical parameters: Non-Parallelity Error (NPE) and Mean Average Deviation (MAD). Molecule, UGA- NPE with respect MAD with respect state SSMRPT2 to MRCI +Q to MRCI +Q and basis scheme (in a.u.) (in a.u.) LiH 11Σ+A 4.87 ×10−41.60×10−4 aug-cc-pVTZ P 4.87 ×10−41.60×10−4 LiH 21Σ+A 4.58 ×10−41.52×10−4 aug-cc-pVTZ P 4.58 ×10−41.52×10−4 Be211Σ+ g A 2.44 ×10−32.85×10−4 aug-cc-pVQZ P 2.45 ×10−32.95×10−4 BeH2+12Σ+A 5.60 ×10−52.13×10−5 aug-cc-pVDZ P 5.60 ×10−52.13×10−5 BeH2+22Σ+A 4.47 ×10−42.26×10−5 aug-cc-pVDZ P 4.47 ×10−42.26×10−5 BeH2+32Σ+A 4.46 ×10−42.05×10−5 aug-cc-pVDZ P 4.46 ×10−42.05×10−5 BC 14Σ−A 7.84 ×10−32.19×10−3 cc-pV5Z(-h) P 7.63 ×10−32.08×10−3TABLE II. Comparison of the distance at which avoided crossing occurs for MRCI +Q and UGA-SSMRPT2. UGA-SSMRPT2 MRCI +Q Molecule States (in a.u.) (in a.u.) BeF2+12Π, 22Π 22.23 20.55 BeH2+22Σ+, 32Σ+52.05 51.80 BC 22Π, 32Π 2.75 2.65 We find from Table IV that our computed dissociation energies tally very well with the corresponding experimental values, and this further validates the rigorous size-extensivity present in our formu- lation. The somewhat larger error for the dissociation energy of the Be2molecule will be discussed in Subsection III A 2. This gives us an indication that the UGA-SSMRPT2 is a good low-scaling candi- date for gaining insight into dissociation and bonding phenomena without sacrificing too much numerical accuracy. Keeping these observations in mind, we proceed to describe the molecular states studied by us in greater detail. A. Molecular states exhibiting strong to moderately strong avoided crossings 1. LiH LiH bond dissociation has always been the first test case that comes to mind for most electronic structure theories. The main rea- sons are as follows: (i) it shows a reasonable multireference character as soon as it is stretched away from equilibrium, and it is in full force as we approach the fragmentation limit; (ii) it is easy to have a Full Configuration Interaction (FCI) comparison with reasonably large bases; and (iii) the first excited state shows reasonably strong avoided crossing with the ground state along with considerable variation in dipole moment. TABLE III. Comparison of the energy gap between the two states at the avoided crossing for MRCI +Q and UGA-SSMRPT2. Molecule States UGA-SSMRPT2 (in eV) MRCI +Q (in eV) LiH 11Σ+, 21Σ+1.23 1.19 BeH2+22Σ+, 32Σ+1.60×10−41.99×10−4 BC 22Π, 32Π 0.19 0.14 TABLE IV. Dissociation energy (D e) for the computed states with UGA-SSMRPT2. UGA-SSMRPT2 MRCI +Q Expt. Molecule States (in eV) (in eV) (In eV) LiH 11Σ+2.522 2.522 2.51578 LiH 21Σ+1.075 1.079 1.07678 Be2 11Σ+ g 0.138 0.129 0.11679 BeF2+12Π 1.93 2.03 1.9280 BC 14Σ−4.554 4.375 4.59776 J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp To study the moderately strong avoided crossing between the ground and first excited1Σ+states, a (2,2) CAS containing the CSFs {σ2,σσ∗, andσ∗2} is sufficient. The ground and first excited states are dominated by the ionic and covalent configurations, respec- tively, before this avoided crossing, and the relative weightages of the configurations swap after that point. However, the correct fragmentation limit of the first excited 1Σ+state is not ionic but a covalent H (2S) and Li (2P) channel. This involves passing through a second avoided crossing with a higher-lying state, which cannot be described by the (2,2) CAS, and this phenomenon can only be described well using an appropriately large basis78along with the inclusion of 2p and 3d orbitals of Li in the CAS, essentially necessitating a (2,10) CAS.81We display the two avoided crossings in Fig. 1. All orbitals were correlated in our computation. From Fig. 1, we find that the gradient changes in our UGA- SSMRPT2 curve follow closely with those in the corresponding MRCISD+Q curve. Here, we again stress the fact that for MRCISD +Q, we get both states as the roots of the same Hamiltonian, while in our theory, the two states are solutions of two different effective Hamiltonians. The non-parallelity error with respect to MRCISD +Q is of the order of 10−4hartree, as seen in Table I, and the gap between the two states taken at 7.50 bohrs is very much comparable between the two methods as seen in Table III. From Table IV, we note that the dissociation energies computed by us differs from the experimental values78by a few milli-eV. 2. Be 2 Be2has a very unusual bonding that is hard to take account of, at a non-correlated level. It has been the topic of research for many decades both theoretically and experimentally. Although the bond is weaker than an average chemical bond, it is far stronger than one caused by van der Waal dispersion forces. Be 2has two distinct features in its dissociation curve: (i) a deep minimum at short FIG. 1. Ground and first excited1Σ+states of LiH computed using the aug-cc- pVTZ basis and a (2,10) CAS.distance that can be described theoretically only after the inclusion of dynamical electron correlation and (ii) a significant gradient change, owing to a very strong avoided crossing with an energetically much higher non-bonding van der Waal state, before fragmentation into two Be(1S) atoms. The unexpected bonding minimum in Be 2is attributed to the promotion of an s electron to a p orbital, resulting in a Be(3P) state.82The avoided crossing between the non-bonding state con- sisting of two 1s22s2Be atoms and the bonding state consisting of two 1s22s12p1Be atoms results in the deep bonding minimum of the ground state around 4.50 bohrs. There is also a strong angular correlation from the 3d πorbitals to this sp hybrid state. Thus, the binding in Be 2can be attributed to the dynamical correlation effect, which is very rare in the case of similar dimer systems. This is also why both the RHF and the CASSCF PEC of Be 2show no bonding minimum. The change in slope of the ground state PEC around 6.0 a.u. is ascribed to the contribution of 3d and, additionally, the presence of f and g orbitals. This is exemplified by the basis set dependence of the ground state PEC, which has been studied previously by Schmidt et al.79and Kalemos.83It has been shown that the inclusion of upto p functions in the basis does not give the deep minimum at all in the CI curves. The inclusion of up to d functions gives not only the bonding well but also a second shallow minimum as an artifact (see Fig. 2). The experimental PEC is reproduced only after including the diffuse f and g functions in the basis for Be, which now shows a slight change in gradient around 6.0 bohrs giving the PEC a shoulder-like appearance in that region79,83(see Fig. 3). We study the aforementioned PEC features and basis set effects with our perturbation theory using the minimal CAS (4,4) and delo- calized canonical CASSCF orbitals. We employ both the aug-cc- pVDZ (containing upto d functions) and the aug-cc-pVQZ (con- taining upto g functions) bases. All orbitals were correlated in our computation. FIG. 2. Ground1Σ+ gstate of Be 2computed by UGA-SSMRPT2 using the aug-cc- pVDZ basis and a (4,4) CAS. The emergence of the shallow minimum is clearly manifest. J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. Ground1Σ+ gstate of Be 2computed by UGA-SSMRPT2 using the aug-cc- pVQZ basis and a (4,4) CAS. From Figs. 2 and 3, we see that even at a much lower order of correlation, the UGA-SSMRPT2 has reproduced all the distinctive features in the PEC of Be 2along with the pronounced basis set effects on it. The non-parallelity between our computed PEC and that by MRCISD+Q is of the order of millihartrees, as seen in Fig. 4 and in Table I. Figure 4 further demonstrates this difference with respect to the Be–Be internuclear distance. The dissociation energy computed, using the delocalized (4,4) CASSCF orbitals, differs from the experimentally obtained value by 0.022 eV, as seen in Table IV. This difference is somewhat larger than the rest of the reported D e’s due to the intrinsic lack of orbital invariance present in our wave operator ansatz. As noted FIG. 4. Energy difference for ground1Σ+ gstate of Be 2between UGA-SSMRPT2 and MRCISD +Q using the aug-cc-pVQZ basis and a (4,4) CAS.before, the minimal CAS was enough for our theory to simulate all the essential characteristics of the ground state PEC on which we are focusing in this paper. However, to achieve a proper size consistent fragmentation limit and to improve upon the value of dissociation energy, an (8,8) CAS containing all the s,p orbitals on each atom should be employed. We have verified that this ensures proper localization of the MOs on each Be atom, which, in turn, results in the size consistency error to be of the order of 10−1μH. This issue of size consistency in homonuclear diatomic systems has been exclusively discussed in an earlier paper from our group.66 3.H2S+ The first two excited2A′states of an asymmetric H 2S+ cation have been an interesting test case for multireference meth- ods and have been previously studied by Li and Huang84and Datta and Mukherjee.85It has been studied as a function of change in the ∠H–S–H bond angle and exhibits a moderately weak avoided crossing between the two states, giving rise to two minima in the 12A′state and one global minimum in the 22A′ state. In our UGA-SSMRPT2 computation, we use a (3,2) CASSCF function state-averaged over the first two2A′states. We employ the Dunning cc-pVQZ basis for both H and S atoms. The bond lengths of H–S are fixed at 1.595 and 1.399 Å. Five core orbitals were kept frozen during the computation of dynamic correlation. From Fig. 5, we see that the change in gradient with the change in the bond angle ∠H–S–H and the moderately weak avoided crossing between the two states is neatly reproduced by our theory. B. Interlacing within a manifold of states of same space-spin symmetry in BC The potential energy profiles of the ground state4Σ−symme- try manifold of BC along with its excited states of varying space-spin FIG. 5. Ground and first excited2A′states of asymmetric H 2S+computed by UGA-SSMRPT2 using cc-pVQZ basis and a (3,2) CAS. J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6. The4Σ−manifold of BC computed by UGA-SSMRPT2 and MRCISD +Q using the cc-pV5Z(-h) basis and a (7,8) CAS. symmetries provide a comprehensive test case for our theory to be able to reproduce the various interlacing and avoided crossings in each manifold. We follow the work done by Mavridis and Tzeli76,77 where they have computed the MRCISD( +Q) profiles of each man- ifold using a high level basis set, cc-pV5Z(-h). A (7,8) CAS was employed for each manifold, and the starting function was a dynam- ically weighted state averaged CAS function over all the states com- puted in an individual space-spin symmetry manifold. The Πstates were computed using the C 2subgroup of the actual symmetry point group in order to preserve the xy degeneracy of the πMOs. The two 1s-dominant molecular orbitals were kept frozen during all our computations. Figures 6–11 demonstrate the accurate reproduction of the multiple interlacing and avoided crossing behavior in each space- spin symmetry manifold of states. The computed curves differ from MRCISD+Q values computed by Mavridis and Tzeli76,77by a FIG. 7. The2Πmanifold of BC computed by UGA-SSMRPT2 and MRCISD +Q using the cc-pV5Z(-h) basis and a (7,8) CAS. See Figs. 8 and 9 for a zoomed-in view of the avoided crossings between 12Π, 22Πand 22Π, 32Πstates, respectively. FIG. 8. The zoomed-in view of the avoided crossing region between the ground and first excited2Πstates of BC computed by UGA-SSMRPT2 using the cc-pV5Z(-h) basis. few millihartrees in terms of non-parallelity error, as is shown in Table I. The moderately weak avoided crossing in the2Πmanifold also occurs at a distance very close to that seen in the MRCISD +Q com- putations along with a comparable energy gap in that region, as can be seen in Tables II and III. The experimental dissociation energy was available only for the4Σ−ground state of BC, and the value computed by us differs from it by just 0.04 eV, as shown in Table IV. C. Molecular states exhibiting very weak avoided crossings 1.BeF2+ BeF2+is one of the few diatomic dications which is thermody- namically stable80in its ground2Πstate. This system is well-suited FIG. 9. The zoomed-in view of the avoided crossing region between the first and second excited2Πstates of BC computed by UGA-SSMRPT2 using the cc-pV5Z(-h) basis. J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 10. The2Σ−manifold of BC computed by UGA-SSMRPT2 and MRCISD +Q using the cc-pV5Z(-h) basis and a (7,8) CAS. to test our theory for weakly avoided crossings. In fact, the ground state has a low energy barrier of dissociation to Be+and F+due to a very weak avoided crossing with the first excited2Πstate. We employ the Dunning cc-pVDZ basis and a (5,4) CAS to generate the ground and first excited2Πstates based on a state- averaged CASSCF function. All orbitals were correlated in our com- putation. The C 2subgroup of the actual symmetry point group was used in order to conserve the x–y degeneracy of the πorbitals. Thus, our starting function had to be state-averaged over two pairs of degenerate states to obtain the “symmetry-pure” state. The origin of the thermodynamic stability of BeF2+in its ground state is well predicted by our computations, as seen in Fig. 12. The very weak avoided crossing region is shown further in a zoomed-in view of the PEC in Fig. 13. From Table II, we find that the avoided crossing between the two states computed by our theory occurs about 1.60 bohrs farther than that in the MRCISD +Q curves, FIG. 11. The4Πmanifold of BC computed by UGA-SSMRPT2 and MRCISD +Q using the cc-pV5Z(-h) basis and a (7,8) CAS. FIG. 12. Ground and first excited2Πstates of BeF2+computed using the cc-pVDZ basis and a (5,4) CAS. See Fig. 13 for a zoomed-in view of the very weak avoided crossing. which indicates that a higher order PT or a non-perturbative treat- ment is warranted to improve upon this feature of avoided crossing. However, the overall gradient changes for both of the states are still adequately described by UGA-SSMRPT2. The dissociation energy of the ground state differs by 0.01 eV from the experimental value, as seen in Table IV. 2.BeH2+ BeH2+is a short-lived, metastable dication that has been stud- ied both theoretically and experimentally in the last decade.86The ground2Σ+state shows a moderately strong avoided crossing with the first excited2Σ+state at short distance, resulting in rapid dis- sociation to the repulsive Be+(2S) and H+fragments (Fig. 14). The first excited2Σ+state seems to have a stable dissociation limit, but at a very large distance ( ∼50 a.u.), it encounters an extremely weak avoided crossing with the second excited2Σ+state giving rise to the FIG. 13. The zoomed-in view of the avoided crossing region between the ground and first excited2Πstates of BeF2+computed by UGA-SSMRPT2 using the cc-pVDZ basis. J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-11 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 14. Ground and first excited2Σ+states of BeH2+computed by UGA- SSMRPT2 using the aug-cc-pVDZ basis and a (1,4) CAS. repulsive Be+(2P) and H+fragments. We study both these avoided crossings with our theory using the Dunning aug-cc-pVDZ basis and employing a (1,4) CASSCF starting function, which is state-averaged over the first three2Σ+states. All orbitals were correlated in our computation. The MRCISD +Q data points have been shown only in the zoomed-in view of Fig. 15, that is, in Fig. 16 for clarity of presen- tation. As is evident from Table I, all three doublet states of BeH2+ differ by a few 10−5hartree from the corresponding MRCISD +Q. This is further exemplified by the distance at which the avoided crossing occurs between states 2 and 3 of the manifold for the two methods, as can be seen in Table II. FIG. 15. First and second excited2Σ+states of BeH2+computed by UGA- SSMRPT2 using the aug-cc-pVDZ basis and a (1,4) CAS. See Fig. 16 for a zoomed-in view of the very weak avoided crossing region. FIG. 16. The zoomed-in view of the avoided crossing region between the first and second excited2Σ+states of BeH2+computed using the aug-cc-pVDZ basis. 3. Weak ionic-covalent avoided crossings in LiF and BN a. LiF. The very weak avoided crossing between the ionic- covalent states of LiF has been a major challenge for all SS type of multireference theories. Relaxation of the starting coefficients is crucial due to the significant difference in the contribution of the constituent configurations throughout the dissociation profile, before and after the inclusion of dynamic correlation. It is well known87that a state-specific model for a multireference perturba- tion theory always fails to describe this particular avoided crossing. Although we have already established the efficacy of UGA-SSMRPT2 in describing very weak avoided crossings in BeF2+ and BeH2+in Secs. III C 1 and III C 2, our theory cannot ameliorate the “double crossing” problem near the weak avoided crossing region between the ground and first excited states of LiF (Fig. 17). FIG. 17. Ground and first excited1Σ+states of LiF computed by UGA-SSMRPT2 using the cc-pVDZ basis (Li) and aug-cc-pVDZ basis (F). See Fig. 18 for a zoomed- in view of the double crossing between the two states. J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-12 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 18. The zoomed-in view of the double crossing between the ground and first excited1Σ+states of LiF computed by UGA-SSMRPT2 using the cc-pVDZ basis (Li) and aug-cc-pVDZ basis (F). The zoomed-in view of the double crossing is shown in Fig. 18. We discuss the rationale behind this observation in the concluding remarks of this sub-section. We employ the cc-pVDZ for the Li atom and aug-cc-pVDZ basis for the F atom, and the minimal (2,2) CAS has been used for the starting CASSCF function. All orbitals were correlated in the SSMRPT computation. b. BN. We test our theory for another weakly avoided ionic- covalent curve crossing between the first and second excited3Π states of BN. It has been established in recent years88,89that the ground state of BN is3Πand is separated by ∼200 cm−1from the 1Σ+state. The first and second excited3Πstates have the same asymptotic limit,90B(2P) and N(2D), unlike the situation we stud- ied in LiF. The (2)3Πstate is formed by excitation of a bonding electron to the non-bonding orbital of B, essentially resulting in a sort of charge transfer from N to B. The (3)3Πstate arises from aπtoπ∗excitation, thus keeping the overall charge of the state neutral. To demonstrate our rationale behind the failure of state-specific perturbation theories in LiF weakly avoided ionic-covalent crossing, we show that the same problem arises when describing the first and second excited3Πstates in BN (Fig. 19) The zoomed-in view of the double crossing region is shown in Fig. 20. We employ a cc-pVTZ basis and use the full valence (8,8) CAS as our starting function. All orbitals were correlated in our computation. 4. Analysis of the double crossing phenomenon The inability to reproduce the weakly avoided ionic-covalent crossing in LiF has been encountered by all state-specific PTs.58,87,91–95These authors have circumvented the spuriosity, man- ifested in the form of a double crossing, by constructing an inter- mediate Hamiltonian consisting both of the same symmetry states, thus forcing the two roots to be explicitly dependent on each other. There is no such constraint with a state-specific approach, as the two states are solutions of two different SS-effective Hamiltonians. FIG. 19. irst and second excited3Πstates of BN computed by UGA-SSMRPT2 using the cc-pVTZ basis. See Fig. 20 for a zoomed-in view of the double crossing between the two states. Nevertheless, as shown in the above applications, our SS theory accurately reproduces the transition between various configurations in a manifold of same symmetry states in most of the systems studied. This includes the very weakly avoided crossings in BeF2+ and BeH2+. The failure to describe the weak ionic-covalent avoided crossings in LiF and BN may be attributed to the insufficient description of (i) anionic (N +1 electron) correlation in the asymp- totic region and (ii) orbital relaxation effects caused by polariza- tion of the charged species, at a second order perturbative level. If the asymptotes are for differently charged sectors, the insufficient inclusion of dynamical correlation will distort and enhance the differences in gradients of the two states concerned as is the case in FIG. 20. The zoomed-in view of the double crossing region between the first and second excited3Πstates of BN computed by UGA-SSMRPT2 using the cc-pVTZ basis. J. Chem. Phys. 155, 014101 (2021); doi: 10.1063/5.0054731 155, 014101-13 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp LiF. On the other hand, as in BN, the asymptotes of the two states are the same, and here, the insufficient inclusion of state-specific orbital relaxation is the main contributor to the “double crossing” artifact. The polarization effects on the molecular orbitals are best described via the action of one-body cluster operators and its var- ious powers on the model functions, an effect we would like to call a Thouless-type relaxation.73The reason we want to distinguish this with the true Thouless relaxation is because we use a normal ordered exponential of a one body operator to induce the orbital relaxation, whereas in the Thouless relaxation, a simple exponen- tial of a one-body operator was used. Despite the difference in the structure of the relaxation operator, the physics induced by them is very similar, although our normal ordered ansatz gives a more compact formulation. At the second order of energy, only the lin- ear power of T 1is present in the first order perturbed wavefunction, which poorly describes these orbital relaxation effects for a charged species. When the orbital relaxation is due to a change in the charge distribution induced by the electron correlation (via a modification of the relative weights of the different components of the wavefunc- tion), Brillouin’s theorem imposes that at first order in the wave function (i.e., second and third order in the energy), this effect is not accounted for. This is a manifestation of an orbital relaxation induced by the electron correlation, as described by Angeli,96an effect that requires higher order of PT or a non-perturbative treat- ment. The combined effect of the insufficient correlation for the anionic species and lack of proper relaxation in the anionic frag- ment of the molecule results in an artifact manifested via the two same space-spin symmetry potential energy curves crossing each other twice, as a way of joining continuously with their behav- ior away from the avoided crossing zone. A lucid analytic demon- stration of this “double crossing” phenomenon using a three state model problem via a low order correlation theory was presented by Spiegelmann and Malrieu.91It is important to mention here that we have already studied the LiF system with the fully non- perturbative UGA-SSMRCC where the weakly avoided crossing was seen to be perfectly reproduced. The results would be demonstrated in a forthcoming publication.72This bolsters our rationale that an all order state-specific orbital Thouless relaxation is crucial to properly reproduce such ionic-covalent weak avoided crossings, and a single- root non-perturbative formalism could be successful to describe this phenomenon. In contrast, for systems such as BeF2+and BeH2+, the two states of interest have orbitals already optimized for a cationic species and thus do not require much orbital relaxation for the different cationic fragments in the two states. Moreover, the dynamical cor- relation for a cationic (N −1)/(N−2) species is described well even at a perturbative level by the N-electron H 0. This allows for a proper description of the very weak avoided crossings in these manifolds of PEC using a second order perturbative correlation theory. IV. SUMMARY AND FUTURE OUTLOOK We have studied the performance of the UGA-SSMRPT2 in the description of various intricate modulations, such as inter- lacing and avoided crossings between PEC of states belonging to the same space-spin symmetry. The accurate depiction of these featuresin a PEC using the simple and computationally inexpensive pertur- bative theory, despite the fact that the solutions belong to unrelated state-specific effective Hamiltonians, encourages its extension into the qualitative understanding of bonding and reaction mechanisms of chemically relevant systems. We have also demonstrated with specific examples the rationale behind the limitations of a state-specific perturbation theory in the cases of weak avoided crossings between ionic and covalent curves. We are hoping to present a rigorously size-extensive, multi-state ver- sion of UGA-SSMRPT2 in the near future, which should amend this pitfall. Finally, we have successfully validated the sufficiency condi- tion we had suggested in our earlier work33by implementing the rigorous projection scheme to solve for cluster amplitudes. The per- formance of both the schemes has been shown to be very simi- lar with each other, and as such, one can safely proceed with the cheaper alternative imparted by our sufficiency conditions. We shall investigate whether the aforementioned equivalence in performance between these two schemes also holds true for its non-perturbative counterpart, the UGA-SSMRCC, in a forthcoming publication. SUPPLEMENTARY MATERIAL See the supplementary material for the data points used in the construction of the PECs. DEDICATION D.M. dedicates this paper to the memory of Werner Kutzelnigg, a dear friend and a close collaborator for many years. ACKNOWLEDGMENTS D.C. and R.K. acknowledge UGC, and K.H. acknowledges his CSIR grant for financial assistance. D.M. thanks the S.N. Bose National Centre for Basic Sciences, where he was affiliated dur- ing the initial stages of this work, for the S.N. Bose Chair Pro- fessorship funds. We thank Professor Ankan Paul and Professor Satrajit Adhikari for providing laboratory and computational facil- ities at IACS. The insightful discussions with Dr. Avijit Sen and Dr. Sangita Sen were extremely helpful for starting the pro- grammatic renovation. The authors also thank Professor Aristides Mavridis and Professor Demeter Tzeli for kindly providing the data points for BC used in their papers.76,77 DATA AVAILABILITY The data that support the findings of this study are available in tabular form in the supplementary material. REFERENCES 1D. I. Lyakh, M. Musiał, V. F. Lotrich, and R. J. Bartlett, Chem. Rev. 112, 182 (2012). 2P. G. Szalay, T. Müller, G. Gidofalvi, H. Lischka, and R. Shepard, Chem. Rev. 112, 108 (2012). 3F. A. Evangelista, J. Chem. Phys. 149, 030901 (2018). 4D. Mukherjee, R. K. Moitra, and A. Mukhopadhyay, Mol. 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5.0053896.pdf

APL Materials ARTICLE scitation.org/journal/apm Field-free spin–orbit torque driven multi-state reversal in wedged Ta/MgO/CoFeB/MgO heterostructures Cite as: APL Mater. 9, 071108 (2021); doi: 10.1063/5.0053896 Submitted: 12 April 2021 •Accepted: 1 July 2021 • Published Online: 15 July 2021 Dong Li,1 Baoshan Cui,2,3Xiaobin Guo,4Zhengyu Xiao,1,5Wei Zhang,1,5Xiaoxiong Jia,1,5Jinyu Duan,1,5Xu Liu,1,5 Jie Chen,2,3 Zhiyong Quan,1,5 Guoqiang Yu,2,3,a) and Xiaohong Xu1,5,a) AFFILIATIONS 1Key Laboratory of Magnetic Molecules and Magnetic Information Materials of Ministry of Education, Research Institute of Materials Science, Shanxi Normal University, Linfen 041004, People’s Republic of China 2Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, People’s Republic of China 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 4School of Physics and Optoelectric Engineering, Guangdong University of Technology, Guangzhou Higher Education Mega Centre, Guangzhou 510006, People’s Republic of China 5School of Chemistry and Materials Science, Shanxi Normal University, Linfen 041004, People’s Republic of China Note: This paper is part of the Special Topic on Emerging Materials for Spin–Charge Interconversion. a)Authors to whom correspondence should be addressed: guoqiangyu@iphy.ac.cn and xuxh@sxnu.edu.cn ABSTRACT We report a current-induced four-state magnetization reversal under zero magnetic field in a wedged Ta/MgO/CoFeB/MgO heterostructure with a perpendicular magnetic anisotropy. Anomalous Hall effect and magneto-optical Kerr effect microscopy measurements were performed to demonstrate that the field-free multi-level reversal is jointly determined by the spin–orbit torque effective field that originates from the lack of the lateral inversion symmetry in the wedged stacking structure and the current-induced Oersted field. Moreover, the creation of robust intermediate Hall resistance states in the multi-state switching strongly depends on the current-induced Joule heating. Our results provide a route for the field-free multi-level state reversal, which is significant for fabricating the non-volatile and energy-efficient multi-level memories or artificial neuron devices. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0053896 In recent years, current-induced spin–orbit torque (SOT), act- ing as an alternative means to effectively manipulate the mag- netization reversal, has drawn considerable attention due to the potential advantages of the simple structure configuration,1opti- mized reading and writing path,2efficient power consumption,3 etc. In general, SOT is believed to be originated from the bulk spin Hall effect of the heavy metal (HM) with strong spin–orbit coupling (SOC) and/or the interfacial Rashba effect due to the inversion asymmetry at the HM/ferromagnetic metal (FM) inter- face.3,4When an in-plane charge current passes through the HM layer, the converted pure spin current exerts a torque on the mag- netization of the adjacent FM layer. To date, many efforts havebeen devoted to enhancing the SOT efficiency in different materials and structure systems, such as Pt/Co,3,5Ta/CoFeB,6Au 1−xPtx/Co,7 BixSe(1−x)/CoFeB,8SrRuO 3/SrIrO 3,9and WTe 2/Py.10Meanwhile, in order to obtain 180○magnetization reversal in perpendicularly magnetized structures, which is essential to the binary storage, an in-plane bias magnetic field along the charge current direction is required to break the symmetry of the magnetization.3However, this evidently goes against the objective for reducing the complexity of the device design and the demand for low-power dissipation in practical commercial applications. At present, many methods have been developed to realize field-free magnetization reversal, such as breaking the lateral symmetry via a wedged structure,11adding a APL Mater. 9, 071108 (2021); doi: 10.1063/5.0053896 9, 071108-1 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm ferromagnetic layer with an in-plane magnetic anisotropy,12uti- lizing the exchange bias effect of an antiferromagnet,13applying an electric field,14local laser annealing,15and designing a tilted magnetic easy axis.9,16 In addition to the application for binary storage, SOT-driven multi-level memory has also been extensively studied because it could further increase the memory density and simulate the function of synapses for neuromorphic computing.17,18The SOT- driven multi-state switching has been demonstrated in He+irra- diated W/CoFeB/MgO Hall bars,19Pt/Co/Pt/Co/Pt structures with two magnetic layers,20Pt/[Pt/Co] 3/Ru/[Co/Pt] nsynthetic antifer- romagnets,21and Pt/Co/Ir 20Mn 80exchange biased systems.22It is worth noting that an in-plane magnetic field is still required for device operation in these systems. In this study, the SOT- driven four-state magnetization reversal in the absence of an exter- nal magnetic field was achieved in a perpendicularly magnetized Ta/MgO/CoFeB/MgO structure (each layer was orderly grown from left to right), in which the quantitative manipulation of SOT effi- ciency has been reported in our previous work.23Here, we take advantage of the simultaneous measurement of the anomalous Hall resistance and magnetic domain pattern to explore the multi-state switching mechanism at different current pulses and the direct current-induced Joule heating effect on the stability of multi-level Hall resistance states. The wedged film with the structure of Ta (5)/MgO (twedged )/CoFeB (1)/MgO (2)/Ta (2) (thickness in nm) was deposited on a thermally oxidized silicon substrate at room temperature via a magnetron sputtering system with a base pressure of below 1.0×10−8Torr. The top MgO layer is used to induce the perpendic- ular magnetization anisotropy (PMA). The bottom MgO interlayer grown by the oblique sputtering method has a wedge-shaped structure, in which the thickness (denoted as twedged ) varies from 0.10 to 0.50 nm within the length of ∼5 cm. The schematic diagram of the film stack is depicted in Fig. 1(a). Subsequently, the film was annealed at 250○C for 30 min in a vacuum with a background pressure of 3.0 ×10−4Pa to enhance the PMA. Then, the annealed stack was patterned into a series of Hall bar devices with a width of 20μm using standard photolithography and dry etching techniques. Finally, the magnetization state of the devices was characterized by electric transport measurements [see Fig. 1(b)] based on the anomalous Hall effect and Kerr differential images using a MOKE microscope with a polar configuration. All the measurements were carried out at room temperature. Figure 1(c) shows the Hall resistance ( RHall) as a function of the out-of-plane magnetic field ( Hz) for the devices with different MgO thicknesses. All the measured devices show sharp square-shape loops, indicating the existence of PMA. The loops gradually become narrow with twedged increasing, indicating the decrease in PMA. As shown in Fig. 1(d), the current-induced magnetization reversal was measured at the corresponding twedged value by applying a series of current pulses with a duration of 1 ms and an interval of 2 s. After each pulse, a small reading current of 0.1 mA was used to detect the Hall resistance, aiming at minimizing the Joule heating effect (see supplementary material S1 for the magnetization switching with dif- ferent reading currents). Full magnetization switching under zero magnetic field was achieved for the samples with MgO thicknesses of 0.5 and 0.1 nm. The switching is driven by the Rashba spin–orbit coupling induced perpendicular effective field ( Heffz), which depends FIG. 1. (a) Schematic diagram of the Ta/MgO (wedged)/CoFeB/MgO film stack. (b) Image of a patterned Hall bar and Hall resistance measurement setup. (c) Anomalous Hall resistance vs magnetic field along the zaxis for the samples with representative wedged MgO interlayer thicknesses. (d) Pulse current-driven field- free magnetization switching curves with different MgO thicknesses. A four-state magnetization reversal is obtained for the sample with twedged =0.25 nm. on the electron’s momentum and the symmetry breaking induced in-plane effective electric field.23The switching polarity for these two samples is opposite, indicating the direction reversal of Heffzfor a given current direction. The spin Hall effect induced damping- like SOT cannot explain the direction reversal of Heffz, as has been discussed in the previous work.23Interestingly, the multi-level and stable Hall resistance states were created at various current pulses fortwedged =0.25 nm. To further explore the origin of the multi- state feature, we performed the simultaneous measurement of the anomalous Hall resistance and magnetic domain pattern. Figure 2 shows the MOKE images of domain configura- tions corresponding to different Hall resistance states as shown in Fig. 1(d), which are manipulated by pulse currents without any external magnetic fields. First, a large pulsed out-of-plane mag- netic field ( Hz=−200 Oe) was applied to initialize a “down” single domain state, and then the image was chosen as the background. The pulse current was swept back and forth between +3.3×1011and −3.3×1011A/m2, and the Hall resistances and MOKE images were simultaneously recorded. As shown in image ①in Fig. 2, the reversed domain was nucleated at the top edge of the racetrack when a pos- itive pulse current was applied along the +xaxis. This relatively stable intermediate state can be regarded as “State 1,” which cor- responds to site ①in the RHall−Jeloop [see Fig. 1(d)]. When a negative pulse with a current density of about −0.9×1011A/m2 was applied, the top edge nucleated domain disappeared, and RHall APL Mater. 9, 071108 (2021); doi: 10.1063/5.0053896 9, 071108-2 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm FIG. 2. MOKE images show the field-free magnetization reversal process of the racetrack with twedged =0.25 nm after applying a series of current pulses along the ±xaxis. The dark and gray regions in the racetrack refer to the “up” and “down” magnetic domains, respectively. reached the positive maximum, corresponding to a “down” magne- tized state. It is referred to as “State 2,” corresponding to site ②in the loop. When continuing to increase the current to −3.0×1011A/m2, RHallstarted to decrease and then increase negatively. Image ③in Fig. 2 shows the domain pattern at site ③in the RHall−Jeloop. One can see that an “up” domain was first nucleated at the bottom edge and gradually extended orientated to the top edge. Then, the resistance state became robust in the negative current density range, which was regarded as “State 3.” Moreover, as shown in image ④, when the current density was scanned to +0.9×1011A/m2, the dark “up” domain was expanded to the whole racetrack. Meanwhile, RHall also reached the negative maximum, which was viewed as “State 4.” When the current density was further increased to +3.0×1011A/m2, the “down” nucleated domain at the bottom edge appeared and fur- ther expanded toward to the top-edge region as shown in image ⑤. Images ④and⑤show the domain at sites ④and⑤in the RHall− Jeloop, respectively. One can see that RHallat site⑤almost coin- cides with “State 1”, and finally, one cycle scanning was finished.The MOKE images of the whole reversal clearly show the four dif- ferent domain patterns corresponding to the four Hall resistance states. Considering the favored domain nucleation at the two edges of the racetrack, it is suggested that the current-induced Oersted field (HOersted ) plays an important role. Next, we discuss the mechanism of the formation of the multi- level Hall resistance states. In the top panels of Figs. 3(a)–3(d), we plot the sketch of HeffzandHOersted for currents along ±xdirections. The bottom panels show the evolution of the domain pattern under these two effective fields, in which the ⊙and⊗symbols refer to the “up” and “down” magnetic domains, respectively. One can see that the current-induced HOersted mainly distributes at the two edges of the racetrack and the orientation is labeled by blue dashed arrows. We calculated the averaged magnitude of HOersted based on the equa- tionHOersted =±Jt[3+2 lnw/t]/4π, considering that the thickness ( t) of the device is much less than its width ( w).24The estimated HOersted value at the device edge is about 34.6 Oe for a current density of +3.0 ×1011A/m2, assuming the current uniformly flows in the conduct- ing layers. The Heffzvalue at the same current density is calculated to be around 9 Oe using the efficiency of the SOT-induced perpendic- ular effective field at twedged =0.25 nm [ ∼0.3 Oe/(1010A/m2)], which is determined by the current-induced hysteresis loop shift method.23 TheHOersted values generated at the two edges are opposite to each other. Since the maximum HOersted value is larger than Heffz, the net equivalent field directions at the two edges of the racetrack are oppo- site, which results in a significant impact on the current-induced magnetization reversal. As shown in the bottom panel of Fig. 3(a), the magnetization was initialized to the −zdirection via a pulsed reset magnetic field. For a positive current, Heffzpoints down and HOersted points down and up in the bottom and top areas, respec- tively. Since HOersted is higher than Heffzin the top area marked by orange five-pointed stars, the net equivalent field is along the +z direction and is able to induce the magnetization reversal in this area. By contrast, both HOersted andHeffzhave the same direction in the bottom area of the device and the net equivalent field is along the −zdirection. As a result, the magnetization in this area remains in its original state. This description can well explain the feature of image ①in Fig. 2. When the current flows along the −xdirection, HOersted and Heffzchange their directions as well, as shown in Fig. 3(b). HOersted andHeffzare still opposite to each other in the orange star area, and the net field is along the −zdirection, which tends to drive the magnetization reversal and makes the nucleated domain disap- pear. If the current is not too large (e.g., current density of −0.9 ×1011A/m2), the net field orientated to the +zdirection in the bottom area is insufficient to induce the domain nucleation due to the insufficiently large effective field, and hence, the single domain remains stable, as shown in image ②in Fig. 2. With the increase in the negative current, the Oersted field starts to dominate and drives the nucleation of the “up” domain in the bottom area. The nucleated reversed domain grew up due to the net positive field and reached the magnetic state shown in image ③in Fig. 2. When the current density was swept back to +0.9×1011A/m2, the domain wall moves to the top edge and forms the “up” domain state under the net equiv- alent field along the +zdirection as shown in image ④of Fig. 2. Similarly, as shown in image ⑤in Fig. 2, the reversed domain in the bottom area is nucleated and is further expanded to the top edge due to the enhanced net field along the −zdirection [see Fig. 3(c)] when APL Mater. 9, 071108 (2021); doi: 10.1063/5.0053896 9, 071108-3 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm FIG. 3. Schematic illustration of the domain nucleation and expansion for a current J. The top panels show the sketch of current-induced effective fields, HeffzandHOersted , for the current along the+xdirection ( +J) for (a) and (c) and the−xdirection ( −J) for (b) and (d). The bottom panels show the domain pat- terns (denoted as black dashed blocks) accordingly. HeffzandHOersted at both sides of the racetrack are illustrated as red solid arrows and blue dashed arrows, respectively. The ⊙and⊗symbols refer to the “up” and “down” magnetized states, respectively. further increasing the current density to +3.0×1011A/m2. Based on the above analyses and discussions, the combined effect of HOersted andHeffzcan well explain the special magnetic domain configurations and emerging multi-level Hall resistance states at the pulse currents. In addition, the direct current-induced magnetization reversal for the sample with twedged =0.25 nm was performed to further inves- tigate the Joule heating stability of multi-level Hall resistance states. As shown in Fig. 4(a), only two relatively stable maximum Hall resistance states were obtained, which reveals that the direct current- induced Joule heating may hinder the formation of the intermediate states. As displayed in Fig. 4(b), the MOKE images of magnetic domain configurations corresponding to the Hall resistance states at sites ①–⑥of the loop shown in Fig. 4(a) were also collected to visualize the magnetization switching process. The magnetization was first set into the “down” state by a negative reset field pulse FIG. 4. (a) The direct current-induced field-free magnetization switching loop with twedged =0.25 nm, showing the current-induced Joule heating effect on magnetiza- tion reversal. (b) MOKE images show the field-free magnetization reversal process of the racetrack at different direct currents along the ±xaxis, corresponding to the domains at sites ①–⑥of the loop in (a). The dark and gray regions in the race- track refer to the “up” and “down” domains or magnetized states, respectively. The magnetic field pulse Hz=−200 Oe was applied to the initial racetrack.(Hz=−200 Oe). When a small current density of −0.07×1011 A/m2was applied, a “down” single domain that corresponds to the maximum positive Hall resistance state [see site ①in Fig. 4(a)] was observed in image ①. With a further increase in the current den- sity to −1.0×1011A/m2, a tortuous nucleated domain appeared and rapidly grew up at the current density of −1.4×1011A/m2(see images ②and③). As shown in image ④, the “up” single domain corresponding to the maximum negative Hall resistance state was formed again at +0.07×1011A/m2. When continuing to increase the current density to +1.0×1011and+1.5×1011A/m2(see images ⑤and⑥), the case is similar to that for the negative current. Obvi- ously, the domain setups with the random nucleated feature in the direct-current mode are not in accord with that in the pulse-current case. This may be ascribed to the strong Joule heating from the direct current. Herein, the Joule heating-induced sample temperature was estimated at different direct currents. The critical current density of ∼1.2×1011A/m2can raise the device temperature to ∼830 K (see supplementary material S2). Hence, the formation of stable inter- mediate states not only depends on the combined effect of HOersted andHeffzbut also is related to the Joule heating. Consequently, the field-free multi-state switching in our wedged stacks can be manipu- lated only by the pulse current. It should be noted that if the Hall bars are made narrower, the field-free multi-state switching behavior may disappear due to the reduction of the Oersted field and nucleation sites.25Further research on the scalability of the structure needs to be investigated in the following experiments. In conclusion, the current-induced four-state magnetization reversal under zero magnetic field was obtained in a wedged Ta/MgO/CoFeB/MgO structure with a perpendicular magnetic anisotropy. The electric transport measurements based on the anomalous Hall effect combined with the magnetic domain imag- ing via magneto-optical Kerr effect microscopy were carried out to demonstrate that the current-induced Oersted field and perpendic- ular effective field play an essential role in the field-free multi-level states reversal. In addition, the stable four-state switching can be greatly affected by the direct current-induced Joule heating. These APL Mater. 9, 071108 (2021); doi: 10.1063/5.0053896 9, 071108-4 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm results provide an efficient route to explore the non-volatile multi- state magnetic random-access memories or spin neuron devices. See the supplementary material for the demonstration of the direct current-induced Joule heating and the magnetization switch- ing with different reading currents. AUTHORS’ CONTRIBUTIONS D.L. and B.C. contributed equally to this work. This work was supported by the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2020A1515110553), the National Natural Science Foundation of China (Grant Nos. 52001190 and 11904056), the China Postdoctoral Science Founda- tion (Grant Nos. 2020M670499 and 2020M680734), the Youth Sci- ence Foundation of Shanxi Province (Grant No. 201901D211405), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2020L0234), the Pro- gram of Collaborative Innovation Center for Shanxi Advanced Per- manent Materials and Technology (Grant No. 2019-04), and the Natural Science Foundation of Shanxi Normal University (Grant No. ZR1814). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. 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The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp Nuclear–electronic orbital methods: Foundations and prospects Cite as: J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 Submitted: 8 April 2021 •Accepted: 17 May 2021 • Published Online: 15 July 2021 Sharon Hammes-Schiffera) AFFILIATIONS Department of Chemistry, Yale University, New Haven, Connecticut 06520, USA a)Author to whom correspondence should be addressed: sharon.hammes-schiffer@yale.edu ABSTRACT The incorporation of nuclear quantum effects and non-Born–Oppenheimer behavior into quantum chemistry calculations and molecular dynamics simulations is a longstanding challenge. The nuclear–electronic orbital (NEO) approach treats specified nuclei, typically pro- tons, quantum mechanically on the same level as the electrons with wave function and density functional theory methods. This approach inherently includes nuclear delocalization and zero-point energy in molecular energy calculations, geometry optimizations, reaction paths, and dynamics. It can also provide accurate descriptions of excited electronic, vibrational, and vibronic states as well as nuclear tunnel- ing and nonadiabatic dynamics. Nonequilibrium nuclear–electronic dynamics simulations beyond the Born–Oppenheimer approxima- tion can be used to investigate a wide range of excited state processes. This Perspective provides an overview of the foundational NEO methods and enumerates the prospects for using these methods as building blocks for future developments. The conceptual simplic- ity and computational efficiency of the NEO approach will enhance its accessibility and applicability to diverse chemical and biological systems. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0053576 INTRODUCTION A wide range of chemical and biological processes rely on the quantum mechanical properties of nuclei, such as zero-point energy effects, vibrationally excited states, and nuclear tunneling. Many of these processes also involve a breakdown of the Born–Oppenheimer separation between electrons and nuclei. A prime example is proton- coupled electron transfer (PCET), which is essential for photosyn- thesis, respiration, DNA synthesis, and many electrocatalytic reac- tions.1–3A meaningful theoretical description of PCET reactions often requires the quantum mechanical treatment of the transferring hydrogen nuclei as well as the electrons and a rigorous treatment of hydrogen tunneling and nonadiabatic effects.1Vibronically nonadi- abatic PCET rate constants and kinetic isotope effects can be calcu- lated in terms of nonadiabatic transitions between electron–proton vibronic states.4Moreover, simulating the nonequilibrium dynamics of PCET reactions typically entails nonadiabatic molecular dynam- ics methods. Incorporating nuclear quantum effects and non-Born– Oppenheimer behavior into quantum chemistry calculations and molecular dynamics simulations is challenging. In conventional Born–Oppenheimer approaches, the potential energy surfaceis generated by solving the electronic Schrödinger equation for fixed nuclear configurations. The nuclei can be propagated either classically or quantum mechanically (i.e., with wavepacket or path integral methods) on the resulting adiabatic potential energy surface. In some cases, the potential energy surface can be generated on-the-fly with ab initio electronic structure methods during the nuclear dynamics. Nonadiabatic dynamics approaches, such as the Ehrenfest,5surface hopping,6multiple spawning,7,8multiconfigura- tional time-dependent Hartree,9,10and nonadiabatic ring polymer molecular dynamics11–14methods, allow nuclei to move on multiple potential energy surfaces. These nonadiabatic dynamics methods invoke different types of approximations and include nuclear quantum effects to varying degrees.15 In contrast to these methods, the nuclear–electronic orbital (NEO) approach16,17treats specified nuclei quantum mechani- cally on the same level as the electrons using molecular orbital techniques without invoking the Born–Oppenheimer separation between the electrons and the quantum nuclei. In this approach, the mixed nuclear–electronic time-independent or time-dependent Schrödinger equation is solved with wave function or density func- tional theory (DFT) methods. Typically, the quantum nuclei are protons, although other nuclei and other types of particles, such J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 155, 030901-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp as positrons, can also be treated quantum mechanically within this framework. At least two nuclei are usually treated classically to avoid problems with translations or rotations, which alternatively could be removed or mitigated while treating all nuclei quantum mechanically.18 An advantage of the NEO approach is that nuclear quan- tum effects such as proton delocalization and zero-point energy, as well as non-Born–Oppenheimer effects between electrons and protons, are inherently included during geometry optimizations, reaction paths, and dynamics. The non-Born–Oppenheimer effects between the quantum and classical nuclei, as well as the elec- trons and classical nuclei, have been found to be negligible for equilibrium molecular properties, as determined by analysis of the NEO diagonal Born–Oppenheimer corrections.19For nonequi- librium processes, these non-Born–Oppenheimer effects can be incorporated through nonadiabatic dynamics methods, such as Ehrenfest and surface hopping approaches for the classical nuclei moving on the NEO electron–proton vibronic surfaces. A variety of related multicomponent wave function and DFT methods have been explored,20–27and exact factorization methods28,29provide an alter- native promising solution to these problems. This brief overview does not cover the vast array of non-Born–Oppenheimer or nona- diabatic methods, and more comprehensive reviews are provided elsewhere.15,17 This Perspective briefly reviews the foundational NEO meth- ods and provides a road map for how these methods can be used as building blocks for future developments and applications. Electron–proton correlation is particularly important because of the attractive electron–proton Coulomb interaction.16,17Two different paths have been explored for including correlation in NEO calcula- tions of ground states: wave function methods, such as coupled clus- ter (CC),30and multicomponent DFT.31–33These paths have been extended to equation-of-motion (EOM)34,35and time-dependent DFT (TDDFT)36,37methods for the description of excited electronic, vibrational, and vibronic states. Furthermore, the combination of the NEO real-time TDDFT38and Ehrenfest dynamics39,40meth- ods allows the simulation of nonequilibrium nuclear–electronic dynamics beyond the Born–Oppenheimer approximation. Finally, the NEO multistate DFT (NEO-MSDFT) method41enables the description of hydrogen tunneling systems. The capacity to describe hydrogen tunneling and nonequilibrium nonadiabatic dynamics in a conceptually straightforward and computationally efficient manner opens up many avenues for exploration. MULTICOMPONENT HARTREE–FOCK In the NEO framework, the system is divided into electrons, quantum nuclei, and classical nuclei. Herein, the quantum nuclei are assumed to be protons, but extensions to other types of nuclei and even positrons are straightforward. The NEO Hamiltonian, HNEO, includes the kinetic energies of the electrons and quantum protons, the interactions of the electrons and quantum protons with the classical nuclei, and the electron–electron, proton–proton, and electron–proton Coulomb interaction terms. The simplest NEO method is based on the NEO-Hartree–Fock (NEO-HF) wave func- tion,16which is the product of electronic and protonic Slater deter- minants, ΨNEO−HF(xe,xp)=Φe(xe)Φp(xp). (1)Here, xeandxpare the collective spatial and spin coordinates of the electrons and quantum protons, respectively. The electronic and protonic Slater determinants, Φe(xe)andΦp(xp), are composed of electronic and protonic spin orbitals, which are expanded in electronic and protonic Gaussian basis sets.42 Minimizing the Hartree–Fock energy with respect to the elec- tronic and protonic spin orbitals leads to two sets of coupled Hartree–Fock–Roothaan equations for the electrons and protons,16 FeCe=SeCeεe, FpCp=SpCpεp.(2) Here, Fe,Ce,Se, and εeare the electronic Fock matrix, orbital coef- ficient matrix, overlap matrix, and orbital energy matrix, respec- tively, and the protonic matrices are defined analogously. These two equations are strongly coupled because the electronic and pro- tonic Fock matrices each depend on both the electronic and pro- tonic orbital coefficients. As a result, these two equations must be solved self-consistently. Typically each quantum proton is repre- sented by a set of electronic and protonic basis sets with the same center, which is optimized variationally during the self-consistent field (SCF) procedure. Due to the lack of electron–proton corre- lation, the NEO-HF method produces proton densities that are much too localized, preventing the accurate description of molec- ular properties such as geometries, energies, and frequencies.16,17 Nevertheless, it serves as the starting point for the correlated NEO methods. MINIMUM ENERGY PATHS IN THE NEO FRAMEWORK The implementation of analytical NEO gradients and Hes- sians43has enabled the optimization of geometries and transition states, as well as the generation of minimum energy paths (MEPs). The NEO potential energy surface depends on only the classical nuclear coordinates, with the underlying assumption that the basis function centers associated with the quantum protons are optimized variationally for each configuration on this potential energy surface. The MEP is generated by starting at a transition state on the NEO potential energy surface, defined to be a first-order saddle point, and using a steepest descent algorithm to step down to the reactant and product minima, optimizing the quantum proton basis func- tion center positions at each step to remain on the NEO potential energy surface. Figure 1 depicts a hydride transfer reaction between two carbon atoms and the corresponding MEP computed with the conventional electronic HF and NEO-HF methods.43For this sys- tem, the NEO-HF energy barrier is lower than the conventional HF energy barrier because the zero-point energy of the transfer- ring hydrogen nucleus is inherently included in the NEO energy. The analytical NEO Hessian, which depends only on the classical nuclear coordinates, enables the inclusion of the zero-point energy and entropic contributions from the classical nuclei. The entropic contributions from the quantum nuclei can be included with the vibrational analysis methods discussed below,43,44thereby provid- ing thermochemical properties that include the anharmonic effects of the quantum nuclei. Analysis of the single imaginary normal mode at the transi- tion state, as well as the geometries along the MEP, provides insight into the dominant nuclear motions driving this hydride transfer J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 155, 030901-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 1. Hydride transfer reaction (upper left), minimum energy path calculated with NEO-HF and conventional HF (right), and protonic and electronic orbitals along this MEP (lower left). The protonic orbital is represented as the purple isosurface, and the reactive electronic intrinsic bond orbital is represented as blue (positive) and red (negative), all with the same isovalue. Adapted from Schneider et al. , J. Chem. Phys. 154, 054108 (2021) with the permission of AIP Publishing. reaction. In the conventional electronic HF case, the imaginary normal mode at the transition state is dominated by the motion of the transferring hydrogen, which also contributes significantly to the intrinsic reaction coordinate (IRC) along the MEP. In the NEO- HF case, this hydrogen nucleus is treated quantum mechanically and therefore cannot contribute to the normal mode or the IRC. Instead, the imaginary normal mode at the transition state is dom- inated by the tetrahedral-to-planar rearrangement about the two central carbon atoms as they switch between sp3and sp2hybridiza- tions. The geometries along the MEP confirm that this tetrahedral- to-planar rearrangement represents the dominant contribution to the IRC and thus drives the hydride transfer reaction. Interest- ingly, this same motion also contributes to the imaginary mode at the transition state for the conventional electronic HF case, even though it is overwhelmed by the transferring hydrogen motion. If the contribution from the transferring hydrogen is eliminated from the conventional HF imaginary normal mode, the resulting vector is nearly identical to the NEO-HF imaginary normal mode after renormalization. Thus, the NEO MEP provides fundamental insights into the nuclear motions that drive hydrogen transfer reactions, including proton, hydride, and hydrogen atom transfer, as well as PCET. This type of analysis is analogous to analyses of electron transfer reac- tions that identify the nuclear motions driving electron transfer. In this case, however, the nuclear motions are driving the transfer of a quantum mechanical hydrogen nucleus as well as the rearrange- ment of electrons. These nuclear motions correspond to the inner- sphere reorganization contributing to the hydrogen transfer reac- tion. Furthermore, analysis of the reactive electronic intrinsic bond orbitals45,46and protonic orbitals along the MEP provides insight into whether the electron and proton are transferring synchronously or asynchronously. For the specific reaction depicted in Fig. 1, such an analysis indicates that this hydride transfer reaction entails synchronous electron and proton transfer. These types of mech- anistic insights are useful for understanding and tuning chemical reactions.MULTICOMPONENT COUPLED CLUSTER AND CONFIGURATION INTERACTION METHODS Correlation effects can be included within the NEO framework with the multicomponent configuration interaction (CI) and cou- pled cluster (CC) approaches.23,30,47Analogous to the conventional electronic counterparts,48,49the CI and CC wave functions can be expressed as ΨNEO−CI=(1+ˆT)ΨNEO−HF, ΨNEO−CC=eˆTΨNEO−HF.(3) Here, the cluster excitation operator ˆTgenerates all single, double, triple, and further excited determinants among the electrons and protons by acting on the NEO-HF reference state. For example, at the singles and doubles (SD) level, ˆT=ˆTe 1+ˆTp 1+ˆTe 2+ˆTp 2+ˆTep 2 includes electronic and protonic single excitations, electronic and protonic double excitations, and mixed electronic–protonic double excitations. The NEO-CISD and NEO-CCSD methods, which include all single and double excitations, have been derived and implemented.30 Moreover, the NEO-BCCD50and NEO-OOCCD51methods, which use the coupled cluster with doubles ansatz in conjunction with opti- mized Brueckner orbitals or variationally optimized orbitals, respec- tively, have also been implemented. The NEO-CISD methods are not sufficiently accurate due to the absence of orbital relaxation.30In contrast, the NEO-CCSD, NEO-BCCD, and NEO-OOCCD meth- ods have been shown to provide accurate proton densities, proton affinities, and optimized geometries for the chemical systems stud- ied.30,50,51More recently, the NEO unitary coupled cluster (UCC) ansatz has been combined with the variational quantum eigensolver algorithm and formulated in the qubit basis to enable calculations of ground state energies and wave functions of multicomponent systems on quantum computers.52 An illustrative example is the application of the NEO-CCSD method to protonated water tetramers, treating all nine protons J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 155, 030901-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp quantum mechanically.53This application became computationally feasible by combining the NEO-CCSD method with density fit- ting53,54for approximating the four-center two-particle integrals. The inclusion of zero-point energy contributions has been shown to change the relative stabilities of the four protonated water tetramer isomers at the conventional electronic CCSD with perturbative triples [CCSD(T)] level of theory, as shown in Fig. 2 (black and blue symbols).55This effect is observed when the zero-point ener- gies are computed via diagonalization of the mass-weighted Hessian matrix within the harmonic approximation and when anharmonic corrections are included with the vibrational second-order pertur- bative (VPT2) approach.53,55,56NEO-CCSD single-point energy cal- culations on the optimized CCSD(T) geometries predict the cor- rect ordering of the isomer energies,53as shown in Fig. 2 (purple symbols). The NEO-CCSD method inherently includes the zero- point energy contributions from the quantized protons, and the con- tributions from the oxygen atoms to the relative zero-point energies of these isomers have been shown to be negligible.53Thus, the NEO- CCSD method provides accurate results for this system without requiring the computationally expensive calculation of the Hessian matrix. The NEO orbital optimized second-order Møller–Plesset per- turbation theory (NEO-OOMP2) method51offers a more com- putationally efficient alternative that approaches the accuracy of the NEO-CCSD method. Additional computational efficiency arises from the scaled-opposite-spin (SOS) strategy,57where the opposite-spin and same-spin components of the second-order electron–electron perturbative term are scaled differently, thereby offering the opportunity to eliminate the computationally expensive same-spin component. The multicomponent extension of this strat- egy,51denoted SOS′, also applies a scaling factor to the second-order FIG. 2. Relative energies of protonated water tetramer isomers calculated with the conventional electronic CCSD(T) method without any zero-point energy (ZPE) contributions (black), the conventional CCSD(T) method with anharmonic zero- point energy (ZPE) contributions (blue), and the NEO-CCSD method with density fitting (purple). The lines are included as a visual guide. Reproduced with permis- sion from F. Pavo ˇsevi´c, Z. Tao, and S. Hammes-Schiffer, J. Phys. Chem. Lett. 12, 1631 (2021). Copyright 2021 American Chemical Society.electron–proton perturbative term. Although the NEO-OOMP2 method is stable and qualitatively reasonable for the systems stud- ied, the NEO-SOS′-OOMP2 method is advantageous in that it produces proton densities and proton affinities that are nearly as accurate as the NEO-CCSD results.51Moreover, the NEO-SOS′- OOMP2 method can be implemented in a manner that leads to N4scaling rather than the N6scaling of NEO-CCSD, where Nis a measure of the system size. The NEO-CCSD and NEO-SOS′-OOMP2 methods provide the foundation for additional developments of NEO ground state wave function methods. The implementation of analytical gra- dients and Hessians will enable geometry optimizations and the calculation of zero-point energy contributions, thermochemical properties, and MEPs at these correlated levels of theory. An advantage of the NEO-CCSD method is that it can be improved systematically by including higher-order excitations, starting with triple excitations. However, this approach will become computa- tionally prohibitive for larger systems, even with density fitting methods. In these cases, the NEO-SOS′-OOMP2 method will be a more practical option, and reparameterization of the scaling fac- tors for specific types of systems could provide additional flexibil- ity. Embedding methods58in which only a small portion of the system is treated at the NEO-CCSD level provide another viable option. MULTICOMPONENT EQUATION-OF-MOTION METHODS FOR EXCITED STATES Analogous to conventional electronic equation-of-motion CCSD (EOM-CCSD) methods,59,60the NEO-EOM-CCSD method34 has been implemented for calculating excited electronic, protonic, and mixed electron–proton vibronic excitation energies in both the frequency and time domains.35Figure 3 illustrates the appli- cation of this approach to the HCN molecule, where all elec- trons and the hydrogen nucleus are treated quantum mechani- cally. This approach produces qualitatively reasonable energies and intensities for the fundamental proton vibrational excitations (i.e., the hydrogen stretch and bend), as well as the combination band corresponding to simultaneous excitation of both bending modes and the overtone associated with the bending mode, as depicted in Fig. 3 (left side). In addition, this method can describe dou- ble excitations corresponding to the simultaneous excitation of an electron and a proton, such as an excited proton vibrational state within an excited electronic state, as shown in Fig. 3 (right side).35However, the energies of these double excitations are much too high, presumably mainly due to the truncated coupled cluster ansatz. An advantage of the NEO-EOM methods is that they can be improved systematically by including higher-order excitations, starting with triples, but the computational expense becomes pro- hibitive with current computational capabilities. Again, embed- ding methods may render the NEO-EOM methods computationally viable. Moreover, these methods play an important role for bench- marking other excited state NEO methods. Additionally, a NEO- EOM method has been formulated in the qubit basis to enable the quantum computation of excitation energies for multicomponent systems.52 J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 155, 030901-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 3. Transition densities for proton vibrational states of the HCN molecule computed with the NEO-EOM-CCSD method (left) and schematic depiction of double excitations corresponding to the simultaneous excitation of the fundamental proton bending or stretching mode with the electronic excitation (e 1p100and e 1p001) observed with the NEO-EOM-CCSD method (right). The excitation energies and oscillator strengths for these transitions are provided in Ref. 35. Adapted with permission from Pavo ˇsevi´c et al. , J. Phys. Chem. Lett. 11, 6435 (2020). Copyright 2020 American Chemical Society. MULTICOMPONENT DFT In multicomponent DFT,22,25,61,62more than one type of par- ticle is treated quantum mechanically. Here, we discuss multicom- ponent DFT for electrons and protons. In the multicomponent Hohenberg–Kohn theorems,63the total energy is a functional of the one-particle electronic and protonic densities. In the multicompo- nent Kohn–Sham formalism,22,25,61,62the reference wave function is the product of electronic and protonic Slater determinants com- posed of electronic and protonic Kohn–Sham orbitals, respectively, as in Eq. (1). Optimization of the NEO-DFT energy with respect to the electronic and protonic orbitals leads to two sets of Kohn–Sham equations for the electrons and protons, respectively, as in Eq. (2). These equations are solved self-consistently, as described above for the NEO-HF method. Within the NEO-DFT framework, the total energy functional is expressed as E[ρe,ρp]=Eext[ρe,ρp]+Eref[ρe,ρp]+Eexc[ρe] +Epxc[ρp]+Eepc[ρe,ρp], (4) where ρeand ρpare the electronic and protonic densities, respec- tively. Here, Eext[ρe,ρp]is the interaction of the electronic and protonic densities with the external potential created by the clas- sical nuclei, Eref[ρe,ρp]includes the noninteracting kinetic ener- gies of the electrons and quantum protons, as well as the classical electron–electron, proton–proton, and electron–proton Coulomb energies, and the last three terms correspond to different types of functionals. The electron–electron exchange–correlation functional,Eexc[ρe], can be chosen from the many available electronic function- als.64The proton–proton exchange–correlation functional, Epxc[ρp], is typically chosen to be the Hartree–Fock exchange energy. For molecular systems, the proton densities are highly localized, and therefore, proton–proton exchange and correlation energies are neg- ligible,17although the diagonal exchange terms must be included to eliminate self-interaction terms. The electron–proton correlation functional, Eepc[ρe,ρp], has required the development of new types of functionals. The epc family of functionals was developed by extending the Colle–Salvetti formalism65to multicomponent systems. The mul- ticomponent electron–proton wave function ansatz is the prod- uct of electronic and protonic wave functions that include all electron–electron and proton–proton exchange–correlation effects, respectively, multiplied by an explicit electron–proton correla- tion factor. This electron–proton correlation factor depends on an inverse correlation length parameter, which is chosen to be the geo- metric mean of the inverse Wigner–Seitz radii for an electron and a proton for the epc17 and epc19 functionals32,33,66and the arithmetic mean of these radii for the epc18 functional.64The epc17 functional has the following form:32 Eepc[ρe,ρp]=−∫drρe(r)ρp(r) a−b[ρe(r)]1/2[ρp(r)]1/2+cρe(r)ρp(r), (5) where a,b, and care parameters that can be fit to properties of interest. The epc17 and epc18 functionals32,33,64rely on the local den- sity approximation (LDA) and depend on only the electron and J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 155, 030901-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp proton densities. They have been shown to be of similar accuracy, and the epc17 functional has been adopted for most applications because it is mathematically simpler. The epc19 functional66relies on the generalized gradient approximation (GGA) and depends on the gradients of the electronic and protonic densities as well as the densities themselves. Thus, the epc19 functional is analogous to the well-known LYP functional67for electron–electron correla- tion. The epc17 and epc19 functionals depend on three and five parameters, respectively, which were determined by fitting to pro- ton densities and energies. In particular, the epc17-2 functional33 has been parameterized to produce an effective balance between accurate proton densities and energies. The NEO-DFT method in conjunction with these functionals has been shown to produce accu- rate proton densities, proton affinities, and optimized geometries for the chemical systems studied.32,33,64,66These epc functionals have also been shown to be transferable across a wide range of electronic functionals.64 The implementation of NEO-DFT analytical gradients enables geometry optimizations. The geometries of protonated water tetramers were optimized using the NEO-DFT method with the B3LYP electronic functional67,68and the epc17-2 electron–proton correlation functional, treating all nine protons quantum mechan- ically. The relative stabilities of the protonated water tetramer iso- mers computed at the NEO-DFT level are consistent with the NEO-CCSD results, as well as the conventional CCSD(T) results including zero-point energy contributions shown in Fig. 2. The lack of NEO-CCSD analytical gradients prevented geometry optimiza- tions at that level of theory, and the NEO-DFT geometry opti- mizations allowed further analysis of geometric effects for this sys- tem. In terms of geometry optimizations, the constrained NEO- DFT (cNEO-DFT) method69produces the same stationary points as those of the NEO-DFT method but also allows the calculation of an extended NEO potential energy surface that depends on the expecta- tion values of the quantum nuclear positions. This type of extended NEO potential energy surface may provide useful conceptual insights. Thus, the NEO-DFT method is computationally efficient and reasonably accurate for computing ground state molecular proper- ties. The implementation of NEO-DFT analytical Hessians allows the calculation of transition states, zero-point energy contributions, thermochemical properties, and MEPs. In contrast to the wave func- tion methods, however, the systematic improvement of the NEO- DFT method is not straightforward. The main source of uncertainty is the electron–proton correlation functional. Although the epc17 and epc19 functionals perform well for many applications, they still have limitations in terms of predicting properties that are sensitive to the subtle, quantitative aspects of the proton densities. The devel- opment of more accurate and robust electron–proton correlation functionals is an important direction to explore. Possible strategies include the multicomponent extension of the SCAN functional,70 which was designed to satisfy exact constraints, or machine learning approaches to functional development.71 MULTICOMPONENT TDDFT Excited states may be studied with the NEO-TDDFT method,36,37which is analogous to its conventional electroniccounterpart.25The linear response of the NEO Kohn–Sham sys- tem to perturbative external nuclear and electronic fields leads to a matrix equation for the linear-response NEO-TDDFT method.36 The solution of this matrix equation provides the excitation energies and corresponding transition amplitudes for electronic, protonic, and mixed electron–proton vibronic excitations in a single calcula- tion. Because the derivation relies on the adiabatic approximation, where the kernel is assumed to be independent of frequency, this method captures only single excitations, which can be described as linear combinations of products of electronic and protonic deter- minants with only one determinant singly excited in each term. In addition to the excitation energies, the transition densities, tran- sition dipole moments, oscillator strengths, and intensities can be calculated to characterize each excitation.37 The low-lying electronic and protonic excitations are separa- ble for predominantly electronically adiabatic systems but become mixed for higher excitations and for nonadiabatic systems. The electronic excitation energies computed with NEO-TDDFT for the lower electronic states are similar to those computed with conven- tional electronic TDDFT,36,37but higher electronic states exhibit differences due to vibronic mixing.38The key advantage of the NEO- TDDFT method is that it also describes proton vibrational excita- tions and, in some cases, mixed electron–proton vibronic excita- tions. The proton vibrational excitation energies for the fundamental vibrational modes (i.e., the bend and stretch modes of a molecule such as HCN) have been shown to be accurate, often within 20 cm−1 of nearly numerically exact grid-based reference values computed for the ground state potential energy surface generated with the same electronic structure method.37This accuracy carries over to systems with multiple protons, such as C 2H2, where the six fundamental pro- ton vibrational modes are linear combinations of single excitations associated with each proton.72In contrast to the NEO-EOM-CCSD method,35the combination bands and overtones are not described accurately, presumably due to the limitations of the electron–proton correlation functionals, and double excitations are absent due to the underlying adiabatic approximation. The implementation of NEO-TDDFT analytical gradients allows the optimization of excited state geometries and the calcula- tion of adiabatic excitation energies.73Within the NEO framework, the adiabatic excitation energy is defined to be the energy difference between geometries optimized on the ground and excited state NEO electron–proton vibronic potential energy surfaces and thus implic- itly includes the zero-point energies and associated anharmonic- ities of the quantum protons. The NEO-DFT analytical Hessian and the NEO-TDDFT semi-numerical Hessian can be used to com- pute the zero-point energy contributions from the classical nuclei within the harmonic approximation for the ground and excited state NEO vibronic surfaces. A combination of these zero-point energy contributions and the adiabatic excitation energies produces the 0–0 adiabatic excitation energies, which can be compared directly to experimental data.73The NEO-TDDFT analytical gradients also enable the simulation of adiabatic dynamics on excited state vibronic surfaces and provide the foundation for nonadiabatic surface hop- ping simulations. The NEO-DFT(V) method44was developed to compute molec- ular vibrational frequencies that can be compared to experimental spectra. Because the NEO potential energy surface depends only on the classical nuclei, the NEO Hessian produces modes that depend J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 155, 030901-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp only on the classical nuclei. The NEO-DFT(V) method couples the NEO-DFT Hessian for the classical nuclei with NEO-TDDFT vibra- tional excitation energies and transition dipole moments for the quantum nuclei to generate the molecular vibrational modes com- posed of both quantum and classical nuclear motions. This method entails the calculation of an extended Hessian that depends on the classical nuclear coordinates and the expectation values of the quantum proton coordinates, where the diagonal block matrix of the Hessian associated with the quantum protons is obtained from NEO-TDDFT calculations. Diagonalization of this extended Hes- sian provides the coupled vibrational modes depending on both classical and quantum nuclear coordinates. A schematic illustra- tion of this approach for HCN is given in Fig. 4. The NEO-DFT(V) method has been shown to incorporate the significant anharmonic effects associated with the quantum protons into the molecular vibrational frequencies, leading to good agreement with experi- mental data as well as calculations that include anharmonic effects perturbatively.44,72 The NEO-TDDFT method has been shown to provide elec- tronic, protonic, and vibronic excitations in a computationally effi- cient manner. As discussed above in the context of the NEO- DFT method, the development of more reliable electron–proton correlation functionals could improve the quantitative accuracy of the combination bands and overtones as well as the fundamen- tal vibrational modes. For some systems, double excitations can be computed by combining the NEO- ΔSCF and NEO-TDDFT meth- ods,17,74although complications with spin contamination, triplet instabilities, and unstable solutions may arise. The exploration of alternative approaches for computing double excitations within the NEO-TDDFT framework is an important future direction. In terms of the NEO-DFT(V) method for calculating molecular vibrational frequencies, the anharmonic effects associated with the quantum nuclei are included through the NEO-TDDFT calculations, but the modes dominated by classical nuclei are still calculated within the harmonic approximation. As discussed below, this issue is addressed by the NEO-Ehrenfest dynamics approach for computing vibra- tional frequencies.39 FIG. 4. Schematic depiction of the NEO-DFT(V) procedure for HCN. The quantum proton is represented by a red mesh, and the classical nitrogen and carbon nuclei are represented by blue and gray spheres, respectively. The single vibrational mode obtained from the NEO Hessian (upper left) is coupled to the NEO-TDDFT proton vibrational excitations for the stretch and doubly degenerate bend with fixed classical nuclei (lower left) to obtain the full molecular vibrational frequen- cies (right). Reproduced with permission from Yang et al. , J. Phys. Chem. Lett. 10, 1167 (2019). Copyright 2019 American Chemical Society.NONEQUILIBRIUM DYNAMICS BEYOND THE BORN–OPPENHEIMER APPROXIMATION An alternative to the linear-response NEO-TDDFT approach is the real-time NEO-TDDFT (RT-NEO-TDDFT) approach.38In this case, substitution of the nuclear–electronic wave function of the form given in Eq. (1) into the time-dependent Schrödinger equation leads to the following time-dependent equations: i̵h∂ ∂tCe(t)=Fe(t)Ce(t), i̵h∂ ∂tCp(t)=Fp(t)Cp(t).(6) These electronic and protonic equations are strongly coupled because the electronic and protonic Fock matrices each depend on both the electronic and protonic orbital coefficients. After some rearrangements, these two sets of equations are propagated numer- ically to obtain the orbital coefficients as a function of time. The electronic and protonic excitation energies are computed from the Fourier transform of the time-dependent electronic and protonic dipole moments. The resulting excitation energies are in agreement with the linear-response NEO-TDDFT results.38 An advantage of the RT-NEO-TDDFT method is that it provides the nonequilibrium dynamics of the electronic–protonic system without invoking the Born–Oppenheimer separation. For example, an electric field pulse can be applied to a molecular sys- tem to simulate ultrafast vibrational excitation by a laser. The appli- cation of this approach to the HCN molecule illustrates that a resonant driving field leads to energy absorption that produces time- dependent oscillations of the dipole moment. These types of calcu- lations could assist in the interpretation of time-resolved infrared spectroscopy.38 The RT-NEO-TDDFT method has also been applied to excited state intramolecular proton transfer (ESIPT) in o-hydroxybenzaldehyde (oHBA).38In this system, photoexcitation induces intramolecular proton transfer between two oxygen atoms. Photoexcitation to the excited electronic state, corresponding to an S 0→S1excitation, is modeled by an electronic transition from the highest occupied molecular orbital to the lowest unoccupied molecular orbital. In the initial studies, the transferring hydrogen nucleus was represented by three sets of electronic and protonic basis functions spanning the initial and final positions of the hydrogen nucleus. When these RT-NEO-TDDFT calculations were performed at the equilibrium ground state geometry, the proton did not transfer because the classical nuclei were fixed and unable to reorganize. When these calculations were performed at a geometry optimized in the excited state with the proton constrained to be bound to the donor oxygen atom, the proton transferred to the acceptor oxygen atom. These calculations highlight the strong coupling between the transferring proton and the other nuclei as well as the importance of excited state structural relaxation in these types of photoinduced reactions. To simulate the nonadiabatic dynamics of the classical nuclei within the NEO framework, the RT-NEO-TDDFT method can be combined with the Ehrenfest dynamics approach.39,40In this case, the quantum subsystem composed of the electrons and quan- tum protons is propagated according to Eq. (6), while the classical nuclei are propagated on the average surface determined by the J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 155, 030901-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp time-dependent electronic–protonic Kohn–Sham wave function MI¨rc I=−∇ I⟨ΦeΦp∣HNEO∣ΦeΦp⟩, (7) where MIandrc Iare the masses and coordinates, respectively, of the classical nuclei. The use of fixed electronic and protonic basis function centers for the quantum protons requires a large number of basis function centers to describe a proton transfer reaction and prior knowledge of its trajectory. To address this issue, a semiclassi- cal traveling proton basis function approach has been developed.39 In this approach, the basis function centers associated with the quan- tum protons follow the same equations of motion as the classical nuclei. Additional terms related to the time derivatives of the pro- ton basis function centers must be added to the protonic equations of motion given in Eq. (6). The semiclassical traveling proton basis function method will converge to the exact result within the RT- NEO-TDDFT Ehrenfest treatment of the dynamics as the basis set approaches completeness, although it is approximate for incomplete basis sets. The molecular vibrational frequencies computed with the RT-NEO-TDDFT Ehrenfest method are consistent with those com- puted with the NEO-DFT(V) method for the chemical systems stud- ied,39although the NEO Ehrenfest approach includes anharmonic effects associated with all nuclei. The application of the RT-NEO-TDDFT Ehrenfest method to ESIPT in oHBA enables the nonequilibrium dynamical simula- tion of the electrons and all nuclei beyond the Born–Oppenheimer approximation.40The nonadiabatic effects between the electrons and quantum protons are included via the RT-NEO-TDDFT method, and the nonadiabatic effects between the classical nuclei and the electron–proton quantum subsystem are described with Ehrenfest dynamics. In this case, photoexcitation at the equilibrium ground state geometry leads to proton transfer because the classi- cal nuclei are allowed to reorganize in the excited electronic state. A comparison of Ehrenfest dynamics with a quantum or classical treatment of the transferring hydrogen nucleus shows that the quan- tization of the hydrogen nucleus within the NEO framework leads to faster proton transfer, as shown in Fig. 5. The delocalization of the quantized proton allows the transfer to occur at a longer dis- tance between the donor and acceptor oxygen atoms, thereby requir- ing less movement of the oxygen atoms prior to proton transfer. Moreover, the kinetic isotope effect for this proton transfer reac- tion is slightly larger with a quantum treatment of the transferring hydrogen or deuterium. The simulation of nonadiabatic dynamics within the NEO framework opens up a wide range of opportunities. In addition to Ehrenfest dynamics, the NEO method can also be combined with other nonadiabatic dynamics methods, such as trajectory sur- face hopping.6In this case, the fewest switches surface hopping algorithm could be applied to trajectories associated with the clas- sical nuclei moving on the adiabatic vibronic surfaces computed with linear-response NEO-TDDFT. In principle, the NEO approach could also be combined with the multiple spawning method.7,8Var- ious technical challenges, such as the calculation of the nonadiabatic coupling vector between NEO-TDDFT vibronic states, would need to be addressed for these types of simulations. In addition, more robust strategies for moving the proton basis function centers could improve the accuracy and energy conservation of the nondiabatic dynamics, and spin-flip methods may be useful. The description FIG. 5. Distance between the transferring proton and the donor oxygen (O D) and acceptor oxygen (O A) as a function of time for ESIPT in oHBA computed using the NEO Ehrenfest dynamics approach with the transferring proton quantized (top) and the conventional Ehrenfest dynamics approach with all nuclei treated classi- cally (bottom). For the NEO calculations, the distances are determined using the proton position expectation value. Adapted with permission from Zhao et al. , J. Phys. Chem. Lett. 12, 3497 (2021). Copyright 2021 American Chemical Society. of hydrogen tunneling within these nonadiabatic dynamics simula- tions also necessitates further developments, as discussed below. HYDROGEN TUNNELING The description of hydrogen tunneling requires the calcula- tion of bilobal, delocalized hydrogen vibrational wave functions. The variationally optimized proton wave function or density computed at the NEO-HF or NEO-DFT level for a proton moving in a sym- metric double-well potential is localized in one well even when the transferring hydrogen is represented by two basis function centers (i.e., a set of electronic and protonic basis functions centered at each minimum).75Thus, a multireference approach is required to pro- duce bilobal wave functions in this situation. Although the NEO complete active space SCF (NEO-CASSCF) approach was developed nearly two decades ago,16extremely large active spaces are required to describe such bilobal wave functions because dynamical as well as non-dynamical correlation is important.75,76Similarly, NEO mul- tireference CI (MRCI) calculations require a very large number of determinants in the expansion. As a computationally practical alternative, the NEO multistate DFT (NEO-MSDFT) method41includes both dynamical and non- dynamical correlation and has been shown to be capable of describ- ing hydrogen tunneling systems. In the NEO-MSDFT approach, two localized electronic–protonic wave functions are obtained with the NEO-DFT method using two basis function centers to rep- resent the transferring hydrogen. A nonorthogonal configuration interaction approach is used to mix these two localized wave func- tions to produce bilobal, delocalized wave functions, as shown in Fig. 6. This method only requires the diagonalization of a 2 ×2 J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 155, 030901-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 6. NEO-MSDFT procedure for describing hydrogen tunneling in malonalde- hyde. The NEO-DFT method is used to compute two localized electronic–protonic wave functions (left), and a nonorthogonal configuration interaction approach is used to mix these two localized wave functions to produce bilobal, delocalized wave functions (right). H,D,S, and Ecorrespond to the Hamiltonian, coefficient, overlap, and energy matrices in the basis of the two localized NEO-DFT states. Adapted with permission from Q. Yu and S. Hammes–Schiffer, J. Phys. Chem. Lett. 11, 10106 (2020). Copyright 2020 American Chemical Society. Hamiltonian matrix in the basis of the two localized NEO-DFT states, ΨIandΨII, H=[ENEO−DFT I HI,II HI,II ENEO−DFT II], (8) where the diagonal matrix elements are the NEO-DFT energies of the localized states.The off-diagonal element can be approximated by a physically motivated form inspired by the conventional electronic MSDFT method.77,78In the NEO framework, this element is expressed as HI,II=⟨ΨI∣HNEO∣ΨII⟩+1 2SI,II(ENEO−DFT I−ENEO−HF I +ENEO−DFT II−ENEO−HF II). (9) Here, the first term is the off-diagonal matrix element computed at the NEO-HF level with the NEO Kohn–Sham determinants, SI,II is the overlap between the localized Kohn–Sham wave functions, and the terms in parentheses correspond to the NEO-DFT and NEO-HF energies of the localized states. A correction function is applied to SI,IIin order to account for limitations of the electron–proton cor- relation functionals and the approximate form of the off-diagonal matrix element.41This correction function requires two parameters that were determined for simple model systems and fixed thereafter. The NEO-MSDFT method has been shown to produce quantita- tively accurate hydrogen tunneling splittings with errors less than ∼6 cm−1for systems such as malonaldehyde, acetoacetaldehyde, and the benzyl-toluene radical complex.41 Although the NEO-MSDFT method is computationally effi- cient and quantitatively accurate in its current form, systematic improvement is not straightforward. More rigorous multirefer- ence wave function approaches, such as NEO-CASSCF or MRCI methods,16,76,79,80are systematically improvable but are significantly more computationally expensive and may not be easily applied to large molecular systems. The importance of dynamical correlation most likely necessitates a method such as the analog of the conven- tional electronic second-order perturbative (CASPT2) method81to FIG. 7. Schematic depiction of NEO methods that have been implemented. Some of the dynamics methods are cur- rently under development. In addition to these methods, the NEO-HF, MP2, CIS, CISD, and TDDFT-TDA methods have also been implemented but are not as accurate and therefore are of limited util- ity within the NEO framework. J. Chem. Phys. 155, 030901 (2021); doi: 10.1063/5.0053576 155, 030901-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp obtain quantitatively accurate hydrogen tunneling splittings. In con- junction with analytical gradients, multireference methods such as NEO-MSDFT and NEO-CASSCF can also be used to generate MEPs that include hydrogen tunneling effects. In addition, such multiref- erence methods could be combined with the NEO Ehrenfest or sur- face hopping algorithms for nonadiabatic dynamics simulations of hydrogen tunneling processes. As discussed in the context of other methods, embedding approaches may be helpful for applications of these multireference methods to larger systems. CONCLUDING REMARKS This Perspective outlines the foundational NEO methods, including NEO-HF, NEO-CI, NEO-CCSD, NEO-OOMP2, NEO- EOM-CCSD, NEO-DFT, NEO-TDDFT, RT-NEO-TDDFT, NEO- Ehrenfest, NEO-MSDFT, and NEO-CASSCF, as depicted in Fig. 7. In each section, the limitations of the methods and the opportuni- ties for future developments are discussed. Most of these methods are implemented in the Q-Chem 5.4 software package82or a devel- opment branch. The real-time NEO methods are implemented in a development branch of the Chronus Quantum software package.83 Some of the NEO methods are also available in other quantum chemistry software packages. These NEO methods will serve as the building blocks for con- structing a framework of strategies aimed at diverse applications. For example, many of these NEO methods could be extended to peri- odic systems, allowing the study of nuclear quantum effects in solids and at interfaces. In principle, all nuclei could be treated quantum mechanically within the NEO framework,84as long as the trans- lations and rotations are treated appropriately. The advantages of including all nuclear quantum effects at the same level may war- rant the added complexity. However, the treatment of only spec- ified nuclei within the NEO framework is appealing for its sim- plicity, accessibility, and computational efficiency. Solvation effects can be included using either implicit or explicit solvation models. A particularly promising strategy relies on embedding85or hybrid approaches that treat a relatively small portion of the system at a correlated NEO level and treat the environment at a lower level of quantum mechanics or with a molecular mechanical force field. Exploring these directions will open up possibilities for simulating a wide range of nonadiabatic processes in complex environments. ACKNOWLEDGMENTS The NEO work discussed in this Perspective would not have been possible without key contributions from the following group members: Simon Webb, Chet Swalina, Michael Pak, Ari Chakraborty, Tanner Culpitt, Kurt Brorsen, Yang Yang, Fabijan Pavo ˇsevi´c, Patrick Schneider, Coraline (Zhen) Tao, Ben Rousseau, and Qi Yu. The real-time NEO work was performed in collabora- tion with Xiaosong Li’s group, with key contributions from Luning Zhao and Andrew Wildman. I am also grateful for many useful dis- cussions with John Tully. 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5.0051331.pdf

J. Chem. Phys. 155, 024103 (2021); https://doi.org/10.1063/5.0051331 155, 024103 © 2021 Author(s).Accurate density functional made more versatile Cite as: J. Chem. Phys. 155, 024103 (2021); https://doi.org/10.1063/5.0051331 Submitted: 24 March 2021 . Accepted: 16 June 2021 . Published Online: 08 July 2021 Subrata Jana , Sushant Kumar Behera , Szymon Śmiga , Lucian A. Constantin , and Prasanjit Samal ARTICLES YOU MAY BE INTERESTED IN Chemical physics software The Journal of Chemical Physics 155, 010401 (2021); https://doi.org/10.1063/5.0059886 Requirements for an accurate dispersion-corrected density functional The Journal of Chemical Physics 154, 230902 (2021); https://doi.org/10.1063/5.0050993 Impact of quantum-chemical metrics on the machine learning prediction of electron density The Journal of Chemical Physics 155, 024107 (2021); https://doi.org/10.1063/5.0055393The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Accurate density functional made more versatile Cite as: J. Chem. Phys. 155, 024103 (2021); doi: 10.1063/5.0051331 Submitted: 24 March 2021 •Accepted: 16 June 2021 • Published Online: 8 July 2021 Subrata Jana,1,a) Sushant Kumar Behera,1 Szymon ´Smiga,2 Lucian A. Constantin,3 and Prasanjit Samal1 AFFILIATIONS 1School of Physical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar 752050, India 2Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Toru ´n, Poland 3Istituto di Nanoscienze, Consiglio Nazionale delle Ricerche CNR-NANO, 41125 Modena, Italy a)Author to whom correspondence should be addressed: subrata.jana@niser.ac.in and subrata.niser@gmail.com ABSTRACT We propose a one-electron self-interaction-free correlation energy functional compatible with the order-of-limit problem-free Tao–Mo (TM) semilocal functional (regTM) [J. Tao and Y. Mo, Phys. Rev. Lett. 117, 073001 (2016) and Patra et al. , J. Chem. Phys. 153, 184112 (2020)] to be used for general purpose condensed matter physics and quantum chemistry. The assessment of the proposed functional for large classes of condensed matter and chemical systems shows its improvement in most cases compared to the TM functional, e.g., when applied to the relative energy difference of MnO 2polymorphs. In this respect, the present exchange–correction functional, which incorporates the TM technique of the exchange hole model combined with the slowly varying density correction, can achieve broad applicability, being able to solve difficult solid-state problems. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0051331 I. INTRODUCTION The remarkable efficiency of the density functional theory (DFT)1in the field of condensed matter physics and quantum chem- istry, including nanostructured solids, supra-molecular chemistry, and several other fields, is possible due to the advent of the accu- rate semilocal exchange–correlation (XC) energy density functional approximations (DFAs), Exc[ρ↑,ρ↓].2–6Because of its low computa- tional cost, the semilocal DFT becomes an undisputed tool to per- form the electronic structure calculations of large systems. Despite its success, there are challenges in which the XC semilocal functional does not perform as expected.7The search for an efficient yet accu- rate universal method continues to be a developing part of DFT with several new prospects. The first three rungs of the “Jacob’s ladder” DFAs8are known as the semilocal variants depending solely on the spin densities ( ρσ), their gradients ( ∇ρσ) and Laplacian ( ∇2ρσ), and/or Kohn–Sham (KS) kinetic energy densities ( τσ=1 2∑i∣∇ψi,σ∣2) expressed via single- particle KS orbitals ψi,σ, withσ=↑,↓. The semilocal DFAs have the following general form:9–12 Exc[ρ↑,ρ↓]=∫drρ(r)ϵDFA xc(ρσ,∇ρσ,∇2ρσ,τσ), (1)whereϵDFA xc is the XC energy per particle defined by a given DFA. Here,ρ=ρ↑+ρ↓is the total density. Beyond the generalized gra- dient approximations (GGAs),13–21the higher rung XC semilocal functionals, i.e., meta-GGA,11,12,22–42are quite feasible to recog- nize the covalent, metallic, and weak bonds.22,43Additionally, var- ious benchmark tests proved that the meta-GGA functionals can bring quite good accuracy for condensed matter physics31,44–54and quantum chemical property31,54–62calculations. Recent advances of the meta-GGA functionals show that the accuracy and uni- versality of these DFAs can be improved by satisfaction of many quantum mechanical constraints.12,30We also recall that func- tionals proposed in an empirical way (such as M06-L63and revM06-L31) are also quite accurate for many quantum chemi- cal and solid-state properties.54,64,65However, the most successful way to construct a non-empirical density functional by satisfying known quantum mechanical constraints came through the strongly constrained and appropriately normed (SCAN) meta-GGA func- tional,12which is quite an accurate method for broad classes of solid- state properties47,51,53,66–73although some limitations52,74–79have also been recognized. To solve some deficiencies, several SCAN vari- ants have been developed,32,33,39also by using the deorbitalization technique.80–82 J. Chem. Phys. 155, 024103 (2021); doi: 10.1063/5.0051331 155, 024103-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Another functional with similar accuracy for most of the solid- state and quantum chemical systems51,56,58is the one proposed by Tao and Mo (TM).30The TM functional is based on the novel tech- nique of the exchange hole model coupled with the slowly varying density gradient expansion.30Despite its considerable success,49,56,58 recent investigation showed that it possesses the order-of-limit prob- lem41that hinders the functional performance, e.g., for the structural phase transition of the solid-state systems.41In Ref. 38, an order-of- limit problem-free TM functional (regTM) has been proposed based on a simple modification of the TM exchange. The exchange part was combined with the revised Perdew–Burke–Ernzerhof (PBE) GGA correlation,27which is not one-electron self-interaction-free. The aim of this article is to construct a reliable one-electron self-interaction-free revised regTM (named rregTM) functional that improves over the TM broad performance and corrects the TM limitations for challenging solid-state systems and properties. The rregTM meta-GGA respects important paradigm densities of quantum chemistry (one electron densities) and condensed mat- ter physics (slowly varying density) but is an unbound functional being fundamentally different from the SCAN meta-GGA, which is tightly bound. From this point of view, the rregTM can be important not only for practical calculations but also for theoretical reasons because its exchange (and XC) energy density gauge may be closer to the exact conventional one.83,84 II. THEORY Our starting point is the regTM exchange, which is con- structed from the TM exchange by modifying the meta-GGA iso-orbital indicator z=1/(1+(3/5)p/α)toz′used in the interpolation function w30,38asz′=1/(1+3 5[α p+f(α,p)]), where p=s2=∣∇ρ∣2/[4(3π2)2/3ρ8/3]is the reduced density gradient and α=(τ−τW)/τunifis the Pauli kinetic energy density85–89enhance- ment factor. Note that z′becomes zupon considering f(α,p) =(1−α)3 (1+(dα)2)3/2e−cp=0, where dandcare constants.38For slowly vary- ing density correction of solids, α≈1 and f(α,p)≈0. Therefore, the proposed f(α,p)correctly recovers the slowly varying density correction of the TM functional. For one-electron or two-electron singlet state, α=0,z′=z=1. Hence, all the known constraints of the TM functional remain intact in the regTM functional form, and additionally, regTM removes the singularity of the exchange enhancement factor as38 lim p→0[lim α→0[FregTM x(p,α)]]=lim α→0[lim p→0[FregTM x(p,α)]]=1.1132. (2) As stated in Ref. 41, there is no simple connection between zand α. Therefore, the z′has been constructed to preserve the properties of the interpolation function w,38correcting only when p→0 and α→0. In Fig. 1, we show the exchange enhancement factor of a stretched Li 2molecule. The order-of-limit problem of the TM func- tional is visible at the position of the bond center, which is eliminated by its revised version regTM exchange.38 Now, we turn to the construction of the correlation energy functional compatible with the regTM exchange, which is devel- oped by satisfying exact constraints: It becomes (i) one-electron self- interaction-free, (ii) correct for slowly and rapidly varying density FIG. 1. The exchange enhancement factor Fxof the TM and regTM functionals for the stretched Li 2molecule (the distance between Li atoms is 2 r0, with r0=5.051 bohrs). The order-of-limit problem of the TM functional is visible at r=0 (position of the bond center), being eliminated by the regTM functional. The self-consistent PBE orbitals in the aug-cc-pVQZ basis set are used to compute Fx. In the inset, Fxis plotted for−0.5≤r≤0.5 for clear visualization of the order-of-limit problem. limits, and (iii) correct dependence on the relative spin-polarization ζ=(ρ↑−ρ↓)/(ρ↑+ρ↓)in the low-density or strong-interaction limit. To do so, we propose the following rregTM correlation energy per particle: ϵrregTM c=ϵ0 cf1(α)+ϵ1 cf2(α), (3) withϵ1 c=ϵLSDA 1 c+HPBE 1 andϵ0 c=(ϵLDA 0 c+HSCAN 0)Gc(ζ). Here,ϵLSDA 1 c is the PW92 local spin density approxima- tion (LSDA) correlation energy density90andϵLDA 0 cGc(ζ) is the local correlation energy density form of SCAN functional with ϵLDA 0 c=−b1c/(1+b2cr1/2 s+b3crs),Gc(ζ)= {1–2.3631 [dx(ζ)−1]}(1−ζ12), and dx(ζ)=[(1+ζ)4/3 +(1−ζ)4/3]/2. It was shown that ϵLDA 0 c is accurate for two electron systems.12,91ForHSCAN 0 andHPBE 1, we use the same forms as given for SCAN and PBE correlations [see Eq. (S14) of the sup- plementary material of Ref. 12 and Eq. (7) of Ref. 13], respectively, withβ(rs)=0.066 725 (1+0.1rs)/(1+0.1778 rs)being the density dependent second-order coefficient of Hu and Langreth.25,92 Finally, the interpolation functions f1(α)and f2(α)are cho- sen such that the correlation functional respects the following con- ditions: For (i)uniform electron gas limit ( α=1) to slowly vary- ing density limit ( α≈1),ϵrregTM c →ϵ1 c, and for (ii)α=0 (recog- nized as one-electron or two-electron singlet state), ϵrregTM c →ϵ0 c. Therefore, we consider a simple extrapolation function as f2(α) =3[g(α)]3 1+[g(α)]3+[g(α)]6and f1(α)=1−f2with g(α)=(1+γ1)α γ1+α. The parameterγ1=0.2 is obtained from the atomization energies of the AE6 molecules.93,94In this sense, the constructed correlation func- tional is empirical. For the details of the functional construction, see the supplementary material. Here, we want to mention that to remove the sensibility of the grid, we use a very fine grid in rregTM calculations. The details of the grid specification are given in the supplementary material. J. Chem. Phys. 155, 024103 (2021); doi: 10.1063/5.0051331 155, 024103-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I. Error statistics [mean absolute error (MAE in mHa) and mean absolute rel- ative error (MARE in %)] of the correlation energy per electron ( Ec/N) for several atoms and ions. Full results are reported in the supplementary material. Error SCAN TM regTM rregTM MAE 3.7 6.8 3.7 3.7 MARE 11.3 23.8 12.9 11.3 III. RESULTS AND DISCUSSIONS A. Model systems To assess the impact of the correlation for high-density limit, we compare the correlation energies per electron (given in the supplementary material) (shown in Table I) of a few atoms and ions as obtained using the Hartree–Fock analytic orbitals.95We observe that TM correlation underestimates the correlation energies. By construction, the rregTM and regTM correlation functionals agree well for larger atoms, but we observe a significant improvement of rregTM in comparison with regTM, for the He and Li isoelectronic series, due to a superior description of the one- and two-electron density regions. In the first two panels of Fig. 2, we show the XC energy of the TM, regTM, and rregTM functionals in the low-density (or strong-interaction) limit ( rs→∞) for spin-polarized (i) one- electron hydrogen (H) density and (ii) Gaussian (G) density. As expected, the regTM deviates most for all cases, while rregTM XC performs quite well. Note that the H and G density mod- els (atζ=1) are also important for atomization energies of molecules.96Next, we calculate the jellium surface XC energies ( σxc), which are reported in Fig. 2(c). The rregTM functional exhibits similar per- formance as SCAN meta-GGA. Comparison to the diffusion Monte Carlo (DMC) results97additionally reveals that the rregTM performs better than TM for the whole range 2 ≤rs≤6. We also test the functional’s performance for the harmonium atom98by calculating the relative error (RE).42,99,100This model example is quite important for strongly correlated (small values ofω) and tightly bounded (large values of ω) systems. These two regimes are often encountered in the condensed matter problems. We observe an excellent agreement for the rregTM correction energy functional for small ω. For large ω, the rregTM and SCAN correction behave similarly. The TM and regTM functionals are far from accurate, indicating the compatibility of the constructed correction with regTM exchange. Next, in Fig. 3 (left panel), we compare the SCAN and rregTM XC enhancement factors Fxc(ζ,α,s,rs)for the spin-unpolarized choice (ζ=0) and slowly varying density region ( α=1), considering the high density (or exchange-only) limit ( rs=0), the metallic den- sity case ( rs=2), and the low density case ( rs=10). For exchange- only, the rregTM is a monotonically (unbound) increasing func- tion, while SCAN ( Fx≤1.174) is tightly bound. For all rs, rregTM and SCAN are quite different, with the exception of the region 0≤s≤0.5, where they agree closely because they both recover the gradient expansions of exchange and correlation. In Fig. 3 (right panel), we show a comparison between SCAN and rregTM corre- lation energies per particle for Ne atom. Near the nucleus, which is recognized as α≈0, the rregTM recovers ϵ0 c(as SCAN), while in the core, where the density is slowly varying ( α≈1), it recovers ϵ1 c. As the PBE correlation is used in ϵ1 c, the tail (α→∞) of correlation energy density ends more rapidly for rregTM than SCAN. FIG. 2. The XC energy in the low-density or strong-interaction limit ( rs→∞) for spin-polarized one-electron (a) hydrogen density (H) and (b) Gaussian density (G). (c) The jellium surface XC energies com- pared with the reference Diffusion Monte Carlo (DMC) ones. (d) Relative error on the total energies of harmonium atoms for various values of the confinement strengthω. J. Chem. Phys. 155, 024103 (2021); doi: 10.1063/5.0051331 155, 024103-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. (Left panel) rregTM (solid lines) and SCAN (dashed lines) XC enhancement factors Fxcas a function of the reduced gradient s in the spin-unpolarized ( ζ=0), slowly varying limit ( α=1) for several values of the bulk parameter rs. (Right panel) rregTM and SCAN correlation energy per particle ϵcvs radial distance rfor Ne atom. B. Performance for solids and molecules 1. Solids First, we turn into the assessment of the constructed XC func- tional for the general purpose solid-state properties. To do so, we consider the 20 bulk solids, which include six simple metals (Li, Na, Ca, Sr, Ba, and Al), four transition metals (Cu, Rh, Pd, and TABLE II. Error statistics (MAE and MARE) for equilibrium lattice constants (LC20), bulk moduli (BM20), and cohesive energies (COH20) of a set of 20 bulk materials compiled in Ref. 46; surface energies of (111) surfaces of Cu, Ir, Pd, Pt, and Rh metals; and adsorption energies of CO molecules on these surfaces. The surface energies and adsorption energies of the SCAN functional are taken from Ref. 36. Rest of the results are calculated in this work. Test set Errors SCAN TM rregTM Lattice constants a0 MAE (mÅ) 26 32 33 MARE (%) 0.55 0.70 0.74 Bulk moduli B0 MAE (GPa) 4.5 4.0 3.1 MARE (%) 6.1 5.3 4.8 Cohesive energies ϵcoh MAE (eV/atom) 0.159 0.266 0.211 MARE (%) 4.833 8.804 6.326 Surface energy ϵsur MAE (J/m2) 0.48 0.18 0.26 MARE (%) 22.8 7.87 10.60 Adsorption energy ϵads MAE (eV) 0.51 0.32 0.17 MARE (%) 42.1 26.68 15.30Ag), five semiconductors (C, Si, Ge, SiC, and GaAs), and five ionic solids (LiF, LiCl, NaF, NaCl, and MgO). All reference geometries and reference values are taken from Ref. 46. The error statistics of all the considered solids are reported in Table II, where we report the mean absolute errors (MAEs) and mean absolute relative errors (MAREs). To start with, we consider the equilibrium lattice constant test (LC20). Note that the accurate prediction of lattice constants is of prime importance for several other solid-state and material appli- cations. The statistical analysis shows that rregTM performs similar to TM and slightly worse than SCAN. In the case of the bulk mod- ulus (BM20) test, we observe that rregTM improves over TM and SCAN functionals. For the cohesive energy (COH20) test, rregTM improves by about 0.5 eV/atom with respect to TM, still being worse by about 0.5 eV/atom than SCAN. For broader assessment, we also consider the surface energies and surface adsorption energies of CO on metal surfaces. All calcu- lations are performed for the (111) surface of Cu, Ir, Pd, Pt, and Rh. For bothϵsurandϵads, TM and rregTM are noticeably better than SCAN. Now, we turn into the structural phase transition of the semi- conducting solids and the results are reported in Table III. The TABLE III. Tabulated are the phase transition pressures ( Pt) (in GPa) of highly sym- metric phases. All values are without temperature corrections. The TM values except SiO 2are taken from Ref. 38. Solids Expt.aSCANaTM rregTM Si 12.0 14.5 3.9 10.7 Ge 10.6 11.3 6.7 7.3 SiC 100.0 74.1 52.5 63.1 GaAs 15.0 17.1 8.2 14.0 Pb 14.0 16.4 10.0 14.2 C 3.7 4.6 2.17 4.8 BN 5.0 2.8 −1.2 4.9 SiO 2 7.5 4.6 0.94 1.52 aSee Ref. 101 and all references therein. J. Chem. Phys. 155, 024103 (2021); doi: 10.1063/5.0051331 155, 024103-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp improved description of the structural phase transition pressure ( Pt) of the meta-GGA functional is associated with the order-of-limit problem of the functional.27As already shown in Ref. 38, the order- of-limit-free regTM functional improves considerably over TM in this respect. Here, further investigation of the rregTM functional shows that it slightly underestimates Ptwith respect to regTM and, overall, it improves over regTM for Si, Pb, and BN (see Ref. 38 for regTM values). Our next assessment is the structural phase transition of TiO 2, FeS 2, and MnO 2, where SCAN fails for two of these solids (TiO 2and FeS 2). The results are shown in Fig. 4. The TM functional fails to predict the MnO 2phases ( R→γ) correctly, which is identified from its order-of-limit problem. The MnO 2problem for TM is resolved by order-of-limit-free rregTM. In addition, the overestimation ten- dency of the energies of different phases of the MnO 2as obtained from regTM (see Ref. 79) is reduced by rregTM. Note also that the correct energy ordering of TM for TiO 2and FeS 2is also main- tained in the rregTM method. Overall, the rregTM functional can be considered as a promising meta-GGA functional for different challenging solid-state systems. Next, we consider the magnetic properties of the itinerant fer- romagnetic solids. It is shown in Refs. 52 and 74–77 that SCAN does not perform as a promising method for the magnetic moments of known ferromagnets. To assess the performance of rregTM, we consider the volume ( V0), bulk moduli ( B0), derivative of B′ 0, energy difference of magnetic and non-magnetic systems, and magnetic moments of bcc Fe, fcc Ni, and hcp Co. All results are reported in the supplementary material. We relax the systems corresponding to each method to calculate the properties. As observed, the volume pre- dicted by the SCAN functional is quite close to the experimental one for bcc Fe, while TM based methods perform closely to each other. For other properties also, the TM based methods show very close performance. For a clear visualization, in Fig. 5, we plot the magnetic energy ( ΔEmag=Epara−Eferro) and spin magnetizations of itinerant ferromagnetic solids. From this figure, it is clear that the magnetic moment (μB) predicted by the SCAN functional deviates most. The problem with the SCAN functional and its different resolutions are FIG. 4. Energy differences of several phases of FeS 2, TiO 2, and MnO 2poly- morphs. We do not consider zero-point energy (ZPE) corrections. FeS 2, TiO 2, and MnO 2energies are taken from Refs. 68 and 79. FIG. 5. Calculated magnetic energy ( ΔEmag=Epara−Eferro) vs spin magneti- zations of bcc Fe, hcp Co, and fcc Ni as obtained from different methods. All calculations are done by relaxing the structure in each method. also proposed.52,76We recall that the SCAN-L76shows improved performance over SCAN in predicting the magnetic moments and the problem with the SCAN method is identified from the iso-orbital indicatorα. In particular, predicting the energy differences ( ΔEmag), the TM based methods also lower the energies in comparison to SCAN. We recall that TM slightly underestimates the ΔEmagbecause of this order-of-limit problem. Turning to the results for the other itinerant metals such as Ni and Co, we observe the same tendency of results as observed for Fe. Next, we consider the binding energy curve of the Cr 2dimer and the band-structure of anti-ferromagnetic (AFM) graphene by considering the spin-symmetry breaking. Both of these cases are crit- ical examples where SCAN fails drastically.78The SCAN method underbinds the Cr 2. However, for AFM construction of graphene, it predicts spurious magnetism,78which opens the bandgaps. This wrong treatment of the SCAN functional does not follow in the case of rregTM (and TM also) methods, and we observe correct prediction of the results from Fig. 6. Moreover, the rregTM bind- ing energy curves are close to the experimental one. This particu- lar result exhibits that rregTM based methods perform better than SCAN for the graphene problem and better than both SCAN and TM for Cr 2binding energies. 2. Molecules Now, the results of different molecular test cases are also reported in Table IV. Considering the main group thermochemistry, rregTM improves over TM in most cases. Notably, the improve- ment of atomization energies from the rregTM functional over TM is noticeable. Next, for barrier heights, all methods perform simi- larly. Finally, we also consider non-covalent interactions. In all these cases, the TM and rregTM perform similarly and improves almost in all cases than SCAN. Next, we consider 96 molecular bond lengths ( re) with 86 neu- tral molecules and 10 molecular cations. This test set (T-96R) is compiled in Ref. 112. Our results of different methods for bond lengths show that SCAN is quite accurate for it, while TM and rregTM perform similarly. For all methods, the largest individual J. Chem. Phys. 155, 024103 (2021); doi: 10.1063/5.0051331 155, 024103-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6. Shown are the binding energy curves of Cr 2dimer (left panel) and band structure of spin-symmetry break graphene (right panel) as obtained from SCAN, TM, and rregTM methods. We observe similar performance from TM and rregTM for the graphene problem. TABLE IV. Mean absolute errors (MAEs) of SCAN, TM, and rregTM functionals for molecular test cases. The least MAEs are denoted in boldface values. Test set Description Reference Basis set SCAN TM rregTM Main group thermochemistry (MGT) (kcal/mol) AE6 6 atomization energies 54, 93, and 94 def2-QZVP 3.4 4.5 3.7 G2/148 148 molecular atomization energies 102 def2-QZVP 3.7 6.5 5.1 W4-11 140 molecular atomization energies 103 def2-QZVP 3.5 7.4 6.4 G21IP 21 ionization potentials 104 and 105 def2-QZVP 4.98 5.34 4.84 G21EA 21 electron affinities 104 and 105 def2-QZVPD 3.92 3.96 3.05 PA26 26 proton affinities 104, 106, and 107 def2-QZVP 3.06 3.06 3.49 TMAE 3.76 5.12 4.43 Barrier heights (BHs) (kcal/mol) HTBH38 38 hydrogen transfer barrier heights 54 and 108 def2-QZVP 7.31 7.25 7.30 NHTBH38 38 non-hydrogen transfer barrier heights 54 and 108 def2-QZVP 7.88 8.86 9.08 TMAE 7.59 8.05 8.19 Non-covalent interactions (NCIs) (kcal/mol) HB6 6 hydrogen bonds 54 and 109 def2-QZVP 0.76 0.23 0.20 DI6 6 dipole interactions 54 and 109 def2-QZVP 0.53 0.40 0.36 CT7 7 charge transfer complexes 54 and 109 def2-QZVP 2.99 2.87 2.78 PPS5 5 π−πsystem dissociation energies 54 and 109 def2-QZVP 0.72 0.74 0.87 WI7 7 weekly interacting systems 110 def2-QZVP 0.07 0.04 0.09 S22 22 non-covalent interaction test set 111 def2-QZVP 0.92 0.61 0.63 TMAE 0.99 0.81 0.82 TMAE (MGT +BH+NCI) 4.11 4.66 4.48 Bond lengths (BL) (mÅ) T-96R 96 bond lengths 112 6-311 ++G(3df,3pd) 7 12a12 Vibrational frequencies (cm−1) T-82F 82 vibrational frequencies 112 6-311 ++G(3df,3pd) 34.6 29.7a34.2 Dipole moments (D) Dipole 200 benchmark dipole moments 113 def2-QZVP 0.097 0.133 0.160 Polarizabilities (Å3) Polarizability 132 static polarizabilities 114 def2-QZVP 1.205 1.196 1.178 aReference 58. J. Chem. Phys. 155, 024103 (2021); doi: 10.1063/5.0051331 155, 024103-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp deviations are observed for Li 2(0.070 Å for SCAN, 0.045 Å for TM, and 0.051 Å for rregTM). We also calculate 82 vibrational frequen- cies (T-82F)112at the equilibrium geometries of respective function- als. We observe SCAN and rregTM perform similarly, while TM gives the least mean absolute error. Next, we calculate the dipole moments and polarizabilities of 200113and 132114benchmark test sets, respectively. For dipole moments, we consider the coupled- cluster single double triple [CCSD(T)] values as in Ref. 113. For dipole moments, the SCAN functional performs considerably bet- ter than TM and rregTM with an MAE of 0.097 D. For 132 static polarizabilities, all methods perform similarly. As in our previous study,115,116we have computed the verti- cal ionization potentials (VIPs) by considering the energy difference between the cation and neutral species [ IP=E(N−1)−E(N)] or as the opposite of the HOMO energy ( IP=−εHOMO ) using the same computational setup from Ref. 117 for all investigated meta-GGAs. For comparison, we also report the ab initio exact exchange (OEPx) and correlated OEP2-sc115,118results. We remark that for the exact DFT calculations, these two quantities shall be the same. Thus, the analysis and comparison of these two results are particularly inter- esting from the standpoint of assessment to judge the quality of an XC functional and potential.117,119–121This is reported in Fig. 7 and Table IV of the supplementary material. One can note that the VIPs obtained from the energy difference are much better than those obtained from the HOMO energy. This failure can be linked to the self-interaction error7and the wrong asymptotic behavior of the XC potential.120,122,123We see that in the case of ab initio DFT functionals, both methods give much similar results.115 Finally, we also calculate the 74 excitation energies from TM and rregTM within the linear-response time-dependent density functional theory (LR-TDDFT).124We consider the test set com- piled in Ref. 125 and later used in Ref. 126. It consists of 8 molecules with 74 total excited states, including singlet, triplet, valence, and Rydberg excited states. The full results are reported in the supplementary material. Our calculation shows that rregTM is slightly worse than TM for LR-TDDFT excitation energies. The MAEs of the singlet and triplet states are obtained as 0.47 and 0.51 eV from rregTM compared to the 0.42 and 0.48 eV obtained from TM. In Ref. 126, it is shown that TM exchange with TPSS correlation is slightly better than TM. The difference between TM, TM-TPSS, and rregTM for excitation energies comes from correla- tion energies, where TPSS correlation is more enhanced than TM and rregTM. FIG. 7. Shown are the MARE (%) of the ionization potentials and its difference from the highest occupied molecular orbital energies as obtained from different methods.IV. CONCLUSIONS To conclude, we have proposed the one-electron self- interaction-free meta-GGA correlation energy functional, compat- ible with the order-of-limit-free regTM meta-GGA exchange. The proposed rregTM functional performance is measured for general- purpose quantum chemical systems and solid-state test cases. The comparative assessment of the rregTM with TM shows that it improves most properties compared to its predecessor. Our assessment has showed that rregTM performs better than SCAN for magnetic moments of ferromagnetic systems, being also accurate for the difficult cases of Cr 2dissociation and band structure of spin-symmetry break graphene. Moreover, the rregTM is correct in predicting the structural phase stability of the challenging solid- state systems, such as FeS 2and TiO 2, for which most of the density functionals, including SCAN, fail badly. The rregTM is also energet- ically better than regTM for predicting the relative energies of the MnO 2polymorphs for which TM fails. Finally, one can conclude that this is an important step toward the construction of the accurate semilocal method based on the TM functional that has several important properties. In fact, we have proved that the rregTM functional has a good overall accuracy and can solve difficult problems such that it can be useful for many condensed matter electronic calculations. SUPPLEMENTARY MATERIAL See the supplementary material for the details of the exchange–correlation functional construction and computational details; correlation energies of atoms and ions; energy differences of different phases of FeS 2, TiO 2, and MnO 2polymorphs; ioniza- tion potentials; orbital energies; time-dependent excitation energies; structure files for the relaxed structure of CO adsorption of different metal surfaces; relaxed structure file for Fig. 5; and structure file for the right panel of Fig. 6. ACKNOWLEDGMENTS S.J. acknowledges the NISER for partial financial support. S.K.B. acknowledges the NISER for financial support. S. ´S. acknowl- edges the National Science Centre, Poland, for financial support under Grant No. 2020/37/B/ST4/02713. All simulations were per- formed in the KALINGA and NISERDFT High Performance Com- puting (HPC) Facility, NISER. S.J. and P.S. would like to thank Q-Chem, Inc. and developers for providing the source code. 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The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Exact-two-component block-localized wave function: A simple scheme for the automatic computation of relativistic ΔSCF Cite as: J. Chem. Phys. 155, 014103 (2021); doi: 10.1063/5.0054227 Submitted: 15 April 2021 •Accepted: 14 June 2021 • Published Online: 2 July 2021 Adam Grofe,1 Jiali Gao,2 and Xiaosong Li1,3,a) AFFILIATIONS 1Department of Chemistry, University of Washington, Seattle, Washington 98195, USA 2Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, Shenzhen 518055, China; Department of Chemistry and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA; and Beijing University Shenzhen Graduate School, Shenzhen 518055, China 3Pacific Northwest National Laboratory, Richland, Washington 99354, USA a)Author to whom correspondence should be addressed: xsli@uw.edu ABSTRACT Block-localized wave function is a useful method for optimizing constrained determinants. In this article, we extend the general- ized block-localized wave function technique to a relativistic two-component framework. Optimization of excited state determinants for two-component wave functions presents a unique challenge because the excited state manifold is often quite dense with degen- erate states. Furthermore, we test the degree to which certain symmetries result naturally from the ΔSCF optimization such as time- reversal symmetry and symmetry with respect to the total angular momentum operator on a series of atomic systems. Variational optimizations may often break the symmetry in order to lower the overall energy, just as unrestricted Hartree–Fock breaks spin sym- metry. Overall, we demonstrate that time-reversal symmetry is roughly maintained when using Hartree–Fock, but less so when using Kohn–Sham density functional theory. Additionally, maintaining total angular momentum symmetry appears to be system depen- dent and not guaranteed. Finally, we were able to trace the breaking of total angular momentum symmetry to the relaxation of core electrons. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054227 I. INTRODUCTION Block-localized wave function (BLW) is a technique for con- strained optimization of molecular orbitals in self-consistent field (SCF) calculations. In this method, molecular orbitals can be strictly localized within a molecular fragment or on the basis of an imposed symmetry,1,2providing a variational procedure to partition the total energy into specific components in energy-decomposition analysis (EDA).3–5BLW-based EDA has been widely used to understand the origin of Lewis acid–base bonding, cation– πinteractions, and transition metal-ligand charge-transfer effects.6Recently, we intro- duced a generalized approach, namely, generalized block-localized wave function (gBLW), to optimize a determinant wave function that can be specifically constrained in molecular orbital space,7,8 which further expands the scope of its application in EDA9–11as well as local and charge-transfer excited states in molecular complexes.12 The generalized block-localized wave function (gBLW) method can be used to optimize configuration states to be used in nonorthogonal configuration interaction, such as that used in effec- tive valence-bond (VB) theory13and multistate density functional theory (MSDFT),14or approximations to excited states.8Effective VB type of diabatic states can be useful to model chemical reac- tions such as the S n2 reaction,15,16charge-transfer reactions,17–21 and excited energy transfer processes in photoreceptor proteins.22,23 In particular, Mo and co-workers investigated hyperconjugation,24 aromaticity,25and the anomeric effect.26We have employed gBLW to optimize electronic excited states in the ΔSCF framework.8The transformation of molecular orbitals into specific blocks in gBLW avoids the risk of variational collapse, and the computational cost J. Chem. Phys. 155, 014103 (2021); doi: 10.1063/5.0054227 155, 014103-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp is similar to that of the ground state SCF optimization by a direct inversion of the iterative subspace (DIIS) scheme that takes into account both block-localized and the full system densities; the method is complementary to other approaches such as the max- imum overlap method27–30or time-dependent density functional theory.30–33 In this work, we extended the gBLW approach to the two- component relativistic electronic structure framework. In the two- component theory, the relaxation of spin axis allows for the inclu- sion of spin–orbit couplings in a variational approach. However, it also creates a challenging scenario for the gBLW as orbitals that belong to the same term (e.g.,2P3/2and2D5/2) can lead to variational collapse of the gBLW due to their spatial overlap. Additionally, since the two-component excited state manifold is more dense, it would be more difficult to guarantee that all of the states within a degenerate manifold be adequately represented. In this regard, it is necessary to construct all configuration states if this method is to be used to generate the determinant basis for nonorthogonal configuration interaction. This article is organized as follows: In Sec. II, we introduce exact-two-component (X2C) wave functions along with the gBLW algorithm. We will explore strategies to allow the gBLW to obtain all excited states that are split due to spin–orbit coupling in the ΔSCF process. Then, in Sec. IV, we apply our method to atomic systems to explore their zero-field-splitting and the degree to which ΔSCF can maintain Kramers’ and total angular momentum symme- tries naturally (i.e., without resorting to extraordinary algorithmic measures). II. THEORETICAL BACKGROUND Throughout this work, we use the following notation: ●μ,νindex atomic spinor orbitals (AO’s); ●i,jindex molecular orbitals (MO’s); ●A,Bindex blocks/subspaces. A. X2C Hamiltonian In the exact-two-component (X2C) method,34–53the electronic and positronic components of the four-component Dirac equation are decoupled by a unitary transformation Uthat block-diagonalizes the four-component Hamiltonian, H=(V T T W −T), (1) U†HU =(H+02 02H−). (2) In Eq. (1), the matrix form of the Dirac four-component Hamil- tonian is written in the restricted-kinetically balanced (RKB) con- dition.54,55Usually, only the two-component Hamiltonian corre- sponding to electronic solutions, H+, is needed for determining the properties of chemical interest. In Eq. (1), VandTare the non- relativistic potential energy and kinetic energy matrices, respectively. Wμν=1 4c2⟨χμ∣(σ⋅ˆp)V(σ⋅ˆp)∣χν⟩gives rise to relativistic corrections and spin couplings in an atomic/molecular system. Here, Vis the nuclear attraction, σis a vector of Pauli spin matrices, and ˆpis the momentum operator.The solutions of the four-component RKB Dirac Hamilto- nian56in Eq. (1) are a set of bi-spinor (four-component) molecular orbitals with the large and small components ( ψL iandψS i) expanded in a set of atomic spinors centered on different nuclei ( RA), ψL i=∑ A∑ μcL,A iμχA μ(r−RA), (3) ψS i=∑ A∑ μcS,A iμχA μ(r−RA), (4) whereχA μ(r−RA)is a basis function centered on nucleus A. The orbital coefficients that correspond to positive and negative energy solutions can be written in a matrix form, C=(CL,+CL,− CS,+CS,−). (5) In X2C, the transformation matrix takes the form U=(12−Y† Y 12)((12+Y†Y)−1/202 02 (12+YY†)−1/2). (6) In Eq. (6), the matrix Yis calculated from the orbital coefficients as Y=CS,+(CL,+)−1. (7) In this work, we employ the one-electron X2C approach, which is a one-step procedure to construct a transformation matrix through the diagonalization of the one-electron four-component core Hamil- tonian. Thus, the one-electron spin–orbit coupling is included in the Hamiltonian in the self-consistent field process. However, since the two-electron spin–orbit terms contribute with an opposite sign to the one-electron spin–orbit terms, we used the Boettger factors to scale the one-electron spin–orbit effect to semi-empirically treat the two-electron spin–orbit terms.57 B. Generalized block-localized wave function In the generalized block-localized wave function (X2C-gBLW) approach, we first partition the basis functions into M subspaces according to specific constraints of interest in a given appli- cation; the basis functions can be the primitive atomic spinor orbitals or a linear combination thereof. The molecular orbitals are defined as ψA,i=∑ μνCA,μiTA,νμχν, (8) where CAis the coefficient matrix for the Ath subspace and TAis a rectangular transformation matrix that enforces the subspace local- ization.8In the X2C-gBLW method, bases {χμ}are atomic spinors and the resulting CAis subject to the “picture change” or the X2C transformation from the four-component orbitals. As in the con- ventional BLW approach, molecular orbitals between different sub- space blocks in the X2C-gBLW framework can be nonorthogonal, although the atomic spinors can be fully shared by different blocks in the latter method. Thus, the molecular orbitals may no longer be block-localized in the coefficient matrix in the atomic spinor basis. J. Chem. Phys. 155, 014103 (2021); doi: 10.1063/5.0054227 155, 014103-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Importantly, the separation into different subspaces permits us to place different constraints on electronic configurations. For a prob- lem with Mnumber of subspaces, the gBLW approach leads to M eigenvalue problems that can be solved separately. The molecular spinor coefficients are determined by solving a modified Roothaan–Hall equation for each subspace, ˜FA˜CA=˜SA˜CA˜ϵA, (9) where ˜FAand ˜SAare the projected Fock and overlap matrices, and ˜ϵAis the diagonal matrix of energy eigenvalues. The projected Fock and overlap matrices are computed using ˜FA=T† AP† AFPATA, (10) ˜SA=T† ASPATA, (11) where Fis the Fock matrix in the full basis space, Sis the overlap matrix, and PAis the projection matrix of subspace A. The projection matrix partitions the full system into subspaces so that the coefficients can be solved separately but also couples the subspaces together both in terms of energy and orthogonality. This results in a wave function that is a variational minimum sub- ject to the constraints of the optimization. The projection matrix is defined as PA=1−DS+DAS, (12) where Dis the full density matrix and DAis the density matrix for subspace Athat has been transformed from the subspace basis into the atomic orbital basis, DA=TA˜DAT† A. (13) Because there is no guarantee that the molecular orbitals will be orthonormal, the density matrices are evaluated using the nonorthogonal formula, D=C[C†SC]−1 C†, (14) ˜DA=˜CA[˜C† ASA˜CA]−1˜C† A. (15) Here, SAis the overlap matrix of the subspace basis ( SA=T† ASTA). The molecular orbitals for the total system ( C) can be assembled by transforming the subspace molecular orbitals ( CA) into the atomic orbital basis and concatenating them together. C. Automated X2C- ΔSCF The X2C-gBLW method presented in Sec. II B allows for the facile optimization of determinants corresponding to electronically excited configurations, which, in certain applications, may be con- sidered as approximations to localized diabatic excited states. When molecular orbitals of a reference state are used as the subspace transformation matrix, it is possible to isolate a given orbital or a group of orbitals into a subspace that is not mixed with orbitals inother blocks in X2C-gBLW optimization. Consequently, this tech- nique is particularly powerful in resolving degenerate states. For instance, spin–orbit coupling splits the six-fold degenerate2Pterm into a two-fold2P1/2and four-fold2P3/2levels. With the X2C-gBLW approach, spin–orbit coupling is included variationally in both the molecular orbitals or the reference state and the X2C-gBLW opti- mized orbitals. We can exclude spinor molecular orbitals that are not the subject of interest but may cause variational collapse due to mixing in the same symmetry group. Thus, the excited or degen- erate state determinant becomes a minimum in the constrained space.8 Consider the sodium atom frontier excitations (Fig. 1). The electrons in the 1 s, 2s, and 2 porbitals comprise the core electrons, while the electron in the 3 sorbital is the excited electron. The first excitation roughly corresponds to a spin-flip excitation of the 3 s electron to yield an energy degenerate doublet state. Then, 3 s→3p excitations give rise to the2Pterm. We define two subspaces for the core electrons and excited electron, respectively. The first subspace contains all ten core electrons and includes all molecular orbitals except the target (the destination of the excited electron) orbital that defines an excited configuration. The second subspace consists of only one electron, the orbital occupied by the excited electron (tar- get orbital), and all unoccupied molecular orbitals that are higher in energy than the target orbital. To prevent the admixture of degen- erate orbitals (e.g., the three 3p orbitals of the sodium atom), only the target orbital where the excited electron resides is included in the second subspace, whereas the other degenerate orbitals are excluded. Therefore, the subspace partition is dependent on the specific excited FIG. 1. A scheme depicting the coefficient matrices in the reference molecular orbital basis for the excited state determinants that defines the2Sspin-flip excited state along with two of the six states of the2Pterm. For each 3 s→3pexcitation, the electrons and molecular orbitals are partitioned into two subspaces [e.g., core (blue) and excited (orange)]. The basis functions that are occupied in the refer- ence configuration are displayed in the solid color, while virtual orbitals are given in the striped color. In gBLW optimization, orbitals in each subspace can admix to yield a variational state. The gray areas indicate basis functions excluded in a par- ticular subspace. Note that the lowest unoccupied basis orbital in the blue region corresponds to the 3 sorbital. J. Chem. Phys. 155, 014103 (2021); doi: 10.1063/5.0054227 155, 014103-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp configurations. Since orbitals with lower energy than the target orbital of the excited configuration are excluded from the second subspace, optimization collapse to the ground state configuration is prevented.8A schematic representation of the coefficient matrices for each of the subspaces is given in Fig. 1 for the first few excited determinants in sodium. We used an arbitrary cutoff of 0.01 hartree of the orbital energy to determine degeneracy. III. COMPUTATIONAL DETAILS The procedure described above is a form of ΔSCF algorithm that has been augmented with the X2C-gBLW Hamiltonian and automated for convenience of application in the Chronus Quantum software package.58,59A series of atomic systems were used to test the algorithm. Calculations on sodium were performed with the Sap- poro quadruple zeta basis set with diffuse functions.60Along with Hartree–Fock (wave function theory),61the BHandH62and PBE063 density functionals were used. For group 13 atoms (gallium, indium, and thalium), the Sapporo triple zeta basis set was used along with HF,61BHandH,62B3LYP,64and PBE0.63For the scandium 2 +cation (Sc2+), a series of basis sets were used to test the effect that the basis set has on the optimization. The following basis sets were used: 6-31G,656-31G(D),66cc-pVDZ, cc-pVTZ, cc-pVQZ,67Sapporo- DZP, Sapporo-TZP, and Sapporo-QZP.60 For all of the atomic systems, a positively charged state was used to define the transformation matrix in Eq. (8) to yield the molecular orbital basis for partition into the core and excited con- figuration subspaces. For sodium, gallium, indium, and thalium, the +1 cationic state was used, and for scandium, the 3 +cation was used. For these states, the core electrons are closed shells. Since our goal is to optimize a set of degenerate states, we found that the molecular orbital basis from the closed-shell cation calcula- tions provides a good description, whereas open-shell optimiza- tion of the atoms results in orbitals biased toward the ground state. IV. RESULTS AND DISCUSSION In this work, we focus on analyzing the quality of X2C-gBLW on computing atomic zero-field splittings and Kramers’ (time- reversal) symmetry recovering, compared to experimental and rel- ativistic MCSCF results. A. Sodium One of the simplest cases is the sodium atom, which has a small but measurable relativistic effect. The molecular orbital dia- gram is given in Fig. 1. There is a single electron in the 3 sorbital that gives rise to the2S1/2doubly degenerate ground state. The2Pmani- fold, from the 3 s→3pexcitation, is split into2P1/2and2P3/2states. The2P1/2and2P3/2states have two-fold and four-fold degeneracies, respectively. The computed energies of each of the states are given in Table I. Overall, X2C-gBLW optimization of the block-localized deter- minant wave functions (denoted by HF) yields the corresponding degenerate states for each of the term symbol groups. However, we did not observe the same using density functional theory (DFT) with block-localized Kohn–Sham orbitals except for the2S1/2states,TABLE I. Computed X2C-gBLW state energies (eV) for the sodium atom. These calculations were performed with the Sapporo quintuple zeta basis set.60 State HF BHandH PBE0 Exp.a 2S1/20.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2P1/21.9814 2.1433 2.03722.10231.9814 2.1440 2.0375 2P3/21.9834 2.1455 2.0395 2.10441.9834 2.1457 2.0403 1.9834 2.1569 2.0678 1.9834 2.1569 2.0678 MUEb0.1209 0.0450 0.0554 aReference 68. bMean unsigned error. which are degenerate for all methods. For the2P1/2states, we observe two states that are roughly degenerate with an energy difference of 0.0007 and 0.0003 eV for BHandH and PBE0, respectively. Mean- while, for the2P3/2states we observed that the two highest energy states are degenerate, but these are not degenerate with the low- est energy states. Thus, the2P3/2manifold is split into two sets of Kramers pairs. The Kramers pairs are close in energy, which sug- gests that the time-reversal symmetry is roughly maintained. The states within the2P3/2atomic term symbol are different compo- nents of the total angular momentum operator ( J) with values of −3/2,−1/2, 1/2, and 3/2 h. The 3/2 and −3/2 components along with the 1/2 and −1/2 components are connected by time-reversal symmetry, respectively. Given the absence of a magnetic field, these components are completely degenerate for the exact wave func- tion. In this article, we will refer the symmetry of the total angu- lar momentum operator as J-symmetry. Overall, this suggests that the time-reversal symmetry is roughly maintained, but J-symmetry is not. A simple method for quantifying the J-symmetry breaking is to compute the energy range within each atomic term symbol. In Fig. 2, we show the range (the difference between highest and low- est energies) within the2Patomic term symbol in a logarithmic scale. In wave function theory, the range is on the order of 10−6eV, whereas it increases to 10−2–10−3eV using DFT. Overall, we observe a smaller deviation in energy degeneracy for the2P1/2states than that for the2P3/2states for all methods. Since BHandH and PBE0 include different (50% vs 25%) Hartree–Fock exchange, the obser- vation of roughly similar performance suggests that the amount of Hartree–Fock exchange does not have a significant impact on the results. In comparison with the experimental excitation energies, there is a smaller mean unsigned error (MUE) for all methods with DFT than that from wave function theory (WFT) calculations. Inclusion of dynamic correlation in X2C-gBLW calculations roughly reduces the MUE by two third. However, the zero-field splitting between 2P1/2and2P3/2exhibits a different trend. The energy difference between the2P1/2and2P3/2states is 2.1 meV experimentally. For comparison, our computational results are 2.0, 7.6, and 16.5 meV for WFT, DFT/BHandH, and DFT/PBE0, respectively. The ener- gies for each term symbol are obtained by averaging over energies J. Chem. Phys. 155, 014103 (2021); doi: 10.1063/5.0054227 155, 014103-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. The range within degenerate states for sodium using several methods to optimize the determinants. The Sapporo-QZP basis set with diffuse functions was used. within each term group. It is surprising that the energy splitting due to spin–orbital coupling is significantly overestimated using DFT. B. Group 13 fine structure splitting We computed the2Patomic transitions in gallium, indium, and thalium. These are all group 13 elements with a single electron in the set of porbitals. As with the2Pstates of sodium, spin–orbit coupling splits the six configurations into two-fold (2P1/2) and four-fold (2P3/2) degenerate states. Spin–orbit coupling increases as the atomic number increases, resulting in greater energy differ- ences between the two groups of states. The results are given in Table II. We observe very good degeneracy among the2P1/2states for all methods, which suggests that time-reversal symmetry is maintained. Interestingly, the time-reversal symmetry is significantly better for these atoms when using DFT with an energy difference on the order of 10−5eV. However, similar to sodium, we observe the2P3/2states split roughly into Kramers pairs for all methods. The energy ranges in a logarithmic scale for the2P3/2states using each method and atom are plotted in Fig. 3. Overall, we do not observe a clear trend in breaking of energy degeneracy with respect to the atom number, and the energy range goes from 10−1to 10−3. BHandH and B3lYP appear to give the smallest range for all of the methods, which contrasts with the sodium results, where Hartree–Fock displays the smallest range. To understand the origin that causes the deviation from energy degeneracy, we optimized the determinants with a frozen core. Here, only the valence porbital is allowed to relax, which is the orbital that contains the excited electron. We use the energy range of the 2P3/2states as a metric for breaking J-symmetry, and the results are given in Table III. We observe no breaking of J-symmetry for the frozen core. Then, we allowed core electrons with spe- cific angular momenta to relax individually from the frozen state. Thus, each set of optimizations begins from the reference state.TABLE II. Computed X2C-gBLW and experimental frequencies (eV) for group 13 elements. The calculations were performed with the Sapporo triple zeta basis set.60 Atom State HF BHandH B3LYP PBE0 Exp.a Ga2P1/20.000 00 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 2P3/20.095 56 0.116 81 0.116 11 0.114 23 0.1020.095 58 0.116 81 0.116 11 0.114 25 0.102 77 0.124 07 0.118 24 0.105 67 0.102 74 0.124 05 0.118 27 0.105 67 In2P1/20.000 00 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 2P3/20.243 96 0.300 23 0.315 79 0.293 56 0.2740.243 98 0.300 24 0.315 78 0.293 56 0.238 10 0.291 40 0.284 73 0.274 51 0.238 11 0.291 40 0.284 75 0.274 51 Tl2P1/20.000 00 0.000 00 0.000 00 0.000 00 0.000 00 0.000 01 0.000 00 0.000 01 2P3/20.784 99 0.879 75 0.855 34 0.872 62 0.9660.784 99 0.879 74 0.855 35 0.872 61 0.817 68 0.877 90 0.856 82 0.855 61 0.817 69 0.877 90 0.856 82 0.855 60 aReference 69. Here, we observe that allowing the sorbitals to relax causes the largest breaking of J-symmetry, followed by the dorbitals. Allow- ing the core porbitals to relax displayed the lowest breaking of symmetry. Kasper et al. recently showed that the state averaged com- plete active space self-consistent field (SA-CASSCF) is able to main- tain both time-reversal symmetry and J-symmetry to at least 10−4 eV in the energy range.70In this case, the core orbitals share a single set of orbitals between all states. Since ΔSCF has a dif- ferent set of core orbitals for each state, it is possible that the FIG. 3. The logarithm of the range within the2P3/2states for group 13 atoms using several methods to optimize the determinants. J. Chem. Phys. 155, 014103 (2021); doi: 10.1063/5.0054227 155, 014103-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III. Energy range of2P3/2states when different parts of the core electrons are allowed to relax. Hartree–Fock was used to optimize the determinants. The valence p electron was allowed to relax in all determinants. Relaxed core Atom Range NoneGa 0.000 00 In 0.000 00 Tl 0.000 00 sGa 0.007 28 In 0.041 63 Tl 0.030 89 pGa 0.000 00 In 0.000 01 Tl 0.000 01 dGa 0.000 14 In 0.001 42 Tl 0.002 52 algorithm breaks J-symmetry to reach a lower variational minimum in the same way that the unrestricted Hartree–Fock breaks spin symmetry. In order to compare with experiment, the2P3/2frequencies of Table II are averaged. The results are given in Table IV. Over- all, the mean unsigned error compared to experiment ranged from 0.067 to 0.040 for Hartree–Fock and PBE0, respectively. Addition- ally, the DFT methods show a lower mean unsigned error compared to Hartree–Fock. Hu et al. showed that dynamic correlation reduces the mean unsigned error using multireference configuration interac- tion singles and doubles (MRCISD).51Overall, using DFT reduced the mean unsigned error but not as much as was observed with MRCISD. However, X2C-gBLW with DFT is significantly cheaper than MRCISD. C. Scandium We tested X2C-gBLW on the scandium 2 +cation. Here, there is a single electron in the d-manifold, and spin–orbit coupling splits thed-manifold into2D3/2and2D5/2states.2D3/2are four-fold degen- erate and the2D5/2are six-fold degenerate in the absence of a mag- netic field. Overall, this is a more difficult system since the level of degeneracy is higher. Additionally, we computed the excitation to the 4 smanifold, resulting in doubly degenerate2S1/2excited states. On this system, we tested the effect due to the size of basis sets on the optimization. The X2C-gBLW energies using Hartree–Fock and theTABLE V. The X2C-gBLW and experimental energies (eV) for the Sc2+cation using Hartree–Fock and the Sapporo basis sets. State DZP TZP QZP Exp.a 2D3/20.0000 0.0000 0.0000 0.0002 0.0021 0.0000 0.0070 0.0068 0.0143 0.0080 0.0070 0.0143 2D5/20.0218 0.0184 0.0174 0.02450.0220 0.0217 0.0199 0.0261 0.0316 0.0360 0.0262 0.0338 0.0366 0.0340 0.0347 0.0691 0.0341 0.0355 0.0703 2S1/23.0428 2.8849 2.92993.16643.0428 2.8849 2.9299 aReference 69. Sapporo bases are given in Table V. The remaining frequencies not displayed in Table V are provided in the supplementary material. Overall, X2C-gBLW yielded two-fold degeneracy for all2S1/2 excited states regardless of the basis set following time-reversal symmetry. Meanwhile, the2Dstates roughly optimized into Kramers pairs. Interestingly, for the2Dmanifolds, as the size of the basis set increases, the energy difference between the sets of Kramers pairs within each term symbol also increases. As the size of the basis set increases, so does the variational free- dom, and the forces that lead to J-symmetry breaking are only encouraged. We expanded this survey to include other basis set fami- lies, such as the Pople and correlation consistent basis sets. In Fig. 4, we present the energy range in a logarithmic scale for each of the2Dterm symbols and compute the same using den- sity functional theory. The correlation consistent basis set (using Hartree–Fock) displayed similar performance to the Sapporo basis sets, and we similarly observed the same trends in the energy range within each atomic term symbol as in Table V. For both the correlation consistent and Sapporo basis set families, the2D3/2 states usually have smaller ranges than the2D5/2states. How- ever, for the Pople basis set, the trends are less obvious. With regards to DFT, BHandH displayed similar trends to HF in that as the basis set gets larger, so does the energy range, but there is no clear trend using the PBE0 functional. However, the TABLE IV. Average X2C-gBLW frequencies for the2P1/2to2P3/2transition compared to X2C-SA-CASSCF(3,8), X2C-MRCISD, and experiment. The X2C-gBLW energies were computed with the Sapporo triple zeta basis,60while the SA-CASSCF and MRCISD calculations were computed with the X2C-TZPAll-2c basis.71 Atom HF BHandH B3LYP PBE0 SA-CASSCFaMRCISDaExp.b Ga 0.099 0.120 0.117 0.110 0.090 0.097 0.102 In 0.241 0.296 0.300 0.284 0.239 0.254 0.274 Tl 0.801 0.879 0.856 0.864 0.863 0.903 0.966 MUE 0.067 0.042 0.050 0.040 0.048 0.029 ⋅⋅⋅ aReference 51. bReference 69. J. Chem. Phys. 155, 014103 (2021); doi: 10.1063/5.0054227 155, 014103-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. The range of energy spread within degenerate states for Sc2+using several methods and basis sets to optimize the determinants. energy range for PBE0 is generally larger than Hartree–Fock or BHandH. To understand where the J-symmetry breaking originates, we froze the core electrons and only optimized the delectron. Then, we relaxed core electrons of the same angular momenta to show which set of orbitals causes a greater change in the energy range within the same J-symmetry. The results are presented in Table VI. Consistent with findings in group 13 elements, we observed exact degeneracy when the core orbitals are frozen. Additionally, the energy range increased slightly when the sorbitals are relaxed. However, relax- ation of the porbitals produces a significantly larger increase in the energy range. To further understand this, we optimized the entire core except for the p manifold directly beneath the dorbital (i.e., the 2 pmani- fold). Here, the energy range was only slightly perturbed relative to the frozen-core case. This small change suggests that the largest con- tribution to the splitting of the Kramers pairs is due to polarization of the porbital manifold. The effect of coupling between basis functions of differ- ent angular momenta was tested by block-localizing the basis TABLE VI. The energy range (eV) for the2Dstates of the Sc2+cation when differ- ent angular momenta of the core were allowed to relax using Hartree–Fock and the Sapporo triple zeta basis set. Here, the d electron was allowed to relax in all states. Relaxed core State Range None2D3/2 0.0000 2D5/2 0.0000 s2D3/2 0.0001 2D5/2 0.0014 p2D3/2 0.0182 2D5/2 0.0298 s, 1p2D3/2 0.0002 2D5/2 0.0015functions with respect to angular momenta. Thus, there is a block fors,p, and dfunctions, respectively. The results are presented in Table VII. Using Hartree–Fock, both the J-symmetry and time- reversal symmetry are significantly improved. Overall, this sug- gests that the coupling between basis functions of different angu- lar momenta is the primary cause for the breaking of J-symmetry and, consequently, time-reversal symmetry. However, this is not observed with the density functional theory results, which show little improvement. Additionally, the BHandH functional main- tains symmetry significantly better than PBE0, perhaps due to the larger amount of Hartree–Fock exchange. Overall, this suggests that the nonlinear nature of the exchange correlation functional makes it more difficult to obtain correct symmetry even in the best of circumstances. TABLE VII. Sc2+optimized states where the coupling between basis functions of different angular momenta is prevented. Calculations were performed with the Sap- poro triple zeta basis set. BHandH and PBE0 energies are ordered according to the corresponding HF states. HF BhandH PBE0 Exp.a 2D3/20.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0011 0.0163 0.0001 0.0011 0.0162 2D5/20.0263 0.0350 0.0572 0.02450.0263 0.0349 0.0565 0.0265 0.0348 0.0148 0.0265 0.0348 0.0155 0.0267 0.0110 0.0227 0.0267 0.0113 0.0228 2S1/22.8047 3.1346 3.40203.16642.8047 3.1346 3.4020 Range2D5/2 0.0004 0.0241 0.0424 aReference 69. J. Chem. Phys. 155, 014103 (2021); doi: 10.1063/5.0054227 155, 014103-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VIII. Mean unsigned error (eV) for Sc2+with respect to the basis set. Reference 69 was used as the reference energy. HF BhandH PBE0 6-31G 0.0489 0.1355 0.2141 6-31G(d) 0.0569 0.1380 0.1946 cc-pVDZ 0.0167 0.0634 0.1127 cc-pVTZ 0.0519 0.0133 0.0645 cc-pVQZ 0.0483 0.0143 0.0640 Sapporo-DZP 0.0241 0.0531 0.0943 Sapporo-TZP 0.0521 0.0062 0.0748 Sapporo-QZP 0.0523 0.0114 0.0758 Furthermore, because these results are constrained, we are able to make a one-to-one comparison between the states using various methods. The DFT energies in Table VII are listed in the order of corresponding states to Hartree–Fock. Particularly, for PBE0, there is a2D5/2Kramers’ pair that is lower in energy than the high energy 2D3/2Kramers’ pair. Overall, this suggests that exchange correla- tion functional approximations with low amounts of Hartree–Fock exchange lack the precision to adequately differentiate the2D3/2and 2D5/2states. Similar to spin symmetry in the non-relativistic case, time- reversal and J-symmetry of the wave function can be con- strained from the beginning72in an analogous way to restricted open-shell Hartree–Fock (i.e., Kramers-restricted). While there are merits to both approaches, the Kramer-unrestricted formal- ism (used in this article) yields the variational minimum and Kramers-restricted methods maintain the symmetry of the wave function. In Table VIII, we compare the mean unsigned error across basis sets and methods using the unconstrained optimizations. Over- all, there is no systematic dependence of the basis set on relative energies. Additionally, the BHandH and PBE0 functionals appeared to perform best when using the correlation consistent and Sap- poro bases, while Hartree–Fock yielded roughly equal errors for all basis sets except for cc-pVDZ. In this data set, it was not clear that DFT performed better than Hartree–Fock. Examining the data further, we found that the largest errors in all methods were for the2S1/2states. We found that this frequency was overestimated using DFT for all basis sets (these frequencies are provided in the supplementary material). Meanwhile, for HF, the Pople basis sets overestimated the frequency, while the Sapporo and correlation con- sistent bases tended to underestimate the frequency. The Sapporo- DZP cc-pVDZ basis sets yielded the lowest error for the2S1/2states, which were also the basis sets where the mean unsigned error was the lowest. V. CONCLUSIONS In this article, we introduced an integration of the general- ized block-localized wave function (gBLW) method within the rel- ativistic X2C framework. We carried out X2C-gBLW calculations on a number of atomic systems consisting of a single unpaired elec- tron. Using the gBLW algorithm, it is possible to constrain orbital mixing within a group of generalized orbital (molecular orbital) basis functions. By excluding lower energy orbitals in the excitedblock, we showed that variational collapse to the ground state can be voided. Furthermore, the specific partition scheme of subspace construction allows all of the complementary configurations within a degenerate manifold to be fully optimized. Overall, the computed excitation energies have mean unsigned errors of about 0.1 and 0.04 eV using Hartree–Fock and DFT, respectively. Furthermore, we observed that the optimized orbitals within a J-symmetry group can roughly be grouped into Kramers pairs that yield a range of energies instead of exactly degenerate states. This suggests that the total angular momentum symmetry is not maintained. The self- interaction error73–76is a common deficiency to all density func- tional approximations, and X2C-gBLW does not limit or remove that. Because the self-interaction error is difficult to analyze, it is unclear how the self-interaction error could be affecting the results. However, the energy spread from degeneracy for all the systems that have been examined in this work is an order of magnitude smaller than the energy splitting due to spin–orbit coupling. We found that the largest contribution to the splitting of the Kramers pairs within an atomic term symbol is due to the relaxation of the core electrons in the orbital manifold directly beneath the unpaired electron of interest. Investigations that employ the constrained excited configurations as the basis state in nonorthogonal config- uration interaction calculations will be reported in a forthcoming study. SUPPLEMENTARY MATERIAL See the supplementary material for the ΔSCF frequencies of Sc2+using Hartree–Fock, BHandH, and PBE0 with a variety of basis sets. ACKNOWLEDGMENTS X.L. acknowledges support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences, in the Heavy- Element Chemistry program (Grant No. DE-SC0021100), for the development of relativistic electronic structure methods. The development of excited state methods was supported by the Computational Chemical Sciences (CCS) Program of the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division in the Center for Scalable and Predictive methods for Excitations and Cor- related phenomena (SPEC) at the Pacific Northwest National Labo- ratory. The development of the open source software package was supported by the U.S. National Science Foundation (Grant Nos. OAC-1663636 and CHE-1856210). Work carried out at the Shen- zhen Bay Laboratory was supported by a grant from the Shenzhen Municipal Science and Technology Innovation Commission (Grant No. KQTD2017-0330155106581 to J.G.). 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The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Spectra and nature of the electronic states of [1]Benzothieno[3,2-b][1]benzothiophene (BTBT): Single crystal and the aggregates Cite as: J. Chem. Phys. 155, 034504 (2021); doi: 10.1063/5.0057202 Submitted: 18 May 2021 •Accepted: 30 June 2021 • Published Online: 16 July 2021 Irena Deperasi ´nska, Marzena Banasiewicz, Paweł Gawry ´s, Olaf Morawski, Joanna Olas, and Boleslaw Kozankiewicza) AFFILIATIONS Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland a)Author to whom correspondence should be addressed: kozank@ifpan.edu.pl ABSTRACT Absorption, fluorescence, and phosphorescence spectra of single crystals of [1]benzothieno[3,2-b][1]benzothiophene (BTBT) and BTBT dispersed in frozen n-nonane, n-hexadecane, and dichloromethane matrices were studied at 5 K. Observation of a new absorption band and related changes in the fluorescence to phosphorescence intensity ratio, when the concentration of BTBT in the matrix increased above 10−4M, indicated the presence of BTBT aggregates. Quantum-chemistry calculations performed for the simplest aggregate, isolated dimer, showed that its structure is similar to the “herringbone” element in the BTBT crystal unit cell and the lowest electronic excited singlet state of the dimer has the intermolecular charge-transfer character. A qualitatively different nature of this state in dimers and in crys- tals, when compared with the situation in BTBT monomer [locally excited (LE) state], is associated with a decrease in the intersystem crossing yield. The structured vibronic structure of phosphorescence spectra in the studied systems indicated LE character of the triplet states. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0057202 INTRODUCTION Organic semiconductors have been receiving widespread atten- tion owing to their foreseen application in future electronics.1 Among various organic semiconductors based on thienoacenes, a family of compounds consisting of benzene and thiophene rings arranged in the linear fashion are among the most frequently studied. The reason is their usually convenient packing, advanta- geous position of the HOMO levels, and high thermal stability. An example of the family are derivatives of [1]benzothieno[3,2- b][1]benzothiophene (BTBT), which are characterized by excellent environmental stability and high mobility of holes.2–7Performance of possible devices is crucially dependent on motifs of the solid state packing; thus, the main effort of the works has been concentrated on developing new derivatives of BTBT and of the procedures of deposition of their films.8,9It is noteworthy that in BTBT and in bigger derivatives of thienoacenes, a convenient packing between the molecules is the result of concomitant π-stacking forces between aromatic moieties and lipophilic interactions of the long alkyl chainsat the terminal positions.10–12Recently, significant attention has been paid to the alkoxy functionalized BTBT molecules.13–15Prop- erties of BTBT derivatives may remain convenient in both thin films and in the crystal phase even when BTBT is substituted with bulky groups, such as t-butyl.16–18Substitution of BTBT with alkylam- onido groups results in a material useable as a component in organic photovoltaic devices.19Unsubstituted BTBT crystals were also tested in the organic field effect transistor (OFET) configuration, and the device exhibited a mobility of 0.032 cm2V−1s−1.20Most recently, two component crystals of BTBT-TCNQ were also characterized.21 In contrast to the studies of charge transport properties,22,23 much less effort has been devoted to understanding of the photo- physical properties of parent BTBT. Absorption and fluorescence studies of BTBT dissolved in several solvents identifies low fluo- rescence quantum yields and specific solute–solvent interactions.24 BTBT crystallizes in the herringbone pattern, and the absorption and fluorescence spectra of the crystal are red shifted with respect to the spectra of this compound in solutions.7All of the previous opti- cal studies were performed at room temperature and the observed J. Chem. Phys. 155, 034504 (2021); doi: 10.1063/5.0057202 155, 034504-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp spectra were broad, lacking any well resolved vibrational structure. Therefore, our aim was to perform experiments at cryogenic tem- peratures with the aim of understanding the origin of the lowest excited electronic states of BTBT and collecting information about their photophysical properties. Our spectroscopic studies of single crystals of BTBT, performed in the temperature range between 5 and 300 K, were supported by the study of BTBT dispersed in frozen rigid matrices of n-nonane, n-hexadecane, and dichloromethane. Experimental studies were supported with the aid of quantum-chemistry calculations. EXPERIMENTAL Sample preparation BTBT was synthesized according to the procedure described in the supplementary material. The obtained material was then zone- refined (about 100 passes) until the pure solid BTBT had white color. Single crystals of BTBT were grown by slow sublimation under ∼0.2 bar of argon gas at temperature of ∼200○C (some 10–20○below the melting point of BTBT, 217–218○C). The obtained high quality single crystals, used in our experiments, were in the form of thin “plates,” few tens of micrometers thick and surface area of about 10 mm2. Crystal “plates” were inserted into an optical cryostat whose temperature was controlled between 5 and 300 K. Samples of BTBT dispersed in low temperature matrices were prepared as follows: The solutions of BTBT in liquid solvents (con- centrations 10–5– 5×10–4M) were poured into a homemade cuvette, which in the case of absorption studies consisted of two quartz win- dows separated by a 1.5 mm Teflon ring or in the case of emission studies was in the form of a small cylinder-shaped fused silica glass (inner diameter 4 mm). The cuvette, filled at room temperature with liquid solution, was quickly immersed into liquid nitrogen (77 K) in order to prevent aggregation, and the frozen sample was then inserted into a cold cryostat. Optical studies Absorption spectra of BTBT crystals, or of solid solutions of BTBT at 5 K, were recorded in a single-beam configuration by monitoring transmission of light emitted by a xenon arc lamp. For emission studies, the excitation source was either a Lambda Physik LPX100 excimer laser (308 nm line, repetition rate 10 Hz) or a Lambda Physik FL3001 dye laser (pumped by the above mentioned excimer laser, lasing with p-terphenyl dye within the 335–346 nm range). Transmission and/or emission light were dispersed with the aid of a McPherson 207 spectrograph/monochromator and detected either with a Hamamatsu H10721-20 photomultiplier (operating in the photon counting mode) or with an Andor DU424A-BR-DD camera (cooled to 180 K). Phosphorescence decay curves were accumulated using a H10721-20 photomultiplier and a Stanford Research SR-430 multichannel scaler. Fluorescence decay curves were monitored with the aid of the “time correlated” single photon counting technique (in the inverted time mode). Excitation pulses were provided by second harmon- ics of a mode-locked Coherent’s Mira-HP picosecond laser pumped by a Verdi18 laser. The original repetition rate of a Mira-HP laser,76 MHz, was reduced to 3.8 MHz with the aid of an APE pulse selector. The fluorescence photons were dispersed with a McPher- son 207 monochromator, and the decays monitored with the aid of a Becker & Hickl GmbH system composed of a HPM-100-07 hybrid detector and a SPC-150 module inserted into a personal computer (PC). Fluorescence decay times were determined by using a deconvolution program, which iteratively fitted a theoreti- cal curve to the experimental decay with the aid of the least-squares Levenberg–Marquardt algorithm. Quantum chemistry calculations Quantum chemistry calculations at the density functional theory (DFT) level of theory were performed using the Gaussian 16 package.25The electronic states of the isolated molecule and of different aggregates of BTBT were determined using several func- tionals ranging from B3LYP and M06 to CAM-B3LYP. The obtained results (Table S4a of the supplementary material) indicate compara- ble results for the different functionals, which give reasonable ioniza- tion potential values of BTBT [i.e., similar to the experimental data (Ref. 4)]. The B3LYP functional, widely used to describe photophys- ical properties of organic semiconductors,26–30,45led to acceptable consistency between the calculated and the experimental transition energy values (Tables S4a and S4b in the supplementary material). It was used for the DFT, TDDFT, and UDDFT B3LYP/6-31G(d,p) optimization of the structures of the isolated BTBT molecule and BTBT dimers in their electronic ground, S 0, excited singlet, S 1, and triplet, T 1, states. Solvent effects were introduced within the pulsed- code modulation (PCM) procedure. Vibrational structures of the electronic spectra were obtained using the procedure for calculation of the Franck–Condon (FC) factors31included in the Gaussian 16 package. RESULTS AND DISCUSSION Experimental results Emission spectra of a single crystal of BTBT, obtained with the excitation wavelength of 308 nm (high energy excitation) at several temperatures between 5 and 300 K, are shown in Fig. 1. Fluorescence was observed in the spectral range between 367 and 480 nm and comprised of three broad vibrational bands with maxima at 375, 395, and 418 nm. At 5 K, the onset of fluorescence (defined as the wavelength at which the intensity reaches 10% of the band maximum) was positioned at 367.3 nm and coincided with the onset of the crystal absorption edge. The structured phosphorescence spectrum was observed at wavelengths longer than 483 nm, and its decay time at 5 K was 215 ms. This long-lived emission was observed only at temperatures below 40 K, and it originated from at least two triplet traps, which were successively thermally depopulated with increasing tempera- ture. A shallow trap (with the origin line at 483 nm) was depopulated at temperatures below 20 K, whereas a deeper trap (with the origin line at 487 nm) was depopulated at temperatures below 40 K. In order to delve deeper into the origin of emission in BTBT, we studied also BTBT dispersed in frozen n-nonane (the length of the long axis of the n-none molecule is similar to that of BTBT, sug- gesting that this solvent can provide preferred conditions for a good J. Chem. Phys. 155, 034504 (2021); doi: 10.1063/5.0057202 155, 034504-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1. Emission (fluorescence and phosphorescence) spectra of a BTBT single crystal at temperatures 5, 20, 30, 40, and 300 K for λexc=308 nm. Absorbance, measured at 5 K, is shown as the blue solid line. Narrow lines observed at 616 nm are second harmonics of the excitation line. Shpol’skii matrix),32n-hexadecane, and dichloromethane. Concen- trations of BTBT dissolved in the matrices were in the range between 10−5and 5 ×10−4M, which allowed us to vary the concentration ratio between the BTBT monomer and its aggregates. Absorption spectra of BTBT dispersed in an n-nonane matrix at 5 K for different dopant concentration are shown in Fig. 2. Sharp absorption lines, with maxima at 329, 332.6, and 336.9 nm, are attributed to the BTBT monomer, whereas the broad spectrum spreads over longer wavelengths, with the weakly marked maxima at around 344, 349, and 358 nm, to BTBT aggregates (predominantly dimers). It could be concluded that at the lowest concentrations of BTBT, we observe the monomer, whereas aggregates contribute to the spectra at higher BTBT concentrations. There are some differ- ence in the shape of the aggregate spectrum, the band with the max- imum at 344 nm dominates at concentration of 8 ×10−5M, whereas at concentrations of 2 ×10−4and 4 ×10−4M, the most intense band has a maximum at 349 nm. Such observations might suggest contri- bution of more than one aggregate. We should stress, however, that we were not able to completely control the freezing process (which FIG. 2. Absorption spectra of BTBT dispersed in the n-nonane matrix at 5 K. Spectra were recorded in a 1.5 mm long cuvette. Concentrations of BTBT (given in M/l units) are indicated on the right-hand side of the spectra. was done by fast immersion of a cuvette in liquid N 2) and thus to evaluate the concentration effect quantitatively. Emission spectra were recorded for two different concentra- tions of BTBT in the matrix and for two wavelengths of excita- tion, 337 and 343 nm. The former led to predominant excitation of the BTBT monomer, whereas the latter to selective excitation of aggregates. The results are presented in Figs. 3 and 4. The emission spectrum of a sample with a BTBT concentration of 10−5M was successfully recorded only with the 337 nm excita- tion, and it exhibited structured phosphorescence (phosphorescence decay time τph=280 ms). Fluorescence emission was practically absent in this spectrum. The trial to record the spectrum with the 343 nm excitation failed because of insufficient concentration of dimers and higher aggregates, in agreement with the absorption spectrum presented in Fig. 2. In the case of a more concentrated sample (see Fig. 4 for BTBT concentration of 8 ×10−5M), we succeeded in recording spectra with both excitation wavelengths. J. Chem. Phys. 155, 034504 (2021); doi: 10.1063/5.0057202 155, 034504-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. Emission spectrum of BTBT dispersed in the n-nonane matrix at 5 K, concentration 10−5M, and excitation wavelength 337 nm. Concentration of BTBT aggregates was too low to record any emission when an excitation wavelength of 343 nm was used. The situation was clear for the 343 nm excitation when aggregates were selectively excited (see the absorption spectrum presented in Fig. 2). The spectrum of aggregates comprised both weakly structured fluorescence, with onset at about 360 nm, and weakly structured phosphorescence. The situation in the case of the 337 nm excitation was more complex, with an excited monomer and also partly of aggregates. Therefore, the recorded spectrum had contri- butions from sharply structured phosphorescence (monomer) and fluorescence (contribution from aggregates). FIG. 4. Emission spectra of BTBT dispersed in the n-nonane matrix at 5 K, concen- tration 8 ×10−5M, recorded with two excitation wavelengths, 343 nm (excitation of BTBT aggregates) and 337 nm (predominant excitation of BTBT monomer). In the case of the 343 nm excitation, the narrow lines at 446, 480, 548, and 583 nm were Raman lines of the n-nonane matrix.Our conclusion was, therefore, that the main channel of depop- ulation of the excited S 1state in the case of the BTBT monomer was the intersystem crossing (ISC) S 1→T1, manifested by T 1→S0 phosphorescence. In the case of BTBT aggregates, at least for those that contribute to the spectrum, the excited S 1state was depopu- lated by both channels, intersystem crossing S 1→T1→S0, leading to weakly structured phosphorescence, and direct S 1→S0transition, manifested by fluorescence. Quantum-chemistry calculations Our low temperature experiments clearly indicated that we observed an aggregate or aggregates of BTBT in the n-nonane matrix, as evidenced by the absorption bands in the region between 340 and 360 nm. Below, we present the results of calculation for the isolated monomer and the simplest aggregate, i.e., the BTBT dimer. Coordinates of atoms, bond lengths, and vibration frequencies in the isolated BTBT molecule, optimized in its S 0, S1, and T 1states, are collected in Tables S5–S8 of the supplementary material. Table S9 describes the vibronic structure of the S 0→S1, S1→S0, and T 1→S0 transitions of the BTBT molecule. Similar calculations performed for the BTBT dimer are given in Tables S10–S12 of the supplementary material. The bond lengths in BTBT optimized in the electronic ground state, S 0(see Table S6), are quite close to the experimental data, determined by the single crystal x-ray diffraction.7Calculated wave- lengths and oscillator strengths of the electronic S 0→S1, S1→S0, and T 1→S0transitions for the isolated BTBT molecule, and BTBT dispersed in the n-nonane matrix are given in Table I. The calculated T1→S0transition wavelength, ∼483 nm, well reproduces the experi- mentally observed phosphorescence onsets, at 483 and 475.5 nm for single crystal and in the n-nonane matrix, respectively. Less accu- rate but still not far from the experimental values were the calculated absorption, ∼317 nm, and fluorescence, ∼346 nm, transitions in the n-nonane matrix. The experimental and calculated energies differ by less than 700 cm−1. TABLE I. The calculated wavelengths (in nm) and oscillator strengths (in parentheses) of absorption, fluorescence, and phosphorescence for the BTBT molecule and for the optimized BTBT dimer. S0→S1 S1→S0 νfl(0,0) T 1→S0 Isolated 315.0 (0.154) 350.2 (0.227) 340.8 483.3 n-nonane 317.3 (0.260) 356.6 (0.378) 345.9 482.9 Isolated 332.7 (0.010) 427.1 (0.001) 392.1 483.5 n-nonane 329.3 (0.021) 408.5 (0.002) 377.9 483.1 J. Chem. Phys. 155, 034504 (2021); doi: 10.1063/5.0057202 155, 034504-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The diagram of the electronic excited states for the BTBT monomer is presented on the left-hand side of Fig. 6. Excitation from the electronic ground S 0state to the lowest excited S 1state is described as a nearly pure HOMO →LUMO transition. The inter- system crossing (ISC) process in the BTBT monomer (point group symmetry C i) occurs from the S 1(Au) state to the lower energy T3(Ag) state. This transition [as in the case of T 1(Au)→S0(Ag)] is forbidden only by spin, and one can expect that its probabil- ity is determined by direct spin–orbit coupling in the expansion of the perturbation theory.33,34On taking also into account the heavy atom effect, connected with the presence of sulfur atoms in the BTBT structure, we may expect efficient intersystem crossing between these two states. Consequently, population of the S 1state relaxes, via T 3, to the lowest triplet T 1(Au) state, and the fluores- cence quantum yield is low (the estimated value in n-nonane at room temperature was 2.4%). The optimized structure of the simplest aggregate, the isolated BTBT dimer, whose binding energy in the electronic ground state is∼340 cm−1, is graphically presented in Fig. 5 (and in Tables S10 and S11 of the supplementary material). The vibrational spectrum of the optimized dimer (given in Table S12 of the supplementary material) was calculated after the optimization process. Lack of any imaginary frequencies was used as a proof that the optimized dimer geometry corresponded to the true minimum of the potential energy surface. In the course of optimization of the dimer structure, we used about ten initial dimer geometries, including different relative orientation of the two BTBT molecules, analogous to those in the crystal UC (among which there were different pairs of BTBT with parallel planes). Most successful optimizations ended with the struc- ture shown in Fig. 5, which corresponds to the true stationary point. We tested also a “linear dimer” starting from the BTBT pair struc- ture like that arranged along the crystallographic axis a(see Fig. S8). Optimization of such a dimer structure was, however, not success- ful because the calculated vibrational spectrum had one negative frequency. Oblique and “side-by-side” orientation of the molecular planes in the S 0state of the optimized dimer is similar to that in the “herringbone” structure of the crystal unit cell (UC), shown in Fig. S8 of the supplementary material. Calculated wavelengths and oscillator strengths of the electronic S 0→S1, S1→S0and T 1→S0 transitions for the BTBT dimer are given in Table I. The diagram of the electronic excited states for the BTBT dimer is presented in the center of Fig. 6. A characteristic feature of the lowest excited singlet state of the dimer is its intermolecular charge-transfer (CT) FIG. 5. Structure of the optimized isolated BTBT dimer. Side view (a) and view along the long axes of the BTBT molecules (b). FIG. 6. Diagram of the electronic excited states for the BTBT monomer (A and B) and the dimer (center) and their HOMO and LUMO orbitals. The lowest excited singlet state in the dimer has intermolecular “charge-transfer” (CT) character. In this state, electronic charge has been transferred from one BTBT molecule to its partner. Intermolecular CT transition (being the property of the dimer as a whole) was already discussed in the description of electronic excited singlet states of pentacene aggregates.35–37 Each monomer local state splits in the dimer to a pair of states. The relation E(S 1(CT))<E(T 7(CT)) is a well-known characteristic feature of CT systems.38 character, where the electron localized on the HOMO orbital of molecule A is transferred to the LUMO orbital localized on molecule B. Charge transfer is also related to small oscillator strength of the transition between the S 0and S 1states. The calculated energy of the S1(CT) state of the dimer is located ∼2000 cm−1below the energy of the locally excited (LE) S 2and S 3states. In the manifold of the dimer triplet states, the counterpart of the singlet S 1(CT) state is the triplet T 7(CT) state, with a slightly higher energy. It is a well-known property of the electron–donor–acceptor systems that the singlet and triplet CT states of the same orbital configuration are nearly isoenergetic and there is no intersystem crossing between them. It should be noted that all six triplet states of the BTBT dimer, which have lower energy than that of the S 1(CT) state, are combinations of the triplets localized on the monomers, i.e., they have LE character (see Fig. 6). Thus, the intersystem cross- ing S 1(CT)→Ti(LE) in the dimer is connected not only with the spin flip but also with intermolecular charge transfer. It has already been found experimentally in similar systems that the matrix element for the spin-forbidden electron-transfer process was 3000 times smaller than the matrix element for the spin-allowed process.39Intersystem crossing from the S 1state of the BTBT dimer has a qualitatively dif- ferent character and much lower efficiency when compared with the situation in the isolated BTBT molecule. Thus, the assumption that the dimer is the fluorescence emit- ting object explains the experiment result qualitatively. A question arises as to the reliability of such a result. Table II shows the calcu- lated energies of the S 0→S1electronic transitions for different BTBT molecular systems (monomer, dimers, tetramer, pentamer, and UC) in the geometries taken from the crystallographic data. Further com- putational results can be found in Table S14 of the supplementary J. Chem. Phys. 155, 034504 (2021); doi: 10.1063/5.0057202 155, 034504-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. The calculated wavelengths of the S 0→S1and S 0→T1transitions of different systems of BTBT, extracted from the BTBT crystal UC (see Fig. S8). The CT character indicates that intermolecular charge transfer takes place in the system. All S 0→T1transitions are of the LE type. Dimer (a) denotes that both component molecules are arranged along the crystallographic axis a. TD DFT B3LYP/6-31G(d,p) E(S 1) (nm) f Character E(T 1) (nm) Monomer 314.3 0.1237 LE 455.9 Dimer (a) 315.6 0.3212 LE 456.1 Dimer (b) 326.6 0.0000 LE 456.6 Dimer (c) 314.9 0.0000 LE 455.9 Dimer herringbone 350.8 0.0019 CT 455.8 Trimer 347.6 0.0024 CT +LE 457.0 Tetramer (bc) 327.1 0.0093 LE +CT 456.9 Pentamer 347.9 0.0079 CT +LE 456.6 UC 349.8 0.0113 CT +LE 457.3 material. Comparing the results obtained for the considered dimers, it was obvious that the herringbone pattern creates favorable condi- tions for the formation of the CT state, and this property is retained in larger aggregates. Energies of the CT states in BTBT aggregates are only weakly dependent on their size. The results obtained by us are similar to the results of calcula- tions performed for low-lying excited states of pentacene oligomers, considered in Refs. 35–37, 40, and 41. Pentacene is the well-known organic semiconductor, which, like BTBT, crystallizes in the her- ringbone pattern. Phosphorescence spectra, recorded for the crystal and BTBT inn-nonane matrix, showed a far-reaching similarity. As presented in Fig. 7, these spectra can be well fitted with the vibrational struc- ture calculated for an isolated BTBT molecule. This is not surprising in the light of our calculations, which indicate locally excited (LE) character of the lowest triplet state. FIG. 7. Phosphorescence spectra of the trap state in the BTBT crystal (black) and BTBT in n-nonane matrix (10−5M, blue), both at 5 K. The (0,0) lines are located at the zero of the energy scale. Calculated FC factors are given in the form of vertical red lines. Green lines correspond to molecules with the same vibrational structure but with the (0,0) line shifted to −840 cm−1. We have, as yet, no confi- dent interpretation for this new set of lines. Calculated frequencies and FC factors attributed to experimentally observed vibrations are collected in Table S13 of the supplementary material.CONCLUSIONS In this work, we concentrated on photophysical properties of BTBT at low temperatures. The most important result of the work is experimental detection of the S 0→S1electronic transition in BTBT aggregates, characterized by the energy and oscillator strength dif- ferent than in the case of monomer in solution. According to the results of calculations, this electronic excitation is connected with intermolecular charge-transfer (CT). Creation of a CT state is closely related to the herringbone arrangement of BTBT molecules in the optimized dimer, which is the basis of the structure in the crystal UC. The dissimilar nature of the lowest excited S 1state in the BTBT monomer and in dimers is not only the result of the model adopted in the calculations but is also manifested experimentally as a change in the effectiveness of the intersystem crossing. Recently, there has been a renaissance in the study of the role of CT interactions in organic semiconductors21,35,42–49in which reference is made to earlier spectroscopic models.50–52In relation to these works, we made preliminary calculations, which are presented in Fig. S11 of the supplementary material. It is clearly apparent from these calcula- tions that the excited state of the system of ten BTBT molecules, which form the crystal UC, is connected with intermolecular trans- fer of charge, where as many as six BTBT molecules are involved in the creation of the excited CT state. A complete treatment of this issue, in relation to high mobility of charges, would require separate research, which is outside the scope of the present work. Due to the specific system of BTBT states, this molecule turned out to be an exceptionally interesting research material which allows us to distinguish the nature of the excited states in the herringbone systems. In the BTBT monomer, the LE singlet S 1state is efficiently depopulated by the intersystem crossing channel due to the heavy atom effect, and phosphorescence dominates over fluorescence. In BTBT aggregates and in the crystal, the lowest excited S 1state is of high CT character, and such a state is less efficiently depopulated by the intersystem crossing channel. Consequently, both emissions, flu- orescence and phosphorescence, were observed. Low-temperature studies have proved to be particularly advantageous when the CT states are in proximity with the LE states. SUPPLEMENTARY MATERIAL See the supplementary material for the procedure of synthesis of BTBT, additional spectroscopic and kinetic results, and results of quantum-chemistry calculations. ACKNOWLEDGMENTS The work was funded by the National Science Centre, Poland, under the QuantERA program (Project No. 2017/25/Z/ST2/03038). Theoretical calculations were performed at the Interdisciplinary Centre of Mathematical and Computer Modelling (ICM) of the Warsaw University under Computational Grant No. G-32-10. There are no conflicts to declare. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. J. Chem. Phys. 155, 034504 (2021); doi: 10.1063/5.0057202 155, 034504-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp REFERENCES 1C. Wang, H. Dong, L. Jiang, and W. Hu, Chem. Soc. Rev. 47, 422 (2018). 2M. Alkan and I. Yavuz, Phys. Chem. Chem. Phys. 20, 15970 (2018). 3M. Mohankumar, B. Chattopadhyay, R. Hadji, L. Sanguinet, A. R. Kennedy, V. Lemaur, J. Cornil, O. Fenwick, P. Samorì, and Y. Geerts, ChemPlusChem 84, 1263 (2018). 4K. Takimiya, I. Osaka, T. Mori, and M. Nakano, Acc. Chem. Res. 47, 1493 (2014). 5K. Takimiya, T. Yamamoto, H. Ebata, and T. Izawa, Thin Solid Films 554, 13 (2014). 6K. Takimiya, S. Shinamura, I. Osaka, and E. Miyazaki, Adv. Mater. 23, 4347 (2011). 7V. S. Vyas, R. Gutzler, J. Nuss, K. Kern, and B. V. Lotsch, CrystEngComm 16, 7389 (2014). 8C. Wang, M. 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5.0052109.pdf

AIP Advances ARTICLE scitation.org/journal/adv Continuous tunable lateral magnetic anisotropy in La 0.67Ca0.33MnO 3/SrRuO 3superlattices by stacking period-modulation Cite as: AIP Advances 11, 075001 (2021); doi: 10.1063/5.0052109 Submitted: 29 March 2021 •Accepted: 8 June 2021 • Published Online: 1 July 2021 Lili Qu,1 Da Lan,1 Kexuan Zhang,1 Enda Hua,1Binghui Ge,2Liqiang Xu,2Feng Jin,1Guanyin Gao,1 Lingfei Wang,1,a) and Wenbin Wu1,3,a) AFFILIATIONS 1Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China 2Institutes of Physical Science and Information Technology, Anhui University, Hefei 230601, China 3High Magnetic Field Laboratory of the Chinese Academy of Sciences, Hefei 230031, China a)Authors to whom correspondence should be addressed: wanglf@ustc.edu.cn and wuwb@ustc.edu.cn ABSTRACT Effective control of magnetic anisotropy is important for developing spintronic devices. In this work, we performed a case study of stacking periods ( N)-mediated reorientation of lateral magnetic anisotropy in ultrathin La 0.67Ca0.33MnO 3/SrRuO 3superlattices. As Nincreases from 1 to 15, the magnetic easy-axis switches from the orthorhombic [010] to [100]-axis. The maximum anisotropy constant of the superlattice (SL) (N=15) reaches −1.83×105erg/cm3. X-ray absorption spectroscopy and x-ray linear dichroism further suggest that the observed changes in lateral magnetic anisotropy are driven by in-plane orbital polarization. For SLs with small N, anisotropic strain-induced orbital polarization along the b-axis can result in the [010]-oriented magnetic easy axis. For SLs with large N, the dimension crossover from 2-dimension to 3-dimension could enhance the hybridization of Ru t2gand Mn dx2−y2orbitals, which can compete with the strain effect and switch the magnetic easy axis to [100]. Our results suggest a potential strategy for engineering magnetic anisotropy through the cooperation of strain engineering and interfacial orbital engineering. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0052109 I. INTRODUCTION Magnetism in low-dimensional epitaxial oxide systems has attracted continuous and considerable research attention due to the potential in spintronic memory and logic devices.1,2In particular, magnetic anisotropy (MA), including lateral magnetic anisotropy (LMA) and perpendicular magnetic anisotropy (PMA), is one of the most important properties and tuning knobs in oxide heterostructures with ferromagnetism. MA can compete with other magnetic interactions and thus determine the ground states of these magnetic systems.3For instance, strong PMA can cooperate with the Dzyaloshinskii–Moriya interaction and stabilize spin chiral domains and magnetic skyrmions.4–7In addition, the cooperation of uniaxial-LMA and interlayer exchange coupling in manganite/ ruthenate superlattices can stabilize synthetic antiferromagnets with a layer-resolved spin-flip.1,8Thus, enhancing the tunability ofmagnetic anisotropy of oxide heterostructures should be essential for developing all-oxide-based spintronic memory and logic devices. Rare-earth-doped manganite is a typical strongly correlated electron system with intimate coupling between charge, orbital, lat- tice, and spin degrees of freedom. Thus, it could be an ideal platform for systematically exploiting the tunability of magnetic anisotropy.3,9 MA of manganites is correlated with Mn 3 dorbital occupation and polarization, which can be modulated by the Mn–O bond lengths and Mn–O–Mn bond angles.10Consequently, previous reports have demonstrated that strain-induced magneto-elastic coupling is an effective route for determining MA. Taking LCMO epitaxial thin film as an example, the magnetic moments always preferentially align along the tensile-strained axis, which consists of a larger Mn–O bond length and Mn–O–Mn bond angle.11,12Moreover, recent reports suggest that structural inversion-symmetry-breaking, the interfacial coupling of oxygen octahedral distortion, orbital AIP Advances 11, 075001 (2021); doi: 10.1063/5.0052109 11, 075001-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv reconstruction, and interlayer exchange couplings also play impor- tant roles in determining the MA of oxide heterostructures.13–15 However, the cooperation and competition of these mechanisms in determining the magnetic anisotropy have not been clearly under- stood. In addition, the associated multi-parameter controllability of MA in manganite-based thin-films has not been realized. In this paper, using the La 0.67Ca0.33MnO 3/SrRuO 3(LCMO/ SRO) superlattice (SL) as an example system, we investigated the repetition number ( N)-driven modulation of LMA. As Nincreases from 2 to 4, reorientation of the magnetic easy axis from the orthorhombic [010] to [100] axis occurs. In addition, the LMA increases further with N. To understand such N-dependent LMA, we have systematically characterized the structural and electronic properties of these heterostructures by a variety of spectroscopy and microscopic methods. We found that the samples with small and large Nhave distinct in-plane orbital polarizations, which could originate from the anisotropic-strain-induced Mn dx2−y2 orbital polarization and hybridization with the Ru dxz/dyzorbital. Such an N-driven evolution of orbital polarization and the asso- ciated spin-orbit couple effect could be the main driving force of the observed changes in LMA. Our findings can provide use- ful guidance for controlling MA and designing spintronic devices through the cooperation of strain engineering and interfacial orbital reconstruction. II. MATERIALS AND METHODS LCMO/SRO ultrathin SLs and single-phase films were grown on a (001)-oriented NdGaO 3[NGO(001)] substrate via pulsed laser deposition. During deposition, ceramic targets LCMO and SRO syn- thesized by the solid-state reaction method sintered at 1350○C for 24 h were alternately ablated with a pulsed KrF excimer laser ( λ =248 nm) (Coherent COMPexPro 205F). The laser fluence on the targets, oxygen partial pressure, and the substrate temperature were set to be 1.5 J cm−2, 40 Pa, and 650○C, respectively. After depo- sition, to improve the epitaxial quality, all those as-deposited films were annealed in situ for 20 min and then cooled down to room tem- perature in a 2 ×103Pa oxygen atmosphere. The stacking structure was denoted as [LCMO( tLC)/SRO( tSRO)]N, where tLC,tSRO, and N are the thickness of the LCMO layer, the thickness of the SRO layer, and the repetition number of the bilayers, respectively. As schemat- ically shown in Fig. 1(a), we fixed the tLCand tSRO values in the SL as 2.4 and 0.8 nm, respectively, and we varied Nfrom 1 to 15. We also fixed the topmost and bottommost layers as SRO to ensure that every LCMO layer has the same structural and magnetic bound- ary conditions. According to our previous studies,16a sizable charge transfer could occur from SRO to LCMO, which can compensate the depleted egelectrons in ultrathin LCMO layers (2.4 nm thick) and enhance the double exchange interaction. Therefore, all of our LCMO/SRO SLs with different Nvalues show a high TCof up to 300 K and large saturated magnetizations ( MS) of∼3.0μB/Mn. We cannot observe any trend of dead layer formation in the SLs. The epitaxial structure of those SLs was characterized by high-resolution x-ray diffraction (XRD, Panalytical X’pert, λ =1.5406 Å) in theω–2θlinear scan mode and off-specular reciprocal space mapping (RSM). The microscopic atomic struc- ture of the SL was further investigated by scanning transmission electron microscopy in the high-angle annular dark-field mode FIG. 1. (a) Schematic illustration of the N-modulated [LCMO(2.4 nm)/ SRO(0.8 nm)] NSLs. (b) XRD ω-2θlinear scans of these N-engineered SLs. The main reflections and satellite peaks are marked as “0” and “ ±1,” respectively, indicating flat and sharp interfaces and surfaces. (c) The XRD reciprocal space mapping (RSM) on the (116) main reflection of the N=15 SL, where we index the main reflection as SL(0) and the satellite peaks as SL( ±1), and the main diffrac- tion of the NGO substrate is marked by the purple arrow. The same Q//[110] vector confirms that the SL is coherently strained. (HAADF-STEM) on a double Cscorrector-JEOL 2010 microscope. The magnetic properties were characterized by a Magnetic Pro- perties Measurement System (MPMS-VSM, Quantum Design) with an in-plane magnetic field ( H). The electrical transport properties were measured in a four-probe geometry using a Physical Properties Measurement System (PPMS, Quantum Design). The x-ray absorp- tion spectra (XAS) of the Mn- L2,3edge were recorded at 300 K in the total electron yield mode (TEY). The beamline incident angle was fixed at 0○and 60○from the sample normal, which present E∣∣ (in-plane polarization, I//) and E/⊙◇⊞(out-of-plane polarization, I/⊙◇⊞), respectively. The in-plane polarized spectra along the a- and b-axis were both collected. The x-ray linear dichroism spectrum (XLD) of Mn 3 dx2−y2orbitals, defined by I//b−I//a, was the intensity difference in normalized XAS along the two polarizations of the b- and a-axis. It provides information about the preferential orbital occupancy of Mn-3 dstates in the in-plane axis.13 III. RESULTS AND DISCUSSION We first analyze the epitaxial quality and strain state of the LCMO/SRO SLs. As shown in Fig. 1(b), the ω–2θlinear scans of the AIP Advances 11, 075001 (2021); doi: 10.1063/5.0052109 11, 075001-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv TABLE I. Crystal parameters of bulk SRO, LCMO, and NGO. Pbnm (a−a−c+) a(Å) b(Å) c(Å) a/b SRO 5.5670 5.5304 7.8446 1.007 LCMO 5.4717 5.4569 7.7112 1.003 NGO 5.4332 5.5034 7.7155 0.987 SLs with various Nvalues show well-defined Laue fringes and satel- lite peaks, signifying the high epitaxial quality and sharp interfaces. As shown in Fig. 1(c), the RSMs confirm that the in-plane lattice constants ( aandbaxes) of both LCMO and SRO layers are identical to those of the NGO(001) substrate, confirming the coherent strain state in all of the SLs with N=1–15. In bulk, LCMO, SRO, and NGO have the same orthorhombic Pbnm symmetry and octahedral rota- tion pattern ( a−a−c+), but unequal lattice parameters (summarized in Table I), which results in distinct orthorhombicity ( a/b) for these three oxides.17,18Specifically, lattice constants of the bulk LCMO and SRO along the orthorhombic b-axis are smaller than those along the a-axis ( a/b>1), while the NGO substrate shows an opposite trend (a/b<1). To gain further insight into the local structural change in the LCMO/SRO SL, we performed atomically resolved STEM character- ization of the N=10 SL. The HAADF-STEM image [Fig. 2(a)] shows well-defined SRO/LCMO interfaces and uniform film thickness. The fast Fourier transform (FFT) patterns of the HAADF-STEM image are shown in Figs. 2(b) and 2(c). The {020} points obtained at the SL and NGO(001) substrate regions share the same recipro- cal space positions.19This result further suggests that the LCMO and SRO layers are coherently and axis-to-axis stacked. Specifically, in the orthorhombic notation, the epitaxial relationship should be [100] SRO∥[100] LCMO ∥[100] NGO and [010] SRO∥[010] LCMO ∥[010] NGO. The origin of this epitaxial relationship arises from the requirement FIG. 2. (a) Low magnification (20 nm) HAADF-STEM image of the [LCMO(2.4 nm) /SRO(0.8 nm)] 10SL. (b) and (c). The diffractograms from the SL region and the 001-oriented NGO substrate are marked by red squares in (a). It shows that the SL has the same orthorhombic structure as the substrate. The blue dotted circle in (b) is just a super-diffraction point from the bilayers in the SL.of interfacial octahedral connectivity in the out-of-plane direction.20 According to this epitaxial relationship and the lattice parameters shown in Table I, both strains of the LCMO and SRO are anisotropic. Specifically, the LCMO layers are strained compressively ( −0.70%) along the a-axis while along the b-axis, the layers undergo a tensile strain ( +0.85%). We then measured the magnetic field-dependent magne- tization ( M-H) curves from the LCMO single-phase film and LCMO/SRO SLs with various Nvalues at 175 K. The LCMO single- phase film [Fig. 3(a)] shows a rectangular hysteresis loop with a small saturation field ( HS) of 119 Oe as His parallel to the tensile-strained b-axis ( H//[010]) while a slim and gradually increased loop when H//[100]. This result suggests a typical uniaxial LMA with [010]- oriented easy axis, which can be explained by magneto-elastic cou- pling.18For the SL with N=1 [Fig. 3(b)], the uniaxial LMA is greatly reduced. As Nincreases to 2 and 3, the M–Hcurves measured with H//[100] and H//[010] display very similar shapes, implying negligi- ble LMA. By further increasing Nfrom 3 to 15 [Figs. 3(c)–3(h)], the HSof the M-Hloop with H//[100] ( H//[010]) decreases (increases), implying a gradually enhanced LMA with the [100]-oriented easy- axis. That is to say, increasing Ncan result in reorientation of the magnetic easy-axis from the b-axis to the a-axis. Note that these M–Hhysteresis loops were recorded at 175 K, higher than the bulk TC(∼160 K) of SRO.17Thus, the observed MA in our SL should solely come from the LCMO thin layer. Additional cooling-field dependent M-Tcurves further attest to the reorientation of the magnetic easy-axis [see Figs. S1(a) and S1(b) of the supplementary material]. To further evaluate the symmetry of MA, we also performed anisotropic magnetoresistance (AMR) measurements. As schemat- ically depicted in the upper panel of Fig. 3(i), the longitudinal resistance is measured along the baxis as a 1 T external Hrotates in-plane. We define θas the angle between Hand the b-axis. The bot- tom panel of Fig. 3(i) displays the polar plots of AMR from SLs with N=1 and 10. Both AMR curves show a twofold symmetry, demon- strating the uniaxial nature of the LMA for both SLs. In addition, a 90○phase difference between AMRs of N=1 and 10 SL further confirms the N-driven reorientation of the magnetic easy-axis. To quantify the LMA of the LCMO/SRO SL and LCMO single-phase film, we calculated the effective MA constant ( KMAE) of the LCMO film and LCMO/SRO SLs. We first measured the M–H curves during the magnetization process with H//[100] and H//[010]. Then, we integrated the area enclosed between the two M–Hcurves and obtained KMAE (details are shown in Figs. S2 and S3 of the supplementary material).3The magnitude of KMAE sig- nifies the strength of the uniaxial LMA. In addition, the sign of KMAE represents the orientation of the magnetic easy axis: posi- tive (negative) KMAE corresponds to the [010]-([100]-)oriented easy- axis. The N-dependent KMAE curves of the LCMO/SRO SL and LCMO film are summarized in Fig. 4(a). For N=1 and 2, KMAE is positive but much smaller than that of the LCMO/NGO(001) film. Starting from N=3,KMAE becomes negative and mono- tonically increases with N. The largest KMAE observed in the N=15 SL can reach −1.83×105erg/cm3. The amplitude is com- parable to that of the LCMO/NGO(001) film. The T-dependent KMAE curves of LCMO film and LCMO/SRO SLs ( N=1 and 10) are summarized in Fig. 4(b). The LCMO thin-film displays a monotonically increased positive KMAE over the entire Trange. AIP Advances 11, 075001 (2021); doi: 10.1063/5.0052109 11, 075001-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. N-engineered magnetic anisotropy of LCMO/SRO SLs. (a) to (h) Magnetic hysteresis loops ( M-H) of 24 nm-thick LCMO film and [LCMO(2.4 nm)/SRO(0.8 nm)] N SLs ( N=1, 2, 3, 4, 6, 10, 15) along in-plane directions: [100]-axis(blue) and [010]-axis(red), performed at 175 K. (i) Schematic and polar plots of the in-plane AMR. The current is along the [010] axis with 1 T magnetic field rotated in the plane, and the θ=0○corresponds to the [010] direction. The polar plots demonstrate a 90○phase shift between N=1 and 10. Here, the AMR value of N=10 is magnified ten times for comparison. This result is consistent with previous reports.21,22For the LCMO/SRO SL with N=1, the T-dependent KMAE shows a similar trend to the LCMO film, but the amplitude becomes much smaller. For the LCMO/SRO SL with N=10, by contrast, KMAE shows a non-monotonic Tdependence [Fig. 4(a)]. As Tdecreases from 300 to 120 K, the negative KMAE first increases and reaches the maxi- mum value of −0.9×105erg/cm3. By further decreasing T,KMAE first decreases and then reverses its sign to positive. This result sug- gests that there two competing LMAs in the LCMO/SRO SLs. The additional one with the [100]-oriented easy-axis could dominate the strain-induced one in the temperature range from 300 to 50 K. As mentioned before, the strain state of the LCMO layers in theLCMO/SRO SL is almost identical to the LCMO/NGO(001) films. On this basis, the SRO layers and LCMO/SRO interface should play crucial roles in inducing such a distinct LMA.6 For manganite-based thin films, MA is usually governed by Mn-3 d e gorbital occupancy (3 dx2−y2and 3 dz2−r2).23,24Previous works have proved that the LMA is dominated by a preferential 3dx2−y2orbital occupancy while the PMA is dominated by a pref- erential 3 dz2−r2orbital occupancy.13,25Under a biaxial strain, the polarization of the 3 dx2−y2orbital is isotropic along the two in-plane axes.13Hence, the observed uniaxial LMA implies anisotropic dx2−y2 orbital polarization along the two in-plane axes. To address this special orbital occupancy, we carried out x-ray absorption FIG. 4. (a) Stacking period ( N)-dependent KMAEof [LCMO(2.4 nm)/SRO(0.8 nm)] NSLs, extracted from the M-Hloops. Noting that the N=0 corresponds to 24 nm-thick LCMO film. The positive KMAEindicates the easy-axis along the [010] while the negative sign shows a 90○shift to [100]-oriented axis, as schematically shown in the inset. (b)KMAEas a function of temperature for 24 nm-thick LCMO film and N=1 and N=10 SLs, extracted from the temperature-dependent M-Hloops. AIP Advances 11, 075001 (2021); doi: 10.1063/5.0052109 11, 075001-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv spectroscopy (XAS) and x-ray linear dichroism (XLD) at the Mn L-edge from the N=1 and 10 SLs. For each sample, two XAS curves were recorded with photon polarization parallel to the in-plane a-axis ( E//a,Ia) and b-axis ( E//b,Ib) [Figs. 5(a) and 5(b)]. Then, the in-plane XLD is calculated as IXLD=Ib−Ia.13The integral of IXLD around the Mn- L2peak (648–660 eV) can provide the most reliable information of egorbital polarization.25–27A positive signal signifies the enhanced electron occupancy along the a-axis while the negative one indicates that along the b-axis.13For SL with N=1, the inte- gral of IXLDis negative, signifying a preferential orbital polarization along the b-axis [Fig. 5(c)], consistent with the [010]-oriented mag- netic easy axis. For the SL with N=10, the integral of IXLDbecomes positive, implying an enhanced orbital polarization along the a-axis [Fig. 5(d)]. Meanwhile, the magnetic easy axis also reorients to the [100] direction. This finding strongly suggests that the polarization of the Mn 3 dorbital can be strongly modulated by N, which can induce the N-modulated LMA.13 Let us now turn to discuss the possible mechanism of the N-driven modulations of LMA. We would like to first clarify the [010]-oriented easy axis in LCMO/NGO(001) films. As mentioned above, the NGO(001) film imposes a tensile strain along the b-axis and a compressive strain along the a-axis. As depicted in Fig. 6(a), such an anisotropic strain state can enhance the Jahn–Teller distor- tion of the MnO 6octahedron and induce a preferential polarization of the dx2−y2orbital along the b-axis, thus leading to a [010]-oriented magnetic easy axis.27For our LCMO/SRO SLs with various Nvalues, according to the structural analyses, the strain state remains the same as that of LCMO/NGO(001) films. Specifically, the strain-induced orbital polarization should not be the main cause of N-modulated LMA. On this basis, we suggest that the SRO interlayers may trig- ger considerable changes in orbital polarization and result in the reorientation of the magnetic easy axis. Unlike the LCMO films, in SRO, only the t2gorbitals ( dxy,dxz, anddyz) are occupied by four electrons.28,29The Ru 4 dorbitals are rather spatially extended compared to the Mn 3 dorbitals. There- fore, in the LCMO/SRO SLs, the Ru t2gorbitals may hybrid with the Mn 3 dorbitals. In bulk SRO, the dxy,dxz, and dyzorbitals havesimilar energy, and all of them are partially occupied.30The Ru dxy orbitals are preferentially confined within the (001) plane. Thus, it has little hybridization with the Mn dx2−y2orbital. By contrast, the Ru dxzanddyzorbitals are spatially expanded along the [100] and [010] axes, respectively. These two orbitals could strongly hybrid with the Mndx2−y2orbital. Furthermore, the bulk SRO has an orthorhom- bicity of a/ b>1,31,32which indicates that the RuO 6octahedron could elongate along the a-axis (see Fig. S5 in the supplementary material for details). This octahedral deformation should make both thedxzanddyzorbitals preferentially polarized along the a-axis. For the SRO films grown on the NGO(001) substrate, the compressive strain along the a-axis can be accommodated by enhanced octahe- dral tilting along the b-axis.31,33Thus, the a-elongated octahedral deformation and preferential polarization of dxz/dyzorbitals should persist. Note that the Ru 4 dorbitals can induce stronger spin-orbit coupling than the Mn 3 dorbitals.34,35In the LCMO/SRO SLs, the hybridization between Ru t2gorbitals and the Mn dx2−y2orbital may significantly alter the LMA of LCMO ultrathin layers. For the N=1 case, the 0.8 nm-thick SRO layers are present at the surface and bottom interfaces only. The strong 2-dimensional (2D) nature of these two SRO layers should enforce a preferential dxyorbital occupation while suppressing the out-of-plane extension ofdxz/dyzorbitals.28According to Si et al. ,30the 2D SRO shows half- occupied dxz/dyzorbitals with a gap opening near the Fermi surface (EF) and a fully occupied dxyorbital. As schematically depicted in Figs. 6(b) and 6(c), the laterally expanded dxyorbital should have lit- tle impact on the orbital polarization and LMA of the LCMO layer. In addition, the Ru dxz/dyzorbitals with negligible density of states near EFshould inevitably weaken the hybridization with the Mn dx2−y2orbital. Therefore, the N=1 SL still shows a [010]-oriented magnetic easy axis, similar to the LCMO/NGO(001) film. The weak hybridization between the Ru dxz/dyzand Mn dx2−y2orbital only reduces the strength of LMA a bit [Fig. 4(a)]. For the SL with large N values, the multiple SRO layers, although separated by LCMO lay- ers, could crosstalk with each other through the spatially extended Ru 4 dorbitals. Due to the 3-dimensional (3D) nature, these SRO layers are expected to show a similar orbital configuration to bulk FIG. 5. Electronic configurations of N-engineered SLs. (a) and (b) The Sketch of the polarized XAS measure- ments on the Mn 3 dx2−y2orbital unoccu- pied states along the b-axis and a-axis in the TEY mode, respectively. (c) and (d) The normalized XAS and XLD signal defined by Ib−Ia(bottom panel) at the Mn-L2,3edge of N=1 and N=10 SLs in-plane polarization, respectively, which were conducted at 300 K. AIP Advances 11, 075001 (2021); doi: 10.1063/5.0052109 11, 075001-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 6. N-driven orbital polarization and lateral magnetic anisotropy in SLs. (a) and (b) Orbital configurations of Mn 3 dx2−y2and Ru t2gorbitals on the corresponding MnO 2 (RuO 2) sheet in the N=1 SL. (c) The proposed spin configuration of the N=1 SL still maintains the [010]-oriented easy-axis since the laterally expanded Ru dxyorbital has little effect on the Mn 3 dx2−y2orbital polarization. (d) and (e) Orbital configurations of Mn 3 dx2−y2and Ru t2gorbitals on the MnO 2(RuO 2) sheet in the N=15 SL, respectively. (f) The proposed spin configuration in N=15 SL, which is assisted by the enhancement of hybridization between the Mn 3 dx2−y2orbital and Ru dxz/dyzorbital along the a-axis; thus, the magnetic easy-axis gradually switches from the [010] to [100] direction as Nincreases continuously. SRO. Specifically, as schematically shown in Figs. 6(d)–6(f), the vertically expanded Ru dxz/dyzorbitals in SRO could strongly hybrid with the Mn dx2−y2orbital and enhance its polarization along the a- axis. We suggest that this interfacial orbital hybridization-induced LMA could compete with the strain-induced magnetoelastic effect, thus triggering the N-driven evolutions of LMA. As Nincreases, a dimension crossover from 2D to 3D should occur in the SRO layers. As a result, the preferential orbital occupation of SRO lay- ers changes from dxytodxz/dyz. The gradually enhanced interfa- cial orbital hybridization will first weaken the strain-induced dx2−y2 orbital polarization along [010] and then induce a [100]-oriented orbital polarization in the LCMO layers. The changes in orbital polarization further switch the magnetic easy-axis from [010] to [100]. IV. CONCLUSION In conclusion, we systematically investigated the LMA of ultra- thin [LCMO/SRO] NSLs with various stacking periods N. As N increases from 1 to 15, the magnetic easy-axis of SLs gradually switches from [010] to [100]. The maximum effective anisotropy constant can reach −1.83×10−5erg/cm3. We found that the LMA ofthe SLs with small Nis determined by the anisotropic strain-induced orbital polarization along the [010] axis, while the LMA of the SLs with large Nvalues is dominated by the hybridization between Ru dxz/dyzand Mn dx2−y2orbitals. These two competing modulations of the orbital polarization LCMO layer should trigger the N-driven modulations of LMA. Our work could pave an effective way to continuously modu- late the LMA of oxide heterostructures. Note that the requirements of MA could vary dramatically for different spintronic devices. For instance, thin film microwave devices require free rotation of magnetic moment in the film plane, i.e., weak LMA.36,37On the contrary, the ferromagnetic layers in magnetic tunnel junctions should have uniaxial LMA with different strengths. Accordingly, our [LCMO/SRO] NSLs can meet all these requirements by simply tuning the Nvalue, which enables broad and versatile application potential for all-oxide-based spintronic devices. SUPPLEMENTARY MATERIAL See the supplementary material for the additional magnetiza- tion measurements and the additional analyses of the x-ray absorp- tion (XAS) spectra as well as the schematic of the bulk SRO in the orthorhombic Pbnm structure. AIP Advances 11, 075001 (2021); doi: 10.1063/5.0052109 11, 075001-6 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv ACKNOWLEDGMENTS We appreciate the beamtime provided by the Shanghai Syn- chrotron Radiation Facility (SSRF) beamline BL08U1A under Project No. 2020-SSRF-PT-012091 and the Hefei National Syn- chrotron Radiation Facility (NSRF) XMCD beamline (BL12B) under Project No. 2020-HLS-PT-002654. This work was sup- ported by the National Basic Research Program of China (Grant Nos. 2016YFA0401003 and 2020YFA0309100), the National Natu- ral Science Foundation of China (Grant Nos. 11974326, 12074365, U2032218, 51872278, and 11804342), the Fundamental Research Funds for the Central Universities (Grant Nos. WK2030000035 and WK2340000102), and the Hefei Science Center of Chinese Academy of Sciences. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1B. Chen, H. Xu, C. Ma, S. 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5.0045203.pdf

J. Chem. Phys. 154, 154302 (2021); https://doi.org/10.1063/5.0045203 154, 154302 © 2021 Author(s).First-principle study of the structures, growth pattern, and properties of (Pt3Cu)n, n = 1–9, clusters Cite as: J. Chem. Phys. 154, 154302 (2021); https://doi.org/10.1063/5.0045203 Submitted: 24 January 2021 . Accepted: 25 March 2021 . Published Online: 15 April 2021 Carlos Daniel Galindo-Uribe , Patrizia Calaminici , Heriberto Cruz-Martínez , Domingo Cruz-Olvera , and Omar Solorza-Feria COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical Chemistry ARTICLES YOU MAY BE INTERESTED IN Homogeneous water nucleation in carbon dioxide–nitrogen mixtures: Experimental study on pressure and carrier gas effects The Journal of Chemical Physics 154, 154301 (2021); https://doi.org/10.1063/5.0044898 Accurate semiempirical potential energy curves for the a3+-state of NaCs, KCs, and RbCs The Journal of Chemical Physics 154, 154304 (2021); https://doi.org/10.1063/5.0046194 Gas-phase studies of chemically synthesized Au and Ag clusters The Journal of Chemical Physics 154, 140901 (2021); https://doi.org/10.1063/5.0041812The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp First-principle study of the structures, growth pattern, and properties of (Pt 3Cu)n, n = 1–9, clusters Cite as: J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 Submitted: 24 January 2021 •Accepted: 25 March 2021 • Published Online: 15 April 2021 Carlos Daniel Galindo-Uribe,a) Patrizia Calaminici,a) Heriberto Cruz-Martínez, Domingo Cruz-Olvera, and Omar Solorza-Feria AFFILIATIONS Departamento de Química, CINVESTAV, Av. Instituto Politécnico Nacional 2508, San Pedro Zacatenco, Gustavo A. Madero, CP 07360 Mexico City, Mexico Note: This paper is part of the JCP Special Collection in Honor of Women in Chemical Physics and Physical Chemistry. a)Authors to whom correspondence should be addressed: carlosd.galindo@cinvestav.mx and pcalamin@cinvestav.mx ABSTRACT In this work, a first-principles systematic study of (Pt 3Cu) n, n = 1–9, clusters was performed employing the linear combination of Gaussian- type orbital auxiliary density functional theory approach. The growth of the clusters has been achieved by increasing the previous cluster by one Pt 3Cu unit at a time. To explore in detail the potential energy surface of these clusters, initial structures were obtained from Born– Oppenheimer molecular dynamics trajectories generated at different temperatures and spin multiplicities. For each cluster size, several dozens of structures were optimized without any constraints. The most stable structures were characterized by frequency analysis calculations. This study demonstrates that the obtained most stable structures prefer low spin multiplicities. To gain insight into the growing pattern of these systems, average bond lengths were calculated for the lowest stable structures. This work reveals that the Cu atoms prefer to be together and to localize inside the cluster structures. Moreover, these systems tend to form octahedra moieties in the size range of n going from 4 to 9 Pt3Cu units. Magnetic moment per atom and spin density plots were obtained for the neutral, cationic, and anionic ground state structures. Dissociation energies, ionization potential, and electron affinity were calculated, too. The dissociation energy and the electron affinity increase as the number of Pt 3Cu units grows, whereas the ionization potential decreases. Published under license by AIP Publishing. https://doi.org/10.1063/5.0045203 .,s I. INTRODUCTION Transition-metal nanoparticles are attracting a lot of attention in different fields of nanoscience and nanotechnology due to the fact that at this size, various properties such as melting point, color, ionization potential (IP), hardness, catalytic activity, and magnetic properties change considerably with respect to the properties of bulk materials and to the properties of systems formed with a few atoms.1–3Among these nanoparticles, platinum based nanoparticles are commonly used for the catalysis in diverse reactions. However, this metal is very expensive and has low availability on our planet. Therefore, the study of more economical and efficient nanocata- lysts is necessary. The use of PtM alloy based nanoparticles, with M being a transition metal such as Fe, Co, Ni, and Cu among others,has turned out to be a good strategy to reduce the cost of catalysts because these alloys present higher catalytic activity and stability with respect to the ones of the most expensive catalyst.4–8 In between many types of alloys formed with Pt and Cu, for the Pt 3Cu alloy, interesting experimental results have been obtained, demonstrating that these materials have better stability than the one of pure Pt.9In addition, the Pt 3Cu nanoparticles with icosahe- dral and octahedral shapes have demonstrated to possess about six times more surface-specific activity and almost ten times more mass activity in comparison with Pt/C materials for the oxygen reduc- tion reaction (ORR).10In addition, dendritic shaped Pt 3Cu alloys demonstrated a superior activity in the methanol and formic acid oxidation reactions with respect to pure Pt, showing the potential- ity of this material as a catalyst for these reactions.11To test the J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp stability of this type of systems, a recent study of Pt 3Cu nanoframes using TiN as support has shown 14.4 times larger mass activity in comparison with the one of commercial Pt/C toward ORR and with a minimum activity loss after 10 000 cycles of cyclic voltammetry.12 These studies demonstrated the potentiality of the Pt 3Cu material as a high active and durable catalyst, and therefore, this alloy has gained a lot of interest. Pt3Cu nanoparticles have attracted our attention because in recent years, synthesis of Pt 3Cu nanoparticles in different morpholo- gies such as amorphous,9octahedra,10icosahedra,10nanowires,11 and nanoframes12was performed. The nanoparticle synthesis gen- erally needs a copper precursor, a platinum precursor, a tensoactive agent, and a solvent. Any modification of these components leads to different a morphology, composition, and activity. It is not clear if there exists any connection between the type of synthesized struc- tures and the changes that might occur in their morphologies. More- over, it is also not clear which nanoparticle morphology is the most stable one. In this sense, a theoretical study that maintains the Pt 3Cu composition with an increase in the cluster size, as the one aimed in this work, could help to understand if structural changes might occur and which are the most stable structures. As stated above, Pt3Cu nanoparticles have been synthesized in different structures and sizes showing a variety in catalytic activity. However, there is no information available about how these nanoparticles are formed. Moreover, to the best of our knowledge, no systematic study to elu- cidate these structures at an atomistic level exists. This information is crucial if one aims to understand the reasons of the superior cat- alytic activity of these systems. To date, a few theoretical studies of mixed Pt and Cu based systems are available, which provide some structural information.13–19 Chaves et al. investigated the clusters of Pt nCumin which n, m = 0–14 to obtain and characterize their most stable structures.13 In addition, in another study, they used the most stable clusters with composition Pt 13, Pt 7Cu 6, and Cu 13obtained in their previous study for the investigation of absorption of small molecules.14 To find the most favorable distribution of atoms, Guedes- Sobrinho et al. modeled cluster alloys of different compositions formed with Pt with Fe, Co, Ni, Cu, and Zn obtaining various clus- ters, all of those built with 55 atoms.15They found that Cu has negative segregation energy, and hence, the Cu atoms would have a preference to allocate themselves at surface sites with respect to the interior atom positions of the cluster.15However, these conclusions are in contradiction with the research of Takagi et al. who in a study of clusters of Cu 38−nMn, n = 1, 2, and 6, suggested that the segrega- tion energy is negative for Pt atoms but small. Due to this, the Cu atoms tend to take interior rather than surface positions of the clus- ter, suggesting that Cu–Pt core–shell nanoparticles are stable.16This conclusion is further confirmed by Nair and Pathak in their study of octahedron shaped Pt 60M19, M = Pt, Ti, Sc, V, Cr, Mn, Fe, Co, Cu, and Ni core–shell clusters, where the authors showed that all clusters are stable, with Pt 60Cu 19being the system with the lowest binding energy in between all studied clusters.19 Theoretical investigations demonstrated to be very important in the design of new nanocatalysts.20–22In particular, auxiliary den- sity functional theory (ADFT) based studies have been very success- ful in the elucidation of structures and properties of various pure complex transition metal clusters as well as a big variety of transi- tion metal based alloys.20,23–39Therefore, ADFT is a very powerfultool to investigate the structures and properties of novel transition metals based alloys of interest, such as Pt 3Cu. As already mentioned, to the best of our knowledge, to date, no systematic theoretical studies on (Pt 3Cu) nalloys are available in which the growing mechanism and the structures of this type of sys- tems are investigated. Here, the growing mechanism refers to the determination of the lowest energy structures depending on the clus- ter size. In this way, knowledge of the thermodynamic equilibrium structures at sufficiently low temperatures would be achieved. How- ever, a few studies have focused on the structure determination of the (Pt 3Cu) 1cluster. Gálvez-González et al. used four-atom clus- ters of different compositions of Pt and Cu to study the adsorption reaction of CO 2.17Paz-Borbón et al. studied clusters of five atoms with different compositions of Pt and Cu supported on a CeO 2(111) surface.18In addition, Chaves et al. in their earlier mentioned work studied this same cluster.13In all of these publications, the most sta- ble structure of the Pt 3Cu 1cluster results in a C 3vsymmetry. For the larger sized clusters, the study of Chaves et al. also includes the (Pt 3Cu) x, with x = 1–3, clusters that are of our interest. However, this research does not present an exploration of different poten- tial energy surfaces (PESs). This is a very important issue for these systems as it is very difficult to predict a priori the spin multiplic- ity of these clusters. Moreover, this study does not concentrate on the composition Pt:Cu = 3:1 so that no clear information about the growth pattern of the clusters with this particular composition exists. Due to this situation, different questions might arise: What are the most favorable energy structures for clusters formed with Pt 3Cu units ranging from one of these units to the nanometric size? What is the trend of the growing mechanism of the clusters formed by increasing the cluster size of one Pt 3Cu unit at a time? Are the Pt 3Cu units maintained as the cluster grows? What are the most favorable distribution of the atoms over the clusters surface? How do various properties change in these clusters by increasing the cluster size? The main aim of this work is to answer these questions. This manuscript focuses on a systematic theoretical study of (Pt 3Cu) n, with n = 1–9, clusters employing ADFT. For these clus- ters sizes, the most stable isomers are found, and different energy, structural, and magnetic properties are calculated. This manuscript is organized as follows. In Sec. II, the compu- tational details are given. In Sec. III, the results of the validation of the used methodology as well as the results obtained for the clusters of our interest are presented and discussed. Finally, in Sec. IV, the conclusions are summarized. II. COMPUTATIONAL DETAILS All the calculations have been performed using a linear com- bination of Gaussian-type orbitals within auxiliary40density func- tional theory (LCGTO-ADFT), as implemented in the deMon2k program.41,42The integration of the exchange–correlation potential was performed numerically on an adaptive grid.43In all calculations, the grid accuracy was set to 10−8a.u. The variational fitting pro- cedure proposed by Dunlap, Sabin, and Connolly was employed to calculate the Coulomb energy.44 The combination of the Perdew–Burke–Ernzerhof generalized gradient approximation (GGA) exchange functional and the revised J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Perdew–Burke–Ernzerhof correlation functional (RPBE-PBE)45,46 has been employed. The combination of these exchange and cor- relation functionals has been previously successfully used for the determination of structures and properties of other complex transi- tion metal based clusters (see, for example, Ref. 38). The Cu atom was described employing a triple zeta valence polarized basis set optimized for GGA functionals (TZVP-GGA),47whereas for the Pt atom, a double zeta basis set from the Los Alamos National Lab- oratory, with an 18-electron quasi-relativistic effective core poten- tial (QECP| LANL2DZ), was used.48The auxiliary function set GEN-A2∗has been employed in all calculations.49 In order to generate initial structures for successive geome- try optimization and to better explore the potential energy sur- face for each cluster size, Born–Oppenheimer molecular dynamics (BOMD) were performed for all cluster sizes at different tempera- tures, such as 2000, 2500, and 2800 K, respectively. The BOMD tra- jectories were recorded considering simulation timings up to 20 ps. A Nosé–Hoover thermostat was used to maintain the average tem- perature.50,51For each cluster size, several dozens of isomers have been optimized in many spin multiplicities. A similarity analysis was performed to discriminate duplicated structures.52This approach has been proven to be appropriate for the study of a large variety of complex finite systems such as alkali metal clusters and transition metal clusters in between the others.28,30,31,52 A frequency analysis was carried out to characterize the opti- mized minimum structures. The second derivatives were calculated by the analytic second differentiation of the ADFT energy with respect to nuclear displacements at the optimized geometry.53The harmonic frequencies were obtained by diagonalizing the mass- weighted Cartesian force constant matrix. In order to obtain the average bond lengths (ABLs) for all cluster structures, a program in Octave developed by the authors was used. For the three smallest cluster sizes, the ABLs were obtained manually in order to calibrate the program, and with these distances, then an interval between 2 and 3 Å was determined. Using this interval and the atomic coordi- nates, the ABLs for the largest clusters were calculated. The approxi- mate size of all the studied clusters, calculated as the longest possibledistance between pairs of atoms for each lowest energy structure, was also obtained employing the same program. The electronic spin density plots were obtained considering iso- surfaces of 0.005 a.u. and using visual molecular dynamics (VMD) molecular graphics.54For these plots, an all-electron triple zeta basis set was used for the Pt atoms considering the obtained lowest energy structures.55,56 III. RESULTS AND DISCUSSION A. Validation of the methodology In order to test the accuracy of the employed methodology, bond lengths, dissociation energy (D 0), ionization potential (IP), and electron affinity (EA) were calculated for the Cu 2, the PtCu, and the Pt2diatomic molecules, for which reliable experimental data in the gas phase are available. The selection of these molecules is moti- vated by the fact that they represent characteristic bonds for the clusters of our interest. In Table I, we compare our results with the available experimental data.57–62Bond lengths and energy proper- ties are reported in Å and eV, respectively. As it can be seen from Table I, for all bond lengths, the calculated values are in good agree- ment with experiment within 0.05 Å being the best agreement for the PtCu molecule. Concerning the calculated energy properties, the computed D 0values for the Pt 2and Cu 2molecules are in excellent agreement with the experimental values (see Table I). Although no D0experimental data for the PtCu molecule are available, it is well known that this value should be in between the D 0value of the cop- per dimer and the D 0value of the platinum dimer. Therefore, the dissociation energies of the diatomic molecules considered in the validation follow the tendency D 0, Cu 2<D0, PtCu<D0, Pt 2due to the following displacement reaction:62 PtCu + Pt/leftr⫯g⊸tl⫯ne→ Pt2+ Cu. (1) As it can be seen from Table I, the obtained D 0result for the PtCu molecule is in concordance with this tendency. For all molecules, the calculated IPs and EAs are also in fair TABLE I . Calculated and experimental bond lengths, dissociation energy, ionization potential, and electron affinity for the Cu2, PtCu, and Pt 2diatomic molecules. Bond lengths are reported in Å, and energy properties are given in eV. D0 IP EA Bond length Cu 2 Theo. 2.269 2.05 8.12 0.90 Expt. 2.219 7 ±0.002 0572.02±0.02587.899 ±0.007580.842 ±0.01059 PtCu Theo. 2.329 2.53 8.37 1.36 Expt. 2.335 ±0.00158. . . 8.26±0.0758. . . Pt2 Theo. 2.373 3.12 8.86 1.95 Expt. 2.332 97 ±0.000 44603.14±0.02618.68±0.02571.898 ±0.00857 J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp agreement with the experiment within around 0.2 and 0.06 eV, respectively. Moreover, we note that the calculations reproduce the experimental trends for all considered spectroscopic and energy properties. Therefore, the obtained results make us confident that the here applied methodology is adequate for the study of struc- tures and energy properties of the (Pt 3Cu) n, n = 1–9, clusters of our interest. B. Lowest energy structures and frequencies Our study indicates that many low-lying energy structures of the (Pt 3Cu) n, with n = 1–9, clusters exist. In Figs. 1 and 2 are illus- trated the four most stable structures we obtained for the (Pt 3Cu) n clusters with n = 1–5 and n = 6–9, respectively. Red spheres represent the copper atoms, and gray spheres represent the platinum atoms. In these figures, for each structure, the spin multiplicity and the sym- metry point group are indicated, as well as the relative energy (in eV) of each isomer with respect to the most stable structure are given. The found ground state structures for each cluster size are depicted at the left-hand side of Figs. 1 and 2 and are indicated with the letter a. The first row of Fig. 1 illustrates the four lowest energy iso- mers found for the Pt 3Cu cluster. As it can be observed from Fig. 1, isomer 1(a) is found on the doublet potential energy surface (PES) and is characterized by a pyramidal geometry in C 3vsymmetry. We note that this result is in agreement with the most stable struc- ture found previously by different research groups.13,17,18With the here employed methodology for this structure, Cu–Pt bond lengths of 2.58 Å and Pt 2bond lengths of 2.60 Å are obtained, which are around 0.22 Å longer of the experimental bond lengths of the CuPt and Pt 2molecules in gas phase. The found lowest energy structure is followed at 0.15 eV by isomer (1b). This isomer has a planar structure with C 2vsymmetry and, similar to isomer (1a), is found on the doublet PES (see Fig. 1). The structure of the (1c) isomer is similar to the structure of isomer (1b) but is found on the quar- tet PES and lies at 0.20 eV from the lowest energy isomer. Iso- mer (1d) is characterized by a deformed pyramidal structure. It is found on the quartet PES and lies 0.58 eV above the lowest energy isomer. In the second row of Fig. 1, the four lowest energy isomers for the (Pt 3Cu) 2cluster are presented. The most stable structure, iso- mer (2a), is characterized by having triplet spin multiplicity and low Cssymmetry. We note that the two copper atoms are positioned at opposite sides of a rhomboidal structure fragment present in the structure (see Fig. 1). Similar to isomer (2a), isomers (2b) and (2c) are found on the triplet PES. These structures are very similar and have very a similar relative energy of 0.12 and 0.13 eV, with respect to the lowest energy structure. It is important here to underline that isomer (2c) has been previously reported by Ref. 13 as the lowest energy structure for this cluster. This result indicates the impor- tance of using an efficient working strategy to better explore the PES of these systems, such us the BOMD simulations used here to take initial structures along BOMD trajectories generated at differ- ent temperature and considering various temperatures for successive geometry optimization. Finally, isomer (2d) has a very similar struc- ture to the one of isomer (2b). This isomer, however, has higher spin multiplicity and is 0.21 eV energetically less stable with respect to the found lowest energy isomer (see Fig. 1).The third row of Fig. 1 illustrates the most stable structures found for the (Pt 3Cu) 3cluster. The structure of the two lowest energy isomers, (3a) and (3b), can be seen as very similar to the one of a truncated tetrahedron with C ssymmetry. Isomer (3a) has quar- tet spin multiplicity, and isomer (3b) is found on the doublet PES, being the last one at 0.21 eV above the lowest energy isomer. Iso- mers (3c) and (3d) are also structurally very similar between them. These two isomers are both in C 1symmetry and are characterized by quartet and doublet spin multiplicity, respectively. Their rela- tive energies with respect to the lowest found isomer are 0.34 and 0.37 eV, respectively, as it is shown in Fig. 1. In the fourth row of Fig. 1, the lowest energy isomers of the (Pt 3Cu) 4cluster are graphically depicted. We note that these struc- tures are quite different from each other. The most stable isomer, (4a), is found on the triplet PES. This isomer presents a square- based pyramid fragment in its structure and a central triangular fragment formed with three copper atoms. The relative energies of isomers (4b) and (4c) with respect to the most stable isomer, (4a), are 0.02 and 0.10 eV, respectively. Isomer (4b) is found on the triplet PES, whereas spin multiplicity of isomer (4c) is septet. Finally, isomer (4d) has relative energy of 0.12 eV with respect to the most stable isomer and is found on the triplet PES. We noted that these four isomers have all C 1symmetry and present a much smaller relative energy between them with respect to the one computed between different isomers of clusters with a smaller size. The most stable isomers found for the (Pt 3Cu) 5cluster are depicted in the last row of Fig. 1. All these structures possess an octa- hedra fragment formed with 19 atoms (see Fig. 1). The most stable isomer has quartet spin multiplicity. This isomer is characterized by a cross-type arrangement of the five Cu atoms that lies in a plane made by a layer of 3 ×3 atoms in the octahedra type fragment. As already mentioned, isomer (5b) is structurally very similar to iso- mer (5a) but has doublet spin multiplicity and lies 0.07 eV above the most stable structure. Isomers (5c) and (5d) have C 1symmetry and are found on the doublet and quartet PES, respectively. Those isomers have a relative energy of 0.17 and 0.19 eV, with respect to isomer (5a), respectively. The first row of Fig. 2 presents the lowest energy isomers we found for the (Pt 3Cu) 6cluster. We note that these isomers conserve the 19-atom octahedra fragment as well as the cross pattern of the Cu atoms present in the previous cluster size, and from them, the cluster grows to form what appears to be an incomplete, bigger octahedron. All these isomers present a similar geometry. The most stable iso- mer, isomer (6a), has C 1symmetry and singlet spin multiplicity. Isomers (6b) and (6c) are characterized by triplet spin multiplicity, and their structures are found in C ssymmetry. Their relative ener- gies with respect to the lowest energy isomer are 0.01 and 0.06 eV, respectively. Finally, isomer (6d) is found on the triplet PES, it has a similar structure as isomer (6a), and it lies at 0.09 eV above it (see Fig. 2). The four most stable structures of the (Pt 3Cu) 7cluster are depicted in the second row of Fig. 2. All these structures can be seen as formed by a two twinned octahedron fragment (Fig. 2). With the here employed methodology, isomers (7a) and (7b) result to be isoenergetic. They are characterized by the same atomic dis- tribution arrangement and have the same symmetry group (C s). However, they differ in spin multiplicity, with isomer (7a) being J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Structures, spin multiplicities, symmetry point groups, and relative energy (in eV) of the four most stable structures of (Pt 3Cu) n, n = 1–5, clusters. J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Structures, spin multiplicities, symmetry point groups, and relative energy (in eV) of the four most stable structures of (Pt 3Cu) n, n = 6–9, clusters. found on the quartet PES and isomer (7b) being found on the doublet PES. Structures (7c) and (7d) are both in C 1symmetry. They are found on doublet and quartet PESs and have a relative energy of 0.17 and 0.19 eV with respect to the most stable isomer, respectively. In the third row of Fig. 2, the most stable isomers found for the (Pt 3Cu) 8cluster are presented. All structures possess the twinned octahedra structure moieties found in the (Pt 3Cu) 7cluster isomers,expanded in one of the triangular faces. All these isomers have low symmetry. Isomers (8a) and (8b) are isoenergetic, and both of them have the same structure with spin multiplicity of a quintet and triplet, respectively. In addition, these clusters have some of the Cu atoms in the pattern of a cross, similar to the structures of the (Pt 3Cu) 5and (Pt 3Cu) 6clusters. Isomers (8c) and (8d) have quin- tet and triplet spin multiplicity and a relative energy of 0.01 and 0.02 eV, with respect to the most stable isomer, respectively. The J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp atomic arrangement of the Cu atoms in all these isomers is very similar, in which an alternated arrangement can be appreciated (see Fig. 2). We note that these four isomers lie in a very small energy window of only 0.02 eV. Finally, the last row of Fig. 2 illustrates the most stable iso- mers of the (Pt 3Cu) 9cluster. Isomers (9a), (9c), and (9d) present similar structural characteristics with C 2symmetry and are found on the doublet, quartet, and sextet PESs, respectively. The energy separation in between these isomers is very small, with isomer (9a) being more stable than isomers (9b) and (9c) of only 0.02 and 0.03 eV, respectively. Finally, isomer (9d) is found at 0.08 eV with respect to the lowest energy isomer (see Fig. 2). We note that the Cu arrangements found in these structures resemble to the copper arrangements previously discussed. In Tables II and III, the calculated frequencies of the nor- mal modes of the found most stable isomers of the (Pt 3Cu) n, n = 1–9, clusters are presented in cm−1. As can be seen from these tables, in all cases, the calculated first frequencies are real. This is indicative that the reported structures are minima on the corresponding PES. These calculated frequencies could be used as a guide for future experimental investigation on these clusters.C. Structural growing pattern, magnetic moments, and spin density Figure 3 shows the most stable structures of (Pt 3Cu) nclusters, with n = 1–9, along with their spin multiplicity, to their symmetry point group and to their computed approximate size. As it can be seen from Fig. 3, the size of the studied clusters grows from about 0.26 nm, for the smallest cluster, up to about 1.2 nm, for the cluster built with 9 P 3Cu units. To the best of our knowledge, this is the first time that nanometric structures of bi-metallic clusters with compo- sition Pt 3Cu have been determined with a first-principle electronic structure method. A detailed analysis of Fig. 3 reveals that the struc- ture of the Pt 3Cu cluster unit is not conserved in the clusters formed with 2–9 of these units and that the obtained ground state struc- tures result from a rearrangement of all atoms involved in the system under study. Another important point we could observe in the struc- tural growing pattern of these clusters is related to the square-based pyramid fragment present in the (Pt 3Cu) 4cluster, which then devel- ops, from the cluster (Pt 3Cu) 5cluster, as a full octahedron fragment formed with 19 atoms. In the (Pt 3Cu) 6cluster, this octahedron frag- ment is conserved, and the structure of the clusters continues to grow forming an incomplete, bigger octahedron fragment pattern TABLE II . Calculated harmonic frequencies (in cm−1) of the ground state structures of the (Pt 3Cu) n, n = 1–9, clusters. n Energy 1 91.2 91.6 137.4 137.7 178.1 231.5 2 28.3 38.5 43.1 56.4 75.8 76.2 80.0 92.5 111.6 125.8 128.4 145.3 163.3 177.6 195.5 224.5 236.4 253.8 3 6.1 37.3 57.5 58.4 61.5 73.3 74.7 78.6 81.5 86.2 93.5 98.3 102.8 106.5 117.0 117.9 133.2 140.4 144.2 150.4 156.7 159.3 171.1 176.6 181.6 187.9 196.4 219.3 228.5 244.6 4 19.9 35.0 40.1 44.7 48.7 51.2 52.2 57.6 65.6 70.9 75.6 79.1 84.2 85.6 88.8 96.3 98.7 102.3 105.0 112.2 114.0 119.3 122.9 125.4 126.5 131.6 143.5 146.9 155.3 157.9 168.0 172.9 180.5 186.0 187.6 200.1 205.9 210.1 220.4 224.5 238.1 240.8 5 38.8 45.8 49.4 51.5 53.1 54.6 57.4 62.6 65.2 68.8 71.9 74.2 74.9 77.3 78.5 81.0 82.6 86.3 92.1 94.3 96.9 100.2 103.7 105.0 106.1 108.5 109.3 112.8 113.9 124.2 133.4 135.3 139.5 140.3 146.3 150.3 157.9 161.5 162.7 162.8 166.1 168.0 171.4 172.7 179.7 181.1 186.4 200.4 201.9 203.4 212.8 219.2 222.7 224.8 6 23.8 29.8 39.3 41.5 45.3 50.2 51.4 54.9 56.5 60.5 62.9 64.4 67.8 68.9 70.4 72.8 75.9 78.8 81.5 83.6 85.1 86.3 91.4 93.3 94.4 99.3 101.4 103.3 105.7 108.1 108.9 113.6 115.0 116.7 119.0 121.9 123.3 124.3 130.7 131.4 134.1 134.6 136.6 141.0 141.1 152.3 155.2 162.6 165.6 169.2 172.4 173.7 179.2 180.8 185.8 187.4 189.6 199.9 207.3 210.9 213.4 215.2 219.6 221.4 227.9 234.5 J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Continuation of Table II. n Energy 7 36.2 38.9 40.5 42.4 43.6 45.6 49.8 51.0 55.2 58.9 61.5 65.8 67.5 68.5 71.0 72.5 73.9 77.3 77.8 79.0 80.9 82.6 83.3 83.7 85.5 86.9 89.1 89.8 91.1 91.7 92.7 95.6 96.6 98.4 100.0 102.6 103.7 105.4 106.9 110.9 111.9 115.8 116.7 118.8 121.0 124.4 129.7 131.2 132.9 135.6 136.1 138.7 148.0 154.3 156.5 156.6 159.0 160.7 163.6 166.8 167.8 169.6 171.4 178.1 179.0 181.3 182.0 182.2 188.8 195.3 195.4 201.3 207.0 209.0 220.5 221.3 222.5 224.8 8 27.9 28.1 33.1 38.6 39.9 44.5 47.3 50.4 52.2 53.1 55.7 57.7 61.3 62.6 63.9 64.7 66.8 68.3 69.0 70.8 73.3 74.1 76.2 78.0 78.5 79.8 81.3 82.7 84.3 86.4 89.2 89.5 90.7 91.9 93.3 95.3 95.4 96.8 99.8 101.1 103.5 103.9 104.1 106.5 107.4 110.4 111.6 113.9 115.3 117.3 117.7 122.0 126.6 128.7 130.1 132.8 134.5 139.1 139.8 143.4 146.1 148.5 153.8 155.6 161.0 161.5 162.6 164.4 166.7 167.4 172.6 174.7 175.4 176.5 181.5 182.4 183.6 192.9 197.3 199.0 201.9 204.4 204.7 206.0 211.4 214.4 214.8 218.6 220.5 228.5 9 8.7 21.5 21.9 28.8 34.3 34.9 37.2 39.4 45.2 45.9 50.7 53.4 54.4 56.9 58.4 59.4 60.4 62.6 63.8 65.9 66.2 68.6 71.4 72.7 73.6 76.4 77.3 77.9 79.3 80.3 81.1 83.5 84.0 85.3 86.6 87.3 88.4 89.7 90.3 91.7 92.8 93.2 95.0 97.0 99.9 100.1 103.0 104.5 105.4 107.8 111.1 111.6 113.1 113.6 116.0 118.8 121.9 122.8 125.5 130.9 134.7 135.5 137.6 139.2 140.6 142.1 144.8 145.2 150.4 150.5 152.7 155.4 157.7 160.5 162.8 163.8 165.1 168.7 169.4 171.6 172.3 173.2 175.1 179.9 180.3 184.9 188.5 189.8 192.2 192.5 196.4 199.4 202.1 205.9 206.9 208.4 210.6 212.6 213.4 215.8 215.9 219.2 (see Fig. 3). To try to rationalize this growing behavior, we could consider a concept proposed in the literature and known as the con- cept of “geometric magic numbers,” in which 19 atoms form an octa- hedra.63This concept also predicts that the next most stable clusters should possess octahedra fragments, too, and that if this tendency continues, a 44-atom octahedra could be formed (which would be also a geometric magic number63) for the (Pt 3Cu) 11cluster structure. Our results from Fig. 3 indicate, however, that the (Pt 3Cu) n, with n = 6–9 units, clusters prefer to grow in a ⟨110⟩like plane that inter- sects the edges of the octahedron fragment, instead of becoming an incomplete octahedron fragment (see Fig. 3). An alternative way to explain the very peculiar growing pattern behavior observed in these clusters is that to overlap two or three 19-atom octahedra to form a “twinned octahedron” structure. This kind of twinned structure was already found in an experimental work by Sun et al. , although those nanoalloys were formed with twinned icosahedra structures instead than with twinned octahedra.10We note that the structures of the (Pt 3Cu) n, with n = 4–9, clusters are in line with this growing pattern, therefore opening the possibility that this growing behavior could be favored instead of forming a single octahedron, as predicted following the magic numbers concept.From Fig. 3, we can also see that, as it was previously discussed, in the structures of the (Pt 3Cu) nclusters with n = 5 and 6, some copper atoms form a cross-type moieties. We note that these cross- type moieties, however, are no longer present in the larger clusters formed with 7, 8, and 9 units of Pt 3Cu. In fact, in these clusters, instead triangular-type arrangements made by the Cu atoms, sim- ilar to the one present in the (Pt 3Cu) 4cluster, can be observed (see Fig. 3). Finally, we note that, in general, the (Pt 3Cu) nclusters starting from n = 3 present central copper atoms in all structures. This behav- ior, in combination with the cross-type copper and the alternated- type copper patterns, could indicate a preference for the Cu atoms to occupy the center sites and neighboring positions to other Cu atoms. If this tendency continues, adding more Pt 3Cu type units to the system, a Cu@Pt core@shell type nanoparticle could be formed. Once the most stable clusters were obtained, the average bond lengths (ABL) in Å were calculated for each diatomic bond type present in these clusters. To understand the differences in struc- tures and properties of the positive and negative charged species with respect to the neutral clusters, the corresponding lowest energy structures for the cationic and anionic (Pt 3Cu) n, n = 1–9, systems J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Most stable structures, spin mul- tiplicities, symmetry point groups, and approximated size (in nm) of (Pt 3Cu) n, n = 1–9, clusters. were studied, too. Figure 4 presents the calculated ABL for each bond type for the (Pt 3Cu) n, n = 1–9, most stable cluster structures in their neutral, cationic, and anionic states, plotted vs the num- ber n of Pt 3Cu units with which each cluster is initially formed. The obtained ABL values for the Pt–Pt bond type are plotted on the top of Fig. 4, whereas the ABL results for the Pt–Cu and Cu–Cu bonds type are graphically displayed in the middle and at the bottom of Fig. 4,respectively. From the analysis of Fig. 4, we can see that the Pt–Pt ABL values have an oscillating behavior starting from clusters with n = 2. On the other hand, the calculated Pt–Cu and Cu–Cu ABLs do not follow a clear pattern, as it can be seen from Fig. 4. Comparing the ionic clusters with their neutral counterparts, the larger differ- ences are observed for the (Pt 3Cu) n, n = 1 and 2, clusters. For the larger cluster sizes, smaller differences and an asymptotic behavior J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Average bond length (ABL) of the Pt–Pt (a), Pt–Cu (b), and Cu–Cu (c) bond types present in the (Pt 3Cu) n, n = 1–9, clusters. All values are in Å. The ABL values calculated for the cationic species are plotted with dashed lines, whereas the values for the neutral and anionic clusters are reported with continuous lines and dotted lines, respectively. are observed, in agreement with what would expect at the bulk limit (see Fig. 4). Concerning the spin multiplicity, we note that, in general, these clusters prefer low spin multiplicity (see Fig. 3). Going from n = 1 to n = 3, the spin multiplicity increases slowly. It then quenches at n = 6. We note that, in general, for the clusters formed with an odd number of Pt 3Cu units, the preferred spin multiplicity is a quartet, with the exception of the (Pt 3Cu) n, with n = 1 and 9, clusters, whereas mostof the clusters formed with an even number of Pt 3Cu units are found on the triplet PES (see Fig. 3). The magnetic moment per atom (MMA) was calculated in Bohr magneton units for the most stable neutral clusters. The obtained results are displayed in Fig. 5 with respect to the number n of Pt 3Cu units in the clusters. As it can be seen from Fig. 5, a decrease in the MMA is observed as the number of Pt 3Cu units increases from n = 1 to n = 6. As expected, at this stage, the MMA drops to zero. For the (Pt 3Cu) n, with n = 7 and 8, clusters, the MMA then increases and falls again for n = 9 (see Fig. 5). In order gain more insight about where the unpaired electrons of the studied clusters are concentrated, plots of the spin density of the most stable structures in the cationic, neutral, and anionic states have been obtained. The obtained results are depicted in Figs. 6 and 7. In Figs. 6 and 7, the calculated spin density for the cationic clus- ters are presented on the left-hand side, whereas the results for the neutral and cationic clusters are plotted in the middle and in the right-hand side of these figures, respectively. In Figs. 6 and 7, the spin density contributions are plotted with gray color on the top of the corresponding optimized cluster structures. From Figs. 6 and 7, it can be appreciated that the structures of neutral and ionic clus- ters are very similar. In Figs. 6 and 7, the spin multiplicity of the ionic clusters is also indicated. It can be noted from these figures that, similar to the neutral species, the positive and negative charged (Pt 3Cu) n, n = 1–9, clusters also prefer low spin multiplicities. Figures 6 and 7 show that, in general, the spin density contributions in these clusters are allocated on the Pt atoms. D. Energy properties With the aim of gaining more insight into the stability of the studied systems and to provide theoretical information for future experimental studies on these systems, different energy properties such as binding energies per atom (D 0), binding energy per Pt 3Cu unit ( D0, Pt 3Cu), vertical and adiabatic ionization potentials (IPs), vertical and adiabatic electron affinities (EAs), and the difference between IP and EA denoted as Δfor the (Pt 3Cu) n(n = 1–9) clusters were calculated considering the obtained ground state structures. All energy values are reported in eV. To the best of our knowledge, no direct experimental BE, IP, and EA values have been so far reported for (Pt 3Cu) n(n = 1–9) clusters. Therefore, here we will focus our discussion only on the trends of the results we have obtained. FIG. 5 . Magnetic moment per atom (MMA) (in μB) of the (Pt 3Cu) n, n = 1–9, clusters. J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Spin density plots of the (Pt 3Cu) n, n = 1–5, clusters calculated in their cationic (left), neutral (middle), and anionic (right) ground states. The binding energy necessary to extract one Cu or Pt atom ( D0,at) from the clusters of our interest was calculated as D0,at=1 4n(E(Pt3Cu)n−nECu−3nEPt−ZPE). (2)Moreover, to observe the stability effect of adding a Pt 3Cu unit to each cluster size, we calculated D0,Pt3Cuas the energy needed to break the clusters of superior size into Pt 3Cu cluster units as follows: D0,Pt3Cu=E(Pt3Cu)n−nEPt3Cu. (3) J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Spin density of the (Pt 3Cu) n, n = 6–9, clusters calculated in their cationic (left), neutral (middle), and anionic (right) ground states. In the above equations, E(Pt3Cu)ndenotes the energy of the clus- ter of n units, E Cuand E Ptare the energy of the atoms of Cu and Pt, respectively, and ZPE is the calculated zero-point energy of the clusters.In Fig. 8, the results of the calculated binding energy are pre- sented. In this figure, the D0,atvalues are illustrated with diamonds and the D0,Pt3Cuvalues are displayed with squares. As shown in Fig. 8, the two dissociation energies grow monotonically with an increase J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Binding energy per atom, D0,at(diamonds), and binding energy per Pt 3Cu units, D0,Pt3Cu(squares), of the (Pt 3Cu) n, n = 1–9, clusters. All values are in eV. in the cluster size, although the increase in D0,atis less pronounced (see Fig. 8). This result indicates that the clusters became more stable as more Pt 3Cu units are added to the system. In addition, we note that for the larger cluster sizes, an asymptotic behavior is observed in both dissociation energies, indicating that as the cluster size increases, the binding energy tend to reach the bulk limit. FIG. 9 . Vertical and adiabatic ionization potential, IP (dots), vertical and adiabatic electron affinity, EA (diamonds), and difference between the IP and EA, Δ(trian- gles). Vertical properties are given in panel (a), and adiabatic properties are given in panel (b). All values are in eV.In Fig. 9, the vertical and adiabatic IP (shown with dots), the vertical and adiabatic EA (shown with diamonds), and the corre- sponding Δvalues (shown triangles) of the most stable structures of (Pt 3Cu) nclusters with n = 1–9 are presented. As it can be observed from Fig. 9, the calculated IPs decrease, and the computed EAs increase with the increase in the cluster size. By consequence, the calculated Δvalues decreases. This behavior is in agreement with the nature of a metallic system, where Δis expected to approach zero in the bulk limit. IV. CONCLUSIONS In this work, LCGTO-KS-ADFT calculations for the structural and energetic properties of (Pt3Cu)n, n = 1–9, clusters were pre- sented. The employed GGA methodology was first validated study- ing diatomic molecules whose bonds are characteristic for the stud- ied clusters and for which reliable experimental data in gas phase are available. To explore in detail the PES of each cluster, several dozens of structures, taken along BOMD trajectories at different temper- atures, were fully optimized in different spin multiplicities for the various cluster sizes. The optimized structures for the clusters with n = 6 and 7 have an approximate size very close to 1 nm. The approx- imated size for the clusters formed initially with 8 and 9 Pt 3Cu units is larger than 1 nm. The lowest energy structures were characterized by analytic frequency analysis to discriminate them between minima and transition states. The computed frequencies could be used as a guide for future experiments on these clusters, which we hope will be coming. This study represents the first extensive study ever pre- formed for these clusters of nano-dimensions with first-principles methods. It was observed that, in general, the most stable structures prefer low spin multiplicities. The analysis of the growth pattern shows that the (Pt 3Cu) n, n = 2–9, clusters do not conserve the struc- ture of the Pt 3Cu unit. The octahedra geometric magic number con- cept can explain the formation of the 19-atom octahedron fragments observed in the (Pt 3Cu) 5and (Pt 3Cu) 6cluster, as well as the geome- try observed in the (Pt 3Cu) 4as an incomplete octahedra of 19 atoms. The (Pt 3Cu) n, with n = 6–9, clusters show a progressive construction of twinned octahedra structures. This is a very important finding, as the twinned octahedral geometries are predicted to be the most sta- ble geometries for the formation of Pt 3Cu based nanoparticles. The approximate size grows as more Pt 3Cu cluster units are added to the systems. The most stable structures of the clusters formed with 6 and 7 units have an approximate size already very close to a nanometer. The larger cluster studied in this work, initially formed with 9 Pt 3Cu units, has an approximate size of around 1.2 nm, being one of largest transition metal based clusters studied with ab initio first-principles methods. The magnetic moment per atom (MMA) decreases from the smallest cluster until the (Pt 3Cu) 6cluster, in which it drops to zero. An abrupt increment of the MMA is then observed going to the (Pt 3Cu) 7and (Pt 3Cu) 8. The spin density shows a pronounced ten- dency to allocate itself on the Pt atoms. The values obtained for the binding energy indicate that these systems become more and more stable with the increase in the Pt 3Cu units. The other calculated ener- getic properties, in general, follow the tendency expected in metallic systems in which the IP and Δdiminish and the EA increases as more Pt 3Cu units are added to the system. It was observed that the structures with n = 7–9 Pt 3Cu moieties present a kind of twinned octahedra fragments, whereas for the clusters possessing from 4 to J. Chem. Phys. 154, 154302 (2021); doi: 10.1063/5.0045203 154, 154302-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 6 of these units, simple octahedra moieties dominate. In addition, the most thermodynamically stable structures are calculated at every size. This information is important because it serves as a guide for the experimental scientists to try to find a suitable combination of synthesis components that results in the most stable nanoparticles. ACKNOWLEDGMENTS C.D.G.-U. acknowledges CONACYT for the doctoral fellow- ship No. 864427. Support by CONACYT through the Project No. 252658 is gratefully acknowledged. 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5.0057863.pdf

Current-induced magnetization switching at charge-transferred interface between topological insulator (Bi,Sb) 2Te3and van der Waals ferromagnet Fe 3GeTe 2 Cite as: Appl. Phys. Lett. 119, 032402 (2021); doi: 10.1063/5.0057863 Submitted: 25 May 2021 .Accepted: 11 June 2021 . Published Online: 19 July 2021 Reika Fujimura,1Ryutaro Yoshimi,2,a) Masataka Mogi,3Atsushi Tsukazaki,4 Minoru Kawamura,2 Kei S. Takahashi,2Masashi Kawasaki,1,2and Yoshinori Tokura1,2,5 AFFILIATIONS 1Department of Applied Physics and Quantum Phase Electronics Center (QPEC), University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan 2RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan 3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan 5Tokyo College, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan a)Author to whom correspondence should be addressed: ryutaro.yoshimi@riken.jp ABSTRACT Ferromagnetic two-dimensional van der Waals materials attract enormous interest as a platform to explore spin-related quantum phenom- ena, especially in conjunction with other quantum materials. Topological insulator is one such candidate to form the junction, because thespin-polarized nature of the surface or interface Dirac states enables the highly efficient spin-charge conversion. Here, we report the current- driven magnetization switching in the bilayer film of a van der Waals ferromagnetic semimetal Fe 3GeTe 2(FGT) and a topological insulator (Bi1/C0xSbx)2Te3(BST). We observed the current-induced magnetization switching via the Edelstein effect in a wide temperature range, whose threshold current density is as small as that reported for the heterostructure of FGT with a Pt layer. By analyzing the transport properties inheterostructures with different Fermi level ( E F) positions in the BST layer, we found that the EFposition of the charge-transferred interface Dirac states causes the significant variation of the threshold current density with a Bi/Sb ratio. The present result may promise spintronic phenomena in heterostructures of 2D van der Waals ferromagnets with topological insulators/semimetals. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0057863 Two-dimensional (2D) ferromagnets among van der Waals (vdW) materials1–4have gained growing interest from the viewpoints of strong spin fluctuation, magnetic phases, and associated topologicalphenomena. Fe 3GeTe 2(FGT) is one of such compounds, having a hexagonal layered crystal structure of Fe and Ge sandwiched by the Telayer [ Fig. 1(a) ]. 5FGT shows the out-of-plane spontaneous magnetiza- tion with a relatively high transition temperature TCof 200 K,6which is reported to be increased to room temperature.7FGT has a nodal- line band structure near the Fermi level, which opens an energy gap bythe spin–orbit interaction and generates a sizable Berry curvature.Creating a heterostructure of vdW materials allows us to explore afunctionality. For example, the current-induced magnetization switch-ing has been demonstrated in the heterostructure of FGT with the Ptlayer, in which the large spin Hall conductivity of Pt produces the functioning spin current. 8–10 Three-dimensional topological insulators (TIs) are appropriate candidates for forming a heterostructure with FGT to endow the spin-tronic properties. 11TIs have an inverted bulk band structure with spin-polarized surface Dirac states. The spin-polarized nature of thesurface state offers spintronic properties due to the high-efficiencyspin-charge conversion. 12Edelstein effect is one such example; apply- ing an electric field to the spin-polarized bands produces non-equilibrium spin accumulation. 13The current-driven magnetization switching due to the Edelstein effect has so far been established in sev-eral materials systems such as interfaces with metals and oxides, bulkstates of inversion-broken semiconductors, and surface states of Appl. Phys. Lett. 119, 032402 (2021); doi: 10.1063/5.0057863 119, 032402-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplTIs.14–19TIs manifest themselves as archetypal materials endowed with a large spin Hall angle. In this Letter, we report on current-induced magnetization switching in the heterostructures of vdW ferromagnetic semimetalFGT in conjunction with a TI (Bi 1/C0xSbx)2Te3(BST). Exploiting the Edelstein effect in the spin-polarized interface Dirac states, we per- formed the current-induced magnetization switching in a wide tem-perature range below T C. The threshold current density for the magnetization switching is 5.8 /C2106and 1.7 /C2106A/cm2at 100 and 180 K, respectively, which are comparably small with the case of the heterostructure with the Pt layer [ /C249.2/C2106A/cm2at 100 K (Ref. 9) and/C241.9/C2107A/cm2at 180 K (Ref. 10)]. We examined the Fermi level position dependence at the interface Dirac state by changing the Sb composition xof BST. The threshold current density for magneti- zation switching shows a significant variation with the Fermi levelposition. By carefully analyzing the transport properties, we found thatthe charge transfer effect through the hetero-interface causes theobserved x-dependence. We fabricated FGT thin films on semi-insulating InP (111)A substrates by molecular beam epitaxy (MBE). We introduced a 1-nm-thick Sb 2Te3buffer layer beneath the FGT, which assists the epitaxial growth of FGT. We grew thin films at 380/C14C at a base vacuum pres- sure of approximately 1 /C210/C07Pa. All fluxes of elemental Fe, Ge, and Te were supplied from Knudsen cells. The equivalent beam pressures,P Fe,PGe,a n d PTe, were monitored by a beam flux monitor. During thin film growth, PFe,PGe,a n d PTewere kept at 1.0 /C210/C06,2 . 5/C210/C06, and 1/C210/C04Pa, respectively. We evaluated the crystal structure and the film thickness by x-ray diffraction and x-ray reflectivitymeasurement. The magnetic field Bdependences of Hall resistivity qyx and magnetization M[Figs. 1(b) and1(c)] demonstrate the out-of-plane spontaneous magnetization of the FGT layer with an apparent hysteresis. The saturation magnetization of the MBE-grown 6-nm-thickthin film is /C240.8l B/Mn, which is comparable with the results of the previous study on a bulk crystal of FGT.5The temperature dependence of anomalous Hall resistivity qyxAHE,d e fi n e db yt h e qyxvalue at zero magnetic fields, shows a good agreement with that of M, representing a typical ferromagnetic behavior. The TC/C24200 K is also close to the pre- viously reported values for the few-layer and bulk crystal of FGT.5,20 We grew the bilayer films of 8-nm-thick topological insulators (Bi1/C0xSbx)2Te3(BST) and 6-nm-thick FGT on InP (111) substrates [Fig. 2(a) ]. Several films were prepared with changing the Sb content FIG. 2. (a) A schematic of Fe 3GeTe 2/(Bi 1/C0xSbx)2Te3bilayer structure. (b) An optical microscope image of a Hall bar device with an illustration of the measurement setup. The scale bar is 10 lm. (c) Perpendicular magnetic field Bdependence of sheet Hall conductivity rxyin the x¼0.3 bilayer at various temperatures, T¼180, 150, 100, and 60 K. (d) Magnetization switching with changing pulse current densityJ Cpulseunder the in-plane magnetic field of Bx¼0.1 T (red curve) and /C00.1 T (blue curve) in the x¼0.3 bilayer, as measured by the rxyatT¼180, 150, 100, and 60 K. The triangles (red and blue) in the 60 K data indicate the switching thresholdcurrent density J Cth. (e)Tdependence of the JCthand the coercive field HCin the x¼0.3 bilayer. FIG. 1. (a) A schematic of the crystal structure of Fe 3GeTe 2. (b) Perpendicular magnetic field Bdependence of Hall resistivity qyxin a 6-nm FGT thin film at vari- ous temperatures ( T¼2, 10, 50, 100, 150, 180, and 200 K). (c) Perpendicular mag- netic field Bdependence of magnetization M(red) and qyx(blue) at T¼10 K. qyx is normalized by the saturation value qyxsatatB¼5 T. (d) Temperature depen- dence of Mand the anomalous Hall effect qyxAHE.qyxAHEis defined as the remnant value of qyxatB¼0.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032402 (2021); doi: 10.1063/5.0057863 119, 032402-2 Published under an exclusive license by AIP Publishingto control the Fermi level in the BST layer.21,22We fabricated Hall-bar devices by the UV lithography technique to perform the current-induced magnetization switching experiment. Figure 2(b) shows a schematic illustration of measurement configuration, including a top- view photograph of the Hall-bar device. The width of the Hall-bar device is 10 lm. We first characterize the charge transport properties of the bilayer films. The Bdependence of sheet Hall conductivity r xy for the x¼0.3 bilayer [ Fig. 2(c) ] shows a clear hysteresis with the coer- cive field monotonically increasing as decreasing temperature; this cor- roborates that the FGT retains ferromagnetism also in the bilayer. Theobserved magnitude of r xyin the bilayer is even larger than that of the single-layer FGT (see Sec. 1 in the supplementary material ), as will be discussed later [ Fig. 4(b) ]. Here, we calculate sheet Hall conductivity rxyusing the following formula: rxy¼qyx qxx2þqyx2; (1) where qxxrepresents the sheet longitudinal resistivity. To induce the magnetization switching, we injected current pulses with a 100- ls duration by a current pulse generator (Keithley: model 6221). We alsoapplied a small in-plane magnetic field (0.1 T) smaller than the coer- cive field ( H C/C240.2 T at T¼100 K) parallel or anti-parallel to a pulse current direction to make the final states of magnetization determinis-tic. We set the maximum amplitude of the current pulse at 8 mA inthis experiment to avoid damaging the sample by Joule heating. Afterevery current pulse injection, we measured Hall resistivity q yxby the low excitation current of 7 /C2103A/cm2to see how large out-of-plane magnetization component is induced by the pulse current. We convertq yxtorxyby using Eq. (1)and the qxxvalue at a zero magnetic field. We repeated this measurement while changing the amplitude of the current pulses. As shown in Fig. 2(d) , we observed the clear hysteresis ofrxywith respect to JCpulseup to 180 K. The switched- Mpolarity is reversed when we reverse the direction of the in-plane magnetic fieldB x(red trace for Bx¼þ0.1 T and blue for /C00.1 T), corroborating the current-driven magnetization switching by the current-induced spin–orbit torque.8As shown in Fig. 2(e) ,t h e T-dependence of the threshold current density JCthis consistent with that of HC,w h i c h increases with decreasing T, confirming that the perpendicular direc- tion of ferromagnetic moment flips via spin torque.15 We performed the same set of experiments for five bilayers with different Bi/Sb compositions; x¼0, 0.3, 0.5, 0.7, and 1. As shown in Fig. 3(a) , we demonstrate the current-driven magnetization switching for all the bilayers with different xvalues. JCthshows a considerable increase with increasing x[Fig. 3(b) ]. To discuss this, we first elucidate thex-dependence of the EFposition by analyzing the transport proper- ties. In Fig. 4(a) ,w es h o wt h e x-dependence of longitudinal sheet con- ductivity for FGT/BST bilayers ( rFGT =BST xx ), a 6-nm FGT single-layer (rFGT singleðÞ xx ,x-irrelevant; see Sec. S1 in the supplementary material ), and an 8-nm BST single layer ( rBSTðsingleÞ xx ).rFGT =BST xx is larger than rFGTðsingleÞ xx especially for x/C210.5, which indicates that the conduction of the BST layer has to be taken into account for x/C210.5.rFGT =BST xx shows an increasing trend from x¼0.3 to 0.7, whereas rBSTðsingleÞ xx does from x¼0.9 to 1. The sharp increasing in rBSTðsingleÞ xx of BST indicates that the EFenters the bulk band.21We attribute the whole shift of the x-dependence of these longitudinal sheet conductivities [shown by ahorizontal arrow in Fig. 4(a) ] to the charge transfer effect through the interface, which results from the reconfiguration of band alignment atthe heterointerface. Such an effect would be more apparent for theBST layer with lower carrier density, hence with smaller sheetFIG. 3. (a) Magnetization switching with pulse current density JCpulse, measured by rxyfor the x¼0, 0.3, 0.5, 0.7, and 1 bilayers at T¼100 K under the in-plane mag- netic field of B¼0.1 T (red curve) and /C00.1 T (blue curve). (b) xdependence of the switching threshold current density JCth. FIG. 4. (a)xdependence of rxxin the FGT/BST bilayer (red) and the single-layer 8-nm BST thin film (blue) at T¼2K .rxxfor the single-layer 6-nm FGT thin film is shown in a green dotted line. The horizontal arrow represents the whole shift of thexdependence of r xxfrom the BST single layer to the FGT/BST bilayer. (b) xdepen- dence of rxyfor the FGT/BST bilayer (red) at B¼0 T and T¼2K .rxyfor the single-layer 6-nm FGT thin film is shown in green dotted line. (c) Illustrations of the bulk and surface bands of the BST for the single-layer BST ( x¼0, leftmost) film and the FGT/BST ( x¼0–1) bilayers. EFpositions are indicated with a dotted red line for the BST single layer and with solid red lines for the FGT/BST bilayers.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032402 (2021); doi: 10.1063/5.0057863 119, 032402-3 Published under an exclusive license by AIP Publishingconductivity. In Fig. 4(b) , we show the x-dependence of sheet Hall conductivity for the FGT/BST bilayer ( rFGT =BST xy ) and the FGT single layer ( rFGTðsingleÞ xy ).rFGT =BST xy is larger than rFGTðsingleÞ xy for all the xand reaches a maximum around x¼0.5. The significant enhancement of rFGT =BST xy points to the presence of the magnetic proximity effect that opens an exchange gap at the Dirac point of the BST interface state, generating the additional Berry curvature.23,24 From the observations and discussions above, we can deduce the position of EFrelative to the bulk and interface Dirac states, as shown inFig. 4(c) . Compared with the case of the x¼0s i n g l e - l a y e rB S T (Bi2Te3)t h i nfi l m ,t h e EFin the bilayer shifts downwards due to the charge transfer effect. The EFlocates in the BST bulk bandgap for x¼0 and 0.3, while for x/C210.5, the EFis likely buried in the BST bulk valence band top, which results in the significant increase in rxx [Fig. 4(a) ]. These hypotheses can also elucidate the x-dependence of JCthof magnetization switching. The spin-charge conversion via the Edelstein effect occurs in the spin-polarized interface-Dirac state.19 The in-plane charge current exerts the spin-torque more effectively for x/C200.5, where EFlies within the bulk bandgap of TI and hence most of the current flows through the surface state; this appears to lead tothe smaller J CthandrFGT =BST xx forx/C200.5. In conclusion, we have observed the current-induced magnetiza- tion switching via the Edelstein effect in the bilayers of the topological insulator (Bi,Sb) 2Te3and the vdW ferromagnetic semimetal Fe 3GeTe 2. The threshold current density for the magnetization switching is assmall as 5.8 /C210 6A/cm2at 100 K in the optimal case. The charge transfer effect through the hetero-interface causes the shift of theFermi level on the BST surface state and hence causes the significant variation in the threshold current density with a Bi/Sb ratio in the BST layer. The present result demonstrates the importance of the Fermilevel tuning and the bandgap alignment for the spintronic functional-ity in the heterostructure of a vdW ferromagnet and a topologicalinsulator. See the supplementary material for magnetotransport properties of a single-layer FGT thin film. This work was partly supported by the Japan Society for the Promotion of Science through JSPS/MEXT Grant-in-Aid for Scientific Research (Nos. 18H01155 and 19J22547) and JST CREST(Nos. JPMJCR16F1 and JPMJCR1874). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature 546, 265 (2017).2M. Mogi, A. Tsukazaki, Y. Kaneko, R. Yoshimi, K. S. Takahashi, M. Kawasaki, and Y. Tokura, APL Mater. 6, 091104 (2018). 3B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P.Jarillo-Herrero, and X. Xu, Nature 546, 270 (2017). 4L. D. Casto, A. J. Clune, M. O. Yokosuk, J. L. Musfeldt, T. J. Williams, H. L. Zhuang, M. W. Lin, K. Xiao, R. G. Hennig, B. C. Sales, J. Q. 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Commun. 4, 2944 (2013). 18Y. Fan, P. Upadhyaya, X. Kou, M. Lang, S. Takei, Z. Wang, J. Tang, L. He, L. T. Chang, M. Montazeri, G. Yu, W. Jiang, T. Nie, R. N. Schwartz, Y. Tserkovnyak, and K. L. Wang, Nat. Mater. 13, 699 (2014). 19K. Kondou, R. Yoshimi, A. Tsukazaki, Y. Fukuma, J. Matsuno, K. S. Takahashi, M. Kawasaki, Y. Tokura, and Y. Otani, Nat. Phys. 12, 1027 (2016). 20Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu, D. H. Cobden, J. H. Chu, and X. Xu, Nat. Mater. 17, 778 (2018). 21J. Zhang, C. Z. Chang, Z. Zhang, J. Wen, X. Feng, K. Li, M. Liu, K. He, L. Wang, X. Chen, Q. K. Xue, X. Ma, and Y. Wang, Nat. Commun. 2, 574 (2011). 22R. Yoshimi, A. Tsukazaki, Y. Kozuka, J. Falson, K. S. Takahashi, J. G. Checkelsky, N. Nagaosa, M. Kawasaki, and Y. Tokura, Nat. Commun. 6, 6627 (2015). 23M. Mogi, T. Nakajima, V. Ukleev, A. Tsukazaki, R. Yoshimi, M. Kawamura, K. S. Takahashi, T. Hanashima, K. Kakurai, T. Arima, M. Kawasaki, and Y.Tokura, Phys. Rev. Lett. 123, 016804 (2019). 24R. Watanabe, R. Yoshimi, M. Kawamura, M. Mogi, A. Tsukazaki, X. Z. Yu, K. Nakajima, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Appl. Phys. Lett. 115, 102403 (2019).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032402 (2021); doi: 10.1063/5.0057863 119, 032402-4 Published under an exclusive license by AIP Publishing

5.0053586.pdf

Appl. Phys. Lett. 119, 032408 (2021); https://doi.org/10.1063/5.0053586 119, 032408 © 2021 Author(s).Control of antiferromagnetic resonance and the Morin temperature in cation doped α- Fe2-xMxO3 (M = Al, Ru, Rh, and In) Cite as: Appl. Phys. Lett. 119, 032408 (2021); https://doi.org/10.1063/5.0053586 Submitted: 09 April 2021 . Accepted: 08 July 2021 . Published Online: 21 July 2021 Kensuke Hayashi , Keisuke Yamada , Mutsuhiro Shima , Yutaka Ohya , Teruo Ono , and Takahiro Moriyama ARTICLES YOU MAY BE INTERESTED IN Positive correlation between interlayer exchange coupling and the driving current of domain wall motion in a synthetic antiferromagnet Applied Physics Letters 119, 032407 (2021); https://doi.org/10.1063/5.0056056 Magnetization switching and deterministic nucleation in Co/Ni multilayered disks induced by spin–orbit torques Applied Physics Letters 119, 032410 (2021); https://doi.org/10.1063/5.0050641 Temperature dependence of the picosecond spin Seebeck effect Applied Physics Letters 119, 032401 (2021); https://doi.org/10.1063/5.0050205Control of antiferromagnetic resonance and the Morin temperature in cation doped a-Fe2-xMxO3 (M5Al, Ru, Rh, and In) Cite as: Appl. Phys. Lett. 119, 032408 (2021); doi: 10.1063/5.0053586 Submitted: 9 April 2021 .Accepted: 8 July 2021 . Published Online: 21 July 2021 Kensuke Hayashi,1,a)Keisuke Yamada,1 Mutsuhiro Shima,1Yutaka Ohya,1Teruo Ono,2 and Takahiro Moriyama2,b) AFFILIATIONS 1Department of Materials Science and Processing, Graduate School of Natural Science and Technology, Gifu University, Yanagido, Gifu City, Gifu 501-1193, Japan 2Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan a)Present address: International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044, Japan. b)Author to whom correspondence should be addressed: mtaka@scl.kyoto-u.ac.jp ABSTRACT Antiferromagnets are one of the few candidate materials that can work at THz frequency and are, therefore, a potential material for THz technology recently attracting interest from high-speed communication and sensing applications. In this work, we investigate an antiferro-magnetic resonant frequency ( x r) tunability for cation doped a-Fe2O3. It is found that, with various cation dopants (Al, Ru, Rh, and In), the resonant frequency can be tuned in a range between a millimeter-wave band and a THz band. We also complementally discuss the mecha-nism of the Morin temperature ( T M) shift upon cation doping by the temperature and dopant dependence of xr. A good frequency tunability shown in this work suggests that cation doped a-Fe2O3is a useful material for millimeter-wave and THz applications. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0053586 Negligible net magnetization, small magnetic susceptibility, and ultrafast magnetization dynamics are key characteristics of antiferro-magnets and are considered advantageous in the emerging antiferro-magnetic spintronics. 1Recent investigations have led to various interesting results including magnetoresistance,2–6spin torque effect,7,8 and spin current transmission9–15in antiferromagnets, which could make them superior alternative to the ferromagnets typically used inthe conventional spintronics. Yet, what makes antiferromagnetic spintronics so fascinating is the ultra-high frequency magnetic resonance, which can reach up to aTHz range. Antiferromagnetic resonance occurs at a frequencyx r¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi2HEHAp,w h e r e c¼1.76/C21011T/C01s/C01is the gyromagnetic ratio, HEis the exchange field, and HAis the anisotropy field. Because HEis quite large and typically several hundred Tesla, antiferromag- netic resonant frequency can be much higher ( /C24THz) than the ferromagnetic resonance ( /C24GHz), which is just equal to cHA. Antiferromagnets are one of the few candidate materials that can beused in ultra-high frequency devices and components for next genera-tion high speed communications. 16For those applications, it is important to have a good and a wide controllability of the resonant frequency. A commonly availablestrength of the external magnetic field is not effective for a wide rangecontrol of the THz resonant frequency as an external magnetic field H excan only shift the resonance frequency on the order of cHex.( F o r 1T ,cHexis only 0.028 THz.) Therefore, unlike the case for ferromag- netic resonance, there have been different approaches to control theresonant frequency. 17–20 Since both HEand HAare material dependent parameters, a material approach has been commonly taken to control the resonantfrequency. For instance, the antiferromagnetic resonant frequency of NiO that occurs at 1 THz can be controlled by modifying H EandHA with a cation doping (Mgþ,L iþ,a n dM n2þ) in a range of 0.7–1.1 THz.20For a lower frequency range, it has been reported that e-Fe2O3and their cation doped compounds can cover a frequency range of 0.05–0.2 THz.17,18 Another approach is to take advantage of the temperature depen- dence of HEandHA. It is known that the HEandHArapidly decrease in the vicinity of the Neel temperature.19Therefore, a dynamic control Appl. Phys. Lett. 119, 032408 (2021); doi: 10.1063/5.0053586 119, 032408-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplof the resonant frequency is possible by a temperature control, which could be more desirable for device applications. In this work, in order to achieve wider frequency tunability, we investigate the antiferromagnetic resonance in a-Fe2O3, which has a characteristic temperature dependence of the magnetic properties associated with the so-called Morin temperature ( TM) at 260 K. Furthermore, the effect of the cation-doping (Al3þ,R u3þ,R h3þ,I n3þ) on the resonant frequency, which is known to shift TM;21–23is investi- gated. We demonstrate that the significant temperature dependence of the magnetic anisotropy around TMallows a wide tunability of the res- onant frequency. We also discuss some insights into the magneticanisotropy of a-Fe 2O3, which causes the Morin transition and a varia- tion of the antiferromagnetic resonance. a-Fe2O3has a corundum structure [ Fig. 1(a) ]w i t ht h eN e e lt e m - perature TN¼950 K. Magnetic moments residing in Fe3þsites construct two magnetic sublattices with M1andM2coupled antiferro- magnetically.24When temperature T<TM,M1andM2are aligned along the c axis and point in the opposite directions as shown in Fig. 1(b) .A tT>TM, they lie in the a-b plane with an emergence of a small net magnetization in the a–b plane due to the Dzyaloshinskii– Moriya interaction (DMI) as shown in Fig. 1(c) . This magnetic phase transition at T¼TM, the Morin transition, is known to result from a competition of different sources of the magnetic anisotropy and their temperature dependences. The main players for the Morin transition are theoretically predicted to be the magnetic dipole anisotropy (MDanisotropy) and the single ion anisotropy (SI anisotropy), the detail ofwhich will also be discussed later in this paper. 25 Considering all the relevant energies in a-Fe2O3,t h em a g n e t i cf r e e energy of the system is written as F¼JM1/C1M2 ðÞ /C0 D/C1M1ð/C2 M2Þ /C0K1=2c o s h2 1þcosh2 2/C0/C1 /C0K2=2c o s h4 1þcosh4 2/C0/C1 ,w h e r e Jis the exchange integral, D(¼jDj)i st h eD M Ic o n s t a n t , M1andM2are the magnetic moment of the magnetic sublattices in a-Fe2O3,K1 and K2are the magnetic anisotropy constant in the first and the second order, respectively, and h1;2is the angle between M1;2and the c axis as indicated in Fig. 1(c) . The resonant frequency derivedf r o mt h ef r e ee n e r g yc a nb ew r i t t e ni nt e r m so ft h ee f f e c t i v efi e l d s ; HE¼/C0 JM,HD¼DM,HK1¼K1=M,a n d HK2¼K2=Mwith M¼M1jj ¼M2jj as24 xr¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2HEHK1þ2HK2 ðÞ /C0H2 Dq at T <TM (1) and xr¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2 D/C02HEHK1q at T >TM: (2) HK2does not appear in Eq. (2)since the K2term in the free energy is neglected by an approximation h1;2/C25p=2a n dc o s h2 1;2/C250o nr e s o - nance above TM. On the other hand, as far as T<TMis concerned, the spin-flop field Hccan be written as H2 C¼2HEHK1þHK2 ðÞ /C0H2 D:24 T h em a g n e t i cm o m e n t sc a nb efl o p p e db e t w e e nt h eca x i sa n dt h ea - b plane without need of an external field at T¼TM. We, therefore, have ar e l a t i o n s h i p2 HEHK1þHK2 ðÞ ¼H2 Dwith Hc¼0a tT¼TM,w h i c h leads to xr¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2HEHK2pat T ¼TM: (3) We see that xrdecreases toward the Morin temperature and has a minimum at T¼TM. It is obvious that xrat a given temperature (e.g., room temperature) can vary by shifting TM. It is now clear that the magnitude of K1and K2is a variable parameter controlling xr.F o r a-Fe2O3,K1andK2are considered as the sum of MD anisotropy KMDand SI anisotropy KSI1and KSI2 (the first and the second order terms), respectively, written as K1 ¼KMDþKSI1þKSI2andK2¼7 8KSI2:25These relationships are ana- lytically derived from the anisotropic spin Hamiltonian25of the MD and SI anisotropy. As it is noticeable, both MD anisotropy and SIanisotropy contribute to K 1and only SI anisotropy does to K2. MD anisotropy is caused by the dipole field given by the sur- rounding magnetic moments. SI anisotropy is caused by a spin–orbitinteraction associating with the electronic states. There have been anumber of reports showing how T Mchanges when K1(KMD,KSI1,a n d KSI2) is altered by doping of various cations,22–24which conclude that the change in TMis mainly due to the change in SI anisotropy, which is rather indirectly deduced by a fact that MD anisotropy is assumedto be invariant when the amount of dopants is insignificant. Our temperature dependent antiferromagnetic resonance study can indeed directly give some insights of how the cation doping changes SI anisotropy since H K2, which is solely due to SI anisotropy, can be estimated using Eq. (3). a-Fe2-xMxO3pellets are made from mixtures of Fe 2O3,A l 2O3, RuO 2(purity 99.97%), and RhCl 3-aq (Rh 1.000 g/l), and In 2O3.A l 2O3 and In 2O3powders are synthesized by heating Al(NO) 3/C19H2O (purity 98%) and In(NO) 3/C13H2O(purity 99.99%). The composition is con- trolled by adjusting the amount of the powders to the target ratio. Thepowders are mixed in a mortar for 30 min and pre-fired at 1473 K for3 h. The samples are milled again in a mortar for 30 min and uniaxiallydie-pressed with a force of 750 kgf applied onto 100 mm 2to form a /C245-mm-diameter and /C243-mm-thick pellets. The pellets are then sintered at 1573 K for 2 h in air to solidify. For control samples, a purea-Fe 2O3pellet and a single crystal a-Fe2O3are also prepared. The crystal structure for the a-Fe2-xMxO3pellets is characterized by x-ray diffractometry (XRD) using Cu-K aradiation. TMfor the a-Fe2-xMxO3pellets is characterized by the temperature dependence of FIG. 1. (a) Crystal structure of a-Fe 2O3. Magnetic structure of a-Fe 2O3(b) below the Morin temperature and (c) above the Morin temperature. The dotted lines areindicating the a-b plane having hexagonal symmetry.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032408 (2021); doi: 10.1063/5.0053586 119, 032408-2 Published under an exclusive license by AIP Publishingt h em a g n e t i cs u s c e p t i b i l i t y .W ed e fi n e TMas the temperature at which the high temperature susceptibility starts to drop. Antiferromagnetic resonance is characterized by a frequency domain continuous wave THz spectroscopy system19,20capable of scanning in the range of 50 GHz–2 THz. THz transmission through the sample pellet is mea-sured as a function of frequency in the temperature range of 80–460 K. The antiferromagnetic resonance is characterized as an absorptionpeak appearing in a particular frequency. No external magnetic field is applied during the measurement. Figures 2(a)–2(d) show the XRD patterns for the a-Fe 2-xMxO3 pellets. The diffraction peaks corresponding to the corundum struc- ture are observed for all the a-Fe2-xMxO3pellets. The lattice volume estimated from the diffraction peaks is summarized in Fig. 2(e) .F o r the Al doping, the lattice volume decreases with increasing x. For the FIG. 2. XRD patterns for (a) a-Fe 2-xAlxO3, (b)a-Fe 2-xRuxO3, (c)a-Fe 2-xRhxO3, and (d) a-Fe 2-xInxO3. (e)xdependence of the lattice volume. The dotted lines are a guide for the eye.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032408 (2021); doi: 10.1063/5.0053586 119, 032408-3 Published under an exclusive license by AIP PublishingRu, Rh, and In doping, it increases with increasing x. Considering that Fe3þ HSis substituted by Al3þ,R u3þ LS,R h3þ LS,a n dI n3þ(Refs. 23 and24) (subscripts “HS” and “LS” stand for high and low spin states) with the ionic radii of 64.5, 53.5, 68.0, 66.5, and 80.0 pm, respectively,26 the volume contraction and expansion observed in this study is consis-tent with the size of the dopant radii. Figures 3(a)–3(d) show the temperature dependence of the susceptibility D.W ed e t e r m i n e T Mfrom the temperature at which vshow a rapid change. The obtained TMis shown in Fig. 3(e) .TM for the pure a-Fe2O3is determined to be 255 K, which agrees with previous reports ( TM/C24260 K).27,28It is found that the Al and In doping decreases TMand the Ru and Rh doping increases TM. While TMwith the Al, Rh, and In doping almost linearly changes with respect to x, the Ru doping shows a non-linear dependencewith respect to x. These trends also agree well with previous studies.21–23 The temperature dependences of xrf o rv a r i o u ss a m p l e sa r e shown in Fig. 4 . We observed clear antiferromagnetic resonance absorption spectra for the single crystal a-Fe2O3in the whole measure- ment temperature range. For the other samples, however, the absorp- tion spectra tend to be smeared out at low temperatures, which result in lack of xrdata in a low temperature range (see the supplementary material for extensive data). Reason for the disappearance of the absorption spectra is not clear at this point but could be related to thecrystallinity of the sample which affects the resonance linewidth. 19It is found that, at T>TM, the Al and In doping increase xrand the Rh doping decreases xr.A t T<TM, the Ru doping increases xr. Considering that xratT>TMshould increase as TMshifts to a lower FIG. 3. The temperature dependence of susceptibility Dfor (a) a-Fe 2-xAlxO3, (b)a-Fe 2-xRuxO3, (c)a-Fe 2-xRhxO3, and (d) a-Fe 2-xInxO3. (e)xdependence of TM. The dotted lines are a guide for the eye.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032408 (2021); doi: 10.1063/5.0053586 119, 032408-4 Published under an exclusive license by AIP Publishingtemperature and vise versa, changes in xrare consistent with the shift ofTMupon doping. We now discuss mechanisms of the changes in TMandxrby cation doping. It is obvious from our experimental results as well as from the antiferromagnetic resonance model [Eqs. (1)–(3) ] that the modification of TMandxressentially due to the modification of HK1 and/or HK2upon the cation doping. We consider these anisotropy fields in terms of MD anisotropy and SI anisotropy. MD anisotropy caused by the dipole field can be varied by the surrounding moments as well as the distance to them,which could be modified by the cation doping. Considering the limit-ing case for the In: x¼0.10 doping where the magnetic moment reduction and the interatomic distance expansion are maximum in this study, the increase in the interatomic distance by 0.5% character-ized by XRD and the decrease in the magnetic moment by 5% (notethat In: x¼0.10 is indeed replacing 5% of the Fe cation) lead to the decrease in the dipole field, or MD anisotropy, by /C246% using an equa- tion H d¼m=r3,w h e r e Hdis the dipole field, mis the magnetic moment, and ris the interatomic distance. This /C246% change is an order of magnitude smaller than the SI anisotropy changes ( >60%) as estimated in the following paragraph. We, therefore, neglect the MDanisotropy change in our analysis. In regard with SI anisotropy, it is difficult to predict the doping effect on SI anisotropy as the modulation of the spin–orbit interactionupon the cation doping cannot be so straightforward. However, asK 2¼7 8KSI2;25experimentally obtained HK2would hint the behavior of SI anisotropy. HK2is obtained by the following analysis based on Eqs.(2)and(3).We assume that HDandHEdo not change in the measurement temperature range, which is sufficiently lower than the Neel tempera-ture ( T N¼950 K).24ForHK1,w ec o n s i d e rt h e Tdependent and x dependent part separately as HK1¼H0 K1ðxÞH00 K1ðTÞ.W i t h AxðÞ¼cHDðÞ2,BxðÞ¼/C02c2HEH0 K1ðxÞ, we rewrite Eq. (2)as xrx;TðÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AxðÞþBxðÞH00 K1TðÞq : (4) We then expand H00 K1ðTÞaround TMasH00 K1TðÞ¼C0T/C0TM ðÞ þD0T/C0TM ðÞ2…. Taking up to the second order in T, we obtain a fitting function for xratT>TMas xrx;TðÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AxðÞþBxðÞ½CT/C0TM ðÞ þDT/C0TM ðÞ2/C138q :(5) With the known TMfrom the susceptibility measurements, the coefficients CandDare determined by fitting the temperature depen- dence of the xrof the control sample (the single crystal a-Fe2O3). With the determined Cand D,w efi n d AxðÞand BxðÞfora- Fe2-xMxO3by the fitting as shown in Fig. 5(a) . With the determined AxðÞ,BxðÞ,C,a n d D,E q . (5)is extrapolated to find xr(T¼TM)a s indicated by the triangle markers in Fig. 5(a) .HK2is then estimated using Eq. (3)by the xr(T¼TM)a n d HE. Since the cation doping can modify TNby 6% at the maximum (for the Al doping at x¼0.10) based on the previous work,22we take HEconstant and use the value for a pure a-Fe2O3(HE¼920 T).24xdependence of HK2shown in Fig. 5(b) indicates that HK2decreases with increasing the Al and In doping, and it increases with the Rh doping. We note that HK2 decreases quite a lot by >60% for the In doping ( x/C210.03). These FIG. 4. Temperature dependence of xrfor (a) a-Fe 2-xAlxO3, (b)a-Fe 2-xRuxO3, (c)a-Fe 2-xRhxO3, and (d) a-Fe 2-xInxO3.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032408 (2021); doi: 10.1063/5.0053586 119, 032408-5 Published under an exclusive license by AIP Publishingbehaviors of the HK2can translate to the same trend in SI anisotropy to have some insight into the change in SI anisotropy due to the cation doping from the variation in both TMandxr(T¼TM). The Al and In doping, therefore, decrease both the SI anisotropy and TM.O nt h e other hand, the Rh doping increases both SI anisotropy and TM.T h i s correlation between the change in SI anisotropy and the TMshift endorses the previous reports discussing that the SI anisotropy varia- tion is the main contribution to the change in TMas well as xr:22–24 In summary, we prepared cation doped a-Fe2O3pellets by solid-phase synthesis and investigated the Morin temperature TM and the antiferromagnetic resonant frequency xrwith respect to cation dopants (Al, Ru, Rh, and In) and their composition. By the cation doping, we show a tunability of xrin the range between 0.21 and 0.95 THz associating with the shift of TM.T h et e m p e r a - ture and dopant dependence of xrsuggests that the modification in SI anisotropy is responsible for TMshift upon the various cation doping. Our results show that cation doped a-Fe2O3is a candidate material that covers frequency bands for millimeter waves andbeyond, i.e., THz waves. See the supplementary material for extensive data of antiferro- magnetic resonance measurements. This work was supported in part by JSPS KAKENHI Grants Nos. 21H04562, 20J12063, and 19K21972, the Mazda Foundation,the Kyoto University Foundation, ISHIZUE 2020 of Kyoto University Research Development Program, and the Collaborative Research Program of Institute for Chemical Research, KyotoUniversity (Grant Nos. 2020-95 and 2021-96). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1V. Baltz, A. Mamchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018). 2X. Marti, I. Fina, C. Frontera, J. Liu, P. Wadley, Q. He, R. J. Paull, J. D. Clarkson, J. Kudrnovsk /C19y, I. Turek, J. Kune /C20s, D. Yi, J. Chu, C. T. Nelson, L. You, E. Arenholz, S. Salahuddin, J. Fontcuberta, T. Jungwirth, and R. Ramesh,Nat. Mater. 13, 367 (2014).3T. Moriyama, N. Matsuzaki, K.-J. Kim, I. Suzuki, T. Taniyama, and T. Ono, Appl. Phys. Lett. 107, 122403 (2015). 4G. R. Hoogeboom, A. Aqeel, T. Kuschel, T. T. M. Palstra, and B. J. van Wees, Appl. Phys. Lett. 111, 052409 (2017). 5M. Kimata, T. Moriyama, K. Oda, and T. Ono, Appl. Phys. Lett. 116, 192402 (2020). 6T. Iino, T. Moriyama, H. Iwaki, H. Aono, Y. Shiratsuchi, and T. Ono, Appl. Phys. Lett. 114, 022402 (2019). 7P. Wadley, B. Howells, J. Elezny, C. Andrews, V. Hills, R. P. Campion, V. Novak, K. Olejnik, F. Maccherozzi, S. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y. Mokrousov, J. Kune, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds, B. L. Gallagher, and T.Jungwirth, Science 351, 587 (2016). 8T .M o r i y a m a ,K .O d a ,T .O h k o c h i ,M .K i m a t a ,a n dT .O n o , Sci. Rep. 8, 14167 (2018). 9H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev. Lett. 113, 097202 (2014). 10T. Moriyama, S. Takei, M. Nagata, Y. Yoshimura, N. Matsuzaki, T. Terashima, Y. Tserkovnyak, and T. Ono, Appl. Phys. Lett. 106, 162406 (2015). 11L. Frangou, S. Oyarz /C19un, S. Auffret, L. Vila, S. Gambarelli, and V. Baltz, Phys. Rev. Lett. 116, 077203 (2016). 12S. Takei, T. Moriyama, T. Ono, and Y. Tserkovnyak, Phys. Rev. B 92, 020409R (2015). 13Z .Q i u ,J .L i ,D .H o u ,E .A r e n h o l z ,A .T .N ’ D i a y e ,A .T a n ,K .U c h i d a ,K .S a t o ,S .Okamoto, Y. Tserkovnyak, Z. Q. Qiu, and E. Saitoh, Nat. Commun. 7, 12670 (2016). 14T. Moriyama, M. Kamiya, K. Oda, K. Tanaka, K.-J. Kim, and T. Ono, Phys. Rev. Lett. 119, 267204 (2017). 15R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati, J. Cramer, A. Brataas, R. A. Duine, and M. Kl €aui,Nature 561, 222 (2018). 16K. David and H. Berndt, “6G vision and requirements,” IEEE Veh. Technol. Mag. 13, 72 (2018). 17S. Ohkoshi, S. Kuroki, S. Sakurai, K. Matsumoto, K. Sato, and S. Sasaki, Angew. Chem. Int. Ed. 46, 8392 (2007). 18A. Namai, M. Yoshikiyo, K. Yamada, S. Sakurai, T. Goto, T. Yoshida, T. Miyazaki, M. Nakajima, T. Suemoto, H. Tokoro, and S. Ohkoshi, Nat. Commun. 3, 1035 (2012). 19T. Moriyama, K. Hayashi, K. Yamada, M. Shima, Y. Ohya, and T. Ono, Phys. Rev. Mater. 3, 051402 (2019). 20T. Moriyama, K. Hayashi, K. Yamada, M. Shima, Y. Ohya, and T. Ono, Phys. Rev. Mater. 4, 074402 (2020). 21A. H. Morrish and J. A. Eaton, J. Appl. Phys. 42, 1495 (1971). 22E. Svab and E. Kren, J. Mag. Mag. Mater. 14, 184 (1979). 23J. Z. Liu and C. L. Fan, Phys. Lett. A 105, 80 (1984). 24A. H. Morrish, Canted Antiferromagnetism: Hematite , 1st ed. (World Scientific Publishing, Singapore, 1994). 25L. M. Levinson, M. Luban, and S. Shtrikman, Phys. Rev. 187, 715 (1969). 26R. D. Shannon, Acta. Cryst. 32, 751 (1976). 27F. J. Morin, Phys. Rev. 78, 819 (1950). 28L. Neel and R. Pauthenet, C. R. Acad. Sci. 234, 2172 (1952). FIG. 5. (a) The experimental data (markers) and the fitting (solid line) for temperature dependence of xrfora-Fe 2-xMxO3. (b)xdependence of HK2atTMfor the Al, Rh, and In doping. The dotted lines are a guide for the eye.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032408 (2021); doi: 10.1063/5.0053586 119, 032408-6 Published under an exclusive license by AIP Publishing

5.0046217.pdf

Voltage-control of damping constant in magnetic-insulator/topological-insulator bilayers Cite as: Appl. Phys. Lett. 118, 252402 (2021); doi: 10.1063/5.0046217 Submitted: 1 February 2021 .Accepted: 3 June 2021 . Published Online: 21 June 2021 Takahiro Chiba,1,a) Alejandro O. Leon,2 and Takashi Komine3 AFFILIATIONS 1National Institute of Technology, Fukushima College, 30 Nagao, Kamiarakawa, Taira, Iwaki, Fukushima 970-8034, Japan 2Departamento de F /C19ısica, Facultad de Ciencias Naturales, Matem /C19atica y del Medio Ambiente, Universidad Tecnol /C19ogica Metropolitana, Las Palmeras 3360, ~Nu~noa 780-0003, Santiago, Chile 3Graduate School of Science and Engineering, Ibaraki University, 4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511, Japan a)Author to whom correspondence should be addressed: t.chiba@fukushima-nct.ac.jp ABSTRACT The magnetic damping constant is a critical parameter for magnetization dynamics and the efficiency of memory devices and magnon transport. Therefore, its manipulation by electric fields is crucial in spintronics. Here, we theoretically demonstrate the voltage-control of magnetic damping in ferro- and ferrimagnetic-insulator (FI)/topological-insulator (TI) bilayers. Assuming a capacitor-like setup, we formu-late an effective dissipation torque induced by spin-charge pumping at the FI/TI interface as a function of an applied voltage. By using realis-tic material parameters, we find that the effective damping for a FI with 10 nm thickness can be tuned by one order of magnitude under thevoltage of 0.25 V. Also, we provide perspectives on the voltage-induced modulation of the magnon spin transport on proximity-coupled FIs. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0046217 Voltage or electric-field control of magnetic properties is funda- mentally and technologically crucial for energetically efficient spin- tronic technologies 1,2such as magnetic random-access memories (MRAMs),3spin transistors,4,5and spin-wave-based logic gates.6In these technologies, voltage-control of magnetic anisotropy (VCMA) inthin ferromagnets 7–9promises energy-efficient reversal of magnetiza- tion by a pulsed voltage10–12and manipulation of propagating spin waves with lower power consumption.13The control of magnetic damping is also highly desirable to increase the performance of spin- tronic devices. For instance, a low magnetic damping allows small crit- ical current densities for magnetization switching and spin-waveexcitation by current-induced spin-transfer 14and spin–orbit tor- ques.15,16On the other hand, a high magnetic damping can be benefi- cial in reducing the data writing time in MRAM devices. Formagnonic devices, magnetic damping is a key factor, because it gov- erns the lifetime of spin waves or magnons as information carriers. 17 Even if the magnetic damping is a vital material parameter that governs magnetization dynamics in several spintronic devices, itsvoltage-control is not fully explored except for a few experiments withferro- and ferrimagnets. 18–22 The main origin of magnetic dissipation is the spin–orbit interac- tion (SOI), which creates relaxation paths of the spin-angular momen-tum into conduction electrons and the lattice. Hence, potentialcandidates to achieve the voltage-control of magnetic damping are magnetic materials and/or strong SOI systems. Three-dimensional topological insulators (3D TIs), such as Bi 2Se3, are characterized by band inversion due to a strong SOI23,24and possess an ideally insulat- ing bulk and spin-momentum locked metallic surface states. Recently,Bi 2/C0xSbxTe3/C0ySey(BSTS)25and Sn-doped Bi 2/C0xSbxTe2S26have been reported to be ideal 3D TIs with two-dimensional (2D) Dirac electronson the surface and a highly insulating bulk. For spintronics, the inter-face between a ferromagnet and a TI can enhance the magnitude of both spin and charge currents. 27,28Some experiments reported29–32 the spin-charge conversion at room temperature33,34in a bilayer of TI/ ferro- and ferrimagnetic-insulator (FI) such as Y 3Fe5O12(YIG) with a very low Gilbert damping constant ( a). An essential feature of the FI/ TI bilayer is that the TI bulk behaves as a semiconductor, enabling thecontrol of the surface carrier density by a voltage. 35Also, magnetically doped TI exhibits VCMA.12,36Hence, TIs are a promising candidate to achieve the voltage-control of magnetic damping. In this work, we theoretically demonstrate the voltage-control of magnetic damping in FI/TI bilayers. We formulate an effective dissipa-tion torque induced by spin-charge pumping at the FI/TI interface asa function of a gate voltage V G. Our main result is that the voltage changes the effective damping by one order of magnitude for a FI witha perpendicular magnetization configuration and 10 nm thickness. Appl. Phys. Lett. 118, 252402 (2021); doi: 10.1063/5.0046217 118, 252402-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplAlso, we provide perspectives on the modification of magnon scatter- ing time in a FI-based magnonic device. To study the effective damping torque, we consider 2D massless Dirac electrons on the TI surface with the magnetic proximityeffect, 29–32i.e., coupled to the magnetization of an adjacent FI. The exchange interaction between the surface electrons and the FI magne- tization is modeled by a constant spin splitting along the magnetiza-tion direction with unit vector m¼M=M s(in which Mis the magnetization vector with the saturation magnetization Ms).37Then, the following 2D Dirac Hamiltonian provides a simple model for the FI/TI interface state:38 ^H¼vFr/C1^p/C2^zðÞ þDr/C1m; (1) where /C22his the reduced Planck constant, vFis the Fermi velocity of the Dirac electrons at zero applied voltage, ^p¼/C0i/C22h$is the momentum operator, f^x;^y;^zgare the unit vectors along the respective Cartesian axes, r¼ðrx;ry;rzÞis the vector of Pauli matrices for the spin, and Dis the exchange interaction constant. For simplicity, we ignore here the particle-hole asymmetry and the hexagonal warping effect in the surface bands. Also, DandvFare assumed to be temperature indepen- dent.40Note that we operate in the weak magnetic coupling limit, and therefore, self-consistent treatment for the induced gap ( D)39is not necessary. Let us begin by calculating the dissipation torque induced by the spin-charge pumping28,41–43of a dynamic magnetization in FI/TI bilayers. A precessing magnetization, driven by ferromagnetic reso-nance (FMR), as shown in Fig. 1 , can be regarded as an effective vector potential A effðtÞ¼D=ðevFÞ^z/C2mðtÞwith the electron charge /C0eðe>0Þ, which drives a charge current via an effective electric field Eeff¼/C0 @tAeff(see the supplementary material ), i.e., JP¼D evFrAH@m @t/C0rL^z/C2@m @t/C18/C19 ; (2) where rLandrAHare longitudinal and transverse (anomalous-Hall) conductivities, respectively, and depend on the z-component of the magnetization ( mz).38From the Hamiltonian (1), the velocity operator ^v¼@^H=@^p¼vF^z/C2rdepends linearly on r. Therefore, the non- equilibrium spin polarization lP(in units of m/C02) is a linear function of the charge current JPon the TI surface, i.e., lP¼^z/C2JP=ðevFÞ. This nonequilibrium spin polarization lPexerts a dissipation torque on the magnetization, TSP¼/C0cD=ðMsdÞlP/C2m,n a m e l y , TSP¼/C0 aAHmzþaLm/C2 ðÞ@m @t/C0@mz @t^z/C18/C19 ; (3) with aLðAHÞ¼cD2 e2v2 FMsdrLðAHÞ; (4) where cis the gyromagnetic ratio and dis the thickness of the FI layer. Equation (3)is equivalent to the charge-pumping-induced damping- like torque that Ndiaye et al. derived using the Onsager reciprocity relation for a current-induced spin–orbit torque.43The first term in Eq. (3) originates from the magnetoelectric coupling (the Chern–Simons term)37,44and renormalizes the gyromagnetic ratio. By using parameters listed in Table I ,D¼40 meV and d¼10 nm,aAH/C2510/C04is estimated even by using rAHat 0 K as the upper value.38 Thus, we disregard the renormalization of the gyromagnetic ratio. In contrast, the second term in Eq. (3)stems from the Rashba–Edelstein effect due to the spin-momentum locking on the TI surface41and con- tributes to magnetic damping. Since we are interested in voltage-control of magnetic damping, we hereafter focus on a Lin this study. FIG. 1. (a) Schematic geometry (side view) of a capacitor-like device comprising a ferromagnetic insulator (FI) film (with thickness d) sandwiched by a TI and a normal metal as a top electric gate with VG. The yellow line corresponds to the TI surface state. The red arrow denotes the precessional magnetization directions in the FMRdriven by static ( B eff) and oscillating ( hðtÞ) magnetic fields. (b) Schematic geometry of a transistor-like device comprising a FI film sandwiched by a TI and a normal metal. The red wave arrow represents a magnon current driven by the difference inspin accumulation ( l L/C0lR) in the attached left and right heavy metal (HM) leads. TABLE I. Material parameters for the TI/FI bilayer. Symbol Value Unit BSTS Fermi velocityavF 4:0/C2105ms/C01 BSTS bulk bandgapa2Ec 300 meV YIG gyromagnetic ratiobc 1.76/C21011T/C01s/C01 YIG Gilbert damping constantba 6.7/C210/C05 YIG saturation magnetizationbMs 1.56/C2105Am/C01 YIG relative permittivityc/C15=/C150 15 aReference 25. bReference 33. cReference 49.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252402 (2021); doi: 10.1063/5.0046217 118, 252402-2 Published under an exclusive license by AIP PublishingAccording to Eq. (4), the electric field effect on the conductivity rLcan be used to control the magnetic dissipation. Namely, the voltage-induced change of the interfacial density of states in rLrenders the TI a more or less efficient spin sink. The damping enhancement aL depends on the chemical potential l, measured from the original band-touching (Dirac) point. At room temperature, or below it, the thermal energy is much smaller than the Fermi one, kBT/C28EF,w i t h Tbeing the temperature and kBbeing the Boltzmann constant. Then, we can use the following Sommerfeld expansion of the chemicalpotential l: lðTÞ/C25E F1/C0p2 6kBT EF/C18/C192"# ; (5) with the voltage-dependent Fermi energy,12EF¼lð0Þ,g i v e nb y EFðVGÞ¼/C22hvFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pnintþD2 4p/C22hvFðÞ2þ/C15 edVG !vuut; (6) where /C15is the permittivity of a FI and n int¼ðE2 Fð0Þ/C0D2Þ=ð4p/C22h2v2 FÞ is the intrinsic carrier density, i.e., at VG¼0. Note that we can define a voltage-dependent surface electron density nVðVGÞ/C17nintþ/C15VG= ðedÞthat shows the underlying mechanism behind the voltage-control of interfacial phenomena in insulating bilayers with surface carriers, which goes beyond topological materials. Namely, a voltage increases or decreases the effective electron density and therefore enhances or weakens all effects that depend on this density, including isotropic45 and anisotropic46exchange interactions, emergence of magnetization in metals,47perpendicular magnetic anisotropy,3,9and spin–orbit tor- ques.12The voltage-generated change in the surface density is equiva- lent to an interfacial Fermi energy shift. In this work, we predict that the spin-charge pumping efficiency is also modulated, an effect that may also appear in usual FI jnormal metal bilayers since the spin- mixing conductance depends on the electronic density.48 We investigate the effect of electric-gate on the effective damping aLso that we assume hereafter that the low-energy Dirac Hamiltonian (1) is an accurate description for a momentum cut kc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2 c/C0D2q =ð/C22hvFÞ,i nw h i c h2 Ecis the bulk bandgap of TIs50[see Fig. 2(c) ]. Sufficiently far from the Dirac point ( /C22hs=EF/C281a n d sisthe transport relaxation time), the electron scattering can be treated by the first Born approximation.51With this, the longitudinal conductiv- ity reads52 rL¼e2 2hðEc /C0EcdEkEksðEk;TÞ /C22hE2 k/C0D2m2 z E2 kþ3D2m2 z/C0@fFD @Ek/C18/C19 ; (7) where fFD¼½expðEk/C0lÞ=ðkBTÞ/C8/C9 þ1/C138/C01is the Fermi–Dirac dis- tribution, the energy Ekis the eigenvalue of Eq. (1),a n d sðEk;TÞis the transport relaxation time of massless Dirac electrons within the Bornapproximation for impurity and phonon scatterings. By applying theMatthiessen rule 1 sðEk;TÞ¼EkaþbkBT ðÞ ; (8) where a¼nV2 0=ð4/C22h3v2 FÞ(in units of eV/C01s/C01) parameterize contribu- tion of the impurity scattering,53,54nis the impurity concentration, andV0is the scattering potential. Also, contribution to the transport relaxation time from the phonon scattering53,54can be approximated byb¼D2 0=ð4/C22h3v2 Fqtsv2 LÞ(in units of eV/C02s/C01), where qis the mass density of the quintuple layer (QL) in the TI crystal structure, tsis the thickness of one atomic layer in 1 QL of TIs, vLis the longitudinal phonon velocity, and D0is the deformation potential constant. Figures 2(a) and 2(b) show the VGand Tdependence of the effective damping enhancement aLfor out-of-plane ( mz¼1) and in- plane ( mz¼0) magnetization configurations, respectively. Also, Fig. 2(c)illustrates the voltage modulation of EFin TI. The bulk damping constant can be influenced by material and device parameters such asSOI and magnetic anisotropies. 34However, we predict the voltage- modulation of the damping enhancement by spin-charge pumping. Therefore, our results are independent of the intrinsic dissipationmechanisms. At the FI/TI interface, orbital hybridization between TIand the 3 dtransition metal in FI, such as YIG, deforms the TI surface states, which might shift the Dirac point to the lower energy and liftupE F,55so that we consider relatively high value EFðVG¼0Þ¼140 meV with the corresponding carrier density of the order of 1012cm/C02. Also,Dis used within the values reported experimentally in FI- attached TIs.56,57For impurity parameters, we use n¼1011cm/C02and V0¼0:15 keV A ˚2based on an analysis of the transport properties of a TI surface.52We could not find estimates of the phonon scattering FIG. 2. Effective damping enhancement aLof a TI/FI bilayer as functions of VGandTforEFðVG¼0Þ¼ 140 meV: (a) mz¼1 and (b) mz¼0. (c) Voltage modulation of EFin a TI. Insets represent schematic of massless (dashed line) and massive (solid line) surface state dispersions in the bulk bandgap. In these graphs, we use parameters listed in Table I ,D¼40 meV and d¼10 nm for a FI thickness. The details of the calculations are given in the text.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252402 (2021); doi: 10.1063/5.0046217 118, 252402-3 Published under an exclusive license by AIP Publishingfor BSTS in the literature so that we adopt those of nonsubstituted Bi2Te3being vL¼2:9/C2105ms/C01;D0¼35 eV, ts¼0:16 nm, and q¼7:86/C2103kg m/C03in Ref. 58. These scattering parameters describe a relatively clean interface with the sheet resistance /C241kX, which is one order less than that of experiments. In Figs. 2(a) and 2(b),aLmonotonically decreases with increasing Tat a fixed VGwhile i th a sp e a k sf o rc h a n g i n g VGat a fixed T>0 (see also the inset of Fig. 3 ). This feature reflects thermal excitation of surface carriers into the bulk states ( Ek>Ec), reducing the spin-charge-pumping contri- bution. With the out-of-plane configuration, aLcan be tuned by one order of magnitude under the voltage, while aLchanges by less than a factor two with the in-plane state, which suggests that the out-of-plane configuration is superior in controllability. The calculated Tdepen- dence of damping enhancement at VG¼0 for the in-plane configura- tion agrees with a few experiments with the FI/TI bilayer.31,59Note that at much lower than EFðVG¼/C0250 mV Þ/C2540 meV, our calcula- tion with the in-plane configuration breaks down because of the finite level broadening due to the higher-order impurity scattering.60The VG-dependent FMR is characterized by the Landau–Lifshitz–Gilbert theory in the supplementary material . The electric manipulation of magnon spin transport is a rele- vant topic in spintronics. For example, in YIG with an injector and a detector Pt contact, changes of the magnon spin conductivity canbe obtained by using a third electrode that changes the magnondensity, 61–63potentially providing a functionality similar to the one of field-effect transistors. Damping compensation by current- driven torques64,65in magnetic heterostructures also influences magnon transport. Here, we provide a perspective on the electric-field-induced modulation of magnon scattering time, s m. Magnons can be injected and detected by their interconversion with charge currents in adjacent heavy metals (HMs) through the direct and inverse spin-Hall effects.33Similar to charge transport induced by an electrochemical potential gradient, a magnon spin current canbe driven by the gradient of a magnon chemical potential injectedby an external source. 66,67Magnon transport through a FI can becontrolled by the gate voltage that modulates the effective damping in Eq. (4). So far, the magnon spin transport in the FI/TI bilayer lacks microscopic theory with few exceptions.68,69However, from the bulk of magnon spin transport,66the control of smresults in the modifica- tion of all transport properties, including the magnon spin conductiv- ity. In the presence of a TI contact, interfacial magnons are scattered by conducting Dirac electrons on the TI surface.70Considering a very thin ferromagnet that can be modeled by a 2D magnet, the inset ofFig. 3 shows that the damping enhancement is at least one order of magnitude larger than that of the bulk one of YIG. 33,34Accordingly, let us assume that interfacial magnons are absorbed by transferringtheir energy and angular momentum to Dirac electrons at a rate1=s m/aL.66While there is no know microscopic expression for the magnon spin conductivity in the present system, bulk magnon trans- port obeys the relationship rm/sm,61,62where rmis the magnon spin conductivity. In our case, the scattering time smis dominated by the magnon-relaxation process into the FI/TI interface. To estimate aneffect of electric-gate on the magnon spin transport, we define the modulation efficiency g m¼smðVGÞ/C0smðVmaxÞ smðVmaxÞ¼aLðVmaxÞ aLðVGÞ/C01; (9) where Vmaxð/C25 /C0 68 mV for Fig. 3 ) gives the maximum value of aL (and therefore the minimum value of sm). In principle, smdepends on VGthrough not only aLbut also via magnon dispersion relation, /C22hxq,66including a VG-dependent magnetic anisotropy. However, this VG-dependence is quite small even for a FI with 2 nm thickness (see thesupplementary material ), so that we disregard the influence of the magnon gap in the following calculation. Figure 3 shows VG-depen- dence of the modulation efficiency at room temperature, in which the strongly nonlinear behavior is interpreted as follows. Down to VG /C25/C0130 mV ;aLis affected by the thermal excitation of surface car- riers, which makes a peak around VG/C25/C070 mV. From /C0130 to /C0250 mV, the thermal excitation is suppressed, so that aLmonotoni- cally decreases with jVGjdue to the reduction of the Fermi surface. Hence, in this regime, one can effectively modulate the magnon spintransport by the voltage. In summary, we have theoretically demonstrated the voltage- control of magnetic damping in ferro- and ferrimagnetic insulator (FI)/ topological insulator (TI) bilayers. Assuming a capacitor-like setup, weformulate an effective damping torque induced by spin-charge pump-ing at the FI/TI interface as a gate voltage function. The presence of a perpendicular electric field results in a shift of the Fermi level or, equiv- alently, a modified interfacial electron density, increasing or decreasingthe efficiency of the pumping process. We studied the consequences ofthis damping enhancement using realistic material parameters for FI and TI. We found that the effective damping with the out-of-plane magnetization configuration can be modulated by one order of magni-tude under the voltage with 0.25 V. The present results motivate anapplication: the magnon scattering time can be tuned by a gate voltage,potentially allowing for a magnon transistor type of application. A complete quantitative description of the latter requires a microscopic theory of magnon spin transport in FI/TI bilayers, which might remainan unexplored issue. The voltage-control of magnetic damping pavesthe way for low-power spintronic and magnonic technologies beyond the current-based control. FIG. 3. Modulation efficiency ( gm) as a function of the gate voltage. Inset shows the corresponding behavior of the effective damping enhancement aL. In these graphs, we use parameters listed in Table I ,D¼40 meV and d¼2 nm for a FI thickness. We also set EFðVG¼0Þ¼ 140 meV and T¼300 K.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252402 (2021); doi: 10.1063/5.0046217 118, 252402-4 Published under an exclusive license by AIP PublishingSee the supplementary material for the calculation of the spin- charge pumping in FI/TI bilayers, the characterization of the FMR under several values of the applied voltage, and the influence of VG- dependence of the anisotropy in the magnon dispersion. We thank Camilo Ulloa and Nicolas Vidal-Silva for fruitful discussions. This work was supported by Grants-in-Aid for Scientific Research (Grant Nos. 20K15163 and 20H02196) from theJSPS and Postdoctorado FONDECYT 2019 Folio No. 3190030. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, “Electric-field control of ferromagnetism,” Nature 408, 944 (2000). 2C. Song, B. Cui, F. Li, X. Zhou, and F. 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5.0053234.pdf

APL Photonics ARTICLE scitation.org/journal/app Strong multipolar transition enhancement with graphene nanoislands Cite as: APL Photon. 6, 086103 (2021); doi: 10.1063/5.0053234 Submitted: 6 April 2021 •Accepted: 19 July 2021 • Published Online: 2 August 2021 Gilles Rosolena) and Bjorn Maes AFFILIATIONS Micro and Nanophotonic Materials Group, Research Institute for Materials Science and Engineering, University of Mons, Place du Parc 20, 7000 Mons, Belgium a)Author to whom correspondence should be addressed: gilles.rosolen@umons.ac.be ABSTRACT For a long time, the point-dipole model was a central and natural approximation in the field of photonics. This approach assumes that the wavelength is much larger than the size of the emitting atom or molecule so that the emitter can be described as a single or a collection of elementary dipoles. This approximation no longer holds near plasmonic nanostructures, where the effective wavelength can reach the nanometer-scale. In that case, deviations arise and high-order transitions, beyond the dipolar ones, are not forbidden anymore. Typically, this situation requires intensive numerical efforts to compute the photonic response over the spatial extent of the emitter wavefunctions. Here, we develop an efficient and general model for the multipolar transition rates of a quantum emitter in a photonic environment by computing Green’s function through an eigen permittivity modal expansion. A major benefit of this approach is that the position of the emitter and the permittivity of the material can be swept in a rapid way. To illustrate, we apply the method on various forms of graphene nanoislands, and we demonstrate a local breakdown of the selection rules, with quadrupolar transition rates becoming 100 times larger than dipolar ones. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0053234 I. INTRODUCTION Usually the quantum emitter is approximated as a point emit- ting a dipolar electric field. This is a legitimate approximation when the wavelength of the emitted light is much larger than the size of the atom or molecule. In that case, high-order transitions (transitions beyond the dipolar one, such as two-photon processes and elec- tric and magnetic multipolar transitions) are negligible and termed forbidden. However, these transitions are of particular importance in spectroscopy, photochemistry, quantum information, and many other fields. Here, we develop a numerical method that computes the multipolar transition rates of a quantum emitter in a general photonic environment, and we show that graphene nanoislands can locally break the conventional selection rules of a hydrogen-like emitter. The transitions are governed by the coupling between the charged constituents of the emitter and the electromagnetic field. In free space, the latter is a plane wave, with expansion exp (ik⋅x) ≈1+i(k⋅x)−0.5(k⋅x)2+⋅⋅⋅ in the limit of k⋅x→0. Through Fermi’s golden rule, each term, which corresponds to a num- ber of gradients of the electromagnetic field (0, 1, 2, ...), can betraced back to a particular multipolar transition.1For visible light, the wave vector is ∣k∣=107m−1, and for hydrogen-like atoms, ⟨∣x∣⟩≈10−10m. It directly shows that the first term in the Tay- lor expansion, attributed to the dipolar transition, dominates by 3 orders of magnitude compared to the linear order, attributed to the quadrupolar transition, and by 6 orders of magnitude com- pared to the quadratic term, attributed to the octupolar transition.2 Therefore, stronger field gradients over the spatial extent of the wave function of the emitter are necessary to enhance higher-order transitions.1 One route toward non-negligible higher-order terms is, there- fore, to consider larger emitters, as shown for quantum dots3and Rydberg excitons.4Another route consists of enhancing the wave vector magnitude ∣k∣by confining light in a nanophotonic structure. The wave vector can be written as k=η0ω/c, with the confinement factorη0being the ratio between the vacuum and the effective wave- length. In this case, higher-order transitions are enhanced by a factor η0to the power of the considered order (for example, the octupolar transition is enhanced by a factor η2 0).5 Under these conditions, plasmonic nanoantennas are ideal can- didates to enhance higher order transitions. For instance, in noble APL Photon. 6, 086103 (2021); doi: 10.1063/5.0053234 6, 086103-1 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app metals, forbidden quadrupolar transitions are enhanced for emit- ters close to tips,6interfaces,7nanowires,8nanogaps,9arrays,10and gold dimers.2,11,12The strongly confined graphene plasmons ( η0 =150–300, depending on the absorption losses13,14) form an excel- lent platform for high-order transitions, which can occur efficiently, even similar to dipolar transitions.5In the case of extremely high confinement ( η0>500) of plasmons in a two-dimensional mate- rial sheet, higher-order transition rates can surpass lower order transitions, hence breaking the conventional selection rules.5,15 Generally, accessing high-orders allows us to probe a much larger range of the electronic energy level structure of an emitter, finding a way to a multiplex and broadband spectroscopy plat- form.5,11These higher-order transitions already play an important role in spectroscopy of many relevant chemical species, from indi- vidual atoms16,17to larger molecules with high symmetry, such as dihydrogen, carbon dioxide, methane, and benzene.18,19In pho- tochemistry, enhancing the magnetic dipole transition in oxygen is interesting for photochemical reactions.20Finally, interference effects between multipolar orders can occur: the possibility of com- plete suppression of a certain transition through interference is required for many applications in the context of quantum comput- ing, quantum storage, and quantum communication.3,21 Despite its high potential and these developments, the field is currently limited by the difficulty in computing the electromag- netic environment of the emitter. Indeed, computing the sponta- neous emission rates of a quantum emitter requires the knowl- edge of the electromagnetic field profile over the spatial extent of the wavefunctions of the emitter.22Usually, the problem is solved for absorption rates: in that case, a plane wave excites a nanophotonic structure and the near-field is extracted.1,6,9,10This is a straightforward routine for conventional numerical methods, such as the finite-element method23,24or the finite-difference time- domain method.25For spontaneous emission, however, the knowl- edge of the vacuum field is essential. As a first approximation, one can resort to symmetric problems26or consider only the rel- evant (properly quantized) modes of the structure for the pro- cess.2,8,11The complete resolution, however, requires knowledge of Green’s function, which is analytical only for uniform media and for simple geometries.5,15Numerical evaluation is very demanding with conventional numerical methods, as repeated simulations for different positions and orientations of a point dipole source are necessary.12 In order to compute advanced photonic structures, a modal- based approach is very useful: a single simulation that determines the modes (e.g., of a cavity) is required to know the full spatial vari- ation of Green’s function.27The eigen permittivity modal expansion is particularly suited for the spontaneous emission of an emitter for which the emission frequency is fixed. Eigen permittivity modes have a permittivity eigenvalue that pertains only to a scattering ele- ment, which spans a finite portion of space. As a result, the nor- malization is trivial. Furthermore, they are orthogonal and appear to form a complete set.28,29Once computed for a scatterer at a fixed wavelength, they straightforwardly give the optical response for any material constituting this scatterer. These modes have been derived during the 1970s in the quasi-static approximation, and were used to derive bounds for scattering problems,30to study spasers,31disor- dered media,32and second harmonic generation.33The formalism (called GENOME for GEneralized NOrmal Mode Expansion) wasrecently extended beyond the quasistatic approximation by comput- ing the electromagnetic fields and the associated Green’s function of open and lossy electromagnetic systems, in particular for general nanoparticle configurations using commercial software (COMSOL Multiphysics).29 In this paper, we derive a general method to compute the tran- sition rate of a quantum emitter influenced by its electromagnetic environment in the weak coupling regime (Sec. II). We apply the macroscopic QED formalism, which separates the electromagnetic environment obtained from the classical Maxwell equations (we use GENOME to determine Green’s function) from the quantum description of the emitter embodied by its wavefunction, which in this case is a hydrogen-like emitter. Then, we apply this method to compute the electric dipolar, quadrupolar, and octupolar transition rates of the emitter in the vicinity of graphene nanoislands with dif- ferent geometries (triangle, square, and crescent), showing strong enhancement of the transition rates (Sec. III A). Afterward, we show that the graphene doping can be tuned to select particular transitions in Sec. III B, before demonstrating a local breakdown of the selection rules in Sec. III C. II. METHOD We consider the spontaneous emission of atomic hydrogen- like emitters into plasmons given by the minimal coupling Hamiltonian,34,35 H=Ha+Hem+Hint, (1) Ha=∑ ip2 i 2me−e2 4πε0r+He−e+HSO, (2) Hem=∑ j=x,y,z∫dr∫dω̵hω[f† j(r,ω)fj(r,ω)+1 2], (3) Hint=∑ ie 2me(pi⋅A(ri)+A(ri)⋅pi)+e2 2meA2(ri)+e̵h 2meσi⋅B(ri), (4) with pi,ri, andσibeing the impulsion, position, and spin of the ith electron, ebeing the electronic charge, mebeing the electron mass, AandBbeing the vector potential and magnetic field, HSO being the spin–orbit coupling, and He–ebeing the electron–electron interaction. f† j(r,ω)and fj(r,ω)are the creation and annihilation operators, respectively. For the interaction Hamiltonian, we neglect the ponderomotive potential ( A2term) and the Bterm as the latter is negligible for non- magnetic structures.36Note that in the Coulomb gauge, ∇ ⋅A=0 except at an interface: with the atom–interface distance we consider, and the rapid decay of the atom wavefunctions, the contribution of this term will be negligible. Writing the vector potential with Green’s function of the system, and applying the Fermi’s golden rule, one finds (for details, see Ref. 5) APL Photon. 6, 086103 (2021); doi: 10.1063/5.0053234 6, 086103-2 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app Γ=2π ̵h2e2̵h3 πε0m2ec2∬ drdr′ψ∗ e(r)ψe(r′)∇ψg(r) ×Im¯¯G(r,r′,ω0)⋅∇ψ∗ g(r′), (5) whereε0is the vacuum permittivity, cis the speed of light, and ψg andψeare the atomic wavefunctions of the ground and excited states of the emitter.¯¯G(r,r′,ω0)is Green’s function of the Maxwell equa- tions and satisfies ∇×(∇ׯ¯G)−ω2 c2εr(r,ω)¯¯G=¯¯Iδ(r−r′), withεr being the relative permittivity and δbeing the delta function.37 Equation (5) computes the transition rates of any emit- ter (via the atomic wavefunction) within any photonic envi- ronment (described by Green’s function) in the weak coupling regime. As mentioned, the spatial variation of Green’s function is known analytically for uniform media and for simple geome- tries. However, more complex structures need to be evaluated numerically, with high computational cost.38Equation (5) is also resource demanding since the integration is performed over six dimensions ( randr′). In order to resolve these two issues, we resort to GENOME,29with the advantage that one modal com- putation allows the knowledge of the complete spatial Green’s function. In GENOME, the problem is written at a fixed frequency, and the mode-related eigenvalue is the permittivity. This formulation suits well the determination of spontaneous emission rates since the emission frequency is determined by the emitter. The modes Em(r) of the scatterer are solved with a commercial finite-element based software (COMSOL Multiphysics),38and Green’s function becomes ¯¯G(r,r′)=¯¯G0(∣r−r′∣)+1 k2∑ mεi−εb (εm−εi)(εm−εb)Em(r)⊗E† m(r′), (6) where mis the mode number, kis the vacuum wave vector, εmis the eigen permittivity, εiis the permittivity of the scatterer, εbis the permittivity of the background material, and¯¯G0(∣r−r′∣)is Green’s function of vacuum, which has an analytical form.37 Inserting Eq. (6) in Eq. (5), we immediately see that the rate is the sum of two contributions Γ=Γ0+Γs, with Γ0being the decay rate in vacuum [based on the contribution of¯¯G0(∣r−r′∣)], and Γs depending on the modes and, hence, the nanophotonic structure. Focusing on this Γscontribution, we can write Γs=2π k2e2̵h πε0m2ec2∬ drdr′ψ∗ e(r)ψe(r′)∇ψg(r) ×(Im∑ mγmEm(r)⊗E† m(r′))∇ψ∗ g(r′), (7) where we defined γm=εi−εb (εm−εi)(εm−εb). Note that the adjoint field ( E†) is the transposed vector and there is no complex conjugate.29 Since we can choose the wavefunctions to be real, the com- plex conjugate for the wavefunctions disappears and we can inte- grate separately for randr′. Both integrations give the same value, leading to Γs=2π k2e2̵h πε0m2ec2∑ mIm[γm(∫ψe(r)Em(r)∇ψg(r)dr)2 ]. (8)Finally, the transition rates are obtained with a three-dimensional integration over the wavefunctions and the mode profiles, with a sum that can be truncated once the convergence is sufficient (40 modes in our case, see the supplementary material for more details on the implementation). Note that when the integral is computed, the rate can be known for any material constituting the scatterer, enclosed in the parameter γm. In that regard, graphene is the per- fect candidate as it can be tuned to match a particular resonance (see Sec. III B). In this work, the graphene nanoislands are modeled with an effective thickness of t=1 nm. The graphene permittivity ( εi) is deduced from the surface optical conductivity ( σ=̃σintra+̃σinter) withεi=1+iσ/ωε0t. The optical conductivity is derived within the local random-phase approximation model39,40and is the sum of the following two contributions: ̃σintra=2ie2kBT ̵h2π(ω+iτ−1g)ln[2 cosh(EF 2kBT)], (9) ̃σinter=e2 4̵h[1 2+1 πarctan(̵hω−2EF 2kBT)] −e2 4̵h[i 2πln(̵hω+2EF)2 (̵hω−2EF)2+(2kBT)2], (10) with T=300 K being the temperature, kBbeing the Boltzmann con- stant, and EFbeing the doping level of graphene. The scattering lifetime of electrons in graphene depends on the doping and is given byτg=μEF/ev2 F≈10−12s for EF=1 eV, with the impurity-limited DC conductivity μ≈10 000 cm2/(V s) and vF=106m/s being the graphene Fermi velocity.41,42 The integration of Eq. (8) is successfully compared to direct simulations of dipolar and quadrupolar transitions in the supple- mentary material, showing great convergence with only 40 modes (1% relative error). In Sec. III, we implement Eq. (8) to compute the rate of a H-like atom close to graphene nanoislands of varying geometry. III. RESULTS AND DISCUSSION We apply our method to compute the electric dipolar (E1), quadrupolar (E2), and octupolar (E3) transition rates of a H-like atom close to a graphene sheet with triangle, square, and crescent geometries. We consider the transition series 6p, d, f →4s, which are E1, E2, and E3 transitions, respectively. We suppose that the angu- lar magnetic number remains m=0 during the transition, and we rotate the emitter wavefunctions to match the corresponding classi- cal point-dipole orientation. The free-space wavelengths of the tran- sitions are all λ=2.63μm, and in the whole paper, the emitter is situated 5 nm above the graphene surface. We then discuss the rate dependence on the graphene doping (Sec. III A) and we demonstrate the advantage of graphene tun- ability for multipolar transitions (Sec. III B). Finally, we optimize a configuration where the conventional selection rules break down, i.e., when the quadrupolar transition rate dominates the dipolar one (Sec. III C). APL Photon. 6, 086103 (2021); doi: 10.1063/5.0053234 6, 086103-3 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app A. Transition rates Figure 1 shows the dipolar ( ΓD), quadrupolar ( ΓQ), and octupo- lar (ΓO) transition rates of a H-like emitter in the vicinity of a graphene nanoisland for three geometries: square, triangle, and crescent shape. The rates are normalized by the dipole emission rate in free space, ΓD0=4.484×105s−1. The latter is obtained by integrating Eq. (5) in free space and is in perfect agreement with the experimental values43(for more information, see the supplementary material). One can see, for example, that the octupo- lar rate is strongly enhanced with respect to vacuum: it is up to 300 times stronger than the dipolar rate in free space for the triangle geometry. One observes that the strongest quadrupolar and octupolar rate enhancements appear along the edges and corners of the geome- tries. This is a consequence of the strong field gradients appear- ing along the graphene edge.1,44Second, for all geometries, themaximum quadrupolar rate is two orders of magnitude smaller than the maximum dipolar rate. This two-order magnitude difference compares fairly with the rate comparison obtained in Ref. 5 for a H- like emitter close to a non-structured two-dimensional material sup- porting plasmons confined with a factor η0≈35–50 (corresponding to doping between 0.7 and 1 eV). The four-order magnitude differ- ence between the dipolar and octupolar transition rates is also in agreement with the literature.5 With the graphene nanoislands, we break the in-plane trans- lational symmetry and the conventional dominance of the dipolar transition rate over the quadrupolar transition rate. From the spa- tial maps, we observe that the maxima of the quadrupolar rate do not coincide with the maxima of the dipolar rate: by moving the emitter, one can find a position where the quadrupolar rate domi- nates the dipolar rate, breaking the conventional selection rules (see Sec. III C). FIG. 1. Dipolar, quadrupolar, and octupolar transition rates of an x-oriented emitter close to graphene nanoislands of various geometries. The dipolar (left), quadrupolar (center), and octupolar (right) transition ( λ=2.63=μm) rates as a function of the emitter position, which is 5 nm above the graphene nanoislands: 50 nm side length square (up), 50 nm side length triangle (middle), and 80 nm height crescent (bottom). The geometry boundaries are represented by a solid white line, and the rates are normalized by the dipolar emission rate in free space ΓD0. For all geometries, the background permittivity is vacuum but graphene doping varies: for the triangle, EF=0.98 eV, for the square, EF=0.72 eV, and for the crescent, EF=0.88 eV. APL Photon. 6, 086103 (2021); doi: 10.1063/5.0053234 6, 086103-4 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app Note that the z-oriented emitter (out-of-plane direction) shows stronger rate enhancement, but the dipolar, quadrupo- lar, and octupolar transition maxima coincide: they all show the maximum enhancement at the same position (see the supplementary material). This implies the conservation of selec- tion rules for the z-oriented emitter, as the dipolar rate always dominates. B. Graphene tunability In a spontaneous emission process the emission wavelength is fixed via the considered transition. Hence, as the frequency of the source is not a variable, a tuning knob is offered by the environment, e.g., the permittivity of the scatterer. The considered mode expan- sion is particularly well suited for this context as the permittivity is the eigenvalue of the problem. As a consequence, the permittivity of the scatterer only appears as a multiplicative constant [ γmfactor in Eq. (8)] of the three-dimensional integration over the plasmonic modes ( Em) and the wavefunctions of the emitter. Tuning the per- mittivity, therefore, allows selecting the mode resonating with the targeted transition.Figures 2(a)–2(c) show the transition rates’ dependence on the permittivity of the material, for a square two-dimensional material with variable permittivity and with a side length of 50 nm. The emit- ter is y-oriented at the position (14.4; 24.5) nm, 5 nm above the material [green arrow in Figs. 2(d)–2(f)] and the transition wave- length remains λ=2.63μm. The transition rate map is characterized by horizontal lines of enhanced transition rates, appearing at partic- ular relative permittivities of the material [Re (εr)]. Each line directly corresponds to a plasmonic mode of the structure, which consti- tutes the dominant decay route for this transition. For example, the dipolar transition couples with mode A [represented in Fig. 2(d)] and the quadrupolar and octupolar transitions couple with modes B and C [represented in Figs. 2(e) and 2(f)]. Note that modes A and B are different, but their eigenvalues are close (resp. εm=−15.65 and εm=−15.45). Their proximity implies that the dipolar rate (ΓD/ΓD0=1.1×105) dominates the quadrupolar rate ( ΓQ/ΓD0 =1.5×104) for the considered position of the emitter, considered emission wavelength, and for a material permittivity close to that particular value [Re (εr)≈−15.5]. Mode C is very interesting since it couples strongly with the quadrupolar transition ( ΓQ/ΓD0=1.2×104) and weakly with FIG. 2. Choosing the material of the scatterer to enhance particular transition rates. The (a) dipolar, (b) quadrupolar, and (c) octupolar transition ( λ=2.63μm) rate enhancement in color scale as a function of the relative permittivity of the scatterer (real and imaginary parts). The imaginary part of the optical conductivity is also represented for more generality. The rates are normalized by the dipolar emission rate in free space. The white line represents the permittivity range covered by graphene upon doping (white dots at 0.5, 0.7, and 1 eV doping) for a DC conductivity of 10 000 cm2/(V s), while the dashed line corresponds to 3000 cm2/(V s). The horizontal green dashed lines indicate the modes contributing the most to the considered transition. Their mode profiles ( ycomponent of the electric field) are represented in (d) εm=−15.65, (e)εm=−15.45, and (f) εm=−23.4. The green dot and arrow represent the position and orientation of the emitter. APL Photon. 6, 086103 (2021); doi: 10.1063/5.0053234 6, 086103-5 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app the dipolar transition ( ΓD/ΓD0=0.6×103), as shown in Figs. 2(a) and 2(b). This conclusion corresponds with the field profile in Fig. 2(f) at the position of the dipole (green dot). The emitter is placed at a position where the field has a low value (weak enhance- ment of dipolar transition), but near the edge, where the field gra- dient is the strongest (strong enhancement of quadrupolar and octupolar transitions). In the horizontal direction of Figs. 2(a)–2(c), the imaginary part of the permittivity of the material broadens the resonance peaks, consequently reducing the maximum value of all transition rates. Therefore, selecting a square two-dimensional material (of this size, and at λ=2.63μm) for a particular permittivity can enhance a particular transition. For example, in order to produce an electric octupolar rate 50 times stronger than the dipolar rate in free space, one can choose a material with real relative permittivity of −23.4 (which corresponds to mode C). The white lines in Figs. 2(a)–2(c) represent the permittivity range covered by graphene at this wavelength via doping. A fine tuning of the doping thus allows us to select the plasmonic mode that will dominate the transition, and hence the transition order. Note that in the case of lower quality graphene samples, the scat- tering is enhanced and consequently the DC conductivity can be lowered to μ≈3000 cm2/(V s).42The dashed lines in Figs. 2(a)–2(c) show the permittivity of graphene in this case. The curve is shifted to the right [compared to a DC conductivity is μ≈10 000 cm2/(V s)], which leaves the conclusion unchanged: fine tuning of the doping allows us to select the transition enhancement rates, even if the dipo- lar, quadrupolar, and octupolar rates are all equivalently reduced by 30%. In Sec. III C, we show a particular doping of graphene where the quadrupolar rate dominates the dipolar transition rate, consequently breaking the conventional selection rules.C. Local breakdown of conventional selection rules At particular positions of the emitter, the quadrupolar tran- sition rate overcomes the dipolar transition rate. This breakdown occurs at ultra-strong plasmon confinement ( η0>500) for pla- nar two-dimensional materials,5which is experimentally achievable with graphene, but at the cost of considerable absorption losses.13 The shape of the graphene nanoislands provides another degree of freedom to mold the field profile and break the selection rules. We focus on the triangular graphene nanoisland of 50 nm side length, for which we computed the normalized dipolar and quadrupolar rates of an emitter 5 nm above its surface (Fig. 1). In Fig. 3(a), we plot the maximum of the ratio ΓQ/ΓD(scanned over all positions of the emitter), as a function of the doping. This shows that the quadrupolar transition rate can be up to 100 times stronger than the dipolar transition rate at particular positions, breaking locally the conventional selection rules (the value is converged for 40 modes, as shown in the supplementary material). In Figs. 3(b) and 3(c), for the x- and y-oriented emitters, respec- tively, we observe enhancement where the field demonstrates strong gradients, i.e., at the corner of the triangle or along the edge. On the contrary, as observed in Fig. 3(a) for the z-oriented emitter, the dipolar rate always dominates the quadrupolar one (the maximum rate enhancement of each order appears at the same position, as discussed in Sec. III A). Note that the maximum is not a consequence of an inhibited dipolar transition: the quadrupolar rate is strongly enhanced. For example, for an x-oriented emitter [Fig. 3(b)] at the left corner of the triangle, the dipolar transition remains enhanced ( ΓD/ΓD0=0.51 ×103), but its rate is weaker than the quadrupolar rate, which is 5.3×104times the dipolar transition in free space. Other areas further away from graphene seem to demonstrate a strong quadrupolar enhancement [for example, position ( −9; 35) nm FIG. 3. Local breakdown of the selection rules with the triangle graphene nanois- land. (a) The maximum of the quadrupo- lar rate ( ΓQ) over the dipolar rate ( ΓD) for a H-like emitter 5 nm above the tri- angular graphene nanoisland in vacuum, evaluated for varying graphene doping and emitter orientation. The z-oriented emitter does not demonstrate a break- down of the selection rules (ratio always smaller than 1). A maximum is obtained for a graphene doping of EF=0.98 eV. At this doping, (b) shows the logarithmic value of the ratio for an x-oriented emitter and (c) a y-oriented emitter. The bound- aries of the triangular graphene nanois- land are represented by a solid white line and the transition wavelength is λ=2.63μm. APL Photon. 6, 086103 (2021); doi: 10.1063/5.0053234 6, 086103-6 © Author(s) 2021APL Photonics ARTICLE scitation.org/journal/app in Fig. 3(b)]. However, these are regions where the dipolar transi- tion is poorly enhanced ( ΓD/ΓD0=16) as well as the quadrupolar transition ( ΓQ/ΓD0=161). IV. CONCLUSIONS AND PERSPECTIVES We develop a numerical method based on Fermi’s golden rule that evaluates the multiple transition orders of a molecule. The molecule is described by its wavefunction, while the photonic environment is implemented through Green’s function. The lat- ter is expanded in eigen-permittivity modes leading to a simpli- fied formula [Eq. (8)] that shows the deep mechanism of strong multipolar enhancement. Indeed, the following two main terms play a role in the sum: the global term γmand the local term ∫ψe(r)Em(r)∇ψg(r)dr. Together, they show that each mode m contributes to the transition rate. The global term states that the permittivity of the material constituting the nano-island ( εi) should match the mode eigen-permittivity ( εm) to contribute to the tran- sition rate. Hence, graphene is an excellent platform to fit εitoεm owing to its optical parameter tunability. The local term shows the primordial importance of the mode field profile locally at the posi- tion of the emitter. If the field profile Emis constant over the spa- tial extent of the wavefunctions of the emitter ( psieandψg), we return to the dipole approximation (or long wavelength approxi- mation), commonly employed in free space, and the higher-order transitions are deemed forbidden. In our case, due to the strong confinement of the field near a graphene nanoisland, the integral is no longer negligible. The order of magnitude of the maximum dipolar, quadrupolar, and octupolar rates compares with the rates obtained for an unstructured graphene sheet5and is in perfect agree- ment with direct simulations. Finally, we demonstrate a breakdown of the selection rules, with the quadrupolar transition rate, forbid- den in free space, becoming 100 times stronger than the dipolar transition rate for an H-like emitter in the vicinity of a triangu- lar graphene nanoisland. These results uncover interesting perspec- tives for applications in spectroscopy, photochemistry, and quantum technologies. Here, we apply the method to a single nano-island in free-space, but the method has a large flexibility and can be applied to more realistic structures reachable in experiments. Indeed, GENOME also allows the determination of Green’s function for more complex structures. For instance, the method can account for a substrate or a background permittivity different from 1.29It can also resolve Green’s function of non-uniform scatterers.45Recently, the proce- dure has been developed to find Green’s function of an assembly of nano-island (cluster) and finite periodic structures.38For exper- imental observations, the coupling of produced photons to the far- field is important. As an example, combining the near-field results (e.g., Fig. 2) with the far-field out-coupling efficiency of the domi- nant mode allows us to select the graphene doping necessary to reach sufficient far-field emission. Such an analysis was carried out for two-photon emission processes near graphene nanoislands.26Other structures may be envisaged to enhance the coupling of a plane wave with a quadrupolar transition.46 Since the transition of the emitter fixes the operating wave- length, the control of the emitted photons goes through the opti- mization of the structure and the permittivity of the material. Here, we consider graphene for its tunable properties, alreadyallowing to target particular transitions (Fig. 2). Strong doping of 1.2 eV has been achieved with electrostatic doping with ionic gel47 or with ionic glass mobility48and a moderate doping of 0.5 eV with chemical (N-doped) doping.49It is challenging to reach high Fermi level values, as a specific structure considering the gate and the substrate should be designed. Other materials can also be considered for their strong plasmonic response, such as thin gold films,50or for their strong phononic response, such as hBN or SiC. In parallel, our method allows for the computation of larger atoms and complex molecules by combining GENOME with time- dependent density functional theory techniques. Hence, controlling the emission rate of quantum dots3,6and Rydberg excitons4in com- plex electromagnetic environments is within reach. Such emitters are larger than the H-like atoms considered here, so placing those in the gaps of clusters should ensure a strong field gradient over their orbital extent to promote the higher order rate enhancements. Fur- thermore, since the dipolar and quadrupolar rates compete, destruc- tive interference effects can be observed and lead to suppression of particular transition channels,3,21leading to diverse quantum applications, such as quantum computing, quantum storage, and quantum communication.3,21 SUPPLEMENTARY MATERIAL See the supplementary material for details on the method implementation. It contains the integration convergence; a verifi- cation of the dipolar transition rate in free-space; a comparison with direct simulations; and additional figures that illustrate the per- mittivity dependence, the quadrupolar transition rate dominance over the dipolar one for the square graphene nanoisland, and rate calculations for various emitter orientations. ACKNOWLEDGMENTS The authors thank Ido Kaminer and Nicholas Rivera for fruitful discussions and Parry Yu Chen and Yonatan Sivan for their support with GENOME. The authors acknowledge support from the FRS- FNRS (Research Project No. T.0166.20 and Grant No. FC 95592) and the Actions de Recherche Concertées, Project No. ARC-21/25 UMONS. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Sanders, A. May, A. Alabastri, and A. 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AIP Advances ARTICLE scitation.org/journal/adv Two-step treatment to obtain single-terminated SrTiO 3substrate and the related difference in both LaAlO 3film growth and electronic property Cite as: AIP Advances 11, 085303 (2021); doi: 10.1063/5.0053323 Submitted: 6 April 2021 •Accepted: 22 July 2021 • Published Online: 2 August 2021 J. J. Peng,a) C. S. Hao, H. Y. Liu, and Y. Yana) AFFILIATIONS Beijing Engineering Research Center of Advanced Structural Transparencies for the Modern Traffic System, Beijing Institute of Aeronautical Materials, Beijing 100095, China a)Authors to whom correspondence should be addressed: pjj.csu.mat@163.com and yue.yan@biam.ac.cn ABSTRACT A two-step treatment, first chemical etching then thermal treatment, is proposed to achieve an atomically flat and thermally stable TiO 2- terminated SrTiO 3substrate. LaAlO 3films were then grown on those TiO 2-terminated and as-received substrates. LaAlO 3films on the TiO 2-terminated SrTiO 3substrate maintained the layer-by-layer growth mode with a sharp interface, while films on the as-received substrates easily underwent reconstruction adverse to the sharp interface. Both LaAlO 3/SrTiO 3interfaces displayed metallic conductive behavior, while the difference in magnetotransport properties indicated the difference in origin for interface conductivity. Large positive magnetoresistance implied that the LaAlO 3/as-received substrate interface was a 3D conductive interface dominated by oxygen vacancies. However, the annealed-LaAlO 3/treated-substrate interface preserved intrinsic quasi-2D interface magnetism as evidenced by large negative magnetoresistance. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0053323 I. INTRODUCTION Novel phenomena at artificial oxide heterointerfaces are attracting extensive scientific attention in both condensed matter physics and materials science areas. The representative discovery of the complex oxide interface is an observation of two-dimensional electron gas (2DEG) at the LaAlO 3/SrTiO 3(LAO/STO) heterointer- face with unique properties.1–3While great progress has been made in the characterization and exploitation of the physical phenom- ena,4–7the mechanisms have remained a subject of intense debate. Controversies focus on whether the phenomenon is determined solely by polar catastrophe, oxygen vacancies, or intermixing at the atomic scale at the interfaces.8–10 Polar catastrophe, referring to polar discontinuity induced charge transfer across the interface, is a plausible explanation for intriguing properties at the interface. To return to its origin, atomic scale control is critical to the solution of the problem, which involves the substrate treatment to obtain a single terminated atomically flatsurface and heteroepitaxial growth of oxide films. SrTiO 3consists of an alternating stack of SrO and TiO 2planes. If the terminating layer is not well defined, a mixed state of SrO and TiO 2planes would be present at the surface that will degrade interfacial properties due to chemical and electronic uncertainty on one unit cell scale. Since atomically abrupt interfaces can be achieved only when substrates are atomically flat, atomically flat single terminated surfaces of sub- strates are indispensable to well-characterized heterostructures with intriguing electronic properties. Generally speaking, the surface of the SrTiO 3substrate is a mixed state of SrO- and TiO 2-terminated. The termination state determines the atom sequence at the interface as displayed in Fig. S1. SrO and TiO 2terminations correspond to TiO 2–SrO–AlO 2+–LaO−and SrO–TiO 2–LaO−–AlO 2+separately, which possesses different interface properties. Proper treatment is essential for obtaining both atomically flat and singly terminated surfaces of SrTiO 3substrates. As TiO 2is more chemically stable compared with SrO, large amounts of research effort have been devoted to exploring effective methods for the preparation of TiO 2 AIP Advances 11, 085303 (2021); doi: 10.1063/5.0053323 11, 085303-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv terminated substrates.11–14SrO and other probable Sr compounds formed on the surface [Sr(OH) 2, SrCO 3, SrO] are soluble in acids,14 which is the basis condition for achieving TiO 2termination. The most widely used acid is buffered-hydrofluoric acid (BHF), which is the same chemical etching procedure used in silicon semiconductor research and industry. A TiO 2termination with a step-and-terrace surface can be obtained by NH 4F–HF (BHF) etching. A solution with a low pH often leads to unit cell deep holes in the terraces and deep etch-pits, which hampers thin film growth. In addition, the effect of acid etching was not reproducible and was extremely sen- sitive to the pH value of the acid. Substrates with different surface states may require acids with different pH values. Therefore, there is an urgent need for a reliable method to produce reproducible, perfect, and stable TiO 2termination without sensibility to acid pH values. As acid induced holes or defects could recover during high temperature annealing due to rapid migration of surface atoms,13a combination of a chemical treatment and a thermal treatment would be desirable for perfectly crystalline TiO 2-terminated surfaces. On the foundation of substrate treatment, direct comparison in sub- sequent film growth and electronic properties would be capable of providing further insight into the intriguing physical phenomena at the interface. In the present work, a two-step treatment (first chemical etch- ing then thermal treatment) to obtain TiO 2-terminated surfaces is proposed. The morphology after chemical and thermal treatment of the surface is carefully examined by atomic force microscopy. After that, LaAlO 3films are grown on those treated and untreated substrates for comparison. Difference in LaAlO 3film growth is observed, as the film on the as-received substrate is much eas- ier to undergo an island-like growth due to the reason that dif- ferently terminated domains form on the surface resulting from different growth kinetics. It is also worthwhile to note that the interface with TiO 2-terminated SrTiO 3substrates exhibits not only conductivity but also magnetic features, while the interface with the as-received SrTiO 3substrates only displays oxygen vacancy related surface conductivity. II. MATERIALS AND METHODS Commercial SrTiO 3substrates (Crystal GmbH, Berlin, Ger- many) were first ultrasonically cleaned and degreased in acetone and alcohol for 5 min each, and then rinsed in de-ionized water for 5 min, after which the substrates were blown dry with pure nitrogen gas. After that, they were treated by buffered HF (BHF) solutions. BHF solutions with different pH values were made by mixing com- mercial HF solution (Sinopharm Co., Ltd.) with NH 4F solutions (Sinopharm Co., Ltd., Beijing, China). We tested the pH of each solution by using a pH meter (Sinopharm Co., Ltd., Beijing, China), which gave us accurate pH values. The chemical-etched substrates were then annealed at 950○C for 1 h in a flowing oxygen atmosphere in an oven (OTF-1200X-S, Hefei Kejing Materials Technology Co., Ltd., Hefei, China). After that, they were slowly cooled down to room temperature at a rate of 60○C/h. The morphology after chemical and thermal treatment of the surface was carefully examined by atomic force microscopy (AFM; Bruker Co., Edge, Karlsruhe, Germany) with a tapping mode. A new AFM tip (OTESPA-R3) was used each time when doing the experiments to avoid unexpected disturbance. Strikingly different surface states of substrates were obtained afterchemical etching and thermal treatment. The as-received substrate had a mixed state of SrO- and TiO 2-terminated. The substrates after chemical and thermal treatment (referred to as treated substrates) were TiO 2-terminated. After pumping down to a base pressure of 10−8Torr, LaAlO 3films were prepared by pulsed laser deposition (PLD, assembled by NBM company, America) from a single crystal LaAlO 3target (Hefei Kejing Materials Technology Co., Ltd., Hefei, China) on those SrTiO 3(001) substrates. LaAlO 3films were grown using optimized parameters (800○C @ 10−6Torr oxygen pressure, 2.0 J/cm2laser energy at 1 Hz repetition rate). Single crystal LaAlO 3 targets were chosen to ensure the stoichiometric ratio of LaAlO 3 films, though higher laser energy was required due to the trans- parency of single crystal targets compared with ceramic polycrystal targets. The layer-by-layer growth mode was identified by reflection high-energy electron diffraction (RHEED, CB801420, STAIB, Ger- man), one oscillation of which corresponded to one unit cell (u.c.). The samples were cooled to room temperature under different oxy- gen pressures. The patterned Hall bars (width: 50 μm and length: 300μm) in the structured samples were wirebonded with Al wire by a wirebonder (CWS3200) to form Ohmic contacts for electri- cal transport measurement. The resistance ( Rs), magnetoresistance MR=[Rs(H)−Rs(0)]/Rs(0), was measured by applying magnetic fields perpendicular to the patterned Hall bars. The conductivity and magnetic resistance were characterized by the four-probe method using a Physical Property Measurement System (PPMS, Quantum Design, San Diego, America). III. RESULTS AND DISCUSSIONS A. Morphology of SrTiO 3(STO) surface after different treatments The surface morphology of a typical as-received single-crystal SrTiO 3substrate was imaged by AFM. The root-mean-square roughness was calculated over the scanned surface, which is dis- played along the image. As shown in Fig. 1(a), the surface was smooth with no obvious terraces, indicating the co-existence of the two possible surface terminations (SrO and TiO 2planes). The sur- face of the mixed state had a roughness of 0.31 nm, which was not desirable for film growth. We soaped those substrates in BHF solutions with different pH values (Table I) each for 1 min. As we find out, BHF solutions with the pH value being 5 can hardly etch the SrO clearly [Fig. 1(b)], while BHF solutions with a pH as low as 3 would result in a large number of etch-pits [Fig. 1(c)]. As shown in Fig. 1(d), the substrate treated with BHF solutions (pH 3.8) showed a step-and-terrace surface. No crystal defect was found on ten equally treated substrates. However, it is worthwhile to note that the edge of the terrace is curved and the width of the terrace was not evenly distributed, which was not optimal for film growth. After chemical etching, we performed high temperature annealing (950○C for 1 h in a flowing oxygen atmosphere) on the chemical-etched (pH being 3 and 3.8) substrate. The AFM image of the annealed substrates showed a nearly perfect single-terminated surface with a very straight terrace ledge as depicted in Figs. 1(e) and 1(f). It is surprise to find out that the thermal-treated substrates displayed no big difference though vast difference was observed just after etching as shown in Figs. 1(c) and 1(d). The holes caused by etching in Fig. 1(c) disappeared after annealing. Thus, the effect of AIP Advances 11, 085303 (2021); doi: 10.1063/5.0053323 11, 085303-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 1. (a) AFM images of the as-received substrate, (b) AFM image of the sub- strate treated by BHF solutions with the pH being 5, (c) AFM image of the substrate treated by BHF solutions with the pH being 3, (d) AFM image of the substrate treated by BHF solutions with the pH being 3.8, (e) AFM image of the etched (pH being 3.0) substrate after high temperature annealing, and (f) AFM image of the etched (pH being 3.8) substrate after high temperature annealing. annealing could be summed up to two points: (1) annealing caused rapid migration of surface atoms to recover some of the defects dur- ing the etching process and (2) annealing improved the quality of the surface by rearrangement of the atomic configuration to form a TABLE I. Dependence of the surface state on BHF solution’s pH values. pH values Substrate state 3.0 Step-and-terrace surface with etch-pits 3.8 Step-and-terrace surface 4.5 Incomplete etch 5.2 No etchregular array of straight step edges. Thus, it is natural to conclude that a combination of chemical etching and thermal treatment pro- vided a route to produce perfect TiO 2termination substrates with- out less sensibility to the acid pH value. The result was reproduced on many equally treated substrates. Since the oxide thin film usually grew at a high temperature and under high vacuum, it was essential to test whether the well-defined surface was stable under such conditions. We put the thermal- treated substrates at the PLD chamber, which were then heated to a high temperature of 1000○C under high vacuum (10−8Torr). After stabilizing at 1000○C for half an hour, the substrate was cooled down to room temperature. The surface (shown in Fig. S2) had no much difference with the annealed ones, indicating that the step and terrace structure was thermally stable. To further examine the characteristics of the surface, we per- formed section analysis to get a height profile using AFM. Scanning electron microscopy cannot be used to detect the surface state as the surface is highly insulting and the terrace is as small as 0.4 nm in height, which was far below the detection limit of the machine. Closer views are shown in Fig. 2. The corresponding height profiles along the lines are shown along it. Regularized spaced terraces are visible, the width of which was determined by the miscut angle of the crystal. If the substrate was in a mixed state of SrO- and TiO 2- terminated, the step height of a half unit cell would be present. All the step height was 0.394 nm, which corresponds to one unit cell of the STO crystal. The result strongly hinted that the surface is singly terminated. As SrO was etched by BHF solution, only the TiO 2-terminated state could exist at the surface. In conclusion, a two-step treatment (first chemical etching then thermal treatment) led to atomically flat, TiO 2-terminated, and thermally stable sur- faces, which was a necessary prerequisite for heteroepitaxial growth of oxide thin films. B. LaAlO 3film growth Next, we moved on to investigate the roles of substrate termi- nations in the growth behaviors for perovskite film materials. The growth dynamics was investigated by monitoring the intensity of reflection high-energy electron-diffraction (RHEED). RHEED mea- surements allowed us to characterize and control the growth surface to the atomic-level accuracy. Figure 3 shows representative RHEED patterns and RHEED intensity oscillations recorded during the growth of a 20 unit cell (u.c.) LaAlO 3(LAO) film on treated substrates and as-received sub- strates. LAO films were grown using the same parameters. High- quality surfaces were revealed by the presence of Bragg spots along the zeroth and the first Laue circles. Clear Bragg spots before film growth, as displayed in Fig. 3(a), indicated a perfectly crystalline and atomic-level flat substrate surface. After the growth of the 20 u.c. LAO film, the position of the diffraction spots remained the same as can be seen in Fig. 3(b). This implies that the LAO film can be grown epitaxially on atomically TiO 2-terminated sub- strates. The diffraction spots of Fig. 3(b) were streaky and a lit- tle bit faint in intensity, indicating the atomic-scale smooth sur- face with surface roughness comparable to that of the treated substrate. Figure 3(c) displays the RHEED intensity oscillations obtained during the growth of the LAO film on TiO 2-terminated substrates. At the initial growth period, the RHEED intensity AIP Advances 11, 085303 (2021); doi: 10.1063/5.0053323 11, 085303-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. AFM images analyzed by the section function. decreased dramatically, owing to the increased surface roughness by the uneven distribution of the deposited film on the surface. The RHEED intensity at the lowest level corresponded to the deposition of the half-unit-cell film. However, after growth of the one-unit- cell thick LAO film, the surface roughness recovered to the origin state. As a result, the RHEED intensity recovered to a high level. The RHEED oscillations of each unit cell growth could be seen very clearly, suggesting that the LAO film can be grown epitaxially on atomically TiO 2-terminated substrates. X-ray diffraction patterns of the sample with 50 u.c. (20 nm) LAO films are shown in the sup- plementary material (Fig. S3) to indicate the epitaxial growth of LAO films on the STO substrate. Figure 3(d) shows RHEED pat- terns of the as-received substrates. Compared with Fig. 3(a), only the center diffraction spot was clear and the side diffraction spots were much fainter in intensity. This suggested that the surface of the as-received substrates was not well defined as that of TiO 2- terminated substrates. After the growth of 20 u.c. LAO films, the position of the diffraction spots changed [Fig. 3(e)], which were of surface reconstruction feature. Figure 3(f) displays the RHEED intensity oscillations obtained during growth of the LaAlO 3film on the as-received substrates. The first four oscillations were obvious, while the oscillation amplitude damped out with later film growth. The damping out phenomenon could be understood. As the as- received substrates with larger roughness were not so good for film growth, the increased surface roughness with the deposition of each layer caused the dropped RHEED intensity and finally the damping out of oscillation.From the above observations, LAO film growth on TiO 2- terminated substrates and as-received substrates exhibited vast dif- ferent growth characteristics, probably originating from different surface termination on the surface. LAO films on the as-received substrates with larger roughness and different terminated domains were much easier to undergo an island-like growth. C. Difference in transport properties As described in Sec. II, LAO films were grown using the same parameters (800○C @ 10−6Torr oxygen pressure, 2.0 J/cm2laser energy at 1 Hz repetition rate) on the as-received substrates and treated substrates. The treated substrates were referred to those chemical-etched and thermal-treated ones with TiO 2-terminated. Some of the films after growth were cooled down at the deposition atmosphere to room temperature without annealing at the deposi- tion temperature, which were referred to as the LAO/as-received- STO and LAO/treated-STO interface according to the substrates that they were grown on. Some of them were cooled down to room temperature at 50 Torr oxygen pressure after annealing at 800○C with the same oxygen pressure for half an hour. The annealed sam- ples made up two types of interface according to the substrates that they are grown on, namely, annealed-LAO/as-received-STO and annealed-LAO/treated-STO. Conductivity and magnetic resistance were characterized by the four-probe method as depicted in Fig. S3. The temperature dependence of the sheet resistance RSis shown in Fig. 4 with summarized tables in the supplementary material. The LAO/STO AIP Advances 11, 085303 (2021); doi: 10.1063/5.0053323 11, 085303-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. Representative RHEED patterns and intensity oscillations of 20 u.c. LAO films on different substrates. (a)–(c) are RHEED patterns before growth, after growth and intensity oscillations recorded during growth for films on TiO 2-terminated substrates, (d)–(f) are RHEED patterns before growth, after growth, and intensity oscillations recorded during growth for films on the as-received substrates. interface with the LAO film thickness above 4 u.c. exhibited metallic behavior down to 2 K in agreement with previous reports.1,8,15The LAO/treated-STO interface displayed metallic conductive behavior with a minimum resistance of 24 Ω, and the LAO/as-received-STO FIG. 4. Temperature dependence of Rsfor different types of interface.interface had the same metallic conductive behavior but with resis- tance several magnitudes larger. Both samples had resistance much lower than the quantum of resistance16h/e2=25.8 kΩ, suggesting no strong localization existing in our samples. Considering that the films were grown at rather low oxygen pressure, a large number of oxygen vacancies might exist at the interface. Annealing in an oxygen atmosphere was an effective method to remove oxygen vacancies. We annealed the films at 50 Torr oxygen pressure at the deposition temperature after growth. The resistance of the annealed-LAO/treated-STO interface was elevated to a high level but still displayed metallic conductive behavior, while the annealed-LAO/as-received-STO interface was insulting (not shown). As annealing removes oxygen vacancies from the inter- face,4it is natural to come to the conclusion that oxygen vacan- cies were the origin of conductivity at LAO/as-received-STO inter- faces. For LAO/treated-STO interfaces, conductivity was intrinsic considering the heteroepitaxial growth of oxide thin films on the treated substrates. A sharp interface was expected. The transferred charge from the LAO film was the intrinsic origin of conductivity for LAO/treated-STO and annealed-LAO/treated-STO interfaces. AIP Advances 11, 085303 (2021); doi: 10.1063/5.0053323 11, 085303-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv Magnetoresistance MR=[Rs(H)−Rs(0)]/Rs(0) was measured by applying magnetic fields perpendicular to the film surface at various temperatures. The MRcurves are shown in Fig. 5, with summarized tables in the supplementary material. As shown in FIG. 5. Magnetoresistance M R, defined as [R s(H)−Rs(0)]/R s(0), with magnetic field perpendicular to the film surface at different temperatures of (a) LAO/as- received-STO interfaces, (b) LAO/treated-STO interface, and (c) annealed- LAO/treated-STO interface.Fig. 5(a), the LAO/as-received-STO interface showed large asym- metric positive MR. It is also worthwhile to note that the magnitude ofMRwas exceeding 41.70% at 2 K and −4 T. The positive MR could be due to the large number of oxygen vacancies present at the interfaces, as reported by many research works.17Given that oxy- gen vacancies existed in large numbers at the LAO/as-received-STO interface, it implies that oxygen vacancies induced the conducting region homogeneously extending over hundreds of μm inside the STO substrate. For the LAO/treated-STO interface [Fig. 5(b)], MR was only about 8.64% at 2 K and −4 T, indicating that oxygen vacan- cies were not as many as that on the as-received substrates. Given that metallic conductivity was present at the interface as evidenced by temperature-dependent resistance, it was supposed that quasi-2D conductivity was present for films on treated substrates. When we come to annealed-LAO/treated-STO, much more interesting phenomena appeared as displayed in Fig. 5(c). The annealed-LAO/treated-STO interface showed negative magnetore- sistance in a butterfly shape for large magnetic fields at 2 K, which was a signal of quasi-2D interface magnetism.15,18An additional suppression around zero-field occurred, which suggested additional spin/domain reorientation effects. The suppression was similar to that observed in granular and spin-valve giant magnetoresistance systems and the Kondo effect in quantum dots in the presence of ferromagnetism.15,19In our case, a quasi-2D magnetic plane at the interface was considered to be responsible for the observed nega- tive magnetoresistance. As we further increased the temperature to above 20 K, negative magnetoresistance disappeared and positive magnetoresistance occurred, which implied that ferromagnetism of a quasi-2D plane only existed at extremely low temperatures in good agreement with other authors.2,3,18The charges transferred from LAO doped SrTiO 3with a mixture of Ti4+and Ti3+ions. The mag- netic Ti3+state at the interface, with fractional occupancy of an additional electron state (3d1) in the t2gorbital band,2was the origin of interface ferromagnetism. In this work, we proposed that two kinds of doping mecha- nisms were involved in the formation of the electron gas in our LAO/STO interface. The first type was extrinsic. The LAO/as- received-STO interface, with a larger number of oxygen vacancies, was typical of extrinsic doping. The high concentration of oxygen vacancies in the STO substrate tended to form a three-dimensional distribution, i.e., 3D interface. The most eminent feature of extrinsic doping was that it was very sensitive to growth oxygen pressure and annealing oxygen pressure. As oxygen vacancies could be reduced by annealing at high oxygen pressure, the annealed-LAO/as-received- STO interface was proved to be almost insulting. In addition, the extrinsic doped LAO/STO interface never displayed ferromag- netism. Another type was intrinsic doping with charge transfer from LAO to STO. In our annealed-LAO/treated-STO interface, charge carriers were generated by the electron transfer from the LAO valence band to the STO conduction band. In addition, the magnetic Ti3+state at the interface, with fractional occupancy of an additional electron state (3d1) in the t2gorbital band, just as reported by Lee,2 was the origin of interface ferromagnetism. IV. CONCLUSIONS In the present work, a two-step treatment, first chemical etching and then thermal treatment, was proposed to achieve an AIP Advances 11, 085303 (2021); doi: 10.1063/5.0053323 11, 085303-6 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv atomically flat and thermally stable TiO 2-terminated SrTiO 3sub- strate. LaAlO 3films on the TiO 2-terminated SrTiO 3substrate main- tained the layer-by-layer growth mode and formed a sharp inter- face with the substrate, while films on the as-received substrates easily underwent reconstruction, which was adverse to the sharp interface. Differences in interface magnetotransport properties were also observed. Large positive magnetoresistance indicated that the LAO/as-received-STO interface was 3D conductive interfaces dom- inated by oxygen vacancies. However, annealed-LAO/treated-STO displayed negative magnetoresistance with ferromagnetic feature. Different doping mechanisms are responsible for the different mag- netoresistance. The LAO/as-received-STO interface is dominated by extrinsic doped with a high concentration of oxygen vacan- cies. The other type of interface was intrinsic doped with charge transfer from the TiO 2-terminated SrTiO 3substrate. The charge transfer induced magnetic Ti3+state with fractional occupancy of an additional electron state (3d1) was the origin of interface ferromagnetism. SUPPLEMENTARY MATERIAL See the supplementary material for the following contents: (1) how the termination states of substrates determine the atom sequence of deposited films, (2) the thermal stability of the treated substrate, (3) XRD patterns of LaAlO 3films, (4) Hall bar prepara- tions and measurements, and (4) summarized tables of transport properties. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos. 51802297 and 52072354). The authors declare no conflict of interest. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request.REFERENCES 1D. A. Dikin, M. Mehta, C. W. Bark, C. M. Folkman, C. B. Eom, and V. Chandrasekhar, Phys. Rev. Lett. 107, 056802 (2011). 2J.-S. Lee, Y. W. Xie, H. K. Sato, C. Bell, Y. Hikita, H. Y. Hwang, and C.-C. Kao, Nat. Mater. 12, 703–706 (2013). 3I. Pallecchi, N. Lorenzini, M. A. Safeen, M. M. Can, E. Di Gennaro, F. M. Granozio, and D. Marré, Adv. Electron. Mater. 7, 2001120 (2021). 4H. Yan, J. M. Börgers, M. A. Rose, C. Baeumer, B. Kim, L. Jin, R. Dittmann, and F. Gunkel, Adv. Mater. Interfaces 8, 2001477 (2020). 5H. Yan, Z. Zhang, S. Wang, X. Wei, C. Chen, and K. Jin, ACS Appl. Mater. Interfaces 10, 14209–14213 (2018). 6A. M. R. V. L. Monteiro, D. J. Groenendijk, N. Manca, E. Mulazimoglu, S. Goswami, Y. Blanter, L. M. K. Vandersypen, and A. D. Caviglia, Nano Lett. 17, 715–720 (2017). 7Y. Frenkel, N. Haham, Y. Shperber, C. Bell, Y. Xie, Z. Chen, Y. Hikita, H. Y. Hwang, and B. Kalisky, ACS Appl. Mater. Interfaces 8, 12514–12519 (2016). 8A. Janotti, L. Bjaalie, L. Gordon, and C. G. Van de Walle, Phys. Rev. B 86, 241108(R) (2012). 9Y. Li, S. N. Phattalung, S. Limpijumnong, J. Kim, and J. Yu, Phys. Rev. B 84, 245307 (2011). 10Y. Z. Chen, D. V. Christensen, F. Trier, N. Pryds, A. Smith, and S. Linderoth, Appl. Surf. Sci. 258, 9242–9245 (2012). 11M. Kawasaki, A. Ohtomo, T. Arakane, K. Takahashi, M. Yoshimoto, and H. Koinuma, Appl. Surf. Sci. 107, 102–106 (1996). 12Y. Xue, C. Geng, and Y. Guo, ACS Appl. Mater. Interfaces 12, 3134–3139 (2019). 13M. Lippmaa, M. Kawasaki, A. Ohtomo, T. Sato, M. Iwatsuki, and H. Koinuma, Appl. Surf. Sci. 130–132 , 582–586 (1998). 14T. Ohnishi, K. Shibuya, M. Lippmaa, D. Kobayashi, H. Kumigashira, M. Oshima, and H. Koinuma, Appl. Phys. Lett. 85, 272 (2004). 15N. Banerjee, M. Huijben, G. Koster, and G. Rijnders, Appl. Phys. Lett. 100, 041601 (2012). 16T. Hernandez, C. W. Bark, D. A. Felker, C. B. Eom, and M. S. Rzchowski, Phys. Rev. B 85, 161407(R) (2012). 17X. Wang, W. M. Lü, A. Annadi, Z. Q. Liu, K. Gopinadhan, S. Dhar, T. Venkatesan, and Ariando, Phys. Rev. B 84, 075312 (2011). 18M. Ben Shalom, C. W. Tai, Y. Lereah, M. Sachs, E. Levy, D. Rakhmilevitch, A. Palevski, and Y. Dagan, Phys. Rev. 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5.0041081.pdf

Appl. Phys. Lett. 118, 152403 (2021); https://doi.org/10.1063/5.0041081 118, 152403 © 2021 Author(s).Spin–orbit torque controllable complete spin logic in a single magnetic heterojunction Cite as: Appl. Phys. Lett. 118, 152403 (2021); https://doi.org/10.1063/5.0041081 Submitted: 18 December 2020 . Accepted: 27 March 2021 . Published Online: 12 April 2021 Y. N. Dong , X. N. Zhao , X. Han , Y. B. Fan , X. J. Xie , Y. X. Chen , L. H. Bai , Y. Y. Dai , S. S. Yan , and Y. F. Tian ARTICLES YOU MAY BE INTERESTED IN Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Field-free spin–orbit torque induced magnetization reversal in a composite free layer with interlayer exchange coupling Applied Physics Letters 118, 132402 (2021); https://doi.org/10.1063/5.0041310 Field-free and sub-ns magnetization switching of magnetic tunnel junctions by combining spin-transfer torque and spin–orbit torque Applied Physics Letters 118, 092406 (2021); https://doi.org/10.1063/5.0039061Spin–orbit torque controllable complete spin logic in a single magnetic heterojunction Cite as: Appl. Phys. Lett. 118, 152403 (2021); doi: 10.1063/5.0041081 Submitted: 18 December 2020 .Accepted: 27 March 2021 . Published Online: 12 April 2021 Y. N. Dong, X. N. Zhao, X.Han, Y. B. Fan, X. J.Xie,Y. X. Chen, L. H. Bai, Y. Y. Dai,S. S.Yan,a) and Y. F. Tiana) AFFILIATIONS School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, People’s Republic of China a)Authors to whom correspondence should be addressed: shishenyan@sdu.edu.cn , and yftian@sdu.edu.cn ABSTRACT The realization of complete spin logic within a single nonvolatile memory cell is a promising approach toward next-generation low-power stateful logic circuits. In this work, we demonstrate that all 16 Boolean logic functions can be realized within a single four-state nonvolatileIrMn/Co/Ru/CoPt magnetic heterojunction, where controllable field-free spin–orbit torque switching of the perpendicularly magnetizedCoPt alloy is obtained, relying on the interlayer exchange coupling and exchange bias effect. By assigning different values to four variables of the four-state memory, that is, the initial control current pulse, the initial control magnetic field, and the input electrical potential of two ter- minals, in sequentially three steps, the complete Boolean logic functions are realized, while the anomalous Hall voltage of the devices is con-sidered as logic output. The coexistence of nonvolatile four-state memory and complete spin logic functions holds promising application forfuture computing systems beyond von Neumann architecture. Published under license by AIP Publishing. https://doi.org/10.1063/5.0041081 Current computer systems store information in memory units and process information in logic units. This means that the intercon- nect delays due to the data transfer between memory and logic unitslead to low computation speeds and high energy consumption—theso-called von Neumann bottleneck. The key issue causing this situa-tion is that binary based conventional logic units are volatile. Hence, using multi-state nonvolatile memory instead of binary volatile logic units to offer logic functions could break the von Neumann bottleneck.This would allow the development of architectures suitable for futurecompact and brain-like computing. Here, we report the realization ofall 16 Boolean logic functions in a nonvolatile four-state IrMn/Co/Ru/ CoPt memory device by adjusting four input variables. Moreover, the realization of the complete spin logic functions in a single multi-statememory unit cell could greatly reduce computation complexity 1and increase integration density. This is because at least two transistors areneeded for each CMOS (complementary metal oxide semiconductor)logic gate. Previous reports have shown that a logic operation can be achieved via different approaches, 2–10such as using resistance switch- ing,11,12magnetization switching,2,5domain wall motion,13–15mag- netic skyrmion motion,16spin wave propagation,17and spin–orbit torque (SOT) based logic.18–21Among these, spin–orbit torque based logic and its related reconfigurable spin logic devices are of particularinterest because of their compatibility with current CMOS technology,high operation speed, low power consumption, and good scalabil- ity. 18–21For example, a programable logic operation with five logic gates is realized in a single device based on the spin Hall effect.19 Moreover, using oxidation-state modification at a ferromagnet–oxideinterface and magnetic-anisotropy manipulation by an electrical field,complementary spin logic operation has been achieved. 20Three logic gates, including AND, NAND, and NOT, are reconfigured in Pt/Co/ Ru/Co/Pt heterojunctions with crossed anisotropy-induced chiralitycontrollable spin–orbit torque switching. 21However, until now, the realization of complete spin logic functions within a single multi-statememory unit cell with controllable spin–orbit torque switching has not been experimentally demonstrated. In this work, the studied IrMn(8)/Co(2)/Ru(0.8)/CoPt(3.3) (thickness in nanometers) heterojunctions were deposited on ther- mally oxidized Si (001) substrates coated with 500 nm thick SiO 2.T h e bottom Co layer and top CoPt alloy layer were antiferromagneticallyexchange coupled via a thin Ru spacer. It should be pointed out thatferromagnetic interlayer exchange coupling could be achieved bychanging the thickness of Ru, which can also be used to achieve spin–orbit torque controllable complete spin logic. A 2 nm Ru buffer layer was first grown on the coated SiO 2to improve the sample qual- ity, and a 2 nm MgO capping layer was grown on the CoPt layer toenhance its perpendicular magnetic anisotropy and to prevent thedevices from being oxidized. The top CoPt alloy single layer has a Appl. Phys. Lett. 118, 152403 (2021); doi: 10.1063/5.0041081 118, 152403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplnominal structure of Pt(0.7)/Co(0.3)/Pt(0.5)/Co(0.5)/Pt(0.3)/Co(1). Due to atomic diffusion during sputtering deposition, CoPt alloy with a composition gradient in the thickness direction was obtained. To promote the exchange bias between IrMn and Co, the as-prepared samples were annealed at 300/C14C under an in-plane magnetic field of 5000 Oe. The magnetic properties of the studied samples were charac- terized using a superconducting quantum interference device (SQUID, Quantum Design, MPMS-XL 7) and a magnetic optical Kerr effect (MOKE) microscope of an EVICO system. For electrical transport measurements, the samples were patterned with conventional Hall barstructures with a channel width of 5 lm and length of 70 lmu s i n g optical lithography and Ar-ion beam etching. Four-point Hall mea- surements were conducted using an Oxford physical property mea- surement device, in which substantial and repeatable field-free spin–orbit torque switching of the perpendicularly magnetized CoPt was observed, as shown in the supplementary material , Figs. S1 and S2. It is believed that bulk spin–orbit torque switching in the perpen-dicularly magnetized CoPt alloy sublayer plays the dominant role in the observed SOT switching, which is realized by introducing a com- position gradient in the thickness direction to break the crystal inver- sion symmetry of CoPt. The bulk SOT-induced effective magnetic field on the magnetic domain walls leads to domain wall motion and magnetization switching of perpendicularly magnetized CoPt alloy. The basic concept underlying complete spin logic in a single four-state nonvolatile memory is schematically illustrated in Fig. 1(a) . By assigning different values to four input variables in three sequential steps, all 16 Boolean logic functions can be realized within a single IrMn/Co/Ru/CoPt magnetic heterojunction. In the heterojunction, the top CoPt alloy layer has perpendicular magnetic anisotropy, while thebottom Co layer has in-plane magnetic anisotropy. For convenience, a two-bit binary number is used to represent the four different magnetic configurations, that is, “00,” “01,” “10,” and “11,” in which the first bit “0” (“1”) means that “the magnetization of the bottom Co layer (red arrow) points to the left (right);” while the second bit “0” (“1”) meansthat “the magnetization of the top CoPt layer (white arrow) is down- ward (upward).” Moreover, the “00” and “01” (“10” and “11”) states have a negative (positive) Hall voltage, corresponding to output logic “0” (logic “1”). Thanks to interlayer exchange coupling through the thin Ru layer, magnetic field-free perpendicular magnetization switch- ing caused by the bulk spin–orbit torque in the composition gradient CoPt alloy is achieved. In addition, the field-free SOT switching polar-ity can be changed from clockwise (red curves) to counterclockwise (blue curves); this is achieved by changing the direction of the exchange bias effect between antiferromagnetic IrMn and Co as sche- matically shown in Fig. 1(b) . Hence, a transition between four different magnetic states can be obtained by controlling the exchange bias and spin–orbit torque switching, enabling complete spin logic functions in the studied nonvolatile four-state memory. Measurements shown in Fig. 2 confirm that the designed IrMn/ Co/Ru/CoPt heterojunctions can function as a nonvolatile four-state memory. Here, longitudinal and polar MOKE measurements are per-formed to reveal the four different magnetic configurations—the lon- gitudinal MOKE measurement is sensitive to the magnetization component along the x-axis (mainly the Co layer), while the polar MOKE measurement is sensitive to the magnetization component along the z-axis (mainly the CoPt layer). Figure 2(a) shows that the exchange bias field is þ150 Oe for states I and II after the applicationof current pulses of 616 mA with H¼/C0 100 Oe, while it is /C0150 Oe for states III and IV after the application of current pulses of 616 mA with H¼þ 100 Oe. Here, the applied magnetic field His also along the current direction x-axis. On the one hand, these results indicate that the magnetization of the bottom Co layer is along the - x-axis direction for states I and II. The magnetic moment of Co is set along theþx-axis direction for states III and IV after the application of I¼616 mA with H¼þ 100 Oe. On the other hand, these results indicate that both the positive and negative current pulses can lead to a reversal in the exchange bias effect under a fixed magnetic field.Hence, it is believed that Joule heating plays a significant role in the reversal of the exchange bias field by heating the antiferromagnetic IrMn layer above its N /C19eel temperature with the assistance of an exter- nal magnetic field. The polar MOKE results shown in Fig. 2(b) indicate that the magnetic moment direction of perpendicularly magnetized CoPt depends on the direction of both the current pulse and the magneticfield. The CoPt magnetic moment is upward for H¼/C0100 Oe, I¼/C016 mA and H¼þ100 Oe, I¼þ16 mA, while it is downward forH¼/C0100 Oe, I¼þ16 mA and H¼þ100 Oe, I¼/C016 mA. Hence, as schematically shown in the left of Fig. 2(b) , four different nonvolatile magnetic configurations can be obtained in a single hetero- junction by controlling the applied H¼6100 Oe and I¼616 mA. It should be pointed out that the hysteresis loops measured by SQUID also confirm that the top CoPt alloy layer has perpendicularanisotropy and the bottom Co layer has in-plane anisotropy in the FIG. 1. Working principles demonstration. (a) Realization of complete spin logic based on a single four-state nonvolatile memory unit by assigning different valuesto four input variables. (b) Transition between different magnetic configurations in the IrMn/Co/Ru/CoPt heterojunction. By controlling the exchange bias effect between IrMn and Co through the simultaneous application of a current pulse andan external magnetic field ( H ext), the field-free spin–orbit torque (SOT) switching polarity of CoPt can be changed into clockwise (red curves) or counterclockwise (blue curves). For convenience, a two-bit binary number is used to represent the four different magnetic configurations. The first purple bit, “0” (“1”), means that “themagnetization of the bottom Co layer (red arrow) points to the left (right),” while thesecond orange bit, “0” (“1”), means that “the magnetization of the top CoPt layer (white arrow) is downward (upward).”Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 152403 (2021); doi: 10.1063/5.0041081 118, 152403-2 Published under license by AIP PublishingIrMn/Co/Ru/CoPt heterojunctions. The observation of the horizontal shift of the hysteresis loop confirms the existence of an exchange bias between IrMn and Co as shown in the supplementary material ,F i g .S 3 . The status of four different magnetic configurations can be detected not only by MOKE measurements, as shown in Figs. 2(a) and 2(b), but also by electrical measurements, a quality particularly impor- tant for practical applications. The magnetization state of perpendicu- larly magnetized CoPt can be detected directly by using the anomalous Hall effect, which is proportional to the average z-component of CoPt magnetization. The most important challenge is how to determine the in-plane magnetic moment of Co using electrical measurements. Figure 2(c) shows that the direction of in-plane Co magnetization can be deduced from the field-free SOT switching polarity of the IrMn/ Co/Ru/CoPt heterojunctions. For example, the Co magnetization is along the - x-axis direction at zero magnetic field in states I and II. Because of antiferromagnetic interlayer exchange coupling, the effec- tive field felt by the CoPt alloy layer is along the þx-axis direction, where clockwise field-free SOT switching is obtained, as shown at the top of Fig. 2(c) . Once the exchange bias field is reversed for states III and IV, the magnetic moment of the bottom Co layer is reversed, and hence, the effective bias field felt by the top CoPt alloy layer is also reversed. Correspondingly, the field-free SOT switching polarity of the IrMn/Co/Ru/CoPt heterojunctions changes from clockwise switching to counterclockwise switching, as shown at the bottom of Fig. 2(c) .I n other words, clockwise and counterclockwise field-free SOT switching means that the in-plane Co magnetization points along the - x-axis and theþx-axis, respectively. In such a way, a purely electrical readout of the four different magnetic configurations is possible, which is benefi- cial for the development of nonvolatile four-state memory. Equipped with four different nonvolatile states and controllable switching between different states, the unique IrMn/Co/Ru/CoPt het- erojunctions provide us a suitable platform to develop logic functional- ity in nonvolatile memory, to achieve logic-in-memory architectures. In order to implement complete spin logic operation in a single IrMn/ Co/Ru/CoPt heterojunction, four logic variables are needed, i.e., theinitial control current pulse ( W), the initial control magnetic field ( E), and the input voltage of two terminals UABandUBA(AandB). As summarized in Fig. 3(a) , each variable has only two choices, which represent two logic inputs “0” and “1.” There are in total three steps to perform a logic operation in the studied nonvolatile memory. The first step, “initialization,” is shown in Fig. 3(b) . By applying a current pulse of616 mA under H¼61 0 0 O e ,t h ed e v i c ei ss e tt oo n eo ft h ef o u r nonvolatile states, after the current pulse and magnetic field are turned off, regardless of its previous state. Here, the positive direction for both WandEis from electrode Bto electrode A, as schematically shown in Fig. 3(b) . For instance, after the application of /C016 mA ( W¼0) under H¼/C0100 Oe ( E¼0), the magnetization of the bottom Co l a y e ri ss e ta l o n gt h e /C0x-axis direction. Hence, the effective bias field felt by the top CoPt layer is along the þx-axis direction. This is because of the antiferromagnetic interlayer exchange coupling between Co and CoPt, resulting in clockwise field-free SOT switching of theCoPt alloy layer, as shown at the top of Fig. 2(c) . The second step is “writing,” where parameter A(B) represents the voltage drop U AB(UBA) from electrode A(B)t oe l e c t r o d e B(A), as schematically shown in Fig. 3(b) .W h e n A¼B(UAB¼UBA), there is no current flow from the device, and thus, the detected Hall voltage (the status of the CoPt layer) remains unchanged, as in the “initialization” states. When A6¼B, the total voltage through the device is either þ12 V or /C012 V, corresponding to a current of either þ16 mA or /C016 mA, which is larger than the critical switching current of/C2410 mA, as shown in Fig. 2(c) . In this case, the detected Hall volt- age (the status of the CoPt layer) is changed or not changed, depend-ing on the initial state, the switching polarity of the CoPt layer, and the direction of current flow. In fact, the logic operation is already finished and the result is stored in the device after this step. The third and last step is reading out the Hall voltage of the device by applying a small current of 0.1 mA. It is worth mentioning that after “initialization,” the four nonvol- atile states are twofold degenerate from an electrical transport detec- tion point of view. The device has a positive Hall voltage (output logic FIG. 2. Nonvolatile four-state characterization. In-plane hysteresis loops measured by using the longitudinal MOKE (a) and the polar MOKE images (b) of the IrMn/Co/Ru/CoPt heterojunctions in different states, as schematically shown in the inset. All the measurements are conducted after removing the applied current pul se and magnetic field, that is,I¼/C0 16 mA ( I¼þ 16 mA) with H¼/C0 100 Oe for state I (state II); I¼/C0 16 mA ( I¼þ 16 mA) with H¼þ 100 Oe for state III (state IV). (c) Clockwise (counterclockwise) current induced magnetic field-free switching of the device with the positive (negative) exchange bias field between IrMn and Co layers for states I and II (states III and IV).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 152403 (2021); doi: 10.1063/5.0041081 118, 152403-3 Published under license by AIP Publishing“1”) when W¼0,E¼0(W/C1E¼1) and W¼1,E¼1 (W/C1E¼1), and a negative Hall voltage (output logic “0”) when W¼1,E¼0(W/C1E¼1) and W¼0,E¼1(W/C1E¼1). However, the field-free SOT switching polarity is opposite for E¼0 (clockwise) and E¼1 (counterclockwise). Hence, different current pulses are needed to switch the Hall voltage from positive to negativeor vice versa. To switch the Hall voltage from positive to negative, aþ12 V ( A/C1B¼1) voltage is needed in the W/C1E¼1 state, while a /C012 V ( A/C1B¼1) voltage is needed in the W/C1E¼1 state. Similarly, to switch the Hall voltage from negative to positive, a /C012 V (A/C1B¼1) voltage is needed in the W/C1E¼1 state, while a þ12 V (A/C1B¼1) voltage is needed in the W/C1E¼1s t a t e .H e n c e ,t h er e l a - tionship between logic output Land the four variables can be described as L¼W/C1E/C1A/C1BþW/C1E/C1A/C1BþW/C1E/C1A/C1BþW/C1E/C1A/C1B: Accordingly, by assigning different values to the four variables, as listed in Fig. 3(c) , we can realize all 16 Boolean logic gates in a single device using “ p” and “ q” as two input variables of Boolean logic. In the following paragraph, the NOR logic operation is illustrated in detail as an example of complete spin logic operation. NOR is uniquebecause it is one of the two functionally complete Boolean logic func-t i o n si na l l1 6B o o l e a nl o g i cg a t e s . 22,23The logic output is always “0” for NOR logic, except when the inputs are both logic “0,” for which thelogic output is “1.” According to Fig. 3(c) , the four input variables W,E, A,a n d Bare set to “ p,” “0,” “0,” and “ q,” respectively. After the “initialization” step, the magnetization of the bottom Co layer alwayspoints in the - x-axis direction, as schematically shown in Fig. 4(a) , because H¼/C0100 Oe is applied ( E¼0) during initialization. In Fig. 4(c),W¼p¼0a n d B¼q¼0 mean that the device is set to apositive Hall voltage in the first step, and no voltage is applied in the sec- ond step. In this case, the Hall voltage remains unchanged in the second step, and hence, the logic output is still “1.” Similarly, W¼p¼0a n d B¼q¼1 mean that a total voltage of þ12 V is applied in the second step. As a result, the magnetization of the CoPt layer is switched from upward to downward [ Fig. 2(c) ]. Hence, the logic output is “0,” as shown in Fig. 4(d) . On the other hand, W¼p¼1(E¼0) means that the device is set to logic “0” in the first step. In this case, B¼q¼0a n d B¼q¼1w i t h A¼0 mean the application of a total voltage of 0 V [Fig. 4(e) ]a n dþ12 V [ Fig. 4(f) ] in the second step, where CoPt magne- tization remains unchanged, and the logic output is “0.” As summarized inFig. 4(b) , the function of NOR logic is fully realized. The other 15 Boolean logic functions can also be realized within three logic steps (supplementary material Figs. S4–S18). The above experimental results clearly demonstrate the realiza- tion of complete spin logic functions in a single nonvolatile four-statememory by assigning different values to four variables in three sequen- tial steps. There are several advantages of the studied system compared to common logic devices: (1) Since at least two transistors are needed for each CMOS logic gate, the integration density can be effectively increased by integrating all 16 Boolean logic functions into a single device. (2) The operation speed can be increased to the gigahertz regime—previous investigations have shown that spin–orbit torqueinduced magnetization switching can be achieved within 0.1–1 ns. 19,24 (3) The logic output is nonvolatile and can be repeatedly read outwithout refreshing, which is suitable for parallel computing. 13(4) There is no physical limit to scaling down the device to 100 nm or below, and so the integration density can be further increased and the energy consumption further reduced. In conclusion, using nonvolatile multi-state memory units to realize complete spin logic operations provides a promising route for FIG. 3. Complete spin logic illustration. (a) Assignment table for the four input variables. Each parameter has two choices, corresponding to logic “0” or “1 .” (b) Three steps needed to accomplish a logic gate. (c) Assignment of the four variables to realize all 16 Boolean logic functions.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 152403 (2021); doi: 10.1063/5.0041081 118, 152403-4 Published under license by AIP Publishingdeveloping electronic devices that surpass von Neumann architecture. In a single IrMn/Co/Ru/CoPt magnetic heterojunction with four dif- ferent magnetic configurations, by assigning different values to four input variables in three sequential steps, all 16 Boolean logic functionscan be realized. The concept of using multi-state nonvolatile memory to realize complete spin logic in a single device could be further extended to systems with controllable switching between differentstates. However, from an applications point of view, eliminating the magnetic field needed in the “initialization” step would equate to a sig- nificant development in purely electrical field controllable spin logic.Considering that great progress has been made in both field-free spin–orbit torque induced magnetization switching 25–28and electrical field control of exchange bias,29it is very likely that this significant development will occur soon. Our study should promote much research that focuses on using nonvolatile multi-state memory to achieve logic-in-memory, paving the way for the commercialization ofspin–orbit torque based stateful logic circuits. See the supplementary material (Figs. S1–S18) for the spin–orbit torque switching and magnetic properties of the studied IrMn/Co/Ru/CoPt heterojunctions and for a demonstration of 15 Boolean logic functions based on the IrMn/Co/Ru/CoPt heterojunctions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11774199, 11774016, and 51871112), the 111 Project B13029, and the major basic research project of Shandong Natural Science Foundation (Grant No.ZR2020ZD28). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. X. Shen, P. P. Lu, D. S. Shang, and Y. Sun, Phys. Rev. Appl. 12, 054062 (2019). 2A. Ney, C. Pampuch, R. Koch, and K. H. Ploog, Nature 425, 485 (2003). 3J. Borghetti, G. S. Snider, P. J. Kuekes, J. J. Yang, D. R. Stewart, and R. S. Williams, Nature 464, 873 (2010). FIG. 4. NOR logic function. (a) Operation sequence to accomplish a NOR logic operation. (b) Truth table of the NOR logicoperation. (c)–(f) Experimental results forNOR logic operation, where positive and negative Hall voltages are used as output logic “1” and “0.”Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. 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5.0061749.pdf

The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ab initio study of the O 3–N2complex: Potential energy surface and rovibrational states Cite as: J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 Submitted: 29 June 2021 •Accepted: 20 July 2021 • Published Online: 5 August 2021 Yulia N. Kalugina,1 Oleg Egorov,1,2 and Ad van der Avoird3,a) AFFILIATIONS 1Laboratory of Quantum Mechanics of Molecules and Radiative Processes, Tomsk State University 36, Lenin Ave., Tomsk 634050, Russia 2Laboratory of Theoretical Spectroscopy, V. E. Zuev Institute of Atmospheric Optics SB RAS 1, Akademician Zuev Sq., Tomsk 634055, Russia 3Theoretical Chemistry, Institute for Molecules and Materials, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands a)Author to whom correspondence should be addressed: A.vanderAvoird@theochem.ru.nl ABSTRACT The formation and destruction of O 3within the Chapman cycle occurs as a result of inelastic collisions with a third body. Since N 2is the most abundant atmospheric molecule, it can be considered as the most typical candidate when modeling energy-transfer dynamics. We report a newab initio potential energy surface (PES) of the O 3–N 2van der Waals complex. The interaction energies were calculated using the explicitly correlated single- and double-excitation coupled cluster method with a perturbative treatment of triple excitations [CCSD(T)-F12a] with the augmented correlation-consistent triple-zeta aug-cc-pVTZ basis set. The five-dimensional PES was analytically represented by an expansion in spherical harmonics up to eighth order inclusive. Along with the global minimum of the complex ( De=348.88 cm−1), with N 2being perpendicular to the O 3plane, six stable configurations were found with a smaller binding energy. This PES was employed to calculate the bound states of the O 3–N 2complex with both ortho- and para-N 2for total angular momentum J=0 and 1, as well as dipole transition probabilities. The nature of the bound states of the O 3–oN 2and O 3–pN 2species is discussed based on their rovibrational wave functions. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0061749 I. INTRODUCTION In depth studies of collisions between O 3and other atmo- spheric species are mandatory for understanding the formation and destruction of O 3under atmospheric conditions. According to Chapman’s cycle, the formation of atmospheric ozone is a multi-step collision process in the presence of a third body M, which must be a sufficiently abundant atmospheric species. The interaction of ozone and nitrogen molecules plays a key role in these processes. The energy-transfer Lindemann mechanism is believed to be the dominant process at low pressures and stratospheric tempera- tures. It can be partitioned in two steps: the formation of metastable O3∗molecules in excited rovibrational states (scattering resonances) O+O2+M→O3∗+M, (1)followed by the stabilization by collisions with an energy absorbing partner M—most likely a nitrogen molecule, N 2, O3∗+M→O3+M∗. (2) At higher pressures, the Chaperon radical complex mechanism1,2 may also play an important role. One of the “strange and unconventional” features3–6of the ozone formation is the anomalous isotope effect observed both in the stratosphere7and in laboratory measurements.8Contrary to most molecules, where isotope enrichment scales according to relative mass differences, ozone formation shows a strong deviation from this rule with large enrichments in the heavy isotopes of oxygen. This phenomenon is known as “mass independent fractionation” almost equal for17O and18O containing species. J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 155, 054308-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Chapman’s cycle has been modeled in many studies with statistical models,3,4classical trajectories,9–13and quantum mechan- ical methods.2,14–19However, a full understanding of the observed reaction rates from first-principles is still lacking: despite significant progress in theory, the calculated reaction rates do not yet provide an accurate explanation of the experimental results.20,21 During the last few decades, much effort has been devoted to ab initio PES calculations of the free ozone molecule (see, for exam- ple, Refs. 22–27 and references therein). It was shown that the cor- rect shape of the PES in the transition state region25,26,28plays a very important role, both for the calculation of absorption spectra29,30 and for modeling the isotopic atom exchange reaction31–37 aO+bOcO→(aObOcO)∗→aOcO+bO, (3) which is one of the steps related to the anomalous isotope effect in ozone formation. Here, superscripts a, b, and c stand for the atomic masses: 16, 17, and 18. The efficiency of reaction (3) depends on the lifetimes of scattering resonances (OOO)∗, which, in turn, depend on the potential energy surface (PES) of the ozone molecule undergoing this dynamical process. Advanced ab initio calculations for the free ozone molecule have shown that a submerged barrier on the minimum energy path toward dissociation (“reef structure”), which appeared in earlier works,38,39should be replaced by a kind of smooth shoulder.25,26,28 The corresponding improved PESs25,26have permitted explaining a negative temperature dependence of the exchange rate31,37for reac- tion (3) that is consistent with experimental data.40–42Quantitative agreement for the reaction rates was reported in Ref. 36. The long-lived metastable states play a crucial role in the for- mation reaction because a sufficiently long lifetime is necessary for O3∗to wait for the colliding partner M in the stabilization step (2). Many factors have an impact on the lifetime, including the shape of the electronic ground state ozone PES37,43and the spin–orbit coupling between the energy levels of the ground singlet and the low-lying triplet electronic states.44A manifestation of the effect of the spin–orbit coupling on the lifetimes of the triplet states has been observed in high-resolution spectra as a significant variation of line broadening45–49that was larger than pressure broadening. The intersection of the excited electronic state PESs in the dissocia- tion channel of the ground electronic state additionally complicates accurate simulations of the dynamics.44Another interesting feature of the ozone PES is the existence of three identical potential wells due to a Jahn–Teller effect.50–52The interaction between the wells results in a delocalization of wave functions for high energy vibra- tional states and in shifts in vibrational band origins53consistent with recent laser spectroscopic experiments near the dissociation threshold.54Anharmonic coupling also induces qualitative changes in the vibrational modes.55Accurate ab initio calculations for the isotopologues of the free ozone molecule have been validated using extensive experimental data for band origins, vibrational depen- dence of rotational constants, and line intensities (see Refs. 25, 27, 30, 53, 54, and 56–60 and references therein). This helped improv- ing spectroscopic databases61–64of absorption line parameters in the microwave and infrared ranges. Interaction potentials are the key ingredients for modeling col- lision dynamics. Recently, O 3–Ar PESs have been constructed65,66and the energies of the rovibrational levels of the corresponding complex have been computed using the MCTDH method.65Transi- tion frequencies extracted from the calculated energy levels were in good agreement with the measurements.67Scattering cross sections calculated in Ref. 68 resulted in larger total rates of state-changing collisions for asymmetric isotopic species of Cspoint group symme- try, such as16O16O18O, in comparison with the main isotopologue 16O3ofC2vsymmetry. This can be explained by symmetry breaking that increases the density of states because some levels in symmetric 16O and18O containing species have zero spin weights.69 Thermal rate coefficients of O 3collisions are necessary for simulating radiative heat transfer in out of local thermodynamic equilibrium (non-LTE) conditions.70,71In non-LTE models, using, for example, “radiative transfer and molecular excitation” (RATRAN)72and “Atmospheric Radiative Transfer Simulator” (ARTS)73computational codes, the population of states depends on the rates for collisions between atmospheric species. Information on collisionally determined populations of the vibrational energy levels is most valuable due to the larger gaps between them in comparison with rotational level spacings. An efficient approach to simulate the energy-transfer process for vibrational states is based on the Infinite Order Sudden (IOS) approximation with vibrational close-coupled channels (VCCs).14,15The application of the VCC-IOS method to the case of the O 3–N 2collision system looks promising to get collision-induced vibrational transitions in O3typical for atmospheric conditions. To our knowledge, no full-dimensional PES for the O 3–N 2 complex is currently available in the published literature. Actually, an argon atom was most often used as the collider when simulating O3+M collisional dynamics (see Refs. 9, 13–15, and 68 and refer- ences therein). In some works, simplified “atom–atom” Lennard- Jones potentials were applied to calculate relaxation rates and line broadening of O 3by N 2.74–76The O 3–Ar complex was easier to study because the corresponding PES has fewer degrees of freedom without inner rotation of the collider. However, the intermolecu- lar interaction between O 3and N 2is stronger than that between O3and Ar, as the global minimum of the O 3–N 2complex is 1.5 times deeper. Due to the existence of rovibrational bound states and dipole allowed transitions, the O 3–N 2complex can contribute to the absorption of radiation in the stratosphere. In this work, we report a five-dimensional PES for the O 3–N 2complex computed using high level ab initio methods with frozen internal vibrational degrees of freedom for the monomers. Section II describes the com- putation and the analytical modeling of the O 3–N 2PES, as well as the analysis of stationary geometrical configurations. Section III is devoted to symmetry considerations, calculation of the rovibrational bound states for J=0 and J=1, and dipole transition probabilities. This work is summarized in Sec. IV. II. POTENTIAL ENERGY SURFACE A.Ab initio calculations The ab initio calculations were carried out for geometries defined in two frames (1 and 2) at the centers of mass of O 3and N2, respectively. Similar to our previous work on the H 2O–CO complex,77the O 3molecule lies in the xz-plane of frame 1 with its twofold symmetry axis on the z axis. The vector Rconnects the J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 155, 054308-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1. The definition of the polar angles of the O 3–N2PES. origins of frames 1 and 2 and the spherical angles θ1andφ1describe the orientation of this vector. The angles θ2andφ2describe the orientation of N 2relative to frame 2, which is parallel to frame 1. Thus, the interaction potential V(R,θ1,φ1,θ2,φ2) is a function of five variables (Fig. 1). As in previous work on the simpler O 3–Ar complexes,65,66 the internal vibrational degrees of freedom for O 3were frozen. The intramolecular distances between the atoms were fixed at the ground vibrational state average distance rN1N2=2.0785 a0of N278and to the equilibrium geometry for O 3:rO1O2=2.4095 a0, ∠(O1O2O3)=116.78○.25Following the strategy of many previous studies65,66,77,79of non-covalent interactions for complexes composed of two closed-shell molecules, we used coupled cluster theory, which usually assures good convergence and accuracy.79 Multi-reference methods, such as MRCI (multi-reference config- uration interaction) or AQCC (multireference averaged quadratic coupled cluster),80,81would be necessary to build a full nine- dimensional PES involving excited vibrational states of the ozone molecule up to the dissociation threshold of O 3(≈8560 cm−1).39,82 This task is actually too demanding in terms of both CPU time and determination of the electronic structure of O 3in the full electronic configuration space of the complex and can be considered as a challenging project for future work. However, it is known that single-reference methods work well for nuclear geometries of ozone near the equilibrium structure, which is the subject of the present study. Ab initio energies were calculated using the explicitly corre- lated single- and double-excitation coupled cluster approach with a perturbative treatment of triple excitations [CCSD(T)-F12a]83 with the augmented correlation-consistent triple-zeta aug-cc-pVTZ basis set84implemented in the MOLPRO2019 package.85The expo- nentβin the correlation factor F12was set to 1.3. The interaction energy is the difference between the energies of the complex and the monomers. The energies of the monomers were computed in the basis set of the full complex in order to take the basis set super- position error (BSSE) into account.86CCSD(T)-F12a is a highly accurate method with relatively low computational costs. To demon- strate the performance of the CCSD(T)-F12a method for the O 3–N 2 complex, we made the 1D cuts of the PES at the most informative angular orientations. According to Fig. 2, the CCSD(T)-F12a method gives energy curves that are very close to those obtained with Complete Basis Set (CBS) limit extrapolation using the CCSD(T) method.87The interaction energy was calculated for 94 770 geometries. For each distance Rbetween the centers of mass of O 3and N 2, we chose 3645 orientations ( θ1,φ1,θ2,φ2) on a quadrature grid. The distance Rwas increased with a constant increment of 0.25 a0start- ing from its minimal value of Rmin=5a0up to 9.5 a0. To build an asymptotic long range part of the PES, the values of R=10, 11, 12, 16, 20, 25, 30 a0were additionally included in the grid of nuclear geometries. B. Analytical description The 5D PES is analytically represented in the form of the following expansion: V(R,θ1,φ1,θ2,φ2)=∑ l1,m1,l2,lvl1m1l2l(R)⋅tl1m1l2l(θ1,φ1,θ2,φ2), where tl1m1l2l(θ1,φ1,θ2,φ2)are normalized spherical tensors that include two spherical harmonics and Wigner 3 j-symbols. The reader can find a detailed definition of tl1m1l2lin previous work.88–90The tensor rank indices l1and l2correspond to the θ1andθ2angu- lar dependences of the PES, while the m1index is related to angle φ1. The lindex obeys the triangular relation ∣l1−l2∣≤l≤l1+l2. Due to the symmetry of the PES with respect to the reflection in the y1z1-plane of the ozone molecule (Fig. 1), m1takes only even values 0, 2, 4, . . .. The homonuclear symmetry of N2also requires only even values for l2. The expansion coefficients were obtained by integration (as described in Ref. 91) using Gauss–Legendre and Gauss–Chebyshev quadratures. Anisotropic expansion coefficients up to l1=l2=8 and l=16 were included. The final set included 1395 expansion coeffi- cients vl1m1l2l(R). The root mean square (rms) deviation between the ab initio data and the potential calculated with the expansion was within 1 cm−1at all attractive energies, including the potential well and the long range part. A smooth interpolation of the radial dependence of the expan- sion coefficients was constructed with the reproducing kernel Hilbert space (RKHS) method.92The RKHS interpolation is widely used for analytically describing ab initio potential energy surfaces of polyatomic molecules calculated on a regular grid.93The leading long range term in the O 3–N 2potential is the dipole–quadrupole interaction proportional to R−4and the RKHS parameter m, which makes the potential decay as R−m−1beyond the largest Rvalue for which it was calculated, was chosen to be m=3, 4, 5, or 6 for different terms in the expansion. The smoothness parameter nwas set to 2. To calculate the bound states of the O 3–N 2complex, we chose a body-fixed (BF) dimer frame. The zaxis of this frame coincides with the direction of the vector R. The orientation of the O 3monomer with respect to the BF frame is described by three Euler angles, ωA≡(αA,βA,γA), while two polar angles, ωB≡(βB,αB), define the orientation of the N 2frame relative to the BF frame. Thus, the final potential in the BF frame is a function of four angles: V(R,βA,γA,βB,α), whereα=αB−αAis the dihedral angle. There is the following relationship between the BF angles and the polar ones, which were used for constructing of the PES (Fig. 1): the polar angles (θ1,φ1)are equal to the angles (−βA,−γA)or, equivalently, (βA,π−γA); the two other polar angles (θ2,φ2) J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 155, 054308-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. Comparison of various versions of ab initio calculations for the radial dependence of the interaction energy of the O 3–N2complex at fixed angular orientations ( θ1, φ1,θ2,φ2): (a) (0, 0, 0, 0), (b) (90, 0, 0, 0), (c) (180, 0, 0, 0), and (d) (90, 90, 0, 0). are transformed with the aid of Euler rotation matrices. The conversion can also be done by using a direct relation between different expansion functions of the PES for the angular coordinates considered.94C. Stationary points The next step was the localization of possible minima in the constructed O 3–N 2ab initio PES. The stationary points of the PES TABLE I. The polar (θ1,φ1,θ2,φ2) and body (βA,γA,βB,α) fixed angles and the depth of the localized minima Vminin the interaction potential energy surface of the O 3–N2 complex.a No. R(a0) θ1,βA(O3) φ1,γA(O3) θ2,βB(N2) φ2,α Vmin(cm−1) 1 6.6927 70.9506, 70.9506 90.0, 90.0 55.2201, 15.7305 90.0, 0.0 −348.8775 2 6.3323 51.3038, 51.3038 90.0, 90.0 0.0,51.3038 0.0,0.0 −320.9698 3 6.0284 68.0416, 68.0416 90.0, 90.0 90.0, 90.0 0.0,90.0 −258.6047 4 7.6290 139.0611, 139.0611 0.0,0.0 101.4770, 37.5841 0.0,0.0 −188.6907 5 6.8825 0.0, 0.0 0.0,0.0 90.0, 90.0 90.0, 90.0 −175.8966 6 7.3604 141.0464, 141.0464 90.0, 90.0 0.0,–38.9536 0.0,0.0 −167.3471 7 8.1372 63.7411, 63.7411 0.0,0.0 90.0, 90.0 90.0, 90.0 −146.3047 aAngles are given in degrees. J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 155, 054308-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. View of the global minimum of the O 3–N2PES in the polar coordinates defined in Fig. 1: (a) ( θ1,φ1) and (b) (θ2,φ2). The Rvalue is fixed here to 6.6927 a0(see Table I). FIG. 4. 2D cuts of the O 3–N2PES in BF coordinates through the global minimum: (a) ( α,R), (b) (γA,βA), (c) (βB,βA), and (d) (γA,α). J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 155, 054308-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp were searched with a modified Levenberg–Marquardt algorithm implemented in the MINPACK library.95A system of nonlinear equations is based on the first derivatives of the PES with respect to the(R,θ1,φ1,θ2,φ2)coordinates. To start the optimization process, the polar angles were varied on a dense grid of geometries to obtain an initial guess for the roots. Finally, we localized up to 23 stationary configurations. However, only seven of them had a confident minimum along all five coordinates, while other ones were saddle points (Table I). The global minimum corresponds to a configuration where N 2 is perpendicular to the O 3plane with the following values of the coordinates: R=6.6927 a0,θ1=70.9506,φ1=90.0,θ2=55.2201, and φ2=90.0. This minimum has a depth of about −349 cm−1, which is∼1.5 times deeper than the minimum for the O 3–Ar complex.65,66 According to Figs. 3 and 4, the global minimum is well localized. The configurations of the local minima may play a role in higher bound states of the complex. III. BOUND STATES OF THE COMPLEX A. Computational method and details The method used to calculate the bound rovibrational states of the O 3–N 2complex was described, in general, in Ref. 94 and was previously applied to other asymmetric rotor–linear molecule complexes: H 2O–H 2,91H2O–CO,77,96and H 2O–HF.97The wave functions of the complex are expanded in a product basis of radial wave functions, symmetric rotor functions for O 3(with total angu- lar momentum j1and projection k1on the twofold symmetry axis), and spherical harmonics for N 2(with total angular momentum j2). This product basis is Clebsch–Gordan coupled and multiplied with a basis of symmetric rotor functions for the overall rotation of the complex.91The symmetry properties of the O 3–N 2complex can be described with the permutation-inversion or molecular symmetry group G8≡D2h(M). The group G8is generated by three operations:98 permutations P12andP34and inversion E∗. Permutation P12inter- changes the equivalent outer O nuclei in the O 3molecule and P34 interchanges the N nuclei in N 2. Since16O and14N nuclei are bosons with integer spin quantum numbers, I=0 and I=1, respectively, the wave functions of O 3and N 2should be symmetric under P12and P34. The nuclear spin wave functions of N 2are even/odd under P12 for ortho/para configurations, while the rotational wave functions are even/odd for even/odd values of j2. The nuclear spin wave func- tion of O 3with I=0 is invariant under P12, and its rotational wave functions are even/odd for even/odd k1. Therefore, only functions with even k1are physically allowed. Since the twofold symmetry axis of O 3is the principal baxis and k1≡kb, this corresponds with even values of ka+kcin terms of the asymmetric rotor quantum num- bers. The nuclear spin weights of the allowed O 3–oN 2and O 3–pN 2 states are 6 and 3, respectively. Although we calculated the bound rovibrational states for all irreducible representations of G8, we will discuss only the allowed states. The total angular momentum Jand the parity p=+/−under E∗ are good quantum numbers. Instead of by using parity p, the bound states will be classified by their spectroscopic parity ε=e/fdefined by the relation p=ε(−1)J. For the rotational constants of O 3and N2, we used the experimental values: 3.553 67, 0.394 75, 0.445 28, and 1.989 50 cm−1, respectively. After performing a series ofconvergence tests, the maximum values of j1and j2in the basis were taken to be 18 and 11. The radial basis is a contracted discrete variable basis with Rranging from 5 to 20 a, in 157 equal steps. We finally used five radial basis functions, optimized as described in Ref. 91. We estimate that with this basis, the energy levels are converged to about 0.1 cm−1, the vibrational transition frequencies are converged to about 0.05 cm−1, and the rotational transition frequencies are converged to about 0.005 cm−1. B. Energy levels The bound energy levels for the o/p-configurations of the O3–N 2complex calculated for total angular momenta J=0 and 1 are listed in Tables II and III. Each energy level has spectroscopic parity eorfand can be characterized with an approximate quantum number K, which is the projection of the total angular momentum Jonto the intermolecular axis R. Actually, we use the absolute value ofK, and the even/odd combinations of functions with +Kand−K are labeled with their spectroscopic parity e/f. States with K=0 and 1 are indicated as ΣandΠ. Both O 3–oN 2and O 3–pN 2have aΣground state with par- ityeand energies, −220.8517 and −220.8515 cm−1, that are very similar. The energy gap between these two ground states may be regarded as a tunneling splitting since the wave functions of O 3–oN 2 are even under P34and those of O 3–pN 2are odd. This tunneling splitting cannot directly be observed, however, because transitions between levels of O 3–oN 2and O 3–pN 2are not allowed. The observation that this tunneling splitting is minute indicates that the complex is a relatively rigid van der Waals molecule. Since the ground state energy of pN2with j2=1 has energy 3.9790 cm−1, the dissociation energies of O 3–oN 2and O 3–pN 2are slightly different: D0=220.85 cm−1and D0=224.83 cm−1. They are about 2/3 of the binding energy De=348.88 cm−1, so the vibrational zero-point energy is about 1/3. As shown below, the near rigidity of the com- plex is confirmed by wave functions being well localized around the global minimum of the PES. In addition, the end-over-end rotational constants ( B+C)/2 of O 3–oN 2and O 3–pN 2extracted from the TABLE II. Rovibrational levels of O 3–oN 2(in cm−1) relative to the J=0 ground state energy, E0=−220.8517 cm−1; dissociation limit =0,D0=220.8517 cm−1. Parity e Parity f No. J=0 J=1 J=0 J=1 1Σ 0.0 0.1315 Π 0.4657 Π 0.4573 2 Σ 23.2515 23.3779 Π 23.4529 Π 23.4663 3Σ 23.8102 23.9446 Π 24.5316 Π 24.5277 4Σ 42.9776 43.1108 Π 43.4021 Π 43.3926 5 Σ 45.0389 45.1681 Π 45.4739 Π 45.4835 6Σ 47.3908 47.5179 Π 47.8794 Π 47.8719 J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 155, 054308-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III. Rovibrational levels of O 3–pN 2(in cm−1) relative to the J=0 ground state energy, E0=−220.8515 cm−1; dissociation limit =3.979 cm−1,D0=224.8305 cm−1. Parity e Parity f No. J=0 J=1 J=0 J=1 1Σ 0.0 0.1315 Π 0.4653 Π 0.4570 2 Σ 23.2304 23.3571 Π 23.4329 Π 23.4461 3Σ 23.8063 23.9406 Π 24.5208 Π 24.5168 4Σ 43.1435 43.2756 Π 43.5615 Π 43.5510 5 Σ 44.9728 45.1028 Π 45.4005 Π 45.4098 6Σ 47.4256 47.5528 Π 47.9046 Π 47.8971 Σground state levels with J=0 and 1 are nearly the same: 0.065 75 and 0.065 75 cm−1. Another indication of the near rigidity of the O 3–N 2com- plex follows from the J=1 levels with energies 0.1315, 0.4573, and 0.4657 cm−1for O 3–oN 2(state 1 in Table II) and nearly the same energies for O 3–pN 2(state 1 in Table III). Calculation of the iner- tia tensor of O 3–N 2in its equilibrium geometry specified in the first line of Table I and diagonalization of this tensor yield a set of rotational constants of the complex: A=0.3963, B=0.0717, and C=0.0629 cm−1. The J=1 rigid-rotor levels calculated with these rotational constants are 0.1346, 0.4592, and 0.4680 cm−1. These energies are very similar to the energies of the J=1 states from the full 5D calculations in Tables II and III and, just as these states, they have characters Σ,Πe, andΠf. This clearly confirms the near rigidity of the complex and shows that these J=1 levels belong to the ground vibrational state of the complex. In floppier van der Waals complexes, such as H 2O–H 291and NH 3–H 2,99,100the inter- nal motions and the overall rotation are more strongly coupled, and states with different values of Kexhibit different internal motions and have more different energies. Table II also lists five excited states of O 3–oN 2. They all have Σcharacter and start at J=0: three of them have parity eand two have parity f. Table III shows the levels of O 3–pN 2, which have the same character as the corresponding O 3–oN 2levels and very similar energies. After the analysis of the ground state J=1 levels discussed above, it becomes clear that also the J=1 levels of each excited state are just rotationally excited with respect to the corresponding J=0 levels. They show the same rotational structure as the ground state, with characters Σ,Πe, andΠf. The end-over-end rotational split- tings between the Σstates with J=0 and J=1 are similar to the ground state splitting, but the larger splittings between the Πstates with J=1 and the J=0 states are different for the different excited states and also different from the ground state. In other words, the rotational constant Athat is related to the moment of inertia of the complex about the intermolecular axis is more strongly affected by excitation of the internal motions than the end-over-end rotational constant ( B+C)/2. FIG. 5. Cut of the ground state wave function (state 1 in Table II), with the angles γ(O3)=90○andα=0○fixed at their equilibrium values and R=6.75 a0. In van der Waals complexes, such as H 2O–H 2,91the character of the states could be analyzed by considering the contributions of the different radial and angular basis functions, with different nR and different monomer quantum numbers j1,k1, and j2. In addition, the individual monomer excitations contributing to the Π(K=1) states could be analyzed by considering the projection quantum numbers m1andm2(K=m1+m2) of the angular momenta j1and j2on the intermolecular axis R. The contributions of these approx- imate quantum numbers can be extracted from the calculated wave FIG. 6. Wave function of the excited state of parity fat 23.25 cm−1(state 2 in Table II), with the angles β(O3)=70.95○andβ(N2)=15.73○fixed at their equilibrium values and R=6.75 a0. J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 155, 054308-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7. Wave function of the excited state of parity eat 23.81 cm−1(state 3 in Table II), with the angles γ(O3)=90○andα=0○fixed at their equilibrium values andR=6.75 a0. functions by considering the squared eigenvector components. In the case of O 3–N 2, we learned from this analysis that many differ- ent monomer rotational states contribute to the wave functions of the complex up to large values of j1andj2. This might be expected since we saw already that the states of the O 3–N 2complex are rather well localized, and many free-rotor functions are needed to represent such angularly localized states. We will show contour plots of the FIG. 8. Wave function of the excited state of parity eat 42.98 cm−1(state 4 in Table II), with the angles β(O3)=70.95○andβ(N2)=15.73○fixed at their equilibrium values and R=6.75 a0. FIG. 9. Wave function of the excited state of parity fat 45.04 cm−1(state 5 in Table II), with the angles β(O3)=70.95○andβ(N2)=15.73○fixed at their equilibrium values and R=6.75 a0. wave functions, which illustrate that in the present case, the angu- lar excitations may be considered as bend, wagging, or twist modes rather than as weakly hindered internal rotations. C. Wave functions The character of the ground and excited states listed in Tables II and III is illustrated by the plots of their wave functions in Figs. 5–10. FIG. 10. Wave function of the excited state of parity eat 47.39 cm−1(state 6 in Table II), with the angles β(O3)=70.95○,β(N2)=15.73○, andγ(O3)=90○fixed at their equilibrium values. J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 155, 054308-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We show only the wave functions of O 3–oN 2because those of the states of O 3–pN 2are almost identical. In addition, we show only those of the Σstates with J=0 because the corresponding J=1 states have the same internal motion character and are just rotation- ally excited. Figures 5–10 contain two-dimensional cross sections of the wave functions that are most informative for the specific charac- ter of each of the states, and the other coordinates are frozen at or close to their equilibrium values. Figure 5 shows that the ground state is quite well localized around the global minimum of the potential at β(O3)=70.95○and β(N2)=15.73○. The first excited state at 23.25 cm−1in Fig. 6 has parity f, which implies that its wave function changes sign when both αandγ(O3) change sign. It might be characterized as a torsionally (α) excited state, although the angle γ(O3) by which O 3rotates about its twofold symmetry axis is also involved. The second excited state at slightly higher energy, 23.81 cm−1, in Fig. 7, has parity eand is clearly a mode in which both the anglesβ(O3) andβ(N2) are involved. The motion of O 3in theβ(O3) angle might be called wagging, while the N 2motion inβ(N2) is clearly a bend. The third excited state at 42.98 cm−1with parity e, shown in Fig. 8, is the overtone of the torsional mode with frequency 23.25 cm−1in Fig. 6. The excited state of parity fat 45.04 cm−1in Fig. 9 also involves the anglesαandγ(O3), but its nodal plane is clearly more horizon- tal than it is in Fig. 6. Therefore, we characterize this state as being excited in the twist angle γ(O3). The state of parity eat 47.39 cm−1in Fig. 10 is clearly excited in the intermolecular stretch coordinate R. D. Transition intensities The wave functions illustrated in Sec. III C were used to com- pute transition line strengths and the squares of transition dipole moments. We assumed that the dipole of the O 3–N 2complex is simply the permanent dipole of the O 3monomer, expressed in the FIG. 11. Stick spectra representing transition frequencies and intensities at T=5 K calculated with the bound states of the O 3–oN 2and O 3–pN 2complexes in Tables II and III: (a) global view; (b)–(d) detailed figures for the regions of 0–2, 18–25, and 42–48 cm−1. The symbols with the sticks denote the following transitions: (1) pure rotational, (2) torsional, (3) wagging-bend β(O3) andβ(N2), (4) torsional overtone, (5) O 3twistγ(O3), and (6) intermolecular stretch. The red sticks represent hot bands. Detailed assignments can be found in the supplementary material. J. Chem. Phys. 155, 054308 (2021); doi: 10.1063/5.0061749 155, 054308-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp coordinates of this monomer in the complex and the orientation of the complex in a space-fixed (laboratory) frame. In addition, we assumed that the O 3permanent dipole is unity since one is usually interested in relative intensities. Transition intensities were calcu- lated by including the Boltzmann factors for the populations of the states at T=5 K. The energies of the states were taken relative to the ground state energies (values in Tables II and III). According to the stick spectrum in Fig. 11(a), there are three absorption regions: 0–2, 18–25, and 42–48 cm−1. Since the ener- gies of the bound states of the ortho- and para-modifications of the O 3–N 2complex are very similar, the discussions of the O 3–oN 2 transitions given below can also be applied to O 3–pN 2. The strongest lines are those in the 0–2 cm−1region [Fig. 11(b)], which involve pure rotational transitions among the four rotational levels listed in Table II. They include two transitions of the Qbranch (ΔJ=0) and two of the Rbranch (ΔJ=1). For the first two lines centered at 0.0084 ( ΔJ=0) and 0.1315 cm−1(ΔJ=1), the initial and final rotational levels have the same projection K of the total angular momentum J. The other two lines centered at 0.3258 (ΔJ=0) and 0.4657 cm−1(ΔJ=1) start with the Σlevels (K=0) and end at the Πlevels ( K=1). There are two types of hot band transitions in this region: transitions within the rovibrational levels of a certain vibrational state and transitions between rovi- brational levels of different vibrational states with nearly the same energy. Next, the 18–25 cm−1region includes cold transitions associ- ated with the rovibrational levels of the torsional and wagging-bend excited states [Fig. 11(c)]. The spectral lines of the wagging-bend band have generally higher intensities than those of the torsional band: the ratio of their strongest intensities exceeds 3.5 at T=5 K. Similar to the pure rotational region, most of the spectral lines belong to QandRbranches and there are one and two lines of the Pbranch (ΔJ=−1) in the torsional and wagging-bend bands, respec- tively. The hot bands calculated for this region involve transitions in which the torsional and wagging-bend excited states are the lower states, while the torsional overtone, twist, and intermolecular stretch are the upper states. The 42–48 cm−1region starts with several lines of the torsional overtone band, which are the strongest in this region. At higher frequencies, there are three lines due to the twist band and two lines of the intermolecular stretch band [Fig. 11(d)]. The intensities of the twist band are the weakest among all other cold bands. It should be kept in mind, however, that the dipole function used in these calcula- tions is an approximate one, as explained above. In order to calculate more accurate intensities, one needs a full dipole moment surface of the O 3–N 2complex from ab initio calculations. IV. CONCLUSIONS A five-dimensional potential energy surface (PES) of the O 3–N 2 complex was constructed with high level ab initio calculations. The analytical form of this PES shows a relatively deep global mini- mum ( De=348.88 cm−1) with the N 2axis nearly perpendicular to the O 3plane. Bound states were calculated for total angular momenta J=0 and 1 for both the ortho and para nuclear spin species of the nitrogen molecule: O 3–oN 2and O 3–pN 2. The rovibrational energy levels obtained were very similar for the O 3–oN 2and O 3–pN 2complexes and so are their end-over-end rotational constants: 0.065 75 and 0.065 75 cm−1. This indicates that O 3–N 2is a relatively rigid van der Waals complex. In addition, the comparison of the J=0 →1 excitation energies with rigid-rotor energies calculated with the inertia tensor of the complex in its equilibrium geometry shows that the complex is nearly rigid. The character of the bound states was analyzed by plotting the calculated wave functions. Five excited vibrational states were iden- tified by looking at the nodal planes in the wave functions: four angular vibrations (torsional, wagging-bend, torsional overtone, and twist) and one intermolecular stretch mode. All the bound state wave functions, especially the wave function of the ground vibrational state, are quite well localized around the global minimum of the PES. This confirms the near rigidity and relative stability of the O 3–N 2 van der Waals complex and, hence, its possible capacity for playing a role in stratospheric processes. The transitions between the calculated bound states are clearly seen in the predicted stick spectra. In terms of line intensities at T=5 K, the pure rotational band located in the 0–2 cm−1region is the strongest. Next are the torsional and wagging-bend bands (18–25 cm−1), while the torsional overtone, twist, and intermolecu- lar stretch bands (42–48 cm−1) are the weakest. 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5.0054600.pdf

Strain dependent spin-blockade effect realization in the charge-disproportionated SrCoO 2.5thin films Cite as: Appl. Phys. Lett. 119, 021901 (2021); doi: 10.1063/5.0054600 Submitted: 20 April 2021 .Accepted: 22 June 2021 . Published Online: 12 July 2021 Sourav Chowdhury, R. J. Choudhary,a) and D. M. Phase AFFILIATIONS UGC DAE Consortium for Scientific Research, Indore 452001, India a)Author to whom correspondence should be addressed: ram@csr.res.in ABSTRACT The spin-blockade phenomena occur in the charge-ordered systems because of the non-conservation of the spin-states of the charges across the charge hopping process. Here, we have investigated the validation of the spin-blockade phenomena in a different kind of charge-orderedsystem, viz. charge-disproportionated SrCoO 2.5thin film, where O-2 phole is the key parameter for such charge-ordering. It is observed with the help of the polarization-dependent O K-edge x-ray absorption spectroscopy that the spin-blockade increases with the decrease in charge- disproportionation vis- /C18a-vis O-2 phole density in the SrCoO 2.5films. The spin-blockade has been realized in terms of the lowest energy charge fluctuation energetics in the SrCoO 2.5films. Our results provide a fundamental understanding of the spin-blockade phenomena in both the usual charge-ordered and unusual charge-disproportionated systems, which would lead to a path forward toward novel materialdesigning. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054600 For the past couple of decades, charge-ordered systems have attracted huge attention owing to their tremendous prospects in dielec-tric and thermoelectric applications. 1–4To facilitate the performance of such materials, the fundamental understanding of the consequences ofthe charge-ordering phenomena is quite crucial. Among them, the spin-blockade phenomenon is quite interesting, which occurs as a result ofthe non-conservation of the spin-states of the respective charges in thecharge hopping process. 5–7It was first observed in quantum dot sys- tems8–10and later generalized for charge-ordered cobaltates to explain its thermoelectric properties.5–7The spin-blockade effect facilitated the performance of such systems in various applications, such as quantum information transmission, spintronics, thermoelectricity, and ferroelec-tricity. 11–15However, a charge-disproportionated (or charge skipping) system, where the O-2 phole is the central ingredient for the charge- ordering, is not yet explored for the spin-blockade effect. In such sys-tems, the charge-disproportionation can be tuned via an easily tunableparameter, the O-2 phole density. 16Thus, it can be used for material designing via modulating the spin-blockade effect in the charge-disproportionated systems, which is not the case for the conventionalcharge-ordered systems. Therefore, it is stimulating to probe the occur- rence of the spin-blockade effect in such O-2 phole facilitated charge- disproportionated systems.To probe this aspect, we have taken a charge-disproportionated SrCoO 2.5(SCO) thin film as a model system, where a considerable amount of O-2 phole is present in the ground state of the film and the system reveals the charge-disproportionation phenomena owing to its strain induced negative charge transfer energy ( D).16The structure of bulk SCO is orthorhombic brownmillerite (BM), which depicts aG-type antiferromagnetic insulating ground state below the Neel temperature 570 K. 16,17Moreover, SCO has recently drawn consider- able interest among the scientific community for its huge prospectsas a fuel cell, catalyst, switching device, and smart window applications. 18–20 In this report, we have investigated the band aspects of the spin- blockade phenomena in the SCO thin films by taking advantage of thepolarization-dependent O K-edge x-ray absorption spectroscopy (XAS) technique. The spin-blockade effect is found to increase with the decrease in charge-disproportionation vis- /C18a-vis O-2 phole density in the SCO films, implying the role of O-2 phole density in tuning the functionality of such charge-disproportionated systems. The spin- blockade effect is understood from the lowest energy charge fluctua-tion energetics in the SCO films. The study was done on strained 30 nm and partially relaxed 70 nm SCO thin films deposited on SrTiO 3(001) substrates. The Appl. Phys. Lett. 119, 021901 (2021); doi: 10.1063/5.0054600 119, 021901-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldetails of the thin film depositions and their structural characteriza- tions can be found elsewhere.16The polarization-dependent XAS mea- surements across the O K-edge were carried out at room temperature (300 K) in the surface-sensitive total electron yield (TEY) mode at thebeamline BL-01, Indus-2 synchrotron source, RRCAT, Indore, India.The experimental resolution was estimated to be /C24250 meV. The pho- ton energy was calibrated using the known energy position of the O K-edge of the Co 3O4compound. The isotropic [i.e., the angle between the electric field vector of the synchrotron beam (E) and the sample surface is 45/C14]OK-edge XA spectrum at 300 K of the 30 nm thick SCO film is displayed in Fig. 1(a) .T h eO - 1 sabsorption spectrum corresponds to the transition of an electron from the O-1 score level to the unoccupied O-2 pstates hybridized with the different bands of Co and Sr. The first broad fea-ture at the position /C24530.4 eV, assigned as A, is the O 2 p-Co 3 d hybridized band and is the prime focus of this study. The second fea- ture at /C24534.9 eV is the O 2 p-Sr 4 dhybridized band, and a broad fea- ture (/C24537–546 eV) is attributed to the O 2 p-Co 4 spand O 2 p-Co 4 s hybridized states, 21–23which are successively assigned as B, C, and D, respectively. The Co L-edge XA spectrum of the 30 nm thick film [ Fig. 1(b) ] reveals that the formal charge-state Co3þis disproportionated into high-spin (H.S) Co2þand low-spin (L.S) Co4þcharge-states. The fea- tures S1 and S2 in the experimental Co L3-edge appear due to the charge-disproportionation, which can be generated in the theoreticalspectrum by only considering the Co 2þ(H.S)þCo4þ(L.S) configura- tion.16,24Thus, the lowest energy charge fluctuation in the SCO film across the Fermi-level (E F)i sb e t w e e nt h eH . SC o2þband [which forms the highest occupied energy state in the valence band (VB)] andL.S Co 4þband [which forms the lowest unoccupied energy state in the conduction band (CB)], mediated via O-2 pband.7,24For better visuali- zation, we have shown the combined VB–CB spectra in Fig. 1(c) .T h e EFalignment with respect to the VB and CB is described elsewhere.16 The uppermost region in the VB can be considered as the occupiedH.S Co 2þband and the lowermost region in the CB as the unoccupied L.S Co4þband, and the lowest energy charge fluctuation is governedby the bands of these two states, as shown in Fig. 1(c) . Therefore, fea- ture A in Fig. 1(a) is providing the information of the unoccupied states of the Co4þband. To investigate the crystallographic axis direction dependency of the unoccupied Co4þstates in the SCO film, we performed the polarization-dependent O K-edge XAS, as shown in Fig. 2(a) .T h e spectrum corresponding to E parallel (and perpendicular) to the film surface is assigned as I ab(and I c). The measurement geometry is sche- matically shown in Fig. 2(b) . Here, we are interested only in feature A, which is involved in the lowest energy charge fluctuation process,without bothering much on the other features of the spectra. The fea- ture A in I abis giving the information of O 2 p-Co 3 dhybridized orbital along the in-plane ( ab-plane) direction, defined as (O 2 p-Co 3 d)jj,a s schematically shown in Fig. 2(b) . Similarly, feature A in I cis giving the information of O 2 p-Co 3 dhybridized orbital along the out-of-plane (c-axis) direction, defined as (O 2 p-Co 3 d)?[Fig. 2(b) ]. It can be seen from the inset of Fig. 2(a) that the spectral intensity of feature A is larger in I abthan in I c,i n d i c a t i n g( O2 p-Co 3 d)jjis stronger relative to (O 2 p-Co 3 d)?. This also suggests that the O-2 phole density, which is a measure from feature A,25–27is larger in the in-plane direction than the out-of-plane direction. The observation can be understood from the structural viewpoint of the SCO film. The octahedral ( Oh) and the tetrahedral ( Td) networks are stacked alternatively along the out-of- plane direction in the SCO film because of its BM structure, as sche-matically shown in Fig. 2(b) . Thus, the O 2 p-Co 3 dhybridization is stronger either between O horTdunits itself along the in-plane direc- tion with respect to that between Ohand Tdunits along the out-of- plane direction.16Furthermore, in the SCO film (30 nm), the in-plane pseudo-tetragonal lattice parameters ( a¼b¼3.905 A ˚)<the out-of- plane lattice parameter ( c¼3.94 A ˚), as observed from the x-ray dif- fraction analysis,16which corroborates the stronger (O 2 p-Co 3 d)jj than (O 2 p-Co 3 d)?. Importantly, we note that the (O 2 p-Co 3 d)jjis located at the lower energy side than the (O 2 p-Co 3 d)?by 0.6 eV [inset of Fig. 2(a) ], which reveals that the CB bottom is of (O 2 p-Co 3 d)jjchar- acter. This also implies that the stronger in-plane O 2 p-Co 3 d FIG. 1. (a) Isotropic O K-edge XA spectrum of a 30 nm thick SCO film. Inset is showing the close view of feature A of 30 and 70 nm thick SCO films. (b) The isotropic Co L- edge XA spectrum of a 30 nm thick SCO film along with the theoretically simulated spectrum for the linear combination of H.S Co2þand L.S Co4þcharge-states. Inset is show- ing the zoomed view of the L3-edge maxima of 30 and 70 nm thick SCO films. The different separations between the charge-disproportionated features S1 and S2 are shown by the vertical solid lines. Co3þreference position is shown by the vertical dotted line, taken from our earlier studies.16,24(c) The combined VB–CB spectra of a 30 nm thick SCO film. The schematic representation of the lowest energy charge fluctuation involved in the SCO film between the H.S Co2þand the L.S Co4þbands. The uppermost region in the VB and the lowermost region in the CB are the representation of the H.S Co2þand L.S Co4þbands, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 021901 (2021); doi: 10.1063/5.0054600 119, 021901-2 Published under an exclusive license by AIP Publishinghybridization pulls the (O 2 p-Co 3 d)jjband at a lower energy position with respect to the (O 2 p-Co 3 d)?band. In other words, it can be stated that the Co2þto Co4þcharge fluctuation energy cost in the in- plane direction ( ejj) is lower than that in the out-of-plane direction (e?), as schematically shown in Fig. 2(b) . It is known that in the spin-blockade phenomena, the spin-states do not conserve across the charge fluctuation in charge-ordered sys-tems, which causes large energy costs for charge fluctuation and is thereason for showing the highly insulating nature of such compound. 5–7 We envisage the SCO film to show the spin-blockade effect since it shows the charge-disproportionation, involving H.S Co2þ(t2g5eg2)a n d L.S Co4þ(t2g5eg0) ions. However, the mechanism of such charge- disproportionation is different from the conventional charge-orderingsince the former occurred in the negative Dinsulators due to the large O-2phole density; ground state: d nþdLd(L:O - 2 phole), which is not t h ec a s ef o rt h el a t t e r( g r o u n ds t a t e : dn).28–31Thus, the lowest energy charge fluctuation involved in such insulators is between the O-2 p bands, i.e., a p-ptype of charge fluctuation,32,33depicted as dnþdLdþdnþdLd!dnþdL2dþdnþd: (1) The charge-disproportionation can be visualized in such insulators from the right-hand side of the above equation since a charge differ-ence of 2 de /C0(e/C0: electronic charge) can be obtained between the neighboring sites. In the present case of a 30 nm thick SCO film, thevalue of dequals to /C241, as the observed charge difference between the Co 2þand Co4þcharge-states is of 2 e/C0. As the ground state of the negative Dmaterial is of O-2 phole dominated, we have considered the O-2 pto O-2 ptype of lowest energy charge fluctuations ( p-p type)32,33in which similar Co-3 dsymmetric orbitals hybridized with the O-2 pstates participate.24,34–36Thus, the two-electron removal (N !N/C02) and addition (N !Nþ2) process in the lowest energy charge fluctuation looks likeCo2þH:S;t2g5eg2/C0/C1 !Co4þL:S;t2g5eg0/C0/C1 ; (2) Co4þL:S;t2g5eg0/C0/C1 !Co2þH:S;t2g5eg2/C0/C1 : (3) The lowest energy charge fluctuation energetics is also schematically shown in the left panel of Fig. 3(a) .S t e p - 1i n Fig. 3(a) is before charge fluctuation, and step-2 is after change fluctuation, considering the two-electron removal (N!N/C02) and addition (N !Nþ2) process. Thus, from Eqs. (2)and (3)and the schematic picture, it is clear that the spin-states remain intact across the charge fluctuation. It should be noted that in these discussions,we have taken the spin-state configurations of Co ions corresponding the O hsites only. However, considering Tdsites also provide the conservation of spin-states of the respective charge-states as follows: Co2þH:S;eg4t2g3/C0/C1 !Co4þL:S;eg4t2g1/C0/C1 ; (4) Co4þL:S;eg4t2g1/C0/C1 !Co2þH:S;eg4t2g3/C0/C1 : (5) Thus, it is inferred that the spin-blockade effect is not observable in the 30 nm SCO thin film, wherein the charge-disproportionation involves Co2þand Co4þ. It should be noted that with the increase in the SCO film thick- ness from 30 to 70 nm, the separation between the S1 and S2 features (signature of the charge-disproportionation) in the Co L3-edge decreases [inset of Fig. 1(b) ], suggesting that the difference between the Co charge-states decreases in a 70 nm SCO film.16Equation (1) divulges that the difference between the charge-states decreases by decreasing the O-2 phole density (that is the value of d). The spectral weight of the O 2 p-Co 3 dhybridized feature in the O K-edge (feature A ) ,w h i c hi st h em e a s u r eo ft h eO - 2 phole density, is found to decrease in the 70 nm thick film relative to the 30 nm thick film [inset of Fig. 1(a) ]. Therefore, the proportional relationship between the O-2 p hole density and the charge-state separation (that is the separationbetween the S1 and S2 feature; the value of d) is validated in the FIG. 2. (a) The polarization-dependent XA spectra of the SCO film (30 nm). The blue and the red spectra, assigned as I aband I c, respectively, are corresponding to the mea- surement geometries as shown, respectively, in the lower right corner and the upper left corner of (b). Inset is showing the zoomed view of feature A. Di stinct peak positions are shown by the vertical solid lines. (b) The charge fluctuation process from Co2þto Co4þalong the different crystallographic directions in the SCO film (30 nm) is shown schematically. The Co2þto Co4þcharge fluctuation energetics along the in-plane (and the out-of-plane) direction can be estimated from the (O 2 p-Co 3 d)jj{and (O 2 p-Co 3d)?}, which is probed via the XA spectrum for the measurement geometry as shown in the lower right corner (and the upper left corner) of the figure. The Co2þto Co4þ charge fluctuation energies along the in-plane and the out-of-plane directions are assigned as ejjande?, respectively. The schematic crystal structure of the SCO thin film is shown in the lower-left corner of the figure.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 021901 (2021); doi: 10.1063/5.0054600 119, 021901-3 Published under an exclusive license by AIP Publishingstudied SCO thin films. A similar proportional relationship between the charge-disproportionation and the O-2 phole density is also observed in another negative Dsystem, CaFeO 3thin films.29 Therefore, the charge-states corresponding to the S1 and S2 features in a 70 nm SCO film would be toward the non-integer Co2.5þand Co3.5þ configuration, respectively, a deviation from the integer Co2þand Co4þconfiguration for the 30 nm SCO film, as obtained from the the- oretical simulation [ Fig. 1(b) ].16,24 Now, we explore the charge fluctuation mechanism involving Co2.5þand Co3.5þ. This non-integer Co2.5þand Co3.5þcharge-state configurations can be visualized by considering subsidiary O-2 phole density [ d¼0.5 in Eq. (1)] .F o rt h i sc o n fi g u r a t i o n ,1 / 2 e/C0has to be removed from the H.S Co2þsite and the same has to be added to the L.S Co4þsite, as shown in the right panel of Fig. 3(a) . Therefore, the electron configurations of Co2.5þand Co3.5þcharge-states are t2g4.5eg2 and t2g5.5eg0, respectively. Thus, the one-electron removal (N !N-1) and addition (N !Nþ1) configurations involved in the lowest energy charge fluctuation process are, respectively, Co2:5þt2g4:5eg2/C0/C1 !Co3:5þt2g4:5eg1/C0/C1 ; (6) Co3:5þt2g5:5eg0/C0/C1 !Co2:5þt2g5:5eg1/C0/C1 : (7) Therefore, it is evident that the spin configurations of the Co2.5þ and Co3.5þcharge-states do not remain the same across the charge fluctuation [right panel of Fig. 3(a) ], in sharp contrast to the charge fluctuation mechanism involving the integer Co2þand Co4þ charge-state configuration. This explicitly divulges the spin- blockade effect for such a non-integer Co2.5þand Co3.5þcharge- state configuration. The implication of the spin-blockade in a system can be seen in its CB owing to its narrow band insulating character.7Figure 3(b) dis- plays feature A of I abspectra for the representation of the CB bottom of the 30 and 70 nm thick SCO films. Feature A in I abcan be used to get information about the bandwidth of the system, as this is ameasure of the CB bottom. It is observed that feature A is getting nar- rowed with the increase in the film thickness from 30 to 70 nm. This isan important finding since the narrowing of the band indicates towardthe charge fluctuation being impeded in the 70 nm film, highlightingthe appearance of the spin-blockade effect in the system, consistentwith the analysis as shown in Fig. 3(a) . The experimental findings obtained from Fig. 3(b) are schematically illustrated in Fig. 3(c) , which depicts that the electron fluctuations or the electron con-ductions are impeded more in the 70 nm film because of band-width narrowing arising due to the spin-blockade effect relative tothe 30 nm film. To summarize this, with the increase in the filmthickness from 30 to 70 nm, the charge-states of Co ions deviatemore from the integer configuration [Co 2þand Co4þ]t o w a r dt h e non-integer configuration (Co2.5þand Co3.5þ)[Fig. 3(a) ] due to t h ed e c r e a s ei nt h eO - 2 phole density, and consequently, the spin-blockade effect emerges in the system. This divulges theadvantages of the O-2 phole facilitated charge-disproportionated systems over the usual charge-ordered systems, in which thespin-blockade can be used for the material designing through theO-2phole density modulation via strain engineering or applying electric field. The proposed mechanism of spin-blockade in thisstudy would help in inducing the metal–insulator transition viamodulating the spin-blockade effect in other such charge-disproportionated systems. In summary, the spin-blockade phenomena have been experi- mentally addressed in the charge-disproportionated SCO thin films byrealizing the lowest energy charge fluctuation energetics, which was sofar an open question. It is envisioned that the charge fluctuations inthe Co 2þand Co4þcharge-disproportionated SCO system do not lead to the spin-blockade effect. However, by maneuvering the O-2 phole density via strain relaxation in thicker film, the same system revealsmore toward Co 2.5þand Co3.5þcharge-disproportionated states, across which the charge fluctuations lead to the spin-blockade effect.Our finding provides a path forward to investigate the spin-blockade FIG. 3. (a) The lowest energy charge fluctuation configurations are shown for the integer Co2þand Co4þcharge-state configurations (left panel) and the non-integer Co2.5þand Co3.5þcharge-state configurations (right panel). The step-1 is the initial configurations, before charge fluctuation, and step-2 is the final configurations , after charge fluctuation. For the integer configurations, the spin configurations of the Co2þand Co4þcharge-state remain intact; no spin-blockade. For the non-integer configurations, the spin configura- tions of the Co2.5þand Co3.5þcharge-state changes; spin-blockade occur. Moving from a 30 to 70 nm thick film, we are approaching toward the non-integer configuration from the integer configuration. (b) The zoomed view of feature A in the O K-edge XAS of the 30 and 70 nm thick SCO films for the E// ab-plane geometry for the representation of the CB bottom. (c) The schematic illustration of the band narrowing and hence impeding charge fluctuations owing to the spin-blockade effect in the 70 nm fil m relative to the 30 nm film.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 021901 (2021); doi: 10.1063/5.0054600 119, 021901-4 Published under an exclusive license by AIP Publishingeffect in other such charge-disproportionated insulators, which would improve the fundamental understanding for the novel material designing. The authors acknowledge A. Wadikar and R. Sah for helping in XAS measurements. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1X. Hao, J. Adv. Dielectr. 3, 1330001 (2013). 2Q. Li, F.-Z. Yao, Y. Liu, G. Zhang, H. Wang, and Q. Wang, Annu. Rev. Mater. Res.48, 219 (2018). 3D. Champier, Energy Convers. Manage. 140, 167 (2017). 4S. B. Riffat and X. Ma, Appl. Therm. Eng. 23, 913 (2003). 5A. Maignan, V. Caignaert, B. Raveau, D. Khomskii, and G. Sawatzky, Phys. Rev. Lett. 93, 026401 (2004). 6A. A. Taskin and Y. Ando, Phys. Rev. Lett. 95, 176603 (2005). 7C. F. Chang, Z. Hu, H. Wu et al. ,Phys. Rev. Lett. 102, 116401 (2009). 8D. Weinmann, W. 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5.0059257.pdf

J. Chem. Phys. 155, 044103 (2021); https://doi.org/10.1063/5.0059257 155, 044103 © 2021 Author(s).Conservation laws in coupled cluster dynamics at finite temperature Cite as: J. Chem. Phys. 155, 044103 (2021); https://doi.org/10.1063/5.0059257 Submitted: 07 June 2021 . Accepted: 01 July 2021 . Published Online: 26 July 2021 Ruojing Peng , Alec F. White , Huanchen Zhai , and Garnet Kin-Lic Chan COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Selfconsistent random phase approximation methods The Journal of Chemical Physics 155, 040902 (2021); https://doi.org/10.1063/5.0056565 Constructing tensor network influence functionals for general quantum dynamics The Journal of Chemical Physics 155, 044104 (2021); https://doi.org/10.1063/5.0047260 Cluster many-body expansion: A many-body expansion of the electron correlation energy about a cluster mean field reference The Journal of Chemical Physics 155, 054101 (2021); https://doi.org/10.1063/5.0057752The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Conservation laws in coupled cluster dynamics at finite temperature Cite as: J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 Submitted: 7 June 2021 •Accepted: 1 July 2021 • Published Online: 26 July 2021 Ruojing Peng,a) Alec F. White, Huanchen Zhai, and Garnet Kin-Lic Chan AFFILIATIONS Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA a)Author to whom correspondence should be addressed: rppeng@caltech.edu ABSTRACT We extend the finite-temperature Keldysh non-equilibrium coupled cluster theory (Keldysh-CC) [A. F. White and G. K.-L. Chan, J. Chem. Theory Comput. 15, 6137–6253 (2019)] to include a time-dependent orbital basis. When chosen to minimize the action, such a basis restores local and global conservation laws (Ehrenfest’s theorem) for all one-particle properties while remaining energy conserving for time- independent Hamiltonians. We present the time-dependent Keldysh orbital-optimized coupled cluster doubles method in analogy with the formalism for zero-temperature dynamics, extended to finite temperatures through the time-dependent action on the Keldysh contour. To demonstrate the conservation property and understand the numerical performance of the method, we apply it to several problems of non- equilibrium finite-temperature dynamics: a 1D Hubbard model with a time-dependent Peierls phase, laser driving of molecular H 2, driven dynamics in warm-dense silicon, and transport in the single impurity Anderson model. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0059257 I. INTRODUCTION The simulation of real-time electronic dynamics in molecules and materials is a challenge due to the many-body nature of the Hamiltonian combined with the need to propagate for many time steps in order to see the relevant phenomena. The problem is fur- ther complicated when finite-temperature effects are important, as is the case in many condensed-phase systems. Inab initio electron dynamics, a variety of methods have been applied to simulate various phenomena, such as multiple photon processes,1high harmonic generation,2and ultrafast laser dynamics3 in atomic and molecular systems. Among them are time-dependent density functional theory4and wavefunction based methods, such as time-dependent Hartree–Fock,5configuration interaction (CI),6–9 complete-active-space self-consistent field (CASSCF),2multiconfig- urational time-dependent Hartree–Fock,10–12density matrix embed- ding theory (DMET),13and coupled cluster (CC) methods.14–18 Ab initio electron dynamics in materials has primarily been carried out at the time-dependent density functional theory level, although studies with low-order diagrammatic approximations have begun to appear.19In more correlated materials, there are many studies involving model Hamiltonians, for which methods such as den- sity matrix renormalization group (DMRG)20–24and diagrammatic Monte Carlo25–29are popular.Here, we are interested in the dynamics of ab initio electronic Hamiltonians at finite temperature in both condensed-phase and molecular systems. We include finite temperature from the out- set to address certain classes of problems. For instance, thermal effects are already important at the spin exchange energy scale in correlated materials with low-temperature electronic phase transi- tions.30,31Temperature also plays a role in the coupling of elec- trons with lattice vibrations.32–34Finally, the study of matter under extreme conditions such as those found in planetary cores35also necessitates finite-temperature methods. Various wavefunction methods traditionally used for ab initio zero-temperature quantum chemistry have been extended into the finite-temperature regime. For example, the equilibrium finite- temperature coupled cluster (FT-CC) equations have been previ- ously derived via the thermal cluster cumulant (TCC) method36–39 and more recently through a time-dependent diagrammatic approach.40,41CI42and CC43,44methods have also been extended into the finite-temperature regime by means of the thermofield dynamics language. In a similar manner, there are also several formalisms to generalize equilibrium finite-temperature theories to non-equilibrium dynamics. Within a time-dependent diagram- matic language, non-equilibrium dynamics corresponds to mov- ing time integration from the imaginary axis to the Keldysh (or Kadanoff–Baym) contour in the complex time plane.45,46In the J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp context of coupled cluster theory, this diagrammatic extension leads to the Keldysh coupled cluster (Keldysh-CC) theory.47 In traditional Feynman diagrammatic approximations, a set of sufficient conditions, known as the “conserving” conditions,48,49 are known. When diagrammatic approximations for the self-energy are constructed to satisfy these conditions, the resulting approxi- mate single-particle Green’s function dynamics satisfies the physical conservation laws that, for example, relate the time derivative of local single-particle observables such as the density and momentum density to their corresponding currents. However, the Keldysh-CC method is not constructed as a conserving approximation, and this can lead to unphysical results when propagated for long times. It is this issue which we aim to correct. It is known from time-dependent wavefunction theories that the use of time-dependent orbitals that obey an equation of motion derived from a time-dependent variational principle (TDVP)50–52 leads to the conservation of one-particle properties in the sense that Ehrenfest’s theorem d dt⟨A⟩=−i⟨[A,H]⟩+⟨∂A ∂t⟩ (1) is satisfied for the one-particle reduced density matrix (1-RDM). At zero temperature, this type of orbital dynamics has been used for time evolution with various ab initio methods, including time- dependent CI,53CASSCF,2DMET,13and CC.17,18,54,55Starting from this same idea, we can extend the Keldysh-CC method to include the variational orbital dynamics via a finite-temperature TDVP. This restores the consistency and local conservation laws for all one-particle properties and further maintains the global conser- vation law for the energy present in the original Keldysh-CC theory. In Sec. II, we review ground-state and finite-temperature cou- pled cluster theory for systems in and out of equilibrium. We discuss some of the deficiencies of the finite-temperature, non- equilibrium Keldysh-CC theory presented in Ref. 47, and we show how we can remedy some of these problems with orbital dynamics in analogy with ground-state theories. This leads to the derivation of the Keldysh orbital-optimized coupled cluster doubles (Keldysh- OCCD) method. In Sec. IV, we apply the method to the model prob- lem of a two-site Hubbard model with a Peierls phase as well as an ab initio model of laser driven warm-dense silicon discussed in Ref. 47. In both, we demonstrate the exact conserving behavior of the theory. We then further assess the numerical behavior of the method in an application to laser driven molecular H 2, as well as non- equilibrium transport in the 1D single impurity Anderson model (SIAM). The performance of Keldysh-OCCD on SIAM is compared to a numerically accurate benchmark result from finite-temperature DMRG. II. THEORY A. Coupled cluster dynamics at zero temperature The ground-state coupled cluster method is defined by an exponential wavefunction Ansatz , ∣ΨCC⟩=eT∣Φ0⟩, (2)where Φ0is a reference Slater determinant. The Toperator is defined in a space of excitation operators indexed by ν, T=∑ νtνˆν†, (3) where ˆν†excites the reference determinant such that ∣Φν⟩=ν†∣Φ0⟩. (4) This space of excitations is usually truncated based on the excitation level. For example, constraining Tto the space of single and double excitations from the reference yields the commonly used coupled cluster singles and doubles (CCSD) method. The coupled cluster energy and amplitudes are determined from a projected Schrodinger equation, ⟨Φ0∣¯H∣Φ0⟩=ECC, (5) ⟨Φν∣¯H∣Φ0⟩=0. (6) Here, we have written the similarity-transformed Hamiltonian as ¯H≡e−THeT. (7) Properties are computed from response theory, which is, in prac- tice, accomplished by solving for Lagrange multipliers, λν, associated with the solution conditions (6). These appear in the Lagrangian L=⟨Φ0∣¯H∣Φ0⟩+∑ νλν⟨Φν∣¯H∣Φ0⟩. (8) The Lagrange multipliers may be associated with elements of a de- excitation operator, Λ=∑ νλνˆν, (9) such that ⟨Φ0∣(1+Λ)is the left eigenstate of the projected Schrodinger equation. The coupled cluster model can be extended to treat dynamics by allowing TandΛto become functions of time. The appropriate equations of motion may be obtained from an action, S=∫tf tidt′⟨Φ0∣(1+Λ)e−T(H−i∂t)eT∣Φ0⟩, (10) by making it stationary with respect to variations in the TandΛ amplitudes. The resulting equations of motion are given by i˙tν=⟨Φ0∣ν¯H∣Φ0⟩, (11) −i˙λν=⟨Φ0∣(1+Λ)e−T[H,ν†]eT∣Φ0⟩. (12) Here,ν†is an excitation operator and νis the corresponding de- excitation operator. The properties of such coupled cluster dynamics have been discussed in many works over the years.56–61In partic- ular, we note that conservation laws are not generally obeyed and Ehrenfest’s theorem [Eq. (1)] is not generally satisfied.55,59,60,62How- ever, there are some special cases in which Ehrenfest’s theorem will be satisfied. Energy will be conserved for a time-independent Hamiltonian, and also the particle number will be conserved. J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp B. Coupled cluster dynamics with time-dependent orbitals It has been shown that making the above action [Eq. (10)] stationary with respect to time-dependent orbitals leads to dynam- ics that satisfy Ehrenfest’s theorem for all one-electron opera- tors.2,17,18,53,55This can be important because, as pointed out by Pedersen et al. ,55it leads to local conservation and gauge invariance in one-electron properties. Orbital dynamics can be included by adding a unitary, orbital- rotation operator to the wavefunction Ansatz , ∣R⟩≡eκeT∣Φ0⟩,⟨L∣≡⟨Φ0∣(1+Λ)e−Te−κ. (13) Here,κis an anti-Hermitian, time-dependent one-electron operator. A Lagrangian of the form L=⟨R∣H−i∂t∣L⟩ (14) will lead to a biorthogonal approach termed OATDCC (orbital- adaptive time-dependent coupled-cluster) by Kvaal.54Alternatively, one may use a Lagrangian given by L=1 2[⟨R∣H−i∂t∣L⟩+c.c.], (15) which has the advantage of being real and leads to orthonormal orbital equations. This is the approach taken by Pedersen et al. and later by Sato et al.17We take the analogous approach in this work. To derive the equations of motion for such a theory, it is convenient to define a one-electron operator X≡e−κ∂teκ. (16) Up to this point, we have assumed a fixed set of reference orbitals, but it is easier to represent the equations using a picture where the orbitals of the reference, and corresponding representation of opera- tors, are time-dependent. In this representation, the matrix elements ofXare related to the time derivative of the molecular orbitals C so that ˙Cpq(t)=∑ rCpr(t)Xrq(t). (17) We find equations for the amplitudes, i˙tν=⟨Φ0(t)∣ν(¯H−ie−TXeT)∣Φ0(t)⟩, (18) −i˙λν=⟨Φ0(t)∣(1+Λ)[¯H−ie−TXeT,ν†]∣Φ0(t)⟩, (19) and a linear equation for the orbital-rotation parameters ( X), ⟨Φ0(t)∣(1+Λ)e−T[H−i˙T−iX,α]eT∣Φ0(t)⟩ +i⟨Φ0(t)∣˙Λe−TαeT∣Φ0(t)⟩+cc=0, (20) whereαis a single excitation operator. The occupied–occupied and virtual–virtual rotations are found to be arbitrary so that we only need to consider occupied–virtual rotations.17,63 In practice, including coupled cluster singles amplitudes in the Ansatz introduces a degree of redundancy that can lead to numerical problems.54For this reason, we will now specialize our discussion tothe case of only doubles amplitudes. This is the TD-OCCD method of Sato et al.17With this simplification, the terms involving Xin Eqs. (18) and (19) vanish, leading to amplitude equations that are the same as those of TD-CCD but with time-dependent orbitals. We may write the orbital equation as a simple linear equation, ∑ bjAia,bjRbj=bia, (21) where i,jlabel occupied orbitals and a,blabel virtual orbitals, and we have defined a Hermitian operator R≡−iX. (22) TheA-matrix has a simple form in terms of the symmetrized 1-RDM computed from the TandΛamplitudes, Aia,bj=δabdj i−da bδji, (23) and the right-hand side of the equation is given by bia=1 2[⟨L∣[H,i†a]∣R⟩+⟨R∣[H,i†a]∣L⟩], (24) where i†adenotes the operator for the de-excitation a→i. As mentioned previously, a consequence of including the sta- tionary action orbital dynamics is that Ehrenfest’s theorem is satis- fied for all one-electron operators. In Appendix A, we show that this is true for a very general class of wavefunction Ansätze . Kadanoff and Baym introduced the notion of conserving approximations for many-body theories based on Green’s functions. These have the property that the self-energies satisfy the self-consistent Dyson equation or, equivalently, are obtained as the derivative of approx- imate Luttinger–Ward functionals.64In Appendix B, we briefly review such approximations. A consequence of this construction is that the resulting Green’s function dynamics satisfies important single-particle conservation laws, such as for the particle density and momentum density, as well as conserves energy. Although the truncated coupled cluster methods are not constructed as conserv- ing approximations, the particle density and momentum density are one-particle properties. Thus, satisfying Ehrenfest’s theorem via orbital dynamics makes the zero-temperature coupled cluster dynamics “conserving” for these properties. C. Finite-temperature coupled cluster Here, we review the finite-temperature coupled cluster theory presented in Refs. 40 and 41, which forms the basis of this work. This theory can be viewed as a realization of the TCC theory and is somewhat different from the thermal coupled cluster method described in Ref. 43. We intend to discuss the precise relationship between various formulations of finite-temperature CC in a future work. The theory is most compactly expressed using the imaginary time Lagrangian [Eq. (5) of Ref. 41] L≡1 β∫β 0dτE(τ)+1 β∫β 0dτλν(τ) ×[sν(τ)+∫τ 0dτ′eΔν(τ′−τ)Sν(τ′)]. (25) J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We note that this quantity is analogous to the action defined in Eq. (10) via its structure as a time integral. The E and S ker- nels resemble the ground-state energy and amplitude equations, respectively, and are given in Appendix A of Ref. 41. The amplitude equations are given by sν(τ)=−∫τ 0dτ′eΔν(τ′−τ)Sν(τ′), (26) while the equations for the Λamplitudes are given by λν(τ)=−Lν(τ). (27) The L kernel is also given in Appendix A of Ref. 41. Defining ˜λν(τ)=∫β τdτ′eΔν(τ−τ′)λν(τ′), (28) we may rewrite the Lagrangian in the form L≡1 β∫β 0dτE(τ)+1 β∫β 0dτλν(τ)sν(τ)+˜λν(τ)Sν(τ). (29) The boundary conditions of s and ˜λare clear from the integral equations [Eqs. (26) and (28)] s(0)=0, ˜λ(β)=0. (30) In practice, this means that the s(τ)amplitudes are propagated with the differential equation ∂sν ∂τ=−[Δνsν(τ)+Sν(τ)], (31) starting from τ=0 toτ=βalong the imaginary time axis. Once the s(τ)are known in the interval [0,β], the ˜λamplitudes are computed according to ∂˜λν ∂τ=[Δν˜λν(τ)+Lν(τ)] (32) fromτ=βback toτ=0. Using these differential relations, it is insightful to re-express the Lagrangian as L≡1 β∫β 0dτE(τ)+1 β∫β 0dτ[(−∂τ+Δν)˜λν(τ)]sν(τ)+˜λν(τ)Sν(τ) =1 β∫β 0dτE(τ)+1 β∫β 0dτ˜λν(τ)[(∂τ+Δν)sν(τ)+Sν(τ)], (33) where we have integrated by parts and used the boundary condi- tions in Eq. (30), and the latter expression is analogous to the form of action in Eq. (10), with ˜λνplaying the role of the Λamplitudes in that equation. As with the zero-temperature theory, the finite-temperature coupled cluster is not “conserving” in the sense of Kadanoff and Baym.48,49The equilibrium theory nonetheless provides a frame- work to compute properties as derivatives of the grand poten- tial. Details relating to this response formulation of properties are discussed in Ref. 41.D. Coupled cluster dynamics at finite temperature FT-CC theory can be extended to treat out-of-equilibrium sys- tems using the Keldysh formalism as discussed in Ref. 47. The key idea is to analytically continue the imaginary time formalism of Sec. II C onto the real axis using a contour like the one shown in Fig. 1. Computation of the response density matrices along this contour provides observables of a thermal system driven out of equilibrium as discussed in Appendix A of Ref. 47. Extending the equilibrium Lagrangian onto the Keldysh contour yields L≡i β∫CdtE(t)+i β∫Cdtλν(t)×[sν(t)+i∫C(t) C(0)dt′eiΔν(t′−t)Sν(t′)], (34) where the integral over t′is understood to proceed along the con- tour. This Lagrangian leads to the same differential equations, now expressed along the real axis, ˙sν(t)=(−i)[Δνsν(t)+Sν(t)], (35) ˙˜λ(t)=i[Δν˜λν(t)+Lν(t)]. (36) These equations are the analytic continuation of Eqs. (31) and (32). Whenνis restricted to singles and doubles, these equations define the Keldysh-CCSD method. Using the differential relations, we can similarly rewrite the Lagrangian as L≡i β∫CdtE(t)+i β∫Cdt˜λν(t)[(−i∂t+Δν)sν(t)+Sν(t)]. (37) In practical calculations, there is a degree of freedom in choosing the position of the real branch of the contour relative to the imaginary- time integration limits. This corresponds to the choice of σin Fig. 2. Furthermore, once s and ˜λare known along the whole of the imaginary branch of the contour, they can be propagated simulta- neously in real time, starting from t=−iσ. In effect, sis propagated forward along the forward branch of the contour and ˜λis propagated FIG. 1. The Keldysh contour including the imaginary branch. The propagation proceeds forward in real time, backward in real time, and in imaginary time, respec- tively. These three branches are labeled (f), (b), and (i). Contours like this which include the imaginary time branch first appeared in the work of Konstantinov and Perel’.65 J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. The contour used in this work (red) with propagation directions for s and ˜λshown in blue and green, respectively. Note that some branches of the contour are drawn slightly away from the actual contour so that all branches can be shown clearly. backward along the reverse branch of the contour so that at each time, the density matrices for the non-equilibrium system can be constructed. The Keldysh-CC method has two deficiencies which appear when propagating for longer times. (i) As is the case for zero- temperature CC dynamics, Keldysh-CC truncated to singles and doubles does not, in general, satisfy Ehrenfest’s theorem. (ii) The time-dependent properties are not stationary under propagation with the same time-independent Hamiltonian that generates the equilibrium density matrix. (However, the energy remains station- ary under propagation by a time-independent Hamiltonian, for the same reasons that it is conserved in zero-temperature coupled clus- ter dynamics.) (i) was discussed already in Ref. 47, which, in particu- lar, showed analytically and numerically that approximate Keldysh- CC dynamics violates global particle number conservation. As sug- gested by previous arguments in Sec. II B, (i) can be addressed by introducing stationary orbital dynamics. (ii) is a different problem, and we will discuss the “stationarity of equilibrium approximations” in Sec. II F. E. Orbital-dependent FT-CC dynamics We now extend the formulation of Sec. II D to a time- dependent orbital basis with the goal of satisfying Ehrenfest’s theorem for all one-particle observables at finite temperature. As in the zero-temperature case, we re-express the time-dependence of the orbitals via the time-dependence of the matrix elements of operators in the orbital basis. For example, the representation of the one-particle part of the Hamiltonian undergoes dynamics as hpq(t)=∑ rshrsC∗ rp(t)Csq(t), (38) where rand s refer to the starting orbitals. In analogy with the zero- temperature theory, we define an Xoperator [see Eq. (17)] for non- redundant pairs, Xai(t)=−Xia(t)∗. In the finite-temperature case, recall that i,j,..., and a,b,..., are used to refer to orbitals with hole and particle characters, respectively, but these orbitals will span the full space in general. (In principle, we can introduce the matrix ele- ments in the pure-hole and pure-particle sector, Xij(t),Xab(t), but as shown in Appendix A, by the choice of Lagrangian below, suchelements vanish). As in the zero-temperature case, the introduction of orbital rotations eliminates the need for singles amplitudes. We will refer to the resulting doubles theory as Keldysh-OCCD. To derive the equations, we define a symmetrized Lagrangian, Q≡1 2(L+L∗)+Ω(0)+i β∫CdtE(1)(t). (39) The Lagrangian is symmetrized to ensure that the Xmatrix is anti- Hermitian, and the second and third terms correspond to the ther- mal Hartree–Fock contribution. Lhas a similar form to Eq. (37) [or equivalently Eq. (34)], but we choose to remove the term containing Δν(further discussed below), L≡i β∫CdtE(t)+i β∫Cdt˜λν(t)[(−i∂t)sν(t)+Sν(t)]. (40) In addition, the kernels E [s(t)], Sν[s(t)], and Lν[s(t),˜λ(t)]are modified as follows: (i) All Hamiltonian tensors are now time- dependent. (ii) The time derivative of the orbitals results in a modification of the one-electron integrals17,63 hai→hai(t)−iXai(t) (41) and similarly for hia. As in Ref. 41, our notation includes factors of the square root of the occupation numbers in the definition of the tensors. For example, hai≡√ ni¯na⟨a∣h∣i⟩. (42) (iii) The Fock matrix becomes time-dependent and is computed from the time-dependent Hamiltonian tensors, and unlike in Keldysh-CC, the diagonal is not subtracted (further discussed below), fij=hij+⟨ik∥jk⟩−√ninjδijεi, Keldysh −CC, →fij(t)=hij(t)+⟨ik∥jk⟩(t), Keldysh −OCC,(43) fab=hab+⟨ak∥bk⟩−√ ¯na¯nbδabεa, Keldysh −CC, →fab(t)=hab(t)+⟨ak∥bk⟩(t), Keldysh −OCC,(44) where in the Keldysh-CC formulation, εa,εiare orbital energies, and ni,¯na=1−naare orbital occupancies. In the Keldysh-CC equations, the Δνterm in the Lagrangian and the corresponding subtraction of occupancy weighted eigen- values from the Fock operator both originate from the choice of zeroth order Hamiltonian, and together ensure that the diagrams of Keldysh-CC correspond to a well-defined time-dependent pertur- bation theory. The mathematical effect of these terms is to introduce eigenvalue time-dependent phases on the indices of the amplitudes during the propagation, as seen from the amplitude Eqs. (35) and (36). In the orbital-optimized Keldysh-CC, such time-dependent phases are fully determined by stationarity of the Lagrangian with respect to the rotation elements Xij,Xab. Thus, we can remove both terms involving the zeroth order eigenvalues discussed above while J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp obtaining the same result at stationarity. As shown explicitly in Appendix A, this choice leads to Xij,Xab=0. To obtain the amplitude equations, we set the variations of Q with respect to the amplitudes to zero, which is equivalent to set- ting the amplitude variations of Lto zero. Since all terms contain- ing singles amplitudes vanish, fiaand faido not enter the kernels directly. Therefore, as in the analogous zero-temperature theory, the appearance of Xaiinhai(t)does not affect the Keldysh-OCCD amplitude equations. This leads to Eqs. (35) and (36) but with the S and L kernels containing time-dependent matrix elements and the Δνcontribution missing ˙sν(t)=−iSν(t), (45) ˙˜λ(t)=iLν(t). (46) Precise equations for the kernels are given in Appendix D. To obtain an equation for X, we vary Qwith respect to the orbital parameters and set the resulting expression to zero. As in the zero-temperature case, this yields a linear equation of the form ∑ bjAia,bjRbj=bia, (47) where we again use the Hermitian Rdefined in Eq. (22). The elements of these tensors are found to be Aia,bj=δabdij−dbaδji (48) and bia=Fai−F∗ ia. (49) Here, we have used dpqto represent elements of the symmetrized reduced density matrices and Fpq≡dprhrq+1 2∑ sudps uv⟨uv∥qs⟩, (50) where dps uvis an element of the two-particle reduced density matrix (2-RDM). As in the zero-temperature case, the inclusion of such orbital dynamics leads to the satisfaction of Ehrenfest’s theorem for all one-particle properties, as shown in Appendix A. F. Stationarity of equilibrium approximations In general, there is no guarantee that an approximate equi- librium density matrix will be stationary when propagated in real time with the equilibrium Hamiltonian. This property holds for con- serving approximations because the approximate Luttinger–Ward functional is expressed in terms of Feynman diagrams, where each diagram implicitly includes all time orderings of the interactions. For a more general class of theories, this suggests that a necessary condition for stationarity is that all time-ordering counterparts of a particular time-ordered diagrammatic contribution to the density matrix should be included in the theory. This is clearly violated in approximate coupled cluster theory. Perturbation theory at finite order includes all time orderings at a particular order, which means that the density defined strictly as the derivative of the action with respect to an external potential has this property (see Appendix C FIG. 3. Diagrammatic contributions to the PT2 1-RDM for a one-particle problem (a), including the contribution from the response of the reference (b). for a more detailed demonstration). For example, consider perturba- tion theory at second order (PT2) with a one-particle perturbation. In Fig. 3(a), we show the diagrammatic contributions to the PT2 one-particle density matrix obtained by differentiating the energy expression with respect to the applied potential. The density matrix defined in this way includes both time orderings of the relevant dia- gram and has the stationary property. However, one often includes a contribution from the response of the reference via terms like those in Fig. 3(b). In this case, not all time orderings of the same dia- grammatic contribution are included and the density is no longer stationary when propagated in real time. In the case of perturbation theory, this property can be restored by including some higher order terms. For coupled cluster theory with limited excitations (such as truncation to singles and dou- bles), this is not generally possible, and as we have stated previously, none of the methods discussed in this work have this stationarity property. For short or moderate propagation times, this can be cor- rected by subtracting the anomalous dynamics. To be precise, given a Hamiltonian of the form H(t)=H0+V(t) [V(0)=0], (51) we can define a corrected density matrix ρc(t)=ρ(t)+ρ(0)−ρ0(t), (52) whereρ(t)is the density propagated with the full H(t)andρ0(t)is the density propagated with the time-independent Hamiltonian H0. III. IMPLEMENTATION In Sec. IV, the Keldysh-CCSD results are obtained using the implementation as described in Ref. 47. The Keldysh-OCCD results are obtained as follows: (i) the equilibrium amplitudes s(τ),˜λ(τ) are first computed as described in Sec. II C. In particular, the dif- ferential form of the s-amplitude Eq. (31) is first propagated from τ=0 toτ=βalong the imaginary time contour. The s(τ)ampli- tudes are then used in Eq. (32) to propagate ˜λ(t)fromτ=βback to τ=0. These equilibrium amplitudes are computed within the cou- pled cluster doubles approximation, using fixed orbitals, and with- out any truncation of the occupied or virtual space. The fourth order Runge–Kutta scheme is used to propagate the differential equations. (ii) The equilibrium amplitudes at τ=β/2 are taken as the initial t=0 amplitudes for the subsequent dynamics. This corresponds to a choice of contour where the real part extends from −iβ/2 in Fig. 2. (iii) The dynamical s(t)and ˜λ(t)amplitudes are computed as described in Sec. II D. In particular, Eqs. (35) and (36) are simultane- ously propagated from t=0 tot=tfon the real contour where the kernels S,Lare modified to include orbital dynamics as described in J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Sec. II E. (iv) At each update of s(t)and ˜λ(t), the orbital Eq. (47) is solved to obtain R, which is used to update the integrals accord- ing to, e.g., Eq. (38). We use the fourth order Runge–Kutta (RK4) scheme to update both the amplitudes and the orbitals in a coupled manner as described below. The amplitude Eqs. (35) and (36) and orbital coefficient Eq. (17) are of the form dy dt=f(t,y(t),h(C(t))), dC dt=CX(t,d(y(t)),h(C(t))),(53) where y∈{s(t),˜λ(t)},h(C(t))denotes the Hamiltonian integral matrix elements in the time-dependent orbital basis C(t), and d(y(t))denotes the reduced density matrices computed from the amplitudes at time t. In RK4, the time step from t→t+δtis assem- bled from intermediate amplitudes yiand orbitals Cias well as their respective finite difference δyiandXiat intermediate times tifor i=1...4. The intermediate quantities are computed as δyi=f(ti,yi,h(Ci)), Xi=X(ti,d(yi),h(Ci)),(54) where ti=t+aiδt, yi=y(t)+aiδtδyi−1, Ci=C(t)eaiδtXi−1,(55) where aiare standard fourth order Runge–Kutta weights {0,1 2,1 2, 1}. Note that all quantities for i=1 correspond to their values at time t. After all intermediate quantities are obtained, the amplitudes and orbitals are updated as y(t+δt)=y(t)+1 6δt(δy1+2δy2+2δy3+δy4), X(t+δt)=1 6(X1+2X2+2X3+X4), C(t+δt)=C(t)eδtX(t+δt).(56) The present Keldysh-OCCD theory has the same scaling as the pre- vious Keldysh-CCSD theory in terms of computational cost and memory. The amplitude update cost scales as N6for each real or imaginary time step, where Nis the size of one-particle basis. Furthermore, for each real time step, Keldysh-OCCD additionally involves solving a linear set of equations for N2variables in the orbital Eq. (47), and an integral update which scales as N5. IV. RESULTS A. Numerical demonstration of Ehrenfest’s theorem We first demonstrate the satisfaction of Ehrenfest’s theorem for the Keldysh-OCCD method for the two-site time-dependent Peierls–Hubbard model. This was previously studied with Keldysh- CCSD in Ref. 47. The time-dependent Hamiltonian is given by H(t)=−tH∑ iσ[eiA(t)a† iσa(i+1)σ+h.c.]+U∑ ini↑ni↓, (57)where the first term is the Peierls driving term, which mimics the effect of the coupling of an underlying nuclear lattice on an external laser pulse. Here, the driving takes the form A(t)=A0e−(t−t0)2/2σ2 cos[ω(t−t0)]. (58) We use U=1.0, for which a chemical potential of μ=0.5 gives half- filling at equilibrium, a temperature T=1.0, and pulse parameters ofσ=0.8,t0=2, andω=6.8. We show the Keldysh-CCSD and Keldysh-OCCD results for the population difference between the two sites ( L,R) defined as nL−nR, where nL=nL,↑+nL,↓, along with the exact result computed by propagating the density matrix in the full Liouville space. As shown in Fig. 4, the deviation from the exact result increases for both Keldysh-CCSD and Keldysh-OCCD with increasing pulse amplitude A0. However, the orbital dynamics in Keldysh-OCCD ensures that it is much closer to the exact result for all pulse parameters. As discussed in detail in Ref. 47, Keldysh-CCSD does not, in general, conserve global symmetries, such as the total particle num- ber, whereas such one-particle quantities are conserved both locally and globally by Keldysh-OCCD. In Fig. 5, we show the change in the total particle number of Keldysh-CCSD and Keldysh-OCCD for the two-site Hubbard model for μ≠U/2 where the particle–hole sym- metry is no longer present. As expected, we see that as μis decreased from half-filling, the Keldysh-CCSD total particle number begins to FIG. 4. Population difference nL−nRas a function of time for the two-site Hubbard model at half-filling. The solid line is the exact result, the dashed line is the Keldysh- OCCD result, and the dotted line is the real part of the Keldysh-CCSD result. In the lower panel, we show the difference of Keldysh-OCCD and Keldysh-CCSD results from the exact result. J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5. Change in total particle number Nin the two-site Peierls–Hubbard model with different chemical potentials μ. The solid line is the Keldysh-OCCD result, and the dotted line is the real part of the Keldysh-CCSD result. Keldysh-OCCD satisfies the global particle number conservation law. deviate from the equilibrium value at longer times, whereas the total particle number remains conserved for Keldysh-OCCD for each μ. In Fig. 6, we demonstrate the stronger condition of conserva- tion of the local particle number for the population of the left site. From Ehrenfest’s theorem, d dt⟨nL⟩=i⟨[H,nL]⟩, (59) FIG. 6. Time derivative of the Keldysh-OCCD left-site population d⟨nL⟩/dtin the two-site Peierls–Hubbard model for μ=0. The red curve in the upper panel is the population derivative computed from the flux i⟨[H,nL]⟩, and the green curve is the numerical time derivative using a finite difference time step of dt=5×10−3. The blue curve is the difference between the flux expression and the finite difference time derivative; this should be zero if Ehrenfest’s theorem is satisfied. The lower panel plots the same but for two time steps of dt=5×10−3anddt=2.5×10−3. These results show that Keldysh-OCCD satisfies Ehrenfest’s theorem for the local particle number and thus the local particle number conservation law.where the expression on the right-hand side is the contribution from the right-site flux. The red curve in the upper panel shows the local population change computed with the flux. The green curve gives the time derivative of nL, computed by the finite-difference expres- sion(⟨nL⟩(t+δt)−⟨nL⟩(t))/δt, with a time step of 5 ×10−3. The two curves are on top of each other; thus, we also plot the differ- ence between the r.h.s. and the l.h.s. quantities. This stays close to 0 at all times, demonstrating the local conservation law. In the lower panel, we also plot the difference for a smaller time step of 2.5 ×10−3 where the quantity is reduced by a factor of 2. This illustrates that Ehrenfest’s theorem will be fully satisfied for an infinitesimal time step. B. Comparison with Keldysh-CCSD: Warm-dense Si We next compare the performance of Keldysh-CCSD and Keldysh-OCCD for the field-driven Si system in Ref. 47. This treats a single primitive cell of Si in a minimal basis (SZV66with GTH-Pade pseudopotentials67,68) and with the ions frozen at the experimental lattice constant (3.567 Å). The matrix elements were obtained from the PySCF software package using plane-wave density fitting.69–71 We use the dipole approximation in the velocity gauge where the coupling to an external field is of the form 1 mc⃗p⋅⃗A(t), (60) where⃗pis the momentum operator, and we choose a pulse shape described by 1 c⃗A(t)=A0⃗ze−(t−t0)2/2σ2 cos[ω(t−t0)]. (61) We choose parameters of σ=2.0, t0=15.0,ω=0.975 29, and A0=0.6752 at a temperature T=0.2. This corresponds to a maxi- mum laser intensity of 2.36 ×1016W/cm2. Figure 7 again demonstrates that, as expected, Keldysh-OCCD conserves the total particle number in contrast to Keldysh-CCSD. In Fig. 8, we show the change in the population of the valence and conduction bands induced by the laser. While the Keldysh- CCSD results show an unphysical divergence for t>25, the Keldysh- OCCD results are stable. The two theories largely agree at short times where we would expect both approximations to be valid. How- ever, at longer times, different physics is predicted: Keldysh-CCSD yields a sharp drop in the valence to conduction population transfer FIG. 7. Change in the number of electrons per unit cell for the Si system as a function of time. J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8. Difference in population of the valence (solid line) and conduction band (dotted line) for Keldysh-CCSD and Keldysh-OCCD as a function of time. The shape of the electric field as a function of time is plotted in the lower panel. Additional discussion in the text. at the point where there is a strong violation of total particle num- ber conservation, while Keldysh-OCCD predicts that the valence to conduction population transfer continues at a reduced rate after the pulse ends. C. Molecular H2in a laser field We further apply the Keldysh-OCCD theory to a field-driven molecular system. The purpose of this calculation is to provide a simple benchmark so that future implementations of Keldysh- OCCD and similar theories can be easily tested. The total molecular Hamiltonian is described in Ref. 3, where H(t)=H(0)−⃗μ⋅⃗A(t). (62) H(0)is the time-independent molecular Hamiltonian, and ⃗μ=−N ∑ i⃗ri+NA ∑ AZA⃗RA (63) is the molecular dipole operator for Nelectrons and NAnuclei. We use a field of the same form as in Eq. (61). In Fig. 9, we show the x-component of the dipole moment for H2in the STO-3G basis72with the molecule aligned along the x-axis. We use field parameters t0=15.0,σ=2.0,ω=1.0,⃗z=(1.0, 1.0, 1.0 ), and T=1.0,μ=0. The raw data for μx(t)are provided in the supplementary material. FIG. 9.μxfrom Keldysh-OCCD as a function of time for a field-driven molecular H2benchmark. The shape of the electric field as a function of time is also plotted in the lower panel. D. Single impurity Anderson model Finally, we apply the Keldysh-OCCD method to transport in the single impurity Anderson model, with a central impurity (“dot”) coupled to two one-dimensional leads. We use a Hamiltonian of the form ˆH=ˆHdot+ˆHleads+ˆHdot−leads+ˆHbias, (64) where ˆHdot=Vgnd+Und↑nd↓, (65) ˆHleads=−tleads∑ pσ(a† LpσaLp+1σ+a† RpσaRp+1σ+h.c.), (66) ˆHdot−leads=−thyb∑ σ(a† L1σadσ+a† R1σadσ+h.c.), (67) ˆHbias=V 2∑ pσ(a† LpσaLpσ−a† RpσaRpσ), (68) as used previously in Ref. 13. Here, the impurity is associated with fermion operators a(†) dand its Hamiltonian is parameterized by a gate voltage Vgand Hubbard interaction U, while the left and right leads are associated with fermion operators a(†) Lpanda(†) Rp, respec- tively, and are described by tight-binding Hamiltonians. In the fol- lowing calculations, the equilibrium state is generated by the Hamil- tonian with parameters tleads=1.0,thyb=0.4, and zero bias V=0 for J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp various specified temperatures and values of U. For all temperatures, the total system has the same number of particles as the number of sites with total Sz=0. A Hartree–Fock calculation is performed at zero temperature, and these orbitals are used in the equilibrium CC calculations. The dynamics is then generated by applying a small bias, V=−0.005, and the other parameters are kept fixed. The dynamics can be characterized by the time-dependent current across the dot, which we compute as the average of the current between the dot and its closest left and right neighbors J(t)=(JL(t)+JR(t))/2, where JL(t)=−ithyb∑ σ⟨a† L1σadσ−a† dσaL1σ⟩, (69) JR(t)=−ithyb∑ σ⟨a† dσaR1σ−a† R1σadσ⟩, (70) and the bracket denotes the expectation value with respect to the Keldysh-OCCD density matrix. Figure 10 shows an example of the time-dependent current divided by the bias for several different temperatures at U=1.0. As discussed, for example, in Ref. 73, the current will quickly reach its steady state in the infinite-size limit, but for finite leads, the finite system size produces an oscillatory behavior. In the case of 16 sites, we propagate for up to a time of t=10.0 corresponding to half the oscillation period, which is sufficient to extract the physics of the system pertaining to the infinite-size steady state. Figure 11 shows the conductance Gas a function of gate volt- ageVgfor the 16 site model and U=1.0 at various temperatures as computed from Keldysh-OCCD, as well as from reference density matrix renormalization group (DMRG) results. The conductance is computed as the current divided by bias averaged over the plateau region from t=2.0 to t=8.0. The zero-temperature Keldysh-OCCD result is computed using a zero temperature implementation similar to that described in Ref .17. The reference DMRG results at both zero- and finite temperature are computed using a time-step tar- getting time-dependent DMRG method74–77as implemented in the PyBlock3 software package78,79interfaced with the HPTT library.80 The “Kondo peak” in the low-temperature limit can be observed in the shape of a high conductance plateau, arising from many-body effects. As described elsewhere (see, e.g., Ref. 81), the Kondo res- onance marks the increase in the density of states of the impurity FIG. 10. Time-dependent current at different temperatures for a gate of Vg=−U/2. The solid lines are systems with 16 sites, and the dashed line is a system with 32 sites. FIG. 11. Conductance as a function of Vgfor various temperatures for 16 sites. The solid lines are Keldysh-OCCD results at the indicated temperatures, and the dashed lines are the DMRG results at the corresponding temperatures. around the Fermi surface of the leads due to the spin interaction between a particle near the Fermi-surface of the leads and a parti- cle on the impurity, and this resonance results in a high tunneling probability. The Kondo effect is only observed at temperatures below TK∼e−1/jρ, where jis the (typically small) effective exchange cou- pling between a particle in a lead state and a particle on the impurity andρis the lead density of states at the Fermi-energy. We find that the plateau at T=0 does not reach the predicted G=1/πunitary value at −U/2. The deviation from the unitary limit has been seen in previous calculations and can be attributed to finite-size errors73[for example, Fig. 6(b) of Ref. 13 plots the conductance at Vg=−U/2 as a function of system size, demonstrating the convergence to the uni- tary value G=1/π(or more precisely, for the unit used in that work, 2e2/h) as the system size approaches infinity]. Figure 12 shows a more detailed comparison of the Keldysh- OCCD results and DMRG results for the 16-site model at T=0.2 with interaction U=1.0 in the top panel and vanishing interac- tion U=0.0 in the lower panel. For both the interacting and non- interacting cases, the Keldysh-OCCD results are propagated with time step dt=0.01, while the DMRG results are obtained using dt=0.1 and bond dimension M=2000 (we used a larger time step in the DMRG to reduce the cost). We make note of two numeri- cal aspects: (i) The conductance is very sensitive to the window over which the current is averaged. This can be seen from computing the conductance in two different ways, i.e., as an average over the current divided by bias from t=2.0 to t=8.0, shown in red, and as the cur- rent divided by the bias value at t=2.0, shown in green. As shown in the top panel, different averaging windows result in both an overall vertical shift of the conductance and a small change in the shape of the curve. (ii) For the non-interacting case in the lower panel, where the Keldysh-OCCD method is exact, there is still a difference between the Keldysh-OCCD conductance and the DMRG conduc- tance, coming from the bond dimension truncation. Thus, given the sensitivity of Gto the averaging window of the current, as well as the DMRG bond dimension truncation error, the Keldysh-OCCD and DMRG results in Figs. 11 and 12 are in very good agreement. J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 12. Conductance as a function of VgatT=0.2 for 16 sites. In both panels, red is the conductance computed as an average of J/Vin the window t=2 to t=8 and green is the conductance taken as the J/Vvalue at t=2.0. Solid lines are the Keldysh-OCCD results, and dashed lines are the DMRG results. The top panel corresponds to an interaction strength U=1.0. The lower panel shows test results with U=0.0 where Keldysh-OCCD is exact and DMRG is not due to the finite bond-dimension. The temperature-dependence of the conductance at Vg=−U/2 (U=1.0) is shown in Fig. 13, where the inset plots the Keldysh-OCCD and DMRG results with data points marked and the temperature axis on a log-scale. Analytic treatments indicate that the peak conductance has a logarithmic dependence on temperature,81which is consistent with our data shown in the inset. Thus, our results show that at moderate interaction strength, Keldysh-OCCD is able to correctly capture the physics of the single impurity Anderson model, including the non-equilibrium Kondo physics and its temperature-dependence. FIG. 13. Conductance from Keldysh-OCCD and DMRG as a function of tempera- ture at Vg=−U/2 for 16 sites, U=1.0. The inset shows the Keldysh-OCCD data with the temperature on a log scale.V. CONCLUSIONS In this work, we present a modification of the Keldysh cou- pled cluster theory that restores local and global conservation of one-particle quantities via an optimal orbital dynamics. On a vari- ety of models and simple ab initio systems, we have demonstrated that such conservation laws are indeed obeyed within the Keldysh orbital-optimized coupled cluster doubles approximation (Keldysh- OCCD), with a concomitant improvement of the predicted dynam- ics, especially at longer times. In the single impurity Anderson model, we qualitatively reproduce the temperature dependent trans- port physics, including that associated with the Kondo plateau. We believe this will be useful in the ab initio modeling of Kondo transport in the future. However, there remain important challenges in the practical application of the Keldysh coupled cluster formalism: (i) at the doubles level, the cost and memory scaling remains a barrier to many interesting applications. To decrease the computational cost, a potential future direction is to replace the optimal orbital dynamics by the orbital dynamics of time-dependent Hartree–Fock. This will not exactly preserve Ehrenfest’s theorem for one-particle dynamics as does the present Keldysh-OCCD approximation. However, for weakly interacting systems, the Hartree–Fock orbital update should still give better results than Keldysh coupled cluster approximations without orbital dynamics. (ii) At a formal level, the fact that the state generated by the finite-temperature coupled cluster theory is not, in general, a stationary state of the dynamical theory leads to an ambiguity in the definition of the equilibrium state. SUPPLEMENTARY MATERIAL See the supplementary material for the x-component of the dipole moment of the laser driven H 2example. AUTHORS’ CONTRIBUTIONS R.P. and A.F.W. contributed equally to this work. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Science, via Grant No. DE-SC0018140. Benchmarks gen- erated by DMRG used PyBlock3, a code developed with support from the U.S. National Science Foundation under Grant No. CHE- 2102505. G.K.C. thanks Emanuel Gull for discussions. G.K.C. is a Simons Investigator in Physics and is part of the Simons Collabora- tion on the Many-Electron Problem. APPENDIX A: ORBITAL ROTATIONS AND EHRENFEST’S THEOREM AT ZERO AND FINITE TEMPERATURE Here, we show that Ehrenfest’s theorem is restored for one- particle properties by including the optimal orbital dynamics into a zero-temperature time-dependent wavefunction Ansatz . Let Ψ(yν,p)be the time-dependent wavefunction Ansatz , where yν(t) are the variational parameters (such as the CI coefficients) and p(t) is the orbital basis. The equation of motion of yν(t)andp(t)can be determined from a time-dependent variational principle2,13,17,53,63by J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-11 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp making the action S[Ψ]=∫T 0dt⟨Ψ∣H−i∂t∣Ψ⟩, (A1) stationary with respect to small variations ∣δΨ⟩=δyν∂yν∣Ψ⟩+Δ∣Ψ⟩, (A2) where the variation with respect to the orbitals, Δ∣Ψ⟩, is parame- terized by an anti-Hermitian 1-body operator, Δ.13SettingδS=0 leads to 0=∂y∗ ν⟨Ψ∣(H∣Ψ⟩−i∣˙Ψ⟩), (A3) 0=i⟨Ψ∣Δ∣˙Ψ⟩+i⟨˙Ψ∣Δ∣Ψ⟩+⟨Ψ∣[H,Δ]∣Ψ⟩, (A4) where solving Eq. (A4) for each orbital pair is equivalent to enforcing Ehrenfest’s theorem for the one-particle density matrix elements and hence for any one-particle property. Furthermore, the energy is conserved: the time-dependence of Ψcan be expressed as ∣˙Ψ⟩=˙yν∂yν∣Ψ⟩+X∣Ψ⟩, (A5) where Xis anti-Hermitian and parameterizes the time-dependence of the orbitals.13Then, the energy derivative is d dt⟨H⟩=⟨˙Ψ∣H∣Ψ⟩+⟨Ψ∣H∣˙Ψ⟩ =(˙yν∂y∗ ν⟨Ψ∣)H∣Ψ⟩−⟨Ψ∣XH∣Ψ⟩ +⟨Ψ∣H(∂yν∣Ψ⟩˙yν)+⟨Ψ∣HX∣Ψ⟩ =i(˙yν∂y∗ ν⟨Ψ∣)∣˙Ψ⟩−i⟨˙Ψ∣(∂yν∣Ψ⟩˙yν) −i⟨Ψ∣X∣˙Ψ⟩−i⟨˙Ψ∣X∣Ψ⟩ =i(⟨˙Ψ∣)∣˙Ψ⟩−i⟨˙Ψ∣(∣˙Ψ⟩)=0, (A6) where we use Eqs. (A3) and (A4) for the third equality and Eq. (A5) for the fourth equality. At finite temperature, the idea is analogous: orbital optimiza- tion based on an action principle can be used to satisfy Ehrenfest’s theorem. We first derive the conditions satisfied by the stationary orbital dynamics. The Lagrangian in Eqs. (39) and (40) can be written as Q=1 2(L+L∗)+i β∫CdtE(1)+Ω(0), (A7) where E(1)=hii+Rii+1 2⟨ij∥ij⟩=⟨H−iX⟩0 (A8) and L=i β∫Cdt(E(t)+˜λν(t)Sν(t)) +i β∫Cdt˜λν(t)(−i∂tsν(t)) (A9) =i β∫Cdt⟨H−iX⟩CCN +i β∫Cdt˜λν(t)(−i∂tsν(t)), (A10)where the term −iXcomes from the modification of the integrals as in Eq. (41). We use the notation ⟨⋅ ⋅ ⋅⟩0to denote an expectation value with respect to the mean-field 1-RDM ppqin Eq. (D6) and 2-RDM ppq rs=pprpqs−ppspqrand use ⟨⋅ ⋅ ⋅⟩CCNto denote an expecta- tion value with respect to the coupled-cluster normal-ordered 1- and 2-RDMs defined in Eqs. (D7)–(D15). As in the zero-temperature case, the orbital variation can be parameterized by an anti-Hermitian matrix Δ, δCpq=CprΔrq, (A11) and then, the variation of the modified Hamiltonian is δ(hpq−iXpq)=(hpr−iXpr)Δrq−Δpr(hrq−iXrp), (A12) δ⟨pq∥rs⟩=⟨pq∥xs⟩Δxr+⟨pq∥rx⟩Δxs −Δpx⟨xq∥rs⟩−Δqx⟨px∥rs⟩, (A13) which can be compactly denoted as [H−iX,Δ]. In the orbital vari- ation of the action Q, the variation of terms (A8) and (A10) comes from the variation of the modified Hamiltonian tensors δE(1)=⟨[H−iX,Δ]⟩0, δ(E(t)+˜λν(t)Sν(t))=⟨[H−iX,Δ]⟩CCN,(A14) and the variation of term (A10) gives −i(δ˜λν)˙sν+i˙˜λνδsν=−iTr(˙dΔ), (A15) where ˙dis the time derivative of the CC 1-RDM in Eq. (D4). Thus, the orbital gradient of Qis δQ=⟨[H,Δ]⟩CC−i⟨[X,Δ]⟩CC−iTr(˙dΔ), (A16) where⟨⋅ ⋅ ⋅⟩CCdenotes the expectation value computed using the CC 1- and 2-RDMs given in Eqs. (D4) and (D5). Setting ΔQ=0 for each orbital pair Δuvgives ˙dvu+∑ qdquXvq−∑ pdvpXpu=i[Fvu−F∗ uv], (A17) where the Fmatrix is defined in Eq. (50). For Keldysh-OCCD, only the “occupied–occupied” and “virtual–virtual” blocks of the 1-RDM are non-zero. This means that the only non-zero blocks of Eq. (A17) are ˙dij+∑ kdkjXik−∑ kdikXkj=i[Fij−F∗ ji], (A18) ˙dab+∑ cdcbXac−∑ cdacXcb=i[Fab−F∗ ba], (A19) ∑ bdbaXib−∑ jdijXja=i[Fia−F∗ ai], (A20) ∑ jdjiXaj−∑ bdabXbi=i[Fai−F∗ ia]. (A21) J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-12 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp For an anti-Hermitian matrix X, the final two equations are complex conjugates of each other and we only need to satisfy one of them, which we rewrite as ∑ jdjiRaj−∑ bdabRbi=[Fai−F∗ ia], (A22) where we have defined R≡−iX. This is the precise form of the orbital equation previously shown in Eq. (47) that comes from the stationarity of the Lagrangian. Furthermore, it can be shown that Eqs. (A18) and (A19) trivially hold for any occupied–occupied and virtual–virtual rotations, and hence, those rotations can be set to zero. We can write Ehrenfest’s theorem in a rotating basis at finite temperature in terms of the reduced density matrices and matrix elements of the operator Ain the orbitals defined in Eq. (17), ∑ uvAuv˙dvu+∑ uvqdquAuvXvq−∑ uvpdvpAuvXpu =i∑ uv[Fvu−F∗ uv]Auv. (A23) Since this equation should hold for any operator, it must be sepa- rately satisfied for each uvpair. This immediately yields Eq. (A17), where the reduced density matrices correspond to their definition in coupled cluster theory. Thus, we see that the stationarity of the cou- pled cluster Lagrangian with respect to orbital variations leads to the satisfaction of Ehrenfest’s theorem. APPENDIX B: CONSERVING APPROXIMATIONS There are several equivalent ways to define a conserving approximation in the theory of Green’s functions. To draw a close correspondence with the approach in this work, we define a con- serving approximation to be one where the Green’s function on the Keldysh contour Gpq(t1,t2)=iTC⟨ap(t1)a† q(t2)⟩(where Tcindi- cates contour time ordering) makes the Luttinger–Ward functional Astationary, defined as A[G]=−1 βTr[log(−G−1 0+Σ)+ΣG]+Φ[G], (B1) where Σ[G]is the self-energy; G0is the zeroth order Green’s func- tion; the underlined notation indicates that the Green’s function and self-energy elements GpqandΣpqare themselves 2 ×2 matrices, with row/columns labeling pairs of contour indices along the for- ward and backward contours; and Tr integrates over contour time as well as sums over the contour and orbital indices. Φ[G]is a sum of closed diagrams of Gand the two-particle interaction, and Σ[G]qp(t2,t1)=sgn⋅δΦ[G]/δGpq(t1,t2), where sgn introduces the appropriate sign for different pairs of contour indices in the ele- ments of Σ. For a more detailed explanation of the terminology, see, e.g., Ref. 82. We neglect some subtleties related to the conver- gence on the real-time contour discussed in Ref. 83. For an equi- librium problem, the Luttinger–Ward functional evaluates to the thermodynamic grand potential; thus, it is an analog of the finite- temperature and Keldysh coupled cluster Lagrangians described here.As we have argued in the main text, Ehrenfest’s theorem arises when the dynamics is stationary under orbital variations. The local conservation laws for one-particle quantities, such as the density and momentum density, are a consequence of Ehrenfest’s theorem for one-particle quantities. In the case of the coupled cluster Lagrangian, variations with respect to TandΛ(i.e., changing the values of the amplitudes) do not completely capture the space of variations when the underlying orbitals are changed. (In other words, even when the coupled cluster Lagrangian is stationary with respect to T,Λ, under an orbital rotation that changes H→eiϵRHe−iϵR, there is not a small change in the values of the amplitudes which completely cancels this rotation.) However, in the case of the Luttinger–Ward functional, stationarity with respect to the Green’s function implies stationar- ity with respect to the underlying orbitals because a small change in H→eiϵRHe−iϵRcan be canceled by a corresponding small change in the Green’s function (with a small abuse of notation, G→e−iϵRGeiϵR) since all quantities in the action correspond to closed diagrams of H andG. APPENDIX C: STATIONARITY OF PERTURBATION THEORY Here, we briefly show that finite-temperature time-dependent perturbation theory yields stationary equilibrium observables. Con- sider a Hamiltonian H(λ)=h+λV, where his the zeroth order piece. The time-dependent observable Oand its equilibrium value are identical under propagation by the equilibrium Hamiltonian since Z−1tre−βH(λ)eiH(λ)TOe−iH(λ)T =Z−1tre−iH(λ)Te−βH(λ)eiH(λ)TO =Z−1tre−βH(λ)e−iH(λ)TeiH(λT)O =Z−1tre−βH(λ)O, (C1) where Zis the partition function, the second line follows from cyclic invariance, and the third line follows from commuting operators, which does not require the trace. The above is an identity which holds for all λ; therefore, it is true order by order in λ, and this means that the perturbation expansion of the left-hand side and right-hand side must agree. The need to include all time orderings to obtain stationarity in an approximate theory is because the above result relies on commuting the imaginary and real-time propagations past each other, which is equivalent to changing the time ordering of interactions on those branches. APPENDIX D: COUPLED CLUSTER EQUATIONS The kernels which precisely determine the Keldysh-OCCD method closely resemble the zero-temperature OCCD equations. The E kernel is given by E(t)=1 4∑ ijab⟨ij∣∣ab⟩sab ij(t). (D1) The S and L kernels which determine the equations of motion for thesand ˜λamplitudes, respectively, are given by J. Chem. Phys. 155, 044103 (2021); doi: 10.1063/5.0059257 155, 044103-13 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Sab ij(t)=⟨ab∣∣ij⟩+P(ab)∑ cfbcsac ij(t)−P(ij)∑ kfkjsab ik(t)+1 2∑ cd⟨ab∣∣cd⟩scd ij(t)+1 2∑ kl⟨kl∣∣ij⟩sab kl(t)+P(ij)P(ab)∑ kc⟨kb∣∣cj⟩sac ik(t) +1 4∑ klcd⟨kl∣∣cd⟩scd ij(t)sab kl(t)+1 2P(ij)P(ab)∑ klcd⟨kl∣∣cd⟩sac ik(t)sdb lj(t)−1 2P(ab)∑ klcd⟨kl∣∣cd⟩sca kl(t)sdb ij(t)−1 2P(ij)∑ klcd⟨kl∣∣cd⟩scd ki(t)sab lj(t), (D2) Lij ab(t)=⟨ij∣∣ab⟩+P(ab)∑ c˜λij ac(t)fcb−P(ij)∑ k˜λik ab(t)fjk+1 2∑ cd˜λij cd(t)⟨cd∣∣ab⟩+1 2∑ kl˜λkl ab(t)⟨ij∣∣kl⟩+P(ij)P(ab)∑ kc˜λik ac(t)⟨cj∣∣kb⟩ −P(ij)1 2∑ klcd˜λik ab(t)⟨jl∣∣cd⟩scd kl(t)−P(ab)1 2∑ klcd˜λij ac(t)⟨kl∣∣bd⟩scd kl(t)+P(ij)P(ab)∑ klcd˜λik ac(t)⟨lj∣∣db⟩scd kl(t)−P(ab)1 2∑ klcd˜λkl ca(t)⟨ij∣∣db⟩scd kl(t) −P(ij)1 2∑ klcd˜λki cd(t)⟨lj∣∣ab⟩scd kl(t)+1 4∑ klcd˜λkl ab(t)⟨ij∣∣cd⟩scd kl(t)+1 4∑ klcd˜λij cd(t)⟨kl∣∣ab⟩scd kl(t). (D3) The density matrices which appear in the orbital Eq. (50) are also used to compute properties. They can be obtained from the deriva- tive of the Lagrangian with respect to the potential, dpq=1 2[(dN)pq+(dN)∗ qp]+ppq, (D4) dpq rs=1 2[(dN)pq rs+(dN)rs pq∗]+pprdqs+pqsdpr−ppsdqr−pqrdps −pprpqs+ppspqr. (D5) We have used pto indicate the mean-field density matrix, pij=δij,pia=pai=pba=0, (D6) anddNfor the coupled cluster contributions, (dN)ia=0, (D7) (dN)ba=1 2∑ ikc˜λki cb(t)sca ki(t), (D8) (dN)ji=−1 2∑ akc˜λkj ca(t)sca ki(t), (D9) (dN)ai=0, (D10) (dN)ij ab=˜λij ab(t), (D11) (dN)aj ib=∑ cksac ik(t)˜λjk bc(t), (D12) (dN)cd ab=1 2∑ ijscd ij(t)˜λij ab(t), (D13) (dN)kl ij=1 2∑ absab ij(t)˜λkl ab, (D14)(dN)ab ij=sab ij(t)+P(ij)P(ab)1 2∑ klcdsac ik˜λkl cd(t)sbd jl(t) −P(ab)1 2∑ klcdsad kl(t)˜λkl cd(t)scd ij(t) −P(ij)1 2∑ klcdscd il(t)˜λkl cd(t)sab kl(t) +1 4∑ klcdsab kl(t)˜λkl cd(t)scd ij(t). (D15) Here and throughout this work, we have assumed that the matrix elements include factors of the square root of the occupation num- bers as in Eq. (42). 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5.0053950.pdf

First principles and atomistic calculation of the magnetic anisotropy of Y 2Fe14B Cite as: J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 View Online Export Citation CrossMar k Submitted: 13 April 2021 · Accepted: 21 June 2021 · Published Online: 8 July 2021 Ramón Cuadrado,1,2,a) Richard F. L. Evans,1,b) Tetsuya Shoji,3Masao Yano,3Akira Kato,3Masaaki Ito,3 Gino Hrkac,4Thomas Schrefl,5 and Roy W. Chantrell1 AFFILIATIONS 1Department of Physics, University of York, York YO10 5DD, United Kingdom 2Computational Systems Chemistry, School of Chemistry, University of Southampton, Southampton SO17 1BJ, United Kingdom 3Toyota Motor Corporation, Toyota City, 471-8572, Japan 4College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, United Kingdom 5St. Pölten University of Applied Sciences, Matthias Corvinus Str. 15, St. Pölten, A-3100, Austria a)Author to whom correspondence should be addressed: R.Cuadrado@soton.ac.uk b)E-mail: richard.evans@york.ac.uk ABSTRACT We present a study of the effects of strain on the magnetocrystalline anisotropy energy and magnetic moments of Y 2Fe14B bulk alloy. The study has been performed within the framework of density functional theory in its fully relativistic form under the generalized gradientapproximation. We have studied seven different in-plane alattice constant values ranging from 8.48 up to 9.08 Å with an increment of δa¼0:1 Å. For each avalue, we carried out an out-of-plane cparameter optimization, achieving the corresponding optimized lattice pair (a,c). We find a large variation in the site resolved magnetic moments for inequivalent Fe, Y, and B atoms for different lattice expansions and a negative contribution to the total moment from the Y sites. We find a strong variation in the magnetocrystalline anisotropy with the c=aratio. However, the calculated variation when coupled with thermodynamic spin fluctuations is unable to explain the experimentally observed increase in the total magnetic anisotropy, suggesting that a different physical mechanism is likely to be responsible in contrast withprevious interpretations. We show that opposing single- and two-ion anisotropy terms in the Hamiltonian gives good agreement with theexperiment and is the probable origin of the non-monotonic temperature dependence of the net anisotropy of Y 2Fe14B bulk alloy. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0053950 I. INTRODUCTION Rare-earth (RE) transitional metal permanent magnetic mate- rials play a critical role in hybrid and electric vehicles and electric power generation.1Recent concern about the impact of climate change has renewed interest in understanding and optimizing these materials to improve the energy efficiency of these key technologies. The most technologically important Nd –Fe–B magnet consists of 2:14:1 phase Nd 2Fe14B. This Nd –Fe–B magnet has the highest energy product among all known permanent magnet materials.1 The magnetocrystalline anisotropy (MAE) is a key factor for under-standing the high coercivity of RE 2TM 14M (RE = rare earth, TM = transition metals, and M = B, C, N) permanent magnets.2 These elements form stoichiometric compounds in the 2:14:1phase of rare earths, transition metals, and metalloids, respectively,which allows the study of different magnetic couplings RE –RE and RE–TM between the different sites. 3–5Related with the MAE is the effect of strain on these materials since the manufacturing process could promote variations in their lattice parameters and therefore some residual strain.6–8 In addition, rare earths have localized 4 felectrons, adding further complexity to their study. These felectrons are relatively insensitive to their environment in contrast with 3 delectrons that are quite sensitive to lattice changes due to their itinerancy. As pro- posed by Torbatian et al. ,8the compound Y 2Fe14B is suitable for the study of Fe –3delectrons since Y does not have felectrons, being a prototypical f0rare earth element keeping the same geo- metrical structure as the other RE-TM bulk materials. Due to the structural symmetry of RE 2Fe14B series alloys, studies of Y 2Fe14B play an essential role in understanding theJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 130, 023901-1 Published under an exclusive license by AIP Publishingcontribution of the Fe sublattice to the overall MAE for the case of more technologically relevant alloys including Nd or Dy. In the majority of RE 2Fe14B series alloys, the rare earth ions dominate the MAE at low temperatures.9However, the combination of the strong temperature dependence of the RE sublattice magnetization andhigher order contributions to the MAE lead to a strong temperature dependence of the MAE, so that at room temperature, the Fe sub- lattice contributes a significant fraction of the total MAE. 9 A common feature of the so-called “non-magnetic ”RE2Fe14B alloys, where RE = Y, Ce, Th is an increase of the anisotropy field with increasing temperature, in direct contrast to the usual expecta- tion of a reduction of the MAE due to spin fluctuations.10–12 Bolzoni et al.13suggested that the origin of this anomalous increase of the MAE may be due to asymmetric temperaturedependent lattice expansion, 14causing an increase of the effective MAE at elevated temperatures. Torbatian et al. calculated the MAE in Y 2Fe14B for two cases: an equilibrium and a compressed lattice, finding that the MAE is enhanced upon lattice compression.However, understanding the influence of lattice changes on theMAE in Y 2Fe14B for a wider range of aand cvalues is essential to ascertain whether temperature-induced lattice expansion can account for the observed increase in the anisotropy field at elevated temperatures. Here, we systematically investigate the effects of the lattice parameters aand con the magnetic properties of Y 2Fe14B, including changes in the local magnetic moments, electronic structure, and MAE. The paper is structured as follows.In Sec. IIis presented in brief the theoretical tools employed to perform the calculations. In Sec. III A ,ag e o m e t r i ca n a l y s i so f the optimized bulk structures is presented. The projected density of states on each atomic species and site is presented in Sec. III B together with the charge transfer between different atomic species. The analysis of the magnetic moments is pre-sented in Sec. III C , and the magnetic anisotropy calculated values are presented in Sec. III D . The dependence of the anisot- ropy with the temperature is shown in Sec. IV. Finally, in Sec. V we summarize the main conclusions of the work. II. THEORETICAL METHODS We have undertaken geometrical, electronic, and magnetic structure calculations of the Y 2Fe14B bulk alloy by means of DFT using the SIESTA code.15,16To describe the core electrons, we have used fully sep arable Kleinmann –Bylander17and norm- conserving pseudopotentials (PPs) of the Troulliers –Martins18 type. As exchange correlation (XC) potential we have employed the generalized gradient approximation (GGA) following the Perdew, Burke, and Ernzerhof (PBE) version.19To have a better description of magnetic systems, pseudocore (pc) correctionswere used to include in the XC terms not only the valence chargedensity but also the core charge. 20In general, the correction will only be significant in the range where valence and core charges overlap. As a basis set, we have employed double- ζpolarized (DZP) strictly localized numerical a t o m i co r b i t a l s( A O ) .T h ee l e c t r o n i c temperature —kT in the Fermi-Dirac distribution —was set to 50 meV. Real space integrals are computed over a three-dimensional grid with a resolution of 1600 Ry, a mesh fine enough to ensure convergence ofthe electronic/magnetic properties. To obtain the charge distribution, we have used the Mulliken partitioning scheme.21 The MAE is defined as the difference in the total self- consistent energy between hard and easy magnetization directions.MAE values are commonly of the order of meV and hence it isnecessary to perform an accurate calculation through the conver- gence of relevant DFT parameters such as the number of k-points. For Y 2Fe14B, we find that the MAE values are particularly sensitive to the k-point sampling. We therefore carried out an exhaustive analysis of the MAE convergence in order to achieve a total energytolerance below 10 /C05eV. We employed more than 500 kpoints in the calculations for each geometric configuration. To obtain the MAE, we have used a recent implementation of the off-site spin – orbit coupling (SOC)22–25in the SIESTA code. This approximation takes into account not only the local SOC contributions to the totalenergy but also the neighboring interactions between atoms to obtain the total self-consistent energy. Figure 1 shows a schematic view of the crystal structure of Y 2Fe14B. It is composed of a tetragonal unit cell (UC) with a space group of P42/mnm and the UC has two different Y sites ( f,g), six distinct Fe sites ( j1,j2,k1,k2,eand c), and one B site, with a total of 68 atoms in the unit cell. In this work, we have performed the study of seven in-plane lattice values: a¼8:48 A/C14þk/C1δa, with δa¼þ0:10 Å and k¼0, ..., 6. Variations of the alattice param- eter will also naturally lead to a variation of the c=aratio, and so it was necessary to optimize the tetragonal out-of-plane distortions for each value of a. Accordingly, c=avalues are found to be: 1.45, 1.42, 1.40, 1.37, 1.34, 1.31, and 1.29, respectively. III. RESULTS A. Geometry optimization All of the geometric configurations studied in the present work were subject to an optimization process. For the range ofstudy, we chose in-plane lattice constants around the experimentalroom temperature value of 8.75 Å (see the end of Sec. II). 14The optimization process was carried out for fixed in-plane lattice cons- tant aby performing self-consistent energy calculations for a series of out-of-plane cvalues. Plotting the total energy as a function of c we are able to obtain local minima, geometrically characterizingeach unit cell in terms of the aand cparameters. In this respect, the ground state (GS) energy value was achieved for the pair (a,c)¼(8:78 A/C14,1 2 :00 A/C14). As noted in Sec. II, we are using the generalized gradient approximation as the XC functional and sothe bonds between atoms will be slightly larger than the experi-mental ones. 5,14However, our optimized lattice value is within 0.03 Å of the values obtained from previous studies of Y 2Fe14B. As an example of this process, we show in Fig. 2 three different opti- mization curves that show the energy dependence with respect tothe out-of-plane cfor fixed a. All the values have been shifted with respect to the lowest energy value pair (filled black dot curve), which shows clearly the energy difference of /difference0:25 eV between the GS and the other values. Figure 3 shows the radial distribution function (RDF) of the bond distances between the Fe atoms located in the upper part of the unit cell and those (denoted Fe ’) located in the lower side, A and B, respectively. A visualization of the atom groupings isJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 130, 023901-2 Published under an exclusive license by AIP Publishingshown in the inset in Fig. 3 . The plotted range represents small bond values, below 4.5 Å, ensuring that only nearest and nextnearest neighbors are taken into account. In order to encompass the geometrical evolution of the UC for different in-plane lattice values, we have chosen a representative avalue for each zone: 8.48, 8.78, and 9.08 Å, small, intermediate, and large, respectively.On each graph two different peaks appear around /difference2:5 Å and/difference4:25 Å. Clearly, these peaks represent bonds between the first and second neighbors of a specific Fe atom, respectively. It is beyondthe scope of this work to discriminate inequivalent Fe distances between different groups located in the upper and lower UC. FIG. 2. Y2Fe14B bulk self-consistent total energy values as a function of the out-of-plane lattice parameter for a fixed 8.68 (empty squares), 8.78 (filled black dots), and 8.88 (filled squares) Å in-plane lattice constant a. The energies have been subtracted from the GS energy value, and the dashed lines joining dotsare a guide for the eye. FIG. 3. Lorentzian broadening of the bond distances between Fe species for differ- ent groups of atoms, d Fe/C0Feand d Fe0/C0Fe0, A and B graphs, respectively. T o the right on each graph is shown the schematic representation of Y 2Fe14B UC which clarifies, between black dashed lines, each one of the Fe groups atoms under consideration. FIG. 1. Schematic representation of the crystal structure of the tetragonal Y 2Fe14B phase. Each atomic occupancy has been depicted by a color scale and on the right it is shown the Fe, Y , and B average magnetic moment peratom of the optimized bulk structure when it has takeninto account the in-plane avalue of 8.78 Å.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 130, 023901-3 Published under an exclusive license by AIP PublishingHowever, after careful inspection of each configuration, we are able to give a qualitative understanding of the geometric expansion/con- traction of the bulk Y 2Fe14B alloy. We find different structural properties for Fe atoms at different points in the unit cell, corre-sponding to Fe and Fe ”sites in Fig. 3 . In particular, the small bond-length range represents bonds mainly between Fe atoms at different out-of-plane level and the longer to those located almost in the same horizontal plane. The overall trend for d Fe/C0Feand dFe0/C0Fe0is quite similar, having a small positive deviation around each peak as aincreases. For example, focusing on Fig. 3(a) we observe that the shift is more pronounced around 4.25 Å than for the smaller bond-length, with the Fe –Fe and Fe ’–Fe”bond dis- tances increasing by 0.2 Å with aexpansion. Then, avariations cause the Fe atoms located at the same plane level ( /difference4:2 Å) to lie further away from each other while those Fe species at differentplanes are positioned almost at the same distance. In summary, we conclude that locally the variation of apromotes that the bonds between different groups of Fe atoms behave differently on whetherthey are located at the lower or upper part of the UC. However, theoverall out-of-plane contraction when aincreases is mainly due to the upper and lower Fe blocks approaching. This makes that the UC keeps constant its volume. B. Density of states and charge transfer InFig. 4 (left), the spin-resolved projected density of states (PDOS) is shown for three different in-plane lattice configurations.In each column, the Fe DOS were projected to show the contribu-tions from different sites, namely, k,jand c,e(see Fig. 1 forinequivalent Fe sites within the unit cell). In general, after inspec- tion of these different Fe sites, we observe that only the k 1and k2 DOS share a similar form. The j,c, and eFe sites have their d-band peaks at different energy positions, for example, the Fe( j1) up-states (solid line) have two peaks at 0.8 and 2 eV and in ( j2) (dashed line) these peaks have shifted to the lower energy values of 3.5 and 2 eV, promoting a population of the Fe( j2)-dband. We can see this behavior in detail in Fig. 5 where the Fe( j1) (empty circles) have lost more charge than Fe( j2), with respect to the isolated atoms. In the same fashion, we are able to explain the charge transfer ofd-bands in Fe( e) and Fe( c) (last column). In a column, the shape of the DOS curves shares a common trend: as aincreases, upper to lower panels, the d-band width becomes slightly smaller and higher, having the states concentrated in a smaller energy range.For instance, for Fe( k), the energy extends from zero to almost /C06 eV for the smaller aand this width is reduced for bigger lattice up to /C05 eV. In other cases, even though the broadness is similar for small and large lattice, the center of the d-band tends to increase, concentrating more states in the range of /C03t o/C02e V . It is also interesting to note in Fig. 4 (right) the two peaks around /C08 and /C09e V i n F e ( e) and Fe( k 1), which permit interaction with the Y and B states. The PDOS for the Y( g), Y( f), and B( g) species are depicted in Fig. 4 (right). As we will see in Sec. III C , Y and B are antiferromag- netically coupled to Fe, having an excess of down-states in the popu- lation. In general, the in-plane lattice constant expansion for these species does not significantly affect the shape of the Y PDOS.However, around the Fermi level, both Y sites differ in height, imply-ing a slightly different amount of up-charge contributing to different FIG. 4. (Left) Spin-resolved density of states for k1=2,j1=2and c,edifferent Fe sites, first second and third column, respectively. In a row is depicted the PDOS for a= 8.48, 8.78, and 9.08 Å lattice values; (right) spin-resolved density of states for the Y( g), Y( f), and B atomic species, thick solid, dashed, and thin solid black lines, respectively. From top to bottom, the smaller ( a¼8:48 Å), intermediate ( a¼8:78 Å), bigger ( a¼9:08 Å) size lattices are presented.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 130, 023901-4 Published under an exclusive license by AIP Publishingimbalance in the up-/down-populations. As previously mentioned, there are two peaks; at /C08a n d /C09 eV belonging to the B species, mainly of scharacter, that permit the hybridization of Fe( k1)a n dF e (e) with the B states. We note that these Fe atoms are located close to B species allowing this interaction. To obtain further insight into the charge distribution on each atomic species in Fig. 5 , we analyze the increase or reduction of the total charge (not spin-resolved) on each atom compared to its iso-lated counterpart. A negative value means that the species has lostcharge and positive means that the atom has been charged. Theupper panel shows how the charge on the different Fe sites changes with the lattice spacing. Only Fe( k 1) is charged positively, however its net charge tends to be, for bigger avalues, close to that of an isolated atom. The remaining Fe sites have transferred charge tothe neighbors by different amounts and, depending on the site thatthey occupy in the unit cell, their net charge will be almost constant or will decrease with the lattice expansion. This is exactly what happens at Fe( c) that reduces its charge by /difference0:20e/at when a changes from 8.48 to 9.08 Å. Fe( k 2), Fe( j1) and Fe( j2) remain nearly constant in all this arange. Analysis of the Y( f) and B( g) charge (lower panel in Fig. 5 ) shows an enhancement of their net charges mainly due to the Fe neighbors and from the B species as well.C. Total and localized magnetic moments We now present the total and local magnetic moment (MM) values of the Y 2Fe14B bulk alloy for different ( a,c) pairs. Our total MM predictions assign bigger MM values to a¼9:08 Å and smaller to a¼8:48 Å, depicting an enhancement of the MM as the volume of the UC expands. The complete MM/f.u. values sequencefor each ( a,c) pair is: 28.70, 29.77, 30.60, 30.99, 31.24, 31.43, and 31.79, the first one corresponding to the smaller in-plane a. Therefore, our GS configuration (8.78 Å) presents a total MM of 30.99 μ B/f.u., in good agreement with other experimental and theo- retical works.4,5,8In addition, it is of huge importance to under- stand how different Fe, Y, and B sites contribute to the total MM. InFig. 6(a) , we show the average magnetic moment (MM) for different Fe sites for a common lattice value. The MM has been cal- culated as MM M=NM/NM=NM, where N M=NMare the total number of magnetic (M) or non-magnetic (NM) species. All the Fe sites havea clear tendency of their MM values to increase as the in-planelattice constant increases. The largest growth (of 0.55 μ B/at) takes place for the Fe( j1) and the smallest variation is for Fe( c). The remaining sites exhibit an increase to a greater or lesser extent. We are able to observe this trend qualitatively by inspecting thespin-resolved projected DOS shown in Fig. 4 (left). For example, in Fe(j 1) (black solid curve), the total up-states increase with a; this reflects that the initial small shoulder at the Fermi level displaces inwards toward the valence band allowing an increase in the spin up charge. Conversely, the down-states act in a different manner,reducing the down charge. As a result, the imbalance in the up/down charge will be more pronounced for the expanded lattices and will promote an enhancement of the on-site Fe( j 1) MM values when a¼9:08 Å. Through the j2PDOS curves, it is also easy to understand why they have the higher MM values among all of theFe sites. They present bigger imbalance in their up-/down-states incomparison with the other Fe sites. The Y( f), Y( g), and B( g) MM values are presented in Fig. 6(b) . The net MM per atom of these two species are negative for all the lattice constant values [labeled in Fig. 6(b) by filled and empty squares, triangles, circles and empty diamonds]. The Y ’s MM values exhibit different behavior depending on whether the lattice constant changes seen by its vertical dispersion between /C01:05 and /C00:8μ B/at. Both, Y(f) and Y(g) location sites, tend to behave similarly. Regarding the variation with a, the tendency is to increase the net MM with lattice expansion, by around /C00:3μB/at for both Y( f) and Y( g). On the other hand, for the Y atoms located at different sites, the MM are slightly bigger for Y( f) than for Y( g), the difference being at most of 0.04 μB/at. B MM values behave in a different manner since they do not change with the latticeexpansion having a common value of /C00:20μ B/at. In general, the down-states population exceeds that of the up-states, resulting in a negative MM for the Y and B atoms. We also note the presence ofa significant magnetic moment on the Y sites, in contradiction tothe usual assumption of a non-magnetic RE. 9 D. Magnetic anisotropy energy We now present the calculations of the total magnetic anisot- ropy energy. FIG. 5. Charge difference between each atomic species in its bulk phase and the isolated atoms as a function of the in-plane lattice constant a, ΔQ¼qbulk/C0qisol. A positive (negative) value means charge adsorption (reduc- tion) with respect to the isolated case.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 130, 023901-5 Published under an exclusive license by AIP PublishingAs was pointed out in Sec. II, the MAE values are of the order of a few meV. Consequently, the calculation of the self-consistent total energies involved have to be sufficiently accurate. In the same fashion as Kitagawa and Asari,5we performed convergence tests for the k-points sampling and other relevant DFT parameters, resulting in sufficient accuracy for the quantities under study. InFig. 7 is shown the MAE convergence for the Y 2Fe14B bulk alloy with increasing k-points. The dashed black lines indicate our required accuracy in the MAE values, of 5 /C210/C05eV. We observethat in the range from 567 k-points (indicated in red in Fig. 7 ) to 1400 all calculated values lie within the required tolerance. Consequently, we decided to use 567 k-points to perform the calcu- lation of the MAE for all the configurations. It is worth noting that,although Isao et al. used a code based on a linear combination of pseudo-atomic-orbital, similar to the SIESTA code, the convergence of each code will depend not only on the scheme/formalism to solve the Kohn –Sham equations but also on the pseudopotential and many other parameters involved. InFig. 8 , we show the MAE as a function of the optimized c=aratio. As was pointed out at the end of Sec. II, an increase FIG. 6. (a) Average magnetic moment per atom for each kind of Fe site. The different symbols represent all the lattice constants values studied in the present w ork. The straight line is a guide for the eye; (b) average magnetic moments per atom for the Y( f), Y( g), and B. The legend depicts the lattice constant values. FIG. 7. Magnetocrystalline anisotropy of the optimized Y 2Fe14B bulk alloy as a function of the number of the k-points. The interval between the two dashed lines shows the magnetic anisotropy dispersion for the higher k-point values. The red arrow represents the number of k-points chosen in order to have accu-rate MAE values. FIG. 8. Magnetic anisotropy energy as a function of the out-of-plane c=a parameter.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 130, 023901-6 Published under an exclusive license by AIP Publishing(reduction) of ameans a reduction (increase) of the out-of-plane constant, thus from right to left along the X axis will mean an in-plane lattice expansion. The MAE ’s dispersion over the studied c=arange is of 2.8 meV. Specifically, the MAE tends to zero when a increases and hence c=adecreases while it presents closer values to 2.8 meV when the in-plane aconstant decreases promoting larger c=avalues. The ground state geometry configuration depicts a MAE value of 1.74 meV/u.c., that is, in reasonable agreement withpreviously reported works of Torbatian et al 8of 2.6 meV/u.c., Miura et al.26of 0.622 meV/u.c. and Isao et al.53.2 meV/u.c. The present trend of the bulk MAE of Y 2Fe14B shown here is similar to the FePt –L10on MgO, where the MAE reduces its value as the in-plane expansion takes place.27In FePt, the tetragonality of the system contributes to the MAE, and we argue that the similarqualitative behavior seen for Y 2Fe14B is due to the same physical effect due to the anisotropy of the local electronic environment. IV. TEMPERATURE DEPENDENCE OF THE ANISOTROPY: ATOMISTIC MODEL CALCULATIONS Finally, we turn to the motivation of this work, which is to understand the unusual (non-monotonic) temperature dependence of the MAE in Y 2Fe14B. For clarity, we refer to the MAE as the intrinsic anisotropy value from DFT calculations and the anisot-ropy as the observed value at non-zero temperature. The latter isthe free energy of the system, which can have two contributions:first, the thermodynamic decrease driven by spin fluctuations, 11 and second any change in the MAE arising from lattice expansion.Clearly, this can potentially lead to non-monotonic temperaturedependence of the anisotropy and as such is the first option consid-ered here. For this calculation, we employ an atomistic model, which introduces the thermally driven spin fluctuations which, in the absence of a temperature dependent intrinsic anisotropy, areresponsible for the reduction in the observed anisotropy value atnon-zero temperatures. Using the atomistic model, we calculate the temperature varia- tion of the magnetisation using conventional Metropolis MonteCarlo methods and the temperature dependence of the anisotropyusing a constrained Monte Carlo algorithm 12with adaptive updates,28both implemented in the VAMPIRE software package.29,30 First, we consider whether the non-monotonic behavior of the measured anisotropy can be explained through our predictedchange in the MAE due to changes in the c=aratio coupled with the thermally induced spin fluctuations. However, our calculations(not shown) give a conventional monotonic decrease of anisotropy with temperature. We can interpret this as follows. We can estimate the reduc- tion in anisotropy over a given temperature range using the changein magnetization and the Callen –Callen law scaling of anisotropy with M nwith exponent n¼3 for uniaxial anisotropy. From the temperature dependence of the magnetization, shown in Fig. 9 it can be seen that there is approximately a 20% reduction in M(T)a t T¼300 K. Consequently, to compensate for the corresponding decrease in anisotropy due to spin fluctuations we require at least a factor 2 increase in the MAE. This is the minimum required to give rise to an increase of the anisotropy with temperature. Qualitativelyfrom the DFT calculations, we find an increase in the intrinsic MAE with increasing c=a. However, the experimental range14for thec=aratio in the bulk is typically 1.37 –1.38 up to TC, which sug- gests a relatively modest increase in the MAE from our calculations.Therefore, while we obtain the correct qualitative behavior, the cal-culations are so far unable to explain the cause of the large increase in MAE. In particular, we note that the temperature variation of the MAE is governed by a combination of spin fluctuations, leadingto a decrease in MAE, coupled with any change in the intrinsicMAE due to the lattice expansion as investigated here. While ourcalculations predict an increase of MAE with lattice expansion, the consequent increase with Tis not sufficient to overcome the effects of spin fluctuations, leading to a monotonic decrease in the MAE,in contrast to experiments. Thus, our calculations show that theprevious interpretation where the increase is attributed to latticeexpansion effects 13,14seems to be unlikely. We finally consider the possibility of a combination of oppos- ing single-ion and two-ion anisotropy as the origin of the largeincrease in anisotropy. We note that the two-ion anisotropy isessentially an exchange anisotropy which is not accessible to our DFT calculations. Recent investigations 31suggest that the origin of an increase in anisotropy with temperature may be due to a compe-tition between single-ion and exchange anisotropy. To extract thetwo-site anisotropy, one uses a generalized tensorial form ofthe exchange H exc¼/C0X i=jSiJijSj, (1) from which one obtains a two-site (exchange) anisotropy as Eintersite ¼/C01 2X i=j(Jxx ij/C0Jzz ij): (2) Our ab initio calculations cannot access the tensorial exchange. Consequently, for this investigation, we introduce a Hamiltonian FIG. 9. Simulated temperature dependent magnetization for Y 2Fe14B.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 130, 023901-7 Published under an exclusive license by AIP Publishingincluding a phenomenological two-site anisotropy while consider- ing the 2-14-1 crystallographic positions of the Fe sites,32 H¼/C0X i,jJij(r)Si/C1Sj/C0X i,jkijSz iSzj/C0X ikiSz i/C0/C12: (3) Equation (3)has three terms; the first being the Heisenberg exchange term and the second and third terms representingtwo-ion and single-ion anisotropies, respectively. The exchangeinteractions are parameterized with an exponential distance depen-dence with a cutoff of 5 Å and normalized to achieve a Curie temperature of 560 K in agreement with experimental measurements. We note that the single- and two-ion terms havedifferent temperature dependences. Specifically, in terms of thetemperature scaling K(T)=K(T¼0)¼(M(T)=M(T¼0)) n, the exponent n¼2:28, 3 for the two- and single-ion terms, respec- tively.11,31Thus, for kijand kiof opposite sign, a non-monotonic variation K(T) of the net anisotropy becomes possible. Figure 10 shows the simulated temperature dependence of the effective magnetic anisotropy of Y 2Fe14B using the constrained Monte Carlo algorithm12with adaptive updates28implemented in the VAMPIRE software package.29,30We apply temperature rescaling to achieve a correct de scription of spin fluctuations and magnetization with temperature.33The optimized anisotropy constants to achieve agreement with the experimental data are kij¼þ1:835 46 /C210/C022J/atom and ki¼/C01:743 687 /C210/C022J/atom, noting the large values and competing signs of the anisotropyconstants. While the agreement between the simulation andexperimental data is good, there are some small systematic differ-ences that may be accounted for with strain effects determined from our ab initio simulations, or indeed due to the temperature dependence of the Hamiltonian parameters themselves.We wish to note the existence of previous explanations of the increasing anisotropy with temperature from crystal-field effects suggested by Bolzoni et al. 13This explanation was not based on a microscopic description but assumed arbitrary higher order anisot-ropy terms that are fictitious in nature and the model could notperfectly explain the experimental behavior. The crystal-field analy- sis by Miura et al. 35concludes that the underlying origin of anisot- ropy in the Fe sublattice remains an unresolved problem. Incomparison, our analysis of mixed anisotropy is based on a micro-scopic model and provides a close quantitative agreement with theobserved increase in the magnetic anisotropy energy. Further ab initio calculations may be able to determine the relative contribu- tions from two-ion and single-ion anisotropy, 36though these calcu- lations are still excessively expensive in terms of computationalcomplexity due to the large cell size and number of electrons. V. CONCLUSIONS We have found that tuning the unit cell geometry of the Y 2Fe14B bulk alloy, specifically changing the in-plane/out-of-plane constants aand c, leads to a significant change in the magnetocrys- talline anisotropy energy. In the present work, we have undertakena geometric, electronic, and magnetic study of seven different con-figurations of the Y 2Fe14B alloy where the in-plane arange was varied from 8.48 to 9.08 Å and the corresponding out-of-plane c parameter was optimized for each case. The ground state configura-tion has an in-plane value of 8.78 Å with c¼12:00 Å and its total MM is 30.99 μ B/f.u. in good agreement with previous experimental and theoretical results. Breaking up the total MM for individual Fe, Y, and B inequivalent sites, we have demonstrated that the higher MM value is for the Fe( j2) sites and the higher dispersion acts on Fe(j1), having a lower value for smaller aincreasing for the larger a value. The Y and B species are antiferromagnetically coupled andonly the Y changes similarly to the pair ( a,c), decreasing their net MM value as the lattice is compressed. It is clear that the strain has an important impact on the magnetic anisotropy of these alloys sothat a volume reduction of the UC promotes higher values of theMAE. The MAE dispersion value for the present work is around 2.8 meV, having almost zero for the bigger unit cell and 2.8 meV for the (8.48 Å,1.46) pair. Our calculations of the variation ofthe MAE with lattice expansion is not sufficient to explain thenon-monotonic variation of MAE with temperature observedexperimentally. Atomistic simulations including contributions from competing two-ion and single-ion anisotropies are able to repro- duce the observed increase in anisotropy with increasing tempera-ture and offer an alternative explanation for the phenomenon inRE-TM intermetallic alloys. The electronic origins of anisotropyrequire further investigation due to the essential role played by Y 2Fe14B in understanding the contribution of the Fe sublattice to the overall MAE for the case of more technologically relevant alloysincluding Nd or Dy. ACKNOWLEDGMENTS This work is based on results obtained from the future pio- neering program “Development of magnetic material technology for high-efficiency motors ”commissioned by the New Energy and Industrial Technology Development Organization (NEDO). FIG. 10. Atomistic simulation of the temperature dependent effective anisotropy of Y 2Fe14B with a mixture of 2-ion and single ion anisotropies (line) in compari- son with the experimental data of Grossinger et al.34(points). The simulation and experimental data show very good agreement, suggesting that mixedanisotropies are an important component of anisotropy in R 2Fe14B alloys.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 130, 023901-8 Published under an exclusive license by AIP PublishingDATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1O. Gutfleisch, M. A. Willard, E. Brack, C. H. Chen, S. G. Sankar, and J. P. Liu, “Magnetic materials and devices for the 21st century: Stronger, lighter, and more energy efficient, ”Adv. Mater. 23, 821 (2011). 2J. M. D. Coey, “Hard magnetic materials: A perspective, ”IEEE. Trans. 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Mendive-Tapia, L. Petit, and J. B. Staunton, “Rare-earth/transition-metal mag- netic interactions in pristine and (Ni, Fe)-doped YCo 5and GdCo 5,”Phys. Rev. Mater. 1, 024411 (2017).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 023901 (2021); doi: 10.1063/5.0053950 130, 023901-9 Published under an exclusive license by AIP Publishing

5.0056182.pdf

J. Chem. Phys. 155, 014102 (2021); https://doi.org/10.1063/5.0056182 155, 014102 © 2021 Author(s).Internal conversion of singlet and triplet states employing numerical DFT/MRCI derivative couplings: Implementation, tests, and application to xanthone Cite as: J. Chem. Phys. 155, 014102 (2021); https://doi.org/10.1063/5.0056182 Submitted: 07 May 2021 . Accepted: 16 June 2021 . Published Online: 02 July 2021 Mario Bracker , Christel M. Marian , and Martin Kleinschmidt ARTICLES YOU MAY BE INTERESTED IN Chemical physics software The Journal of Chemical Physics 155, 010401 (2021); https://doi.org/10.1063/5.0059886 Exact-two-component block-localized wave function: A simple scheme for the automatic computation of relativistic ΔSCF The Journal of Chemical Physics 155, 014103 (2021); https://doi.org/10.1063/5.0054227 Two-dimensional electronic–vibrational spectroscopy: Exploring the interplay of electrons and nuclei in excited state molecular dynamics The Journal of Chemical Physics 155, 020901 (2021); https://doi.org/10.1063/5.0053042The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Internal conversion of singlet and triplet states employing numerical DFT/MRCI derivative couplings: Implementation, tests, and application to xanthone Cite as: J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 Submitted: 7 May 2021 •Accepted: 16 June 2021 • Published Online: 2 July 2021 Mario Bracker, Christel M. Marian, and Martin Kleinschmidta) AFFILIATIONS Institute of Theoretical and Computational Chemistry, Heinrich-Heine-University Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany a)Author to whom correspondence should be addressed: Martin.Kleinschmidt@hhu.de ABSTRACT We present an efficient implementation of nonadiabatic coupling matrix elements (NACMEs) for density functional theory/multireference configuration interaction (DFT/MRCI) wave functions of singlet and triplet multiplicity and an extension of the V IBESprogram that allows us to determine rate constants for internal conversion (IC) in addition to intersystem crossing (ISC) nonradiative transitions. Following the suggestion of Plasser et al. [J. Chem. Theory Comput. 12, 1207 (2016)], the derivative couplings are computed as finite differences of wave function overlaps. Several measures have been taken to speed up the calculation of the NACMEs. Schur’s determinant complement is employed to build up the determinant of the full matrix of spin-blocked orbital overlaps from precomputed spin factors with fixed orbital occupation. Test calculations on formaldehyde, pyrazine, and xanthone show that the mutual excitation level of the configurations at the reference and displaced geometries can be restricted to 1. In combination with a cutoff parameter of t norm=10−8for the DFT/MRCI wave function expansion, this approximation leads to substantial savings of cpu time without essential loss of precision. With regard to applications, the photoexcitation decay kinetics of xanthone in apolar media and in aqueous solution is in the focus of the present work. The results of our computational study substantiate the conjecture that S 1 T2reverse ISC outcompetes the T 2↝T1IC in aqueous solution, thus explaining the occurrence of delayed fluorescence in addition to prompt fluorescence. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0056182 I. INTRODUCTION Density functional theory/multireference configuration inter- action (DFT/MRCI) is a very efficient method for computing electronically excited states of extended molecular systems.1–3It combines density functional theory (DFT), which is well suited for describing dynamic electron correlation, with multireference configuration interaction (MRCI), which provides non-dynamic correlation. To avoid double counting of electron correlation contributions, the Hamiltonian is parameterized against experimen- tal excitation energies, and the configuration space is subjected to extensive selection procedures. The combination of these mea- sures results in high computational efficiency in conjunction with root mean square deviations from experiment typically smaller than 0.2 eV for organic molecules and metal organic complexes with aclosed-shell electronic ground state.1,3–7The related spin–orbit cou- pling kit, S POCK,8–10forms the basis for computing spin-dependent properties, such as rate constants for phosphorescence, inter-system crossing (ISC), and its reverse (rISC), with good accuracy.11–14So far, the ability to compute rate constants for internal conversion (IC) of arbitrary multiplicity states at the DFT/MRCI level of theory is missing. This work aims to fill this gap. Derivative couplings, also addressed as nonadiabatic coupling matrix elements (NACMEs), play an essential role in studying spin-allowed excited-state decay kinetics. Analytical approaches for computing NACMEs are available, e.g., for time-dependent density functional theory (TD-DFT),15–18coupled-cluster (CC) methods,19–21multiconfiguration self-consistent field (MCSCF) approaches,22–26complete active space self-consistent field with perturbative second-order correction (CASPT2) methods,27and J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ab initio multi-reference configuration interaction singles and dou- bles (MRCISD) schemes.28–30Among these methods, only TD- DFT and MCSCF with small active spaces are applicable to extended molecular systems. The individual configuration selec- tion inherent to the DFT/MRCI method destroys the invari- ance of the energy with respect to orbital rotations and thus prevents an easy formulation of analytic energy gradients and derivative couplings. Hence, a numerical ansatz for evaluating NACMEs appears more appropriate in conjunction with the DFT/MRCI. In principle, there are the following three major routes that can be pursued for obtaining numerical derivative couplings between two wave functions Ψa(R)andΨb(R), 1. indirect calculation making use of the Hellmann–Feynman theorem fab=⟨Ψb∣∇H∣Ψa⟩(Ea−Eb)−1; 2. indirect calculation by application of an approximate diabati- zation procedure; and 3. direct calculation of the desired overlap matrix element fab=⟨Ψb(R)∣∇Ψa(R)⟩. The Hellmann–Feynman-type expression31(route 1) appears to be rather straightforward, but it requires the evaluation of addi- tional one-electron potential energy derivatives in static approaches. This route has been followed, e.g., by Peng et al . for com- puting IC rate constants at the TDDFT level of theory.32In fewest-switches surface-hopping molecular dynamics simulations, the nonadiabatic coupling has been approximated numerically using a finite difference method to evaluate the derivative of the wave function with respect to time.33–35Iterative block diabati- zation (route 2) is typically pursued in quantum dynamics cal- culations where precomputed adiabatic potentials along selected coordinates are available.36In conjunction with DFT/MRCI, the latter approach has been applied, e.g., to study the nonadiabatic excited-state dynamics of the ISC and rISC of a spiro-conjugated donor–acceptor emitter with multi configuration time-dependent Hartree (MCTDH) methods.37Recently, Plasser et al.38suggested a simple ansatz for approximating the NACMEs in a direct approach (route 3) by computing finite differences of wave function overlaps, fab(⃗R0)=⟨Ψb(⃗R)∣∇RΨa(⃗R)⟩∣⃗R0 =lim ϵ→0∑ κSba(⃗R0,⃗R0+ϵ⃗eκ)−Sba(⃗R0,⃗R0−ϵ⃗eκ) 2ϵ⃗eκ. (1) Herein, ⃗eκis a unit displacement vector in the κdirection, and ϵis the modulus of the displacement. The challenge lies in the efficient computation of a large number of overlaps between con- figurations in terms of nonorthogonal sets of molecular orbitals. In their paper, Plasser et al.38reported benchmark studies and applications at the CASSCF and MR-CIS(D) levels of theory. An implementation for DFT/MRCI wave functions was pre- sented lately by Neville et al.39who basically followed Plasser’s algorithm. These authors used the overlaps of the electronic wave functions at neighboring nuclear geometries in MCTDH quantum dynamics calculations for a propagative block diagonaliza- tion diabatization.The aim of the present work is to extend this approach to DFT/MRCI wave functions of higher multiplicity and to apply the method after careful validation to compute IC rate constants in xan- thone. In this aromatic ketone, T 2↝S1ISC followed by delayed flu- orescence is observed in aqueous solution.40,41The reason why this spin-forbidden transition outcompetes the spin-allowed T 2↝T1IC is not known so far and will be elucidated in this work. II. THEORY A. IC and ISC rate constants Following Henry and Siebrand,42–44pure spin Born– Oppenheimer molecular states are chosen as starting points for all further considerations. In the harmonic oscillator approximation, they are given by Ξa,S,MS,v(r,Qa)=Ψa,S,MS(r,˜Qa)3N−6 ∏ ιva,ι(Qa,ι), (2) where alabels the electronic state, Sis its total spin quantum number, MSis its magnetic spin quantum number, and rand Qarepresent the electronic and mass-weighted vibrational normal coordinates, respectively. The tilde in ˜Qaindicates a parametric dependence of the electronic wave functions on the nuclear coordi- nates. In this basis, the perturbation causing IC and ISC transitions is given by the perturbation operator ˆV=ˆTN+ˆHSO=−̵h2 2∇† Q∇Q+ˆHSO(r,˜Q). (3) While ultrafast nonradiative transitions at the subpicosecond time scale are best modeled by nonadiabatic dynamics computations, slower processes may be treated perturbatively employing the Fermi’s golden rule approximation. In the statistical limit, the proba- bility of a nonradiative transition from a manifold of thermally pop- ulated initial vibronic states to a quasi-continuum of final vibronic states subject to a perturbation ˆVcan be characterized by a rate con- stant. To simplify the resulting expressions, we omit the coordinate dependencies of the electronic and vibrational wave functions in the following and label the manifolds of vibrational states, {va j}and {vbk}, respectively, by an index symbolizing the related electronic state and a common vibrational index irrespective of the normal coordinate. Wherever needed, the coordinate dependencies will be reintroduced. According to Fermi’s golden rule, a Boltzmann pop- ulation of the initial vibrational states corresponding to the tem- perature Tgives rise to an expression for the rate constant of the nonradiative transition (in atomic units), kNR ab=2π Z∑ j∑ ke−Ea j/kBT×∣⟨Ψb,S′,M′ S,{vb k}∣ˆV∣Ψa,S,MS,{va j}⟩∣2 ×δ(ΔEab+Ea j−Eb k), (4) where Z=∑je−Ea j/kBTis the canonical partition function for vibra- tional motion in the initial electronic state, kBis the Boltzmann constant, and Eajis the energy of the vibrational level jin the elec- tronic state Ψa,S,MS. The Dirac delta function ensures that only isoen- ergetic transitions are allowed, i.e., the vibrational energy in the final electronic state Ψb,S′,M′ S,Ebk, must match the adiabatic energy J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp difference between the electronic states ΔEabplus the vibrational energy in the initial state. In first order, perturbations due to ˆTNand ˆHSOdo not give mixed terms. Hence, the selection rules δS,S′δMS,M′ Sapply for IC in this order. The major perturbations driving the IC are the non- adiabatic coupling terms, VIC ab=⟨Ψb,S′,M′ S,{vb k}∣∇† Qa∇Qa∣Ψa,S,MS,{va j}⟩ ×δS,S′δMS,M′ S(5) ≈3N−6 ∑ κ⟨{vb k}∣⟨Ψb∣∂ ∂Qa,κ∣Ψa⟩∂ ∂Qa,κ∣{va j}⟩ (6) =3N−6 ∑ κ⟨{vb k}∣fab(κ)∂ ∂Qa,κ∣{va j}⟩. (7) In the latter expression, diagonal nonadiabatic corrections have been omitted, retaining only the linear derivative coupling terms fab(κ). The symmetry properties of ˆHSOallow ISC between states of different spin multiplicities, namely, S′−S=0,±1 for S′+S≥1, and different spin magnetic quantum numbers, namely, M′ S−MS =0,±1 unless for MS=0 where matrix elements vanish for com- ponents with ΔMS=0. Hence, spin–orbit coupling matrix elements (SOCMEs) are preferentially evaluated for Ms=Swave functions, and SOCMEs for all symmetry allowed combinations of MSquan- tum numbers are then generated by means of the Wigner–Eckart theorem.45ˆHSOdepends parametrically on Qand does not act directly on the vibrational wave functions. The dependence of the SOCME on the nuclear coordinates is therefore often expanded in a Taylor series about a fixed nuclear arrangement, typically the equilibrium geometry of the initial state ( Q0). Because this expan- sion closely resembles the Herzberg–Teller (HT) expansion of the radiative electric dipole transition, the Q-dependent SOC terms have been dubbed Herzberg–Teller spin–orbit coupling (HT SOC) terms,13,14,46 VISC a,S,MS,b,S′,M′ S=⟨Ψb,S′,M′ S,{vb k}∣ˆHSO∣Ψa,S,MS,{va j}⟩ (8) ≈⟨Ψb,S′,M′ S∣ˆHSO∣Ψa,S,MS,⟩∣Qa0×⟨{vb k}∣{va j}⟩(9) +3N−6 ∑ κ∂⟨Ψb,S′,M′ S∣ˆHSO∣Ψa,S,MS⟩ ∂Qa,κ⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪Qa0 ×⟨{vb k}∣Qa,κ∣{va j}⟩+⋅⋅⋅ . (10)Note that the HT SOC terms indirectly account for admix- tures of other electronic structures upon nuclear displacements and are not easily told apart from spin–vibronic terms aris- ing from second-order couplings involving ˆTNas well as ˆHSO.14 In the linear HT SOC approximation, the ISC rate is a sum of three contributions due to (1) a Franck–Condon term kISC, FC ab, (2) a mixed FC/HT term kISC, FC/HT ab, and (3) a quadratic HT/HT term kISC, HT/HT ab.13 VIBESis an in-house computer program that determines rate constants for nonradiative ISC transitions in harmonic approxi- mation according to Fermi’s golden rule. In addition to FC terms, HT-type expressions can be evaluated. While the original imple- mentation was based on the computation of discrete vibrational overlaps,46,47later versions make use of a generating function approach.13,48,49Generating function approaches for the calcula- tion of the transition rate constants are based on a transformation of the Fermi golden rule approximation into the Heisenberg pic- ture.50They are particularly attractive for multidimensional molec- ular systems because they completely avoid the explicit summation over the vibrational states in Eq. (4). Analytic expressions for time- correlation functions that depend on two sets of normal coordinates QaandQbcan be derived employing Mehler’s formula.51,52Formu- las for computing thermal ISC rate constants in FC and HT approxi- mations using the generating function approach in conjunction with a Duschinsky transformation Qb=JQa+D (11) between the normal coordinates Qbof the final state and Qaof the initial state had been implemented in the V IBESprogram before.13,49 Herein, Jis the Duschinsky rotation matrix, and Dis the coordi- nate displacement. The generating function approach did not only extend the application range of the V IBESprogram to larger molecules but also allowed the inclusion of temperature effects on the cal- culated properties. The latter feature turned out to be extremely valuable for computing rate constants of thermally activated reverse ISC (rISC) processes that play a key role in modern TADF OLEDs.14 In the present work, formulas for IC rate constants have been worked out in analogy. Once, the NACMEs fab(κ) with respect to all normal mode coordinates Qa,κof the ini- tial electronic state Ψa[cp. Eq. (7)] are known, the gener- ating function of the transition can be set up in the time domain, Ga↝b IC(t)=(2π)−N√ det(S−1aS−1 bΩaΩb)N ∑ κ=1N ∑ ¯κ=1fab(κ)f∗ ab(¯κ)∫e−1 4[(Qb+¯Qb)†ΩbBb(Qb+¯Qb)+(Qb−¯Qb)†ΩbB−1 b(Qb−¯Qb)] ×∂2 ∂Qa,κ∂¯Qa,¯κe−1 4[(Qa+¯Qa)†ΩaBa(Qa+¯Qa)+(Qa−¯Qa)†ΩaB−1 a(Qa−¯Qa)]dNQadN¯Qa, (12) J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Herein, Qadenominates the set of normal coordinates of state Ψawith harmonic vibrational frequencies {ωa,κ}, and Qband{ωb,κ}are the corresponding entities in the electronic stateΨb. Barred symbols have been introduced in accordance with the nomenclature of Islampour and Miralinaghi53to stress that the quadrature of the coupling matrix elements VIC ab[Eq. (7)] produces diagonal as well as off-diagonal terms, and hence, the sums over κand ¯κrun independently. S,B, andΩarediagonal matrices with elements (Sa)κκ=sinh[(β−it)ωa,κ], (Sb)κκ=sinh(iωb,κt),(Ba)κκ=tanh[ωa,κ(β−it)/2],(Bb)κκ =tanh(iωb,κt/2),(Ωa)κκ=ωa,κ, and (Ωb)κκ=ωb,κ,β=1/kBT is the inverse temperature, iis the imaginary unit, and tis the time. Expressing the set of normal mode coordinates of the final state, Qb, in the basis of the initial state normal coordi- nates, Qa, by means of the Duschinsky transformation (11) results in Ga↝b IC(t)=(2π)−N√ det(S−1aS−1 bΩaΩb)N ∑ κ=1N ∑ ¯κ=1fab(κ)f∗ ab(¯κ)×∫e−1 4[(JQa+D+J¯Qa+D)†ΩbBb(JQa+D+J¯Qa+D)+(JQa+D−J¯Qa−D)†ΩbB−1 b(JQa+D−J¯Qa−D)] ×∂2 ∂Qa,κ∂¯Qa,¯κe−1 4[(Qa+¯Qa)†ΩaBa(Qa+¯Qa)+(Qa−¯Qa)†ΩaB−1 a(Qa−¯Qa)]dNQadN¯Qa. (13) We now can carry out the differentiation of the exponential function. Employing coordinate transformations X=Qa+¯Qaas well as Y =Qa−¯Qa, as proposed by Islampour and Miralinaghi,53yields Ga↝b IC(t)=(4π)−N√ det(S−1aS−1 bΩaΩb)e−D†ΩbBbDN ∑ κ=1N ∑ ¯κ=1fab(κ)f∗ ab(¯κ)1 2∫{[(ΩaB−1 a)κκ−(ΩaBa)κκ]δκ¯κ+(ΩaBa)κκ(ΩaBa)¯κ¯κXκX¯κ −(ΩaBa)κκ(ΩaB−1 a)¯κ¯κXκYκ′+(ΩaB−1 a)κκ(ΩaBa)¯κ¯κYκX¯κ−(ΩaB−1 a)κκ(ΩaB−1 a)¯κ¯κYκY¯κ} ×e−1 4X†(ΩaBa+J†ΩbBbJ)X−X†J†ΩbBbDe−1 4Y†(ΩaB−1 a+J†ΩbB−1 bJ)YdXNdYN. (14) At this point, the evaluation of the generating function is reduced to N-dimensional Gaussian integrals, which can be determined analytically. Finally, the double sum over κand ¯κis rewritten in terms of vector–matrix–vector products, and Eq. (14) is cast into a form that is closely related to the quadratic HT/HT term of ISC,13 Ga↝b IC(t)=⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪det(S−1aS−1 bΩaΩb) det(ΩaBa+J†ΩbBbJ)det(ΩaB−1a+J†ΩbB−1 bJ){1 2tr[F(Ω−1 aBa+J†Ω−1 bBbJ)−1]−1 2tr[F(Ω−1 aB−1 a+J†Ω−1 bB−1 bJ)−1] +D†ΩbBbJ((ΩaBa+J†ΩbBbJ)−1)†ΩaBaFΩaBa(ΩaBa+J†ΩbBbJ)−1J†ΩbBbD} ×eD†(ΩbBbJ(ΩaBa+J†ΩbBbJ)−1J†ΩbBb−ΩbBb)D, (15) whereFcontains products corresponding to all possible combina- tions of NACMEs fab(κ)andfab(¯κ). B. Numerical determination of nonadiabatic derivative couplings Computing the wave function overlaps in a straightforward manner without further approximations would require massive amounts of computational power. The MO overlap determinant of every Slater determinant of the reference state with every Slater determinant of the distorted state has to be evaluated. It is therefore mandatory to reduce this enormous cost to a reasonable amount, especially as we are aiming at larger molecules than xanthone. This can be achieved by 1. reducing the number of overlap determinants to be calculated and 2. speeding up the computation of individual overlap determinants.A reduction in the number of overlap determinants to be calculated has been achieved by 1. exploiting the spin blocking when computing all Slater deter- minants of a given pair of configurations (see Secs. II B 1 and II B 3), 2. estimating the value of the determinant by Hadamard’s inequality (Sec. II B 6), 3. identifying small determinants by excitation class (Sec. II B 5), and 4. using the fact that the value of the determinant will be multiplied by two CI coefficients, which may be very small (Sec. II B 4). Other authors reduce the number of determinants by expand- ing CIS-type wave functions in terms of natural transition orbitals.54 This strategy is not transferable to our case because the DFT/MRCI method, at least in its current formulation, strictly requires the use of canonical Kohn–Sham MOs as one-particle basis.1,3 J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The calculation of the remaining overlap determinants is accel- erated by 1. applying a caching strategy for spin blocks, as the same spin blocks may occur in multiple pairs of configurations (Sec. II C), 2. using the Schur’s complement method for generating spin factors in open-shell cases (Sec. II B 2), and 3. precomputing some of the entities in Schur’s identity (Sec. II C). 1. Spin blocking of the determinants In the case that the Hamiltonian is spin-independent and the CI expansion is given in a basis of Slater determinants, the overlap matrix can be block diagonalized because their αandβspin parts do not interact.38Let the unbarred entities ϕandϕ′denote spin orbitals occupied by one of the nαelectrons with αspin and the barred entities ¯ϕand ¯ϕ′denote those occupied by one of the nβ=n−nα electrons with βspin. The overlap of two Slater determinants ⟨Φk∣and∣Φ′ l⟩, defined in the basis of the spin orbitals {ϕ,¯ϕ} ⟨Φk∣=∣ϕk(1)...ϕk(nα)¯ϕk(nα+1)...¯ϕk(n)∣ (16) and{ϕ′,¯ϕ′} ∣Φ′ l⟩=∣ϕ′ l(1)...ϕ′ l(nα)¯ϕ′ l(nα+1)...¯ϕ′ l(n)∣, (17) respectively, is then given by38 ⟨Φk∣Φ′ l⟩=⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⟨ϕk(1)∣ϕ′ l(1)⟩...⟨ϕk(1)∣ϕ′ l(nα)⟩ ⋮...⋮ ⟨ϕk(nα)∣ϕ′ l(1)⟩...⟨ϕk(nα)∣ϕ′ l(nα)⟩⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪ ×⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⟨¯ϕk(nα+1)∣¯ϕ′ l(nα+1)⟩...⟨¯ϕk(nα+1)∣¯ϕ′ l(n)⟩ ⋮...⋮ ⟨¯ϕk(n)∣¯ϕ′ l(nα+1)⟩...⟨¯ϕk(n)∣¯ϕ′ l(n)⟩⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪ ≡Sα kl×Sβ kl. (18) As already noted by Plasser et al. ,38these spin determinantal fac- tors are not unique to a specific combination of Slater determinants ⟨Φk∣and∣Φ′ l⟩but can be reused for all pairs of determinants with an equal occupation pattern either in the Sαblock or in the Sβblock. Note further that this factorization scheme does not require the two spin blocks to be of equal dimension, i.e., it is applicable to deter- minants with an arbitrary Msquantum number. For corresponding orbital occupations of spin restricted wave functions, the work load can be additionally reduced by exchanging the factors SαandSβ. For example, in the two-open-shell case with equal numbers of αandβ spin electrons ( MS=0), ⟨ϕi¯ϕj∣ϕ′ i¯ϕ′ j⟩=⟨¯ϕiϕj∣¯ϕ′ iϕ′ j⟩, (19) ⟨ϕi¯ϕj∣¯ϕ′ iϕ′ j⟩=⟨¯ϕiϕj∣ϕ′ i¯ϕ′ j⟩. (20)2. Reuse of the fixed spin determinantal subblocks As shown in Sec. II B 1, the computation of numerical deriva- tive couplings of DFT/MRCI wave functions essentially reduces to the calculation of very many spin-blocked determinants of orbital overlaps. The efficient evaluation of these determinants is therefore crucial for keeping the task manageable. As described in more detail in the following, we employ a combination of LU decomposition and the Schur’s complement method for this purpose. Given an n×n-dimensional matrix A=(aij), one possibility to calculate its determinant ∣A∣is to carry out a Laplace expansion, detA=n ∑ i=1(−1)i+jaijAij=n ∑ j=1(−1)i+jaijAij, (21) where Aijis the determinant of the (n−1)×(n−1)submatrix, obtained by removing row iand column jfrom A. Problematic about the Laplace expansion is its factorial scaling [O(n!)], which makes it impossible to be applied straightforwardly to DFT/MRCI wave functions. A Laplace expansion can be useful, however, in conjunction with other techniques. If, e.g., only a single MO is exchanged in a Slater determinant, precomputed and stored cofac- tors ˜aij=(−1)i+jAijcould be reused to calculate the determinant of the full matrix at the cost of a scalar product of two ndimensional vectors.38Note that an extension to level 2 minors, as introduced by Sapunar et al.54for CIS-type wave functions, requires the precom- putation and storage of n2(n−1)2/4 unique level 2 minors, which can become very demanding. As outlined in more detail below, we employ the Schur’s determinant complement method instead. For computing the initial determinants, an LU decomposition is performed because of its advantageous2 3n3+O(n2)scaling.55The principle idea of an LU decomposition is to factorize a matrix into a lower Land an upper Utriangular matrix, A=LU. (22) Due to the vanishing off-diagonal elements, computing the corre- sponding determinant det Aconsequently reduces to products of main diagonal entries of the matrices LandU, detA=detLdetU=∏ ilii∏ iuii. (23) Although the number of active orbitals is not a limiting factor in DFT/MRCI calculations, in practical applications, many electrons reside in frozen or inactive orbitals that are common to all Slater determinants of the CI expansion. Hence, large parts of the overlap matrices SαandSβare independent of the specific determinant pair under consideration. Let Fbe an f×fsquare matrix corresponding to the fixed part and Vbe the v×vsquare matrix corresponding to variable part of the overlap matrix. Let further PandOdenote off- diagonal blocks with dimension v×fand f×v, respectively. The contribution of the fixed part to the determinant det (S)can then be precomputed and reused in the evaluation of every determinant pair employing the Schur’s determinant identity detS=∣V P O F∣=detFdet(V−PF−1O⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ S/F), (24) where S/Fdenotes the Schur complement. J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In open shell cases, several determinants share the same spa- tial occupation numbers and differ only in the spin parts of the open shells. Here, the matrix Fcan be extended to include all doubly occu- pied orbitals of this configuration. The inversion of the matrix F, an extra step scaling like2 3f3+O(f2), pays off already if the number of unique determinantal pairs to be evaluated exceeds two, i.e., for configurations with more than two open shells. 3. Choice of the magnetic spin quantum number DFT/MRCI variationally determines wave function coefficients for configuration state functions (CSFs) built up from Slater deter- minants with a maximal spin magnetic quantum number, i.e., MS =S. In this way, the number of determinants is kept minimal. For example, the MS=1 CSF of a two-open-shell triplet configuration consists of a single Slater determinant, whereas two Slater determi- nants are required to form an MS=0 triplet wave function. Another argument in favor of an MS=1 triplet wave function is the fact that SOCMEs of two triplet states can be determined in subsequent SPOCK calculations, whereas the corresponding matrix elements of the MS=0 sublevels vanish by symmetry. The eigenvalues of a spin-free Hamiltonian do not depend on the MSquantum number. There- fore, ladder or tensor operators can be applied to the eigenvec- tors of the CI matrix with MS=1 to generate solutions with the MS=0 quantum number. In the example given above, calculation of the nonorthogonal overlap of the configurations in a determinantal basis requires the evaluation of a single pair of MS=1 determinants as opposed to four pairs in an MS=0 expansion. The number of pairs grows rapidly with an increasing number of open shells. For a configuration with four open shells, six determinants with MS=0 are obtained, yielding 36 determinant pairs in contrast to 4 ×4=16 pairs for MS=1 wave functions. A six-open-shell triplet will produce 225 determinant pairs in the MS=1 case as opposed to 400 pairs in theMS=0 case. On the other hand, the spin symmetry blocking of determinant overlaps [Eq. (18)] is optimal only when the number of αandβspin electrons is equal, i.e., in the MS=0 case. As shown in Eqs. (19) and (20), only two of the four pairs in a triplet configuration with two open shells have unique determinantal overlaps. Likewise, the 36 pairs of a four-open-shell wave function can be reduced to 18 unique pairs, and the 400 pairs of the six-open-shell case can be contracted to 200 unique expressions. Right from the beginning, it is therefore not clear which of the following routes is more efficient: 1. Application of ladder operators on the two triplet wave func- tions to down shift their MSvalues from 1 to 0, followed by the computation of overlaps between many pairs of determi- nants with optimal use of symmetry relations between Sαand Sβblocks. 2. Evaluation of the minimal number of determinant overlaps while accepting that for each pair of determinants, the use of spin blocking symmetry relations is suboptimal. Similar considerations hold true for quintet states. Among other parameters, the preference for either route will depend on the weights of configurations with many open shells in the wave func- tion. If their contributions can be neglected, the choice MS=Sis expected to be more favorable. If, on the other hand, the reuse ofprecomputed information is decisive, things could be different. A comparison of the timings for evaluating triplet derivative couplings will be presented in Sec. III B 3. 4. Truncation of the wave function expansion An obvious approach to save computing time is the trunca- tion of the CI expansion.38,39This is accomplished by introducing a threshold t norm for the desired residual norm of the truncated CI vector ∥˜Ψa∥2=⟨˜Ψa∣˜Ψa⟩=kt ∑ k=1d2 ak≥1−tnorm. (25) Herein, ktis the smallest index of the sorted CI vector that fulfills the inequality [Eq. (25)]. Alternatively, the magnitude of the individual CSF coefficients can be used as a selection criterion t cutoff . To con- serve spin multiplicity, all determinants of a given configuration are included irrespective of their individual contributions. The norm of the truncated CI vector typically is smaller than 1, and therefore, the truncation of the CI expansion will, in general, lead to an underesti- mation of the overlap. Assuming that the direction of the derivative coupling is approximately conserved, i.e., ⟨˜Ψb∣˜Ψ′ a⟩ ∥˜Ψb∥∥˜Ψ′a∥≈⟨Ψb∣Ψ′ a⟩ ∥Ψb∥∥Ψ′ a∥, (26) the overlap Sabcan be approximated as Sab≈˜Sab=⟨˜Ψb∣˜Ψ′ a⟩ ∥˜Ψb∥∥˜Ψ′a∥. (27) In Sec. III C, extensive tests and timings regarding the choice of the wave function truncation parameters will be presented. 5. Restriction of the excitation class The kinetic energy operator of the nuclei, ˆTN, acts only on the one-electron terms of the electronic Hamiltonian. Nonvanish- ing linear derivative coupling terms fab(κ)are therefore obtained only if ΨaandΨbin Eq. (7) are singly excited with respect to one another. This property transfers to a certain extent to approximate expressions for fab(κ)as well. For small displacements ϵ, there is usually a good correspon- dence between the MOs at the reference and displaced geometries. This relation implies that ⟨Φi∣Φ′ j⟩≈δij. The analysis of numerous test calculations has shown that every orbital replacement typically reduces the nonorthogonal overlap of the Slater determinants at the reference and displaced geometries approximately by two orders of magnitude, irrespective of their CI coefficients. A double replace- ment of orbitals therefore generates nonorthogonal determinant overlaps of the order of 10−4, etc. Given a close correspondence between the MO sets {Φi}and{Φ′ i}, it might therefore be possible to restrict the mutual excitation between ΨaandΨ′ bto at most one replacement in the orbital occupation vector (1-exc. approximation) without significant loss of precision. Note that this selection crite- rion does not necessarily mean that these configurations are singly excited with respect to the ground-state configuration. It is their mutual excitation level that counts. A similar approach was pursued by Lee et al.56to speed up the evaluation of the determinants. They proposed to introduce an J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp order parameter λrepresenting the relative magnitudes of orbital overlaps and to truncate the direct determinant evaluation by the Leibniz formula according to this order parameter. In combination with determinant factorization, improvements of the performance up to five orders were achieved for spin flip-TDDFT and linear response-TDDFT in that work. 6. Hadamard’s inequality Neville et al.39suggested estimating the size of det Sby Hadamard’s inequality ∥detS∥2≤s ∏ i=1∥xi∥2(28) as a means for reducing the number of SαandSβfactors to be eval- uated. Here, xirepresent the column vectors of the matrix Swith dimension s. They proposed to set the screening threshold δHfor the vector norms in Eq. (28) to values between 10−6and 10−4. In their pyrazine test case, even the relatively loose screening threshold of 10−4led to a negligible degradation of the accuracy while saving about 70% of the computation time. As we will see below, Hadamard screening is less efficient when applied in combination with the 1- exc. approximation (Sec. II B 5) because the latter filters out already many small spin factors. C. Implementation strategies: The program Delta Unlike the procedure in MD simulations, where the computa- tion of wave function overlaps at neighboring points of the trajectory is required at every time step t,38,39,57the number of points for an IC rate constant in the Fermi’s golden rule approximation is limited to the number of symmetry-allowed vibrational coupling modes in the harmonic oscillator model. Under these particular conditions, the computational effort of evaluating nonorthogonal DFT/MRCI wave function overlaps can be significantly reduced with respect to more general cases. In the current version of the DFT/MRCI program, all exci- tations are defined with respect to the so-called anchor determi- nant, which therefore plays a prominent role for understanding the algorithm implemented in D ELTA. For electronic states with an even number of electrons (singlets, triplets, ...), the anchor determinant is the closed-shell determinant used for generating the restricted Kohn–Sham (RKS) DFT orbital basis. In that case, the fixed sub- blocks FofSare identical for αandβelectrons, i.e., Fβ=Fα. There- fore, only the αblocks of the matrices Fand the corresponding inverse matrices (Fα)−1need to be processed explicitly. In princi- ple, the program branch for computing overlaps of determinants with the magnetic spin quantum number MS=S(vide infra ) could be employed equally well for radicals with an odd number of elec- trons (doublets, quartets, ...). In that case, the anchor determinant would be a restricted open-shell Kohn–Sham (ROKS) determinant with one unpaired electron. The workflow of our current implementation is sketched in Fig. 1. To speed up the computation of the nonorthogonal over- laps, the expansion lengths of the wave functions can be reduced in a preparatory step. Application of the selection criterion [Eq. (25)] toΨaandΨ′ bin conjunction with the user-defined cutoff thresh- old t norm yields the truncated expansions ˜Ψaand ˜Ψ′ b, respectively. For triplet states, the spin-symmetry adapted expansions of Slater FIG. 1. Workflow of the algorithm implemented in the D ELTAprogram for computing nonorthogonal overlaps of DFT/MRCI wave functions. determinants with the magnetic spin quantum number MS=1 are transformed to MS=0. For each normal mode, the outermost loop runs over con- figurations m of the wave function ˜Ψaat the reference geometry (unprimed symbols). At this level, the annihilation and creation operators become known, which are needed for generating a con- figuration of the (truncated and renormalized) DFT/MRCI wave function from the RKS anchor determinant at the reference point. Their application exchanges rows in the overlap determinant. Later, application of the annihilators in the displaced configurations will exchange columns and thereby define the size of the Fblock for use in conjunction with the Schur’s complement method [Eq. (24)]. As a large number of displaced configurations will use the same Fblock, it is advantageous to determine all Fblocks already at this point. To this end, for Ff×fblocks of decreasing size ( f=nα−i), the inverses (F(nα−i)×(nα−i))−1, the determinants det F(nα−i)×(nα−i), and the prod- uctsPi×(nα−i)(F(nα−i)×(nα−i))−1are precomputed and stored. Subse- quently, the loop over the configurations nof the displaced wave function ˜Ψ′ bis processed. In case the configurations m and ndiffer by more than a mutual single excitation, the displaced configura- tionnis skipped (1-exc. approximation). In the implementations of numerical derivative couplings by Plasser et al.38and Neville et al. ,39 all spin factors SαandSβare precomputed and cached. This caching substantially accelerates the computations of the wave function over- laps in the nonadiabatic dynamics simulations, on the one hand, but leads to huge memory requirements, on the other hand. We tried to find some middle ground and store only spin factors that correspond to single excitations with respect to the anchor determinant. As the single excitation may take place independently in the Sαand Sβ subblocks, partially even spin factors for higher excited Slater deter- minants can be retrieved from memory. Due to the high memory demand, we refrained from storing the spin factors for all possible combinations of double and higher excitations. These determinants have to be (re)constructed on-the-fly. Before the Schur comple- ment S/F=V−P(F)−1Ois actually formed, Hadamard’s inequality [Eq. (28)] can be applied piecewise to estimate the size of det S, ∥detS∥2≤s ∏ i=1∥xi∥2=s−f ∏ i=1∥xi∥2s ∏ i=s−f+1∥xi∥2. (29) J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Herein, fis the dimension of the Fblock. Once, the blocks PandF are known in the outer reference loop, the product of their row vec- tor norms can be precomputed and stored. When the much smaller Vblock and the Oblock, both carrying information about the excita- tions at the displaced geometry, become available in the inner loop, the precomputed product needs to be extended by just a few terms corresponding to the rank of V. The threshold δHfor the upper bound of Hadamard’s estimate can be set by the user. Values of the order of 10−6were found to be sufficient for determining derivative couplings with a precision of five decimal places. Finally, the product of the resulting spin factors, SαSβ, is weighted by the CSF coefficients of the DFT/MRCI wave functions and by the expansion coefficients of the Slater determinants within the respective CSFs. III. VALIDATION OF THE ALGORITHM AND THE IMPLEMENTATION The present version of the D ELTA program is executed sequen- tially, and all reported cpu times were measured on a single core of a 2.8 GHz AMD EPYC “ROME” 7402 CPU. A. Comparison with analytic MCSCF derivative couplings To validate the implementation of our finite-difference algo- rithm, we computed analytic CASSCF derivative couplings of small model systems by means of the Columbus package.58–60Two dif- ferent ab initio CASSCF setups were prepared for investigating NAMCEs within the triplet manifold of the formaldehyde molecule. To ensure that the technical parameters and wave function char- acteristics match as closely as possible, a double zeta (DZ) basis without dfunctions was chosen to avoid normalization and phase issues when using the Columbus-generated CASSCF MOs in sub- sequent Turbomole one- and two-electron integral routines. Ana- lytic NACMEs were determined at the C2v-symmetric ground-state geometry with all nuclei lying in the XZplane. For technical details, see Sec. S1 A of the supplementary material. The first test case com- prises the coupling between the A1-symmetric T 2(πHOMO→π∗ LUMO) and the A2-symmetric T 1(n→π∗ LUMO)pair of states in a CAS(4,4) setup with MOs optimized for state-averaged electron densities of these two states. The second test case is purely artificial and was chosen because the NACME between the A2-symmetric T 8(πHOMO →σ∗)and the B2-symmetric T 6(πHOMO→σ∗′) states of the CAS(6,6) space has a very large value although the wave functions contain non-negligible contributions of multiply excited determi- nants. In this case, a state-averaging of six triplet densities was performed in the orbital optimization step. Ab initio MRCI calculations were performed such that the inac- tive orbitals were frozen, and a full CI expansion was obtained in the respective reference space while all excitations to external orbitals were prohibited. Energies and wave function coefficients of the cor- responding triplet states from Columbus and MRCI are listed in Tables S2–S6 of the supplementary material. The entries closely resemble one another, but they are not exactly equal because the MRCI program employs two-electron integrals in the resolution-of- the-identity (RI) approximation,1,61,62whereas Columbus uses the full four-index two-electron integrals.The displacement of one H-atom in the Y-direction (perpen- dicular to the molecular plane) is capable of coupling the T 2(πH →π∗ L)and T 1(n→π∗ L)pair of states. A step size of 0.01 a 0was chosen for evaluating the numerical derivative. Its ⟨T2∣⃗∇Y(H1)∣T1⟩ component (0.225 867 a−1 0) agrees with the analytical reference value (0.225 591 a−1 0) within 3 decimal places (see Table S7 of the sup- plementary material for details). In this particular case, the leading terms alone give rise to a NAMCE that overshoots the full value by less than 1%. It should be noted, however, that the CI expansions comprise merely 15 CSFs. The CI expansion lengths of the T 8(πHOMO→σ∗)and T6(πHOMO→σ∗′)states amounts to 189 CSFs without symmetry blocking. In their case, an in-plane distortion of the carbon atom in the Xdirection (perpendicular to the C2axis) was chosen for comparison with the corresponding component of the analytical NACME (3.851 424 a−1 0). The numerical procedure yields a slightly larger value of 3.853 828 a−1 0for the derivative coupling. Because of the higher percentage of multiple excitations in their electronic wave functions, the magnitude of this NACME is more sensitive with respect to the truncation of the CI expansion. Restricting the computation of the non-orthogonal overlap to configurations that are singly excited with respect to the anchor configuration yields a coupling of 4.058 911 a−1 0. Admixture of doubly excited configura- tions (3.842 277 a−1 0) brings the analytical and numerical couplings into agreement within 0.2%. If the modulus of the CI coefficients (∥di∥) is chosen as a truncation criterion, the cutoff has to be set to values lower than 0.01 to reach an accuracy of better than 1%. With regard to timings, truncation of the MRCI wave function leads to substantial speed-ups. As may be anticipated, the cpu time required for calculating the NACME correlates nearly linearly with the number of spin factors because their computation is the time determining step. B. The pyrazine test cases Pyrazine [Fig. 2(a)] is a D2h-symmetric molecule in the elec- tronic ground state. Due to its moderate size and strong vibronic coupling effects in the singlet excited state manifold, it has ever since been a guinea pig of the theoretical chemistry commu- nity for testing the validity of their vibronic coupling methods and algorithms.39,63–69The much lower vibronic activity in the T 1 ←S0absorption spectrum has been explained by the fact that this spin-forbidden transition borrows its intensity from optically bright singlet transitions by a direct spin–orbit mechanism without the necessity to invoke spin–vibronic coupling.69,70 Due to its high molecular symmetry, pyrazine is an ideal test case for verifying the proper implementation of our numerical FIG. 2. The pyrazine (a) and xanthone (b) molecules. J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp algorithm for computing DFT/MRCI NACMEs as well. The preci- sion of numerical zeros for symmetry-forbidden couplings and of the NACMEs for symmetry-allowed couplings offers the possibil- ity to check the dependence of the derivative couplings on various wave function parameters, such as the convergence thresholds of the underlying KS calculation and of the iterative Davidson diag- onalization in the DFT/MRCI step. These parameters set aside, the computational protocol for generating molecular orbitals and DFT/MRCI wave functions of pyrazine proceeds along the lines of previous work performed in our laboratory, which focused on vibronic traces in the S 1←S0and T 1←S0absorption spectra caused by HT-type couplings.69Technical details may be found in Sec. S1 of the supplementary material. The wave function of the first excited singlet state of pyrazine, S1(nπ∗), transforms according to the B3uirreducible representation (irrep) at the ground-state geometry. Energetically, the S 1(nπ∗)state is well separated from the B2usymmetric S 2(ππ∗)state. According to the symmetry selection rules, the derivatives along all normal coor- dinates must vanish, save for b1gmodes. There is only one mode of the appropriate symmetry type, namely, mode 7 ( ˜ν=941 cm−1), which corresponds to ν10ain the common nomenclature introduced by Innes et al.63The situation is slightly more complicated in the triplet manifold. The spacing between the T 13(nπ∗)excitation ( B3u symmetry) and the T 2(ππ∗)state is much smaller than for their singlet counterparts. Moreover, the T 2and T 3states swap order upon geometry relaxation.69In the FC region, which is relevant in the present case study, the B2usymmetric T 2(ππ∗)is energetically favored, and its nonadiabatic coupling with T 1will be investigated here as the second test example. In the following, unprimed state designations refer to DFT/MRCI wave functions at the S 0reference geometry ⃗Rof pyrazine, whereas primed entities symbolize wave functions calculated at a displaced geometry ⃗R′. 1. Dependence on orbital and wave function convergence parameters When the usual SCF convergence thresholds (10−8Ehfor the SCF energy and 10−7for the SCF density matrix), recommended for subsequent DFT/MRCI computations, are employed in the dscf iter- ative eigenvalue solver of the Turbomole program,71a value for the only symmetry-allowed ⟨T2∣∂ ∂Q10a∣T1⟩∣ S0,minNACME is obtained, which is∼2% lower than the fully converged value (Table S9 of the supplementary material). This means that the standard SCF con- vergence thresholds are too sloppy for computing this DFT/MRCI NACME with a higher precision than the first decimal place. Reduc- ing the convergence thresholds by a factor of 10 yields a NACME for the coupling of the T 1and T 2states by ∂/∂Q10athat agrees with the value obtained for an even tighter convergence threshold in the third decimal place. We therefore recommend to set the convergence thresholds in the KS orbital optimization step to at least 10−10Eh for the SCF energy and 10−9for the SCF density matrix in con- junction with the numerical determination of DFT/MRCI deriva- tive couplings. In contrast, minuscule variations of the NACME are encountered when the default convergence threshold (0.5 ×10−5Eh) of the iterative Davidson solver of the DFT/MRCI secular equation is tightened (Table S9 of the supplementary material). However, the quality of the numerical zeros for the symmetry-forbidden deriva- tive couplings improves in this case. While a value of ≈1.3×10−5is obtained for the ⟨T2(⃗R)∣T′ 1(⃗R′)⟩overlap for a displacement in the direction of mode 4 when the standard convergence threshold of 0.5×10−5Ehis used in the Davidson procedure, the numerically determined value of this symmetry-forbidden coupling drops to ≈1.9×10−7when a threshold of 0.5 ×10−7Ehis employed instead. For all other 22 symmetry-forbidden coupling modes, the precision of the numerical zero is higher than for mode 4 (Table S10 of the supplementary material). Summarizing, we find the impact of the orbital convergence parameters on the precision of the computed derivative couplings to be more pronounced than the influence of the Davidson convergence parameter. Nevertheless, we recommend tightening the SCF and CI convergence thresholds by a factor of 0.01 in relation to the default values. 2. Step size of the normal mode displacement The second set of test calculations on pyrazine investigated the range of deflections for which the magnitude of the nonorthogo- nal overlaps of the3B2uT2DFT/MRCI wave function of pyrazine at the S 0reference geometry ⃗Rand the3B3uT′ 1wave function at a dis- placed geometry ⃗R′=⃗R+ϵ⃗eQ10ascales approximately linearly with the step size ϵ. TheνQ10amode is the only out-of-plane vibration that has the proper b1gsymmetry for coupling these states. The results as well as the overlaps of the corresponding singlet wave functions ⟨S2(⃗R)∣S1(⃗R′)⟩are depicted graphically in Fig. 3. The confidence interval for which the nonorthogonal overlap may be considered to increase linearly with increasing displacement is approximately ϵ=±0.01, in agreement with observations for other molecules. Unless noted otherwise, all finite difference calculations [Eq. (1)] have been carried out using this ϵvalue. 3. Performance of the Delta program for different triplet sublevels Test calculations for comparing the outcomes and timings ofMS=0 and MS=1 triplet wave functions were performed for the ⟨T2(⃗R)∣T1(⃗R′)⟩overlap of the pyrazine molecule where FIG. 3. Nonorthogonal overlaps of the1B3uS1′(⃗R′)and the1B2uS2(⃗R)pair of states (black) as well as of the3B3uT1′(⃗R′)and3B2uT2(⃗R)(red) pair of states in pyrazine determined for varying displacements along the mass-weighted nor- mal coordinate of the ν10a(b1g)vibrational mode. ⃗Rdenotes the S 0reference geometry, ⃗R′=⃗R+ϵ⃗eQ10adenotes the displaced geometry. J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ⃗R′=⃗R+0.1⃗eQ10aand the unit vector ⃗eQ10apoints along the normal coordinate of the only symmetry-allowed vibrational coupling mode ν10a. At the ground state geometry ( ⃗R), the T 1and T 2DFT/MRCI wave functions are composed of 6663 CSFs corresponding 3065 con- figurations. Further computational details may be found in Sec. S1 B of the supplementary material. Two conclusions may be drawn from the data entries in Table S11. First, corresponding matrix elements employing either MS=0 orMS=1 are identical to at least 14 decimal places. As different parts of the program are addressed in either case, we consider this agreement a confirmation that the down shift of the MSquantum number and the computation of the spin factors Skl[Eq. (18)] are coded properly. Second, convergence of the ⟨T2(⃗R)∣T1(⃗R′)⟩matrix element up to 10−4is reached for a cutoff parameter t cutoff=10−3. This parameter steers the length of the DFT/MRCI wave function and is chosen such that configurations are discarded if none of its CSF coefficients difulfills the condition ∣di∣≥tcutoff . To converge the coupling matrix element to ±10−5a−1 0requires the wave function cutoff parameter to be reduced to t cutoff=10−6at least. Table S12 displays the number of spin factors Skl[Eq. (18)] and the timings for the two approaches and a series of cutoff parameters t cutoff . Because of the smaller number of Slater deter- minants, the total number of spin factors Sklneeded for com- puting⟨T2(⃗R)∣T1(⃗R′)⟩is, of course, smaller in the MS=1 case in comparison with MS=0. Counterintuitively, however, the tim- ings tell a different story. Because the percentage of cached spin factors that can be retrieved from memory and reused instead of being computed on-the-fly is much larger if the number of α andβelectrons is identical, the MS=0 setup has a clear advan- tage, in particular, when the number of open-shell configurations increases. C. Truncation of the wave function expansion: Precision and timings Various actions can be taken to speed up the computation of the numerical derivatives. Aside from making use of symmetry selec- tion rules, the most obvious one is the truncation of the MRCI expansion because the timings increase quadratically with the num- ber of Slater determinants to be processed. As described in detail in Sec. II B 4, we defined a cutoff parameter for the residual norm of the truncated wave function [Eq. (25)]. Alternatively, the magni- tude of individual CSF coefficients was used as a criterion. More- over, we have investigated the performance of the numerical proce- dure with regard to precision and timings for cases in which only the overlaps of singly excited pairs of configurations are evaluated (1-exc. approximation, Sec. II B 5), either as a stand-alone crite- rion or in combination with truncated wave function expansions. Finally, Hadamard’s inequality [Eq. (28)] has been used in addition to the 1-exc. approximation to speed up the numerical evaluation of NACMEs. Many of the tests have been performed for the spin-allowed derivative couplings between the lowest1,3(nOπ∗ L)and1,3(πHπ∗ L) states of xanthone in vacuum. The physical relevance of these cou- plings will be discussed at length in Sec. IV. Here, we focus on the technical aspects of the NACME evaluation. Using C2vpoint-group symmetry, the (πHπ∗ L)states transform according to the totally sym- metric irreducible representation A1, whereas the (nOπ∗ L)areA2symmetric. For this reason, only the 9 a2-symmetric normal coor- dinates exhibit the appropriate point-group symmetry to produce nonvanishing NACMEs. The 1-exc. approximation results in a substantial speed-up of the calculation without essential loss of precision. As shown in Table S13 of the supplementary material, the NACMEs of the full DFT/MRCI expansion and the 1-exc. approximation agree to within 3–4 decimal places while cpu times are reduced on the average from about 24 h to 7 min per mode in the triplet case and from about 48 h to 11 min per mode in the singlet case. A further reduction in computation time can be achieved if the wave function expan- sions are truncated. Setting the cutoff parameter t norm to 10−11yields essentially the same IC rate constant as the full expansion in 1-exc. approximation but diminishes the cpu time by another factor of 6 (Tables S14 and S15 of the supplementary material). The entries of these tables and the graphs in Figs. S1 and S3 of the supplementary material suggest that setting t norm to values>10−7is not meaningful due to loss of precision. Application of the Hadamard screening as an additional measure for reducing the computation time appears to be less effective in comparison to the truncation of the wave function expansion, at least in our implementation. A Hadamard screening parameter of 10−5for the overlap of the spin factors leads to a saving of merely 20% in comparison to employing all overlaps in the 1-exc. approximation (Tables S16 and S17 of the supplementary material). Summarizing the trade-off between precision and timing, we recommend to restrict the mutual excitation level of the configura- tions at the reference and displaced geometries to 1 (1-exc. approx- imation) in combination with a cutoff parameter of t norm=10−8for the DFT/MRCI wave function expansion. IV. APPLICATION TO EXCITED-STATE PROCESSES IN XANTHONE Xanthone [Fig. 2(b)] is a good triplet sensitizer in apo- lar and mildly polar environments, whereas it is fluorescent in water.40,41,72,73In apolar media such as hexane, the lowest triplet state exhibits3(nOπ∗ L)character, whereas this changes to3(πHπ∗ L) character in polar environments such as acetonitrile. In the singlet manifold, this transition occurs at much higher dielectric permit- tivities.12,74,75While the S 1state retains its1(nOπ∗ L)character in ace- tonitrile and methanol, a strongly polar protic solvent, such as water, reverses the energetic order of the1(nOπ∗ L)and1(πHπ∗ L)electronic structures. In the following, we will investigate the probabilities of IC vs ISC decay in two different environments: gas phase and aqueous solution. A. Gas phase and in apolar media In hexane, a biphasic rise of triplet–triplet absorption was observed experimentally after photoexcitation of the S 21(πHπ∗ L) state. Cavaleri et al.72assigned the fast component with a time con- stant of 0.70 ps to the El-Sayed allowed S 21(πHπ∗ L)↝T13(nOπ∗ L) ISC and the slower process with a time constant of 10 ps to the alternative decay cascade S 11(nOπ∗ L)↝T23(πHπ∗ L)ISC, including the subsequent T 23(πHπ∗ L)↝T13(nOπ∗ L)IC. For the S 21(πHπ∗ L) ↝S11(nOπ∗ L)IC, a time constant of 2.3 ps was determined in that solvent. These findings and their interpretation raise the question why the spin-forbidden S 2↝T1channel and not the spin-allowed J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp S2↝S1dominates the decay of the S 2population of xanthone in non-polar solvents. DFT/MRCI calculations in our laboratory con- firmed the assumed energetic scheme but found the vibronically induced S 21(πHπ∗ L)↝T23(πHπ∗ L)ISC to be about two orders of magnitude faster than the direct S 21(πHπ∗ L)↝T13(nOπ∗ L)ISC in Condon approximation.12It should be noted, however, that the geometry optimization of the S 21(πHπ∗ L)state using TDDFT leads to a very shallow C2-symmetric double minimum potential con- nected by a planar C2v-symmetric saddle point. At the DFT/MRCI level of theory, the C2v-symmetric structure is lower in energy and represents the minimum along this reaction coordinate. The artifi- cial occurrence of symmetry breaking deformations at the TDDFT level has been observed in various aromatic ketones46,76and flavin derivatives.77,78In these cases, we typically use the vibrational coor- dinates of the saddle point and replace the imaginary frequency by a reasonable real value corresponding to the curvature of the outer branches of the TDDFT profile. Peculiar in the case of the xan- thone S 21(πHπ∗ L)is that the T 23(πHπ∗ L)and T 13(nOπ∗ L)states are energetically almost degenerate ( ΔE≈0.01 eV) at the C2-symmetric S21(πHπ∗ L)structure and exhibit strongly mixed wave functions. Even small coordinate displacements can and do lead to a reversal of the order of triplet states. The determination of HT-SOC terms by a finite difference procedure at this point in coordinate space is therefore extremely error-prone and needs close monitoring of the dominant wave function characteristics. In our earlier work,12 the rate constant of the El-Sayed allowed S 21(πHπ∗ L)↝T13(nOπ∗ L) transition was determined in the Condon approximation at the C 2- symmetric minimum geometry of the S 2state, which appears inap- propriate in a retrospective analysis. Moreover, due to the lack of nonadiabatic derivative couplings, IC rate constants could not be determined at the DFT/MRCI level at that time, and therefore, the kinetic model remained fragmentary. Here, we include vibronic terms and employ the C2v- symmetric structure of the S 21(πHπ∗ L)state as the reference point for our (re-)investigation of the S 2ISC and IC processes while care- fully testing the sensitivity of the computed rate constants withrespect to the vibrational frequency of the A2-symmetric defor- mation mode. The adiabatic energy difference between the C2v- symmetric S 21(πHπ∗ L)structure and the S 11(nOπ∗ L)minimum amounts to 0.71 eV at the DFT/MRCI level of theory. Zero-point vibrational energy (ZPVE) corrections (Table S18 of the supple- mentary material) are automatically accounted for in the V IBESpro- gram. Varying the vibrational frequency of the A2-symmetric defor- mation mode between 50 and 500 cm−1yields rate constants of kIC S2S1=3−4×1011s−1for the S 21(πHπ∗ L)↝S11(nOπ∗ L)IC in vac- uum (Table S19 of the supplementary material). These values agree very well with the time constant of 2.3 ps observed by Cavaleri et al.72 in hexane solution. (Choosing the C 2-symmetric structure as refer- ence point instead gives a rate constant of kIC S2S1=4.3×1011s−1for this process, which fortuitously exactly matches the experimental time constant.) The comparatively slow IC in the singlet manifold is considered a consequence of the substantial energy separation between the adiabatic S 1and S 2states and a lack of an energetically accessible conical intersection between them. Both low-lying triplet states, T 23(πHπ∗ L)and T 13(nOπ∗ L), have planar minimum geometries at the DFT/MRCI level of theory. In contrast, TDDFT finds one imaginary frequency at the C2v- symmetric T 1structure, which is why we followed Rai-Constapel et al.12in employing the nuclear arrangement and vibrational modes of the S 11(nOπ∗ L)state as a proxy for the corresponding triplet state properties. Because of the much smaller energy gap between the two triplet states (0.14 eV), the rate constant for the T 23(πHπ∗ L) ↝T13(nOπ∗ L)IC is significantly larger ( kIC T2T1>×1013s−1) than for the corresponding singlet transition. We may therefore conclude that the T 23(πHπ∗ L)↝T13(nOπ∗ L)IC is not the rate determin- ing step for the population of the T 13(nOπ∗ L)state, monitored experimentally through time-resolved triplet excited-state absorp- tion spectroscopy. Combining the knowledge on IC and ISC rate constants acquired in our present study with the experimental infor- mation provided by Cavaleri et al. ,72the following schematic pic- ture emerges for the decay kinetics following photoexcitation of xanthone to the S 21(πHπ∗ L)state in apolar media [Fig. 4(a)]. FIG. 4. Theoretically determined energy level scheme and rate constants (300 K) for IC and ISC transitions following the photoexcitation of xanthone in vacuum (a) and aqueous solution (b). In vacuum, ZPVE-corrected adiabatic transition energies amount to 0.63 eV for IC in the singlet domain, 0.09 eV for IC in the triplet domain, and 0.01 eV for the S 1↝T2ISC. In aqueous solution, the (nOπ∗ L)states are blue shifted relative to the (πHπ∗ L)states. To better match with experimental data, their solvent shifts were reduced by 0.11 eV in comparison to the hybrid model employed by Rai-Constapel et al.12In this way, the vibrational ground states of the T 2and S 1potentials become isoenergetic in aqueous solution while energy separations of 0.07 eV between S 2and S 1and of 0.61 eV between T 2and T 1are obtained. J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-11 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp S2can be considered the starting point for two spin- forbidden and one spin-allowed nonradiative transitions. The direct S21(πHπ∗ L)↝T13(nOπ∗ L)ISC is an El-Sayed-allowed process, but the adiabatic energy separation between these states is substantial. The two-step process involving the T 23(πHπ∗ L)state as an interme- diate is El-Sayed forbidden and appears unattractive at first sight. Nevertheless, both ISC channels are investigated here because a con- ical intersection of the T 2and T 1potentials lies close to the S 2min- imum. While intuitively the one-step process is expected to prevail, the calculations actually find the cascade S 21(πHπ∗ L)↝T21(πHπ∗ L) ISC followed by T 23(πHπ∗ L)↝T13(nOπ∗ L)IC to be much more effective than the direct S 21(πHπ∗ L)↝T13(nOπ∗ L)process when spin–vibronic interactions are included. A further productive channel for populating the T 1state on the picosecond time scale is a cascade starting with the spin- allowed S 21(πHπ∗ L)↝S11(nOπ∗ L)IC. Due to the energetic prox- imity of the S 1and T 2states and their substantial mutual SOC (⟨T23(πHπ∗ L)∣ˆHSO∣S11(nOπ∗ L)⟩=−50.75 cm−1at the S 1minimum), the S 11(nOπ∗ L)↝T21(πHπ∗ L)ISC is ultrafast. This ISC step is fol- lowed by an even faster T 23(πHπ∗ L)↝T13(nOπ∗ L)IC. Thus, the rate determining step in this cascade is the S 2↝S1IC. In summary, the photoexcitation decay of xanthone in apo- lar media is best described by a unified kinetics model comprising branched and consecutive pathways involving the S 11(nOπ∗ L)and T21(πHπ∗ L)states as intermediates [Fig. 4(a)]. Note further that the equilibration between S 1, T 2, and T 1by ISC and thermally acti- vated rISC processes, on the one hand, and by IC and the rIC up- conversion from T 2 T1, on the other hand, are much faster than the prompt S 11(nOπ∗ L)→S0fluorescence in apolar media, which is therefore quenched. B. Water solution In addition to prompt fluorescence, xanthone emits delayed fluorescence in aqueous solution. Heinz et al. concluded that the T23(nOπ∗ L)state rapidly equilibrates with the near degenerate bright S 11(πHπ∗ L)state and that the backtransfer of population via the El-Sayed-allowed T 2↝S1ISC is the origin of this delayed fluo- rescence.41This interpretation implies that the spin-allowed T 2↝T1 IC cannot compete against the spin-forbidden T 2↝S1nonradiative transition in aqueous solution. Because T 13(πHπ∗ L)and S 1have similar electronic structures and are separated by about 0.60 eV, thermally activated T 1↝S1rISC was considered highly improbable. The quantum chemical investigation by Rai-Constapel et al.12cor- roborated the near degeneracy of the S 1and T 2states in water and reported a S 1↝T2rate constant of 2.5 ×108s−1in Condon approx- imation, but the rate constant neither for the T 2↝S1rISC process nor for the T 2↝T1IC was computed. Here, we use a similar com- putational setup as described before and set out for determining the missing rate constants. A complication arises due to the flexibility of the solvent envi- ronment. In polar solvents, nπ∗excited states typically experience strong blue shifts, while ππ∗excited states are moderately red shifted.77The electrostatic solvent–solute interaction can be well described by continuum models, such as COSMO. In contrast, the shifts brought about by hydrogen bond formation in protic sol- vents require a hybrid model including explicit solvent molecules and an electrostatic surrounding. To mimic an aqueous medium,Rai-Constapel et al.12used a hybrid model consisting of a hydrogen- bonded complex of xanthone and two water molecules surrounded by a COSMO electrostatic field. Reorientation of the loosely bonded water molecules upon geometry relaxation in the excited states leads to large amplitude displacements of the corresponding normal coor- dinates, thus preventing a meaningful Duschinsky transformation (11) of the normal coordinates. To account for solvent–solute inter- actions, we use here an ansatz that closely resembles the energy level shift procedures in two-step spin–orbit coupling calculations.79,80 The impact of the aqueous solution is taken care of implicitly by introducing state-specific energy shifts in the Davidson diagonaliza- tion procedure of the DFT/MRCI program. To this end, the diagonal elements of the Hamiltonian matrix Hsolv ij=Hvac ij+nroots ∑ K=1ΔEsolv K∣Ψvac K⟩⟨Ψvac K∣ (30) are modified. In accord with the results of previous work,12the energy shifts ΔEsolv K tabulated in Table S21 of the supplemen- tary material were employed for the states under consideration. Test calculations reveal that truncated DFT/MRCI vectors includ- ing only terms with ∥di∥>10−5can be used to construct the projection operators without loss of accuracy but speeding up the Davidson diagonalization of the modified Hamiltonian matrix significantly. Including spin–vibronic interactions in the computation of rate constants for the El-Sayed allowed processes1(πHπ∗ L)↭3(nOπ∗ L) do not change lot. While the reverse ISC is predicted to proceed at a rate constant of about 2.3 ×1010s−1, the computed rate con- stant of 1.8 ×108s−1for the forward ISC is too small to explain the fast S 1↭T2equilibration observed experimentally by Heinz et al.41For an ISC process that is dominated by direct SOC, the electronic SOCME and the Franck–Condon weighted density of vibrational states at the energy of the initial state determine the tran- sition probability. Because of the close energetic proximity of the 1(πHπ∗ L)and3(nOπ∗ L)states in water, already small changes of the relative energy separation of their vibrational ground states ( ΔE0) can have large impact on the magnitude of the vibrational overlaps and hence on the transition rates. Considering that our rather crude solvent model might overestimate the blue shifts of the (nOπ∗ L)state energies, we therefore varied this parameter. The entries of Table S22 in the supplementary material reveal that a reduction in their solvent shift by 0.11 eV, which makes the S 1and T 2states nearly degenerate, increases the rate constant of the thermally activated S1↝T2transition by a factor of about 80 while changing the rate constant of the reverse S 1 T2process only slightly. This solvent model appears to give a more balanced description of the forward and backward S 1↭T2ISC kinetics of xanthone in aqueous solu- tion. For this reason, we applied the same shift to the1(nOπ∗ L)state potential and investigated the sensitivity of the calculated S 2↝S1and T2↝T1IC rate constants with respect to the applied shift as well. An overview of the kinetic constants obtained for this model is shown in Fig. 4(b). At first sight, it is surprising that the S 2↝S1IC rate constant is reduced from about 8 ×1012s−1to about 7×1011s−1when their ΔE0separation changes from 0.18 to 0.07 eV (Table S26 of the supplementary material). To understand the variation of the IC rate constants with the applied shift, we have to be aware that two J. Chem. Phys. 155, 014102 (2021); doi: 10.1063/5.0056182 155, 014102-12 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp factors influence the magnitude of the nonradiative transition rate constants if vibronic coupling is accounted for. In addition to the FC overlaps, the NACME itself is sensitive with respect to the energy separation between the coupling states. In principle, both factors can increase or decrease with diminishing ΔE0. In the weak cou- pling limit, characterized by nested potentials, the FC overlaps drop exponentially with growing adiabatic energy gap, whereas an inverse energy gap relation is expected for strong coupling cases character- ized by large displacements of the minimum geometry parameters.81 In contrast, the magnitude of the NACME is primarily related to the vertical energy difference of the interacting states ΔEvertical , which is easily seen when inspecting the Hellmann–Feynman-type expres- sion31fab=⟨Ψb∣∇H∣Ψa⟩(Ea−Eb)−1. While the1(πHπ∗ L)state is adiabatically the lowest excited singlet state of xanthone in water, it lies vertically above the1(nOπ∗ L)state at the1(nOπ∗ L)mini- mum geometry. Counterintuitively, therefore, a global red shift of the1(nOπ∗ L)potential by small amounts leads to an increase in the vertical energy separation ΔEvertical between these states and a strong reduction of the NACME (Fig. S5 of the supplementary material), leading, in turn, to a decrease in the IC rate constant (Table S26 of the supplementary material) by a factor of about 10. The contrary is true for the reverse IC process because the 1(πHπ∗ L)lies below the1(nOπ∗ L)state at the1(πHπ∗ L)minimum. An additional red shift of the1(nOπ∗ L)state therefore brings the two states into closer proximity and increases the rIC rate con- stant by two orders of magnitude. The IC in the triplet manifold is less affected by small variations of the relative3(πHπ∗ L)–3(nOπ∗ L) energy separation because these states lie further apart in aqueous solution. V. CONCLUSIONS AND OUTLOOK In this paper, we have presented a new program for comput- ing numerical NACMEs for DFT/MRCI wave functions of triplet and singlet multiplicity. The implementation makes extensive use of the Schur’s determinant identity to improve the scaling of the computation of spin factors. To keep the task manageable with respect to memory demand, only the spin factors of singly excited determinants are cached, while those of double and higher excita- tions with respect to the ground-state determinant are generated on-the-fly. This flexibility and the application of suitable selec- tion schemes allow the computation of NACMEs even for large molecules. Although the current implementation is restricted to sin- glet and triplet states, the foundation has been laid to extend our approach to states of arbitrary multiplicity. The recommended value for the step size in numerical calcu- lations of derivative couplings is ϵ=±0.01 in units of dimension- less mass weighted normal coordinates. The derivative couplings are found to be quite sensitive to the convergence of the SCF proce- dure. To obtain numerically stable results, the convergence thresh- olds should be set to at least 10−10Ehfor the KS energy and to 10−9 for the SCF density. Restriction of the mutual excitation class of configurations at the reference and displaced geometries to single replacements (1-exc. approximation) in combination with a trun- cation of the wave function expansion (t norm=10−8) substantially accelerates the numerical evaluation of nonorthogonal DFT/MRCI wave function overlaps without significant loss of precision. On-topapplication of Hadamard’s inequality for estimating the contribu- tion of spin factors does not lead to marked additional savings of computation time. Formulas for computing IC rate constants in a generating function approach including Dushinsky transformations have been worked out and implemented into the V IBESprogram. This new program version has been used in conjunction with DFT/MRCI NACMEs to provide a better understanding of the complex pho- toexcitation decay kinetics of the xanthone chromophore in vacuum as well as aqueous solution. SUPPLEMENTARY MATERIAL See the supplementary material for technical parameters of the calculations and data supporting the analyses and conclusions presented in this work. ACKNOWLEDGMENTS This research was funded by the Deutsche Forschungsgemein- schaft (DFG, German Research Foundation)—No. 396890929/GRK 2482. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Grimme and M. Waletzke, J. Chem. Phys. 111, 5645 (1999). 2M. Kleinschmidt, C. M. Marian, M. Waletzke, and S. Grimme, J. Chem. 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5.0052312.pdf

The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Multireference calculations on the ground and lowest excited states and dissociation energy of LuF Cite as: J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 Submitted: 30 March 2021 •Accepted: 3 June 2021 • Published Online: 23 June 2021 Nuno M. S. Almeida, Timothé R. L. Melin, and Angela K. Wilsona) AFFILIATIONS Department of Chemistry, Michigan State University, East Lansing, Michigan 48864, USA Note: This paper is part of the JCP Special Collection in Honor of Women in Chemical Physics and Physical Chemistry. a)Author to whom correspondence should be addressed: akwilson@msu.edu ABSTRACT High level multireference calculations were performed for LuF for a total of 132 states, including four dissociation channels Lu(2D)+F(2P), Lu(2P)+F(2P), and two Lu(4F)+F(2P). The 6 s, 5d, and 6 porbitals of lutetium, along with the valence 2 pand 3 porbitals of fluorine, were included in the active space, allowing for the accurate description of static and dynamic correlation. The Lu(4F)+F(2P) channel has inter- system spin crossings with the Lu(2P)+F(2P) and Lu(2D)+F(2P) channels, which are discussed herein. To obtain spectroscopic constants, bond lengths, and excited states, multi-reference configuration interaction (MRCI) was used at a quadruple- ζbasis set level, correlating also the 4 felectrons and corresponding orbitals. Core spin–orbit (C-MRCI) calculations were performed, revealing that 13Π0−is the first excited state closely followed by 13Π0+. In addition, the dissociation energy of LuF was determined at different levels of theory, with a range of basis sets. A balance between core correlation and a relativistic treatment of electrons is fundamental to obtain an accurate description of the dissociation energy. The best prediction was obtained with a combination of coupled-cluster single, double, and perturbative triple excitations /Douglas–Kroll–Hess third order Hamiltonian methods at a complete basis set level with a zero-point energy correction, which yields a dissociation value of 170.4 kcal mol−1. Dissociation energies using density functional theory were calculated using a range of functionals and basis sets; M06-L and B3LYP provided the closest predictions to the best ab initio calculations. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0052312 I. INTRODUCTION The accurate description of ground and excited state prop- erties of lanthanides provides a route toward understanding their fundamental chemical reactivity. The high density of states and partially filled 4 fand 5 dorbitals are hurdles that need to be properly addressed in order to achieve such predictions. The use of multireference methods in lanthanide electronic structure calculations is of paramount importance and allows for an accu- rate description of static and dynamic correlation. Additionally, an appropriate choice of methods to account for correlation and spin–orbit effects is necessary for both the ground and excited states. Lutetium, the last element in the lanthanide series, is also gener- ally regarded as the first element of the sixth period transition metals due to its full 4 fand partially filled 5 dorbitals. Recently, interest inlutetium has grown, with one of its main applications in the radio- pharmaceutical industry, more specifically, with the use of177Lu as a radionuclide.1Small molecules, such as peptides and steroids, have been radiolabeled with177Lu in the treatment of a number of diseases. For example,177Lu-labeled DOTA-Tyr3-octreotate, which is a somatostatin analog peptide, is currently being used to treat neuroendocrine tumors.1Lutetium also has been linked to astro- physics. It has been discovered in the composition of the metal-poor stars CS 31062-050 and CS 22892-052 and in the enriched star BD +17 3248.2–5The Lu+spectra has been investigated by Hartog and co-workers revealing the presence of an excited state at 28 503.16 cm−1, which corresponds to a 6 s6p,3P1configuration.6 Lanthanide species, in general, are also being used and considered in a broad range of applications, such as in electrodes and opti- cal telecommunications (i.e., NaLuF 4). With such a wide range of applications, it is important to better understand lutetium at a J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp fundamental level and the methodologies needed to describe its complex electron manifold. In considering the ground and excited state properties of LuF, the available experimental data are from the 1960s, 1970s, and 1980s. In 1968, Zmbov extrapolated the dissociation energy of LuF from other lanthanide monofluorides by means of mass spectroscopy and obtained 136 ±12 kcal mol−1.7The authors estimated the dissocia- tion energy of lutetium fluoride using both the heats of sublimation and the enthalpies of other lanthanide fluorides. Their estimation came from fluorine-exchange reactions of Sm, Eu, Gd, and Dy, and Er. Kaledin et al. predicted the dissociation energy of LuF to be 124 kcal mol−1. The authors used ligand field theory and extrapo- lated the dissociation energy, utilizing a fitting model and experi- mentally determined ionization potentials for other lanthanide fluo- rides.8Since the 1970s, several experimental studies have targeted the vibrational and rotational spectrum of lutetium fluoride.9–12 D’Incan et al. and Effantin et al. reported dissociation energies for LuF (105 kcal mol−1) and assigned the lowest lying electronic excited states for LuF. The symmetry and spin were labeled either1Σor 1Πfor all the excited states.9,10,12These results were later compiled by Huber and Herberg in an extensive review of molecular spec- tra.11In the 1980s, Rajamanickam and Narasimhamurthy and Reddy et al. obtained experimental dissociation energies of 96.0 ±2.4 and 79 kcal mol−1, respectively.13,14These authors used the experimental spectroscopic constants of the ground state ( ωe,ωeχe, etc, obtained from the work of Effantin et al.12), calculated the vibrational poten- tial energy curve (PEC), fitted it with different empirical formulas, and calculated the dissociation energy. Theoretical studies are useful in describing the spectroscopic properties of lanthanides. There are a number of recent studies on lanthanide monohalides (LnX, X =F, Cl, Br, I).15–22In the 1990s, a number of theoretical studies focused on the spectroscopic prop- erties of lanthanides and actinides. Wang et al. and Küchle et al. studied diatomics, lanthanide, and lanthanide and actinide contrac- tions and were the first to use density functional theory (DFT) along with coupled cluster (CC) methods to calculate ground state prop- erties and bond lengths for some of these molecules.23,24Cooke et al. investigated the rotation spectra of LuF and used DFT to compare with their ground state experimental values. Their theo- retical prediction of the dissociation energy of 96.6 kcal mol−1was based on a statistical average of orbital potentials.25Density func- tional theory with scalar-relativistic ZORA and Douglas–Kroll–Hess approaches have been used by Hong et al. to calculate the dissoci- ation energy of LuF. The authors obtained values in the range of 167–176 kcal mol−1.26In 2016, Grimmel et al. determined for the Ln54 set, a set of 54 enthalpies of formation and bond dissociation energies of small lanthanides, using 22 different DFT functionals and employing the Douglas–Kroll–Hess Hamiltonian in combination with a triple- ζlevel basis set [Sapporo-Douglas–Kroll–Hess third order Hamiltonian (DKH3)-TZP-2012 for Ln and cc-pVTZ-DK or cc-pV(T +d)Z-DK for the ligands], resulting on average, overall energy errors for the set on the order of 1 eV, even with the most popular and well-utilized functionals for the lanthanides.27 Aebersold et al. reexamined the energies of the Ln54 set using the same functionals employed by Grimmel and co-workers, con- sidering the several impacts including the introduction of effec- tive core potential (ECP) and DKH3 approaches.27,28In terms of ab initio studies, the equation of motion completely renormalizedcoupled-cluster single, double and perturbative triple excitations [CCSD(T)] [EOM-CR-CCSD(T)] was used in a study of NdF and LuF.29The authors reported that the use of a full valence shell rather than the traditional frozen core approximation can result in a dramatic change in the dissociation energy of LuF (a change of ∼35 kcal mol−1).Ab initio composite methods have also been employed in the prediction of ground state properties of lan- thanides. Solomonik and Smirnov calculated the bond dissocia- tion of LuF as 169.7 kcal mol−1and Qing computed the same as 172.4 kcal mol−1,30,31which are near to our recent prediction of 170.2 kcal mol−1in a large scale study of lanthanides.32In consid- ering the prior experimental and theoretical studies, as overviewed, there are substantial differences in the predictions. It is important to note that the dissociation energies reported from experiments are not direct measurements but are instead based on empirical models.7–11 In terms of excited states, a complete understanding of the potential energy surface of LuF and its bonding patterns allows for the probing of possible chemical reactivity routes using excited state dissociation channels. Toward this goal, in 2009, Hamade et al.33 used CASSCF (complete active space self-consistent field) and MRCI (multi-reference configuration interaction), for the first low-lying excited states of LuF, using a pseudopotential for lutetium of 60 electrons. The authors determined 26 electronic states, including the spectroscopic constants and bond lengths for each state; however, these calculations did not account for spin–orbit effects. The authors assigned the first and second excited states as3Πand3Δinstead of the1Σand1Πstates, respectively, previously assigned in the litera- ture.9–11In 2019, Assaf et al. used multireference methods (CASSCF and MRCI +Q) to calculate spectroscopic constants and bond lengths for ground and excited states.34The authors considered a 28 electron pseudopotential (ECP28MWB), which allowed for a more accurate treatment of electron correlation. In addition, sub-valence electrons (4 f) were also correlated, though not included in the active space. The latter step enables the prediction of bond lengths within 0.1 Å of experiment. The active space utilized in this study did not include the bonding orbitals of fluorine, which are important in the construction of the full potential energy curves. However, spin–orbit effects were considered, and spectroscopic constants were calculated for the low-lying excited states using the Breit–Pauli Hamiltonian. Although there have been a number of studies on lutetium fluoride, detailed insight about its dissociation channels and bind- ing patterns have not yet been provided. For this work, 132 states were investigated using multireference methods and double-, triple-, and quadruple- ζlevel basis sets. The results herein provide impor- tant insight into the higher energy channels that play a role in the excited state surface of LuF. MRCI calculations were performed to recover dynamic correlation of the system beyond what CASSCF can obtain. Valence, sub-valence, and inner core levels of correlation were probed, detailing their effects on the energetics of the ground and excited states. The second part of this work (See Sec. III C) focuses on the dissociation energy (D 0) using a range of DFT functionals and also ab initio methods, including coupled-cluster and CASSCF. Complete basis set (CBS) extrapolation was also considered for the ab initio methods. II. COMPUTATIONAL DETAILS Multireference calculations were performed using MOLPRO 2018.35As MOLPRO does not use full linear molecule J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp symmetries, the C 2vpoint group symmetry was utilized and the molecular orbitals were optimized using CASSCF. For this step, the active space used was composed of eight electrons and fifteen orbitals (8, 15). The 15 orbitals correspond to 6 a 1[5dz2, 5dx2−y2, 6s, 6pz(Lu), 2 pz, 3pz(F)], 4 b 1[5dxz, 6px(Lu), 2 px, 3px(F)], 4 b 2[5dyz, 6px(Lu), 2 px, 3px(F)], and 1 a 2[5dxy(Lu)], which correspond to the 6 sand 5 dof lutetium and to 2 pand 3 pof fluorine. The inclusion of the additional 3 porbitals of fluorine was deemed necessary to obtain smooth potential energy curves (PECs). MRCI and MRCI +Q were employed to calculate spectro- scopic constants.36–39Harmonic vibrational frequencies, anhar- monicities, and ΔG1/2values were calculated solving the rovi- brational Schrödinger equation numerically using the Dunham approach.40Due to the computational cost, the 2 pand 3 porbitals of fluorine were not included in the active space and thus were not optimized at the CASSCF level, within the MRCI calculations. The active space for MRCI consists of the following orbitals: 4 a 1[5dz2, 5dx2−y2, 6s, 6pz(Lu)], 2 b 1[5dxz, 6px(Lu)], 2 b 2[5dyz, 6px(Lu), and 1 a2(5dxy(Lu)]. However, the 2 porbitals of fluorine were included in the MRCI calculations as “core” (per MOLPRO 2018) by allowing the electrons to be promoted to the active and virtual spaces through single and double excitations. Considering the CI vectors, for the equilibrium bond region, there are not significant contributions that correspond to the promotion of electrons from the 2 porbitals of fluorine. In addition, for the MRCI calculations, sub-valence cor- relation effects were also described by including the 4 f14orbitals of Lu by also allowing single and double excitations to the active and virtual spaces. Since a pseudopotential was considered for the metal (see next paragraph), the remaining 52 electrons (9 from fluorine and 43 of lutetium) were also correlated for MRCI calculations. The Davidson correction or MRCI +Q as implemented within MOLPRO was used to account for size extensivity issues.36–39To account for spin–orbit coupling, the Breit–Pauli Hamiltonian was diagonalized in the basis of the MRCI wavefunction. For this step, two levels of correlation were considered for inclusion in the core: 4 f14(Lu) and 2p5(F) orbitals, and 4 d105s25p64f14(Lu) and 2 s22p5of (F) orbitals. The latter describe the effects of inner-shell correlation. For CASSCF calculations, a segmented contracted basis set along with a pseudopotential (ECP28MWB) developed by Cao and Dolg was employed (triple- ζlevel).41,42For fluorine, the aug- cc-pVTZ basis set was utilized.43For MRCI and spin–orbit calcu- lations, the def2-QZVPP basis set was employed for lutetium with a pseudopotential (ECP28MWB), while fluorine was described with aug-cc-pVQZ.42–45 For the second part of this work (See Sec. III C), the geome- try optimization step was carried out with CCSD(T) in combination with a contracted basis set by Cao and Dolg, which was used for lutetium, and the aug-cc-pVTZ basis set for fluorine.41–43The fre- quency was also obtained at the same level to ensure a minimum at the potential energy surface. The geometry was then used to evaluate dissociation energies at different levels of theory, and the energy was corrected for the zero-point vibrational energy (ZPE). CCSD(T) and the completely renormalized [CR-CCSD(T)] approach with DKH3 in combination with Sapporo double-, triple-, and quadruple- ζbasis set for lutetium and fluorine have been utilized.46The effect of a four-component Hamiltonian on the dissociation energy was also probed with CCSD(T) using a Dirac–Coulomb (DC) Hamiltonian. In addition, the Perdew-Burke-Ernzerhof (PBE),47the Becke,3-parameter, Lee -Yang -Parr (B3LYP),48,49the Minnesota 2006 local functional (M06-L)50and the Tao, Perdew, Staroverov, Scuseria (TPSS)51functionals were utilized to predict dissociation energies, employing a DKH3 Hamiltonian. These functionals were chosen as they are either widely utilized or were among the better functionals for the prediction of enthalpy of formation and dissociation energies for lanthanide complexes.27,28 Moreover, these functionals will provide some level of compar- ison between the generalized gradient approximation (GGA): PBE; meta-GGA: TPSS, M06-L; and hybrid-GGA: B3LYP on the predic- tion of the dissociation energy. The double-, triple-, and quadruple- ζ level Sapporo basis sets for lutetium and fluorine were used (noted Sap-nz) and the Dyall augmented double-, triple-, and quadruple- ζ(noted Dyall- nz) for the Dirac–Coulomb Hamiltonian where n=D, T, Q.52 The dissociation energy was calculated using the methods described above and at each level of basis set as well. Extrapolations of the total energies to the complete basis set limit were performed using a mixed exponential/Gaussian three point scheme developed by Peterson,53 En=ECBS+Be−(n−1)+Ce−(n−1)2, (1) where BandCare constants determined in the scheme, nis the basis set level ( n=D, T, Q), E nrepresents the energy for each basis set level, and E CBSrepresents the energy at the CBS limit. Unfortunately, it was not possible to obtain values at a quadruple- ζbasis set for CCSD(T) and MP2 with the Dirac–Coulomb Hamiltonian due to the very high computational cost. Thus, the complete basis set limit using the following two-point extrapolation (Dyall.dz and Dyall.tz) scheme by Martin54was used: E=ECBS+B (n+0.5)4. (2) This scheme has been shown to provide reliable extrapolated ener- gies for molecules containing lighter elements when compared to experiment.55,56The final dissociation energy is calculated by adding the zero-point vibrational energy to the final energy. The 95% confi- dence limit has been investigated, and results (Table S1) are given in the supplementary material. In addition to evaluating the 95% confidence intervals, the error from basis set superposition (BSSE) was calculated utilizing Boys and Bernardi’s counterpoise correction approach (Sec. III C).57 Due to the large number of electrons, it is important to consider different frozen-core spaces, i.e., the number of electrons explic- itly correlated. Thus, two frozen-core spaces have been considered: (FC)-val and FC-subval. FC-val corresponds to a space where only the valence electrons (6 sand 5 dof Lu and 2 sand 2 pof F) are treated and the rest is frozen. The FC-subval describes the space where the valence and sub-valence electrons are explicitly treated (5 s, 5pof Lu). All calculations using the DKH3 Hamiltonian were performed with NwChem 6.1,58while the Dirac–Coulomb calculations were done using DIRAC18.59 III. RESULTS AND DISCUSSION A. Electronic structure calculations (CASSCF, MRCI, and MRCI +Q) The PECs calculated at the CASSCF level are displayed in Figs. 1 and 2. The former portrays the Lu (2D; 5d16s2)+F(2P) and J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1. CASSCF PECs of LuF with respect to the Lu–F distance. Lu (2P; 6s25p1)+F(2P) channels, and the latter shows the two upper binding Lu (4F; 5d16s25p1)+F(2P) channels. In Fig. 3, MRCI +Q curves are provided with selected states spanning the equilibrium bond region. The zero of the energy scale in Figs. 1–3 is taken as the energy of the lowest energy asymptote Lu (2D)+F(2P). In Table I, detailed spectroscopic information of the ground and 22 excited states is shown, which includes spectroscopic constants, harmonic vibrational frequencies, ωeχe,ΔG1/2, and T e(excitation energies). In Table II, the CI vectors of the studied ground and excited states are shown. For the calculations, CASSCF, MRCI, and MRCI +Q were used, and for states that were deemed to be single FIG. 2. Example of intersystem crossing from upper dissociation channels of LuF at the CASSCF level. FIG. 3. MRCI+Q PECs of LuF with respect to the Lu–F distance. reference in nature, CCSD(T) was employed. For the first part of this work (Fig. 1), state averaged CASSCF was used for the 132 states, which aids in describing intersystem crossings that come from upper channels and merge with the Lu (2D)+F(2P) channel. This is the first time such a level of detail is considered for LuF, providing insight into how the dissociation channels are formed and describ- ing some of the higher energy, upper channel intersystem cross- ings. In addition, no evidence of the presence of the ionic channel (Lu++F−) was found in the MCSCF calculations. In addition, from the 132 states studied, none of them converged to Lu++F−at infin- ity, demonstrated by its CI vectors. The orbital pictures included in the active space at an equilibrium bond length (1.92 Å) and at 6 Å are shown in Figs. 4 and 5, respectively. At 6 Å, the orbitals resemble atomic ones, with no mixing between fluorine and lutetium, providing insight into dissociation. The radial distribution using CR-CCSD(T) is plotted in Fig. 6. The large orbital overlap near the equilibrium bond length (1.92 Å) shows that the 3 porbitals need to be included at the CASSCF level to describe the full dissociation channels from infin- ity to equilibrium smoothly. An active space with 15 orbitals in the calculation of full potential energy curves for LuF was deemed necessary to obtain smooth curves. Accounting for the irreducible representation for each spin generates hundreds of thousands of configuration state functions (CSFs), increasing both the complexity of the calculations and the computational time. According to the Witmer–Wigner angular momentum cou- pling rules, the four channels generate the following manifolds of states: first—Lu (2D)+F(2P):1,3[Σ+(2),Π(3),Δ(2),Φ,Σ−]; second—Lu (2P)+F(2P):1,3[Σ+(2),Π(2),Δ,Σ−]; third and fourth—Lu (4F)+F(2P):3,5[Σ+(2),Π(3),Δ(3),Φ(2),Γ,Σ−]. The calculations show that the ground state is a well separated1Σ+, a closed shell singlet, which is in agreement with J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I. Computational method, bond length R eq(Å), harmonic vibrational frequen- ciesωe(cm−1), anharmonicity ωeχe(cm−1),ΔG1/2(cm−1) values, and excitation energy T e(cm−1) for the lowest electronic excited states of175Lu19F. MRCI, MRCI +Q and CCSD(T) calculations were performed using an ECP28MWB/Def2-QZPP for Lu and aug-cc-pVQZ for F and CASSCF with ECP28MWB/ANO-TZ for Lu and aug-cc-pVTZ for F. States Methodology R eqωeωeχeΔG1/2 Te X1Σ+Exp.111.9171 611.79 2.54 ⋅ ⋅ ⋅ 0 Exp.121.9165 611.79 2.54 0 CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 MRCI 1.916 613.9 2.67 608.6 0 MRCI +Q 1.914 611.7 2.82 606.1 0 CCSD(T) 1.917 610.4 2.51 605.4 0 CCSD(T) 1.917 610.8 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 MRCI +Q341.913 618.9 2.5 ⋅ ⋅ ⋅ 0 MRCI +Q331.922 606.6 3.3 ⋅ ⋅ ⋅ 0 13ΔExp.111.9319 587.95 2.58 ⋅ ⋅ ⋅ 16 165 Exp.121.9313 587.95 2.58 16 153 CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 18 000 MRCI 1.947 573.4 2.54 568.3 14 917 MRCI +Q 1.945 570.5 2.45 565.6 14 676 MRCI +Q341.947 576.0 2.7 ⋅ ⋅ ⋅ 14 927 MRCI +Q331.952 596.2 3.0 ⋅ ⋅ ⋅ 17 904 13ΠExp.111.9361 576.08 2.5 ⋅ ⋅ ⋅ 16 800 Exp.121.933 581.3 2.6 16 785 CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 17 155 MRCI 1.928 570.8 3.88 563.0 15 630 MRCI +Q 1.930 570.0 3.75 562.5 15 805 CCSD(T) def2 1.943 574.8 2.50 569.6 18 528 MRCI +Q341.933 579.2 2.7 ⋅ ⋅ ⋅ 15 959 MRCI +Q331.923 567.1 2.6 ⋅ ⋅ ⋅ 16 165 13Σ+Exp.9,10605.5 2.5 ⋅ ⋅ ⋅ 18 894 CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 19 900 MRCI 1.957 600.4 2.47 595.4 17 947 MRCI +Q 1.958 590.5 2.53 585.4 18 181 MRCI +Q341.961 559.6 2.5 ⋅ ⋅ ⋅ 18 856 MRCI +Q331.953 567.1 2.6 ⋅ ⋅ ⋅ 19 131 11ΔExp.111.948 569.7 2.5 20 048 Exp.12⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2.6 20 027 CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 21 612 MRCI 1.954 567.0 2.14 562.7 19 392 MRCI +Q 1.953 564.4 2.11 560.2 19 060 MRCI +Q341.955 567.7 2.8 ⋅ ⋅ ⋅ 19 471 MRCI +Q331.956 555.0 2.5 ⋅ ⋅ ⋅ 21 634 11ΠExp.111.9584 543.42 2.28 24 474 Exp.121.9584 543.42 2.28 24 440 CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 27 049 MRCI 1.966 554.0 2.42 549.1 23 371 MRCI +Q 1.969 546.7 2.53 541.7 23 065 MRCI +Q341.972 525.3 2.2 ⋅ ⋅ ⋅ 23 708 MRCI +Q331.945 544.7 2.6 ⋅ ⋅ ⋅ 25 538TABLE I. (Continued. ) States Methodology R eqωeωeχeΔG1/2 Te 21Σ+Exp.111.9520 555.59 2.6 25 832 Exp.121.9514 560.8 2.6 25 806 CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 29 240 MRCI 1.959 548.5 4.41 539.7 25 628 MRCI +Q 1.957 543.2 3.82 535.5 25 292 MRCI +Q341.959 553.0 2.5 25 932 MRCI +Q331.947 563.8 2.8 26 524 23ΠCASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 34 583 MRCI 1.983 570.9 2.90 565.1 29 091 MRCI +Q 1.978 559.9 1.67 556.5 28 870 MRCI +Q341.981 577.7 2.3 29 354 MRCI +Q331.995 525.7 3.4 30 681 21ΠExp.111.951 599.1 2.6 ⋅ ⋅ ⋅ 33 226 CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 38 511 MRCI 1.948 593.9 3.09 587.7 32 809 MRCI +Q 1.944 606.4 3.12 600.1 32 517 MRCI +Q341.951 614.7 2.9 32 968 MRCI +Q331.961 579.3 2.5 33 378 13ΦCASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 38 846 MRCI 1.944 565.2 −0.33 565.9 33 566 MRCI +Q 1.944 571.2 0.28 570.7 33 499 MRCI +Q341.944 570.5 2.7 34 248 MRCI +Q331.942 570.8 3.2 36 401 33ΠCASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 42 188 MRCI 1.960 554.9 2.54 549.8 36 422 MRCI +Q 1.959 543.7 2.63 538.4 36 123 MRCI +Q341.956 545.0 2.8 ⋅ ⋅ ⋅ 36 896 MRCI +Q331.957 552.4 3.2 ⋅ ⋅ ⋅ 39 048 23ΔCASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 42 211 MRCI 1.974 573.3 5.50 562.3 36 674 MRCI +Q 1.974 592.2 7.18 577.8 36 323 MRCI +Q341.976 540.8 3.0 ⋅ ⋅ ⋅ 37 162 MRCI +Q331.969 541.8 2.3 ⋅ ⋅ ⋅ 39 569 13Σ−CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 41 126 MRCI 1.974 534.3 0.34 533.6 36 683 MRCI +Q 1.974 522.2 0.24 521.8 36 338 MRCI +Q341.973 544.0 2.6 ⋅ ⋅ ⋅ 37 338 MRCI +Q331.949 551.3 3.6 ⋅ ⋅ ⋅ 39 216 21ΔCASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 46 419 MRCI 1.955 567.0 1.85 563.3 40 151 MRCI +Q 1.956 557.6 1.47 554.6 39 524 MRCI +Q341.956 558.5 2.6 40 954 MRCI +Q331.946 566.6 3.3 45 661 31Σ+CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI +Q ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI +Q341.942 550.1 3.0 ⋅ ⋅ ⋅ 42 847 MRCI +Q331.917 588.9 2.8 ⋅ ⋅ ⋅ 42 763 J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I. (Continued. ) States Methodology R eqωeωeχeΔG1/2 Te 11Σ−CASSCF 46 100 MRCI 1.953 565.3 2.40 560.5 43 049 MRCI +Q 1.959 557.9 2.33 553.2 41 310 MRCI +Q34⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI +Q33⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 11ΦCASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 51 158 MRCI 1.952 566.9 2.41 562.1 43 048 MRCI +Q 1.951 562.5 2.41 557.7 41 767 MRCI +Q341.942 564.2 2.7 ⋅ ⋅ ⋅ 43 231 MRCI +Q331.950 567.7 2.4 ⋅ ⋅ ⋅ 45 152 23Σ−CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI 1.983 522.9 2.12 518.6 42 275 MRCI +Q 1.983 510.5 2.16 506.2 41 714 MRCI +Q34 MRCI +Q33 31ΠCASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 52 481 MRCI 1.963 600.1 2.79 594.5 44 083 MRCI +Q 1.955 555.6 2.04 551.5 42 790 MRCI +Q341.941 550.4 2.9 ⋅ ⋅ ⋅ 44 678 MRCI +Q331.944 574.2 2.8 ⋅ ⋅ ⋅ 45 319 43ΠCASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 49 935 MRCI 1.968 553.4 4.25 544.9 44 648 MRCI +Q 1.972 545.9 4.06 537.8 44 453 MRCI +Q341.972 553.5 3.0 ⋅ ⋅ ⋅ 44 849 MRCI +Q331.957 553.4 3.2 ⋅ ⋅ ⋅ 45 454 31ΔCASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 53 789 MRCI 1.996 517.3 −3.92 525.1 45 578 MRCI +Q 1.982 525.52 2.79 519.9 44 774 MRCI +Q341.975 546.6 3.3 ⋅ ⋅ ⋅ 43 806 MRCI +Q331.965 540.3 2.1 ⋅ ⋅ ⋅ 47 006 21Σ−CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI 1.980 529.5 2.38 524.6 45 660 MRCI +Q 1995 500.6 −6.47 513.59 45 461 MRCI +Q34⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI +Q33⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 23Σ+CASSCF ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI +Q ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MRCI +Q341.900 556.4 2.7 ⋅ ⋅ ⋅ 47 316 MRCI +Q331.871 664.6 2.1 ⋅ ⋅ ⋅ 43 031 experiment.9–11In the ground state, the unpaired 5 d1(Lu, at infin- ity) electron couples with the unpaired electron on the 2 pzorbital of fluorine (see Table II). The spectroscopic constants calculated with CCSD(T) and MRCI/MRCI +Q are all within 1 cm−1of experiment. The next two states were assigned as either3Πor3Δin previous liter- ature. Hamade et al. predicted the3Πto be the first excited state, andAssaf et al. predicted the3Δas the first excited state.33,34According to our calculations, for CASSCF, 13Πis followed by 13Δand their separation is 845 cm−1. However, for MRCI and MRCI +Q, the3Δ is the first excited state followed by the3Π. The separation of states for MRCI and MRCI +Q is 713 cm−1and 1129 cm−1, respectively (see Table II). Both states are a product of electron promotion from the lutetium 6 s(at infinite separation) to its 5 dorbitals (see Figs. 4 and 5). In order to generate the 13Δstate, an electron populates the 5 dx2−y2(Lu), while for3Π, it occupies the 5 dxz(Lu). These two states are very close in energy and both were assigned a different spin and symmetry in previous experimental data. In the present work, the two experimental values from the literature were assigned to 13Δand 13Π.9–12Previous theoretical data from Hamed et al. and Assaf et al. do not compare the first experimental excited state energy with their first calculated excited state.11,12,33,34Assaf et al. assigns their second excited state to Aand Bfrom the literature, 1Σ+and1Π, respectively.11,12,34The 13Πis in good agreement with experiment for bond lengths and spectroscopic constants, but the 3Δis∼1000 cm−1below the experimental value. However, when both 13Δand 13Πare corrected for spin–orbit effects (see Sec. III B), the range of Ω-state energies spans over 3000 cm−1(Table III). The next excited state is 13Σ+, which corresponds to a promotion of an electron from the 6 sof lutetium to the 5 dz2. In fact, electronic excitations from 6 s→5dorbitals occur until ∼33 000 cm−1, as per Table II. States 11Δand 11Πare the corresponding open-shell singlets of 13Δand 13Π, respectively, and are 4384 and 7260 cm−1 above the aforementioned, according to MRCI +Q. 11Δand 11Πare also 1000 cm−1below their assigned experi- mental states, but their bond length is within 0.01 Å from experi- ment. The next three states, 21Σ+, 23Π, and 21Π, also correspond to the promotion of an electron from the 6 s(Lu) into the 5 dorbitals (Lu). 13Φis 33 566 and 33 499 cm−1above the ground state accord- ing to MRCI and MRCI +Q, respectively, and it is the first excited state that has two electrons promoted from the 6 s(Lu) into 5 d and 6 p(Lu) orbitals. There is a ∼3000 cm−1gap in which there are no populated states, but in the 36 000 cm−1region, there are three excited states within 200 cm−1of one another according to MRCI +Q (33Π, 23Δ, and 13Σ−). From 36 000 to 50 000 cm−1, there is a large agglomeration of states, which show mixing from the first two dissociation channels. In this 14 000 cm−1or 30 kcal mol−1 region, nine states overlap each other. The first state in this region is 21Δ, followed by 31Σ+. The latter belongs to the next binding channel, Lu (2P; 6s25p1)+F(2P) (see Fig. 2). This channel is not displayed in Fig. 1 due to the very large mix of states from upper channels, so only the binding region (2.7–1.4 Å) is plotted. The other states displayed in Fig. 1, which belong to the Lu (2P; 6s25p1) +F(2P) channel, are 23Σ−, 43Π, 31Δ, and 21Σ−. The last states that belong to the first binding channel are 11Σ−, 11Φ, and 23Σ+. The first 1Σ−state undergoes intersystem crossings, as shown in Fig. 2. There is a range of singlet and triplet states that couple together after 45 000 cm−1(∼175 kcal mol−1) from three different dissoci- ation channels, which originate multiple avoided and intersystem crossings. B. Spin–orbit calculations Spin–orbit calculations were performed on the ground state and the first eight excited states of LuF, which cover a region of J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. CI vectors at equilibrium bond length for LuF were obtained through CASSCF using the ECP28MWB/ANO-TZ for Lu and aug-cc-pVTZ for F. States Coeff 1 πz1σ1δx2−y21δz22πz3πz1πx1δxz2πx3πx1πy1δyz2πy3πy1δxy X1Σ+0.91 2 2 0 0 0 0 2 0 0 0 2 0 0 0 0 13Δ 0.94 2 α α 0 0 0 2 0 0 0 2 0 0 0 0 13Π 0.91 2 α 0 0 0 0 2 α 0 0 2 0 0 0 0 13Σ+0.96 2 α 0α 0 0 2 0 0 0 2 0 0 0 0 11Δ0.59 2 β α 0 0 0 2 0 0 0 2 0 0 0 0 −0.59 2 α β 0 0 0 2 0 0 0 2 0 0 0 0 11Π−0.35 2 β 0 0 0 0 2 α 0 0 2 0 0 0 0 0.35 2 α 0 0 0 0 2 β 0 0 2 0 0 0 0 −0.50 2 β 0 0 0 0 2 0 α 0 2 0 0 0 0 0.50 2 α 0 0 0 0 2 0 β 0 2 0 0 0 0 21Σ+−0.65 2 β 0α 0 0 2 0 0 0 2 0 0 0 0 0.65 2 α 0β 0 0 2 0 0 0 2 0 0 0 0 23Π 0.86 2 α 0 0 0 0 2 0 α 0 2 0 0 0 0 21Π−0.50 2 α 0β 0 0 2 0 0 0 2 0 0 0 0 0.50 2 β 0α 0 0 2 0 0 0 2 0 0 0 0 13Φ0.66 2 0 α 0 0 0 2 α 0 0 2 0 0 0 0 0.66 2 0 0 0 0 0 2 0 0 0 2 α 0 0 α 33Π−0.54 2 0 0 0 0 0 2 0 0 0 2 α 0 0 α 0.54 2 0 α 0 0 0 2 α 0 0 2 0 0 0 0 23Δ 0.94 2 0 α α 0 0 2 0 0 0 2 0 0 0 0 13Σ− 0.64 2 0 α 0 0 0 2 0 0 0 2 0 0 0 α −0.65 2 0 0 0 0 0 2 α 0 0 2 α 0 0 0 21Δ0.59 2 0 β α 0 0 2 0 0 0 2 0 0 0 0 −0.59 2 0 α β 0 0 2 0 0 0 2 0 0 0 0 31Σ+0.56 2 0 2 0 0 0 2 0 0 0 2 0 0 0 0 0.57 2 0 0 0 0 0 2 0 0 0 2 0 0 0 2 −0.30 2 0 0 0 0 0 2 0 0 0 2 2 0 0 0 −0.30 2 0 0 0 0 0 2 2 0 0 2 0 0 0 0 11Σ−−0.69 2 0 α 0 0 0 2 0 0 0 2 0 0 0 β 0.69 2 0 β 0 0 0 2 0 0 0 2 0 0 0 α 11Φ0.43 2 0 α 0 0 0 2 β 0 0 2 0 0 0 0 −0.43 2 0 β 0 0 0 2 α 0 0 2 0 0 0 0 0.43 2 0 0 0 0 0 2 0 0 0 2 α 0 0 β −0.43 2 0 0 0 0 0 2 0 0 0 2 β 0 0 α 23Σ−0.71 2 0 α 0 0 0 2 0 0 0 2 0 0 0 α −0.34 2 0 0 0 0 0 2 0 α 0 2α 0 0 0 −0.34 2 0 0 0 0 0 2 α 0 0 2 0 α 0 0 0.45 2 0 0 0 0 0 2 α 0 0 2 α 0 0 0 31Π−0.30 2 0 0 β 0 0 2 α 0 0 2 0 0 0 0 0.30 2 0 0 α 0 0 2 β 0 0 2 0 0 0 0 J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. (Continued. ) States Coeff 1 πz1σ1δx2−y21δz22πz3πz1πx1δxz2πx3πx1πy1δyz2πy3πy1δxy 43Π0.80 2 0 0 α 0 0 2 α 0 0 2 0 0 0 0 0.30 2 0 α 0 0 0 2 α 0 0 2 0 0 0 0 −0.30 2 0 0 0 0 0 2 0 0 0 2 α 0 0 a 31Δ0.49 2 0 0 0 0 0 2 α 0 0 2 β 0 0 0 −0.49 2 0 0 0 0 0 2 β 0 0 2 α 0 0 0 21Σ−0.48 2 0 0 0 0 0 2 α 0 0 2 0 β 0 0 −0.48 2 0 0 0 0 0 2 β 0 0 2 0 α 0 0 −0.48 2 0 0 0 0 0 2 0 α 0 2β 0 0 0 0.48 2 0 0 0 0 0 2 0 β 0 2α 0 0 0 23Σ+0.94 2 α 0 0 α 0 2 0 0 0 2 0 0 0 0 ∼100 kcal mol−1or∼36 000 cm−1. The first nine2S+1Λstates split into the Ω-states as follows: X1Σ+→X1Σ+ 0+; 13Δ→3Δ1,3Δ2,3Δ3; 13Π→3Π0−,3Π0+,3Π1,3Π2; 13Σ+→3Σ+ 0+,3Σ+ 1; 11Δ→1Δ2; 11Π →1Π1; 21Σ+→21Σ+ 0+, 23Π→3Π0−,3Π0+,3Π1,3Π2, 21Π→1Π1; and 13Φ→3Φ2,3Φ3,3Φ4. For singlet states, Λ=0 is expected to be minimal. The C-MRCI spin–orbit PECs are depicted in Fig. 7 (spin–orbit states with the same Ωvalue have the same color), and MRCI spin–orbit are depicted in supplementary material (Fig. S1. and Table S2). The bond lengths and spectroscopy constants are included in Table III, and the decomposition of the spin–orbit states is included in Table IV. The ground state of LuF (X1Σ+), 11Δ, 11Π, and 21Σ+remain almost unaffected due to zero first order spin–orbit effects. With- out spin–orbit effects, the 13Δis the first excited state followed by 13Π, which is ∼1200 cm1higher in energy according to MRCI +Q. However, with spin–orbit correction, the ordering of Ω-states is more complex to assess due to the closeness of the energetics gaps. The3Δand3Πstates, spin–orbit corrected at the MRCI and C-MRCI level, follow the same ascending order:3Π0−,3Π0+,3Π1, 3Π2and3Δ1,3Δ2,3Δ3. According to C-MRCI, the 13Π0−is the first excited state followed by 13Π0+, which is ∼400 cm−1above in energy. However, for MRCI, the 13Δ1, is the second excited fol- lowed by 13Π0+. The effect of the core orbitals is also felt on the bond lengths of 13Π0−, 13Π0+, and 13Δ1, which drop by ∼0.01 Å when using C-MRCI. For C-MRCI, the third excited state is3Δ1, followed by3Π1,3Δ2,3Π2, and3Δ3. The Ω-states of 13Δand 13Π span over a range of more than 3000 cm−1, which shows a large spin–orbit contribution and the importance of including inner core correlation. When comparing this work with Assaf et al. , their state order- ing is different, and the 13Δ1is their first excited state followed by 13Δ2and then 13Π0−. These differences can be attributed to the use of a more state specific approach in the CASSCF and MRCI calculations, a higher level basis set in the present study. The inner orbitals of lutetium and fluorine were not considered in their calculations, but only the lutetium sub-valence 4 f14was included along with the 2 p5of fluorine. The use of inner core orbitals results in significant differences in bond lengths and spectroscopic constants.In terms of composition (Table IV), Ω-states =1, 2 for 13Δand 13Πare heavily mixed, but 13Δ3can only mix with 13Φ3. The next excited is3Σ+, which splits into3Σ+ 0+and3Σ+ 1. The bond length dropped ∼0.07 Å when using C-MRCI, and the T eis∼500 cm−1 for both Ωstates above MRCI. The next three states have min- imal spin–orbit effects, but the inclusion of the core orbitals for C-MRCI changed their bond lengths by almost 0.1 Å, and the T e is∼1000 cm−1above MRCI. The last three states considered in Fig. 3 are 23Π, 21Π, and 13Φ. The 23Πfollows the same ordering for its Ω states as the 13Π. When comparing MRCI and C-MRCI, the bond length for this state only varies 0.02 Å on average. C-MRCI still is∼1000 cm−1above MRCI. 21Π1is in between the 13Φ Ω states. 13Φ2is a heavily mixed state as reported in Table IV. 13Φ3can only mix with 13Δ3, but 13Φ4is a pure state. For the 13Φsplitting, C-MRCI also drops the bond length by almost ∼0.1 Å for the three Ωstates. The T efor C-MRCI is also on average 1000 cm−1above MRCI. C. Dissociation energy Dissociation energies calculated in this work as well as those reported previously from both theoretical and experimental studies are included in Table V. For the correlation, two approaches to the valence space were considered: FC-val, which includes only valence electrons (6 s2, 5d1 of Lu and 2 s2, 2p5of F), and FC-subval, which includes sub-valence orbitals (5 s2, 5p6of Lu). In addition, the effects of using a full rel- ativistic Hamiltonian and ECPs (28 electrons) were probed. For ab initio calculations, CCSD(T), CR-CCSD(T), and MP2 were uti- lized. For DFT, a variety of functionals were considered: PBE, TPSS, M06-L, and B3LYP. The dissociation energy difference between the Sapporo-DZ and Sapporo-TZ for CR-CCSD(T) is 2 kcal mol−1with FC-val, while between Sapporo-TZ and Sapporo-QZ basis sets, the energy dif- ference drops to 0.03 kcal mol−1, which implies that the energy is almost converged at the triple- ζlevel. The same trend is observed for CCSD(T), where the energy at the triple- ζlevel is almost converged. When the sub-valence electrons from Lu are added (FC-subval results), the dissociation energy with CR-CCSD(T)/Sapporo-DZ J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. Molecular orbitals for LuF at 1.92 Å. J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5. Molecular orbitals for LuF at 6.0 Å. dropped by 14 kcal mol−1and by ∼13 kcal mol−1at the CBS limit. At the CCSD(T) level of theory, the difference between FC-val and FC-subval dissociation energies is ∼14 and ∼12 kcal mol−1with the Sapporo-DZ and at the CBS limit, respectively. Such a large differ- ence arising from the choice of valence indicates that the electron correlation arising from the sub-valence electrons is important in the overall energy.The basis set superposition error has been investigated by using the counterpoise method suggested by Boys and Bernardi for CCSD(T) and CR-CCSD(T) at the CBS limit for FC-val and FC-subval.57For both FC-val calculations, considering CCSD(T) and CR-CCSD(T), the BSSE extrapolated to the CBS limit using a mixed exponential/Gaussian by Peterson is 0.87 kcal mol−1.53 For CCSD(T) and CR-CCSD(T) using sub-valence electrons, 0.81 J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6. Radial distribution functions at the CR-CCSD(T) level. TABLE III. Methodology, bond length R eq(Å), harmonic vibrational frequencies ωe (cm−1), anharmonicity ωeχe(cm−1),ΔG1/2(cm−1) values, and excitation energy T e (cm−1) for the lowest electronic excited states of175Lu19F at the spin–orbit level. The states are ordered according to C-MRCI energetics. MRCI and C-MRCI calculations were performed using an ECP28MWB/Def2-QZPP for Lu and aug-cc-pVQZ for F. States Methodology R eqωeωeχeΔG1/2 Te X1Σ+ 0+MRCI 1.917 614.9 2.24 610.4 0 C-MRCI 1.913 618.2 2.52 613.2 0 MRCI341.914 619.4 2.53 ⋅ ⋅ ⋅ 0 13Π0−MRCI 1.933 584.7 2.72 579.3 13 831 C-MRCI 1.924 595.6 2.59 590.4 14 377 MRCI341.938 573.3 2.58 ⋅ ⋅ ⋅ 14 629 13Π0+MRCI 1.928 590 2.62 584.9 14 270 C-MRCI 1.919 601.3 2.57 596.2 14 788 MRCI341.935 577.3 2.69 ⋅ ⋅ ⋅ 15 003 13Δ1MRCI 1.952 567.2 2.81 561.6 13 866 C-MRCI 1.939 565.0 1.62 561.8 14 943 MRCI341.949 572.4 2.57 ⋅ ⋅ ⋅ 13 513 13Π1MRCI 1.935 585.9 1.76 582.4 15 142 C-MRCI 1.932 601.3 3.29 594.7 15 844 MRCI341.938 571.3 2.59 ⋅ ⋅ ⋅ 15 600 13Δ2MRCI 1.953 569.3 2.30 564.7 14 781 C-MRCI 1.943 571.3 2.88 565.6 15 890 MRCI341.949 572.7 2.82 ⋅ ⋅ ⋅ 14 435 13Π2MRCI 1.929 589.8 2.48 584.8 16 774 C-MRCI 1.922 601.0 2.62 595.7 17 313 MRCI341.931 580.5 2.66 ⋅ ⋅ ⋅ 16 884 13Δ3MRCI 1.947 576.4 2.43 571.5 16 748 C-MRCI 1.941 579.7 2.20 575.2 17 641 MRCI341.946 576.4 2.53 ⋅ ⋅ ⋅ 16 170 13Σ+ 1MRCI 1.952 574.6 2.51 569.5 18 955 C-MRCI 1.945 581.6 2.52 576.5 19 520 MRCI341.956 567.7 2.80 ⋅ ⋅ ⋅ 19 101TABLE III. (Continued. ) States Methodology R eqωeωeχeΔG1/2 Te 13Σ+ 0−MRCI 1.952 575.9 2.74 570.4 19 238 C-MRCI 1.945 581.6 2.48 576.7 19 782 MRCI341.955 567.9 2.77 ⋅ ⋅ ⋅ 19 352 11Δ2MRCI 1.953 570.6 2.07 566.5 19 902 C-MRCI 1.946 576.8 2.38 572.0 21 180 MRCI341.954 570.2 2.54 ⋅ ⋅ ⋅ 19 702 11Π1MRCI 1.967 534.1 2.00 530.1 24 403 C-MRCI 1.954 544.1 2.03 540.1 25 493 MRCI341.969 528.65 2.31 ⋅ ⋅ ⋅ 23 839 21Σ+ 0−MRCI 1.955 562.7 2.71 557.3 26 211 C-MRCI 1.946 572.5 2.82 566.9 27 042 MRCI341.959 553.1 2.60 ⋅ ⋅ ⋅ 26 037 23Π0−MRCI 1.993 532.1 ⋅ ⋅ ⋅ 556.6 28 782 C-MRCI 1.991 523.8 ⋅ ⋅ ⋅ 566.08 29 920 MRCI341.984 573.6 2.61 ⋅ ⋅ ⋅ 28 744 23Π0+MRCI 1.991 536.8 ⋅ ⋅ ⋅ 557.5 28 818 C-MRCI 1.989 523.4 ⋅ ⋅ ⋅ 568.3 29 959 MRCI341.984 574.5 2.58 ⋅ ⋅ ⋅ 28 730 23Π1MRCI 1.988 570.2 ⋅ ⋅ ⋅ 556.8 29 369 MRCI 1.986 567.8 ⋅ ⋅ ⋅ 556.0 30 461 C-MRCI341.982 577.5 2.64 ⋅ ⋅ ⋅ 29 291 23Π2MRCI 1.981 656.3 ⋅ ⋅ ⋅ 586.9 30 345 C-MRCI 1.979 611.7 ⋅ ⋅ ⋅ 564.2 31 373 MRCI341.978 583.2 2.66 ⋅ ⋅ ⋅ 30 095 13Φ2MRCI 1.948 512.8 ⋅ ⋅ ⋅ 536.9 31 774 C-MRCI 1.939 531.6 ⋅ ⋅ ⋅ 553.4 33 444 MRCI341.951 560.6 2.41 ⋅ ⋅ ⋅ 31 877 21Π1MRCI 1.949 577.15 ⋅ ⋅ ⋅ 579.3 32 891 C-MRCI 1.946 584.1 ⋅ ⋅ ⋅ 590.5 33 812 MRCI341.957 594.72 2.35 ⋅ ⋅ ⋅ 32 921 13Φ3MRCI 1.946 597.7 ⋅ ⋅ ⋅ 586.9 34 013 C-MRCI 1.937 608.9 ⋅ ⋅ ⋅ 595.5 35 587 MRCI341.946 565.9 2.41 ⋅ ⋅ ⋅ 33 965 13Φ4MRCI 1.942 571.8 ⋅ ⋅ ⋅ 568.7 36 287 C-MRCI 1.934 583.0 ⋅ ⋅ ⋅ 578.7 37 762 MRCI341.949 494.93 2.53 ⋅ ⋅ ⋅ 36 218 and 0.59 kcal mol−1were obtained, respectively, for BSSE cor- rections at CBS. As an example, for CCSD(T)/FC-subval at a double-, triple-, and quadruple- ζbasis set levels, the BSSE is 6.82, 3.52, and 1.21 kcal mol−1, respectively, which at CBS yields 0.81 kcal mol−1. In addition, the dissociation energy of LuF was evaluated using the ECP28MWB pseudopotential and Def2-TZVPP, Def2-QZVPP J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-11 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7. Core-spin–orbit MRCI (C-MRCI) PECs of LuF with respect to the Lu–F distance. (Lu) and aug-cc-pVTZ, aug-cc-PVQZ (F) basis sets. The value obtained at the quadruple- ζlevel is very close to DKH3 predic- tions mentioned earlier, while the triple- ζresult is slightly higher than the DKH3 dissociation reported. The pseudopotential used for lutetium accounts for relativistic effects arising from the inner-coreelectrons. To evaluate the spin–orbit contribution to the ground state, the Dirac–Coulomb (DC) four component Hamiltonian was utilized. CCSD(T), MP2, and HF were probed for this step. The utility of the double- and triple- ζCBS extrapolation by Martin54 has been considered for CCSD(T)/FC-subval/DKH3. This double-, triple-ζCBS extrapolation scheme results in a dissociation energy of 171.1 kcal mol−1, while considering a two-point scheme extrap- olation with triple- ζand quadruple- ζbasis sets, 170.4 kcal mol−1 is obtained. Considering the unextrapolated triple- ζbasis set, the value obtained is 169.3 kcal mol−1. For CCSD(T)/FC-subval/DKH3, the double-, triple- ζCBS extrapolated energy is closer to the triple-, quadruple- ζextrapolated energy than the unextrapolated triple- ζ energy. This shows that the spin–orbit contribution is small to the ground state, which is expected for a1Σ+. In terms of calculations at the Hartree–Fock level, the necessary electron correlation is not present, so its dissociation energy prediction is very far from the best estimate. Finally, CASSCF was also used to calculate the dis- sociation energy by using the state-averaged wavefunction utilized to construct Fig. 1. The prediction is 159.74 kcal mol−1at a triple-ζ level, which is ∼9 kcal mol−1from the CCSD(T)/DKHH3/FC-subval dissociation energy. CR-CCSD(T) and CCSD(T) results obtained in this study are in good agreement with other theoretical dissociation energies from the literature. When comparing the current results with Solomonik and Smirnov, a difference of 2 kcal mol−1is obtained when using a sub-valence space correlation.30Solomonik’s dissociation energy was obtained with a composite scheme based on CCSD(T)/CBS with TABLE IV. Spin–orbit composition at the C-MRCI level (1.92 Å) for the lowest excited states of175Lu19F. State Composition X1Σ+ 0+ 99.87% X1Σ+, 0.06% 13Π, 0.08% 23Π 13Π0− 89.24% 13Π, 10.76% 13Σ+ 13Π0+ 98.44% 13Π, 1.49% 21Σ+,0.06% X1Σ+ 13Δ1 42.49% 13Δ, 53.19% 13Π, 3.81% 13Σ+, 0.26% 11Π, 0.16% 21Π,0.08% 23Π 13Π1 45.12% 13Π, 46.13% 13Δ, 7.32% 13Σ+, 1.34% 11Π, 0.07% 23Π, 0.02% 21Π 13Δ2 69.16% 13Δ, 25.20% 13Π, 5.30% 11Δ, 0.24%3Φ, 0.08% 23Π 13Π2 74.0% 13Π, 25.83% 13Δ, 0.12% 11Δ, 0.02% 23Π, 0.02% 13Φ 13Δ3 99.95% 13Δ, 0.05% 13Φ 13Σ+ 1 86.83% 13Σ, 11.78% 13Π, 1.14% 11Π, 0.24% 13Δ, 0.01% 23Π 13Σ+ 0− 89.22% 13Σ+, 10.76% 13Π, 0.02% 23Π 11Δ2 93.78% 11Δ, 4.84% 13Δ, 0.78% 13Π, 0.36% 23Π, 0.24 13Φ% 11Π1 97.21% 11Π, 2.03% 13Σ+, 0.57% 13Π, 0.14% 13Δ, 0.06% 23Π 21Σ+ 0+ 98.32% 21Σ+, 1.50% 13Π, 0.18% 23Π 23Π0− 99.98% 23Π, 0.02% 13Π, 0.01% 13Σ+ 23Π0+ 99.74% 23Π, 0.19% 21Σ+, 0.07% X1Σ+ 23Π1 94.6%, 5.26% 11Π, 0.08%, 0.01% 23Π2 99.28% 23Π, 0.46% 11Δ, 0.14% 13Φ, 0.13% 13Δ 13Φ2 99.52% 13Φ, 0.34% 11Δ, 0.10% 13Π, 0.05% 13Δ 21Π1 94.56% 21Π, 5.19% 23Π, 0.22% 13Δ, 0.03% 13Π 13Φ3 99.95% 13Φ, 0.06% 13Δ 13Φ4 100% 13Φ J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-12 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE V. Dissociation energy of LuF in kcal mol−1with different levels of theory and a range of basis sets. Relativistic Method Frozen-core treatment D0(dz) D0(tz) D0(qz) D0CBS CR-CCSD(T) FC-val DKH3 177.93 180.35 180.38 180.3 CR-CCSD(T) FC-subval DKH3 163.01 167.20 167.66 167.9 CCSD(T) FC-val DKH3 178.99 182.16 182.40 182.3 CCSD(T) FC-subval DKH3 164.16 169.25 169.96 170.4 CCSD(T) FC-val ECP28-Def2 ⋅ ⋅ ⋅ 172.35 169.47 167.9 CCSD(T) FC-subval DC 158.33 165.63 ⋅ ⋅ ⋅ MP2 DC 158.78 167.19 ⋅ ⋅ ⋅ HF DC 134.15 137.49 137.74 137.8 CASSCF(8,15) ECP28-ANO ⋅ ⋅ ⋅ 159.74 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ PBE DKH3 174.40 172.29 176.04 ⋅ ⋅ ⋅ TPSS DKH3 170.64 167.87 171.73 ⋅ ⋅ ⋅ M06-L DKH3 169.02 169.77 171.93 ⋅ ⋅ ⋅ B3LYP DKH3 167.72 166.09 169.97 ⋅ ⋅ ⋅ Other theoretical values Composite30169.7 Composite31173.32 PP-CCSD(T)24Valence ECP60 173 PP-MRACPF24Valence ECP60 175 DFT: SOAP2596.6 ⋅ ⋅ ⋅ PBE26ZORA 174 DFT27,aDKH3 195.3–161.6 EOM-CR-CCSD(T)29Valence DKH3 171.3 EOM-CR-CCSD(T)29Full DKH3 139.6 Experimental value Mass spectroscopy7136±12 Ligand field theory8124 Fitting PES1479 Fitting PES9105 Fitting PES1396.0±2.4 aDFT functionals used are SVWN, BP86, BLYP, PW91, PBE, B97-D, SSB-D, M06-L, TPSS, PBE0, B3LYP, BHLYP, B3P86, MPW1K, B97-1, X3LYP, M06, M06-2X, TPSSh, M11, CAM-B3LYP, and B2PLYP. core–valence correlation energy, spin–orbit, and scalar relativistic effects. The CCSD(T)/CBS results herein are in very good agreement with previous work from Lu.31A composite scheme utilizing the Feller–Peterson–Dixon scheme renders a value of 173.32 kcal mol−1, which is only ∼3 kcal mol−1from our best CCSD(T)/CBS results and 5 kcal mol−1from CR-CCSD(T). Küchle et al.24used the multireference averaged coupled-pair functional (MRACPF), and their dissociation energy is 4 and 7 kcal mol−1higher than the results obtained in the CCSD(T)/CBS and CR-CCSD(T)/CBS predictions herein, respectively. However, both CCSD(T)/CBS and CR-CCSD(T)/CBS dissociation energies are quite distant from reported experimental values. In Table IV, the smallest difference in dissociation energy between experiment and our predictions was obtained by mass spectroscopy (Zmbov and Margrave,7 136 kcal mol−1). The other experimental values presented in Table IV have large energetic differences from our calculated val- ues, with a maximum ΔE of∼90 kcal mol−1. This shows the large discrepancy between experiment and theory. Additionally, the potential utility of several DFT functionals in the determination of the dissociation energy of LuF has been con- sidered. The PBE, TPSS, M06-L, and B3LYP functionals have beenused, along with a DKH3 Hamiltonian and the Sap- nz basis set. The PBE dissociation energy obtained in this study is in agreement with the one predicted by Hong et al.26using PBE and the ZORA Hamil- tonian. The dissociation energies obtained with the functionals are in a range between 166 and 176 kcal mol−1. B3LYP at the triple- ζ level results in the lowest dissociation energy (166.09 kcal mol−1), while PBE with the quadruple- ζbasis set leads to the largest disso- ciation (176.04 kcal mol−1). These results largely compare with the DFT dissociation energies reported by Grimmel et al.27However, in their study, a larger range of functionals were used, with SVWN leading to the largest dissociation energy at 195.3 kcal mol−1and BHLYP resulting in the lowest energy at 161.6 kcal mol−1. Moreover, from the prior effort, B97-1 predicted a dissociation energy that is the closest to our CCSD(T)/CBS with DKH3/FC-subval dissociation energy. Finally, when comparing PBE, TPSS, M06-L, and B3LYP dis- sociation energies from our work and Grimmel et al. , PBE has the largest dissociation energy among the four functionals and B3LYP the lowest. (To note, the differences between the Grimmel study and the present one are the use of a larger basis set (quadruple- ζ) in this study as well as a different type of basis set for the ligand.) J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-13 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp IV. CONCLUSION The bond lengths, spectroscopic constants, energetics, and potential energy curves are reported, which include four dissocia- tion channels and detailed information concerning intersystem and avoided crossings. In addition, spin–orbit effects are calculated at a level of correlation that can aid experimentalists in further pur- suits of the description of the ground and excited states and their spectroscopic data. The use of sub-valence orbitals at spin–orbit demonstrated that they are necessary to recover the necessary cor- relation to obtain results that are in agreement with experiment, especially for the low-lying excited states. The first excited state of LuF at spin–orbit C-MRCI is 13Π0−, followed by 13Π0+and 13Δ1, which shows the importance of considering sub-valence and inner core orbitals to calculate spectroscopic constants and bond lengths. In the second part of this work, the sub-valence orbitals are of paramount importance for predicting dissociation energies and can shift the dissociation energy by up to ∼13 kcal mol−1. CR- CCSD(T) and CCSD(T) at the CBS limit estimate the dissociation energy as 167.9 and 170.4 kcal mol−1, respectively. Utilizing a four- component Hamiltonian (Dirac–Coulomb) resulted in a dissocia- tion energy ∼2 kcal mol−1lower than the DKH3 calculations. The DFT calculations are overall in good agreement with our best esti- mate (from ∼1 to ∼6 kcal mol−1to 170.40 kcal mol−1). Due to the large discrepancies between the results in this study as well as other theoretical data and the experiment, the experimental dis- sociation energy might need to be revisited. Finally, while in this case DFT gave similar dissociation than ab initio methods, a study of an open-shell molecule with a multi-reference character at the ground state might need more robust methods, such as ab initio methods. Overall, lanthanide species are difficult to investigate from both theoretical and experimental perspectives. The high density of states, which can be very close in energy (herein, 132 states, most of which are bound and in a ∼55 000 cm−1range, just below the dissociation energy), the effect of spin–orbit on the ground and excited states, as well as the influence of the sub-valence elec- trons are effects that should be included in a detailed analysis. Ab initio methods, as utilized herein, are vital to the description of the complex electronic manifold. Already for diatomics, such an analysis is significantly demanding and requires judicious selec- tion of the active space, the electron correlation space, and the method. SUPPLEMENTARY MATERIAL MRCI spin–orbit potential energy curves along with the cor- responding energies and distances are provided in the supplemen- tary material. As well, an analysis of an error estimate in the CBS extrapolation scheme is provided, considering the 95% confidence limit. ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant No. CHE-1900086. This work uti- lized computational facilities at the Center for Advanced Scientific Computing and Modeling (CASCaM) at the University of NorthTexas, which, in part, were supported by the NSF (Grant No. CHE- 1531468), as well as the Institute for Cyber-Enabled Research (ICER) at Michigan State University. We also gratefully acknowledge sup- port from The Extreme Science and Engineering Discovery Environ- ment (XSEDE) supercomputer, which is supported by the National Science Foundation (Grant No. ACI-1548562). The XSEDE staff request the acknowledgment of the following publication: DOI: https://doi.org/10.1109/MCSE.2014.80. The authors declare no competing financial interests. DATA AVAILABILITY The data that support the findings of this study are avail- able within the article and its supplementary material and from the corresponding author upon reasonable request. REFERENCES 1S. Banerjee, M. R. A. Pillai, and F. F. (Russ) Knapp, Chem. Rev. 115, 2934 (2015). 2S. G. Hacker, Astrophys. J. 83, 140 (1936). 3J. A. Johnson and M. Bolte, Astrophys. J. 605, 462 (2004). 4C. Sneden, J. J. Cowan, J. E. Lawler, I. I. Ivans, S. Burles, T. C. Beers, F. Primas, V. Hill, J. W. Truran, G. M. Fuller, B. Pfeiffer, and K. L. Kratz, Astrophys. J. 591, 936 (2003). 5I. U. Roederer, C. Sneden, J. E. Lawler, and J. J. Cowan, Astrophys. J. Lett. 714, L123 (2010). 6E. A. Den Hartog, J. J. Curry, M. E. Wickliffe, and J. E. Lawler, Sol. Phys. 178, 239 (1998). 7K. F. Zmbov and J. L. 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Nayak, P. Norman, G. Olejniczak, J. Olsen, J. M. H. Olsen, Y. C. Park, J. K. Pedersen, M. Pernpointner, R. di Remigio, K. Ruud, P. Sałek, B. Schimmelpfennig, A. Shee, J. Sikkema, A. J. Thorvaldsen, J. Thyssen, J. vanStralen, S. Villaume, O. Visser, T. Winther, and S. Yamamoto, available at https://doi.org/10.5281/zenodo.2253986, see also http://www.diracprogram.org, 2018. J. Chem. Phys. 154, 244304 (2021); doi: 10.1063/5.0052312 154, 244304-15 Published under an exclusive license by AIP Publishing

5.0053482.pdf

Temperature Dependent Luminescence Behaviour Rf Cerium'oped Silica (Sio 2:Ce) Nanophosphor I.M. Nagpure Department of Physics, National Institu te of Technology, Uttarakhand, India –246174 Corresponding author: indrajitnagpure@gmail.com Abstract. The temperature dependent luminescence behavior of Cerium (Ce) doped Silica (SiO 2:Ce) nanophosphor was studied by using Laser excitation source. The porous SiO 2:Ce nanophosphor has been synthesized by using urea assisted combustion method and further annealed at 1000oC for 1h in reducing environment. XRD and TEM analysis was carried out for the confirmation crystallinity and morphology of the nanophosphor. Laser excited temperature dependent luminescence of SiO 2:Ce3+nanophosphor shows blue emission due to 4f05d1→ 4f1(2FJ) transitions of the Ce3+ions. Keywords: XRD- X –ray Diffraction; LEL –Laser excited luminescence; TEM –Transverse electron microscopy. INTRODUCTION The SiO 2prevalently well known as silica, exist in various forms and most abundant material of the earth. It occurs predominantly in porous form with amorphous nature. It has very long history of theoretical and experimental investigations. SiO 2in various forms has been widely studied due to various application in the field of construction industry, hydraulic fracturing, production of glass, sedative, anti –caking agents, pharmaceutical tablets, anti –reflection coating and in toothpaste [1 –2]. Along with said applications several researchers also repo rted that the luminescence behavior of rare earth and some metals element doped silica are useful in the optical industry [3]. It is reported by various researchers that the SiO 2is a proficient host for luminescence ap plication due to its high transparency, composition adoptability and easy mass production [4]. Silica phosphor is reproducible in the form of single or binary host by means various chemical rou tes. The chemical routes such as sol –gel, hydrothermal and precipitation method are widely used. The combustion route is also employed for the synthesis of SiO 2[5]. SiO 2show minimum losses by the excitation of laser source due to its porosity. Laser photon has higher penetration depth in doped SiO 2due to its porosity and lower density, it transforms the luminescence behaviour of the material to bulk dependent phenomena. Also, it becomes easy to excite maximum possible active centres present in SiO 2host. In the rare earth family, cerium is one of the most extensi vely used activators for the optical applications. Furthermore, the Ce ion exists in two different ox idation states depends on the present host, defect levels and employed synthesis method. The luminescent trivalent Ce3+state is due to the allowed 5d →4f transition and the non– luminescent Ce4+ due to 4d10→4f0transition which rarely show luminescence be havior [5]. The radiative transition from Ce3+state is the origin of the violate –blue emission. However, non –radiative Ce4+state via defect level present in the host. Among the lanthanides, Ce3+species are very popular as an activator in different solid host lattice like single crystals, polycrystalline glassy powders, ceramics and recently in binary oxides [2, 5]. The luminescence features of Ce3+and Ce4+in crystalline and amorphous SiO 2matrix was investigated from both experimental and theoretical points of view [5]. Cathodeluminescence and photoluminescence behavior of SiO 2:Ce nanophosphor was recently reported by our group [5]. Herein, our attempt is to investigate the temperature de pendent laser excited lumine scence (LEL) of spherical nanosized SiO 2nanophosphor and Ce doped SiO 2(silica) nanophosphor at room temperature and also at 185K by using NeCu –30 laser of 248 nm source. The nanophosphors were prep ared by using urea assisted combustion method. Advanced Materials and Radiation Physics (AMRP-2020) AIP Conf. Proc. 2352, 040040-1–040040-5; https://doi.org/10.1063/5.0053482 Published by AIP Publishing. 978-0-7354-4105-7/$30.00040040-1An annealing in reducing atmosphere was used to achieve the better crystallinity and homogeneous distribution of luminescence Ce3+ions in the SiO 2nanophosphor. EXPERIMENTAL Synthesis Method The requisite raw materials with stoichiometric com positions were taken as mentioned in reaction –1. They mixed together in the ceramic mortar. The mixture was th en transferred to a muffle furnace preheated at 550 –600oC. The porous silica was obtained as fine porous powder and noted as as –prepared sample. The sample was further annealed at 1000oC in reducing environment of charcoal for 1hr and noted as annealed sample. The chemical reaction of combustion process is described as follow: SiO 2·nH 2O+2 0 N H 2CONH 2+ 20 NH 4NO 3→ SiO 2+ 80N 2↑ + 29CO 2↑ + 151H 2O↑ –––– (1) The as –prepared and annealed sample was characterized by using X –ray powder diffraction (XRD) and TEM. Laser excited luminescent (LEL) was carried out by using a NeCu –30 as an excitation source of 248 nm source. RESULTS AND DISCUSSION X-ray Diffraction And TEM Analysis X–ray diffraction pattern of as –prepared and annealed SiO 2nanophosphor is shown in figure –1. The XRD figure is taken from our previous report [5] as our group is reported recently. The XRD peaks were assigned by comparing the XRD profile with the respective ICCD data file No. 86 –1565. The X –ray diffraction pattern indicates that formation host matrix is a low –quartz’s glass phase of SiO 2glassy matrix. The major broad h k l peaks at (1 0 0), and other prominent peak at (0 1 1) indicates that it is crystalline low quartz’s phase. The improvement in crystallinity due to clear aapreance of all pha ses of the low quartz’s peaks in the nanophosphor which was annealed at 1000oC for 1 hr was seen from the inset of figure –1. The formation of broad XRD peaks of as –prepared and annealed SiO 2 nanophosphor also indicates that is porous in natu re. The morphology image of the annealed SiO 2nanophosphor was recorded by using TEM analysis and is shown in the figure –2. The formation of porous nanophosphor was confirmed from TEM image. The particles are sp herical in nature with estimates average particle size are around 3 –8 nm in diameter [5]. FIGURE 1XRD pattern of as –prepared SiO 2, annealed SiO 2 nanophosphor and standard JCPDS data file for the comparison [5]. FIGURE 2TEM image of annealed SiO 2nanophosphor. 040040-2Laser Excited Luminescence (LEL) Analysis Laser excited luminescence (LEL) spectrum for annealed SiO 2and, as –prepared and annealed SiO 2:Ce 2m% nanophosphor is shown in the figure –3 (a and b). Laser excited luminescence (LEL) from the nanophosphors was recorded under the excitation of NeCu –30 laser of 248 nm source at room temp erature (RT) and at 185 K. The laser excited luminescence from annealed SiO 2nanophosphor at room temperature was observed as a broad band with maximum emission intensity around 470 nm as shown in figure –3 (a). The board band emission from annealed SiO 2 nanophosphor is attributed to the –O–O–type defect, an O 2–intrinsic defect and an irradiation defect present in the nanophosphor [6]. However, marginal increases in the LE L emission intensity with shift towards lower wavelength for annealed SiO 2nanophosphor was seen at 185 K as shown in figure –3 (b). The observed LEL broad band emission of SiO 2nanophosphor was clearly shown in the inset of figure –3 (a and b). The increase in emission intensity of annealed SiO 2nanophosphor at low temperature may be due to stabilization of shallow defects and improvement in the crystallinity caused by annealing effect [7]. It may pos sible that increase of temperature around the nanophosphor due to high energy incident laser photons was decreases at lo w temperature resulting to reduction in heat energy which converted to increased emission output. In the case of as –prepared SiO 2:Ce 2%nanophosphor at RT, the maximum emission intensity was observed at 390 nm and shoulder around 403 nm as shown in figure –3(a). However, when the measurement temperature decreases to 185 K the increase in the emission intensity was seen maximum at 395 nm and shoulder around 408 nm as shown in figure –3 (b). In the case of annealed SiO 2:Ce 2% nanophosphor at RT, the maximum emission intensity was observed at 400 nm and shoulder around 416 nm as shown in figure –3(a). However, when the measurement temperature decreases to 185 K the maximum emission peak was shifted at 405 nm along with shoulder at 422 nm and also increase in the emission intensity was seen as shown in figure –3 (b). It has been reported in the earlier literatures [8] that at high temperatures all ex citons bound with shallow impurities are dissociated, and free exciton then vanishes at a temperature above 100 K resultin g to higher emission intensity. This is consistent with our results obtained for SiO 2:Ce 2%nanophosphor at 185 K. FIGURE 3 (a and b) Temperature dependent Laser excited luminescence (LEL) spectra of annealed SiO 2 and, as –prepared and annealed SiO 2:Ce 2m%nanophosphor obtained with a NeCu –30 Laser source. 040040-3Furthermore, the observed broad band emission remains unal tered in both the cases as it arises from dopant Ce ion of SiO 2:Ce 2% nanophosphor. The observed red shift in the emissi on wavelength at lower temperature may be due to stoke shift. The observed increase in the emission intens ity at low temperature tells that the stabilization of defects occurs and reduction in the heat ene rgy of incident laser photons play an important role during the process. The prominent emission intensity in the case of SiO 2:Ce 2%nanophosphor between 375 –450 nm attributed to 2D(5d) → 2F5/2,7/2electronic transition of Ce3+ions present in the host matrix [9]. Th e weak PL emission in the case of as – prepared SiO 2:Ce 2%nanophosphor as compared to annealed nanophosphor may be due to lack Ce3+ions. The nanophosphor which was annealed in charcoal environment helps to improve emission intensity due to improved crystallinity and possibly conversion of non –luminescent Ce4+to Ce3+ion. The broad band emission from Ce3+ions being regarded as the electric dipole allowed transition. The observed Ce3+doublet emission is attributed to spin –orbit splitting of the 2F5/2and 2F7/2state. It has earlier been reported that porous silica assist the luminescence active centre to act as a photo carrier and helps in the improvement of emission intensity as compared to its dense nanophosphor [10]. The comparative Laser excited luminescence (LEL) intensity between annealed SiO 2and, as –prepared and annealed SiO 2:Ce 2m%nanophosphor obtained under excitation of NeCu –30 Laser source of 248nm source is shown in the figure –4. The comparative graph as shown in figure –4 shows that the maximum emission intensity was obtained for the nanophosphor when was measured at 185 K as compared to room temperature. FIGURE 4Comparative Laser excited luminescence (LEL) intensity spectra of annealed SiO 2and, as –prepared and annealed SiO 2:Ce 2m%nanophosphor. CONCLUSION The urea assisted combustion method is used for the synthesis of SiO 2and SiO 2:Ce 2m%nanophosphor and later annealed at 1000oC in charcoal environment for 1hr. It has been concluded from the results that the prepared nanophosphors are partly crystalline porous in nature ha ving spherical size particles with average diameter of 3 –8 nm. The annealing under charcoal environment is employed for the homogeneous distribution and conversion of non- luminescent Ce4+to luminescence Ce3+ions. The temperature dependent laser excited luminescence (LEL) was measured for SiO 2 and SiO 2:Ce nanophosphor. The broad band doublet emission around 375 –425 nm from SiO 2:Ce nanophosphor was obtained due to dopant Ce3+ ion with enhancement in emission intensity at 185 K as compared to room temperature measurement. The annealing effect under charcoal environment helps for improvement in the emission intensity along with decrease of temperature. REFERENCES 1. D. Panayotov, P. Kondratyuk, J.T. Yates, Langmuir 20, 3674 –3678, (2004). 2. P.G. Jeelani, P. Mulay, R. Venkat, C. Ramalingam, Silicon 12, 1337 –1354 (2020). 040040-43. I. Tanahashi, H. Inouye, K. Tanaka, A. Mito, Jpn. J. Appl. Phys. 38, 5079 –5082 (1999). 4. W.J. Salcedoa, M.S. Bragab, R.F.V.V. Jaimesc, J Lumin. 199, 109 –111 (2018). 5. I.M. Nagpure, S.S. Pitale, K.G. Ts habalala, V. Kumar, O.M. Ntwaeaborwa, J.J. Terblans, H.C. Swart, Mater. Res. Bull. 46, 2359 –2366 (2011). 6. A.P. Barabana, S.N. Samarina, V.A. Prokofieva, V.A. Dmitrieva, A.A. Selivanova, Y. Petrova, J Lumin. 205 102– 108 (2019). 7. A.S. Lenshin, P.V. Seredin, V.M. Kashkarov, D.A. Minakov, Mater Sci Semicond Process 64, 71 –76 (2017). 8. Y. Isokawa, H. Kimura, T. Kato, N. Kawaguchi, T. Yanagida, Opt. Mater. 90187–193 (2019). 9. X. Yang, K.Y. Law, L.J. Brillson, J. Vac. Sci. Technol. A 15880–885 (1997). 10. J. M. Knaup, P. Deák, Th. Frauenheim, Physical Review B 72, 115323 –9 (2005) 040040-5

5.0046766.pdf

APL Mater. 9, 041105 (2021); https://doi.org/10.1063/5.0046766 9, 041105 © 2021 Author(s).Composition dependence of spin– orbit torques in PtRh/ferromagnet heterostructures Cite as: APL Mater. 9, 041105 (2021); https://doi.org/10.1063/5.0046766 Submitted: 06 February 2021 . Accepted: 18 March 2021 . Published Online: 01 April 2021 Guoyi Shi , Enlong Liu , Qu Yang , Yakun Liu , Kaiming Cai , and Hyunsoo Yang COLLECTIONS Paper published as part of the special topic on Emerging Materials for Spin-Charge Interconversion ARTICLES YOU MAY BE INTERESTED IN Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Spin–orbit torque characterization in a nutshell APL Materials 9, 030902 (2021); https://doi.org/10.1063/5.0041123 Highly efficient charge-to-spin conversion from in situ Bi 2Se3/Fe heterostructures Applied Physics Letters 118, 062403 (2021); https://doi.org/10.1063/5.0035768APL Materials ARTICLE scitation.org/journal/apm Composition dependence of spin–orbit torques in PtRh/ferromagnet heterostructures Cite as: APL Mater. 9, 041105 (2021); doi: 10.1063/5.0046766 Submitted: 6 February 2021 •Accepted: 18 March 2021 • Published Online: 1 April 2021 Guoyi Shi,1,2Enlong Liu,2 Qu Yang,2Yakun Liu,2Kaiming Cai,2 and Hyunsoo Yang1,2,a) AFFILIATIONS 1NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore 119077, Singapore 2Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore Note: This paper is part of the Special Topic on Emerging Materials for Spin-Charge Interconversion. a)Author to whom correspondence should be addressed: eleyang@nus.edu.sg ABSTRACT We experimentally study the spin–orbit torque (SOT) in PtRh/heterostructures by varying the composition of PtRh alloy. By performing dc- biased spin-torque ferromagnetic resonance and second-harmonic measurements in Pt xRh1−x/ferromagnet heterostructures, we find that the effective damping-like spin-torque efficiency and spin Hall conductivity are 0.18 and 3.8 ×105̵h/2eΩ−1m−1for Pt 0.9Rh0.1, respectively, with a low resistivity of 46.9 μΩcm. Furthermore, current induced SOT switching in PtRh/Co is investigated. The critical current density for SOT switching decreases with an increase in the Rh composition of the PtRh alloy, which can be understood by domain wall assisted switching. Due to a large spin Hall conductivity, a relatively low resistivity, and sustainability of the high temperature process, the PtRh alloy could be an attractive spin source for SOT applications. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0046766 I. INTRODUCTION Spin currents generated by the spin Hall effect and/or inter- facial spin–orbit coupling effect can exert torque on the magne- tization. This so-called spin–orbit torque (SOT) has been demon- strated to effectively switch the magnetization,1–6drive domain wall motion,7–9and cause precession of magnetization.10The conver- sion efficiency of charge current to spin current is characterized by the effective spin-torque efficiency or spin Hall conductivity, which can be extracted by spin-torque ferromagnetic resonance (ST-FMR), inverse spin Hall, magneto-optic Kerr, second-harmonic, and pla- nar Hall effects.11–19So far, various materials have been utilized to study spin–orbit torques. First of all, heavy metals such as Pt, Ta, W, and Hf have been extensively utilized as a spin source as the strength of spin–orbit coupling is expected to be stronger in heav- ier elements.1–9Second, interfacial Rashba systems such as Bi/Ag20 and Bi 2Se3/Ag21have been studied. Third, topological insulators are proposed to have a large effective spin-torque efficiency originated from spin-momentum locking and can even switch the magneti- zation.22–28Recently, two-dimensional materials such as transition metal dichalcogenides WTe 2, MoTe 2, PtTe 2, and TaS 2have been explored.29–33From a technological point of view, the spin–orbit torque mag- netic random access memory (SOT-MRAM) is a candidate of next generation non-volatile memory because of its fast speed and low power consumption.34The power consumption for switching the magnetization per unit volume is proportional to ρSS⋅j2 SS, where ρSS is the resistivity of the spin source and jSSis the critical current density flowing through the spin source, which is inversely pro- portional to the effective spin-torque efficiency of the spin source. Therefore, a spin source with a high conductivity and large effec- tive spin-torque efficiency, that is, a large spin Hall conductivity, is desirable. Studies have been carried out in alloying heavy met- als such as Bi, Pb, Pt, Ir, and Ta to increase the effective spin- torque efficiency.35–39In particular, PdPt and AuPt alloys have been demonstrated to have a large effective spin-torque efficiency.40,41 Realizing deterministic SOT switching of the magnetization with perpendicular anisotropy requires inversion symmetry breaking that is typically realized by applying an external magnetic field along the current direction, which hinders the real application of SOT devices. Several techniques such as exchange-bias field,42,43lateral structure asymmetry,44out-of-plane polarized spin currents,29,30,45 and some other techniques were proposed to realize field-free SOT switching.46–49 APL Mater. 9, 041105 (2021); doi: 10.1063/5.0046766 9, 041105-1 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm In this work, we systematically study the SOT in the PtRh /ferromagnet heterostructures. Pt has a large intrinsic spin Hall conductivity and relatively low resistivity. Rh is a 4 dtransition metal with a face-centered cubic structure and has a relatively low resis- tivity as well. We first use the dc-biased ST-FMR technique11,33,50to quantify the effective damping-like and field-like spin-torque effi- ciency. Then, we perform second-harmonic and current-induced SOT switching measurements. We find a large spin Hall conduc- tivity of 3.8 ×105̵h/2eΩ−1m−1for Pt 0.9Rh0.1with a relatively low resistivity of 46.9 μΩcm, indicating that PtRh could be a useful spin source. II. EXPERIMENTAL DETAILS The multilayer structure of Si substrate/Ta (2)/Pt xRh1−x(6)/Py (2)/MgO (2)/Ta (1.5) (thicknesses in nanometers, x =0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1), where Py is Ni 81Fe19, is deposited for ST-FMR measurements by magnetron sputtering at a base pressure of 5×10−9Torr. Pt xRh1−xis deposited by co-sputtering Pt and Rh targets. For the Pt concentration x ≥0.5, the power of Pt is fixed at 60 W, and the power of Rh is varied to adjust the composition. For the Pt concentration x ≤0.4, the power of Pt is tuned, while the power of Rh is fixed at 60 W. The films are patterned into rectangular strips with a dimension of 70 ×10μm2by lithography and Ar ion etching. The coplanar waveguide (CPW) with the gap between the signal and the ground electrode of 15 μm is fabricated by sputtering and lift-off processes. In order to investigate the second-harmonic based spin-torque efficiency and the SOT switching properties, we also deposit per- pendicular magnetic anisotropy (PMA) Co on top of the PtRh alloy. The film structure of thermally oxidized Si substrate/Ta (2)/Pt xRh1−x (6)/Co (0.8)/Pt (0.3)/MgO (2)/Ta (1.5) (thicknesses in nanometers, x=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1) is prepared. Then, the films are annealed at 300○C with an external field of 5 kOe applied perpendicular to the films for 0.5 h in order to enhance the PMA of the Co layer. After that, the films are fabricated into Hall- bar devices with a channel width of 10 μm using optical lithography and Ar etching, and the electrodes are patterned using lithography and lift-off processes.III. RESULTS Figure 1(a) shows the XRD characterization results of the thermally oxidized Si substrate/Ta (2)/Pt xRh1−x(20) (thicknesses in nanometers, x =0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1). With the increase in Rh concentration, the (111) peak of the PtRh alloy shifts from 39.5○to 41.1○and the peak inten- sity also decreases. Utilizing Bragg’s law, the lattice constant ( c) of Pt and Rh are determined to be 3.951 and 3.804 Å, respec- tively, which is comparable to the lattice constant of bulk Pt (3.924 Å) and Rh (3.803 Å). The resistivity of 6 nm thick PtRh films as a function of the Pt composition ( x) measured by the four probe method is shown in Fig. 1(b). Figure 2(a) shows the schematic of ST-FMR devices and the measurement configuration. A rf current is applied to devices, and an external field is simultaneously swept in the xyplane with an angle of 38○with respect to the xdirection.27,30,37The rf charge current generates a transverse rf spin current in Pt xRh1−x, which is injected into the adjacent Py layer and exerts oscillating SOTs on the Py mag- netic moments. At the same time, the additional dc current can also generate spin currents and exert SOT on Py, which modifies the res- onance field Hresand effective damping αeff=∣γ∣/(2πf)⋅(W−W0). Here, γis the gyromagnetic ratio, fis the rf current frequency, Wis the linewidth, and W0is the inhomogeneous linewidth broadening. These torques together with the Oersted field cause the precession of Py magnetization, and the anisotropic magnetoresistance of the device also oscillates with the same frequency. As a result, the oscil- lating resistance and charge current produce a dc mixing voltage (Vmix), which is detected by a lock-in amplifier. We apply rf cur- rents with a fixed frequency of 4–6.5 GHz and a nominal microwave power of 13 dBm to ST-FMR devices while sweeping the external field ( Hext) from−1.1 to 1.1 kOe. Typical ST-FMR dc mixing voltages for Pt 0.9Rh0.1/Py samples are shown in Fig. 2(b). Figure 2(c) shows that the Hresand ffol- lows the Kittel equation f=γ 2π√ Hres(Hres+Meff), where Meffis the effective magnetization. We extract the Meffof 0.665 T. The effect of dc current on the ST-FMR data is shown in Fig. 2(d), and the linewidth decreases (increases) and the resonance field increases (decreases) after applying 2 mA ( −2 mA) dc current. Figure 2(e) FIG. 1. (a) XRD data of Si substrate/Ta (2)/Pt xRh1−x(20) (thicknesses in nanometers) with different Pt concentrations. (b) Resistivity of 6 nm thick PtRh films vs Pt concentration, x. APL Mater. 9, 041105 (2021); doi: 10.1063/5.0046766 9, 041105-2 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm FIG. 2. (a) Schematic of the ST-FMR measurement setup. (b) ST-FMR data of Ta (2)/Pt 0.9Rh0.1(6)/Py (2)/MgO (2)/Ta (1.5) (thicknesses in nanometers) by applying rf currents. (c) Frequency vs resonance field Hres. The solid line represents the Kittel fitting. (d) ST-FMR data under ±2 mA dc currents. Linewidth W(e) and resonance field normalized by field direction φ(f) vs dc current ( IDC). The effective damping-like spin-torque efficiency ξDL(g) and the effective field-like spin-torque efficiency ξFL(h) vs Pt concentration, x. APL Mater. 9, 041105 (2021); doi: 10.1063/5.0046766 9, 041105-3 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm FIG. 3. (a) Schematic of the Hall bar device. (b) AHE curves of Ta (2)/Pt xRh1−x(6)/Co (0.8)/Pt (0.3)/MgO (2)/Ta (1.5) (thicknesses in nanometers) samples with the external field along the zandxdirections. (c) Extracted coercivity ( Hc) and AHE resistance ( RAHE) for samples with different Pt concentrations, x. The first (d) and second (e) harmonic voltage for the Pt 0.9Rh0.1/Co sample. (f) Calculated damping-like effective field ( HDL) of the Pt 0.9Rh0.1sample under different ac current densities ( j). (g) Effective damping-like spin-torque efficiency ( ξDL) and spin Hall conductivity ( σSH) vs Pt concentration from harmonic measurements. APL Mater. 9, 041105 (2021); doi: 10.1063/5.0046766 9, 041105-4 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm depicts the linear relationship between the linewidth Wand dc cur- rent. The effective damping-like spin-torque efficiency can be cal- culated by ξDL=2∣e∣ ̵h(Hres+Meff/2)μ0MstF sinφ∣Δαeff ΔjPtRh∣,11,33,50where eis the elec- tron charge, μ0is the vacuum permeability,̵his the reduced Planck constant, MsandtFare the saturation magnetization and layer thick- ness of Py, φdenotes the external field direction with respect to the xdirection, and jPtRhis the current density passing through the PtRh layer. We find that the effective damping-like spin-torque efficiency of Pt 0.9Rh0.1is 0.081±0.008. The field-like torques generated by the dc current together with the dc current induced Oersted field shift the resonance field as shown in Fig. 2(f). The dc current induced Oersted field is calcu- lated by Ampère’s law, HOe=0.5jPtRhtPtRh, where tPtRh is the thick- ness of PtRh. By subtracting the contribution from the Oersted field, the effective field-like spin-torque efficiency can be obtained by ξFL =2∣e∣ ̵hμ0MstF∣HFL jPtRh∣, where HFLis the field-like effective field extracted from the shift of the resonance field by subtracting the contribution of the Oersted field.50We find that the effective field-like spin-torque efficiency of Pt 0.9Rh0.1is 0.017±0.002. The calculated ξDLand ξFL are shown in Figs. 2(g) and 2(h), respectively, with different com- positions. For pure Rh and Pt, the ξDLis 0.038±0.006 and 0.074 ±0.006, respectively. The ξDLof the PtRh alloy increases up to 0.081 ±0.008 for Pt 0.9Rh0.1. The ξFLincreases from 0.005 ±0.004 for pure Rh to 0.018 ±0.003 for pure Pt.We then investigate the second harmonic based ξDLof PtRh/Co. The device structure and measurement setup are depicted in Fig. 3(a). Utilizing the anomalous Hall effect (AHE), we can detect the PMA of the samples. We apply the current of 0.5 mA and measure the Hall voltage while sweeping the external field along the zand xdirections. As shown in the left column of Fig. 3(b), for samples with x ≥0.3, the square AHE loops obtained by sweeping the external field along the zdirection ( Hz) indi- cate good PMA of the Co layer. On the other hand, samples with x≤0.2 show in-plane anisotropy (IPA). When the in-plane mag- netic field ( Hx) is large enough to align the magnetization on the xyplane, the AHE resistance approaches zero in the right col- umn of Fig. 3(b), as the AHE resistance is proportional to the zcomponent of Co magnetization. The saturation field with Hx can be used to estimate the anisotropy field, which increases with an increase in the Pt composition. We extract the coercive field (Hc) and AHE resistance ( RAHE) summarized in Fig. 3(c). With the increase in Pt concentration, both Hcand RAHE tend to increase overall. Second-harmonic measurements are exploited to determine theξDLof PtRh. We apply the ac current along the xdirection and record the in-phase first and out-of-phase second-harmonic Hall voltages while sweeping the external field in the xzplane with 2○tilted with respect to the xaxis. Corresponding results for the Pt0.9Rh0.1/Co sample are shown in Figs. 3(d) and 3(e). The first FIG. 4. SOT switching of Pt 0.9Rh0.1/Co under different amplitudes of positive (a) and negative (b) external magnetic fields. (c) Msvs concentration x. (d) Calculated normalized Jcand extracted critical switching current density jcunder an external field of 250 Oe for samples with different Pt concentrations. APL Mater. 9, 041105 (2021); doi: 10.1063/5.0046766 9, 041105-5 © Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm TABLE I. Comparison of reported effective spin-torque efficiency ( ξDL), resistivity ( ρ), and spin Hall conductivity ( σSH) of various spin sources. Material ξDL ρ(μΩcm) σSH(×105̵h/2eΩ−1m−1) Reference Cu 0.25Pt0.75 0.08 59 1.4 37 Cu 0.16Ta0.84 0.04 200 0.2 39 Pd0.25Pt0.75 0.26 57.5 4.5 40 Au 0.25Pt0.75 0.3 83 3.6 41 Pt(high- ρ) 0.16 44.2 3.6 This work Pt(low- ρ) 0.06 20 3 11 β-Ta 0.15 190 0.8 3 β-W 0.33 260 1.3 5 Bi2Se3 2–3.5 1754 1.1–2 22 WTe 2 0.8 580 1.4 30 PtTe 2 0.15 750 0.2 31 TaS 2 0.25 16.9 14.9 33 Pt0.9Rh0.1 0.18 46.9 3.8 This work harmonic voltage ( V1ω xy) in Fig. 3(d) shows a typical parabolic curve for the Co magnetization pointing up or down, while the second- harmonic signals ( V2ω xy) in Fig. 3(e) show straight lines for the up and down Co magnetization. The damping-like effective field ( HDL) can be determined by HDL=−2∂V2ω xy ∂Hx/∂2V1ω xy ∂Hx2.13Here, we note that the planar Hall correction is not considered in our analysis as it might overestimate the spin-torque efficiency, as discussed in Ref. 40. We estimate the HDLfor different applied ac currents. As shown in Fig. 3(f), HDLlinearly increases with the current density and the slope can be used to extract the effective damping-like spin- torque efficiency through ξDL=(2eM st/̵h)⋅HDL/jPtRh. The spin Hall conductivity ( σSH) is calculated using σSH=(̵h/2e)ξDL/ρPtRh, where ρPtRh is the resistivity of the PtRh alloy. The ξDLand σSHusing the second-harmonic measurements for the PtRh alloy are summarized in Fig. 3(g). As the Pt concentration increases, both ξDLand σSH increase overall. The largest ξDLis 0.18±0.01 and the largest σSH is 3.8±0.12×105̵h/2eΩ−1m−1for Pt 0.9Rh0.1. It is worth mention- ing that the ξDLmeasured by the second-harmonic measurements is larger than that of dc-biased ST-FMR. This might be due to the inter- face transparency.51We also measure harmonic signals by sweep- ing the external field in the yzplane with 2○tilted with respect to theyaxis in order to extract the field-like torque, but the signal is negligible, as shown in the inset of Fig. 3(e). We next study the SOT switching of PtRh/Co heterostructures. A current pulse with a pulse width of 150 μs and different amplitudes is applied along the xdirection of the Hall bar devices, and the Hall voltage is probed while a fixed magnetic field is applied along the current direction to break the symmetry. Figures 4(a) and 4(b) show the deterministic magnetization switching of Pt 0.9Rh0.1/Co samples. For positive external fields, the Co magnetization favors down for positive currents and favors up for negative currents, manifesting clockwise switching, whereas for negative external fields, the switch- ing is anticlockwise, which is a typical SOT switching feature of posi- tive spin-torque efficiency materials. As expected, the critical current density ( jc) reduces as the amplitude of external fields increases. We extract the jcfor devices with different compositions under the exter- nal field of 250 Oe and summarize the results in Fig. 4(c). With anincrease in the Pt composition (x), jcincreases. For micrometer size devices, SOT switching is governed by SOT-assisted domain wall propagation, and thus, the critical current density is proportional to MsHc/ξDL.7,39,52TheMsdata of samples with different Pt concentra- tions are shown in Fig. 4(c). It increases and then slightly decreases as the Pt concentration increases. Using the measured Ms,Hc, and ξDLvalues, we normalize the value of MsHc/ξDLwith different com- positions and define as Jc. As shown in Fig. 4(d), the Jcfollows the same trend as the measured jc. Table I shows a comparison of the ξDL,ρ, and σSHof various materials. The σSHof Pt 0.9Rh0.1is larger than most of the spin sources and close to that of Pd 0.25Pt0.75, and theρof Pt 0.9Rh0.1is smaller than that of Pd 0.25Pt0.75and close to the value of pure Pt. IV. CONCLUSION We have studied the SOT in the PtRh alloy. Utilizing dc- biased ST-FMR and second-harmonic measurements, we determine the effective spin-torque efficiency and spin Hall conductivity of the PtRh alloy. We find that the spin Hall conductivity of Pt can be enhanced by alloying with Rh. The spin Hall conductivity is 3.8×105̵h/2eΩ−1m−1for Pt 0.9Rh0.1with a relatively low resistivity of 46.9 μΩcm. The critical SOT switching current density increases with an increase in the Pt composition, which can be explained by SOT-assisted domain wall propagation. With a large spin Hall con- ductivity and small resistivity, PtRh could be a promising spin source for SOT applications. AUTHORS’ CONTRIBUTIONS G.S. and H.Y. designed the experiments. G.S. and Q.Y. prepared the film samples. G.S. and Y.L. fabricated the devices. G.S., E.L., and K.C. carried out the ST-FMR, second harmonic, SOT switching, and XRD measurements. G.S. and H.Y. analyzed data and wrote the manuscript. H.Y. supervised the research. 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J. Chem. Phys. 154, 224116 (2021); https://doi.org/10.1063/5.0050902 154, 224116 © 2021 Author(s).Low communication high performance ab initio density matrix renormalization group algorithms Cite as: J. Chem. Phys. 154, 224116 (2021); https://doi.org/10.1063/5.0050902 Submitted: 19 March 2021 . Accepted: 21 May 2021 . Published Online: 14 June 2021 Huanchen Zhai , and Garnet Kin-Lic Chan COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Requirements for an accurate dispersion-corrected density functional The Journal of Chemical Physics 154, 230902 (2021); https://doi.org/10.1063/5.0050993 Chemical physics software The Journal of Chemical Physics 155, 010401 (2021); https://doi.org/10.1063/5.0059886 Many-body van der Waals interactions beyond the dipole approximation The Journal of Chemical Physics 154, 224115 (2021); https://doi.org/10.1063/5.0051604The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Low communication high performance ab initio density matrix renormalization group algorithms Cite as: J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 Submitted: 19 March 2021 •Accepted: 21 May 2021 • Published Online: 14 June 2021 Huanchen Zhaia) and Garnet Kin-Lic Chanb) AFFILIATIONS Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA a)Electronic mail: hczhai@caltech.edu b)Author to whom correspondence should be addressed: gkc1000@gmail.com ABSTRACT There has been recent interest in the deployment of ab initio density matrix renormalization group (DMRG) computations on high per- formance computing platforms. Here, we introduce a reformulation of the conventional distributed memory ab initio DMRG algorithm that connects it to the conceptually simpler and advantageous sum of the sub-Hamiltonian approach. Starting from this framework, we further explore a hierarchy of parallelism strategies that includes (i) parallelism over the sum of sub-Hamiltonians, (ii) parallelism over sites, (iii) parallelism over normal and complementary operators, (iv) parallelism over symmetry sectors, and (v) parallelism within dense matrix multiplications. We describe how to reduce processor load imbalance and the communication cost of the algorithm to achieve higher efficiencies. We illustrate the performance of our new open-source implementation on a recent benchmark ground-state calculation of benzene in an orbital space of 108 orbitals and 30 electrons, with a bond dimension of up to 6000, and a model of the FeMo cofactor with 76 orbitals and 113 electrons. The observed parallel scaling from 448 to 2800 central processing unit cores is nearly ideal. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0050902 I. INTRODUCTION The Density Matrix Renormalization Group (DMRG) algo- rithm1,2is established as a method to obtain highly accurate low- energy eigenstates of ab initio quantum chemistry Hamiltonians.3–15 While multiple techniques can now solve for low-energy eigen- states to high precision in problems that are formally beyond the reach of full configuration interaction,16–20the DMRG provides a unique capability to treat problems with a large number of open shells.21,22Consequently, it is particularly useful in active space prob- lems where a large fraction of the orbitals have open-shell character, for example, as found in molecular clusters with multiple open- shell transition metal centers.23–28In many such problems, teasing out the relevant chemistry requires not only a single ground-state energy calculation but also the characterization of many compet- ing low-energy states. For such applications, improving the scala- bility and efficiency of current DMRG implementations is highly desirable.Over the last two decades, many different strategies have been proposed to parallelize the DMRG algorithm in quantum chemistry. These include the following: (i) Parallelism within dense matrix multiplications.29,30This is a fine-grained parallelism that is effective when the size of the dense matrices is sufficiently large [namely, when a large Matrix Product State (MPS) bond dimension Mis used]. It can be implemented simply by linking the code to a paral- lelized shared-memory math library. (ii) Parallelism over symmetry sectors,11,30which is available when the DMRG is implemented with symmetry restric- tions. Typically, particle number, total spin or projected spin, and spatial symmetry are used in ab initio DMRG implementations. (iii) Parallelism over normal and complementary operators.31,32 This is often considered the largest source of parallelism for typical ab initio problems. J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (iv) Parallelism over a sum of sub-Hamiltonians.33This is a coarse-grained parallelism with a very low communication cost and is easy to express in a Matrix Product Operator (MPO) description of the DMRG. (v) Parallelism over sites.34For a large number of sites, this is an additional source of coarse-grained parallelism. Such an implementation relies on the transformation of the MPS to a form with multiple canonical centers. Recently, Brabec et al. reported a non-spin-adapted massively parallel implementation of DMRG for quantum chemistry using strategies (ii) and (iii).35We also note promising recent progress in graphics processing unit (GPU) accelerated parallel DMRG algo- rithms.36–38However, to the best of our knowledge, there has not been an implementation that utilizes all five sources of parallelism in a scalable DMRG code for ab initio problems. This may be partly ascribed to the fact that strategies (iv) and (v) are most conve- niently implemented in a DMRG code39,40that is structured using an MPO/MPS formalism,33,41while many other efficient ab initio DMRG implementations42,43using strategies (i), (ii), and (iii) are organized around the construction and transformation of renormal- ized operators.6 In this work, we first reformulate strategy (iii) for a distributed memory model using the sum of sub-Hamiltonian language. This demonstrates that a low communication version of strategy (iii) can, in fact, also be viewed as a variant of strategy (iv). This analysis constitutes Secs. II A and II B. In Sec. II C, we discuss how the load imbalance that arises in strategy (v) can be alleviated via the dynamical determination of connection sites. In Secs. II D–II F, we briefly introduce the shared-memory parallelism strategies (i), (ii), and (iii) used in this work. In Sec. II G, we briefly discuss a few lower level implementation details. Next, in Sec. III, we illustrate the computational performance of our new implementation of the parallel DMRG for a recent ground-state benzene benchmark20in a polarized valence double zeta basis.44Although not a correlated or open-shell system that is particularly suited to the DMRG, the size of the calculation serves to illustrate the scalability of our algorithm. For a correlated electron problem with many open shells that is more suited for the DMRG, we also consider a model of the FeMo cofac- tor system45and observe that similar scaling can be achieved. Finally, the conclusions are given in Sec. IV. II. THEORY Rather than reintroduce the DMRG formalism here, we summarize the background theory and notation for the serial DMRG algorithm1,2and the SU(2) (spin-adapted) ab initio DMRG algorithm32,42,46in Appendixes A and B, respectively. We encour- age readers unfamiliar with the standard DMRG algorithm and terminology to first consult Appendixes A and B. A. Parallelism over renormalized operators In most parallel implementations of ab initio DMRG, the most important source of distributed memory parallelism comes from distributing the left–right renormalized operator decomposition of the Hamiltonian, as discussed in Ref. 31. In this approach, “normal” and “complementary” renormalized operators (see Appendix B for definitions) are assigned to different processors according to their orbital indices.The leading communication cost per sweep in the approach described in Ref. 31 is O(16M2K2logPhamil)from the blocking step, where Mis the MPS bond dimension, Kis the number of sites, and Phamil is the total number of processors (processor cores) at this par- allelism level. The sub-leading term in the communication cost is O(M2K2logPhamil)from the transformation (rotation) step. In order to achieve better scalability, it is desirable to reduce the communication cost. For this purpose, we note that the leading and sub-leading terms in the communication cost in the above approach mainly come from the accumulation of the RL/R[1 2] i operators [defined in Eq. (B7)]. Therefore, the communication cost can be greatly reduced by never accumulating RL/R[1 2] i . Specifically, we can arrange for each processor to compute and store a partial contribu- tion to RL/R[1 2] i for all indices i. Compared to the original scheme, this new scheme only needs to communicate when accumulating the wavefunction, with a communication cost of O(16M2KlogPhamil) per sweep. However, since all partial components of the RL/R[1 2] i operators have to enter into the solving (Davidson) step, the com- putational cost for the solving step increases from O(M3(K3 +K2)/Phamil)toO(M3K3/Phamil+M3K2)per sweep. The total disk storage cost also increases from O(M2(K3 +K2))toO(M2K3+M2K2Phamil). For the typical case where , the increase in the sub-leading term of the storage is not a large concern. We note that in this new scheme, the communication of renor- malized operators is completely removed. In other words, each pro- cessor performs blocking, solving, and transformation steps for a part of the Hamiltonian—i.e., a sub-Hamiltonian—independently, and only wavefunctions from the solving step are communicated. This motivates a more general picture where we can develop low communication algorithms that are formulated in terms of sub-Hamiltonians, rather than the left–right decomposition of the Hamiltonian. B. Parallelism over sub-Hamiltonians For this purpose, we write the ab initio Hamiltonian Eq. (B1) as ˆH=ˆH(1)+ˆH(2)+⋅⋅⋅+ ˆH(Phamil), (1) where ˆH(p)is the sub-Hamiltonian assigned to processor p. To describe this assignment, we can write ˆH(p)=1 2∑ ij,σ[proc(p,i)+proc(p,j)]tij,σa† iσajσ +1 2∑ ijkl,σσ′proc(p,i,j,k,l)vijkl,σσ′a† iσa† kσ′alσ′ajσ, (2) where proc (p,⋅⋅⋅)defines the mapping from orbital indices to pro- cessor rank p(p=1, 2,⋅⋅⋅,Phamil). There is clearly much freedom in choosing the definition of these mappings. A possible definition of proc (p,⋅⋅⋅)is J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp proc(p,i)=⎧⎪⎪⎨⎪⎪⎩1 p≡i(mod Phamil) 0 otherwise,(3) proc(p,i,j)=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩1 p≡(j−1)j 2+i(mod Phamil)andi≤j 1 p≡(i−1)i 2+j(mod Phamil)andi>j 0 otherwise, and proc (p,i,j,k,l)has the same value for any permutation of parameters i,j,k,l, namely, proc(p,i,j,k,l)=proc(p, sorted: i,j,k,l). (4) As discussed above, we can think of a modified version of the normal–complementary operator parallelism as arising from a particular decomposition into sub-Hamiltonians. In particular, the normal/complementary (NC) renormalized operator partition [Eq. (B5)] corresponds to proc(p, sorted: i,j,k,l)=⎧⎪⎪⎨⎪⎪⎩proc(p,j), j=k proc(p,i,j), otherwise,(5) while the complementary/normal (CN) renormalized operator par- tition [Eq. (B6)] corresponds to proc(p, sorted: i,j,k,l)=⎧⎪⎪⎨⎪⎪⎩proc(p,j), j=k proc(p,k,l), otherwise.(6) Here, the notation proc (p, sorted : i,j,k,l)means that i≤j ≤k≤l. Based on the above definitions, the symmetry condition proc(p,i,j)=proc(p,j,i)is satisfied. This is important for efficiency, since the operator symmetry conditions used for efficient DMRG algorithms [Eq. (B8)] can still be used on each processor without any communication. To see how this assignment of Hamiltonian terms gives the cor- rect scaling for multiple processors, we note that in the NC partition in Eq. (B5), the summation over two-index operators is over indices in the left block of sites L, which are the small indices i,jin the tuple i,j,k,l, and thus, in Eq. (5), the indices i,jare used for the proces- sor assignment. For similar reasons, the large indices k,lare used in the CN partition case. This ensures that the total number of terms in the left–right decomposition of the effective Hamiltonian on each processor is roughly O(K2/Phamil)(if ). It is worth noting that Eq. (1) is in the same spirit as the sum of MPO formulation first introduced in Ref. 33. This is often con- sidered a different strategy from the strategy of parallelism over renormalized operators. The two methods, indeed, have a very dif- ferent origin and motivation. However, our new formulation of the low communication version of the parallelism over renormalized operators establishes a clear connection between the two methods. In addition, we find that this new formulation inherits the most important advantages from both methods:(i) Low communication time.33Since each sub-Hamiltonian can be manipulated completely independently, only the com- munication of the (small) wavefunction obtained from the DMRG solving step is required. (ii) Simple implementation. To parallelize a serial ab initio DMRG code, one only needs to start with a distributed inte- gral file, where for each processor, some integrals tijandvijkl are set to zero according to Eq. (2). A single communica- tion step then needs to be added to accumulate wavefunctions from all processors. No other part of the code needs to be changed significantly. (iii) Compatibility with both the exact renormalized operator and compressed MPO DMRG formalisms. Because the descrip- tion of our algorithm does not rely on specific definitions and choices of normal and complementary operators,3one has great freedom to decompose each sub-Hamiltonian. For example, we have presented examples that correspond to the conventional exact NC or CN renormalized operator parti- tions6and their corresponding MPOs, but other MPO rep- resentations of the sub-Hamiltonians, including compressed representations,33can be used. In this work, we will only use exact MPO representations of the sub-Hamiltonians. (iv) Compatibility with index-symmetry conditions. We note that the previous description of sub-Hamiltonians in Ref. 33 was based on splitting the Hamiltonian based on single- site indices. This has the disadvantage that it becomes dif- ficult to use the two-index symmetry conditions Eq. (B8) to reduce the computational cost associated with each sub- Hamiltonian in a distributed setting, because a single- site-index based processor assignment can easily assign index-symmetry related operators to different processors. The current two-index based splitting does not have this problem since ˆOijand ˆOjiare always assigned to the same processor. (v) Load balance between processors. The two-index based assignment assigns roughly equal amounts of work to differ- ent processors, if . (vi) Compatibility with the sum of sub-Hamiltonians and auto- mated MPO construction. In Ref. 33, it was demonstrated that by expressing the ab initio Hamiltonian as a sum of Ksub- Hamiltonians, we can work with KMPOs each with bond dimension O(K), instead of one MPO with bond dimension O(K2). The advantage is that this captures the primary spar- sity within the single MPO representation of the Hamilto- nian. This means that a simple MPO construction of the sub- Hamiltonians, which uses only dense matrices, produces the correct serial computational cost of O(M3K3+M2K4), rather than the naïve (and incorrect) cost of O(M3K3+M2K5)aris- ing from a single dense MPO representation of H, making the correct scaling of the ab initio implementation very easy to achieve. In particular, this is attractive when combined with various automated MPO construction approaches, which then do not need to implement sparse tensor algebra.33,40,41,47 The two-index based sum of sub-Hamiltonians retains this attractive feature, but has a further advantage that the computational prefactor (e.g., from the sub-MPO bond dimensions) is smaller, when compared with the previous one-index decomposition. J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In this work, we combine the low communication scheme based on sub-Hamiltonians with a mixed NC/CN partition11to achieve high efficiency. The mixed NC/CN partition introduces additional costs for computation and communication at the middle site of the sweep. The details are discussed in Sec. III A. C. Parallelism over sites A more recent approach to coarse-grained parallelism in the DMRG is the “real space parallel DMRG” approach introduced by Stoudenmire and White,34,48which has been shown to give near- ideal scaling in some calculations with model Hamiltonians and very recently for quantum chemistry Hamiltonians.49An implementa- tion of this approach for (non-spin-adapted) quantum chemistry Hamiltonians can also be found in the QCM AQUIS code.40 The approach relies on a representation of the MPS with mul- tiple canonical centers.34Each extra canonical center can be intro- duced by first performing a singular value decomposition (SVD) on the effective wavefunction [given in Eq. (A11)] at the original canonical center k, Ψ[k]eff=L[k]S[k]R[k]. (7) Then, we can write Ψ[k]eff=Ψ1[k]effS[k]−1Ψ2[k+1]eff, (8) where Ψ1[k]eff=L[k]S[k], Ψ2[k+1]eff=S[k]R[k](9) are the two new canonical centers at sites kand k+1. Once we have two canonical centers in the MPS, two partial DMRG sweeps, namely, a backward sweep starting from site kand a forward sweep starting from site k+1, can be performed simultaneously by sepa- rate processors. The above approach can be invoked iteratively to generate Psitecanonical centers in the MPS, where Psiteis the total number of (groups of) processors at this level of parallelism. Matrix S[k]−1(termed the connection matrix) is used after a round of for- ward and backward partial sweeps to merge the updated Ψnew 1[k]eff andΨnew 2[k]effto yield an approximation to the updated Ψnew[k]eff, Ψnew[k]eff=Ψnew 1[k]effS[k]−1Ψnew 2[k+1]eff. (10) Merging the two separately optimized portions of the MPS using this connection matrix does not change the MPS when it has reached its variational optimum. The two partial sweeps over sites ⋅⋅⋅,k andk+1,⋅⋅⋅cannot update the MPS bond between the sites kand k+1. Therefore, a sweep iteration at the connection site is per- formed, where Eq. (A12) is solved for the merged wavefunction Ψnew[k]eff. The solution of Eq. (A12), denoted as Ψ′new[k]eff, is then split according to Eq. (8) to generate the updated connection matrix Snew[k]−1. In a typical ab initio application, the amount of computation is not distributed homogeneously among different groups of sites (see Fig. 1) because of the boundary effects of the MPS and the different block sizes from different truncations at different sites. In addition, the total number of sites available for this level of parallelism is lim- ited. If the same number of sites is assigned to different processors, FIG. 1. MPO tensor bond dimensions (dashed lines) and wall time cost (solid lines) at each site for the NC and mixed NC/CN approaches for the benzene system. (Unsimplified refers to the bond dimension obtained without account- ing for zero integrals and symmetry conditions, given by 2 +2K+6k2.) The performance data are from M=4000 calculation with the parallelism scheme Psite=1,Phamil=16,Top=28, and Tdense=1. one observes a significant load imbalance, which negatively impacts the scalability.49 To alleviate this problem, similar to the dynamic boundary strategy used in Ref. 49, we have added an additional step to dynami- cally determine the position of the canonical center (connection site) to improve load balancing. After all processors finish their partial sweeps, the total computational cost is measured for each proces- sor for the partial sweep and all its sweep iterations. From this, it is possible to estimate whether changing the connection site from k tok+2 (for example) reduces the cost discrepancy between the two processors connected at site k. If this is the case, then the connection site is moved to k+2, with the hope that this helps in reducing the degree of load imbalance during the next sweep. We note that changing the position of the connection site between sweeps is not an operation with negligible cost, since not only the MPS tensors but also the renormalized operators need to be transformed [see Eq. (A15)]. Consequently, in our implementa- tion, we have limited the distance between the old connection site and the new connection site to at most two sites. This ensures that the operation itself does not consume a significant amount of time. In practice, we can start the DMRG algorithm with an arbitrary set of connection sites. After several sweeps with dynamical adjustment of the connection sites, we observe that we can often achieve a sta- ble set of connection sites and a well-balanced workload amongst the processors. The performance of the parallelism over sites with the dynamical adjustment of connection sites is discussed in Sec. III C. Comparing to the strategy very recently introduced in Ref. 49, our approach does not directly reduce the waiting time of the cur- rent sweep; instead, the performance statistics of the current sweep are accumulated to determine the position of the connection sites for the next sweep. In contrast, the approach introduced in Ref. 49 completely removes the waiting time at each sweep, but introduces J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp an extra projection error in the wavefunction initial guess [Eq. (10)] when the connection site is changed (which is larger for ab initio systems compared to spin systems, according to Ref. 49). This extra error in the wavefunction transformation may increase the number of Davidson iterations. In order to improve single-node performance, we have also considered the fine-grained strategies for shared-memory paral- lelism.29Most of them can be easily implemented in an ab initio DMRG code with minor modifications. These are discussed here. D. Shared-memory parallelism over normal and complementary operators The left–right decomposition of Hamiltonians [Eqs. (B5) and (B6)] is a sum of products of normal and complementary operators. For the ab initio sub-Hamiltonians, there are O(K2/Phamil+K)terms in the summation. Therefore, for the matrix–vector multiplication, ∣Ψ′[k]eff⟩=ˆH[k]eff∣Ψ[k]eff⟩, (11) invoked during the Davidson procedure, we can divide the work among Topthreads, namely, ˆH[k]eff=ˆH[k]eff (1)+ˆH[k]eff (2)+⋅⋅⋅+ ˆH[k]eff (Top). (12) The partial contribution to ∣Ψ′[k]eff⟩is computed on every thread tas ∣Ψ′[k]eff (t)⟩=ˆH[k]eff (t)∣Ψ[k]eff⟩. (13) Finally, a reduction step is performed to obtain ∣Ψ′[k]eff⟩, as ∣Ψ′[k]eff⟩=∣Ψ′[k]eff (1)⟩+∣Ψ′[k]eff (2)⟩+⋅⋅⋅+∣Ψ′[k]eff (Top)⟩ (14) with a small additional computation cost of O(16M2KlogTop)per sweep. E. Shared-memory parallelism over symmetry sectors In addition, every term in Eq. (13) is implemented as a block- sparse matrix–matrix multiplication, which can be further decom- posed into dense matrix–matrix multiplications over independent symmetry sectors. Instead of using nested threaded parallelism over normal and complementary operators and symmetry sectors, we can collapse the two thread parallelism levels to one level35to achieve a better load balance and reduce the overhead from creating threads. F. Shared-memory parallelism within dense matrix multiplication Thread-level parallelism in dense matrix multiplication can be easily introduced by using a threaded math library.29The effective- ness of this lowest level of parallelism is analyzed in Sec. III B. G. Numerical implementation We have used many low-level performance optimizations in the numerical implementation of the DMRG algorithm. We have reused many of the ideas in the S TACK BLOCK implementation,50whichhas been developed in our research group over many years.6,31,42 Specifically, we store renormalized operators using stack memory to reduce memory fragmentation, we delay tensor contraction in the blocking step, and we use specialized routines for operations involving identity matrices or matrices with only a single non-zero element. In the new B LOCK 2 code,51we also allow developers to switch off all these efficiency related optimizations so that one can implement new methods without considering the effects of low-level optimizations. In addition, we have introduced several improvements. (i) We use symbolic algebra methods to construct and manipu- late a symbolic MPO, where the elements of the MPO matrices are operator symbols (e.g., 1, a2,ˆRL[1 2] 3, etc.; for a more com- plete list of symbols, see the left–right decomposition formu- las in Appendix B). After the MPO construction, symbolic simplification can be used to generate optimal blocking for- mulas to multiply out the MPO, where related operator terms (e.g., a1a2=−a2a1) are merged, and elements in the partially contracted MPO that do not contribute to the final Hamil- tonian are eliminated. In addition, the symbolic algebra sys- tem supports the lazy contraction of renormalized operators during the DMRG algorithm. (ii) Multiplication and tensor products of block-sparse matrices are handled lazily by building a list of the associated GEMM operations to be later dispatched using shared-memory par- allelism. The list of GEMMs is generated once for each site along the sweep. In addition, because many matrices share the same block-sparse skeleton, the list of block indices used during a contraction can be generated once for every combi- nation of matrix skeletons. The delayed contraction, together with the use of stack memory, helps reducing the overhead of switching central processing unit (CPU) contexts. Using this approach, we see a significant speed-up relative to the S TACK - BLOCK code for small bond dimensions ( M<1000). At larger M, the performance difference between the two implemen- tations on a single node is small, since most time is spent in the math library. However, the reduced overhead in the BLOCK 2 code is beneficial when scaling to thousands of CPU cores. (iii) For large scale DMRG calculations, the total required stor- age and file input/output (IO) speed can be bottlenecks. To address this, we directly dump and load stack mem- ory chunks to avoid the CPU overhead of serialization and deserialization. In addition, we use floating point compres- sion during the disk IO. The compression introduces a small user-defined per-number absolute error (10−14in this work) in the stored renormalized operator matrix elements. We observed a 40% reduction in required storage for the benzene system discussed below and no discernible effect on accu- racy, while the total IO time was similar with or without compression. III. RESULTS As a first benchmark, we assess our parallel DMRG imple- mentation in a ground-state energy calculation of benzene using a cc-pVDZ basis44with an orbital space comprising 108 orbitals and J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 30 electrons.20Although the benzene system is a closed shell sys- tem and thus does not showcase the strengths of the DMRG algo- rithm, it, nonetheless, serves as an example in the literature, where a DMRG calculation with a large bond dimension and a relatively large number of orbitals have been recently reported. For the benzene calculation, we use particle number, SU(2) (spin), and Cspoint group symmetry to reduce the overall cost of the calculation. The same orbitals, integrals, and orbital ordering as in Ref. 20 were used in this work. The DMRG correlation energy atM=6000 (plus ∼200 states to represent the low-weight quantum numbers) obtained in this work is −859.1 mE H. Given the differ- ences in implementation that give rise to small differences in bond truncations across many sweeps, this is in excellent agreement with the DMRG correlation energy ( −859.2 mE H) reported in Ref. 20 at M=6000. In addition, we demonstrate the performance of our DMRG implementation in a calculation on the FeMo cofactor system, using a model with 76 orbitals and 113 electrons in the active space recently proposed by Li et al. in Ref. 45. This is an example of a system with multiple transition metal centers, where the strengths of the DMRG algorithm can, in principle, be demonstrated. We use the integral file provided in Ref. 45 without any further orbital reordering. The state with total spin S=3/2 is targeted. All calculations in this work use the two-site DMRG algorithm with perturbative noise.52Five sweeps were performed at each MPS bond dimension M. To measure the wall time per sweep, we used the average wall time for the last four sweeps for each M. For the ben- zene system, to alleviate the problem of losing quantum numbers, we kept at least one state for each quantum number after the normal decimation process.31This makes the bond dimension Min the cal- culation slightly larger than its specified target value. For example, when Mwas set to 6000, the observed actual Mwas typically about 6200. We denote different parallelism schemes by a set of numbers Psite,Phamil,Top, and Tdense indicating the number of groups of pro- cessors, processors, or threads used in the four levels of parallelism. Specifically, Psitedenotes the parallelism over sites, Phamil denotes the distributed parallelism over sub-Hamiltonians, Topis for the joint shared-memory parallelism over normal and complementary operators and symmetry sectors, and Tdense is for the thread-level parallelism in the dense matrix multiplications. The total number of CPU cores for a specific parallelism scheme is given by Ncore =PsitePhamilTopTdense. The calculations were executed on nodes with 28-core Intel Cascade Lake 8276 CPUs (2.20 GHz), made available via the Cal- tech high performance computing facility. Each node has 56 CPU cores and 384 GB memory. A. Mixed NC/CN approach As discussed in Appendix B, in conventional DMRG imple- mentations, there are two possible ways to write the left–right decomposition of the ab initio Hamiltonian at each site k. The NC scheme corresponds to an MPO with tensor dimensions that increase from the left to the right, while the CN scheme corresponds to an MPO with tensor dimensions that decrease from the left to the right.33Typically, efficient DMRG implementations use a mixed NC/CN approach,11where the NC decomposition is used for sitesk<K/2 and the CN decomposition is used for sites k≥K/2, which gives a significantly smaller “MPO” bond dimension. However, a transformation from the normal to complementary (two-index) operators is required near the middle site in this mixed NC/CN approach. The time complexity for this transformation is O(K4M2). Additionally, for parallelism over sub-Hamiltonians, since we use different processor assignments for the NC and CN schemes, an extra reduction step for all the two-index complementary opera- tors is required. The communication cost is O(K2M2logPhamil). The extra computation and communication cost mean that the middle site of the sweep is significantly more expensive than the other sites. Consequently, for parallelism over sites, we consider only odd Psiteand use a non-uniform division of the sweep ranges so that the high cost of computation at the middle site (included in the sweep range of the processor group p=⌈1 2Psite⌉) is amor- tized among all Psitegroups of processors. In this work, we tested Psite=1, 3, 5. In Fig. 1, we show the distribution of MPO bond dimensions (for each MPO tensor) and the corresponding wall time cost at each site, for the NC and mixed NC/CN partitions of the benzene sys- tem. For a spin-adapted DMRG algorithm, the bond dimension of the MPO tensor at site kis 2+2K+6k2(blue dashed line), if the NC scheme is used without any optimization or additional simplifica- tions. Using the symmetry conditions Eq. (B8), the bond dimension can be reduced to approximately 2 +2K+2k2(green dashed line), and the sudden decrease in the MPO bond dimension near the right- most site in Fig. 1 is due to the removal of complementary operators with vanishing integrals. The mixed NC/CN approach gives a much better distribution of bond dimensions (black dashed line), with the maximal value D=5996 appearing near the middle site of the test system. From Fig. 1, we can see that the time cost near the mid- dle site of the mixed approach is approximately two times as large as that of the NC approach, but the mixed approach gives a much smaller total wall time per sweep (with M=4000 and Phamil=16, tmixed=13 071 s) as compared to the NC approach ( tnc=19 356 s), mainly due to the smaller MPO bond dimensions for k≥K/2. The speed-up tnc/tmixed is∼148% based on the data in Fig. 1. Due to this, subsequent calculations in this work all use the mixed approach. B. Parallelism within dense matrix multiplication As discussed in earlier studies,29using thread parallelism in the dense matrix multiplications is not very effective in the DMRG, compared with the other parallel strategies. Table I shows that this is also true for our implementation of ab initio DMRG. For M=2500 and 3000, parallelism schemes with Tdense=4 are∼60%–70% slower than the scheme with Tdense=1. Therefore, for production calcula- tions in this work, this level of parallelism was not utilized (i.e., we used only Top=28 and Tdense=1). C. Parallelism over sites It is sometimes argued that when parallelism over sites is used, the convergence of the DMRG energy as a function of the num- ber of sweeps is slower than that of the standard DMRG approach, if the same sweep schedule is used.34In Fig. 2, we compare paral- lelism schemes with different Psitefor MPS bond dimensions up to M=6000 (data for M=6000 with Psite=3 and Phamil=8 could not be obtained due to memory constraints) for the benzene system. We J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I. Wall time per sweep (in seconds) in the benzene calculation for MPS bond dimensions M=2500 and 3000 using parallelism schemes with different Tdense . Parallelism scheme Wall time per sweep (s) Psite Phamil Top Tdense M=2500 M=3000 5 14 28 1 893 1291 5 14 7 4 1521 2106 5 7 14 4 1493 2079 can see that in our test system, convergence is only slightly affected by increasing Psitefrom 1 to 5. When five sweeps were performed for each M, the energy obtained from the last sweep for each M was almost the same with different Psite, up to M=5000. Although we started the calculation from the same initial MPS (with a single canonical center) for different Psite, forPsite=3 and 5, the initial MPS is re-canonicalized to introduce extra canonical centers. During this canonicalization step, some low-weight single-state quantum num- bers were discarded, which makes the Psite=3 and 5 DMRG energy atM=2500 (artificially) higher in Fig. 2. To examine the effect of the dynamical connection sites, we have compared the estimated performance using dynamical, fixed, and uniform connection sites in Fig. 3. For our test benzene sys- tem with 108 sites and Psite=5, four connection sites are required. From the dotted lines in Fig. 3, we see that selecting connection sites based on a uniform division of sweep ranges (namely, Kconn ={21, 43, 64, 86 }) gives a large load imbalance among the five pro- cessors. At the last sweep, the longest processor task consumed 388% more time than the shortest processor task, which can be mainly attributed to the highly non-uniform distribution of computational effort among sites (see the solid black line in Fig. 1). In this work, we found that Kconn={33, 49, 57, 73 }(obtained from using dynam- ical connection sites for small bond dimensions) gave much better performance. This corresponds to the dashed lines in Fig. 3. With FIG. 2. Sweep energies for parallelism schemes with different Psiteand different MPS bond dimensions Mfor the benzene system. For each M, five sweeps are performed. FIG. 3. Estimated wall time per processor for parallelism over sites (as a percent- age of the sum of wall times for all processors), when the connection centers are dynamically adjusted (solid lines), fixed (dashed lines), and uniformly distributed (dotted lines). The performance data are from the Psite=5 and Phamil=20 ben- zene calculations with the MPS bond dimension increasing from M=2500 to M=6000. this fixed set of connection sites, the longest task consumed 53% more time than the shortest task. If we allow the set of connec- tion sites to be dynamically adjusted between the sweeps, we end up with a slightly altered set of connection sites Kconn={33, 48, 59, 74 }. Using this, in the last sweep, the longest task then consumed only 26% more time than the shortest task. D. Parallel scaling In Table II, we list the average wall time per sweep with MPS bond dimensions from M=2500 to M=6000 for the benzene sys- tem, when parallelism schemes with different Psiteand Phamil are used. The speed-up relative to the Psite=1 and Phamil=16 (Ncore =448) case is plotted in Fig. 4. In Fig. 4, we see that, when Psite=3, increasing Phamil from 12 to 18 only reduces the wall time slightly, while nearly ideal speed-up is observed for different (Psite,Phamil)when increasing from (1, 16) to (3, 12) and from (3, 12) to (5, 14). This illustrates that a combination of different DMRG parallelism strategies is essential to achieve good scaling across thousands of CPU cores. The better-than-ideal speed- up for(Psite,Phamil)when increasing from (1, 16) to (3, 8) is also due to the change in parallelism strategy. Ideally, the Psite=Phamil=1 case should be used as the reference point for computing the speed- up. However, this is not feasible in our test system due to the large maximum MPO bond dimension D=5996 and when using a large MPS bond dimension M=6000. Here, we need Phamil≥10 to ensure that the memory cost per node is less than 384 GB. This is an impor- tant reason to use larger Phamil rather than Psitein certain systems, since increasing Psitedoes not reduce the memory cost per processor group. Finally, we note that the speed-up for M=2500 appears to be significantly less than the other cases with the larger M. This is likely related to the fact that for the Psite=3 and Psite=5 cases, an J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. Average wall time per sweep (in seconds) of the benzene calculation for different MPS bond dimensions using parallelism schemes with different PsiteandPhamil.Top=28 and Tdense=1 were used for all parallelism schemes. Parallelism scheme Average wall time per sweep (s) Psite Phamil Ncore M=2500 M=3000 M=4000 M=5000 M=6000 1 16 448 3145 5253 12 740 22 212 35 451 3 8 672 1855 2542 6 158 11 335 12 1008 1529 2107 5 079 18 1512 1487 2049 4 379 5 14 1960 894 1291 3 051 5 317 8 696 20 2800 816 1105 2 539 4 526 7 419 initial M=2500 MPS with an artificially higher energy was used (see Fig. 2). For the largest calculation considered in this work with Psite =5,Phamil=20, and M=6000 for the benzene system, the aver- age communication and idle time among the Phamil processors constituted ∼15% of the total wall time for each group of Phamil processors, and the average idle time among the Psitegroups of processors was ∼10% of the total wall time. The Davidson step FIG. 4. Speed-up of average wall time per sweep relative to the Ncore=448 case for different MPS bond dimensions using parallelism schemes with different Psite andPhamil.Top=28 and Tdense=1 are used for all parallelism schemes. TABLE III. Average wall time per sweep (in seconds) of the FeMo cofactor calculation for different MPS bond dimensions using parallelism schemes with different Psiteand Phamil.Top=28 and Tdense=1 were used for all parallelism schemes. Parallelism scheme Average wall time per sweep (s) Psite Phamil Ncore M=2000 M=2500 M=3000 1 16 448 10 596 19 464 50 677 3 8 672 6 380 12 191 31 496 5 16 2240 3 262 5 156 12 499(including communication) constituted 60%–70% of the total wall time for each processor. Reading/writing disk files cost ∼5% of the total wall time. In Table III, we show that a similar scaling can be observed for the FeMo cofactor system. When increasing (Psite,Phamil)from (1, 16)to(5, 16), for a sufficiently large MPS bond dimension (M=3000), we obtain a speed-up of 4.05, which is close to the ideal speed-up (5). Note that the worse-than-cubic scaling with respect to Mfor the M=2500 and M=3000 cases shown in Table III is mainly due to the difference in the Davidson convergence criteria used for different M. Finally, we note that using a similar number of CPU cores (Ncore≈2000) and the same bond dimension ( M=3000), the FeMo cofactor system wall time per sweep was 10 times larger than that for the benzene system. Although a slightly higher point group symmetry ( Cs) was used in the benzene calculations, the main rea- son for this difference appears to be the strength of correlation in the FeMo cofactor. This leads to more Davidson iterations at each site, associated with the increased DMRG truncation error (7×10−4for the FeMo cofactor vs 1 ×10−5for benzene), which means that the quality of the initial Davidson guess wavefunction is poorer, when moving from site to site in the sweep. IV. CONCLUSIONS In this work, we introduced a modification of the conventional strategy for distributed memory parallelism in ab initio DMRG algorithms that reduces the computation to the manipulation of independent sub-Hamiltonians, together with a small wavefunction communication step. This formulation, thus, combines the concep- tual advantages of the sum of sub-Hamiltonian approach introduced in earlier work, with the greater parallelizability and lower prefactor of the conventional distributed memory DMRG algorithm. In addi- tion, we carried out a comprehensive examination and implementa- tion of four other sources of parallelism in the DMRG, introducing techniques for load balancing via dynamic connection sites in site- based parallelism and collapsing tasks to maximize thread efficiency in the shared-memory parallelism. Finally, we showed that the com- bination of different DMRG parallelism strategies using both dis- tributed and shared-memory models was essential to achieve near- ideal speed-ups for a benchmark calculation with 108 orbitals and a DMRG bond dimension of M=6000, scaling from 448 to 2800 CPU cores. J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The DMRG implementation in the B LOCK 2 code used in this work is open-source and can be freely obtained.51In addition to the ground-state DMRG described here, it also supports several other MPS algorithms for ab initio systems, including finite-temperature DMRG, imaginary and real time evolution, and reduced density matrix and transition density matrix evaluation. Applications using these algorithms will be explored in future work. ACKNOWLEDGMENTS This work was supported by the U.S. National Science Founda- tion (NSF) (Grant No. CHE-2102505). H.Z. thanks Seunghoon Lee for providing the integrals and reference DMRG outputs for the ben- zene system, and Henrik R. Larsson, Zhi-Hao Cui, and Tianyu Zhu for helpful discussions. The computations presented in this work were conducted on the Caltech High Performance Cluster, partially supported by a grant from the Gordon and Betty Moore Foundation. APPENDIX A: THE SERIAL DMRG ALGORITHM To establish a notation for the DMRG algorithm, consider a quantum lattice system with Ksites. Each site is associated with a Hilbert space spanned by a site-basis {∣nk⟩}. A complete basis of the system Hilbert space can be defined as the tensor product of Ksite- bases, {∣n1n2⋅⋅⋅nK⟩}={∣n1⟩⊗∣n2⟩⊗⋅⋅⋅⊗∣nK⟩}. (A1) The goal of the DMRG algorithm is to optimize a variational wave- function in this Hilbert space, whose amplitudes can be written as a product of matrices, ∣Ψ⟩=∑ {n}A[1]n1A[2]n2⋅⋅⋅A[K]nK∣n1n2⋅⋅⋅nK⟩, (A2) where each A[k]nk(k=2,⋅⋅⋅,K−1)is an M×Mmatrix and the leftmost and rightmost matrices are 1 ×Mand M×1 vectors, respectively. The dimension Mis known as the bond dimension of the MPS ∣Ψ⟩. Within the MPS ansatz, variational minimization of the energy, formally written as E0=min ∣Ψ⟩⟨Ψ∣ˆH∣Ψ⟩ ⟨Ψ∣Ψ⟩, (A3) where ˆHis the system Hamiltonian and E0is the ground-state energy, can be performed iteratively by optimizing the parameters of a single matrix at a time in the MPS, while the parameters in the remaining matrices are kept constant. This corresponds to the one-site DMRG algorithm. A common variant, designed to improve the ability to escape local minima, optimizes a single larger matrix A[k]nknk+1that describes the variational space of two sites at a time. This formally takes one outside the single-site MPS variational space, and thus, the solution must be decimated back to the standard MPS form. This corresponds to the two-site DMRG algorithm. The same idea can be generalized to psites, but in this work, we mainly consider the p=2 case. The iterative process in a serial DMRG algorithm is struc- tured as a series of sweeps along a fixed one-dimensional order- ing of the Ksites. Each sweep alternates between the forward andbackward directions, consisting of K+1−p sweep iterations . In thekth(k=1,⋅⋅⋅,K+1−p)sweep iteration of a forward sweep, the parameters in the current matrix being optimized (associated with dadjacent sites, A[k]nk⋅⋅⋅nk+d−1) are updated, while in a back- ward sweep, the matrices are updated in reverse order. The lat- tice can then be conveniently divided into 2 +d blocks (or sets of sites){Lk−1,Sk,⋅⋅⋅,Sk+d−1,Rk+d}in the kth sweep iteration (of a for- ward sweep, for example): a left block (or the system )Lk−1for sites 1,⋅⋅⋅,k−1, the individual sites whose matrices are being optimized Sk⋅⋅⋅Sk+d−1, and the right block (or the environment )Rk+dfor sites k+d,⋅⋅⋅,K(see Fig. 5). In each sweep iteration, we consider a left–right decomposi- tion of the system Hamiltonian as the sum of tensor products of operators defined in blocks LkandRk+1, ˆH[k]=ˆHLk⊗ˆ1Rk+1+ˆ1Lk⊗ˆHRk+1+∑ iˆhLk iˆhRk+1 i, (A4) where a bipartition of the lattice {Lk,Rk+1}has been used. A conve- nient way to construct this left–right decomposition for any kis to first write the system Hamiltonian in a so-called MPO form, ˆH=∑ {n,n′}W[1]n1n′ 1W[2]n2n′ 2⋅⋅⋅W[K]nKn′ K ×∣n1n2⋅⋅⋅nK⟩⟨n′ 1n′ 2⋅⋅⋅n′ K∣, (A5) where each W[k]nkn′ k(k=2,⋅⋅⋅,K−1)is a D×D′matrix and the leftmost and rightmost matrices are 1 ×D′andD×1 vectors, respectively. The maximal dimension Damong these matrices will be called the bond dimension of the MPO. FIG. 5. The left block (system), right block (environment), and the individual sites being optimized in a given sweep iteration of the two-site DMRG algorithm. J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The left–right decomposition of the MPS can be defined as (in two-site DMRG, for example) ∣Ψ[k]⟩= ∑ αk−1αkαk+1,nknk+1A[k]nkαk−1αkA[k+1]nk+1αkαk+1 ×∣αL k−1⟩⊗∣nknk+1⟩⊗∣αR k+1⟩, (A6) where the left and right renormalized basis vectors are ∣αL k⟩=∑ {n1⋅⋅⋅nk}[A[1]n1⋅⋅⋅A[k]nk]αk∣n1⋅⋅⋅nk⟩, ∣αR k⟩=∑ {nk+1⋅⋅⋅nK}[A[k+1]nk+1⋅⋅⋅A[K]nK]αk∣nk+1⋅⋅⋅nK⟩.(A7) Using the MPO form, the decomposition equation (A4) can be constructed as ˆH[k]=∑ βkˆH[k]L βk⊗ˆH[k]R βk, (A8) where ˆH[k]L βk=∑ {n1⋅⋅⋅nk,n′ 1⋅⋅⋅n′ k}[W[1]n1n′ 1⋅⋅⋅W[k]nkn′ k] βk ×∣n1⋅⋅⋅nk⟩⟨n′ 1⋅⋅⋅n′ k∣, ˆH[k]R βk= ∑ {nk+1⋅⋅⋅nK,n′ k+1⋅⋅⋅n′ K}[W[k+1]nk+1n′ k+1⋅⋅⋅W[K]nKn′ K] βk ×∣nk+1⋅⋅⋅nK⟩⟨n′ k+1⋅⋅⋅n′ K∣. (A9) Each sweep iteration of the two-site DMRG algorithm is divided into three main steps (see Fig. 6):6 (i) Blocking, where we compute the matrix representation of ˆH[k]L βkand ˆH[k]R βkEq. (A9) in bases ∣αL k−1nk⟩and∣nk+1αR k+1⟩ FIG. 6. The (a) blocking, (b) transformation, and (c) solving steps in each sweep iteration of the two-site DMRG algorithm.33from the renormalized operators represented in bases ∣αL k−1⟩ and∣αR k+1⟩, respectively, ⟨αL k−1nk∣ˆH[k]L βk∣α′L k−1n′ k⟩ =∑ βk−1W[k]nkn′ k βk−1βk⟨αL k−1∣ˆH[k−1]L βk−1∣α′L k−1⟩, ⟨nk+1αR k+1∣ˆH[k]R βk∣n′ k+1α′R k+1⟩ =∑ βk+1W[k+1]nk+1n′ k+1 βkβk+1⟨αR k+1∣ˆH[k+1]R βk+1∣α′R k+1⟩. (A10) (ii) Solving, where we update the wavefunction in the renormal- ized basis ∣αL k−1nk⟩⊗∣nk+1αR k+1⟩[Eq. (A6)], given by Ψ[k]eff αk−1nk,nk+1αk+1=∑ αkA[k]nkαk−1αkA[k+1]nk+1αkαk+1, (A11) by solving the eigenvalue problem H[k]effΨ[k]eff=E[k]Ψ[k]eff, (A12) where the matrix elements of the effective Hamiltonian H[k]effare given by H[k]eff αk−1nk,nk+1αk+1;n′ k+1α′ k+1,α′ k−1n′ k =∑ βk⟨αL k−1nk∣ˆH[k]L βk∣α′L k−1n′ k⟩ ×⟨nk+1αR k+1∣ˆH[k]R βk∣n′ k+1α′R k+1⟩. (A13) Since the Hamiltonian is sparse, the eigenvalue problem is normally solved using an iterative method such as the David- son algorithm.54 (iii) Decimation and transformation. Once the optimized wave- function Ψ[k]effis determined, the new A[k]andA[k+1] can be found by decomposing the wavefunction using the density matrix, or via a singular value decomposition (SVD). After the decomposition, the matrix dimensions of A[k]and A[k+1]are truncated to bond dimension Mby discarding small singular values or eigenvalues. The truncated A[k]and A[k+1]are then used to construct new renormalized bases ∣αL k⟩and∣αR k⟩, in a forward and backward sweep iteration, respectively, as ∣αL k⟩=∑ αk−1A[k]nkαk−1αk∣αL k−1nk⟩, ∣αR k⟩=∑ αk+1A[k+1]nk+1αkαk+1∣nk+1αR k+1⟩.(A14) The operators formed in the blocking step [Eq. (A10)] are also transformed to the new renormalized basis J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ⟨αL k∣ˆH[k]L βk∣α′L k⟩=∑ αk−1nk;α′ k−1n′ kA[k]nkαk−1αkA[k]n′ k α′ k−1α′ k ⟨αL k−1nk∣ˆH[k]L βk∣α′L k−1n′ k⟩, ⟨αR k∣ˆH[k]R βk∣α′R k⟩= ∑ nk+1αk+1;n′ k+1α′ k+1A[k+1]nk+1αkαk+1A[k+1]n′ k+1 α′ kα′ k+1 ×⟨nk+1αR k+1∣ˆH[k]R βk∣n′ k+1α′R k+1⟩ (A15) APPENDIX B: NOTATION FOR THE SU(2) SPIN-ADAPTED AB INITIO DMRG For the ab initio DMRG implemented in this work, we asso- ciate each site k(k=1, 2,⋅⋅⋅,K) with a spatial orbital. The ab initio Hamiltonian is written as32 ˆH=∑ ij,σtij,σa† iσajσ+1 2∑ ijkl,σσ′vijkl,σσ′a† iσa† kσ′alσ′ajσ, (B1) where tij,σ=∫dxϕ∗ iσ(x)(−1 2∇2−∑ aZa ra)ϕjσ(x), vijkl,σσ′=∫dx1dx2ϕ∗ iσ(x1)ϕ∗ kσ′(x2)ϕlσ′(x2)ϕjσ(x1) r12,(B2) with the following symmetry conditions: tij,σ=tji,σ, vijkl,σσ′=vjikl,σσ′=vijlk,σσ′=vklij,σ′σ.(B3) With SU(2) spin symmetry, we additionally have42 tij=tij,α=tij,β, vijkl=vijkl,αα=vijkl,αβ=vijkl,βα=vijkl,ββ.(B4) In conventional ab initio DMRG, the left–right decompo- sition of the Hamiltonian [Eq. (A4)] is written in terms of normal and complementary operators.3One can choose to use two-index complementary operators only with the right block [the Normal/Complementary (NC) partition] or only with the left block [the Complementary/Normal (CN) partition]. The SU(2) spin- adapted left–right decomposition of the Hamiltonian using the NC and CN partitions is, respectively,42 ˆH[k]NC[0]=ˆHL[0]⊗[0]ˆ1R[0]+ˆ1L[0]⊗[0]ˆHR[0] +2∑ i∈L(a†[1 2] i⊗[0]ˆRR[1 2] i+a[1 2] i⊗[0]ˆRR†[1 2] i) +2∑ i∈R(ˆRL†[1 2] i⊗[0]a[1 2] i+ˆRL[1 2] i⊗[0]a†[1 2] i) −1 2∑ ij∈L(ˆA[0] ij⊗[0]ˆPR[0] ij+√ 3ˆA[1] ij⊗[0]ˆPR[1] ij +ˆA†[0] ij⊗[0]ˆPR†[0] ij+√ 3ˆA†[1] ij⊗[0]ˆPR†[1] ij) +∑ ij∈L(ˆB[0] ij⊗[0]ˆQR[0] ij+√ 3ˆB[1] ij⊗[0]ˆQR[1] ij) (B5)and ˆH[k]CN[0]=ˆHL[0]⊗[0]ˆ1R[0]+ˆ1L[0]⊗[0]ˆHR[0] +2∑ i∈L(a†[1 2] i⊗[0]ˆRR[1 2] i+a[1 2] i⊗[0]ˆRR†[1 2] i) +2∑ i∈R(ˆRL†[1 2] i⊗[0]a[1 2] i+ˆRL[1 2] i⊗[0]a†[1 2] i) −1 2∑ ij∈R(ˆPL[0] ij⊗[0]ˆA[0] ij+√ 3ˆPL[1] ij⊗[0]ˆA[1] ij +ˆPL†[0] ij⊗[0]ˆA†[0] ij+√ 3ˆPL†[1] ij⊗[0]ˆA†[1] ij) +∑ ij∈R(ˆQL[0] ij⊗[0]ˆB[0] ij+√ 3ˆQL[1] ij⊗[0]ˆB[1] ij), (B6) where the superscript and subscript [S]are used to indicate the total spin quantum number for the spin tensor operator and the result- ing spin tensor operator obtained from the tensor product, respec- tively, and the block Hamiltonian ˆHL/R[0], normal operators ˆA[S] ij, ˆB[S] ij, and complementary operators ˆRL/R[1 2] i ,ˆPL/R[S] ij , and ˆQL/R[S] ij are defined by ˆRL/R[1 2] i=√ 2 4∑ j∈L/Rtija[1 2] j+∑ jkl∈L/Rvijkl(a†[1 2] k⊗[0]a[1 2] l)⊗[1 2]a[1 2] j, ˆA[0/1] ij=a†[1 2] i⊗[0/1]a†[1 2] j, ˆB[0/1] ij=a†[1 2] i⊗[0/1]a[1 2] j,(B7) ˆPL/R[0/1] ik=∑ jl∈L/Rvijkla[1 2] j⊗[0/1]a[1 2] l, ˆQL/R[0] ij=∑ kl∈L/R(2vijkl−vilkj)a†[1 2] k⊗[0]a[1 2] l, ˆQL/R[1] ij=∑ kl∈L/Rvilkja†[1 2] k⊗[1]a[1 2] l, with the following symmetry conditions (when i≠j):11 ˆA[S] ij=(−1)SˆA[S] ji, ˆB[S] ij=(−1)S(ˆB[S] ji)† , ˆP[S] ij=(−1)SˆP[S] ji ˆQ[S] ij=(−1)S(ˆQ[S] ji)† .(B8) The corresponding MPO for the NC and CN partitions can be constructed based on the blocking formulas for the spin tensor operators, and Eqs. (B5) and (B6), respectively. Since these operators have at most two spatial orbital indices, the MPO bond dimension D∼K2. The blocking formulas explicitly yield only the non-zero elements of the MPO, and thus, using the blocking formulas in the DMRG algorithm can be viewed as implementing sparse tensor contraction with the MPO. Alternatively, there are procedures to automatically construct the elements of the MPO tensors by matrix decomposition (and J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-11 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp other algorithms) simply given the list of two-electron integrals. Examples of these automated MPO construction approaches are the fork-merge approach,40the SVD approach,33the delinearization approach,41and the bipartite approach.47Note that some of these procedures work with a dense matrix representation of the MPO tensors (even if the matrices have exact zeros). As discussed in the main text, the sum of sub-Hamiltonians allows for the correct scaling of implementations, which use such MPO construction techniques without explicit implementation of sparse tensor algebra. Thus, the strategies in this work, when used with automated MPO construc- tion techniques, achieve both the correct serial cost as well as have a low communication overhead for parallel scaling. DATA AVAILABILITY The performance data presented in this work can be repro- duced using the B LOCK 2 code51and the integral files53provided in Refs. 20 and 45. REFERENCES 1S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. 69, 2863 (1992). 2S. R. 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B 72, 180403 (2005). 53The integral file for the benzene calculation can be found in https://github. com/seunghoonlee89/SI-benzene-paper-DMRG. The integral file for the FeMo cofactor calculation can be found in https://github.com/zhendongli2008/Active- space-model-for-FeMoco. 54E. R. Davidson, “The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices,” J. Comput. Phys. 17, 87–94 (1975). J. Chem. Phys. 154, 224116 (2021); doi: 10.1063/5.0050902 154, 224116-13 Published under an exclusive license by AIP Publishing

5.0053297.pdf

The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Electron binding energies and Dyson orbitals of O nH2n+1+,0,−clusters: Double Rydberg anions, Rydberg radicals, and micro-solvated hydronium cations Cite as: J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 Submitted: 6 April 2021 •Accepted: 3 June 2021 • Published Online: 17 June 2021 Ernest Opoku,a) Filip Pawłowski,b) and Joseph Vincent Ortizc) AFFILIATIONS Department of Chemistry and Biochemistry, Auburn University, Auburn, Alabama 36849-5312, USA a)ezo0009@auburn.edu b)filip@auburn.edu c)Author to whom correspondence should be addressed: ortiz@auburn.edu ABSTRACT Ab initio electron propagator methods are employed to predict the vertical electron attachment energies (VEAEs) of OH 3+(H2O)nclusters. The VEAEs decrease with increasing n, and the corresponding Dyson orbitals are diffused over exterior, non-hydrogen bonded protons. Clusters formed from OH 3−double Rydberg anions (DRAs) and stabilized by hydrogen bonding or electrostatic interactions between ions and polar molecules are studied through calculations on OH 3−(H2O)ncomplexes and are compared with more stable H−(H2O)n+1isomers. Remarkable changes in the geometry of the anionic hydronium–water clusters with respect to their cationic counterparts occur. Rydberg electrons in the uncharged and anionic clusters are held near the exterior protons of the water network. For all values of n, the anion–water complex H−(H2O)n+1is always the most stable, with large vertical electron detachment energies (VEDEs). OH 3−(H2O)nDRA isomers have well separated VEDEs and may be visible in anion photoelectron spectra. Corresponding Dyson orbitals occupy regions beyond the peripheral O–H bonds and differ significantly from those obtained for the VEAEs of the cations. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0053297 I. INTRODUCTION Transport, reactions, and structural characteristics of molecular ions in aqueous environments play important roles in many chem- ical and biological processes.1–3Seeking understanding of struc- tural, electronic, and dynamic properties of protonated water clus- ters with cationic, neutral, or anionic charges (O nH2n+1+,0,−) is also motivated by the involvement of such species in the formation of clouds and haze and in solvent effects in aqueous systems.4Con- cepts of acid–base chemistry are often defined in terms of pro- ton transfer. Therefore, understanding the behavior of protons and other ions in water networks is a perennial concern in physical chemistry. However, hydrogen bond networks, such as protonated water clusters, present several structural and dynamical complexities which render them challenging to study.5Notwithstanding their ubiquitous presence in nature, protonated water clusters are elusivemolecular entities that are extremely difficult to probe with direct experiments. Owing to the anomalously high mobility of protons in water, a detailed understanding of structures, transport mech- anisms, and solvation in H+(H2O)nclusters remains a challeng- ing goal. Considering the difficulty in direct experimental investi- gations, the chemistry of protonated water clusters and associated molecular ions stands to greatly benefit from quantum-chemical calculations. Attachment of an electron to a protonated water cluster may give rise to a Rydberg molecule that departs from traditional con- cepts of chemical bonding. Rydberg molecules consist of a geometri- cally and electronically stable cationic core that binds a non-bonding electron held in a diffuse orbital.6In the case of double Rydberg anions (DRAs), a stable ground-state molecular cation binds a pair of diffuse electrons that are distributed over the periphery of the cationic core.7The electrons that bind to a cationic core in DRAs J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and Rydberg neutrals may occupy orbitals with unusual amplitudes. The first DRA to be observed and characterized was tetrahedral NH 4−.7–9Tetrahedral NH 4−was anticipated in preceding computa- tional studies.10,11Subsequent experimental7,12and theoretical13–15 investigations have considered several isomers of N nH3n+1−DRAs that arise from hydrogen bonding or electrostatic interactions. Theoretical studies have predicted and identified many DRAs without nitrogen atoms, such as OH 3−and O 2H5−, that are authen- tic minima in potential energy surfaces (PESs) and that have pos- itive vertical electron detachment energies (VEDEs).6,15–20Melton and Joy provided the first experimental and theoretical evidence for the existence of OH 3radicals through the irradiation of water vapor with ionizing electrons.21Evidence from the kinetics of water radi- olysis was provided for the existence of OH 3as an intermediate.22,23 Andersen et al.24reported that the mechanism of generating water molecules in interstellar clouds occurs through a dissociative recom- bination reaction of OH 3+with electrons. In this mechanism, Ryd- berg states of OH 3appear as intermediates in agreement with later ab initio investigations.25–29Since their report,24the OH 3radical and its solvated derivatives have attracted serious consideration in computational and experimental studies. Brozek-Pluska et al.30 observed from femtosecond spectroscopy experiments that solvated oxonium radicals play a significant role in the domain of liquid phase reactivity. Early computational and experimental studies on OH 3radi- cals focused on analysis of electronic structure and ground-state stability. Some successive studies provided extensive details on Ryd- berg excited states and transition intensities of OH 3and other radicals.31–37Recent studies38,39have combined ab initio electron propagator theory (EPT) and coupled-cluster (CC) methods with the molecular quantum defect orbital (MQDO) formalism to study the electronic absorption spectra of NH 4and OH 3Rydberg rad- icals. A few years later, cross sections for the ionization of NH 4 and OH 3from their outermost shells were analyzed in terms of the photoelectron energy.40Chen and Davidson41studied the electronic structure of OH 3, NH 4, and FH 2Rydberg molecules using second-order Møller–Plesset perturbation (MP2) theory.42,43 They reported the relative stabilities of NH 4, OH 3, and FH 2and their isotopic congeners and constructed PESs for these Rydberg neutrals. In the case of hydrated proton clusters, past studies44 have focused more on understanding the vibrational spectra of H+(H2O)n,n=2–28. There are two key structural proposals that pertain to the hydrated proton. Eigen45suggested a O 4H9+complex in which an OH 3+core is strongly hydrogen-bonded to three H 2O molecules. Zundel46later proposed the formation of a O 2H5+com- plex, in which the proton is shared between two H 2O molecules. Early ab initio and density functional theory (DFT) quantum- chemical investigations on H+(H2O)nclusters ( n=1–8),47–52vibra- tional spectroscopy experiments,53and molecular dynamics sim- ulations54–57have led to the conclusion that the hydrated proton H+(H2O)nforms a fluxional defect in the hydrogen-bonded net- work, with both H 9O4+and H 5O2+occurring in the sense of limiting or ideal structures.58These works suggest that the rate-limiting step for proton transfer may not be an actual proton transfer through a hydrogen bond between OH 3+and a neighboring water molecule. Instead, it may involve rupture of a hydrogen bond between the first and second solvation shells of OH 3+induced by fluctuationsthat reduce the coordination number of a water molecule in the first solvation shell.5,54 Recently, Verma et al.59also reported that electron impact ioni- zation of He droplets doped with clusters of water molecules yields water and O 2H5+cations implanted in the droplets. They obtained infrared spectra by measuring the intensity of the cations released from the droplets upon laser excitation. The OH −stretch spectra of water (H 2O+) and Zundel (O 2H5+) cations in helium droplets in the 3μm region were reported. The authors59observed a significant dif- ference in the intensity of the two components of the asymmetric vibrational bands in contrast with the results of a prior computa- tional study that predicted very similar intensities under effective D2dsymmetry.60 Some recent theoretical studies on vibrational spectroscopy of hydrated proton clusters H+(H2O)nrelied on vibrational second- order perturbation theory (VPT2),61–66with the electronic energies typically obtained from density functional theory (DFT) employing the B3LYP67,68functional. CC methods are less frequently used for the studies of larger solvated oxonium clusters due to their compu- tational expense.69To address electron correlation, MP2 methods are generally used for geometry optimization as demonstrated by Xantheas70in the study of various isomers of OH 3+(H2O)20. It has also been found that empirical or semiempirical potentials that ade- quately describe proton transfer processes can be used to probe the structure and dynamics of larger solvated oxonium systems.71–73Shi et al.4used a comprehensive genetic algorithm combined with den- sity functional theory (CGA-DFT) to search the PES of H+(H2O)n clusters with n=10–17 and simulated the IR spectra of the low- est energy structures with anharmonic corrections using the VPT2 method. Likewise, there have been theoretical studies on structure, vertical detachment energies, and electronic spectra of water cluster anions (H 2O)n−.74–78 Rydberg neutrals and DRAs arising from OH 3+have been stud- ied since 1990.6,15Ortiz20presented structural and spectral predic- tions of the OH 3−DRA. Electron propagator calculations combined with larger diffuse basis sets were used to obtain vertical electron binding energies for the OH 3−DRA at the geometry of the anionic minimum. A later study39provided predictions of structures, vibra- tional frequencies, relative total energies, and electron binding ener- gies on isomers of OH 3−and O 2H5−DRAs in comparison to NH 4− and N 2H7−. No studies on H−(H2O)n,n≥3 isomers have been per- formed. A careful search of the literature reveals no experimental evidence of the H 3O−DRA or any of its higher clusters. However, the H−(H2O) isomer of the OH 3−DRA has been observed exper- imentally with a VEDE of 1.53 eV.79The H−(H2O) isomer can be considered to be an H−anion solvated by one water molecule. Con- siderable experimental efforts have been made to produce H 3O− DRA with no success, perhaps because this species is much higher in energy than the H−(H2O) isomer.39At the same time, the metastable Rydberg radicals, OH 3and OD 3, have been identified with neutralized ion beam spectroscopy experiments by Porter and co-workers.80–82Sobolewski and Domcke58studied the structures and spectroscopy of cationic and neutral hydronium–water clusters arising from the Eigen-like model H 3O+(H2O)nand the Zundel- like geometries O 2H5+(H2O)nat the MP2, complete-active-space self-consistent field, complete-active-space perturbation theory of second order, and DFT/B3LYP levels of theory. Okumura and co- workers53also reported combined experimental and computational J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp studies on vibrational spectroscopy of the hydrated hydronium clus- ter ions OH 3+(H2O)n(n=1–3). These authors found that the Zundel cation O 2H5+may have C 2and C Sisomers, with the C 2 geometry being more stable. Similarly, the O 3H7+structures were assigned to the C Sand C 2vpoint groups, with the C Sstructure being the most stable. The Eigen cation O 4H9+was assigned to the C 3v and D 3hpoint groups. The C 3vstructure was reported as the most stable. Despite all this progress, a thorough search of the chemical literature reveals no studies on the O 2H5Rydberg radical and cor- responding higher H(H 2O)nclusters. In the case of DRAs, we found studies only on the OH 3−and O 2H5−species,39prompting the need for theoretical studies on larger clusters to offer correlative guid- ance for future experiments. Techniques of EPT provide reliable predictions for the discovery and identification of novel DRAs9,39 and related solvated-electron precursors.83–85To extend the scope of double Rydberg concepts to oxygen-based systems, stabilization of the pyramidal H 3O−DRA by hydrogen bonding or electrostatic interactions ought to be considered. Herein, we have employed high-accuracy, ab initio EPT and CC methods with large, diffuse basis sets to study the hydronium OnH2n+1Rydberg molecules, O nH2n+1−DRAs, and related solvated species. In completing the first solvation shell, both the Eigen-like H3O±(H2O)nstructures and Zundel’s H 5O2±(H2O)n−1models are considered. Predictions of structures, vibrational frequencies, rel- ative total energies, vertical electron binding energies, and Dyson orbitals associated with Rydberg neutrals and DRAs are investi- gated. Isomers that arise from electrostatic intermolecular interac- tions or hydrogen bonding to form hydride- and DRA-molecule complexes are all considered. Calculations on isomers of O nH2n+1± complexes with only their first solvation shells provide Aufbau rules for electrons that occupy the periphery of these complexes. II. THEORY AND COMPUTATIONAL DETAILS A. Summary of electron propagator theory EPT86–88is a quantum-chemistry method that allows for cal- culating electron binding energies (e.g., electron attachment and detachment energies of molecular ions) directly, without any explicit reference to final state wave functions and their total energies. The electron attachment and detachment energies occur at poles of the electron propagator matrix, G(E), Grs(E)=lim η→+0[Attachment ∑ P⟨0∣ar∣P⟩⟨P∣a† s∣0⟩ E+E0−E+ P+iη +Detachment ∑ P⟨0∣a† s∣P⟩⟨P∣ar∣0⟩ E−E0+E− P−iη], (1) where the N-electron state, ∣0⟩, and the ( N±1)-electron states, ∣P⟩, and their energies ( E0andE± P, respectively) are the solutions to the Hamiltonian eigenvalue problem, H∣0⟩N=E0∣0⟩N, H∣P⟩N±1=E± P∣P⟩N±1,(2) aranda† sare the annihilation and creation operators, respectively, with the indices, rands, referencing a basis (a finite, spin–orbitalbasis of a reference, Hartree–Fock state is assumed in the following, which implies that only discrete electron attachment, E=E+ P−E0, and detachment, E=E0−E− P, energies are obtained), and the infinitesimal factor ηensures the correct Fourier transformation between the time-dependent definition of the electron propagator and its energy representation in the spectral form in Eq. (1). While electron binding energies occur at the poles (i.e., van- ishing denominators) of Eq. (1), the corresponding residues (i.e., numerators pertaining to the vanishing denominators) are products of Feynman–Dyson amplitudes, CrP=⎧⎪⎪⎨⎪⎪⎩⟨0∣ar∣P⟩ ( electron attachment ) ⟨P∣ar∣0⟩ ( electron detachment ),(3) and their complex conjugates. The Feynman–Dyson amplitudes are expansion coefficients of non-normalized Dyson spin–orbitals in the orthonormal, molecular spin–orbital basis {χr}M r=1(where Mis the size of the basis), ϕDyson P=M ∑ r=1χrCrP. (4) The probability, ΓP, pertaining to the electron detach- ment/attachment process described by the non-normalized Dyson spin–orbital, ϕDyson P , therefore reads (recall that the basis {χr}M r=1is orthonormal) ΓP≡∫∞ −∞(ϕDyson P)∗ ϕDyson P dx=M ∑ r=1∣CrP∣2, (5) where xdenotes the spatial and spin coordinates of the detached/attached electron (the x-dependence of ϕDyson P has been suppressed in the notation). The probability factor, ΓP(referred to as a pole strength), may thus be used to normalize the Dyson spin–orbital ϕDyson P such that the unit-normalized Dyson spin–orbital reads φDyson P=Γ−0.5 PϕDyson P . (6) Dyson spin–orbitals describe the change in the electronic struc- ture between the N-electron wave function, ΨN, representing the initial state ∣0⟩, and the ( N±1)-electron wave function, ΨN±1,P, representing the final state ∣P⟩, i.e., for electron detachment, ϕDyson P=N1/2∫dx2dx3⋅ ⋅ ⋅dxN ×ΨN(x1,x2,x3,⋅ ⋅ ⋅,xN)Ψ∗ N−1,P(x2,x3,⋅ ⋅ ⋅,xN), (7) whereas for electron attachment, ϕDyson P=(N+1)1/2∫dx2dx3⋅ ⋅ ⋅dxN+1 ×ΨN+1,P(x1,x2,x3,⋅ ⋅ ⋅,xN+1)Ψ∗ N(x2,x3,⋅ ⋅ ⋅,xN+1). (8) J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Dyson spin–orbitals thus convey information about correlated motion of electrons embedded in many-electron wave functions ΨN andΨN±1,P(which may, in principle, be made exact) and yet retain an easily interpretable, one-electron character [as the integration in Eqs. (7) and (8) is carried out over all spin–spatial, electronic coor- dinates, xi, but one— x1—on which the Dyson spin–orbital depends] analogous to that of the non-correlated, Hartree–Fock theory. As electron binding energies occur at poles of G(E), they may conveniently be obtained by seeking zero eigenvalues of the inverse electron propagator matrix, G−1(E), and the Dyson spin–orbitals are then determined by the eigenvectors corresponding to those zero eigenvalues. When the canonical Hartree–Fock reference state and the Møller–Plesset partitioning of the Hamiltonian are employed, the inverse electron propagator matrix takes the form G−1(E)=EI−ε−σ(E), (9) where εis a diagonal matrix of Hartree–Fock orbital energies and σ(E)is a self-energy matrix which contains a part that does not depend on Eand describes the correlation of the initial N-electron state∣0⟩and a part that depends on Eand describes differential (with respect to ∣0⟩) correlation and orbital relaxation effects in the final, (N±1)-electron state ∣P⟩. Retaining only the diagonal part in G−1(E)is equivalent to the following choice of the Feynman–Dyson amplitudes: CrP=δrPC (10) (where Cis, in general, a complex number), and it leads to a partic- ularly simple form of the electron binding energy pertaining to the final state ∣P⟩[cf. Eq. (9)], EP=εP+σPP(E). (11) The corresponding, non-normalized Dyson spin–orbital becomes just the normalized, canonical molecular spin–orbital (with the spin–orbital index equal to the index of the final state due to the diagonal approximation) rescaled by the pole strength [cf. Eqs. (4)–(6) and (10)], ϕDyson P=Γ0.5 PχP, (12) where the pole strength may be obtained from ΓP=(1−dσPP(E) dE∣ E=EP)−1 . (13) Systematic, perturbative approximations to EPof Eq. (11) may be devised by collecting terms through a given order in the fluctua- tion potential. As zeroth- and first-order contributions to σPP(E) vanish, the zeroth-order approximation to EPis identical to the orbital energy, εP, and the corresponding Dyson orbital is identi- cal to the canonical, Hartree–Fock orbital, χP[i.e., the zeroth-order approximation yields the Koopmans theorem (KT)89result], and the first-order approximation to EPvanishes. Collecting all terms through second order in Eq. (11) leads to the diagonal, second- order method (D2),87,90and collecting all terms through third order gives the diagonal, third-order method (D3).87,90,91A scaled version of D3 has dubbed the outer-valence Green’s function (OVGF)92–94 method and comes in three variants—A, B, and C. Numerical crite-ria have been introduced to automatically choose among these vari- ants in a given calculation.92,94,95A careful analysis of the physical importance, numerical stability, and computational cost of third- order contributions has allowed for proposing alternatives to D3, namely, the partial third-order (P3)96method and its renormalized counterpart (P3 +).87 OVGF, P3 and, particularly, P3 +have been highly success- ful97–99in describing valence detachment energies and electron affinities of systems for which KT constitutes a valid zeroth- order approximation. This validity of KT, and hence the appli- cability limit of the diagonal, perturbative approximations may be quantified by the pole strength, ΓPin Eq. (13), for if its value drops below 0.85,100the results of the diagonal, perturba- tive methods should be treated with caution. This usually hap- pens for systems with strong correlation (many unpaired elec- trons) or for inner-shell ionization. In such cases, one needs to resort to non-diagonal methods, in which the eigenvalue problem of the super-operator Hamiltonian is solved in a truncated mani- fold of field operators. Among the non-diagonal methods, Brueck- ner doubles (BD)-T191,101–103has proven particularly successful. In BD-T1, all blocks of the super-operator Hamiltonian that arise in the hole (h), particle (p), two-hole-one-particle (2hp), and two- particle-one-hole (2ph) operator manifold are considered except the second-order couplings between 2ph and 2hp operators. BD- T1 uses the Brueckner doubles (BD), rather than Hartree–Fock, reference state. BD-T1 not only improves upon the results of the diagonal methods, in particular in cases where the pole strength is less than 0.85, but, as any other non-diagonal method, also employs Eq. (4), rather than Eq. (12) (used in diagonal methods), to produce Dyson spin–orbitals, which will therefore have contribu- tions from all canonical, Hartree–Fock spin–orbitals. Thus, in addi- tion to providing high-accuracy, quantitative information on bind- ing energies, BD-T1 is also capable of improving the description of corresponding orbitals that may sometimes alter the qualitative pic- ture provided by the canonical molecular orbitals, as is, for example, the case for a copper chloride trimer, Cu 3Cl3.103 B. Geometry optimizations, vibrational frequencies, vertical electron binding energies, and excitation energies Optimizations of geometries of all the cationic and anionic iso- mers of the O nH2n+1(n=1–6) species were performed, and their energies were determined. For cationic structures, initial guesses were obtained by constraining specific internal coordinates of the cations (bond lengths, bond angles, and dihedral angles) and opti- mizing the remaining internal coordinates. The resulting geome- tries were then used as an input for full geometry optimizations without any geometry or symmetry constraints. Previous litera- ture39,53,58,69,77was consulted in arriving at initial guess input struc- tures for the cations. The fully optimized cationic geometries were employed as initial structures for full anionic geometry optimiza- tions. Initial guess structures for the remaining anionic geome- tries (hydride-molecule complexes and electrostatic structures with ionic and polar-molecule fragments) were obtained in analogy with nitrogen-based N nH− 3n+1DRAs.14The work of Melin and Ortiz39 also provided useful hints about the H−molecule complexes and the electrostatic geometries. For n =1–5, the global minima for J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the cationic geometries strongly resemble the second most stable isomers of the anions. For n =6, the Eigen-like geometry E2 is the second most stable isomer. The full structural optimizations were initially executed at the Hartree–Fock, MP2, DFT/CAM-B3LYP,104and coupled-cluster singles-and-doubles (CCSD)105levels with a wide range of basis sets to identify an appropriate level of theory. The augmented, triple- ζ, correlation-consistent basis (aug-cc-pVTZ)106,107was found to be sufficient for cations and anions; similar conclusions for n =1–2 were reached by Melin and Ortiz.39Generally, MP2 results obtained with the aug-cc-PVTZ basis set106,107were found to be in close agreement with those of the coupled-cluster single double perturba- tive triple [CCSD(T)]/aug-cc-pVTZ level (a similar observation was made in Ref. 69). We therefore used MP2/aug-cc-pVTZ to optimize geometries for all the hydronium–water clusters, except for n=1 andn=2, where we employed the CCSD(T)/aug-cc-pVTZ struc- tures. Vibrational frequency calculations were carried out for all the geometries to ensure that all minima are characterized by all real vibrational frequencies. MP2 and CCSD(T) frequency calculations were carried out using analytical and numerical Hessians, respec- tively. Single-point calculations on the O nH2n+1radicals were per- formed at the equilibrium geometry of cationic/anionic O nH2n+1+,− complexes to determine vertical ionization energies. Vertical electron affinities of ground-state cations (H nO2n+1+) that correspond to various final states of O nH2n+1and VEDEs from the anions (H nO2n+1−) were calculated with electron propagator methods described in Sec. III A using large basis sets augmented with Rydberg-type diffuse orbitals (described in Sec. III C). Vertical exci- tation energies were inferred from differences between the vertical electron affinities. C. Basis sets To determine an optimal basis set for EPT calculations (and hence for an appropriate description of the ground and excited states of all the radicals considered in this work), primitive, diffuse Gaus- sian functions were added to the standard aug-cc-pVTZ basis.106,107 Exponents of these additional Gaussians were obtained by scaling the smallest exponents of the s,p,d, and fprimitives of the orig- inal aug-cc-pVTZ basis set with the factor of 0.5. Convergence of electron binding energies with respect to the s,p,d, and fpri- mitive Gaussian functions was obtained with an additional 4s4p4d diffuse set on all H atoms and a 4s4p4d4f set on all O atoms. At the geometry of cations and anions, several basis sets were tested for cal- culation of electron binding energies to ascertain a balance between accuracy and computational cost. For selected systems, we tested the aug-cc-pV XZ (X=T, Q, 5, 6), d-aug-cc-pVTZ, t-aug-cc-pVTZ, and q-aug-cc-pVTZ basis sets (the latter three basis sets contain, respec- tively, two, three, and four sets of diffuse, primitive functions with exponents obtained in the even-tempered manner starting from the exponents of the diffuse set of the original aug-cc-pV XZ basis).108 The results were compared against those obtained using aug-cc- pVTZ plus the above-described, additional diffuse 4s4p4d/4s4p4d4f set. We found the d-aug-cc-pVTZ basis to be the optimal choice for both cations and anions. Therefore, except where stated other- wise, all the EPT calculations were executed with the d-aug-cc-pVTZ basis.D. Software All geometry optimizations and KT, D2, D3, OVGF, P3, and P3+EPT calculations were performed with Gaussian 16.109The BD-T1 calculations were carried out with a development version of Gaussian. Dyson orbitals were plotted with GaussView110using a cube file with an enhanced edge size generated with Molden.111 The Dyson orbitals were plotted with an iso-value of 0.08 a.u. for the cations and 0.004 a.u. for the anions. Red and green surfaces of Dyson orbitals displayed in Figs. 1–14 correspond to opposite phases of a real-valued wave function. The surfaces are semi-transparent (to display the underlying molecular framework), and therefore, the inner, nodal surfaces of the Dyson orbitals are also visible. FIG. 1. n=0: Optimized geometry and the Dyson orbital of electron attachment to the OH 3+cation. FIG. 2. Optimized geometry and the Dyson orbital of electron attachment to O2H5+. FIG. 3. n=3: Optimized geometries and Dyson orbitals of electron attachment to O3H7+cations. J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. n=3: Optimized geometries and Dyson orbitals of electron attachment to O4H9+cations.III. RESULTS AND DISCUSSION A. O nH2n+1Rydberg molecules This subsection presents the optimized ground-state geome- tries of closed-shell O nH2n+1+cations. For each cluster with a given n, relative energies are discussed along with geometric parameters. Vertical electron attachment energies (VEAEs) obtained with the KT, D2, D3, OVGF, P3, P3 +, and BD-T1 methods are discussed. Subsection III B considers radical excitation energies of selected OnH2n+1clusters. Tables I–III and Figs. 1–6 present numerical and graphical data on the O nH2n+1+,0isomers. Electron binding energies presented in Table I for all O nH2n+1+ isomers do not vary much between the EPT methods. In all instances, pole strengths are above 0.85, indicating the adequacy of the diagonal, perturbative EPT methods. Therefore, unless stated otherwise, the discussion in the remainder of this subsection is focused on P3 +VEAEs. 1. OH 3+ The optimized pyramidal ground-state structure of OH 3+ belongs to the C 3vpoint group. OH 3+can be considered as the pyra- midal C 3vanalog of the sodium cation. All the O–H bond distances are equal (0.979 Å) at the CCSD(T)/aug-cc-pVTZ level (Fig. 1). OH 3+has a bound VEAE of 5.209 eV and a pole strength of 0.985, which suggests that electron correlation is not qualitatively impor- tant. The Dyson orbital for the lowest VEAE is a totally symmetric s-type function whose amplitude extends beyond the H nuclei. The two radial nodes in this Dyson orbital suggest resemblance to a 3 s function in the united atom limit. 2. O 2H5+ Coordination of a water molecule to OH 3+through hydrogen bonding leads to the Zundel O 2H5+cation, which is generally con- sidered to have a symmetric hydrogen bridge with C 2symmetry.39,69 FIG. 5. n=4: Optimized geometries and Dyson orbitals of electron attachment to O5H11+cations. J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6. n=5: Optimized geometries and Dyson orbitals of electron attachment to O 6H13+cations. FIG. 7. Dyson orbitals of first three excited states of the Eigen O 4H9radical. The bridging hydrogen atom is positioned at the distance of 1.20 Å from the oxygen nuclei of the two water molecules. Initial guesses corresponding to C 2v, D2h, and D 2dsymmetries converged to transi- tion state geometries. Zundel O 2H5+has a bound VEAE of 3.969 eVand a pole strength of 0.986. The near-unity pole strength con- firms the adequacy of the one-electron picture of the VEAE. There- fore, the qualitative features of the Dyson orbital are identical to a Hartree–Fock virtual orbital. The totally symmetric diffuse Dyson FIG. 8. Dyson orbitals of first three excited states of the Z1 O 6H13radical. J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9. n=1: Optimized geometries and Dyson orbitals of electron detachment from anions. FIG. 10. n=2: Optimized geometries and Dyson orbitals of electron detachment from anions. FIG. 11. n=3: Optimized geometries and Dyson orbitals of electron detachment from anions. orbital of the lowest electron attachment has its largest amplitudes beyond the periphery of the H nuclei. 3. O 3H7+ Two structures pertain to n=3. The C 1and C sgeometries are nearly identical and only differ in the relative orientation of the H nuclei as noted elsewhere.69The C sgeometry is only 0.004 eV more stable than the C 1isomer (Fig. 3). Both isomers have a bound VEAE of 3.33 eV. Pole strengths of 0.987 imply that correlation effects are not significant, and the qualitative features of the Dyson orbitals are valid. The Dyson orbitals in both cases are diffuse s-like functions that spread over regions beyond the external non-bonding H nuclei. The Dyson orbital amplitudes are largest near non-bonding H in the OH 3+fragment. 4. O 4H9+ Addition of three water molecules to OH 3+through hydro- gen bonding leads to four minima. Structures and Dyson orbitals for the lowest VEAE are displayed in Fig. 4. The Eigen isomer is the global minimum, whereas the trans -Zundel isomer is the least energetically stable. In the Eigen O 4H9+isomer, three water molecules are engaged in hydrogen bonding with the H nuclei of J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 12. n=4: Optimized geometries and Dyson orbitals of electron detachment from O 4H9−anions. FIG. 13. n=5: Optimized geometries and Dyson orbitals of electron detach- ment from O 5H11−anions. the OH 3+fragment in a C 3arrangement. A proposed D 3hgeome- try53is found to be a transition state with an imaginary frequency of −173.33i cm−1. Sobolewski and Domcke58proposed a C 3visomer that differs marginally by 1.0 kcal/mol from the C 3isomer. A thorough search for the proposed C 3visomer at B3LYP, MP2, and CCSD levels of theory in conjunction with several basis sets con- verged to transition state geometries in all instances. Note that the Eigen isomer has a completed solvation sphere for the central pyra- midal OH 3+fragment, for there are three H–O–H bridges with sur- rounding water molecules through hydrogen bonding. The Eigen isomer has a bound VEAE of 2.917 eV, the least among isomers ofthe same composition. The VEAE corresponds to a diffuse Dyson orbital (see red contours) whose largest amplitudes occur beyond the protons at the periphery. Orthogonalization to valence sfunctions on the oxygen atoms contributes to the red contours. In the ring isomer, there is a discernible O 3H7+fragment with an acceptor H 2O molecule, which is shared by two of the water molecules from the O 3H7+fragment. This isomer is ∼0.10 eV less stable than the global minimum and possesses the largest VEAE among the O 4H9+isomers. The nearly symmetric Dyson orbital is spread over the cationic O 4H9+core with its largest lobes occurring on the acceptor water molecule. J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 14. n=6: Optimized geometries and Dyson orbitals of electron detachment from O 6H13−anions. Coordination of two water molecules to a Zundel O 2H5+ion leads to the cis- and trans -Zundel isomers with the C 2point group. The structural features of the Zundel O 2H5+ion fragment are retained in both isomers, for the symmetric hydrogen is positioned at 1.20 Å from the oxygen nuclei of the two neighboring water molecules. In both isomers, the Dyson orbital is diffused beyond the periphery of the external hydrogens. The green contours that arise from orthogonalization to the valence sfunction of O’s present cis- and trans -like shapes, respectively. Both isomers have nearly the same VEAE of 2.94 eV, while the cis-isomer is∼0.12 eV energetically more stable than the trans . D2, D3, OVGA, P3, and P3 +results are in close agreement in all instances. Pole strengths of ∼0.98 indicate that the diagonal self-energy approximations are reliable for qualitative predictions. 5. O 5H11+ Forn=5, there are five isomeric structures, which are displayed in Fig. 5 along with their Dyson orbitals. In the B1 and B2 isomers, a water molecule starts filling the second solvation shell of the Eigen O4H9+cation. In both instances, the Eigen O 4H9+ion fragment is discernible, while an H 2O molecule enters the secondary solvation shell through hydrogen bonding. The two geometries are almost identical, and the positions of external H’s constitute the chief dif- ference. B1 is only 0.001 eV more stable than B2. The two isomershave nearly equal VEAEs of 2.65 eV. In both instances, the Dyson orbital is diffused over the Eigen O 4H9+ion fragment. Minor delo- calization onto the valence functions in the H 2O molecule within the secondary sphere can be seen. In the C O 5H11+isomer, an O 3H7+ion fragment is discernible. Two water molecules form hydrogen bonds with the H 2O fragments in the O 3H7+ion. In other words, the C O 5H11+isomer can be considered to be a chain of water molecules with a non-symmetric hydrogen nucleus. A diffuse s-like function whose largest lobes occur on non-bonding H in the OH 3+ion fragment is the chief feature of the Dyson orbital. Orthogonalization to the valence sfunctions of the heavy atoms is also visible. Interaction of a water molecule through hydrogen bonding to cis-Zundel O 4H9+gives rise to the PR (pentagonal ring) O 5H11+ isomer, which is 0.005 eV less stable than B1. An H 2O molecule is shared by two water molecules from the cis-Zundel O 4H9+ion frag- ment and forms a pentagonal ring geometry. This isomer has the largest VEAE, 2.899 eV, among the set. Only the PR isomer features the Zundel cation among the O 5H11+isomers. The Dyson orbital spreads over the cation, and its largest amplitudes are visible on H’s of the acceptor water molecule. The remaining isomer (R isomer) can be regarded as an Eigen O 4H9+cation coordinated to an acceptor water molecule to form a four-member ring framework. This isomer has a VEAE of J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I. OnH2n+1+isomerization and vertical electron attachment energies (VEAEs). Pole strengths are in parentheses. All energies are in eV. Calculations were executed with the d-aug-cc-pVTZ basis set. n Isomer ΔE KT D2 D3 OVGF P3 P3 + 1 OH 3+⋅ ⋅ ⋅ 4.691 5.211 (0.986) 5.209 (0.984) 5.186 (0.985) 5.209 (0.985) 5.209 (0.985) 2 O 2H5+⋅ ⋅ ⋅ 3.527 3.969 (0.987) 3.969 (0.985) 3.960 (0.986) 3.969 (0.986) 3.969 (0.986) 3 C sO3H7+0.000 2.941 3.329 (0.989) 3.333 (0.987) 3.330 (0.988) 3.333 (0.987) 3.333 (0.987) C1O3H7+0.004 2.937 3.331 (0.989) 3.335 (0.987) 3.333 (0.988) 3.335 (0.987) 3.334 (0.987) 4 Eigen O 4H9+0.000 2.560 2.912 (0.990) 2.922 (0.988) 2.920 (0.989) 2.917 (0.988) 2.917 (0.988) Ring O 4H9+0.101 2.727 3.141 (0.988) 3.144 (0.986) 3.141 (0.987) 3.144 (0.986) 3.144 (0.986) Cis-Zundel O 4H9+0.083 2.572 2.932 (0.989) 2.937 (0.987) 2.940 (0.988) 2.937 (0.987) 2.937 (0.988) Trans -Zundel O 4H9+0.202 2.584 2.930 (0.990) 2.935 (0.988) 2.935 (0.989) 2.935 (0.988) 2.935 (0.988) 5 B1 O 5H11+0.000 2.315 2.638 (0.990) 2.645(0.989) 2.650 (0.989) 2.645 (0.989) 2.644 (0.989) B2 O 5H11+0.001 2.317 2.643 (0.990) 2.649 (0.988) 2.655 (0.989) 2.649 (0.989) 2.649 (0.989) C O 5H11+0.006 2.321 2.655 (0.990) 2.661 (0.988) 2.667 (0.989) 2.661 (0.988) 2.660 (0.988) PR O 5H11+0.005 2.505 2.894 (0.988) 2.899 (0.986) 2.894 (0.987) 2.899 (0.986) 2.899 (0.986) R O 5H11+0.002 2.421 2.786 (0.989) 2.792 (0.987) 2.790 (0.988) 2.792 (0.987) 2.791 (0.987) 6 C1 O 6H13+0.135 2.439 2.836 (0.988) 2.840 (0.986) 2.834 (0.987) 2.840 (0.986) 2.840 (0.986) E1 O 6H13+0.047 2.108 2.402 (0.991) 2.410 (0.989) 2.418 (0.990) 2.410 (0.989) 2.409 (0.989) E2 O 6H13+0.051 2.112 2.410 (0.991) 2.417 (0.989) 2.425 (0.990) 2.417 (0.989) 2.417 (0.989) P1 O 6H13+0.042 2.279 2.628 (0.989) 2.635 (0.987) 2.630 (0.988) 2.635 (0.987) 2.634 (0.988) T1 O 6H13+0.050 2.188 2.519 (0.990) 2.526 (0.988) 2.528 (0.989) 2.526 (0.988) 2.525 (0.988) T2 O 6H13+0.046 2.215 2.549 (0.990) 2.556 (0.988) 2.557 (0.989) 2.556 (0.988) 2.556 (0.988) T3 O 6H13+0.059 2.248 2.588 (0.990) 2.595 (0.988) 2.593 (0.989) 2.595 (0.988) 2.594 (0.988) T4 O 6H13+0.048 2.242 2.581 (0.990) 2.588 (0.988) 2.585 (0.989) 2.588 (0.988) 2.587 (0.988) Z1 O 6H13+0.000 2.121 2.416 (0.991) 2.432 (0.989) 2.432 (0.990) 2.423 (0.989) 2.423 (0.989) 2.791 eV. In analogy to the PR O 5H11+and ring O 4H9+isomers, the Dyson orbital is a diffuse, symmetric function occurring over peripheral H’s with its largest amplitudes on the acceptor water molecule.6. O 6H13+ Forn=6, nine distinctive geometric isomers were located. Their structures and Dyson orbitals are displayed in Fig. 6. The C1 TABLE II. 20 lowest electron affinities (EAs) and radical excitation energies (EEs) for Eigen O 4H9+. All energies are in eV. Calculations were executed with the d-aug-cc-pVTZ basis set. Final state KT D2 D3 OVGF P3 P3 + P3+EEs OVGF EEs 12A1 2.560 2.912 2.922 2.920 2.917 2.917 0.000 0.000 12E 2.026 2.259 2.267 2.266 2.266 2.266 0.651 0.654 22A1 1.838 1.954 1.964 1.962 1.960 1.959 0.958 0.958 22E 1.372 1.458 1.463 1.462 1.463 1.462 1.455 1.458 32A1 1.300 1.387 1.392 1.391 1.391 1.391 1.526 1.529 32E 1.288 1.353 1.359 1.359 1.358 1.358 1.559 1.561 42A1 1.288 1.353 1.359 1.359 1.358 1.358 1.559 1.561 52A1 1.079 1.215 1.219 1.219 1.217 1.217 1.700 1.701 42E 0.916 1.032 1.035 1.034 1.034 1.034 1.883 1.886 62A1 0.854 0.930 0.936 0.935 0.932 0.932 1.985 1.985 72A1 0.619 0.678 0.682 0.681 0.683 0.683 2.234 2.239 82A1 0.537 0.601 0.608 0.607 0.607 0.607 2.310 2.313 52E 0.529 0.602 0.607 0.606 0.605 0.605 2.312 2.314 62E 0.518 0.565 0.569 0.569 0.568 0.568 2.349 2.351 92A1 0.459 0.555 0.559 0.558 0.557 0.557 2.360 2.362 72E 0.437 0.507 0.513 0.512 0.513 0.512 2.405 2.405 J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-11 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III. 20 lowest electron affinities (EAs) and radical excitation energies (EEs) for Z1 O 6H13+. All energies are in eV. Calculations were executed with the d-aug-cc-pVTZ basis set. Final state KT D2 D3 OVGF P3 P3 + P3+EEs OVGF EEs 12A1 2.121 2.416 2.432 2.432 2.423 2.423 0.000 0.000 22A1 1.780 2.000 2.010 2.009 2.008 2.007 0.416 0.423 32A1 1.727 1.909 1.920 1.918 1.917 1.916 0.507 0.514 42A1 1.674 1.826 1.838 1.836 1.833 1.832 0.591 0.596 52A1 1.357 1.481 1.487 1.486 1.487 1.487 0.936 0.946 62A1 1.252 1.334 1.341 1.339 1.340 1.339 1.084 1.093 72A1 1.206 1.278 1.286 1.284 1.283 1.283 1.140 1.148 12E 1.178 1.236 1.243 1.243 1.241 1.241 1.182 1.189 82A1 1.029 1.145 1.151 1.150 1.147 1.147 1.276 1.282 92A1 0.853 0.935 0.939 0.939 0.938 0.937 1.486 1.493 102A1 0.830 0.886 0.891 0.890 0.890 0.889 1.534 1.542 112A1 0.816 0.868 0.873 0.872 0.872 0.871 1.552 1.56 122A1 0.742 0.851 0.855 0.855 0.854 0.854 1.569 1.577 132A1 0.719 0.830 0.834 0.834 0.834 0.834 1.589 1.598 142A1 0.666 0.758 0.763 0.762 0.763 0.763 1.660 1.67 152A1 0.604 0.667 0.672 0.671 0.670 0.670 1.753 1.761 162A1 0.532 0.637 0.643 0.642 0.641 0.641 1.782 1.79 172A1 0.486 0.544 0.551 0.550 0.549 0.549 1.874 1.882 182A1 0.439 0.521 0.529 0.528 0.525 0.525 1.898 1.904 192A1 0.416 0.468 0.475 0.475 0.474 0.474 1.949 1.957 O6H13+isomer is a three-dimensional (3D) cage structure. In this isomer, there is a discernible Eigen O 4H9+fragment whose external H’s engage in hydrogen bonding with two H 2O molecules result- ing in a 3D cage geometry. This structure has the largest VEAE of 2.840 eV among the isomers of the same composition, and it is the least energetically stable isomer. The Dyson orbital forms a cloud of diffuse symmetric sfunctions distributed over the 3D water network. Note that the Dyson orbital does not touch the OH 3+fragment core. The E1 and E2 O 6H13+isomers result from the Eigen O 4H9+ ion fragment with two water molecules that are engaged in hydrogen bonding. Note that two water molecules are in the secondary coordi- nation sphere through hydrogen bonding with the peripheral hydro- gens from the Eigen O 4H9+ion fragment. In E1, the H 2O molecules in the secondary shell approach the O 4H9+ion fragment in a cis-like fashion, whereas the trans -orientation is observed in E2. The E2 iso- mer is 0.003 eV more energetically stable than E1. Both isomers have VEAEs near 2.4 eV. The symmetric Dyson orbitals are diffused over exterior H’s on the O 4H9+ion fragments with minor delocalization onto the neighboring water molecules in the secondary sphere. The red and green contours are the key distinctive features in E1 and E2, respectively. Note that the Dyson orbital closely resembles that of the O4H9+ion. Addition of a water molecule through hydrogen bonding with PR O 5H11+results in the P1 O 6H13+isomer. A diffuse Dyson orbital spread over H’s in the PR O 5H11+fragment is observed. Coordi- nation of a water molecule to R O 5H11+results in four isomers, T1–T4. In T1, the acceptor water molecule in the R O 5H11+fragment engages in hydrogen bonding with an H 2O molecule. The diffuse Dyson orbital has its largest lobes around the regions of the acceptorwater dimer. The isomer T2 of O 6H13+contains a discernible Eigen O4H9+cation and forms a tetramer ring wherein a water molecule enters the second solvation sphere of the R O 5H11+fragment. The Dyson orbital is distributed beyond the R O 5H11+fragment with its largest amplitude on the acceptor water molecule in the tetramer ring. In the T3 and T4 O 6H13+isomers, an H 2O forms a hydro- gen bond with water in the R O 5H11+fragment, which does not participate in any secondary hydrogen bonding. This results in the formation of a water dimer in the second shell in both isomers. The chief difference between the T3 and T4 isomers pertains to the rela- tive orientations of the water dimer. In both cases, the Dyson orbital is diffused over the R O 5H11+fragment ion. The Z1 O 6H13+isomer encompasses the Zundel O 2H5+cation fragment. Each of the exterior protons in the Zundel O 2H5+ion fragment engages in hydrogen bonding with an H 2O molecule to complete the first solvation shell in a C 2symmetry arrangement. All structural features in the original Zundel O 2H5+ion fragment are retained. The new hydrogen bonds occur at 1.67 Å. This isomer is the global minimum among the nine O 6H13+isomers. It also has the smallest VEAE of 2.430 eV. Given the stability of this isomer, a peak near 2.4 eV is expected to dominate the photoelectron spectrum of O6H13isomers. Its Dyson orbital can be seen to spread over exte- rior H’s to form a symmetric function. Orthogonalization to valence sfunctions of O’s results in the red contours. B. Radical excitation energies Vertical electron affinities of cations and radical excitation energies for several electronic states of micro-solvated Eigen O 4H9 J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-12 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and Z1 O 6H13are shown in Tables II and III, respectively. Pole strengths for all the considered states are larger than 0.95, and hence, the diagonal approximation of the self-energy, wherein the Dyson orbitals are proportional to Hartree–Fock virtual orbitals (see Sec. II A), is qualitatively valid. Generally, there is good agreement between methods for electron affinities. Therefore, vertical excita- tion energies from the2A1final states to higher electronic states may be inferred from P3 +and OVGF results shown in Tables II and III. Orbital relaxation and correlation corrections to KT results become smaller for higher final electronic states of Eigen O 4H9and Z1 O 6H13, which have unpaired electrons in increasingly diffuse Rydberg orbitals. For Eigen O 4H9, the first 20 excitation energies fall within 0.7 and 2.4 eV. In Z1 O 6H13, the excitation energies lie within 0.4 and 2.0 eV. Hence, the radical excitation energies are larger for Eigen O 4H9than for Z1 O 6H13. For both cations, excited states of the radicals correspond to diffuse Dyson orbitals (Figs. 7 and 8). The second and third elec- tron affinities of Eigen O 4H9denoted as EA 1and EA 2correspond to a doubly degenerate (P xand P y)2E state with diffuse p-like orbitals (Fig. 7). Note the orthogonalization of the contours to the valence functions of O’s. The higher P zlevel corresponding to EA 3is also diffused with orthogonalization to the O–H bond- ing functions. However, EA 1, EA 2, and EA 3electron affinities of Z1 O 6H13+correspond to non-degenerate final states with diffuse, p- like orbitals (Fig. 8). In all radical excited states considered presently, Dyson orbitals with diffuse p-like and d-like nodal patterns emerge. C. O nH2n+1−double Rydberg anions Geometry optimizations performed at the CCSD(T)/aug-cc- pVTZ (for n=1, 2) and MP2/aug-cc-pVTZ (for n=3–6) lev- els with default convergence criteria for all isomers converge to minima. A change in convergence criteria to 300 maximum cycles is necessary for trans -(H 2O)O 2H5−(H2O) and for Z1 O 6H13−. To judge the reliability of EPT methods for calculating VEDEs, a preli- minary comparison with ΔCCSD(T) results was conducted for the OH 3−DRA. The results in Table IV were obtained with various basis sets to evaluate the effectiveness of each method in describing elec- tron correlation and orbital relaxation corrections to the KT results. Only the lowest VEDE was considered. Compared to ΔCCSD(T), KT results are too small, whereas D2, D3, P3 +, and OVGF self- energy approximations are too large. D2 and OVGF have lower pole strengths. The BD-T1 result is the closest to ΔCCSD(T). A com- parison of BD-T1 results obtained with various basis sets shows that BD-T1/d-aug-cc-pVTZ is the optimal level for computational accuracy and cost. Data from Table IV suggest that for DRAs, BD- T1 is the best EPT method. Of the remaining EPT methods, P3 + generally performs better than OVGF with respect to VEDEs and pole strengths. The data in Table IV show that improvements cor- responding to the t-aug-cc-pVTZ, q-aug-cc-pVTZ, aug-cc-pVQZ, and aug-cc-pV5Z basis sets are not needed. Hence, unless stated otherwise, BD-T1/d-aug-cc-pVTZ results are discussed in the text. 1. OH 3−isomers Figure 9 displays the optimized geometries and Dyson orbitals for the lowest vertical electron detachment energies (VEDEs) for the hydride–water complex H−(OH 2) and the OH 3−DRA. Numerical results for relative energies and electron binding energiesTABLE IV. Comparison of VEDEs of the OH 3−DRA calculated at different levels of theory. Pole strengths are in parentheses. All energies are in eV. OH 3−VEDE KTa0.362 D2a0.633 (0.830) D3a0.620 (0.874) P3+a0.622 (0.870) OVGFa0.481 (0.772) BD-T1a0.539 (0.862) ΔCCSD(T)/d-aug-cc-pVTZ 0.536 BD-T1/aug-cc-pVTZ 0.431 (0.894) BD-T1/d-aug-cc-pVTZ 0.537 (0.864) BD-T1/t-aug-cc-pVTZ 0.539 (0.862) BD-T1/q- aug-cc-pVTZ 0.539 (0.862) BD-T1/aug-cc-pVQZ 0.458 (0.889) BD-T1/aug-cc-pV5Z 0.486 (0.884) aCalculation performed with the aug-cc-pVTZ basis augmented with an additional set of diffuse functions: 4s4p4d on all H atoms and 4s4p4d4f on all O atoms. are shown in Table V. The hydride–molecule complex is more sta- ble than the OH 3−DRA. In the former anion, the VEDE of the free hydride (0.75 eV) increases by 0.9 chiefly due to electrostatic inter- actions with the water molecule. The H 2O molecule approaches the free hydride with one of its protons in a C sgeometry. There is a minor increase in the length of the O–H bond that approaches H−. A structure in which the hydride anion forms a symmetric bond- ing arrangement with the molecule’s protons is a transition state.20,39 The Dyson orbital is chiefly composed of diffuse sfunctions, which display no bonding relationship with valence functions on the clos- est O–H bonds. A totally symmetric Dyson orbital whose amplitudes are beyond the hydrogen nuclei is observed for the lowest VEDE of the OH 3−DRA. This Dyson orbital smoothly correlates to the 3s orbital of Na−in the united atom limit, for it has two radial nodes. The lowest VEDE of 0.537 eV nearly equals that of Na−(0.55 eV). The Dyson orbital for the lowest VEDE closely resembles that of the first electron attachment energy of the cationic core OH 3+, for both electron binding energies corresponding to the same final uncharged doublet OH 3state. The O–H bond lengths are almost the same in the cation, neutral, and anion and imply the insignificance of the anti- bonding features in the Dyson orbital. The present results can be considered as an upper bound to previous calculations.6,19,20,38,39 2. O 2H5−isomers There are three O 2H5−isomers, and their optimized structures and Dyson orbitals are illustrated in Fig. 10. The most stable isomer is the hydride anion complex, which has two water molecules that weakly coordinate to the H−via attractions to their protons. A peak around 2.5 eV is expected to dominate the photoelectron spectrum of O 2H5−considering its high stability. The new water molecule increases the VEDE of the H−(OH 2) by 0.911 eV, and the Dyson orbital is localized on the hydride’s nucleus. The asymmetric hydrogen-bridged isomer has discernible OH 3−and H 2O molecular fragments. The binding of the same two fragments by electrostatic interactions results in the least sta- ble isomer. In the latter isomer, the Dyson orbital’s amplitudes are largest outside the three H nuclei of the OH 3−fragment. Minor J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-13 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE V. OnH2n+1−isomerization and vertical electron detachment energies (VEDEs). Pole strengths are in parentheses. All energies are in eV. Calculations were executed with the d-aug-cc-pVTZ basis set. n Isomer ΔE KT D2 D3 OVGF P3 P3 + BD-T1 1 H−(OH 2) 0.000 2.009 1.522 (0.881) 1.695 (0.901) 1.732 (0.899) 1.505 (0.906) 1.506 (0.904) 1.659 (0.864) OH 3−1.540 0.338 0.658 (0.867) 0.494 (0.833) 0.409 (0.750) 0.585 (0.831) 0.591 (0.834) 0.537 (0.864) 2 H−(H2O)2 0.000 8.159 5.446 (0.858) 7.501 (0.923) 6.925 (0.900) 6.656 (0.901) 2.401 (0.911) 2.570 (0.819) OH 3−(H2O) 2.238 0.500 0.873 (0.863) 0.726 (0.851) 0.700 (0.844) 0.837 (0.847) 0.840 (0.849) 0.803 (0.881) O2H5−1.323 0.244 0.603 (0.865) 0.435 (0.825) 0.336 (0.696) 0.529 (0.822) 0.536 (0.826) 0.519 (0.860) 3 H−(OH 2)3 0.000 3.715 3.131 (0.908) 3.392 (0.912) 3.394 (0.909) 3.159 (0.911) 3.156 (0.911) 3.352 (0.899) OH 3−[OH 2]2 2.454 0.685 1.105 (0.871) 0.974 (0.866) 0.961 (0.867) 1.089 (0.864) 1.090 (0.865) 1.066 (0.924) O2H5−(H2O) 2.607 0.371 0.781 (0.851) 0.607 (0.833) 0.572 (0.818) 0.736 (0.826) 0.740 (0.828) 0.731 (0.879) O3H7−2.491 0.209 0.609 (0.863) 0.432 (0.823) 0.341 (0.700) 0.540 (0.817) 0.547 (0.821) 0.537 (0.865) 4 H−(H2O)4 0.000 4.303 3.669 (0.909) 3.947 (0.911) 3.936 (0.910) 3.711 (0.911) 3.707 (0.911) 3.899 (0.900) [OH 2]2OH 3–H 2O 2.704 0.198 0.597 (0.844) 0.407 (0.809) 0.327 (0.706) 0.528 (0.798) 0.535 (0.803) 0.535 (0.867) Eigen O 4H9−2.629 0.200 0.605 (0.857) 0.430 (0.820) 0.360 (0.736) 0.540 (0.812) 0.549 (0.817) 0.574 (0.874) Ring O 4H9−2.780 0.157 0.610 (0.871) 0.437 (0.830) 0.342 (0.684) 0.551 (0.822) 0.557 (0.828) 0.521 (0.882) Cis-(H 2O)O 2H5−(H2O) 2.789 0.151 0.541 (0.856) 0.364 (0.809) 0.244 (0.491) 0.465 (0.800) 0.472 (0.806) 0.450 (0.840) Trans -(H 2O)O 2H5−(H2O) 2.638 0.151 0.541 (0.856) 0.364 (0.809) 0.244 (0.491) 0.465 (0.800) 0.599 (0.809) 0.596 (0.879) 5 H−(H2O)5 0.000 4.655 3.984 (0.910) 4.270 (0.912) 4.250 (0.912) 4.042 (0.912) 4.036 (0.910) 4.218 (0.901) B1 O 5H11−2.802 0.194 0.574 (0.833) 0.389 (0.801) 0.314 (0.708) 0.506 (0.788) 0.513 (0.793) 0.527 (0.905) B2 O 5H11−2.745 0.179 0.544 (0.835) 0.358 (0.802) 0.281 (0.698) 0.475 (0.788) 0.482 (0.794) 0.484 (0.905) C O 5H11−2.860 0.160 0.502 (0.823) 0.308 (0.788) 0.208 (0.599) 0.424 (0.773) 0.432 (0.779) 0.424 (0.848) PR O 5H11−2.946 0.165 0.595 (0.838) 0.383 (0.793) 0.269 (0.521) 0.515 (0.777) 0.523 (0.785) 0.499 (0.845) R O 5H11−2.884 0.146 0.576 (0.854) 0.397 (0.810) 0.296 (0.617) 0.508 (0.800) 0.515 (0.806) 0.506 (0.853) 6 H−(H2O)6 0.000 5.233 4.514 (0.910) 4.774 (0.909) 4.755 (0.908) 4.558 (0.908) 4.553 (0.908) C1 O 6H13−3.025 0.101 0.522 (0.868) 0.379 (0.816) 0.275 (0.566) 0.379 (0.816) 0.469 (0.818) E1 O 6H13−2.878 0.154 0.490 (0.15) 0.292 (0.782) 0.208 (0.639) 0.414 (0.764) 0.422 (0.770) E2 O 6H13−2.868 0.160 0.548 (0.833) 0.361 (0.795) 0.267 (0.623) 0.477 (0.780) 0.484 (0.780) P1 O 6H13−2.906 0.154 0.549 (0.821) 0.339 (0.77) 0.225 (0.430) 0.464 (0.759) 0.472 (0.765) T1 O 6H13−3.174 0.161 0.567 (0.814) 0.337 (0.765) 0.224 (0.311) 0.473 (0.744) 0.482 (0.751) T2 O 6H13−3.026 0.165 0.620 (0.848) 0.445 (0.814) 0.376 (0.734) 0.565 (0.801) 0.570 (0.807) T3 O 6H13−2.997 0.118 0.490 (0.842) 0.322 (0.799) 0.210 (0.493) 0.421 (0.785) 0.428 (0.792) T4 O 6H13−2.898 0.159 0.573 (0.845) 0.397 (0.810) 0.313 (0.685) 0.512 (0.796) 0.518 (0.802) Z1 O 6H13−3.028 0.123 0.500 (0.826) 0.295 (0.776) 0.181 (0.284) 0.411 (0.759) 0.421 (0.767) delocalization onto the coordinated water molecule can be seen. The VEDE of the asymmetric hydrogen-bridged isomer of O 2H5−dif- fers by∼0.3 eV from OH 3−(H2O). The Dyson orbital is a nearly symmetric sfunction with its largest lobes occurring on the pro- tons. Plots of lower iso-values give credence to a Dyson orbital which predominantly accumulates on the OH 3−fragment with minor delo- calization to regions near the water molecule’s H nuclei as observed in a preceding study39and in isomers of the N 2H7−DRAs.13,14 3. O 3H7−isomers Interaction of a water molecule with the O 2H5−isomers gives rise to four different anionic O 3H7−clusters. The optimized struc- tures and associated Dyson orbitals are displayed in Fig. 11. The hydride–molecule complex is the most energetically stable. This complex has a C 3axis and a perpendicular plane through the hydride’s nucleus. Two water molecules coordinate to H−(OH 2) with their protons. The coordination of an H 2O molecule increasesthe VEDE by 0.782 eV compared to H−(H2O)2. The Dyson orbital is dominated by delocalized sfunctions on the hydride, whereas an antibonding relationship with the neighboring O–H bonds is seen. Given the high stability of this isomer, a peak near 3.4 eV is expected to dominate the photoelectron spectrum of O 3H7−. Other isomers containing the OH 3−and O 2H5−DRAs are energetically unstable and have VEDEs that are higher than the O 3H7−DRA. The OH 3−[OH 2]2isomer arises from addition of a water molecule to OH 3−(H2O). The new water molecule forms a hydrogen- bonded water dimer, denoted as [OH 2]2with the H 2O fragment in OH 3−(H2O). The newly coordinated water increases the VEDE by 0.263 eV compared with OH 3−(H2O). The Dyson orbital is retained on the OH 3−kernel with negligible red contours on the surrounding water dimer. Addition of a water molecule to the O 2H5−DRA through weak electrostatic interactions leads to the O 2H5−(H2O) isomer. The water molecule interacts with the hydrogen nuclei. The Dyson J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-14 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp orbital closely resembles that of O 2H5−, with its largest amplitudes at the peripherical hydrogens on the O 2H5−fragment. Negligible antibonding relationships between the Dyson orbitals and valence functions on the water molecule are seen. This isomer is the least energetically stable among all the O 3H7−isomers. The VEDE of O2H5−(H2O) is 0.212 eV above that of O 2H5−. In the O 3H7−DRA, two water molecules interact with two of the OH 3−protons through hydrogen bonding. This structure can be considered as an O 2H5− DRA in which one of the non-bonding hydrogens in OH 3−forms a hydrogen bond with the water molecule. The VEDE of the O 3H7− isomer is 0.018 eV larger than that of the O 2H5−DRA. The Dyson orbital accumulates on the peripheral non-bonding hydrogens, with its largest amplitudes on non-bonding H’s in the OH 3−core and water fragments. The O 3H7−DRA has almost the same VEDE as OH 3−. 4. O 4H9−isomers Forn=4, six isomers were found on the PES. The graphi- cal results are displayed in Fig. 12. The hydride–molecule complex is the global minimum. In this H−(H2O)4structure, each of the three coordinating water molecules interacts through one of its H nuclei with the hydride anion in H−(H2O)3, increasing the VEDE of the latter anion complex by 0.547 eV. Delocalization of sfunc- tions on H−is the most important feature of the Dyson orbital, with minor green and red contours on three of the neighboring water molecules. Addition of a water molecule to the O 3H7−DRA leads to the [OH 2]2OH 3−(H2O) isomer. The new water molecule forms a hydrogen-bonded water dimer with one of the H 2O molecules in O 3H7−. This isomer is relatively unstable energetically and has a VEDE of 0.535 eV. In the Eigen O 4H9−isomer, three water molecules are hydrogen-bonded with the OH 3−protons. It is the most energeti- cally stable isomer after the hydride ion complex. The OH 3−frag- ment imposes a rigid threefold geometry on this cluster. The relative stability of this isomer suggests that a micro-solvated OH 3−clus- ter with a complete solvation shell is more stable than other isomers. The Dyson orbital is a localized function represented by red contours that spread over all the exterior non-bonding H nuclei in the water network. The next isomer is the ring form of O 4H9−, in which the O3H7−kernel is connected to a bridging water molecule that forms a four-member ring. The dominance of green contours on the accep- tor water molecule is the characteristic feature of the Dyson orbital. Delocalization of the Dyson orbital over the remaining unshared H’s can also be seen. The last two structural isomers originate from the typical Zun- del cationic structures where the proton is shared by two water molecules. For these isomers, each of the water molecules in the O2H5−fragment forms a hydrogen bond with one water molecule in the cis- and trans -orientations. In the cis-isomer, the new water molecule forms a water dimer. At the initial cationic geometry, the trans -anionic isomer converges to a structure that closely resembles thecis-isomer except for the orientation of H nuclei in the perip- hery. The structural distortion arises from the rotation of one [H2O]2about the bridging H nuclei. The trans -isomer is more ener- getically stable than the cisby 0.151 eV. In addition, the VEDE of the trans -isomer is higher than that of the cis-isomer by 0.146 eV. The Dyson orbitals of both anions correspond to diffuse, red contours whose largest amplitudes occur on the exterior H nuclei. Minorantibonding relationships with the neighboring valence functions can also be seen. 5. O 5H11−isomers The coordination of a water molecule to the O 4H9−isomers through hydrogen bonding or electrostatic interactions leads to six O5H11−isomers. Graphical results for the O 5H11−isomers are dis- played in Fig. 13. H−(H2O)5is at least 2.7 eV more energetically stable than the DRAs. The VEDE of H−(H2O)4increases by 0.329 eV upon the addition of an H 2O through electrostatic interactions. The Dyson orbital remains localized on the hydride’s nucleus with slight delocalization onto water H’s. Addition of a hydrogen-bonded water dimer denoted by [H 2O]2to the O 3H7−DRA results in the B1 O5H11−isomer whose VEDE has decreased by 0.01 eV. In other words, the B1 O 5H11−isomer is an Eigen O 4H9−ion in which one of H’s in the OH 3−fragment engages in further hydrogen bond- ing. The Dyson orbital accumulates on the external hydrogens. The B2 O 5H11−structure consists of an Eigen-like anionic fragment in which two of the H’s in the OH 3−fragment engage in further hydro- gen bonding with a water molecule. The three water molecules that form hydrogen bonds with the H nuclei of OH 3−DRA do so at 1.55 Å, suggesting typical hydrogen bonds. The water in the sec- ond solvation shell forms a hydrogen bond at 1.84 Å, suggesting a relatively weak hydrogen bond compared with those in the first sol- vation shell. There is a decrease in the VEDE by 0.09 eV compared to the Eigen O 4H9−DRA. The C O 5H11−isomer has the smallest VEDE among the O5H11−structures. The parent cationic core forms a linear chain of water molecules with two H nuclei in the OH 3+unit. After opti- mization, the cationic chain O 5H11+rearranges to form a book-like anionic isomer. Three water molecules form hydrogen bonds with the OH 3−framework to complete the primary solvation shell, while the remaining water enters the secondary solvation shell. The Dyson orbital is diffused over the non-bonding H nuclei with its largest amplitudes on the exterior water H nuclei. The PR O 5H11−isomer also undergoes structural rearrangement upon optimization at the cationic geometry to form another book-like isomer. In this iso- mer, an O 3H7−DRA fragment can be seen. Each of the two water molecules forms hydrogen bonds with the H 2O molecules originally bonded to H nuclei in the OH 3−fragment. This isomer has the lowest VEDE of 0.499 eV. The positive red contour of the Dyson orbital densely populates the regions between the non-bonded H nuclei in the OH 3−fragment those from the two acceptor water molecules. Small, delocalized, and green contours are seen on the remaining non-bonding H nuclei. In the remaining ring O 5H11−iso- mer, the parent cationic core forms a tetramer ring through hydro- gen bonding with one acceptor H 2O. The optimized anionic geo- metry structurally looks like the parent cationic core. It has the low- est VEDE of 0.506 eV. The Dyson orbital spreads over the H nuclei of the acceptor water molecule with some negligible delocalization on neighboring non-bonding protons. 6. O 6H13−isomers There are ten geometric isomers of the anionic hydronium–water hexamer clusters. The optimized geome- tries and Dyson orbitals of the lowest VEDE are illustrated in Fig. 14. Only P3 +electron propagator calculations were carried out for these isomers, for data in Table V show that P3 +generally gives J. Chem. Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-15 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp good VEDEs compared to BD-T1, albeit with lower pole strengths. Hereafter, only P3 +results are discussed. The hydride anion–molecule complex is the most energetically stable with the lowest VEDE of 4.553 eV. All the water molecules have their protons oriented toward the H−anion. The Dyson orbital accumulates on the hydride anion and has antibonding relation- ships with valence functions on the neighboring O–H bonds. The remaining nine O 6H13−isomers were obtained from their corre- sponding parent cationic cores. The cage C1 O 6H13−isomer strongly retains its cationic geometry orientation with a VEDE of 0.469 eV. The diffuse Dyson orbital is spread chiefly over H’s in the water network. Two structures pertain to the Eigen-like configurations E1 O 6H13−and E2 O 6H13−, for their corresponding cationic cores contain the O 4H9+Eigen fragment. The two new water molecules enter the second solvation shell through hydrogen bonding. The optimized anionic structures are distorted when compared with their cationic counterparts, for the two water molecules originally in the second solvation shell of the cationic core form a water dimer acceptor molecule in the molecular anion. The E1 O 6H13−isomer has a smaller VEDE than the E2 O 6H13−. In both E1 O 6H13−and E2 O6H13−, the nearly symmetric diffuse Dyson orbitals are localized on H’s of the water network. The P1 O 6H13−isomer arises from the parent cationic core in which the new water molecule enters the secondary solvation shell from the ring O 5H11+isomer. A water dimer is formed in the sec- ond coordination shell of the anionic P1 O 6H13−isomer. A near symmetric s-like Dyson orbital whose largest amplitudes occur over the peripheral water network can be seen. Four isomers arise from parent cationic cores that contain water tetramer rings. In the T1 O6H13−isomer, an Eigen-like O 4H9−can been seen with a water molecule forming a bridging hydrogen bond with two of the water molecules in O 4H9−. The bridging water molecule forms an addi- tional hydrogen bond with one of its protons to form a water dimer. A diffuse Dyson orbital spreads over the H nuclei of the water network. In the T2 O 6H13−isomer, the bridging water molecule does not interact with the water in the second solvation shell as it forms a hydrogen bond with one of the water molecules directly bonded to the OH 3−fragment. The Dyson orbital is diffused over the peripheral hydrogens with its largest amplitudes on the bridg- ing water molecule and the one in the secondary shell. There are low amplitudes near the lone pairs on the oxygens. The T3 and T4 O 6H13−isomers have book-like structures. The Dyson orbital in both instances spreads over the regions of the water network with its highest amplitudes on the acceptor water molecules. The last isomer is the Z1 O 6H13−isomer, which arises from the Zundel-like solvated cationic core. The anionic geometry contains the O 2H5−fragment with a non-symmetric proton. Each of the exterior hydrogens in the O 2H5−fragment was anticipated to form a hydrogen bond with a water molecule to complete the first solvation shell of the Zundel-like anionic isomer. However, the optimized geometry shows that one of the water molecules leaves the second solvation shell to form a water dimer with one of the exterior water molecules. This observation indicates that anionic hydronium–water clusters prefer the Eigen O 4H9−geometry over the Zundel O 6H13−structure. Sobolewski and Domcke58observed the same trend in H 3O(H 2O)nclusters composed of a hydronium radical and water molecules. The Dyson orbital of electron attach- ment is diffused over the O 6H13+core.IV. CONCLUSIONS Micro-solvated H 3O+can exist in both Eigen-like H9O4+(H2O)nand Zundel-like H 5O2+(H2O)nstructures. The VEAEs of these cations decrease with increasing cluster size. The corresponding Dyson orbitals show that with increasing cluster size, the diffuse electron migrates from the closed-shell H 3O+core to form a cloud in the network of water molecules. Solvation of the OH 3−and O 2H5−DRAs through electro- static interactions or hydrogen bonding with their solvation shells results in remarkable changes in geometry with respect to cationic antecedents. Interaction of water molecules with the OH 3−DRA leads to several cluster isomers in which the Dyson orbital is held at the periphery of the hydrogens in the water network. It is seen that the stable H 3O−unit imposes a rigid structure on these complexes, especially for isomers with closed solvation shells, such as the Eigen H3O−(H2O)3structure. The O 2H5−fragment ion also imposes a rigid geometry on the Zundel O 2H5−(H2O)4isomer. For all these clusters, the Rydberg electrons are held by the exterior protons from the water network. Several isomers of oxygen-based OH 3−DRAs may occur through hydrogen bonding, as OH 3−(H2O)nstructures or as elec- trostatic H−(H2O)ncomplexes. 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Phys. 154, 234304 (2021); doi: 10.1063/5.0053297 154, 234304-18 Published under an exclusive license by AIP Publishing

5.0050596.pdf

J. Chem. Phys. 154, 224308 (2021); https://doi.org/10.1063/5.0050596 154, 224308 © 2021 Author(s).Redox states of dinitrogen coordinated to a molybdenum atom Cite as: J. Chem. Phys. 154, 224308 (2021); https://doi.org/10.1063/5.0050596 Submitted: 16 March 2021 . Accepted: 20 May 2021 . Published Online: 11 June 2021 Maria V. White , Justin K. Kirkland , and Konstantinos D. Vogiatzis ARTICLES YOU MAY BE INTERESTED IN Machine learning meets chemical physics The Journal of Chemical Physics 154, 160401 (2021); https://doi.org/10.1063/5.0051418 Accurate frozen core approximation for all-electron density-functional theory The Journal of Chemical Physics 154, 224107 (2021); https://doi.org/10.1063/5.0050296 All-order explicitly correlated relativistic computations for atoms and molecules The Journal of Chemical Physics 154, 224110 (2021); https://doi.org/10.1063/5.0051237The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Redox states of dinitrogen coordinated to a molybdenum atom Cite as: J. Chem. Phys. 154, 224308 (2021); doi: 10.1063/5.0050596 Submitted: 16 March 2021 •Accepted: 20 May 2021 • Published Online: 11 June 2021 Maria V. White, Justin K. Kirkland, and Konstantinos D. Vogiatzisa) AFFILIATIONS Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, USA a)Author to whom correspondence should be addressed: kvogiatz@utk.edu ABSTRACT Chemical structures bearing a molybdenum atom have been suggested for the catalytic reduction of N 2at ambient conditions. Previous com- putational studies on gas-phase MoN and MoN 2species have focused only on neutral structures. Here, an ab initio electronic structure study on the redox states of small clusters composed of nitrogen and molybdenum is presented. The complete-active space self-consistent field method and its extension via second-order perturbative complement have been applied on [MoN]nand[MoN 2]nspecies ( n=0, 1±, 2±). Three different coordination modes (end-on, side-on, and linear NMoN) have been considered for the triatomic [MoN 2]n. Our results demonstrate that the reduced states of such systems lead to a greater degree of N 2activation, which can be the starting point of different reaction channels. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0050596 I. INTRODUCTION The catalytic activation of the N–N bond is an important indus- trial process for the formation of ammonia, a raw material for the synthesis of fertilizers and other nitrogen-containing molecules and materials, but the dinitrogen functionalization under ambient con- ditions still remains a challenge due to the thermodynamic stability of the N 2molecule. The Haber–Bosch process persists to this day as the main industrial procedure for the production of ammonia, a pro- cess that requires high pressure and high temperature conditions. On the other hand, biological enzymes can fix nitrogen from the air under ambient conditions. Nitrogenase contains an eight-metal active site, the FeMo-cofactor, where the N 2bond is reduced and converted into ammonia and other nitrogen compounds. The com- position of the active site is known to be Fe 7MoS 9C.1While the exact binding site of the substrate is still a topic of active research, the Mo unit potentially plays a role in the activation of dinitrogen. This bio- logical process has inspired interest in the development of synthetic catalysts that could reduce N 2at ambient conditions. Since the syn- thesis of the first transition metal complex bound to a dinitrogen ligand,2[Ru(NH 3)5N2]2+, much effort has been expended in the development of coordinated N 2complexes to assist in the synthesis of ammonia.3–13 Molybdenum is one of the most azophilic metals of the d- block, and the synthesis and characterization of catalytic Mo-basedcomplexes that can fix N 2has been the topic of numerous stud- ies.5,14–17Moderate ammonia yields have been attained under ambi- ent conditions utilizing complexes bearing molybdenum metal centers, and intriguing reactivities have been shown for Mo com- plexes under different ligand environments.18–24Eizawa et al.22 reported the formation of up to 230 equivalents of ammonia uti- lizing dimolybdenum complexes bearing phosphine PCP-pincer lig- ands and N-heterocyclic carbenes. In addition, they showed with a detailed density functional theory (DFT) analysis that PCP-pincer ligands serve as σ-donors and π-acceptors, allowing for the strong link between molybdenum and nitrogen atoms. Sita et al. devel- oped a chemical cycle using metal mediated complexes composed of molybdenum and tungsten for the production of isocyanates through the activation of dinitrogen.18Isocyanates are exemplary synthetic targets because they help us to offset the free energy associated with the breaking of the dinitrogen bond. Using a tri- dentate phosphine molybdenum system, Liao et al. studied the transition-metal-catalyzed formation of silylamine.19Furthermore, they showed for the first time that the N–N bond of the hydrazido complex could be split via reduction to form the nitrido complex, with the formation of bissilylamide. The Chatt cycle suffers from the presence of anionic coligands, which cause disproportionation in the first molybdenum stage.25New multidentate phosphine lig- ands have been developed by Hinrichsen et al. in order to remedy this problem.20Other methods are being examined such as ammonia J. Chem. Phys. 154, 224308 (2021); doi: 10.1063/5.0050596 154, 224308-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp synthesis via electrocatalytic nitrogen reduction reaction (NRR). This method appears to be a prominent alternative to the cur- rent industrial technology. Finally, the first electrocatalytic NRR under ambient conditions using nitrogen-doped porous carbon with anchored single Mo atoms was recently reported.23 In addition to experimental studies, substantial work has been performed on molybdenum-based complexes for N 2activation on a theoretical level. Smaller systems, such as MoN x, have been pre- viously analyzed in the gas phase at a theoretical level in order to understand their electronic properties. Computational studies of the neutral states of MoN and MoN 2have been performed via dif- ferent levels of theory in order to elucidate the electronic effects that promote the dissociation of the triple bond of N 2between the constituents.26–36An early multireference configuration-interaction (MRCI) study discussed the low-lying states of MoN and highlighted the triple bond between Mo and N.34Pyykkö and Tamm performed calculations for the end-on and side-on isomers of MoN 2in order to elucidate multiple minima on the potential energy surfaces (PESs) of the species under consideration.31The authors applied density func- tional theory (DFT) together with a variety of post-Hartree–Fock methods, and they provided a detailed molecular orbital analysis related to the N 2binding and activation. Three independent DFT studies focused on the reaction of a neutral Mo with N and N 2 and reported binding energies and harmonic frequencies of different electronic and spin states.32,33,35 The aforementioned studies concluded that binding and acti- vation of N 2on a neutral Mo atom is unfavorable. The energetically most stable isomer of the MoN 2system has a linear geometry with N2weakly bound on Mo through noncovalent interactions, while a side-on isomer with a dissociated N 2molecule is about 20 kcal/mol less stable. Here, we further expand these studies by examining the electronic structure of ionic states of [MoN 2]n, where n=0,±1,±2. Our aim is to examine the N 2activation channels from charged Mo sites by means of multiconfigurational quantum chemical calcula- tions, which elucidate the underlying electronic structure effects of metal–N 2reactivity. Thus, the analysis and conclusions presented in this work can be used as the basis for future computational and experimental studies that bridge the electronic structure of the bare metal–N 2clusters with the coordination chemistry of a molec- ular complex. For example, gas-phase mass spectrometry37,38and computational studies39,40on the activation of small molecules by mono-ligated metal centers provide vital insights into the reactiv- ity of the more complex counterparts. In addition, the redox states of molybdenum nitride [MoN]n(n=0,±1,±2) were also included. All species considered in this study are shown in Fig. 1. Section II presents the computational details of the multiconfigurational meth- ods applied on this study. Our results are presented in Sec. III, and a short discussion is provided in Sec. IV. FIG. 1. Coordination modes between nitrogen and molybdenum considered in this study.II. METHODS The state-specific and state-average complete active space self-consistent field (CASSCF)41,42and its extension through second-order perturbation theory (CASPT2) were employed in this study.43,44All calculations were performed with the open- source OpenMolcas program package45using the ANO-RCC- VTZP triple- ζrelativistic basis set for all atoms (Mo: 7 s6p4d2f1g and N: 4 s3p2d1f).46,47In all calculations, scalar relativistic effects were included using a second-order Douglas–Kroll–Hess Hamilto- nian.48,49In the CASPT2 step, an Ionization Potential - Electron Affinity (IPEA) shift of 0.25 and an imaginary shift of 0.20 a.u. were applied.50,51All calculations were performed under the C2v point group, including the linear NMoN ( C2vwas preferred instead of the higher Abelian point group for consistency with respect to the other molecular species). For the linear molecules [MoN]n, [NMoN]n, and end-on [MoN 2]n(n=0,±1,±2), the LINEAR key- word of OpenMOLCAS was used. The CAS( n,m) nomenclature is followed throughout this arti- cle for the definition of the selected active space, where nis the number of electrons and mis the number of active orbitals. For the molybdenum nitride species, the valence 2 porbitals of the nitro- gen and the valence 5 s4dorbitals of the molybdenum were included within the active space, which give rise to a CAS( n,9). The number of electrons nvaries based on the total charge of the diatomic molecule. In the case of the MoN 2end-on, side-on, and linear NMoN species, the three 2 porbitals of the additional nitrogen were included for the formation of a CAS( n,12). The potential energy curves (PECs) of the diatomic Mo–N were constructed by placing the Mo atom at the origin of the Cartesian coordinate system while stretching N along the z-axis for a given set of distances. For the neutral surface, energies were computed from RMo–N =1.45 until 10.00 Å with displacement steps of 0.25 Å from 1.45 until 2.175 Å, by 0.1 Å from 2.20 to 5.50 Å, and then at 6.00, 8.00, and 10.00 Å. For the redox cases, energies were com- puted for RMo–N =1.45–4.00 Å by steps of 0.05 Å (1.45 until 2.00 Å) and steps of 1.0 Å (2.00 until 4.00 Å). For the end-on MoN 2iso- mer, the potential energy surface (PES) was generated by placing the central N atom at the origin and stretching Mo in the −z direc- tion and the remaining nitrogen in the +z direction. For the neu- tral species, 0.5 Å displacement points for RMo–N =1.00–10.00 Å and 0.05 Å for RN–N=0.80–1.50 Å were considered and additional points were added closer to the equilibrium distance ( RMo–N =1.90 –2.50 Å by 0.05 Å and RN–N=1.10–1.80 Å by 0.01 Å). All redox states of the end-on MoN 2were computed between the intervals RMo–N =1.00–10.00 Å (0.5 Å displacement) and RN–N=0.80–1.50 Å (0.05 Å displacement). The PES for the side-on MoN 2isomer was constructed by stretching the dinitrogen molecule along the y-axis and the Mo atom along the z-axis, with RMo–N 2=1.00−10.00 Å (0.5 Å displacement) and RN–N=0.80–3.00 Å (0.1 Å displacement), for both neutral and redox species. Finally, potential energy curves of the linear N–Mo–N complex were computed by the symmetric stretch of the Mo–N bond distances between 0.80 and 5.00 Å (0.05 Å displacement) for both neutral and redox species. III. RESULTS A. MoN The potential energy curve for the dissociation of neutral MoN species was generated using a CAS(9,9) active space. The CASSCF J. Chem. Phys. 154, 224308 (2021); doi: 10.1063/5.0050596 154, 224308-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. Natural orbitals and their occupation number (in parentheses) included in the (9,9) active space of the4Σ−ground state of [MoN]0. natural orbitals of the ground state of MoN are shown in Fig. 2. These orbitals were obtained at the equilibrium distance (1.645 Å) of the4Σ−ground state. Different electronic states across a range of spin states were evaluated with CASPT2(9,9). The potential energy curves of the ground state (4Σ−) and the three most stable excited states of [MoN]0are given in Fig. 3. Excitation energies along with their respective equilibrium internuclear dis- tances, equilibrium bond distances RMo–N , dissociation energies, and rotational–vibrational constants for the four low-lying states are summarized in Table I. The4Σ−ground state equilibrium bond dis- tance is at 1.645 Å, in agreement with the experimental value of 1.648 Å36and the MRCI value of 1.636 Å of Shim and Gingerich.34 FIG. 3. Potential energy curve of the4Σ−ground state and the first excited states of[MoN]0at the CASPT2 level.TABLE I. Equilibrium bond distances RMo–N (in Å), dissociation energies DeandD0 (in eV), vibrational constants ωeandωeχe(in cm−1), rotational constants Be(in cm−1), and excitation energies ΔE(in eV) for the four low-lying states of [MoN]0 obtained at CASPT2(9,9). State RMo–N De D0ωeωeχeBe ΔE 4Σ−1.645 5.10 5.03 1031.1 5.3 0.508 0 2Δ 1.629 5.71 5.65 1084.2 4.0 0.519 1.19 4Π 1.651 9.25 9.19 994.6 5.1 0.505 2.19 2Π 1.637 4.13 4.06 859.9 −19.2 0.503 2.90 General valance bond (GVB) calculations performed by Allison and Goddard27indicated a bond distance of 1.60 Å. We have computed a dissociation energy of 5.10 eV at the CASPT2(9,9) level, which is also in agreement with the experimentally refined value (5.12 eV)52and the MRCI value by Stevens et al. (5.17 eV).34A previous DFT36study has analyzed the low-lying states of the MoN diatomic molecule. The authors found that the ground state is4Σ−and the first excited doublet state2Δis 0.79 eV higher (BP86 functional, QZ4P basis set). Their results are in qualitative agreement with the new results presented in this study. CASPT2(9,9) identified4Σ−and2Δas the most stable states, but with a relative energy difference of 1.19 eV. Figure 3 also includes the atomic term symbols at the dissociation limit. The ground state dissociates into the7Sand4Sterms of the Mo and N atoms, respectively. The Mo atom has the neutral elec- tronic configuration 5 s14d5, and the N atom has a 2 p3electronic configuration. The doubly degenerate2Δexcited state has an exci- tation energy of 1.19 eV and dissociates into the5Sand4Sterms for Mo and N atoms, respectively. A 5 s24d4electronic configuration is observed for Mo where the electron of the 4 dx2−y2atomic orbital is promoted to the 5 sorbital. The 2 p3electronic configuration remains consistent for the N atom. The same dissociation terms are observed for the doubly degenerate2Πexcited state with an excitation energy of 2.90 eV. In this case, the electron is promoted to the 5 sorbital of Mo from the 4 dyzatomic orbital. Finally, the4Πstate with an exci- tation energy of 2.19 eV dissociates into the6Sand3Pterms of the Mo and N atoms, respectively. The electronic configuration of Mo at the dissociation limit pertains to 5 s04d5, while the nitrogen atom is reduced and its configuration becomes 2 p4. The redox states of molybdenum nitrate [MoN]n(n=1±, 2±) were analyzed by means of multiconfigurational methods. The ground electronic state, the relative energy difference ΔEfrom the most stable species ( [MoN]1−,vide infra ), the equilibrium distance RMo–N , and spectroscopic constants for each of the five species are given in Table II. The number of electrons in the active space depends on the total charge, while the number of orbitals was kept constant. The dominant electronic configuration for the ground states of each of the five species with different total charge together with their respective active spaces is shown in Fig. 4. The triple bond between Mo and N for the neutral [MoN]0is clearly shown on the molecular orbital diagram. Three bonding orbitals formed between the three porbitals of the N atom and the dz2,dxz, and dyzorbitals of the molybdenum are doubly occupied, while the three non- bonding orbitals 5 s, 4dx2−y2, and 4 dxyremain singly occupied. For the3Σ−ground state of [MoN]1−, the singly occupied non-bonding J. Chem. Phys. 154, 224308 (2021); doi: 10.1063/5.0050596 154, 224308-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. Equilibrium bond distances RMo–N (in Å), dissociation energies DeandD0 (in eV), vibrational constants ωeandωeχe(in cm−1), rotational constants Be(in cm−1), and excitation energies ΔE(in eV) for [MoN]n(n=0, 1±, 2±)obtained at CASPT2(9,9). State RMo–N DeD0ωeωeχeBeΔE [MoN]1− 3Σ−1.658 5.66 5.60 1034.4 4.9 0.500 0 [MoN]0 4Σ−1.645 5.10 5.03 1031.1 5.3 0.508 0.88 [MoN]2− 4Π 1.660 2.58 2.51 1110.6 10.2 0.500 4.81 [MoN]1+ 3Σ−1.613 4.77 4.70 1079.8 5.3 0.529 8.30 [MoN]2+ 2Δ 1.606 4.54 4.47 1076.6 5.8 0.534 24.73 5sorbital becomes doubly occupied, thereby stabilizing the complex by 0.88 eV with respect to the neutral [MoN]0diatomic molecule. Similarly, the dissociation energy increases for the anionic species from 5.10 eV (neutral) to 5.64 eV. Further reduction of [MoN]2− yields a doubly degenerate4Πground state, wherein the additional electrons occupy both the non-bonding dxyanddx2−y2orbitals of Mo and one of the π-antibonding orbitals. For the oxidized [MoN]1+ and[MoN]2+species, electrons are removed from the non-bonding orbitals, which give rise to3Σ−and2Δground states, respectively. TheΔEenergies relative to the equilibrium energy of the most stable diatomic molecule ( [MoN]1−) show that the ionization of [MoN]0 is an energetically favorable process. B.MoN 2end-on Next, we examined the interaction of dinitrogen with molyb- denum. The addition of another nitrogen atom expands the active space from CAS(9,9) to CAS(12,12) to accommodate the additional 2porbitals of the second nitrogen. This CAS is used for the end- on, side-on and linear NMoN isomers of the [MoN 2]ntriatomic molecule ( n=0, 1±, 2±). For the end-on isomer, a two-dimensional potential energy surface was constructed by considering the N–N and Mo–N bond stretch. Table III summarizes the CASPT2(12,12) results for the neutral [MoN 2]0, i.e., excitation and dissociation energies, as well as the equilibrium bond distances RMo–N andRN–N. The discussion is focused on the most stable state per different spinTABLE III. Excitation and dissociation energies (in eV) and equilibrium bond dis- tances RMo–N andRN–N(in Å) for the spin states of the end-on [MoN]0obtained from CASPT2(12,12). State Excitation energy Dissociation energy RMo–N RN–N 7Σ+0 0.01 4.50 1.10 5Π 0.77 0.75 1.90 1.15 3Π 1.94 1.92 1.90 1.16 1Π 2.37 2.35 1.90 1.14 (S=0, 1, 2, 3). These computations revealed that the ground state is7Σ+, which corresponds to zero binding (0.01 eV) between N 2 and molybdenum, in agreement with previous studies.31The doubly degenerate quintet state5Πis 0.77 eV higher than the7Σ+ground state and shows a weak N 2binding (dissociation energy of 0.75 eV or 17.3 kcal/mol). CASPT2 results reported by Pyykkö and Tamm are also in close agreement with our calculations (excitation and dis- sociation energies of 21.2 kcal/mol and 0.92 eV, respectively). N 2 is found to be weakly bound to Mo on the quintet state via bond- ing between the 4 dxzorbital of Mo and the 2 pxantibonding orbital of the center nitrogen displaying a bond length of 1.15 Å between the nitrogen atoms. In addition, a second molecular orbital displays the electron density between the 4 dxzorbital of the Mo and both of the nitrogen atoms. This molecular orbital is mostly polarized toward the nitrogen atoms. The same kind of electron polarization is observed between the 4 dyzorbital of the Mo and the nitrogen atoms. The triplet doubly degenerate3Πstate was found to be 1.94 eV higher than the septet ground state, which has a strong N 2binding character (dissociation energy of 1.92 eV). A similar bonding behavior is observed, although π-backbonding was found between the 4 dyzorbital of the Mo and the central nitrogen atoms. The results collected for the ionic states of [MoN 2]n (n=1±, 2±)are summarized in Table IV. For the 1 +charged species, three states were analyzed utilizing state-average CASSCF and multi-state CASPT2. Excitation energies ( ΔE) and dissociation energies ( Ediss) are shown in Table IV. For the end-on species, the lowest energy state at the dissociation for each given charge was also FIG. 4. Molecular orbital diagrams together with the dominant electronic configuration in the neutral and charged [MoN]n(n=0, 1±, 2±)diatomic molecules. J. Chem. Phys. 154, 224308 (2021); doi: 10.1063/5.0050596 154, 224308-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV. Electronic configurations of the ground state of end-on [MoN 2]n,n=0, 1±, 2±, percent weight of the dominant configurations, equilibrium bond distances RMo–N andRN–N(in Å), and energies (in eV) obtained at CASPT2(12,12) relative to the minimum of the lowest energy state (1 −).ΔEcorrespond to relaxed energies. State Active space Configuration % weight ΔE E diss RMo–N RN–N [MoN 2]1− 6Σ+(13,12) (σ)2(π)4(2s)2(dxz)1(dyz)1(dxy)1(dz2)1(dx2−y2)1(σ∗)0(π∗)093.1 0 0 - 1.10 [MoN 2]0 7Σ+(12,12) (σ)2(π)4(5s)1(dxz)1(dyz)1(dxy)1(dz2)1(dx2−y2)1(σ∗)0(π∗)093.5 0.31 0.01 4.50 1.10 [MoN 2]2− 5Π (14,12) (σ)2(π)4(2s)2(dxz)1(dyz)1(dxy)1(dz2)1(dx2−y2)1(σ∗)1(π∗)053.2 3.87 0.58 10.0 1.20 [MoN 2]1+ 6Σ+(11,12) (σ)2(π)4(dxz)1(dyz)1(dxy)1(dz2)1(dx2−y2)1(5s)0(σ∗)0(π∗)093.3 6.88 0.59 2.50 1.10 [MoN 2]2+ 5Σ+(10,12) (σ)2(π)4(dxz)1(dyz)1(dxy)1(dx2−y2)1(5s)0(dz2)0(σ∗)0(π∗)093.1 21.68 1.95 2.00 1.10 the ground state species. No bonding is observed in the charged species. Although the sextet (1 −) is lowered in energy than the neutral septet, no binding was observed between Mo and N 2. The electronic configuration of the anionic states shows doubly occupancy for the 2 sorbital of the nitrogen. We believe that this orbital was introduced into our active space due to radial correla- tion. The 2 −complex displays a N–N bond length of 1.2 Å. This is due to the occupancy of one of the antibonding orbitals of the nitro- gen as displayed in the electronic configuration. On the contrary, the oxidized [MoN 2]1+and[MoN 2]2+have a binding character, but they are 6.88 and 21.68 eV less stable than the [MoN 2]1−species, respectively. C.MoN 2side-on The side-on [MoN 2]nisomer ( n=0, 1±, 2±) was examined by means of multiconfigurational methods. A CAS(12,12) was used for all CASPT2 calculations (see Fig. 5). Table V includes the most sta- ble electronic states per spin multiplicity as well as the excitation and dissociation energies Ediss. In addition, the distance at the min- imum geometry between Mo–N and N–N is also shown. For the neutral species, the7A1state was determined as the ground state geo- metry where no overlap is found between Mo and N 2. The CASPT2 potential energy surface of the first excited state,5B2, is shown in Fig. 6. The5B2state is only 0.62 eV higher than the septet ground FIG. 5. Natural orbitals and their occupation number (in parentheses) included in the (12,12) active space of the5B2state of the side-on [MoN]0.state (Table V). The internuclear distance of the Mo and nitrogen atoms is 2.09 Å, and the bond distance of the dinitrogen is 1.2 Å. This indicates activation of the N 2bond due to the elongated bond, in agreement with the CASPT2 study of Pyykkö and Tamm. On the contrary, a previous DFT study32reported a reverse order for these two states, but the authors commented that this could be due to overbinding of weakly bound systems predicted by standard DFT functionals. Although our geometries for the singlet state agree with the DFT work, again the relative order is reversed. CASPT2(12,12) estimates that the1A1neutral state is more stable than the triplet3B2 by 0.12 eV. We now turn our attention to the ionic states of the side-on [MoN 2]n(n=1±, 2±). Table VI includes the relative energy dif- ferences ΔEwith respect to the most stable species of the side-on [MoN 2]1−molecule, which was found again to be more stable than their neutral equivalent. The dominant electronic configuration for the ground states of each of the five species with different total charge together with their respective active spaces is also displayed on Table VI. Dinitrogen is reduced in the case of 1 −as seen from its2A1 ground state, while molybdenum is in oxidation state V. We report this electronic configuration as “Mo(V) +(N−3)2.” The dissociated dinitrogen bond exhibits a RN–N distance of 2.80 Å (Fig. 7). Reduc- tion of the neutral species lowers the energy by 0.83 eV. Molybde- num is found in the Mo(IV) oxidation state upon further reduction ([MoN 2]2−), but this complex is 3.23 eV less stable than the most sta- ble[MoN 2]1−case. Dissociation energies display the same pattern as for the end-on species, increasing in energy from 0.02 (neutral) to 0.55 eV (1 −) and to 1.82 eV for the 2 −electron reduced species. For the two oxidized species [MoN 2]1+and[MoN 2]2+, electrons are removed from the 5 sanddz2non-bonding orbitals, yielding6A1 and5A1ground states, respectively. The ΔEenergies relative to the most stable side-on species [MoN 2]1−also show that the ionization of[MoN 2]0is an energetically intense process. TABLE V. Excitation and dissociation energies (in eV) and equilibrium bond distances RMo–N andRN–N(in Å) for the spin states of the side-on [MoN]0obtained from CASPT2(12,12). State Excitation energy Ediss RMo–N RN–N 7A1 0 0.02 4.53 1.10 5B2 0.62 0.60 2.09 1.20 1A1 0.85 0.83 1.68 2.70 3B2 0.97 0.95 1.72 2.80 J. Chem. Phys. 154, 224308 (2021); doi: 10.1063/5.0050596 154, 224308-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6. Relative CASPT2(12,12) energies (in eV) of the5B2state of the side- on[MoN]0species. The X mark indicates the minimum of the potential energy surface. D. Linear NMoN For the sake of completeness, the linear isomer [NMoN]n, n=0, 1±, 2±, was examined. In this case, the molybdenum atom is inserted into the dinitrogen molecule and forms a high-energy isomer. All energies, potential energy curves, spectroscopic data, electronic configurations, and molecular orbitals of the CAS(12,12) are presented in detail in the supplementary material. The cen- tral molybdenum atom forms two sets of bonding, antibonding, and non-bonding molecular orbitals, σ,σ∗,nσandπ,π∗,nπ, respec- tively. For the neutral [NMoN]0species, the3Σ− gground state has a(σ)2(π)4(dx2−y2)1(dxy)1(nπ)4(5s)0(nσ)0(π∗)0(σ∗)0domi- nant configuration. The lowest quintet (5Πu), singlet (1Δg), and septet (7Σ− u) states are 0.52, 0.71, and 0.93 eV less stable than the ground state. The minimum for all spin states varies in the range of RMo–N =1.70–1.85 Å, with RMo–N =1.70 Å for the triplet3Σ− uground state. We turn now our attention to the redox states. Similar to the end-on and side-on species, the anionic [NMoN]1−was found to be more stable than the neutral [NMoN]0counterpart by 2.14 eV. The quartet4Σ− gspin state is the ground state for the [NMoN]1− FIG. 7. CASPT2(12,12) potential energy surface (in eV) of the2A1state of the side-on[MoN]1−. The X mark indicates the minimum of the potential energy surface. case, where the additional electron occupies the 5 satomic orbital of molybdenum. The doubly anionic state5Σ− ulies 4.15 eV higher than the4Σ− ustate, while the cationic2Σ− uand3Δustates of [NMoN]1+ and[NMoN]2+species, respectively, are more than 10 eV higher than4Σ− g(11.32 and 27.35 eV, respectively). IV. DISCUSSION The interaction between neutral and ionic states of molyb- denum and dinitrogen was investigated. Three different binding modes were considered, the end-on and side-on modes of MoN 2and a fully dissociated dinitrogen that forms the linear NMoN, and all results are summarized in Table VII. For all charges included in this study, the linear NMoN complex is less stable than the other two isomers, with relative energies that range from 2.23 until 6.38 eV. For that reason, it will be excluded for the rest of the discussion. The 2A1state of the anionic side-on [MoN 2]1−isomer was found to be the lowest state among all isomers, charges, and spin states consid- ered in this study. The other two anionic isomers with a 1 −charge (end-on [MoN 2]1−and linear [NMoN]1−) are less stable by 0.53 and 2.23 eV, respectively. Since the minimum of the2A1state involves an TABLE VI. Electronic configurations of the ground state of side-on [MoN 2]n,n=0, 1±, 2±, percent weight of the dominant configurations, equilibrium bond distances RMo–N andRN–N(in Å), and energies obtained at CASPT2(12,12) relative to the minimum of the lowest energy state (1 −).ΔEcorrespond to relaxed energies. State Active space Configuration % weight ΔE E diss RMo–N RN–N [MoN 2]1− 2A1 (13,12) Mo(V) +(N−3)2 79.3 0 0.55 1.72 2.80 [MoN 2]0 7A1 (12,12) (σ)2(π)4(5s)1(dxz)1(dyz)1(dxy)1(dz2)1(dx2−y2)1(σ∗)0(π∗)093.5 0.83 0.02 4.53 1.10 [MoN 2]2− 1A1 (14,12) Mo(IV) +(N−3)2 74.0 3.23 1.82 1.72 2.80 [MoN 2]1+ 6A1 (11,12) (σ)2(π)4(dxz)1(dyz)1(dxy)1(dz2)1(dx2−y2)1(5s)0(σ∗)0(π∗)093.5 7.83 0.17 2.56 1.10 [MoN 2]2+ 5A1 (10,12) (σ)2(π)4(dxz)1(dyz)1(dxy)1(dx2−y2)1(5s)0(dz2)0(σ∗)0(π∗)093.4 23.19 0.77 2.56 1.10 J. Chem. Phys. 154, 224308 (2021); doi: 10.1063/5.0050596 154, 224308-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VII. Relative energy differences between the end-on MoN 2, side-on MoN 2, and NMoN species, for the neutral and redox states, computed at CASPT2(12,12). The relative energy differences from the ground state of the side-on [MoN 2]1−are shown in parentheses. MoN 2side-on MoN 2end-on NMoN Charge State ΔE State ΔE State ΔE 07A10 (0.83)7Σ+0 (0.83)3Σ− g3.54 (4.37) 1−2A10 (0)6Σ+0.53 (0.53)4Σ− g2.23 (2.23) 2−1A10 (3.23)5Π 1.16 (4.40)5Σ− u3.14 (6.38) 1+6A10 (7.83)6Σ+−0.42 (7.41)2Σ− u5.71 (13.54) 2+5A10 (23.19)5Σ+−0.99 (22.20)4Δu6.38 (29.57) activated dinitrogen with RN–N=2.80 Å, we believe that the reduced species can lead to a dissociation channel relevant to N 2function- alization. On the contrary, no N 2activation was observed for the anionic sextet state6Σ+of the end-on isomer. Upon further reduc- tion ([MoN 2]2−), N 2activation was observed for both coordination modes. The5Πand1A1states were determined as the most stable states for the end-on and side-on isomers, respectively, with the1A1 state being more energetically favorable by 1.16 eV. The singlet state displayed the same geometry as the doublet state of [MoN 2]1−, while the5Πstate displayed a further elongation of the N 2bond due to the additional electron occupancy of the antibonding orbital of the dinitrogen. The two neutral septet states7A1and7Σ+of the side-on and end-on[MoN 2]0, respectively, were found to be 0.83 eV less stable than the global minimum. The CASPT2(12,12) results indicate that these two species are isoenergetic with a relative energy difference of less than 0.1 eV (0.0035 eV). Therefore, the two isomers have similar stabilities since the energy difference falls within the error of the CASPT2 method, a result that suggests a polytopic van der Waals interaction.53,54The relative energy difference between the 7Σ+state of the end-on and the most stable end-on MoN 2isomer with an activated dinitrogen molecule (5Πof end-on, Table III) is 0.77 eV, in agreement with the result reported by Pyykkö and Tamm (0.92 eV).31 We have also considered the energetically less favorable isomers for the 1 +and 2+oxidation states. The relative energy differences displayed in the results allowed us to conclude that such states could be attained only under the presence of a ligand field around a Mo(I) or Mo(II) center. We are currently examining the Mo ligand field effects on the N 2activation and fixation, which will be the topic of a separate study. For the cationic species, the end-on species were more stable than the side-on isomer by 0.42 and 0.99 eV for the 1+and 2+cases, respectively. This conclusion is in contrast to the anionic cases, where the side-on isomer was always more stable than the side-on equivalent. Their electronic configurations of the6A1 (side-on) and6Σ+(end-on) were identical, where Mo(I) has five singly occupied 4 datomic orbitals and an unoccupied 5 sorbital. The Mo–N bond distance differed by 0.06 Å between the end-on and side-on isomers, while the N–N remained at its natural bond length. Since the electronic configurations and the bond lengths were found to be nearly identical, the high energy difference can be associated with the ability of Mo to bind easily to one nitrogen rather than to two nitrogen atoms in the angled position. Finally, the quintetspin states were found to be the most stable states for both modes of the 2+charged species, with a relative energy difference of 0.99 eV. The end-on isomer displayed lower energy at a Mo–N bond distance of 2.00 Å. The Mo–N distance of the side-on isomer was 2.56 Å. Both states displayed a loss of electron density from the 5 sand 4 dz2 orbitals of the molybdenum with respect to the neutral species. V. CONCLUSIONS A multiconfigurational electronic structure study was per- formed on different coordination modes of Mo to N and N 2. These species were studied in their neutral and redox states, and we found that the reduced states lead to N 2activation. The results presented in this study can provide a computational first approach for the under- standing of N 2activation channels when ligand effects of the tran- sition metal are introduced, from mono-ligated to fully coordinated Mo in molecular complexes. Such studies are currently underway in our research group. SUPPLEMENTARY MATERIAL See the supplementary material for electronic energies and relative energy differences for all electronic states considered in this study. 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Phys. 154, 224308 (2021); doi: 10.1063/5.0050596 154, 224308-8 Published under an exclusive license by AIP Publishing

PT.3.4794.pdf

Asteroids in the inner solar system Sarah Greenstreet Citation: Physics Today 74, 7, 42 (2021); doi: 10.1063/PT.3.4794 View online: https://doi.org/10.1063/PT.3.4794 View Table of Contents: https://physicstoday.scitation.org/toc/pto/74/7 Published by the American Institute of Physics ARTICLES YOU MAY BE INTERESTED IN How two planets likely acquired their backward orbits Physics Today 74, 12 (2021); https://doi.org/10.1063/PT.3.4742 Macroscopic systems can be controllably entangled and limitlessly measured Physics Today 74, 16 (2021); https://doi.org/10.1063/PT.3.4789 Ernest Rutherford’s ambitions Physics Today 74, 26 (2021); https://doi.org/10.1063/PT.3.4747 Science and technology of the Casimir effect Physics Today 74, 42 (2021); https://doi.org/10.1063/PT.3.4656 Muon measurements embolden the search for new physics Physics Today 74, 14 (2021); https://doi.org/10.1063/PT.3.4765 Lead-208 nuclei have thick skins Physics Today 74, 12 (2021); https://doi.org/10.1063/PT.3.4787Sarah Greenstreet Observations and computer simulations of their orbits and interactions with planets yield insights into the asteroids’ dynamic lives. 42PHYSICS TODAY |JULY 2021 About 500 meters wide, Bennu orbits our Sun in the Apollo population of near-Earth asteroids. It was visited and photographed by spacecraft OSIRIS-REx in December 2018.NASA/GODDARD/UNIVERSITY OF ARIZONAJULY 2021 |PHYSICS TODAY 43Sarah Greenstreet is a senior researcher with the B612 Asteroid Institute in Mill Valley, California, and a research scientistwith the Dirac Institute at the University of Washington in Seattle. The gravitational influence of the planets over small bodies, particularly those in the solar system’s inner regions, has mod-ified many asteroid orbits in quite dramatic ways. For example,the interactions can push asteroids from nearly circular orbitsin the main asteroid belt between Mars and Jupiter to highlyelliptical orbits that cross those of all the terrestrial planets.Eventually, those perturbations can move the asteroids’ perihelia— their closest orbital distance from the Sun— to withinthe star’s radius. They are known as Sun- grazing orbits. Theasteroids’ transformation from main- belt orbiters to Sun graz-ers can take place in the surprisingly short time scale of a mil-lion years. That’s less than 0.02% the age of the solar system. In addition to decreasing the asteroids’ perihelia, gravita- tional interactions can also decrease their aphelia— their far-thest orbital distance from the Sun— and push them to increas-ingly smaller orbits. That movement progressively nudgesasteroids onto hard- to- reach orbits that are closer to the Sunthan either Earth’s orbit or Venus’s orbit; the asteroids areknown as Atiras and Vatiras, respectively. The orbital evolutionof those rare Atira and Vatira asteroids is a reminder that theirtrajectories can change dramatically over their lifetimes andtake them throughout the inner solar system. That evolutioncan tell us where the asteroids have likely been and where theywill likely go as their orbits continue to evolve. More practi-cally, it can tell us where to point our telescopes to find thoseelusive objects. From main- belt to Sun- grazing orbits More than 525000 numbered asteroids with well- known orbitsand in sizes ranging over several orders of magnitude— fromhundreds of meters to a thousand kilometers— currently in-habit the main asteroid belt. Several basic parameters describetheir orbits around the Sun. The semimajor axis refers to an as-teroid’s average orbital distance from the Sun. That distance isoften measured in astronomical units, with 1 AU defined as themean Earth– Sun distance. The eccentricity is the asteroid’s or-bital ellipticity. It equals 0 for a circular orbit and 1 for a para-bolic orbit, a trajectory whose energy is the minimum required for an aster-oid to become unbound from the Sunand escape the solar system. For all val-ues betw een those extremes, the orbit is more or less elongated. The inclina-tion refers to the orbit’s angular tiltrelative to the plane of the solar sys-tem in which the planets lie. Figure 1 shows two commonly used projections of the known main- belt asteroids between Mars and Jupiter. Compared with theroughly circular planetary orbits, main- belt asteroid orbits are more elliptical and are inclined by as much as 30°. The firstperson to arrange the ever- growing number of discovered asteroids by average distance from the Sun was mathemati-cian and astronomer Daniel Kirkwood. Upon arranging the asteroids that way in 1866, Kirkwood noticed sharp drops, now called Kirkwood gaps, in the number of asteroids located at specific semimajor axes. Few asteroids reside in the Kirk-wood gaps, and they span large ranges in eccentricity andinclination. Kirkwood identified the most obvious gaps in the asteroid population at 2.50, 2.82, 2.95, and 3.27 AU as locations of the3:1, 5:2, 7:3, and 2:1 orbital resonances, respectively, with Jupiter. 1 An orbital resonance occurs when an asteroid’s orbital periodis an integer multiple of a planet’s. For example, the 2:1 reso-nance with Jupiter occurs when an asteroid orbits the Sun ex-actly twice for every orbit of Jupiter. Resonances occur at spe-cific semimajor axes because, as Kepler’s third law tells us, thesquare of an object’s orbital period is proportional to the cubeof its semimajor axis. Thus, because those resonances occur forspecific orbital periods, they are located at the correspondingsemimajor axes. Why are so few asteroids located in the resonances associ- ated with the Kirkwood gaps? The main belt hosts numerousorbital resonances with Mars, Jupiter, and Saturn, and severalof them overlap in the gaps. The overlapping resonances causethe orbits of the asteroids in the region to be unstable, and theinstability leads to an excitation, or increase, in asteroid eccen-tricities. The semimajor axis of an asteroid in a resonance can-not itself change, so as the asteroid’s orbit evolves, its eccentric-ity follows a vertical path from low to high values in the plotof eccentricity versus semimajor axis (see figure 1b). Eventually,the eccentricity reaches values larger than the moderate eccen-tricities seen in the main belt. As an asteroid’s eccentricity increases, its orbit becomes increasingly less circular and more elliptical. The elongationeople tend to think of the solar system as a static environment, in which the orbits of the planets, asteroids, and comets have remained the same over its lifetime. But although its current architecture has existed for roughly the past 4.5 billion years, the solar systemis far from the unvarying environment that we imagine. P44PHYSICS TODAY |JULY 2021ASTEROIDS causes the asteroid’s perihelion to decrease and its aphelion to increase, eventually putting the asteroid on a terrestrial planet- crossing orbit. If the orbit becomes so highly elongated thatthe perihelion drops to within the solar radius, the asteroidreaches a Sun- grazing orbit and incinerates during its next peri-helion passage. Over time, the resonant- eccentricity- excitationprocess has nearly emptied the Kirkwood gaps of asteroids asthey are transported from the main belt to the inner solar system. How long does it take for an asteroid orbit to be altered from main-belt to Sun-grazing? A landmark study performed byPaolo Farinella and colleagues in 1994, when Farinella was a visiting professor at the Nice Observatory in France, foundthat low- eccentricity asteroids located in a resonance with aplanet can evolve onto Sun- grazing orbits in as little as 1 mil-lion years. 2Over the past 4.5 billion years of solar system his- tory, asteroids located along the borders of those resonanceshave slowly diffused into the resonances and supplied at asteady rate the sunward transportation of asteroids from the main belt. That slow diffusion most frequently occurs through gravi- tational close encounters with the planets, which can changean asteroid’s semimajor axis. Just as a spacecraft can gain orlose speed by passing closely behind or in front of a planet, socan an asteroid. Because orbital speed is inversely proportionalto orbital period and the period squared is proportional to thesemimajor axis cubed, an increase in orbital speed causes a de-crease in semimajor axis and vice versa. Planetary interactionscan thus change the semimajor axis of an asteroid located justoutside the border of a resonance enough to move it into theresonance. Near- Earth asteroids Not all asteroids in resonances reach Sun- grazing orbits. Oncean asteroid’s eccentricity increases enough, a gravitational closeencounter with a planet can move the asteroid out of the reso-nance. The interaction leaves the asteroid on a planet- crossingorbit that is no longer on a resonant path to a Sun- grazing orbit.Because orbital periods close to the Sun are quite short, the as-teroid experiences frequent planetary close encounters once itreaches terrestrial planet- crossing orbits. Each encounter causesa small change in the asteroid’s semimajor axis, and frequentencounters scatter asteroids throughout the inner solar system.The process feeds the population of what are called near- Earthasteroids (NEAs). That population is defined to have periheliasmaller than 1.3 AU. Traditionally, NEAs are divided into four dynamical subpopulations— Amors, Apollos, Atens, and Atiras— whose orbits are categorized relative to Earth’s orbit. Amors follow or-bits that are always farther from the Sun than is Earth’s. Atiras,by contrast, follow orbits that are always closer to the Sun thanis Earth’s. Apollos and Atens are both on Earth- crossing orbits.Based on a rare, dynamical subset of orbits found in our sim-ulations of the NEA subpopulations, in 2012 I and my colleaguesBrett Gladman and Henry Ngo, then all at the University of British Columbia, proposed the addition of a fifth sub- population we called the Vatiras. 3That asteroid class is similar in nature to Atiras, but they follow orbits that keep them closerto the Sun than Venus’s orbit— hence their name as a play onVenus and Atira. Figure 2a shows a schematic of sample orbitsfor each of those five sub populations.Our NEA dynamical model and an updated model produced by the University of Helsinki’s Mikael Granvik and colleaguesin 2016 independently predict that the Amor, Apollo, Aten,Atira, and Vatira subpopulations contain roughly 39%, 55%,4%, 1%, and less than 1% of NEAs, respectively, at any giventime. 3,4Together, the Amors and Apollos make up the vast ma- jority (94%) of NEAs . That’s partly the result of the much larger volume of near- Earth space they cover. The curves shown infigure 2b mark the boundaries of each subpopulation and follow the perihelia and aphelia of Earth, Venus, and Mercury.For example, while all NEAs must have perihelia that are lessthan 1.3 AU, Amors must have perihelia that are larger thanEarth’s aphelion (1.017 AU) to remain farther from the Sun thanEarth’s orbit is. Apollos have perihelia that are smaller than Earth’s aphelion and semimajor axes that are greater than the semimajor axis ofEarth (1 AU). Atens have semimajor axes that are less than 1 AUand aphelia that are larger than Earth’s perihelion (0.983 AU). INCLINATION (degrees) SEMIMAJOR AXIS (AU) SEMIMAJOR AXIS (AU)ECCENTRICITY40 0.5 0.430 0.320 0.210 0.10 02.0 2.02.5 2.53.0 3.03.5 3.5a b FIGURE 1 . ORBITAL DISTRIBUTIONS of the roughly 525 000 asteroids in the main asteroid belt. The data come from the MinorPlanet Center Orbit Database, a public list of computed orbits for allknown small bodies in the solar system. The semimajor axis measuresthe average orbital distance from the Sun relative to the Earth– Sundistance (1 AU). (a)Inclination measures the angular tilt of the orbit out of the planetary plane. (b)Eccentricity measures the orbital ellipticity. Main- belt asteroids sit between the orbits of Mars andJupiter on moderately elliptical and inclined orbits. So- called Kirkwood gaps can be seen at specific semimajor axes, where orbital resonances exist and the number of asteroids drops sharply.(Image by Sarah Greenstreet.)JULY 2021 |PHYSICS TODAY 45Their orbital parameters make Apollos and Atens Earth- crossing. Atiras must have aphelia that are smaller than Earth’s perihe-lion to stay closer to the Sun than Earth’s orbit is at all times.Lastly, Vatiras have aphelia that are inside Venus’s perihelion(0.718 AU) and outside Mercury’s perihelion (0.307 AU). Anyasteroids with aphelia inside Mercury’s perihelion would re-main closer to the Sun than Mercury’s orbit is. Another reasonAmors and Apollos are the most numerous NEAs is becausethey overlap the resonances where asteroids enter the NEApopulation. That overlap greatly enhances their number overthe Atens, Atiras, and Vatiras, which lie much closer to the Sunthan those resonances. Generally, asteroids can reach the Aten, Atira, and Vatira sub- populations only through a series of planetary close encountersthat cause their semimajor axes to jump to increasingly smallervalues. The close encounters are more frequent in the innersolar system because of higher orbital speeds. But for the en-counters to push asteroids into the three innermost NEA pop-ulations, they must occur in such a way that they cumulativelydecrease, not increase, an asteroid’s semimajor axis. As you canimagine, that process becomes increasingly difficult as asteroids reach smaller orbits and encounter fewerplanets. Thus a large drop-off in asteroid populationoccurs between the Apollos and the Atens. Likewise, it becomes more difficult to gravitation- ally scatter asteroids onto orbits decoupled from the planets— that is, orbits that are no longer planet- crossing— at the increasingly smaller orbits entirely in-terior to the orbits of Earth and Venus. For those rea-sons, Atiras are rare and Vatiras even rarer among theNEAs. And although it is theoretically possible, aster-oids almost never reach orbits completely interior toMercury’s orbit. Dynamical behavior Despite the rarity of Atiras and Vatiras, they providea unique glimpse into the dynamic environment ofthe innermost regions of our solar system. Any givenasteroid that becomes a Vatira will have passedthrough the Amor, Apollo, Aten, and Atira popula-tions to reach its eventual small orbit. Each Atira andVatira will have taken a unique path from the mainbelt, often over tens of millions of years, and will havespent varying amounts of time in each populationalong the way. Asteroids do not remain in the Atira and Vatira populations for long. Detailed dynamical simulations of Atiraasteroids performed by Anderson Ribeiro of the Geraldo DiBiase University Center in Brazil and his colleagues indicate thatthe very planetary close encounters required to enter those hard- to- reach orbits are responsible for continually scattering the as-teroids into and out of the Vatira and Atira populations manytimes during their lifetimes. 5Those events keep the asteroids from lingering in either of the planet- decoupled populations. As Gladman, Ngo, and I discovered in our simulations,3 the asteroids typically spend only a couple million years— integrated over their lifetimes— as Atiras and a few hundredthousand years as Vatiras. It is possible for asteroids to enterthe Vatira region and to remain there for more than a millionyears before leaving. However, such long- lived Vatiras are ex-tremely rare. If most of those asteroids don’t remain in the Atiraand Vatira populations, where do they go? The frequent plan-etary close encounters generally push the asteroids outward,over tens of millions of years, back onto Venus- and Earth- crossing orbits. (The asteroids become Atens and Apollos.)Because the vast majority of Atiras and Vatiras do not remain Apollos Amors Earth AtensMarsSunVatiras AtirasVenusa ECCENTRICITY SEMIMAJOR AXIS (AU)1.00 0.75 0.50 0.25 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Perihelion = 1.017 AU Amors (39%)Apollos (55%)Aphelion = 0.307 AUAphelion = 0.718 AUAphelion = 0.983 AU Semimajor = 1 AU Vatiras (< 1%) Atiras (1%)Atens (4%)b Perihelion = 1.3AUFIGURE 2 . SCHEMATIC ORBITS of near- Earth- asteroid classes. (a)Those of Apollos and Atens cross Earth’s orbit. Amors orbit the Sun entirely outside Earth’s orbit, and Atirasorbit the Sun completely inside Earth’s orbit. Vatiras are on orbits entirely inside Venus’s orbit. (b)This projection shows the inner solar system in a plot of near- Earth asteroideccentricities versus their semimajor axes. At the lower rightare the known main- belt asteroids. The vast majority (Apollos and Amors) in the inner solar system reside outsideEarth’s semimajor axis (1 AU). The fraction of near- Earth asteroids in each subpopulation and the limits on theiraphelia or perihelia are shown by their names. (Image bySarah Greenstreet.)46PHYSICS TODAY |JULY 2021decoupled from Earth and Venus, most eventually collide with one of those planets or with Mercury. Any that remain are frequently pushed onto Sun- grazing orbits, and some are evenscattered back out to Mars- crossing orbits and potentially be-yond the asteroid belt. Dynamical simulations reveal the orbits in which Atiras and Vatiras spend most of their time and thus where it is best tolook in the night sky to find them. Their rarity has made themof particular interest to telescopic surveys focused on aster-oid discovery. Many of those surveys have dedicated time to searching for asteroids near the Sun, and some telescopes, suchas the space- based Near- Earth Object Surveillance Satellite (NEOSSat ), 6were specifically designed to discover new Atens, Atiras, and Vatiras. Not only are Atiras and Vatiras a rare part of the NEA pop- ulation because of their difficulty in gravitationally scatteringto small, planet- decoupled orbits, they are also challenging ob-jects to observe in the night sky because of their close proximityto the Sun. The asteroids are never farther from the Sun thanis Earth, so ground- based telescopes— which make most aster-oid discoveries— must aim near the horizon during a brief pe- riod of time shortly after sunset and shortly before dawn tohave any hope of capturing one in an image. Considering thatobservational limit, when new discoveries of such asteroids aremade, it is quite exciting. Atira discoveries To date, 23 known asteroids orbit the Sun in the Atirapopulation. They range in size from 50 m to 5 km. Both theGranvik and Greenstreet models estimate the existence of 10 Atiras 3,4with diameters larger than a kilometer. Beyond the six known Atiras in that size range, few large ones are likelyleft to be discovered. The number of Atiras with increasinglysmaller diameters is much greater, so many more remain tobe found. The first confirmed Atira asteroid was discovered in 2003 by the Lincoln Near- Earth Asteroid Research program at theMIT Lincoln Laboratory near Socorro, New Mexico. Called163693 Atira, it was named after the Pawnee goddess of Earth.The asteroid follows a highly inclined orbit with an aphelionjust inside Earth’s perihelion— the cutoff for orbits interior toEarth’s. Because the asteroid’s aphelion is so close to Earth’sperihelion, it is possible for the Atira to have close encounterswith Earth. In January 2017 one such encounter occurred whenthe asteroid passed close enough for the Arecibo Observatoryto capture it in a series of radar images. FIGURE 3. THE ZWICKY TRANSIENT FACILITY is located at the Palomar Observatory’ s 48- inch Samuel Oschin Telescope. The facility’s twilight observing program is responsible for finding the three near- Earth asteroids with the shortest orbital periods known to date.(Image courtesy of Palomar/Caltech.)ASTEROIDSJULY 2021 |PHYSICS TODAY 47Radar astronomy uses reflected microwaves from nearby solid targets to constrain the shape,size, and spin state of an asteroid. As reported by Edgard Rivera- Valentín and colleagues, all atArecibo Observatory at the time, the radar mea -surements revealed an unexpected finding— that163693 Atira is a binary system. 7It consists of two objects, a primary and a smaller secondary, thatorbit each other. The diameter of the primary wasmeasured at 4.8 ± 0.5 km with an elongated andvery angular shape; the diameter of the secondarywas 1.0 ± 0.3 km. The semimajor axis of the binarywas fit at near 6 km with an orbital period ofroughly 16 hours. Astronomers are not lucky enough to get radar measurements of many asteroids, but telescopic observationsreveal many of their features. The asteroids’ brightness, distance,and reflectivity reveal their size. The periodicity at which thatbrightness changes reveals their rotation periods. And spectralanalysis reveals their surface composition. Using dynamicalsimulations of an observed orbit, astronomers can learn wherean asteroid likely came from, how long it is likely to stay on itscurrent orbit, and what its most likely future trajectory will be.Using our dynamical model of the NEA orbital distribution, 3 we can say that 163693 Atira probably entered the NEA popu-lation through the inner portion of the main asteroid belt andlikely took tens of millions of years to scatter down to its cur-rent region. It will probably scatter into and out of the Atirapopulation several times with an integrated lifetime in theAtira region of a couple million years before most likely collid-ing with a terrestrial planet. Over time, more Atira- class asteroids have been discov- ered that have increasingly smaller orbits. In 2019 astronomers found two Atiras that have the smallest semimajor axes knownand aphelia that put them near the Atira– Vatira boundary (Venus’s perihelion). The first, 2019 AQ3, was discovered on 4 January by the twilight observing program at the ZwickyTransient Facility 8at Palomar Observatory, shown in figure 3. The second, 2019 LF6, was discovered five months later, on 10 June. The first Vatira Within a year of the discoveries of 2019 AQ3 and 2019 LF6, thefirst Vatira- class asteroid was spotted by the same program atthe Zwicky Transient Facility that had discovered the two Ati-ras. Figure 4 shows the asteroid appearing as a tiny dot in thenight sky four days after its discovery. Designated 2020 AV2,the asteroid has an aphelion well inside the Venus perihelion cut- off, which makes its orbit entirely inside Venus’s orbit. Asestimated by Marcel Popescu of the Astronomical Institute ofthe Romanian Academy and colleagues, 2020 AV2 is roughly1.5 km in diameter. 9And to judge by the Greenstreet and Granvik dynamical models,3,4it is likely one of two Vatiras of that size currently in existence. Popescu and colleagues classified the composition of 2020 AV2 as one that dominates the inner main belt.9The composi- tion is consistent with our model prediction that the asteroidmost likely originated at the inner edge of the main belt beforemaking the long journey to the Vatira population. 3,10It will likely remain a Vatira for a few hundred thousand years be-fore scattering back out through a Venus- crossing Atira orbit to the Earth-crossing region, where it will most likely collidewith Venus or Earth several million years from now. 10The cur- rent orbit of 2020 AV2 puts it very close to the 3:2 resonancewith Venus. 10,11Unlike the resonances located in the main- belt Kirkwood gaps, that resonance with Venus is relatively stable,given the scarcity of resonances in the innermost portion of thesolar system. Vatiras can remain in the resonance for millionsof years, making the Venus resonance a likely place whereother Vatiras are lurking and thus a good hunting ground fordiscovering more asteroids in that class. Asteroid surveys, such as the twilight observing program at the Zwicky Transient Facility and the space- based NEOSSat , are ongoing. In addition, two large observing programs— theVera C. Rubin Observatory’s Legacy Survey of Space and Timeand NASA’s Near- Earth Object Surveillance Mission— are up-coming. Those new programs are likely to vastly increase thenumber of known NEAs, particularly given the new softwarebeing developed to prepare for the deluge of data from thenew surveys. For example, using a program 12built by the Uni- versity of Washington’s Dirac Institute, the cloud- based AsteroidDecision Analysis and Mapping platform will allow the B612Asteroid Institute to extract even the trickiest- to- find asteroidsamong the data. REFERENCES 1.D. Kirkwood, Meteoric Astronomy: A Treatise on Shooting- Stars, Fire Balls, and Aerolites , J. B. Lippincott (1867). 2.P . Farinella et al., Nature 371, 314 (1994). 3.S. Greenstreet, H. Ngo, B. Gladman, Icarus 217, 355 (2012). 4.M. Granvik et al., Icarus 312, 181 (2018). 5.A. O. Ribeiro et al., Mon. Not. R. Astron. Soc. 458, 4471 (2016). 6.A. R. Hildebrand et al., Earth, Moon, Planets 95, 33 (2004). 7.E. G. Rivera- Valentín et al., Cent. Bur. Electron. Teleg. 4347 , 1 (2017). 8.Q. Ye et al., Astron. J. 159, 70 (2020). 9.M. Popescu et al., Mon. Not. R. Astron. Soc. 496, 3572 (2020). 10.S. Greenstreet, Mon. Not. R. Astron. Soc. Lett. 493, L129 (2020). 11.C. de la Fuente Marcos, R. de la Fuente Marcos, Mon. Not. R. Astron. Soc. Lett. 494, L6 (2020). 12.J. Moeyens et al., http://arxiv.org/abs/2105.01056. PT FIGURE 4 . A VATIRA ON FILM. The Virtual Telescope Project took this image on 8 January 2020. It shows the average of 14 60- secondexposures. They were combined to track the motion of 2020 AV2— the white dot, marked by an arrow— across the sky to reveal the asteroid as a point source, against which the stars streak. (Imagecourtesy of Gianluca Masi, Virtual Telescope Project.)

4.0000102.pdf

Ultrafast strong-field dissociation of vinyl bromide: An attosecond transient absorption spectroscopy and non-adiabatic molecular dynamics study Cite as: Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 Submitted: 25 March 2021 .Accepted: 24 May 2021 . Published Online: 15 June 2021 Florian Rott,1Maurizio Reduzzi,2 Thomas Schnappinger,1 Yuki Kobayashi,2 Kristina F. Chang,2 Henry Timmers,2Daniel M. Neumark,2,3 Regina de Vivie-Riedle,1,a) and Stephen R. Leone2,3,4,a) AFFILIATIONS 1Department of Chemistry, LMU Munich, 81377 Munich, Germany 2Department of Chemistry, University of California, Berkeley, California 94720, USA 3Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 4Department of Physics, University of California, Berkeley, California 94720, USA a)Authors to whom correspondence should be addressed: regina.de_vivie@cup.uni-muenchen.de andsrl@berkeley.edu ABSTRACT Attosecond extreme ultraviolet (XUV) and soft x-ray sources provide powerful new tools for studying ultrafast molecular dynamics with atomic, state, and charge specificity. In this report, we employ attosecond transient absorption spectroscopy (ATAS) to follow strong-field-initiated dynamics in vinyl bromide. Probing the Br M edge allows one to assess the competing processes in neutral and ionized molecularspecies. Using ab initio non-adiabatic molecular dynamics, we simulate the neutral and cationic dynamics resulting from the interaction of the molecule with the strong field. Based on the dynamics results, the corresponding time-dependent XUV transient absorption spectra are calculated by applying high-level multi-reference methods. The state-resolved analysis obtained through the simulated dynamics and relatedspectral contributions enables a detailed and quantitative comparison with the experimental data. The main outcome of the interaction withthe strong field is unambiguously the population of the first three cationic states, D 1,D2, and D3. The first two show exclusively vibrational dynamics while the D3state is characterized by an ultrafast dissociation of the molecule via C–Br bond rupture within 100 fs in 50% of the analyzed trajectories. The combination of the three simulated ionic transient absorption spectra is in excellent agreement with the experi- mental results. This work establishes ATAS in combination with high-level multi-reference simulations as a spectroscopic technique capableof resolving coupled non-adiabatic electronic-nuclear dynamics in photoexcited molecules with sub-femtosecond resolution. VC2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http:// creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/4.0000102 I. INTRODUCTION Since the demonstration of attosecond pulses, in the extreme ultraviolet (XUV) region of the electromagnetic spectrum (10–124 eV), via high-order harmonic generation (HHG), these pulses have been exploited for time-resolved investigations of ultrafast photo- initiated processes in atoms, molecules, and solids. Due to the high associated photon energy, capable of easily removing a valence elec- tron from the sample under consideration, attosecond pulses allowone to create electron wave packets extremely well localized in time. This property, which makes them very sharp photo-triggering tools in a “pump-probe” scenario, is at the basis of the early attosecond streak- ing technique. 1Originally devised as a temporal characterization methodology for the attosecond pulses themselves,2“streaking” hasbeen exploited as a genuine spectroscopic tool, allowing for the deter- mination of attosecond delays in photoemission both in atomic3,4and solid-state5,6systems. With the same spirit, attosecond pulses have been used in the last decade to create electron wave packets in highlyexcited cationic states of molecules, leading to the discovery of effectssuch as electron localization in diatomic molecules 7and, later, of purely electronic charge migration in biomolecules.8On the other hand, the broad bandwidth of attosecond pulses makes them particu-larly valuable probing tools, because of the element, charge, andelectronic state sensitivity gained by accessing the inner valence (in theXUV) and the core level states (in the soft x-ray region, and, possiblyin the future, in the hard x ray) of elements. Along these lines, after theseminal work of Goulielmakis et al. , 9attosecond XUV pulses have Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 8, 034104-1 VCAuthor(s) 2021Structural Dynamics ARTICLE scitation.org/journal/sdybeen exploited to take snapshots of ultrafast processes in atoms,10,11 molecules,12,13and solid-state materials,14–17triggered by the strong- field interaction of the sample with an ultrashort few-cycle pulse. While this scheme is very convenient from the implementation view- point [carrier-envelope phase (CEP) stable few-cycle pulses are idealdrivers for the generation of isolated attosecond pulses, and thus, a replica for excitation purposes can be easily derived] and grants exquisite temporal localization of the initially prepared excited wave packet due to the extremely short pulses employed, there are two main bottlenecks historically ascribed to this methodology, limiting therange of applications—on the one hand, the non-resonant and non- perturbative nature of the excitation process; on the other, the compli- cations in data interpretation due to multiplet effects in probing inner valence states (usually M edges) rather than genuine core level states (K and L edges). Some of the main recent directions in attoscience and, in general, of table-top ultrafast soft x-ray spectroscopy solve these methodological issues, moving toward single photon excitationschemes (from strong field to weak field) 18,19and core level probing (from 800-nm-driven XUV HHG to soft x-ray ponderomotively scaled HHG).20–22Because of the great spectroscopic interest in the water window spectral region (containing the K absorption edges of carbon, nitrogen and oxygen as well as the L edges of calcium, scan- dium, titanium, and vanadium) for chemistry, biology, and materials science, these developments are crucial for the future of the field andhold promise to elucidate photoinduced processes of paramount importance, such as UV radiation damage of DNA and light-induced phase transitions from each individual atom’s perspective. Still, the above-mentioned difficulties can be greatly diminished by an appro- priate theoretical ansatz. For the complete treatment of such time-resolved spectroscopy experiments in a theoretical framework, three different problem sets need to be addressed. As a first step, one has to describe the interaction of the system with the strong near infrared (NIR) field. Although it is possible 23–25to simulate the strong-field ionizations and excitations, it is not the focus of this work and we assume an instantaneous ioniza- tion or excitation. Second, one has to adequately describe the ultrafast processes the system undergoes after irradiation with the strong few- femtosecond NIR pump pulse. In our work, this is done by performing ab initio non-adiabatic molecular dynamics (NAMD) for the relevant electronic states. This makes it possible to resolve both the changes in the electronic structure and the nuclear motion over time. The thirdproblem set is the simulation of the time-resolved XUV absorption spectrogram. In order to tackle this task, we use the restricted active space self-consistent field (RASSCF) theory followed by a perturbative inclusion of the dynamic electron correlation (RASPT2) to calculate the XUV absorption along each trajectory. Besides the high-quality description of the absorption itself, the RASPT2 technique also allows one to decompose the entire spectro- gram into its state-specific components. The RASPT2 ansatz to calcu- late the XAS spectra was first proposed by Josefsson and co-workers 26 to calculate the L-edge spectra of transition metal complexes. Due to the versatility of the RASSCF, it was used by many groups in varying cases,27–42but, in general, there are a variety of methods reported in the literature to calculate x-ray absorption (XAS) or XUV absorption spectra. First, simulations were done with the static exchange (STEX) approximation.43–46Later, Stener and co-workers47proposed an ansatz based on the restricted excitation window time dependentdensity functional theory (REW-TDDFT), which was successfully uti- lized in a number of different cases.20,48–51More recently, the core- valence separation (CVS) approximation52was used in conjunction with coupled-cluster (CC) theory.53–57Also, Hait and co-workers suc- cessfully utilized the Restricted Open-Shell Kohn–Sham (ROXS)theory 58,59in the prediction of core-level spectra.60,61A good overview of the various methods is given by Zhang and co-workers62and, more recently, in the reviews by Norman and Dreuw63and Lundberg and Delcey.64 Here, we propose a joint experimental and theoretical approach and apply it to follow the strong-field-initiated dynamics of vinyl bro-mide (C 2H3Br), exposed to a few-cycle NIR field. The ultrafast photo- physics of vinyl bromide, similar to other substituted ethylenes, hasbeen historically studied in connection to photoactivated cis/trans- isomerization. 65In vinyl halides, in particular, the halogen substituent (F, Cl, Br, I) introduces conical intersections between the pp/C3state and np/C3/nr/C3states, leading to efficient ultrafast photodissociation of the halogen–carbon bond.66We employ attosecond transient absorp- tion spectroscopy (ATAS) to simultaneously follow neutral and cat-ionic states dynamics of the molecule after the interaction with thestrong NIR field by probing the Br M edge. The experiment leverages a previous investigation, 67performed at coarser tens-of-femtoseconds temporal resolution, highlighting the great amount of information anddetail accessed by attosecond spectroscopies. That work assigned fea-tures based on conventional intuition known at the time; in hindsightfrom the new theoretical work here, numerous assignments of the fea-tures in the earlier experimental spectra are reassessed. II. METHODS The output of a carrier-envelope phase (CEP) stable Ti:sapphire system ( k¼800 nm), delivering 27 fs pulses with an energy of 2 mJ per pulse at 1 kHz, is spectrally broadened in a Ne-filled stretched hollow-core fiber. The pulses are then compressed using both a set ofbroadband chirped mirrors and a 1-mm-thick ammonium dihydrogenphosphide (ADP) crystal, 68yielding a compressed pulse duration of sub-4 fs. The pulses are then divided into a pump and probe arm usinga 50:50 beam splitter. The employed ATAS pump-probe scheme isdepicted in Fig. 1 . The XUV-probe pulse is generated via HHG by focusing the few-cycle NIR pulses into a 3-mm-long gas cell backed with 30 Torr of Ar. Isolated attosecond pulses, covering the energyrange between 55 and 73 eV, are obtained by the amplitude gatingtechnique, 69as shown in Fig. 1 . The residual NIR light is filtered out of the probe arm with a 200-nm-thick Al filter. According to previousstreaking measurements, 68the XUV pulse duration is estimated to be 170 as. A toroidal mirror is used to focus the radiation into a static gas cell (3 mm interaction length) filled with 10 Torr of C 2H3Br (Sigma- Aldrich) at room temperature (298 K). The NIR-pump pulse travels through a piezoelectric delay stage and is collinearly recombined with the XUV-probe with a hole mirrorplaced before the focusing toroidal mirror. On target, the intensity of the NIR-pump pulse, I¼2/C210 14Wc m/C02, is high enough to trigger various competing processes, both in the neutral molecule (via frus-trated tunnel ionization and multiphoton excitation) and in its cation(via strong-field ionization). The NIR-pump pulse is finally removedfrom the beam path after the interaction gas cell using a second 200-nm-thick Al filter. The XUV radiation transmitted through the inter- action gas cell is analyzed with a home-built spectrograph, consistingStructural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 8, 034104-2 VCAuthor(s) 2021of a dispersive, gold-coated, aberration-corrected concave XUV grat- ing (Hitachi) and a back-illuminated x-ray-sensitive cooled CCD cam- era (Princeton Instruments). The energy resolving power of thespectrograph is estimated to be E=dE¼1000 from previous experi- ments, 11yielding a resolution of 70 meV at 70 eV photon energy. A beam shutter is programmed to block the NIR pump arm in order to reference the C 2H3Br XUV absorption spectrum with and without the NIR pump pulse and measure the differential optical density, definedas the natural logarithm of the ratio between the spectra when the pump is on and off. III. COMPUTATIONAL DETAILS A.Ab initio level of theory The electronic states of vinyl bromide were computed using the state-average complete active space self-consistent field method 70,71 including five states in the state-averaging procedure (SA5-CASSCF). All calculations were carried out with the OpenMolcas72,73program package using a modified ATZP basis set (m-ATZP). For more detailsabout the modification and the validation of its use, see the supple- mentary material Sec. II B. 108An active space (AS) including eight electrons in seven orbitals [AS(8,7)] was employed. It includes thecarbon–carbon double bond ( p 1,p/C3 3)(Fig. 2 ), the carbon–bromine single bond ( r1,r/C3 2) as well as both remaining bromine 4 porbitals. One nonbonding orbital is part of a second porbital ( p2); the other one forms the lone-pair n1. To stabilize the active space, we also had to include one additional virtual orbital ( p/C3 4), which has significantRydberg (Ryd.) contributions. The complete AS is shown in Fig. S7 in the Supporting Information, and the main parts of it, except for the p/C3 4/Ryd. orbital, are also in Fig. 2 . An overview of the excited states of both neutral and cationic vinyl bromide, their vertical excitation ener- gies, and electronic character is given in Table I . The states are ordered FIG. 1. (Top) Schematic of the employed ATAS setup. After being delayed by a piezo-driven stage and collinearly recombined (not shown), the 800 nm fs pumpand XUV attosecond (as) probe pulses are focused with a toroidal mirror into a static absorption gas cell, filled with vinyl bromide. The transmitted XUV light, after removal of the copropagating 800 nm radiation by an aluminum filter, is spectrallydispersed with a grating and imaged with a CCD camera. (Bottom) The XUV spec-trum of the generated attosecond probe pulses. The sharp decrease at 72.7 eV is due to the employed aluminum filter. FIG. 2. Diagram of the relevant excitations for an x-ray/XUV absorption spectrum. The RAS2 depicted in blue allows us to describe the valence excited states as well as cationic states after the interaction with the strong field. The RAS1 includes all five 3 dbromine core orbitals and is shown in red. The combination of both RASs allows us to treat the x-ray/XUV absorption and simulate the XAS. The orbitals areshown with an isovalue of 0.02 and oriented according to the geometry at the bottom. TABLE I. Excited state electronic character and vertical excitation energies DEof neutral and cationic vinyl bromide. The excitation energy is given in units of eV rela- tive to the neutral ground state ( S0) at the Frank–Condon point. For the character of the cationic vinyl bromide, the orbitals that are partially occupied are listed. Neutral Cationic State Character DEðeVÞ State Character DEðeVÞ T1 p2!p/C3 3 4.08 D1 p2 8.81 T2 p2!r/C3 2 6.26 D2 n1 9.72 T3 n1!r/C3 2 6.61 D3 p1 11.28 S1 p2!r/C3 2 6.67 D4 r1 12.94 S2 p2!p/C3 3 7.34 D5 n1;p2;p/C3 3 14.26 S3 n1!r/C3 2 7.50 T4 n1!p/C3 3 7.53 S4 n1!p/C3 3 7.77 T5 p2!p/C3 4 8.16Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 8, 034104-3 VCAuthor(s) 2021with increasing energy and are labeled according to their adiabatic state number at the Frank–Condon (FC) point. B. Dynamics simulation T h eN A M Do fv i n y lb r o m i d ew e r es i m u l a t e du s i n gt h es u r f a c e hopping including arbitrary couplings (SHARC) program pack-age. 74–76A set of 400 initial conditions (geometries and velocities) were generated based on a Wigner distribution77,78at 0 K computed from harmonic vibrational frequencies in the optimized ground state minimum. The underlying optimization and frequency calculationswere performed at the closed-shell coupled-cluster level of theory,including singles, doubles, and perturbative contributions of triplesusing the aug-cc-pVTZ basis set 79–83[CCSD(T)/aug-cc-pVTZ]. For the neutral species, a subset of 300 trajectories was randomly chosena n dp r o p a g a t e ds t a r t i n gi nt h eb r i g h t pp /C3state (at the FC point S2). To relate to the excitation process, the initialization for the neutral specieswas done in the diagonal representation, where the spin-mixed, fully adiabatic states were obtained by diagonalizing the electronic Hamiltonian matrix, which itself is built up from five singlet and trip-let states for the neutral species. In addition, all relevant spin–orbitinteractions are included in the Hamiltonian matrix. To ensure excita-tion only to the bright pp /C3state, the transition dipole moment for each initial geometry is calculated as the selection criterion. For thecation, the trajectories were set up in the molecular CoulombHamiltonian (MCH) representation, where the states are representedin the basis of the eigenfunctions of the molecular Coulomb Hamiltonian, consisting of five doublet and quartet states. Sets of 50 randomly chosen trajectories were started each in the D 1,D2,a n d D3 ionic states. In both cases, the propagation itself was performed in the diagonal representation. The necessary energies, gradients, spin–orbitand non-adiabatic couplings (SOC, NAC) were calculated on-the-flyat the SA5-CASSCF level of theory using the Molcas/OpenMolcasinterface included in SHARC. The non-adiabatic transitions weretreated within Tully’s fewest switch trajectory surface hopping (TSH)algorithm 84as implemented in SHARC.76The integration of the nuclear motion is done using the Velocity-Verlet algorithm with a maximal simulation time of 100 fs using a time step of 0.5 fs. The sub-sequent analysis of the trajectories of both species is done in the MCHrepresentation. C. Simulation of x ray/XUV absorption spectra The following describes, first, the general procedure how we cal- culated the static x-ray/XUV absorption spectrum (further abbreviatedas only XAS) based on the RASPT2 ansatz and second, how we usedthe NAMD simulation to generate the transient XAS of vinyl bromide. The algorithm for the calculation of the XAS is mainly based on the works of Josefsson and co-workers 26and Wang, Odelius, and Prendergast.42Here, both the valence space of vinyl bromide and the excitation processes of the core electrons are described in the same the-oretical framework of the restricted active space self-consistent field(RASSCF) method. 85,86RASSCF is an extension to the general CASSCF approach, where the active space is further partitioned intothree subspaces RAS1, RAS2, and RAS3, with additional constraintsapplied to their occupation. In general, they can be systematicallylabeled RAS( n;l;m;i;j;k), where, i,j,a n d kare the number of orbitals in the RAS1, RAS2, and RAS3 subspaces, respectively, nis the totalnumber of electrons in the active spaces, lthe maximum number of holes allowed in the RAS1, and mthe maximum number of electrons allowed in RAS3. For the RAS2, all possible configurations are allowed, making it analogous to the AS within CASSCF. For vinyl bromide, the RAS2 was built up similarly to the AS utilized in the NAMD. It alsoincluded the carbon–carbon double bond ( p 1,p/C3 3), the carbon– bromine single bond ( r1,r/C3 2) as well as both remaining bromine 4 p orbitals ( p2,n1) and one additional virtual orbital with significant Rydberg contributions. For the RAS1, we included all five 3 dorbitals of bromine and allowed for one hole. To enhance convergence withinthe RASSCF and to suppress unwanted orbital rotation with the valence space, the core orbitals were kept frozen to their shape from the Hartree–Fock calculation. The RAS3 was not utilized, thus result- ing in RAS(18 ;1;0;5;7;0) for neutral and RAS(17 ;1;0;5;7;0) for cationic vinyl bromide. The diagram in Fig. 2 summarizes all relevant excitations and orbitals involved for the computational setup. As bro- mine is already subject to spin–orbit effects, 87relativistic effects were included and treated in two steps, both based on the Douglas–Kroll Hamiltonian.88,89In the first step, within each spin symmetry the states were optimized in a state-averaged RASSCF procedure carried out with the relativistic atomic natural orbital basis (ANO-RCC),90–94 contracted to ATZP quality (ANO-RCC-ATZP). This is followed by aperturbative inclusion (PT2) of the dynamic electron correlation within the multistate RASPT2 method, 95–99where the default ionization-potential electron-affinity (IPEA) shift100of 0.25 Hartree was used. An additional imaginary shift101of 0.2 Hartree was applied to reduce problems with intruder states. In the second step, these “spin-free” states were used as a basis in the restricted active space state interaction (RASSI) method,102–104where the spin–orbit coupling (SOC) was treated with the use of the atomic mean-field spin–orbit integrals (AMFI).105 For the simulation of the transient XAS discussed further below, we utilized the geometric information from the on-the-fly trajectories. For every 3 fs snapshot, the XAS spectrum is calculated. Note that themulti-configurational wave function from the dynamics simulation could not be reused as it did not include all orbitals necessary for the proper description of the XAS spectrum. The spectrum requires a new active space including the five bromine 3 dcore-orbitals and the extended ANO-RCC-ATZP basis set. For each single XAS calculation, seven singlet and eight triplet valence excited states for the neutral and five-doublet and three- quartet valence excited states for the cationic species are calculated. Additional 40 core excited states for both species, and all four multi- plicities, were calculated by enforcing one hole in RAS1. This results in a total of 95 states for the neutral and 88 states for the cationic species. The interaction between all these states, including SOC, was treatedwithin in the RASSI method. Note that since the valence and core excited states are obtained from different calculations, their corre- sponding wave functions are not inevitably orthogonal. This was already mentioned by Wang, Odelius, and Prendergast. 42Following their proposed procedure, we applied the original correction to theoverall Hamiltonian as introduced in Ref. 104. The next step is to match the valence states from the XAS calculation with the valence states from the trajectory calculations to obtain the transient XAS spectrum for the active state. In the case of the neutral vinyl bromide, we used the dipole moment vector of the active state of the trajectory for the matching. For the cationic case, we found it to be sufficient toStructural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 8, 034104-4 VCAuthor(s) 2021use the state number (index) of the active state. The oscillator strengths that are related to the matched valence excited state were then used tosimulate the transient XAS. The vertical transition energies wereobtained from the spin–orbit RASSI states (SO-RASSI) and no energy shift was applied. An overall Gaussian convolution with r¼0:1e V was applied to broaden the calculated spectrum. For each time step,the convoluted spectra were averaged over all geometries and thennormalized to the maximum of all calculated time steps. Furthermore, the XAS of the S 0ground state at time t¼0 fs was subtracted to simu- late the ground state bleach in the energy range upwards of 70.0 eV. Aflow chart of the complete procedure as well as example Molcas inputsfor the necessary calculations can be found in the supplementarymaterial Sec. II G. 108 IV. RESULTS AND DISCUSSION A. Spectroscopic assignments of the transient absorption spectra The transient absorption spectrogram, recorded by collecting the differential optical density at different pump-probe delays with 2.3 fsstep size, is shown in Fig. 3 . Further details on data processing are pro- vided in Sec. I A of the supplementary material. 108The spectral range of the attosecond XUV-probe pulse covers the pre-edge region of the M4,5edge of Br (located at 75.3 eV), corresponding to the excitation of aB r3 delectron. The applied NIR pump pulse launches multiple strong-field-initiated dynamics in vinyl bromide. The main processtriggered is strong-field ionization, but neutral valence excited-state vinyl bromide can also be formed by a multiphoton process. As shown by closely related studies on methyl iodide, 12methyl bromide,13and vinyl bromide,67both processes can occur under these conditions, leading to a rich transient absorption spectrum. The bleaching signals at 71.0 and 71.9 eV are due to the depletion of the ground state of the molecule after the interaction with the strong NIR field. They appear as a doublet, as is the case for features discussed below, due to the 0.9 eV spin–orbit splitting of 3 d5=2and 3 d3=2states in Br.106The main doublet appearing at 65.5 and 66.4 eV as well as the weaker signals up to 70 eV has been assigned to ionic states includingalso the dication.67Following the previous assignment, the main dou- blet corresponds to the excitation of the 3 dBr electron of the ionic vinyl bromide in the D2state (for details of the nomenclature see Table I ). At long temporal delays ( t>100 fs), sharp atomic Br lines at 64.4 and 65.0 eV indicate an ultrafast dissociation of the molecule or ion. Pronounced, periodic, delay-dependent spectral modulations of the absorption are visible both in the ground state bleaching and cat- ionic state signals. In both cases, they reflect vibrational wavepacket motion, as already pointed out in a number of closely related investiga- tions.12,13The delay-dependent spectral first-moment of the absorp- tion signal in the range 70.3–71.5 eV reveals a clear oscillation (see Sec. I B of the supplementary material108) with a period of 52.5 fs. The corresponding frequency of 635 cm/C01agrees well with the C–Br eigen- mode /C233in the ground electronic state of neutral C 2H3Br (see Table S7 in the supplementary material108and Ref. 107). Given the initial phase of /0¼/C0p(cosine-like wave), the wavepacket is likely to be launched by a bond softening mechanism, as detailed by Wei et al.12 This fine mapping in the energy domain of the vibrational wavepacket motion is one of the unique aspects of ATAS. A more prominent example of this resolving capability is provided by the dynamics of the cationic D2state. The spectral first- moment of the signal in the range 65.2–65.7 eV is in this casecomposed of the sum of two dephased sinusoidals (see Sec. I B of the supplementary material 108) with frequencies of 480 and 1220 cm/C01, corresponding to the /C233(C–Br, stretching) and /C237(CCH, bending) eigenmodes, respectively (see Table S7 in the supplementary mate- rial108). The activation of the /C233and/C237modes in the ionization pro- cess to the D2s t a t eo fc a t i o n i cv i n y lb r o m i d eh a sa l r e a d yb e e n observed and understood in terms of the displacive nature of the verti- cal Franck–Condon excitation.107Indeed, the equilibrium geometry of theD2cationic state features two main differences compared to the neutral ground state: an increase in C–Br bond length (by 0.06 A ˚) a n di nC C Ha n g l e( b y9 . 1 0 % )( s e eT a b l eS 9i nt h es u p p l e m e n t a r y material108). In comparison with a spectral domain spectroscopy method, here, ATAS allows one to completely map in space and timethis multidimensional motion, giving access to the phase of the differ- ent contributions, relative to the instant and to each other. In the energy region of 64.0 and 65.0 eV, a previously not reported short-lived ( t<100 fs) feature centered at 64.3 eV at t¼0f s is observed. A zoom of the transient absorption spectrogram showing this new feature can be found in Fig. 4 . It is immediately evident that, compared to the oscillatory nature of the vibrational coherencesexplained above, the temporal behavior of the feature is more compli- cated. After an initial energy upshift of the absorption feature, from 64.30 to 64.45 eV within the first 25 fs, the signal bifurcates around t¼30 fs. The main branch downshifts in energy by 0.35 eV in the fol- lowing 10 fs, reaching the asymptotic value of the atomic Br M 4,5tran- sition106P1=2!D5=2at 64.1 eV at t¼40 fs. It is important to note that the 1 =2t o5 =2 transition is strictly dipole forbidden in the atomic limit due to spin–orbit selection rules, but, in the molecule, because of spin–orbit coupling among electrons, the restriction is removed. In fact, the signal at 64.1 eV vanishes within the following 40 fs, in con- comitance with the birth and rise to asymptotic intensity of the allowed atomic Br transitions P3=2!D5=2(at 64.4 eV) and P1=2! D3=2(at 65.0 eV). The transient behavior, combined with the delayed rise of the atomic Br dissociation products, clearly indicates that we FIG. 3. XUV transient absorption spectrogram of C 2H3Br exposed to strong-field interaction with a sub-4 fs NIR pulse centered at k¼800 nm, with a peak intensity ofI¼2/C21014Wc m/C02. The transient is recorded by collecting the differential optical density at different pump-probe delays, with a step size of 2.3 fs. For nega-tive delays, the XUV pulse arrives first, and for positive delays the NIR pulse arrivesfirst.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 8, 034104-5 VCAuthor(s) 2021are observing an ultrafast dissociation process. The observation appears in close connection to the recently investigated ATAS ofmethyl bromide. 13There, the feature was assigned to a neutral excited state populated by multiphoton process and the bifurcation was attrib-uted to the non-adiabatic passage through a conical intersection. Inthe case of vinyl bromide, it is not straightforward to distinguish in thespectroscopic data whether this signal originates from an ionic or neu-tral dissociation process. In the critical range between 64.1 and65.0 eV, the previous investigation 67of vinyl bromide predicted the appearance of a signal due to the second-excited cationic state ( D3) according to Koopman’s theorem, which could, however, not beresolved at that time. Applying the same simple Koopman’s picture onneutral vinyl bromide, a signal in the same range is expected, whichcould be assigned to an electronically excited pp /C3state of the neutral. In fact, the first UV absorption band of the molecule is composed of a very broad ( >0:5 eV) feature centered around 6.5 eV;107this is in range with the distance between the signal under consideration andthe ground state bleach doublet. In the present strong-field pumpscenario, such an excited state could be reached in principle via afour-photon-process. Noteworthy, an ultrafast C–Br bond-rupturechannel of the neutral has been inferred from 193 nm photodissocia-tion studies. 65In order to disentangle between the two possibilities, neutral vs ion as well as to explain the complete transient absorptionspectrogram, we simulated the transient XAS spectra for neutral andionic vinyl bromide. B. Excited state dynamics of the neutral vinyl bromide In order to initiate the neutral dynamics, we approximate the multiphoton process as a one-photon excitation. We start 300 trajecto-ries in the bright excited state (for more details see Sec. III B). Using the default selection criteria provided by SHARC, 189 trajectories weretaken into account for the analysis of the excited state dynamics andcalculation of the XAS spectrum. The initial and final populated adia-batic states are summarized in Table II . The excitation to the bright pp /C3state corresponds to an initial population of 38% in the S2state and 34% in the S3state. Since Br is a heavy atom, where spin–orbit effects arise, we also see a slightlysmaller initial population of 16% in the T 3state. After excitation, all trajectories show non-adiabatic as well as spin–orbit transitionswithin the simulation time of 100 fs. The overall change inpopulation for all states is shown in Fig. S9 in the supplementary material. 108In the first 20 fs, population is mainly transferred from the three initial states into the triplet states T2toT5. After this ini- tial period of 20 fs, also, the population of the S0and T1states increases, where T1is populated faster. Over the next 80 fs, the general trend of population transfer to the triplet states stays thesame. Only the population of the S 0state increases to about 8%. At the end of the simulation, most of the population (about 75%) can be found in one of the triplet states. The remaining 25% is distrib-uted over all five singlet states. In each multiplicity, the first twostates ( S 0,S1,T1,T2) are populated the most. No matter in what electronic state the trajectories end up, their final geometries show a quite uniform picture. For elucidation, the temporal evolution of both the C ¼C double bond and the C–Br bond is shown in Fig. 5 . All analyzed trajectories are of dissociative nature, showing on average a doubling of the initial C–Br bond length after about 50 fs. No oscillation in the C–Br bond and only a very weak one in the C ¼C double bond are observed. The fragments reveal that only homolytic dissociation occurs. Two types of electronic configurationare present, which differ only in the position of the unpaired electroni nt h ev i n y lr a d i c a l .I nt h ec a s eo ft h e S 0/C02and T1/C03states, the unpaired electron resides in the 2 porbital of carbon that was part of the former rC–Br bond (H 2C¼C•H). For states S3;4and T4;5,t h e electron is in the p2orbital of the C ¼C bond, with the carbon 2 p orbital now being doubly occupied (H 2C•–CH). For both channels, FIG. 4. Zoom of the transient absorption spectrogram showing the new feature in the energy range between 64.0 and 65.0 eV.TABLE II. Population of the ten calculated neutral states at the start and end point of the simulation. All percentages are given with respect to the total number of analyzed trajectories. State Start End State Start End S0 0% 8% T1 0% 18% S1 4% 8% T2 0% 18% S2 38% 5% T3 2% 16% S3 34% 2% T4 16% 11% S4 3% 2% T5 3% 12% FIG. 5. Temporal evolution of the CC double bond and the CBr bond for the 189 analyzed trajectories of neutral vinyl bromide.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 8, 034104-6 VCAuthor(s) 2021the electron on the Br radical sits in one of the three now degenerate 3porbitals. Combining the data from the analyzed trajectories with our RASPT2 ansatz, we simulated the transient XAS of neutral vinylbromide. The resulting time-resolved spectrum is plotted in Fig. 6 . First one has to note that the simulated spectrum matches theexperimental energy range quite well, keeping in mind that we didnot apply a shift to the excitation energies. The energetic position of the ground state bleach at 71.0–72.5 eV matches very well. The temporal modulation is missing, since we simply subtracted theXAS of the S 0ground state at time t¼0 fs as mentioned in Sec. III C . In the range of 64.0–65.0 eV, prominent features occur that are predicted by Koopman’s theorem. All of them are shiftingapproximately 1 eV toward lower energies within the first 20 fs.Thereafter, two sharp features, a stronger one at 64.3 eV and a sig- nificantly weaker one at 65 eV, appear. After another 10–20 fs, all signals remain constant in energy and the feature at low energybecomes more and more pronounced. Since nearly all trajectoriesare dissociated at this point in time, these two remaining constantsignals are the spin-allowed transitions of atomic bromine. Fromboth transitions, the lower lying Br ðP 3=2Þ! BrðD5=2Þis more dominant. Comparison to the experimental spectrum reveals thatthe signals from the neutral species alone cannot explain the experimentally observed pattern in this energy range. Thus, signal contributions from the ionic species are required in this region aswell as evidently also in the higher energy part above 65.5 eV. C. Dynamics of the cationic vinyl bromide Next, we examined the excited state dynamics of cationic vinyl bromide. As discussed in the previous work, 67the first three ionic states ( D1,D2,a n d D3) are accessible with the pump pulse used. Accordingly, we started 150 trajectories equally distributed in each of these states. For the analysis, 82 out of the 150 calculated trajectorieswere taken into account, using the same selection criteria as for theneutral case. The 82 trajectories break down to 11 that started in theD 1state, 26 in the D2state, and 45 in the D3state. The overall change in population for all 82 trajectories is shown in Fig. S17 in the supple-mentary material. 108All three states are bound states, with, however,decreasing dissociation energies, reflecting the increasing weakening of the C¼C double bond with ionization energy. Contrary to the neutral case, their dynamics differ depending on the initial state and will bediscussed individually in the following. 1. Starting from the D 1state The electronic character of the D1state is characterized by an ionization from the p2orbital (see Figs. 2 and S7). The complete popu- lation remains in the D1state for the entire simulation time. There is no interaction with either the other doublet or quartet states (see Fig. S11). None of the analyzed trajectories dissociate. This is in goodagreement with the result of a relaxed scan along the C–Br bond thatpredicts a barrier of about 1.5 eV for the dissociation (see Fig. S28 andthe computational details Sec. V B in the supplementary material1 108). We observe an oscillation of the C ¼C bond with a period of about T¼25 fs and of the C–Br bond with a period of T¼50 fs (see Fig. S12). The launching of both oscillations is reasonable, since theaffected p 2orbital has contributions from both the C ¼C double bond as well as one of the bromine 4 porbitals. 2. Starting from the D 2state For the D2state, the electron hole is generated in the n1lone-pair orbital of the bromine. Again, most of the population remains in theinitial state D 2, with only a small part, less than 5%, being transferred into the D1state. As for the D1state, there is no further interaction with the other electronic states (see Fig. S13) and none of the analyzedtrajectories dissociate. The barrier toward direct dissociation is about1 eV (see relaxed scan Fig. S28). Since ionization from orbital n 1 mainly affects the C–Br bond, a pronounced oscillation along this bond with a period of about T¼75 fs (see Fig. S14) is observed. The bond elongates from the initial 1.9 to roughly 2.08 A ˚.O n l yaw e a k oscillation, with a period of about T¼20 fs, that dampens over the simulation time, is present in the C ¼C bond. This bond is hardly affected and its bond length shortens only marginally to about 1.31 A ˚. In summary, the first two ionic states exclusively show bound state vibrational dynamics. 3. Starting from the D 3state For the D3s t a t e ,t h eh o l ei sc r e a t e di nt h eC ¼Cb o n d i n g p1 orbital (see Figs. 2 and S7), which has contributions from the C ¼C double bond and the bromine 4 porbital. The C ¼C double bond as well as the C–Br bond are now strongly weakened. Accordingly, therelaxed scan (see Fig. S28) predicts a small barrier of only 0.2 eV for adirect dissociation. The temporal evolution of the D 3population (see Fig. S15) shows a slow but steady decay due to population transfer totheD 2andD1state after the first 20 fs. There is also a slight intermedi- ate interaction with the D4state between 20 and 60 fs, but again, there is no further significant interaction with the remaining electronicstates. After 100 fs, more than 80% of the population is still in the D 3 and about 10% in each of the D2and D1states. In the barrier region, transitions to the D4state via conical intersections are possible (see Sec. V of the supplementary material108for the optimized structure). After the barrier, the D3andD4state switch character. For the new character, ther1orbital of the C–Br bond is only singly occupied leading to a repulsive potential correlating with the dissociation channel of the D1 FIG. 6. Simulated transient XAS of the neutral trajectories starting in the bright pp/C3 state. On the left and right, the spectrum for time t¼0 and t¼99 fs, respectively, is shown in blue overlaid with the experimental transient absorption spectogram inblack at delays d¼6.8 and d¼107:6 fs.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 8, 034104-7 VCAuthor(s) 2021orD2state. Judging from the small barrier compared to both other cat- ionic states, one would expect most of the trajectories to dissociate. And indeed, the dynamics starting in the D3state reveal a more com- plex situation regarding the temporal evolution of the C–Br bond (see Fig. 7 ). Three types of dynamics were recognized, the pure vibrational (green), the slow dissociative (yellow), and the fast dissociative (red). The classification was done with regard to the length of the C–Br bond at the end of the simulation. Trajectories with a bond length longer than 3.5 A ˚were classified as fast. The range 3.5–2.2 A ˚was classified as slow and trajectories with shorter bond length as vibrational. The final distribution regarding these three types of dynamics is listed in Table III . About half of the analyzed trajectories dissociate within the simulation time of 100 fs, with the other half showing purely vibra- tional dynamics. For all trajectories, the C ¼C bond shows oscillation with a period of about T¼25 fs. The purely vibrational trajectories indicate a slow oscillation with about T¼100 fs in the C–Br bond. All of them stay in the D3state, not being able to overcome the barrier. For the fast dissociating trajectories, at about 40 fs and a C–Br bond length of about 2.59 A ˚, we can see a clear separation from the rest of the trajectories. In the case of the slow dissociating trajectories, the sep- aration happens later at 60 fs at a bond length of about 2.37 A ˚. In general, the dissociation in the D3state is considerably slower compared to the neutral vinyl bromide, where we observed the onset of dissociation after 50 fs. For the cationic states, the bond cleaves het- erolytically, leaving a neutral bromine radical and a vinyl cation. For the dissociation channel of all three states, D1,D2,o rD3,t h eu n p a i r e d electron is located in one of the three now degenerate Br 3 porbitals. Again using the described ansatz in Sec. III C and the data from the 82 analyzed trajectories, we simulated the complete transient XAS for the cationic states, which are depicted in Fig. 8 .C o m p a r i n gi tt o the experimental spectrum in Fig. 3 , the overall resemblance in the measured energy range is strikingly good. To assign the different fea- tures of the spectrum, we additionally simulated the XAS only for a subset of the trajectories depending on their starting state, D1 (Fig. S21), D2(Fig. S22), and D3(Fig. S23). In the case of the D3,i ti sfurther separated according to the bound and dissociative dynamics (Figs. S24–S26). In the following, we discuss the significant features of the spec- trum in the context of the corresponding dynamical processes of vinylb r o m i d e .S t a r t i n gf r o mt h et o p ,w eo b s e r v et h es a m eg r o u n ds t a t e bleach as for the neutral states, but this time the intensity distribution is modulated by contributions from the D 1signal. The signals in the energy range 67–69 eV can be attributed to the D1state. Although it is difficult to see the peaks in the complete spectrum, Fig. S21 clearly shows the pronounced doublet signal at 66.7 and 67.7 eV, the formerwith a higher, about twice, intensity compared to the latter. This dou-blet arises from the already mentioned spin–orbit splitting of the 3 d 5=2 and 3 d3=2orbitals of Br. Both signals oscillate with a period of about T¼50 fs, which is in good agreement with the observed oscillation of the C–Br bond in the trajectories. In correspondence to the shortening of the C–Br bond during the oscillation, the signals shift to higher energies, reaching a maximum of about 67.7 and 68.7 eV in a halfperiod. This shift can be explained by the electronic character of the D 1state. It has a hole in the p2orbital, which has anti-bonding charac- ter with respect to the C–Br bond. Hence, shortening the bond desta-bilizes the p 2orbital and results in higher XUV excitation energies. The next features at 65.5 and 66.5 eV in Fig. 8 are the most intense ones of the spectrum and can be attributed mainly to the D2 state by comparison to the D2XAS (see Fig. S22). Here, the bromine- induced doublet structure shows most clearly and the energetically higher signal is less intense. The slight modulations with periods of FIG. 7. Temporal evolution of the C ¼C double bond and the C–Br bond for the 45 analyzed trajectories that started in the D3state of the cationic vinyl bromide. For the C–Br bond, the corresponding trajectories are color coded according to their observed behavior. Only vibrational trajectories are shown in green, slow dissociat-ing ones in yellow, and (fast) dissociating trajectories in red. The average of theC–Br bond length was only calculated for the subset of the trajectories.TABLE III. Observed dynamics of the 82 analyzed trajectories for cationic vinyl bro- mide. In the table heading, Dissociation is abbreviated as Diss. Initial state Vibrational Slow Diss. Fast Diss. D1 11 0 0 D2 26 0 0 D3 21 8 16 FIG. 8. Simulated transient XAS of the cationic trajectories starting in the D1,D2, and D3states. On the left and right, the spectrum for time t¼0 and t¼99 fs, respectively, is shown in blue overlaid with the experimental transient absorption spectrogram in black at delays d¼6.8 and d¼107:6 fs. Additional overlay plots at times 24, 48, and 72 fs as well as separated XAS of the different starting statesand different observed dynamics are depicted in the supplementary material,Sec. IV.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 8, 034104-8 VCAuthor(s) 2021about T¼80 and T¼2 5 f sc a nb ea s s i g n e dt ot h eC – B ra n dC ¼C bond vibrations in the D2trajectories. With the elongation of the C–Br bond the doublet shifts toward lower energies. The elongation stabil- izes the n1lone-pair orbital and thus the D2state, resulting in lower core excitation energies. This shift of about 0.3 eV is considerably smaller compared to the 1 eV of the D1signal. Summarizing, the sig- nals from both the D1andD2states show clear oscillations, but neither reproduce the characteristic splitting of the intense 65.5 eV peak at about 50 fs nor the weak band around 64 eV. These features must come from the D3state. The combined XAS for all D3trajectories is shown in Fig. S23. The weak signals between 66 and 70 eV correspond to the core excita-tion into the r /C3orbital of the C–Br bond. Again, the spectrum is dom- inated by the distinct doublet, now at 64.3 and 65.2 eV. In contrast to the previously discussed oscillating features, these signal traces become constant after about half an oscillation period at 40 fs and split up into four contributions at 64.0, 64.5, 65.0, and 65.5 eV with different inten- sities. They can be assigned to the various processes observed in the D3 trajectories. The XAS for the purely vibrational bound state dynamics (see Fig. S24) shows weak signals between 66 and 70 eV and the prominent doublet peak at 64.3 and 65.2 eV, both oscillating with the period of the C–Br bond. The phase shift originates from the different final orbi- tals that are excited to. Bond elongation stabilizes the r/C3orbital result- ing in a down-shift of the weak signal. Simultaneously, the singly occupied p1orbital is destabilized leading to an up-shift of the doublet signal. For the XAS of the fast dissociating trajectories (see Fig. S26), the same doublet splitting at 64.3 and 65.2 eV occurs. After the initial first half of the oscillation period, the signal changes drastically. Both peaks become constant at an energy of 64.0 and 65.0 eV, and a strong increase in intensity for the energetically higher signal at 65.0 eV is observed while the other one is fading. This change in intensity andshape reflects the transition from the “molecular” bromine to the atomic bromine. With increasing distance between the bromine and the vinyl cation, the Br character becomes progressively more atom- like. From the two spin-allowed transitions Br ðP 3=2Þ! BrðD5=2Þand BrðP1=2Þ! BrðD3=2Þ, dominantly the higher lying transition BrðP1=2Þ! BrðD3=2Þsurvives. Thus, in the cationic dissociation, spin–orbit excited Br ðP1=2Þatoms are mostly generated, which is in excellent agreement with the experiment. From the discussion of the XAS from the bound and the fast dis- sociating D3trajectories, one would expect to see a mixture of their distinct signals in the XAS of the slow dissociating ones. Indeed, in the first 50 fs, Fig. S25 only shows the oscillating signals of the bound tra- jectories. In the time between 50 and 80 fs, the signals from the dissoci- ating trajectories start to overlap converting into the constant signals of the atomic bromine at 64.0 and 65.0 eV. In summary, the prominent features between 64 and 66 eV of the total XAS ( Fig. 8 ) originate from overlapping signals of the cationic vinyl bromide. The signal is composed of spectral features of the non- dissociative dynamics in the D2and D3states and of the spin–orbit excited atomic Br generated in the dissociation of D3. This assignment relies on the fact that in vinyl bromide, the dynamics in the neutral states can be well distinguished from the dynamics in the cationic states, which can also be resolved in the XAS spectrum. In the accessi-ble neutral state, a pp /C3excitation is immediately followed by atransition from the pp/C3to the nr/C3configuration initiating barrierless dissociation. In contrast, for the cationic case, the hole is generated indifferent orbitals for the different states. The first two cationic statesare bound due to being well separated in energy from dissociation channels. However, for D 3a slow transfer of the hole from the pinto therorbital weakens the C–Br bond and leads to dissociation. V. CONCLUSION We introduced a joint experimental and theoretical approach to follow in real-time the electronic structure change in molecules via attosecond transient absorption spectroscopy. It is then applied to study the ultrafast dynamics of vinyl bromide, after strong-field excita-tion from a few-cycle pulse centered at 800 nm. The remarkable agree-ment between the experimental and the calculated ATAS traces provesthe high fidelity of the RASPT2 ansatz in combination with NAMD.Thus, it is possible to remove one of the two main bottlenecks histori-cally ascribed to the employed methodology: the difficulty in datainterpretation due to multiplet effects in probing inner valence ratherthan genuine core level states. Based on the NAMD trajectories, wecalculated the state-specific XAS spectra and their temporal evolution.This allows for detailed insights into the electronic-state-resolveddynamics. In particular, we were able to clearly assign the experimen- tally observed spectrum to the dynamics of the first three electronic states of the cation, D 1,D2,a n d D3. The first two states remain bound, allowing us to time-resolve the ensuing multidimensional vibrationaldynamics with high sensitivity. The second excited cationic state D 3, instead, presents richer dynamics. In addition to the pure vibrationalmotion, fast and slow dissociation channels also appear, leading toultrafast rupture of the halogen–carbon bond in 50% of the calculatedtrajectories. No significant ultrafast relaxation channel from D 2toD1 has been observed, which is different from the very recent results on strong-field ionized ethylene.22Each of the above-mentioned channels has a clear fingerprint in the ATAS spectrum, and they are unambigu-ously assigned thanks to the theoretical comparison. Extensions of thedescribed methodology are imaginable both on the experimental and theoretical side. The combination of attosecond absorption and charged-particle-based spectroscopies (angle-integrated, via time-of-flight, or even angle-resolved, via velocity-map-imaging spectroscopy)would allow for an even greater insight into the different relaxationand fragmentation channels, providing a multidimensional space ofcorrelated observables. In order to circumvent the issue of the non-resonant and non-perturbative nature, one could explicitly simulatethe interaction of the molecule with the strong NIR field. A suitableapproach would be the recently developed R-matrix ansatz 23,24or the B-Spline Restricted Correlation Space - Algebraic DiagrammaticConstruction approach. 25Overall, the platform here presented can be readily applied to single-photon-excitation studies and to the soft x- ray spectral region, where the presence of carbon, nitrogen, and oxy- gen edges allows for molecular ultrafast studies with unprecedentedtime resolution. AUTHORS’ CONTRIBUTIONS F.R. and M.R. contributed equally to this work. ACKNOWLEDGMENTS This work was supported by the National Science Foundation (Grant Nos. CHE-1951317 and CHE-1660417) (M.R., K.F.C., Y.K.,Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 034104 (2021); doi: 10.1063/4.0000102 8, 034104-9 VCAuthor(s) 2021and S.R.L.) and the U.S. Army Research Office (Grant no. W911NF-14-1-0383) (K.F.C., Y.K., H.T., D.M.N., and S.R.L.). Y.K. additionally acknowledges the financial support from the Funai Overseas Scholarship. F.R., T.S., and R.d.V.-R. acknowledgefunding from the DFG Normalverfahren VI 144/9-1. We would liketo thank Andrew Attar for fruitful discussions on the interpretationof ATAS signals and Michael Odelius for assistance with aspects ofthe calculations and some Molcas-specific settings. The authorsgratefully acknowledge the computational and data resourcesprovided by the Leibniz Supercomputing Centre ( www.lrz.de ). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001). 2J. Itatani, F. Qu /C19er/C19e, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88, 173903 (2002). 3M. Schultze, M. Fieß, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. 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5.0056230.pdf

Photoconductivity effect in SnTe quantum well Cite as: Appl. Phys. Lett. 119, 032104 (2021); doi: 10.1063/5.0056230 Submitted: 7 May 2021 .Accepted: 13 July 2021 . Published Online: 22 July 2021 G. R. F. Lopes,1S.de Castro,2,a) B.Kawata,3P. H. de O. Rappl,3 E.Abramof,3 and M. L. Peres1 AFFILIATIONS 1Instituto de F /C19ısica e Qu /C19ımica, Universidade Federal de Itajub /C19a, Itajub /C19a, CEP 37500-903 Minas Gerais, Brazil 2Universidade do Estado de Minas Gerais, Divin /C19opolis, CEP 35501-170 Minas Gerais, Brazil 3Laborat /C19orio Associado de Sensores e Materiais, Instituto Nacional de Pesquisas Espaciais, S ~ao Jos /C19e dos Campos, CEP 12201-970 S ~ao Paulo, Brazil a)Author to whom correspondence should be addressed: marcelos@unifei.edu.br ABSTRACT We investigated the photoconductivity effect observed in a p-type SnTe quantum well in the temperature range of 1.9–100 K. The negative photoconductivity effect is observed for temperatures below 4 K, and it is strongly dependent on the light wavelength. A systematic analysis of the photoconductivity indicates that the origin of the negative photoconductivity is not related to the topological surface states but ratherto the reduction of carrier mobility when the SnTe quantum well is illuminated with energies above 2 eV. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0056230 The phenomenon of photoconductivity provides an important tool for the investigation of carrier dynamics and its interactions with the band structure of a material. The observation of effects such as negative photoconductivity (NPC), in which the material undergoes a drop in the conductivity in the presence of light, and persistent photo- conductivity (PPC), in which the conductivity takes a longer time to return to its initial value after illumination is removed, constitute a set of tools that enables the quantitative determination of the mechanisms responsible for the dynamics of carriers, e.g., helping to locate defect states and to calculate trap ionization energy. These effects have been widely explored in different materials such as Cu 2Se photodetectors,1 Cd3As2nanowires,2Cs4PbBr 6single crystals,3Ga2O3photodetectors,4 and carbon nanotubes networks.5 Regarding the photoconductivity effect for IV–VI compounds, a more detailed analysis is found in the literature for lead-based com- pounds such as the pseudo-binaries Pb 1/C0xEuxTe and Pb 1/C0xSnxTe.6–8 Tavares et al.6showed that for the alloy Pb 1/C0xSnxTe, with x/C240.44, the NPC effect is present even at room temperature. This behavior is asso- ciated with defect states that alter the carrier dynamics via generation and recombination rates when the sample is under illumination. In this case, a competition between the generation and recombination mechanisms is observed as a function of temperature and a level of disorder of the system, introduced by the Sn atoms. On the other hand, both the NPC and PPC effects are observed in a p-type Pb1/C0xEuxTe film,7,8where these phenomena are also related to the presence of defect states in the bandgap, which alters the generation and recombination rates and becomes more evident at lower tempera- tures since trapping is more effective as temperature decreases.The narrow gap semiconductor SnTe is a binary compound that possesses promising physical properties with potential application for the development of optoelectronic devices, broadband photovoltaic detectors,9thermoelectric devices,10,11solar cells,12and photodetec- tors.13A few years ago, SnTe was identified as a topological crystalline insulator (TCI), where the metallic surface states are protected by crys- tal symmetries thanks to the rock salt structure of this compound.14 Due to the TCI phase, quantum coherent transport was observed inSnTe thin films, 15as well as other quantum effects such as Shubnikov–de Haas oscillations and weak-antilocalization.16,17It also presents a positive photoconductivity effect, investigated in a SnTe:Si film.13 In this paper, we present a detailed study of the electrical trans- port properties and the photoconductivity effect in a 30 nm-wide SnTe quantum well (QW). It was observed that the QW was sensitive to broadband light, ranging from infrared (IR) to ultraviolet (UV) light in a wide range of temperatures. Also, it was observed a transition from positive to negative photoconductivity as temperature is reduced below 4 K, when the sample is illuminated with energies above 2 eV. We show that a simple classical model is able to explain the transition from positive photoconductivity to negative photoconductivity consid- ering the variations of carrier concentration and carrier mobility when the sample is under illumination. We show that the reduction of car- rier mobility leads to the NPC effect. We also found that there is no indication of contribution from topological surface states to the electric transport and photoconductivity in the SnTe QW. The p-type SnTe quantum well was grown by molecular beam epitaxy (MBE), using a Riber 32P system, on (111) BaF 2 Appl. Phys. Lett. 119, 032104 (2021); doi: 10.1063/5.0056230 119, 032104-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplmonocrystal substrates. The sample structure is a 30 nm-wide SnTe layer embedded with a 1.5 lmP b 0.9Eu0.1Te buffer and a 300 nm Pb0.9Eu0.1Te cap layer. The Eu content x/C240.10 provides the quantum confinement since barriers are completely insulators for this Eu con- centration.18The electric contact preparation follows the van de Pauw geometry [see the inset in Fig. 1(a) ] using gold wires soldered with indium pellets. The indium diffuses through the heterostructure and crosses the layers (barriers and the well). The electric measurements were carried out using a physical property measurement system (PPMS) from Quantum Design composed of a He-cooled supercon- ducting system with the magnetic field up to 9 T and an operating temperature of 1.9–400 K. The optical excitation was provided by light-emitting diodes (LEDs) with wavelengths k¼398, 449, 568, 591, 634, and 940 nm and a constant excitation current of 7 mA. Figure 1(a) shows the metallic behavior of R(T) under dark and light (UV LED) conditions indicating that the transport occurs in the QW in both situations. The schematic illustration of the fabricated contacts and the LED position is an inset of Fig. 1(a) . The Ohmic contacts were tested before the photoconductivity measurements.Figure 1(b) shows the I–V curves of the sample measured at 300, 100, and 4.2 K, and the linear dependences indicate that contacts are Ohmic. Figure 1(c) represents the sample structure, indicating the sub- strate, the buffer, the QW, the cap layer, and the applied magnetic field in the (111) direction, which will be used to further discussion later in the text. Figure 2(a) presents the time dependence of normalized photo- conductivity r=r 0;where r0is the conductivity in the dark conditions, for temperatures ranging from 1.9 to 100 K, when the sample is illumi- nated by IR light. All the curves present the positive photoconductivity effect in the whole range of temperatures, i.e., conductivity increases under illumination and r=r0>1. The illumination is switched onand off in the positions indicated by the arrows in this figure. In addi- tion, a strong PPC effect is observed for lower temperatures. The maxi- mum amplitude of the photoconductivity, rM, increases as the temperature decreases down to 10 K, but starts to decrease below thistemperature, as observed in Fig. 2(c) . This behavior reveals the pres- ence of trap levels that are active in this temperature region and are also responsible for the persistent effect. 19Figure 2(b) shows the values ofr=r0as a function of time when the sample is illuminated by UV light. Similar to the case when the sample is illuminated with IR light, the maximum amplitude increases as the temperature decreases from 100 to 10 K. However, a transition from positive to NPC takes place for lower temperatures. This effect is more clear in Fig. 2(d) , where around 3.9 K r=r0becomes smaller than 1 (see arrow). The drastic variation of therMis observed and may also be related to trapping levels. Information about trap levels can be obtained from the analysis of decay curves of the photoconductivity once they follow the relation r¼r0exp/C0t s/C0/C1,w h e r e sis the recombination time. For this sample, the decay curves are well fitted for the whole temperature range if we consider a combination of two exponentials r¼r01expð/C0t s1Þ þr02expð/C0t s2Þ, suggesting the presence of two trap levels.20The fit- tings using this combination are presented in the inset of Figs. 3(a) and 3(b) at 75 K. From the fittings, two recombination times are extracted, one shorter ( s1Þ, that can be from shallower traps, and one longer ( s2), due to deeper traps. The activation energy Deof the trap level, i.e., the barrier energy that carriers have to overcome to recom- bine, can be obtained using the expression s¼s0expð/C0De kBTÞ.Figures 3(a) and3(b) show the natural logarithm plots of sas a function of 1=kBTwhen the sample is illuminated with IR and UV light, respec- tively. In Fig. 3(a) , two regions can be observed and the trap energies obtained are De1/C240:2460:02 ðÞ meV, at lower temperatures, and FIG. 1. (a) Electrical resistance as a function of temperature showing metallic behavior indicating that the transport occurs in the well, the inset shows th e schematic of the con- tacts of the sample. (b) I–V curves for Ohmic contacts of the SnTe QW at 300, 100, and 4.2 K. (c) Structure of the quantum well and the applied magnetic field along the (111) direction.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032104 (2021); doi: 10.1063/5.0056230 119, 032104-2 Published under an exclusive license by AIP PublishingDe2/C247:160:8 ðÞ meV at higher temperatures. The temperatures for which the thermal energy kBTcorresponds to the trapping levels De1 andDe2areT/C242.8 and 81.2 K, respectively, i.e., above these tempera- tures the trapping mechanism becomes less effective. For the case when the sample was illuminated by UV light, we found De3/C243266 ðÞ meV, see Fig. 3(b) . The temperature for which thermal energy corresponds to 32 meV is T/C24371 K. This temperature is much higher than the temperature for which the photoconductivityeffect was observed in the SnTe QW [see Fig. 2(b) ]w h e nl i g h ti s switched off, which explains well why the sample presents the high persistent effect after being illuminated by the UV LED. Figure 3(c) displays the time-dependent photoconductivity curves for UV and IRlight at 100 K. In fact, even at 100 K, one can observe that the sampleexhibits a strong persistent effect after being illuminated by UV lightwhen compared to the situation where the sample is illuminated with IR light. This indicates that De 3is located inside the band above 2 eV inLpoint for the SnTe band structure since this trap level is accessed only when the sample is illuminated by UV light with /C243.4 eV. This observation is further corroborated by the fact that when the sample was illuminated by IR light, De3is not observed. Figure 3(d) shows a schematic representation of the QW structure, where the wider gaps correspond to the barriers, and the narrow gap corresponds to SnTe. In the region of the QW, the valence and conduction bands areinverted (see inverted colors), representing the band inversion caused by the strong spin–orbit coupling in this compound. Under IR illumi- nation, carriers are photogenerated from the valence band to the con-duction band (process 1), which causes the positive photoconductivity. The photogenerated carrier remains in the conduction band and, after FIG. 2. Time dependence of the normalized photoconductivity in the SnTe QW in the temperature range of 1.9–100 K under IR radiation in (a) and UV irradiation in (b). The arrows indicate the moment of switching on and off illumination. (c) Maximum amplitude of the photoconductivity normalized due to IR, and (d) UV illum ination as a function of temperature.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032104 (2021); doi: 10.1063/5.0056230 119, 032104-3 Published under an exclusive license by AIP Publishinga given time, it recombines in the defect level with energy barrier De2 (processes 2 and 3). The trapping in this level causes the persistent effect and, after a certain time, the carrier recombines back in the valence band (process 4). Figure 3(e) represents the QW structure illu- minated by UV light. Under illumination, the electrons are photogen-erated from the valence band to the conduction band, where theyremain from a certain time (processes 1 and 2). After that, electrons decay to the defect level with energy barrier De 3(process 3) leading to t h ep e r s i s t e n te f f e c tw h e nt h es a m p l ei su n d e rU Vl i g h ti l l u m i n a t i o n .Later, electrons recombine back to the valence band (process 4). To quantitatively investigate the origin of the NPC effect observed under UV light, as shown in Fig. 2(b) , Hall measurements were performed on the SnTe QW structure in the range of 1.9–300 K,under UV illumination (on) and dark conditions (off). Figure 4(a) shows the carrier concentration as a function of temperature. In dark conditions, one observes that the carrier concentration increases as temperature decreases and reaches a maximum value around 100 Kand then drops slightly as temperature is lowered. For temperaturesbelow /C24100 K, carriers cease to be thermally promoted to the valence band and are trapped in the level instead, which causes the decrease in the hole concentration observed. In fact, 100 K corresponds to the thermal energy ( k BT) of 8.6 meV, which is close to the value of the trap level De2/C247:160:8 ðÞ meV obtained from the photoconductiv- ity results presented earlier. However, when the sample is under illu-mination by the UV LED (open squares), an increase in the hole concentration is observed below 100 K due to the photogenerated FIG. 3. Arrhenius plot for the SnTe QW. The relaxation time swas obtained from decay curves after switched off the (a) IR and (b) UV LED. The dashed lines represent a fit- ting using two exponentials. The insets show the photoconductivity decay at 75 K together with the fitting using two exponentials. (c) The photocurren t measurement as a func- tion of time in the presence of UV and IR light at 100 K. Pictorial representation of the quantum well illuminated by (d) IR and (e) UV lights. The colors sh ow the inversion of the conduction and valence bands in the well. Open and dark circles represent the holes and electrons, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032104 (2021); doi: 10.1063/5.0056230 119, 032104-4 Published under an exclusive license by AIP Publishingcarriers from traps and from the valence band to the conduction band. ForT>100 K, the photogenerated carriers are negligible due to the high recombination rate and do not contribute to the total carrier con- centration. Figure 4(b) presents the carrier mobility as a function of temperature under light and dark conditions. Without illumination, adecrease in temperature leads to an increase in carrier mobility asexpected. Under illumination, the behavior of the mobility curve for T>100 K is basically the same as the one in the dark condition. However, for temperatures below 100 K, the carrier mobility becomessmaller than the mobility values in dark conditions. Charge carriermobility depends on its effective mass and scattering time. Since thetheory has indicated that both of these quantities can be strongly dependent on the carrier density in semiconductors, we believe that the increase in the carrier concentration for T<100 K can lead to a decrease in the carrier mobility under light conditions. Such behaviorwas also observed in the PbTe QW at low temperatures. 21 The basic processes that govern the magnitude of the photocon- ductivity are the generation of electrons and holes through theabsorption of incident photons and their recombination. In the pres- ence of trap levels, a significant fraction of photogenerated carriersmay become immobilized. The photoexcitation may change both thecarrier density and the carrier mobility, thus the variation of the elec-trical conductivity under illumination can be described by a simpleclassic model as Dr¼ql 0Dpþp0þDp ðÞ qDl; (1) where qis the electronic charge, p0andl0are the carrier density and mobility in the dark, respectively, Dp¼plight/C0p0,a n d Dl¼llight /C0l0. This model predicts that the NPC effect can be observed in the case of Dp<0a n d / o r Dl<0. Figure 4(c) shows Drobtained from Fig. 2(b) (open circles) together with Drcalculated from Eq. (1)(solid line) using the Hall data. According to this figure, the experimental values are close to the onescalculated from Eq. (1)for temperatures above 4.2 K. Hence, the model predicts a transition from positive to negative photoconductivity at FIG. 4. (a) Hole concentration and (b) Hall mobility in dark conditions (open circles) and under illumination by a UV LED (open squares) of the SnTe QW measured as a func- tion of temperature. (c) and (d) show the comparison between experimental and theoretical values of Dras a function of temperature when the sample is illuminated by the UV LED and the IR LED, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032104 (2021); doi: 10.1063/5.0056230 119, 032104-5 Published under an exclusive license by AIP Publishing4.2 K by taking into account the carrier density and mobility variations. In this case, the model shows that the NPC is mainly caused but areduction of carrier mobility under illumination ( Dl<0). However, theDramplitude calculated by the theoretical model deviates from the experimental values for the temperature range of T¼3.5–1.9 K. The same deviation between theoretical and experimental values is observed when the sample is under IR illumination, see Fig. 4(d) . The origin of this discrepancy at lower temperatures indicates that Eq. (1)does not take all effects present in this temperature region. We have also investigated the magnetic field ( B) dependence of the photoconductivity to verify its influence on the NPC effectobserved in the SnTe quantum well under UV light. Figure 5(a) shows the longitudinal resistance ( R xx) as a function of the magnetic field B applied perpendicularly to the sample surface under dark and lightconditions at 3 K. At a low magnetic field, the dark resistance presentsa sharp increase in R xxthat characterizes the presence of the weakantilocalization (WAL) effect. We also verify the possible contribution from topological surface states to the NPC effect investigating the pres-ence of quantum oscillations. 17The presence of quantum oscillations can be verified if the second derivative of the longitudinal resistance with respect to the magnetic field is applied. The inset in Fig. 5(a) shows the curve of d2Rxx/dB2as a function of Bat 3 K, where there is no presence of beating patterns. The absence of oscillations in themagnetoresistance curves indicates that the topological surface statesdo not contribute to the NPC effect. Moreover, a transition from nega- tive to positive photoconductivity has also been observed at 0.7 T, and it can be better observed in Fig. 5(b) (indicated by the arrow). These results reveal that the NPC effect can be easily suppressed by the mag-netic field, which can be interesting from the viewpoint of applicationsin devices. This interesting behavior of suppression of negative photo- conductivity by the magnetic field may be correlated with the electron mobility, which influences the recombination rates of the carriers. FIG. 5. (a) Longitudinal resistance ( Rxx) as a function of the magnetic field applied perpendicularly to the p-type SnTe QW sample surface. The inset shows the curve of d2Rxx/ dB2as a function of Bat 3 K, where there is no presence of beating patterns. In this figure, it is clear the presence of WAL. (b) Time dependence of the normalized photocon- ductivity under different magnetic fields at 3 K. We can see that the magnetic field influences the amplitude of the NPC effect. (c) Evolution of the photoc onductivity with differ- ent wavelengths of incident light for sample A2 at 3.5 K. (d) Pictorial representation of the band structure of SnTe. The horizontal lines represent th e energy of the LEDs.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 032104 (2021); doi: 10.1063/5.0056230 119, 032104-6 Published under an exclusive license by AIP PublishingIt has been reported that in systems with magnetic or ionized impuri- ties, an increase in the mobility upon magnetic field can be associated with the decrease in the spin-flip scattering.22,23Since the recombina- tion rate should decrease with increasing mobility, it is expected areduction of the NPC effect with the increase in the magnetic field.Such behavior is clearly observed in Fig. 5(b) , where a transition from NPC to PPC occurs when the magnetic field increases from 0 to 2.0 T. Despite the change in carrier concentration being the origin of the photoconductivity effect observed in the SnTe QW, in the present case,our results indicate that a small change in mobility made the major con-tribution to the negative photoconductivity. Furthermore, since the NPC was observed only when the sample was illuminated by UV light, we believe that the band above 2 eV is also responsible for the effectobserved. To investigate both the influence of the band above 2 eV andthe mobility in the NPC mechanism, a detailed study of the photocon-ductivity effect was performed in another SnTe QW (sample A2) that presents slightly higher mobility, l 2,t h a ns a m p l eA 1( l1¼51:4a n d l2¼56:9c m2/V s). Figure 5(c) illustrate the photoconductivity behav- ior of sample A2 under LED excitations at 398, 449, 568, 591, 634, and950 nm at 3.5 K. First, we observe that the positive photoconductivitydecreases gradually by reducing the light wavelength from 950 nm (1.3 eV-IR light) to 634 nm (1.9 eV-red light). The positive photocon- ductivity can be understood by the light-induced electron–holes pairs,which increase the carrier concentration when the light energy increasesfrom IR to red. The increase in the carrier concentration leads to adecrease in the mobility of the carriers, which means changes in the intensity of the photoconductivity. From 591 (yellow LED-2.1 eV) to 398 nm (UV light-3.1 eV), the NPC effect was observed at the beginningof the illumination, and then the effect becomes positive. In order to illustrate this assumption, Fig. 5(d) presents a sche- matic representation of the band structure of the SnTe, where the energies’ positions according to the LED energies are indicated by thesolid lines. This diagram illustrates that when the LED energyapproaches to the energies around 2 eV, the NPC manifests itself,which is in accordance with Fig. 5(c) . We have investigated the origin of the NPC effect observed on thep-SnTe QW sample. We showed that a strong dependence of the photoconductivity on the incident radiation wavelength is observedand that it is possible to tune the photoconductivity from positive tonegative. A systematic analysis of the photoconductivity indicates that the origin of the NPC is not related to the topological surface states, but it is probably due to the dynamic between the carrier mobility andconcentration when the sample is under illumination. In addition, thestrong persistent effect observed can be explained due to the influenceof the trap level in the dynamics of the recombination rates. The authors thank CAPES for financial support. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1S. C. Singh, Y. Peng, J. Rutledge, and C. Guo, “Photothermal and Joule-heating- induced negative-photoconductivity-based ultraresponsive and near-zero-biased copper selenide photodetectors,” ACS Appl. Electron. Mater. 1(7), 1169–1178 (2019). 2K. Park, M. Jung, D. Kim, J. R. Bayogan, J. H. Lee, S. J. An, J. Seo, J. Seo, J. Ahn, and J. Park, Nano Lett. 20, 4939 (2020). 3J. Cha, J. H. Han, W. 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5.0058693.pdf

Novel modeling approach for fiber breakage during molding of long fiber-reinforced thermoplastics Cite as: Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 Submitted: 1 June 2021 .Accepted: 21 June 2021 . Published Online: 26 July 2021 Abrahan Bechara,1,a) Sebastian Goris,1 Angel Yanev,2Dave Brands,2 and Tim Osswald1 AFFILIATIONS 1Department of Mechanical Engineering, University of Wisconsin Madison, Madison, Wisconsin 53705, USA 2Global Application Technology, SABIC, AH Geleen 6160, The Netherlands Note: This paper is part of the special topic, Celebration of Robert Byron Bird (1924-2020). a)Author to whom correspondence should be addressed: bechara@wisc.edu ABSTRACT Long fiber-reinforced thermoplastics (LFTs) are an attractive design option for many engineering applications due to their excellent mechan- ical properties and processability. When processing these materials, the length of the fibers inevitably decreases, which ultimately affects themechanical performance of the finished part. Since none of the existing modeling techniques can accurately predict fiber damage of LFTs during injection molding, a new phenomenological approach for modeling fiber attrition is presented. First, multiple controlled studies employing a Couette rheometer are performed to determine correlations between processing conditions, material properties, and fiber lengthreduction. The results show shear stress and fiber concentration impact fiber damage. Based on these findings, a phenomenological model topredict breaking rate and unbreakable length of a fiber under giving conditions is developed. The model is based on the beam theory withdistributed hydrodynamic stresses acting on a fiber. Fiber–fiber interactions are accounted for and correlated with the fiber volume fraction via a fitting parameter. The model tracks both the number-average and weight-average fiber length during processing, which can in turn be used to extract the fiber length distribution. VC2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http:// creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/5.0058693 I. INTRODUCTION The use of injection-molded long-fiber thermoplastic composites (or LFTs) has been rapidly increasing in the past decade, mainly due to the market push for more fuel-efficient vehicles as well as electric vehicles. While the transportation industry constitutes 80% of LFTs’world usage, their use is also increasing in applications, such as durableconsumer appliances, electronics, and sporting goods. 1,2Since their advantage is the high aspect ratio of the fibers, a primary concern for manufacturers is preserving the fibers’ length throughout the moldingprocess. 3Initial fiber length in LFT pellets typically ranges from 10 to 15 mm. Fiber length measurements suggested a remaining average fiber length in the molded part in the 1–3 mm range.4–8 The impact of processing variables on fiber attrition at different stages of the molding process has been the focus of numerous studies.Bailey and Kraft found that most damage occurs during plastication,as did Lafranche et al. and others. 4–7Most notably, they found higher fiber lengths in the core region than the skin region in molded parts,also observed in other studies, which suggests that the characteristic flow regime during mold filling causes uneven fiber breakage.5,9These findings go hand in hand with the recent findings on the inhomoge- neous fiber density distribution in injection molding of LFT.10 Rohde et al. performed a full factorial DOE varying processing parameters, such as injection speed, screw back pressure, holding pres- sure, and screw speed, where their results clearly show a significant impact of back pressure on fiber length.7Most processing variables had no significant statistical impact on fiber length, while an increase in backpressure from 50 to 80 bar reduced the fiber length in the moldcavity by approximately 30%. Nevertheless, the authors state that it is difficult to isolate the mechanisms that cause fiber attrition due to the complex phase change and shear history present in the injection mold- ing process. Inoue et al. found that the screw design in the compression zone h a sas u b s t a n t i a li m p a c to nfi b e rb r e a k a g e . 11They observed that an optimized screw design (Dulmadge12with a variable pitch) could Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-1 VCAuthor(s) 2021Physics of Fluids ARTICLE scitation.org/journal/phfreduce the fiber breakage compared to a standard screw. Von Turkovich et al. conducted compounding experiments with short- fiber thermoplastic composites (or SFTs) and found that most of the fiber damage occurred in the compression zone.13Based on their results, they concluded that fiber concentration and initial fiber disper- sion had no visible impact on the fiber length. More fundamental studies on fiber motion were conducted by Forgacs and Mason, employing a Couette device and subjecting single fibers to simple shear flow.14,15They used Burger’s formulation along with Euler’s beam theory to derive an equation to estimate the critical product _cgat which a fiber would buckle under compressive stresses.16 By increasing either the aspect ratio of the fibers or the product _cg, Forgacs and Mason identified various orbits of rotation; Fig. 1 summa- rizes their findings. Using a similar device, Salinas and Pittman performed experi- ments with reinforcing fibrous materials, including glass fibers with aspect ratios ranging from 280 to 680.17They found that fibers frac- tured in the snake orbit regime when the minimum radii of curvature were reached, and that only very stiff or brittle materials (carbon fiber, asbestos) were predicted to break in the springy regime. They derivedan empirical expression for the critical product _cgthat would cause sufficient deformation to cause rupture. These early studies on fiber motion and failure are the basis for most modeling approaches to the fiber attrition phenomena. Shon et al. evaluated various combinations of mixing elements and their impact on fiber length reduction as a function of axial dis-tance in a counter-rotating twin screw extruder. 18They introduced a kinetic model in which an average fiber length, L;decreases exponen- tially toward a residual, or unbreakable length, L1, as follows: dL dt¼kfL/C0L1 ðÞ ; (1) where kfis a breaking rate coefficient. They did not attempt to derive expressions for the parameters ( L1;kf) in their model but approached the problem by simply fitting the curve to the experimen- tal data. Bumm et al. later extended the work of Shon et al. and derived expressions for both parameters based on Euler’s buckling theory.19Durin et al. and Phelps et al. proposed similar models for fiber attrition during compounding and injection molding, respectively.20,21 Rather than tracking the average fiber length changes over time, theywrote an equation of conservation for the complete fiber length distri-bution (FLD) based on a breakage probability. Following Forgacs andMasons’ analysis, fiber buckling is determined to be the mechanism bywhich fibers break. A dimensionless buckling number, Bi,d e fi n e st h e critical conditions under which a fiber would have the highest possibil- ity of buckling and eventually failing. This number is a function of theproduct _cg, the fiber’s stiffness and aspect ratio, given by Bi¼ 4fgml4 i p3Efd4 f; (2) where gmis the matrix viscosity, Efis the fiber’s Young’s modulus, andlianddfare the fiber’s length and diameter. The variable fis the fiber’s dimensionless drag coefficient and is used as a fitting parameter.These models introduce a normal or Weibull probability distributionthat determines the location along the fiber’s axis where failure is likelyto occur. Kang et al. , proposed a model similar to Durin’s. 22In their approach, fiber buckling leads to damage, which is based on the purebending theory of simply supported beams. They compared the pre-diction capability to fiber breakage experiments performed with glassfiber-reinforced polypropylene (PPGF) in a parallel-plate rheometer.The experiments were purposely conducted with low fiber concentra-tion (10 wt. %) to minimize the effect of fiber–fiber interactions onfiber damage. Malatyali et al. also employed a similar approach to pre- dict breakage of carbon fibers along a twin screw extruder. 23For model verification, 35 wt. % carbon fiber-reinforced polypropylene was used.The common denominator between these models is that fiber bucklingis presumed to be the cause or initiator of failure. Also, none of thesemodels consider fiber–fiber interactions to have influence on the fiberbreakage phenomena. The model of Phelps et al. is currently the stan- dard method for predicting fiber damage during injection molding incommercial software. The model of Durin et al has also been imple- mented in commercial software for twin-screw extrusion. Chen et al. studied the breakage on a capillary rheometer and proposed a breakage model based on the fibers’ semi-flexible orienta-tion. 24Following Salinas and Pittman’s conclusions, they argue fibers break due to large deformations when the minimum radius ofcurvature is reached. Their model indirectly accounts for fiber–fiberinteractions by using the isotropic rotary diffusion parameter C iin the orientation model.25Moritzer et al. conducted fiber breakage experiments under simple shear employing a Couette device andpre-compounded polypropylene with short fibers. 26Their results corroborate Shon et al.’ s proposed kinetics and showed that fiber con- centration influenced both the breakage rate and the residual lengthL 1. After performing a dimensional analysis, they proposed a phe- nomenological model that accounts for hydrodynamic stresses, fiberconcentration, and fiber properties. In recent years, direct particle simulation approaches have been used to study in detail the fiber breakage mechanisms. Sasayama et al. simulated fiber breakage under simple shear flow considering fiber–fiber interaction. 27Their results show that the fiber length decreases with the product _cg. They present an interesting depiction of how fiber–fiber interactions can lead to breakage. Chang et al. built a failure FIG. 1. Types of fiber motion in simple shear flow as flexibility increases from (a) to (e): (a) springy rotation, (b) and (c) snake orbit, and (d) and (e) coiled rotation(adapted from Ref. 17).Physics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-2 VCAuthor(s) 2021criterion based on a critical radius distribution obtained via loop test measurements, similar to Salinas and Pittman.28The introduction of this failure distribution in their direct particle simulation improved their fiber length prediction compared to Couette experiments. These approaches are somewhat limited due to their computational cost, so it is unlikely they will be used to predict fiber length in real compo- nents. However, they have great potential as numerical rheometers since the coupled effects of fiber orientation, fiber length, and fiber concentration can be studied. The main objective of this study is to develop a new model to predict fiber length during injection molding of long fiber-reinforced thermoplastics. First, a set of fiber damage experiments were con- ducted under highly controlled conditions employing a Couette rhe- ometer. Important process and material variables are studied, and t h e i ri m p a c to nfi b e rd a m a g ei sm e a s u r e d .B a s e do nt h ee x p e r i m e n t a l observations, this work derives a constitutive equation for fiber break- age kinetics, and expressions for the model parameters based on the hydrodynamic stress and mechanical failure are deduced. II. MATERIAL AND METHODS A. Material The material used in this work is a commercially available long glass fiber-reinforced polypropylene. The material is supplied as coated fiber pellets with a length of 15 mm, nominal glass fiber content of 20%, 30%, and 40% by weight (PPGF20, PPGF30, and PPGF40). Due to their fiber length, these materials are widely used for automotive panels and multi-wall structured panels in the construction industry. Table I lists the pertinent material properties. B. Experimental Investigating the underlying physics of fiber breakage experimen- tally in an injection molding process or during compounding in a twin-screw extruder is challenging because of the complex and chang- ing flow conditions that the fibers are exposed to in the various stages of the process. In this work, a Couette rheometer was used to study fiber breakage under simple shear and controlled processing condi- tions.32T h eC o u e t t er h e o m e t e rd i m e n s i o n sw e r es e l e c t e dt ob ec h a r a c - teristic of the conditions during plastication in injection molding, with the inner cylinder diameter being 35 mm, the annular gap being 5 mm, and the length of the annular volume being 80 mm ( Fig. 2 ). The temperature is controlled through insulated heater bands surrounding the outer cylinder and a thermocouple measuring the melt tempera- ture. The inner cylinder is driven by a Plasti CorderTMtorquerheometer (C.W. Brabender Instruments Inc., Hackensack, NJ), con- trolling the rotational speed. Based on the fiber breakup model suggested by Shon et al.,18as e to f studies were conducted with the Couette rheometer to understand the kinetics of fiber length degradation and to isolate the effects that process- ing conditions have on the steady-state fiber length ( L1)a n dt h eb r e a k - a g er a t ec o e f fi c i e n t( kf). After each experiment, the material was removed from the Couette, and the fiber length was measured employing a tech-nique developed at the Polymer Engineering Center at the University of Wisconsin-Madison, which is based on the method developed by Kunc et al. to determine the fiber length distribution in LFT materials. 33,34 1. Length decay over time The first study explored the fiber length degradation over time for different processing speeds while the fiber content and melt tem- perature were kept constant ( Table II ). The material was sheared for increasing intervals of time, starting with 20 s until 300 s. This set of experiments allow the characterization of fiber length degradation as afunction of residence time. Hence, the overall kinetic can be observed and general conclusions can be drawn. 2. Steady-state length The second set of experiments aimed to study the different impact variables have on the residual fiber length, that is, the length at which no additional amount of shearing time will cause additional fiber breakage, or as described by Shon et al., the fiber length at which there is no more buckling. For each variant in this study, the residence time and processing speed were selected to ensure L1was reached based on the results of the first experimental plan (300 s). This study TABLE I. Typical long fiber PPGF material properties. Material property Value Nominal fiber length Pellet length Fiber diameter ( lm) 14–2429 Density of fibers (g/cm3) 2.5630 Density of polypropylene (PP) (g/cm3) 0.930 Modulus of fibers (GPa) 74–8031 Ultimate strength of fibers (MPa) 2000–250031 FIG. 2. Illustration of the Couette rheometer setup for the study of fiber breakage. TABLE II. Experimental plan 1: impact of residence time and shear rate on length decay. Variable Levels Fiber content (wt. %) 30 Melt temperature (/C14C) 250 Rotational speed (rpm) 50, 100Residence time (s) 20–300Physics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-3 VCAuthor(s) 2021was designed as a full factorial DOE with three factors and three levels, as shown in Table III . The melt temperature was varied between 220/C14C and 280/C14C, representing the limits of the processing tempera- ture range suggested by the material supplier. Hence, the obtained measurements represent the most severe impact that can be expected from this factor. 3. Attrition rate This third set of experiments aim to identify the influence proc- essing conditions have on the initial breakage rate. Based on the expo- nential decay kinetics observed in previous studies, the most significant fiber length change occurs early in the shearing cycle. Hence, fiber length measurements at the start of shearing would pro- vide the best insights into the dynamic behavior. The goal of modeling fiber attrition in this work is aimed for a homogeneous suspension. Consequently, the fact that the fibers and matrix are not mixed in the initial pellet introduces additional variabil- ity into the study. To illustrate this fact, Fig. 3 shows a sequence of x- ray micro-computed tomography ( lCT) images of PPGF40 pellets before shearing and after 5 s, 12 s, and 20 s of shearing. It is evidentthat perhaps a heterogeneous mixture of fiber bundles and polymer matrix is present before 12 s rather than a suspension as observed at20 s. To address the issue of initial heterogeneity, a pre-dispersion step was introduced to disperse the bundle of fibers without causing exces-sive damage. This step involved shearing the polymer melt for 2 s with a shear rate of 50 s /C01. After pre-dispersion, the sample was sheared for two additional seconds at higher shear rates as part of the actual exper-iment. The experiments were performed using a high acceleration rate Couette rheometer at SABIC Technology Center in Geleen, The Netherlands. With this equipment, high deformation rates could beimposed on the polymer melt for short periods. The temperature remained constant, while fiber content and processing speed were var- ied, as shown in Table IV . III. RESULTS The results are presented in three sections corresponding to the three experimental studies. When evaluating the results, fiber length is presented in terms of the weight-average L W.T h en u m b e r - a v e r a g e fiber length ( LN) is also needed to reconstruct the FLD; however, as both variables show the same trends with respect to the processing parameters, the results are presented in terms of LWas it is more rep- resentative of the length variable in LFTs. A. Average length decay over time Figure 4 shows the fiber length reduction as a function of resi- dence time for PPGF30 with a melt temperature of 250/C14C at differentTABLE III. Experimental plan 2: impact of process variables on equilibrium length. Variable Levels Fiber content (wt. %) 20, 30, 40 Melt temperature (/C14C) 220, 250, 280 Rotational speed (rpm) 50, 100, 150Residence time (s) 300 FIG. 3. lCT slices of fiber dispersion for PPGF40 exposed to a simple shear flow at 50 s21for different residence times.TABLE IV. Experimental plan 3: impact of shear rate and fiber concentration on attri- tion rate. Variable Levels Fiber content (wt. %) 20, 30, 40 Melt temperature (/C14C) 250 Shear rate (s–1) 300, 500, 700 Residence time (s) 2 FIG. 4. Fiber length reduction over time for PPGF30 at 250/C14C.Physics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-4 VCAuthor(s) 2021shearing speeds. For these processing conditions, LWdecreased from 15 mm down to 1.6 and 0.75 mm with Couette rotational speeds of 50 and 100 rpm, respectively. Overall, the results confirm the expected exponential decay with a severe length reduction of the fibers occur- ring within the first 50 s of processing. As expected, an increment in processing speed increases the fibers break rate and decreases the residual/equilibrium length.18,26 B. Steady-state length The investigation of the processing conditions and fiber concen- tration on the unbreakable fiber length, L1,a l l o w sd i r e c ta n a l y s i so f the mechanisms that drive fiber damage because it excludes the tran- sient attrition at lower residence times. Figure 5 presents the results of the DOE showing the unbreakable length as the weight-average fiber length. An analysis of variance was applied to the measurements to determine differences in means between the factors and the level of the DOE factors. Figure 6 shows the main effect plots with respect to LW1. All three factors influence LW1and have a statistically signifi- cant impact on the process-induced fiber breakage.The melt temperature affects the suspension viscosity and, conse- quently, the stresses that the fibers are exposed to during processing. This dependency is evident in the obtained results, as the increase in temperature results in longer equilibrium length across all other fac- tors. Additionally, processing speed also had the expected effect on the equilibrium length as the rate of deformation increases the hydrody- namic stress experienced by the fibers, forcing the equilibrium length to go down. In all trials, an increased fiber concentration resulted in reduced LW1. An increment in fiber content implies an increase in one of the breakage mechanisms, namely, fiber–fiber interactions. This result is essential since it indicates that fiber concentration is an essen- tial variable in the kinetics of fiber attrition, and the controlled condi- tions of the experimental setup allow for a clean incorporation into the modeling. C. Attrition rate Figure 7 s h o w st h ed e c r e a s ei nl e n g t hf o ras h o r tp e r i o df o r PPGF20, PPGF30, and PPGF40. Here, the pre-dispersion step’s impact is evident; the length was reduced from its initial value of 15 to 7 mm approximately. However, it is still constant for each FIG. 5. Outcome of the DOE for the steady -state length showing the obtained weight-average fiber length, LW1. FIG. 6. Result of the statistical analysis showing the main effect plots on LW1.Physics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-5 VCAuthor(s) 2021fiber concentration which is necessary for the subsequent analysis. As observed in the first study, the attrition rate increases with the rate of deformation, and the fiber length after shearing the sample at 700 s/C01is nearly half of the length after shearing at 300 s/C01.A l t h o u g h the slope depends on the initial length and the initial length changes between fiber concentrations, it can be observed that the lines are nearly parallel, which suggests that the fiber content might not have a strong impact on the fibers’ initial break rate. Plotting the absolute value of the average initial slope as a func- tion of the shear rate shows a linear proportionality between the varia- bles, especially when including the data point at the origin; since if the fibers are subjected to no shear, there will be no damage, and the slope will be zero ( Fig. 8 ). IV. MODEL DEVELOPMENT Based on the analysis of the experimental results and reviewed previews work, the main arguments for the development of a new modeling approach for stress-induced fiber damage are presented.I. Von Turkovich et al. conducted studies on the compound- ing of SFTs employing a single screw extruder.13One main conclusion was that fiber content does not have a significanteffect on fiber attrition. Partly based on their conclusion,Phelps et al. neglected fiber interactions as a source of dam- age and defined buckling as the sole mechanism for trigger- ing failure. 20,35However, high stresses and complex flow conditions present during the extrusion process make it hard to establish direct correlations between fiber length and individual factors confidently. II. While the hydrodynamic stresses will cause buckling, which might lead to failure in short fibers; on average, much larger deformations are needed to cause breakage of the E-glass fibers commonly used in LFTs (D /C2520lm). Salinas and Pittman show that fibers can reach large deformations, farpast the point of buckling before finally breaking. 17 Additionally, critical radii measurements via loop test for E-glass fibers showed that very small curvatures could bereached before a fracture occurs. 28 III. The experiments under controlled processing conditions employing the Couette rheometer showed a significant cor- relation between fiber volume fraction and the steady-statelength. 26While Moritzer concluded that fiber content impacts both the attrition rate and the steady-state length, the experimental result of this work suggests the fiber vol- ume fraction has a weak impact on the attrition rate. We postulate the average fiber length reduction over time follows an exponential decay toward an equilibrium value as suggested byShon et al. 18A state equation can then be written for each length aver- ageLNand LW. This will allow for the reconstruction of the fiber length distribution at any point in time, dLN dt¼/C0kN;fLN/C0LN1 ðÞ ; (3) dLW dt¼/C0kW;fLW/C0LW1 ðÞ : (4) We now move on to deriving expressions for both parameters of the state equation L1andkf. FIG. 7. Outcome of experimental plan 3 showing the effect of shear rate on fiber attrition rate. FIG. 8. Average initial slope as a function of shear rate. Dashed line shows linear fit with R2¼0.998.Physics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-6 VCAuthor(s) 2021A. Equilibrium length Describing the fundamental attrition mechanism as a single fiber undergoes bending deformation caused by the hydrodynamic drag ofthe moving melt, the fiber could be assumed as a cantilever beam under a distributed load. This fiber held by perhaps a group of neigh- boring fibers or the walls of the barrel or screw can be at the brink ofrupture at the length, L 1, due to the distributed load, as depicted in Fig. 9 . The hydrodynamic load exerted by the melt flow on the cylindri- cal body is based on Stokes law by approximating the cantilever length as a chain of spheres with equal diameter.36The load w per unit length is calculated as w¼3pgmUo; (5) where gmis the matrix viscosity and Uois the uniform speed of the melt. This leads to the maximum hydrodynamic stress at the base of rmax¼48gmUoL2 d3 f; (6) with Landdfas the cantilever length and fiber diameter, respectively. When the stress reaches the fiber’s ultimate strength, failure occurs.This leads to the following expression for the critical or unbreakablelength L 1, L1¼rutd3 f 48gmUo !1=2 : (7) The characteristic shear rate can be defined as _c¼U0=h,w h e r e his introduced to describe a characteristic distance, such as channel depth,runner size, or cavity thickness, L 1¼rutd3 f 48gm_ch !1=2 : (8) Assuming that his proportional to df,h/df,s ot h a tL1/rutd2 f 48gm_c !1=2 : (9) The proportionality can be resolved by introducing a dimensionless constant, k, which gives L1¼krud2 f gm_c !1=2 : (10) The coefficient kis a material-dependent property and a measure of fiber interactions that cause fiber attrition during processing. The parameter is assumed to capture the effects of fiber concentration (fiber–fiber interactions) and fiber–wall interactions. Figure 10 shows the interaction coefficient kas a function of fiber content as obtained from the measured LW1values of the Couette rheometer experiments. B. Breakage rate The parameter kfcan also be calculated from the Couette rheom- eter experiments. As seen in Fig. 11 , an analysis of variance was applied to kfto determine its variation with respect to fiber concentra- tion and shear rate. While the fiber concentration does not have a sta- tistically significant impact on the breakage rate coefficient, the shear rate does. In the current modeling approach, kfrepresents how often fibers will reach the critical conditions described in the formulation of L1. Similar to the fiber–fiber interaction in the Folgar–Tucker model,37it can be argued that this frequency is proportional to the amount of fiber motion, kf/_c, caused by the flow as described by Forgacs and Salinas.17The linear correlation between _candkfshown in Fig. 11 leads to a straightforward expression for the breakage rate coefficient kf¼n_c; (11) where nis a scale factor for the rate of deformation. In the work by Wolf and Gupta et al., fiber length was measured along the melting zone of a single screw extruder.38,39Both researchers observed that a large population of very short fibers originated from FIG. 9. Illustration of a fiber experiencing distributed load due to drag force. FIG. 10. Average interaction coefficient kas function of fiber concentration.Physics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-7 VCAuthor(s) 2021the damage occurring at the interface between the bed of solids and the melt pool. In contrast, moderate fiber damage was observed insidethe melt pool where fibers were already dispersed and fully surrounded by the matrix. This points to the presence of different damage mecha- nisms during plastication. When full bundles are present, fiber damage happens mainly at the ends of these bundles, when the drag flow shears off the tips of the fibers [ Fig. 12(a) ]; but when fibers are fully surrounded by matrix, drag forces cause deformation, which leads to damage [ Fig. 12(b) ]. The attrition rate coefficient, k f, is a measurement of how regu- larly breakage occurs. Therefore, in the presence of two different dam- age mechanisms, the attrition rate coefficient kfwill undergo a transition as the flow condition evolves from a heterogeneous mix of bundles and matrix, to a fully dispersed suspension. To evaluate the effect fiber dispersion can have on the breakage rate, the resultsobtained from the Couette experiments are used. First, the value for n Wcan be recovered from the attrition rate experiment, where the pre-dispersion step ensures damage occurred predominantly due to fiber motion and deformation as illustrated in Fig. 12(b) .T h i sr e s u l t s in a value of nW¼4.66/C210/C04. Similarly, by fitting Eq. (1)to the experimental data presented in Fig. 4 ,t h ev a l u ef o r kf;WandnWcan be calculated. In this case, the shearing process started when the fiberbundles were undispersed, mainly leading to the type of damageillustrated in Fig. 12(a) . This results in a value of n W¼1.21/C210/C03. This simple comparison suggests that fiber breakage occurs muchmore often when the fibers are not well dispersed. C. Constitutive model The proposed model uses two fitting parameters: n,w h i c hs c a l e s _cin the breakage rate formulation and the interaction coefficient k, which encapsulates the effects of fiber interactions and is a function of the fiber concentration. An independent set of fitting parameters must be used for the constitutive equations in terms of L NandLW. The defi- nition of both parameters in terms of processing conditions and mate- rial properties allows for the constitutive equations to be written in terms of the total derivative and implemented in a cavity filling simula- tion where the flow field information is used to calculate the model c o n s t a n t st op r e d i c tt h efi b e rl e n g t h , dLN dtþu/C1rðÞ LN¼/C0kf;NLN/C0LN;1 ðÞ ; (12) dLW dtþu/C1rðÞ LW¼/C0kf;WLW/C0LW;1 ðÞ : (13) To test the performance of the proposed model, its prediction can be compared to fiber length measurements from the Couette studies. Since the fiber length measurements from various Couette studieswere used to find the fitting parameters kð/Þandn, an independent set Couette experiments, similar to those presented in Fig. 4 will be used as reference for comparison. However, since the initial sample is composed of undispersed bundles, the parameters n NandnWused in this validation correspond to the values determined from the experi-ments where the fibers were initially undispersed at the start of the shearing cycle. The processing conditions for the reference Couette experiments are listed in Table V . These processing conditions are rather extreme since the temper- ature is near the lowest value recommended by the material supplier and the rotational speed was set to the higher limit of the Brabender Plasti Corder TMtorque rheometer. This places the processing condi- tions right at the bound of the design space from the fiber breakage FIG. 11. Result of the statistical analysis showing main effect plots on kf: FIG. 12. Illustration of fiber breakage mechanisms under different dispersion stages: (a) undispersed fiber bundles and (b) fully dispersed fibers.Physics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-8 VCAuthor(s) 2021DOE. When modeling the flow in the Couette rheometer, the velocity profile is constant, and there are no changes in the z and hdirections; therefore, the convective terms in Eqs. (12)and(13)are not included and the length decay follows Eqs. (3)and(4). The comparison between experimental and predicted fiber length is presented in Fig. 13 . As expected, the equilibrium length is well cap- tured by the model for both length averages LNandLW.I ti sd i f fi c u l t to establish comparisons in the dynamic portion of the length decay since there are a few experimental points. However, it can be observedthe rate of attrition shows a good agreement for L N, but it seems to be overpredicted for LW.N o r m a l l y , LNmeasurements have lower stan- dard deviation than LW, and this is reflected in the estimation of the fitting parameter nW. Nevertheless, the model captures the overall behavior well. Using LNandLWas two moments of the fiber length distribu- tioin function, a predicted fiber length distribution is computed usinga lognormal probability density function (PDF) given by fxðÞ¼1 xrffiffiffiffiffi 2pp explnx/C0l ðÞ2 2r2/C18/C19 : (14) The resulting PDF is presented in Fig. 14 .T h ew o r kb yN g u y e n et al. shows that this PDF can closely reproduce the FLD and yield elastic constants very close to those obtained from experimental measure-ments. 40However, the authors noted that when using this probability distribution, the quantity of mid-range fiber lengths is oftenoverpredicted. This can also be observed in our results, as the continu- ous distribution diverts form the experimental data near the 2 -mm mark. This discrepancy has little to no effect in the predicted elastic constants; however, it can have more impact in the prediction of theultimate strength and impact strength. V. CONCLUSIONS Fiber breakage under simple shear experiments was conducted by employing a Couette rheometer and investigating the influence of fiber concentration, rate of deformation, and viscosity in breakagekinetics. Based on the experimental observations, a continuum approach is developed to predict the fiber length averages L NandLW. A model for the equilibrium length ( L1) is deduced, using the concept that fibers break by bending under hydrodynamically induced forces and fiber–fiber interactions. The breakage rate coefficient ( kf)i sa s s o - ciated with the fiber motion and, therefore, modeled proportionally to the rate of deformation. Finally, both length averages are used torecover the fiber length distribution employing a Weibull or lognormal PDF. Some of the obtained experimental results contradict previous assertions on fiber–fiber interactions’ role in fiber breakage phenom- ena. Bailey and Kraft observed an increase in fiber length with a 20% wt increment in fiber concentration in moldings with PA66 and PP, while von Turkovich et al. concluded the fiber concentration had no impact on fiber damage. Using a Couette device to impose simple shear on the suspension, this work looks to remove the complexities present during molding and extrusion processes, allowing us to isolateindividual factors’ role on fiber breakage. However, it is important to identify the different breakage mechanisms present as the fibers dis- perse into a homogeneous suspension. There are similarities between the model presented in this work and the model by Phelps et al. In their work, they derive an unbreak- able length, L 1(orLub), L1¼p3Efd4 f 4fgm_c"#1=4 : (15)TABLE V. Process conditions of Couette experiment for model validation. Parameter Value Melt temperature (/C14C) 222 Residence time (s) 30–300Fiber concentration (wt. %) 30Speed (rpm) 160 FIG. 13. Comparison between fiber length decay obtained with the Couette rheom- eter and fiber predicted with the proposed model. FIG. 14. Recovered fiber length distribution employing lognormal PDF vs experi- mental fiber length distribution.Physics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-9 VCAuthor(s) 2021This expression represents an equilibrium between internal resis- tance of the fiber ( Efdf) and external stresses ( gm_c), as does the expression of Eq. (10). However, the model presented in this work assigns more weight to the product gm_con account of the exponent 1=2vs1=4in their model. In their model, the overall rate of reduction of fiber length scales with CB_c, which is equivalent to the expression kf¼n_cused in the model presented in this work. Furthermore, the values obtained for nafter fitting kfto the Couette results (1/C210/C03–4/C210/C04) fall within the range suggested for CBin previous work (2 /C210–2–2/C210/C04).20,35 Reducing the number of fitting parameters introduced when developing a model is beneficial since this makes the approach more robust and potentially reduces the number of experiments needed to determine such parameters.41Modeling LNandLWindependently for this case, each constitutive equation has its set of two fitting parame- ters that are determined from the experimental data. Additionally, a single length measurement provides data for both sets of modelparameters. The proposed model in this work can be implemented into a flow solver for either injection molding or extrusion compounding. Implementation in a mold filling simulation using COMSOLMultiphysics, experimental validation and comparison with other modeling approaches will be the subject of a following publication. ACKNOWLEDGMENTS The authors wish to thank the National Science Foundation for financially supporting this work (Award No. 1633967). Theauthors also thank SABIC Global Technologies B.V. for their financial support, technical input, and for providing the materials used in this work. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. The data are not publicly available due to company restrictions. REFERENCES 1H .N i n g ,N .L u ,A .A .H a s s e n ,K .C h a w l a ,M .S e l i m ,a n dS .P i l l a y ,“ Ar e v i e wo f Long fibre thermoplastic (LFT) composites,” Int. Mater. Rev. 65, 164–188 (2020). 2J. Markarian, “Long fibre reinforced thermoplastics continue growth in automotive,” Plast. Addit. Compd. 9, 20–24 (2007). 3U. Gandhi, S. Goris, T. A. Osswald, and Y. Song, Understanding Discontinuous Fiber Reinforced Composites - Automotive Applications (Hanser, 2020). 4R. Bailey and H. 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Vergnes, “A matricial approach of fibre breakage in twin-screw extrusion of glass fibres rein-forced thermoplastics,” Composites, Part A 48, 47–56 (2013). 22J. Kang, M. Huang, M. Zhang, N. Zhang, G. Song, Y. Liu, X. Shi, and C. Liu, “An effective model for fiber breakage prediction of injection-molded long fiber reinforced thermoplastics,” J. Reinf. Plast. Compos. 39, 473–484 (2020). 23H. Malatyali, V. Sch €oppner, N. Rabeneck, F. Hanselle, L. Austermeier, and D. Karch “A model for the carbon fiber breakage along the twin-screw extruder,” SPE Polym. 2, 145–152 (2021). 24H. Chen, M. Cieslinski, P. Wapperom, and D. G. Baird, “Long fiber (glass) breakage in capillary and contraction flow,” in the 2016 SPE Annual Technical Conference (ANTEC) (Curran Associates, Inc., 2015), pp. 546–550. 25J. H. Phelps and C. L. Tucker, “An anisotropic rotary diffusion model for fiber orientation in short- and long-fiber thermoplastics,” J. Non-Newton Fluid Mech. 156, 165–176 (2009). 26E. Moritzer and H. Gilmar, “Fiber length degradation of glass fiber reinforced polypropylene during shearing,” in SPE ANTEC (2016), pp. 647–651. 27T. Sasayama, M. Inagaki, and N. Sato, “Direct simulation of glass fiber breakage in simple shear flow considering fiber-fiber interaction,” Composites, Part A 124, 105514 (2019). 28T.-C. Chang, A. Bechara Senior, H. Celik, D. Brands, A. Yanev, and T. Osswald, “Validation of fiber breakage in simple shear flow with direct fiber simulation,” J. Compos. Sci. 4, 134 (2020). 29See L. Nippon Electric Glass Co., https://www.neg.co.jp/en/product/e-roving_- list/for “E Glass Fiber—Direct Roving Product List” (2021) (last accessed May 6, 2021). 30See NetComposites, https://netcomposites.com/calculator/volume-weight-frac- tion-calculator/ for “Volume-Weight Fraction Calculator” (2019) (last accessed May 6, 2021). 31See 3B-the fibreglass company, https://www.3b-fibreglass.com/3b-e-glass for “Typical 3B E-CR Glass Properties (n.d.)” (last accessed May 6, 2021). 32S. Goris, S. Simon, C. Montoya, A. Bechara, M. V. Candal, D. Brands, A. Yanev, and T. A. Osswald, “Experimental study on fiber attrition of long glass fiber-reinforced thermoplastics under controlled conditions in a Couette flow,” in Annual Technical Conference—ANTEC, Conference Proceeding (2017). 33S. Goris, T. Back, A. Yanev, D. Brands, D. Drummer, and T. A. Osswald, “A novel fiber length measurement technique for discontinuous fiber-reinforcedPhysics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-10 VCAuthor(s) 2021composites: A comparative study with existing methods,” Polym. Compos. 39, 4058–4070 (2018). 34V. Kunc, B. Frame, B. N. Nguyen, C. L. Tucker, and G. 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Part I: Reduction in fiberlength during processing,” Polym. Compos. 10, 8–15 (1989). 40B. Nghiep Nguyen, S. K. Bapanapalli, V. Kunc, J. H. Phelps, and C. L. Tucker, “Prediction of the elastic—plastic stress/strain response for injection-molded long-fiber thermoplastics,” J. Compos. Mater. 43, 217–246 (2009).. 41F. Dyson, “A meeting with Enrico Fermi,” Nature 427, 297 (2004).Physics of Fluids ARTICLE scitation.org/journal/phf Phys. Fluids 33, 073318 (2021); doi: 10.1063/5.0058693 33, 073318-11 VCAuthor(s) 2021

5.0055606.pdf

Appl. Phys. Lett. 119, 012404 (2021); https://doi.org/10.1063/5.0055606 119, 012404 © 2021 Author(s).Spin-lattice dynamics of surface vs core magnetization in Fe nanoparticles Cite as: Appl. Phys. Lett. 119, 012404 (2021); https://doi.org/10.1063/5.0055606 Submitted: 30 April 2021 . Accepted: 16 June 2021 . Published Online: 08 July 2021 Gonzalo dos Santos , Robert Meyer , Romina Aparicio , Julien Tranchida , Eduardo M. Bringa , and Herbert M. Urbassek ARTICLES YOU MAY BE INTERESTED IN Magnetic nanohelices swimming in an optical bowl Applied Physics Letters 119, 012406 (2021); https://doi.org/10.1063/5.0058848 Ultrafast non-thermal and thermal switching in charge configuration memory devices based on 1T-TaS 2 Applied Physics Letters 119, 013106 (2021); https://doi.org/10.1063/5.0052311 Electrical tuning of the spin–orbit interaction in nanowire by transparent ZnO gate grown by atomic layer deposition Applied Physics Letters 119, 013102 (2021); https://doi.org/10.1063/5.0051281Spin-lattice dynamics of surface vs core magnetization in Fe nanoparticles Cite as: Appl. Phys. Lett. 119, 012404 (2021); doi: 10.1063/5.0055606 Submitted: 30 April 2021 .Accepted: 16 June 2021 . Published Online: 8 July 2021 Gonzalo dos Santos,1 Robert Meyer,2 Romina Aparicio,1 Julien Tranchida,3 Eduardo M. Bringa,1,4 and Herbert M. Urbassek2,a) AFFILIATIONS 1CONICET and Facultad de Ingenier /C19ıa, Universidad de Mendoza, Mendoza 5500, Argentina 2Physics Department and Research Center OPTIMAS, University Kaiserslautern, Erwin-Schr €odinger-Straße, D-67663 Kaiserslautern, Germany 3Multiscale Science Department, Sandia National Laboratories, P.O. Box 5800, MS 1322, Albuquerque, New Mexico 87185, USA 4Centro de Nanotecnolog /C19ıa Aplicada, Facultad de Ciencias, Universidad Mayor, Santiago 8580745, Chile a)Author to whom correspondence should be addressed: urbassek@rhrk.uni-kl.de .URL: http://www.physik.uni-kl.de/urbassek/ ABSTRACT Magnetization of clusters is often simulated using atomistic spin dynamics for a fixed lattice. Coupled spin-lattice dynamics simulations of the magnetization of nanoparticles have, to date, neglected the change in the size of the atomic magnetic moments near surfaces. We show that the introduction of variable magnetic moments leads to a better description of experimental data for the magnetization of small Fe nano- particles. To this end, we divide atoms into a surface-near shell and a core with bulk properties. It is demonstrated that both the magnitudeof the shell magnetic moment and the exchange interactions need to be modified to obtain a fair representation of the experimental data.This allows for a reasonable description of the average magnetic moment vs cluster size, and also the cluster magnetization vs temperature. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0055606 Magnetic nanoparticles (NPs) are fundamental components for applications such as catalysts or biomedical materials. 1–3Developing numerical atomistic tools allowing for accurate predictions of the tem- perature dependence of the NPs’ magnetization dynamics, as well as other relevant quantities of interest such as heat capacity, is of practicaldesign interest. 4Such tools could bring insight into the fundamental processes at stake and help tailor NPs by locating optimum properties,including size and shape, for technological 5and biomedical applications.6 Our numerical effort is based on the classical spin-lattice dynam- ics (SLD) approach.7SLD is an atomistic method assigning classical spins to the magnetic atoms in the simulated system.8–10The potential energy is computed by combining a purely mechanical potential witha Heisenberg exchange interaction. Through the interatomic depen- dence of the exchange interaction, the magnetic spins are coupled to the atomic potential energy. Thus, the corresponding moleculardynamics simulation of such a system allows us to calculate simulta-neously the influence of (spin-derived) forces on the atoms’ positionsand the changes in the precession of the spins by the magnetic field setup by the surrounding atoms. Such spin-lattice simulations have beenused until now to describe the temperature dependence of magneto-mechanical properties, phase transitions, phonon dispersion, demag- netization experiments, and other phenomena. 11–15 Dos Santos et al. showed that SLD could be used to compute the temperature dependence of the magnetization for iron NPs.16In this former study, the influence of thermal fluctuations on the total magne-tization was accounted for both the lattice and the spin systems, sincetheir coupling is evaluated in a time-dependent way by the SLD simu-lations. However, this work used fixed size for the atomic moments, 16 thus ignoring their norm fluctuations due to surface effects.17Those results yielded good qualitative agreement for the NPs’ magnetizationvs temperature trends, but quantitative agreement with experimentswas obtained only if the magnetization was re-normalized. Since thetotal magnetization of NPs is a strong function of the NP diameter forsmall sizes, this situation is unsatisfactory. In this work, we present an approach allowing us to account for surface effects on the magnetic moments. We explore its effectson the magnetization of iron NPs and display good quantitativeagreement with experimental measurements. We also show thatour approach can be straightforwardly used to compute the heat- capacity of magnetic NPs, accounting for both lattice and magnetic contributions. Appl. Phys. Lett. 119, 012404 (2021); doi: 10.1063/5.0055606 119, 012404-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplWe investigate spherical NPs with a diameter dbetween 1 and 8 nm. Considered NP diameters are small compared to typicaldomain-wall thickness measured in iron, 18allowing us to treat them as single-domain magnets. The spheres are cut out from bcc Fe crystalsand relaxed for 10 ps. In our SLD implementation, this means that theatoms’ positions are allowed to thermally expand and keep pressure tozero, while the spins respond to the changed environment. The atomsinteract through forces computed from the spin Hamiltonian and theChamati et al. potential 19(see the supplementary material for more details). This potential was designed to accurately compute surfaceenergies, and former studies leveraged it to investigate finite size systems. 20,21 Initially, each classical spin vector si(of unit length) points in the [001] direction. The total spin Sof a NP is given by the magnitude of the vectorial sum over all atomic spins sias S¼1 N/C12/C12/C12/C12X isi/C12/C12/C12/C12: (1) Thus, at temperature T¼0,S¼1. Each atom icarries a local magnetic moment (whose magnitude is determined by the atom’s environment) of size l iand direction si. The average magnetic moment per atom in the NP is then given by hli¼1 N/C12/C12/C12/C12X ilisi/C12/C12/C12/C12; (2) which gives us the magnetization of the considered NP. The interaction between the atomic spins may change the spin direction and also influence the atomic motion. It is dominated bythe exchange interaction, /C0Jðr ijÞsisj. The space dependence of the exchange interaction is described by a Bethe–Slater function, fit to theresults by Pajda et al., 25since it was proven to describe well the magne- tism of Fe NPs.16J(r) is the only position dependent term in this inter- action and is responsible for energy exchange between the spin andthe atomic systems. Cubic magneto-crystalline anisotropy is alsoincluded as in Ref. 26, and details are given in the supplementary material [see Eqs. (S2) and (S3)]. Our time step is set to 1 fs. We use Langevin thermostats to equil- ibrate our NPs at the desired temperatures for a time of 0.5 ns. Afterthis time, the magnetic properties are measured as an average over0.3 ns. Further details on our SLD approach are provided in Ref. 16. We assign a magnetic moment to each iron atom depending on its local environment. Figure 1(a) displays the magnetic moment of Fe atoms in compressed or expanded bcc lattices, and how magnetismvanishes at high atom densities and increases toward the free atom value(4l B) for expanded lattices.27In a zero-pressure bcc lattice, lequ/C252:154lB. We thus use the local atomic volume in order to assign local magnetic moments to Fe atoms.28O u rw o r ko n l yu s e st h el o c a lv o l - ume, but more detailed information about the local atomic coordinationmight be needed for a more precise assignment of local moments. 29We also note that thermal expansion is more pronounced in NPs than in thebulk, and the average atomic volume increases strongly with decreasingdiameter (see Fig. S1 in the supplementary material ). The local electronic density—as obtained from the EAM-type potential—carries information about the local atomic volume, see Fig. 1(b). We thus have an algorithmic mapping between the local elec- tronic density (readily available at each step of the SLD simulation)and the size of the local moments. We fit our data to a function of the form 24,28 li¼C1/C0ffiffiffiffiffiqi qcr !c : (3) For the Chamati et al. potential,19we obtain C¼ð3:45760:259ÞlB and c¼0:41460:073; qcdenotes the critical electron density at which ferromagnetism vanishes. Figure 2 exemplifies our results on the radial dependence of mag- netic moments lðrÞfor a NP of diameter 2.3 nm. The data can be divided in two categories, (i) a core region extending up to r/C259A˚, in which the mag- netic moment is close to the bulk value and (ii) a shell of width Dr/C252:5A˚, in which the moments increase linearly with rup to values of 2.6–2.65 lB. This feature holds also at elevated temperature: at 600 K, the atomic moments calculated through Eq. (3)a r eo n l ys l i g h t l yh i g h e r (but within the uncertainty limits), since thermal expansion increasesthe atomic volume, as seen in Figs. S1 and S3 of the supplementary material . We note that this separation into core and shell regions was found for all NP sizes investigated here; in particular, the shell width isalways equal to 2.5 A ˚. This is plausible, since it corresponds to a roughly monatomic shell. The second outer atomic shell feels anFIG. 1. Electron density (a) and local magnetic moment (b) dependence on the atomic volume. Data computed for a homogeneously expanded/contracted bcc structure. Ab initio data obtained by Moruzzi et al.22and Herper et al.23(taken from Ref. 24). The cross marks the equilibrium magnetic moment, lequ¼2:154lB,a t the equilibrium atomic volume of bcc Fe, 11.78 A ˚3.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012404 (2021); doi: 10.1063/5.0055606 119, 012404-2 Published under an exclusive license by AIP Publishingenvironment that resembles bulk Fe, and hence the magnetic moments take values close to bulk atoms.30–32All SLD calculations are performed leveraging the SPIN package of LAMMPS.26,33This SLD implementation considers normalized spin vectors when computing the exchange interaction. Therefore, in order to take into account theinfluence of the atomic moments’ distribution, which varies withrespect to the atomic volume and the neighboring spin configura-tion, 17,34atoms were divided into two separate groups: (i) the core with moment lcoreand (ii) the shell with moment lshell, where the thickness of the shell has been fixed to 2.5 A ˚. As can be seen in Fig. 2 , this only allows us to approximate the continuous magnetic momentfluctuations. We choose l core¼2:15lBand lshell¼2:45lBas the average over the core and shell atoms, respectively, in agreement withFig. 2 . Both the shell width and average shell magnetic moments proved valid for all cluster sizes ( /C208 nm) and temperatures ( T/C20900 K) (see Figs. S2 and S3 in the supplementary material ) .N P sl a r g e rt h a n8 n m are expected to behave similarly, see Fig. S3. This grouping influences the spin-lattice dynamics only via the exchange interaction. While core spins interact via Jðr ijÞsisj,t h ef a c t o r Jis scaled by lshell=lcorefor core–shell spin interactions, and by ðlshell=lcoreÞ2for shell–shell interactions. Former studies have been reporting bulk exchange interaction calculations for iron,35,36but we are not aware of similar computations for cluster shells. Experimental data are available in Ref. 37for Fe NPs at 120 K with a number of atoms below N¼800, corresponding to diameters below 2.6 nm. Figure 3 shows that our SLD simulation results— denoted by lshell¼2:45—underestimate the experimental values by up to approximately 25%. While the magnetic moment increases with decreasing the NP size, it only leads to magnetic moments of around2:25l B, even though the number of core atoms is smaller than the number of shell atoms for these small clusters, N<100. An analysis of our data shows that this is caused by the misorientation of the atomspin for these small clusters: while the core spin assumes values of S¼0.94, the shell spin is below 0.92, and hence the total magnetization of NPs does not increase much at T¼120 K. Note that a simulation with all magnetic moments fixed to their bulk value l equshows a size independent magnetic moment of around 2 :0lB’0:92lequ16(open circles in Fig. 3 ). Experimental data reach moments up to 3 lBfor small NPs (N/H11351100), whereas our larger magnetic moment values are approxi- mately 2 :8lB, obtained for atoms in edge positions on the NP surface.This demonstrates that our model underestimates the magnetic moment of shell atoms. Even deviations from spherical shapes (whichmay occur in experiments) would not produce larger magnetic moments in our approach. Those high moments in small clusters have been investigated using tight-binding calculations 39and have been assigned to structural changes in small NPs favoring fcc-like and icosa- hedral coordinations connected to a re-ordering of atomic orbitals and associated spin changes. Such structural changes are beyond the scopeof this work (where a bcc bulk-like structure is always assumed). We therefore tested a second model, in which l shell¼3:0lB. Figure 3 shows that this calculation gives better agreement with the experimental data (reducing the error to approximately 10% for thesmaller NPs). Figure 3 also shows reasonable agreement between our simulations and other experimental results for 1.6 nm clusters. 38 However, even for the smallest clusters, the magnetization remainsbelow 3 :0l Bfor the reasons discussed above: the core moments are only 2 :1lBand the thermal fluctuations disorder the low-coordinated shell spins at 120 K. A discussion of the temperature dependence of the magnetization brings us further insight. Figure 4(a) reproduces experimental data for 2.3 nm diameter clusters ( N¼533). Our core–shell simulations with lshell¼2:45lBare in fair agreement with the experiments and only slightly overestimate the magnetic moment for intermediate tempera- tures ( T¼400/C0600 K). However, the calculations with lshell¼ 3:0lBstrongly overestimate the magnetic moment for all tempera- tures, indicating that the spin-spin coupling is too strong in this model. Motivated by the results of Billas et al.37(a large magnetic moment and a relatively low magnetization), we set a third model [denoted by J/C3ðrÞ], in which we retain the high value of the shell moments, lshell¼3:0lB, but reduce the exchange coupling of shell atoms with other shell or core atoms to half their value. Thus, theFIG. 3. Average magnetic moment and experimental data by Billas et al.37as a function of the number of atoms in the NP. Theoretical results obtained by our SLD approaches, compared to the data published by Dos Santos et al.16The experi- mental point by Margeat et al.38is also included. FIG. 2. Atomic magnetic moment dependence on the distance from NP center. Data are for a 2.3 nm Fe NP, at two different temperatures. Dashed lines indicate the approximate shell region.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012404 (2021); doi: 10.1063/5.0055606 119, 012404-3 Published under an exclusive license by AIP Publishingfunction J(r)b yP a j d a et al.25is scaled by1 2lshell=lcorefor core–shell spin interactions, and by1 2ðlshell=lcoreÞ2for shell–shell interactions. When compared to our simulation results, this indicates that exchange between shell–shell and shell–core spins is weaker than we assumed before. Here, we find that the 1/2 re-scaling factor of the shell–shell and shell–core J(r) couplings gives excellent agreement with experiments. An alternative approach could tune the exchange coeffi- cients, and advanced fitting methods could be employed.40Previous simulations use the same exchange for the core and the shell41,42but, given the current lack of theoretical or experimental guidance we take surface exchange as a free parameter, as in previous studies.43,44 Figure 4(a) demonstrates that this choice nicely reproduces the experimental data for the temperature dependence of the 2.3 nm diameter clusters. Compared to the calculations with unchanged J,t h e magnetic moments of NPs at 120 K have been reduced for all NP sizes, thus deteriorating the comparison with experiments, see Fig. 3 .A sd i s - cussed above, this disagreement is likely caused by structural changes in the NPs, and therefore out of scope of the present calculations. Locating the Curie transition on magnetic nanoclusters is chal- lenging. Heat capacity ( Cp) measurements give an indirect determina- tion of this phase-transition temperature.45,46Cpis computed by taking the internal energy’s derivative with respect to temperature. (Details are provided in the supplementary material .) Our simulation results for the 2.3-nm NPs, Fig. 4(b) ,s h o w Cpp e a k sn e a r5 0 0 K(associated with the ferromagnetic-paramagnetic phase-transition). This is a considerable shift compared to the bulk value, consistent with our magnetization vs temperature results. The Cpcurve for the NP with homogeneous values of land J(r) shows a similar peak, but shifted to higher temperatures since those shell spins require more energy to disorder than the shell–core case with J/C3.W en o t et h a t frozen-lattice spin dynamics simulations give different positions and shapes of the specific maximum, indicating that coupled spin-lattice simulations are required to correlate Cpand the Curie temperature. These results could be especially relevant for biomedical applications, s i n c eap r e c i s ee s t i m a t i o no f Cpis essential to evaluate heating of NPs for applications like magnetic induction hyperthermia.47This is partic- ularly important as lattice-only simulations do not account for mag- netic degrees of freedom and thus cannot recover this intermediate temperature Cppeak.48Most “atomistic spin dynamics” (ASD) simu- lations use a fixed perfect lattice, ignoring the thermal motion of the atoms. Dos Santos et al.16studied the importance of moving atoms for 3 nm NPs and found that frozen-lattice simulations lead to unphysical compressive stresses of 0.2 GPa. Figure S4 in the supplementary material compares our magnetic-moment results to those obtained by a frozen-lattice calculation and shows better agreement with the exper- iment for the moving lattice case. Figure 5 displays how the NP spins become disordered with tem- perature. The total spin Saveraged throughout the NP, and over the shell and the core only are plotted for 2.3 nm diameter clusters. The distinction between the core and shell atoms and their different mag- netic moments and couplings has a strong influence on the spin disor- dering. The model with lshell¼3:0lBshows a strong coupling of the shell and core atoms, which is not unexpected, since for this cluster size, more than 51% of all atoms are in the shell. In this model, the low coordination of surface atoms is compensated with higher mag- netic exchange coupling among them. On the other hand, spins show a clearer distinction between the shell and core values for the lshell¼2:45lBmodel and even more for the J/C3ðrÞmodel, demon- strating the decreased coupling between the moments in these models. Because of their higher coordination, the core atoms remain magnetically stiffer to higher temperatures than shell atoms. In previ-ous work, 16it was shown that the main influence of the atomic ther- mal motion on the magnetization is felt at temperatures beyond 400 K and leads to a substantial decline in magnetization; this influence is included in our present calculations. We used SLD to perform a core–shell investigation of magnetic NPs. Calculations of the influence of the atomic volume on the mag- netic moment based on a local bcc-like coordination show that the surface shell, in which atomic moments deviate from their bulk value, is only 2.5 A ˚thick, and atomic moments there assume an average value of lshell¼2:45lB. We greatly improved agreement with available experimental data37by increasing the shell moment to lshell¼3:0lBbut decreasing the exchange interaction among shell atoms and between the shell and core atoms by 50%. We showed that a core–shell model has sufficient flexibility to represent the magnetiza- tion of Fe NPs with SLD and to accurately compute quantities of prac- tical design interest such as the temperature and the NP size dependence of the magnetization or heat-capacity. We believe a major approximation of our classical model con- sisted in treating the exchange coupling J(r) as temperature indepen- dent; including this dependence might lead to more precise FIG. 4. Temperature dependence of the (a) average magnetic moment and (b) heat capacity for a 2.3 nm diameter NP. Experimental data by Billas et al.37are com- pared to our theoretical results.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 012404 (2021); doi: 10.1063/5.0055606 119, 012404-4 Published under an exclusive license by AIP Publishingpredictions,36,49,50improving agreement in Fig. 3 ,w h i l er e t a i n i n gt h e excellent agreement in Fig. 4(a) . In addition, we have considered the magnetic cubic anisotropy identical for all spins,16although it should vary near the surface due to the coordination reduction.51,52Different values for NPs having different core/shell fractions should be considered (with an expected reduction of about 10% when halving the clustersize 53). Former studies also leveraged the N /C19eel pair interaction to repro- duce magneto-elastic and surface anisotropic effects15,54but using it for SLD simulations of NPs remains to be investigated. In future work, the possibility of assigning arbitrary magnetic moment values for individual atoms during the calculation of exchangeinteraction will be investigated. This would allow us to reproduce the moment fluctuations displayed in Fig. 2 a n ds h o u l di m p r o v eo u ra g r e e - ment with experimental predictions. Complementary ab initio compu- tations of finite size effects on both the size of moments and theeffective exchange interaction between atoms for small clusters couldalso be designed. See the supplementary material for additional information on the dependence of the average atomic volume of NPs on their size, thecubic magneto-crystalline anisotropy, a comparison of our calculationsto frozen lattice simulations, the Hamiltonian used, and heat capacity calculation details. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project No. 268565370–TRR 173 (Project A06). Simulations were performed at the High Performance Cluster Elwetritsch (Regionales Hochschulrechenzentrum, TUKaiserslautern, Germany). E.M.B. thanks support from SIIP-UNCuyo06/M104 and ANPCyT PICTO-UUMM-2019-00048 grants. J.T. acknowledges that Sandia National Laboratories is a multimission laboratory managed and operated by the National Technology andEngineering Solutions of Sandia, LLC., a wholly owned subsidiary ofHoneywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE- NA-0003525. This paper describes objective technical results andanalysis. Any subjective views or opinions that might be expressed inthe paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1Y. Zhu, L. P. Stubbs, F. Ho, R. Liu, C. P. Ship, J. A. Maguire, and N. S. Hosmane, “Magnetic nanocomposites: A new perspective in catalysis,”ChemCatChem 2, 365–374 (2010). 2Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. 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5.0042058.pdf

J. Chem. Phys. 154, 134112 (2021); https://doi.org/10.1063/5.0042058 154, 134112 © 2021 Author(s).Thermally activated delayed fluorescence: A critical assessment of environmental effects on the singlet–triplet energy gap Cite as: J. Chem. Phys. 154, 134112 (2021); https://doi.org/10.1063/5.0042058 Submitted: 28 December 2020 . Accepted: 15 March 2021 . Published Online: 05 April 2021 Rama Dhali , D. K. Andrea Phan Huu , Francesca Terenziani , Cristina Sissa , and Anna Painelli COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical Chemistry ARTICLES YOU MAY BE INTERESTED IN Impact of secondary donor units on the excited-state properties and thermally activated delayed fluorescence (TADF) efficiency of pentacarbazole-benzonitrile emitters The Journal of Chemical Physics 153, 144708 (2020); https://doi.org/10.1063/5.0028227 Intramolecular-rotation driven triplet-to-singlet upconversion and fluctuation induced fluorescence activation in linearly connected donor–acceptor molecules The Journal of Chemical Physics 153, 204702 (2020); https://doi.org/10.1063/5.0029608 An exact solution in the theory of fluorescence resonance energy transfer with vibrational relaxation The Journal of Chemical Physics 154, 134104 (2021); https://doi.org/10.1063/5.0045008The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Thermally activated delayed fluorescence: A critical assessment of environmental effects on the singlet–triplet energy gap Cite as: J. Chem. Phys. 154, 134112 (2021); doi: 10.1063/5.0042058 Submitted: 28 December 2020 •Accepted: 15 March 2021 • Published Online: 5 April 2021 Rama Dhali, D. K. Andrea Phan Huu, Francesca Terenziani, Cristina Sissa, and Anna Painelli AFFILIATIONS Department of Chemistry, Life Science and Environmental Sustainability, University of Parma, Parma, Italy Note: This paper is part of the JCP Special Collection in Honor of Women in Chemical Physics and Physical Chemistry. a)Author to whom correspondence should be addressed: anna.painelli@unipr.it ABSTRACT The effective design of dyes optimized for thermally activated delayed fluorescence (TADF) requires the precise control of two tiny ener- gies: the singlet–triplet gap, which has to be maintained within thermal energy, and the strength of spin–orbit coupling. A subtle interplay among low-energy excited states having dominant charge-transfer and local character then governs TADF efficiency, making models for environmental effects both crucial and challenging. The main message of this paper is a warning to the community of chemists, physicists, and material scientists working in the field: the adiabatic approximation implicitly imposed to the treatment of fast environmental degrees of freedom in quantum–classical and continuum solvation models leads to uncontrolled results. Several approximation schemes were proposed to mitigate the issue, but we underline that the adiabatic approximation to fast solvation is inadequate and cannot be improved; rather, it must be abandoned in favor of an antiadiabatic approach. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0042058 .,s I. INTRODUCTION Thermally activated delayed fluorescence (TADF) occurs in fluorescent systems where triplet states sit very close in energy to the emissive singlet state. Triplet states, which are populated upon intersystem crossing (ISC) following photoexcitation or directly via charge recombination in electrically operated devices, can undergo a thermally activated reverse intersystem crossing (RISC) process and be converted into emissive singlet states. TADF emitters thus typically show a fast emission regime (prompt fluorescence) and a slow emission regime (delayed fluorescence). First observed in 1961,1TADF gained wide popularity in 2011 when Adachi pro- posed its exploitation to harvest triplets in organic light-emitting devices (OLEDs).2Indeed, using TADF-based materials, it is pos- sible to increase the theoretical internal quantum efficiency of an OLED from 25% to 100%, maintaining a high color purity of emis- sion.3–6Almost immediately, it was recognized that organic charge transfer (CT) dyes are good candidates as TADF emitters, as long asthe conjugation between electron-donor (D) and electron-acceptor (A) moieties is low, to guarantee for a small energy gap between sin- glet and triplet CT states. This condition is easily met in systems where D and A moieties are arranged almost orthogonally.7The inherent synthetic flexibility of organic compounds made it possi- ble to synthesize a large collection of dyes, which differ not just in the nature of the D and A units but also in the ways these units are connected: other than dipolar emitters, quadrupolar and octupo- lar emitters have been synthesized and studied, as well as more exotic systems with through-air CT interaction or conjugated DA structures.7–15 Quantum chemical calculations can help the work of synthetic chemists and material scientists: the systematic in silico study of a large amount of novel chemical structures can indeed reduce the expensive and time-consuming work in the experimental laboratory, allowing the experimentalists to focus on just the most promising structures. To this effect, cheap, fast, and reliable computational approaches are needed. Time-dependent density functional theory J. Chem. Phys. 154, 134112 (2021); doi: 10.1063/5.0042058 154, 134112-1 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (TD-DFT) arguably represents one of the most effective computa- tional tools in this respect, thanks to the favorable trade-off between accuracy and computational cost. However, modeling CT transi- tions is non-trivial in TD-DFT, the relevant results being strongly dependent on the adopted functional. Progress in this direction is offered by the development of tuned range-separated hybrid func- tionals, where the proper amount of exact exchange for each system is selected, without the need for a comparison with experimental data.16–18 Once the proper functional is selected, reliable TD-DFT results can be obtained for isolated ( gas phase ) dyes, but material scien- tists need to address the properties of the dyes in condensed phases (either in solution or in a matrix). In the following, we adopt the generic term solvent to address either the liquid solvent surround- ing the dye in solution or the solid matrix surrounding the dye, as, e.g., in a device. Models where both the dye and the surrounding sol- vent are treated quantum-mechanically are clearly impractical, and approximation strategies must be devised to separate the solute and solvent problems. In this perspective, effective solvation models are introduced where a quantum mechanical Hamiltonian is defined for the solute, implicitly accounting for the effects of the surrounding medium. When constructing an effective solvation model, a hierarchy of approximations must be considered. The first step is the sepa- ration of the solute and solvent problems, relying on the different timescales of relevant degrees of freedom (DoF). Specifically, being interested to model the solute optical spectrum in the visible-near UV regions, we can safely assume that polar solvation, related to the orientational motions of polar solvent molecules around the solute, represents a slow motion and can be treated adiabatically. On the opposite, electronic solvation accounts for the rearrangement of the electronic clouds of solvent molecules in response to the charge distribution in the solute: the corresponding DoF have typical fre- quencies far in the ultraviolet and are therefore much faster than the solute DoF. The adiabatic approximation must then be abandoned in favor of the antiadiabatic (AA) approximation.19Of course, cases may occur where the timescales of solute and solvent motions are comparable. In these special cases, effective solvation models cannot be reliably defined. Once the framework for the solute–solvent separation is set, models for the solute and solvent and for their interaction must be defined. As for the solute, a vast variety of quantum-mechanical models is possible, ranging from parametric models accounting for just few electronic degrees of freedom, semiempirical models, first- principle DFT and TD-DFT models, high-quality ab initio , etc. The choice of the model Hamiltonian and of the relevant basis, of course, heavily affects the quality of the results and their reliabil- ity. The solvent, in turn, can be described as a continuum dielectric medium, linearly responding to electrostatic perturbations (elas- tic medium).20,21Alternatively, one can rely on atomistic pictures for the solvent, in MM or MD approaches.22–24Quite interestingly, in these mixed approaches, a number of solvent molecules can be included into the portion of system treated quantum-mechanically, and when this number is large enough, the limit of a full quantum mechanical treatment of the solute and solvent is reached.25 The solute–solvent interaction can be simplified to a dipolar interaction in an approach that can be extended to multipolar terms. In more refined approaches, the solute is contained in a cavity carvedin the solvent whose shape and dimensions are defined according to several approximation schemes with variable degrees of details. In continuum solvation models, the solute generates charges at the cav- ity surface, which, in turn, affect the potential felt by the solute. In atomistic models, the solute affects the orientation of surrounding molecules and (in polarizable models) also their charge distribu- tion. In turn, the charges on the surrounding molecules affect the potential felt by the solute. Fixing all the details in the approximation ladder leads to a pro- liferation of effective solvation models, which cannot be reviewed here. However, the first approximation, related to the separation of solute and solvent degrees of freedom, leads to two qualitatively dif- ferent approaches to effective solvation. In the adiabatic approxima- tion, the molecular Hamiltonian is diagonalized for a fixed value of the potential generated by the surrounding medium (that, depend- ing on the model, means fixed charges on the surrounding molecules or on the surface cavity or a fixed reaction field). The calculation can be repeated for different values of the potential, typically fixing it at the equilibrium value relevant for each state (hence, leading to state- specific approaches). In any case, in the adiabatic approximation, each Hamiltonian is defined and diagonalized for a fixed poten- tial. The adiabatic approach to solvation closely resembles the adia- batic approach adopted to separate electronic and vibrational DoF in molecular systems (most often in the so-called Born–Oppenheimer scheme).26It is well known that the adiabatic approximation can be reliably applied to separate electronic and vibrational DoF when the nuclear dynamics is much slower than the electronic dynam- ics. Analogously, the adiabatic approximation applied to separate solvation degrees of freedom works well when solvation charges (and the resulting potential) move slowly with respect to the solute DoF of interest. However, when dealing with electronic solvation, we are considering fast DoF: the adiabatic approximation must be abandoned since it relies on a molecular Hamiltonian where the charges in the surrounding solvent are considered frozen, while they actually move faster than the solute DoF. Rather, an AA approx- imation can be invoked, assuming an instantaneous rearrange- ment of the solvent charges and of the resulting potentials to the charge fluctuations in the solute. A single AA Hamiltonian is thus obtained whose diagonalization leads in a single shot to all molecular eigenstates. In this paper, with reference to TADF dyes, we show how cur- rent implementations of continuum solvation models do not prop- erly address environmental effects on the singlet–triplet gap, with results that widely depend on the adopted approximation scheme and lead, in some cases, to an inversion of the order of the lowest sin- glet and triplet states. In Sec. II, we briefly introduce the three avail- able implementations of the polarizable continuum model (PCM) in the Gaussian package. In Sec. III, we report results obtained with the three approaches on six different TADF dyes. In Sec. IV, we demon- strate that the observed inversion of the singlet and triplet states is indeed a spurious result obtained imposing the adiabatic approx- imation to fast solvation. Finally, Sec. V describes main results in perspective. II. COMPUTATIONAL APPROACH In this work, we consider three dipolar emitters ( A1,B1, and C1) and their quadrupolar counterparts ( A2,B2, and C2), J. Chem. Phys. 154, 134112 (2021); doi: 10.1063/5.0042058 154, 134112-2 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp respectively, as shown in Fig. 1. For each emitter, single point TD- DFT calculations on the optimized ground state geometry are per- formed to obtain excitation energies (in the Tamm–Dancoff approx- imation),27both in gas phase and accounting for non-equilibrium solvation in PCM. All DFT and TD-DFT calculations are performed using Gaussian 16 B.01.28The optimized ground state structures of A2,B2, and C2are obtained at the B3LYP/6-31G(d) level. Ground state geometries for A1,B1, and C1are obtained substituting one of the donor units with a hydrogen atom. TD-DFT calculations are performed at the M06-2X/6-31G(d) level (the choice of the func- tional is addressed in the supplementary material), imposing the Tamm–Dancoff approximation. As discussed in Ref. 19, the ground state properties of the solute are not properly addressed when the adiabatic approximation is adopted to fast solvation. Since in current PCM implementations, the ground state geometry is optimized in this approximation, lead- ing to unreliable results, all data presented here are obtained for the optimized geometry in the gas phase. Moreover, in order to exclude any contribution from polar solvation, we consider cus- tom non-polar solvents, setting the static dielectric constant equal to the squared refractive index (results for a few natural solvents are available in the supplementary material). Calculations are repeated for different values of the refractive index, η. The results are dis- played as a function of f(η2) = (η2−1)/(2η2+ 1), the region cor- responding to most organic solvents and polymeric hosts covering the 0.175<f(η2)<0.225 interval. Three different implementations of PCM are currently avail- able in the Gaussian package,28named linear response (LR), cor- rected linear response (CLR), and external iteration (EI). In all cases, the calculation starts with a reference (initial) state with FIG. 1 . Molecules considered in this work: ( A1) PO-TXO2. ( A2) DPO-TXO2. ( B1) 2-PTZ-DBTO2. ( B2) DPTZ-DBTO2. ( C1) PTZ-DBTO2. ( C2) 3,7-DPTZ-DBTO2. In all molecules, red and blue colors refer to the donor and acceptor groups, respectively.equilibrated fast and slow solvent DoF and a final state, defined in different ways in the three approaches.29,30LR represents the default approach in TD-HF and TD-DFT calculations. In LR,31excitation energies are determined directly as singularities of the frequency- dependent linear-response functions of the solvated molecule in the ground state, avoiding explicit calculations of the excited state wavefunctions, leading to a fast and computationally convenient approach. Specifically, defining the frozen-solvent transition energy as the transition energy calculated maintaining the fast and slow solvent DoF equilibrated to the reference state (the ground state for absorption, the excited state for emission), LR corrections are applied that only depend on the transition dipole moment between the reference and the final state. While computationally convenient, LR does not account for the variation of the charge distribution in the solute upon excitation, and therefore, its use for CT transitions is not recommended.31 State-specific approaches were then proposed, accounting for the variation of the solute charge distribution upon excitation. Specifically, in EI, the fast DoF of the solvent are equilibrated to the excited state charge density, in a self-consistent procedure.32,33The non-equilibrium transition energy is then computed as the differ- ence between the energy of the final state and of the initial state, both states being obtained with the fast solvent DoF equilibrated for the relevant state (for polar solvents, slow solvent DoF are maintained fixed to the equilibrium value for the ground state, when referring to absorption processes, and to the excited state when referring to emis- sion). It is important to underline that in EI, two different potentials for the ground and the excited states are considered in an effort to account for the fast relaxation of the solvent DoF. However, the approach is still strictly adiabatic, as each Hamiltonian is defined and diagonalized for a specific constant potential. Moreover, since tran- sitions are computed between eigenstates obtained from the diago- nalization of different Hamiltonians, the calculation of fundamental spectroscopic properties such as the transition dipole moments is precluded.32,33 CLR bridges the gap between LR and EI and represents a per- turbative approximation to EI.30,34As in LR, the zero-order tran- sition energy is calculated as the frozen-solvent transition energy. Corrections are then applied that depend on the variation of the charge distribution upon excitation. According to Ref. 35, the cor- rection is computed by considering the orbital response to the exci- tation of interest, in turn, obtained as the solution of the Kohn– Sham Z-vector equations (relaxed density). CLR relies on a first order perturbative approach so that corrections only apply to the energies, while wavefunctions are not affected. Transition dipole moments are therefore accessible and indeed coincide with those obtained in LR. However, CLR represents just a linear pertur- bative approximation to the complete EI calculation, and apart from computational convenience, it is unclear why a linear per- turbative treatment should be used rather than a nominally exact calculation. III. COMPUTATIONAL RESULTS A. PO-TXO2 ( A1) and DPO-TXO2 ( A2) A1 and A2 are TADF emitters with dipolar (D–A) and quadrupolar (D–A–D) structures, respectively, where A is J. Chem. Phys. 154, 134112 (2021); doi: 10.1063/5.0042058 154, 134112-3 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 9,9-dimethylthioxanthene-S,S-dioxide (TXO2) and D is the phe- noxazine (PO) group. The optimized ground state structure has the D and A moieties almost orthogonal. Both A1andA2have a negligible permanent dipole moment. Figure 2 shows the f(η2)- dependence of the transition energies for the first few excitations of both molecules, calculated in the different implementations of PCM, discussed above. The nature of each state is defined with ref- erence to the natural transition orbital (NTO), displayed in Figs. S3 and S5. In the gas phase, the lowest triplet excitation of A1at 3.389 eV is fully localized on the donor and has a negligible dipole moment; we call it3LED. The second triplet at 3.487 eV and the lowest singlet at 3.504 eV are instead almost pure CT states, labeled3CT and1CT, respectively, and have a large permanent dipole moment oriented along the CT axis (see Table S2). Increasing f(η2), LR excitation energies marginally increase due to the solvent stabilization of the ground state, without any significant effect on the energies of the excited states. CLR and EI give qualitatively different results from LR: indeed, already in non-polar solvents, both approaches point to a different nature of the lowest excited triplet that becomes a CT state rather than an LE state. This has enormous spectroscopic con- sequences,8,9,36and it is important to realize that LR, the default approach to solvation, gives the wrong order of excited states for TADF dyes. In fact, not accounting for the large charge reorganiza- tion upon CT excitation, LR does not capture the large stabilization FIG. 2 . Results for A1(left) and A2(right) molecules. Top panels: excitation ener- gies vs f(η2) for states1CT (blue),3CT (red), and3LED(green). Bottom panels: the calculated energy gap between the lowest singlet and triplet states. In all panels, solid lines refer to LR, symbols refer to CLR, and dashed lines refer to EI.of CT states when going from gas-phase to condensed phases, lead- ing to unreliable results already in non-polar media. On the other hand, CLR and EI lead to widely different results, with energy dif- ferences≈0.5 eV for typical f(η2)values for organic media. CLR and EI results for the energy gap between the lowest singlet and triplet states, ΔEST, are similar, even if largely different from the LR result. InA2, the number of relevant excited states doubles with respect to A1, as symmetric and antisymmetric CT and LE Dstates enter into play. In the gas phase, the lowest triplets ( ≈3.39 eV) are two degenerate states localized on the donors,3LED, while CT states are at higher energies: a pair of almost degenerate triplets,3CT, at ≈3.42 eV and a pair of singlets,1CT, at 3.428 and 3.436 eV. Despite the different structure and higher number of excited states, the dependence of LR, CLR, and EI transition energies on f(η2) (Fig. 2, right panel) can be explained in a similar way as for A1, with the caveat that EI and CLR corrections are due to the variation of the molecular quadrupolar moment of A2upon excitation. Once again, ΔESTresults from CLR and EI calculations are similar but largely different from LR results. B. 2-PTZ-DBTO2, 2,8-DPTZ-DBTO2 ( B1andB2) and 3-PTZ-DBTO2, 3,7-DPTZ-DBTO2 ( C1andC2) B1andB2have been extensively studied both from a theo- retical and experimental perspective.8,9The D and A units [phe- nothiazine (PTZ) and dibenzo[b,d]thiophene 5,5-dioxide (DBTO), respectively] are connected, as shown in Fig. 1. In the optimized ground state, D and A moieties lie on nearly orthogonal planes. The results for B1andB2are displayed in Fig. 3. Several states must be considered for these systems. In fact, the gas phase NTO and MO analyses (Figs. S6 and S7) reveal that B1lowest triplet (3.493 eV) has a predominant CT character so that we dub it as3CT, but with a non- negligible contribution from a local state. The next triplet,3LEA, at 3.604 eV, is almost entirely localized on the A unit. The lowest sin- glet state at 3.607 eV,1CT, is a pure CT state, with a large permanent dipole moment aligned approximately along the DA axis. The third triplet at 3.753 eV is a localized excited state on the D unit,3LED state, with a non-negligible CT character. As mentioned before, the LR corrections to the excitation energies are minor for all states, in view of the very small transition dipole moments of relevant excita- tions. On the opposite, CT states are largely stabilized in CLR and EI, but, as mentioned before, the two approaches yield very different results. B2is the quadrupolar counterpart of B1, and more states enter into play. However, the nature and relative energies of the states in the gas phase are similar in B1and B2. The lowest triplets (≈3.41 eV) are mostly3CT but have a non-negligible LE compo- nent, as shown from the NTO analysis (Fig. S8). Interestingly, the low energy triplet in B2has a larger CT character than in B1. The next triplet at 3.580 eV is localized on A. The pair of degenerate 3LEDstates at≈3.70 eV has a non-negligible CT component. The lowest singlets,1CT, at 3.473 and 3.484 eV are essentially pure CT states. As already discussed, LR corrections are negligible due to the very small transition dipole moments in TADF dyes. In CLR, cor- rections to the3LEAand3LEDstates are also negligible. On the other hand,3CT and1CT states are stabilized as the transitions occur with J. Chem. Phys. 154, 134112 (2021); doi: 10.1063/5.0042058 154, 134112-4 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Results for B1(left) and B2(right) molecules. Top panels: calculated exci- tation energies vs f(η2) for states1CT (blue),3CT (red),3LEA(black),and3LED (green). Bottom panels: the calculated energy gap between the lowest singlet and triplet states. In all panels, solid lines refer to LR, symbols refer to CLR, and dashed lines refer to EI. a significant change in the charge distribution. However, another serious problem emerges: both CLR and EI show an inversion in the order of the lowest singlet and triplet states. In other terms, accord- ing to these calculations, the lowest excited state of both B1andB2 dissolved in an organic non-polar medium would correspond to a singlet and not to a triplet state. As discussed below, this result origi- nates again from the mishandling of fast solvation. In B1, the lowest triplet has dominant CT character but with a sizable contribution from the triplet excitation localized on A, while the lowest singlet state is an almost pure CT state. The variation of the charge distri- bution upon excitation is therefore larger for the lowest singlet than for the lowest triplet excitation, leading to a larger stabilization of the singlet state with respect to the triplet state, with an effect that is most apparent in CLR. Indeed, in CLR, the nature of the states is frozen, while in EI, the nature of the states changes in the iterative process. Specifically, in our case, during the EI iterations, the weight of the LE component in the lowest triplet state decreases, reducing ΔEST, which stays small but negative. In any case, the three implementa- tions of the solvation model lead to very different values for ΔST. Due to the larger CT component in3CT states, in B2, with respect to the same state in B1, the singlet–triplet inversion occurs at larger f(η2)values. C1andC2are very similar to B1andB2, respectively, as they share the same D and A units, even if connected in a different way. The results in Fig. 4 are self-explanatory now. NTOs (see Figs. S9 and S11) show a smaller mixing of local and CT triplet states than FIG. 4 . Results for C1(left) and C2(right) molecules. Top panels: calculated exci- tation energies vs f(η2) for states1CT (blue),3CT (red),3LEA(black), and3LED (green). Bottom panels: the calculated energy gap between the lowest singlet and triplet states. In all panels, solid lines refer to LR, symbols refer to CLR, and dashed lines refer to EI. observed in B1/B2. Accordingly, for both C1andC2,3CT states have a larger weight of CT character than for B1andB2, resulting in larger charge separation. However, negative ΔESTare observed again with most prominent effects in CLR. IV. AN ANTIADIABATIC APPROACH The scattering of the results obtained in the three current PCM implementations available in the Gaussian package and the impos- sibility of calculating the transition dipole moment in the formally exact EI approach, addressed by limiting the analysis to first order perturbation theory in CLR, clearly point to some fundamental problem in solvation models, which can be traced back to the adi- abatic approximation, as discussed in Ref. 19. To also demonstrate that the singlet–triplet inversion calculated in CLR and EI for some dyes in non-polar solvents is a spurious effect resulting from the adiabatic approximation to fast solvation, we focus on B1dye and compare adiabatic and AA results. At present, AA implementations of PCM are not available; therefore, following Ref. 19, we adopt a simplified model for the sol- vated molecule that relies on the dipolar approximation to describe the solute–solvent interaction and on the choice of a small elec- tronic basis. With these approximations, we build a model than can be solved both in the adiabatic and in the AA approximation to fast solvation, allowing for a stringent comparison of the two approaches. J. Chem. Phys. 154, 134112 (2021); doi: 10.1063/5.0042058 154, 134112-5 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the dipolar approximation, the solute dissolved in a non- polar solvent feels the electric field ⃗Fgenerated by the polarization of the surrounding solvent molecules. At the equilibrium, the field is proportional to the molecular dipole moment through a constant r that, assuming a spherical shape for the cavity occupied by the solute, reads37,38 r=2 4πε0a3f(η2)=2 4πε0a3η2−1 2η2+ 1, (1) where ais the cavity radius and ε0is the vacuum permittivity. A quadratic potential energy is associated with the field, with the force constant fixed to the inverse of the corresponding r, to guarantee for the proportionality between the reaction field and the solute dipole moment at equilibrium.39The effective Hamiltonian for the solvated molecule then reads H=Hgas+[⃗F2 2r+ˆT−ˆ⃗μ⋅⃗F], (2) where Hgasis the gas phase molecular Hamiltonian, ˆ⃗μis the molec- ular dipole moment operator, and ˆTis the kinetic energy associated with the electronic polarization. In the adiabatic approximation, the kinetic energy is neglected so that⃗Fbecomes a classical variable and the problem is solved for fixed⃗F. In other terms, in the adiabatic approximation, the poten- tial generated by the medium on the solute is frozen. In the AA approximation, instead, the medium responds instantaneously to the charge fluctuations in the solute, and it is not possible to define a molecular Hamiltonian at frozen field since each state feels its own reaction field. A single effective Hamiltonian is obtained in the AA approximation that reads19 ˆHAA=ˆHgas−1 2relˆμ2. (3) The diagonalization of the AA Hamiltonian gives in a single shot all molecular eigenstates, properly renormalized to account for the effects of fast solvation. To address the AA problem, we define a few state molecular model, writing the Hamiltonian in Eq. (3) on the basis of the eigen- states of the gas-phase Hamiltonian. Specifically, we neglect spin– orbit coupling and consider two independent subspaces formed by the first nsinglets and the first mtriplets, as obtained from the TD- DFT calculation for the gas-phase molecule. On this basis, Hgasis clearly diagonal. The dipole moment matrices were obtained using the MULTIWFN software.40The results, of course, depend on the number of states included in the basis sets, and since the diagonaliza- tion is performed independently in the singlet and triplet subspaces, it is important to consistently choose the number of states in the two subspaces. Setting the same small number of states in both sub- spaces (see Fig. S12 for details), indeed, gives rise to the crossing of singlet and triplet states. The reason for this result is easily rec- ognized in a basis that spans a much wider energy interval for the singlet vs the triplet subspace. Increasing the number of triplet states, so that the same energy window is roughly spanned in both sub- spaces, leads to more reliable results. Data in Fig. S12 show that span- ning a range of ∼6 eV with 17 singlets and 26 triplets leads toward convergence.The right panels of Fig. 5 collect AA results for B1, obtained set- ting the cavity radius to Onsager’s radius, a= 5.44 Å. These results clearly point to an excitation spectrum where the transition ener- gies for the state with CT character (either singlet or triplet) are lowered due to the medium polarizability, while LE states are less affected. As expected, LR results are completely off for CT states. On the other hand, EI largely overestimates the stabilization of CT states and CLR underestimates it (see Fig. 3). At variance with EI and CLR, AA results point to a normal order of excited states, with the lowest excited state having a triplet nature. Comparing AA results in the right panels of Fig. 5 with PCM results in Fig. 3 may, however, be misleading due to the approxi- mations introduced to build the few-state models adopted to run AA calculations. For a stringent comparison of AA and adiabatic approximations, left panels of Fig. 5 show results obtained in the adiabatic approximation (and specifically, in its LR, CLR, and EI variants; see the supplementary material for relevant equations) for precisely the same model adopted for the AA approach (the same basis set and dipole moment matrices). The first observation is that adiabatic results in Fig. 5 com- pare favorably with PCM results in Fig. 3, suggesting that the adopted approximations capture most of the relevant physics. More important is, however, the comparison between adiabatic and AA results in Fig. 5, relevant to the same model. Solvation effects on LE states are marginal, but, as for CT states, neither EI nor CLR properly captures the stabilization of either the singlet or triplet FIG. 5 . Comparison between adiabatic and AA results (left and right panels, respectively) for B1in the few state models accounting for 17 singlet and 26 triplet states. Top panels: calculated excitation energies vs f(η2) for states1CT (blue), 3CT (red),3LEA(black), and3LED(green). Bottom panels: the calculated energy gap between the lowest singlet and triplet states. In left panels (adiabatic results), solid lines refer to LR, symbols refer to CLR, and dashed lines refer to EI. J. Chem. Phys. 154, 134112 (2021); doi: 10.1063/5.0042058 154, 134112-6 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp states with differences in the estimated transition energies of sev- eral tenths of eV at f(η2)∼2, as relevant to common organic media. Moreover, the singlet–triplet gap decreases considerably as a result of the medium polarizability, but at variance with CLR and EI results, it stays positive. Quite irrespective of the quality of the proposed molecular model, results in Fig. 5 unambiguously demon- strate that the adiabatic approach, when applied to describe the spectroscopic effect of the medium polarizability, leads to unreliable results. V. DISCUSSION AND CONCLUSIONS TADF dyes are particularly delicate to model since the sub- tle interplay between localized and CT states makes environ- mental or matrix effects crucial in the definition of the tiny energies, the singlet–triplet gap, and the spin–orbit coupling, which define the system performance.8,9,23,36,41–45Explicit-solvent quantum–classical approaches23,24,46–53are applied in several papers to investigate matrix effects in TADF-dyes. Even more popular are continuum solvation models, with LR,42–44,54–62CLR,63and EI implementations.46,47,62,64–67As extensively discussed here, none of these approaches properly accounts for the electronic polar- izability of the medium, leading to results that need a careful consideration. Two main approaches are possible to separate the relevant DoF of the solute from the solvent DoF: an adiabatic and an antiadia- batic approach. Both approaches rely on the distinctively different dynamics of solute and solvent DoF. In the adiabatic approximation, one separates the relevant system (in our case, the low-lying elec- tronic excitations of the solute, typically in the visible and near-UV spectral region) from slow solvent DoF so that the relevant Hamil- tonian may be defined while maintaining the slow environmental DoF fixed. Accordingly, one diagonalizes several Hamiltonians as relevant to the different configurations of the slow DoF. This is, indeed, what is done in EI, where different molecular Hamiltoni- ans are defined with the solvent DoF specifically equilibrated to each state, in an approach that is perfectly adequate to deal with polar solvation. As the name suggests, the antiadiabatic approximation applies to the opposite case, i.e., when the DoF to be renormalized away are much faster than the relevant ones. In this approxima- tion, one assumes that the solvent DoF readjust instantaneously to the motion of the relevant DoF. Therefore, a single Hamiltonian is defined for the relevant system in the antiadiabatic approximation.19 This is very well apparent if one works with a basis of diabatic states, as in Ref. 68: the antiadiabatic Hamiltonian describes a system where the fast DoF of the solvent are equilibrated to each diabatic basis state. Here, a fairly simple implementation of the AA approximation is introduced to demonstrate that the anomalous results obtained when the adiabatic approximation is applied to fast DoF are quite naturally solved when the proper approximation scheme is adopted. To implement an AA calculation, a model is introduced, rely- ing on a limited electronic basis and describing the solute–solvent interaction in the dipolar approximation. Moreover, the molec- ular geometry is always maintained fixed at the gas-phase equi- librium. Therefore, AA results in the left panel of Fig. 5 must be taken with care and we do not pretend that they offer anaccurate description of the system. Yet, the comparison with adi- abatic results obtained for precisely the same model is solid and unambiguously demonstrates that the adiabatic approximation, implicitly adopted in all effective solvation models, leads to unre- liable results. Several variants of continuum solvation models are discussed in the literature,20,21,29–33which face the problem of fast solvation from slightly different perspectives; however, with the notable exception of early attempts,69,70all approaches rely on the diagonalization of the molecular Hamiltonian obtained for a fixed potential from envi- ronmental charges. Whatever choice is made for the definition of the excited states of interest for absorption and emission processes, these methods are bound to fail since the actual molecular states for a molecule in a polarizable environment should all be obtained diagonalizing a single Hamiltonian where the environmental polar- izability affects in different ways the energy of the states of the sys- tems and their coupling. Indeed, the adiabatic approximation leads to an incorrect description of the molecular ground state itself.19 Just as an example, in polar dyes, with a largely neutral ground state, the adiabatic approximation underestimates the increase in the ground state dipole moment due to the polarizability of the environment, simply because the equilibrium reaction field for a largely neutral ground state is small and cannot account for the large stabilization of polar charge fluctuations.19Similarly, quantum– classical approaches with explicit solvent models do not properly account for the solvent polarizability, even when a polarizable envi- ronment is considered. In fact, in polarizable models, one allows the charges on the solvent molecules to reorganize in response to the solute perturbation, but the molecular Hamiltonian is always defined accounting for a frozen potential generated by the surrounding charges. While the adiabatic approximation can never be applied to elec- tronic solvation whose dynamics is faster than the relevant solute DoF, the AA approximation works well when the solvent degrees of freedom are much faster than the solute ones. The AA approxi- mation therefore should be considered with care when the solvent excitation spectrum comes very close in energy to the solute spec- trum, as it is the case for some matrices used in TADF applications. For common solvents and polymeric matrices used in spectroscopy, the UV-cutoff is typically larger (and often much larger) than 4 eV. Moreover, it must be recognized that the UV cutoff signals the frequency where the solvent absorption starts, the relevant absorp- tion bands being located at much larger energy (just as an example, the water cutoff is at 6.5 eV, but the absorption spectrum peaks at ∼15 eV,71with a large UV tail that moves the central frequency to ∼24 eV19). In systems where the AA approximation to fast solva- tion breaks down due to similar timescales of the solute and solvent motions, the adiabatic approximation does not represent a viable alternative. Rather, solute and solvent degrees of freedom cannot be disentangled, and one must resort to a full quantum mechanical approach to the solute and the solvent. Along these lines, the work reported in Ref. 25 for water solvated dyes offers another indepen- dent demonstration of the failure of the adiabatic approximation to fast solvation. In that work, a QM-MM approach is adopted, where the potential generated in the QM region by the charges on water molecules in the MM region is described (as usual) in the adiabatic approximation. In order to get reliable results, the solvation sphere described by QM must include a large number of water molecules J. Chem. Phys. 154, 134112 (2021); doi: 10.1063/5.0042058 154, 134112-7 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (of the order of at least 200, depending on the solute and on the state of interest). Of special concern here is the inversion of the lowest singlet and triplet states calculated in the adiabatic CLR and EI implemen- tations of PCM for some TADF-dyes. Indeed, the breaking of the Hund rule was reported in some very special molecular systems, typically with highly symmetric structures and a very characteris- tic spatial separation of HOMOs and LUMOs that are delocalized on the whole molecular structure.72–76The molecules discussed here do not show these characteristics. Moreover, the singlet triplet inver- sion in these very special structures was only observed in high quality ab initio calculation, involving at least double excitations.72,73Quite interestingly, the inclusion of a standard TADF dye in a polarizable environment was also suggested as a possible origin for singlet– triplet inversion.23,72However, these results were obtained and dis- cussed treating the medium polarizability in the adiabatic approx- imation and deserve a careful reconsideration, either adopting the more adequate AA approximation or possibly addressing both the solute and its surrounding medium in a fully quantum mechanical approach. In the early 1990s, antiadiabatic approaches were proposed for fast solvation, but never gained traction.69,70Most probably, this is related to the choice of a wrong name for the approximation that was called Born–Oppenheimer rather than antiadiabatic. Indeed, the Born–Oppenheimer approximation is a specific flavor of the most general adiabatic approximation that allows us to separate slow DoF from relevant electronic DoF, through the definition of an electronic Hamiltonian that parametrically depends on slow coordi- nates.26It is also true that slow DoF are finally treated in the adiabatic approximation, but this is only possible after the adiabatic electronic Hamiltonian (defined for frozen slow coordinates) is diagonalized. Using the name Born–Oppenheimer to address an antiadiabatic approximation, where, instead, a single electronic Hamiltonian is defined, was therefore unfortunate and may be the reason why the strategy was not recognized until very recently as the only viable approach to renormalize out the problem of the DoF related to fast solvation. The term antiadiabatic, borrowed from the physics community working on polarons and superconductivity,77was used in the con- text of fast solvation by one of the authors of this paper in 1999,39 with reference to semiempirical model Hamiltonians, and was pro- posed again in the context of quantum chemical approaches.19Other authors have also recently recognized the value of the antiadiabatic approach to treat fast solvation.78Unfortunately, they stick on the Born–Oppenheimer notation, which obscures the qualitatively dif- ferent nature of the antiadiabatic approach with respect to the adia- batic approximation. As extensively discussed in Ref. 19, the antia- diabatic approximation can be applied to solute DoF slower than the electronic DoF of the solvent, typically located deep in the UV (ener- gies much larger than 6 eV): applying it to all electronic excitations in the solute is bound to fail, as also demonstrated in Ref. 78. How- ever, a clever choice of the basis states can be made as to renormalize only relevant DoF, and as the results in Fig. S12 show for a specific example, converged antiadiabatic results can be obtained working in an energy window well within the critical threshold for common solvents. Effective solvation models are of paramount importance in material science since molecular properties are largely affected bythe local molecular environment. Treating the active molecule and its environment on the same foot is a formidable task. Quantum– classical and continuum solvation models are therefore widely adopted in the community of computational chemists, physicists, and material scientists. The main message of this work is a warn- ing to these communities: the adiabatic approximation implicitly assumed in all these approaches to deal with fast solvation, i.e., to account for the medium polarizability, yields uncontrolled results, exemplified here by the prediction of a singlet excited state lying at lower energy than the lowest triplet state. The adiabatic approx- imation, of course, works very well to deal with slow solvation DoF, including, e.g., polar solvation. However, it cannot be applied to fast solvation: there is no way to improve on it. A different scheme, based on the antiadiabatic approximation, must rather be adopted. SUPPLEMENTARY MATERIAL See the supplementary material for additional information and results on TD-DFT calculations (NTO/MO analysis, ω-tuning func- tional) and for the details about the convergence of AA calcula- tions. ACKNOWLEDGMENTS This project received funding from the European Union Hori- zon 2020 research and innovation program under Grant Agreement No. 812872 (TADFlife) and benefited from the equipment and sup- port of the COMP-HUB Initiative, funded by the “Departments of Excellence” program of the Italian Ministry for Education, Univer- sity and Research (MIUR, 2018–2022). We acknowledge the sup- port from the HPC (High Performance Computing) facility of the University of Parma, Italy. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. A. Parker and C. G. Hatchard, Trans. Faraday Soc. 57, 1894 (1961). 2A. Endo, K. Sato, K. Yoshimura, T. Kai, A. Kawada, H. Miyazaki, and C. Adachi, Appl. Phys. Lett. 98, 083302 (2011). 3H. Nakanotani, T. Higuchi, T. Furukawa, K. Masui, K. Morimoto, M. Numata, H. Tanaka, Y. Sagara, T. Yasuda, and C. Adachi, Nat. Commun. 5, 4061 (2014). 4E. Zysman-Colman, Nat. 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5.0054662.pdf

Interface-driven electrical magnetochiral anisotropy in Pt/PtMnGa bilayers Cite as: Appl. Phys. Lett. 118, 252403 (2021); doi: 10.1063/5.0054662 Submitted: 20 April 2021 .Accepted: 4 June 2021 . Published Online: 21 June 2021 K. K. Meng,a) J. K. Chen, J.Miao, X. G. Xu, and Y. Jianga) AFFILIATIONS School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China a)Authors to whom correspondence should be addressed: kkmeng@ustb.edu.cn and yjiang@ustb.edu.cn ABSTRACT Nonreciprocal charge transport, which is frequently termed as electrical magnetochiral anisotropy (EMCA) in chiral conductors, touches the most important elements of modern condensed matter physics. Here, we have investigated the large EMCA in Pt/PtMnGa (PMG) bilayers,which can be attributed to the nonreciprocal response of an interface-driven chiral transport channel. Different from the traditional linearcurrent-dependent EMCA, for Pt/PMG bilayers, higher-order EMCA coefficients should be phenomenally added especially for the small current region. This unusual behavior has been explained based on both quantum transport and semiclassical transport models. Furthermore, a combination of asymmetrical electron scattering and spin-dependent scattering furnish the PMG thickness-dependent chiraltransport behaviors in Pt/PMG bilayers. The dramatically enhanced anomalous Hall angle of PMG further demonstrates the modified surfacestate properties by strong spin–orbit coupling. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054662 Nonreciprocal response that excitations can only propagate in one direction of the materials system with broken inversion providesrich physics and functionalities. 1–4Two well-known examples are the integer quantum Hall effect (QHE) that low-energy excitations canpropagate only clockwise or counterclockwise, and the diode in semi-conductor p–n junction. 5Such nonreciprocal response can occur ubiquitously in various systems with noncentrosymmetry when thetime-reversal symmetry is further broken by applying a magnetic fieldor with spontaneous magnetization, which is frequently termed aselectrical magnetochiral anisotropy (EMCA) for electronic and mag-netotransport in chiral conductors. 6–14The EMCA has been phenom- enologically discussed by Rikken, who provided a description for thedirectional transport in the noncentrosymmetric system by generaliz-ing Onsager–Casimir reciprocal relations as follows: 2 RI;BðÞ ¼R01þbB2þcI/C1B/C0/C1 ; (1) where R0,I,a n d Brepresent the resistance at zero magnetic field, the electric current, and magnetic field, respectively. The parameter b describes the normal magnetoresistance that is allowed in all conduc-tors, and cis a coefficient tensor, representing the magnitude of the nonreciprocal resistance. From the viewpoint of not only fundamentalphysics but also applications, the nonreciprocal transport has beeninvestigated in various kinds of quantum materials, i.e., molecular con-ductor, 6chiral magnets,7,8polar semiconductor,9superconductor,10noncentrosymmetric oxide interfaces,11heavy metal/ferromagnet (HM/FM) bilayers,12and nonmagnetic/magnetic topological insulator bilayers.13,14 Recently, Mn-based compounds have attracted much attention since a wealth of underlying and interesting physics have beenexplored, such as large anomalous Hall effect (AHE) in noncollinearantiferromagnet Mn 3Sn,15current-induced magnetization switching in single antiferromagnet CuMnAs,16Weyl semimetal phase in ferro- magnet Co 2MnAl,17skyrmions in MnSi, and antiskyrmions in Mn–Pt–Sn.18,19All the above phenomena illustrate the unique merits of Mn-based compounds for various kinds of applications, especiallyfor spintronics devices. Here, we have used Pt/PtMnGa (PMG) hetero-structures to investigate the EMCA, which can be attributed to the nonreciprocal response in a chiral transport channel at the Pt/PMG interface due to the combination of strong spin–orbit coupling (SOC)of Pt and PMG with strong magnetic anisotropy. In addition to cI/C1B, higher-order dependence terms should be taken into account inEq.(1), and all the coefficients depend on the current amplitude. To explain this unusual nonlinear current-dependent EMCA, we have introduced the fluctuation theorems based on the quantum transport model and the emergence of Lorentz force based on the semiclassicaltransport model. Furthermore, a combination of asymmetrical elec-tron scattering and spin-dependent scattering furnishes thePMG thickness-dependent nonreciprocal transport behaviors in the Appl. Phys. Lett. 118, 252403 (2021); doi: 10.1063/5.0054662 118, 252403-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplPt/PMG films, exhibiting distinct angular dependences. Through AHE measurements, we have found that both extrinsic skew scattering and intrinsic terms have been dramatically altered, which results in anenhanced anomalous Hall angle in the PMG capped with the Pt layerand further demonstrates the modified quantum geometrical nature at the Pt/PMG interface. Sample preparation, structural characterization, magnetic prop- erties, and the discussion of possible strong interfacial Dzyaloshinskii–Moriya interaction (DMI) are shown in supplementary material Note 1. A combination of the strong magnetic anisotropic PMG layer that breaks time-reversal symmetry, the noncentrosymmetric interfaces,and the strong spin–orbit coupling (SOC) introduced by the Pt layer may bring out a chiral transport channel at the Pt/PMG interface. 20,21 We thus anticipate that EMCA could emerge at the Pt/PMG interface in the presence of the magnetic field. To verify it, we have carried outthe harmonic measurements with lock-in techniques. Figure 1(a) shows the schematic for longitudinal ( R 2x) and transverse ( RH 2x)s e c - ond harmonic magnetoresistance measurements with rotating Bin x–y(a),z–y(b), and z–x(h) planes. In our experiments, an AC I¼I0sinxtof frequency x/2p¼125 Hz was applied, and the AC longitudinal ( V) and transverse ( VH) voltages were recorded, respectively. The longitudinal second harmonic resistance consists of three contributions, namely, the EMCA ( REMCA 2x), the magnetothermal effects due to the temperature gradients induced by Joule heating(R rT 2x), and the spin–orbit torque (SOT) induced modulation of the total magnetoresistance ( RSOT 2x), R2x¼REMCA 2xþRrT 2xþRSOT 2x: (2)D u et ot h es y m m e t r i cb e h a v i o ro ft h em a g n e t o r e s i s t a n c ew i t h respect to the x–yplane, the out-of-plane oscillations driven by the damping-like SOT do not contribute to R2x, while only Oersted and field-like effective fields contribute to R2x.A c c o r d i n gt ot h e work of Avci et al. ,12by taking into account the geometrical factor of the effective fields, the angular form of the SOT term is RSOT 2x/ðsinacos2aÞ.Figure 1(b) shows the in-plane ( x–y)fi e l d - angle ( a) dependence of R2xfor Pt (5 nm)/PMG (6 nm)/MgO (001) films with varying current from 1 to 5 mA and applying mag-n e t i cfi e l d9 Ta t5K .T h ed a t as h o was u p e r p o s i t i o no f nth power of cosine function, and nequals to a positive integer. As increasing current, the data show more cosine-like field-angle dependence with a period of 360 /C14.T h es i n acos2acontributions are negligible for all current range, indicating that the SOT term has not contrib- uted to R2x. In addition, we have found that RrT 2xis negligible (supplementary material Note 3). Therefore, it seems that the I/C1B¼IBcosaterm dominates the harmonic signals for larger cur- rent, indicating the emergence of EMCA in this film system. On the contrary, we have not found such transport behaviors in SiO 2 (2 nm)/PMG ( dnm)/MgO (001) films, indicating the important role of Pt. According to the EMCA model of Rikken et al. ,IandB should be integrally related to each other as ðI/C1BÞ,a n dt h e cI/C1Bis a small term. We, therefore, infer that the two-terminal resistance of Pt/PMG films can be expressed as a polynomial of ðI/C1BÞ, RðI/C1BÞ¼Rð0Þþc1I/C1Bþ1 2!c2ðI/C1BÞ2þ1 3!c3ðI/C1BÞ3þ/C1/C1/C1 ;(3) where Rð0Þrepresents the resistance that does not depend on ðI/C1BÞ. For harmonic measurements, one has the longitudinal voltage as FIG. 1. (a) Schematic for longitudinal (R2x) and transverse ( RH 2x) second har- monic magnetoresistance measurements with rotating Binx–y(a),z–y(b), and z–x (h) planes. The AC is applied along x- direction. (b) Angle ( a) dependence of the longitudinal second harmonic resistance R2xwith varying applied AC in Pt (5 nm)/ PMG (6 nm)/MgO (001) films under 9 T at5 K. Solid lines are fits to the experimentaldata according to Eq. (4). (c) Magnetic field dependence of R 2xwith varying applied AC at 5 K. Solid lines are fitted tothe experimental data according to Eq.(4). The inset shows the enlarged curves with 5 mA at around zero magnetic field. (d) AC amplitude dependence of R 2x under 9 T at 5 K.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252403 (2021); doi: 10.1063/5.0054662 118, 252403-2 Published under an exclusive license by AIP PublishingVIðÞ¼IR; ¼R0ðÞI0sinxtþc1I2 0Bcosasin2xt þ1 2!c2I3 0B2cos2asin3xtþ1 3!c3I4 0B3cos3asin4xtþ/C1/C1/C1 ¼R0ðÞI0sinxtþc1I2 0Bcosasin2xt þ1 2!c2I3 0B2cos2asinxt1/C0cos2xt ðÞ þ1 3!c3I4 0B3cos3asin2xt1/C0cos2xt ½/C138 þ/C1/C1/C1 /C25R0ðÞþ1 2!c2I2 0B2cos2a/C20/C21 I0sinxt þc1Bcosaþ1 3!c3I2 0B3cos3a/C18/C19 I2 0sin2xtþ/C1/C1/C1 /C25R0ðÞþ1 2!c2I2 0B2cos2a/C20/C21 I0sinxt þ1 2c1Bcosaþ1 3!c3I2 0B3cos3a/C18/C19 I2 01þsin 2 xt/C0p 2/C18/C19 /C20/C21 þ/C1/C1/C1 : (4) The phases of lock-in amplifiers were set to 0/C14and/C090/C14for the first and second harmonic signal measurements, respectively. Then the first and second harmonic resistances are /C24ðRð0Þþ1 2!c2I2 0B2cos2aÞ and /C241 2c1I0Bcosaþ1 3!c3I3 0B3cos3a/C0/C1, respectively. Considering Rð0Þ/C291 2!c2I2 0B2cos2a, the first harmonic resistance only reveals com- mon anisotropic magnetoresistance ( supplementary material Note 4). It should be noted that one can add much higher power terms into thesecond harmonic resistance, but only odd terms contribute accordingto our calculation. Then the data can be fitted well if the R 2xwas expressed as a superposition of odd power of cosine function.However, this conjecture cannot fully explain the phenomena sincehigher power terms become more important for smaller current ( sup- plementary material Note 5). Therefore, we should note that all the coefficients c iði¼1;2;3;…Þare functions of current. Based on this conjecture, the R2x-Bdata with sweeping magnetic field parallel to current directions ( x) at 5 K can also be fitted well as shown in Fig. 1(c) . It should be noted that in low field range (/C02T<B<2T ) , R2xare almost zero, deviating from the fitted lines. As mentioned in the supplementary material Note 1, we have specu- lated an interfacial strong D12/C1ðS1/C2S2Þinduced EMCA at the Pt/ PMG interface due to the strong SOC of Pt and PMG film with strongmagnetic anisotropy. However, it should be noted the formation oflong-range ordered noncoplanar magnetic configurations ( S 1/C2S2) may need a large magnetic field according to the in-plane M–Bcurves as shown in the supplementary material Fig. S2. Therefore, evident EMCA in the Pt/PMG films (nonzero R2x) needs applying a large magnetic field and a finite magnetization ( supplementary material Note 6). A similar nonmonotonic magnetic field dependence of R2x has also been observed in chiral magnets MnSi and CrNb 3S6,7,8and the current dependence of R2xshowed linear behaviors, which were ascribed to chiral spin fluctuations at specific temperature-magneticfield-pressure regions. On the other hand, as mentioned above, the spin correlation with vector spin chirality in chiral magnets scatters electrons asymmetrically, resulting in the EMCA. 20,21As i m i l a rs c a t - tering picture could also be depicted at the Pt/PMG interfaces. On theother hand, considering the EMCA is an interface-driven phenome- non, the increased current shunting effect for the thicker PMG films could quench the EMCA ( supplementary material Note 7). However, why high-power terms become more evident for the smaller current? One possible reason for this unusual nonlinear current-dependent EMCA is the emergence charge fluctuations in the chiral transport channel at the Pt/PMG interface. Generally, the non- equilibrium fluctuation theorems can provide the basis of the nonrecip- rocal transport beyond the Onsager–Casimir reciprocal relations.22,23 The linear response theory founded on Onsager–Casimir reciprocal relations provides a powerful tool to describe a variety of physical sys- tems.24–27However, such relations are justified only when the systems are close to thermodynamic equilibrium. In the regime of nonequilib- rium (e.g., in nonreciprocal transport), the fluctuation theorem is pro- posed to play a more generalized role.28–30It can give the relation between the nonreciprocal transport coefficients and current noise in quantum transport of a mesoscopic conductor.31–33Both the current I and current noise power Spassing through a conductor can be expressed as a polynomial of the bias voltage VasI¼G1Vþ1 2!G2V2 þ1 3!G3V3þ/C1/C1/C1 and S¼S0þS1Vþ1 2!S2V2þ/C1/C1/C1 .G e n e r a l l y ,i n mesoscopic experiments, large bias voltages can easily induce nonlinear transport, and average current does not satisfy the Onsager–Casimirrelation. On the contrary, for the Pt/PMG films, higher-order coeffi- cients are much more crucial for lower current. According to the fluc- tuation theorem, Sis minimal at a nonzero voltage since the coefficient is nonzero in general. Therefore, the current noise could be infinite by applying a very small voltage. On the other hand, for purely elastic transport, the scattering matrix is a function of both the energy of car- riers and the potential landscape of the conductor. Away from equilib-rium, the potential should depend on the voltages applied to the leads. However, the calculation of quantum scattering problem is in general a very difficult problem. Based on our experiments, we can only give phe- nomenological conclusion that the fluctuation theorems provide the basis of current dependent c i. However, we want to mention that it is hard to give an accurate explanation that why the charge fluctuation or current noise was so large in the low current region of Pt/PMG films.The length of our Hall device is /C24100lm, which should be much l a r g e rt h a nt h em e a nf r e ep a t h ,s ot h em e s o s c o p i cv i e wo fe l e c t r o n i c motion is that it should be “diffusive”! 34,35Therefore, the applications of fluctuation theorems to the bulk transport phenomena need more theoretical and experimental works. Another possible explanation could be carried out according to the semiclassical transport model. The presence of both electric and magnetic fields means that the force on an electron is now f¼/C0eE/C0ev/C2B,s oi n k-space, one can get /C22hdk dt¼/C0eE/C0ev/C2B. When the current is parallel to the magnetic field, denoted as a¼0i n our work, there is no Lorentz force. However, when the sample is rotated, there should be a component of Bthat is perpendicular to the direction of electrons motion. Therefore, there will be a combinationof rectilinear and circular motion (in x–zplane) of the electrons in the chiral transport channel as shown in Fig. S10 of the supplementary material , and the cyclotron frequency is x c¼eB m/C3,w h e r e m/C3is the effective mass of electrons. Therefore, the circular motion will enhance the motion path and the probability of random scattering, resulting in a suppressed EMCA. When the current is very small, which means smaller electric field in the x-direction or smaller kinetic energy 1 2m/C3v2, the component of rectilinear motion becomes weaker, and theApplied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252403 (2021); doi: 10.1063/5.0054662 118, 252403-3 Published under an exclusive license by AIP Publishingrandom scattering during the circular motion results in a suppressed EMCA as shown in Fig. S9(a) of the supplementary material .O nt h e other hand, the existence of Lorentz force can also explain the nonlin- earR2x/C0Bcurves as shown in Fig. 1(c) . The Coulomb scattering due to electrons or ionized impurity will lead to a transverse electron motion in a plane (such as y–zplane) that is perpendicular to the mag- netic field along x-direction.36The cyclotron radius ris proportional to 1=B, so the smaller magnetic field will result in a larger r,l e a d i n gt o a longer motion path and an enhanced random scattering. Therefore, the cyclotron motion in this plane will also suppress the EMCA. It can also explain why the R2x/C0BinFig. 1(c) can be roughly considered as a nearly linear relationship at large magnetic field, while the nonlinear phenomenon is more evident in small field range. The smaller mag- netic fields suppress the EMCA more dramatically due to larger r.O n the other hand, if one applied larger current, the electrons with higher kinetic energy have long momentum relaxation times, because rapidly moving electrons are deflected less for Coulomb scattering mecha- nisms.36Therefore, although this kind of cyclotron motion will lead to an o n l i n e a r R2x/C0Bcurve, the appearance of EMCA needs smaller magnetic field for larger current as compared with the smaller current case as shown in Fig. 1(c) . Both quantum transport and semiclassical transport model may influence the phenomenon in our work, and more theoretical works are needed. Furthermore, nonlinear planar Hall effect is another proof to ver- ify the emergence of nonreciprocal charge transport, though different determined mechanisms have been reported and remain elusive.14,37 Figure 2(a) shows the in-plane ( x–y)fi e l d - a n g l e( a) dependence of sec- ond transverse harmonic magnetoresistance RH 2xfor Pt (5 nm)/PMG (6 nm)/MgO (001) films with varying current from 1 to 5 mA and applying magnetic field 9 T at 5 K. Similar with second longitudinal harmonic magnetoresistance, the nonreciprocal transport coefficients ofRH 2xalso depend on current amplitudes and higher-order terms become more crucial for smaller current. Meanwhile, it can also prove that the SOT contribution in Pt (5 nm)/PMG (6 nm)/MgO (001) films is negligible since SOT will be quenched by applying large magnetic fields. On the contrary, RH 2xwith varying current shows nonmono- t o n i cm a g n e t i cfi e l dd e p e n d e n c ea t5 Ka ss h o w ni n Fig. 2(b) ,s i m i l a r with R2x–Brelationships. On the other hand, Fig. 2(c) shows that RH 2x also has nonlinear relationship with current at 5 K. Therefore, neither asymmetric magnon-mediated scattering nor concerted actions of spin-momentum locking and time-reversal symmetry breaking areapplicable to the nonlinear planar Hall effect in Pt (5 nm)/PMG (6 nm)/MgO (001) films.13,14,38In order to gain further insight into the EMCA of the Pt/PMG films, we have also investigated the AC transport properties in Pt (5 nm)/PMG (2 nm)/MgO (001) films, in which contribution from unidirectional magnetoresistance (UMR) should be added ( supplementary material Note 7). A combination of asymmetrical electron scattering and spin-dependent scattering fur- nishes the PMG thickness-dependent chiral transport behaviors in Pt/ PMG bilayers. Microscopically, in chiral transport systems, the nonreciprocal transport phenomena are frequently encoded by the Berry phase and t h ea s y m m e t r i cs c a t t e r i n g ,4which can be analyzed through discussing the AHE contributions.39Therefore, we have further investigated the AHE in Pt ( tnm)/PMG ( dnm)/MgO (001) films. By subtracting the ordinary Hall term from the total Hall resistivity, we have obtained the anomalous Hall resistivity qHof all the multilayers under perpen- dicularly applied magnetic field Bin the temperature range from 5 to 400 K ( supplementary material Note 9). It should be noted that assum- ing that each film in the multilayers acts as a parallel resistance path,40 both anomalous Hall resistivity qHand longitudinal resistivity qxx have been expressed as those of the PMG layers. Figure 3(a) shows the representative relationships between qHand the square of longitudinal resistivity q2 xxin Pt ( tnm)/PMG (6 nm)/MgO films, in which the sam- ple of t¼0n m r e f e r s t o S i O 2(2 nm)/PMG (6 nm)/MgO films. It is found that the experimental data can be described by scaling law qH¼aqxx0þbq2 xx,w h e r e qxx0is the residual resistivity induced by impurity scattering, aand bdenote extrinsic skew scattering and intrinsic mechanism, respectively41We have found that qHis signifi- cantly increased after capping Pt layers, indicating an enhanced anom- alous Hall angle. The Pt thickness dependent aandbin Pt ( tnm)/ PMG (6 nm)/MgO films is shown in Fig. 3(b) .B o t h aand absolute value of bincrease sharply when tis varied from 0 to 1.5 nm, and then the values keep almost constant with further increasing t.I ti n d i c a t e s that both skew scattering and intrinsic mechanism related to the Berry phase contributions have been enhanced after capping Pt layers, which should be determined by the strong interface SOC introduced by Pt. This behavior has also been found in Pt ( tnm)/PMG (2 nm)/MgO (001) films as shown in Figs. 3(c) and3(d), but the enhancement of qHis much more significantly especially for low-temperature regions as compared with Pt ( tnm)/PMG (6 nm)/MgO (001) films. Therefore, the Pt layer has dramatically modified the surface state properties of FIG. 2. (a) Angle ( a) dependence of the transverse second harmonic resistance RH 2xwith varying applied AC in Pt (5 nm)/PMG (6 nm)/MgO (001) films under 9 T at 5 K. (b) Magnetic field dependence of RH 2xwith varying applied AC at 5 K. (c) AC amplitude dependence of RH 2xunder 9 T at 5 K.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252403 (2021); doi: 10.1063/5.0054662 118, 252403-4 Published under an exclusive license by AIP PublishingPMG. Combinated with PMG with strong magnetic anisotropy, the strong SOC may induce a chiral transport system at the interface. In conclusion, large EMCA has been found in Pt/PMG films. In the presence of large magnetic field, the coefficients of EMCA in interface-driven chiral transport channels are functions of current. With varying the PMG thickness, the EMCA in the Pt/PMG films exhibits distinct nonreciprocal transport behavior that is ascribed to the competition of asymmetrical electron scattering and spin- dependent scattering. The modified Berry phase and scattering proper- ties due to the strong SOC of Pt at the interface have also been demonstrated through the AHE measurement. Our results suggest this improved explanation of EMCA may be applicable to the chiral trans- port properties in a variety of quantum materials. See the supplementary material for a more detailed discussion. This work was partially supported by the National Key Research and Development Program of China (No. 2019YFB2005801), the National Natural Science Foundation of China (Grant Nos. 51971027, 51731003, 51971023, 51927802, 51971024, and 52061135205), Beijing Natural Science Foundation Key Program (Grant No. Z190007), and the Fundamental Research Funds for the Central Universities (Grant Nos. FRF-TP-19-001A3, FRF-MP-19-004, FRF-BD-20-06A, and FRF-BD-19-010A). DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1G. L. J. A. Rikken and E. 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Jpn. 12, 570–586 (1957). FIG. 3. (a)qH/C0q2 xxin Pt ( tnm)/PMG (6 nm)/MgO (001) films with different t. The sample of t¼0 nm refers to SiO 2 (2 nm)/PMG (6 nm)/MgO (001) films. Solid lines are fitted to the experimental dataaccording to scaling law q H¼aqxx0 þbq2 xx. (b) Pt thickness ( t) dependence of coefficients aand bin Pt ( tnm)/PMG (6 nm)/MgO (001) films. (c) qH/C0q2 xxin Pt (tnm)/PMG (2 nm)/MgO (001) films with different t. The sample of t¼0n m refers to SiO 2(2 nm)/PMG (2 nm)/MgO (001) films. Solid lines are fitted to theexperimental data according to the abovescaling law. (d) Pt thickness ( t) depen- dence of coefficients aandbin Pt ( tnm)/ PMG (2 nm)/MgO (001) films.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252403 (2021); doi: 10.1063/5.0054662 118, 252403-5 Published under an exclusive license by AIP Publishing28K. Saito and Y. Utsumi, Phys. Rev. B 78, 115429 (2008). 29Y. M. Blanter and M. B €uttiker, Phys. Rep. 336, 1–166 (2000). 30D. 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cjcp2009170.pdf

Chin. J. Chem. Phys. 34, 273 (2021); https://doi.org/10.1063/1674-0068/cjcp2009170 34, 273 © 2021 Chinese Physical Society.Huge tunneling magnetoresistance in magnetic tunnel junction with Heusler alloy Co2MnSi electrodes Cite as: Chin. J. Chem. Phys. 34, 273 (2021); https://doi.org/10.1063/1674-0068/cjcp2009170 Submitted: 24 September 2020 . Accepted: 22 October 2020 . Published Online: 21 July 2021 Yu-jie Hu , Jing Huang , Jia-ning Wang , and Qun-xiang Li ARTICLES YOU MAY BE INTERESTED IN Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Recent progress and challenges in magnetic tunnel junctions with 2D materials for spintronic applications Applied Physics Reviews 8, 021308 (2021); https://doi.org/10.1063/5.0032538 Quantum-spin-Hall phases and 2D topological insulating states in atomically thin layers Journal of Applied Physics 129, 090902 (2021); https://doi.org/10.1063/5.0029326CHINESE JOURNAL OF CHEMICAL PHYSICS VOLUME 34, NUMBER 3 JUNE 27, 2021 ARTICLE Huge Tunneling Magnetoresistance in Magnetic Tunnel Junction with Heusler Alloy Co 2MnSi Electrodes Yu-jie Hua;Jing Huangb∗;Jia-ning Wanga;Qun-xiang Lia∗ a. Department of Chemical Physics &Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China b. School of Materials and Chemical Engineering, Anhui Jianzhu University, Hefei 230601, China (Dated: Received on September 24, 2020; Accepted on October 22, 2020) Magnetic tunnel junction with a large tunneling mag- netoresistance has attracted great attention due to its importance in the spintronics applications. By performing extensive density functional theory calculations combined with the nonequilibrium Green's function method, we explore the spin-dependent transport properties of a magnetic tunnel junction, in which a non-polar SrTiO 3barrier layer is sandwiched between two Heusler alloy Co 2MnSi electrodes. Theoretical results clearly reveal that the near perfect spin-ltering eect appears in the parallel magnetization conguration. The transmission coecient in the parallel magnetization conguration at the Fermi level is several orders of magnitude larger than that in the antiparallel magnetization conguration, resulting in a huge tunneling magnetoresistance ( i.e.>106), which originates from the coherent spin-polarized tunneling, due to the half-metallic nature of Co 2MnSi electrodes and the signicant spin-polarization of the interfacial Ti 3d orbital. Key words: Magnetic tunnel junction, Spin-dependent transport, First-principles, Tunnel- ing magnetoresistance I. INTRODUCTION Magnetic tunnel junction (MTJ), in which a non- polar band insulator is sandwiched between two ferro- magnetic electrodes, has attracted great attention for years since it plays an important role for applications in spintronics. Generally, the tunneling resistance varies as a function of the relative magnetic conguration of the electrodes, resulting in an eect called tunneling magnetoresistance (TMR) [1]. The TMR eect is of Authors to whom correspondence should be addressed. E-mail: jhuang@ustc.edu.cn, liqun@ustc.edu.cngreat importance in spintronics, such as magnetic sen- sor and magnetic random access memory [2, 3]. Previ- ous experimental and theoretical investigations mainly focused on magnetic metals ( i.e. Fe, Co, Ni) and mag- netic alloys like CoFeB as electrodes in various magnetic tunnel junctions [4{10]. Note that one central issue in MTJs is how to eciently inject spin polarized elec- trons from the ferromagnetic electrodes into the sand- wiched insulating layer. The low spin polarizability of ferromagnetic metal will lead to low spin injection ef- ciency. Therefore, one eective way to improve the performance of TMJs is using the half-metallic ferro- magnets (HMFs), i.e. Fe 3O4[11] and La 0:7Sr0:3MnO 3 [12], as electrodes [13{15], since the HMFs carry cur- DOI:10.1063/1674-0068/cjcp2009170 273 c⃝2021 Chinese Physical Society274 Chin. J. Chem. Phys., Vol. 34, No. 3 Yu-jie Hu et al. rent in only one spin channel, leading to complete spin polarization at the Fermi level [16], greatly enhancing the spin injection eciency. For example, in experi- ments, the MTJ with manganite La 0:7Sr0:3MnO 3elec- trodes and SrTiO 3barrier layer has achieved a large TMR ratio of 1800% at low temperature [17]. As one kind of half-metallic materials, Heusler com- pounds (especially, Co-based full-Heusler alloys) have been widely used as electrodes in MTJs, which display attractive performance, i.e. the high TMR eect of sev- eral hundred percent [18{20]. Due to the high Curie temperature ( 985 K) above room temperature and the nearly perfect spin polarized around the Fermi level [21], half-metallic Co-based full-Heusler alloys hold the most potential for applications in spintronics devices, and they have been widely conducted on TMR measure- ments for various MTJs with AlO xor MgO barrier layer and Co-based Heusler alloys, such as, Co 2Fe(Al 0:5Si0:5), Co2MnSi, Co 2MnAl, and so on [22{26]. Recently, Rout et al. have successfully grown L21- type Co 2MSi (M=Mn and Fe) on a variety of semicon- ductors and oxide dielectrics ( i.e. SrTiO 3) and mea- sured two-dimensional electron-gas-like charge trans- port at the interface between a Heusler alloy Co 2MSi (M=Mn or Fe) and SrTiO 3in their experiments [27]. Nazir et al. recently performed spin-polarized den- sity functional theory calculations on the structural and charge transfer at the TiO 2terminated interfaces between the magnetic Heusler alloys Co 2MSi (M=Ti, V, Cr, Mn, and Fe) and the non-polar band insulator SrTiO 3(barrier layer) [28]. However, theoretical spin- dependent transport investigation of these correspond- ing MTJs is lacking so far. In this work, we explore the spin-dependent trans- port properties of Co 2MnSi/SrTiO 3/Co 2MnSi MTJs with dierent interfaces by performing extensive den- sity functional theory (DFT) calculations combined with the nonequilibrium Green's function (NEGF) tech- nique. According to the calculated zero-bias transmis- sion spectra of the MTJ with the MnSi-TiO 2interface, we nd that the transmission coecient at the Fermi level in the parallel magnetization conguration (PC) is several orders of magnitude larger than in the an- tiparallel magnetization conguration (APC), resulting in a huge TMR ( i.e.>106), which is signicantly larger than previous reports for MTJs with Co 2MnSi elec- trode or SrTiO 3barrier layer [14, 26, 29, 30]. More- over, the nearly perfect spin-ltering eect is observedin the examined MTJ in the PC. Through analysing the projected density of states (DOS) of MTJ, the par- tial DOS of the atoms in the interface, and the in-plane wave vector dependence of transmission spectra, we nd that the huge TMR originates from the coherent spin- polarized tunneling, due to the half-metallic nature of Heusler alloy Co 2MnSi electrodes and the signicant spin-polarization of the interfacial Ti 3d orbital. II. COMPUTATIONAL METHODS The geometric optimizations and electronic struc- ture calculations in this work are carried out by using DFT within Perdew-Burke-Ernzerh (PBE) of the gen- eralized gradient approximation (GGA) exchange cor- relation functional implemented in the Vienna ab initio simulation package (VASP) [31{33]. We adopt a plane- wave basis adjusted by expanding the Kohn-Sham or- bitals with a 520 eV kinetic energy cuto. A k-mesh of 10101 is used, and the Hellmann-Feynman forces acting on each atom are less than 0.02 eV/ A for geo- metric optimization. The spin-dependent transport properties of these ex- amined MTJs are explored by performing DFT calcula- tions combined with the NEGF technique, implemented in the ATK package [34, 35]. In our calculations, the GGA in the PBE form is used to describe the exchange and correlation energy, the double-zeta polarized ba- sis sets are adopted for all atoms, a Monkhorst-Pack k-mesh of 10 10100 is used to converge the density matrix, a 100 100k∥meshes is adopted to calculate transmission coecient, and a cuto energy is set to be 160 Ry for the real-space grid. The spin-dependent con- ductance per unit cell is given by the Landauer-B uttiker formula, G=e2 h∑ k∥;jT+(k∥; j) (1) here, stands for the spin-up ( ") and spin-down ( #) channels, jrepresents the Bloch state for a given value ofk∥=kx+ky, and T+(k∥; j) represents the transmission probability of an electron at the Fermi level with spin () and the Bloch wave vector ( k∥). Then, the TMR ratio at zero bias voltage is dened as TMR =GPCGAPC GAPC100% (2) where, GPCand GAPC stand for the conductance of DOI:10.1063/1674-0068/cjcp2009170 c⃝2021 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 34, No. 3 Huge Tunneling Magnetoresistance in Junction 275 FIG. 1 (a) Atomic lattice of bulk Co 2MnSi, and the top view of Co 2-terminated and MnSi-terminated surfaces. (b) Atomic lattice of bulk SrTiO 3, and the top view of SrO-terminated and TiO 2-terminated surfaces. (c) The partial DOS of Co 3d and Mn 3d orbitals in bulk Co 2MnSi, labeled with the red and black lines, respectively. (d) The partial DOS of Ti 3d (red line) and O 2p (black line) orbitals in bulk SrTiO 3. magnetic tunnel junction in the parallel magnetization conguration, and antiparallel magnetization congu- ration, respectively. III. RESULTS AND DISCUSSION Before exploring spin-dependent transport proper- ties of Co 2MnSi/SrTiO 3/Co 2MnSi MTJs, we rstly ex- amine the geometric and electronic properties of bulk Co2MnSi and SrTiO 3. FIG. 1(a) shows the optimized geometric structure of bulk Co 2MnSi. The optimized lattice constant is about 3.974 A, corresponding to a0/p 2 of cubic L21-type Co 2MnSi. The spin-polarized parital DOS of Co 3d and Mn 3d orbitals of bulk Co2MnSi are plotted in FIG. 1(c) with the red and black lines, respectively. Clearly, one can observe half- metallicity for bulk Co 2MnSi. That is to say, the spin- up channel, contributed by the Co 3d and Mn 3d or- bitals, is metallic, while the spin-down channel is insu- lating with a band gap of 0.5 eV. The atomic magnetic moments of Co and Mn atom are predicted to be 1.0 and 3.0B. While for bulk SrTiO 3, as shown in FIG. 1(b), the lattice constant is 3.914 A. The calculated partial DOS of Ti 3d and O 2p orbitals plotted in FIG. 1(d) show that SrTiO 3is a nonmagnetic semiconductor with a band gap of 1.7 eV, since the spin-up DOS coincides exactly with the spin-down DOS. These structural pa- rameters and electronic structures of bulk Co 2MnSi and SrTiO 3agree well with previous experimental and the-oretical reports [36{38]. There are two dierent terminations for Co 2MnSi and SrTiO 3along cdirection (transport direction). Namely, Co2-terminated and MnSi-terminated for Heusler al- loy Co 2MnSi electrode, SrO-terminated and TiO 2- terminated for SrTiO 3barrier layer, are illustrated in FIG. 1 (a) and (b), respectively. Therefore, we con- struct four possible Co 2MnSi/SrTiO 3/Co 2MnSi MTJs with the MnSi-TiO 2, MnSi-SrO, Co 2-SrO, and Co 2- TiO 2interfaces, which are named with MTJ1, MTJ2, MTJ3 and MTJ4 for short, respectively. Due to the slight dierent lattice constants, there is a small in- plane lattice mismatch of 1.5% between Co 2MnSi elec- trodes and SrTiO 3barrier layer. To nd the most en- ergetically stable junction, we calculate the interfacial energy ( Eint), which is dened as Eint=Etot∑ Nii (3) here, Etotstands for the total energy, Niis the atomic number of each element, and iis the atomic chemical potential. The optimized interfacial distances and the calculated interfacial energies are summarized in Ta- ble I. It is clear that MTJ1 is the most stable junction with the MnSi-TiO 2interface, in which Mn atoms sit on the hollow site of TiO 2layer with the relaxed ver- tical distance of 1.8 A, as shown in FIG. 2(a). This kind of MSi-TiO 2(M=Fe or Mn) interface has been ex- amined by scanning transmission electron microscopy DOI:10.1063/1674-0068/cjcp2009170 c⃝2021 Chinese Physical Society276 Chin. J. Chem. Phys., Vol. 34, No. 3 Yu-jie Hu et al. FIG. 2 (a) The optimized atomic structure of MTJ1, in which a SrTiO 3barrier layer (4.5 unitcells) is sandwiched between two Heusler alloy Co 2MnSi electrodes with the MnSi-TiO 2interface. (b) The zero-bias transmission spectra of MTJ1 in the PC (parallel magnetization conguration) and APC (antiparallel magnetization conguration), labeled with the red and black lines, respectively. The spin-resolved transmission spectra of MTJ1 in (c) PC and (d) APC, here, the blue and black lines stand for the spin-up and spin-down electrons, respectively. imaging [27]. In our calculations, the proposed MTJ can be divided into three parts: the central scattering region, left and right Co 2MnSi electrodes. The scatter- ing region consists of 9 atomic layers of SrTiO 3and 13 atomic layers of two electrodes. Then, we take MTJ1 as an example to explore the spin-dependent transport properties of Co 2MnSi/SrTiO 3/Co 2MnSi MTJs. To investigate the spin-dependent transport proper- ties of MTJ1, we calculate their zero-bias transmission spectra and plot them in FIG. 2(b), here, the red and black lines stand for the transmission spectra for the PC and APC, respectively. It is clear that the zero- bias transmission spectra of two magnetization cong- urations show remarkable dierent feature. The trans- mission curve of the PC (red line) is smooth around the Fermi level, and the transmission coecients are sev- eral orders of magnitude larger than that of the APC (black line) in the relative wide energy range, i.e. from 0.4 eV to 0.4 eV. At the Fermi level, the transmission coecients of the PC and APC are predicted to be 0.04 and 1.18 10−7, respectively. This remarkable trans- mission coecient dierence between the PC and APC results in a huge TMR ratio of 3.08 107%, according to Eq.(2). It should be pointed out that this TMR ratio is generally larger than the experimental results for the similar junctions [17, 26]. One most possible reason isthat in experiments the existence of defects ( i.e. oxygen vacancy, substituting, and doping) cannot be avoided, which aects the performance of the MTJ. For exam- ple, the introduction of oxygen vacancies destroys the half-metallicity of HMFs, then reduces spin injection eciency [27, 39]. FIG. 2 (c) and (d) show the zero-bias transmission spectra of MTJ1 in the PC and APC, respectively, here, the red and black lines stand for the spin-up and spin- down electrons. Clearly, we observe a nearly perfect spin-ltering eect in MTJ1 with the PC. In the energy range from 0.3 eV to 0.5 eV, the spin-up transmis- sion coecients are signicantly larger than that of the spin-down electrons. At the Fermi level, the transmis- sion coecient for the spin-down electrons is close to zero, i.e. less than 10−8, while for the spin-up channel, the transmission coecient is about 0.04. That is to say, the spin transport properties of MTJ1 is dominated by the spin-up electrons. This low-bias transport proper- ties governed by the spin-up electrons have been ob- served in various MTJs, i.e. MgO barrier-based MTJs and half-metal electrodes-based MTJs [7, 11]. As for MTJ1 in the APC, the shapes of the spin-up transmis- sion spectra almost coincides with that of the spin-down electrons. The transmission coecient of the spin-down electrons is about 5.89 10−8at the Fermi level, which DOI:10.1063/1674-0068/cjcp2009170 c⃝2021 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 34, No. 3 Huge Tunneling Magnetoresistance in Junction 277 FIG. 3 Spin-resolved projected DOS of MTJ1 in (a) PC and (b) APC along the transport direction, here, left and right panels stand for the spin-up and spin-down electrons, respectively, and the white dashed line labels for the Fermi level for clarity. is signicantly less than that of the spin-up electrons in the PC (0.04). To explore the nature of the above dramatically dif- ferent spin-dependent transport process in MTJ1 with two dierent magnetization congurations, we calcu- late the spin-resolved projected DOS along the trans- port direction, as shown in FIG. 3 (a) and (b) for the PC and APC, respectively. As for the PC case, the right and left electrodes have spin-up electronic states at the Fermi level, but for the spin-down electrons, no electronic states of two electrodes appear, due to the half-metallic nature of Heusler alloy Co 2MnSi electrode. Moreover, one can see that the Fermi level crosses spin- up electronic states of SrTiO 3barrier layer. These ob- servations show that the transport process is dominated by the spin-up electrons, meaning that the spin injec- tion eciency is close to 100%. In FIG. 3(b), for the APC, there are several spin-up and spin-down electronic states at the Fermi level, note that, the spin-down elec- tronic states appear in the left electrode, which provides the spin-down electrons, but there are not spin-down electronic states in the right electrode, which can re- ceive the spin-down electrons from the left electrode. At the same time, the left electrode cannot provide the spin-up electrons for the right electrode to receive. So, the asymmetry distribution of the PDOS of the left and right electrodes, as shown in FIG. 3(b), hinders the elec- FIG. 4 (a) For MTJ1 in the PC, partial DOS of Co 3d and Mn 3d orbitals at the MnSi-TiO 2interface, labeled with the red and black lines, respectively. (b) Partial DOS of Ti 3d (red line) and O 2p (black line) orbitals. tron transport, which results in the transmission coe- cient in the APC being signicant less than that in the PC, as shown in FIG. 2(b), then a huge TMR appears. Previous experimental and theoretical investigations have shown that the spin transport properties of MTJ strongly depend on the local geometric distortion and electronic structures of the interface between electrode and barrier layer [40, 41], one needs to see the local geometries and electronic structures at the MnSi-TiO 2 interface in MTJ1 in more detail. Here, taking MTJ1 in the PC as an example, we calculate the partial DOS of Co 3d (red line) and Mn 3d (black line) orbitals in the MnSi-TiO 2interface, labeled with a circle in FIG. 2(a), and the partial DOS of Ti 3d (red line) and O 2p (black line) orbitals, and plot them in FIG. 4 (a) and (b), re- spectively. At the interface region, the Co Si bonds are compressed by 0.13 A, and the lengths of Ti O, TiSr, MnSi and Mn Co bonds are elongated by 0.13, 0.04, 0.17, and 0.25 A, respectively, compared to the corre- sponding bond lengths in bulk Co 2MnSi and SrTiO 3. It is clear that the interfacial electronic states are ob- viously modied due to the interface local structural relax. We observe a relative small decrease of the par- tial DOS of Co 3d orbital. The spin-up occupied elec- DOI:10.1063/1674-0068/cjcp2009170 c⃝2021 Chinese Physical Society278 Chin. J. Chem. Phys., Vol. 34, No. 3 Yu-jie Hu et al. tronic states and the spin-down conduction bands of Mn 3d orbital shift to lower energy. Due to the charge transfer (about 0.75 e) from two Co 2MnSi electrodes to SrTiO 3barrier layer through the interface, the mag- netic moment of Mn atoms increases by 1.1 Band the nonmagnetic Ti atoms in SrTiO 3bulk become magnetic in the interface (the Ti atomic moment magnetic mo- ment is about 0.1 B), and then the partial DOS of Ti 3d orbital are obviously spin-polarized, which plays an important role in spintronics applications. Most of these theoretical results for describing the MnSi-TiO 2 interface agree well with previous reports [28, 30]. FIG. 5 (a) and (b) illustrate the in-plane wave vector k||dependence transmission of MTJ1 in the PC at the Fermi level for the spin-up and spin-down electrons, re- spectively. Clearly, a broad peak is observed locating around k||=(0, 0), as shown in FIG. 5(a), indicating that the k||dependence of MTJ1 conductance of the spin-up channel originates from the coherent tunneling. In contrast, for the spin-down electrons, there is no peak atk||=(0, 0), while some spiky peaks appear, as shown in FIG. 5(b), and the corresponding transmission coef- cients are very small. The average transmission coef- cients over the Brillouin zone of the spin-up and spin- down electrons are about 0.04 and 5.37 10−12, respec- tively. This observation again veries that the trans- port behavior of MTJ1 in the PC is governed by the spin-up electrons, consisting with the results shown in FIG. 2(c). Previous investigations have reported that the TMR of MTJ depends on the barrier thickness [7, 42]. Here, we tune the thickness of SrTiO 3barrier layer in MTJ1, and the obtain results are similar to those plotted in FIG. 2(b) (d). Note that, the value of the TMR ratio is changed to be (2.82 107)% and (2.66 106)% for the 7 and 15 atomic barrier layers, respectively. That is to say, the huge TMR oscillates as a function of tuned barrier thickness. Similar oscillated TMR behavior has been experimentally observed in Fe/MgO/Fe TMJ [6], which implies that the huge TMR is coherent spin- polarized tunneling in MTJ1, and the SrTiO 3barrier layer plays a selective ltration role in transport. Finally, the calculated zero-bias transmission spectra of MTJ2, MTJ3 and MTJ4 are presented in FIG. 6, here, the red and black lines stand for the transmis- sion spectra for the PC and APC, respectively. It is clear that the spin-dependent transport properties of Co 2MnSi/SrTiO 3/Co 2MnSi MTJs are sensitive to FIG. 5 In-plane wave vector kjj=(kx,ky) dependence of the spin-up (a) and spin-down (b) transmission of MTJ1 in the PC at the Fermi level. TABLE I Calculated interfacial energies ( Eint) and the in- terfacial distance ( dint) for four examined MTJs. Structure Interface Eint/(eV/nm2) dint/A MTJ1 MnSi-TiO 2 78.94 1.8 MTJ2 MnSi-SrO 76.90 2.0 MTJ3 Co 2-SrO 70.00 2.6 MTJ4 Co 2-TiO 2 77.20 1.9 the interface between electrode and barrier layer. At the Fermi level, the transmission coecients for MTJ2, MTJ3, and MTJ4 in the PC are about 1.77 10−2, 4.7010−5, 1.42 10−3, while they are 1.02 10−8, 3.9010−8, 8.1410−11for the APC case, respectively. It is clear that the values of these transmission coef- cients depend mainly on the vertical distance at the MnSi-SrO, Co 2-SrO, and Co 2-TiO 2interfaces of 2.0, 2.6, and 1.9 A (summarized in Table I), respectively. Namely, a larger vertical distance results in a less trans- mission coecient. Then, the corresponding TMR ratio DOI:10.1063/1674-0068/cjcp2009170 c⃝2021 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 34, No. 3 Huge Tunneling Magnetoresistance in Junction 279 FIG. 6 Zero-bias transmission spectra of (a) MTJ2, (b) MTJ3, and (c) MTJ4 in the PC and APC, which are la- beled with the red and black lines, respectively. Here, the top panel illustrates the corresponding scattering region of three MTJs. is easily calculated to be (3.43 107)%, (9.57 104)%, and (1.31 107)% for MTJ2, MTJ3, and MTJ4. It should be pointed out that these huge TMR ratios are theoretically predicted for Co 2MnSi/SrTiO 3/Co 2MnSi MTJs with ideal single crystal structures, meaning that, to obtain high magnetoresistance, one has to improve the single crystal quality of Heusler alloy Co 2MnSi and SrTiO 3barrier layer in experiments.IV. CONCLUSION In summary, we explore the spin-dependent transport properties of Co 2MnSi/SrTiO 3/Co 2MnSi MTJs with four dierent interfaces via performing extensive DFT calculations within the NEGF technique. We nd that the transport properties of Co 2MnSi/SrTiO 3/Co 2MnSi MTJ strongly depend on the interface between Heusler alloy Co 2MnSi electrode and SrTiO 3barrier layer. The transmission coecient of MTJs with the MnSi-TiO 2 interface in the PC at the Fermi level is several orders of magnitude larger than that of in the APC, which re- sults in a huge TMR. According to the calculated pro- jected DOS of the MTJ1, the partial atomic DOS at the interface, and the in-plane wave vector dependence of transmission, we conclude that the predicted huge TMR originates from the coherent spin-polarized tunneling, due to the half-metallic nature of Co 2MnSi electrode and the signicant spin-polarization of the interfacial metal atomic orbitals. These theoretical ndings sug- gest that Co 2MnSi/SrTiO 3/Co 2MnSi MTJs hold great potential in spintronics if one can improve the single crystal quality of Heusler alloy Co 2MnSi and SrTiO 3 barrier layer in experiments. V. ACKNOWLEDGMENTS This work was partially supported by the National Natural Science Foundation of China (No.21873088 and No.11634011) and the Natural Science Foun- dation of the Anhui Higher Education Institutions (No.KJ2010A061 and No.KJ2016A144). Computa- tional resources have been provided by CAS, Shanghai and USTC Supercomputer Centers. [1]Y. Goto, T. Yanase, T. Shimada, M. Shirai, and T. Nagahama, AIP Adv. 9, 085322 (2019). [2]N. Kudo, M. Oogane, M. Tsunoda, and Y. Ando, AIP Adv. 9, 125036 (2019). [3]L. Lang, Y. Jiang, F. Lu, C. Wang, Y. Chen, A. D. Kent, and L. Ye, Appl. Phys. Lett. 116, 022409 (2020). [4]I. I. Oleinik, E. Y. Tsymbal, and D. G. Pettifor, Phys. Rev. B 62, 3952 (2000). [5]W. H. Butler, X. G. Zhang, T. C. Schulthess, and J. MacLaren, Phys. Rev. B 63, 054416 (2001). [6]S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando, Nat. Mater. 3, 868 (2004). [7]D. Waldron, V. Timoshevskii, Y. Hu, K. Xia, and H. Guo, Phys. Rev. Lett. 97, 226802 (2006). 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5.0024109.pdf

Appl. Phys. Lett. 118, 092401 (2021); https://doi.org/10.1063/5.0024109 118, 092401 © 2021 Author(s).Antiferromagnet-mediated spin–orbit torque induced magnetization switching in perpendicularly magnetized L10-MnGa Cite as: Appl. Phys. Lett. 118, 092401 (2021); https://doi.org/10.1063/5.0024109 Submitted: 04 August 2020 . Accepted: 16 February 2021 . Published Online: 01 March 2021 Xupeng Zhao , Siwei Mao , Hailong Wang , Dahai Wei , and Jianhua Zhao COLLECTIONS Paper published as part of the special topic on Spin-Orbit Torque (SOT): Materials, Physics, and Devices ARTICLES YOU MAY BE INTERESTED IN Field-free and sub-ns magnetization switching of magnetic tunnel junctions by combining spin-transfer torque and spin–orbit torque Applied Physics Letters 118, 092406 (2021); https://doi.org/10.1063/5.0039061 Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Current switching of interface antiferromagnet in ferromagnet/antiferromagnet heterostructure Applied Physics Letters 118, 032402 (2021); https://doi.org/10.1063/5.0039074Antiferromagnet-mediated spin–orbit torque induced magnetization switching in perpendicularly magnetized L10-MnGa Cite as: Appl. Phys. Lett. 118, 092401 (2021); doi: 10.1063/5.0024109 Submitted: 4 August 2020 .Accepted: 16 February 2021 . Published Online: 1 March 2021 Xupeng Zhao,1,2Siwei Mao,1,2Hailong Wang,1,2 Dahai Wei,1,2,3and Jianhua Zhao1,2,3,a) AFFILIATIONS 1State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, China 2Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100190, China 3Beijing Academy of Quantum Information Science, Beijing 100193, China Note: This paper is part of the Special Topic on Spin-Orbit Torque (SOT): Materials, Physics and Devices. a)Author to whom correspondence should be addressed: jhzhao@red.semi.ac.cn ABSTRACT Current-induced magnetization switching plays an essential role in spintronic devices exhibiting nonvolatility, high-speed processing, and low-power consumption. Here, we report on the spin–orbit torque-induced magnetization switching in perpendicularly magnetized L10- MnGa/FeMn/Pt trilayers grown by molecular-beam epitaxy. An antiferromagnetic FeMn layer is inserted between the spin current generat- ing Pt layer and spin absorbing MnGa layer. Due to the exchange bias effect, the trilayers show field-free spin–orbit torque switching. Overall, the spin transmission efficiency decreases monotonically as the FeMn thickness increases. It is found that the spin current can betransmitted through an 8 nm-thick FeMn layer as evidenced by partial switching of the L1 0-MnGa. The damping-like spin–orbit torque effi- ciency shows a peak value at tFeMn¼1.5 nm due to the enhanced interfacial spin transparency and crystalline quality of the FeMn. These results help demonstrate the efficacy of emerging spintronic devices containing antiferromagnetic elements. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024109 The long-term goal of spintronics is to develop spin-based storage and logic devices with high speed and low-power consumption.1The spin–orbit torque (SOT) effect provides an effective way to manipulatethe magnetization of nanoscale ferromagnets (FMs) and even antiferro-magnets (AFMs), which are key elements in various spintronic devi-ces. 2,3In particular, the SOT effect has been proposed to be an alternative writing mechanism for next-generation magnetic randomaccess memory (MRAM). 4,5For practical applications, ferromagnetic materials with strong perpendicular magnetic anisotropy (PMA) arehighly desirable for high storage density and robust thermal stability.The SOT can engender perpendicular magnetization switching via twodifferent mechanisms, a damping-like torque with the form of s DL ¼m/C2ðr/C2mÞa n dafi e l d - l i k et o r q u ew i t ht h ef o r mo f sFL¼m/C2r, where mandrcorrespond to the unit vectors of the magnetization and spin polarization of the spin current, respectively.6,7 AFM with fully compensated magnetic moments exhibits some favorable properties, such as high stability against the parasitic mag-netic field, minimal stray field, and superior high-frequencydynamics. 8,9Some of the AFM coupled materials have also been dem- onstrated to be transparent to spin current, such as NiO, Fe 2O3,a n d yttrium iron garnet (YIG).10–16The high spin transmission efficiency in AFM is attributed to the specific fundamental quasi-particle excita-tions, which can be further enhanced near the N /C19eel temperature T N.17,18Most interestingly, it is predicted that the spin current can even be amplified within a critical thickness.19In contrast, the spin current is expected to dephase rapidly in a FM, resulting in a shortspin-diffusion length of several angstroms. 20On the other hand, SOT switching has been realized in AFM/FM bilayers without the assistanceo ft h ei n - p l a n ee x t e r n a lfi e l db yu t i l i z i n gt h ee x c h a n g eb i a se f f e c t . 21 Therefore, a SOT device containing an AFM insertion layer not onlygives rise to field-free electrical switching but also enables long-distance preservation and even amplification of spin signals.Nevertheless, there have been few studies on AFM layer-mediatedSOT switching and a detailed investigation is needed. 15,16FeMn is a prototypical metallic AFM with a non-collinear spin structure andmoderate N /C19eel temperature ( T N/C24500 K).22,23It is free of heavy-metal Appl. Phys. Lett. 118, 092401 (2021); doi: 10.1063/5.0024109 118, 092401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apl(HM) elements and shows a negligibly small spin-Hall angle hSH (/C240.008).24Hence, the intensity of spin dephasing in FeMn is expected to be small due to its weak spin–orbit coupling. In this Letter, we report on a systematic investigation of SOT- induced magnetization switching in perpendicularly magnetizedL1 0-MnGa mediated by an FeMn insertion layer. The harmonic Hall voltage measurements are performed to characterize SOT-induceddamping-like effective field. It is found that the spin current can betransmitted efficiently through the AFM FeMn and drive the magneti-zation switching in the L1 0-MnGa. The samples were grown on buffer smoothed GaAs (001) sub- strates in a molecular-beam epitaxy (MBE) system with two intercon-nected growth chambers. Reflection high-energy electron diffraction(RHEED) was applied in situ to monitor the surface reconstruction during the growth. The crystalline structure was further investigatedby x-ray diffraction (XRD) measurements. The magnetic property andanomalous Hall effect were measured using a Quantum Design super-conducting quantum interference device (SQUID) magnetometer anda physical property measurement system (PPMS), respectively. TheSOT switching and harmonic Hall voltage measurements were per-formed by utilizing PPMS combined with external test instruments including a Keithley 6221 current source and a 2182A nanovoltmeter. Figure 1(a) shows the schematic stacking structure of the L1 0- MnGa/FeMn/Pt trilayers. The quadratic-type (3Q) AFM spin struc-ture is the ground state of the bulk c-FeMn, which has attracted much attention for its rich physical phenomena. 25,26An FeMn layer with the nominal thickness from 0.6 to 11 nm is inserted between L10-MnGa and Pt layers. Among various PMA materials, the L10-MnGa alloy exhibits many attractive properties, such as large intrinsic PMA, smallGilbert damping constant, and high spin-polarization. 27,28The L10-MnGa film was epitaxially grown on a GaAs(001) substrate with the epitaxial relationship of MnGa(001)[100]//GaAs(001)[110].27The HM Pt layer serves as the source of spin current as well as the cappinglayer to prevent oxidation. Figure 1(b) further presents the XRD h-2h scan of the L1 0-MnGa (3 nm)/FeMn(6 nm)/Pt (5 nm) trilayers. Both the superlattice (001) and fundamental (002) peaks of the L10-MnGa film are observed at 2 h¼23.4/C14and 49.8/C14, respectively. According to Bragg’s law, the out-of-plane lattice constant cof the L10-MnGa is cal- culated to be 3.69 A ˚, which is close to the bulk value (3.67 A ˚).27It should be noted that the lattice constant a(3.64 A ˚)o fF e M ni sv e r y close to cofL10-MnGa. Moreover, the FeMn film is epitaxially grown onL10-MnGa at first but gradually becomes polycrystalline due to the in-plane lattice mismatch. Therefore, the diffraction peaks of the FeMn film are indistinguishable. Figure 1(c) shows the out-of-plane hysteresis loops of the sample with tFeMn¼6 nm measured at 5 and 300 K. A near-square hysteresis loop was observed with a large coer-cive field ( H c) of 7.2 kOe at 300 K, indicating a large PMA. An asym- metry in the hysteresis loop appears at 5 K, which is consistent with our previous results and can be attributed to different anti-phase domain nucleation centers and unequal domain wall motion pro- cesses.29As shown by the inset, the L10-MnGa thin film has a moder- ate saturation magnetization Msof 287 emu/cm3and large effective anisotropy field Heff kof 5.1 T. Hence, the uniaxial PMA constant Ku can be estimated to be 7.8 Merg/cm3based on Ku¼MsHeff k=2 þ2pM2 s; the value is slightly smaller than the value of the thick films.27 Here, the perpendicular exchange bias field Heband coercive field Hc can be defined as Heb¼/C0 ð Hc1þHc2Þ=2a n d Hc¼jHc1/C0Hc2j=2; where Hc1andHc2are the coercive forces of left and right branches, respectively. According to Fig. 1(d) ,b o t h Heband Hcshow a strong temperature dependence. Moreover, the observed relatively small Heb of 36.5 Oe reveals that the FeMn film is still in the AFM phase at 300 K, which is close to its practical TN. Considering that spin trans- mission efficiency tends to get enhanced near TN, AFM FeMn could be a promising candidate for investigating spin-transport behavior atroom-temperature. 17–19 Figure 2(a) is a schematic of the Hall bar devices, which are fabri- cated by using photolithography and ion-beam etching and have achannel width of 10 lm. As plotted in Fig. 2(b) , the anomalous Hall curve for the sample with an FeMn thickness of 1.5 nm exhibits nearly p e r f e c ts q u a r e n e s sa t3 0 0 Kw i t has m a l l H ebof 24 Oe. According to the in-plane anomalous Hall curve shown in the inset, the effective anisotropy field Heff kis consistent with the value of the film obtained by SQUID measurements. We then performed measurements of SOT- induced magnetization switching in the L10-MnGa (3 nm)/FeMn (1.5 nm)/Pt (5 nm) heterostructure. Specifically, to electrically switchthe perpendicular magnetization of L1 0-MnGa, a series of longitudinal current pulses of 50 ls duration are applied under in-plane fields Hx. The current is mostly confined in the Pt layer, and due to the spin-Hall effect, the electrons with opposite spin polarities flow to and accumulate at opposite interfaces of the Pt layer, which leadst ot h ep u r es p i nc u r r e n t J sinto the FeMn and MnGa. Thus, the effective spin-current injected into the FeMn can be expressed as Js¼ð/C22h=2eÞTHM =AFM int hHM SHJe,w h e r e THM =AFM int is the interfacial spin transparency.30Subsequently, the spin current may be transmitted through the FeMn layer via conducting electrons and magnons. As shown in Figs. 2(c) and2(d), abrupt current-induced magnetization switchings are observed with anticlockwise (clockwise) switching FIG. 1. (a) Schematic depiction of the stacking structure of the samples. (b) X-ray diffraction pattern of a L10-MnGa (3 nm)/FeMn (6 nm)/Pt (5 nm) trilayer grown on the GaAs (001) substrate. (c) The out-of-plane hysteresis loops of the sample with tFeMn¼6 nm measured at 300 and 5 K. The inset shows the in-plane hysteresis loop at 300 K. (d) The temperature dependence of coercive field Hcand exchange bias field Hebof the sample with tFeMn¼6 nm.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 092401 (2021); doi: 10.1063/5.0024109 118, 092401-2 Published under license by AIP Publishingpolarities under negative (positive) Hx, which is consistent with the picture of the SOT effect and the positive spin Hall angle of Pt.2 Generally, the role of the Hxin SOT switching is to overcome the domain wall chirality originated from the Dzyaloshinskii–Moriya interaction (DMI).2,3Particularly noteworthy is a clear SOT-induced switching in the absence of external field with relatively small RH change, suggesting the existence of in-plane effective field Heffinduced by the exchange bias effect. It is found that the magnitude of RH reaches the maximum value at þ1000 Oe, which is decreased com- pared to that of the anomalous Hall measurement. That is, magnetic domains in the L10-MnGa film are not completely reversed in the SOT switching process. Therefore, the proportion of flipped domainscan be defined as the ratio of R AHobtained by the current and mag- netic field-induced magnetization switching. To investigate the AFM layer thickness dependence of spin transmission in the L10-MnGa/FeMn/Pt heterostructure, the current-induced magnetization switching measurements are per- formed in a series of samples with tFeMn varying from 0 to 11 nm, as illustrated in Figs. 3(a)–3(d) . In the absence of the FeMn inser- tion layer, the magnetization of L10-MnGa shows a clear reversible switching under negative in-plane fields Hx.A s tFeMn increases, the samples still retain a similar feature of deterministic SOT switch-ing, indicating high spin-transmission efficiency. Notably, the criti- cal switching current density J cin the Pt layer is calculated to be 3.21/C2107A/cm2under Hx¼/C01000 Oe when tFeMn¼6 nm, which is close to that in other soft magnetic PMA materials and much smaller than that of 1 /C2108A/cm2in the previous work on L10- MnGa/HM bilayers.31–33The decreased Jccan be attributed to the improved interfacial and crystalline quality of trilayers grown by MBE.In conventional FM/HM heterostructures, the damping-like tor- que generated by the bulk spin-Hall effect in HM is considered to be the dominant contribution to the magnetization switching. To quan- tify the damping-like effective field, the harmonic Hall voltage mea- surements were performed. The first and the second harmonic voltages ( V1xandV2x) can be obtained by applying an AC with low frequency (133 Hz) through the Hall devices with a varying in-plane field along the x-direction in order to exert periodic SOT. Figures 4(a) and4(b)show the V1xandV2xcurves as functions of Hxmeasured at 300 K under 5 mA AC for a sample of tFeMn¼0.9 nm, respectively. Herein, the SOT-induced damping-like effective field HDLcan be obtained by the following formula:34,35 HDL¼/C02@V2x=@Hx @2V1x=@H2 x: (1) Overall, the SOT-induced effective field shows a linear relation- ship with the applied current density. In this sample, the damping-likeeffective field H DLis calculated to be 40.7 Oe under a driving current density Jeof 8.24 /C2106A/cm2in Pt. Thus, the strength of the damping-like effective field can be deduced to be 4.94 Oe/(106A/cm2). Moreover, the effective spin Hall angle nin the heterostructures can be obtained by n¼2eMst /C22hHDL je; (2) where /C22his the reduced Planck’s constant, eis the electron charge, and t is the thickness of the L10- M n G afi l m .H e n c e ,t h ee f f e c t i v es p i nH a l l angle nis calculated to be /C240.095 for tFeMn¼0.9 nm, which is consis- tent with the previous results in the Pt/FM bilayer system.30Figure 4(c)shows the FeMn thickness dependence of the Hall resistance RAH obtained by the anomalous Hall effect and SOT switching. Due to the shunting effect in the trilayers, both of the two values decrease FIG. 2. (a) Schematic of a L10-MnGa/FeMn/Pt heterostructure-based Hall device. (b) The Anomalous Hall curves of the L10-MnGa/FeMn/Pt heterostructure with tFeMn ¼1.5 nm measured at 300 K. (c) and (d) show the deterministic spin–orbit torque- induced magnetic switching of a L10-MnGa (3 nm)/FeMn (1.5 nm)/Pt (5 nm) hetero- structure under in-plane positive and negative fields Hx, respectively. FIG. 3. (a)–(d) The deterministic SOT-induced magnetization switching curves under external negative in-plane fields for L10-MnGa/FeMn/Pt heterostructures with tFeMn¼0 nm, 0.6 nm, 4 nm, and 6 nm, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 092401 (2021); doi: 10.1063/5.0024109 118, 092401-3 Published under license by AIP Publishingmonotonically with tFeMn. The FeMn thickness dependences of the switching ratio and damping-like torque effective field per current density HDL=Jare further summarized in Fig. 4(d) .B o t ht h es w i t c h i n g ratio and HDL=Jexhibit a decreasing trend with tFeMn, demonstrating that the damping-like torque effective field plays a key role in the deterministic magnetization switching of the L10-MnGa film. However, an interesting peak value appears at tFeMn¼1.5 nm, which suggests that the spin current reaching the L10-MnGa becomes enhanced from 0.9 nm to 1.5 nm. Even at a tFeMn of 8 nm, a SOT- induced RAHloop with a switching ratio of 3% can still be clearly observed. According to our previous work, the perpendicular exchange bias effect in the L10-MnGa/FeMn bilayer system appears at a critical thickness of 1–2 nm for the FeMn layer, which can be interpreted as the critical thickness for the formation of stable AFM domains.29 Generally, the diffusion length for electronic spin current Je sis quite short due to the rapid spin dephasing and Joule heating; thus the mag- n o ns p i nc u r r e n t Jm sprobably dominates the long-range spin transmis- sion process in AFM materials.20When FeMn is only several angstroms thick, the film tends to be partially disordered with a rough interface, leading to magnetic defects and low spin-mixing conduc- tance at the Pt/FeMn and FeMn/MnGa interfaces. For the samples with tFeMn larger than 0.9 nm, the interfaces of FeMn/Pt and L10- MnGa/FeMn become smoother as tFeMn increases. Meanwhile, the crystalline quality and magnetic ordering of the FeMn film are also sig- nificantly improved, which gives rise to stable AFM domains. Therefore, both the spin transparency at the interfaces and spin- diffusion length in the AFM FeMn see significant improvement, resulting in the larger damping-like torque effective field densityHDL=J. For the thicker FeMn film ( tFeMn/C212 nm), the FeMn film gradually becomes polycrystalline due to the release of strain, which leads to the enhancement of the spin-memory loss. When tFeMn is beyond 11 nm, the damping-like effective field drops to zero, i.e., no spin current can be transmitted through the FeMn insertion layer and exert any torque on the magnetization of the L10-MnGa film. Thus, we conclude from the experiments that the spin current generated by the Pt can be effectively transmitted through the FeMn layer with a transport length up to 8 nm, which is considerably longer than that of the conventional FM and shows good agreement with the previous results obtained by spin pumping measurements.14,20The observed high spin transmission efficiency combined with the field-free nature of the SOT switching demonstrated in our system indicates that the AFM FeMn is a viable component in spin-based memory and logicdevices having low-power consumption. In conclusion, room temperature SOT-induced magnetization switching has been demonstrated and systematically investigated inL1 0-MnGa/FeMn/Pt heterostructures with an AFM insertion layer. Due to the exchange bias effect, the partial field-free SOT switching has been observed. In particular, the spin current generated in the Ptlayer is observed to be transmitted through an 8 nm-thick FeMn layer, evidenced by clear magnetization switching in the L1 0-MnGa. The experiments provide clear evidence for long-distance spin transport inthe AFM FeMn. Both the proportion of flipped magnetic domains and damping-like effective field show peak values at t FeMn¼1.5 nm, which is attributed to the enhanced interfacial spin transparency and crystal- line quality of FeMn. These results have provided insights into the SOT effect in HM/AFM/FM structures and could help pave the wayfor high-performance and low-power consumption AFM spintronic devices. We acknowledge the insightful discussions with Professor Peng Xiong at Florida State University. This work was supported by the National Key R&D Program of China (MOST, Grant No.2018YFB0407601), the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant Nos. XDB44000000 and QYZDY-SSW-JSC015, and the National Natural Science Foundation of China under Grant Nos. 11874349 and 11774339. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Fert, Rev. Mod. Phys. 80, 1517 (2008). 2L. Q. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). 3I. M. Miron, K. Garello, G. Gaudin, P. J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011). 4M. Cubukcu, O. Boulle, N. Mikuszeit, C. Hamelin, T. Bracher, N. Lamard, M.-C. Cyrille, L. Buda-Prejbeanu, K. Garello, I. M. Miron, O. Klein, G. De Loubens, V. V. Naletov, J. Langer, B. Ocker, P. Gambardella, and G. Gaudin, IEEE Trans. Magn. 54, 1 (2018). 5K. Jabeur, G. D. Pendina, F. Bernard-Granger, and G. Prenat, IEEE Electron Device Lett. 35, 408 (2014). 6E. Jue, C. K. Safeer, M. Drouard, A. Lopez, and P. 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5.0048833.pdf

J. Appl. Phys. 129, 225105 (2021); https://doi.org/10.1063/5.0048833 129, 225105 © 2021 Author(s).Methods to accelerate high-throughput screening of atomic qubit candidates in van der Waals materials Cite as: J. Appl. Phys. 129, 225105 (2021); https://doi.org/10.1063/5.0048833 Submitted: 26 February 2021 . Accepted: 25 May 2021 . Published Online: 09 June 2021 Rodrick Kuate Defo , Haimi Nguyen , Mark J. H. Ku , and Trevor David Rhone ARTICLES YOU MAY BE INTERESTED IN Magnetic imaging and statistical analysis of the metamagnetic phase transition of FeRh with electron spins in diamond Journal of Applied Physics 129, 223904 (2021); https://doi.org/10.1063/5.0051791 Structural and optoelectronic properties change in Bi/In 2Se3 heterostructure films by thermal annealing and laser irradiation Journal of Applied Physics 129, 223101 (2021); https://doi.org/10.1063/5.0048852 Shape programming lines of concentrated Gaussian curvature Journal of Applied Physics 129, 224701 (2021); https://doi.org/10.1063/5.0044158Methods to accelerate high-throughput screening of atomic qubit candidates in van der Waals materials Cite as: J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 View Online Export Citation CrossMar k Submitted: 26 February 2021 · Accepted: 25 May 2021 · Published Online: 9 June 2021 · Publisher error corrected: 11 June 2021 Rodrick Kuate Defo,1,2,a) Haimi Nguyen,3 Mark J. H. Ku,4 and Trevor David Rhone5 AFFILIATIONS 1Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08540, USA 2John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA 3Department of Chemistry, Columbia University, New York, New York 10027, USA 4Department of Physics and Astronomy and Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716, USA 5Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA a)Author to whom correspondence should be addressed: rkuatedefo@princeton.edu ABSTRACT The discovery of atom-like spin emitters associated with defects in two-dimensional (2D) wide-bandgap (WBG) semiconductors presents new opportunities for highly tunable and versatile qubits. So far, the study of such spin emitters has focused on defects in hexagonal boron nitride (hBN). However, hBN necessarily contains a high density of nuclear spins, which are expected to create a strong incoherent spin- bath that leads to poor coherence properties of spins hosted in the material. Therefore, identification of new qubit candidates in other2DWBG materials is necessary. Given the time demands of ab initio methods, new approaches for rapid screening and calculations of iden- tifying properties of suitable atom-like qubits are required. In this work, we present two new methods for rapid estimation of the zero- phonon line (ZPL), a key property of atomic qubits in WBG materials. First, the ZPL is calculated by exploiting Janak ’s theorem. For finite changes in occupation, we provide the leading-order estimate of the correction to the ZPL obtained using Janak ’s theorem, which is more rapid than the standard method ( ΔSCF). Next, we demonstrate an approach to converging excited states that is faster for systems with small strain than the standard approach used in the ΔSCF method. We illustrate these methods using the case of the singly negatively charged calcium vacancy in SiS 2, which we are the first to propose as a qubit candidate. This work has the potential to assist in accelerating the high-throughput search for quantum defects in materials, with applications in quantum sensing and quantum computing. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0048833 I. INTRODUCTION Research in the field of layered materials has grown quickly since the development of the high-quality sample yielding a scotch-tape exfoliation technique. 1,2For instance, the quantum spin Hall effect up to 100 K was observed in monolayer tungsten ditelluride (WTe 2).3Another important discovery was the observation of a correlated insulator state by tuning the twist degree of freedom inbilayer structures of graphene due to the presence of flat bandsnear zero Fermi energy at a twist angle of about 1 :1 /C14, which upon electrostatic doping yields superconducting states with a critical temperature of up to 1.7 K.4,5Research in the field of point defectqubit candidates has also grown rapidly, particularly since the detection of single negatively charged nitrogen –vacancy (N V/C0) color centers in diamond.6These qubit candidates consist of defects in the crystal structure involving substitutional or interstitial atoms and/orvacancies and act as single-photon sources as well as sources of elec-tronic spin. Desirable characteristics for such qubits include indistin-guishability of emitted photons, negligible spectral diffusion, andlong spin coherence time. State-of-the-art results include second-longcoherence time of N Vcenters, 7km-scale entanglement of two N Vs,8 and the discovery of spectrally stable germanium –vacancy (Ge V) and silicon –vacancy (Si V) point defects.9,10The properties of spinJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-1 Published under an exclusive license by AIP Publishingqubits can be tuned by changing the crystal structure and the nature of the defect. Engineering spin qubits with properties that are desirable for applications in sensing, quantum computing, andquantum communications are at the forefront of research interest.Nevertheless, identifying the right combination of crystal struc-tures and defects is enormously challenging due to the combina- torially large number of potential candidates that need to be explored using experiments or first-principles calculations. The layered material hexagonal boron nitride ( h-BN) is of par- ticular interest among layered materials as it has a wide bandgap, 11 enabling its ability to encapsulate other layered materials and to host a variety of point defects, giving rise to optical transitions lying within the range of its bandgap.2,12–15The issue with h-BN as a host for a qubit candidate employing electronic spins is that theatomic nuclei of boron and nitrogen have spins as well, which cancause spin-decoherence of the electronic spin state. Such an argu- ment invites consideration of SiS 2as a host, which theory has pre- dicted to exist in a layered form between 2.8 and 3.5 GPa for thespace group P2 1/c.16The SiS 2host would be diamond-like or silicon carbide (SiC)-like in that the atoms that constitute it have alow natural abundance of isotopes with nuclear spin, which can be further suppressed via isotope purification in growth, diamond and SiC being systems that have been extensively studied as hosts forqubit candidates. 9,10,17–32The point defect we will investigate in SiS2has the further advantage of exhibiting inversion symmetry, which eliminates the issue of an electric dipole moment making it susceptible to external noise and local fields that cause broadeningof transitions. Given the growing number of experimental investigations, the need to rapidly identify promising point defect qubit candidates is being recognized. Indeed, recent work 33has looked into automat- ing the process of characterizing point defects through a codenamed ADAQ, to aid in the more rapid identification of defect sig-natures from experiments, and a push toward high-throughputpoint defect calculations was also present in the earlier PyCDT code. 34In identifying point defects in experiments, knowledge of the zero-phonon line (ZPL) transition is essential. We presentin this paper two methods that can be incorporated into high-throughput searches for point defect qubit candidates. The first is aquick new method for estimating the error associated with comput- ing the energy of the zero-phonon line (ZPL) transition of a point defect using Janak ’s theorem, and the second is a new method for rapid convergence of excited state self-consistent field (SCF)calculations. These new tools are an important step in accelerating the high-throughput discovery of materials that can support spin qubits for applications in sensing, quantum communication, andquantum computing. Janak ’s theorem states that 35 @E @ni¼ϵi, (1) where Eis the total energy, niis the orbital occupation of the ith orbital, and ϵiis the corresponding eigenvalue of the orbital. This theorem is similar to Koopmans ’theorem for the Hartree –Fock theory36in that it relates energy differences under a change in the number of electrons to orbital energies. The theorem is in fact adensity-functional theory (DFT) version of a theorem originally proved by Slater for his X–αmethod, which was introduced in an early attempt to account for both exchange and correlation in elec-tronic structure calculations. 37,38As Janak ’s theorem does not imply that ΔE¼ϵiΔnifor finite Δni, an estimate of the error asso- ciated with using the theorem to determine excitation energies where orbital occupations change by integer amounts is necessary. Applying the theorem and the method of error estimation is inher-ently faster than computing the energy of the excited state for agiven point defect, as the error calculation amounts to terminatingthe excited state calculation before full convergence. We also show that excited state calculations for systems with small strain converge faster if the charge density is initialized by appropriately mixing thehighest occupied molecular orbital (HOMO) and the lowest unoc-cupied molecular orbital (LUMO) from the ground state calcula-tion, leaving the contribution to the charge density from the remaining orbitals unchanged, as compared to initialization of the charge density from a superposition of atomic charges. In this work, we first outline the computational methods in Sec.II. In Sec. III, we include results related to (i) ZPL estimations for silicon monovacancies in 4 H-SiC (see Sec. III A ) and (ii) the singly negatively charged calcium vacancy in SiS 2(see Sec. III B , which also includes stability calculations) and a discussion of theseresults, ending with a summary of our conclusions. II. THEORETICAL FORMULATION AND APPROACH A. Calculation method The purpose of this work is to extend the notion of a mixing parameter 39to the initialization of the charge density and to bolster the validity of using Janak ’s theorem35to calculate the zero-phonon line (ZPL) by demonstrating success for the case of the singly nega-tively charged calcium vacancy in SiS 2, which we investigate as a potential qubit candidate. Our approach builds upon previous workemploying Janak ’s theorem to calculate the electronic properties of excited states of atoms, molecules, and solids. 40,41In this work, we introduce an innovation by employing Janak ’s theorem for the cal- culation of defect properties and additionally providing a lowestorder estimate of the error associated with using the theorem for anintegral change in the occupation of the single-particle states. We argue that the calculations in excess of the ground state calculation in the ΔSCF method are not always necessary, which directly follows from Janak ’s theorem in the limit where the change in occupation is infinitesimal. To motivate the argument, we computethe error between the ΔSCF method and the approach of using Janak ’s theorem to the lowest order under a change in occupation. We consider the operator from the single-particle equations in DFT, O(n(r))¼/C0 /C22h2 2me∇2 rþV(r)þe2ðn(r0) r/C0r0jjdr0þδExc[n(r)] δn(r0), (2) where n(r) is the particle number density, the first term represents the kinetic energy of noninteracting quasiparticles with electronmass, the second term represents the external potential, the third term represents the Hartree potential for the Coulomb interaction between the quasiparticles, and the last term represents theJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-2 Published under an exclusive license by AIP Publishingexchange-correlation potential recapturing the fermionic and many-body nature of electronic interactions. The single-particle equation for the single-particle state f(n) i with eigenvalue ϵ(n) ithen reads O(n(r))f(n) i¼ϵ(n) if(n) i: (3) Under a change in occupation, let n0(r) be the new number density and let f(n0) ibe the new ith single-particle state such that O(n0(r))f(n0) i¼ϵ(n0) if(n0) i (4) is satisfied. The equation for the total energy from DFT is42 E¼XN iϵ(n) i/C01 2ððn(r)n(r0) r/C0r0jjdrdr0þð n(r)ϵxc[n(r)]/C0μxc[n(r)] ½/C138 dr: (5) If we let F[n(r)]¼/C01 2ððn(r)n(r0) r/C0r0jjdrdr0þð n(r)ϵxc[n(r)]/C0μxc[n(r)] ½/C138 dr (6) and Δϵ¼XNþ1 i¼1(ϵ(n0) i/C0ϵ(n) i)/C0(ϵ(n0) N/C0ϵ(n) N), (7) then we can approximate the ΔSCF result as EΔSCF/C25ϵ(n) Nþ1/C0ϵ(n) NþΔϵþF[n0(r)]/C0F[n(r)]: (8) Therefore, if we only take the difference in ground state eigenval- ues, the associated error is ΔEΔSCF/C25ΔϵþF[n0(r)]/C0F[n(r)]: (9) We can choose to obtain n0(r) at any arbitrary iteration in the full constrained-occupation calculation for the excited state.Therefore, in the limit where n 0(r) is obtained at the final iteration of the full constrained-occupation calculation for the excited state,the error becomes exact. In performing the full constrained- occupation calculation for the excited state, we have also explored initializing the charge density by mixing the HOMO and LUMOby varying amounts. To obtain the defect levels and total energies, we performed first-principles DFT calculations for the various defects in 4 H-SiC and SiS 2using the VASP code43–45and the QUANTUM ESPRESSO code46,47forΔSCF calculations. In VASP, atomic struc- tures were first converged using the generalized gradient approxi-mation (GGA) for the exchange-correlation energy of electrons, asparameterized by Perdew, Burke, and Erzenhof (PBE) 48and then, for the calculation of defect levels of the uncompressed structure,using the screened hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE) with the original parameters (0.2 Å/C01for screen- ing and 25% for mixing).49,50The terms in F[n(r)] were obtained for the ground state and for the state with the changed occupation,where a non-SCF calculation was performed until convergencekeeping the charge density fixed in the latter case. The eigenvalues, ϵ (n0) i, were also provided by the non-SCF calculation. Code from the work by Feenstra et al.51was used to change orbital occupa- tions. In QUANTUM ESPRESSO, we performed ΔSCF calculations to investigate the SiS 2system using PAW pseudopotentials45with a 108-atom supercell with gamma-point integration. Modified sourcecode was used to alter the charge density in QUANTUM ESPRESSO. The different values of strain investigated in this work were for biaxial strain. For strain values other than the uncom-pressed structure, the lattice parameters along the strain directionswere rescaled from the lattice parameters of the uncompressedstructure. The out-of-plane lattice parameter was held fixed. Constant volume relaxations were then performed for the strained structures as described above. Additional computational detailsmay be found in the Appendix . For first-principles phonon calculations, performed using Phonopy, 52the atomic positions in the stoichiometric conventional unit cell were relaxed until the magnitude of the Hellmann – Feynman forces was smaller than 10/C06eV A/C14/C01with a Monkhorst – Pack grid of 6 /C26/C22 k-points. Supercells containing 108 atoms (3/C23/C21 multiple of the conventional unit cell) with appropriately scaled k-point grids and a cutoff energy of 500 eV were then con- structed from the stoichiometric conventional unit cell and used toobtain the force constants to compute the phonon dispersion. B. Formation energies The formation energies of the calcium vacancy in SiS 2in various charge states were calculated according to the formula53,54 Ef(q)¼Edef(q)/C0E0/C0X iμiniþq(EVBMþEF)þEcorr(q), (10) where qdenotes the charge state, Edef(q) is the total energy for the defect supercell with charge state q,E0is the total energy for the stoichiometric neutral supercell, μiis the chemical potential of atom i,niis a positive (negative) integer representing the number of atoms added (removed) from the system relative to the stoichio-metric cell, E VBM is the absolute position of the valence band maximum, EFis the position of the Fermi level with respect to the valence band maximum (generally treated as a parameter), and Ecorr(q) is a correction term to account for the finite size of the supercell when performing calculations for charged defects.55 This correction term does not simply treat the charged defect as apoint charge but rather considers the extended charge distribution. The chemical potentials of all the reference elements used in our cal- culations are listed as follows as a function of their crystal structureand total energy per atom: Si (diamond structure, /C05:42 eV/atom); S (the total energy of a gas phase S 8molecule was calculated and the sublimation enthalpy was then subtracted,56,57/C04:20 eV/atom); and Ca (face-centered cubic structure, /C02:00 eV/atom). Consideration ofJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-3 Published under an exclusive license by AIP PublishingSi-rich or S-rich preparation conditions was made similar to the pre- vious work.30 C. Material screening approach In order to leverage the vast body of existing knowledge in the field, we utilized electronic databases of two-dimensional materialsknown as the 2D Materials Encyclopedia 58in a top-down filtering approach and the Computational 2D Materials Database(C2DB) 59,60to find the most promising host materials. The entries from the databases were first selected based on the presence of cen- trosymmetry in their space groups, yielding 4822 potential candi-dates with duplicates discarded. Fulfillment of this criterion allowsfor the possibility of creating in the host a defect without a dipolemoment that could interact with local fields. The following filters were then applied (the number of candidates at each filtration step, discarding duplicates, is listed in parentheses): 1.PBE bandgap above 2 eV (985). The optical transitions of the qubit candidate implanted in the host must not introduce inter-ference from the electronic states of the host. 21The value is taken to be at least as large as the energy of the optical transitions of well-known point defects such as the N V/C0center in diamond and singly negatively charged silicon monovacancies in 4 H-SiC. Given that the PBE48functional generally underestimates the value of the bandgap, such a value allows for the defect levels for transitions similar in energy to those of negatively charged silicon monovacancies in 4 H-SiC and the N V/C0center in diamond to be separated from the band edges. 2.Exfoliation energy below 80 meV/atom (441). The material must be easy to exfoliate and, therefore, to fabricate. The value is taken to be commensurate with exfoliation energies of common layered materials such as MoS 2with space group P /C226m2. 3.Does not contain an atom with a nuclear spin (6) . Decoherence caused by the interaction of nuclear spins with the electronicones of the qubit candidate must be minimized. We set the cutoff at ,5% natural abundance of isotopes containing a nuclear spin using as reference 4 H-SiC, which is the less restric- tive among the two commonly studied systems of diamond and4H-SiC with defects showing long coherence times at room temperature. The filtering process yielded SiS 2with space group P2 1/c as the host candidate with the lowest decomposition energy. Theother host candidates left at the final stage of filtration were CaOwith space group P4/nmm, SO 2with space group P2 1/c, CO with space group Cmme, Si 2O3with space group P2 1/m, and S 4O9with space group P /C223. To our knowledge, none of these compounds have been used as qubit hosts. We note that similar criteria have beenused in the work of Ferrenti et al. , 61though in that work, the con- dition regarding the natural abundance of isotopes with a nuclearspin was relaxed to a cutoff of ,50% and bulk materials were con- sidered, yielding more known candidate hosts. Given the possibility of employing isotope purification, we invite researchers wishing tofollow up on our work to relax the cutoff if the above candidatesdo not prove readily amenable to synthesis. Working from the literature on defects in diamond and 4H-SiC, we investigated vacancies, germanium –vacancy complexes,and lead –vacancy complexes as possible qubit candidates in the SiS 2host. However, these all break inversion symmetry when they are relaxed with spin polarization. On the other hand, the singlynegatively charged calcium –vacancy complex is able to preserve inversion symmetry upon relaxation with spin polarization due inpart to the large size of the calcium atom. III. RESULTS AND DISCUSSION A. Silicon monovacancies in 4 H-SiC We begin by demonstrating the accuracy of Janak ’s theorem for silicon monovacancies in 4 H-SiC. This system is an example of a success of the theorem though for another prominent system, the NV/C0in diamond, Janak ’s theorem alone is not sufficient to produce good agreement between experiment and theory. The lack of agree-ment in that case can be explained by the significant differencebetween the ground and excited state wavefunctions 62at a location of a significant change in the external or ionic potential (at the N atom in that case), which, from the single-particle equations, will in general lead to larger energy changes under a change in occupationcompared to differences elsewhere. We can motivate this argumentexplicitly using what is referred to as “effective-mass theory, ” 38,63 where the potential introduced by the impurity is considered to be a small perturbation to the crystal potential. Let Vdefect(r)b et h e change in the external or ionic potential of the crystal due to thepresence of the defect and expand the defect wavefunction of interest, fdefect(r), in the basis of the eigenfunctions, ψk(r), of the perfect crystal, fdefect(r)¼X kβkψk(r): (11) The wavefunction will obey the single-particle equation HcrystalþVdefect(r)/C2/C3 fdefect(r)¼ϵdefectfdefect(r), (12) whereHcrystalis the single-particle hamiltonian for the perfect crystal so that Hcrystalψk(r)¼ϵkψk(r): (13) Using the orthogonality of the crystal wavefunctions and the expan- sion for fdefect(r), we then have ϵkβkþX k0,ψkjVdefectψk0j.βk0¼ϵdefectβk: (14) Thus, for perturbations to the defect wavefunction that equally change the projection of the defect wavefunction onto some perfectcrystal eigenfunction (i.e., that result in the same β kfor some k), the perturbation that will, in general, cause the greatest change in energy is the one where the other coefficients are such that the defect wave-function changes most at the location of the defect potential. As wewill see in Sec. III B, the partial charge densities associated with the orbitals changing occupation can be a quick way to identify when Janak ’s theorem is likely to fail.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-4 Published under an exclusive license by AIP PublishingFor the lattice parameters of the stoichiometric hexagonal unit cell of 4 H-SiC using the HSE06 functional, we find a¼3:08 Å and c¼10:04 Å. These values are in good agreement with the experi- mental values64ofa¼3:07 Å and c¼10:05 Å and other theoreti- cal values65ofa¼3:07 Å and c¼10:05 Å. The structure of the VSiis shown in Fig. 1 . We observe that the expected C3vsymmetry is weakly broken from the distances displayed in Table I . From the work of Soykal et al. ,66we know that for a spin- polarized system, the approach of considering holes is equivalent tothe approach of considering electrons. Defect levels calculatedusing the HSE06 functional for the singly negatively charged silicon monovacancy V /C0 Siin 4 H-SiC with S¼3=2 at the two inequivalent hand ksites are illustrated schematically in Fig. 2 and are provided in Tables II andIII. We note that the zero of energy is arbitrary as it depends on the pseudopotential, but, as we considerdifferences in eigenvalues, the value of the zero is not important. From the work of Soykal et al. , 66in the hole picture, the ground state manifold is composed of the states, uexeyþi/C22u/C22ex/C22ey/C13/C13 .=ffiffiffi 2p , uexey/C0i/C22u/C22ex/C22ey/C13/C13 .=ffiffiffi 2p ,uex/C22eyþu/C22exeyþ/C22uexey/C13/C13 .=ffiffiffi 3p ,/C22u/C22exeyþ/C13/C13 /C22uex/C22eyþu/C22ex/C22ey.=ffiffiffi 3p , while the excited state manifold in the hole picture is composed of the states, vexeyþi/C22v/C22ex/C22ey/C13/C13 .=ffiffiffi 2p , vexey/C0i/C22v/C22ex/C22ey/C13/C13 .=ffiffiffi 2p ,vex/C22eyþv/C22exeyþ/C22vexey/C13/C13 .=ffiffiffi 3p , and /C22v/C22exey/C13/C13 þ/C22vex/C22eyþv/C22ex/C22ey.=ffiffiffi 3p . Above, uand vare single-particle orbitals transforming as the A1irreducible representation of the C3vpoint group, while exand eytransform as the xand ycomponents of the two-dimensional Eirreducible representation of the C3vpoint group. The overbar denotes the minority spin state. To calculate FIG. 1. Shown above is the local structure of a VSiin 4H-SiC, which can be located at either of the two inequivalent korhsites. Nearest-neighbor carbon atoms to the vacancy are shown in brown and labeled with numbers, and the silicon vacancy is shown as a blue dashed circle. The VSiis displayed along the [001] direction.TABLE I. Distances in Å, d(X1−X2), between carbon atoms numbered in Fig. 1 and the silicon vacancy for VSiat the kandhsites. Site d(VSi−C1) d(VSi−C2) d(VSi−C3) d(VSi−C4) k 2.05 2.06 2.04 2.03 h 2.04 2.04 2.03 2.02 FIG. 2. Schematic of the majority (without an overbar) and minority (with an overbar) spin energy levels calculated in the ground state using the HSE06 functional for the V/C0 Si(k) and V/C0 Si(h) point defects in 4 H-SiC. The conduction band is indicated in blue and the valence band in red. Single-particle orbitalstransforming as the A 1irreducible representation of the C3vpoint group are rep- resented by uandv, while exandeytransform as the xandycomponents of the two-dimensional Eirreducible representation of the C3vpoint group. The vertical axis of the figure is not drawn to scale. TABLE II. DFT eigenvalues in eV calculated in the ground state using the HSE06 functional for the hole single-particle states of the V/C0 Si(k) point defect in 4 H-SiC. Single-particle state Majority spin Minority spin u 7.553 8.116 v 7.773 10.011 ex 7.828 10.152 ey 7.855 10.225Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-5 Published under an exclusive license by AIP Publishingthe ZPL, we take the lowest energy hole states, which from Tables II andIIIwe see must be the /C22u/C22exeyþ/C22uex/C22eyþu/C22ex/C22ey/C13/C13 .=ffiffiffi 3p (excited) and /C22v/C22exeyþ/C22vex/C22eyþv/C22ex/C22ey/C13/C13 .=ffiffiffi 3p (ground) states. Explicitly, in the following, we denote the eigenvalue for the major-ity spin single-particle orbital γasϵ γand for the minority spin single-particle orbital /C22γasϵ/C22γ. Then, the ZPL we obtain for the k siteV/C0 Siis (2( ϵ/C22v/C0ϵ/C22u)=3þ(ϵv/C0ϵu)=3)¼1:34 eV (the remaining terms cancel), while the ZPL we obtain for the hsite V/C0 Siis (2(ϵ/C22v/C0ϵ/C22u)=3þ(ϵv/C0ϵu)=3)¼1:43 eV (where again, the remain- ing terms cancel) in excellent agreement with the experimentalvalues of 1.35 and 1.44 eV, respectively. 67The slight underestima- tion of the ZPL values may, therefore, be due to the slightly smaller HSE06 bandgap (3.18 eV compared to about 3.2 eV for experiment68). We note, however, that other theoretical calculations yield 1.44 eV fortheksite and 1.54 eV for the hsite using the HSE06 functional. 69We believe that error compensation in taking the difference of many eigenvalues may be causing the greater accuracy of our approximation to the ΔSCF method using Janak ’s theorem than the ΔSCF method itself, though later theoretical work shows better agreement with ourwork and with experiment. 70 We note that, as the many-body states from the work of Soykal et al.66have been defined as Slater determinants of products of single-particle orbitals, the only requirement for the correctnessof the calculation outlined above is that these single-particle orbit-als be orthogonal, which is consistent with the orbitals constructed in the work of Soykal et al. 66Though after performing structural relaxation with the HSE06 functional the expected C3vsymmetry is weakly broken, we have used the assignment and energetic orderingof the orbitals from the work of Soykal et al. 66as such an ordering and assignment is consistent with obtaining the lowest energy spin- conserving excitation of the S¼3=2 states that our calculations produce. Ultimately, these results demonstrate that in cases wherethe wavefunctions for the states that will change occupation do notappreciably differ at locations of a significant change in the ionicpotential and where the change in the ionic potential can be viewed as a small perturbation, there is no need to calculate excited state properties to obtain the ZPL as values derived entirely fromthe ground state calculation can provide excellent agreement withexperiment. These conditions are satisfied for the single vacancy asnone of the defect wavefunctions are appreciable at the location of the removal of the atom and a single vacancy is a perturbation of less than 1% of the integrated ionic potential for the supercell size. Based on the results of the calculations for silicon monovacan- cies in 4 H-SiC outlined above, Janak ’s theorem shows promise for rapid estimation of ZPL values. Indeed, we will see below that the accuracy of Janak ’s theorem is maintained for the strain ( ε) valueswith the two lowest total energies, namely, the uncompressed and theε¼þ2% structures, for the singly negatively charged calcium vacancy in SiS 2and that the error calculations we have outlined in Sec.II A provide a clear indication of when Janak ’s theorem fails. B. Singly negatively charged calcium vacancy in SiS 2 As alluded to above, we now turn to demonstrating the con- tinued accuracy of Janak ’s theorem for the εvalues with the two lowest total energies for the singly negatively charged calciumvacancy in SiS 2. We additionally show that the associated error consistently identifies the larger discrepancies between the results of the theorem and the results of ΔSCF calculations. We also intro- duce results showing faster convergence of excited state SCF calcu-lations when the charge density is initialized by mixing the HOMO and LUMO compared to when it is initialized from a superposition of atomic charges. For the lattice parameters of the stoichiometricunit cell SiS 2with space group P2 1/c using the PBE functional, we find a¼5:93 Å, b¼8:13 Å, α¼90/C14,β¼102:57/C14,γ¼90/C14and a monolayer thickness of 3 :65 Å with a vacuum of 16 :94 Å. Using the HSE06 functional, we find a¼5:88 Å, b¼8:04 Å, α¼90/C14, β¼103:11/C14,γ¼90/C14and a monolayer thickness of 3 :59 Å with a vacuum of 16 :67 Å. The lattice parameters with strain are included inTable IV . Structures for the singly negatively charged calcium vacancy for five strain values are found in Fig. 3 . The values in Table V show that compressive strain tends to destroy the Cisym- metry present in the uncompressed, ε¼þ2% and ε¼þ5% struc- tures, due to buckling of the system. The approximation to the ZPL using Janak ’s theorem and using the ΔSCF method for different in-plane εcan be found in Fig. 4 , where the corresponding eigenvalues can be found in Table VI . The difference between the LUMO and HOMO eigenval- ues is used in the application of Janak ’s theorem. We observe little variation of the approximation to the ZPL with tensile ε, but more variation with compressive ε, which can also distort the structure. Since, due to Poisson ’s ratio, the application of pressure should lead to tensile in-plane ε, we do not expect the ZPL value to change significantly from what we have predicted in experimentallyrealizable structures. The error calculations perform best for the uncompressed (error of 0 :0128 eV) and ε¼þ2% (error of 0:0249 eV) structures, which have the smallest energy differences, obtaining the correct sign of the error as well, but fail to capturethe correct sign and are much too large for the remaining εvalues (errors of /C017:4096 eV, /C00:3898 eV, and /C00:6123 eV for ε¼/C05%,/C02%, and þ5%, respectively). We note that forTABLE III. DFT eigenvalues in eV calculated in the ground state using the HSE06 functional for the hole single-particle states of the V/C0 Si(h) point defect in 4 H-SiC. Single-particle state Majority spin Minority spin u 7.551 8.119 v 7.800 10.137 ex 7.871 10.227 ey 7.906 10.289 TABLE IV . PBE lattice constants in Å and corresponding angles with strain. The monolayer thickness was not computed as the lattice parameters were directly applied to the defect-containing supercell structures, which were then relaxed at constant volume. ε(%) ab c αβγ +5 6.22 8.53 21.09 90/C14102:57/C1490/C14 +2 6.05 8.29 21.09 90/C14102:57/C1490/C14 −2 5.81 7.96 21.09 90/C14102:57/C1490/C14 −5 5.63 7.72 21.09 90/C14102:57/C1490/C14Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-6 Published under an exclusive license by AIP Publishingε¼/C05%, the Ca atom departed from the position that preserved inversion symmetry and that the strain value was, therefore, notsimilar to the other structures. The larger errors nonethelesscorrectly indicate more significant disagreement between the ΔSCF result and the result from using Janak ’s theorem. We note that based on a statistical learning based prediction of the bulkmodulus, 71,72the structures that should have the two lowest total energies given the requirement of an applied pressure of2.8–3.5 GPa are the uncompressed and ε¼þ2% structures, which is verified by QUANTUM ESPRESSO total energy calculations. Naturally, the uncompressed structure has the lower energy of thetwo structures. As shown in Fig. 5 , these structures with the two lowest total energies also responded best to performing the excited state calcu- lation by replacing the initialization of the charge density from asuperposition of atomic charges with the charge density of theground state calculation, where the HOMO contribution is modi-fied according to HOMOj .!βHOMOj .þ(1/C0β) LUMOj ., (15) 0/C20β/C201. We note that a ground state relaxation is needed for both the standard method or default initialization and the approach of initializing the charge density from a mixture of the HOMO and LUMO orbitals for the excited state SCF calculation. This prerequi-site calculation, therefore, does not change the net increase ordecrease in the number of iterations for one method over another. Given that there may be instances where a ground state relaxationis not performed, we report the number of iterations required toconverge the ground state SCF calculation for completeness. Forε¼/C05%,/C02%,þ2%,þ5% and the uncompressed structure, the number of iterations required were 29, 35, 34, 31, and 35,respectively. The value of the bandgap for the uncompressed struc-ture using the PBE functional is 3.55 eV, and the defect has totalspin S¼1=2. The ΔSCF calculations and the differences in ground state eigenvalues were from QUANTUM ESPRESSO. Based on the computational efficiency of our method for determining the error associated with using Janak ’s theorem dem- onstrated in Fig. 6 , we argue that the approach of using Janak ’s FIG. 3. Shown above is the structure of a singly negatively charged calcium vacancy using the PBE functional for ε¼+2,+5 and the uncompressed structure. Silicon atoms are in blue, sulfur atoms are in yellow, and the calcium atom is in cyan. Silicon and sulfur vacancies are shown as blue and yellow dashed ci rcles, respec- tively, for the uncompressed structure. The sulfur atoms closest to the calcium atom in the uncompressed structure are numbered as 1 –4. TABLE V . PBE distances in Å, d(X1−X2), between sulfur atoms numbered in Fig. 3 and the calcium atom for ε= ±2, ± 5 and the uncompressed structure as well as the smallest angles, θ(X1−X2−X3), between the bonds joining the calcium atom to opposite sulfur atoms. HSE06 values are in parentheses. εd(Ca −S1)d(Ca −S2)d(Ca −S3)d(Ca −S4)θ(S1−Ca −S2)θ(S3− Ca−S4) þ5% 2.76 2.76 2.83 2.83 179 :99/C14179:99/C14 þ2% 2.75 2.75 2.80 2.80 179 :99/C14179:98/C14 0% 2.76 (2.75)2.76 (2.75)2.79 (2.79)2.79 (2.79)179:98/C14 (179 :93/C14)179:97/C14 (179 :89/C14) /C02% 2.76 2.76 2.79 2.79 179 :92/C14179:88/C14 /C05% 2.97 2.70 3.03 2.72 150 :28/C14138:39/C14 FIG. 4. ZPL values in eV , calculated using Janak ’s theorem (red) and using the ΔSCF method (green) for the singly negatively charged calcium vacancy in SiS 2for in-plane ε¼+2%,+5% and for the uncompressed structure. The error associated with the result from Janak ’s theorem was calculated as described in Sec. II. The ε¼/C0 5% case caused the Ca atom to depart from the position that preserved inversion symmetry and was, therefore, not similar tothe other structures.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-7 Published under an exclusive license by AIP Publishingtheorem should be acceptable for screening large numbers of potential point defect single-photon emitter candidates for desired ZPL values. Our approach to the initialization of the charge density also shows promise. A key point is that the search would focus onpoint defect candidates with total spin S¼1=2; otherwise, more involved group theoretic arguments would be required to deter- mine the correct many-body hole wavefunction as in the work of Soykal et al. 66Other specific systems, such as triplet systems,73will also be the focus of future work. The bandgap for the uncom-pressed structure using the HSE06 functional is 4.75 eV, and weagain find that the defect has total spin S¼1=2. Using QUANTUM ESPRESSO, the HSE06 calculation yields a ΔSCF result of 0.1705 eV and a difference in ground state eigenvalues of0.2676 eV. The error calculated as described in Sec. IIyields 0.3008 eV. For hosts where the accuracy of hybrid functionals hasbeen demonstrated, clearly, their continued use would be recom- mended. However, one cannot generally state that hybrid function- als will be more accurate than semilocal ones. Indeed, Johari and Shenoy 74find that their PBE bandgap for monolayer MoS 2under- estimates the experimental bandgap of 1.8 eV by just 0.12 eV, whileKuc et al. 75and Ataca and Ciraci76found that the PBE0 and HSE06 hybrid functionals overestimate this bandgap by approxi- mately 1 and 0.45 eV, respectively. In our work, we have, therefore,included both the PBE and HSE06 ZPL transitions for the uncom-pressed structure to reduce the possibility of missing the transition,noting that the transition energy does not vary significantly with strain for low total energy strain configurations. We recommend that any application of our methods using a semilocal functional isalso performed with a hybrid functional. The DFT partial charge densities obtained for ground hole (LUMO) and excited hole (HOMO) states and differences (LUMO /C0HOMO) are shown in Fig. 7 . The differences show the closenessTABLE VI. DFT eigenvalues in eV calculated in the ground state of the negatively charged calcium vacancy in SiS 2using the PBE and HSE06 functionals for various ε values. PBE HSE06 ε −5% −2% Uncompressed +2% +5% Uncompressed HOMO −2.2137 −2.8524 −3.1014 −3.2690 −3.4880 −3.1995 LUMO −1.7557 −2.7062 −2.9211 −3.0855 −3.3124 −2.9319 FIG. 5. Change in the number of iterations to convergence using the charge density initialization presented in Eq. (15) and using the default initialization in QUANTUM ESPRESSO. The zero represents the default initialization or the standard method so that a data point with a value along the vertical axis of /C07 indicates that the particular run converged in seven fewer iterations than thestandard method for the appropriate strain value. In-plane ε¼+2%,+5%, where εdenotes strain, and the uncompressed structure were investigated. The maximum number of iterations was set at 100, which was attained for bothε¼þ 2% and ε¼þ 5% and explains the plateaus. We took 0 /C20β/C201a n d used increments of 0.01. Data points are indicated by “/C2,”and the dashed lines are a guide for the eye. The number of iterations required to converge the ground state SCF calculation is included in the text for all εvalues for completeness. FIG. 6. Runtimes for convergence of the constrained-occupation calculation for the excited state (yellow) for completion of 20 SCF iterations (turquoise) and for calculation of the error associated with using Janak ’s theorem using our method (blue). As our error calculations were done using the V ASP code and theconstrained-occupation calculations for the excited state were done using theQUANTUM ESPRESSO code, we took the time for a single SCF iteration for a given system from V ASP and multiplied by the number of iterations required for the constrained-occupation calculation for the excited state in QUANTUMESPRESSO to converge.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-8 Published under an exclusive license by AIP Publishingof certain LUMO –HOMO pairs. More importantly, Fig. 7 shows that Janak ’s theorem succeeds when the differences are not appre- ciable at the location of the greatest change in the ionic potential of the perfect lattice (at the location of the Ca atom) and does worseotherwise. The structure of the defect is four missing atoms, two silicon and two sulfur, replaced by a single calcium atom such that the resulting point defect is inversion-symmetric, as confirmed bythe partial charge densities in Fig. 7 . This inversion symmetry is broken, however, for in-plane ε¼/C05%. The lowest formation energy as a function of Fermi level is displayed in Fig. 8 for the uncompressed structure. The plot demonstrates that in sulfur-rich preparation conditions, the introduction of a calcium vacancy actu-ally stabilizes the SiS 2structure with space group P2 1/c. Given the fact that the singly negatively charged calcium vacancy only existsfor a very limited range of Fermi level values, it would be necessary to pin the Fermi level at the appropriate value by doping the system. Using a 43Ca atom instead of a40Ca atom could also lead to coupling between the nuclear spin and the electronic spin toimplement a long-lived quantum memory realized with thenuclear spin. 77 We now turn to addressing the possibility of non-radiative decay with phonon emission upon excitation of the singly nega-tively charged calcium vacancy. Unlike in diamond where opticalphonon modes with energies of about 0.17 eV can be found, 78the energies of the phonon modes in SiS 2with space group P2 1/c are all below 20 THz as shown in Fig. 9 , corresponding to an energy of about 0.08 eV. This result indicates that non-radiative decay wouldrequire multi-phonon processes, which would be rare. We note that unlike diamond that is routinely synthesized at high pressure but shows no dynamical instability at ambient condi- tion, our calculations show (see Fig. 9 ) that SiS 2with space group FIG. 7. DFT calculated partial charge densities for a singly negatively charged calcium vacancy in SiS 2using the PBE functional for in-plane ε¼+2%,+5% and for the uncompressed structure and using the HSE06 functional for the uncompressed structure are displayed corresponding to the LUMO (left), the HOMO (middle), and the difference between the total charge densities with andwithout exchanging the occupations of the HOMO and LUMO (right). The valueofεis indicated to the left of the respective panels. Sulfur atoms are in yellow, silicon atoms are in blue, and the calcium atom is in cyan. Charge accumulation (depletion) is indicated by translucent red (green). The isosurface of the chargedensity is 0.0005 e=Å 3for all plots. FIG. 8. Formation energy as a function of Fermi level for the calcium vacancy for a PBE DFT calculated gap Eg¼3:55 eV . The lower line (yellow) corre- sponds to sulfur-rich preparation conditions, while the upper line (blue) corre-sponds to silicon-rich preparation conditions. The integers between the lines indicate the most stable charge state of the defect at the corresponding values of the Fermi level, with each region bounded by vertical dotted lines.Calculations were done for the uncompressed structure.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-9 Published under an exclusive license by AIP PublishingP21/c does exhibit a dynamical instability at ambient condition and would, therefore, need to be maintained at the 2.8 GPa required forits synthesis to be stable. We would like to make the followingpoints: Experiments at high pressure ( .10 2GPa) are routinely performed. In particular, the discovery of room-temperature super- conductors at high pressure (e.g., Snider et al.79at 267 GPa) will undoubtedly propel further technical advances in this area.Experiments of emitters at high pressure (e.g., NV centers indiamond) have also been performed (e.g., Lesik et al. , 80Yipet al. ,81 and Hsieh et al.82), showing such experiments to be within current feasibility. While it is not ideal that the proposed candidate materialrequires high pressure, it does not remove one of the key elementsof the paper —the tools for the rapid screening of defect qubits. In summary, the case of the singly negatively charged calcium vacancy in SiS 2confirms that using Janak ’s theorem and the associated error estimation for ZPL calculations is quick and clearly signalswhen the theorem is applicable. We have also shown that theconstrained-occupation calculation for the excited state can be completed in some cases with fewer SCF iterations by carefully ini- tializing the charge density.IV. CONCLUSION In conclusion, we have developed a novel computational framework for high-throughput screening of the ZPL of atom-likequbits with applications for materials discovery, quantum sensing, and quantum computing. We propose the use of Janak ’s theorem for ZPL calculations and provide a quick new method for estimat-ing the associated error. We also outline a new method for reduc-ing the number of iterations required to converge excited state SCFcalculations, which is compared to the default procedure for carry- ing out such calculations. Our ZPL results are consistent with the previous experimental work where available and suggest that con-sidering only ground state eigenvalues in the hole approach is acomputationally efficient way of calculating optical excitation ener-gies of color centers for screening purposes. Our novel method for initializing the charge density to reduce the number of iterations required for excited state calculations also works well for systemswith small strain. Finally, we propose the new singly negativelycharged calcium vacancy in SiS 2, which has the advantage of being inversion-symmetric and of containing a low density of nuclear spins that would lead to poor coherence properties of spins hosted in the material. However, this new proposed system would need tobe maintained at the 2.8 GPa required for its synthesis to ensurestability. We also note that our rapid screening methods can createtraining data for machine learning and serve in technologies for predicting defect structures with desirable properties. ACKNOWLEDGMENTS We thank Efthimios Kaxiras of Harvard University and Georgios A. Tritsaris for useful discussions. R.K.D. gratefullyacknowledges financial support from the Princeton PresidentialPostdoctoral Research Fellowship. We also acknowledge support by the STC Center for Integrated Quantum Materials (NSF Grant No. DMR-1231319). This work used computational resources of theExtreme Science and Engineering Discovery Environment(XSEDE), which is supported by the National Science Foundation(NSF) (Grant No. ACI-1548562), 83on Stampede2 at TACC through allocation (No. TG-DMR120073), and of the National Energy Research Scientific Computing Center (NERSC), a U.S.Department of Energy Office of Science User Facility operatedunder Contract No. DE-AC02-05CH11231. APPENDIX: COMPUTATIONAL DETAILS We provide in this appendix details of the computations. The atomic positions were relaxed until the magnitude of the Hellmann –Feynman forces was smaller than 10 /C04eV Å/C01on each atom without spin polarization and subsequently until the magni-tude of the Hellmann –Feynman forces was smaller than 10 /C02eV Å/C01on each atom with spin polarization to obtain defect levels, and, for the stoichiometric conventional unit cell, the lattice parameters were concurrently relaxed. The wavefunctions were expanded on a plane wave basis with a cutoff energy of 500 eV forall systems, and a Monkhorst –Pack grid of 6 /C26/C22 k-points was used for integrations in the reciprocal space for SiS 2and a gamma centered grid of 4 /C24/C22 k-points was used for integrations in the reciprocal space of 4 H-SiC. The relaxed lattice parameters of the FIG. 9. Phonon dispersion for the uncompressed P2 1/c structure of SiS 2along the indicated high-symmetry path, as calculated with the phonopy code.52The structure does show imaginary frequencies near the Γpoint and would, there- fore, need to be maintained at the 2.8 GPa required for its synthesis to bestable.16The vectors a/C3,b/C3, and c/C3are reciprocal to the vectors corresponding to the lattice constants a,band the out-of-plane lattice constant, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 225105 (2021); doi: 10.1063/5.0048833 129, 225105-10 Published under an exclusive license by AIP Publishingstoichiometric unit cell were then used for the supercell structures. Formation energies and defect levels were calculated using a super- cell of 108 atoms for SiS 2(3/C23/C21 multiple of the stoichiometric unit cell) with appropriately scaled k-point grids. For 4 H-SiC, a supercell of 576 atoms (6 /C26/C22 multiple of the stoichiometric unit cell) was used. DATA AVAILABILITY The data from DFT calculations that support the findings of this study and the modified source code for our charge densityimplementation are available from the corresponding author upon reasonable request. REFERENCES 1A. K. Geim and I. V. Grigorieva, Nature 499, 419 (2013). 2M. Toth and I. Aharonovich, Annu. Rev. Phys. Chem. 70, 123 (2019). 3S. Wu, V. Fatemi, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, Science 359, 76 (2018). 4Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori,and P. Jarillo-Herrero, Nature 556, 80 (2018). 5Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43 (2018). 6A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. V. 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5.0054409.pdf

J. Appl. Phys. 129, 223906 (2021); https://doi.org/10.1063/5.0054409 129, 223906 © 2021 Author(s).Magnon spin transport around the compensation magnetic field in easy-plane antiferromagnetic insulators Cite as: J. Appl. Phys. 129, 223906 (2021); https://doi.org/10.1063/5.0054409 Submitted: 18 April 2021 . Accepted: 20 May 2021 . Published Online: 09 June 2021 Ka Shen ARTICLES YOU MAY BE INTERESTED IN On the temperature-dependent characteristics of perpendicular shape anisotropy-spin transfer torque-magnetic random access memories Journal of Applied Physics 129, 223903 (2021); https://doi.org/10.1063/5.0054356 Optical phonon modes, static and high-frequency dielectric constants, and effective electron mass parameter in cubic In 2O3 Journal of Applied Physics 129, 225102 (2021); https://doi.org/10.1063/5.0052848 Introduction to antiferromagnetic magnons Journal of Applied Physics 126, 151101 (2019); https://doi.org/10.1063/1.5109132Magnon spin transport around the compensation magnetic field in easy-plane antiferromagnetic insulators Cite as: J. Appl. Phys. 129, 223906 (2021); doi: 10.1063/5.0054409 View Online Export Citation CrossMar k Submitted: 18 April 2021 · Accepted: 20 May 2021 · Published Online: 9 June 2021 Ka Shena) AFFILIATIONS The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China a)Author to whom correspondence should be addressed: kashen@bnu.edu.cn ABSTRACT In this work, we theoretically study the magnon spin transport in easy-plane antiferromagnetic insulators in the presence of an in-plane magnetic field. By exactly calculating the magnon spectrum, we find that the band splitting due to magnetic anisotropy can be fullycompensated by the external field at a particular strength, which makes the dynamics nearly equivalent to an easy-axis antiferromagnet. As a result, the intrinsic magnon spin Hall effect due to the dipole –dipole interaction, previously predicted in easy-axis antiferromagnets, is activated in easy-plane antiferromagnets. The compensation feature also allows the field control of magnon spin lifetime and hence the spindiffusion length. The compensation feature is robust against biaxial anisotropy. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054409 I. INTRODUCTION The magnetic dynamics and transport properties in antiferro- magnetic materials are essential for the performance of antiferromag- netic spintronic devices and have been investigated intensively in thelast decade. 1,2Among these studies, high efficient spin transmission through an insulating antiferromagnetic layer was demonstrated in ferromagnet –antiferromagnet –normal metal (NM) trilayer structures.3–6The lack of itinerant electrons implies that the spin information should be able to transmit across the antiferromagneticinsulator (AFI) in the form of polarized magnons. Non-local mea-surements in different materials revealed that the spin transport distance associated with antiferromagnetic magnons can reach several or even tens of micrometers, 7–9which is already comparable with that in the high quality yttrium iron garnet, a ferrimagnetic materialfamous for its ultralow magnetic damping. 10,11Recently, spin injec- tion into NM from subterahertz magnons was realized in AFI-NM bilayers by spin pumping12,13and optical approach.14These pro- gresses offer new opportunities for promising applications of AFIs. As most of the previous experimental works are about easy- axis AFIs, such as α-Fe2O3(below transition temperature around 260 K in bulk),7Cr2O3,8,12,15MnF 2,13and MnPS 3,9systems with a magnetic easy plane like NiO and α-Fe2O3(above transition tem- perature) are also quite interesting because of their distinctivefeatures. For instance, the U(1) spin-rotational symmetry within the easy plane allows spin superfluidity.16–18Another important advantage for applications, compared with the easy-axis AFIs, is the easy access of magnetization manipulation, because the Néel vector in easy-plane AFIs keeps perpendicular to and, therefore,can be rotated by an in-plane magnetic field. This allows the fieldmodulation of spin transport 19and spin Hall magnetoresistance.20 In contrast to the easy-axis case, where the magnon bands are twofold degenerate, magnetic anisotropy in easy-plane AFIs breaks the symmetry between the in-plane and out-of-plane magnetizationdynamics and results in a band splitting. The lower and higher fre-quency modes correspond to the in-plane and out-of-plane motion of the Néel vector, 21respectively. Such a splitting leads to a coher- ent dynamics of magnon spin polarization19and its modulation by an external magnetic field causes a Hanle-type effect.22,23 Interestingly, the spatial motion of the magnons in AFIs, similar to the mobile electrons in metallic systems, can be corre- lated with their spin polarization, for example, in noncentrosym- metric systems via the Dzyaloshinskii –Moriya interaction (DMI), which provides the possibility to discover electron-like spin –orbit phenomena. Theoretical studies predicted magnon spin Nernsteffect 24,25and magnonic Edelstein effect26,27driven by DMI. Recently, the dipole –dipole interaction (DDI), which was usuallyJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223906 (2021); doi: 10.1063/5.0054409 129, 223906-1 Published under an exclusive license by AIP Publishingignored in antiferromagnets, was shown to be able to manifest itself as an effective spin –orbit coupling (SOC)28,29between magnon states in uniaxial easy-axis AFIs. Such a magnon SOC can also giverise to various spin –orbit phenomena, e.g., an intrinsic magnon spin Hall effect (SHE), 28D’yakonov –Perel ’(DP)-type magnon spin relaxation, and topological surface states.30The role of this DDI-induced mechanism in easy-plane AFIs is yet to be examined. In this work, we calculate the magnon spectrum analytically in easy-plane AFIs by taking into account the exchange interaction,magnetic anisotropy, Zeeman energy due to an in-plane magneticfield and, as interpreted above, the DDI. Since the magnitude of DDI is relatively weak than the splitting between the in-plane and out-of-plane polarized magnon modes under magnetic anisotropy,its influence is negligible in a weak magnetic field regime. Very inter-estingly, as the magnetic field increases, the band splitting is foundto be globally suppressed and, at a compensation magnetic field, the contributions from magnetic anisotropy and Zeeman term cancel with each other exactly in the entire Brillouin zone, making the easy-plane AFI approximately equivalent to an easy-axis one. Physically,this is because the external field introduces additional magneticanisotropy, which behaves as a hard axis along the magnetic field and together with the original natural hard axis defines a hard plane, making the normal direction equivalently an easy axis. The resultingmagnetic anisotropy becomes uniaxial when the strengths in the twohard axes are equal. As a result, the momentum-dependent SOC due to DDI becomes dominant and the magnon spin Hall mechanism is switched on. In the meantime, the DP-type magnon spin relaxation,although it is relevant regardless of the strength of the magneticfield, can be strongly modified around the compensation field.Moreover, the role of DMI and additional magnetic anisotropy within the easy plane will also be addressed. II. MODEL We start from the minimal model for an easy-plane AFI includ- ing magnetic anisotropy and antiferromagnetic exchange interaction between the nearest neighbors. An external magnetic field is applied within the y–zeasy plane to control the orientation of the Néel vector. Without loss of generality, as illustrated in Fig. 1 ,t h emagnetic field is set to be along the yaxis, which leads to a canting of the two antiferromagnetic coupled sublattice magnetizations m 1and m2. The net magnetization m¼m1þm2and the Néel vector n¼m1/C0m2are, therefore, along yandzdirections, respec- tively. The canting angle ( θ) can be determined by minimizing the total energy described by the Hamiltonian H¼X iK(Sx ai)2þK(Sx di)2/C0gμBBSy ai/C0gμBBSy di/C2/C3 /C0X hi,jiJSai/C1Sdj, (1) where the anisotropy coefficient K.0 and the inter-sublattice exchange coupling constant J,0. The subscripts aanddlabel the two sublattices. For a system with 2 Nmagnetic ions, the total energy reads E/C25/C0NzjJjS2cos 2 θ/C02NgμBBSsinθ, (2) and thus the canting angle is determined by sinθ¼ωZ 2ωex: (3) Here, ωZ¼gμBB=/C22hand ωex¼zjJjS=/C22hrepresent the frequency scales of the Zeeman term and the exchange interaction, respectively. The spin operators in Eq. (1)can be expressed under the local equilibrium configuration via a rotation operation Sx ai Sy ai Sz ai0 B@1 CA¼10 0 0 cos θ sinθ 0/C0sinθcosθ0 B@1 CA~Sx ai ~Sy ai ~Sz ai0 B@1 CA, (4) Sx di Sy di Sz di0 B@1 CA¼10 0 0 cos θ/C0sinθ 0 sin θ cosθ0 B@1 CA~Sx di ~Sy di Sz di0 B@1 CA, (5) which leads to H¼X iK(~Sx ai)2þK(~Sx di)2/C0gμBBsinθ(~Sz ai/C0~Sz di)hi /C0X hi,jiJ~Sai/C1~Sdj/C02 sin2θ(~Sy ai~Sy djþ~Sz ai~Sz dj)h /C0sin 2θ(~Sy ai~Sz di/C0~Sz ai~Sy di)i : (6) By performing the standard Holstein –Primakoff transformation31 to the spin operators ~Sz a¼S/C0aya,~Sþ a¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2S/C0ayap a, ~Sz d¼/C0Sþdyd,~Sþ d¼dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2S/C0dydp ,(7) one can write out the quadratic terms in momentum space under FIG. 1. The spin configuration of an easy-plane AFI at equilibrium state in the presence of an in-plane external magnetic field.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223906 (2021); doi: 10.1063/5.0054409 129, 223906-2 Published under an exclusive license by AIP Publishingthe basis of ( ak,dk,ay /C0k,dy /C0k)Tas H0 k,/C0k¼AC k A0Bk/C0Ck Ck AB k/C0CkA0 A0Bk/C0Ck AC k Bk/C0CkA0Ck A0 BB@1 CCA, (8) where A=/C22h¼ω anþωex,A0=/C22h¼ωan,Bk=/C22h¼γkωex, and Ck=/C22h¼γkωδwith ωan¼KS=/C22hand ωδ¼ωexsin2θ. The form factor γk¼(1=z)P δexp(iδ/C1k) averages the phase factor over all z antiferromagnetic coupled neighbors and is real in cubic lattice. A. Magnon dispersion relation In order to compute the dispersion relation analytically, it is convenient to define the magnon operators, according to the sym-metry, as the orthogonal linearly polarized basis f+ k¼(ak+dk)=ffiffiffi 2p (9) and rewrite Hamiltonian (8)under the basis of [ fþ k,(fþ/C0k)y, f/C0 k,(f/C0/C0k)y]Tas ~H0 k,/C0k¼AþCkBþ k/C0Ck 00 Bþ k/C0CkAþCk 00 00 A/C0CkB/C0 kþCk 00 B/C0 kþCkA/C0Ck0 BB@1 CCA, (10) in which B + k¼A0+Bk. Apparently, Hamiltonian (10) can be divided into two individual BdG blocks, both of which can besolved analytically via the Bogoliubov transformation. A straight-forward calculation gives the eigenfrequencies of two linear polar-ized magnon modes, ω + k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (AþB+ k)(A/C0B+ k+2Ck)q =/C22h, (11) and the operators of the eigenstates ψ+ k¼u+ kf+kþv+ k(f+/C0k)y, (12) where the coefficients can be expressed by u+ k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A+Ckþ/C22hω+ k 2/C22hω+ ks , (13) v+ k¼sgn(B+ k+Ck)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A+Ck/C0/C22hω+ k 2/C22hω+ ks : (14) In the long-wavelength limit, k≃0, one has γk≃1 and, therefore, ωþ 0¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωδ(ωexþωan)p ¼ωZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiωexþωan ωexr , (15)ω/C0 0¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωan(ωex/C0ωδ)p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiωan ωex(4ω2 ex/C0ω2 Z)r : (16) Notice that ωþ 0is proportional to the external field and, therefore, vanishes at zero field, whereas ω/C0 0is of finite value and relatively insensitive to the magnetic field. As a consequence, they becomeequal at ω δ¼ωexωan ωexþ2ωan, (17) corresponding to the compensation Zeeman field ωZc¼2ωexffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiωan ωexþ2ωanr : (18) In hematite, the compensation field is around 8 T.22The canting angle at this compensation field is sinθc¼ωZc 2ωex¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωan=ωex 1þ2ωan=ωexs : (19) Since in-plane anisotropy in typical antiferromagnets is much smaller than the exchange energy, i.e., ωan=ωex/C281, the canting angle θat the compensation field is relatively small, retaining col- linear antiferromagnetic configuration approximately. Figure 2 shows the dispersion relations with three typical strengths of the external magnetic field. The gapless linear disper- sive mode ( ωþ k) branch in the absence of the magnetic field corre- sponds to the Néel vector oscillating within the easy plane togetherwith a small net magnetization oscillating out of the plane. In con-trast, the gaped mode ( ω /C0 k) displays a large out-of-plane oscillation of the Néel vector along with a small in-plane magnetization oscil- lation. Since the ωþ kmode is more sensitive to the magnetic field than the ω/C0 kmode as discussed above, the two frequencies at k¼0 become equal at ωZ¼ωZc. Very importantly, according to the middle plot of Fig. 2 and Eq. (11), the two branches at this condi- tion actually become degenerate for any k. The dispersion relation reads ω+ k¼/C22hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiωex ωexþ2ωanr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2ωanþωex)2/C0(γkωex)2q : (20) Under this condition, an arbitrary combination of the two linearly polarized mode remains the eigenmode of Hamiltonian (10), which allows a transform from the linear polarized modes to circularlypolarized modes. This is very similar to the situation in easy-axis AFIs. In other words, the compensation magnetic field drives the easy-plane AFI into a configuration equivalent to an easy-axis AFI.This issue will be discussed further below in Sec. II C. This effect is robust against in-plane anisotropy as will be shown later in thepaper. For a field stronger than ω Zc, the ωþ kbranch is lifted above theω/C0 kone.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223906 (2021); doi: 10.1063/5.0054409 129, 223906-3 Published under an exclusive license by AIP PublishingB. DDI-induced SOC The long-range dipole –dipole interaction includes the cou- pling between any spin pair and it reads Hd¼μ0(gμB)2 2X l=l0jRll0j2Sl/C1Sl0/C03(Rll0/C1Sl)(Rll0/C1Sl0) jRll0j5, (21) where gis the gfactor, μBis the Bohr magneton, and μ0is the vacuum permeability. As mentioned above, the ground spin config-uration remains approximately collinear in the regime we are inter-ested in, due to the small canting angle. Therefore, for the sake of simplicity, we make an approximation S≃~Sto Eq. (21) and apply the Holstein –Primakoff transformation, which results in, under the basis of ( a k,dk,ay /C0k,dy /C0k)T,28 Hd k,/C0k¼Ak γkB* k B*k γkAk γkBk AkγkAk Bk Bk γkAk Ak γkBk γkAk B*k γkB*k Ak0 BB@1 CCA, (22) in which A k¼/C02Sμ0μ2 BX Rll0=0R2 ll0/C03(Rz ll0)2 R5ll0e/C0ik/C1Rll0, (23) Bk¼/C06Sμ0μ2 BX Rll0=01 R5 ll0(Rþ ll0)2eik/C1Rll0: (24) After computing the summation in continuum limit,28,32,33we obtain Bk¼Ake2ifk¼1 2/C22hωmsin2θke2ifk,( 2 5 )with ωm¼γμ0Ms.H e r e , Msrepresents the magnetization of a single sublattice. By projecting Eq. (22) to the magnon particle space ( ψþ k,ψ/C0 k), we obtain Hd k,/C0k¼Δþþ k iΔþ/C0 k /C0iΔþ/C0 k/C0Δ/C0/C0 k/C18/C19 , (26) which shows a coupling between the two linear polarized magnon modes. The coupling parameters are Δþþ k¼<Bk(γkuþ kuþk/C02vþ kuþkþγkvþ kvþ k), (27) Δ/C0/C0 k¼<Bk(γku/C0 ku/C0kþ2v/C0 ku/C0kþγkv/C0 kv/C0 k), (28) Δþ/C0 k¼=Bk(γkuþ ku/C0 kþuþ kv/C0 k/C0vþ ku/C0 k/C0γkvþ kv/C0 k): (29) The total effective non-interacting Hamiltonian Hkunder the basis of (ψþ k,ψ/C0 k) thus becomes Hk¼/C22εkþδεk iΔþ/C0 k /C0iΔþ/C0 k/C22εk/C0δεk/C18/C19 , (30) with /C22εk¼(/C22hωþ kþ/C22hω/C0 kþΔþþ k/C0Δ/C0/C0 k)=2 and δεk¼(/C22hωþ k/C0/C22hω/C0 kþ Δþþ kþΔ/C0/C0 k)=2. One see that magnetic anisotropy supplies a contri- bution to the spin –orbit field via band splitting j/C22hωþ k/C0/C22hω/C0 kjinδεk. Since this band splitting is in the subterahertz region, much strongerthan the dipolar interaction jB kjin the order of gigahertz, the magnon SOC is dominated by magnetic anisotropy, except around the compensation magnetic field where j/C22hωþ k/C0/C22hω/C0 kj≃0. C. Spin polarized representation As discussed above, the two magnon eigenstates given by Eq.(12) are both linearly polarized, meaning that they do not carry net spin. A unitary transformation into circularly polarized basiscan be achieved by α k βk/C18/C19 ¼Aψþ k ψ/C0 k/C18/C19 , (31) where the transformation matrix is defined as A¼cosχ sinχ /C0sinχcosχ/C18/C19 : (32) One can verify that with the parameter χgiven by sin 2χ¼(uþ ku/C0 kþvþ kv/C0 k)/C01, (33) the two modes αandβhave one unit spin, but with opposite sign. In particular, at the compensation magnetic field, we have ωþ k¼ω/C0 k¼ωkandωδ/C28ωex, which lead to uþ k≃u/C0 k≃uk, (34) vþ k≃/C0v/C0 k≃vk, (35) FIG. 2. Dispersion relations of the two magnon modes (the green curve for ω/C0 k and the purple one for ωþ k) with three typical magnetic field strengths. In the cal- culation, we adopt ωan=ωex¼0:01 and the form factor γk¼ cos(kxa=2)cos( kya=2)cos½kza=2/C138is applied along ( kx,0,0) momentum line. The insets illustrate the magnetization dynamics of the two sublattices, with the white curves representing the trajectory of each magnetic moment.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223906 (2021); doi: 10.1063/5.0054409 129, 223906-4 Published under an exclusive license by AIP Publishingand, therefore, sin 2 χ≃(u2 k/C0v2 k)/C01¼1. Under this condition, the transform matrix reduces to A≃1ffiffi 2p1ffiffi 2p /C01ffiffi 2p1ffiffi 2p ! , (36) and the SOC coefficients become Δ++ k≃<Bk(γku2 k/C02vkukþγkv2 k), (37) Δþ/C0 k≃=Bk(γku2 k/C02vkukþγkv2 k): (38) In this spin polarized representation ( α,β), the effective Hamiltonian reads ~Hk¼/C22εk /C0δεkþiΔþ/C0 k /C0δεk/C0iΔþ/C0 k/C22εk ! ¼/C22εkþhk/C1σ, (39) with the effective spin –orbit field hk¼(/C0δεk,/C0Δþ/C0 k,0). Up to the first-order of ωan=ωk, we obtain ~Hk¼/C22hωk/C0ηkB* k /C0ηkBk /C22hωk/C18/C19 , (40) with ηk¼(γku2 k/C02vkukþγkv2 k)/C252γkωan=ωk. Equation (40) is in the same form as the easy-axis case,28because, as aforementioned in Sec. I, magnetic anisotropy and magnetic field together define the effective hard plane ( x–yplane) and the easy axis ( zaxis) normal to the plane. III. MAGNON SPIN TRANSPORT The spin dynamics of magnons can be described by the semi- classical kinetic equation @tρkþi[Hk,ρk]þ1 2{∇kHk,∇ρk}¼Ik, (41) where ρkis defined as a 2 /C22 magnon density matrix and the collision integral Ikshould include all relevant, not only elastic but also inelastic, scattering processes.23,32,34–39By taking into account the large splitting between the two spin bands, the density matrix and Eq. (41) should be written in the representation of ( ψþ k,ψ/C0 k), especially for an accurate computation of the collision integrals.The second and third terms on the left side of Eq. (41) describe sep- arately the coherent (quasi)spin precession due to band splitting and the diffusion owing to spatially inhomogeneous distribution. 23,40 It is important to recall that the density matrix ρk(t) defined under ( ψþ k,ψ/C0 k), however, does not tell spin information directly. In order to extract the spin polarization, one has to project ρk(t) into the spin polarized representation via ~ρk(t)¼Aρk(t)AT: (42)The magnon spin density then can be read out easily from si k(k)¼(1=2)Tr[ ~ρk(k)σi]: (43) Around the compensation field, the two magnon branches are nearly degenerate, in that case, it is more convenient to write andsolve the kinetic equation directly in the ( α k,βk) representation28 @t~ρkþi[~Hk,~ρk]þ1 2{∇k~Hk,∇~ρk}¼~Ik: (44) Strictly speaking, different scattering processes will contribute to the dynamics in different ways, relying on their characteristics about the conservation of particle number, spin polarization,momentum, and so on. 38,39As a simplified treatment, one may apply the relaxation-time approximation as ~Ik¼/C01 τ(~ρk/C0~ρ0 k), (45) where ~ρ0 kandτrepresent the quasi-equilibrium density matrix and relaxation time for a specific scattering mechanism.34,35,38,39,41 A. Magnon spin relaxation After some calculations based on perturbation expansion tech- nique,42we obtain a drift-diffusion equation,23,28 @tSi¼D∇2Siþϵijkhhj kiSk/C01 τi sSi, (46) in which Si¼P ksi kstands for the local spin density and D¼τhv2 ki=3 is the diffusion constant. The first term on the right hand side corresponds to the spin diffusion due to spatial inhomo- geneity of the magnon spin density and the second term describes the spin precession around the net effective spin –orbit field hhki. Here, h:irepresents the average over all thermally occupied magnon states weighted by the Bose distribution. The magnetic-field dependence of hh kiresults in a Hanle-type feature, which has been explicitly discussed in Refs. 22and23. The last term in Eq. (46) is the spin relaxation term, which can be caused by various spin non-conserving scattering processes.28Due to the presence of spin –orbit field, the spin-conserving scatterings can also contribute to the spin relaxation via the DP-type mechanism.43 The spin relaxation time from this mechanism reads (τi s,DP)/C01¼X j=iτ[h(hj k)2i/C0h hj ki2]: (47) In easy-axis AFIs, the magnon spin –orbit field hkis solely from dipole –dipole interaction.28In the present case, magnetic anisot- ropy provides an additional contribution. Although this SOC piece is collinear (with only hx kcomponent), its magnitude varies with frequencies, resulting in a difference between h(hj k)2iandhhj ki2. Accordingly, the relaxation time τshould involve the inelastic scat- terings, such as magnon –magnon and magnon –phonon scatter- ings. As the SOC field due to magnetic anisotropy relies on theJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223906 (2021); doi: 10.1063/5.0054409 129, 223906-5 Published under an exclusive license by AIP Publishingstrength of the external field, the spin relaxation rate given by Eq. (47) also varies with the magnetic field and achieves a minimum at the compensation field. Assuming the diffusion cons-tant Dis insensitive to the magnetic field, the magnon spin diffu- sion length λ s¼ffiffiffiffiffiffiffiDτspwill then also vary sharply around the compensation point, as qualitatively shown in Fig. 3 . B. Magnon (inverse) spin Hall effect To examine the magnon (inverse) spin Hall effect in the pres- ence of band splitting due to magnetic anisotropy, we next calculatethe Berry curvature for the spin Hall effect, 44 Ωz,+ x,y(k)¼/C02=hψ+ kj^vxjψ+ kihψ+ kj^vz yjψ+ ki (ε+ k/C0ε+ k)2, (48) where ε+ kandjψ+ kiare eigenenergies and wave functions. For a general Hamiltonian in the form of Hk¼εkþΔx kσxþΔy kσy, (49) one has ε+ k¼εk+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (Δx k)2þ(Δy k)2q , (50) jψ+ ki¼1ffiffiffi 2p1 +eiwΔk/C18/C19 : (51)The (spin) velocity operators ^vx¼v0 xþvx xσxþvy xσy, (52) ^vz y¼v0 yσz, (53) where v0 x=y¼@kx=yεkand vx=y x¼@kxΔx=y k. The matrix elements in Eq.(48) then can be calculated hψ+ kj^vxjψ+ ki¼+i(vx xsinwΔk/C0vy xcoswΔk), (54) hψ+ kj^vz yjψ+ ki¼v0 y: (55) By substituting these matrix elements into Eq. (48), we obtain Ωz,+ x,y(k)¼+2v0 y(vx xsinwΔk/C0vy xcoswΔk) (Δx k)2þ(Δy k)2: (56) Specifically, for the present case, we have εk¼(/C22hωþ kþ/C22hω/C0 kþΔþþ k/C0Δ/C0/C0 k)=2, (57) Δx k¼/C0δεk¼/C0(/C22hωþ k/C0/C22hω/C0 kþΔþþ kþΔ/C0/C0 k)=2, (58) Δy k¼/C0Δþ/C0 k: (59) Around the compensation field, the approximate expression of the low energy dispersion relation εk/C25ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε2 0þc2 sk2p gives v0 y¼(c2 sk=εk)s i nθksinfk,w h e r e cs¼ωexa=2. According to Eqs. (39) and(40), we have Δx k≃/C0ξk/C0ζksin2θkcos(2fk), (60) Δy k≃/C0ζksin2θksin(2fk): (61) where ξk¼/C22h(ωþ k/C0ω/C0 k)=2 and ζk¼γk/C22hωmωan=ωkare SOC due to magnetic anisotropy and dipole –dipole interactions, respectively. Notice that tanwΔk¼Δy k Δx k¼2ζkkxky k2ξkþζk(k2 x/C0k2 y): (62) Very close to the compensation point, the SOC is dominant by the dipolar interaction, i.e., ζk/C29ξk. Thus, from Eq. (62),w eh a v e wΔk¼2fk. The berry curvature then reads Ωz,+ x,y(k)¼+c2 s εkζk1þkξ0 k ζkcos2fk/C18/C19sin2fk sin2θk, (63) which is globally negative and positive for the upper and lower magnon bands, respectively. This indicates the occurrence of the spin Hall effect. FIG. 3. Magnon spin diffusion length as a function of the external field at temperature kBT¼ωZ.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223906 (2021); doi: 10.1063/5.0054409 129, 223906-6 Published under an exclusive license by AIP PublishingIn the opposite limit, magnetic anisotropy dominates the SOC, i.e., ξk/C29ζk,w eh a v e tanwΔk¼Δy k Δx k≃ζk ξksin2θksin 2fk/C281 (64) and, therefore, sinwΔk≃(ζk=ξk) sin2θksin 2fk, (65) coswΔk≃1, (66) which lead to Ωz,+ x,y(k)≃+c2 s εkξkζk ξkþ/C18kζ0 k ξk/C02ζk ξk/C19 sin2θkcos2fk/C20/C21 /C2sin2θksin2fk: (67) One sees that the Berry curvature reduces with ζk=ξk, meaning the suppression of the spin Hall effect by magnetic anisotropy. This isbecause of the collinear nature of the SOC due to magneticanisotropy. IV. INFLUENCE OF DMI AND IN-PLANE ANISOTROPY In some easy-plane antiferromagnetic magnets like α-Fe 2O3, there is a zero-field magnetization induced by DMI. To examinethe consequence of DMI, we describe it by H DM¼DX hi,ji0^x/C1Sai/C2Sdj, (68) where only those DMI-active bonds are counted in the summation. The DMI then leads to an additional energy EDM¼/C0Nz0DS2sin 2θ, (69) where z0stands for the number of neighboring ions connected by DMI. The condition of the equilibrium canting angle then can bederived by including Eq. (69) into Eq. (2)as ω ZcosθþωDMcos 2 θ¼ωexsin 2θ, (70) with ωDM¼z0DS=/C22h. After some calculations following the techni- ques introduced in Sec. II, we find its contribution to magnon Hamiltonian can be included by the substitutions A!Aþ/C22hωDMtanθ, (71) Ck!/C22hγkωexsin2θ1/C0γ0 kωDM γkωexcotθ/C18/C19 : (72) In reality, only part of the exchange interacting bonds are involved in the DMI, which means in general γk=γ0 k. This makes Ck=γk no longer a constant. As a result, ωþ k¼ω/C0 kis not able to satisfy in the entire Brillouin zone for any magnetic field. In particular, nocompensation field is allowed and the DMI provides an effective spin –orbit field at any external magnetic field, from which it can affect the magnon spin relaxation and spin Hall effect. In biaxial antiferromagnets like the intensively studied mate- rial, NiO,3,4,6there is an easy axis within the easy plane. This effect can be included by in-plane magnetic anisotropy term21,45 Hin¼X iK0(Sz ai)2þK0(Sz di)2, (73) with anisotropy parameter K0,0. For simplicity, we here assume the in-plane easy axis is along z-direction, i.e., perpendicular to the applied field. This term gives an energy Ein¼2NK0S2cos2θ: (74) By taking this term into account, we find the condition of the equi- librium (without DMI) canting angle sinθ¼ωZ 2(ωex/C0ω0 an), (75) with ω0 an¼jK0jS=/C22hand the corrections to the magnon Hamiltonian can be included in Hamiltonian (8)via the replace- ment A!Aþ/C22hω0 an(2/C03 sin2θ), (76) A0!A0þ/C22hω0 ansin2θ: (77) Since the corrections are moment independent, the compensation features will survive. V. SUMMARY In summary, we study the magnon spin transport in easy- plane antiferromagnetic insulators under an in-plane magnetic field. From the analysis on the influence of the magnetic field, wefind that the two magnon branches become degenerate at a com-pensation magnetic field, making the easy-plane antiferromagnetequivalent to the uniaxial easy-axis antiferromagnets. At this com- pensation condition, the magnon spin –orbit coupling due to dipolar interaction results in magnon (inverse) spin Hall effect andD’yakonove –Perel ’-type spin relaxation. The compensation feature is found to survive in biaxial easy-plane systems but will be removed by Dzyaloshinskii –Moriya interaction. Far away from the compensation magnetic field, the magnon spin –orbit coupling is dominated by magnetic anisotropy, where the magnon (inverse)spin Hall effect is suppressed. These results are expected to beapplicable in synthetic antiferromagnets, in which the larger mag- netic moments of the artificial spin elements benefit the enhance- ment of the predicted dipolar-induced spin –orbit effects. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (NNSFC) (Grant No. 11974047).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223906 (2021); doi: 10.1063/5.0054409 129, 223906-7 Published under an exclusive license by AIP PublishingDATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018). 2M. B. Jungfleisch, W. Zhang, and A. Hoffmann, Phys. Lett. A 382, 865 (2018). 3H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev. Lett. 113, 097202 (2014). 4W. Lin, K. Chen, S. Zhang, and C. L. Chien, Phys. Rev. Lett. 116, 186601 (2016). 5Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. N ’Diaye, A. Tan, K. Ichi Uchida, K. Sato, S. Okamoto, Y. Tserkovnyak, Z. Q. Qiu, and E. Saitoh, Nat. Commun. 7, 12670 (2016). 6Y. Wang, D. Zhu, Y. Yang, K. Lee, R. Mishra, G. Go, S.-H. Oh, D.-H. Kim, K. Cai, E. Liu, S. D. Pollard, S. Shi, J. Lee, L. L. Teo, Y. Wu, K.-J. Lee, and H. Yang, Science 366, 1125 (2019). 7R. Lebrun, A. Ross, S. A. 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5.0053615.pdf

J. Chem. Phys. 154, 244101 (2021); https://doi.org/10.1063/5.0053615 154, 244101 © 2021 Author(s).Orbital optimization in nonorthogonal multiconfigurational self-consistent field applied to the study of conical intersections and avoided crossings Cite as: J. Chem. Phys. 154, 244101 (2021); https://doi.org/10.1063/5.0053615 Submitted: 09 April 2021 . Accepted: 03 June 2021 . Published Online: 22 June 2021 Andrew D. Mahler , and Lee M. Thompson ARTICLES YOU MAY BE INTERESTED IN Chemical physics software The Journal of Chemical Physics 155, 010401 (2021); https://doi.org/10.1063/5.0059886 Machine learned Hückel theory: Interfacing physics and deep neural networks The Journal of Chemical Physics 154, 244108 (2021); https://doi.org/10.1063/5.0052857 Nuclear–electronic orbital methods: Foundations and prospects The Journal of Chemical Physics 155, 030901 (2021); https://doi.org/10.1063/5.0053576The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Orbital optimization in nonorthogonal multiconfigurational self-consistent field applied to the study of conical intersections and avoided crossings Cite as: J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 Submitted: 9 April 2021 •Accepted: 3 June 2021 • Published Online: 22 June 2021 Andrew D. Mahler and Lee M. Thompsona) AFFILIATIONS Department of Chemistry, University of Louisville, 2320 South Brook Street, Louisville, Kentucky 40292, USA a)Author to whom correspondence should be addressed: lee.thompson.1@louisville.edu ABSTRACT Nonorthogonal approaches to electronic structure methods have recently received renewed attention, with the hope that new forms of nonorthogonal wavefunction Ansätze may circumvent the computational bottleneck of orthogonal-based methods. The basis in which nonorthogonal configuration interaction is performed defines the compactness of the wavefunction description and hence the efficiency of the method. Within a molecular orbital approach, nonorthogonal configuration interaction is defined by a “different orbitals for different configurations” picture, with different methods being defined by their choice of determinant basis functions. However, identification of a suitable determinant basis is complicated, in practice, by (i) exponential scaling of the determinant space from which a suitable basis must be extracted, (ii) possible linear dependencies in the determinant basis, and (iii) inconsistent behavior in the determinant basis, such as disap- pearing or coalescing solutions, as a result of external perturbations, such as geometry change. An approach that avoids the aforementioned issues is to allow for basis determinant optimization starting from an arbitrarily constructed initial determinant set. In this work, we derive the equations required for performing such an optimization, extending previous work by accounting for changes in the orthogonality level (defined as the dimension of the orbital overlap kernel between two determinants) as a result of orbital perturbations. The performance of the resulting wavefunction for studying avoided crossings and conical intersections where strong correlation plays an important role is examined. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0053615 I. INTRODUCTION Nonorthogonal wavefunction expansions have a long history in electronic structure theory, particularly in the context of valence bond (VB) theory.1,2The restriction of the electron pair to two atomic orbitals was dropped in later developments of VB theory, leading to an extended version of the Coulson–Fischer model,3 which combines VB and molecular orbital (MO) approaches. From the MO perspective, there have been a number of developments in the preceding decades in which different orbitals for differ- ent configurations (DODC) models have been applied to con- struct nonorthogonal wavefunction expansions.4–11The motivation for the development of MO-based nonorthogonal models is the realization that (combinations of) single-determinant wavefunc- tions can provide a compact description of electronic structure,particularly in excited states and away from the equilibrium geome- tries. Particularly through the use of symmetry breaking, MO local- ization and diabatic character of the basis determinants can be intro- duced.7As a result, nonorthogonal MO approaches can also be seen as combining traditional VB and MO pictures from a different perspective to modern VB theory. A popular approach for constructing the nonorthogonal Slater determinant (SD) basis is to use the manifold of solutions to the Hartree–Fock (HF) equations. These determinants have directly been used to describe electronic excited states12–14and provide a reference for post-HF methods to obtain accurate excitation ener- gies.15Hamiltonian and overlap matrix elements in the determinant basis can be constructed to perform non-orthogonal configuration interaction (NOCI) calculations, in which the determinant basis is fixed and the configuration interaction (CI) coefficients are linearly J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp optimized. Important developments in MO-based NOCI include the Löwdin rules for nonorthogonal matrix elements,16generalized Wick’s theorem,17the generalized Slater–Condon rules,7and the developments of Broer and co-workers.5,18,19Optimizing both CI and MO orbital coefficients defines the nonorthogonal multicon- figurational self-consistent field (NOMCSCF) model. NOMCSCF is a generalization of the resonant Hartree–Fock (ResHF) approach proposed by Fukutome and co-workers,4,6which starts from a lin- ear combination of independently optimized self-consistent field (SCF) solutions and was motivated by the presence of multiple near- degenerate solutions to the HF equations in carbon monoxide.20 Subsequently, ResHF has been used to study vertical excitation ener- gies with promising results.10However, the ResHF equations give zero coupling in the orthogonal limit and so must start from a set of nonorthogonal determinants. Determining an initial basis of HF solutions is challenging, as despite algorithmic advances in glob- ally elucidating SCF solution space,21–24the space over which to search to find relevant solutions grows exponentially with system size. Additionally, the set of solutions is not necessarily consistent with changes that arise from external perturbations, such as nuclear geometry changes or time-dependent magnetic or electric fields, although developments such as holomorphic HF may circumvent such issues.9,25,26 To avoid the problems associated with identifying HF solu- tions, an alternative approach is to use a well-defined and repro- ducible heuristic for constructing the wavefunction expansion. Through subsequent reoptimization of orbitals resulting from cou- pling of determinants, it is possible to recover the most suitable determinant basis for a given number of variational parameters. The most straightforward approach for constructing the initial basis is to use an orthogonal determinant expansion, for example, a trun- cated CI or complete active space (CAS) expansion, which is then permitted to become nonorthogonal through reoptimization. This work describes the necessary developments to performing NOMC- SCF without regard as to determinant orthogonality. As a result, we derive the equations to allow for global wavefunction optimization in the framework of the generalized Slater–Condon rules. We then use an implementation of these equations to demonstrate that an orthogonal basis provides a suitable initial basis for NOMCSCF cal- culations. Examination of the energy and wavefunction of the disso- ciation of LiF and the conical intersection in linear water provide an understanding of how the model performs in describing electronic structure that is a challenge for the most well-used computational methods. II. METHODS The NOMCSCF approach approximates the many-electron wavefunction as a linear combination of nonorthogonal SD ∣ΦI⟩, ∣ΨA⟩=∑ IDIA∣ΦI⟩, (1) where A,B,C,. . .are solutions of the NOMCSCF equations and I,J,K,. . .are the set of SCF solutions in which the NOMCSCF wavefunction is expanded with expansion coefficients DIA. From the wavefunction Ansatz in Eq. (1), the energy and electronic gradient can be determined.A. Nonorthogonal multiconfigurational self-consistent field energy First, we provide the derivation of the NOMCSCF energy. The energy of electronic state Ain the NOMCSCF calculation is obtained from EA=∑ μνhμνγA μν+1 2∑ μνητ⟨μν∣∣ητ⟩ΓA μνητ, (2) whereμ,ν,η,τ,. . . are atomic orbital basis functions, hμνand ⟨μν∣∣ητ⟩are the atomic orbital one-electron and two-electron res- onance integrals (2ERIs), respectively, and γA μνandΓA μνητare atomic orbital one-electron and two-electron density matrices for state A, respectively, γA μν=S−1 A∑ IJD∗ IADJANIJρ2IJ μν, (3) ΓA μνητ=S−1 A∑ IJD∗ IADJANIJρ1IJ ημρ2IJ τν. (4) The energy in Eq. (2) is expressed in the atomic orbital (AO) basis instead of the usual MO basis for orthogonal MCSCF because DODC means that in the MO basis, the integrals cannot be factored out of the summation over configuration indices. ΓAcan be consid- ered as the sum of scaled transition density matrix ρouter products, although in reality the rank-4 tensor ΓAis never formed, but the matrix G(ρ)is first formed by contracting one of the transition density matrices with the 2ERIs. The scaling factor is the product of the overlap NIJbetween solutions IandJand the inverse of the NOMCSCF norm SA, SA=∑ IJD∗ IADJANIJ, (5) which can always be set to unity through arbitrary scaling of the CI coefficients DIA. Linear dependencies in the determinant basis can be established through inspection of the eigendecomposition of N, in which NIJare the matrix elements. Typically, in the relatively small expansions used in NOMCSCF calculations ( ≪50 determinants), the entire NOMCSCF Hamilto- nian can be stored in memory and diagonalized in a generalized eigenvalue problem to yield the entire spectrum of NOMCSCF solu- tions. The matrix elements of the metric NIJare computed from the determinant of the occupied–occupied ( oo) overlap matrix NIJ= det(IJM), where IJM=IC† occSJCocc, (6) where Coccis the occupied molecular orbital coefficient matrix and Sis the atomic orbital overlap matrix. The oooverlap matrix gives the projection of orbitals in the bra (ket) determinant/configuration onto orbitals in the ket (bra) determinant/configuration. Note thatIJMis not necessarily symmetric because occupied orbitals in the bra are not necessarily the same as those in the ket. The Hamilto- nian matrix elements HIJare computed using the transition density matrices ρ1andρ2, HIJ=˜NIJ⟨I∣ˆH∣J⟩=˜NIJ(⟨hρ2⟩+1 2⟨ρ1G(ρ2)⟩), (7) where ˜Nis the pseudodeterminant ofIJMand⟨⋅⋅⋅⟩indicates the matrix trace. The pseudodeterminant is the determinant of the J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp image of a matrix (i.e., the product of nonzero eigenvalues), and so NIJcoincides with ˜NIJwhen the matrixIJMis full rank (nonsingu- lar). The form of the transition density matrices depends on the size of the null space (dimension of the kernel) ofIJM. TheIJMnull space indicates components of the bra determinant that span an orthogo- nal portion of the N-electron Hilbert space to the ket determinant. To compute ρ, the null space must be separated, which requires transformation of the orbital basis via singular value decomposition (SVD) ofIJM, except in the obvious case thatIJMis full rank (null space is zero) IJM=UIJΣV†(8) such that in the new basis,I˜C=ICU andJ˜C=JCVand the overlap between two transformed orbitals is given by the matrix element of IJΣ. The transition density matrices can then be computed as7 ρIJ 1=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩JCIJM−1ICorJ˜CIJΣ−1I˜C for dim(ker(IJM))=0 ∑jJ˜CjI˜C† j/Σjj for dim(ker(IJM))=1 J˜CjI˜C† j for dim(ker(IJM))=2 0 for dim(ker(IJM))>2,(9) ρIJ 2=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ρIJ 1for dim(ker(IJM))=0 I˜CiJ˜C† i for 1≤dim(ker(IJM))≤2 0 for dim(ker(IJM))>2,(10) where the diagonal elements ofIJΣare in ascending order, starting with index i(such thatIJΣii=0 andIJΣjj=0 if the nullity ofIJMis two). The value of HIJis the same regardless of how the transition density matrices are computed, but whereIJMis not full rank, com- putation in the transformed basis avoids numerical instabilities that result from an undefined matrix inverse asIJMis singular. Use of SVD introduces a sign indeterminacy problem that must be resolved in order to obtain correct results. Unlike eigenvectors, where the relative phases of all elements in the matrix of eigenvec- tors are fixed even though the overall phase is not, the relative phases of singular vectors are only fixed within each individual vector Ui orVi. Additionally, where Σi≠0, the relative phases of the singular vectors UiandViare also fixed, while in the case that Σi=0, the rel- ative phases of UiandViare independent. Thus, after performing SVD, we rotate all elements of each vector in Uin the Argand space such that the largest element lies in the first quadrant at an angle of 0≤θ≤π/2. If the singular value of a vector is greater than zero, the corresponding vector in Vis rotated by the same amount; other- wise, if the singular value is zero, Viis also rotated so that the largest element lies in the first quadrant in Argand space. To finish this section, we briefly discuss the computational scal- ing associated with the nonorthogonal energy evaluation in com- parison to orthogonal basis methods. Although full diagonaliza- tion has cubic scaling with respect to the number of configurations Nconfig , iterative solvers avoid this step and so building the Hamil- tonian matrix becomes the bottleneck. Calculation of the Hamilto- nian matrix scales quadratically in Nconfig , but linear scaling can be obtained in orthogonal CI as elements between configurations with the substitution level greater than two evaluate to zero. In contrast, to establish the substitution level in nonorthogonal CI, SVD withO(N3 occ)scaling must be performed for each configuration pair, giv- ing quadratic scaling. However, linear scaling can be achieved in evaluating Hamiltonian matrix elements giving a total scaling of at leastO(N2 configN3 occ). The strength of nonorthogonal methods is that the configuration expansion is more rapidly convergent, and so the method avoids the exponential scaling of Nconfig in the orthogonal full CI method. B. Nonorthogonal multiconfigurational self-consistent field electronic gradient Having discussed the relevant terms and intermediates in the NOMCSCF energy, we now differentiate Eq. (2) to obtain the energy change with respect to a perturbation, dEA=∑ muνdhμνγA μν+1 2∑ μνητ⟨μν∣∣ητ⟩ΓA μνητ +2∑ IdD∗ IA⎧⎪⎪⎨⎪⎪⎩∑ JDJAS−1 A(HIJ−NIJEA)⎫⎪⎪⎬⎪⎪⎭ +2∑ ID∗ IA⎧⎪⎪⎨⎪⎪⎩∑ JDJAS−1 A(dHIJ−dNIJEA)⎫⎪⎪⎬⎪⎪⎭. (11) The first and second right-hand side terms contain the atomic orbital integral derivatives, which are zero for perturbation of the wave- function parameters {C,D}. The third term is the gradient with respect to changes in the NOMCSCF CI coefficients D, for which the optimal values can be obtained directly via linear optimization via the generalized eigenvalue equation. The fourth term is the more challenging term as it is the gradient with respect to changes in the MOs of all basis solutions in the NOMCSCF expansion, for which the energy has a nonlinear dependence. The required terms to determine the NOMCSCF orbital rotation gradient are dHIJand dNIJ, which are both differentiable at all coordinates in wavefunction parameter space, and so the gradient is always defined. Using Jacobi’s formula, dNIJis obtained from dNIJ=d(det(IC†SJC))=NIJ⟨IJM+T dIJM⟩, (12) whereIJM+is the Moore–Penrose pseudoinverse ofIJM. The occu- pied MO overlap matrix dIJMis dIJM=dIC†SJC+IC†dSJC+IC†SdJC, (13) where, as discussed previously, the second right-hand-side term involving the integral derivative will be zero when considering elec- tronic perturbations. The use of the pseudoinverse in Eq. (12) avoids numerical instabilities when NIJ→0 but gives the same answer as the regular inverse owing to the presence of the term NIJ, which becomes zero where the pseudoinverse and regular inverse diverge. Thus, dNIJis a continuous function with well-defined derivatives at all coordinates. The term dHIJis obtained by differentiation of Eq. (7), giving dHIJ=d˜NIJ⟨I∣ˆH∣J⟩+˜Nd⟨I∣ˆH∣J⟩. (14) To clarify how the current developments differ from previous ResHF implementations, we emphasize that without properly accounting J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp for changes in the null space dimension, Eq. (14) is undefined when- everNIJ→0 and so cannot work if any configurations in the expan- sion are orthogonal. The overlap pseudodeterminant ˜NIJis obtained from the product of non-zero singular values ˜NIJ=∏ i∀IJΣi≠0IJΣi, (15) and thus, ˜Nrequires the derivative singular values dIJΣ=U†dIJMV, (16) use of which gives d˜NIJ=∑ i∀IJΣi≠0˜NIJdIJΣi IJΣi. (17) Computing d⟨I∣ˆH∣J⟩also depends on the rank ofIJMthrough the transition density matrices d⟨I∣ˆH∣J⟩=∑ μνhμνdρ2IJ μν+1 2∑ μνητ⟨μν∣∣ητ⟩(dρ1IJ ημρ2IJ τν+ρ1IJ ημdρ2IJ τν), (18) where the derivative transition density matrices depend on the rank ofIJM. In the case thatIJMis full rank, ρ1=ρ2and dρ=JCIJM−1dIC†(1−Sρ), (19) where we used the identity dIJM−1=−IJM−1dIJMIJM−1. Here, the null space ofIJMis greater than dimension 2, d⟨I∣ˆH∣J⟩=0. In the case that the nullity ofIJMis 1 or 2, the derivative transition density matrix is most easily computed in the transformed basis dρ1=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩∑jdJ˜CjI˜C† j+J˜CjdI˜C† j IJΣj−dIJΣjJ˜CjI˜C† j IJΣ2 jfor dim(ker(IJM))=1 dJ˜CjI˜C† j+J˜CjdI˜C† j for dim(ker(IJM))=2, (20) dρ2=dJ˜CiI˜C† i+J˜CidI˜C† i. (21) The derivative transformed basis MO coefficients require knowledge of the derivative singular vectors dI˜C=dICU+ICdU, (22) dJ˜C=JCdV, (23) where each derivative singular vector is computed according to27,28 dIJUi=−{IJΣi(IJMIJM†−IJΣ2 i1)+dIJM}Vi −{IJM(IJM†IJM−IJΣ2 i1)+dIJM†}Ui, (24) dIJVi=−{(IJMIJM†−IJΣ2 i1)+IJM†dIJM}Vi −{IJΣi(IJM†IJM−IJΣ2 i1)+dIJM†}Ui, (25) where the vectors {Vi}and{Ui}are rotated as detailed in Sec. II A.The computation of the MO derivatives for each basis solution is performed such that the mean-field form of the solutions is pre- served. Under the mean field approximation, the set of MOs for each solution in the orthogonalized atomic orbital (OAO) basis forms a unitary matrix, and so changes in the MOs can be described using theSU(2Nbasis)Lie group. Using the connection between Lie groups and Lie algebras, the Thouless theorem gives the form of the orbital rotation parametersIˆKthat define the generators of the su(2N) Lie algebra. Spin-symmetry adaption of the generators enables the orbitals to be reoptimized with conservation of spin-symmetries as desired. From an initial set of MO coefficientsIC, a new set of molecular orbitalsIC′can be written as IC′=ICexp ˆK=⎛ ⎝1+∑ giaIkg iaIkg ia+⋅⋅⋅⎞ ⎠IC, (26) where gis the label for the unitary group generator ( ˆkA00,ˆkS00,ˆkAz, ˆkSz,ˆkAx,ˆkSx,ˆkAy, and ˆkSy),i,j,k,. . .are the set of occupied orbitals, anda,b,c,. . .are the set of virtual orbitals. Differentiation of Eq. (26) truncated at first order with respect to changes in the orbital rotation coefficientsIkg iagives ∂ ∂Ikg iaIC=Ikg iaC. (27) As for mean-field SCF, the NOMCSCF equations have a number of different minima, which can be associated with different sets of electronic states. As a result, it is possible to optimize a local mini- mum and avoid having to determine intermediate states if they are not of interest. However, in common with SCF solutions, variational behavior can be guaranteed only under limited circumstances (see the Appendix). Finally, we note that, rather than building the transition den- sity matrices and contracting them with AO 2ERIs, it is possible to transform the 2ERIs and perform the optimization in the MO basis. An AO to MO transformation is required for every determi- nant pair, but for gradients, only 2ERIs containing a single virtual index are required, meaning that the transformation can be per- formed at O(N2 basisN3 occ)scaling. The one-electron derivative for the kA00 iagenerator is dIJH1 dkA00 ia=∑ jσ1σ2IJhaσ1jσ2IJM−1 jσ1iσ2−∑ σ1σ2⟨IJhσ1σ2(dIJM−1 σ1σ2)†⟩, (28) where, as mentioned above, i,j,k,. . .are occupied orbital indices and a,b,c,. . .are virtual orbital indices. The other seven generators differ in the sign with which each term enters the expression. The two- electron contribution for the kA00 iagenerator is dIJH2 dkA00 ia=1 2∑ jklm σ1σ2σ3σ4{IJ⟨jσ1aσ2∣∣lσ3mσ4⟩δik +IJ⟨jσ1kσ2∣∣lσ3aσ4⟩δim}IJM−1 jσ1kσ2IJM−1 lσ3mσ4IJM−1 lσ3mσ4 −∑ jklm σ1σ2σ3σ4⟨jσ1kσ2∣∣lσ3mσ4⟩dIJM−1 jσ1kσ2IJM−1 lσ3mσ4, (29) J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where, as for the one-electron derivative, other generators differ only in terms of the sign with which they enter the expression. From Eq. (29), it can be seen that the use of the MO basis reduces the required contraction of 2ERIs from O(N4 basis)toO(N4 occ). Here, the oooverlap matrix is rank deficient, as with the AO-based algorithm, transformation of the basis to separate the kernel and image space of IJMprovides the most straightforward approach to determine dHIJ. However, the MO integrals in Eqs. (28) and (29) must be trans- formed to the diagonal overlap basis. Fortunately, only ootransfor- mations are involved in the change in the basis, the virtual orbital index does not require transformation, and there are fewer integrals required to construct these matrix element derivatives, which follow from the Slater–Condon rules. The computational cost of the gradient calculation in the MO basis is O(N2 configN2 basisN3 occ)if the integral transformation is the bot- tleneck or O(N2 configN5 occNvirt)if the contractions are the bottleneck. In the AO basis, scaling of the method is O(N2 configN4 basisNoccNvirt). However, this scaling is a worst-case scenario because configura- tion pairs, which differ by a substitution level greater than two, have zero gradient unless the substitution level is changing, giv- ing a minimum scaling of O(N2 configN4 occNvirt). In orthogonal CI approaches, there is a single set of orbitals, rather than many sets as in DODC approaches. For large CI expansions, the CI solver is the bottleneck, but for Hamiltonians of the size considered in this work, the 2ERI transformation is the source of most computational time. C. Resolving discontinuities in matrix elements between orthogonality levels HIJis continuous with respect to changes in wavefunc- tion parameters as can be demonstrated through a perturbative approach, assuming that the spectrum of His analytic.29However, the underlying terms in Eq. (14), ˜NIJand⟨I∣ˆH∣J⟩, are not continu- ous. Both ˜NIJand⟨I∣ˆH∣J⟩show a “jump” discontinuity at coordinates where the rank ofIJMchanges (Fig. 1), in which around the disconti- nuity, there exists a continuous function for which the values on each side of the discontinuity are not equal. The point of discontinuity can be recognized by inspecting the derivative singular values cor- responding to the change in singular values equal to zero. If any of these derivative singular values are non-zero, it implies that the rank ofIJMis changing under the perturbation considered. As a result, a fully analytical formulation cannot be obtained through standard calculus. In order to compute an analytical electronic gradient, an alter- native approach is to employ generalized functions and obtain a weak derivative for d˜NIJandd⟨I∣ˆH∣J⟩. Generalized functions are the formal basis of the Dirac delta function, although only fully devel- oped in the decades after its introduction.30,31As a result, the theory of distributions in combination with the Hilbert space—the rigged Hilbert space—provides the most suitable representation of prob- lems in quantum mechanics, although only of consequence in the description of systems with unbound eigenstates.32In this section, we seek to introduce the use of generalized functions as a tool for deriving analytic NOMCSCF electronic gradients. Our interest in generalized functions is motivated by the property of the deriva- tive of a generalized function dF[ϕ]=F[−dϕ], whereϕ(x)is an FIG. 1. Value of NIJ(green triangles), ˜NIJ(purple saltires), HIJ(blue crosses), and⟨I∣ˆH∣J⟩(red squares) matrix elements computed as a function of orbital rota- tion angle in D4hH4with bond distance 1.5 Å using STO-3G. The ⟨I∣ˆH∣J⟩matrix element shows a singularity at lim NIJ→0and a finite value at NIJ=0 in (a) and (b). infinitely differentiable test function and the square brackets indi- cate a functional, and so the derivative of a generalized function is everywhere defined. Turning first to the derivative overlap pseudodeterminant d˜NIJ, we can write the pseudodeterminant ˜NIJas a single function ˜NIJ=IJ˜N0+(IJ˜N1−IJ˜N0)Θ10+(IJ˜N2−IJ˜N1)Θ21 +(IJ˜N3−IJ˜N2)Θ32+⋅⋅⋅ =IJ˜N0(1−Θ10)+IJ˜N1(Θ10−Θ21)+IJ˜N2(Θ21−Θ32)+⋅⋅⋅ , (30) J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp whereIJ˜Nris the pseudodeterminant when the matrix MIJhas rank randΘrsis the Heaviside function, which has value zero when the IJMrank is rand one when theIJMrank is sand contains a jump discontinuity at the point where the rank changes. Equation (30) can be considered as a continuous function constructed by shifting the values of intervals so that no discontinuity in the function value remains, although derivative discontinuities may still be present. Differentiating Eq. (30) gives d˜NIJ=dIJ˜N0(1−Θ10)−IJ˜N0dΘ10+dIJ˜N1(Θ10−Θ21) +IJ˜N1(dΘ10−dΘ21)+dIJ˜N2(Θ21−Θ32) +IJ˜N2(dΘ21−dΘ32)+⋅⋅⋅ , (31) where the terms dIJ˜Nrcan be determined from Eq. (12), while d Θrs are zero except at the points where the rank ofIJMchanges, in which case it is undefined. However, a weak derivative for dNIJcan be obtained as a distribution by first writing ˜NIJ[ϕ]=∫ˆK˜NIJ(⃗k)ϕ(⃗k)d⃗k, (32) where we have used⃗kto indicate the vector of kg iaparameters in Eq. (26). As ∣NIJ∣can take on values between zero and one over all space, it is a locally integrable function, and so Eq. (32) is a regular distribution. Using the linearity of generalized functions and the property that a(x)F[ϕ(x)]=F[a(x)ϕ(x)], where a(x)is an infinitely differentiable function [ a(x)∈C∞], a weak derivative for˜NIJcan be determined. From distribution theory, d Θ(x)=δ(x), which is the Dirac delta function, resulting in d¯NIJ=dIJ˜N0(1−Θ10)−IJ˜N0δ10+dIJ˜N1(Θ10−Θ21) +IJ˜N1(δ10−δ21)+dIJ˜N2(Θ21−Θ32) +IJ˜N2(δ21−δ32)+⋅⋅⋅ , (33) where the bar indicates the distributional derivative and δrsis zero everywhere except at the point where the rank ofIJMchanges r↔s. The exact value of Θ(x)as a generalized function at x=0 has no sig- nificance but can be considered as a parameter, which can be tuned such that the gradient at the discontinuity more closely resembles the limit of the gradient of one of the functions around the discon- tinuity. The range of possible values defines the set of subderivatives of the discontinuity.33A value of Θ(0)=0.5 gives the derivative as an average of the left and right limits, plus a term to account for the magnitude of shift to remove the discontinuity. Having determined d˜NIJ, we now examine a general approach to computing d⟨I∣¯ˆH∣J⟩. As for ˜NIJ,⟨I∣ˆH∣J⟩can be written for any given set of wavefunction parameters through the use of Heaviside functions. Due to the fact the Hamiltonian comprises up to two- electron operators, HIJ=0, where the rank of NIJis greater than two, and only terms in which dim (ker(IJM))≤2 need to be evaluated, giving ⟨I∣ˆH∣J⟩=⟨I∣ˆH∣J⟩2(Θ32−Θ21)+⟨I∣ˆH∣J⟩1(Θ21−Θ10)+⟨I∣ˆH∣J⟩0Θ10, (34) where⟨I∣ˆH∣J⟩rgives the value of ⟨I∣ˆH∣J⟩when the matrix MIJhas rank r. In contrast to ˜NIJ, singularities can be present in ⟨I∣ˆH∣J⟩,where the rank ofIJMchanges, resulting from the1 Σscaling of den- sity components in Eq. (10). As a result and verified by numerical integration, ⟨I∣ˆH∣J⟩is not locally integrable (not in L1 loc) and must be regularized in order to define a distribution. Using Eq. (34) as a generalized function gives ⟨I∣ˆH∣J⟩[ϕ]=⟨I∣ˆH∣J⟩2[Θ32ϕ]−⟨I∣ˆH∣J⟩2[Θ21ϕ]+⟨I∣ˆH∣J⟩1[Θ21ϕ] −⟨I∣ˆH∣J⟩1[Θ10ϕ]+⟨I∣ˆH∣J⟩0[Θ10ϕ], (35) where, e.g., the fifth term is ⟨I∣ˆH∣J⟩0[Θ10ϕ]=∫ˆK⟨I(⃗k)∣ˆH∣J⟩0Θ10(⃗k)ϕ(⃗k)d⃗k. (36) Continuing to use Eq. (36) as an example (the other terms can be derived in the same manner), the transition density matrix can be written in a form amenable to regularization through separating the finite and singular parts and applying a regulator ε. The approach is most straightforward in the transformed basis in whichIJMis diagonal, ⟨I∣ˆH∣J⟩0[Θ10ϕ]=⎡⎢⎢⎢⎢⎣∑ i∉{Σi}=0hii Σii+1 2∑ i,j∉{Σi}=0⟨ij∣∣ij⟩ ΣiiΣjj⎤⎥⎥⎥⎥⎦Θ10[ϕ] +⎡⎢⎢⎢⎢⎣hkk+1 2∑ i∉{Σi}=0⟨ik∣∣ik⟩ Σii⎤⎥⎥⎥⎥⎦ ×lim ε→0{∫Σkk=∞ Σkk=ε1 Σkkϕ(Σkk)dΣkk}, (37) where we have used the fact that only one overlap Σkkapproaches zero, while all others are non-zero (generalized cases can be obtained by setting, e.g., Θ10=Θ21), as well as using the Heaviside function to determine the limits of the integration. Additionally, the MO coeffi- cients have been used to transform integrals to the MO basis, and we have expressed the integral as a univariate function of Σkk(⃗k)rather than a multivariate integral of the wavefunction parameters⃗k, which is valid as Σkk(⃗k)→0. To remove the singularity as ε→0, Hadamard regularization of the integral can be used to define the pseudofunc- tion Pf(ΘΣ−1), which smoothly connects the functions across the discontinuity, Pf(Θ10 Σkk)[ϕ]=FP∫∞ 01 Σkkϕ(Σkk)dΣkk =lim ε→0{∫∞ ε1 Σkkϕ(Σkk)dΣkk+ln∣ε∣ϕ(0)}, (38) where FP indicates the Hadamard finite part in which the first and second terms in the limit approach ±∞ asε→0 in such a way that the sum remains finite. The derivative of Eq. (37) follows from the discussion of Sec. II B, except for the pseudofunction derivative that is d dΣkkPf(Θ10 Σkk)=−Pf(Θ10 Σ2 kk) =lim ε→0{1 εϕ(0)−∫∞ ε1 Σ2 kkϕ′(Σkk)dΣkk}. (39) Using the results of Eqs. (38) and (39) in practice, the derivative transition density matrices in Eq. (21) are formed with modified J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Σεvalues that avoid singularities, with εset to a suitable value for numerical stability, where the modified matrix element is denoted as⟨I∣ˆHε∣J⟩r. The final derivative is d⟨I∣¯ˆH∣J⟩=d⟨I∣ˆHε∣J⟩2(Θ32−Θ21)+⟨I∣ˆHε∣J⟩2(δ32−δ21) +d⟨I∣ˆHε∣J⟩1(Θ21−Θ10)+⟨I∣ˆHε∣J⟩1(δ21−δ10) +d⟨I∣ˆHε∣J⟩0Θ10+⟨I∣ˆHε∣J⟩0δ10. (40) Finally, multiplication of generalized functions must be performed as shown in Eq. (14). Multiplication of generalized functions is not generally defined because the product of two functions may not be locally integrable, even though the individual functions are in L1 loc. However, HIJis seen to be continuous, differentiable, and locally integrable, indicating that products can be formed of the general- ized functions and derivative generalized functions defined in this section. III. NUMERICAL TESTS In this section, we use the expressions derived in Sec. II to examine the NOMCSCF solutions in the context of avoided cross- ings and conical intersections. First, we study LiF, which has been an important test model in the development of MO-based NOCI meth- ods because coupling of its low-energy SCF solutions reproduces the avoided crossing between ground and excited states, but where solu- tion coalescence prevents the NOCI potential energy surface (PES) from being plotted along the entire stretching mode. Second, we explore the conical intersection of linear water between 11A1and 21A1(using C2virreducible representation labels), which has been used in several studies in order to demonstrate the suitability of methods for determining the topology around conical intersections. The algorithm was implemented in a stand-alone in-house code that utilizes the MQCPack library34and interfacing with a mod- ified version of Gaussian 16.35Comparative post-HF calculations were performed using implementations within both Gaussian 16 and the Molpro package.36The initial wavefunction was constructed from an orthogonal CAS expansion of the HF orbitals. Optimiza- tion of wavefunction parameters was performed using a two-step algorithm in which CI and MO coefficient optimization steps were performed alternately. The optimized CI coefficients can be obtained through solving the CI eigenvalue problem, while MO coefficients used a BFGS quasi-Newton approach, in which an identity matrix was used as the Hessian seed such that the first step is just steepest- descent, while in subsequent steps, the BFGS Hessian updating was used. In each step, a backtracking line-search with cubic interpola- tion was performed along the BFGS optimization direction until the Armijo–Goldstein conditions were satisfied. A. Lithium fluoride avoided crossing Computation of the LiF PES is a model problem for test- ing multireference methodologies owing to the avoided crossing between1Σ+states resulting from rapid switching of wavefunction character from ionic to covalent as the bond is dissociated.7,11,37 To correctly predict the position of the avoided crossing, a suit- able description of dynamic correlation is required. As a result, state-averaged CAS self-consistent field (SA-CASSCF) calculations underestimate the distance at which the avoided crossing occursrelative to full configuration interaction (FCI). Dynamic correlation can be included with perturbation theory, where extended multistate complete active space with second-order perturbation theory (XMS- CASPT2) calculations are able to provide results in close agreement with FCI. Although XMS-CASPT2 is an excellent approach to study LiF with a medium-sized basis, its applicability to larger systems is limited by the factorial scaling of the correlated space. Within NOCI formalisms based on HF solutions, LiF presents an interest- ing problem because the basis determinants required to describe the avoided crossing coalesce and disappear, reducing the dimension- ality of the NOCI problem and leading to energy discontinuities.7 In order to avoid the problem of disappearing HF solutions, Thom and co-workers developed the holomorphic HF approach, in which HF solutions obtained from analytic continuation can be used in the NOCI problem.9,25,26However, a key problem in HF-based NOCI approaches is identifying the relevant basis solutions, although effec- tive approaches for identifying these solutions are being devel- oped.22–24Therefore, the LiF PES provides a good test for orbital optimization in the NOCI framework because we can start from an arbitrary set of determinants that can be readily identified, such as those from a small CAS expansion. In Fig. 2, we compare the results of state-specific CASSCF (SS-CASSCF), state-specific CASPT2 (SS-CASPT2), and state- specific NOMCSCF (SS-NOMCSCF) with full CI for the ground state, in order to first provide an idea of how much dynamic corre- lation can be obtained through DODC when the method only has to describe a single state. All wavefunction expansions were con- structed from two electrons in two σ+orbitals, except for full CI wavefunction in which only the two 1s core orbitals were frozen. As has been previously observed, CASSCF (green triangles) does not provide sufficient flexibility to obtain dynamic correlation around the equilibrium region and so underestimates the dissociation energy by 1.009 eV. In contrast, CASPT2 recovers dynamic correla- tion and so provides a good description of the equilibrium region, although it overestimates the dissociation energy by 0.053 eV. FIG. 2. Potential energy surface of the LiF1Σ+state computed using a 6-31G basis with four determinant CASSCF (green triangles), CASPT2 (purple saltires) and NOMCSCF (red squares). The results are compared to full CI (blue crosses) at the same basis and referenced to each method’s dissociation energy. J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp However, at larger distances, the CASPT2 PES displays issues associ- ated with the perturbative correction, including a noticeable “hump” with respect to the full CI result, and a singularity in the forces at 3.1 Å. The orbital-optimized NOMCSCF result also provides a noticeable improvement over CASSCF in the equilibrium region, although in this case, the dissociation energy is slightly underesti- mated by 0.036 eV. Additionally, at longer distances, the NOMCSCF PES does not deviate from the full CI curve to the same extent as CASPT2 or exhibit a derivative discontinuity. Therefore, the NOM- CSCF approach appears to provide a good balance of dynamic and static correlation and avoid some of the deficiencies associated with a perturbative approach. The origin of the additional correlation in NOMCSCF can be determined by examining the differences in orbitals and CI weights with CASSCF (Fig. 3). At 1.6 Å, the CASSCF reference determinant has a contribution of over 99%, while all other determinants have a weight less than 0.005. In contrast, the determinant with the highest weight in the NOMCSCF wavefunction is only 62%. A comparison of the natural orbitals from HF, CASSCF, and the three most impor- tant NOMCSCF determinants reveals that, while the nearly doubly occupied orbital is similar between all three methods [Figs. 3(a), 3(c), and 3(e)], the nearly unoccupied natural orbital takes quite differ- ent forms in all three methods. The nearly doubly occupied orbital is almost entirely of fluorine pzcharacter (assuming that the z axis is defined by the internuclear axis), indicating the ionic electronic structure of LiF at short internuclear distances. For the HF σ∗orbital [Fig. 3(b)], the lithium 2 sorbital is the only significant atomic orbital contribution, while the NOMCSCF orbital [Fig. 3(f)] has a small contribution from fluorine pz. In contrast, the CASSCF nearly unoc- cupied orbital [Fig. 3(d)] has a very different form, with a lithium pzorbital being the main atomic orbital contributor, although the fluorine pzorbital also contributes as in the NOMCSCF natural orbital. The two other significant NOMCSCF determinants can be examined to understand why the CASSCF nearly unoccupied nat- ural orbital takes such a different form to the HF σ∗orbital [Figs. 3(g)–3(j)]. The natural orbital analysis reveals that the NOM- CSCF determinant with 26% contribution to the wavefunction has an orbital containing 0.05 electrons [Fig. 3(h)] in which there is greater electron density in the internuclear region, although still of lithium 2 scharacter. Similarly, the NOMCSCF determinant that has 13% weight has a natural orbital with greater internuclear density in the internuclear region, but arising from fluorine pzand in both fractionally occupied natural orbitals [Figs. 3(i) and 3(j)]. Addition- ally, this determinant has occupation numbers closer to the diradical character (1.46 and 0.54 electrons), which becomes more important in the electronic structure as the bond distance increases. Thus, the form of the nearly unoccupied CASSCF natural orbital is a conse- quence of trying to capture dynamic correlation by increasing inter- nuclear electron density and also accounting for contributions from the diradical. We now turn from the state specific description of the ground state to explore how NOMCSCF models the avoided crossing between 11Σ+and 21Σ+that results from the wavefunction charac- ter changing from ionic to covalent. Figure 4 illustrates the result of a three state-averaged NOMCSCF (SA-NOMCSCF) calculation (long dashed lines) compared to a 2SA-CASSCF calculation (short dashed lines), with equal weighting for all states in both calculations. FIG. 3. Hartree–Fock HOMO (a) and LUMO (b). CASSCF natural orbitals with frac- tional occupation [(c) and (d)], and NOMCSCF basis determinant natural orbitals with fractional occupation numbers between 1.98 and 0.02 and CI weights greater than 0.005 [(e)–(j)] at a LiF bond distance of 1.6 Å and an isovalue of 0.004. Both calculations used an initial determinant expansion of two elec- trons in two orbitals of σsymmetry. A 3SA-NOMCSCF approach was used to avoid convergence issues that arise from the SD basis, computing the triplet state that lies between the two singlet states of interest, while the CASSCF calculation used a configuration state function basis, and so only required a two-state averaged procedure. Unlike Fig. 2, Fig. 4 shows the absolute energy of each methodol- ogy. CASSCF predicts the avoided crossing to occur at 3.3 Å, which is significantly shorter than 4.0 Å predicted by FCI. Additionally, the 2SA-CASSCF energy gap is 0.054 hartree, compared to 0.032 hartree in FCI. The difference in geometry and sharpness of the J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. Potential energy surface of LiF 11Σ+(red) and 21Σ+(blue) states com- puted using the 6-31G basis with 2SA-CASSCF(2,2) (short dashed lines) and 3SA-NOMCSCF(2,2) (long dashed lines). The results are compared to full CI (solid lines). avoided crossing results from the absence of dynamic correlation. In contrast, the 3SA-NOMCSCF calculation predicts the avoided cross- ing to occur at longer distance (4.7 Å) and the two states to come closer (0.017 hartree). This result implies that orbital optimization in NOMCSCF stabilizes the ionic configuration more than the covalent configuration. Compared to Fig. 2, in the ground state of the state- averaging procedure, the equilibrium region is more stabilized than the stretched-bond region, implying that state-averaging is respon- sible for the greater stabilization of the ionic state. The exact role of including the triplet in the state averaging is not entirely clear, but as the triplet is of the same character as the covalent singlet dia- bat, the state averaging can stabilize the triplet at the expense of the singlet energy. Alternatively, as the level of theory is improved, the avoided crossing moves to longer distances and the gap becomes much closer. Almost-exact calculations give the avoided crossing at 6.9 Å and as close as several mhartree.38As the NOMCSCF approach uses DODC, it would be expected to behave like a calculation per- formed at a larger basis set, and so the avoided crossing at longer distance than the FCI PES may indicate that NOMCSCF captures the behavior of larger basis sets. B. Water conical intersection In this section, we compute the PES of C2vwater in which the two coordinates are the O–H symmetric stretch and the H–O–H bending angle. The interest in computing such a surface arises from the role that linear water plays in the homolytic dissociation of water into hydrogen and hydroxyl radicals via a conical intersection between1A1electronic states. A number of methodologies have been used to compute the conical intersection of linear water, including two-electron reduced density matrix calculations39and constrained density functional theory (DFT) CI.40NOMCSCF and constrained DFT CI have several parallels, although an advantage of NOMCSCF is that there is no requirement for the user to define basis determi- nants. In this work, we seek to understand if NOMCSCF can repro- duce the topology of the conical intersection starting from an initial orthogonal expansion of the wavefunction. Although methods suchas constrained DFT CI can correctly reproduce the PESs around the point of degeneracy, selecting the correct basis states requires both chemical insight and trial-and-error in order to obtain a correct description. Poorly chosen determinants can result in wavefunctions that are qualitatively incorrect, even to the extent that they fail to obtain any degeneracy. Thus, an approach that combines systematic construction of the wavefunction and nonorthogonality in a MO framework has potential to provide a reliable and straightforward model of electronic structure around conical intersections. Figure 5 shows the topology around the linear water conical intersection for several different methods with the cc-pVDZ basis set. The 4SA-NOMCSCF surface shown in Fig. 5(a) was computed from an initial expansion of two electrons in two orbitals (3 a1and 4a1), with the core comprised of 1 a1, 1a1, 1b1, and 1 b2orbitals. As can be seen compared to Figs. 5(b)–5(f), the SA-NOMCSCF calcula- tion is able to qualitatively reproduce the conical intersection com- pared to SA-CASSCF and XMS-CASPT2 models. The most com- plete of the comparative methods in Fig. 5(f), which uses two-state FIG. 5. Potential energy surfaces of 11A1and 21A1states of water around the conical intersection geometry for different state-averaged methodologies and the determinant basis with the cc-pVDZ basis set. In particular, note that (a) and (b) are in the Slater determinant basis and so state-averaging includes the triplet states, while (c)–(f) only include singlet states. J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp average (SA) XMS-CASPT2(8e,15o) in a configuration state func- tion (CSF) basis, differs from NOMCSCF with regard to (1) the loca- tion of the conical intersection along the symmetric stretch coordi- nate and (2) how steep the energy increases in the excited state when moving away from the conical intersection. To establish the origin of the differences between 4SA-NOMCSCF(2e,2o) and 2SA XMS- CASPT2(8e,15o), the PES was computed without the perturbative correction [Fig. 5(e)], with a smaller active space [Figs. 5(c) and 5(d)] and using four-state averaging in a SD basis [Fig. 5(b)]. From these calculations, it was apparent that the difference in conical intersec- tion geometry, which occurred at 1.32 Å in 4SA-NOMCSCF(2e,2o), compared to 1.41 Å in 2SA XMS-CASPT2(8e,15o), resulted from the use of the four-state averaging model used as a result of the SD basis, with SD-4SA-CASSCF(2e,2o), giving a conical intersec- tion geometry of 1.30 Å compared to CSF-2SA-CASSCF(2e,2o) at 1.41 Å. Additionally, comparing SD-4SA-CASSCF(2e,2o) and CSF- 2SA-CASSCF(2e,2o) PESs, the use of 4SA is partly responsible for the steeper ascent away from the conical intersection geometry. However, both increasing the size of the active space and including dynamic correlation also increase the gradient of the excited-state PES, and so the NOMCSCF PES may reflect a contribution from both these effects. Further support for this argument comes from examining the topology of the conical intersection, which qualita- tively changes from sloped to peaked upon increasing the size of the active space, indicating that the peaked intersection produced by the NOMCSCF expansion results from compressing the wavefunction into as few determinants as possible. Examining the natural orbitals of the determinants involved reveals how DODC enables NOMCSCF to capture the conical inter- section topology. At the conical intersection geometry, the two degenerate states are (1) an electronic state in which 3 a1and 1 b1 orbitals are double occupied and 4 a1is unoccupied and (2) an elec- tronic state in which 3 a1and 4 a1are singly occupied and 1 b1is doubly occupied. Note that the conical intersection geometry is D∞hsymmetry in which 1 b1and 3 a1orbitals are degenerate, but theC2virreducible representation labels are maintained for clarity. In the first state, the leading determinant has real restricted (RR) symmetry and accounts for 99.6% of the CI weight. The remaining CI contribution is from the 3 a1→4a1doubly substituted determi- nant, which has an overlap with the leading determinant of 0.000 002. The second electronic state is a spin adapted linear combi- nation of two determinants with 50.0% weight, i.e., linear combi- nations of the two determinants form the MS=0 triplet and sin- glet states. In addition to single occupation of 3 a1and 4 a1orbitals, the fractional occupation numbers from the natural orbital analy- sis reveal that the 1 b2(occupation of 1.91) and the 2 b2(occupation 0.09) orbitals also comprise the correlated space. Again, the over- lap of these two determinants was 0.000 002, which is not simply numerical noise but indicative of a small amount of nonorthogo- nality, providing energy stabilization. Additionally, a slightly larger, but still small, amount of nonorthogonality is observed between the determinants pairs where each determinant contributes to dif- ferent electronic states (0.000 925–0.004 062). Thus, it is clear that the role of permitting DODC is subtle, yet a CASCI expansion of two electrons among the 3 a1and 4 a1orbitals does not result in any electronic degeneracy, while the same CASSCF expan- sion gives a significantly different topology around the conical intersection.IV. CONCLUSIONS In this work, we derived the electronic gradient for MO-based nonorthogonal wavefunction expansions. In deriving the electronic gradients, we identified derivative discontinuities in some of the underlying terms that prevent a straightforward implementation of analytic gradients. In order to resolve issues resulting from discon- tinuities and singularities, we proposed and derived an approach based on generalized functions. The advantage of the derivation is that it is valid across the entire wavefunction parameter space. As a result, it is possible to use an initial orthogonal wavefunction expan- sion, which is easier to construct, and then allow it to optimize to give the best performing set of basis determinants. In order to test the performance of NOMCSCF, numerical tests were performed on LiF and water in order to determine if the region around conical intersections and avoided crossings could be success- fully modeled. Optimization of the LiF ground state showed promis- ing behavior, with NOMCSCF providing a balanced description of correlation-types and so producing a PES that closely mirrored the result from full CI. Additionally, NOMCSCF was able to avoid issues resulting from the use of perturbation theory and so could be a successful method in applications where perturbation theory can- not be used (e.g., linear response). The LiF avoided crossing was reproduced by NOMCSCF, although at longer distances than full CI because of stabilization of the ionic diabat due to state averag- ing and possibly because of the larger effective one-electron basis of NOMCSCF. In water, the conical intersection topology of the linear symmetric structure was studied. As in LiF, the state averaging was seen to have an effect on geometry at the point degeneracy, while the qualitative shape of the excited state PES was affected by both state- averaging, the larger effective basis, and inclusion of some dynamic correlation. NOMCSCF enabled a relatively small basis expansion to correctly predict the qualitative topology of the conical intersection, which otherwise required much larger full-valence active spaces. Having identified promise in the methodology, we turn to dis- cuss further improvements and developments that can be made. First, the use of generalized functions enables fully analytical deriva- tives so that nuclear gradients and other properties can be devel- oped. Additionally, an analytical electronic Hessian can be derived to enable robust second-order electronic optimization. The method- ology derived in this work is entirely general as to the spin sym- metries preserved in the basis SD, although we have only used real unrestricted (RU) wavefunction symmetry in this work. It will be interesting to examine how NOMCSCF in the space of real general (RG) generators differs and establishes useful applications. Addi- tionally, complex orbital determinants may be of interest, although the implementation of derivatives over Crequires further develop- ment. Finally, further investigation is required to establish conver- gence to the full CI limit as a function of the size of the configuration basis. ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the ACS Petroleum Research Fund (ACS-PRF Grant No. 59768-DNI6), the University of Louisville, and an EVPRI Internal Research Grant from the Office of the Executive Vice President for Research and J. Chem. Phys. 154, 244101 (2021); doi: 10.1063/5.0053615 154, 244101-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Innovation. This work was conducted, in part, using the resources of the University of Louisville’s research computing group and the Cardinal Research Cluster. APPENDIX: VARIATIONAL BEHAVIOR OF LOCAL RESONANT HARTREE–FOCK SOLUTIONS In this work, we have optimized both global and local minima of the orbital-optimized NOMCSCF equations. In this appendix, we comment on the relation of the NOMCSCF solutions obtained from local minimization of the NOMCSCF equations and the exact solutions. As for standard SCF procedures, local minima of the NOMCSCF equations can be considered to give approximations to excited electronic states, i.e., the Nroots of the NOMCSCF prob- lem do not have to correspond to the lowest Nelectronic states. As a result, NOMCSCF can be used to target a few states of interest with- out obtaining all intermediate states. Additionally, unlike the SCF approximations, the different electronic states from NOMCSCF are rigorously orthogonal. However, as we illustrate here, such solutions are still not necessarily variational with respect to the exact electronic state they mostly resemble. Cauchy’s interlace theorem41is the theoretical basis for varia- tional behavior of CI methods and states that for two calculations carried out using the linear variational spaces HandH′, i.e., using determinant basis ∣Φ⟩and∣Φ′⟩, such that ∣Φ⟩⊂∣Φ′⟩, E′ A≤EA≤E′ A+(dim(H′)−dim(H)), (A1) where E′ AandEAare the eigenvalues obtained in space H′andH, respectively.42For single determinant SCF optimization, dim (H) =1, and so the only possible value of Ais 1. Therefore, regard- less of dim (H′), all that can be determined based on Eq. (A1) is that the single-determinant energy is variational with respect to the exact ground state. Such an analysis provides an explanation for why mean-field approximations of excited states are not necessarily variational with respect to the excited state solution that they rep- resent. Extending the analysis to NOMCSCF, it is apparent that the NOMCSCF solutions are variational with respect to only the low- est dim(H)states. Therefore, NOMCSCF solutions that represent higher electronic states do not necessarily exhibit variational behav- ior. For the Nth NOMCSCF solution to be variational, a NOMCSCF basis of at least Ndeterminants must be used. In practice, spin or spatial symmetries may be used to permit higher roots to behave variationally. However, as has recently been established for single- determinant SCF solutions, the ability to obtain computationally efficient approximations of excited states without explicitly com- puting all intermediate solutions is likely to be useful, despite the problem of variational behavior. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1P. C. Hiberty and S. Shaik, J. Comput. Chem. 28, 137 (2007). 2P. Su, S. Shaik, and P. C. Hiberty, Chem. Rev. 111, 7557 (2011). 3C. Coulson and M. I. Fischer, Philos. Mag. 40, 386 (1949).4H. Fukutome, Prog. Theor. Phys. 80, 417 (1988). 5R. 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5.0052279.pdf

Appl. Phys. Lett. 119, 013302 (2021); https://doi.org/10.1063/5.0052279 119, 013302 © 2021 Author(s).Band mobility exceeding 10cm2 V−1 s−1 assessed by field-effect and chemical double doping in semicrystalline polymeric semiconductors Cite as: Appl. Phys. Lett. 119, 013302 (2021); https://doi.org/10.1063/5.0052279 Submitted: 30 March 2021 . Accepted: 23 June 2021 . Published Online: 08 July 2021 Masato Ito , Yu Yamashita , Taizo Mori , Katsuhiko Ariga , Jun Takeya , and Shun Watanabe COLLECTIONS Paper published as part of the special topic on Organic and Hybrid Thermoelectrics ARTICLES YOU MAY BE INTERESTED IN Doping of organic semiconductors: Insights from EPR spectroscopy Applied Physics Letters 119, 010503 (2021); https://doi.org/10.1063/5.0054685 Detailed balance analysis of advanced geometries for singlet fission solar cells Applied Physics Letters 119, 013301 (2021); https://doi.org/10.1063/5.0047964 Continuously graded doped semiconducting polymers enhance thermoelectric cooling Applied Physics Letters 119, 013902 (2021); https://doi.org/10.1063/5.0055634Band mobility exceeding 10 cm2V/C01s/C01assessed by field-effect and chemical double doping in semicrystalline polymeric semiconductors Cite as: Appl. Phys. Lett. 119, 013302 (2021); doi: 10.1063/5.0052279 Submitted: 30 March 2021 .Accepted: 23 June 2021 . Published Online: 8 July 2021 Masato Ito,1YuYamashita,1,2 Taizo Mori,1,2 Katsuhiko Ariga,1,2 JunTakeya,1,2,3and Shun Watanabe1,3,a) AFFILIATIONS 1Material Innovation Research Center (MIRC) and Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan 2International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science (NIMS), 1-1 Namiki,Tsukuba 305-0044, Japan 3AIST-Utokyo Advanced Operando-Measurement Technology Open Innovation Laboratory (OPERANDO-OIL), National Institute ofAdvanced Industrial Science and Technology (AIST), 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan Note: This paper is part of the APL Special Collection on Organic and Hybrid Thermoelectrics. a)Author to whom correspondence should be addressed: swatanabe@edu.k.u-tokyo.ac.jp ABSTRACT The assessment of intrinsic carrier mobility in disordered polymeric semiconductors is critical for improving optoelectronic devices; however, it is currently limited. We examined how to accurately determine intrinsic, band mobility in doped, semicrystalline polymers usingthe field-effect and chemical double doping. In particular, chemical doping with a strong molecular oxidant effectively shifts the Fermi energywithin the valence band, and field-effect modulation of the carrier density at the Fermi energy determines the field-effect mobility. Therefore, a band-like field-effect mobility exceeding 10 cm 2V/C01s/C01with a negative temperature coefficient was demonstrated for uniaxially aligned semicrystalline polymeric semiconductors, which indicates that the band description derived from the semiclassical Boltzmann transportmodel is applicable even to semicrystalline polymers with finite structural disorders. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0052279 Carrier mobility is a critical parameter of a semiconductor mate- rial, impacting the electronic performance of variety of electronic devi-ces such as field-effect transistors, solar cells, and thermoelectric devices. 1–3It characterizes how a charge carrier, that is, either a hole or an electron, can move in a solid state under an applied electric field.4 With the band description, the mobility ( l) is proportional to the momentum relaxation time ( s) and inversely proportional to the effec- tive mass ( m/C3), that is, l¼es=m/C3. Hence, an in-depth understanding of mobility with its temperature and field dependence allows a directassessment of the carrier scattering mechanism and the electronic struc- ture in solid-state materials. 4Recently, it has been found that the band- like mobility expression, which is based on the formation of a periodicelectrostatic potential, is applicable even to organic semiconductors (OSCs). In particular, it is applicable to their single crystals, 5–7which details critical factors limiting the mobility of single-crystalline OSCs. It is controversial whether the band mobility description is valid for organic materials with finite structural disorders, particularly inconjugated polymers.5,8Charge transport in such conjugated polymers is likely to undergo the nearest-neighbor hopping between localizedstates. The presence of localized tail states owing to the structural dis-order causes the Fermi energy ( E F) to be pinned deep in the bandgap. Therefore, the conceptual failure of the band description is rooted in the structural disorder inevitably found in conjugated polymers,8 where the mobility is limited by extrinsic factors such as defects,impurities, and grain boundaries. Specifically, in semicrystalline polymers, the observable mobility is limited mainly by microscopic morphology 9and defects caused by imperfections in the orientation of main chains.8From an energetic point of view, such defects pro- duce trap density of states (DOSs) near the edge of the valence band.10When EFis pinned by the trap DOS, the hopping process between localized states dominates the charge transport. Therefore,currently, the assessment of the intrinsic band mobility in semicrys-talline polymers is limited, and it is still disputable what factor microscopically determines the mobility and the extent to which Appl. Phys. Lett. 119, 013302 (2021); doi: 10.1063/5.0052279 119, 013302-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplthe mobility is improved, particularly in doped, semicrystalline polymers.11–18 In this study, we propose a method to modulate carriers by the field-effect and chemical doping simultaneously to estimate the intrin-sic band mobility in semiconducting polymers. The carrier doping u s i n gam o l e c u l a rd o p a n ta c h i e v e sah i g hc a r r i e rd e n s i t yc l o s et oo n e hole per one monomer unit, 11,19which shifts the Fermi energy effec- tively up to 0.8 eV from the valence band edge. Compared to chemical doping, field-effect doping is controllable, albeit weak, sufficient to effectively modulate a certain population of carriers, resulting in anaccurate determination of the field-effect mobility. Thus, especially in uniaxially aligned polymeric semiconductors, a hole mobility higher than 10 cm 2V/C01s/C01and its negative temperature dependence (dlFE=dT<0) from room temperature down to 40 K was observed. These results validate that the band-like transport description is appli- cable to semicrystalline polymers, particularly when the Fermi energyis located within the valence band. Figure 1(a) shows a schematic of the doped organic field-effect transistor (OFET) structure. Unlike conventional OFETs, where intrinsic (undoped) OSCs are employed, an active polymeric semicon-ductor, poly(2,5-bis(3-tetradecylthiophen-2-yl)thieno[3,2- b]thio- phene) (PBTTT) 20was intentionally doped with a radical salt dopant, tris(4-bromophenyl)ammoniumyl bis(trifluoromethylsulfonyl)imide[TBPA-TFSI, see the inset of Fig. 1(a) ]. 19When the undoped PBTTT thin film is exposed to the TBPA-TFSI solution (1.5 mM in acetoni- trile), the TBPA radical cation initiates a one-electron transfer owingto the half-cell reaction; TBPA •þþe/C0!TBPA. This converts TBPA•þinto a neutral state, while leaving TFSI/C0as a counter-anion to maintain charge neutrality with respect to the positively chargedPBTTT. The carrier density achieved by this doping process was evalu-ated to be approximately 1 /C210 21cm/C03( e q u i v a l e n tt oas h e e tc a r r i e r density of 2 /C21014cm/C02) by the Hall effect measurements.19The remarkably high carrier density, which corresponds to one hole permonomer unit, is consistent with an effective EFshift of approximately 0.8 eV from the valence band (HOMO: highest occupied molecular orbital) edge of PBTTT [see the estimated position of EFwith respect to the valence band edge in Fig. 1(b) ], which was verified by photoelec- tron yield spectroscopy.19We performed field-effect doping to deter- mine the field-effect mobility lFEatEF. Because the Debye screening length at room temperature is estimated to be less than 0.1 nm when a carrier density of 1 /C21021cm/C03is assumed, the field-effect can modu- late the carrier density within one molecular layer near the gate dielec-tric.Figure 1(b) shows the energy dependence of the DOS from the HOMO band edge of PBTTT that is determined by density functional theory (DFT) calculations. In conventional OFETs with an undoped semiconductor layer, nearly all accumulated carriers fill the trap DOS in the bandgap, resulting in E Fbeing pinned deep in the bandgap. However, with the present chemical and field-effect double doping, a priori shifted EFby chemical doping can be effectively modulated by the field-effect. Before assessing lFE, we characterized the charge-transport prop- erties of uniaxially aligned PBTTT thin films doped with TBPA-TFSI. Uniaxially aligned films were fabricated by the high-temperature LB method on the subphase of 1-ethyl-3-methylimidazolium bis(trifluor- omethylsulfonyl)imide at 120/C14C.21PBTTT thin films were transferred to a highly doped Si substrate with a 500 nm thick thermally grownSiO 2layer. Gold electrodes were thermally deposited to form the source, drain electrodes, and four-terminal voltage probes. The direc- tion of the polymer main chains was parallel to the channel direction.Thin films were annealed at 180 /C14C. Thereafter, the thin films were patterned to define the channel. Figures 2(a) and2(b) show the cross- polarized optical microscopy images of the aligned film. A change inbrightness with rotation of the polarizer was observed, which confirms the anisotropic nature of the fabricated film. 21–23The thickness was determined to be 12 nm near the channel using an atomic force FIG. 1. (a) Schematic illustration of the doped OFETs structure. The PBTTT thin film was chemically doped with the radical salt dopant TBPA-TFSI via sequential doping.11–18The chemical structures of PBTT and TBPA-TFSI are shown in the inset. (b) Energy diagram and integrated density of states (DOS) of the valenceband (HOMO) of PBTTT evaluated by the DFT calculation. The presence of local- ized tail states owing to the structural disorder, represented as a shaded area, pins E Fdeep in the bandgap. Chemical doping can effectively shift the EFinto the valence band. Upon chemical doping, field-effect doping was performed to deter-mine the field-effect mobility. FIG. 2. (a) and (b) Cross-polarized microscopy images of uniaxially aligned PBTTT thin films. (c) Typical Hall voltage VHallprofiles with respect to sampling time at tem- perature T¼250 K. The Tdependence of (d) conductivity r, (e) Hall carrier density (eRH)/C01, and (f) Hall mobility ( lH).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 013302 (2021); doi: 10.1063/5.0052279 119, 013302-2 Published under an exclusive license by AIP Publishingmicroscope. Thereafter, PBTTT thin films were doped with TBPA- TFSI.19In previous studies,19,21,24diffraction measurements indicated that the uniaxial alignment of the main chains and an edge-on struc- ture was maintained after chemical doping. Temperature dependence measurements of the electrical properties and the Hall effect were per-formed in a He-gas-exchanged cryostat with a superconducting magnet. To understand the transport mechanism and the coherence of carriers, the temperature ( T) dependence of the four-terminal conduc- tivity and Hall effect measurements was performed. Figure 2(c) shows the representative time-domain profile for the Hall voltage ( V Hall)a t T¼250 K for the aligned PBTTT thin film. A clear Hall voltage corre- sponding to an external magnetic B[black lines in Fig. 2(c) ]i s observed over a wide temperature range from room temperature to 2 K; thus, indicating the band-like transport for delocalized charge car-riers. 25The four-terminal conductivities ( r) were evaluated to be approximately 1000 and 500 S cm/C01for the aligned and spin-coated films at near room temperature, respectively, which is consistent with a previous study.19Figure 2(d) shows the temperature dependence of r. The observed positive temperature coefficient of conductivity does not contradict the existence of delocalized carriers; the Hall voltage is likely to be generated by summing over the contributions from thehighly crystalline domain, whereas a fully longitudinal conductivity can be limited by the presence of grain boundaries. For undoped OFETs, a charge transport in polymeric semiconductors is typically dominated by the nearest-neighbor hopping, that is, the thermally activated tunneling process of carriers. For the doped OFETs with the spin-coated thin films presented in this work, the decrease in conduc- tivity with a decrease in temperature is significantly weaker. This tem-perature dependence of conductivity can be reproduced well with the 3D variable-range hopping (VRH) model, that is, the conductivity is proportional to exp ½/C0ðT 0=TÞ1 4/C138(T0is the characteristic temperature), suggesting that the charge carriers hop over a small distance with a high activation energy or hop over a long distance with a low activationenergy. 26For the aligned PBTTT, the temperature dependence of con- ductivity follows a power-law behavior ( /Tb,bis the the exponent). Although the origin of the power-law behavior in the temperature- variant conductivity is still unclear, a similar trend has been observed for various highly doped conducting polymers that are likely to be in a critical regime for the insulator–metal transition.27,28The observation of the weak temperature dependence of conductivity for the alignedPBTTT indicates that the orientation of the main chains in polymeric semiconductors plays a decisive role in band-like charge transport. Figures 2(e) and2(f)show the temperature dependence of the inverse Hall coefficient [( eR HÞ/C01, which is equivalent to the Hall car- rier density] and Hall mobility ( lH). Both the Hall carrier density and Hall mobility show a weak temperature dependence, which is consis- tent with previous studies.11The observed Hall carrier density ðeRHÞ/C01/C241/C01:5/C21021cm/C03corresponds to one-hole per mono- mer unit of PBTTT; a large number of counter-anions (TFSI/C0) /C241021cm/C03are present in the PBTTT thin film as an impurity dop- ant. Note that the decrease in ( eRHÞ/C01owing to a decrease in tempera- ture does not necessarily indicate a reduction in the actual carrier density and is presumably attributed to changes in the carrier coher- ence.25,29Although a finite contribution of localized carriers remains and becomes more dominant at lower temperatures, band-like carriers are the major contributors to the charge conduction.To further assess the band-like carrier conduction, the field-effect carrier modulation upon chemical doping was performed. The appli- cation of a gate bias ( VG) to the gate electrode allows the field-effect modulation of carriers, where the sheet carrier conductivity ( rsheet)a t the dielectric interface can be modulated. The change in rsheetcan be described as Drsheet¼eDnsheetlFE,w h e r e lFEis the field-effect mobil- ity,Dnsheetis the change in the sheet carrier density modulated by VG, andeDnsheet¼CiVG.T h u s , lFEcan be determined by a measure of the dependence of Drsheet. Note that although Dnsheetis estimated to be on the order of 1 :7/C21012cm/C02(with a capacitance value of Ci¼6.9 nF cm/C02and an applied gate voltage of –40 V), which is only 0.86% of the net sheet carrier density achieved by chemical doping (2 /C21014 cm/C02), tiny changes in the VG-variant conductivity are observable. During the measurement, the applied drain voltage (longitudinal volt- age) was maintained as small as 5 mV to ensure that no gradient in the carrier density along the in-plane channel direction is formed. The res-olution of current measurement for the used source measure unit (Keithley 2636B) was high enough (0.1 nA fA) for resolving the change in current (a few nA) due to field-effect doping [see the inset of Fig. 3(a)]. For clarity, with a known carrier density (1 :5/C210 21cm/C03), the thickness of the depletion layer is estimated to be approximately3.0 nm, which corresponds to a quarter of a total thickness of PBTTT thin films (12 nm). Figures 3(a) and3(b) show plots of Dr sheetvsVG at various temperatures for the aligned and spin-coated PBTTT thin films, respectively. Drsheet shows an almost linear response to the applied VG, as expected. From the linear slope, lFEcan be derived at various temperatures [as summarized in Fig. 3(c) ]. The obtained lFE for both the aligned and spin-coated thin films was found to be approximately 2–3 times larger than the Hall mobility, which is theaverage value over the bulk of PBTTT thin films; thus, suggesting that the mobility evaluated at E F(located at 0.8 eV below the valence band edge) is significantly large. Surprisingly, for aligned PBTTT thin films, lFEreaches 10 cm2 V/C01s/C01and increases as the temperature decreases; the signature of delocalized charge transport ( dlFE=dT<0) is observed down to 40 K. Below 40 K, lFEtends to saturate. This may be attributed to the com- pensation between the delocalized and localized transport, both ofwhich may coincidentally provide the temperature-independent mobility at cryogenic temperatures. We would also like to note that there is an alternative scenario. The temperature-independent mobilityhas typically been observed in two-dimensional electron systems such as an inversion layer of Si, 30modulation-doped GaAs,31and oxide interfaces.32In such two-dimensional electron gas (2DEG) systems, the carrier transport is dominated by phonon scattering at elevated temperatures and is consequently dominated by the scattering ofdefect impurities at cryogenic temperatures, causing the saturation of mobility. 33The crossover from phonon scattering at higher tempera- tures to impurity scattering at lower temperatures is consistent with the experimental observation of the mobility trend for aligned PBTTT thin films. This interpretation does not contradict the observation ofthe Hall effect, which is indicative of the Fermi degeneracy, and the fact that a large number of counter-anions (TFSI /C0)/C241021cm/C03are likely to behave as an impurity scattering center. This is different fromthe positive temperature coefficient of mobility obtained for spin- coated PBTTT thin films. This can be attributed to the strong carrier localization typically observed in the existing OSC materials with a finite structural disorder. Thus, the temperature dependence of l FEforApplied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 013302 (2021); doi: 10.1063/5.0052279 119, 013302-3 Published under an exclusive license by AIP Publishingspin-coated PBTTT can be fitted with a thermal activation model with an activation energy ( EA)o f3 0 0 leV;lFE/expð/C0EA=kBTÞ(kBis the Boltzmann constant). The remarkably small EAmay be indicative of the presence of a soft gap at EF.26,34Alternatively, the randomness of the DOS-energy landscape near EFresults in a strong energy depen- dence of the drift mobility and limits the carrier transport for the spin- coated film at low temperatures, which has been often observed in amorphous oxide semiconductors35and amorphous Si.36The critical difference in the temperature dependence of mobility observed for aligned and spin-coated thin films suggests that the precise control of polymer chain alignment is important to fully understand the elec-tronic structure of these systems. We present a possible strategy to improve the carrier mobility, particularly for doped polymers. From the observation of the Hall effect and dl FE=dT<0, the band transport description is applicable even to polymeric semiconductors with structural disorders. The effec- tive mobility measured within the valence band (HOMO band) is derived as l¼es=m/C3, indicating that the effective mass and momen- tum scattering time play a predominant role in determining the mobil- ity, similar to other crystalline semiconductors. Meanwhile, a small effective mass on the order of 0.1 m0along the conjugated polymer chain37,38can be advantageous for increasing the mobility. For exam- ple, with an effective mass of 0.1 m0and a momentum relaxation time of 10 fs, which is typically observed in single-crystalline OSCs,7,39the mobility is estimated to be 176 cm2V/C01s/C01at room temperature. Although such a high mobility has not been verified for polymeric semiconductors to date, covalently bonded polymer main chains facili-tate a reduction in the effective mass. Further studies are required, par-ticularly for an in-depth understanding of the momentum relaxation mechanism, that is, the carrier scattering mechanism along the poly- mer chains. Here, the alignment of the conjugated backbone isimportant. We would emphasize the importance of the electronic structure atE Ffrom the viewpoint of thermoelectric transport. In a degenerated semiconductor, only carriers around EFwithin a few kBTcontribute to both charge and thermoelectric transport. Under the rigid bandapproximation, the Mott formula is often employed to interpret the thermopower S,40,41 S¼/C0p2k2 BT 3edlnrE dE/C20/C21 E¼EF; (1) where rEðEÞdenotes the transport function at aparticular energy level E. The Mott formula provides a reasonable guideline to search for the candidate of thermoelectric materials and can be applied tosemicrystalline conducting polymers. 25,39,42,43The energy deriva- tive of the logarithm of rEðEÞin Eq. (1)suggests that the thermo- power can be improved by steeping rEðEÞnear EF. Polymeric materials, which possess an inherently one-dimensional nature, canbe advantageous in terms of their non-uniform DOS. In particular,the optimization of carrier filling may lead to a large and controlla-ble thermopower, which may be realized by the microscopic order-ing of polymeric chains. In conclusion, we presented a universal strategy for assessing the intrinsic field-effect mobility in disordered polymeric semiconductors.The field-effect and chemical double doping demonstrated in thisLetter allows an accurate determination of the band-like field-effectmobility exceeding 10 cm 2V/C01s/C01with a negative temperature coeffi- cient down to 40 K for uniaxially aligned PBTTT thin films. Theseobservations indicate that the band description derived from the semi-classical Boltzmann transport model can be applied even to semicrys-talline polymers with finite structural disorders. In future, the presentdouble-doping method will contribute to the understanding of theintrinsic carrier transport mechanism, particularly in disordered sys-tems, and to the improvement of electronic devices, in which theenergy-dependent carrier mobility is a critical factor. S.W. acknowledges the support from the Leading Initiative for Excellent Young Researchers of JSPS. This work was also supportedin part by JSPS KAKENHI Grant (Nos. JP17H06123, JP20K15358,JP20H00387, JP20K20562, and JP20H05868) and by JST FORESTProgram, Grant No. JPMJFR2020.FIG. 3. Gate voltage dependence of sheet conductivity Drsheet for (a) uniaxially aligned and (b) spin-coated thin films. The inset shows the current–voltage characteristic mea- sured at 100 K. The change in IDwith respect to VGis found to be approximately 0.3% of IDand is sensible enough by the used source measure unit. (c) Temperature depen- dence of the field-effect mobility lFE.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 013302 (2021); doi: 10.1063/5.0052279 119, 013302-4 Published under an exclusive license by AIP PublishingDATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. M. Sze, Y. Li, and K. K. Ng, Physics of Semiconductor Devices (John Wiley & Sons, 2021). 2H. Sirringhaus, Adv. Mater. 26, 1319 (2014). 3A. Shakouri, Annu. Rev. Mater. Res. 41, 399 (2011). 4C. Hamaguchi and C. Hamaguchi, Basic Semiconductor Physics (Springer, 2010), Vol. 9. 5S. Fratini, M. Nikolka, A. Salleo, G. Schweicher, and H. Sirringhaus, Nat. Mater. 19, 491 (2020). 6T. Kubo, R. H €ausermann, J. Tsurumi, J. Soeda, Y. Okada, Y. Yamashita, N. Akamatsu, A. Shishido, C. Mitsui, T. Okamoto et al. ,Nat. Commun. 7, 11156 (2016). 7J. Tsurumi, H. Matsui, T. Kubo, R. H €ausermann, C. Mitsui, T. Okamoto, S. Watanabe, and J. Takeya, Nat. Phys. 13, 994 (2017). 8R. Noriega, J. Rivnay, K. Vandewal, F. P. Koch, N. Stingelin, P. 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5.0052233.pdf

Recent developments of quantum sensing under pressurized environment using the nitrogen vacancy (NV) center in diamond Cite as: J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 View Online Export Citation CrossMar k Submitted: 30 March 2021 · Accepted: 9 June 2021 · Published Online: 24 June 2021 Kin On Ho,1 King Cho Wong,1Man Yin Leung,1Yiu Yung Pang,1Wai Kuen Leung,1King Yau Yip,1Wei Zhang,1 Jianyu Xie,1Swee K. Goh,1,2 and Sen Yang1,2,a) AFFILIATIONS 1Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China 2Shenzhen Research Institute, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China Note: This paper is part of the Special Topic on Materials, Methods, and Applications of Color Centers with Accessible Spin. a)Author to whom correspondence should be addressed: syang@cuhk.edu.hk ABSTRACT Pressure has been established as a powerful way of tuning material properties and studying various exotic quantum phases. Nonetheless, measurements under pressure are no trivial matter. To ensure a stable pressure environment, several experimental restrictions must beimposed including the limited size of a sample chamber. These have created difficulties in assembling high-pressure devices and conducting measurements. Hence, novel sensing methods that are robust and compatible with high-pressure devices under pressure are highly in demand. In this review, we discuss the nitrogen-vacancy (NV) center in diamond as a versatile quantum sensor under pressure. The excel-lent sensitivity and superior resolution of the NV center enable exciting developments in recent years. The NV center has great potential insensing under pressure, especially beneficial to magnetic-related measurements. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0052233 I. INTRODUCTION High pressure is a clean and systematic tuning parameter without introducing chemical impurities to the material system under investigation. It provides unprecedented opportunities for examining various quantum states that emerge only under extreme conditions. Tuning pressure to explore striking states of physical systems has, therefore, become a central tactic in materials research. For example, it has long been proposed that metallic hydrogen canbe realized under ultrahigh pressure. 1,2Other physical phenomena that can be achieved by applying pressure include surface plasmon resonance3and glass transitions.4Pressure has emerged as a very effective tool in superconductivity research, especially in the quest of room temperature superconductors. Superconductors with a remarkably high superconducting transition temperature Tcof 250 –260 K has been reported in lan- thanum hydride (LaH 10/C0δ) at around 200 GPa5,6and Tcof 203 K has been reported in sulfur hydride (H 3S) pressurized to 155 GPa.7 Tcof 243 K in yttrium hydrides (YH 9) at 201 GPa was alsoreported, and it is predicted that the Tccan be further enhanced in different phases.8In 2020, Snider et al.9found superconductivity in a photochemically transformed carbonaceous sulfur hydride (C–S–H) system. The maximum Tcis found to be 287 :7+1:2K , achieved at 267 +10 GPa. These exciting discoveries define a new landscape for superconductivity research, requiring sophisticatedprobes to extract physical information under demanding experi- mental conditions. For instance, the detection of the Meissner effect, a defining property of a superconductor, is not a trivial mea-surement under such extreme pressures. To detect superconductivity under pressure, multi-coil mutual induction, tunnel diode oscillator (TDO), and four-wire method are some of the commonly used techniques. The four-wire method is a standard way to measure electrical resistivity, whereby fourgold wires are attached to the sample. On the other hand, themulti-coil mutual induction technique and TDO are contactlessmethods that are sensitive to the magnetic susceptibility of the sample. 10–14These techniques have played an important role in the superconductivity research. However, a sensor that can directlyJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-1 Published under an exclusive license by AIP Publishingmeasure the expulsion of the magnetic field from superconductors provides the most direct confirmation of the Meissner effect. Diamond anvil cell (DAC) is a traditional workhorse for high pressure research. With careful design, DAC offers superior opticalaccess to the sample. This offers an opportunity for implementinga method based on the negatively charged nitrogen-vacancy (NV /C0) center in diamond for probing the expulsion of the magnetic field when the sample becomes superconducting. It is widely acknowl-edged that the four-wire technique under pressure is experimentallydemanding. On the other hand, contactless methods based on thepickup coil mentioned above can pick up unwanted signals in the proximity of the sample. Incidentally, the NV /C0-based technique is simpler to prepare —the experimentalist can either implant the NV/C0on the surface of a diamond anvil or disperse small diamond particles around the sample. Furthermore, these sensors can sensethe local magnetic field profile close to the superconductor. Local measurement is particularly important for research works focusing on nanoscale phenomena. Therefore, the NV /C0-based technique might even revolutionize the high pressure research ofnanomaterials.In this review, we first describe the key properties of the NV /C0 center in diamond. We then demonstrate that the NV/C0center is a versatile sensor workable under pressure, making it a powerful toolto probe pressure-induced phenomena as well as the pressure envi-ronment itself. Cutting-edge results in the field and future pros-pects will be discussed. II. THE NV CENTER The NV /C0center is a point defect in the diamond lattice. We refer NV always as NV/C0unless otherwise mentioned. It is com- posed of a vacancy and an adjacent substitutional nitrogen atom, resulting in a C3vsymmetry with respect to the NV axis. Due to an extra electron, the NV center is a spin-1 system with electronicground states ( j 3Ai) and excited states ( j3Ei), both of which are spin triplets. A metastable state also exists in between. A schematic drawing of its atomic structure and energy levels are shown in Figs. 1(a) and1(b). Upon incidence of a green laser beam, the NV center is excited from ground states to excited states under spin conservation. From FIG. 1. (a) Atomic structure of the NV center. It is composed of a vacancy and an adjacent substitutional nitrogen atom, resulting in a C3vsymmetry with respect to the NV axis. Reproduced with permission from Doherty et al. , Phys. Rev. B 90, 041201(R) (2014). Copyright 2014 American Physical Society. (b) Energy level structure of the NV center. It is a spin-1 system with electronic ground states ( j3Ai) and excited states ( j3Ei). A metastable state also exists in between. The spin-state-dependent decay leads to spin-state-dependent fluorescence that allows the ODMR measurement. Reproduced with permission from Doherty et al. , Phys. Rev. B 85, 205203 (2012). Copyright 2012 American Physical Society.16(c) ODMR spectrum of three different diamond samples. The intrinsic splitting is due to the internal strain, which is obviously sample dependent. Reproduced with permission from Gruber et al. , Science 276, 2012 (1997). Copyright 1997 American Association for the Advancement of Science.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-2 Published under an exclusive license by AIP Publishingthe excited states, the system could return to the ground states directly while conserving spin, thus emitting red fluorescence that originates from a zero-phonon line (ZPL) at 637 nm. Peculiarly, theexcited states could also return to the ground states after making adetour through the metastable state, thereby emitting infrared radia-tion. If the system does go through the metastable state, it would preferentially return to the jA,0istate compared to the jA,+1i states. Overall, the excitation-decay cycles under continuous greenlaser incidence favor a higher population at the jA,0istate. As a result, the initialization of the spin system to the jA,0istate is easily achieved by using a green laser alone. This spin-state-dependent decay leads to spin-state-dependent fluorescence that allows the opti- cally detected magnetic resonance (ODMR) measurement. TheODMR measurement allows us to experimentally obtain the reso-nance frequencies of the NV energy level structure. During the exci-tation –decay cycles, if in addition to the green laser, microwave (MW) signals are simultaneously swept and applied to the system, resonances due to transitions from jA,0itojA,+1icould be observed in the fluorescence-frequency spectrum. As shown inFig. 1(c) , the left peak corresponds to jA,0i!j A,/C01itransition and the right peak corresponds to jA,0i!j A,þ1itransition. For more details about the NV center, one could refer to some previous studies. 17–23 The transition frequencies depend heavily on external parame- ters. Considering the ground states only, the effective Hamiltonian of the NV center could be modeled by H¼~S/C1D$ /C1~Sþγe~B/C1~S, (1) where ~Sis the electron spin operator, D$ is the spin –spin coupling tensor, γe/C252:8 MHz/G is the electron gyromagnetic ratio, and ~Brepresents the external magnetic field. If one chooses the NV axis to be the z-axis thereby diagonalizing matrix D,(1)becomes H¼DS z2/C0S(Sþ1) 3/C20/C21 þE(Sx2/C0Sy2)þγe~B/C1~S: (2) The first term is the longitudinal (along NV axis) zero-field split- ting (ZFS) term, which splits the j0istate from the j+1istates. Dis a constant and approximately equals 2 :87 GHz at ambient pressure and room temperature. The second term is the transverse ZFS term that further splits the jþ1iandj/C01istates. Being sample-dependent, Ecould range from negligible to several MHz as shown in Fig. 1(c) . The last term accounts for Zeeman splitting. The research works on the NV center in diamond not only provided a deeper understanding of the solid-state defect as a quantum system but also opened up opportunities for quantuminformation technologies. One can tune the solid-state defect into adesired quantum state for further processing. 24–27On the other hand, fundamental quantum sensing methods using the NV center are often based on exploiting the change in DandEterms as well as the Zeeman splitting term under various circumstances (alteringthe temperature and pressure, and/or applying an external mag-netic field). Some more advanced techniques can be achieved by considering more interaction terms in (2), like the spin-spin coupling term of an electron and a nuclear spin. Including thesecoupling effects allows the NV center to, for example, sense nearby spins. 28–38In this review, we focus on pressure sensing using the NV center in diamond, as well as various sensing protocols underpressure. In the high-pressure community, searching for a robust sensor is a crucial issue since a pressurized environment is too demanding for most of the common devices. The NV center works as an inte- grated versatile sensor that can be used to probe different parame-ters given that a correct protocol is implemented. Previous researchworks have shown that the NV center can be used to probe themagnetic field, 39–42electric field,43and temperature.15,44–46With a careful design of experimental setup, one can, in principle, measure various parameters under pressure. One recent example is the com-bination of NV magnetic microscopy and x-ray diffraction underpressure. 47 III. A ROBUST PRESSURE SENSOR A. Pressure calibration and sensing DAC is a popular tool in high-pressure experiments. The pres- sure is produced by compressing two opposing diamond anvilstoward the pressure medium confined by a gasket. In order to measure the pressure inside a DAC, a calibrated material is placed inside the pressurized chamber. In particular, ruby (Al 2O3:Cr) works well under pressure, and it is commonly used among the highpressure community. 48,49Two resonance lines ( R1,R2)i nt h er u b y fluorescence spectrum are sensitive to both pressure (3.64 Å/GPa) and temperature (0.068 Å/K). When applying pressure to tune materials properties, hydro- staticity is an important, yet rarely addressed issue. The pressuredistribution can induce significant artifacts or even give differentresults. 50–54Nonetheless, detailed mapping of the gradient lacks discussion. One major problem is that the usual sensors are in the bulk size and are not sensitive enough. In contrast, the use of theNV center can be an elegant solution to this problem. Doherty et al. 55were the first to experimentally demonstrate the capability of the NV center to detect pressure. The experimen- tal setup is shown in Fig. 2(a) . They put a bulk diamond together with a ruby inside the sample chamber of a DAC. The DAC gasketwas filled with NaCl/Ne as the pressure medium. A platinum (Pt)wire was used as an MW antenna for ODMR measurement. To perform a pressure calibration for the NV center, the ZFS Dfrom the diamond ’s ODMR spectrum was plotted against the pressure measured by the ruby up to 60 GPa, and they discovered a linearslope of dD=dP¼14:58(6) MHz/GPa [ Figs. 2(b) and 2(c)]. This calibration enabled the NV center to be employed as a pressure sensor. Moreover, a linear shift in ZPL was presented [ Fig. 2(d) ]. To explain their findings, they outlined a preliminary theoreticalpicture. Upon increasing pressure, the unpaired spin density of theNV ground-state level would contract, favoring spin –spin interac- tion and thus increasing D. They further suggested two contribu- tions to the spin density contraction: The compression of the nuclear lattice (i) decreases the distance between atomic orbitalsand (ii) deepens the localizing electrostatic potential of the NVcenter to concentrate the electron density more on the inner neigh- bor shells of the defect. They mathematically demonstrated this physical picture by using a semi-classical molecular orbital model.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-3 Published under an exclusive license by AIP PublishingAssuming hydrostatic pressure, the shift in Ddue to the displace- ment of the atomic orbitals was found to be around 6.2 MHz/GPa.The remaining /difference57:5% in the total shift of 14.58(6) MHz/GPa was attributed to contribution (ii) mentioned above. Meanwhile, ODMR pressure sensitivity of a single NV center at room tempera- ture is estimated to be /difference0:6G P a /ffiffiffiffiffiffi Hzp , analogous to the estimationof field and thermal sensitivities of NV ODMR. 56They further pro- posed to use NV centers for magnetometry under pressure, whichwill be shown in Sec. III B to be a breakthrough in high-pressure instrumentation. Ivády et al. 57later did a comprehensive theoretical study to discuss the results of Doherty et al.55and further elucidate the FIG. 2. (a) The schematic design of a DAC. A bulk diamond together with a ruby were put inside the sample chamber with NaCl/Ne being the pressure medium. A Pt wire was used as an MW antenna for the ODMR measurement. (b) The ODMR spectra of the diamond sample at different pressures. (c) The measured value of Dvs pressure using two different pressure media. A linear slope of dD=dP¼14:58(6) MHz/GPa was discovered. (d) The linear shift of the ZFL position of the NV center with pressure. Reproduced with permission from Doherty et al. , Phys. Rev. Lett. 112, 047601 (2014). Copyright 2014 American Physical Society.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-4 Published under an exclusive license by AIP Publishingphysical origin of the observed NV response to applied pressure. Using ab initio computational methods to simulate a single NV center embedded in a 512-atom diamond supercell, they studiedthe pressure and temperature dependence of the ground-state ZFStensor of the NV center. By varying the size of the supercell at afixed temperature, they investigated the longitudinal ZFS Das a function of external pressure Pover a broad pressure range from /C020 to 210 GPa as shown in Fig. 3(a) .D(P) was found to show alinear relation to the first order, with a slope of 10.30 MHz/GPa from 0 to 50 GPa. This was in fair agreement with the measure- ment of Doherty et al. In the complete simulated solution includ- ing higher-order terms, D(P) exhibited a weak non-linearity in the high-pressure range, which was attributed to microscopic effects.Furthermore, Ivády et al. summarized four different sources contributing to the shifting in Dunder pressure: (i) macroscopic compression of the bulk diamond crystal, (ii) microscopic FIG. 3. (a) Calculated pressure dependence of ZFS Dat zero temperature. The change is linear to the first order. (b) Comparison of the experimental result (A) with differ- ent levels of theoretical methods (B, C, and D). The arrows represent the logic flow of the theory: the most simplified model (B) considers only the effe ct of the macro- scopic compression of the diamond lattice, while the modified model (C) takes into account the effect of the local structural relaxation. Nonetheles s, the wavefunctions are kept fixed. In the fully self-consistent solution (D) both the structure and the orbitals are relaxed. Reproduced with permission from Ivády et al. , Phys. Rev. B 90, 235205 (2014). Copyright 2014 American Physical Society. (c) An image showing the tungsten microchannel. It consists of a series of eight concentric rings w ith 5μm width. (d) Pressure dependence of the ZFS D(I), and the ZFS E, and linewidth (II). This showed the feasibility of their proposed DAC design. Reproduced with the permission from Steele et al. , Appl. Phys. Lett. 111, 221903 (2017). Copyright 2017 AIP Publishing LLC.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-5 Published under an exclusive license by AIP Publishingstructural relaxation at the defect site, (iii) change in spin density on neighbor shells of the NV center, and (iv) change in sphybridi- zation of the dangling bonds. Considering source (i) only, theyfound that dD=dP¼6:26 MHz/GPa at P¼0, which agreed excel- lently with the semi-classical calculation of Doherty et al. However, different from Doherty et al. ’s expectation, Ivády et al. proved rig- orously that source (ii), instead of source (iii), should be the main contributor to the observed pressure dependence of D. In other words, they revealed that the microscopic structural relaxation atthe defect site plays a key role in the pressure-induced shifting ofthe ZFS D.Figure 3(b) compares the experimental result and differ- ent theoretical methods. In addition to the flourishing theoretical understandings, experimentalists have also made great progress since Doherty et al. published their pioneer work. 55Steele et al.58refined the MW transmission scheme in a DAC. In the setup of Doherty et al. ,M W transmission to NV centers in the sample space was achieved by embedding a Pt wire as an MW antenna in a non-metallic gasket.To improve the cell design for high-quality NV sensing, Steeleet al. proposed the concept of a designer anvil for MW transmis- sion, where metallic microchannels [concentric tungsten rings] were lithographically deposited on the culet of the diamond anvil as shown in Fig. 3(c) . In this way, the gasket was not restricted to be non-metallic, and the MW power could also be lowered toreduce undesired heating effects. Besides, in contrast to the bulk diamond used by Doherty et al. , Steele et al. glued some 15- μm diamond particles onto the anvil culet using vacuum grease. WithDaphne oil 7373 as the pressure medium, they measured a linearpressure dependence of D,dD=dP¼11:72(68) MHz/GPa as shown inFig. 3(d) , which was in fair agreement with the data of Doherty et al. This showed the feasibility of their proposed DAC design. Recently, Ho et al. 63conducted a thorough study on the local pressure environment in anvil cells. Their anvil cells consisted oftwo moissanite anvils and a metallic beryllium copper (BeCu)gasket confining the pressure medium (Daphne oil 7373). Inside the gasket, a microcoil was mounted for MW transmission; a ruby was placed for pressure calibration; a large quantity of 1- μm diamond particles was drop-casted either on one anvil culet or on adummy sample. The choice of using smaller diamond particles canminimize the disturbance to the pressure medium and allow the pressure environment to be probed locally. This is one of the best resolutions for mapping the local pressure so far. They built a con-focal microscope featuring a computer-controlled Galvo mirror toperform spatially resolved ODMR spectroscopy, see Fig. 4(a) .A sa result, the pressure inhomogeneity can be probed in detail. Besides the ambient condition, Ho et al. also performed measurements at low temperatures with the help of a Montana cryostation. With thisexperimental setup, the pressure dependence of Dwas calibrated with reference to the ruby first. After that, they showed that the pressure gradient was not radially symmetric but rather a linear pattern. Moreover, the cell design and alignment affected thepressure gradient in a way that misalignment would induce moregradient. Then, at each pressure point, they used diamond particlesto spatially map the local pressure distribution in the pressure medium for different temperatures (295, 10, and 6 K). The mapping measurements were repeated for a few pressure points. Allpresented in Fig. 4(b) . Interestingly, there was a linear pressuregradient parallel to the line joining two screws of the anvil cell, if the Daphne oil already underwent a pressure-induced solidification at room temperature. Nonetheless, no specific pressure pattern wasobserved if the Daphne oil was solidified upon cooling. Thisrevealed that the solidification upon cooling was purely random.Furthermore, the rise of pressure in cryogenic condition was due to the thermal contraction of the cell body. Besides, to fully utilize the information from ODMR spectroscopy, the transverse ZFS E, the linewidth of ODMR peaks, and the standard deviation (SD) ofmeasured pressures at different locations were taken as three inde-pendent indicators to characterize the solidification process. From the critical pressure obtained, the T–Pphase diagram of Daphne oil 7373 was plotted. Last but not least, they examined the temporaldependence of the average pressure and the local pressure distribu-tion in the medium. From the detected pressure relaxation overtime, it was found that it took a day for the pressure medium to stabilize after changing the pressure of the anvil cell. B. Comparison of NV incorporation techniques The works mentioned in Sec. III A employed various types of diamond (with NV centers inside) to achieve their particular sensing goals. In general, there are three types: diamond particles, bulk diamond, and implanted diamond anvil. We shall discuss themajor properties of each of these incorporation techniques as wellas their pros and cons. Diamond particles have a wide range of sizes depending on the manufacturing process, typically ranging from 30 nm to 10 μm scale. The microscopic size allows the particles to serve as pointsensors, provided that a confocal setup is applicable during theexperiment. As the diffraction limit of the confocal microscope isusually much bigger than the particle size, the small size of the par- ticle also gives a better spatial resolution. Meanwhile, a point sensor like this can be immersed into the environment withoutlarge perturbation or disruption, giving a more accurate result.Moreover, the variation in size grants us the freedom to tune the spatial sensing resolution. The non-reactive and bio-compatible nature of diamond also permits it to be implemented in a vastrange of samples unintrusively. It, therefore, works well for materi-als such as metals, superconductors, or even biological samples. Inthis sense, it is an excellent real-time quantum sensor. However, one of the biggest drawbacks is that the orientations of the particles are often ambiguous. Since, unlike bulk diamond and implanteddiamond anvil, the absolute orientation of each particle in the labo-ratory frame is unknown, it is difficult to compare any sensingresults along the NV axes of different particles. In the case of sensing magnetic field or stress field, the angle of the NV axis rela- tive to the surroundings or samples is a crucial parameter. Forinstance, under a uniform magnetic field, individual diamond par-ticles could show different ODMR spectra due to different projec-tions of the field on the NV axes. Careful analysis is usually needed upon receiving signals from different diamond particles. In contrast to diamond particles, bulk diamond is free of this problem. The orientation of NV centers is well-defined for a bulkcrystal since there are only four possible choices of the NV axis in the lab frame. This is why they work so well on magnetometry or other imaging that is derived from it. In particular, bulk diamondJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-6 Published under an exclusive license by AIP PublishingFIG. 4. (a) The confocal setup of the local pressure sensing experiment. APD refers to the Perkin –Elmer avalanche photodiodes. (b) The spatial pressure distribution of the pressure medium at different pressures and temperatures, with a top view picture of an anvil cell. The Daphne oil 7373 solidifies at around 2 GPa.53,54,59–63There was a linear pressure gradient parallel to the line joining two screws of the anvil cell, if the Daphne oil 7373 already underwent a pressure-induced solid ification at room temper- ature. Nonetheless, no specific pressure pattern was observed if the Daphne oil 7373 was solidified upon cooling. Reproduced with permission from Ho et al. , Phys. Rev. Appl. 13, 024041 (2020). Copyright 2020 American Physical Society.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-7 Published under an exclusive license by AIP Publishinggives robust performance at ambient pressure as shown in many recent sensing works.64–69Nonetheless, the bulk size of the diamond is problematic in high-pressure experiments. Usually, thesample chamber is too tiny to accommodate such a bulk sensor.On the other hand, a bulk sensor detects macroscopic averageresponses rather than microscopic local responses. Not to mention, the sensor itself may disrupt the pressure environment. One elegant approach is implanting NV centers at the diamond anvil culet of a DAC. Since diamond is a component ofthe DAC, it is natural to insert NV centers in it for sensing. Likethe bulk diamond, of course, it has four well-defined NV axes and is a great choice for magnetic field imaging. This approach is very powerful when there is the need for wide field imaging. However,unlike diamond particles or bulk diamond, the NV centers areinside the anvils, so they are mainly receiving a uni-axial pressurefrom the pressure medium. In the case of diamond particles that are immersed in the medium entirely, the NV centers experience thorough pressure of the chamber. The difference between sensinga uni-axial pressure and the actual pressure in the medium maypose a significant effect on the high-pressure experiment. After summarizing the NV incorporation techniques, we shall see how some of the recent works employed these techniques on pres- sure sensing experiments. Briefly speaking, recent works employedimplanted diamond anvil and diamond particles to probe magnetictransitions, superconducting transitions, and stress-tensor distribution. IV. SENSING PRESSURE-DRIVEN QUANTUM PHASE TRANSITIONS Pressure is one of the most successful tuning parameters in material research. Nevertheless, a high-pressure experiment is nottrivial at all to implement. One major challenge is the tiny size of the sample chamber, which imposes a huge restriction on the sample and sensor. In particular, compatible magnetic field sensorswith sufficient sensitivity are rare. Moreover, to prevent failure ofthe high pressure device, access to the sample is highly restricted, thus most of the traditional methods can no longer be applied under pressure. Under these demanding experimental conditions,measurements can be extremely difficult to carry out. In 2019, three groups independently demonstrated quantum sensing using the NV center under pressure. 70–72They applied dif- ferent protocols utilizing the excellent spatial resolution and sensi- tivity of the NV center to probe properties of different materialsunder pressure. These provide many possibilities for high-pressureinstrumentation and the study of pressure-induced phenomena. Lesik et al. 71implanted NV centers at the culet of the DAC anvil and used a camera for wide-field imaging of the NV fluores- cence. They successfully probed, under high pressure, the magneti-zation of iron (Fe) at room temperature and the Meissner effect inmagnesium diboride (MgB 2) at low temperature. First, a Fe bead was placed inside the DAC sample chamber, and wide-field ODMR was performed at 24 GPa and under a constant external magnetic field of /difference11 mT. For each family of NV centers (the four possible NV orientations in the implanted anvil), they provided a spatialmap of the splitting of the corresponding ODMR. This splitting was a combined consequence of the non-hydrostatic strain in the anvil, the applied magnetic field, and the stray magnetic fieldgenerated by the Fe bead. Thus, the four maps revealed the inho- mogeneity in the induced magnetic field and the anisotropy of the strain field. To proceed further, they isolated the ODMR effect ofthe sample ’s stray field by referencing to a region far away from the bead. This allowed them to map the magnetic field solely attributedto the Fe bead, and they fitted the experimental data with a mag- netic dipole model where the sample ’s magnetization Mwas the only fitting parameter. In this way, the magnetization was mea-sured for different pressures P, and the α–ϵtransition of Fe was observed in the M–Pplot, shown in Fig. 5(a) . Meanwhile, different M/C0Pplots were obtained by increasing and releasing the pressure of the DAC, showing the expected hysteresis of the structural tran- sition. Next, instead of a Fe bead, they confined a MgB 2sample in the DAC, pressurized it to 7 GPa, and zero-field-cooled it to 18 Kwith a cryostat. A series of wide-field ODMR was then performedupon warming up from 18 to .30 K and under a constant mag- netic field of /difference1:8 mT, where the field direction was chosen such that the four NV orientations showed identical responses. By plot-ting the ODMR resonance splitting against temperature, the criticaltemperature T cof superconducting phase transition was extracted from where the sample ’s diamagnetic response disappeared, see Fig. 5(b) . Their results were in good agreement with the previously reported data.73In summary, their work showcased the adaptability of the NV sensing technique to different experimental conditionsunder extreme conditions. In contrast to using implanted NV centers and wide-field imaging, Yip et al. 70adopted diamond particles and confocal microscopy to probe the Meissner effect of a superconductingmaterial. The experimental setup is similar to Fig. 4(a) .I nt h e i r experiment, they spread 1 μm diamond particles over a type II iron- based superconductor (BaFe 2(As 0:59P0:41)2) and tracked individual particles using a confocal microscope to perform ODMR measure-ments correspondingly at different temperatures and pressures, showninFig. 6(a) . From the Zeeman splitting in the ODMR spectrum, the vector magnetic field projected along an NV axis in the diamond par- ticle could be determined. A signif icant change in the splitting was expected across the critical temperature T c[Figs. 6(b) –6(e)]. This allowed them to measure Tcas a function of pressure and a T–P phase diagram was constructed [ Fig. 6(f) ]. They benchmarked this novel technique by cross-checking results from measuring AC susceptibility, which are commonly applied to probe the bulk response.10,74,75Since the superconducting phase transition is extremely sensitive to temperature, any slight temperature perturbationdue to the heating from the measurement can destroy the supercon- ducting state. Both results agreed well on the value of T c,t h e r e b y proving that ODMR measurement did not cause an observableheating effect. On the other hand, the same technique also enabledthe sensing of the local magnetic fi eld vector near the superconducting sample across the normal state to Meissner state transition. It was clearly shown that an approximately 90 /C14change of the magnetic field vector occurred right above the superconductor upon cooling belowthe critical temperature, picturizing the field expulsion from the super-conductor [ Fig. 6(e) ]. Furthermore, at each fixed pressure, they extracted the critical fields H c1andHc2from the two turning points in the plot of the measured magnetic field against temperature. Then, they could construct the H–Pphase diagram [ Fig. 6(g) ], which is crucial for condensed matte r physics. In particular, Hc1is not trivialJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-8 Published under an exclusive license by AIP Publishingto be measured, so this work opened up a discussion of the supercon- ducting gap function. Hsieh et al.72showcased the versatility of the NV center by performing NV sensing of stress and magnetic fields under high pressure. In their experimental protocol, NV centers wereimplanted at the anvil culet in their miniature DAC, and scanning confocal microscopy was adopted to obtain two-dimensional ODMR maps. Their optical resolution was superior, with a/difference600 nm laser spot illuminating /difference10 3NV centers at a time. One of their big feature is that they resolved the full stress-tensor acrossthe culet surface, see Fig. 7(a) . Similar work was done at ambient pressure by Doherty et al. 76Under an external magnetic field per- pendicular to one selected NV orientation out of the four, the reso-nance pairs from the three other NV orientations were stronglysplit apart by the Zeeman effect, leaving the resonance pair fromthe chosen orientation at around the middle of the entire ODMR spectrum. The stress information corresponding to this particular NV orientation was then extracted, and the same thing wasrepeated for all other NV orientations to compute the fullstress-tensor. By plotting the variation in stress-tensor componentsalong a straight line across the pressurized region, they revealed the emergence of a pressure gradient after the pressure medium entered the glassy phase. This result is important for the high-pressure community. On the other hand, they probed the pressure-driven α–ϵstructural phase transition of Fe. After calibrating the magnetic sensitivity of the implanted NV layer, a polycrystalline Fe pellet was placed inside the DAC sample chamber with a constant external magnetic field being applied to induce magnetization inthe pellet. From the 2D ODMR mapping, they reconstructed thefull vector dipole field due to the pellet ’s magnetization. By increas- ing the DAC pressure, they successfully imaged the depletion of the dipole field after the Fe pellet underwent α–ϵtransition at the criti- cal pressure P¼16:7+0:7G P a [ Fig. 7(b) ]. The hysteresis of this first order transition was also demonstrated through measuring adifferent critical pressure P¼10:5+0:7 GPa upon decompression of the DAC. Furthermore, they integrated their optical setup into a cryogenic system and constructed the magnetic P–Tphase diagram of rare-earth element Gd using the same methodology as the Femeasurement. Last but not least, they demonstrated, under ambientpressure and without external magnetic field, the use of noise spec- troscopy to probe the ferromagnetic Curie transition of Gd. From the kink and the subsequent decrease in the plot of the depolariza-tion time T 1upon entering the ferromagnetic phase, they found that Curie temperature was /difference10 K higher than that observed via DC magnetometry. This discrepancy was claimed to be evidence for their noise spectroscopy being more sensitive to surface physics. These three papers show that the NV center helps us to overcome many prevailing obstacles in magnetometry under high pressure, pre-senting a great breakthrough in high-pressure instrumentation. The NV-based methodologies for stress-tensor reconstruction, magnetic field imaging, magnetic field vector reconstruction, and critical fieldextraction provide powerful tools for material research under high pressure. V. OUTLOOK The above discussion has revealed the increasingly important role of NV technology in high-pressure research. It should be clear thatthere are much potential and advantages in implementing the NVcenter as a quantum sensor under extreme conditions. In the past few years, novel sensing techniques using the NV center have been devel- oped. Although most of them were demonstrated at ambient pressure FIG. 5. (a) The magnetization vs pressure plot of the iron bead sample. Different M–Pplots were obtained by increasing and releasing the pressure of the DAC, showing the expected hysteresis of the structural transition. (b) ODMRsplitting vs temperature plot of the MgB 2sample. The critical temperature Tcof the superconducting phase transition was extracted from where the sample ’s diamagnetic response disappeared. Reproduced with permission from Lesiket al. , Science 366, 1359 (2019). Copyright 2019 American Association for the Advancement of Science.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-9 Published under an exclusive license by AIP Publishingonly, we hereby highlight a few noteworthy examples that can as well be applied under high pressure with a careful experimental design. For the high-pressure community, one of the experimental challenges is to collect fluorescence emitted by NV centers inside aDAC for ODMR measurement, as a low numerical aperture and a long working distance must be ensured in the optics. In 2015, Bourgeois et al.77introduced the photoelectric detection of mag- netic resonance (PDMR) that allows an electrical readout for the FIG. 6. (a) (Left) Photograph of the microcoil with sample on top of the anvil. (Right) Fluorescence image from the confocal scan showing the microcoil and NV centers. Theshape of the sample is traced by the gray line. The loca-tion of three particular diamond particles NV C,N V E, and NVFare marked. NV Cis near the center of the top surface, NV Eis near the edge, and NV Fis far away from the sample and serves as a control sensor (b) ODMRspectra of NV centers in NV Cat different temperatures. (c) Comparison between the ODMR method (red) and AC susceptibility method (black) in determining the transitiontemperature T c. (d) The change of the Zeeman splitting for NV centers in NV C,N V E, and NV Fas a function of temperature. (e) The variation of the local magnetic field vector of NV C,N V E, and NV F. The measurement was performed at 0.83 GPa. (f) The T–Pphase diagram of the BaFe 2(As 0:59P0:41)2sample measured by AC suscept- ibility and ODMR methods. (g) The H–Pphase diagram measured by the ODMR method. This result revealed thatthe NV center is a novel tool to probe H c1. Reproduced with permission from Yip et al. , Science 366, 1355 (2019). Copyright 2019 American Association for the Advancement of Science.70Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-10 Published under an exclusive license by AIP PublishingNV spin states. The experimental setup is summarized in Fig. 8(a) . They used bulk diamond as a sample to demonstrate the PDMRmeasurement on an NV ensemble. Electrodes were deposited onthe diamond surface for direct detection of electrical signals, whilegreen laser and MW were applied for NV excitation and spin-state manipulation, respectively. The green laser could excite the elec- trons of NV centers from the ground states to the excited states,and even further to the conduction band, generating a photocur- rent across the electrodes. The photocurrent was measured as afunction of MW frequency, and it gave a spectrum strikinglysimilar to that obtained from a simultaneous ODMR measurement.The similarity was attributed to the very same spin-state-dependent decay of the NV center. The PDMR method was shown to have an advantage over the conventional ODMR method in terms of signal FIG. 7. (a) The full stress-tensor mapping across the cutlet surface. The two mapping corresponds to loadingstress (left) and mean lateral stress (right). (bottom) Thestress-tensor along the dotted line is presented at two dif- ferent pressures. The inset compares the spatial gradient in the loading stress σ zzat two different pressures. Noticeable pressure gradient is observed in higher pres-sure. (b) Measured dipole moment of the iron pellet as a function of applied pressure at room temperature, for both compression (red) and decompression (blue). Theaverage critical pressure is around 13 :6+0:6G P a . Reproduced with permission from Hsieh et al. , Science 366, 1349 (2019). Copyright 2019 American Association for the Advancement of Science. 72Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-11 Published under an exclusive license by AIP Publishingcontrast. For PDMR, the contrast increased monotonically with the laser power, unlike ODMR in which the contrast started to decrease if the laser power was too high [ Fig. 8(b) ]. This prompted an expec- tation that PDMR can be a good alternative to ODMR in high-pressure experiments, for in many cases, a highly pressurized envi-ronment reduces the signal contrast in ODMR measurements, andPDMR can provide an easy solution where simply raising the laser power will do. Later, research papers on enhancing the photoelec- tric detection 79and demonstrating coherence spin control80as well as a review about PDMR81were published. Recently, Siyushev et al.78refined the PDMR technique to perform Rabi oscillation on a single-defect level [ Fig. 8(c) ] and photoelectric imaging of indi- vidual NV centers. Furthermore, Gulka et al.82applied PDMR to detect the coupling between a single14N spin and the NV electron. All these works convince us that the PDMR method can potentiallyovercome the limitations faced when performing ODMR measure-ment. Hence, PDMR opens up a promising direction for the NV sensing research under high pressure. However, it is worth noting that the PDMR involves multi-photon excitation and ionization,which are based on the level structure of color center in the hostband. Since the level structure of the NV center under high pressure is not yet clear, the efficiency of this PDMR is an open question for further investigation.On the other hand, engineering the electrical feedthrough to the pressurized chamber in a DAC is another difficulty in high- pressure experiments using the conventional NV sensing protocol that requires both laser and MW. Recently, Paone et al. 83proposed an all-optical and MW-free sensing scheme for the NV center anddemonstrated it by probing the Meissner effect of a La 2/C0xSrxCuO 4 (LSCO) thin film with an NV implanted diamond membrane. First, without placing the sample, they measured the NV fluores- cence drop as a function of an applied magnetic field at ambientconditions. Under specified conditions, they found that a strongermagnetic field would quench more fluorescence. The data offluorescence drop vs magnetic field were interpolated to obtain a calibration curve [ Figs. 9(a) and 9(b)]. To benchmark their meth- odology, they carefully investigated the LSCO sample by themutual inductance method and superconducting quantum interfer-ence device (SQUID). Then, under a uniform applied field of4.2 mT and at a temperature of 4.2 K, they measured the NV fluo- rescence drop (relative to zero-field value) across the membrane with the presence of the LSCO sample. They subsequently con-verted the fluorescence drop to magnetic field strength using theircalibration curve [ Fig. 9(c) ]. In this way, they could map the spatial distribution of the Meissner screening from the sample. By fitting the mapping data with Brandt ’s model, 84they extracted the critical FIG. 8. (a) Schematics of the experimental setup of the PDMR measurement. (b) Signal contrast vs green laserpower of ODMR and PDMR. Reproduced from Bourgeois et al. , Nat. Commun. 6, 8577 (2015). Copyright 2015 Author(s), licensed under a Creative Commons Attribution(CC BY) license. (c) Contrast of the photoelectrically (6%)and optically (4%) detected Rabi oscillations. Reproduced with permission from Siyushev et al. , Science 363, 728 (2019). Copyright 2019 American Association for theAdvancement of Science.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-12 Published under an exclusive license by AIP Publishingcurrent density jc, as shown in Fig. 9(d) , and it was in good agree- ment with their SQUID data and the previously reported result.85 This proved the viability of their all-optical and MW-free method, offering a potential way out for the difficulty in inserting an MWantenna into the DAC in high-pressure experiments. Nonetheless, the sensitivity to an off-axis magnetic field may be reduced under certain environments. For instance, the ZFS Dis the dominantterm under strong uni-axial stress, states mixing could be avoided hence reducing the sensitivity to an off-axis magnetic field. Although the NV center is a reliable pressure sensor, cur- rently, there is a limitation on the pressure working range. For themeasurement protocol using ODMR spectroscopy, the maximum working pressure is /difference60 GPa. 55This is limited by the blue shift of the ZPL toward the green optical excitation wavelength (usually FIG. 9. (a) ODMR spectra of an NV ensemble in a diamond membrane under different magnetic fields alongz-direction ( B z) at 4.2 K. (b) Effect of increasing the Bzon the observed NV fluorescence. The Bz, applied by a per- manent magnet, is estimated from the correspondingODMR measurements at ambient conditions. (c)(leftpa-nel) Spatial variation in the photoluminescence (PL) drop at 4.2 K with (blue dots) and without (red dots) supercon- ducting sample. (rightpanel) Estimation of the magneticfields. A sharp peak is observed at the boundary y¼a. Due to the diamagnetic properties of the LSCO sample, the magnetic field flux is screened to the edge. (d) Fitting of the experimental data by using Brandt ’s model. The fitting functions reveal a critical current density j cof 1:4/C2108A/cm2. Reproduced with permission from Paone et al. , J. Appl. Phys. 129, 024306 (2021). Copyright 2019 AIP Publishing LLC.83Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 241101 (2021); doi: 10.1063/5.0052233 129, 241101-13 Published under an exclusive license by AIP Publishingaround 532 nm). Exciting the NV center with laser beam of a shorter wavelength can increase the value by a few factors, but the trade-off would be a poorer contrast due to the complex chargestate dynamics. In fact, with the interesting energy structure andthe charge state dynamics of the NV center, we expect that, in thenear future, different protocols or energy levels can be used for pressure sensing. For instance, the temperature dependence of the 1042-nm ZPL has been revealed. 86This can potentially be a distinct approach to high pressure measurements. Another challenge of implementing NV sensing under pres- sure is the electrical wiring. In most of the experiments, MW exci- tations are required, but the shielding from the gasket will be detrimental. Doherty et al.55and Hsieh et al.72directly embedded Pt wire/foil as the MW antenna. This approach greatly avoided theshielding effect. Moreover, Steele et al. 58deposited microchannels on their designer diamond anvils, while Ho et al.63and Yip et al.70 put a microcoil inside the pressure chamber to minimize eddy current. Interestingly, Lesik et al.71made a slit in the gasket with a coil outside the pressure chamber. The slit-gasket not only reducedthe eddy current but also worked as a converging lens for the MW.In general, metal gaskets are robust under pressure, but the shield- ing effect is detrimental to MW transmission. In contrast, non- metal gasket could eliminate shielding effects. The best configura-tion of gasket and MW antenna is still subject to open studies. In summary, we have presented the NV center in diamond as a unique quantum system, which allows it to be a versatile sensor withstanding a large pressure range. We have also showed variousworks on pressure calibration and sensing with NV centers in bulkdiamond and in small diamond particles. These techniques arefound to be very useful in high pressure experiments, for example, studying detailed pressure distribution inside the medium of DAC and phase transitions of superconducting materials. 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4.0000099.pdf

Real-time interfacial electron dynamics revealed through temporal correlations in x-ray photoelectron spectroscopy Cite as: Struct. Dyn. 8, 044301 (2021); doi: 10.1063/4.0000099 Submitted: 6 March 2021 .Accepted: 21 June 2021 . Published Online: 8 July 2021 Felix Brausse,1 Mario Borgwardt,1 Johannes Mahl,1,2 Matthew Fraund,1 Friedrich Roth,3 Monika Blum,1,4 Wolfgang Eberhardt,5 and Oliver Gessner1,a) AFFILIATIONS 1Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 2Physics Department, Universit €at Hamburg, 22607 Hamburg, Germany 3Institute of Experimental Physics, TU Bergakademie Freiberg, 09599 Freiberg, Germany 4Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 5Center for Free-Electron Laser Science DESY, 22607 Hamburg, Germany a)Author to whom correspondence should be addressed: ogessner@lbl.gov ABSTRACT We present a novel technique to monitor dynamics in interfacial systems through temporal correlations in x-ray photoelectron spectroscopy (XPS) signals. To date, the vast majority of time-resolved x-ray spectroscopy techniques rely on pump–probe schemes, in which the sample is excited out of equilibrium by a pump pulse, and the subsequent dynamics are monitored by probe pulses arriving at a series of well-defined delays relative to the excitation. By definition, this approach is restricted to processes that can either directly or indirectly be initiatedby light. It cannot access spontaneous dynamics or the microscopic fluctuations of ensembles in chemical or thermal equilibrium. Enabling this capability requires measurements to be performed in real (laboratory) time with high temporal resolution and, ultimately, without the need for a well-defined trigger event. The time-correlation XPS technique presented here is a first step toward this goal. The correlation-based technique is implemented by extending an existing optical-laser pump/multiple x-ray probe setup by the capability to record thekinetic energy and absolute time of arrival of every detected photoelectron. The method is benchmarked by monitoring energy-dependent,periodic signal modulations in a prototypical time-resolved XPS experiment on photoinduced surface-photovoltage dynamics in silicon, using both conventional pump–probe data acquisition, and the new technique based on laboratory time. The two measurements lead to the same result. The findings provide a critical milestone toward the overarching goal of studying equilibrium dynamics at surfaces and interfacesthrough time correlation-based XPS measurements. VC2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http:// creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/4.0000099 I. INTRODUCTION The development of time-resolved spectroscopy1has been driven by the goal to study elementary processes in atoms and molecules ontheir natural timescales. Most studies in this field are enabled through the use of the so-called pump–probe schemes. A pump laser pulse excites the system out of equilibrium, while a probe pulse, with vari-able and precisely adjustable delay, interrogates the sample evolution over time. The pump–probe scheme is an extremely powerful tech- nique to resolve dynamics in systems ranging from isolated atoms toextended biological complexes. However, it inherently requires thedynamics to be initiated either directly or indirectly by an optical exci- tation. While many reactions can be triggered by light, the vast major-ity of chemical transformations occurring, for example, in living organisms or during chemical synthesis proceed through thermal acti- vation of molecules in their electronic ground states. The correlation spectroscopy (CS) approach provides a possible pathway to reveal such chemical dynamics. 2,3While time-averaged properties of a system in chemical or thermal equilibrium, such as, for example, the density of molecules on a sample surface, are constant in time, they fluctuate around their mean values on microscopic length Struct. Dyn. 8, 044301 (2021); doi: 10.1063/4.0000099 8, 044301-1 VCAuthor(s) 2021Structural Dynamics ARTICLE scitation.org/journal/sdyscales and on timescales that are defined by reaction rates and molecu- lar mobilities. Reaction rates themselves are time-invariant properties and can be determined through temporal correlations of spectroscopic signals,4which are measured as a function of absolute laboratory time. A particularly prominent example for this approach is fluorescence correlation spectroscopy (FCS),5which exploits temporal correlations in laser-induced fluorescence signals. This technique has been used to measure diffusion constants of the thermal motion of molecules and reaction rates in chemical equilibrium.2,3 To the best of our knowledge, only one example has been reported for translating FCS into the x-ray domain. In a proof-of-principle X-FCS experiment, Wang et al. used hard x-ray FCS to monitor diffu- sion and sedimentation dynamics of colloidal particles suspended in water.6In the x-ray scattering domain, x-ray photon correlation spec- troscopy (XPCS)7is routinely employed to monitor spontaneous dynamics. There are, however, important distinctions. While FCSexploits number-density fluctuations in a small sample volume, which can be probed by incoherent spectroscopic signals, XPCS is based on the formation of speckles by interference of x-rays that propagate through the sample on different paths but reach the detector at the same q-vector. Fluctuations in the speckle pattern enable XPCS to probe random fluctuations in the distribution of scatterers in a mate- rial. By definition, the speckle phenomenon requires coherent x-ray flux, which does not apply to FCS or the technique presented here. Translating CS to a larger variety of inner-shell spectroscopy techniques, such as x-ray photoelectron spectroscopy (XPS), offers a number of opportunities to combine the unique strengths of element specificity and chemical sensitivity of x-ray transitions with the dynamic insight provided by CS. For example, the excellent energy resolution of XPS can be exploited to distinguish molecular species that are chemically very similar, such as molecules chemisorbed on different surface sites or molecules that are part of an ensemble under- going an isomerization reaction, but sufficiently different to affect the local chemical environment of a particular element, leading to a dis- tinct shift in core-level binding energies. This sensitivity is routinely exploited in electron spectroscopy for chemical analysis (ESCA). 8,9 Another opportunity provided by the detection of photoelectrons lies in the extreme surface sensitivity of XPS. In combination with ambient-pressure capabilities, one may envision, for example, the microscopic real-time study of interfacial equilibrium dynamics between gases or liquids and a solid bulk catalyst by time-correlation XPS (TCXPS). Here, we present an implementation of TCXPS based on the event-by-event recording of laboratory-time photoelectron data that include the absolute time of arrival and the kinetic energy of every sin- gle photoelectron, from which temporal and spectral correlations are calculated after the measurement. The test experiment monitors pho- toinduced surface photovoltage (SPV) dynamics in p-doped silicon10 using a laser-pump/x-ray-multiple-probe setup that is able to measure both conventional pump–probe traces and laboratory-time XPS event streams. The test experiment focuses on photoinduced, non- equilibrium dynamics in order to provide a rigorous test of the correlation-based analysis. The recording of pump–probe and correla- tion data with the same setup back-to-back under virtually identical conditions provides the most stringent, quantitative test of the correla- tion approach. To demonstrate the equivalence between both meth- ods, relaxation dynamics derived from the correlation data using basicmodel assumptions are compared to the dynamics observed in pump–probe mode. We note that the use of a reproducible, periodic excitation by the pump laser simplifies the correlation analysis and is only a first step toward future TCXPS studies of spontaneous chemicaldynamics, the feasibility of which still needs to be demonstrated. Yet,the presented results establish important milestones toward this long-term goal under well-controlled conditions that are more challengingto achieve for spontaneous processes. II. EXPERIMENT The time-resolved x-ray photoelectron spectroscopy (tr-XPS) setup used in the experiment has previously been described in detail inRefs. 11and12. Briefly, photoelectrons generated on the sample sur- face are detected in a hemispherical electron analyzer that translates the photoelectron kinetic energy into a specific impact position on thedetector. A fast delay-line detector is used to register the time and posi-tion of each electron impact. In the present experiment, a soft x-raybeam ( /C22hx¼700.8 eV) from beamline 11.0.2 of the Advanced Light Source (ALS) is spatially overlapped with an optical laser beam (wave- length k¼532 nm, fluence 0.49 mJ/cm 2)o na p-doped (100) silicon wafer (resistivity q/C253.5Xcm), as sketched in Fig. 1(a) . The optical pump pulses induce above-bandgap excitations in the Si substrate,leading to transient variations in the electronic band bending towardthe surface, that is, a time-dependent SPV, which is reflected in tran-sient binding energy shifts of the Si 2 pphotolines. 10The/C25127 kHz repetition rate of the optical laser is chosen such that one laser pulse is generated for every 24 x-ray pulses [ Fig. 1(b) ], with the ALS operating in two-bunch mode at an x-ray pulse spacing of 328 ns. Data are acquired in three distinct modes, the pump–probe mode and two TCXPS modes. In the pump–probe mode, the optical laserand the detector are synchronized to the ALS master clock, which we refer to as “fully synchronized.” In the TCXPS modes, the detector hardware is not synchronized to either the optical laser or the synchro-tron pulses. Therefore, measurements in the traditional pump–probemode and in the TCXPS modes have to be taken separately. In oneTCXPS mode, the optical laser and the ALS master clock are syn-chronized to each other to establish a constant phase relationbetween the two. The relative pulse timing is identical to the pump– probe experiment as shown in Fig. 1(b) . Therefore, this operating mode is referred to as a “phase-stabilized” TCXPS. In the other case,the phase lock between the laser and the synchrotron is lifted, caus-ing the laser pulse timing to drift relative to the x-ray pulses. In thiscase, all possible pump–probe delays are sampled over the durationof the experiment, not only the 24 points sketched in Fig. 1(b) . This operating mode is referred to as “free-running” TCXPS. In the fully synchronized (pump–probe) and the phase-stabilized TCXPS modes,the delay between the optical laser pulses and the x-ray pulses is fixedsuch that the laser pump pulse arrives /C24800 ps before the next x-ray pulse. In all experiments, the hemispherical analyzer is operated with constant voltages at a photoelectron kinetic energy of E kin¼600 eV and a pass energy of Epass¼30 eV, focusing on the Si 2 pphotolines with/C2499.6 eV binding energy. Ahead of every data acquisition run, which typically spans 900 s, the silicon wafer is irradiated with a laserfluence of /C2430 mJ/cm 2for 120 s to remove the native SiO 2layer. Additional XPS reference measurements with the optical laser turned off are referred to as steady-state experiments in the following.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 044301 (2021); doi: 10.1063/4.0000099 8, 044301-2 VCAuthor(s) 2021Conventional pump–probe spectra are generated by collecting histograms of the detected photoelectrons inside the detector hard- ware, yielding two-dimensional data sets of detector position (i.e., kinetic energies) vs time-of-arrival relative to the laser trigger, from which the pump–probe delay s[Fig. 1(b) ] can be calculated. With the new TCXPS method implemented here, the time of arrival tiand impact position/kinetic energy eiof each ith detected photoelectron is stored separately, and tiis measured in the laboratory frame; that is, itis referenced to the beginning of the data acquisition period, not the laser trigger. The resulting datasets consist of event tables with Nlines forNdetected photoelectrons. III. DATA ANALYSIS The nature of the single-event data is illustrated in Fig. 2(a) , where ( ti,ei) pairs are shown as individual points. This format repre- sents the maximum information content of the data, which are further FIG. 1. (a) Sketch of the experiment. (b) Temporal structure of the optical laser (tall vertical bars) and x-ray pulse trains (red bars). The dotted green trac es schematically illus- trate the periodic sample response to the laser excitation. Orange and blue horizontal arrows at the bottom of the panel indicate the two different ref erence systems for measur- ing time. In the pump–probe experiment (orange), time is referenced to the pump laser pulses with a maximum timescale corresponding to the spacing bet ween two laser pulses. The correlation experiments (blue) operate in the laboratory timeframe with an arbitrary chosen time zero and a maximum timescale only limit ed by the maximum sam- pling range of the detector electronics. FIG. 2. Illustration of the single-event TCXPS data structure and derived quantities. (a) Recorded pairs of arrival times tiand detector positions ei, shown as individual points. (b) Integration over a range of detector positions yields laboratory time-dependent electron signals, which form the basis of the temporal correlation analysis. (c) Integration over all arrival times yields the conventional, time-averaged photoelectron spectrum. Note that the data in panel (a) only show the first 200 out of /C24107events contained in the spectrum in panel (c).Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 044301 (2021); doi: 10.1063/4.0000099 8, 044301-3 VCAuthor(s) 2021processed to access various observables and their temporal correla- tions. For instance, integration of Fig. 2(a) over a range of detector positions yields the laboratory time-dependent total electron signal within the corresponding energy range [ Fig. 2(b) ]. Similarly, integra- tion over all arrival times yields the conventional, time-averaged pho-toelectron spectrum [ Fig. 2(c) ]. The temporal autocorrelation of a time- and energy-dependent signal f(e,t) may be illustrated as the overlap of the signal with a time- delayed copy f(e, t–Dt)[Fig. 2(b) ]. In general, this overlap may be eval- uated for any two kinetic energies, e 1ande2, resulting in a correlation function, R(e1,e2,Dt), which depends on three variables: the arrival time difference (“lag time”) Dt, and the energies e1and e2.I fe1¼e2, we refer to Ras an autocorrelation function and otherwise as a cross correlation between signals at energies e1and e2. Here, f(e,t)i sm e a - sured on a discrete, equidistant grid in both kinetic energy (throughthe detector positions) and laboratory time (through the temporal res-olution of the detector, here defined as w). Positions along the time axis can be described by integer multiples of the detector resolution: t m¼[0w,1w,2w,… mw,(mþ1)w, …]. Thus, the lag time can be written as Dt¼mwandf(e,t) can be reduced to fm(e). If the intervals span from m¼0 to some maximum m¼M, the correlation is given by13 Re 1;e2;Dt¼kw ðÞ ¼1 M/C0k ðÞXM/C0k m¼0fme1ðÞfmþke2ðÞ fe1ðÞ/C10/C11fe2ðÞ/C10/C11 ; (1) where hf(e)iis the average of fm(e) over all m.F o rc o m p u t a t i o n a le f fi - ciency, correlation functions are calculated from the photoelectronevent data by using a sparse correlation algorithm as described in the supplementary material. 18 IV. RESULTS AND DISCUSSION A. Steady-state correlation experiment Before discussing the TCXPS results for laser-induced dynamics, it is instructive to analyze data recorded in steady-state mode, that is,with the optical laser off. Summation over all electron arrival timesrecovers the conventional Si 2 pphotoelectron spectrum [ Fig. 3(a) ]. Note that the two spin–orbit components Si 2 p 1/2and Si 2 p3/2are not resolved but indicated by a notable asymmetry of the photoelectronpeak. Integrating the signal over the entire detector and calculating the temporal autocorrelation of this energy-integrated signal reveals the periodic trace shown in Fig. 3(b) . The period of this signal corresponds to the pulse-to-pulse spacing T ALS¼328 ns of the ALS in two-bunch operating mode. The periodicity of the correlation function spansmany orders of magnitude in lag time, as exemplified by the correla-tion peaks corresponding to 100 to 10 7ALS bunch periods, shown as blue circles in Figs. 3(c)–3(h) . The autocorrelation trace shown in Figs. 3(b)–3(h) can be quali- tatively understood by recalling that the x-ray pulse duration of /C2470 ps is much shorter than the pulse-to-pulse spacing TALS. Therefore, photoelectron emission occurs in short bursts during theinteraction of the sample with the x-ray pulses, leading to a time-dependent photoemission signal that repeats every integer period pat a lag time of Dt¼p/C1T ALS.U pt o /C24105periods, the shapes of the corre- lation peaks are virtually indistinguishable from each other, except for the very first one at a lag time of 0 [ Fig. 3(b) ]. This first peak contains information about the temporal distribution of the photoelectronarrival times from a single x-ray pulse on the detector, leading to some additional structure within the peak. As this information is not rele-vant for the sample response on /C24ns and longer timescales, we exclude the first correlation peak from the analysis. After /C2410 6periods, the correlation peak shape begins to broaden and an asymmetric pedestal starts to emerge [ Fig. 3(g) ]. A more quantitative analysis of the autocorrelation function is provided by fits to a Gaussian function, GDt;l;r ðÞ ¼1 rffiffiffiffi 2pp expð/C0ðDt/C0lÞ2 2r2Þ, shown as solid lines in Figs. 3(c)–3(h) . For the first four peaks [ Figs. 3(c)–3(f) ], the full width at half-maximum (FWHM) of 32 ns agrees within the precision of the fit ( /C252%), and no indications for peak shape changes are observed within the experimental uncer-tainty. For the peaks in Figs. 3(g) and3(h), the FWHM increases to 36 and 48 ns, respectively, and noticeable deviations from a Gaussian peak shape become apparent. Still, the observed dephasing, that is, thepedestal extending to short lag times in Fig. 3(h) ,i so nt h eo r d e r /C2450 ns at a lag time of 3 s, which corresponds to a relative deviation smaller than 10 /C07. As all lag times correspond to time differences that are recorded at arbitrary absolute laboratory times, correlations in Fig. 3 for long lag times are sensitive to time-averaged, relative changes between the periodicity of the synchrotron pulses and the stability ofthe detector clock. The observed relative dephasing on the order of/C2010 /C07is likely dominated by the detector timing, as typical quartz oscillator drifts are on this order of magnitude. The results suggest that the present implementation of the laboratory-time based data acquisi- tion should enable correlation measurements spanning at least sevenorders of magnitude in lag time. In general, the time resolution of a correlation measurement is determined by three limiting factors: the repetition rate of the x-raysource, the temporal resolution of the spectrometer, and the temporalresolution of the detector. In the present experiment, the time resolu-tion is determined by the 328 ns x-ray pulse spacing. Still, we note that the autocorrelation peak width of 32 ns is more than two orders of magnitude larger than the theoretical lower limit of 100 ps one mayexpect from the autocorrelation of 70 ps long x-ray pulses. The peakbroadening is caused by the finite photoelectron time-of-flight (TOF)spread within the hemispherical analyzer, as discussed in detail inRef.11. In this work, it has been demonstrated that the TOF spread can be reduced to /H113511 ns by a different choice of electron kinetic energies and/or pass energies. In principle, this reduced TOF spread issufficient to exploit the minimum x-ray pulse spacing of 2 ns of theALS in multi-bunch operating mode for correlation experiments.With the current setup, however, the dead time of the delay-line detec- tor (/C2410 ns) would define the overall achievable temporal resolution. In practice, depending on the experimental conditions and samplecharacteristics, a compromise may have to be found between TOFspread and kinetic energy resolution. As the TOF-spread and, there-fore, the correlation peak width is much larger than the x-ray pulseduration, every peak can be integrated over its full width in time with- out losing information relevant to the time-dependent sample response. Thus, in the following, for determining sample dynamicsthrough the correlation analysis, signals are integrated over the lagtime intervals indicated by the vertical dashed lines in Fig. 3(b) . B. Pump–probe reference measurement To quantify the SPV dynamics of the silicon sample, a reference measurement is carried out in the conventional pump–probe dataStructural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 044301 (2021); doi: 10.1063/4.0000099 8, 044301-4 VCAuthor(s) 2021acquisition mode. A typical time-resolved photoelectron spectrum is shown in Fig. 4(a) . Upon sample excitation by the pump pulse, the entire spectrum shifts to smaller binding energies by approximately120 meV. This SPV effect 14is caused by a photoinduced reduction of the intrinsic downward band-bending in p-doped silicon toward the surface.10–12,15When a laser pulse induces superbandgap transitionsin the semiconductor, mobile electron–hole pairs are created that can counteract the dipole fields underlying interfacial band bending, lead-ing to a reduction of the downward band bending (“band flattening”)and a corresponding transient shift of the core levels to smaller bind-ing energies. Through charge-carrier recombination, the SPV gradu-ally decays on a microsecond timescale. FIG. 3. Steady-state TCXPS experiment without optical excitation. (a) Si 2 pphotoelectron spectrum of a p-doped Si (100) substrate. The two spin–orbit components are not resolved but indicated by the peak asymmetry. (b) Temporal autocorrelation function of the signal in (a), integrated over the entire kinetic energy r ange. Vertical dashed lines indicate the integration regions used to determine the total area under the correlation peaks (see the text for details). (c)–(h) Details of the TCXPS peaks (circles) and fits to a Gaussian function (solid line) for lag times Dt¼p/C1328 ns with p¼102(c), p¼103(d), p¼104(e), p¼105(f), p¼106(g), and p¼107(h). Double-headed arrows indicate the full width at half-maximum (FWHM) of the Gaussian fit results.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 044301 (2021); doi: 10.1063/4.0000099 8, 044301-5 VCAuthor(s) 2021InFig. 4(b) , a photoelectron spectrum recorded before the arrival of the pump pulse ( s¼/C00.3ls, blue) is compared to one recorded at the smallest positive pump–probe delay of þ800 ps ( s¼0ls, orange), where the SPV effect is most pronounced. The rigid shift in binding energies leads to strong intensity modulations in spectral range A, which contains the low binding energy wing of the peak, in particular the region of greatest relative change in intensity. In the following,the intensity modulations in this region, shown as relative change in Fig. 4(c) , are used to demonstrate the TCXPS analysis and to compare it with the results of a conventional pump–probe analysis. Note that the traces in Fig. 4(c) are normalized to the signal average over all pump–probe delays, which is most suitable for comparison with the TCXPS results. C. Reconstruction of dynamic trends from TCXPS data For the vast majority of dynamic phenomena, the actual temporal evolution of a system is of primary interest, while auto- or cross-correlations are a means to make inferences about the temporal evolution where it cannot be directly measured. Thus, it is of particular interest to assess, to which degree dynamic trends that are readily available from pump–probe experiments can be reconstructed from aTCXPS measurement. A universal, deterministic transformation from temporal correlations to real-time dynamic trends does not exist. However, basic assumptions regarding the dynamics can often provide sufficient boundary conditions to enable a unique reconstruction of dynamic trends within these model assumptions. Such an approach is illustrated in Fig. 5 . The symbols in Fig. 5(a) are autocorrelation data from region A [Fig. 4(a) ], recorded in the phase-stabilized (blue circles) and free- running (orange diamonds) TCXPS modes. Note that the data showninFig. 5(a) correspond to an average over 1000 laser cycles spanning a total lag time range of /C248 ms. The data were recorded over a data acquisition time of 1 h. Relative standard deviations of the autocorrela-tion function for lag times spanning a single laser cycle are typically on the order of /C246/C210 /C02, translating into a relative standard deviation of/C2410/C03for the cycle-averaged data. Details of the data analysis as well as autocorrelation traces before cycle-averaging are discussed in Sec. II of the supplementary material.18Real-time dynamics are recon- structed based on the assumption that the SPV dynamics underlyingthe measured autocorrelation traces may be described by a bi-exponential decay model, which is frequently the case. 12Within this boundary condition, the autocorrelation traces in Fig. 5(a) can be modeled by five independent parameters: two decay time constants si, two amplitudes Ai, and a constant offset C. A fit procedure for the measured autocorrelation traces is defined by applying a numerical autocorrelation to the bi-exponential SPV decay model function, and iteratively varying the five model parameters toward best agreement ina non-linear, least squares fit approach. Figure 5(a) shows the result of these fits as blue and orange lines compared to the autocorrelation traces for region A measured in phase-stabilized and free-runningmode, respectively. We note that the modulation depth of the free-running data has been scaled by a factor f 2,w i t h f¼0.84. This scaling compensates for an /C2416% difference in SPV amplitudes of data recorded on different days. The variation is likely the result of slightlydifferent sample surface and laser excitation conditions. Sample aging and inhomogeneity effects, such as a varying thickness of the native silicon oxide layer that grows over time on the silicon surface, can sig-nificantly affect the SPV amplitude. Slight day-to-day variations in thelaser intensity and focusing conditions may also be a contributing factor. FIG. 4. Pump–probe reference measure- ment. (a) Time-resolved Si 2 pphotoelec- tron spectrum of a p-doped silicon (100) sample, excited with 532 nm laser pulses.Dashed lines indicate the energy integra-tion region A that is used to compare the pump–probe with the correlation-XPS analysis. (b) Photoelectron spectrarecorded before (blue) and 800 ps after(orange) optical excitation. (c) Time- dependent signals obtained from integrat- ing the time-resolved photoelectron spec-tra in (a) over energy region A. Intensitiesare normalized to signal averages over all pump–probe delays.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 044301 (2021); doi: 10.1063/4.0000099 8, 044301-6 VCAuthor(s) 2021For the phase-stabilized TCXPS measurement, the fit of the auto- correlation trace in Fig. 5(a) leads to decay constants of s1¼1.360.4 ands2¼1362ls for the fast and slow SPV relaxation components, respectively. For the measurement with the free-running TCXPSmode, these time constants are s 1¼2.260.8 and s2¼1163ls. The corresponding bi-exponential model functions are plotted as solidlines in Fig. 5(b) along with the results of the pump–probe measure- ment (red squares). As noted above, the results for the free-running data were scaled by f¼0.84. Very good agreement is found between the dynamic trends measured directly by the conventional pump–p-robe approach and reconstructed from the correlation measurementsin both phase-stabilized and free-running TCXPS modes. A direct fitof the pump–probe data with a bi-exponential model function leads todecay time constants of s 1¼1.260.2 and s2¼13.561.2ls. These timescales agree well with those reconstructed from the TCXPS datawithin the given uncertainty ranges. The results demonstrate that, based on an educated guess, quan- tities like decay constants can be quantitatively recovered from tempo-ral correlations. While some information, such as the relative phases between the fast and the slow components, is lost in the correlation measurement, the physics that underlie the processes under investiga-tion often provide sufficient boundary conditions to compensate forthis loss. Alternative comparisons between the pump–probe and theTCXPS results that concentrate on auto- and cross-correlations, ratherthan on the time-dependent trend itself, are described in Sec. II of thesupplementary material. 18 The 328 ns x-ray pulse spacing in the experiments described herein limits the temporal resolution of the correlation approach.Efforts are under way to employ the multi-bunch operating mode of the ALS with a 2 ns pulse spacing to significantly improve the tempo- ral resolution. We note that, principally, the method should be extend-able into the femtosecond regime using x-ray free electron lasers(XFELs) in combination with a recently developed correlation analysistechnique that exploits ultrafast split-pulse schemes. 13,16While fast detectors can enable /C24ls time-resolved XFEL measurements,17to access even shorter timescales, signals from different x-ray pulses arenot separated by their detection times, but the lag time is defined by awell-controlled double-pulse spacing and the sum of signals from both pulses is detected. As shown by Gutt et al., the two schemes can pro- vide equivalent information. 13The practical implementation of split- pulse XPCS at XFELs with improved temporal resolution has been demonstrated by Roy, Turner, and collaborators.16We expect that a similar transfer of TCXPS to XFEL conditions should be possible, but also note that it will require careful shot-by-shot monitoring of inten- sity fluctuations in each of the x-ray pulses. V. CONCLUSIONS A method is presented for extracting real-time dynamics in XPS experiments through temporal correlations in the detected photoelec- tron signals. The technique is enabled by the capability to separately record the kinetic energy and absolute arrival time of every single detected photoelectron. The successful implementation of the tech- nique is demonstrated in a benchmark experiment on laser-induced transient SPV effects in a Si sample. Excellent agreement is found between the results of the TCXPS and pump–probe approaches. The correlation analysis involves contributions spanning up to seven orders of magnitude in lag-time, providing opportunities for future studies of processes involving a vast range of timescales, simulta- neously monitored in a single TCXPS experiment. The demonstration experiment uses a periodic excitation by the pump laser, which enables direct comparison between TCXPS and pump–probe results and sim- plifies the correlation analysis. Future efforts will focus on developing TCXPS further into a tool for monitoring spontaneous dynamics as previously demonstrated for related techniques, such as FCS and XPCS. We envision a wide range of applications, such as the study of diffusion dynamics of molecules on liquid and solid surfaces, adsorp- tion/desorption kinetics at gas–solid interfaces in and out of equilib- rium, spontaneous chemical dynamics on catalyst surfaces, and spontaneous phase transitions. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical FIG. 5. Reconstruction of SPV dynamics from the TCXPS experiment. (a) Cycle-averaged autocorrelation data (see the supplementary material for details18) for region A (circles and diamonds) compared to least squares fits that derive temporal autocorrelations from bi-exponential model functions of the SPV dynamics (solid lines). Blue and orange colors indicate results for data recorded under phase-stabilized and free-running laser/x-ray conditions, respectively. (b) Time-depend ent intensity modulation in region A as determined by the pump–probe measurement (red squares) and as determined from the fits in panel (a) (blue and orange lines).Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 044301 (2021); doi: 10.1063/4.0000099 8, 044301-7 VCAuthor(s) 2021Sciences, Geosciences and Biosciences Division, and by the Laboratory Directed Research and Development Program ofLawrence Berkeley National Laboratory under U.S. Department ofEnergy Contract No. DE-AC02–05CH11231. The research used resources of the Advanced Light Source, a DOE Office of Science User Facility under Contract No. DE-AC02–05CH11231. Theresearch was partially funded through the program “Structure ofMatter” of the Helmholtz Society of Research Centers in Germany.M.Bo. acknowledges support by the Alexander von Humboldt Foundation through the Feodor Lynen Research Fellowship. We thank Andreas Oelsner and Martin Ellguth from Surface Conceptfor their technical support with the detector hardware and software. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. H. Zewail, J. Phys. Chem. A 104, 5660 (2000). 2D. Magde, E. Elson, and W. W. Webb, Phys. Rev. Lett. 29, 705 (1972). 3E. L. Elson and D. Magde, Biopolymers 13, 1 (1974). 4B. Saleh, Photoelectron Statistics (Springer Berlin Heidelberg, Berlin, Heidelberg, 1978). 5O. Krichevsky and G. Bonnet, Rep. Prog. Phys. 65, 251 (2002). 6J. Wang, A. K. Sood, P. V. Satyam, Y. Feng, X. Wu, Z. Cai, W. Yun, and S. K. Sinha, Phys. Rev. Lett. 80, 1110 (1998). 7O. G. Shpyrko, J. Synchrotron Radiat. 21, 1057 (2014). 8K. Siegbahn, Rev. Mod. Phys. 54, 709 (1982).9H. Siegbahn, J. Phys. Chem. 89, 897 (1985). 10W. Widdra, D. Br €ocker, T. Gießel, I. V. Hertel, W. Kr €uger, A. Liero, F. Noack, V. Petrov, D. Pop, P. M. Schmidt, R. Weber, I. Will, and B. Winter, Surf. Sci. 543, 87 (2003). 11A. Shavorskiy, S. Neppl, D. S. Slaughter, J. P. Cryan, K. R. Siefermann, F. Weise, M.-F. Lin, C. Bacellar, M. P. Ziemkiewicz, I. Zegkinoglou, M. W. Fraund, C. Khurmi, M. P. Hertlein, T. W. Wright, N. Huse, R. W. Schoenlein, T. Tyliszczak, G. Coslovich, J. Robinson, R. A. Kaindl, B. S. Rude, A. €Olsner, S. M€ahl, H. Bluhm, and O. Gessner, Rev. Sci. Instrum. 85, 093102 (2014). 12S. Neppl and O. Gessner, J. Electron Spectrosc. Relat. Phenom. 200,6 4 (2015). 13C. Gutt, L.-M. Stadler, A. Duri, T. Autenrieth, O. Leupold, Y. Chushkin, and G.Gr€ubel, Opt. Express 17, 55 (2009). 14L. Kronik, Surf. Sci. Rep. 37, 1 (1999). 15B. F. Spencer, D. M. Graham, S. J. O. Hardman, E. A. Seddon, M. J. Cliffe, K. L. Syres, A. G. Thomas, S. K. Stubbs, F. Sirotti, M. G. Silly, P. F. Kirkham, A. R. Kumarasinghe, G. J. Hirst, A. J. Moss, S. F. Hill, D. A. Shaw, S. Chattopadhyay,and W. R. Flavell, Phys. Rev. B 88, 195301 (2013). 16M. H. Seaberg, B. Holladay, J. C. T. Lee, M. Sikorski, A. H. Reid, S. A. Montoya, G. L. Dakovski, J. D. Koralek, G. Coslovich, S. Moeller, W. F. Schlotter, R. Streubel, S. D. Kevan, P. Fischer, E. E. Fullerton, J. L. Turner, F.-J. Decker, S. K.Sinha, S. Roy, and J. J. Turner, Phys. Rev. Lett. 119, 067403 (2017). 17F. Lehmk €uhler, F. Dallari, A. Jain, M. Sikorski, J. M €oller, L. Frenzel, I. Lokteva, G. Mills, M. Walther, H. Sinn, F. Schulz, M. Dartsch, V. Markmann, R. Bean, Y. Kim, P. Vagovic, A. Madsen, A. P. Mancuso, and G. Gr €ubel, Proc. Natl. Acad. Sci. U. S. A. 117, 24110 (2020). 18See supplementary material at https://www.scitation.org/doi/suppl/10.1063/ 4.0000099 for a detailed description of the sparse correlation algorithm and additional comparisons between the pump–probe and time-correlation XPS results, which also include temporal cross-correlations between signals from different spectral regions.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 044301 (2021); doi: 10.1063/4.0000099 8, 044301-8 VCAuthor(s) 2021

5.0058020.pdf

AIP Conference Proceedings 2368 , 030003 (2021); https://doi.org/10.1063/5.0058020 2368 , 030003 © 2021 Author(s).Crystal structure and magnetic properties of lithium orthophosphate LiMn0.9Cu0.1PO4 Cite as: AIP Conference Proceedings 2368 , 030003 (2021); https://doi.org/10.1063/5.0058020 Published Online: 29 June 2021 Z. Mohamed , and C. D. Ling ARTICLES YOU MAY BE INTERESTED IN Synthesis and optical properties of borate glass co-doped erbium/vanadium AIP Conference Proceedings 2368 , 030009 (2021); https://doi.org/10.1063/5.0057938 Structural and optical properties of tellurium-based double perovskite Sr 2ZnTeO 6 AIP Conference Proceedings 2368 , 030001 (2021); https://doi.org/10.1063/5.0058018 Detection of low PPM of volatile organic compounds using nanomaterial functionalized reduced graphene oxide sensor AIP Conference Proceedings 2368 , 020004 (2021); https://doi.org/10.1063/5.0057775 Crystal Structure and Magnetic properties of Lithium Orthophosphate LiMn 0.9Cu 0.1PO 4 Z. Mohamed1, a) and C. D. Ling2, b) 1Faculty of Applied Sciences, Universiti Teknologi MARA, UiTM, Shah Alam 40450, Selangor, Malaysia 2School of Chemistry, The University of Sydney, NSW 2006, Australia a) Corresponding author: zakiah626@uitm.edu.my b) chris.ling@sydney.edu.au Abstract. Lithium orthophosphates LiMn 0.9Cu0.1PO 4 were successfully synthesised usi ng a solid-state reaction method and characterised by X-ray powder diffraction and magnetic suscep tibility measurement. Structure refinements against powder diffraction data were conducted in the orthorhombic space group no. 62, that is, Pnma . Magnetic susceptibility measurements showed that LiMn 0.9Cu0.1PO 4 undergo long-range antiferromagnetic ord ering and antiferromagnetic, AFM at low temperature. Cu2+ doped in Mn sites resulted in the Néel temperature, T N = 34 K. INTRODUCTION LiMPO 4, as a cathode battery, has been reported for the first time by Goodenough et al. in 1997 [1]. Olivine family LiMPO 4 has attracted considerable attention for the next generation of lithium-ion batteries because of its low cost, safety, low toxicity and environmental friendliness [2]. Numerous investigations in this structure were conducted due to its capability to deliver good performance as a cathode battery and its use in a wide range of applications, such as battery for cars. This olivine has high theo retical specific capacity of around 170 mAhG í, and LiMn2+PO 4/Mn3+PO 4, LiFe2+PO 4/Fe3+PO 4, LiCo2+PO 4/Co3+PO 4 and LiNiPO 4 UHVSHFWLYHO\DFKLHYHGUHGR[SRWHQWLDOVDURXQGíí DQGí9YHUVXV/L/L+) [3, 4]. The low-temperature structural and magnetic properties of many lithium-ion battery cathode materials remain unknown despite the substantial ongoing research effort to understand their fundamental properties. The lithium-ion batteries ha ve received considerable attention due to their lithium extraction–insertion reactions, which are relatively more efficient compared with those of available cathode materials LiCoO 2 and LiMn 2O4 >@7KHROLYLQH/L0Q32 4, which adopted orthorhombic oliv ine structures in the space group Pnma , uses an olivine 0J 2SiO 4) type structure [7]. This structure can be described as a distorted hexagonal close packing of anion oxygen atoms with OLWKLXPDWRPVORFDWHGLQRFWDKHGUDOVLWHV 4a SRVLWLRQDQGSKRVSKRUXVLQWHWUDKHGUDOVLWHV 4c position). 7KHPDJQHWLFLRQV0 4c SRVLWLRQDUHORFDWHGLQWKHFHQWUHRIVOLJKWO\GLVWRUWHG02 6) octahedra. Therefore, each MO 6 octahedral is connected by corner sharing to other MO 6 octahedra in the bc plane and shares a common edge with LiO 6 octahedra and PO 4 tetrahedra. Researchers have sh own an increased interest in magnetic measurements to study the performance of the cathodes. The magnetic properties of the lithium orthophosphate family LiMPO 4 0  &R0Q)H1LKDYHEHHQSUHYLRXVO\LQYHVWLJDWHGWKURXJKPDJQH WLFVXVFHSWLELOLW\PHDVXUHPHQWV> –11]. All these FRPSRXQGVVKRZDQWLIHUURPDJQHWLFRUGHULQJ$)0DWORZWHPSHUDW XUHV7KH/L0Q32 4 and LiFePO 4 show AFM at ORZDQG1pHOWHPSHUDWXUHVRIDQG.UHVSHFWLYHO\>@  Proceedings of the 2nd Physics and Materials Science International Symposium (PhyMaS 2.0) AIP Conf. Proc. 2368, 030003-1–030003-7; https://doi.org/10.1063/5.0058020 Published by AIP Publishing. 978-0-7354-4106-4/$30.00030003-1To date, no publication has reported the synthesis, structural and magnetic properties of the LiMn 0.9Cu0.1PO 4 with the introduction of Cu content into their lattice. LiMn 0.9Cu0.1PO 4 were synthesised in the present work using solid- state reaction, and the influence of Cu2+ doping in LiMnPO 4 was investigated using XRD and magnetic susceptibility measurements. This study focuses on the magnetisation, which has been studied in less detail thus far. These magnetic measurements are also important methods for the detect ion of impurity in samples at low concentrations. EXPERIMENTAL Lithium orthophosphates LiMn 0.9Cu0.1PO 4 were synthesised by the solid-state method using high-purity lithium carbonate Li 2CO 3 $OGULFK!PDQJDQHVH,,FDUERQDWH0Q&2 3 $OGULFKFRSSHU,,R[LGH$OGULFK DQGDPPRQLXPGLK\GURJHQRUWKRSKRVSKDWH1+ 4+2PO 4 $OGULFK5DZ materials were weighed out in stoichiometric ratios and ball- PLOOHGZLWKDFHUWDLQDPRXQWRIHWK\ODOFRKROIRUKDWUS P7KHVOXUU\PL[WXUHZDV WKHQGULHGLQDLUIRUK7KHV DPSOHVZHUHFDOFLQHGXQGHUIOR ZLQJDUJRQRUQLWURJHQJDVHVDW&IRUKLQD tube furnace after drying to prevent the oxidation of Fe2+ and Mn2+ ions. These grey-coloured samples were reground XVLQJDQDJDWHPRUWDUIRUDWOHDVWKEHIRUHVLQWHULQJDW &RYHUQLJKWDQGFRROLQJE\WXUQLQJRIIWKHWXEHIXUQDFH X-ray pow GHUGLIIUDFWLRQ;5'ZDVXVHGW RFKDUDFWHULVHWKHVWUXFWXUHDQ GSKDVHSXULW\RI/L0Q 0.9Cu0.1PO 4. The VDPSOHVZHUHSXWRQDIODWSODWH ZKLOVWWKHLQWHQVLW\GDWDZHU HFROOHFWHGLQWKHUDQJHșRIWRo with a step size 0.013o using a Panalytical X’per WDQG&X.ܤUDGLDWLRQVRXUFHZLWKZDYHOHQJWKVȜ  ܤcȜ  ܤcDW room temperature. The crystal structure and lattice parame ters were refined with the Rietveld method using GSAS programme [14] and EXP *8,SDFNDJH>@ UHVSHFWLYHO\ Magnetic susceptibility measurements were conducted using Quantum Design physical property measurement system with vibrating sample method in the temperature range of 4 K to 300 K under magnetic field 1000 Oe. Data were collected in zero-field- FRROHG=)&DQGILHOG -FRROHG)&FRQGLWLRQV7KHPDJQHWLVDWLRQPHDVXUHPHQWV0DV DIXQFWLRQRIDSSOLHGILHOG+ZHUHPHDVXUHGXQGHUH[WHUQDOP DJQHWLFILHOGVXSWR2HDWWHPSHUDWXUH. RESULTS AND DISCUSSION Structural Analysis The XRD patterns of LiMn 0.9Cu0.1PO 4 were analysed using the Rietveld refinement technique as shown in Figure 1. The LiMn 0.9Cu0.1PO 4 samples were indexed to a single- SKDVHRUWKRUKRPELFFU\VWDOVWUXFWXUHDEFZLWKVSDFH group Pnma 1RXVLQJWKHVWDUWLQJPRGHOE\2VRULRJXLOOHQHWDO>@ 7KHSHDNVKDSHZDVPRGHOOHGXVLQJWKH pseudo-9RLJWIXQFWLRQUHILQHGWRJHWKHUZLWKFHOOSD rameters, zero factor, scale factor and background function. The atomic positions x, y and z and the isotropic displacement parameter U ,62 were constrained to be equal to the ratio for Cu and Mn due to the site sharing of Cu/Mn ions in octahedral coordination for the intermediate compounds of LiMn 0.9Cu0.1PO 4. Thus, all peaks are remarkably sharp, clear and well-defined, suggesting the effective crystallisation of LiMn 0.9Cu0.1PO 4. The reasonably low weighted profile R- IDFWRU5 wp Ȥ 2 = 1.947) suggests the collection of single-phase LiMnPO 4. The lattice parameters are determined as follows: a c b c c  c7KHXQLWFHOOYROXPHRIc3 at room temperature LVVOLJKWO\ORZHUWKDQc3 of polycrystalline LiMnPO 4. The reduced unit cell volume of the sample is probably due to some replacement of Mn2+ by Cu2+. The average of the atoms Mn–O distance of the MnO 6 octahedral and the P– 2GLVWDQFHUDQJHVIURPcWR cDQGWRcUHVSHFWLYHO\7KH0Q –O distance of the LiMn 0.9Cu0.1PO 4 is in good agreement with that of previous studies [17]. Meanwhile, Li– 2GLVWDQFHVVOLJKWO\LQFUHDVHGEHWZHHQcWRcFRPSD UHGZLWKWKDW of pure LiMnPO 4, in which the average Li– 2GLVWDQFHVDUHHTXDOWRc7DEOH 1 provides a summary of bond distance and angle. 030003-2 TABLE 1. Bond distance and bond angles for LiMn 0.9Cu0.1PO 4 determined by X-ray powder diffraction at room temperature. LiMn 0.9Cu 0.1PO 4 Mn–O (Å) P–O (Å) Li–O (Å) Mn–2 × 1 2.26 P– 2 × 1  Li– 2 × 2  Mn–2 × 1 2.04 P– 2 × 1 1.623 Li–2 × 2 2.041 Mn–2 × 2  P– 2 × 2  Li–2 × 2 2.141 Mn–2¶ × 2  - - - - Mn–O average 2.16 P–O average  Li–O average  FIGURE 1 . Rietveld refinement of LiMn 0.9Cu0.1PO 4 against X- UD\GLIIUDFWLRQ;5'GDWDFR OOHFWHGDWURRPWHPSHUDWXUH7KH observed, calculated and difference profiles are represented by red crosses, black lines and blue lines, respectively. 030003-3 FIGURE 2 . 6WUXFWXUHRIROLYLQHWULSK\OLWH -type lithium metal phosphates. The transition metal octahedra, phosphate tetrahedra and lithium are respectively shown in cyan, purple and green. Magnetic measurement The temperature dependences of the inverse mo lar magnetic susceptibility curves of LiMn 0.9Cu0.1PO 4 samples are VKRZQLQ)LJXUH1RQRWLFHDEOHG LYHUJHQFHRI=)&)&FRQGLWLRQ VLVREVHUYHGJUDSKQRWVKRZQ7KHWHPSHUDWXUH dependence versus )1o mFF curves show a sharp p eak corresponding to T N. LiMn 0.9Cu0.1PO 4 display AFM ordering behaviour below T N, and the value of T N is at 36 K. These findings are cons istent with those of other research, which suggested the dominance of the AFM behaviour in the orthophosphate family. Above T N, the inverse magnetic susceptibility of all samples in the paramagnetic region obeys the Curie– :HLVVODZ(T CT mT F 1   where C, cwT, expP and oF refer to the Curie constant, Weiss temperature, effective magnetic moment and temperature-independent contributions to the susceptibility, respectively. The cwT and expP values obtained by linear fitting are summarised in Table 2. The negative value of șcw indicates VWURQJ$)0LQWHUDFWLRQV>@7KHH[SHULPHQWDO expP values were determined from the Curie–Weiss constant as shown below. BCP P.2exp   These values are larger compared with the predicted spin-only magnetic moment per formula unit for Mn2+ obtained from the equation 1 SSgtheoryP ȝ%FRQVLGHULQJ0Q2+ in the high-spin configuration, g = 2, Mn2+/ 6 7KHH[SHULPHQW DOHIIHFWLYHPRPHQWLVKLJKHUW KDQWKHRU\SRVVLEO\GXHWRWKHSDUWLDOO\TXHQFKHG orbital angular momentum contribution by the crystal field in the octahedral site [19, 20]. This phenomenon is common in the case of elements containing more than half-filled d-orbitals. Figure 4 illustrates the hysteresis curves for LiMn 0.9Cu0.1PO 4 VDPSOHVXQGHUDQDSSOLHGPDJQHWLFILHOGRI2HDW7 . 1RPD[LPXPVDWXUDWLRQZDV observed for LiMn 0.9Cu0.1PO 4 with the application of magnetic field. A linear increase in the magnetisation with no saturation of the magnetic field su ggested a typical pure AFM phase. LiO 6 octahedra PO 4 tetrahedra MnO 6 octahedra 030003-4 TABLE 2: Magnetic Parameter of LiMn 0.9Cu0.1PO 4 samples obtained from the Curie –Weiss Law fitting from 100 –300 K. Sample Curie constant, C (emu K/mol) Weiss Temperature, cwT(K) Effective magnetic moment, effP BP Nèel Temperature, TN (K) Magnetic moment, theory effP LiMn 0.9Cu0.1PO 4 3.0647 -26.1    Temperature (K)0 50 100 150 200 250 300FmFR (mol/emu) 020406080100120 LiMn0.9Cu0.1PO4 C = 3.0647 emu K/mol Tcw= -26.17 K Peff = 4.9508 P% FIGURE 3: ,QYHUVHPDJQHWLFVXVFHSWLELOLW\ versus temperature for LiMn 0.9Cu0.1PO 4 samples under applied magnetic field, H = 1000 Oe. The solid red lines show the fits using the Curie –Weiss law. 030003-5Magnetic Field (Oe)-10000 -5000 0 5000 10000Magnetization ( P%/f.u) -0.10-0.08-0.06-0.04-0.020.000.020.040.060.080.10 LiMn0.9Cu0.1PO4 T = 4K FIGURE 4: 0DJQHWL]DWLRQ0FXUYHVDVDIXQFWLRQRIDSSOLHGPDJQHWLFILHOG +SORWIRU/L0Q 0.9Cu0.1PO 4 at T=4 K CONCLUSION LiMn 0.9Cu0.1PO 4 samples were successfully prepared using the solid-state reaction route. The XRD analysis and Rietveld refinement show the crystallisation pattern of LiMn 0.9Cu0.1PO 4 in an orthorhombic structure Pnma . The magnetic susceptibility demonstrates antiferromagnetic behaviour at T N . ACKNOWLEDGMENTS The main part of this work was performed at the School of Chemistry, The University of Sydney, Australia. Z. Mohamed acknowledges support from the Universiti Teknologi MARA, Malaysia under the project number 600- 50&*55  0). REFERENCES 1. A. K. Padhi, K. S.Nanjundaswamy, C. Masquelier, S. Okada, and J. B. Goodenough, J. Electrochem. Soc. 144  - 2. D. 9DNQLQ-/=DUHVWN\//0LOOHU- -35LYHUDDQG+6FKPLG3K\V5HY B, 65 , 224414  3. S. Y. Chung, J.T. Bloking, and Y. M. Chiang, Nat Mater, 1 - 4. +Ehrenberg, N.N. %UDPQLN$6HQ\VKLQ+)XHVV Solid State Sciences, 11 -  3%DUSDQGD*5RXVVH7<H&' /LQJ=0RKDPHG<.OHLQ $<DPDGD,QRUJDQLFFKHPLV try 52  3334- 6. M. Avdeev, Z. Mohamed, C. D. Ling, Journal of Solid-State Chemistry, 216, 42- 030003-67. E. Dachs, &$*HLJHU96HFNHQGRUIDQG0*URG]LFNL The Journal of Chemical Thermodynamics, 39 906-  D. Arcon, A, Zorko, R. Dominko, Z, Jaglicic, Journal of Physics: Condensed Matter, 16 -  9. M. Sale, M. Avdeev, Z. Mohamed, C. D. Ling, P. Barpanda, Dalton Transactions, 46  - 10. 0$YGHHY=0RKDPHG&'/LQJ-/X07DPDUX$<DPDGD 3%DUSDQGD,QRUJDQLFFKHPLVWU\ 52  - 11. O. Ofer, -6XJL\DPD-+%UHZHU00DQVVRQ.3UVD , E. J. Ansaldo, G. Kobayashi and R. Kanno, Physics Procedia, 30, 160- 12). 12. M. Kope¢, A. Yamada, G. Kobayashi, S. Nishimura, R. Kanno, A. Mauger, Journal of Power Sources, 189 - 13. A. Jena and B. R. K. Nanda, Scientific Reports, 6 . 14. C. Larson, 5%9 Dreele, General Structure Analysis 6\VWHP*6$65HSRUW/$85 -/RV$ODPRV 1DWLRQDO/DERUDWRU\/RV$ODPRV1086$ –  +7RE\(;3*8,-$SSO&U\VWDOORJU 34 16. J. M. Osorio- JXLOOHQ%+ROP5$KXMD , B. Johansson, 6ROLG6WDWH,RQLFV 167-4), 221- 004). 17. S. Geller and J. L. Durand, Acta cryst. , 13  K. Zaghib, A. Mauger, C. M. Julien, Journal of Power Sources, 160-  19. R. P. Santoro and R. E. Newnham, Acta Cryst , 22, 344- 20. S. Nishimura, G. Kobayashi, K. Ohoyama, R. Kanno, M. Yashima, A. Yamada, Nat Mater, 7 -11  030003-7

5.0055515.pdf

AIP Advances 11, 065312 (2021); https://doi.org/10.1063/5.0055515 11, 065312 © 2021 Author(s).Semiconducting character of LaN: Magnitude of the bandgap and origin of the electrical conductivity Cite as: AIP Advances 11, 065312 (2021); https://doi.org/10.1063/5.0055515 Submitted: 04 May 2021 . Accepted: 23 May 2021 . Published Online: 08 June 2021 Zihao Deng , and Emmanouil Kioupakis ARTICLES YOU MAY BE INTERESTED IN Next generation ferroelectric materials for semiconductor process integration and their applications Journal of Applied Physics 129, 100901 (2021); https://doi.org/10.1063/5.0037617 Ferroelectrochemistry APL Materials 9, 051112 (2021); https://doi.org/10.1063/5.0051129 Fully epitaxial ferroelectric ScAlN grown by molecular beam epitaxy Applied Physics Letters 118, 223504 (2021); https://doi.org/10.1063/5.0054539AIP Advances ARTICLE scitation.org/journal/adv Semiconducting character of LaN: Magnitude of the bandgap and origin of the electrical conductivity Cite as: AIP Advances 11, 065312 (2021); doi: 10.1063/5.0055515 Submitted: 4 May 2021 •Accepted: 23 May 2021 • Published Online: 8 June 2021 Zihao Deng and Emmanouil Kioupakisa) AFFILIATIONS Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA a)Author to whom correspondence should be addressed: kioup@umich.edu ABSTRACT Lanthanum nitride (LaN) has attracted research interest in catalysis due to its ability to activate the triple bonds of N 2molecules, enabling efficient and cost-effective synthesis of ammonia from N 2gas. While exciting progress has been made to use LaN in functional applications, the electronic character of LaN (metallic, semi-metallic, or semiconducting) and magnitude of its bandgap have so far not been conclusively determined. Here, we investigate the electronic properties of LaN with hybrid density functional theory calculations. In contrast to previous claims that LaN is semi-metallic, our calculations show that LaN is a direct-bandgap semiconductor with a bandgap value of 0.62 eV at the X point of the Brillouin zone. The dispersive character of the bands near the band edges leads to light electron and hole effective masses, making LaN promising for electronic and optoelectronic applications. Our calculations also reveal that nitrogen vacancies and substitutional oxygen atoms are two unintentional shallow donors with low formation energies that can explain the origin of the previously reported electrical conductivity. Our calculations clarify the semiconducting nature of LaN and reveal candidate unintentional point defects that are likely responsible for its measured electrical conductivity. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0055515 Nitride compounds are a rich class of functional mate- rials. The main-group III-nitrides are important semiconduc- tors that find applications in electronics, optoelectronics, and photocatalysis. Recently, transition-metal and rare-earth nitrides have attracted attention due to their promise in, e.g., piezoelec- tric,1superconducting,2and catalysis applications.3,4In partic- ular, Ye et al. discovered that lanthanum nitride (LaN) facil- itates stable and highly efficient ammonia synthesis through the activation of N 2gas by nitrogen vacancies at the sur- face. LaN shows a catalytic performance that is comparable to ruthenium-based catalysts but at a much lower cost.4Thus, LaN is a promising nitride material for catalysis and other chemical applications. While LaN exhibits great promise in functional applications, one fundamental question that has not been fully addressed is whether LaN has a metallic, semi-metallic, or semiconducting nature. Understanding this fundamental electronic character of LaN is critical for its future applications. Many previous theoretical cal- culations have attempted to elucidate the electronic band structureof LaN. Early density functional theory (DFT) calculations with the augmented plane wave method (APW) showed that the conduction and valence bands of LaN overlap by up to 40 mRy, indicating a semi-metallic nature.5,6Vaitheeswaran et al. studied the electronic properties of LaN using tight-binding linear muffin-tin orbitals with the local-density approximation (LDA) to the exchange-correlation functional and found a metallic nature for LaN. They also estimated the superconducting transition temperature to be 0.65 K.7Later cal- culations with the generalized gradient approximation (GGA) func- tional observed the overlap between valence and conduction band in the band structure and characterized LaN either as metallic or as semi-metallic.8,9However, these results contradict to the calcula- tions using the hybrid screened-exchange local density approxima- tion (sX-LDA) functional where an indirect bandgap of 0.75 eV was found, suggesting that LaN might be a semiconductor.10Recently, more calculations seem to support the semiconducting nature of LaN. Gupta et al. used the LDA functional and found an indirect bandgap of 0.5 eV.11The GGA +USICfunctional was employed by Meenaatci et al. , and an indirect bandgap of 0.65 eV was found.12 AIP Advances 11, 065312 (2021); doi: 10.1063/5.0055515 11, 065312-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv However, by using the LSDA +U functional, Larson et al. discov- ered a small direct bandgap of 0.4 eV for LaN.13Similarly, a direct bandgap of 0.6 eV was recently obtained by the Modified Becke- Johnson Local Density Approximation (MBJLDA) functional for both wurtzite and rocksalt LaN.14,15In addition, Sreeparvathy et al. found a direct bandgap of 0.814 eV using the full potential linearized augmented plane wave (FP-LAPW) method with the TB-mBJ func- tional,16which seems to agree with the experimental bandgap of 0.82 eV measured from optical absorption.17Despite the progress from previous theoretical studies, the nature (direct or indirect) and the magnitude of the bandgap are not conclusively determined for LaN due to different methods and functionals employed in the calculations, which necessitates a re-investigation of the electronic properties of LaN with modern electronic structure calculations. Since LaN is found by the more advanced functionals to exhibit a bandgap and thus to present a semiconducting rather than a semi- metallic character, its electrical conductivity must originate from intrinsic or unintentional dopants. However, to the best of our knowledge, there is no theoretical investigation into the thermo- dynamics of the intrinsic defects and common impurities of LaN, which is the key to understand the origin of its electrical conduc- tivity. This knowledge is also necessary in order to rationally tune its conductivity by controlling the defect formation and doping in experiments. Thus, theoretical insights into the intrinsic defect for- mation and ionization energies are crucial to enable the adoption of LaN in wider electronic and catalysis applications. In this work, we study the electronic properties (band structure, effective masses, dielectric constants, and so on) and defect thermo- dynamics of LaN using first-principles calculations based on DFT with the HSE06 hybrid functional,18,19which predicts accurate elec- tronic properties for a wide range of materials. We find that LaN is a direct-bandgap semiconductor with a bandgap of 0.62 eV at the X point of the Brillouin zone. Our defect calculations attribute the ori- gin of its electrical conductivity to the unintentional formation of N vacancies or substitutional O impurities. Our calculations clarify the semiconducting nature of LaN and reveal candidate defects that are the likely origin of its measured electrical conductivity. DFT calculations were performed using the Vienna ab initio Simulation Package (VASP).20GW-compatible Perdew–Burke– Ernzerhof (PBE) pseudopotentials21for La and N and a plane- wave energy cutoff of 500 eV were employed in all calculations. La 5s25p65d16s2and N 2s22p3were treated as valence electrons. In order to get accurate electronic properties, we used the HSE06 hybrid exchange-correlation functional with a standard mixing parameter of 0.25.18,19The electronic band structures of LaN were calculated with a fully relaxed two-atom rocksalt primitive cell (space group Fm-3m) and a Γ-center 8 ×8×8 Brillouin zone sampling grid.22The special k-point path for plotting the band structure followed the convention of Setyawan and Curtarolo.23 Spin–orbit coupling effects were included in the band structure. The static and high-frequency dielectric constants were calculated by the self-consistent response to the finite electric field at the HSE level using modern theory of polarization.24–27Electron and hole effec- tive masses were extracted by fitting the HSE band structure with the hyperbolic equation, E(k)=∓1±√ 1+4α̵h2k2 2m∗ 2α+E0, (1)where E0is the energy of the band extremum, m∗is the effec- tive mass, and αis a fitting parameter to characterize the non- parabolicity of the band. The band alignment of LaN was obtained by aligning the bulk average electrostatic potential to the vacuum level. This was done by performing HSE calculations for the LaN (100) and (110) slabs without surface relaxation. The alignment results for the two slabs differ by only 70 meV. Defect calculations28 were performed with 2 ×2×2 supercells built from the eight-atom rocksalt unit cell (i.e., 64 atoms), and the Brillouin zone was sam- pled with a Γ-center 2 ×2×2 grid. The defect formation energies and charge-transition levels for the O Ndefect are found to change by less than 0.05 eV for a larger supercell with 128 atoms (Table S1). Thus, the 64-atom supercell was employed for all subsequent defect studies to reduce the computational cost. All defect super- cells were relaxed with HSE06 by allowing ion displacements until the forces on the ions were less than 0.02 eV Å−1. Spin polarization was included in calculations with unpaired electrons. We employed the scheme of Freysoldt et al.29and our calculated static dielectric constant for LaN to correct the artificial periodic charged-defect interactions. The competing phases we considered in the thermo- dynamic analysis of defect formation are La 2O3, NO 2, NH 3, and all stable elemental phases. We first investigate the band structure of LaN to determine its electronic character (metal or semiconductor) and the magni- tude of its bandgap if one exists. Previous DFT calculations using the GGA functional characterized LaN as a semimetal.8,9Indeed, our own PBE band-structure calculations (Fig. S1) reveal that the conduction and valence bands overlap by 115 meV at the X point, which leads to the conclusion that LaN is a semimetal and agrees with many previous theoretical studies. However, calculations with semilocal exchange-correlation functionals such as LDA or PBE severely underestimate the bandgaps of materials and may erro- neously lead to a closed gap in LaN. Moreover, we find that even after one-shot GW corrections on top of PBE, the bandgap is still extremely small (0.05 eV from Table I), which indicates that the metallic PBE state is a poor starting point for GW. Thus, the adop- tion of a non-local functional in the calculations is necessary to give an accurate bandgap and a good starting point for GW cor- rections. Table I shows our calculated bandgap for LaN using the HSE06 hybrid functional with GW and spin–orbit coupling cor- rections, respectively. Under all circumstances, LaN shows a direct bandgap at the X point of the Brillouin zone. The bandgap value is 0.75 eV if we only use HSE in the calculations. Both one-shot GW correction and spin–orbit coupling decrease the bandgap value into the range of 0.6–0.7 eV. Our results are consistent with the most recent studies by Winiarski and Kowalska where they found a direct bandgap of 0.6 eV for LaN using the MBJLDA functional including the relativistic spin–orbit effect.14,15This provide the direct evidence that LaN is a semiconductor rather than a semimetal. We therefore attribute the mis-categorization of LaN as a semimetal in previous studies to the systematic bandgap underestimation problem of LDA and GGA. Figure 1 shows the electronic band structure of LaN calcu- lated with HSE and spin–orbit coupling. The conduction band and the valence band are separated by a direct gap of 0.62 eV at X. The valence band of LaN is mainly derived from N 2p orbitals and the conduction band is derived from the spatially extended unoccupied La 5d orbitals, as can be seen from the orbital projected AIP Advances 11, 065312 (2021); doi: 10.1063/5.0055515 11, 065312-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv TABLE I. Comparison of the LaN bandgap calculated in this work and previous studies. Method E gap(eV) Character PBE (this work) 0 Semi-metallic PBE+GW (this work) 0.05 Direct X–X HSE06 (this work) 0.75 Direct X–X HSE06 +GW (this work) 0.68 Direct X–X HSE06 +SOC (this work) 0.62 Direct X–X APW,5,6LDA,7GGA,8,9HGH110 Semi-metallic sx-LDA100.75 Indirect Γ–X LDA110.5 Indirect Γ–X GGA +U120.65 Indirect Γ–X LSDA +U130.4 Direct X–X MBJLDA14,150.6 Direct X–X FP-LAPW, TB-mBJ160.814 Direct X–X band structure in Fig. S2. Spin–orbit coupling has a profound effect on the structure of the valence band, particularly at the high sym- metry points ( Γ, X, and W). This arises from the contribution of La p orbitals near those points (Fig. S2). After including the spin–orbit effect, the bandgap decreases from 0.75 to 0.62 eV. The three-fold degenerate N p orbitals at Γsplit into a two-fold degenerate p 3/2band and a one-fold p 1/2band. Similarly, the two-fold degeneracy between the second and third valence bands at the X point and between the first and second valence bands at the W point is broken, with energy splittings of 340 and 301 meV, respectively. Table II summarizes the structural and electronic properties of LaN. We obtain a lat- tice constant of 5.282 Å, which is in excellent agreement with the experimental value of 5.284 Å.30The conduction band of LaN is dis- persive with a bandwidth of several eV, leading to small electron masses especially along the transverse X–W and X–U directions. The hole effective masses are comparable to or even lighter than the hole effective masses of GaN.31The energy splitting between FIG. 1. The electronic band structure of LaN calculated with the HSE06 hybrid density functional with (w) and without (w/o) spin–orbit coupling. Instead of a semimetal, LaN is a direct bandgap semiconductor with a bandgap of 0.62 eV at the X point.the two topmost valence bands is 85 meV, which is larger than the thermal energy kBTof 26 meV at room temperature. This indicates a much weaker phonon-mediated hole scattering from the valence band maximum to the second-highest valence band, as in the case of strained BAs.32The reduced scattering rate coupled with light effec- tive masses may lead to a high hole mobility in LaN, although no p-type conductivity has been observed experimentally so far. To investigate the band offsets between LaN and the other III-N materials in heterostructures and to understand its proper- ties for photocatalysis, we calculate its absolute band positions by aligning the bulk average electrostatic potentials to the vacuum region. Figure 2 shows the absolute band alignment for LaN in comparison to other III-N materials. The bandgap and absolute band positions of LaN are similar to ScN, another nitride mate- rial with the rocksalt crystal structure, forming a type-II align- ment at the interface with a relative band offset of 0.3 eV. Inter- estingly, the electron affinity is almost identical for LaN and GaN. This can find potential applications in LaN/GaN heterostructures where electrons can move across the interface without a potential barrier. We then turn to the origin of the electrical conduction in LaN that is observed in experiment.35–37Previous studies attributed the conduction to the semi-metallic character of LaN. However, this explanation is inconsistent with our accurate band-structure results that find LaN to be a semiconductor with a bandgap. An alterna- tive explanation is that the electrical conduction in LaN is caused by unintentional doping by either intrinsic defects or unintentional impurities incorporated during growth. To shed light on this issue, we calculate the formation energy of intrinsic defects (lanthanum vacancies V Laand nitrogen vacancies V N) and common uninten- tional impurities (substitutional O and H interstitials). Figure 3 shows their formation energy as a function of Fermi energy (refer- enced to the VBM) and growth conditions (N-rich or N-poor). Our key finding is that the donor-like defects (V Nand O N) have signif- icantly lower formation energies (even negative) than acceptor-like defects, which identify LaN to be an intrinsically n-type semicon- ductor. Nitrogen vacancies act as negative-U double donors and are stable in the +2 charge state for Fermi levels throughout the majority of the bandgap. O impurities are singly charged shallow donors, with an ionization energy of 50 meV. Therefore, both of these donor-like defects are candidate origins of the measured conductivity. To further confirm the origin of the conductivity, we compare our findings to experimental measurements of the electrical conduc- tivity as a function of temperature measured by four-point probe measurements in the literature. Lesunova et al.36found an exponen- tially increasing electrical conductivity with increasing temperature, which is characteristic of thermal activation of dopants in semicon- ductors. The temperature trend is also in contrast with the typical behavior in metals in which the conductivity decreases with tem- perature due to increased carrier scattering by phonons. We fur- ther extract the donor activation energy by fitting the Arrhenius equation, σ(T)=σ0exp(−EA kBT), (2) to the experimental electrical-conductivity measurements by Lesunova et al. ,36as shown in Fig. 4. The fitted value for the activa- tion energy EAis 40 meV, which is in excellent agreement with our AIP Advances 11, 065312 (2021); doi: 10.1063/5.0055515 11, 065312-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv TABLE II. Structural and electronic properties of LaN evaluated with the HSE06 hybrid functional and spin–orbit coupling. Crystal structure Lattice constant (Å) Rocksalt 5.282 Bandgap (eV) Valence band splitting (meV) 0.62 85 Static dielectric constant, ε0 High frequency dielectric constant, ε∞ 13.32 9.04 Electron and hole effective masses X–G X–W X–U m∗ e/m0 1.211 0.178 0.136 m∗ h1/m0 0.329 0.632 0.376 m∗ h2/m0 1.956 0.246 0.149 m∗ h3/m0 0.720 0.399 0.254 calculated ionization energy for shallow donors (39 meV) evaluated with the Bohr model, EA=13.6 eV m∗ e ε2r, (3) using a directionally averaged electron effective mass of 0.51 me and a static dielectric constant of 13.32. Based on this qualitative and quantitative agreement with the literature-reported experimen- tal conductivity measurements, we conclude that the electrical con- ductivity of LaN originates from unintentional doping and V Nand ONare likely candidate defects. In our calculation, O Nhas a negative formation energy under both N-rich and N-poor conditions. However, this should not be interpreted as a sign of instability for LaN. In fact, the synthesis chemistry of LaN is well understood and LaN has been utilized in catalysts for NH 3synthesis4and for electrodes in supercapacitors.37 Rather, the negative formation energy indicates that substitutional FIG. 2. Band alignment between LaN and other nitride materials. LaN has a type-II alignment with ScN and a type-I alignment with GaN and AlN. The electron affinity of LaN and GaN is almost identical. The band-alignment data of ScN and III-N are taken from the work of Kumagai et al.33and Moses et al. ,34respectively.O is a major source of unintentional impurities, which could pos- sibly lead to degenerate n-type doping for LaN. In comparison, rocksalt ScN and Sc-containing nitride alloys, compounds with sim- ilar chemistry as LaN, have been found to have negative formation energy for O Nin defect calculations,33and O gets unintentionally incorporated during growth38,39and results in degenerate n-type doping.40Thus, an oxygen-free growth environment is necessary to prevent the undesired degenerate doping by substitutional O for LaN. In conclusion, we study the electronic properties of LaN using DFT calculations based on the HSE06 hybrid functional. In con- trast to the semi-metallic nature claimed by most previous studies, we find that LaN is a direct-bandgap semiconductor with a gap of 0.62 eV at the X point. The light electron and hole effective masses and the near-zero conduction-band offset with GaN make LaN promising for electronic and optoelectronic applications. Our defect calculations indicate that LaN is intrinsically n-type and the source FIG. 3. Calculated defect formation energy as a function of Fermi level for the nitrogen vacancy (V N), lanthanum vacancy (V La), hydrogen interstitial (H i), and N- substitutional oxygen impurity (O N) in LaN. Donor-like defects (V Nand O N) exhibit the lowest formation energy for every Fermi level, indicating that LaN is an intrinsic n-type semiconductor and its conductivity originates from V Nor O N. AIP Advances 11, 065312 (2021); doi: 10.1063/5.0055515 11, 065312-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 4. Electrical conductivity of LaN as a function of temperature. The experimen- tal data are taken from the measurements of Lesunova et al.36The dashed line is a least-squares fit to the conductivity data. The donor activation energy extracted from the fitting is 40 meV, in good agreement with the ionization energy for O Nand VN. of the measured electrical conductivity is attributed to unintentional doping by nitrogen vacancies or substitutional oxygen. Our studies clarify the semiconducting nature of LaN and reveal candidate unin- tentional donors that explain the origin of its measured electrical conductivity. See the supplementary material for the convergence of the O N defect formation energy and charge-transition level as a function of simulation cell size, the calculated band structure with the PBE functional, and the orbital-projected HSE band structure. This study was supported by the National Science Founda- tion through Grant No. DMR-1561008. The band-structure and band-alignment calculations used resources of the National Energy Research Scientific Computing (NERSC) Center, a Department of Energy Office of Science User Facility supported under Contract No. DEAC0205CH11231. The defect calculations used Comet and Data Oasis at the San Diego Supercomputer Center (SDSC) through Allo- cation No. TG-DMR200031, an Extreme Science and Engineering Discovery Environment (XSEDE)41user facility supported by the National Science Foundation (Grant No. ACI-1548562). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Mertin, B. Heinz, O. Rattunde, G. Christmann, M.-A. Dubois, S. Nicolay, and P. Muralt, Surf. Coat. Technol. 343, 2 (2018). 2R. Yan, G. Khalsa, S. Vishwanath, Y. Han, J. Wright, S. Rouvimov, D. S. Katzer, N. Nepal, B. P. Downey, D. A. Muller, H. G. Xing, D. J. Meyer, and D. Jena, Nature 555, 183 (2018). 3T.-N. Ye, S.-W. Park, Y. Lu, J. Li, M. Sasase, M. Kitano, and H. Hosono, J. Am. Chem. Soc. 142, 14374 (2020).4T.-N. Ye, S.-W. Park, Y. Lu, J. Li, M. Sasase, M. Kitano, T. Tada, and H. Hosono, Nature 583, 391 (2020). 5A. Hasegawa, J. Phys. 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APL Photonics PERSPECTIVE scitation.org/journal/app Topological photonics in 3D micro-printed systems Cite as: APL Photon. 6, 080901 (2021); doi: 10.1063/5.0058478 Submitted: 30 May 2021 •Accepted: 27 July 2021 • Published Online: 6 August 2021 Julian Schulz,1 Sachin Vaidya,2 and Christina Jörg1,2,a) AFFILIATIONS 1Physics Department and Research Center OPTIMAS, University of Kaiserslautern, D-67663 Kaiserslautern, Germany 2Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA a)Author to whom correspondence should be addressed: christinaijoerg@gmail.com ABSTRACT Topological materials have been at the forefront of research across various fields of physics in hopes of harnessing properties such as scatter- free transport due to protection from defects and disorder. Photonic systems are ideal test beds for topological models and seek to profit from the idea of topological robustness for applications. Recent progress in 3D-printing of microscopic structures has allowed for a range of imple- mentations of topological systems. We review recent work on topological models realized particularly in photonic crystals and waveguide arrays fabricated by 3D micro-printing. The opportunities that this technique provides are a result of its facility to tune the refractive index, compatibility with infiltration methods, and its ability to fabricate a wide range of flexible geometries. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0058478 I. INTRODUCTION Topological insulators are materials that are insulating in their bulk but conduct current along their edges without back-scattering even in the presence of disorder and defects. The first manifestation of topology was shown in the quantum Hall effect1where an elec- tron gas restricted to two dimensions in a magnetic field introduces Landau levels leading to energy gaps. When the Fermi energy lies in such a gap, the conduction only stems from the contribution of edge channels and is pinned to a quantized value that is independent of the amount of disorder. This novel phenomenon can be under- stood within a branch of mathematics called topology that deals with properties of geometric objects that are preserved under continuous transformations. In physical systems, topology manifests as a property of the eigenstates of the governing Hamiltonian. The eigenstates of a peri- odic system that depend on a few independent parameters can be parallelly transported along a closed contour in that space of param- eters. On doing so, the eigenstates may acquire a non-zero measur- able phase that is connected to a topological invariant. According to the bulk-boundary correspondence, the non-trivial topological invariant of the bulk is linked to a range of rich physical phenom- ena that manifest on the boundaries of a finite-sized sample. The robustness of such edge states is the most accessible and, thus, the most important feature for applications of topology since the edgestates persist even in deformed or perturbed systems, as long as the topology of the system is not changed. More recently, it has been shown that the same principles can be applied to photonics to achieve novel and unusual transport properties of light, resulting in the research area of topological photonics.2–6In photonics, topolog- ical robustness or unidirectional transport is sought to be used in optical devices, making them less prone to local fabrication imper- fections, e.g., for robust data transfer7,8or single-mode laser light sources.9–15 Photonic platforms are ideal test beds for theoretical models, and due to the ubiquitousness of topology across systems, many different photonic platforms have emerged. Among these are vari- ous waveguide systems,16photonic crystals,17–20ring resonators,10,21 gyromagnetic rods,22fiber loops,23,24polaritons,25and many more. Adding to that toolbox of platforms, recently, structures fabricated by 3D micro-printing have come into play. This fabrication method uses multi-photon polymerization to create three-dimensional structures on the micrometer scale.26Indus- try solutions for micro-printing include Nanoscribe, Multiphoton Optics, UPnano, and Femtika, to name a few. Furthermore, home- built micro-printing setups can also be found in research groups across various universities. We will henceforth focus on the Nano- scribe Photonic Professional GT2 since it is the most widely used system. APL Photon. 6, 080901 (2021); doi: 10.1063/5.0058478 6, 080901-1 © Author(s) 2021APL Photonics PERSPECTIVE scitation.org/journal/app The Nanoscribe is a commercially available “ready-to-play” instrument that can be found in many universities today. The ability of the Nanoscribe to fabricate versatile structures spanning orders of magnitude has resulted in a wide range of applications (for an exhaustive overview, we direct the reader to the website27). Partic- ularly for topological photonics, the Nanoscribe has been used to fabricate two types of structures: photonic crystals and waveguide systems. Therefore, this paper is organized as follows: We will first give a short introduction into topological insulators (Sec. II) and the working principle of the Nanoscribe (Sec. III). Then, we will review the topological features that have been observed in pho- tonic crystals (Sec. IV) and waveguide systems (Sec. V) fabricated using the Nanoscribe. At the end, we will comment on recent advances, upcoming approaches, and possible future directions (Sec. VI). II. TOPOLOGICAL INSULATORS An insulating system is described by a Hamiltonian Hthat has a bandgap in its eigenvalue spectrum of some generalized energy E(Fig. 1). The eigenmodes of this Hamiltonian can be character- ized by an integer topological invariant that is a global property over the entire momentum space. Under continuous transformation to a different system described by the Hamiltonian H′, this invariant cannot change unless this transformation closes and re-opens the gap. Thus, two Hamiltonians, related by a transformation that keeps the bandgap open, are said to be in the same topological phase. The prototypical example of a topological invariant, the Chern number, can be found in two-dimensional systems in the absence of time- reversal symmetry. At an interface between a topologically trivial and a non-trivial material, the gap must close in real-space at the boundary between the two materials, resulting in the presence of edge states [Figs. 1(a) and 1(b)]. These edge states are prohibited from scattering into the bulk by the gap on both sides of the inter- face and are exponentially confined to the boundary. They are also protected from back-scattering when their slope is monotonic with respect to momentum and when coupling between edge states with different slopes is forbidden. Besides the Chern number, a plethora of topological invari- ants are known to exist, depending on the dimensionality of FIG. 1. Topological phases. (a) In a finite topological system interfaced by a trivial one, the gap closes at the real-space boundary between the two systems in the form of edge states (yellow and red), which cross the bandgap (b), while edge states at the interface between two trivial insulators do not have a monotone dis- persion. (c) Floquet modulations create replicas of the bands, spaced in energy by the driving frequency ω. Overlapping bands “mix and reorder,” which results in band inversions and topologically non-trivial systems with edge states.the system and the presence of additional symmetries.16,28This leads to novel symmetry-protected topological phases that are only distinguishable from trivial phases when the presence of cer- tain symmetries is enforced. Furthermore, some band degenera- cies can also be associated with invariants that result in topological semi-metals. Since the concept of topology is quite universal, it can be applied across many physical systems. Raghu and Haldane trans- ferred the principles of topological protection to photonics, propos- ing a photonic analog of the quantum Hall effect.29While the quan- tum Hall effect relies on the action of a uniform magnetic field on electrons, in photonic systems, a similar action on uncharged pho- tons needs to be found. This can be done employing the so-called synthetic30–32or artificial gauge fields (AGFs).33,34These AGFs con- trol the dynamics of photons such that they behave as if effective external fields were acting on them. AGFs can be engineered by geometric or periodic time-dependent modulations (Floquet modu- lations). Floquet modulations impose periodic boundary conditions on the energy such that the energy bands of the static systems are now being replicated at energies nhω, with nbeing an integer and ω being the frequency of the modulation. This way, states that were separated in energy in the static system can now hybridize when the driving frequency matches their energetic separation [Fig. 1(c)]. Therefore, Floquet modulations can induce topological phases from systems that are topologically trivial in the static case. III. THE NANOSCRIBE The Nanoscribe uses near infrared femtosecond laser pulses to cross-link a liquid photoresist via two-photon polymerization.35–41 Typical parameters of the laser used are a pulse duration of 100 fs at a repetition rate of 80 MHz and 780 nm wavelength. The laser light is focused by an objective into the resist applied onto a sub- strate. Using a 63 ×, NA 1.4 objective with an aperture of 7.3 mm, an average laser power of less than 50 mW at the entrance pupil of the objective is conventionally deployed for printing.42Absorption of two photons within a very short time-span in the laser’s focal volume triggers polymerization of the resist. This causes only the material within a certain volume to be cross-linked and solidified, allowing us to fabricate elaborate three-dimensional structures by moving the laser focus with respect to the resist. Using galvanometric mirrors to move the laser focus allows for high writing speeds of up to 20 mm/s. The cross-linking of the molecules leads to a change in the material’s refractive index and solubility in organic solvents. After washing off the un-cured resist in a development step, solid structures remain. Fabricated structures have a minimum feature size of about 160 nm in the x–ydirection and 1 μm in the vertical direction (when using the 63 ×objective), while mesoscopic options allow for up to 8 mm object height.27 Presently, mainly UV-curable polymer resists of a refractive index of about 1.5 are used, but the development of metal resists,43–46 metal oxides,47and bio-materials48and the integration of optically active substances (e.g., laser dyes, quantum dots, nanodiamonds, etc.49–54) are under way. IV. PHOTONIC CRYSTALS Historically, the Nanoscribe was developed to fabricate 3D photonic crystals.37,39Such crystals are described by using the full APL Photon. 6, 080901 (2021); doi: 10.1063/5.0058478 6, 080901-2 © Author(s) 2021APL Photonics PERSPECTIVE scitation.org/journal/app Maxwell equations with a spatially periodic dielectric function. While a great variety of photonic crystals have been fabricated using the Nanoscribe alone39,55–58and in combination with coating or infiltration techniques,18,59–68only recently have topological struc- tures been examined. 3D photonic crystals host many of the same topological phenomena as conventional solids and are known to host topological degeneracies in momentum space, such as Dirac points, Weyl points, and nodal lines.17 Dirac points and nodal lines are point and line degeneracies, respectively, that occur in the presence of certain symmetries that are required to prevent a gap from opening at the degenerate momenta. In contrast, Weyl points are robust degeneracies that are protected by an integer topological invariant or charge. In other words, any arbitrary periodicity-preserving perturbation merely displaces the Weyl point in momentum space but cannot cause a gap to open at the Weyl point. Furthermore, higher charged Weyl points can exist in the presence of additional spatial symmetries, which cause Weyl points of the same charge to overlap. Charge-2 Weyl points have recently been observed in a chiral woodpile photonic crystal fabricated using the Nanoscribe at low refractive index contrast19 [Figs. 2(a) and 2(b)]. At such a low refractive index contrast, there is no complete bandgap surrounding the Weyl point since the pro- jected bands of a finite structure overlap. Nevertheless, the disper- sion of the bands forming the Weyl point can be directly observed in the reflection spectrum of the photonic crystal. This is possible since s- and p-polarized light couples selectively to the bands, providing a way to map out the feature of interest. Due to the bulk-boundary correspondence, Weyl points are associated with Fermi-arc states that reside on the surface of a finite sample whose dispersion connects Weyl points of opposite charge. A measurement of such Fermi-arc states in photonics would allow for a direct observation of the topological charge of the Weyl point. In the photonic crystal sample made of the photoresist only, projec- tions of other bulk bands do not allow for a clear measurement of the associated Fermi-arc surface states. However, increasing the refrac- tive index using infiltration techniques will likely allow for a clear observation of the surface states in the spectrum. Weyl points can also be found in gyroid photonic crystals.7,17 While many gyroid photonic crystals have been fabricated using the Nanoscribe,69,70Weyl points have not been directly observedin such structures consisting only of the bare photoresist. How- ever, Nanoscribe-made photonic crystals can also be coated with or transformed into high-index material, such as titania65or sili- con,59,60,66and also metals.62,63Several processes have been devel- oped for that including the use of chemical vapor deposition (CVD) and/or atomic layer deposition (ALD).67Peng et al. reported on the fabrication of a single gyroid by direct laser writing, subsequent coating with a thick layer of Al 2O3via ALD, and infiltration with silicon using CVD.71Goiet al. were able to observe the signature of charge-1 Weyl points in direct laser written gyroid crystals coated with antimony telluride by ALD18[Figs. 2(c) and 2(d)]. The same group also developed a protocol to increase the effective refractive index of a gyroid photonic crystal by 40% using charged layer-by- layer deposition of PbS thin films.64Gyroids have also been fabri- cated in chalcogenide glasses72and out of a resist–silver composite by electroless deposition.73 Photonic crystals with Dirac and linear Weyl points exhibit vanishing density of states at the degeneracies and as such can potentially be used to realize large-volume single-mode lasing.74,75 Furthermore, such degeneracies can be used to achieve long-range algebraic interactions between embedded quantum emitters.76In Nanoscribe structures, this can be readily achieved by dissolv- ing dyes and quantum dots in the photoresist before writing.49–54 While the usual dimensions of photonic crystals that can be fab- ricated with the Nanoscribe exhibit features in the near infrared wavelength range, several mechanisms have been examined to shift them to smaller wavelengths. Among these are stimulated emission depletion lithography77and controlled post-printing shrinking.78–80 The aforementioned woodpile photonic crystals with Weyl points can also be realized using air-in-metal designs that are reason- ably described by using a tight-binding model.81More complicated 3D metallic structures can be realized using the Nanoscribe by either printing a template structure, which is then coated with a metal using vapor deposition or inverted using electroplating techniques,63or by direct micro-printing of metallic structures.43–46This allows for tight-binding models with unusual properties such as higher-order topology to be directly implemented using the photonic crystals platform. Losses and gain are inherent to many optical systems, and the effects of non-Hermiticity on the topological properties of photonic FIG. 2. Photonic crystals bearing Weyl points. (a) Scanning electron microscopy image of a chiral woodpile photonic crystal. (b) Angle-resolved transmission spectrum that shows a charge-2 Weyl point (indicated by an arrow). (c) Using ALD and CVD, a photonic crystal fabricated by direct laser writing can be transformed into a photonic crystal with a higher refractive index. (d) Scanning electron microscopy image of the direct laser written gyroid structure (top) and of the structure coated with Sb 2Te3(bottom). Scale bars are 1 μm. [(c) and (d)] Reprinted with permission from Goi et al. , Laser Photonics Rev. 12, 1700271 (2018). Copyright 2018 John Wiley & Sons, Inc. APL Photon. 6, 080901 (2021); doi: 10.1063/5.0058478 6, 080901-3 © Author(s) 2021APL Photonics PERSPECTIVE scitation.org/journal/app crystals are yet to be sufficiently explored. Recently, it was exper- imentally shown that a photonic Dirac point transforms into an exceptional ring in the presence of losses. The experiment showing the existence of this Dirac exceptional ring was performed in a pho- tonic crystal slab made out of silicon nitride;82however, such phe- nomena should exist at any refractive index contrast in the presence of sufficiently large non-Hermiticity. Moreover, the photoresists commonly used with the Nanoscribe exhibit vibrational resonances in the mid-IR, which could be used to further explore topological effects in the vicinity of these lossy resonances. V. WAVEGUIDE SYSTEMS A wide range of topological phenomena has been examined in waveguide systems.16Light propagation through an array of wave- guides can act as a model for quantum mechanical systems due to the mathematical analogy between the paraxial approximation of the Helmholtz equation that governs light propagation in a wave- guide and the Schrödinger equation that describes the evolution of a quantum mechanical state.83,84In solids where the electrons are strongly bound to atoms, the system can be accurately described by a tight-binding Hamiltonian. In photonics, this analogously extends to waveguide arrays where light is strongly confined to the wave- guides due to their higher refractive index than the medium in which they are embedded. The eigenmodes of a cylindrical optical waveguide are trans- verse electromagnetic modes, with a corresponding propagation constantβ. The fundamental mode, i.e., the one with highest β, has a Gaussian shape, and its propagation constant can be writ- ten asβ0=2πneff/λ, with the effective refractive index neffand the wavelength λ. The propagation constant β0thus is proportional to the inverse of the length along zover which the phase of the wave changes by 2 π. It is equivalent to the eigenenergy of an atomic wave function at its own lattice site. In a waveguide, light can be guided inside the core region as long as 2 πn0/λ<β0<2πnwg/λ, with nwg andn0being the refractive index of the core and of the surround- ing material, respectively. Outside the core material, the amplitude of the light is decaying exponentially. If a second waveguide is placed in close vicinity to the first one, it can pick up the decaying field such that the light couples to this second waveguide while propagating along z. This is called evanescent coupling. For an array of Ncoupled equal single-mode waveguides, we can describe the evolution of the light inside the waveguide array by the field amplitudes aq(z)of the eigenmodes in each waveguide qalong the propagation direction zwith Ncoupled differential equations,85,86 i∂ap ∂z=∑ ⟨q⟩NNcp,qaq+β0ap, (1) where the sum is only over nearest neighbors. The coupling cp,q between two waveguides, at distances dxanddyfrom each other in thexandydirection, is given by the overlap of the transverse field modes ⃗Ep(x,y)and⃗Eq(x,y)=⃗Ep(x−dx,y−dy), cp,q=2π λ∬/leftr⫯g⊸tl⫯ne →E∗ q(x,y)Δnq(x,y)/leftr⫯g⊸tl⫯ne →Ep(x,y)dxdy, (2) assuming that at waveguide q, the refractive index contrast Δnq =nwg−n0is small.In such systems, a single waveguide corresponds to an atomic site, while the waveguide axis zacts analogously to the time vari- able in the Schrödinger equation. The energy in electronic systems is replaced by the waveguides’ propagation constants β. For an array of waveguides, we thus obtain a band structure of βover the transverse wavevector k. The intensity distribution along zover waveguide sites is equivalent to the time evolution of the probability density of an electron in the periodic potential of a solid state material. It is pos- sible to map this intensity distribution by exciting a certain state at the waveguide array’s input facet and observing the intensity distri- bution at the output facet. Reciprocal space is also accessible in both addressing the waveguides by the use of a spatial light modulator to define the phase at the input facet and imaging momentum space by introducing an additional lens after the output facet to perform a Fourier-transformation87[Fig. 3(d)]. Existing methods to fabricate waveguide arrays are fs-writing in glass,84,88–90surface plasmon polariton waveguides,91,92optical induction of waveguides in photorefractives,93,94etc. The Nano- scribe not only broadens the range of accessible systems to do exper- iments but also has some unique advantages (see Secs. V A, V B, and VI). So far, two different methods are used to create waveguide arrays with the Nanoscribe, explained in Secs. V A and V B. A. Infiltrated waveguides 1. Fabrication The inverse of a waveguide array is printed with the Nanoscribe [Fig. 3(a)], which leads to an array of empty channels after develop- ment of the uncured photoresist [Fig. 3(b)]. These empty channels are then infiltrated with a second photoresist, e.g., SU8, with the help of capillary forces [Fig. 3(c)]. The sample is baked on a hotplate to solidify the infiltrated resist. Using IP-Dip as a resist during print- ing and SU8 as an infiltration material, a refractive index contrast of about 0.05 between the waveguide core and the surrounding mate- rial is achieved. The big advantage of this method is the flexibility to choose different infiltration materials (see also Sec. VI A) as well as mixing dyes into the resist.51,54Typical single-mode waveguides are ≥1μm in diameter, of lengths up to 1 mm, and have a center-to- center distance to their neighbors of about 1.4 μm. While this fabri- cation method leads to waveguides with a round cross section—since the writing laser focus is only elongated along the zaxis, the prop- agation axis of the waveguides—the structures are very sensitive to parameters during printing. For example, waveguide channels with a diameter smaller than 1 μm or printed using too high laser power tend to clog during development. This is due to the chemical poly- merization process: During printing, photo-initiator molecules that are excited in the area surrounding the empty channels can dissipate into the cross-sectional area of the channels and start polymerization reactions there, even though that area has not been illuminated.96 These reactions can lead to polymer filaments inside the waveguide channels, hindering the infiltration or causing the complete clog- ging of a waveguide. To minimize this effect, the waveguide radius needs to be ≥1μm and the structure is written with a laser intensity close to the polymerization threshold to reduce the concentration of excited photo-initiator molecules. In addition, employing aberra- tion pre-compensation of the laser focus used for two-photon poly- merization via a spatial light modulator97,98leads to some improve- ment, as it further confines the laser focus and concentrates the APL Photon. 6, 080901 (2021); doi: 10.1063/5.0058478 6, 080901-4 © Author(s) 2021APL Photonics PERSPECTIVE scitation.org/journal/app FIG. 3. Fabrication of infiltrated waveguide arrays. (a) The inverse of a waveguide array is printed using the Nanoscribe. After development, the sample consists of empty channels arranged on a lattice: (b) Scanning electron microscopy images from top (i) and side (ii) of typical structures.95These empty channels are infiltrated with a second resist (c). Under the microscope, unfilled waveguide channels look dark as they scatter light into the surrounding material with a higher refractive index. As the channels are infiltrated with a higher index material, they become waveguides and turn bright. (d) Measurements are done by focusing light into one or multiple waveguides at the input facet and observing the intensity pattern at the output facet. The introduction of an additional Fourier-transforming lens (dashed) between objective 2 and the camera allows for the imaging of the state population in momentum space.87 polymerization reaction to a smaller volume. To avoid deformations during the development process caused by the shrinking of struc- tures written with a low laser intensity, additional supports can be written around the structure [Fig. 3(b-ii)]. The combination of SU8 as a waveguide material and IP-Dip as a surrounding material leads to a refractive index contrast of 0.05, as opposed to a contrast of about 7 ⋅10−4for waveguide fabrication in glass.84Therefore, the light is bound more strongly to the wave- guide core, which allows for a tighter bending of helical waveguides. For waveguides with a center-to-center distance of 1.5 μm, the cou- pling is relatively strong with a hopping distance of around 60 μm,95 which reduces the required overall length zof a sample that is neces- sary in order to observe enough of the propagation dynamics. This is to our advantage since the biggest limitation of this fabrication method lies in the realizable propagation length. As the infiltration process relies on capillary forces, the height of a structure is limited to a maximum of 1 mm (for which the parameters during printing need to be exactly fine-tuned to allow infiltration), while a height of 500μm is, in general, easily infiltratable. 2. Implementations using infiltrated waveguide arrays Waveguide arrays fabricated by infiltration were first used to recreate95a photonic Floquet topological insulator (FTI).99The photonic FTI consists of waveguides arranged on a honeycomb lattice [Fig. 3(b)] whose trajectory along zfollows a helical path. The helical spin of the waveguides effectively leads to a phase term in the coupling constant, making it complex, such that the system mim- ics the topological Haldane model.100Mathematically, this phase term can be derived by writing the tight-binding equations in the coordinate system co-rotating with the waveguides to obtain an arti- ficial vector potential,99followed by a Peierls substitution.101The arrangement resembles a graphene lattice in a circularly polarized electric AC-field. Due to the breaking of time-reversal symmetry by this artificial field, the system supports a chiral topological mode that moves along the edge of the array only in one direction (only clockwise or only counter-clockwise). Since the topological edge state has a monotone group velocity at each edge, it cannot back- scatter when it encounters a defect, such as a missing waveguide or a corner. In addition, no scattering to the bulk waveguides is pos- sible since the edge state’s energy lies within a bandgap. To test this robustness against “time”-dependent defects, waveguides with different trajectories along zthan the rest of the sites (e.g., oppo- site helicity or a straight waveguide) were put on the edge to see whether the edge mode still moves around these [Fig. 4(a)]. It was observed that as long as the defect’s rotation-frequency in zis the same as the other waveguides’, the edge mode is robust. FTIs fabri- cated this way were also sought to be used as a sensor of changes APL Photon. 6, 080901 (2021); doi: 10.1063/5.0058478 6, 080901-5 © Author(s) 2021APL Photonics PERSPECTIVE scitation.org/journal/app FIG. 4. Implementations of infiltrated waveguide arrays. (a) A photonic Floquet topological insulator with time-dependent defect at the edge.95(b) Interfaces between different photonic artificial gauge fields lead to refraction and reflection ink-space.87 in the refractive index of the infiltrating liquid in the waveguide cores.102 A one-dimensional implementation used the Su–Schrieffer– Heeger (SSH) model and its associated topological edge states.103In the middle of the SSH chain, a defect is created by interfacing a topo- logically trivial with a non-trivial chain such that the resulting defect waveguide hosts a topological edge state.104The waveguide that con- stitutes that defect is modulated in z(“time”), generating Floquet copies of the edge mode in kz(i.e., “energy”). When the frequency of the defect modulation is such that the kzof a Floquet copy of this edge mode coincides with a bulk band, light guided in the defect waveguide can couple out into the lattice. Otherwise, it is trapped.105 As stated before, topological photonics relies on the creation of artificial gauge fields.33,34AGFs in waveguide systems can be cre- ated by altering the trajectories of waveguides along z. In a wave- guide implementation, it was shown that at a boundary between two regions with different AGFs, refraction and reflection occurs, even though the two regions are otherwise of exactly the same pho- tonic medium. Reflection and refraction at such “gauge interfaces” are governed by a generalized Snell’s law.87One can measure the states’ population in momentum space of the incoming, refracted and reflected light by Fourier-transforming the electric field at the output facet with a lens [Fig. 4(b)]. Despite the range of implementations using infiltrated wave- guide arrays, the infiltration method is not applicable to all systems. Some applications or models require longer propagation lengths. This calls for longer waveguides that are tricky to obtain using this infiltration method due to limited stability for very long structures and an increase in difficulty during infiltration for long structures. Additionally, some implementations require the ability to mod- ulate the refractive index profile of a waveguide along its crosssection or the effective refractive index of a waveguide along z. This is readily possible using the fabrication method described in Sec. V B. B. Direct printing of waveguides by using differing laser power 1. Fabrication With the Nanoscribe, the refractive index of the fabricated structures depends on the amount of cross-linking achieved in the resist during the writing process. In turn, the amount of cross- linking depends on the dose and thus the laser power of the beam that is focused into the resist.108,109This way, one can print the core of a waveguide using a higher laser power than for the surround- ing material and achieve a refractive index difference of up to 0.008. Skipping the development step after printing and post-curing the excess resist under UV light leads to waveguides of very long lengths (up to 8 mm length have been fabricated so far), only limited by the Nanoscribe’s z-drive range. 2. Implementations of directly printed waveguides Using to advantage the large lengths of the waveguides pos- sible with this fabrication method, negative next-nearest neighbor (NNN) coupling was shown to exist in angled 1D arrays of cou- pled waveguides106[Fig. 5(a)]. Adjusting the angle of a zig-zag chain of waveguides, the value and sign of the NNN-coupling c2can be tuned. For positive values of c2, the spreading along the array of a wave packet with a wavevector of kxclose to zero is large com- pared to a wave packet at the edge of the Brillouin zone kx=π/dx (dxbeing the lattice constant), while for negative c2, it is the oppo- site. Interestingly, it was observed that a negative NNN-coupling exists naturally even in a straight 1D array of waveguides fabricated with this method. Since in some systems the NNN-coupling is neces- sary for the creation of topologically non-trivial phases,90,110–112the correct phase of the coupling is very important. In addition, being able to tune c2to zero allows for decreasing the distance between the waveguides to achieve higher nearest neighbor coupling without introducing unwanted effects caused by NNN-coupling. The presented fabrication method also allows us to make waveguides of almost arbitrary cross section and trajectory [see Fig. 6(b)], e.g., multimode waveguides that support modes carry- ing orbital angular momentum (OAM) with ℓ≠0. Beams carrying OAM have a helically shaped phase front and a phase singular- ity in the beam’s center. By encircling the singularity, the electric field amplitude collects a phase of 2 πℓ,ℓ∈Z, which is why ℓis also referred to as the topological charge of the beam. In order to sup- port OAM modes, the OAM mode has to be an eigenmode of the waveguide. The OAM mode of ℓ=±1 is a superposition of the TE01 and TE10 mode, which implies that the waveguides must have a cir- cular shape, as an elliptical shape would lift the degeneracy of the TE01 and TE10 mode, thereby making the OAM mode not a valid eigenmode of the waveguide. In a diamond lattice of directly printed waveguides [Figs. 5(b) and 5(c)], a gauge field depending on the OAM of the input light was demonstrated using the Aharonov–Bohm effect.107Inserting a wave- guide mode with a non-zero OAM, ∣ℓ∣=1, leads to non-vanishing flux through a plaquette and thus to Aharonov–Bohm-caging [lower row in Fig. 5(c)], while excitation with a constant phase profile, ℓ=0, APL Photon. 6, 080901 (2021); doi: 10.1063/5.0058478 6, 080901-6 © Author(s) 2021APL Photonics PERSPECTIVE scitation.org/journal/app FIG. 5. Waveguides printed using different laser power for the core and the surrounding material. (a) Experimental demonstration of negative next-nearest neighbor coupling in an angled 1D chain of waveguides. While for positive values of the next-nearest neighbor coupling c2, light spreads little to the array upon excitation with a wavevector ofkx=π/dx, for negative c2, it spreads significantly.106(b) Waveguides arranged on a diamond lattice for the demonstration of an OAM-dependent gauge field using the Aharonov–Bohm effect. Inserting a waveguide mode with a vortex-shaped phase profile leads to non-vanishing flux through a plaquette and thus Aharonov–Bohm-caging [lower row in (c)] such that the light comes back to the excited waveguide after each 500 μm of propagation distance. In contrast, excitation with a constant phase profile only leads to spreading of light in the lattice [upper row in (c)].107 FIG. 6. (a) Upcoming directions for infiltrated waveguides. [(i) and (ii)] Waveguide structure printed parallel to the substrate, infiltrated with a mixture of SU8 and laser dye Oxazin 1:133Reprinted with permission from J. Schulz, “Herstellung und Charakterisierung evaneszent gekoppelter Wellenleiterstrukturen,” M.S. thesis, TU Kaiserslautern, 2018. (i) Scanning electron microscopy image of the waveguide structure and prisms. (ii) Composite fluorescence image of light propagating through the waveguide array. (iii) Site-dependent infiltration of waveguides with two different materials: white waveguides are infiltrated with SU8, while blue waveguides are infiltrated with a mixture of SU8 and Oxazin 1. Reprinted with permission from A. Bayazeed, “First steps to produce a non-Hermitian system of coupled waveguides by selective infiltration,” B.Sc. thesis, TU Kaiserslautern, 2020. (b) Donut-shaped waveguides fabricated by direct printing of waveguides using different laser powers. (c) Waveguides (blue lines on the three-dimensional gray object) in curved space. Reprinted with permission from Lustig et al. , Phys. Rev. A 96, 041804 (2017). Copyright 2017 American Physical Society. The periodic curvature of space introduces topological phase transitions of the SSH-like model for usinθ<vasxexpands and contracts. During propagation along z, these transitions show in the alternation of light between bulk states and edge states. (d) Nitrogen-vacancy center nanodiamonds embedded in waveguides. Reprinted from Landowski et al. , APL Photonics 5, 016101 (2020) with the permission of AIP Publishing. The electron multiplying CCD (EMCCD) image (i) shows the fluorescence of a single integrated nanodiamond, overlayed in (ii) with the reflected light microscope image. Scale bar is 20 μm. (iii) Tilted scanning electron microscopy image of such a waveguide. APL Photon. 6, 080901 (2021); doi: 10.1063/5.0058478 6, 080901-7 © Author(s) 2021APL Photonics PERSPECTIVE scitation.org/journal/app does not achieve caging but only leads to spreading of light in the lat- tice [upper row of Fig. 5(c)]. This allows us to switch the topology of the lattice from trivial to non-trivial by only changing the OAM of the input beam using the same fabricated structure. VI. UPCOMING NEW DIRECTIONS A. Infiltrated waveguides A key advantage of the fabrication method of waveguides by infiltration is that we are not limited to only one material; infiltration with a wide range of materials is possible. Work-in-progress includes infiltration with materials that show nonlinearity or absorption, by, e.g., mixing dyes into the photoresist.51,113To actually observe light as it propagates through a waveguide array (in a similar way as in Ref. 114), a laser dye can be solved in the infiltration material. The local fluorescence allows the observer to track the path of the light along z. To image the emitted light from the structure, it is help- ful if the waveguides are aligned parallel to the substrate such that the entire propagation length can be imaged with the second objec- tive in Fig. 3(d). Additional prisms were fabricated at the ends of the structure to facilitate in- and out-coupling of light from the waveg- uides [Fig. 6(a-i)]. The 3D micro-printed structure was infiltrated with a solution of SU8 and the dye Oxazin 1 with a concentration of 1 ml SU8 to 5 mg of Oxazin 1. In the composite fluorescence image in Fig. 6(a-ii), the quantum-walk like dispersion of the light in the array of waveguides can be seen, when light with a wave- length of 680 nm is coupled via the prism into one waveguide at the input facet. Using objectives with higher focal lengths would allow us to use the previous version of waveguide structures, with znor- mal to the substrate, removing the need for the in- and out-coupling prisms. Another upcoming direction is to explore the role of non- linearity in topological photonic systems. This is possible in infil- trated waveguides by mixing materials with a strong nonlinear response into the infiltrating resist. One of the main challenges of this approach is the inability of the photoresists to withstand very high laser power115(unlike glass waveguides used in Refs. 116–119). Therefore, there is a need to find materials with sufficiently high nonlinear indices such that only low peak powers are needed. In addition, the mixture must exhibit low absorption and suitable refractive index at operating wavelengths and low viscosity for infil- tration. Besides Kerr nonlinearity, other types of nonlinear inter- actions could also be introduced depending on the choice of the infiltrating material. Yet another possibility to be considered is to have site-dependent nonlinearity by infiltrating different waveguide sites with different materials (as described below). Besides mixing dyes into the infiltration material, they can also be added to the resist before printing.113This has already been exploited to fabricate a waveguide equivalent of Flamm’s paraboloid.120Adding fluorescein to IP-Dip, the authors were able to observe the evolution of light in a structure inspired by black holes. Also very recently, a method has been developed to selec- tively infiltrate different independent waveguides with different materials121[see Fig. 6(a-iii)], even further expanding the degrees of freedom in waveguide arrays. This method can, for example, be employed to realize non-Hermitian topological systems withpatterned gain and/or loss. With loss and gain, the system is inher- ently open and interacts with its environment. As a result of this non-Hermiticity, the time evolution is not unitary as the eigenvec- tors are no longer orthogonal and the eigenvalues can be complex. One application for this is a topological laser10where the imaginary part of the eigenvalue of the topological edge mode is increased so that only in this mode lasing can occur. Introducing gain and loss into the system does not neces- sarily implicate that the eigenvalues will become complex. If an eigenvector of the Hamiltonian is also an eigenvector of the parity- time (PT) operator, its corresponding eigenvalue is real. In this phase, the gain/loss only has a small influence compared to the bandgap. Upon increasing gain/loss, the bandgap shrinks. When the bandgap closes, some eigenvalues become complex conjugate pairs and the PT-symmetry is broken. Further increasing gain/loss such that it becomes dominant, the system reaches the anti-PT- symmetric phase, where the bandgap opens again; however, this time all eigenvalues are purely imaginary.122The point at which this phase transition happens is referred to as an exceptional point, where the eigenvectors of the system coalesce and the eigenvalues switch from being purely real to being complex conjugate pairs. Such non-Hermitian systems harboring exceptional points and their interplay with topology have gained great interest in recent years. For example, it has been recently demonstrated that a Weyl point—a monopole of Berry curvature—expands from a point-degeneracy into an exceptional ring as the non-Hermiticity is increased. The topological charge is preserved but is distributed over the ring of exceptional points.123 To implement such systems in a waveguide structure, the indi- vidual amount of loss in each waveguide needs to be adjustable. This has been done (in fs-laser written waveguide systems) either by exploiting that radiation losses can be tuned by the bending of waveguides124or by introducing scattering points125or breaks123 along the waveguide trajectories. In those cases, the losses are caused by periodically coupling to the continuum of radiating modes. As our infiltration method allows us to use multiple mate- rials, losses could be realized via different amounts of absorption. This design freedom also opens the door to achieve site-dependent nonlinearity. B. Directly printed waveguides Upcoming new directions, facilitated by direct printing of waveguides using different laser powers as described in Sec. V B, include the potential to create extra (artificial) dimensions by the use of multimode waveguides. The radial refractive index profile of the waveguide can be changed almost at will. For example, donut- shaped waveguides can be printed [Fig. 6(b)] that support OAM modes with higher ℓwhile suppressing Laguerre Gaussian modes with knots along their radial coordinate. Furthermore, this allows us to experiment with modes other than the ground mode that does not have an interesting phase profile itself. By printing waveguides with elliptical cross sections, the degeneracy between TE01 and TE10 modes can be explicitly lifted, creating direction-dependent positive or negative coupling. The design freedom in the fabrication process does not end at elliptical or donut-shaped waveguides but allows almost arbitrary cross sections and trajectories of each individual waveguide. APL Photon. 6, 080901 (2021); doi: 10.1063/5.0058478 6, 080901-8 © Author(s) 2021APL Photonics PERSPECTIVE scitation.org/journal/app Thereby, this platform could be used to provide the experi- mental proof of principle for many proposed OAM-converting waveguide structures. The discrete rotation symmetry of the wave- guide cross section and the helical twist are essential features of these proposals, necessary to lift the degeneracy between OAM modes of equal∣ℓ∣.126,127The OAM can also be inverted from ℓto−ℓby cou- pling two helically twisted elliptical waveguides with opposite helic- ity.128These waveguide structures would be harder or impossible to implement in other platforms. Since not only a complex trajectory and freely chosen cross sec- tions but even the refractive index profile can be designed within this fabrication method, this opens the door to fabricate wave- guides with a configurable inner mode structure. For example, some works have transferred the principle of supersymmetry to the opti- cal regime. In optical waveguides, a supersymmetric partner system has eigenmodes with propagation constants matching the ones from the original system, but missing the ground mode. This has so far been demonstrated for systems of coupled waveguides129and its implications on topological states,130but it has not yet been imple- mented by adjusting the refractive index profile, as proposed in Refs. 131 and 132. The ability to tailor the refractive index profile and implement structures with higher mode profiles opens up many new directions of research, and we are looking forward to interesting discoveries in this field. C. Topological photonics in curved space Since the Nanoscribe prints 3D structures, it seems optimally suited to fabricate devices using curved space to control the prop- erties of light120,136–138and exploit its characteristics for topology. Recent works suggested that the interplay between the curvature of space as in general relativity and the topology of the system could lead to new effects in topological physics.134The metric of a curved surface alters the potential139and/or coupling between waveguides printed onto the surface of a three-dimensional body134 [see Fig. 6(c)]. Unique to waveguides on curved surfaces is that the phase front between neighboring waveguides can be kept the same while changing the distance between them (and thus the coupling), which is not possible on flat surfaces. In particular, a “time” (i.e., z)-varying curvature of the surface acts as a metric- dependent gauge field and allows for tuning of topological phase transitions via its (periodic) curvature. It has been theoretically demonstrated that Thouless pumping and curvature-induced delo- calization in the Andre–Aubry–Harper model can be implemented in such systems.134 D. Single photons Upcoming directions include doing topological photonics with actual quantum states, i.e., single photons.140–144Using the Nano- scribe, structures such as waveguides can be printed containing nan- odiamonds with a nitrogen-vacancy (NV) center135,145–147[Fig. 6(d)] to act as embedded single photon sources. While at room tem- perature, NV-centers might not be directly suitable for quantum experiments due to the low yield in un-distinguishable single pho- tons,148other single photon sources, including silicon or germanium vacancy centers, or quantum dots148could be integrated in the same fashion into 3D printed structures.VII. CONCLUSION The Nanoscribe is a flexible instrument that allows for the fab- rication of a multitude of structures with unique topological prop- erties. Due to the design freedom that direct laser writing offers, it is possible to readily fabricate interesting 2D and 3D photonic struc- tures even in curved, periodic, or more complicated geometries. This Perspective provided an overview of the topological effects that have been observed in structures fabricated by the Nanoscribe, ranging from photonic crystals to different types of waveguide structures. The fabrication of waveguide systems benefits from the Nanoscribe’s great flexibility and precision. Waveguide structures fabricated with the infiltration method allow us to selectively introduce and benefit from material properties of the chosen infiltration medium, a fea- ture unavailable for other waveguide platforms. The outlined new fabrication method for waveguides, using the dependence of the refractive index on the laser power, allows us to design the cross section of the waveguides to explore the interplay of topology and higher-order waveguide modes. Besides the flexibility in geometry, we suggested how different material properties, such as fluorescence or single photon emission, could be exploited when introduced in these photonic structures. Since the Nanoscribe is widely available at many universities, we are convinced that more people will be able to contribute to the fascinating field of topological photonics. ACKNOWLEDGMENTS J.S. acknowledges funding from the Deutsche Forschungs- gemeinschaft through CRC/Transregio 185 OSCAR (Project No. 277625399). S.V. acknowledges the support of the U.S. Office of Naval Research (ONR) Multidisciplinary University Research Ini- tiative (MURI) under Grant No. N00014-20-1-2325 on Robust Pho- tonic Materials with High-Order Topological Protection as well as the Packard Foundation under Fellowship No. 2017-66821. C.J. acknowledges funding from the Alexander von Humboldt Founda- tion within the Feodor-Lynen Fellowship program. The authors are grateful to Georg von Freymann and Mikael C. Rechtsman for their comments and suggestions. 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5.0055002.pdf

The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Superposition-state N 2+produced in the intermolecular charge transfer from low-energy Ar+to N 2 Cite as: J. Chem. Phys. 154, 234303 (2021); doi: 10.1063/5.0055002 Submitted: 24 April 2021 •Accepted: 30 May 2021 • Published Online: 16 June 2021 Jie Hu, Jing-Chen Xie, Chun-Xiao Wu, and Shan Xi Tiana) AFFILIATIONS Hefei National Laboratory for Physical Sciences at the Microscale, Collaborative Innovation Center of Chemistry for Energy Materials (iChEM), Department of Chemical Physics, University of Science and Technology of China, Hefei 230026, China a)Author to whom correspondence should be addressed: sxtian@ustc.edu.cn ABSTRACT Molecular electronic or vibrational states can be superimposed temporarily in an extremely short laser pulse, and the superposition-state transients formed therein receive much attention, owing to the extensive interest in molecular fundamentals and the potential applications in quantum information processing. Using the crossed-beam ion velocity map imaging technique, we disentangle two distinctly different pathways leading to the forward-scattered N 2+yields in the large impact-parameter charge transfer from low-energy Ar+to N 2. Besides the ground-state (X2Σg+) N 2+produced in the energy-resonant charge transfer, a few slower N 2+ions are proposed to be in the superpositions of the X2Σg+-A2Πuand A2Πu-B2Σu+states on the basis of the accidental degeneracy or energetic closeness of the vibrational states around the X2Σg+-A2Πuand A2Πu-B2Σu+crossings in the non-Franck–Condon region. This finding potentially shows a brand-new way to prepare the superposition-state molecular ion. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0055002 I. INTRODUCTION Conical intersection is defined as a crossing point between two potential energy surfaces of adiabatic electronic states,1where the molecular nuclear motions in superposition of these two states significantly diversify the reaction pathways.2Such a superposition- state species is short-lived (picoseconds or less), and its experimental preparation is scarcely feasible. On the other hand, a femtosecond or attosecond laser pulse was used to generate a coherent superposition of cationic electronic states in atoms or molecules through photoionization.3,4The electronic states that were initially separated in energy were coherently superimposed in the ultrashort laser pulse field, but the nuclear motions in multiple freedoms subse- quently destroyed the valence electronic coherence within a few femtoseconds.5Much effort has been put into investigations on the properties of superposition-state molecules, arising from not only fundamental interests in chemical dynamics5,6but also the potential applications in quantum information processing.7–9In principle, molecular structure and dynamics, besides electronic and nuclear spins,9offer rich scales that are at the heart of new protocolsin quantum information. Therefore, it is full of meaning and imper- ative to establish practical methods to prepare the superposition- state molecules. Here, we report a completely different approach for this purpose, namely, producing the superposition-state N 2+ion by the intermolecular charge transfer (CT) from low-energy Ar+to neutral N 2. As a model system of the CT reactions, Ar+(2Pj, j=1/2, 3/2) +N2(X1Σg+,ν=0)→Ar(1S0)+N2+(X2Σg+,ν′) has been inves- tigated extensively.10–17It is endothermic to produce N 2+(X2Σg+, ν′=1) by 0.092 eV for Ar+(2P3/2) while exothermic by 0.086 eV for Ar+(2P1/2) when excluding the collision-energy transformation to the internal energy of N 2+in an energy-resonant CT. Moreover, the rotational state distributions of N 2+(X2Σg+,ν′=0, 1) were observed at much lower collision energy, where an intermediate [Ar–N 2]+was possibly observed.10,13,17The vibrational excitations (X2Σg+,ν′>1) of N 2+were found in crossed-beam experiments with the increase in collision energy.11,14,15Furthermore, the elec- tronically excited states, such as A2Πuand B2Σu+of N 2+, could be populated if a sufficient amount of collision energy is transformed J. Chem. Phys. 154, 234303 (2021); doi: 10.1063/5.0055002 154, 234303-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to the internal energy (E int) of N 2+. Recently, the ion velocity map imaging (VMI) technique18was applied in the crossed-beam experiments, providing the state-resolved angular differential cross sections of the ionic product.14,15,19In this work, we employ the crossed-beam VMI apparatus19to discover a unique pathway to the state-mixed N 2+yields in the CT reaction between Ar+and N 2, and furthermore, these N 2+ions are highly possible to be in the superposition state. II. EXPERIMENTAL METHOD AND COMPUTATION A clear velocity image of N 2+can be obtained with the time-sliced VMI method18,19and benefited from the well-confined bunches of the Ar+projectile ion.20The operation procedures can be found in Ref. 19, and a brief description is given here. High-purity (99.99%) samples (Ar and N 2) were available commercially and used without further purifications. The Ar+reactant ions were produced on electron impacts at 30 eV, and a statistical ratio of P 3/2:P1/2in the Ar+ion beam was about 2:1. During the experiments, the reac- tion chamber was evacuated under a steady vacuum condition of 4.2×10−7Torr and the reactions happen in the field-free region (within a volume of about 2 ×2×2 mm3). The N 2targets in a supersonic molecular beam collided with the pulsed Ar+ions. The N2+products were periodically pushed out and expanded as a New- ton sphere in the VMI lens system, and the central slice (about 60 ns thickness) of the Newton sphere of the product N 2+ions wasselectively detected with a set of multichannel plates plus a phosphor screen. A CCD camera was used to record this sliced image. The working frequency of the above cycle was 500 Hz. The kinetic energy or velocity and its spread ( ΔE orΔv) of the reactant were determined prior to the collision experiments, and the energy resolution in the sliced image of the product N 2+was limited primarily due to the kinetic energy resolution ( ΔE/E∼8%) of Ar+. Excluding the spin–orbit coupling effect (because the N 2+ products cannot be distinguished from the present reactions with 2P3/2- and2P1/2-state Ar+), the potential energy surfaces of the (Ar–N 2)+system were calculated with the state-averaged complete- active-space self-consistent field method21and aug-cc-pVTZ basis.22 For ensuring the symmetry consistency of different θangles in the Jacobi coordinate, we used the Cspoint group of the system, and the active space consisted of the valence orbitals 7a′-15a′and 2a′′-4a′′, while the core orbitals were frozen. Two representative potential energy surfaces ( θ=90○and 30○) were scanned, where the N–N bond length r was varied from 0.82 to 1.50 Å with a step of 0.02 Å and R (the distance from Ar to the middle point of the N–N bond) was from 1.20 to 5.12 Å with a step of 0.08 Å. Thereby, each potential energy surface was constituted with 1750 energy points. III. RESULTS AND DISCUSSION The laboratory-coordinate vdistribution of the N 2+products was transformed to the velocity udistribution in the center-of-mass FIG. 1. Velocity transformation from the laboratory to center-of-mass coordinates (a) and time-sliced velocity images of the product N 2+at the center-of-mass collision energies of 1.98 (b), 2.47 (c), 3.73 (d), 6.37 (e), and 9.21 (f) eV. (a) Forward and backward scattering directions are defined, and the ion intensities in each image are normalized independently and scaled in different colors. (b)–(f) The left and right arrows represent the velocity vectors of the reactants N 2and Ar+, respectively. The circle (dashed) denotes the velocity of the N 2+ion produced in the energy-resonant charge transfer from Ar+(2P3/2). J. Chem. Phys. 154, 234303 (2021); doi: 10.1063/5.0055002 154, 234303-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (c.m.) coordinate, as described in Fig. 1(a). The time-sliced images were recorded in the collision-energy range of E c.m.=1.98–9.21 eV and plotted on the ux–uyplane after the coordinate transforma- tion in Figs. 1(b)–1(f). The N 2+products are distributed primarily in the forward scattering direction and concentrated gradually into smaller volume with the E c.m.increase, indicating the large impact- parameter collision mechanism. These yields have velocities close to those available in the energy-resonant CT (denoted with white dashed circles, i.e., the velocity of N 2+is equal to that of the tar- get N 2), implying that the most N 2+yields should be populated in X2Σg+(ν′=1). Comparing the image at E c.m.=2.47 eV and the previous one at 2.5 eV,15our new finding is a weak inner distribution of N 2+ yields in Fig. 1(c). By extending E c.m.to higher values, as exhibited in Figs. 1(d)–1(f), one can find an E c.m.-dependence of this weak dis- tribution. According to the energy conservation, the N 2+state is determined with the E intvalue with respect to the energy level of X2Σg+(ν′=0), Eint(N+ 2)=ΔIP+Ec.m.−KER tot, (1) where ΔIP is the ionization-potential difference between Ar and N2, e.g., ΔIP=0.159 eV for Ar+(2P3/2) and N 2+(X2Σg+,ν′=0) (Ref. 23) and KER totis the total kinetic energy release in the reaction. Considering the momentum conservation in the c.m. coordinate, we can derive the KER totvalue with the measured value of the N 2+ velocity, KER tot=mN+ 2(mN+ 2+mAr) 2m Ar∣uN+ 2∣2, (2) where m represents the mass of the product Ar or N 2+. On the other hand, the potential energy curves of N 2+in Fig. 2(a) (reproduced from Ref. 24) indicate that the electronically excited states A2Πu, B2Σu+or the higher states of N 2+are accessible by an efficient energy transfer from the collision energy. However, the most N 2+ions, as the fastest forward-scattered yields shown in Figs. 1(b)–1(f), are populated in the states nearby X2Σg+(ν′=1) due to the energy-resonant CT process. The slower ions, as the minor yields observed in the inner region, could be attributed to the higher electronic states of N 2+. We also notice that Fig. 2 shows two crossings among the potential energy curves of X2Σg+, A2Πu, and B2Σu+states located outside of the Franck–Condon region. Although photoionization cannot directly set foot in these non- Franck–Condon regions, the vibrational states of different electronic states in the vicinity of these crossings [as shown in Figs. 2(b) and 2(c)] could be populated with assistance of the N–N bond stretching in the CT process. Significant differences between the CT process and the molecular photoionization were frequently observed,19,25,26 but the bond-stretching assisted CT dynamics was never reported before. The E intprofiles are obtained from the images of Figs. 1(b)–1(f) and assigned with the electronic and vibrational states of N 2+for the CT reaction with Ar+(2P3/2) shown in Figs. 3(a)–3(e). Although Ar+(2P1/2) co-exists in the projectile ion beam, its contribution is minor because of the intensity ratio of2P3/2:2P1/2=2:1 and the lower CT cross sections in the present E c.m.range.12Comparing the He I photoelectron spectrum [Fig. 3(f), reproduced from Ref. 23], one FIG. 2. Potential energy curves of N 2+(a) and the enlarged views of X–A and A–B state crossings (b) and (c). (a) Ar+(2P3/2) state lies energetically (noted with a dashed line) between ν′=0 and 1 of X2Σg+, the Franck–Condon region of the N 2photoionization is shaded in yellow, and the state crossings at large and short N–N distances are shaded in different colors. (b) and (c) Some of the vibra- tional states around the crossing point are energetically close or degenerate. The potential energy curves and assignments are reproduced from F. R. Gilmore, R. R. Laher, and P. J. Espy, J. Phys. Chem. Ref. Data 21, 1005–1107 (1992) with the permission of AIP Publishing. can find that the vibrational state distribution of X2Σg+, as the major band in the CT spectrum, significantly deviates from the electron vertical promotion strengths, namely, the Franck–Condon factors. As shown in Fig. 3(f), the population at ν′=0 is highly preferred for the X state in the He I photoionization of N 2. In contrast, owing to the resonant CT process, the ν′=1 state of X2Σg+is predomi- nant at 1.98 and 2.47 eV, and the ν′=1, 2 states contribute jointly to the maximum intensities of the major bands at 3.73, 6.37, and 9.21 eV. The weak band in the E intspectrum also exhibits different pro- files with respect to those of the A2Πuand B2Σu+bands in the pho- toelectron spectrum. Only the low-lying vibrational states of A or B states are accessible in photoionization [see Fig. 3(f)], but they are completely absent in the CT process according to the assignments for the weak band observed in Figs. 3(b)–3(e). The assignments in Figs. 3(b)–3(e) are obtained from the energetics calculation with Eq. (1), indicating the unusual populations of the vibrational states of X2Σg+, A2Πu, and B2Σu+of N 2+. It is surprising that these elec- tronic states or the vibrational states of a certain electronic state are not accessed sequentially with the E c.m.increase. The state popu- lation in such an abrupt manner should rise from some unusual dynamics. Since a few vibrational states around the X2Σg+-A2Πu and A2Πu-B2Σu+crossings are energetically close or degenerate [as shown in Figs. 2(b) and 2(c)], the weak band may be attributed to specific superpositions of the above energy-closing or degen- erate vibrational states. For instance, the weak band (highlighted in dark orange) observed in Fig. 3(b) can be assigned as the state superposition, ∣φ⟩=α∣X2Σ+ g,ν′⟩+β∣A2Πu,ν′⟩, (3) J. Chem. Phys. 154, 234303 (2021); doi: 10.1063/5.0055002 154, 234303-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. Internal energy (E int) distributions of the product N 2+at 1.98 (a), 2.47 (b), 3.73 (c), 6.37 (d), and 9.21 (e) eV and He I photoelectron spectrum of N 2(f). (a)–(e) The vertical dashed line denotes the energy position of resonant charge transfer from Ar+(2P3/2), E intis given with reference to the state X2Σg+(ν′=0), and the superposition- state N 2+yields are presented as colored bands and assigned with the vibrational states that are possibly involved. (f) The He I photoelectron spectrum is from Kimura et al. , Handbook of HeI Photoelectron Spectra of Fundamental Organic Molecules. Copyright 1981 Japan Scientific Societies Press. Reproduced with permission from Japan Scientific Societies Press for comparison with the charge transfer spectra. where ∣α∣2and∣β∣2are the contributions of X2Σg+(ν′) and A2Πu(ν′) states, respectively. ∣α∣2+∣β∣2=1, and each of them is the function of exponential complex e−iωt[whereωis the quantum-state phase of X2Σg+(ν′) or A2Πu(ν′)]. As shown in Fig. 2(b), the vibrational states ν′=7, 8, and 9 of X2Σg+and their energetically close or degenerate states ν′=3, 4, and 5 of A2Πuare likely involved in this state mixing, i.e., ∣φ⟩=1√ 2(∣X2Σ+ g,ν′=7⟩+e−iΔωt∣A2Πu,ν′=3⟩), ∣φ′⟩=1√ 2(∣X2Σ+ g,ν′=8⟩+e−iΔω′t∣A2Πu,ν′=4⟩), and ∣φ′′⟩=1√ 2 (∣X2Σ+ g,ν′=9⟩+e−iΔω′′t∣A2Πu,ν′=5⟩), where two electronic- vibrational states are assumed to have equal weights and Δω,Δω′, orΔω′′is the state-phase difference. Thereby, the band intensity is proportional to the combination of two or three items of ∣∣φ⟩∣2, ∣∣φ′⟩∣2, and ∣∣φ′′⟩∣2. Similarly, the weak band (shaded in sea blue) observed in Fig. 3(d) may be attributed to ∣φ⟩=α′∣A2Πu,ν′⟩+β′∣B2Σ+ u,ν′⟩. (4) According to Fig. 2(c), ν′=11, 14 of A2Πuand their accidental degenerate states ν′=1, 3 of B2Σu+would be preferred in the above state mixing. In a case between formulas (3) and (4), the following superposition states should be responsible for the weak band shown in Figs. 3(c)–3(e):∣φ1⟩=α1∣X2Σ+ g,ν′=11⟩+β1∣A2Πu,ν′=7⟩, (5a) ∣φ2⟩=α2∣A2Πu,ν′=9⟩+β2∣B2Σ+ u,ν′=0⟩, (5b) where the left side (colored in orange) of the weak band corre- sponds to ∣φ1⟩and∣φ2⟩is responsible for the right side (colored in sea blue), indicating a superposition-state transition ∣φ1⟩→∣φ2⟩ shown in Fig. 3(c). Once a specific superposition state is produced, its corresponding N 2+intensity will slightly increase with the E c.m. enhancement [as found from Figs. 3(b) and 3(c) or Figs. 3(d) and 3(e)] because more and more nearly degenerate vibrational states possibly participate in the state mixing. This straightforward mechanism is proposed on the basis of the potential energy curves of the isolated N 2+(Fig. 2) due to the fol- lowing facts: The present CT is accomplished primarily in the large impact-parameter collision between Ar+and N 2. In the asymptotic exit, the departing co-product Ar hardly influences the state dis- tribution of N 2+. To rationalize the internuclear distance effect in the reaction entrance, we calculated the multidimensional potential energy surfaces of the (Ar–N 2)+system in the Jacobi coordinate ( θ, r, R). The results in the perpendicular arrangement ( θ=90○) are shown in a left-side view in Fig. 4(a). The potential energy surfaces atθ=30○show similar features and thus are not presented here. On J. Chem. Phys. 154, 234303 (2021); doi: 10.1063/5.0055002 154, 234303-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. Potential energy surfaces of the (Ar–N 2)+system (a) and electron trans- fers from the occupied outer valence orbitals of N 2to the half-occupied 3p component of Ar+(b) and (c). (a) These potential energy surfaces are obtained within the perpendicular arrangement, the curves at the large Ar–N 2distance represent the adiabatic potential energy curves, and the left and right intersec- tion seams are accessed at low and high E c.m.values, respectively. With the N–N bond elongation (b) or shortening (c), the 3p orbital (differently oriented but energetically degenerate) accepts an electron from 3 σg, 1πu, or 2σuorbitals. account of the important role of the large impact-parameter colli- sion, here, we focus on the profiles of the potential energy surfaces of the (Ar–N 2)+system at the cutoff ( θ=90○, R=5.12 Å). Except for the potential energy surface of the reactants N 2+Ar+, the topologi- cal features are similar to those of the isolated N 2+[Fig. 2(a)]. Mean- while, there should be small energy shifts of the N 2+vibrational states, as predicted for Ar–N 2+and Ar+–N 2in the ground states,17 which might further enhance their energetic degeneracy discussed in Eqs. (3)–(5). At a relatively low or high E c.m., the CT process occurs around the intersection seams [highlighted with two small circles in Fig. 4(a)] where the N–N bond shortening or stretching can be ini- tiated by long-distance charge (Ar+)-induced dipole (N 2) attraction. This electron –vibration coupling is a typical non-Born–Oppenheimer process and is analogous to the entanglement between the electronic and nuclear wave functions in the conical intersection or intersection seam . This homology in dynamics will most likely result in produc- ing N 2+in the superposition state, rather than the individual state or simply mixed state. As illustrated in Fig. 4(b), the electron transfers from 3 σg(the highest occupied molecular orbital, HOMO) and 1 πu(the next HOMO, HOMO −1) of N 2to one half-occupied component of the Ar+3p orbital, together with the N–N bond elongation, resulting in the X2Σg+(ν′)-A2Πu(ν′) superposition-state N 2+. Analogously, as shown in Fig. 4(c), the electron transfers from 1 πuand 2σu(HOMO −2) of N 2to the 3p orbital of Ar+but in the N–N bond shorten- ing lead to the A2Πu(ν′)-B2Σu(ν′) superposition-state N 2+. Note that the symmetries and parities of the σ- and π-type molecular orbitals are unchanged in the bond stretching of this homonuclear diatomic molecule. The same 3p orbital is artificially orientated in two direc- tions merely to match the symmetries, but the electron transfer routes from the bond-elongation 3 σgand 1 πuorbitals (or the bond- shortening 1 πuand 2σuorbitals) are indistinguishable in energy. In principle, the superposition of two electron transfer routes is the same as observed in the double-slit Young’s interference of electrons.IV. CONCLUSION In summary, we report an extraordinary pathway to likely pro- duce the superposition-state N 2+in the CT reactions between the low-energy Ar+and N 2. As observed in Figs. 1(b)–1(f), the ground- state and superposition-state N 2+ions arrive at distinctly different positions of the VMI detector; thus, the superposition-state N 2+will be handily separated in the ion-flying downstream to further evalu- ate the state coherency. Moreover, a pair of nearly degenerate states, such as ν′=3 of A2Πuandν′=7 of X2Σg+, will be two classic states of a quantum bit (qubit) if they are coherently coupled (i.e., Δω=0) for a certain time. The coherent superposition-state molec- ular ions, if long-lived, could be stored and utilized as a multi-qubit device to perform quantum information or algorithms in a cold trap. ACKNOWLEDGMENTS This work was supported by the NSFC (Grant Nos. 22003062 and 21625301) and the CAS (Grant No. YZ201565). We thank Yong Wu for his help with the theoretical calculations. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1E. 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5.0060587.pdf

Revealing the interrelation between C- and A-exciton dynamics in monolayer WS 2via transient absorption spectroscopy Cite as: Appl. Phys. Lett. 119, 051106 (2021); doi: 10.1063/5.0060587 Submitted: 20 June 2021 .Accepted: 25 July 2021 . Published Online: 5 August 2021 Yuanzheng Li,1 Xianxin Wu,2,3Weizhen Liu,1Haiyang Xu,1,a) and Xinfeng Liu2,3,a) AFFILIATIONS 1Key Laboratory of UV-Emitting Materials and Technology of Ministry of Education, Northeast Normal University, Changchun 130024, China 2CAS Key Laboratory of Standardization and Measurement for Nanotechnology, CAS Center for Excellence in Nanoscience,National Center for Nanoscience and Technology, Beijing 100190, People’s Republic of China 3University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China a)Authors to whom correspondence should be addressed: hyxu@nenu.edu.cn and liuxf@nanoctr.cn ABSTRACT Two-dimensional transition metal dichalcogenides (TMDs) are emerging as a promising complement for traditional semiconductor materials in ultrathin optoelectronic device fields. Developing a better understanding of high-energy C-exciton dynamics is essential for effi-ciently extracting hot carriers and building high-performance TMD-based light-harnessing devices; however, insight into the C-exciton dynamics remains scarce. To further understand the C-exciton dynamics, here, we have unraveled the interrelation between C-exciton and band edge A-exciton dynamics in monolayer WS 2by transient absorption spectroscopy. It is found that the band edge A-excitons could effectively generate high-energy C-excitons via the many-body process, and, in turn, the hot carriers relaxing from C-excitons to band edgestates could compensate and slow the decay of the A-excitons. The comprehensive understanding of the interrelation between C-exciton andA-exciton dynamics in monolayer TMDs may trigger the potential applications for future TMD-based light-harvesting devices. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0060587 The carrier dynamics of monolayer transition metal dichalcoge- nides (TMDs), such as MoS 2,W S 2,a n dW S e 2, have been the subject of intensive research recently.1–6Due to the quantum confinement and reduced dielectric screening in contrast to the bulk counterpart, the Coulomb interactions between carriers are significantly enhanced and give rise to tightly bound excitons upon photoexcitation of the mono- layer TMD materials.7,8This leads to the exciton dynamics in mono- layer TMDs, drastically different from the conventional quantum well or bulk materials.9,10In addition, coupled with the renormalized bandgap,11,12the relaxation process of photoexcited electron–hole pairs in the band nesting (BN) region, denoted as the C-excitons, in monolayer TMDs is expected to be distinguishable from its few-layer and bulk counterparts. Due to the BN region with the parallel band structure, the C-exciton could exhibit a strong optical response in theabsorption spectrum and ultrafast intraband relaxation via spontane- ous charge separation in the momentum space. 13,14 Although band edge exciton dynamics in monolayer TMDs, including recombination, transport, and annihilation, have beenextensively studied,15–19very limited work has been done on insight into the high-energy C-exciton dynamics. For instance, Camellini et al. observed an anisotropic relaxation dynamic of the C-exciton in the ripple-shaped sample.20Wang et al. reported a slow relaxation of C-excitons and ascribed it to the transient and complex excited-state Coulomb environment induced by band edge exciton states.21Liet al. also observed a relatively slow relaxation of C-excitons and proposed that the C-exciton relaxation is limited by the intervalley transfer pro- cess.22Moreover, Lee et al. demonstrated that the main annihilation mechanism of C-excitons is the multistep diffusion mediated by exciton–phonon scattering.23However, the completed C-exciton relaxation dynamics and the interrelation between band edge exciton and C-exciton dynamics remain to be clarified. Developing a better understanding of the C-exciton dynamics is essential to efficientlyextract high-energy hot carriers and realize high-performance light- harvesting devices. In this Letter, we have investigated the interrelation between C-exciton and band edge A-exciton dynamics in monolayer WS 2 Appl. Phys. Lett. 119, 051106 (2021); doi: 10.1063/5.0060587 119, 051106-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplsynthesized by the chemical vapor deposition (CVD) method to fur- ther enrich the understanding of the C-exciton dynamics. Assistedby transient absorption (TA) spectroscopy, we found that the decay time of A-excitons under near-resonant excitation of the B-exciton (515 nm) progressively extends rather than being conventionallyshortened with an increase in pump fluences. Such results can beattributed to hot carriers relaxing from C-excitons to band edge, whichcould compensate for the decay of the band edge A-excitons, wherethe C-excitons are generated by an efficient many-body process.Excitation wavelength-dependent TA spectroscopies prove the validityof the many-body process for which the photobleaching signals ofC-exciton can still be observed for excitation below C-exciton. Due tothe C-exciton relaxation with picosecond timescale, the compensationof hot carriers for A-exciton only plays a major role in the carrier–phonon scattering ( /C2410 ps) and direct electron–hole recombination (<100 ps) process. These results could provide us with a further understanding of the C-exciton dynamics and the interrelationbetween it and band edge exciton in monolayer WS 2and other TMDs materials. Figure 1(a) demonstrates the calculated bandgap structure of monolayer WS 2based on density functional theory (DFT), where both the minimum of conduction band and the maximum of valence bandare located at the K point of the Brillouin zone. Due to the spin–orbitalsplitting of the valence band, 24,25there are two direct band edge transi- tions, namely, A-exciton and B-exciton. By contrast, C-exciton is notband edge transition rather than corresponding to band-nesting regions, that is, the midway between the Kvalley and the Chill. With the unique BN effect, the C-exciton could present strong optical absorption as with band edge A- and B-excitons. In the steady-state absorption spectra [ Fig. 1(b) ], a stronger absorption peak at around 420 nm, which corresponds to C-exciton, can be observed comparingwith the band edge A-/B-exciton around 615 and 520 nm. Since theband edge excitonic emission is a direct transition at the K points ofthe Brillouin zone, the A-exciton could present a high yield fluores-cence emission. However, C-exciton with optically inactive cannotrecombine to generate photons because the parallel band structurecould induce self-separation of C-excitons and propel electrons andholes separately relaxing to Kvalley and Chill, leading to the momen- tum mismatching. According to previous works, 13the electrons at the Kvalley and the holes at the Chill will eventually relax to the lowest excited state, namely, the K point, where both electrons and holes dis- tribute in the band edge. Obviously, there is a complex interrelationbetween C-exciton and band edge exciton dynamics and that shouldbe understood adequately. In this regard, broadband transient absorption (TA) spectroscopy (HELIOS Ultrafast Systems, time resolution: /C24100 fs) is carried out to investigate the carrier dynamics. The light source is an opticalparametric amplifier (Coherent, OperA Solo) pumped using 800 nmpulses from a Coherent Astrella regenerative amplifier (80 fs, 1 kHz,2.5 mJ per pulse). The probe beam was generated by focusing a FIG. 1. (a) Theoretically calculated band structure of monolayer WS 2based on DFT. These arrows represent the optical transition of A-exciton (red), B-exciton (blue), and C- exciton (green). E b: exciton binding energy. The green area is the band nesting (BN) region. (b) PL spectra and steady-state absorption spectra of monolayer WS 2. (c) TA spectra probed at different delay times under 515 nm excitation (pump fluence: 12.1 lJc m/C02). (d) Decay dynamics of A-exciton under the pump condition in (c), where the purple solid line is the fitting curve.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 051106 (2021); doi: 10.1063/5.0060587 119, 051106-2 Published under an exclusive license by AIP Publishing1.55 eV laser pulse onto a sapphire crystal to produce the time-delayed white light continuum. First, the band edge exciton dynamics ofmonolayer WS 2are elucidated by broadband TA spectra under 515 nm excitation with a pump fluence of 12.1 lJc m/C02(initial exciton density: 3.96 /C21012cm/C02), as shown in Fig. 1(c) .H e r e i n ,t h eT As p e c - tra are described by the differential optical density ( DO.D.) between with and without pump light excitation, and all TA measurements areperformed on the same monolayer WS 2sample (see supplementary material Fig. S1 for the optical image and atomic force microscope image). Two photobleaching signals can be observed at around 615and 520 nm corresponding to the A- and B-exciton, respectively. Notethat the photoinduced excited-state absorption (positive DO.D.) over- laps the photobleaching signals of B-exciton (negative DO.D.), which results in a reduction in the intensity of photobleaching signals of B-exciton accompanied by a low signal-to-noise ratio. Hence, we haveonly chosen A-exciton as the representative of band edge excitons tocarry out subsequent research. Figure 1(d) shows a typical decay curve of A-exciton that can be well fitted with a triple exponential decayfunction of Ae t/s1þBet/s2þCet/s3, where the three time-constants are 43966f s( s1), 5.860.5 ps ( s2), and 19.3 62p s( s3), respectively. According to the previous report,26,27the three different decay compo- nents can be assigned to surface-defect trapping, carrier–phonon scat-tering, and direct electron–hole recombination processes, respectively.Note that the timescale of the electron–hole recombination of mono-layer WS 2is less than that of monolayer MoS 2, indicating the mono- layer WS 2with a higher radiative recombination rate. Such a result is consistent with that the quantum yield observed in the monolayerWS 2is higher than that of monolayer MoS 2.28 To systematically analyze the band edge A-exciton dynamics, we have conducted pump fluence-dependent TA spectra where the pumpfluence increases from 12.1 to 60.5 lJc m /C02.Figure 2(a) shows pump fluence-dependent TA spectra measured at 0.6 ps after the resonantpump excitation of B-exciton at 515 nm. It can be seen that there is no visible variation in the peak of the A-exciton bleach even though the pump beam increases to the highest fluence of 60.5 lJc m /C02. There seems to be neither charge-carrier-induced Stark effect nor FIG. 2. (a) Pump fluence-dependent TA spectra measured at 0.6 ps after excitation. (b) The peak amplitude of the bleaching signal of A-exciton as a function of p ump fluence (excitation density). (c) Pump fluence-dependent A-exciton dynamics of monolayer WS 2under 515 nm excitation. (d) The fast decay s1, (e) the intermediate decay s2, and (f) the slow decay s3as a function of pump fluence. The weight factor (g) A 1, (h) A 2, and (i) A 3as a function of pump fluence.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 051106 (2021); doi: 10.1063/5.0060587 119, 051106-3 Published under an exclusive license by AIP Publishingband-filling effect, that is, Pauli blocking of the occupied band edge states drives photogenerated excitons leaping into higher-energy states,existing in the A-exciton dynamics. The amplitude of the A-excitonbleach, however, demonstrates a sublinear behavior with the pumpfluence [ Fig. 2(b) ], indicating that the unoccupied states from A- exciton tend to be fully filled, which means Pauli blocking of these already occupied states would push the excitonic transitions to higherunoccupied energies up to a critical saturation density, that is, band-filling effect. However, the corresponding blueshift in the A-excitonpeak position caused by the band-filling effect 29is not observed in Fig. 2(a) . A reasonable explanation may be due to the redshift that is induced by the Stark effect30or bandgap renormalization31,32offset the blueshift caused by the band-filling effect, resulting in no signifi-cant shift in the A-exciton peak position. The fluence-dependentamplitude of the A-exciton bleach can be fitted well based on the satu-rable absorber model as follows: 33jDO.D.j/n/(nþns), where nand nsare on behalf of exciton density and saturation density, respectively. Herein, the saturation density n sis estimated as 6.3 60.6/C21012cm/C02 and the corresponding exciton–exciton distance is around 40 62A˚. Such results indicate the A-exciton dynamics are dominated by thevariations of exciton population, when the exciton density is below 6.360.6/C210 12cm/C02. For the exciton density above the saturation density, other photoexcited many-body species and many-body pro-cesses may be generated, such as biexciton and exciton–exciton anni-hilation (EEA) processes. 34,35Intriguingly, the decay time of A-exciton fails to reduce above this saturation density but becomes longer withthe increasing pump fluence, as shown in Fig. 2(c) . To clarify the origin of this prolonged decay with pump fluences, we have established quantitative fitting for decay curves of A-exciton measured at different pump fluences. These decay curves were also fit- ted by a triexponential function of A 1exp(/C0t/s1)þA2exp(/C0t/s2) þA3exp(/C0t/s3), where A 1,A2,a n dA 3are the weight factors and s1, s2and s3represent the time-constants. Figures 2(d)–2(f) separately show the fast decay s1of 0.4–0.7 ps, the intermediate decay s2of 5–9 ps, and the slow decay s3of 20–80 ps as a function of pump fluences. Different from s2ands3that are monotonically increasing with the pump fluence, the fast decay s1increases first and then remains at a certain level with the increase in the pump fluence. As mentionedabove, the three decay processes (namely, s 1,s2,a n d s3) can be attrib- uted to surface-defect trapping, carrier–phonon scattering, and direct electron–hole recombination, respectively. Hence, such a saturationbehavior of s1with pump fluence can be explained as full filling of the available surface-defect states at high pump fluence limiting the rate ofsurface-defect trapping. 36F o rt h i sr e a s o n ,t h ew e i g h tf a c t o rA 1of the fast decay process tends to gradually reduce after a slight increase asthe pump fluence rises [ Fig. 2(g) ]. However, it is difficult to under- stand the extension of s 2ands3with the increasing pump fluence. In general, high pump fluence will inevitably generate a large amount of heat caused by the photothermal effect in the monolayerWS 2, which could result in the carrier–phonon scattering process to be aggravated with an increase in the number of phonons. Hence, thecorresponding weight factor A 2of the carrier–phonon scattering is monotonically increasing with the pump fluence, as shown inFig. 2(h) . Due to the enhanced carrier–phonon scattering, the nonra- diative recombination rate of A-exciton would also augment, and the weight factor A 3of the direct electron–hole recombination should be reduced as the pump fluence increases. In Fig. 2(i) ,h o w e v e r ,t h e weight factor A 3exhibits a trend of decay only when the pump fluence is less than 28.2 lJc m/C02, and after that, the A 3slightly augments with the increasing pump fluence. Obviously, such unexpected changes inthe carrier–phonon scattering and direct electron–hole recombinationprocesses should be further elucidated. One possible interpretation isthat the relaxation of hot carriers from high-energy states, namely, C-excitons, compensates the decay of carrier–phonon scattering anddirect electron–hole recombination to slow their decay and aggrandizetheir weight factors. If the compensation of hot carriers was the princi- pal reason for the extension of s 2ands3, the precondition must be the relaxation timescale of the hot carriers from C-excitons, which couldmatch that of carrier–phonon scattering and direct electron–holerecombination, that is, few picoseconds to tens of picoseconds. To verify this assumption, we first have carried out broadband TA spectra excited by the 365 nm (3.4 eV) pump beam (pump density:0.8lJc m /C02) that is above the C excitonic feature, as shown in Fig. 3(a) . Besides A- and B-excitons, an obvious negative bleaching sig- nal can be observed at around 430 nm (2.88 eV) corresponding to theC-exciton. Similarly, the A-exciton dynamics excited at 365 nm con-tain three distinct decay processes and can be fitted by a triexponentialattenuation function (red solid line) in which the three time-constantsremain at the same order of magnitude with that excited at 515 nm,respectively [ Fig. 3(b) ]. For the C-exciton, however, the decay curve is only fitted by a biexponential decay function (red solid line) with two different time-components: s 1(5.3 ps) and s2(48.2 ps) [ Fig. 3(c) ]. FIG. 3. (a) TA spectra probed at different delay times under 365 nm excitation (pump density: 0.8 lJc m/C02). Decay dynamics of (b) A-exciton and (c) C-exciton under the pump condition in (a), where the solid lines are the fitting curves.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 051106 (2021); doi: 10.1063/5.0060587 119, 051106-4 Published under an exclusive license by AIP PublishingAccording to the previous work,13the hot carriers from C-excitons first relax to the intermediate energy state Kvalley/Chill and then transfer from the Kvalley/Chill to the lowest energy state K valley/hill via intervalley transfer/scattering [ Fig. 1(a) ]. The two attenuation pro- cesses of C-exciton are mainly caused by the limitation of the interval-ley transfer rates of the hot carriers. 22Importantly, the timescale of the fast and the slow intervalley transfers exactly matches that of carrier–phonon scattering and direct electron–hole recombination, respec-tively. Such results indicate the hot carriers from C-excitons could provide effective compensations for the loss of A-excitons caused by carrier–phonon scattering and direct electron–hole recombinationprocesses in timescale. In this way, the time-components and weightfactors of carrier–phonon scattering and direct electron–hole recombi-nation could be extended and supplemented, respectively. It is worthnoting that the pump fluence-dependent TA spectra [ Fig. 2(c) ]a r e excited at 515 nm (2.4 eV) in which the carriers at the ground state could not be directly excited to form the C-exciton with higher energy (2.9 eV). This means that the band edge excitons need to leap into thehigher-energy state (namely, C-exciton) via upconversion. In themonolayer TMDs, the many-body processes, such as EEA process anddefect-assisted Auger recombination, are the main pathway for upcon-version of band edge A-excitons into higher-energy C-excitons. 21,37 Obviously, an efficient many-body process that exists in band edgeexciton dynamics is the other premise of the assumption, which weproposed is tenable. To validate the many-body process, we have chosen different pump beams whose photon energy is below the C-exciton, including480, 507, 515, and 608 nm (pump density: 1.6 lJc m /C02), to perform TA spectra, as shown in Figs. 4(a)–4(d) . Among them, 515 and 608 nm separately correspond to the near-resonant excitation of theB-exciton and A-exciton. Here, both the TA signals and pump pulsesare collected by the spectrometer, which results in the partial TA sig-nals are covered by the signals of the pump pulse. Even so, the bleach-ing signals of C-exciton remain unaffected and can be observed atthese pump beams excitation, which indicates that the band edge exci- tons could effectively generate the C-excitons via the many-body pro-cess in monolayer WS 2. Furthermore, pump fluence-dependent C-exciton dynamics under near-resonant excitation of the B-exciton and A-exciton are shown in supplementary material Fig. S2, which further supports the conclusion. Intriguingly, the decay dynamics ofC-exciton stay in step with A-exciton and demonstrate three distinctdecay processes rather than previous two processes shown in Fig. 3(c) [Figs. 4(e)–4(h) ]. A sub-picosecond decay process emerges when the photon energy of the pump beam is below the C-exciton. Accordingto the previous work, 32such sub-picosecond decay process can be attributed to the exciton formation from initially photogeneratedunbound charged species. To summarize, it can be concluded that forexcitation below C-exciton, the hot carriers relaxing from C-excitonsthat are generated via many-body process to A-exciton could compen-sate the carrier–phonon scattering and direct electron–hole recombi-nation, thereby slowing the decay of A-excitons. In addition, we haveperformed the pump fluence-dependent and excitation wavelength-dependent TA experiments for monolayer MoS 2films and found simi- lar phenomena and results with WS 2(see the supplementary material Fig. S3). Hence, the proposed conclusion should be valid for other monolayer TMDs materials. Figures 5(a) and5(b) demonstrate the photophysical processes of the carrier relaxation and the many-bodyin monolayer WS 2, respectively. Due to the parallel band structure, the hot carriers (electrons and holes) could rapidly relax to Kvalley/C hill within 500 fs26and then transfer to the lowest energy state K valley/hill via intervalley transfer including a fast process (1–5 ps) anda slow process (10–60 ps). 22The band edge A-exciton dynamics could be described as three different processes, including surface-defecttrapping ( <1 ps), carrier–phonon scattering ( /C2410 ps), and direct electron–hole recombination ( <100 ps). On the other hand, the band edge A-excitons could effectively generate the C-excitons via the many-body process even for the excitation below C-exciton. Subsequently,the hot carriers, transferring from C-exciton to K valley/hill, could FIG. 4. TA spectra probed at different delay times under (a) 480, (b) 507, (c) 515, (d) 608 nm excitation (pump density: 1.6 lJc m/C02). Decay dynamics of C-exciton under (e) 480, (f) 507, (g) 515, and (h) 608 nm excitation. These pink solid lines are the fitting curves.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 051106 (2021); doi: 10.1063/5.0060587 119, 051106-5 Published under an exclusive license by AIP Publishingcompensate the loss of the band edge A-excitons that are mainly caused by carrier–phonon scattering and direct electron–hole recombi- nation and slow the decay of A-excitons. In summary, we have unraveled the interrelation between C-exciton and band edge exciton dynamics in monolayer WS 2based on pump fluence-dependent and excitation wavelength-dependent TAspectroscopy. The band edge A-excitons could effectively generate high-energy C-excitons via the many-body process, and in turn, the hot carriers relaxing from C-excitons to band edge states could com-pensate and slow the decay of the band edge A-excitons. Since the relaxation time of C-excitons is limited by intervalley transfer with picosecond timescale, the compensation of hot carriers for A-excitonsonly plays a major role in the carrier–phonon scattering and direct electron–hole recombination process. Insight into the interrelation between C-exciton and band edge exciton dynamics in monolayerWS 2not only enriches the understanding of ultrafast C-exciton dynamics but also may motivate the potential applications for future TMDs-based light-harvesting devices by effectively collecting the high-energy hot carriers. See the supplementary material for information related to the optical image and atomic force microscope image of monolayer WS 2, pump fluence-dependent C-exciton dynamics of monolayer WS 2,a n d pump fluence-dependent A-exciton dynamics of monolayer MoS 2. AUTHORS’ CONTRIBUTIONS Y.L. and X.W. equally contributed to this work. This work was supported by the Program of National Natural Science Foundation of China (Nos. 51732003, 51872043, 61604037, 11874104, 12074060, 12004069, and 12074086), the National Science Fund for Distinguished Young Scholars (No. 52025022), the NationalKey Research and Development Program of China (No. 2019YFB2205100), Fund from Ministry of Education (No. 6141A02033414), the China Postdoctoral Science Foundation fundedproject (Nos. 2020M681025, 2021M693905, and 2021T140109), the Fundamental Research Funds for the Central Universities(Nos. 2412020QD015, 2412019BJ006, 2412021ZD007, and 2412021ZD012), and the Fund from Jilin Province (Nos. 111865005,YDZJ202101ZYTS049, YDZJ202101ZYTS041, YDZJ202101ZYTS133,JJKH20211273KJ, and JJKH20211274KJ). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1C. Jin, E. Y. Ma, O. Karni, E. C. Regan, F. Wang, and T. F. Heinz, Nat. 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5.0047371.pdf

The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ab initio molecular dynamics modeling of single polyethylene chains: Scission kinetics and influence of radical under mechanical strain Cite as: J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 Submitted: 12 February 2021 •Accepted: 18 June 2021 • Published Online: 8 July 2021 Gary S. Kedziora,1,a) James Moller,2 Rajiv Berry,3 and Dhriti Nepal3 AFFILIATIONS 1Department of Engineering Physics, Air Force Institute of Technology, Wright-Patterson AFB, Ohio 45433, USA 2Department of Mechanical and Manufacturing Engineering, Miami University, Oxford, Ohio 45056, USA 3Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright Patterson Air Force Base, Dayton, Ohio 45433, USA a)Author to whom correspondence should be addressed: gary.kedziora@afit.edu ABSTRACT Ab initio molecular dynamics was used to estimate the response to constant imposed strain on a short polyethylene (PE) chain and a radical chain with a removed hydrogen atom. Two independent types of simulations were run. In the first case, the chains were strained by expanding a periodic cell, restraining the length but allowing the internal degrees of freedom to reach equilibrium. From these simulations, the average force on the chain was computed, and the resulting force was integrated to determine the Helmholtz free energy for chain stretching. In the second set of simulations, chains were constrained to various lengths, while a bond was restrained at various bond lengths using umbrella sampling. This provided free energy of bond scission for various chain strains. The sum of the two free energy functions results in an approx- imation of the free energy of chain scission under various strains and gives a realistic and new picture of the effect of chain strain on bond breaking. Unimolecular scission rates for each chain type were examined as a function of chain strain. The scission rate for the radical chain is several orders of magnitude larger than that of the pristine chain at smaller strains and at equilibrium. This highlights the importance of radical formation in PE rupture and is consistent with experiments. Constant strain results were used to derive a constant-force model for the radical chain that demonstrates a roll over in rate similar to the “catch-bond” behavior observed in protein membrane detachment experiments. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0047371 INTRODUCTION For decades, researchers have sought to understand and char- acterize the fundamental bases for nucleation and propagation of fracture in polymers. Experiments using instrumentation to detect atomic- and nano-scale phenomena (e.g., FTIR, EPR, and x-ray scattering) have provided insight1–7including evidence of chain uncoiling and slip, void formation, chain rupture by bond break- ing,8,9and mechanoradical reactions. One of the important find- ings of Zhurkov2was that a single chain rupture in a macroscale sample would lead to mechanoradicals that are reactive with neigh- boring chains. It was estimated that a single chain rupture leads to tens or hundreds of subsequent reactions. Regarding the sequence of events, Kausch argued that bond breaking follows nucleationmanifested by large local strains.3,8,10With the development of molecular dynamics (MD) methods, detailed atomistic understand- ing of chain dynamics under stress can be achieved.11,12However, these methods are not currently equipped to deal with bond scission or radical reactions. Historically, kinetics-based methods have been used to glean insight into the effect of force on atomistic scale phenomena of poly- mer rupture. These simple models are based on Transition State Theory (TST) originally due to Eyring and Polanyi (originally pub- lished in 1931),13as expressed in a general form by the equation k=κkBT hexp(−ΔG‡ RT)=κkBT hexp(ΔS‡ R)exp(−ΔH‡ RT), (1) J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where ΔG‡is the Gibbs free energy increase associated with the acti- vated complex relative to the reactant, κis the proportionality con- stant, and the termkBT his the frequency of conversion to products at the activated complex. TST in the context of rupture of polymer chains placed in tension due to a shearing motion was studied by Kauzmann and Eyring.14It was put forth that application of a constant force places a downward bias on the energy barrier for bond rupture. In their work, bonds were defined broadly to include non-covalent attrac- tions and entanglements. The adapted rate equation for bond break- ing took the form dN dt=−flNkBT hexp(ΔS‡ R)exp(−D′ RT), (2) where Nis the number of bonds, ΔS‡is the entropy increase between the transition and unreacted states, flis the fraction of loaded bonds, and D′is the activation energy for bond breaking diminished by the presence of a superposed stress. For common inter-atomic potentials, D′generally exhibits a nonlinear dependence on stress. While Eyring’s TST-based rate equation, or more generally the Arrhenius equation, has been demonstrated in many cases as an appropriate and useful representation of experimental trends, it is, nonetheless, challenging to completely reconcile the force- or stress- dependent form with the results of experiments. In its simplest force-dependent form, this equation can be expressed as kf=k0exp(αF kBT), (3) where Fis an external applied force, k0is the equilibrium rate, andαis the activation length (or volume in the stress form). k0 may have an atomistic interpretation based on TST, but it is diffi- cult to understand the activation length or volume from an atom- istic perspective. Kauzmann and Eyring14depicted the effect of a steady force on a bond potential energy by subtracting from a Morse potential function a linearly varying potential. This depic- tion makes sense in cases of mechanical relaxation, for example.15 The effect of the application of stress is to bias the energy land- scape to favor interchain slip motions that lead to placement into lower adjacent energy wells (and, therefore, increased strain). The subtracted potential due to work done on the bond leads to a puz- zling result, however, when applied to individual bond potentials in that the resulting function approaches negative infinity with increas- ing distance. This non-physical result apparently indicates that the nuclei would continue to accelerate with the continued application of an external force. Recently, however, others have developed more general and realistic descriptions of the effect of force on activation energies.16–19 Given the multiple types of events that can be broadly termed bond breaking when a polymer sample is loaded, the resulting rate equations reflecting these collective events show more compli- cated dependencies on stress or force. These types of observations have been made since the beginning of the application of kinet- ics to mechanical properties. For example, even though a bond energy model having a simple dependence on force was assumed, Kauzmann and Eyring14derived a rate constant for bond rup- ture in a polymer under shear in which activation energy has anonlinear force dependence. In another instance, Tobolsky and Eyring,20guided by the results on the tensile strength of fabrics that showed a linear dependence of the logarithm of lifetime on load,21 derived an expression for the fabric thread lifetime. Given a thread with Nbonds per unit area and a tensile stress σ, the rate at which the bond breaks was modeled as dN dt=−NkBT hexp(−ΔG‡ kBT)exp(σλ 2NkBT), (4) where ΔG‡is the activation free energy for breaking the bond and λ is the distance between equilibrium positions. Assuming largeσλ 2NkBT, the thread lifetime is approximated by τ=2N0h σλexp(ΔG‡−σλ/2N0 kBT), (5) where N0is the number of initial bonds per unit area and the activa- tion volume (λ 2N0)is the cross sectional area per bond multiplied by the distance required to break a bond. While there is a term that varies linearly with stress in the exponential, the prefactor also depends on stress. This approach is similar to the one later taken by Bell in modeling the detachment of cells from membranes.22 Subsequently, the term “catch-bond” was coined by Dembo et al. , describing a hypothesized effect of force producing a maximum in detachment rate as a function of force,23which was later observed.24 Nonlinear effects aside, the assumption of linear dependence on force or stress for the effective reduction in activation energy can be effective even in a complex system. Zhurkov and co-workers per- formed experiments on many solids including metals and polymers, showing that an equation potentially consistent with TST, τ=τ0exp(U0−σγ kBT), (6) could be a fit to the data. A review of the work led by Zhurkov is summarized in Ref. 25 by Regel and other members of the labora- tory Zhurkov directed. In the form of Eq. (6), however, the physical meaning of the coefficients remains obscure, which is due, in part, to the inability to measure the atomic positions in the process of fracture and the fact that the equation is used to describe collec- tive phenomena as alluded to above. While they concluded that the activation energy is due to covalent bond breaking, others were not convinced,26–28due to the inability of Eq. (6) to fit data in certain cir- cumstances29and that the activation energy found (27–29 kcal/mol) was at a level nearer to that for melting than the C–C bond energy, 87.9 kcal/mol.30 Some researchers have used the result of pulling experiments of single molecules to infer force dependence of reaction rates. Dudko and co-workers examined the effect of force on reactions of interest in bio-physics16,31–33and showed how exceptions to linear depen- dence on force in the rate exponent can arise. Suzuki and Dudko16 developed a simple and flexible model to describe the effect of force on the activation barrier of a reaction using the assumption that the free energy of the system is the work done on straining the entire system with an external force plus the intrinsic free energy of the reaction path. They offered a simple model for the “catch-bond” J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp phenomenon valid even for a single-dimensional reaction coor- dinate. They found an interplay among the values of the transi- tion state and energy minimum in the molecular extension coordi- nate (i.e., pulling direction) and the reaction coordinate (e.g., bond length) that allows for realistic roll-over behavior in their model. Others have studied the effect of force or strain on reaction energies. For example, ab initio methods have been used to study the effect of strain on bonds in small molecules34and in polyethy- lene (PE) chains.35,36Saitta and Klein studied scission in linear and knotted alkane chains under tensile strain with ab initio MD.37Sim- ilar approaches to the umbrella sampling approach we employ here have been developed.38,39Previous efforts that have studied in detail the effect of force or strain on reaction paths include Refs. 17 and 40. In one of the seminal influential papers on the effect of force on rates, Evans and Ritchie41performed dynamical Monte Carlo simulations based on Kramer’s theory42to study the effect of dynamic loads on the rate of “bond” rupture in biological systems, where the forces required are on the order of 10–100 s of pN. In this case, the “bonds” are collections of van der Waals or electrostatic interactions. This simulation method was used by Grandbois et al.43to model detach- ment of polymers from a silica surface and obtain the force required to rupture the covalent Si–C bond, which was estimated to be about 2 nN. Single-molecule force-clamp AFM experiments have shown that the effect of force on the thiol disulfide exchange reaction low- ers the barrier of the rate-limiting step,44which is consistent with the phenomenological theories. Protein force-clamp experiments have demonstrated disulfide exchange reaction rates that are nonlinearly dependent on force45demonstrating “catch-bond” behavior, where the rate on the enzyme catalyzed reduction of a disulfide bond has a concave curve with a minimum at some intermediate force. This work demonstrates a similar behavior for a constant-imposed force simulation of a model radical PE chain, a much simpler system. Given the findings of Zhurkov and co-workers in PE and other groups in mechanically generated reactions of similar polymers, it is relevant to seek to quantitatively predict the rates of these respective reactions under mechanical loads. For PE, the simplest reactions to consider are the rupture of a pristine chain and rupture of a chain with a hydrogen atom removed. Rupture in a thermo-plastic poly- mer is decidedly more complicated than single chain rupture or a plastic flow in idealized models. However, to begin to unravel the various contributions to polymer rupture in quantitative kinetics or atomistic theories, we need to have a detailed understanding of all the relevant phenomena at their respective scales. In particular, a detailed understanding of how strain and force affect polymer chain scission has been lacking. Here, we show how a force-biased potential can be created using results from constant strain ab initio molecular dynamics simulations, although the strain- based results are simpler and more reliable. The assumption made by Suzuki and Dudko to understand the free energy of a bond under force and the free energy of the larger portion of the system must be considered, which inspires this work and is a key concept that allows a chain-strain-based alternative to Fig. 1 that has a correct asymptotic limit. Ong et al. examined the effect of imposed force on ring opening energy barriers with molecular dynamics similar to this work, but explicitly controlled the force,40whereas we impose strain. We examine the Gibbs free energy landscape for the pristine PE chain and the one from which a hydrogen atom has been removed. The energy barriers for chain rupture for each case are found as FIG. 1. Representation of Eyring and Kauzmann’s concept for the effect of strain on bond. It is a Morse potential for an aliphatic C–C bond using the parameterization of de Boer, subtracting work done for stretching the bond assuming a constant force. Note that the C–C bond energy was not accurate at the time. functions of both the chain strain and bond length. Barrier heights are compared to values derived from experiments and found to be in good agreement. Our results support earlier experimental results that radical reactions are important in PE rupture and support that the activation energy fit by Zhurkov1is related to radical chains. Fur- thermore, we show that the PE radical chain model under varying imposed force has a roll-over behavior similar to the “catch-bond” behavior, where there is a maximum in the rate of scission with respect to force. Finally, and most importantly, we demonstrate a first-principles approach based on constant chain-strain simulations that can be used in multi-scale models for polymers to determine the effect of strain on the mechanical and thermodynamic properties of chains in polymers under extreme loads. COMPUTATIONAL METHODS An uncoiled PE chain was modeled by periodic boundary con- ditions with a single –C 8H16−chain in a cell with adequate space to eliminate most interactions with neighboring periodic images. This low pressure limit, while not a representative of a condensed-phase system, is simple and allows focus on details such as force and work done on a single stretched chain. A corresponding chain with one hydrogen atom removed from carbon atom number 5 (see Fig. 2), leaving a dangling bond, was also simulated. In the first set of sim- ulations, the equilibrium box sizes at 300 K were 10.201 ×7.204 ×7.204 and 10.173 ×7.204×7.204 Å3for the –C 8H16−and –⋅C8H15– systems, respectively. The chains spanned the xdimen- sion, and the cells were stretched incrementally in that direction. Constant NVT ab initio molecular dynamics calculations were run with CP2K46,47using the N12 density functional48with the DZVP- GTH basis set49and the GTH-PBE pseudopotential.50,51We previ- ously showed that N12 can be reliably used to model single bond breaking events in stretched molecules.52,53The xdimension of the equilibrium cell was determined by first using the length from a combined cell and geometry optimization at 0 K. Then, these were equilibrated at 300 K, and the resulting average stress from the first few expanded box calculations was extrapolated to zero stress at 300 K in the strain direction. Next, the effect of strain on J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. Schematic of the radical chain model structure with a hydrogen atom removed from the pristine chain at carbon atom 5 showing the weak beta bonds with red bonds. The thick red bond is the one treated with umbrella sampling. The periodic boundary is denoted by dashed vertical lines, and the chain segment spans the xdimension. the mechanical response of these idealized chains was determined. Beginning with the 0-K chain length, the xdimension was increased in steps of 0.1 Å until a bond broke in less than 50 ps at 300 K. There was only one simulation per strain, and the average stresses were obtained after 4 ps for equilibration. The PE chain (–C 8H16–) broke in under 50 ps at a strain of 0.21, while the radical PE chain (–⋅C8H15–) broke at a strain of 0.17. The –C 8H16– strain is larger than the strain for which the activation energy derived below is equal to zero and is not representative of the strain for a valid theoretical ultimate strength estimate. The simulation time was chosen based on computational convenience. Some initial computational experi- ments with a 100 ps time limit at strains near 0.20 for PE showed histograms that resemble Poisson distributions (see Fig. S1 in the supplementary material). It is interesting to note that 0 K optimiza- tions of internal degrees of freedom of a stretched PE chain did not break until a strain of 0.36 due to a lack of motion in degrees of freedom favorable to bond scission. The Helmholtz free energy, ΔA, for stretching the chains was computed using the well-known relation ΔA(L)≤W(L)=∫L L0σx(l)axdl, (7)where W(L)is the work done by stretching a chain from length L0 toL,σx(l)is the average virial stress54–56at box length l, and axis the cell face area perpendicular to the xcell face ( ax=7.204×7.204 Å2=51.898 Å2), which remains constant during the simulation. The average force on the chain at box length LisF(L)=σx(L)ax, with F(L0)=0. The internal energy change, ΔU, and ΔAas a function of chain strain, ε, are shown in Fig. 3(a), where, for example, A(ϵ)=A(ϵ(L)), (8) with ϵ(L)=(L−L0)/L. (9) The internal energy was computed as the average of the kinetic energy plus potential energy excluding the first 4 ps for each cell to allow for equilibration. This resulted in smooth plots in Fig. 3(a) by simply connecting the average values of internal energy by straight lines, which is true for all the curves in Fig. 3. In order to put the molecule-scale stress in perspective by com- paring with experimental data, we converted the stress at the density of our simulation cell to the stress at the density of crystalline PE, 0.98 g/cm3.57Using the molecular weight of the chain, the mass of the single chain in the simulation cell was calculated and used to get the density. In both the crystal and the simulation cells, the formula for density is ρ=m/V=m/Lax, where mrepresents the mass and V is the volume. Assuming that the weight and the length of the cell are the same in both cases, the following relation is easily derived: ρcrystal ρcell=ax,cell ax,crystal. (10) A theoretical estimate for the stress at the crystalline density for non- interacting chains is then given by σx,crystal=F ax,crystal=F ax,cellρcrystal ρcell. (11) We are neglecting any changes in volume that would occur in the real crystal when strained and are neglecting any changes in the FIG. 3. (a) Free energy ΔA(work), internal energy ΔU(left axis), and external force (right axis) for the –C 8H16– and – ⋅C8H15– chains at various strains; (b) the corresponding stress and strain curves at the crystalline density. The plots are truncated where the chains broke in less than 50 ps. J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp interaction between chains from the simulation. The resulting stress verses strain plot is shown in Fig. 3(b). The periodic images of chains actually very weakly interact in the simulation cell, as can be seen in supplementary material, Table S1, which shows that when the perpendicular cell dimensions are doubled from 7.204 ×7.204 to 14.408×14.408 Å2, the total potential energy changes by about 0.2 kcal/mol, indicating a small attraction between the images in the cell that was used. Since the xcell dimension is 10.2 Å and the rad- ical is fairly well-localized as a dangling bond, we expect a smaller interaction among periodic images of the radical in that dimension. The differences between the internal energy and Helmholtz free energy at rupture are 3.7 and 2.3 kcal/mol for the –C 8H16– and –⋅C8H15– chains, respectively. In order to determine if this dif- ference may be due to an error in the integration, we tested the convergence of the trapezoidal rule used to obtain the Helmholtz free energy by including only half the number of points, and this changed the results by only 0.1 and 0.06 kcal/mol for the –C 8H16– and –⋅C8H15– chains, respectively, demonstrating that the sampling is close to convergence. Furthermore, each of the above-mentioned strained MD simulations began with the 0-K optimized structure and equilibrated quickly. Tests indicated that leaving out various numbers of steps from the beginning of each simulation had a neg- ligible effect on the averages. For example, when skipping the first 4000 time steps (4 ps), the differences in each ΔUcompared for all steps were less than 0.1 kcal/mol. The cell-stretching simula- tions are, in principle, quasistatic because the chains are uncoiled at the starting equilibrium cell; the system is elastic up until a bond breaks; the chains very weakly interact, and the forces of the molecule would cause it to relax to the original structure at the equilibrium cell since it was already an uncoiled chain. To check that these assumptions are correct and that the averages are suf- ficiently converged, the reversibility of the cell-stretching simula- tion of the –C 8H16– chain was checked for hysteresis by compar- ing with a reverse sequential simulation (see the supplementary material). If hysteresis is present in comparing the internal ener- gies of the reverse simulations and the original forward simula- tion, then the simulation is not reversible, and we cannot attribute the difference between ΔUandΔAto entropy effects. We found that there is no hysteresis, and therefore, the differences between ΔUandΔAare due to entropy (see supplementary material, Fig. S2). The entropy is small, and more sampling would be required to obtain an accurate estimate of the entropy change for chain stretching since the statistical errors in the averages are significant compared to TΔS. We now consider the dependence of free energy on bond length. The relative free energy of a molecular system at a given set of internal coordinates is related to the probability that the system be at those coordinates, P(s)∝∫dNpdNqe−H(r,p) kBTδ(f(q1,q2,. . .,qN)−s) ∝⟨δ(f(q1,q2,. . .,qN)−s)⟩, (12) where f(q1,q2,. . .,qN)is acollective variable of the internal coordi- nates q1,q2,. . .,qN; see, for example, Ref. 58. This probability func- tion, which is the ensemble average of the delta function, is inter- preted as a histogram. In order to sample higher energy, or lower probability portions of phase space, a biasing potential is employed.After these sampling methods are used in a simulation, the unbiased probability is reconstructed. This has often been accomplished with the weighted histogram analysis method (WHAM).59,60The class of methods that use biasing potentials to sample simulations are called umbrella sampling methods (see Ref. 61 for a review). Once the probability distribution is determined, the free energy change as a function of sis given by ΔA(s)=−kBTln(P(s)). (13) In this work, we use the bond distance between two carbon atoms as our collective variable, s=r=∥rC3−rC2∥. (14) The free energy for the reaction path for breaking a bond in each chain at various strains was calculated using umbrella integra- tion,61–63which has the advantage of producing smooth free energy curves. As will be described below, this was useful for interpolating a free energy surface as a function of bond distance and chain strain. Umbrella sampling was accomplished using CP2K46with the PLUMED 2 plugin64to provide a restraining potential on the bond length. For the first eight strain increments, a chosen C–C bond restraint using a harmonic bias potential centered at riwas varied from just below equilibrium to dissociation: ri=1.40 to 4.40 Å in 0.05 Å increments. The bias potential for restraint iwas wi(r)=1 2K(r−ri)2, (15) where Kwas chosen to be 750 kcal/(mol Å). For each bias window, a 5000 time step (5 ps) NVT simulation was performed. Before the umbrella sampling runs were performed, in order to get a some- what equilibrated starting geometry for each bias potential center, ri, a constant-velocity bond-distance scan was first run for each box size, where the bond started at 1.35 Å and ended at 4.45 Å in 20 ps, increasing 1.55 ×10−4Å at every step (with a time step of 1 fs). Then, in the bias potential runs, the geometry from the scan with the bond distance that best matched the restraint potential center was chosen as the starting geometry. This allowed for the 60 biased runs to be computed in parallel with efficient sampling. In addi- tion, in these computations, the lateral cell dimensions ( yand z) were changed to 10.0 ×10.0 Å2to better accommodate the lateral expansion of the chain due to the increasing bond distance. In the subsequent umbrella integration averages, the first 100 steps (0.1 ps) were excluded. For the small number of degrees of freedom in these simulations, this was an adequate equilibration time, which was ver- ified by experimenting with the starting time for the averages. In the radical chain calculations, the dangling bond was placed on the fifth carbon atom from the origin, and the bond that was varied was the beta bond (the third bond that is highlighted with a thick red line in Fig. 2), known to be the weakest bond. The fourth bond included the radical carbon atom. Python scripts were written that extracted data from the CP2K output files and computed the free energies with the windows centered at the most probable bond distance from the fitted histograms.63 Free energy variations as a function of the C–C bond length for strains up to ∼0.12 are shown in Figs. 4(a) and 4(b). These curves are the result of adding the work done stretching the chains using J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. (a) The free energy for dissociating a C–C bond in the polyethylene chain at 300 K for various chain strains. (b) The free energy for breaking the beta C–C bond in the radical-defected chain. For both plots, the energy at the equilibrium bond distances was adjusted by the free energy data that were plotted in Fig. 3(a). The spread of the free energies near dissociation represents the statistical error in the simulations. (c) The activation energies for the polyethylene chain, and (d) the activation energies for the radical chain with corresponding fitting quadratic equations. data for the curves in Fig. 3(a) to the free energy change in the bond coordinate, which was set to zero at the equilibrium bond distance for the given strain. This is a more realistic way to view the effect of the strain on a molecule than that presented in Fig. 1 because the curves have a vanishing slope at the asymptotic limit and express the fact that the free energy of the system is a sum of the free energy changes due to stretching the molecule and stretching the bond,ΔA(ΔL,Δr)=ΔAcell(ΔL)+ΔAcoord(ΔL,Δr). (16) The overall process of the free energy is shown schematically in Fig. 5. ΔAcell(ΔL)is the change in free energy due to changing the cell length by ΔLusing Eq. (7) substituting ΔLforL, and ΔAcoord(ΔL,Δr)is the change in free energy at constant cell dimen- sions as a function of Δr, which in this case is a change in bond distance but could be a change in a general collective variable. FIG. 5. Schematic of the process for computing the free energy change for bond scission in a strained PE chain using periodic boundary conditions. J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ΔAcoord(ΔL,Δr)is computed using the umbrella integration algo- rithm,61–63as outlined above and described in more detail in the supplementary material. In principle, the curves for each strain should go exactly to the same limit in Figs. 3(a) and 3(b), but they do not here because of statistical errors in the simulations—both the equilibrium MD simulations for each chain strain and the restrained umbrella inte- gration simulations. To compensate for the statistical variation, the free energy at dissociation is given by the average value of ΔAat 4.40 Å and is 81.1 kcal/mol for PE and 20.8 kcal/mol for radical PE. The standard deviation and maximum absolute deviation are 1.1 and 2.3 kcal/mol, respectively, for PE and 0.5 and 0.7 kcal/mol for radi- cal PE. Because only one run for each bond length and chain length was performed for the umbrella integration curves, we will take the larger values as our error estimates. These are, nonetheless, small compared to the calculated energies. RESULTS AND DISCUSSION Activation energies as a function of chain strain for –C 8H16– and –⋅C8H15– are shown in Figs. 4(c) and 4(d), respectively. They are well fit by the quadratic functions, where ϵis the molecular chain strain. Quadratic dependence on force has been previously noted in other systems.16,18,19For zero strain, the activation energies are 82.6 and 30.5 kcal/mol. The bond-dissociation energy for the pris- tine PE chain is about 2 kcal/mol lower than the activation energy. It appears that there is an entropy contribution that lowers the dis- sociation energy because the internal energy is flat in that region (see Fig. S3). Examination of this matter is beyond the scope of this work as this requires significantly reducing the statistical errors with more sampling. The experimental bond-dissociation enthalpy at 298 K for C 8H18→C3H7+C5H11is 86.7±1.5 kcal/mol and for C 8H18→2 C 4H9is 87.0±0.9 kcal/mol.65The enthalpy from our simulation is estimated to be 90.4 ±0.6 kcal/mol (see Fig. S3), about 3 kcal/mol larger than the experimental estimate. Luo reports for the radical dissociation CH 2CH 2C⋅HCH 2–CH 2CH 2CH 2CH 3 →CH 2CH 2CHCH 2+⋅CH 2CH 2CH 2CH 3an experimental enthalpy estimate of 21.5 ±2 kcal/mol,65while our estimate for a simi- lar bond-dissociation energy in our periodic system is 27.2 ±0.8 kcal/mol. The difference in the length of the chain or position of the radical may be contributing to the larger disagreement, or more likely, our chains are constrained to remain uncoiled, while in experiment, this is not the case. An estimate of the experimental activation energy of the octyl radical dissociation, ⋅C8H17→C4H8 +⋅C4H9, from a careful analysis of shock-tube pyrolysis data is 28.6 kcal/mol.66Our internal energy barrier is 31.5 kcal/mol, and the free energy barrier is 29.9 kcal/mol from a ten run average of the equilibrium box; see Fig. S3. Tsang et al. used the RRKM method to fit the rate constant,66and since this does not have a tempera- ture dependent pre-exponential factor, the activation energy from it can be compared with our free energy of activation, which is 1.3 kcal/mol higher than theirs. Figure S4 shows that the internal energy maximum of the radical is due to an energy increase initially from breaking the C–C σbond and then an energy decrease due to one of the unpaired electrons from the C–C σbond redistributing to form aπbond with the original unpaired electron. To check for errors in the umbrella sampled cells due to peri- odic images, we doubled the xandzdimensions from 10.0 ×10.0to 20.0×20.0 Å2and found energies changed by less than 0.01 kcal/mol. These results are shown in Table S1 in the supplementary material. In Truhlar and Peverati’s paper on N12, their Table 3 shows that the mean unsigned error of the N12 func- tional for alkyl bond energies (ABDE4/05 dataset) is 3.81 kcal/mol,48 which is better than the other functionals considered there and is consistent with our results. The activation energy from the fit of Eq. (6) to time-to-rupture data for semi-crystalline PE and the rate of production of radi- cals in strained PE was reported by Zhurkov and Korsukov to be 27 or 28 kcal/mol.1This is comparable to the rate of pyrolysis of PE25and octyl radicals66and suggests that the rate-limiting step for PE rupture in the experiments of Zhurkov and Korsukov may be due to unstrained or slightly strained radical PE chains, which they supported with EPR data. Using EPR to analyze or develop models of polymer fracture by monitoring radical production has been pursued also by Roylance and co-workers6,67,68and Kausch and Devries.69The types of simulations used here can be used in conjunction with multi-scale models to build on this. We used TST to estimate the rate constant for unimolecular dissociation in the PE and radical PE chain cases. This method to estimate the rate constant leaves out tunneling, recrossing, and other effects such as bulk interactions. Nonetheless, the trends with respect to chain strain and the relative rates between the pristine chain and the radical-defected chain are instructive and likely to be use- ful as the input for multi-scale modeling. For a review of ways to improve rate estimates, see for example, the reviews in Refs. 70–72. TST rates at various chain strains are shown in Fig. 6. At equilib- rium, the dissociation rate is about 1038times greater for the radical beta bond than that for the alkane C–C bond. This ratio decreases with increasing strain. As the strains get larger, the rates become more comparable. For example, at a strain of ∼0.12 for the radical PE chain and 0.16 for the defect-free chain, the scission rate of both chains is about 105s−1. For the TST estimate, scission rates peak at 6.2×1012s−1when the activation energies are zero, which occurs at strains of 0.159 and 0.172 for the radical and pristine chains, respec- tively. Notably, the PE chain has to be strained to ∼0.135 for the rate to be approximately the same as the rate of the radical chain at equilibrium. FIG. 6. Unimolecular PE rate constant at 300 K for the dissociation of the C–C bond in PE and the beta bond in radical PE. J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp With this knowledge of the relative rates for chain scission, we are now in a position to discuss our results in the context of pos- sible PE rupture mechanisms. It is not until the entire system has strained significantly that individual chains experience significant tensile strain, unless for some reason, due to the microstructure, stress becomes concentrated in isolated regions. For example, in recent experiments on pre-oriented high-density PE (HDPE), strain hardening begins around 1.0.73In strain hardening, some chains may become strained to the point of breaking, thus creating a radi- cal for each chain that broke. In addition, friction from non-covalent entanglement between chains and chain branches could tear apart a bond. Once radicals are formed, they can be relatively quickly trans- ferred to other chains by hydrogen abstraction. For example, Tsang et al. found an intra-octyl radical transfer rate 8 orders of magni- tude greater than beta bond scission in gas phase pyrolysis.66The ratio would presumably be larger for adjacent molecules in poly- mers, where reorienting conformers is not as necessary as it is in the octyl radical. If those new radical chains are strained, they will break quickly. If rupture was due mostly to pristine chains break- ing at crystalline density, then it would occur at stresses of ∼30 GPa (see Fig. 3). If rupture was due exclusively to chains moving past one another in the viscous flow, the activation energy would be much lower (around 10 kcal/mol). Zhurkov and Korsukov reported1that the activation energy for the rupture rate is the same as that for the radical production rate from EPR experiments and that activation energy is∼27 to 28 kcal/mol. A re-examination of the available data in that paper by refitting according to Eq. (6), however, shows that activation energy is temperature dependent and at the lowest tem- perature (223 K) is about 26 kcal/mol for HDPE. Smook et al. per- formed constant stress time-to-rupture experiments on cross-linked ultra-high molecular weight PE (UHMWPE) and used the Zhurkov equation and two others.29With the Zhurkov equation, the activa- tion energy was found to be 18 kcal/mol. Due to the large chain lengths and orientation of the chains, we may expect that UHMWPE rupture would depend more on covalent bond breaking than HDPE. However, the activation energy is lower than the radical beta bond activation energy by about 10 kcal/mol. Rupture is, therefore, influ- enced heavily by flaws and chain end slipping, which is supported by theoretical calculations.74Nevertheless, there is ample evidence for radicals being formed in PE stretching and rupture1and in sim- ilar polymers.6,7,67–69DeVries et al. had some success reproducing experimental data of nylon 6 fibers using a stress-dependent rate equation similar to Eq. (6) for individual chains in a kinetics model.6 The activation energy they used, 67 kcal/mol, is consistent with some covalent bond types. It is clear, however, that the activation energies that are derived from experiments are due to a combination of phe- nomena and do not represent a single reaction as the rate-limiting step. We can deduce further evidence that the radical chain scission is an important factor by comparing with results of an idealized model of bulk PE. Using a diameter-dependent Griffith’s type model for the ultimate strength of fibers that assumes some amount of defects in a finite diameter fiber, Smook et al.29fit their data and extrapolated to a zero diameter, which represents an idealized sin- gle chain that does not have macroscopic defects. They obtained a theoretical ultimate strength of 26 GPa for PE. This is similar to our value of 29 GPa for the radical PE chain at crystalline density, which is somewhat of a better match than 30 GPa for the normal PE chain.Here, we used the strains at which the activation energies as a func- tion of strain are zero and not the strain before which the chains broke in the 50 ps MD runs, although for the radical, the strain is the same. The radical chain ultimate strength agrees with Smook’s value slightly better. At 26 GPa, our model has a strain of 0.13. At this strain, the TST rates are 4.3 ×10−12s−1for PE and 3.1 ×106s−1for radical PE. The relationship between rates of individual chain scis- sion and bulk rupture would require solving differential equations similar to those developed by Bell22and Eyring,14which is outside the scope of this work. We now consider how the radical PE chain model would respond to constant applied force. This would be difficult to achieve in reality at this scale because force can fluctuate significantly due to thermal motion. In what follows, we consider how the bond distance and chain end-to-end length (or strain) would need to coordinate to maintain a constant average force while removing the average thermo-statistical effects from other bonds. Even though it would be difficult or impossible to maintain these conditions except in simu- lations, it is instructive to consider because it gives insight into how bond-dissociation rates may be affected by force. Contours of constant force [shown in Fig. 7(a)] were derived from constant strain simulations by smoothing and interpolating the force on the simulation cell in the strained ( x) direction to make the surface a function of beta bond length and chain strain. The force used was the umbrella-averaged virial stress, σx,x, multiplied by the cell face area. Figures S4–S7 in the supplementary material and the corresponding discussion provide details on how the sur- face was derived. Curves of the equilibrium and transition state geometry as a function of the beta bond distance under various constant chain strains were superimposed on the lines of constant force in Fig. 7(a), and the corresponding free energy differences between the points where the equilibrium geometry and transition state geometry curves intersected the lines of constant force pro- vided the constant-force activation energies plotted in Fig. 7(b). The activation energy at forces lower than 750 pN could not be derived because the cell strain dimension would have to be reduced to a value lower than the equilibrium cell distance to maintain a lower force as the bond length was increased at those forces. Let us consider a physical interpretation of this model and its implications. Following the curves of constant chain force in Fig. 7(a), we see that as the beta bond length initially increases, the chain strain does not need to change significantly in order to main- tain a constant force. This is indicated by the path of constant force being roughly vertical. As the beta bond length increases more, the chain strain must increase significantly for a small change in the bond length, which is evident because the paths become skewed toward horizontal. The pulling speed would need to increase in order to maintain a constant force on the chain. When the iso-force curves cross the upper orange curve, the derivative of the free energy is non-zero in the direction of the constant-force path (as can be inferred from Fig. S8). However, the derivative of the free energy in the direction of beta bond length with constant strain is zero at the orange curve [as can be seen in Fig. S6(b)]. With thermal motion randomly perturbing the beta bond length, the bond distance may increase beyond the ridge of the meta-stable region of constant force, which would, in turn, create a force that would push the atoms apart, and the overall force on the chain as well as on the bond would go to zero. In order to maintain the balance of constant force, the J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7. Imposed chain force results: (a) contours of constant force in pN vs bond length and chain strain, the upper horizontal orange line is at the transition state and the lower horizontal blue line is at the equilibrium bond length for each strain, (b) activation energy as a function of imposed chain force, (c) unimolecular radical PE rate constant as a function of imposed chain force, and (d) the corresponding bond lifetime. pulling speed would need to rapidly increase, but ultimately, the situation is unstable near the constant strain transition state, and the chain will dissociate. This justifies the use of the constant strain transition state in defining an approximate constant-force activation energy. Given this construction, we arrive at rates [shown in Fig. 7(c)] that are analogous to rates in the “catch-bond” scenario. The “catch- bond” phenomenon was first described by Dembo et al.23based on an analysis of a kinetics model for cell detachment from a membrane and later expanded on by many authors.16,24,41,75–79Evans et al.80 derived from experimental information a lifetime of two proteins bonded together that increases with force to a maximum and then decreases in a manner qualitatively similar to Fig. 7(d) with a peak at about 40 pN. The mechanism in that case is different than that for covalent bonding. In the “catch-bond” literature, the system being modeled is often comprised of several non-covalent bonds being broken such as in a cell attached to a membrane or a poly-peptide being pulled from a surface. However, in our case, we have a sin- gle covalent bond being broken, where the applied force reduces the bond breakage rate. We note that the distance between the equilib- rium geometry and the transition state becomes smaller as the force gets larger (see Fig. S8), in accordance with theories by Dudko,16 Boulatov,18and Makarov.81 Given the detailed data from the simulations, a physical expla- nation of the nonlinear activation energy trend in Fig. 7(b) is avail- able. To our knowledge, this is the first time a detailed and morerealistic prediction of the effect of constant imposed force on cova- lent bond breaking based on reliable simulations has been presented. Free energy curves of constant strain and constant imposed chain force are shown in Fig. 8. The constant strain curves in Fig. 8 were designed to intersect the designated transition state of the constant- force curves to help guide the eye. The constant-force curves start out at lower strains, where the free energies are lower near the equi- librium bond length. As the bond stretches at constant strain, the force on the chain goes down (see Fig. 9). In order for the force to remain constant, the chain must be strained further. As the sys- tem evolves along the constant-force path, the strain continues to increase. This causes the bond to be in the transition state region at higher strains than where it started, which leads to larger activa- tion energies, as shown in Fig. 7(b). Therefore, imposing a constant force increases the activation energy and reduces the scission rate. This would be the case in any reaction where the force (or stress) on a larger portion of the system goes down as the reaction pro- gresses at constant strain, or in other words, forces relieved due to the reaction are opposed to the force imposed on that portion. In the case of covalent bond breaking considered here, the bond stretching reduces strain in other portions of the chain, which reduces the force on the cell. Let us consider some aspects of the electronic structure for the case of radical PE that are influenced by the mechanical manip- ulations. At the transition state, the radical dangling bond at car- bon C 5has mostly moved to carbon C 3, which will be localized on J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8. Two views of the free energy as functions of constant strain (solid lines) and constant force (dashed lines). The curves of constant strain are at strain values of 0.016, 0.031, 0.042, 0.049, 0.054, 0.064, 0.072, 0.080, 0.087, 0.097, 0.108, and 0.118. The curves of constant force are the same as those in Fig. 7 (875, 1000, 1250, 1500, 1750, 2000, 2500, and 3000 pN). the –H 2C–H 2C–H 2C⋅fragment in the product (the left fragment in Fig. 2). This is supported by a natural resonance theory calcu- lation,82–84showing that at the transition state, the product reso- nance structure has a 71% weight. See Table S2 and Fig. S10 in the supplementary material. As the molecule gets strained, this weight goes down somewhat and resonance structures with the radical on the other fragment increase. Therefore, we can interpret that the effect of strain on the transition state is to reduce the amount of transfer of the radical to the product fragment and that strain reduces the probability of the radical moving to the product state. This is commensurate with an activation free energy increase. Note that the main change due to strain in the geometry at the transi- tion state is increasing the C 3C4C5angle (see Table S2), but the distance between C 3C4, the reaction coordinate, decreases as already noted. Near the equilibrium geometry, the effects of the strain on the molecule increase C–C–C bond angles and C–C bond lengths that are distributed evenly throughout the chain. The strain on these internal coordinates raises the energy of the chain, as shown in Fig. 3. When a constant strain path is followed, the activation energy decreases with increasing strain. Therefore, we can conclude that strain affects the energy of the chain near the equilibrium geom- etry more than that at the transition state because some strain is relieved on other bonds. In constant-force paths, the strain increases as the bond breaks so that the effects of strain are enhanced near the transition state. Finally, we find that the forces from the pristine –C 8H16– chain are not reliable enough to perform a similar analysis. This is due to consequences of bond breaking in the unrestricted Kohn–Sham approach, which is supported by the spin-symmetry breaking seen in Fig. 10, where ⟨S2⟩is plotted along with the force in the strained direction derived from the virial stress, σx,x.54–56Decreases in the force magnitude occur where ⟨S2⟩increases for the intermediate cell lengths, but are not significant for the equilibrium length or the most strained length. The corresponding forces and ⟨S2⟩are shown in Fig. 9 for the radical −⋅C8H15−chain. In this case, the force curvesare relatively smooth and ⟨S2⟩does not change significantly (note the change in scale for ⟨S2⟩). In this case, the dangling bond in the original radical is first responsible for the unpaired electron, but as the beta bond stretches, the radical moves to the other fragment in a singly occupied σ∗orbital (see the HOMO in Fig. S3), and the spin- symmetry breaking is not significant. For the non-radical case, the free energy curves are not pathological but the forces are. A previous study showed kinks in potential energy with respect to stretching molecules for many functionals including hybrid and range-separated functionals.52Generalized gradient approximation (GGA) functionals showed the smoothest potential energy curves, and for most cases, no kinks were observed, especially for the N12 functional. In a couple of cases with the N12 functional, we noted subtle changes in the geometry when the unrestricted Kohn–Sham density functional theory (DFT) solution switches from the sin- glet solution to the spin-symmetry-broken solution even though the kink (discontinuity in the force) was small (e.g., isopropyl amine).52 In the case of –C 8H16–, examination of the internal coordinates shows that when the bond begins to break where ⟨S2⟩becomes non- zero, the dihedral angles deviate from maintaining the back-bone carbon atoms roughly in a plane, the bond angles containing the bond begin to fluctuate more, and the negative force on the cell increases, even though we do not see a notable kink in the potential energy averages or free energy curves. Hait et al. noted pathologies in the second derivative properties of Kohn–Sham density func- tional theory solutions of simple systems with stretched bonds at the Coulson–Fischer point,85where the spin-polarized (spin-symmetry broken) solution is equal to the spin-restricted solution, attributing the non-physical behavior to singularities due to a zero eigenvalue of the orbital Hessian.86The virial stress is the first derivative of the simulation cell total energy with respect to the strain tensor ele- ments, which does not involve a second derivative of the energy with respect to orbital parameters. The problem observed here is due to discontinuities in the forces. It would require further study to under- stand how these discontinuities of the forces with respect to internal J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9. −⋅C8H15−force in the xdirection derived from the virial stress, σx,x, as a function of bond length for each of the stretched cells (blue) along with the length of the y +zcomponents of the force (orange) and with the spin squared expectation value (green). Chain strains are (a) 0.00, (b) 0.02, (c) 0.04, (d) 0.06, (e) 0.08, and (f) 0.10. coordinates influence the virial stress and whether this can be mit- igated by other electronic structure methods, which is outside the scope of this paper. We do note, however, that the umbrella inte- gration method can lead to smooth and fairly accurate curves with a minimal amount of statistical sampling. These desirable properties compensate for small discontinuities in the forces. However, due to the force discontinuity issue and the com- plex dependence of the chain force on the activation energy, it isnot recommended to use constant-force-based ab initio molecu- lar dynamics approaches that depend on the virial stress computed from periodic boundary condition simulations for modeling general covalent bonds. Rather, for these types of simulations, strain-based methods are more reliable. In any case, care must be taken to validate the effect of spin-symmetry breaking on the results when depending on it for localization of the electrons for quantitatively correct bond dissociation. J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-11 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 10. –C8H16– force in the x direction derived from the virial stress, σx,x, as a function of bond length for each of the stretched cells (blue) along with the length of the y +zcomponents of the force (orange) and with the spin squared expectation value (green). The chain strains are (a) 0.00, (b) 0.02, (c) 0.04, (d) 0.06, (e) 0.08, and (f) 0.10. CONCLUSIONS We have demonstrated that ab initio MD with umbrella sam- pling can be used to model the effect of strain and force on small model polymer chains and that this information can be used to help interpret polymer scission rates in tension or be used in larger scale simulations. We present a sufficiently accurate demon- stration of a free energy method for breaking a covalent bond with DFT, which also shows good agreement with experimentalactivation energies and dissociation energies from gas-phase data. Isolated PE chains were used as a proof of principle for using transi- tion state theory for estimating rates. The approach used here could be extended in several ways. For example, a condensed-phase sim- ulation could be designed to show the effect of pressure and inter- action with nearby chains such as entanglements. Rate estimates can be improved using more sophisticated methods that account for recrossing and tunneling. J. Chem. Phys. 155, 024102 (2021); doi: 10.1063/5.0047371 155, 024102-12 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We demonstrated a realistic picture of the effect of strain on the activation energy of breaking a covalent bond in a model PE chain, which varies quadratically with applied strain. We demon- strated that the work done on stretching the chain should be used to adjust the equilibrium energy for a given chain strain according to Eq. (16). This results in the correct asymptotic behavior and is fun- damentally correct and more satisfactory than the traditional picture shown in Fig. 1 because the asymptotic behavior is correct. Theories of mechanical properties of molecules should be based on equations similar to Eq. (16), as has already been noted by Suzuki and Dudko in the case of imposed force applied to simple models.16Here, we have confirmed this viewpoint with realistic simulations of idealized polymer chains. Our results support the importance of radical reactions in poly- mer rupture. The activation energy of breaking a single radical- defected PE chain near equilibrium is similar to the activation energy reported by Zhurkov and Korsukov.1In that study, the activation energy for radical concentration is similar to the acti- vation energy of the lifetime of bulk semi-crystalline PE. The life- time of a defect-free PE chain is several orders of magnitude larger than that of the radical chain. These may break only if a large amount of stress is concentrated in a small number of chains,87 which would introduce radicals into the system, or radicals could be generated by other mechanical means. Once a radical is formed, it can be transferred to nearby chains through hydrogen transfer at a much faster rate than radical chain scission. We have sup- ported this quantitatively by citing gas phase rates.66In PE, chain scission of segments under significant stress or strain will occur after the bulk systems have been strained significantly. The con- tribution of the scission of pristine chains is likely to be relatively rare. The effect of a constant imposed force on a molecular bond is more complicated than Arrhenius-like equations such as Eq. (3) or Eq. (6). These phenomenological equations have been interpreted as applicable to bonds but are more appropriate for bulk behav- ior under certain circumstances. These result from the kinetics of multiple processes and may be derived from a system of equations that include the effect of strain or force at the molecular scale. More research is needed to understand how systems of force- or strain- dependent kinetics equations can lead to simpler functional forms for Arrhenius-like equations. Multi-scale models that include micro- structural features and bulk properties may also help interpret the Eyring, Zhurkov, and Bell equation parameters and would lead to deeper insight into any case. It would seem to be more satisfying to use constant-force simulations in this pursuit, but constant strain simulations are more convenient, and as we have shown, there is a pathological behavior in the virial stress in periodic boundary sim- ulations due to issues from DFT in modeling the breaking of a sin- glet covalent bond. Rates for strained chains can be used in kinetic Monte Carlo simulations such as those in Ref. 88 to model late stages of polymer rupture. SUPPLEMENTARY MATERIAL See the supplementary material for additional figures and tech- nical details.ACKNOWLEDGMENTS The authors gratefully acknowledge funding through Dr. Ming- Jen Pan of the Air Force Office of Scientific Research (Grant No. 18RXCOR060). The authors would like to thank the HPCMP for support under the PET program. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. N. Zhurkov and V. E. Korsukov, J. Polym. Sci., Polym. Phys. Ed. 12, 385–398 (1974). 2S. N. Zhurkov, V. A. Zakrevskyi, V. E. Korsukov, and V. S. Kuksenko, J. Polym. Sci., Part A-2: Polym. Phys. 10, 1509 (1972). 3H. H. K.-B. von Schmeling, Colloid Polym. Sci. 289, 1407 (2011). 4J. Sohma and M. Sakaguchi, Adv. Polym. Sci. 20, 109 (2005). 5H. H. K.-B. von Schmeling and J. Becht, Rheol. Acta 9, 137 (1970). 6K. L. DeVries, D. K. Roylance, and M. L. Williams, Int. J. Fract. Mech. 7, 197 (1971). 7G. S. P. 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The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Adsorbate modification of electronic nonadiabaticity: H atom scattering from p(2×2)O on Pt(111) Cite as: J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 Submitted: 2 June 2021 •Accepted: 30 June 2021 • Published Online: 19 July 2021 Loïc Lecroart,1,2Nils Hertl,1,2 Yvonne Dorenkamp,2Hongyan Jiang,1,2 Theofanis N. Kitsopoulos,2,3,4 Alexander Kandratsenka,1,a) Oliver Bünermann,2,5 and Alec M. Wodtke1,2,5 AFFILIATIONS 1Department of Dynamics at Surfaces, Max-Planck-Institute for Biophysical Chemistry, Am Faßberg 11, 37077 Göttingen, Germany 2Institute for Physical Chemistry, Georg-August University of Göttingen, Tammannstraße 6, 37077 Göttingen, Germany 3Department of Chemistry, University of Crete, 71003 Heraklion, Greece 4Institute of Electronic Structure and Laser–FORTH, 71003 Heraklion, Greece 5International Center for Advanced Studies of Energy Conversion, Georg-August University of Göttingen, Tammannstraße 6, 37077 Göttingen, Germany a)Author to whom correspondence should be addressed: akandra@mpibpc.mpg.de ABSTRACT We report inelastic differential scattering experiments for energetic H and D atoms colliding at a Pt(111) surface with and without adsorbed O atoms. Dramatically, more energy loss is seen for scattering from the Pt(111) surface compared to p(2×2)O on Pt(111), indicating that O adsorption reduces the probability of electron–hole pair (EHP) excitation. We produced a new full-dimensional potential energy surface for H interaction with O/Pt that reproduces density functional theory energies accurately. We then attempted to model the EHP excitation in H/D scattering with molecular dynamics simulations employing the electronic density information from the Pt(111) to calculate electronic friction at the level of the local density friction approximation (LDFA). This approach, which assumes that O atoms simply block the Pt atom from the approaching H atom, fails to reproduce experiment due to the fact that the effective collision cross section of the O atom is only 10% of the area of the surface unit cell. An empirical adiabatic sphere model that reduces electronic nonadiabaticity within an O–Pt bonding length scale of 2.8 Å reproduces experiment well, suggesting that the electronic structure changes induced by chemisorption of O atoms nearly remove the H atom’s ability to excite EHPs in the Pt. Alternatives to LDFA friction are needed to account for this adsorbate effect. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0058789 I. INTRODUCTION The Born–Oppenheimer approximation1has served as a bedrock for theoretical chemistry,2providing a standard model of chemical reactivity involving nuclear motion on an electronically adiabatic potential energy surface.3In constructing an analogous model for interactions of atoms and molecules at metal surfaces and especially for surface chemistry, it is necessary to account for electron–hole pair (EHP) excitations that may accompany nuclear motion.4,5For example, H and D atoms adsorbing at the metal surface embedded in a Schottky diode produce chemicurrents,6–8and diatomic molecules prepared in high initial vibrational states exhibit multiquantum vibrational relaxation9–11even producing emitted electrons12–15when scattered from metals. This behavior is not seen on insulators where vibrational relaxation is inefficient.16,17 Failure of the Born–Oppenheimer approximation in surface chemical dynamics has been a topic of great discussion, and com- peting viewpoints have contributed to progress toward its better understanding.18 H atom scattering from metals proves to be an excellent testing ground for theory, as new experimental methods allow the precise measurement of translational inelasticity, and the simplicity of the J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 155, 034702-1 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp H (or D) atom makes the problem ambitious yet tractable for theory.19Using nearly monoenergetic H and D atom beams whose energy is tunable, differentially resolved H scattering from several metals20,21has been detected by Rydberg atom tagging time-of-flight22,23to obtain high-resolution translational energy distributions at selected incidence and scattering angles. Using full-dimensional potential energy surfaces (PESs) and background electron densities, based on an effective medium theory (EMT)24,25 fitted to density functional theory (DFT) data, electronically non- adiabatic molecular dynamics (MD) trajectories experiencing elec- tronic friction at the level of local density friction approximation (LDFA)26–28reproduce many experimental observations.29This theory was also able to explain the H and D isotope effect and magnitude of experimentally observed chemicurrents.30 In this work, we extend these ideas to investigate electroni- cally nonadiabatic interactions of H and D atoms on a metal surface with a chemically bound adsorbate. We first present experimental observations comparing H scattering from Pt(111) with and without ap(2×2)adsorbed O atom overlayer and show that O atom adsorption dramatically reduces the translational inelasticity by limiting EHP excitation. This is the case despite the fact that O atoms are present at only 0.25 monolayer (ML) surface concentration. We then explore two hypotheses to explain these observations involving: (1) a mechanical blocking LDFA model and (2) an electronically adiabatic sphere model. In the blocking model, the adsorbed O atoms prevent the H atom approach to Pt, limiting EHP excitation possible in the high electron density regions near the Pt atoms. In the adiabatic sphere model, the EHP excitation probability is suppressed within an empirically optimized distance 𝜚adfrom each O atom. While both models reduce translational inelasticity by reducing the probability for EHP excitation, we find that the effective mechan- ical blocking cross section of the adsorbed O atom in collisions with H or D is only 2.84 Å2, a small fraction of the 27.16 Å2area of the surface unit cell. This relatively small modification of scattering dynamics cannot explain the experimental observations. We find, however, that when 𝜚ad=2.8 Å, good agreement with experiment is found. We emphasize that the adiabatic length scale is similar to the O–Pt bonding length scale and that the adiabatic shadow area π𝜚2 ad=24.6 Å2is a large fraction of the unit cell’s area. Thus, these trajectory simulations are only weakly nonadiabatic. This failure of the LDFA leads us to conclude that the changes in the surface electronic structure induced by the adsorption reduce the ability of the H/D atom to excite EHPs in energetic collisions. Application of more sophisticated theories of nonadiabatic dyna- mics such as orbital specific friction31,32or independent-electron surface hopping33may be necessary to explain the adsorbate effect. II. METHODS A. Experimental setup The experimental scattering apparatus has been described elsewhere.19,34Briefly, H atoms were generated by photodissoci- ation of a supersonic molecular beam of hydrogen iodide with pulses of KrF excimer laser light at 248.35 nm, producing H atoms with incidence energies of Ein=1.92±0.02 eV. H atoms travelingnormal to the molecular beam scatter from the room temperature surface and are subsequently excited to a long-lived Rydberg state by two laser pulses, one exciting the 1 s→2ptransition at 121.6 nm and another the 2 p→34dtransition at 365.9 nm. The resulting metastable atoms travel 25 cm without radiative loss and are field-ionized and detected by using a multichannel plate detector. A multichannel scaler records the arrival time, and the calibrated flight length is used to obtain H atom speeds. The Pt(111) surface was cleaned using several cycles of Ar-ion sputtering and annealing at 1000 K. The oxygen-covered Pt(111) surface was prepared in situ by dosing the cleaned Pt(111) surface for 5 min with oxygen (10−6mbar) at room temperature. This proce- dure results in a saturation coverage of 0.25 ML atomic oxygen with a preferential p(2×2)orientation.35,36The surface structure was confirmed by Auger electron spectroscopy and low-energy electron diffraction. B. Potential energy surface We constructed a full-dimensional PES for an H atom at an O-covered Pt(111) surface, V=VEMT+VO–H+VO–Pt, (1) by augmenting VEMT—the H–Pt(111) EMT-PES reported else- where37—with terms accounting for the interaction of the adsorbed oxygen with hydrogen VO–Hand platinum VO–Pt atoms. We neglect the interaction between adsorbed oxygen atoms, which are 5.6 Å from one another (see Fig. 1). The advantages of using an EMT-PES derive from its calcu- lational efficiency and simplicity, while retaining a relatively small root mean square error (RMSE) with respect to DFT. It also provides the background electron density necessary to calculate the electronic friction coefficient.29Most importantly, H atom energy losses cal- culated from MD simulations employing a Langevin propagator to describe LDFA electronic friction on an EMT-PES agree well with the energy losses obtained from the H(D) scattering experiments from transition metals including Pt(111).38 The O–Pt interaction is modeled by an anharmonic potential with a dissociative asymptote,39 UM(r)=D(e−α0(r−r0)−1)2 −D, (2) where ris the O–Pt distance, Dis the dissociation energy, and r0is the equilibrium distance where the curvature is α0. The last term in the rhs of Eq. (1) is then given by VO–Pt=NO ∑ i=1NPt ∑ j=1UM(rij), (3) where rijis the distance between oxygen atom iand platinum atom j andNOandNPtare the number of O and Pt atoms in the simulation box, respectively. J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 155, 034702-2 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1. Thep(2×2)O on the Pt(111) surface as seen by an H atom. (Left panel) The size of the O atoms (red) is 2.84 Å2, representing the area over which repulsive collisions between H and O atoms may occur at E in=2 eV. The unit cell is shown as a parallelogram with an area of 27.16 Å2. The radius of the orange circles 𝜚ad=2.8 Å represents the distance of adiabatic shadowing derived from this work. (Right panel) Pictorial representation of various sites in the surface referred to in the text. The interaction between the H and the O atoms—second term in the r.h.s. of Eq. (1)—is given by VO–H=NO ∑ i=1UEOM(ri,θi), (4) where riis the distance between the H atom and the ith O atom and θiis the angle between the vector connecting H and the ith O and the normal to the surface. UEOM(r,θ)is the extended oriented Morse (EOM) potential, UEOM(r,θ)=UOM(r,θ)+D1(r−ρ)2sin2θ 2e−β(r−ρ), (5) where UOM(r,θ)is an oriented Morse (OM) oscillator, UOM(r,θ)=D0(e−2α(r−r0)−2 cosθe−α(r−r0)). (6) The parameters V0,V1,r0,ρα, andβwere determined by fitting to DFT data as described below. C. Fitting parameters The EMT fitting parameters needed to describe the H–Pt system were determined by Kammler et al.37using a genetic algorithm method to find a global minimum. To find the parameters for the O–Pt interaction potential (3), we used data available in the literature for: (i) the frequency of the hindered O atom translations measured by electron energy loss spectroscopy,40ν=490 cm−1; (ii) the equilibrium Pt–O distance at the fccsite calculated by DFT using a generalized gradient approximation Perdew–Burke–Ernzerhof (GGA-PBE) functional, req=2.04 Å;41and (iii) the adsorption energy of an O atom at the fccsite of Pt(111) obtained from calorimetry42and temperature programmed desorption43—both methods giving Ea=3.4 eV for 0.25 ML O coverage. These three physical properties of the O–Pt(111) system are related tothree parameters of the Morse potential (2) by the following equations: ν=α 2π√ 2D μ, (7) ∂VO–Pt ∂zO∣ zO=zeq=0, (8) VO–Pt(rO=req)=−Ea. (9) Here, Eq. (7) relates the fundamental vibrational frequency to the parameters of the Morse potential in Eq. (2), where μdenotes the reduced mass of the oscillator.39Equation (8) accounts for the fact that when the O atom is in its equilibrium position ( rO=req), the projection of the force on the O atom along the axis znormal to the surface vanishes. Finally, Eq. (9) associates the energy of the O atom adsorbed at the fccsite with the measured O atom adsorption energy. The values of the parameters of the Morse potential deter- mined from the system of algebraic Eqs. (7)–(9) are D=1.06 eV, r0=2.06 Å, and α0=2.59 Å−1. To determine the optimized values of the six parameters of the O–H interaction described by Eq. (4), we first performed spin- polarized DFT calculations (see Sec. II D for details) for the H atom approach to the O–Pt surface at 11 symmetry sites (see Fig. 1) using a step size of 0.2 Å in the H atom distances from the surface. In the vicinity of the O atom, the step size was reduced to 0.025 Å. Here, the Pt and O atoms were held fixed at their relaxed configuration in the absence of the H atom. We then optimized the parameters of Eq. (4) to obtain the minimum possible deviation between the pre- dicted energy of Eq. (1) and the DFT data. The optimized parameter values are D0=1.01 eV,α=4.00Å−1,r0=0.92 Å, D1=90.3 eV Å−2, β=4.56Å−1, andρ=0.70 Å, with a RMSE of 187 meV. The fit (solid line) to the DFT data (empty circles) is shown in Fig. 2 for the bridge site closest to the adsorbed oxygen (b 1in Fig. 1). The PES matches the features of the DFT energy profile well. J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 155, 034702-3 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. A 2D cut (—) through a multidimensional PES developed in this work for H interaction with p(2×2)O on Pt(111) at the b 1site (see Fig. 1). DFT data ( ⋅) are shown for comparison. zis the distance from the H atom and the upper Pt layer. Two fits using simplified functions are also shown. See the text. The PES is fit poorly if simpler fitting functions are employed. With D1set to 0 in Eq. (5), the dotted line in Fig. 2 is obtained. This demonstrates the importance of the exponentially repulsive term in Eq. (5) to correctly represent the DFT data in the vicinity of the adsorbed O atom (region between 0 and 2 Å for the H atom dis- tance to the Pt surface zin Fig. 2). The dashed line in Fig. 2 shows the fit obtained when setting the O–H repulsion radius ρ=0 in the second term in the rhs of Eq. (5). Though the fit looks better than that in the case of D1=0, it is still qualitatively bad in the vicinity of the oxygen. We note that upon O–H bond formation, our PES fails to account for the migration of the adsorbed oxygen from its lowest energy binding site at an fcchollow to a bridge site, the minimum energy binding site of OH diatomic on Pt(111). Of course, it is of interest to correct this deficiency in the PES, for example, to accu- rately describe OH adsorbate formation and equilibration with the solid. However, we do not expect that this aspect of the PES will have significant influence on the subject of this study, since the duration of the H and D atom collisions with the p(2×2)O–Pt(111) surface is about 100 fs or less. In fact, the time scale within which the H atom samples the adsorbtion well is even shorter than this. On these short time scales, the heavy O atom cannot carry out its displacement from fccto the bridge site. D. Details of the DFT calculations The DFT data were generated using VASP5.3.544–47with the GGA-PBE functional.48The optimized lattice constant for an ideal platinum crystal was determined to be a0=3.96 Å. The plane wave cutoff energy was set to 400 eV. The Pt(111) surface was modeled as a 2×2 slab with six layers. To avoid interactions between the slab and its periodic images in the direction normal to the surface,a vacuum layer of 13 Å was incorporated into the simulation cell. The Brillouin zone was sampled by a 4 ×4×1Γ-point centered k-point mesh using Monkhorst–Pack sampling.49The interaction between valence and core electrons was described by the projector- augmented wave method.50Partial electronic occupations were modeled with the Methfessel–Paxton ( N=1) smearing scheme51 with a smearing width of σ=0.2 eV. For the calculations involv- ing the oxygen-covered Pt(111), an oxygen atom was set to the most-stable binding site, the fcchollow, to simulate a coverage of 0.25 ML. The electronic energy calculations were considered to be converged when the energy difference was smaller than 10−4eV between two iteration steps. For calculations including oxygen adsorbates, van der Waals corrections according to the method of Tkatchenko–Scheffler52were taken into account. E. Molecular dynamics simulation details The MD trajectories were simulated with the md_tian2 package available at a public repository.53The package utilizes the propagator as described by Dellago et al.54to solve numerically the Langevin equation. The time step in simulations was set to 0.1 fs. III. RESULTS AND ANALYSIS Figure 3 compares the experimentally obtained translational energy distributions for H scattering from Pt(111) ( +) and p(2×2) O on Pt(111) ( ○) under representative conditions. The former exhibits a broad energy loss distribution with ⟨EH–Pt loss⟩≈0.71 eV, FIG. 3. Comparing H scattering experiments for Pt(111) ( +) with p(2×2) O–Pt(111) (○).Ei=1.92 eV,ϑi=40○, andϑs=35○. The results of the electron- ically nonadiabatic molecular dynamics simulations for H scattering from Pt(111) (—) are also shown. See also Ref. 38. ΔεBCMshows the maximum energy that can be transferred in a single collision between an H and an O atom without EHP excitation. J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 155, 034702-4 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp resulting from EHP excitation and multibounce collisions.38For H scattering from the clean metal, excellent agreement with exper- iment is obtained using a Langevin propagator to account for electronic friction at the level of LDFA24,55(—). The LDFA is popular and relatively easy to implement means to model EHP excitation in atom scattering from metals. It assumes that the friction coefficient depends only on the metal’s electron den- sity at the position of the H atom. Figure 4 shows the electron density calculated from DFT at several surface sites for H interacting with Pt(111) (blue lines) and from p(2×2)O on Pt(111) (red and black lines). At sites far from the adsorbate biding site f 1(see Fig. 1), the electron density is similar to that of Pt(111) (compare blue and red curves); however, near f 1, the electron density is enhanced by the presence of the O atom [compare blue and black curves in Figs. 4(a) and 4(b)]. These calculations suggest that the electron density and, hence, the friction coefficient should certainly not be diminished by O atom adsorption. Figure 3 also shows the experimental energy loss for H scatter- ing from p(2×2)O on Pt, which exhibits a broad, high energy loss feature similar to H scattering from Pt(111). However, the major feature in the energy loss spectrum is a sharp peak at low-energy loss. The average energy loss ⟨EH–O–Pt loss⟩≈0.42 eV is smaller than that for H scattering from Pt(111) under the same conditions. This is clear evidence that the LDFA fails to capture the adsorbate effect, which clearly suppresses EHP excitation. Hence, we modi- fied our model of EHP excitation to try for better agreement with experiment. The sharp peak in the energy loss distribution for H scattering from p(2×2)O on Pt (Fig. 3, open circles) appears to die away at the binary collision limit, ΔεBCM=Ei[1−(mO−mH mO+mH)2 ], (10)which is the maximum energy that can be transferred in a single collision between an H and an O atom without EHP excitation. This observation suggested us a mechanical blocking model, where the H atom interacts with O atoms adiabatically giving rise to the sharp low-energy loss peak and with Pt atoms according to the LDFA producing a broad high energy loss feature. In this picture, the H atom collisions at O atoms occur without electronic friction, so the maximum energy loss might well be expected to coincide with ΔεBCM. The dynamics could then be similar to the case of H on Pt, but the effect of the O atom adsorbate would be to limit the approach of the H atom to regions of high electron den- sity associated with the Pt metal, where electronic friction can be high. The PES obtained by fitting to the DFT data, as described in Secs. II B and II C, is represented in detail in Fig. 5 as 1D cuts, and the Pt electron density function used for calculating electronic friction is shown in Fig. 4 ( ○). In each panel, the newly developed PES for the H atom approaching along the surface normal is com- pared to DFT data both with and without the O atom. Figures 5(a) and 5(e) show the situation for the two fcchollow sites, only one of which can bind to an O atom. The influence of the O adsorbate is striking. At f 1, where the O atom is bound, one sees a binding well (H–O–Pt formation) at z≈2.3 Å and a strong repulsive inter- action blocking the H atom from entering the fcchollow of the Pt(111) surface. Figures 5(b), 5(d), and 5(h) show the interaction potential for the vertical approach to sites with an intermediate distance from the O atom: fh, the nearby b 1, and the ot sites. Figures 5(c), 5(f), and 5(g) show more distant sites. One obser- vation worth noting from the inspection of these potential curves is that the H atom must collide quite closely to the adsorbed O atom to avoid collision with the underlying Pt surface. At the b 1 site, the H–O–Pt binding well can still be seen, but the O atom can no longer fully prevent the H atom’s approach to the Pt surface. FIG. 4. Electron densities experienced by the H atom at different surface sites of Pt(111) with and without adsorbed O atoms. The lines show DFT results along with EMT background densities ( ○). We show two: (a) fcchollow sites, (b) bridge sites, (c) hcp hollow sites, and (d) top sites. Figure 1 shows the sites. J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 155, 034702-5 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5. Cuts through the multidimensional PES for H interacting with p(2×2)O on Pt(111) (—) and Pt(111) (—). DFT data ( ○) used in the fitting procedure are also shown. The H atom approaches: (a) the f 1hollow, where the O atom preferentially binds, (b) the b 1site, (c) the t 1site, (d) the fh site, (e) the f 2site, (f) the b 2site, (g) the hcp hollow site h 1, and (h) the ot hollow site. Sites are identified in Fig. 1. MD simulations of the H atom scattering from O-covered Pt(111) on the constructed PES, where O atom blocking is accounted for and electron densities from the H/Pt system are employed, produce energy loss distributions, as shown inFigs. 6–9. Here, experiment and theory are compared for both H and D scattering at two incidence energies of translation. One sees that under all experimental conditions, the mechanical blocking model (solid lines) exhibits much larger energy loss FIG. 6. Comparison of MD simulations with experimental energy loss distribu- tions (○) in H scattering from p(2×2) O on Pt(111) at Ei=1.92 eV.ϑi=45○. Mechanical blocking model (—), adia- batic results (—), and predictions of the adiabatic sphere model are also shown (⋅). J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 155, 034702-6 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7. Comparison of MD simulations with experimental energy loss distribu- tions in D scattering from p(2×2)O on Pt(111) at Ei=1.87 eV. Other conditions are as in Fig. 6. (through EHP excitation) than is seen experimentally (open circles). To understand how the mechanical blocking model fails, consider Fig. 1, where we show a scale drawing of p(2×2)O on the Pt(111) surface (view from above). The size of the O atoms represents the area over which a repulsive H–O collision may occur, which reflects the idea that the attractive part of the interaction potential between H and O is ineffective at preventing an H–Pt collision. This picture shows that the area of the unit cell (parallelo- gram) blocked by the O atoms (red circles) is small (10%), revealingthat the blocking cross section of the O atom is only sufficient to affect a small fraction of the trajectories. It is immediately clear that such a picture cannot explain the dramatic adsorbate effect seen in Fig. 3. This simple geometric picture provides a realistic explana- tion of our MD trajectory simulations. Figure 10 shows the penetration depth distribution of all the trajectories run in our MD simulation of the blocking model using the PES for H interacting with p(2×2)O on Pt(111). Less than 10% of the trajectories exhibit a penetration depth consistent with an H–O FIG. 8. Comparison of MD simulations with experimental energy loss distribu- tions in H scattering from p(2×2)O on Pt(111) at Ei=0.99 eV. Other conditions are as in Fig. 6. J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 155, 034702-7 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9. Comparison of MD simulations with experimental energy loss distribu- tions in D scattering from p(2×2)O on Pt(111) at Ei=0.94 eV. Other conditions are as in Fig. 6. collision (at 1.8 Å). The vast majority of the trajectories have at least one collision with a Pt atom approaching closer than 1.5 Å. We conclude that mechanical blocking by the adsorbed O atom thereby preventing the H atom from approaching the Pt atoms is an insufficient explanation of the experimental observations. A second hypothesis is more successful. Consider that due to O atom preference to bind at the f 1hollow site, three Pt atoms FIG. 10. Penetration depth probability distribution of the collisions of H atoms on ap(2×2)O on the Pt(111) surface. Ei=1.92 eV.ϑi=45○. Integrated over all values ofϑs.are engaged intimately in bonding, thereby strongly influencing the electronic structure of the Pt atoms within about one lattice con- stant distance (orange circles in Fig. 1). We asked what if changes in the electronic structure of the Pt atoms engaged in bonding to the O atoms reduce electronically nonadiabatic interactions in H atom collisions with this surface. To model this, we reduced to zero the friction coefficient within a distance of 𝜚adfrom the O atom using a cutoff function, fcut=1 2[1+erf(rO–H−𝜚ad σ)], (11) with 𝜚ad=2.8 and σ=0.2 Å. This represents an adiabatic sphere, which casts an adiabatic shadow (shown by the orange circles in Fig. 1) on the Pt surface. Figure 11 shows a representation of the effective friction coefficients present for the adiabatic sphere model; the friction reducing effect of the O atom adsorbate is present at sites quite distant from its binding site. Applying this model, we ran trajectories again and com- pared to experiment. The comparison is shown in Figs. 6–9, where results from the adiabatic sphere model (solid circles) are com- pared to experiment (open circles). The agreement is excellent for both H and D at Ei=1.92 and Ei=1.87 eV, respectively. Although the values of the parameters in the adiabatic sphere model were adjusted to agree with experiment at Ei=1.92 eV, the agreement with experiment at Ei=0.99 eV is still good and, furthermore, superior to the mechanical blocking model. If anything, the adiabatic sphere model appears to overstate the likelihood of EHP excitation at this incidence energy. A purely adiabatic simulation (—) at Ei=0.99 eV compares nearly as well to the data; however, it does not describe the experimental results well atEi=1.92 eV. J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 155, 034702-8 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 11. Friction coefficient experienced by the H atom at different surface sites with and without adsorbed O atoms. Fric- tion coefficients in the blocking model (⋅) and for the adiabatic sphere model (—, —) are shown for: (a) two fcchollow sites, (b) two bridge sites, (c) two hcp hol- low sites, and (d) two top sites. See Fig. 1 for a spatial description of the different sites. IV. DISCUSSION The most important outcome of this work is the persuasive evidence that the LDFA electronic friction treatment of EHP exci- tation cannot account for the adsorbate effect in H/D scattering from p(2×2)O on Pt(111). Figure 4, which shows electron densi- ties calculated from DFT, helps explain the problem. The electron densities for Pt(111) and p(2×2)O on Pt(111) are both shown. Only at the sites f 1and b 1are the electron densities affected by O atom adsorption, and the electron densities appear to be enhanced. For all other surface sites more distant from the O atom binding site, the electron densities are hardly affected by the adsorbate. Thus, within the LDFA, we would expect that most H collisions atp(2×2)O on Pt(111) experience similar or perhaps greater EHP excitation as collisions on Pt(111). This is contradicted by experimental observation and by the success of the adiabatic sphere model, since only a model that dramatically reduces the probability for EHP excitation is able to capture the experimental observations. This means that EHP excitation is sensitive to the nature of the chemical bonding of an adsorbate and not simply to the elec- tron density. DFT calculations of the band structure for p(2×2) O on Pt(111) show that O adsorption lowers the energy of the center of the Pt d-band.56The density of states at the Fermi level, thus, goes down. Furthermore, there is a migration of electron density from the d-band associated with Pt atoms to the ( s,p)-band associated with the O atoms. While it is not surprising that substantial electronic structure changes accompany O atom adsorp- tion, this work shows that these changes in electronic structure influence electronic nonadiabaticity in H/D atom collisions with the surface. The LDFA approach starts with the assumption that only the local electron density determines the nonadiabaticity. In the real system, the electron density alone cannot determine the probability of EHP excitation. For example, an LDFA approach always assumes that there are empty states to which an excited electron can betransferred. This is obviously not the case for electrons far below the Fermi level. If one considers DFT calculations of the Pt band struc- ture, the changes in the electron density at the Fermi level, seen, for example, in the calculations of Ref. 56, are also not large enough to account for the changes seen in our adiabatic sphere model. In fact, the migration of electron density from the d-band associated with Pt atoms to the ( s,p)-band associated with the O atoms is even more substantial above the Fermi level.56 The strong adsorbate effect seen in this work represents a chal- lenge to new theories of electronically nonadiabatic dynamics at surfaces. One approach worth exploring is the independent-electron surface hopping (IESH) model.33Here, an electron first jumps from the solid to the projectile, forming a transient anion and leaving a low-energy hole, a mechanism that has been applied to NO multi- quantum vibrational relaxation in collisions with metals.10,11Sub- sequent electron transfer back to the solid leaves the electron in an excited state, robbing energy from the projectile. The two electron transfer events described by this model are energetically governed by the surface work function and projectile electron binding energy. IESH is certainly capable of describing a strong adsorbate effect as the surface work function can be dramatically altered by the presence of an adsorbate. Thus, the results of this work might be an indica- tion of the failing of the LDFA and possible suitability of the IESH model. Another alternative to LDFA is orbital-dependent friction.31,32 In the LDFA, the electronic friction is computed from a model reference system that has the same local electron density as the system of interest. In the orbital-dependent approach, the friction is described by a tensor, coupling the three translational degrees of freedom of the H atom projectile. Perhaps of greater relevance to the case of H atom scattering from p(2×2)O on Pt, the fric- tional tensor elements depend on the precise electronic structure of the system. Here, the friction tensor is likely to depend on the precise nature of the solid conduction band, e.g., dcharacter vs ( s,p) character. J. Chem. Phys. 155, 034702 (2021); doi: 10.1063/5.0058789 155, 034702-9 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp V. CONCLUSIONS AND OUTLOOK We have observed a strong adsorbate effect, where O atom adsorption to Pt(111) nearly eliminates an H atom’s ability to excite EHPs in energetic collisions. We also developed two semi-empirical descriptions of the adsorbate effect. An LDFA mechanical blocking model, where the adsorbate prevents the H atom’s approach to the metal, fails to explain experimental observations, whereas an adia- batic sphere model, where electronic nonadiabaticity is removed in a sphere around the O atom whose radius is similar to the Pt–O bond length, broadly reproduces experiment. This shows that the electronic structure changes induced by the chemisorption of O atoms nearly remove the H atom’s ability to excite electron–hole pairs. Alternatives to LDFA friction are needed to account for this adsorbate effect. ACKNOWLEDGMENTS We acknowledge support from the Deutsche Forschungsge- meinschaft under Grant No. 217133147, which is part of the Collab- orative research Center 1073 operating project (No. A04). L.L., A.K., and T.N.K. acknowledge the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 833404). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1M. Born and R. Oppenheimer, “Zur Quantentheorie der Molekeln,” Ann. Phys. 389, 457–484 (1927). 2J. Tully, “Perspective on ‘Zur Quantentheorie der Molekeln’ M. Born and R. Oppenheimer, Ann. Phys. 84, 457 (1927),” Theor. Chem. Acc. 103, 173–176 (2000). 3D. J. Auerbach, J. C. Tully, and A. M. Wodtke, “Chemical dynamics from the gas-phase to surfaces,” Nat. Sci. 1, e10005 (2021). 4K. Golibrzuch, N. Bartels, D. J. Auerbach, and A. M. 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5.0049734.pdf

Integrated physics package of micromercury trapped ion clock with 10/C014-level frequency stability Cite as: Appl. Phys. Lett. 119, 044001 (2021); doi: 10.1063/5.0049734 Submitted: 8 March 2021 .Accepted: 22 June 2021 . Published Online: 29 July 2021 Thai M. Hoang,1 Sang K. Chung,1Thanh Le,1John D. Prestage,1LinYi,1Robert L. Tjoelker,1Sehyun Park,2 Sung-Jin Park,2 J. Gary Eden,2 Christopher Holland,3 and Nan Yu1,a) AFFILIATIONS 1Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA 2Laboratory for Optical Physics and Engineering, Department of Electrical and Computer Engineering, University of Illinois, Urbana, Illinois 61801, USA 3Applied Sciences Division, SRI International, Menlo Park, California 94025, USA a)Author to whom correspondence should be addressed: Nan.Yu@jpl.nasa.gov ABSTRACT Mercury trapped ion clocks have demonstrated great long-term frequency stability and robustness. In this paper, we report a demonstration of an integrated 100-cc physics package in an effort to develop a micromercury trapped ion clock with high frequency stability. The physics pack-age consists of a sealed 30-cc vacuum tube with one layer of magnetic shielding, light source, and detector assembly. A field emitter array and a194-nm microplasma lamp were employed together with a microtrap tube to reduce the size and power consumption for a mercury trappedion clock. We show that the 100-cc physics package is capable of providing a fractional frequency stability of 1 /C210 /C011s/C01=2down to 5/C210/C014after a few hours of integration. We also show a set of environmental sensitivity evaluations as well as the clock frequency retrace. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0049734 Precision timekeeping is essential for navigation, communication, and scientific research. The widely popular global positioning system (GPS) is made possible thanks to the atomic clocks onboard its satellites. W h i l eG P Sh a sb e c o m et h eb a c k b o n ef o rm o d e r nn a v i g a t i o ns y s t e m s , GPS signals are prone to disruptions and jamming. In such GPS-denied environments, a small atomic clock at the user end can provide a stablefrequency reference locally to support its navigation system and reduce the impact of disruption. In the past few decades, considerable efforts have been devoted to reducing the size and power and increasing the robustness of atomic clocks using different approaches. 1–39The chip scale atomic clock (CSAC) is an example. It has only 17 cc, 34 g, and 120 mW but offers a frequency stability of 10/C011. However, its fre- quency drifts at time scales greater than 1000 s due to wall collisionshifts in a vapor cell. 8–12The wall collision effects can be eliminated by utilizing trapped atoms or ions, allowing higher stability and accuracy. As a result, trapped ions have become attractive for miniature atomic clocks with enhanced stabilities beyond CSAC.19–21 In this Letter, we report our recent results toward developing the micromercury trapped ion clock (M2TIC). The M2TIC is based on the mercury trapped ion clock approach, which offers a number ofadvantages for miniaturization. The 40.5-GHz clock transition of mer- cury ions has the highest microwave clock frequency used to date, pro- viding a high line quality factor [ Fig. 1(a) ] and less fractional frequency error for the same environmental perturbations. The mer- cury ions can be optically pumped by mercury discharge lamps and buffer gas cooled. The absence of lasers significantly reduces complex-ity in practical implementation and robustness. The relatively high vapor pressure of mercury near room temperature eliminates the need of a heated atom source. Indeed, these features have been demon-strated and realized in the deep space atomic clock (DSAC) mission by NASA and JPL, which was launched into a low Earth orbit in 2019. DSAC is the current state-of-the-art for space clocks with a size,weight, and power (SWaP) of 17 l, 16 kg, and 47 W, 36respectively, and provides a frequency stability at the 10/C015level. The M2TIC aims to reduce size and power significantly while maintaining relatively highperformance with 1 /C210 /C014stability floor. In the following, we describe our approach to miniaturization, key innovations, and experi- mental results from the first generation of the M2TIC physics package.Other major parts of a M2TIC clock would include an oscillator, a microwave synthesizer with a low power CMOS synthesizer, 40and Appl. Phys. Lett. 119, 044001 (2021); doi: 10.1063/5.0049734 119, 044001-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplnecessary electronics that are being actively developed and beyond the scope of this Letter. The basic concept of a mercury trapped ion microwave clock is illustrated in Fig. 1(a) . At the heart of the atomic clock is the atomic physics package where atomic physics is implemented. The physicspackage includes a trap vacuum tube and its RF driver, light source and detection assemblies as well as magnetic bias and shielding. The construction of the vacuum tube follows the general approach for DSAC and microYb trap tubes. 19,41To further reduce the power con- sumption in the trap tube, a field emitter is used instead of a traditional heater-based electron source. We also use a dielectric barrier discharge microplasma lamp42to replace the mercury inductive discharge lamp, which is one of the largest power consumption subsystems in the com- pact mercury ion clocks.41Overall, we have achieved a completely functional physics package of 4 /C24/C26c m3. The main components of the microvacuum trap tube consist of a 5/C25/C220 mm3linear RF trap with four rods and end caps [ Fig. 2(a) ]. A 1.6-MHz RF drive of about 450 V peak-to-peak is applied between the trap rods, where the voltages on the adjacent trap rods have oppo- site phases. A 12 V DC is applied between the trap rods and the end caps to complete the trapping potential well. The ion trap consumes about 250 mW and can confine up to a few million ions. The ion trap is housed inside a 30-cc vacuum tube made of titanium walls andsapphire optical windows [ Fig. 2(b) ]. The current size of the housing is limited by the long feedthrough header, which has been significantly shortened in the latter versions of 15-cc tubes. The vacuum tube is typ- ically baked out at 400/C14C to achieve an initial vacuum level in the 10/C09Torr range. During this process, the nonevaporative getter in the vacuum tube is also activated. There is no active ion pump. The vac-uum tube is then charged with a given amount of mercury and backfilled with about 3 /C210 /C06Torr of helium. After the tube is pinched off and completely sealed, the ion storage time is exceptionally long asshown in Fig. 3(a) a n de x p e c t e dt of u r t h e ri m p r o v eo v e rt i m e . 20Ap a i r of magnetic coils is placed around the housing to generate the needed bias field (C-field) along the z-axis [ Fig. 2(c) ]. A layer of magnetic shielding encloses the entire housing and coil assembly with openingsfor optical and electrical accesses. The field emitter array (FEA) is another key component in mini- mizing the size and power. Mercury ions are generated via electronbombardment. A cathode with an Ohmic filament heater is histori- cally used as the electron source, which consumes up to a few watts. 38 In M2TIC, the FEA43integrated directly into the vacuum trap tube [Fig. 2(d) ] requires about 100 V DC to extract enough electron current from the field emitter tips. Only a few lA of current is needed to completely load mercury ions of about a few million into the trap in afew minutes. Inductively coupled mercury discharge lamps used so far are bulky and consume over 10 W of total electric power. 35,36The physical process of discharge and the inductive coupling do not allow a simplereduction in size. In M2TIC, adapting the approach of the microcavityenhanced dielectric barrier discharge already developed in the lightingindustry, 42,44–46we have developed a small yet efficient 194-nm mer- cury microplasma lamp. The efficiency of 194-nm emission comesfrom several main factors. First of all, the use of helium as the dis-charge gas in the lamp results in direct collisional energy transfer from helium metastable excimers to the excited state of mercury ions responsible for 194 nm emission. 47Second, the small cavity volume allows discharge to operate near atmospheric pressure, which enhan-ces the helium dimer production and increases the electron density atthe same time. Finally, the microcavity enhanced-electric fields in thedischarge cell lead to a high electron density. Overall, we were able toachieve 194 nm surface brightness similar to that of the large inductivelamps 35,36with similar ion optical pumping times [ Fig. 3(b) ], but in a FIG. 2. Photograph of the linear ion trap used (a) in the 30-cc vacuum tube (b) enclosed by a layer of shielding where the referenced axes in the text are indicate d (c). Also shown are an image of the FEA (d) and a microcavity plasma lamp used (e). The integrated 100-cc physics package is shown in (f).FIG. 1. (a) Energy level of199Hgþions. The two hyperfine ground states with m¼0 are the clock states used. (b) Mercury-ion clock concept.199Hgþions are trapped in a linear RF trap. The 194-nm light from a202Hg lamp optically pumps the ions to the lower clock state. 40.5-GHz microwave radiation is used toexcite the clock transition. A photomultiplier tube detects the fluorescence photonsfrom the trapped ions. The servo controller locks the microwave frequency to that of the Hg þclock transition.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 044001 (2021); doi: 10.1063/5.0049734 119, 044001-2 Published under an exclusive license by AIP Publishinglamp size of 12 /C212/C24m m3[Fig. 2(e) ] with an estimated 50 mW of discharge power. Light emitted from the lamp is projected onto the trapped ions from one side of the linear trap and the ion fluorescence collected from another side. Figure 2(f) shows arrangement of the PMT, the lamp, and the trap tube in the integrated 100-cc physics package. Thetotal power dissipation in the physics package, including the micro- plasma lamp (50 mW), the trap RF drive (250 mW), PMT HV supply ( 1 5m W ) ,a n dt h eF E A( <1mW), is estimated to be <320 mW. The dominating power consumption of the trap RF drive is limited by theavailable Q of the tank circuit inductors. To characterize the performance of the physics package, we used both a hydrogen maser as the LO to assess the intrinsic stability pro- vided by the physics package, and a voltage-controlled quartz oscillatorsteered to the ion transition in an operating clock mode. In a typicaloperation, the microwave clock is interrogated with a 1 s Rabi pulse. The fluorescence collection time is about 5 s, during which the trapped ions are optically pumped. A typical ion resonance spectrum is shownin the inset of Fig. 3(c) with an SNR of 7.5. In addition, the FEA is turned on briefly for /C240:5 s to compensate for the ion loss in each interrogation cycle. Figure 3(c) plots the measured frequency stability performance. The red open squares show the intrinsic ion stability measured againsta hydrogen maser. We obtained 7 /C210 /C012s/C01=2averaging trend with 5/C210/C014floor after a few hours of integration. The short-term stabil- ity is primarily determined by the Fourier transform-limited linewidthand the ion SNR. 41Using a quartz LO with a free-running short-term stability of 2 /C210/C012, we were able to servo it to the ion transition and demonstrate 1 /C210/C011s/C01=2that reaches below 1 /C210/C013stability level after averaging for a few hours [blue solid squares in Fig. 3(c) ]. This stability level is among the best performance for a clock physics package of comparable size. In addition, we measured the retrace ofthe physics package by repeating the stability measurements after turn-ing off power to all active components for 24 h. Figure 4(a) shows theclock transition fractional frequencies of three separate measurements against a hydrogen maser, yielding a retrace of less than 10 /C012. Atomic clocks can be sensitive to various environmental pertur- bations; chiefly among them are temperature, magnetic field, andacceleration. We have subjected the physics package as a whole tothese environmental conditions to evaluate its sensitivities. For the temperature sensitivity tests, we placed the physics pack- age in a thermal box with a heater inside. Figure 4(b) shows the mea- sured fractional frequency deviations at different temperatures with the FEA turned off. Each point represents about 30 min of measure-ment time. The temperature was monitored at the trap tube housing.Extrapolating the temperature coefficient by a simple linear fit yieldsð2:164:3Þ/C210 /C014=/C14C. As will be discussed later, the temperature sensitivity is consistent with the ion number loss in the trap over the measurement time. For the magnetic field sensitivity measurements, an external magnetic field was applied along the x-,y-, and z-directions [ Fig. 2(c) ] separately. Figure 4(c) shows the measured fractional frequency devia- tions at the different applied field amplitudes and orientations. In thexand ydirections (perpendicular to the internal bias field), the f r e q u e n c yn e v e rc h a n g e sb ym o r et h a n1 0 /C012over the range of 0.8 G, setting an upper limit of 10/C012/G. The nonlinear increase in the y-direction is not fully investigated, but likely due to some nonzero residual field in the z-direction [see Eq. (1)and later discussions]. In thez-direction, the internal 250 mG bias field makes the second-order Zeeman shift larger. Efforts are being made to reduce the bias field amplitude. With a second layer magnetic field shielding outside the entire clock package planned, the overall magnetic field sensitivitieswill be significantly reduced. Finally, we tested the acceleration sensitivity by performing 2- g flipping measurements. For the standalone integrated physics package,we were able to simply rotate the entire physics package relative to thegravity direction. Figure 4(d) shows the measured fractional frequen- cies at alternative þgand/C0gin the y-direction. Again, the results are FIG. 3. (a) Plot of an ion storage time measurement by observing the ion signal level over time. The open circles, error bars, and solid line represent the measu red ion signal means, standard deviations, and an exponential fit of 137 h 1 =edecay time, respectively. (b) Sample optical pumping curves with a conventional inductive discharge lamp and a microplasma lamp. (c) Allan deviation measurements with a hydrogen (red open squares) and a steered quartz oscillator (blue solid squares). The das hed lines represent thes/C01=2trend. The error bars represent the estimated statistical uncertainty. The last blue square point is only a three measurement average. The inset show s a typical ion line spectrum with a gate time of 5 s.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 044001 (2021); doi: 10.1063/5.0049734 119, 044001-3 Published under an exclusive license by AIP Publishingall within 10/C012. Note that the rotation of the physics package also changes its ambient magnetic field. The measured sensitivity, there- fore, represents an upper limit. In principle, the ions are tightly trapped and held by the trap, and no direct effects are expected due tothe gravity bias itself. While the first M2TIC physics package showed very promising performance results in the small SWaP package, it is instructive to examine the limiting factors of the M2TIC stability achievable. There are two main factors determining the current frequency instability ofM2TIC: the second-order Zeeman shift and the second-order Doppler shift. The second-order Zeeman shift, Df, can be expressed as Df f0¼ððB0þdBÞ2/C0B2 0Þ/C2bZ f0; (1) where B0and dBare the bias magnetic field and its fluctuation, respectively, f0is the clock transition frequency, and bZ¼100 Hz =G2 is the Zeeman shift coefficient. The sensitivity clearly depends on the magnitude of the bias field, the smaller the better. The desired small magnitude of the bias field is limited by the ion secular motions in the presence of the residual magnetic field inhomogeneity due to inducedspin flips and line broadening. 29,48In the current physics package, a relatively high bias field of 250 mG has to be used. The magnetic field gradient in the trap can be improved with a better bias coil design anda more careful choice of nonmagnetic materials used near the trap. With the current bias field amplitude, a total shielding factor of 7000 is necessary against 100 mG of external field variations for a long-termstability floor of 1 /C210 /C014. With the second-order Zeeman shifts properly controlled, the M2TIC clock instability will be primarily determined by the second-order Doppler shift from the motions of the trapped ions. The ion cloud in the RF trap is constantly heated by the driven micromotionsand cooled by the buffer gas, typically reaching an equilibrium temper-a t u r ea ta b o u t4 0 0 /C14C.29This thermal motion velocity is much smaller compared to that of the driven micromotions in the RF trap,29,48 which dominates the Doppler shift of the trapped ions. The micromo- tion amplitude varies with the quadrupole field in the radial directions of the trap, the farther out from the center of the trap, the larger themicromotion. This induces an averaged second-order Doppler shift ofthe ion cloud, given by 29 Df f¼/C0q2 8pe0mc2N L/C18/C19 ; (2) in a linear quadrupole trap, where Nis the total number of ions in the trap, Lis the linear axial length of the trap, mis the ion mass, qis the ionic charge, and e0is the vacuum permittivity. This frequency shift introduces an ion number dependence. With the trap configuration inFig. 2(a) and the mentioned operating conditions, we estimate that a 10 /C014stability requires the ion number stable to /C240:5%. This ion number stability can be readily achieved, especially given the long ion lifetime. The ion number can also conceivably be actively stabilizedover a long period of time. Similarly, the ion number dependence willaffect the clock frequency retrace. This will be a challenge under verydifferent starting temperature environment since the temperature willaffect the mercury vapor pressure, ion loading rate, and hence the totalnumber of ions initially loaded. If the mercury vapor pressure is deter-ministic inside the trap tube with the trap tube temperature, then theFEA electron current can be adjusted accordingly to keep the sameloading rate and a number of ions loaded. Again, an ability to activelyFIG. 4. Retrace measurements (a), temperature sensitivity measurements (b), magnetic field sensitivity measurements (c), and gravity acceleration measu rements (d), all plot- ted in fractional frequency deviations from the hydrogen maser. Dashed lines and shades provide data scattering ranges. Magnetic field orientations should be referenced in the text. The zero points are chosen arbitrarily to illustrate the frequency shifts.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 044001 (2021); doi: 10.1063/5.0049734 119, 044001-4 Published under an exclusive license by AIP Publishingmonitor the ion number will be an ideal solution in the future to guar- antee its long-term stability and retrace. In conclusion, we have demonstrated a 100-cc integrated physics package for the micromercury trapped ion clock. The physics packageincludes a micro-30-cc mercury ion trap vacuum tube with FEA as thelow-power ionizing electron source. It is operated by a 194 nm VUVmicroplasma lamp. We have evaluated the clock functionality and per-formances of the physics package and showed it capable of 10 /C011s/C01=2 fractional frequency stability that averages down to the 10/C014level. Various environmental tests and retrace measurements were also per-formed. The results pave a path to low-SWaP atomic clocks withhighly enhanced long-term stabilities. We would like to thank William Diener for providing support from the frequency standards test laboratory. The research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics andSpace Administration. Support from the Defense AdvancedResearch Projects Agency was acknowledged. The views, opinions,and/or findings expressed are those of the author and should not beinterpreted as representing the official views or policies of theDepartment of Defense or the U.S. Government. Copyright 2021. California Institute of Technology. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1W. J. Riley, in Proceedings of 22nd Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting (Jet Propulsion Laboratory, 1990), pp. 221–227. 2P. T. Fisk, M. J. Sellars, M. A. Lawn, and G. Coles, IEEE Trans. Ultrason., Ferroelectr., Freq. Control 44, 344 (1997). 3W. Riley, in 29th Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting (U.S. Naval Observatory, Washington, DC, 1998), pp. 213–220. 4D. Todd, F. Gerald, P. John et al. , in 34th Annual Precise Time and Time Interval (PTTI) Meeting, Virginia (2002), Vol. 175. 5M. Epstein, G. Freed, and J. 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5.0044933.pdf

Appl. Phys. Lett. 118, 214002 (2021); https://doi.org/10.1063/5.0044933 118, 214002 © 2021 Author(s).Single-hole couplings in GaAs/AlGaAs double dots probed with transport and EDSR spectroscopy Cite as: Appl. Phys. Lett. 118, 214002 (2021); https://doi.org/10.1063/5.0044933 Submitted: 20 January 2021 . Accepted: 28 April 2021 . Published Online: 24 May 2021 J. Ducatel , A. Padawer-Blatt , A. Bogan , M. Korkusinski , P. Zawadzki , A. Sachrajda , S. Studenikin , L. Tracy , J. Reno , and T. Hargett ARTICLES YOU MAY BE INTERESTED IN Self-aligned gates for scalable silicon quantum computing Applied Physics Letters 118, 104004 (2021); https://doi.org/10.1063/5.0036520 Non-polar true-lateral GaN power diodes on foreign substrates Applied Physics Letters 118, 212102 (2021); https://doi.org/10.1063/5.0051552 Inverse spin-Hall effect in GeSn Applied Physics Letters 118, 212402 (2021); https://doi.org/10.1063/5.0046129Single-hole couplings in GaAs/AlGaAs double dots probed with transport and EDSR spectroscopy Cite as: Appl. Phys. Lett. 118, 214002 (2021); doi: 10.1063/5.0044933 Submitted: 20 January 2021 .Accepted: 28 April 2021 . Published Online: 24 May 2021 J.Ducatel,1,2 A.Padawer-Blatt,1,2 A.Bogan,1 M.Korkusinski,1P.Zawadzki,1A.Sachrajda,1S.Studenikin,1,2,a) L.Tracy,3J.Reno,3and T. Hargett3 AFFILIATIONS 1National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada 2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 3Sandia National Laboratories, Albuquerque, New Mexico 87185, USA a)Author to whom correspondence should be addressed: sergei.studenikin@nrc-cnrc.gc.ca ABSTRACT We report a detailed study of the tunnel barriers within a single-hole GaAs/AlGaAs double quantum dot device (DQD). For quantum infor- mation applications as well as fundamental studies, careful tuning and reliable measurements of the barriers are important requirements. In order to tune a DQD device adequately into the single-hole electric dipole spin resonance regime, one has to employ a variety of techniques to cover the extended range of tunnel couplings. In this work, we demonstrate four separate techniques, based upon charge sensing, quantumtransport, time-resolved pulsing, and electron dipole spin resonance spectroscopy to determine the couplings as a function of relevant gatevoltages and magnetic field. Measurements were performed under conditions of both symmetric and asymmetric tunnel couplings to theleads. Good agreement was observed between different techniques when measured under the same conditions. The results indicate that even in this relatively simple circuit, the requirement to tune multiple gates and the consequences of real potential profiles result in non-intuitive dependencies of the couplings as a function of the plunger gate voltage and the magnetic field. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0044933 Single hole spins isolated in gated solid-state quantum dots have recently emerged as promising candidates for scalable spin qubit cir-cuits. 1–7Compared to conduction band electrons, valence band holes have many attractive features both for fundamental studies of few holesystems as well as spin-qubit applications. 1,7,8The ability to tune and characterize barrier tunneling rates is one of the important calibration requirements for quantum dot circuits. It enables one to initialize cir-cuits in regimes for specific experiments and precise measurements.For example, asymmetric tunnel barrier schemes are required forexperiments that take advantage of latching spin-readout 9or hole Electric Dipole Spin Resonance (EDSR) experiments discussed below. We show in this paper that tunneling rate gate dependencies can onoccasion behave counter-intuitively to naive electrostatic intuition. Wespeculate that this is a consequence of realistic confining dot potentialgeometries. Additionally, techniques described in this work may be relevant to the future development of virtual tunnel-rate gates, in anal- ogy to virtual gates, 10,11which would allow one to control each barrier tunneling rate independent of other barriers. In this paper, we employdifferent methods to acquire values of the tunneling rates over a broadrange of couplings in a GaAs/AlGaAs Double Quantum Dot (DQD)device which has been used successfully for various few hole experi- ments. 6,7,9,12The different techniques are used to determine inter-dot tunnel coupling, t, which in the presence of a magnetic field involves both inter-dot spin-conserving and spin-flipping couplings, tNandtF, respectively, as well as the tunneling matrix elements to the right, tR, and left, tL, leads. Specifically, we employ (i) the broadening of the charge transfer line using charge sensing, (ii) low-bias current spec-troscopy of two-dot hybridized states, (iii) single-hole EDSR spectros-copy, and (iv) time resolved pulsing technique. The tunneling rates aretuned by varying gate voltages as well as the strength of the normalmagnetic field. Good agreement is found between the techniqueswhen measured under similar conditions. All the following experiments were performed on a gate defined lat- eral GaAs/AlGaAs DQD device. 12,13The two-dimensional hole gas (2DHG) is generated by an accumulation gate at the GaAs =AlxGa1/C0xAs (x¼0.5) interface located 65 nm beneath the wafer surface. The scanning electron microscopy (SEM) image in Fig. 1(a) shows the layout of the Ti/ Au surface depletion gates used to create and tune the DQD confining potential. The aluminum accumulation gate is separated from the wafersurface by a 110 nm Al 2O3dielectric layer. Yellow squares mark the Appl. Phys. Lett. 118, 214002 (2021); doi: 10.1063/5.0044933 118, 214002-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplpositions of the AuBe alloy Ohmic contacts.14,15Labels IDOTandICSindi- cate the dot device and the charge sensor currents, respectively. Figure 1(b)shows an example of the charge stability diagram measured by the charge sensor, dICS=dVL, as a function of the detuning plunger gates vol- tages, VLandVR, labeled in Fig. 1(a) . Numbers in brackets ( nL,nR)d e n o t e the hole occupation numbers of the left and right dots, respectively. We start with considering the symmetric and asymmetric coupling regimes. The system can be tuned such that the tunneling rate from theleft lead to the left dot, C L, is equal to the one from the right lead to the right dot, CR, i.e., CL/C25CR¼C.[ N o t e ,o f t e nf o rt h e o r e t i c a l discussions, it is more convenient to use tunnel coupling elements tLðRÞ¼hCLðRÞwhere his the Plank constant. We will use these mea- surement parameters interchangeably and employ both scales in Figs. 5 and6.] Such a symmetric regime is characterized qualitatively by a sym- metric transport diagram in a low-bias regime. An example of suchtransport diagram is shown in Fig. 2(a) . The symmetric regime allows us to simplify analysis as it requires a reduced number of variables to describe the system. In certain experiments, asymmetric regimes arepreferable, for example, for optimum initialization in EDSR experi-ments. To change from a symmetric, Fig. 2(a) , to strongly asymmetric regime, all six gate voltages are changed in our device by up to 0 :4V .A transport diagram in the asymmetric regime is presented in Fig. 2(c) . This regime ( C L/C29CR) ensures that holes tunnel much faster into the DQD from the left lead to increase the left lead initialization fidelity.This is discussed later in this paper in the EDSR experiment context.As the first method, we explore the inter-dot coupling, t, using a standard technique based on (1,0)–(0,1) charge transfer line broaden-ing by measuring the charge occupation probability detected by thecharge sensor. Possible variations of tare investigated along the e ? line, connecting the (0,0) and (1,1) triple points. Five traces were recorded through the (1,0)–(0,1) charge transfer line at different valuesofe ?marked with dash lines in Fig. 1(b) . Note that we define detuning asDe¼eL/C0eRwhere eLðRÞi st h eo n s i t ee n e r g yo ft h el e f t( r i g h t )d o t . Figure 1(c) shows line traces of the (0,1) occupation probability P01 determined from the charge sensor current ICSat different e?with a linear background contribution subtracted. The background is due toa cross-capacitance coupling between the detuning plunger gates andthe charge sensor. Twenty one traces are averaged to reduce noise inthe data. Perpendicular detuning values e ?with respect to the (0,0) tri- ple point in leV are calculated using measured gate lever arms.16,17In order to find the inter-dot coupling, the experimental data in Fig. 1(c) are fitted with the following equation18,19with tas the fitting parameter: Pð0;1ÞðDeÞ¼Deffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi De2þ4t2p tanhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi De2þ4t2p 2kBTh ! ; (1) where kBis the Boltzmann constant. The hole temperature, Th¼102 mK, is determined by measuring the Coulomb blockade broadening of lead addition line, (0,0) !(1,0), under weak coupling conditions as a function of the mixing chamber temperature.20,21In this experiment, we assume that the effective hole temperature in the FIG. 1. (a) SEM image showing the gate layout of a DQD device similar to one used in this study. Yellow boxes represent Ohmic contacts used to measure cur- rents IDOTandICS. (b) An example of the charge stability diagram measured as derivative of charge sensor current dICS=dVLfor improved visibility of the charge addition (orange) and charge transfer (dark) lines; ( nL,nR) indicates the number of holes in each dot. The inter-dot coupling at different e?is measured by sweeping along white dashed lines (for details see test). (c) Two individual traces of the charge occupation probability Pð0;1Þfor minimum and maximum e?detuning mea- sured by sweeping through the charge transfer line at different e?detunings with respect to the (0,0) triple point. Traces are offset vertically for clarity (for details see text). (d) The inter-dot coupling, t, as a function of e?obtained from fits similar to those in (c). Error bars indicate a one standard deviation error in the fits. FIG. 2. Transport diagram of IDOTas a function of the plunger gate voltages, VL andVR: (a) in the symmetric ( CL/C25CR) and (c) in the asymmetric ( CL/C29CR) regimes. The transport diagrams are measured at low source-drain voltage,V SD¼50lV. Black dashed lines are inserted along the addition and charge trans- fer lines. (b) The maximum current, IMAX, through the DQD as a function of detuning for the (0,0) triple point in a symmetric regime; (d) same as above but for the asym-metric diagram presented in panel (c). Solid lines are theory fits to the data. Thecrosses in insets in (b) and (d) mark the positions of selected I MAXðVL;VRÞthat are used to calculate IMAXðDeÞin the main panels [(b) and (d)] (for details see text).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 214002 (2021); doi: 10.1063/5.0044933 118, 214002-2 Published under an exclusive license by AIP Publishingdot is the same as in the leads, i.e., we measure the temperature of the corresponding environment coupled to the DQD levels.21The data obtained for the inter-dot coupling tðe?Þare presented in Fig. 1(d) .I t is evident that tdoes not vary with e?except for the last trace which was taken the closest to the second triple point (1,1). This abrupt change is consistent with a shallow DQD potential when compared to the QD charging energy. This effect is also evident in Fig. 2(a) where the current at the (1,1) triple point is significantly larger than at (0,0)and where extended conduction tails along the addition lines revealnoticeable co-tunneling processes. 16 Now, we discuss how we determine tunnel couplings from the transport measurements. We start with the simpler symmetric case, tL/C25tR, as it involves fewer fitting parameters. The procedure will be slightly different for the asymmetric case. The electric current troughthe device, I DOT, around the triple point is described using a two-dot molecular model.22We consider the sequential tunneling of a single hole through the device in a two-level system under the assumptionthat for low bias tunneling only occurs through the ground state (GS) of the system. 22We first identify the maximum of the current, IMAX, for each vertical trace around the (0,0) triple point in Fig. 2(a) along with the corresponding coordinates ðVL;VRÞMAX. The coordinates ðVL;VRÞiof each IMAXare then converted to the detuning values, De, in energy units using the four previously determined lever arms.16,17 The current though the dots in the symmetric regime is described bythe following equation: I MAXðDeÞ¼eVSDC 16kBTh1/C01 1þ4t2=De2/C18/C19 : (2) An example of the extracted IMAXdata vs Dein the symmetric case is plotted in Fig. 2(b) for a center (C) gate voltage, VC¼0:16 V. The data are fitted to Eq. (2)to determine the tunneling rates to the leads, C, and the inter-dot coupling, t, by setting them as fitting parameters. For the data presented in Fig. 2(b) , we obtain t ¼(14.660:2ÞleV,C ¼(22862) MHz. To verify that this kind of measurement is not affected by the bias voltage, we repeat these measurements and fittings(not shown) at different source-drain bias voltages across the DQD.We find that, as expected, the results are not affected by the bias volt-age up to V SD¼100leV. Now, let us consider asymmetric regime, CL/C29CRstarting with discussion of how to extract the inter-dot coupling and the tunneling rates of the left and right leads separately. In this situation, we have more variables to fit. To reduce number of the fitting parameters, wefirst determine the inter-dot coupling using the charge transfer linetechnique described above. We then proceed with a similar approachto the symmetric case above but using the following more general equations: 23 IMAXðDeÞ¼eVSD 4kBThjaðDeÞj2ð1/C0jaðDeÞj2ÞCLCR jaðDeÞj2CLþð1/C0jaðDeÞj2ÞCR ! ;(3) jaðDeÞj2¼1 21/C0Deffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi De2þ4t2p/C18/C19 : (4) An example of the resulting fit in the asymmetric regime is shown in Fig. 2(d) , with an inter-dot tunneling rate 2 :54 GHz ðt¼10:5leVÞ, yielding the tunneling rates CL¼6:9 GHz and CR¼122 MHz. The various dependencies of the tunneling rates will be discussed below.Note, that when a magnetic field is applied, the two-level system splits into a four-level, spin-resolved system6and the inter-dot coupling, t, should be replaced by the spin-conserving coupling element, tN. Now we present a spectroscopic Electric Dipole Spin Resonance (EDSR) technique to extract spin-conserving, tN, and spin-flipping, tF, tunneling elements. We first set the DQD barriers in an optimizedregime for EDSR experiments using the above techniques, as qualita- tively shown in Fig. 3(b) . The external magnetic field is applied along the growth direction. The energy levels alignment for this experiment is shown in Fig. 3(b) .Figure 3(a) shows the high-bias transport dia- gram for this experiment, V SD¼/C00:5 meV. The green dashed line in Fig. 3(a) indicates the sweep trajectory, which allows for a control of Dewhile holding eR#andeR"constant relative to the Fermi level of the right lead. The energy levels of the right dot are kept such that the rightlead Fermi level is positioned between the spin up and spin down lev- els as shown in panel (b), e R#<EF<eR". In the equilibrium condi- tion (with MW off or the MW frequency, fMW, being off resonance), the current is energetically blocked as a hole occupying the ground spin-down state cannot escape to the lead. The blockade is lifted under the resonance condition hfMW¼g/C3lBB,w h e r e his the Plank con- stant, g/C3the effective g-factor, and lBis the Bohr magneton, resulting in small resonant current highlighted by a circle in Fig. 3(a) .6Stepping FIG. 3. (a) An example of a current diagram IDOTat high source-drain bias voltage, VSD¼/C0 0:50 mV, for B¼1:86 T and MW frequency f¼38:3 GHz. The EDSR signal is highlighted with a circle. (b) A qualitative diagram showing the DQD onsiteenergy levels relative to the Fermi energy of each leads during the EDSR experi- ment. The levels are color-labeled same way as in (d). The ground state corre- sponds to the spin down state assuming level hybridization is small. The left dot istuned to have more transparent barrier compared to the right dot, and therefore theDQD system is reliably loaded from the left source lead into the ground state (GS). The applied MW signal (red arrow) promotes holes from GS to ES1 or ES2 excited states resulting in small resonant current, I DOT. (c) The current IDOTEDSR spectrum in the f/C0Despace at constant B¼1:8 T. The detuning sweep is made along a line similar to the green dashed trace in panel (a). A fit of the model6to the data (white dash line) is used to extract the coupling elements tN¼29:4leV and tF¼23:4leV. Line 1 corresponds to the GS-ES1 transition, and line 2 corre- sponds to the GS-ES2 MW-induced transitions marked by arrows in (d) which isthe energy levels of the four-level system6calculated for tN¼30 and tF¼23leV.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 214002 (2021); doi: 10.1063/5.0044933 118, 214002-3 Published under an exclusive license by AIP Publishingthe applied MW frequency and repeatedly sweeping along the green detuning trajectory allows us to recover the EDSR spectrum, example of which at B¼1:8 T is presented in Fig. 3(c) with theoretical fits to the model6plotted as white dash lines. Line 1 corresponds to the GS- ES1 transition, while line 2 corresponds to transition to the secondexcited state GS-ES2 of the four-level system indicated by arrows in Fig. 3(d) . The theoretical fit to the data allows us to extract both spin- conserving and spin-flipping tunneling elements as a function of themagnetic field. 6The observed large spin-flip coupling is related to the p-type hole orbitals and the 3/2 nature of the hole spins.12,24Note that electrons are particles whose microscopic wave function is s-type, while the holes have p-type microscopic wave functions. This results in much stronger spin–orbit interaction for holes. For electrons, tFis one-two orders of magnitude smaller compared with tFfor holes.25 We now describe a direct, time resolved measurement of the tunneling rate for the most opaque right barrier. We use a two-step pulsing protocol shown in Fig. 4(a) . Voltage pulses from an arbitrary waveform generator are applied to the L gate. The initialization stageof the pulse allows for the right dot to be reliably loaded from the leftreservoir due to much shorter combined tunneling times from the leftlead to the left dot, T L¼1 CL, and from the left dot to the right dot, TC¼h t, than from the right dot directly to the right lead, TR¼1 CR(i.e., TLþTC/C28TR). This is qualitatively shown in the left diagram of Fig. 4(a) for a spin unresolved two-level system (note that spin splitting does not play a role in this measurement as we measure spin-averagedtunneling out time to the left lead). During the readout stage of the pulse, right diagram of Fig. 4(a) , the energy levels are adjusted such that the right dot may eject the hole only through the right lead(E F;R<ER<EL), while additional holes are prevented from tunnel- ing from the left lead because EF;L<EL,w h e r e ERðLÞis the hole energy level in right (left) dot. The total duration of the pulse cycle is set to Tcycle¼1ls. The sequence is continuously applied during the mea- surement, generating a current proportional to the tunneling out prob-ability during the readout stage. The duration of the readout stage,t Readout , is varied from 0 to 300 ns to determine the time required for the hole to tunnel out through the right barrier. Experimental results are shown in Fig. 4(b) , where the measured IDOTis converted to the average number of holes transported through the device per pulsecycle, eNh¼IDOTTcycle.T h ed a t ai n Fig. 4(b) are fitted to the following equation: Nh¼A1/C0expð/C0tReadout =TRÞ ðÞ ; (5) where Ais a scaling factor. The fitted value of A¼0.86 indicates that the right dot is reliably loaded from the left reservoir 86% of the time. The remaining 14% is due to an increased inter-dot inelastic tunneling time when the left and right dot energy levels are off resonance.9,24 This does not affect precision of the fitted exponent. From this mea-surement, we obtain the right lead tunneling rate T R¼61:4n s a t B¼1:9 T. This point is plotted in Fig. 5(c) as a red triangle demon- strating a very good agreement with the data obtained from the trans- port diagrams in Fig. 2 . Let us now discuss the tunneling rates dependence on the strength of the external magnetic field and the VCgate voltage. The magnetic field dependencies are plotted in Fig. 5 , panels [(a) and (b)] for symmetric and panels [(c) and (d)] for asymmetric cases. It is evi- dent from panels (a) and (c) that the coupling of each dot to its respec- tive lead, CLðRÞ, decreases exponentially more than an order in magnitude over a perpendicular field ranging from 0 T to 2 :5T i n both the symmetric (a) and asymmetric (c) regimes. The measurement conducted in a strong magnetic field ( B¼1:9 T) using the time- resolved pulsing technique described above is in excellent agreementwith the measurements made using the low-bias current technique as marked by a red triangle data point in Fig. 5(c) . The observed decrease FIG. 4. (a) Schematic energy diagrams (top) of the DQD showing the energy levels of the left and right dots relative to the Fermi energy of each leads for a two-levelsystem during the pulsing cycle (bottom) (for details see text). (b) The average number of holes per pulse cycle through the dots calculated from the transport cur- rent using the total cycle time T cycle¼1ls. Solid red line is a fit to the data with Eq.(5)to extract the right lead tunneling time, TR. The extracted right lead tunneling time, TR¼61:4n sa t B¼1:90 T, is plotted with a red solid triangle in Fig. 5(c) .FIG. 5. Tunability characteristics of tunneling rates vs magnetic field. (a) The lead tunneling rate, C, and (b) inter-dot spin-conserving coupling, tN, in the symmetric regime (i.e., CL/C25CR¼C). (c) and (d) are the tunneling rates for the asymmetric regime (i.e., CL/C29CR). Panel (c) present results for left and right lead tunneling rates,CL(blue stars) and CR(black squares), respectively. (d) tNtuning capability using B. Square and star data show the rates extracted using the low-bias current technique, while purple diamond data show the rates extracted using EDSR spec-troscopy. Red triangle point in (c) shows the right lead rate extracted using the time-resolved scheme. Black circle data in (d) show the rates measured using the charge transfer line technique (see the text for details). The average 1 rstandard deviation error is 4.9% in (a), 4.7% in (b), 8.0% for C L(left lead coupling), 2.6% for CR(right lead coupling) in (c), and 2% in (d).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 214002 (2021); doi: 10.1063/5.0044933 118, 214002-4 Published under an exclusive license by AIP Publishingin coupling is expected due to the diamagnetic shrinking of hole orbi- tals in strong magnetic field leading to an increase in mean distance of the hole wave functions from the corresponding reservoirs. However, the magnetic field dependence of inter-dot spin-conserving coupling is much weaker, which is evident from Fig. 5(b) for symmetric and Fig. 5(d)f o ra s y m m e t r i cr e g i m e s .W en o t et h e s eg r a p h sa r ep l o t t e di nal i n - ear scale in contrast to log scales in panels [(a) and (c)]. This is a non- intuitive experimental result, emphasizing the importance of carefully calibrating the device. The observed dependence may be associatedwith the non-circular shape of the quantum dot confining potential and the specific gate layout. 12Our measurements illustrate the tunabil- ity of the tunnel couplings vs gate voltage and the magnetic field. While in principle more quantitative studies are possible, they are beyond the scope of this paper. Now that the tunneling rates can be reliably measured, we exam- ine their dependence on the center gate voltage, VC,a tB¼0T . W e plot the dependencies of Cand tonVCfor symmetric case in Figs. 6(a) and6(b) respectively. The inter-dot tunneling coupling in Fig. 6(b) decreases exponentially with increasing VC,a se x p e c t e df o r positively charged carriers.18,19However, counter-intuitively the tunneling rate to the leads CinFig. 6(a) increases by nearly two orders of magnitude with increasing VC. This behavior qualitatively indicates the left and right DQD potential minima are pushed closer to the cor- responding lead reservoirs by more positive center gate VC.N o t et h a t the position of the triple-point ( VR,VL)a l s oc h a n g e sw i t h VCself- consistently. This example shows the importance of careful gate- voltage calibration in the device tuning procedure. In conclusion, we performed a detailed study of the tunneling rates and their tunability of a single hole GaAs/AlGaAs DQD device as a function of the magnetic field and the center plunger gate voltage. We employed four different techniques to obtain information about the three tunneling rates of the DQD system to the respective leads and the inter-dot tunnel coupling. We found that some tunneling rates may exhibit counter-intuitive tuning dependence vs the center plunger gate voltage. Good agreement is achieved between different techniques for data when measured under the same conditions. The non-intuitivedependencies of the couplings found in this work provide motivation to develop virtual tunneling barrier protocols which would allowtunnel couplings to be tuned individually. This will be the subject of future work. We thank Bill Coish, Jan Kycia, Louis Gaudreau, Jason Phoenix, and Guy Austing for useful discussions. This work wasperformed, in part, at the Center for Integrated Nanotechnologies, aU.S. DOE, Office of Basic Energy Sciences, user facility. 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Lett. 117, 206802 (2016).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 214002 (2021); doi: 10.1063/5.0044933 118, 214002-6 Published under an exclusive license by AIP Publishing

5.0049513.pdf

Molecular library of OLED host materials—Evaluating the multiscale simulation workflow Cite as: Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 Submitted: 5 March 2021 .Accepted: 18 June 2021 . Published Online: 26 July 2021 Anirban Mondal,1 Leanne Paterson,1Jaeyoung Cho,1,2 Kun-Han Lin,1Basvan der Zee,1 Gert-Jan A. H. Wetzelaer,1Andrei Stankevych,3Alexander Vakhnin,3Jang-Joo Kim,2Andrey Kadashchuk,3,4 Paul W. M. Blom,1 Falk May,5and Denis Andrienko1,a) AFFILIATIONS 1Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany 2Department of Materials Science and Engineering and the Center for Organic Light Emitting Diode, Seoul National University, Seoul, 151-744, South Korea 3Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauky 46, 03028 Kyiv, Ukraine 4IMEC, Kapeldreef 75, B-3001 Leuven, Belgium 5Merck KGaA, 64293 Darmstadt, Germany a)Author to whom correspondence should be addressed: denis.andrienko@mpip-mainz.mpg.de ABSTRACT Amorphous small-molecule organic materials are utilized in organic light emitting diodes (OLEDs), with device performance relying on appropriate chemical design. Due to the vast number of contending materials, a symbiotic experimental and simulation approach wouldbe greatly beneficial in linking chemical structure to macroscopic material properties. We review simulation approaches proposed forpredicting macroscopic properties. We then present a library of OLED hosts, containing input files, results of simulations, and experi- mentally measured references of quantities relevant to OLED materials. We find that there is a linear proportionality between simulated and measured glass transition temperatures, despite a quantitative disagreement. Computed ionization energies are in excellent agree-ment with the ultraviolet photoelectron and photoemission spectroscopy in air measurements. We also observe a linear correlationbetween calculated electron affinities and ionization energies and cyclic voltammetry measurements. Computed energetic disorder corre- lates well with thermally stimulated luminescence measurements and charge mobilities agree remarkably well with space charge–limited current measurements. For the studied host materials, we find that the energetic disorder has the greatest impact on the charge carriermobility. Our library helps to swiftly evaluate properties of new OLED materials by providing well-defined structural building blocks.The library is public and open for improvements. We envision the library expanding and the workflow providing guidance for futureOLED material design. VC2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http:// creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/5.0049513 TABLE OF CONTENTS INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 GLASS TRANSITION TEMPERATURE. . . . . . . . . . . . . . . . . 3CHARGE TRANSPORT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 DENSITY OF STATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ELECTRON AFFINITY AND IONIZATION ENERGY . . . 5CHARGE CARRIER MOBILITY . . . . . . . . . . . . . . . . . . . . . . . 8OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Gas-phase ionization energy and electron affinity. . . . 10Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Coupling elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Reorganization energies . . . . . . . . . . . . . . . . . . . . . . . . . . 11Mobilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Glass transition temperature: Differential scanning calorimetry (DSC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Ionization energies: Photoelectron yield spectroscopy in air (PESA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Energetic disorder: TSL technique . . . . . . . . . . . . . . . . . 11 Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-1 VCAuthor(s) 2021Chemical Physics Reviews REVIEW scitation.org/journal/cprSOFTWARE AND INPUT FILES . . . . . . . . . . . . . . . . . . . . . . 12 SUPPLEMENTARY MATERIAL . . . . . . . . . . . . . . . . . . . . . . . 12AUTHORS’ CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . 12 INTRODUCTION In recent years, display and lighting technologies based on organic light emitting diodes (OLEDs) have steadily increased in pop- ularity, from the automotive industry to smart phones and televisions. Not only do they offer improved image quality via a greater contrast ratio when compared with the liquid crystal display (LCD) technology, but the flexible and transparent possibilities of OLEDs make them an innovative choice in a competitive market. OLEDs usually comprise multiple layers of small organic molecules, sandwiched between two electrodes. Each layer is tailored for a specific functionality to facilitate balanced charge carrier injection and transport to the emissive layer, where the charge carriers recombine to form excitons and conse- quently photons, achieving a desirable wavelength of emission. 1–3 One of the major advantages of using organic molecules is the possibility of manipulating chemical composition to target specific properties, such as emission color and charge carrier transport abili- ties. The aim of OLED material design is to achieve an optimal bal- ance between stability, efficiency, operational driving voltage, and color coordinate of the device. However, molecular design often relies on chemical intuition and with a vast number of potential can- didates for each of the layers, this approach is inherently flawed. Therefore, in order to optimize design a systematic approach which links chemical structure to macroscopic properties would be greatly beneficial.4,5Ideally, this would focus solely on in silico prescreening, prior to synthesis. Predictive computational modeling is, however, not yet sufficiently accurate for this task exclusively6and with experi- mental prescreening being an extensive and costly procedure, a col- lective experimental and in silico approach presents a favorable alternative. The in silico evaluation of organic materials involves accurate prediction of device characteristics from the corresponding molecular building blocks, which requires simulations over a broad range oflength and timescales. The multiscale simulation techniques, as depicted in Fig. 1 , help us to establish links between microscopic and macroscopic material properties. Here, quantum chemical calculationsare used to obtain gas-phase geometries of a neutral molecule, its cat- ion and anion, as well as ionization energies and electron affinities. The material morphology is then simulated by molecular dynamics(MD), with classical force fields describing atomistic interactions. Polarizable force fields are then used to account for electrostatic effects upon charge/exciton transfer. Kinetic Monte Carlo (KMC) is thenused to perform charge transport simulations, whereby it is possible to extract macroscopic observables, such as charge carrier mobilities, by solving the master equation. 5,7–13 It is clear that the morphology is an integral part of the simula- tions, and it is often a challenge to generate morphologies representa- tive of experimental systems. For this task, both the classical forcefields and the more computationally demanding polarizable force fields have to be parameterized. Experimental input at this stage ensures the accuracy of the structural predictions using the classicalforce fields, while polarizable force fields can be parametrized from the first principles. The parametrization of these force fields is a tedious task and impossible for the vast number of organic compoundsrequired for prescreening. 14–16To overcome this, it is possible to use the similarities among the molecules most likely to be experimentally investigated, in order to create molecular fragments or building blocks, including the force field parameters. The concept of an extendable molecular library containing these well-defined building blocksrequired to generate realistic morphologies would then permit the swift characterization of new systems. For this concept to be brought to a realization, a well-defined simulation workflow capable of predicting relevant system propertiesfrom the chemical structure, must first be approved. Therefore, the accuracy and reliability of the various simulation techniques of this workflow, is the subject of the present work. To establish a starting point, simulation results for 12 small organic molecules are summa- rized, the molecular structures of which are shown in Fig. 2 .T h ei n d i - vidual steps of the simulation workflow are scrutinized and directly compared to experimental results for the glass transition temperature, Molecular structure ~ ÅAmorphous morphology ~ 10 nm, 3000 molecules Ground state properties Atomistic and polarisable force fieldsCharge transport rates Master equation and kinetic Monte CarloDevice ~100 nmPixels ~mm FIG. 1. OLED modeling multiscale simulation workflow. Starting from the first principles calculations of an isolated molecule, combined with atomistic forc e fields to generate the amorphous morphology, with the use of molecular dynamics. Polarizable force fields are used to account for the environmental effects on the densit y of states. Site ener- gies, reorganization energies, and electronic coupling elements are computed, followed by the charge transfer rates. Kinetic Monte Carlo is used to solve the master equation, to study charge dynamics (e.g., carrier mobilities), giving macroscopic device characteristics.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-2 VCAuthor(s) 2021energetic disorder, ionization energy in the solid-state, and charge car- rier mobilities. By doing so, the outlined simulation workflow and the force fields used can be validated, allowing for the expansion of thelibrary and further structures to be investigated. While this review focuses on one multiscale simulation proce- dure, it is worth emphasizing that there are several different computa-tional approaches for the consideration of organic semiconductors. One such method highlights the importance of using a fully quantum mechanical approach for charge dynamics to improve on semiclassical Marcus rates. 17Quantum chemical methods can also be used for eval- uating ground and excited state properties of organic electronic mate- rials. However, standard DFT functionals can lead to errors surrounding localization and delocalization of electron and holedensities, prompting the use of tuned long-range corrected hybrid functionals. 18Excited state properties have also been evaluated using many-body Green’s function theory and GW approximation with the Bethe–Salpeter equation (GW-BSE).19–24In terms of charge transport, KMC has been used to study degradation and the sensitivity of OLEDdevice lifetime and efficiency, in relation to material-specific parame- ters, to aid with design strategies with regard to energy levels. 25 Additionally, KMC models with molecular-level resolution have beendeveloped for investigating organic field-effect transistors (OFETs). 26 Computational methods have also been used to investigate chiral com-position–dependent charge mobilities, helping to link variations of molecular chirality to charge transport properties. 27Multiscale simula- tions have also aided with the study of photoluminescence quenching mechanisms in phosphorescent OLEDs.28Therefore, it is clear that predictive multiscale protocols are essential in the search and the design of novel materials for organic electronic devices.4GLASS TRANSITION TEMPERATURE In addition to electronic properties, the thermal stability of an organic material is essential in determining its suitability to be used in OLED devices. To enhance the device’s lifetime, materials which are less susceptible to thermal degradation are targeted, particularly mak- ing use of materials with a high glass transition temperature, Tg. When the device is operational, the flow of current results in Ohmic heating within the organic layers and due to this local heating, materi- als which have a low Tgexperience a higher degree of molecular vibra- tions29and chemical decomposition, resulting in lower stability. Furthermore, high Tgalso prevents morphology changes during post- processing of devices, like unwanted partial re-crystallization. Therefore, accurate predictions of the Tgvalue for OLED materials can serve as a method of screening thermally stable candidates. The Tg for the 12 organic materials was obtained by MD simulations, as out- lined in the Methods section, the values are listed in Table I , and the comparison to experimental values is shown in Fig. 3 . The experimen- tal values are listed as referenced values from previous studies, or asnew experimental values obtained by differential scanning calorimetry (DSC), as outlined in the Methods section. The simulation results show a systematic overestimation of T gfor all systems, in comparison to the experimental values, except for TMBT. This overestimation is expected as the process of obtaining thesimulated morphologies, outlined in the Methods section, involves thermal annealing above T g, followed by a fast-cooling process. The simulated morphologies often disagree with experiment, as the cooling rates are much faster in simulation compared to experiment and reproducing the realistic molecular packing requires long simulation times. Additionally, the relaxation times of typical OLED depositionsBCP MTDATA TCTA 2-TNATATMBT TPBiNBPhenNPB Spiro-TADH3C H3CCH3 CH3 CH3CBP mCBP mCP FIG. 2. Chemical structures of the 12 small organic molecules investigated for the OLED material library: BCP, CBP, mCBP , mCP , MTDATA, NBPhen, NPB, TCTA, TMB T, TPBi, Spiro-TAD, and 2-TNATA.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-3 VCAuthor(s) 2021cannot be achieved by the slow molecular dynamics close to Tg. Furthermore, the deposition conditions can have a significant impact on molecular orientation,39,40such that experimental Tgvalues may also show variation, particularly for systems with low Tg, for example TMBT or also BCP, where Tgwas not detectable (as there was no sig- nificant kink in calorimetric measurements, described in the Methods section) despite a previously reported value in the literature.30For some of the compounds, such as TMBT, it is known that vacuum deposition leads to molecular alignment, i.e., molecular orientations are not random. This effect is not accounted for in our outlined proto-col, as it is difficult to reproduce in simulations; large timescales arenecessary to cover the diffusion process of evaporated materials, mim- icking experimental film-growth rates of 1 A ˚/s. 39,41,42 In order to accurately predict the glass transition temperature from the morphology, it is clear that slower deposition rates must be employed, for example with simulation methods such as coarse grain- ing.39,43An alternative to predict Tgfor OLED materials is a quantita- tive structure–property relationship approach,44with the use of topological indices45and various descriptors. However, this only allows to interpolate within known chemical space. Despite the inaccu- rate prediction of the Tgvalues, the proceeding simulation results will show that this does not have a significant impact on the charge trans-port simulations and mobilities for the materials in this study. CHARGE TRANSPORT In disordered organic materials, due to weak intermolecular cou- pling 46and relatively large energetic disorder, the charges are localized and propagate by a successive hopping process from one molecule in the system to the next. The hopping rate between two sites in a mate- rial is characterized as a thermally activated transport process, where the rate can be expressed in terms of the Marcus rate equation,47–49 xij¼2p /C22hJ2 ijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi4pkijkBTp exp /C0DEij/C0kij/C0/C12 4kijkBT"# : (1) Here Tis the temperature, kBis Boltzmann’s constant, kijis the reor- ganization energy, Jijis the electronic coupling matrix element, and DEij¼Ei/C0Eji st h ed r i v i n gf o r c eo rs i t ee n e r g yd i f f e r e n c eb e t w e e n two neighboring sites, where Eiis the site energy of molecule i.50The site energy includes the internal molecular energy ( Eint i) and interac- tion with the environment, including the electrostatic ( Eelec i)a n d induction ( Eind i) contributions. It is calculated as Ei¼Eint iþEelec i þEind iþqF/C1ri,w h e r e qis the hopping carrier charge, Fis an applied external field, and riis the center-of-mass of molecule i.5For computa- tional efficiency, here we use the Marcus rate expression; the entirescheme can be also adapted to the quantum treatment of vibrational modes. 17The microscopic quantities, such as reorganization energy, electronic coupling, etc., are computed from first principles and serve as an input to the master equation, the solution of which gives charge carrier mobilities.51,52 The methodology of obtaining the reorganization energies and electronic coupling elements is described in the Methods section. The reorganization energies for each material are listed in Table S1 andelectronic coupling elements are shown in Fig. S1, of the supplemen- tary material . Even though the variations of the reorganization ener- gies and electronic coupling elements lead to variations in thesimulated mobility, l, the most significant parameter is the distribu- tion of site energies E iwithin the system, characterized by the ener- getic disorder r. To a certain extent, this is anticipated, as the mobility is exceptionally sensitive to changes in the width of the disorder distri- bution, for example, l/exp½/C0Cðr kBTÞ2/C138.53–55The energetic disorder stems from the disorder on the local electronic states which, as we will see in Fig. 4 , is Gaussian-distributed. DENSITY OF STATES To evaluate the site energies of holes and electrons in the 12 systems, a perturbative scheme was used, as outlined in the Methods section. The distributions of the on-site energy differences, i.e., the dif- ferences between the energies of the system when a selected moleculeTABLE I. Glass transition temperature ( Tg): Comparison between simulation and experiment, including referenced literature (experimental) values for the 12 systems. For BCP, the Tgwas not detectable (ND) experimentally, as there was no significant kink in calorimetric measurements. SystemTg(/C14C) Simulation Experiment Literature BCP 143.6 ND 8930 CBP 133.0 115 10929 mCBP 155.3 93.1 9731 mCP 140.3 66 6032 MTDATA 164.7 78 7533 NBPhen 238.2 /C1/C1/C1 10534 NPB 166.9 99.5 9832 Spiro-TAD 217.6 135 13335 TCTA 184.7 154 15133 TMBT 68.5 57 9536 TPBi 204.8 /C1/C1/C1 12737 2-TNATA 212.2 /C1/C1/C1 11038 250 230 210 190 170 170150 150130 130110 11090 9070 7050 50Simulation Tg (°C) Experiment Tg (°C)NBPhen TPBi MTDATA2-TNATA TCTA mCBP mCBP BCPTCTA Literature TMBTCBP CBP TMBT ExperimentNPBSpiro-TAD Spiro-TAD NPBMTDATA mCP mCP FIG. 3. Comparison of simulation and experiment Tgvalues (blue) with linear corre- lation (blue dashed line) R2¼0:53. Experimental values found in the literature are also compared (red) with linear correlation (red dashed line) R2¼0:22. The linear relationship ( x¼y) is shown by the black dashed line.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-4 VCAuthor(s) 2021is in the anionic/cationic or neutral state, including the constant inter- nal contribution due to the gas-phase electron affinity/ionizationpotential, are displayed in Figs. 4(a) and4(b) for electrons and holes, respectively. The corresponding energetic disorder (widths of site energy distributions) are summarized in Table II .I na m o r p h o u so r g a n i c materials, the energetic disorder is predominantly electrostatic;such electrostatic interaction originates from the potential exertedon a molecule from its specific environment. Therefore, the disor-d e ri sg o v e r n e db yt h em o l e c u l a rs t a t i cm u l t i p o l e s ,a sw e l la st h epositional and conformational order in a given material. On the other hand, the induction contribution stemming from the interac- tion of microscopic dipoles (the distributions of the electrostaticpotential in the ground state are shown in Fig. S2 of the supple- mentary material ) with the localized charge carrier, reduces the energetic disorder. 13,56The electrostatic and induction contribu- tions for each system, for both electrons and holes, are listed inTable II . Apart from energy distributions, the long-range electrostatic interactions of a charge with molecular dipoles can lead to spatialcorrelations of site energies. 55,57To unveil such spatial correlations, we computed these correlations for holes and electrons, which are shown in Fig. S3 of the supplementary material . We found that the hole and electron site energy correlations extend up to 2–3 nm. The correlations exhibit a decay profile with distance, approximately following the r/C01 signature expected for the random dipolar disorder.55Therefore, all of these materials possess a correlated disorder even though their ground state dipole moments are small for a large set of molecules. We now analyze in more detail ionization energies, electron affinities, and the widths of the DOSs, which we consider to be the most important parameters for charge transport. ELECTRON AFFINITY AND IONIZATION ENERGY The gas-phase electron affinities (EA 0) and ionization energies (IE0) were calculated using density functional theory at the M062X/ 6–311g(d,p) as well as IP/EA-EOM-DLPNO-CCSD/aug-cc-pVTZ level of theories, as described in the Methods section and in the supplementary material . The comparison of EA 0and IE 0obtained at various levels of theory is shown in Figs. S4–S6 (see also Table S2) of the supplementary material . In inverse photoemission spectroscopy (IPS) Ionization ener gy (eV) Electron affinity (eV) ProbabilityProbability 0.0 0.5 1.0 1.5 2.0 2.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0BCP CBP mCBP (a) (b) mCP MTDATA NBPhen NPB Spiro-TAD TCTA TMBT TPBi 2-TNATABCP CBP mCBP mCP MTDATA NBPhen NPB Spiro-TAD TCTA TMBT TPBi 2-TNATA FIG. 4. The density of states (distribution of site energies) in the amorphous materials for (a) anion, with solid-state electron affinity (EA tot) shown by the black dashed lines, and (b) cation, with solid-state ionization energy (IE tot) shown by the black dashed lines. Experimental reference lines for ionization energy (IE exp) are shown as red dashed lines. Gas-phase ionization energy (IE 0) values obtained by M062X/6–311 þg(d,p) level of theory are shown using blue dashed lines.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-5 VCAuthor(s) 2021and ultraviolet photoelectron spectroscopy (UPS), the electron affinity and ionization energy are measured as an onset of the spectra. In simulations, to account for the finite width of the den- sity of states ( r), the solid-state electron affinities (EA tot) and ioni- zation energies (IE tot)w e r es h i f t e db y2 r,s u c ht h a tE A ¼aþ2r or IE ¼a/C02r,w h e r e arepresents the mean of the DOS for elec- t r o n so rh o l e s ,r e s p e c t i v e l y .T h eE A totand IE totvalues are summa- rized in Table II for each material. Experimental ionization energies are also listed in the table and shown in Fig. 4(b) ,f o r comparison. Further to this, the comparison of solid-state IE val- ues to the corresponding experimental values are shown in Fig. 5 . The method of obtaining IE values by photoelectron yield spec- troscopy in air (PESA) is described in the Methods section, theUPS values are taken from the literature, with references included inTable II . Both gas-phase and solid-state ionization energies are compared to PESA and UPS values in the supplementary material ,shown in Fig. S7. Additionally, the gas-phase and solid-state ioni- zation energies and electron affinities are compared to experimen- tal cyclic voltammetry (CV) data, shown in Fig. S8 of thesupplementary material , where the experimental method is also described. The IE values obtained from the DOS of the various materials are in good agreement with the experimental IE values, as depicted inFig. 4(b) for the individual systems and Fig. 5 as the total corre- lation. The computed IE totvalues are a combination of the gas- phase IE 0as well as the electrostatic and induction contributions. This is necessary as the gas-phase IE is for a single isolated mole-cule and does not account for the solid-state effects required to accurately determine electronic properties. Interestingly, as shown in Fig. S7 of the supplementary material , the linear correlation to experimental solid-state values is already reproduced from the gas-phase simulations, however, with a proportionality coefficientTABLE II. (Top) energetic disorder ( r, eV) for electron transport and electronic affinities (EA, eV) and (bottom) energetic disorder ( r, eV) for hole transport with experimental values (eV) and ionization energies (IE, eV) in the studied amorphous materials, with experimental values (eV) and references, where available. SCL C: space charge–limited current, TSL: thermally stimulated luminescence, UPS: ultraviolet photoelectron spectroscopy, PESA: photoemission spectroscopy in air. System rsim electronEAsim EA0 EAelec EAind EAtot BCP 0.192 0.39 /C00.30 0.66 1.14 CBP 0.118 0.45 0.003 0.68 1.37 mCBP 0.151 0.41 /C00.19 0.69 1.21 mCP 0.145 /C00.27 /C00.22 0.80 0.61 MTDATA 0.109 /C00.11 /C00.12 0.95 0.93 NBPhen 0.200 1.04 /C00.43 0.58 1.58 NPB 0.098 0.12 /C00.21 0.73 0.83 Spiro-TAD 0.102 0.29 /C00.13 0.69 1.05 TCTA 0.189 0.08 0.01 1.43 1.90TMBT 0.159 0.98 /C00.36 0.70 1.64 TPBi 0.125 0.59 /C00.18 0.69 1.35 2-TNATA 0.187 0.16 /C00.10 1.04 1.47 IE sim System rsim hole rexp holeIE0 IEelec IEind IEtot IEexp BCP 0.190 7.57 0.29 0.59 6.31 6.52/6.5 PESA/UPS58–60 CBP 0.096 0.125/0.10 TSL/SCLC617.10 /C00.05 0.59 6.37 6.07/6.1 PESA/UPS62–64 mCBP 0.122 0.131 TSL 7.32 0.09 0.65 6.34 6.07/6.1 PESA/UPS62 mCP 0.127 0.140 TSL 7.38 0.21 0.77 6.14 5.98/5.9 PESA/UPS60 MTDATA 0.079 5.70 0.12 0.54 4.88 5.13 PESA NBPhen 0.194 0.167 TSL 7.23 0.43 0.55 5.86 5.8 UPS65 NPB 0.087 0.088/0.09 TSL/SCLC616.25 0.20 0.60 5.28 5.47/5.4 PESA/UPS58,63 Spiro-TAD 0.090 0.110/0.09 TSL/SCLC616.23 0.15 0.62 5.28 5.50 PESA TCTA 0.122 0.110/0.10 TSL/SCLC616.63 /C00.02 0.79 5.62 5.71/5.7 PESA/UPS62,66 TMBT 0.141 8.06 0.35 0.77 6.66 6.41 PESA TPBi 0.134 0.150 TSL 7.40 0.17 0.66 6.30 6.2 UPS64 2-TNATA 0.097 0.10 SCLC615.72 0.09 0.50 4.94 5.0 UPS63Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-6 VCAuthor(s) 2021different from unity. The agreement of the computed (IE tot)a n d experimental IE values signifies a reasonable degree of accuracy ofthe polarizable force fields. T h em e t h o dd e p l o y e di nt h ep r e s e n tw o r ka l s oe n a b l e st h ed i s - tinction of different contributions to the EA and IE, e.g., gas-phase,electrostatic, and induction, shown in Table II .F o ra l lc o m p o u n d s ,t h e stabilization due to induction ( E ind) contributes around 0.5–0.8 eV and 0.6–1.4 eV for holes and electrons, respectively. On the otherhand, the electrostatic contribution ( E elec) varies from one system to another (0.01–0.4 eV for both holes and electrons) governed by thedistribution of molecular dipoles in a given system. A closer scan of the numbers in electrostatic contribution ( E elecinTable II ) reflects a salient correlation between energetic disorder and electrostatic interac-tions for molecules such as NBPhen, TMBT, and BCP possessing largecharge-dipole interaction. Here, the magnitudes of E elecare almost two t i m e sh i g h e rt h a nt h er e s to ft h em a t e r i a l s ,a sw e l la st h eh i g h e r rval- ues, shown for both holes and electrons. The energetic disorder for hole transport for each material is taken as the width of the DOS in simulations. These values are directlycompared in Fig. 6 to experimental values from previously reported space charge–limited current (SCLC) measurements, 61and/or newly carried out thermally stimulated luminescence (TSL) measurements. Details of the SCLC measurements can be found in Ref. 61, and the TSL measurements are described in the Methods section. The simula-tions predict a significant variation of the energetic disorder amongthese materials which features an almost twofold change spanning from r¼0:09 eV observed for a weakly disordered NPB, to r¼0:19 eV obtained for highly disordered NBPhen. The experimen- tal results demonstrate a remarkably similar trend ( Fig. 6 ), which testi- fi e st ot h ei n t r i n s i cn a t u r eo ft h el a r g ew i d t ho ft h eD O S( r>0.1 eV) observed experimentally in most of these OLED materials, implying the DOS of a chemically pure disordered material rather than beingsignificantly affected by impurity-related traps. The energetic disorderinferred from TSL normally provides the DOS of shallower trappedcarriers, which are thermally released and recombine with the moredeeply trapped counter charges. This is because TSL is an electrode- free technique, so the number of electrons and holes in the film is always the same to maintain the material neutrality at any time after the carriers were photogenerated. Once a shallower trapped carrier isreleased from its trap, it will recombine with its deeper trapped coun-terpart. Therefore, the latter sort of carriers is expected to already be completely annihilated, and there are simply no carriers left at a tem- perature relevant to their anticipated release from their deeper states. The ionization energies of most of the materials measured in this study are less than, or around 6 eV ( Table II ). Therefore, their hole DOS should not be affected much by extrinsic traps, as the IE values fallwithin an energy window, inside which organic semiconductors nor- mally do not experience charge trapping. 67Materials with an ionization energy above 6 eV will exhibit trap-limited hole transport, similarly, anelectron affinity lower than 3.6 eV will cause electron trapping to limitelectron transport. 67As the materials in this study have electron affini- ties significantly less than 3.6 eV ( Table II ) shown by simulations, the electron transport in these systems is most likely hindered by electrontrapping (as oxygen-related traps). Due to this reasoning and takinginto account that all samples were exposed to air prior to TSL measure- ments, which makes relatively deep electron traps unavoidable, our TSL measurements probe the hole DOS in these materials. Although thesimulated r-parameters for holes and electrons ( Table II ) are not appre- ciably different, it should be mentioned that TSL measures the “effectiveDOS” and even a small concentration of shallow extrinsic traps (depending on the trap depth it can even be at trap concentration level of/C240.1%) can give rise to a notable DOS broadening. This has been previously demonstrated by deliberate doping of photoconductive poly-mers with traps. 68Since the IE of CBP, mCBP, and TPBi slightly exceeds the trap-free window threshold value of 6 eV, a certain extrinsic hole trapping cannot be excluded here. This can explain why TSL mea-surements for these materials yield slightly overestimated energetic dis-order parameters, compared to the simulated values ( Fig. 6 ). Finally, we comment on some differences in r-parameters deter- mined by SCLC and TSL techniques observed for some materials, e.g.,IE experiment (eV) 2-TNATA MTDATASpiro-TAD NPBTCTATCTANBPhen UPS PESAmCBPmCBPTPBiBCPBCPTMBT CBPCBP mCP mCP NPBIE simulation (eV) 7.0 7.06.5 6.56.0 6.05.5 5.55.0 5.04.5 4.5 FIG. 5. Simulated solid-state ionization energies (IE tot) compared to experimental values obtained by UPS: R2¼0:898 (blue) and PESA: R2¼0:900 (green). The linear relationship ( x¼y) is shown by the dashed line.Ener getic disorder experiment (eV) Energetic disorder simluation (eV) 0.20 SCLC NBPhen TCTA 2-TNATA Spiro-TAD NPB NPB CBP CBP mCBPmCP Spiro-TADTCTA TPBiTSL 0.200.18 0.180.16 0.160.14 0.140.12 0.120.10 0.100.08 0.080.06 0.06 FIG. 6. Simulated and experimental energetic disorder (eV) values for the studied systems. SCLC (orange squares) and TSL (blue squares; R2¼0:736). The linear relationship ( x¼y) is shown by the black dashed line, highlighting the correlation of experimental and simulated values.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-7 VCAuthor(s) 2021for CBP. This may be partially due to different film morphologies related to different film deposition techniques used: while films for SCLC measurements were vacuum-deposited, most of the materialsfor TSL measurements were spin-coated from a solution, except NPB, TCTA, and Spiro-TAD, which were vacuum-deposited. Another likely reason might be due to the fact that the SCLC probes the DOS in alarge-carrier-concentration transport regime, for which deep tail states of the DOS (and/or traps) might be filled with carriers during the mea- surements and therefore, JV-characteristics might be governed mostly by a shallower portion of the DOS. On the other hand, carrier concen- tration in TSL experiments is significantly smaller (comparable to time-of-flight experiments) and deep tail states can be well probed bythis technique. One can also attempt to relate the observed energetic disorder to molecular dipoles, since lattice models with randomly oriented dipoles of equal magnitude dand lattice spacing a, yield disorder r/C24d=a. However, in a realistic morphology, both distances and dipoles can vary from molecule to molecule, which then increases the disorder fur- ther. It is evident from the distributions in Fig. S9 that the dipolemoments of individual molecules in an amorphous morphology vary significantly from their equilibrium values (see Table S3 of the supple- mentary material ). Such broad distributions of dipole moments are attributed to the presence of one or more soft dihedrals in the organic materials. At finite temperature, rotation around such soft degrees of freedom leads to multiple conformers. In general, systems with a nar-row distribution of molecular dipoles in their amorphous morpholo- gies possess small fluctuations of electrostatic multipoles in the amorphous matrix which results in smaller energetic disorders. This conclusion is, however, not universal since large local electrostaticpotential variations of TMBT, for example, can result in a sizable dis- order despite a zero molecular dipole moment. The correlation of aver- age dipole moment and energetic disorder for the 12 systems areshown in Fig. S10 of the supplementary material, for both holes andelectrons. Although there is a slightly better correlation for holes com-pared to electrons, a number of outliers remain for both, implying thatit is difficult to predict the trend in sigma, solely from the moleculardipole moment. CHARGE CARRIER MOBILITY Charge transport rates were computed using the high tempera- ture limit of classical charge transport theory 47–49as given by the Marcus rate equation. The master equation can then be solved withkinetic Monte Carlo (KMC), providing the time evolution of the sys-tem, giving a randomly generated trajectory of charge carrier move-ment. This was carried out for one charge carrier (hole or electron) inthe presence of an applied electric field ( F¼1/C210 4V/cm), using a periodic simulation box. Mobilities were extracted as outlined in theMethods section. The extrapolated dependence of mobility on temper-ature is shown in Fig. S11 and Fig. S12 of the supplementary material , for holes and electrons, respectively. The single-carrier mobilities atroom temperature for holes and electrons are summarized in Table III. The experimentally measured mobility and the corresponding experimental techniques used, are also listed for comparison. Table III andFig. 7 show the correlation between simulated and experiment mobility. The remarkable agreement of simulation andexperiment is particularly evident for hole mobilities. On the otherhand, for electron mobilities, a larger deviation is observed betweenexperiment and simulation, where simulated results indicate a TABLE III. Room temperature hole and electron mobility (cm2/V s), achieved from simulations of the amorphous organic materials, with experimentally achieved mobilities and the corresponding techniques used, references included. TOF: time-of-flight experiment, SCLC: space charge–limited current method. System lðsimÞ holelexpðÞ holelsimðÞ elecrtonlðexpÞ electron BCP 6.42 /C210–11/C1/C1/C1 /C1/C1/C1 6.59 /C210–9/C1/C1/C1 /C1/C1/C1 CBP 4.91 /C210–42.2/C210–4SCLC613.64 /C210–5/C1/C1/C1 /C1/C1/C1 5.0/C210–4TOF69 mCBP 9.00 /C210–4/C1/C1/C1 /C1/C1/C1 8.24 /C210–5/C1/C1/C1 /C1/C1/C1 mCP 2.35 /C210–45.0/C210–4TOF701.35 /C210–7/C1/C1/C1 /C1/C1/C1 MTDATA 1.56 /C210–51.3/C210–5TOF718.17 /C210–6/C1/C1/C1 /C1/C1/C1 NBPhen 2.41 /C210–11/C1/C1/C1 /C1/C1/C1 6.22 /C210–10/C1/C1/C1 /C1/C1/C1 NPB 2.04 /C210–42.3/C210–4SCLC614.04 /C210–4(6–9) /C210–4TOF58 2.7/C210–4TOF71 Spiro-TAD 4.99 /C210–43.1/C210-4SCLC613.67 /C210–5/C1/C1/C1 /C1/C1/C1 5.0/C210–4TOF72 TCTA 1.00 /C210–48.9/C210–5SCLC611.61 /C210–9<10–8Ref.74a 2.0/C210–4TOF73 TMBT 7.47 /C210–6/C1/C1/C1 /C1/C1/C1 6.38 /C210–51.2/C210–4TOF75 TPBi 1.18 /C210–7/C1/C1/C1 /C1/C1/C1 1.18 /C210–56.5/C210–5SCLC76 (3–8) /C210–5TOF77 2-TNATA 1.72 /C210–52.7/C210/C05SCLC614.64 /C210–6(1–3) /C210/C04TOF58 (2–9) /C210/C05TOF58 aNot measurable by TOF.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-8 VCAuthor(s) 2021systematic underestimation of experimental measurements. There are several possible explanations to account for these discrepancies. First, due to the much larger energetic disorder for electrons in certain mate- rials ( Table II ) when compared to holes, the electron mobility will be inherently lower. As a result of the large disorder found in 2-TNATA, the simulated and experimental mobilities show large variation. Thismay stem from energetic traps in the simulated morphology, which can lead to lower mobility values. To highlight the significant role of energetic disorder on the simulated mobility, a correlation plot is shown in Fig. 8 , for both holes and electrons. Due to different morphologies, the structural and energetic disor- der can differ significantly between simulation and experiment. Despite the reasonable agreement for energetic disorder for hole trans- port, shown in Fig. 6 , the experimental systems used for the mobility correlation are a collection of referenced values from various studies, with potentially significant variations in disorder. It should also be noted that the simulated morphologies for the 12 systems do not account for the presence of carrier traps formed by s t r u c t u r a ld e f e c t so ri m p u r i t i e ss u c ha sw a t e r ,w h i c ha r et y p i c a l l y unavoidable in reality. However, the inclusion of carrier traps in the simulated morphology would, in fact, lead to larger deviation betweensimulated and experimental mobilities. As previously stated, hole or electron transport have been shown to become trap-limited in materi- als with an IE greater than 6 eV or an EA less than 3.6 eV, respec- tively. 67Therefore, direct comparison of simulation and experiment mobilities may be difficult when considering low EA and high IEmaterials. Finally, the takeaway message here is that the width of the density of states is the key property in determining the mobility ( Fig. 8 ). Hence, accurate predictions of the DOS should be given priority when prescreening OLED hosts. DOS width correlates with some extentwith the molecular dipole, but this correlation has too many outliers, e.g., due to conformational freedom or higher order multipoles and therefore, cannot be used as a reliable descriptor for mobility predictions.OUTLOOK It is clear that a molecular library of OLED hosts would be invaluable, permitting the swift evaluation of new materials. The keyquestion is how accurately and reliable a combination of various simu-lation techniques can predict relevant material properties, to bring pre-screening a step closer. For the simulated morphological properties, the T gcomparison showed large variation to experimental values, suggesting that the cur-rent approach needs to be improved for accurate evaluation of ther- mally stable materials. Reliable methods for predicting T gand accurate atomistic force fields are the key improvements required for moreaccurate simulation results. The morphologies used for charge transport simulations revealed that the inaccuracies in T gpredictions had no significant impact on simulated ionization energies. In this respect, PESA measurementsand UPS data taken from the literature, as well as cyclic voltammetrymeasurements, are in an excellent agreement with simulation results,supporting our confidence in the polarizable force fields used for eval-uation of the solid-state electrostatic contribution. The situation isExperiment mobility log ( m)Simulation mobility log ( m)−2.5 −2.5Hole (TOF)Spiro-TAD Spiro-TADCBP CBP NPB NPB TCTA TCTA TMBT 2-TNATA 2-TNATA 2-TNATA MTDATA TPBi TPBiNPB mCPHole (SCLC) Electron (TOF) Electron (SCLC)−3.0 −3.0−3.5 −3.5−4.0 −4.0−4.5 −4.5−5.0 −5.0−5.5 −5.5 FIG. 7. Room temperature hole and electron mobility ( l) values achieved by simu- lation and compared to experiment values where available. Hole-TOF (red): R2¼0:95 and hole-SCLC (yellow): R2¼0:94; electron-TOF (blue): R2¼0:64 and electron-SCLC (green). The linear relationship ( x¼y) is shown by the black dashed line, highlighting the correlation of experimental and simulated values. log ( m) electronslog ( m) holes (electrons)2 (holes)2 0.045 NBPhen NBPhenBCP BCPTPBi TPBi2-TNATA 2-TNATAMTDATA MTDATATCTA TCTASpiro-TAD Spiro-TADTMBT TMBTmCP mCPmCBP mCBPCBP CBPNPB NPB0.040 0.0350.0300.0250.0200.0150.0100.005 0.000 −11.00 −9.00 −7.00 −5.00 −3.00 −1.00 −11.00 −9.00 −7.00 −5.00 −3.00 −1.000.0450.050 0.040 0.035 0.030 0.0250.0200.0150.0100.0050.000s s(a) (b) FIG. 8. Correlation of simulated mobility ( l) and energetic disorder ( r) for (a) holes and (b) electrons. The dashed lines are lines of best fit for the corresponding data, to serve as a visual aid.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-9 VCAuthor(s) 2021somewhat different for electron affinities, where using different com- putational techniques led to large variation of the gas-phase electron affinity values, even at a computationally affordable variation of the coupled cluster implementation. Moreover, there is no clear bench- mark possible for the solid-state because of the sparse availability of the inverse photoemission spectroscopy measurements. Accurate solid-state energetics allowed us to predict the density of states which, when compared to the thermally stimulated lumines- cence measurements, showed a similar trend. The energetic disorder can be potentially correlated with the distribution of molecular dipoles, but the extent of this will require further investigation to be conclusive. Finally, the simulated charge carrier mobility showed a remark- able agreement with experimental values, particularly for hole trans- port where energetic disorder is typically lower. The accurate prediction of energetic disorder is therefore vital, as it has a significant impact on mobility. Overall, the correlation of simulation and experimental results has been used to validate the accuracy of the force fields and the simu- lation methods, as an initial step toward building a larger molecular library. The agreement of simulation and experiment for the various parameters, particularly mobility, highlights the predictive capability of the outlined methods and the simulation workflow. The next step is to expand this library with further materials, in an effort to draw structure-property conclusions for effective prescreening. METHODS Gas-phase ionization energy and electron affinity For the isolated molecules, density functional theory (DFT)–based electronic structure methods were used to compute gas- phase ionization energy (IE 0) and electron affinity (EA 0) using the Gaussian09 program.78For this, the neutral molecule in the neutral geometry ( EnN), as well as the charged molecule in the charged geome- try (EcC), are computed. The IE 0and EA 0values are then calculated as EcC–EnN,w h e r e EcCrepresents the cationic and anionic state, respectively. The prediction of accurate IEs and EAs remain a challenge in electronic structure theory, primarily due to the self-interaction error79 (SIE) or localization/delocalization80error, inherent to commonly used DFT functionals. In a series of recent reports,18,81–84the impor- tance of considering long-range corrected hybrid functionals has been demonstrated, such that the SIEs in DFT description of molecules can be reduced. Therefore, adiabatic IE 0and EA 0calculations were per- formed by employing a range of DFT functionals: PBEPBE, B3LYP, CAM-B3LYP, xB97X-D, M062X, and LC- xPBE, combined with the basis set 6–311 þg ( d , p ) .T h eI E sa n dE A sa r ec o m p a r e df o re a c ho ft h e 12 organic molecules and each DFT functional, as shown in Fig. S4 of thesupplementary material . Different levels of theory are also com- pared in Fig. S5 and Fig. S6. It is evident that calculations performed using PBEPBE and B3LYP underestimate IE 0compared to other functionals, an observa- tion which is attributed to an underestimation of Kohn–Sham eigen- value by hybrid functionals, leading to significant overscreening of the Coulomb interaction. Overall, the IE 0is better predicted than the EA 0 for the given molecules. This is clear when comparing the Cam-B3LYP, xB97X-D, and M062X functionals, as there is greater varia- tion in the EA 0prediction. The small deviation among these functions, with regard to the IE 0, makes any of the three a suitable choice(M062X is the chosen functional for comparison to experimental IE a n dE Av a l u e si nt h i ss t u d y ) . Molecular dynamics DFT methods were then used to accurately parameterize the empirical OPLS-AA force field,14–16for the 12 chemically diverse mol- ecules. All Lennard–Jones parameters were taken from this force field in combination with the fudge-factor of 0.5 for 1–4 interactions. Atomic partial charges were computed using the ChelpG85scheme for electrostatic potential fitting as implemented in Gaussian09,78employ- ing the ground state electrostatic potential determined at the B3LYP/6–311 þg(d,p) level of theory. In order to generate amorphous morphologies, molecular dynamics (MD) simulations were carried out using the GROMACS simulation package. 86,87The amorphous state was generated by an annealing step, followed by a rapid quenching to lock the molecules ina local energy minimum. This procedure has been previously appliedfor the preparation of amorphous structures of OLED materials. 61,88 The starting configurations used in the MD simulations were preparedby randomly arranging 3000 molecules in a simulation box using thePackmol program. 89These initial structures were energy-minimized using the steepest-descent method and annealed from 300–800 K, fol- lowed by fast quenching to 300 K. Further equilibration for 2 ns and 1 ns production runs were performed at 300 K. All simulations were per-formed in the NPT ensemble using a canonical velocity rescaling ther-mostat, 90a Berendsen barostat for pressure coupling,91and the smooth particle mesh Ewald technique for long-range electrostaticinteractions. A time step of 0.005 ps was used to integrate the equa-tions of motion. Non-bonded interactions were computed with a real-space cutoff of 1.3 nm. To obtain the glass transition temperature of each material, a similar procedure was carried out with a time step of 0.001ps. Thematerial was first equilibrated at 800 K, followed by gradual cooling to0 K at a rate of 0.1 K/ps. The change in density as the material wascooled was used, in combination with two linear fits (the linear fittingprocedure is described in the supplementary material ) to extract the intersection point, giving the T gvalue. Density of states We used MD simulation trajectories to evaluate the site energies of holes and electrons by employing a perturbative scheme. In thisapproach, the electrostatic and induction energies are added to the gas-phase energies (IE 0or EA 0), to obtain the total site energy. The electrostatic contribution is calculated with the use of Coulomb sumsbased on distributed multipoles (obtained from the GDMA pro-gram 92) for neutral and charged molecules in their respective ground states. The polarization contribution is computed using a polarizableforce field based on the Thole model 93,94with isotropic atomic polariz- abilities ( aai)o na t o m s ain molecules i. Aperiodic embedding of a charge method95as implemented in the VOTCA13,43package, was used for these calculations. Coupling elements The transfer integral or coupling elements, Jij¼/i^Hjj/j/C10/C11 , represent the strength of the coupling of the two frontier orbitals j/ii andj/jilocalized on each molecule in the charge transfer complex. ItChemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-10 VCAuthor(s) 2021is highly sensitive to the characteristic features of the frontier orbitals as well as the mutual orientations of the two molecules and follows anexponential decay with distance. The electronic coupling elementsshown Fig. S1 of the supplementary material , were computed for each neighboring molecular pair ( ij)u s i n gap r o j e c t i o nm e t h o d . 96,97 Molecular pairs were added to the neighbor list, with a center-of-mass distance cutoff (between rigid fragments) of 0.7 nm. These calculations were performed at PBEPBE/6–311 þg(d,p) level of theory using the Gaussian0978and VOTCA13,43packages. The frozen core approxima- tion was used with the highest occupied molecular orbitals providing amajor contribution to the diabatic states of the dimer. Reorganization energies The reorganization energy of the system takes into account the charging and discharging of a molecule. When a charge moves frommolecule ito molecule j, there is an intramolecular contribution ( k int ij), due to the internal reorganization of the two molecules and an inter-molecular, known as an outersphere contribution ( k out ij), due to the relaxation of the surrounding environment.13,50The internal reorgani- zation energy is calculated as kint ij¼ðUnC i/C0UnN iÞþð UcN j/C0UcC jÞ, for molecule iandj, where the lowercase represents the neutral (n) or charged (c) molecule and the uppercase represents the neutral (N) orcharged (C) geometry. The reorganization energies are summarizedfor holes and electrons in Table S1 of the supplementary material ,f o r each of the systems. The individual contributions were calculated byDFT using the B3LYP/6–311 þg(d,p) level of theory. In the current work, we ignore the outersphere contribution to the reorganization energy, since in the amorphous solids considered here, the Pekar factor is on the order of 0.01, 13leading to a relatively small contribution to the total reorganization energy. Mobilities Mobilities were extracted as l¼hvi/F,w h e r e hviis the average projection of the carrier velocity in the direction of the field(F¼1/C210 4V/cm). The convergence of simulated mobilities with respect to the system size (i.e., a sufficient number of sites for the simu-lated transport to be nondispersive) due to energetic disorder must be ensured. For this purpose, the critical temperature (within Gaussian disorder model), T c, at which the transition from dispersive to nondis- persive regime takes place, was estimated and is shown in Fig. S11 andFig. S12 of the supplementary material . The mobilities obtained in the nondispersive regimes are then fitted with an empirical temperaturedependence, allowing for extrapolation of the nondispersive chargecarrier mobility at room temperature. Details of such calculations canbe found elsewhere in Refs. 98and99. All charge transport calcula- tions were performed using the VOTCA package. 13,43 Glass transition temperature: Differential scanning calorimetry (DSC) For determination of the glass transition temperature ( Tg)a t Merck KGaA, Darmstadt, Germany, we used differential scanning cal-orimetry (DSC) analyzing samples of 10–15 mg in DSC 204/1/GPh€onix from Netsch. Samples were heated by 5 /C14C/min up to 370/C14C then cooled by 20/C14C/min to 0/C14C and finally heated again by 20/C14C/ min to 370/C14C, where Tgwas determined by the kink in heat flow vs temperature using the temperature corresponding to half the drop inheat flux. Only for BCP and TMBT, this protocol did not yield a signif- icant kink. TMBT was expected to be the lowest Tgmaterial from sim- ulation, so we tried other protocols to measure Tg. We finally used a 5 mg TMBT sample in DSC Discovery from TA Instruments in nitrogen atmosphere and first heated by 20/C14C/min up to 320/C14C, then for cool- ing, quenched the sample by liquid nitrogen and finally heated by20 /C14C/min up to 320/C14C, where the T gwas observed. Other protocols we tried for TMBT without the cooling quench did not lead to obser- vation of Tg. Ionization energies: Photoelectron yield spectroscopy in air (PESA) Photoelectron yield spectroscopy in air (PESA) was performed on 50 nm thick thermally evaporated films at Merck KGaA, Darmstadt, Germany, using the surface analyzer AC-3 from RIKENKEIKI Co., Ltd. The film is exposed to monochromatic light from adeuterium-lamp, with incident photon energy between 4 and 7 eV (increased in steps of 0.05 eV), 100w h i l ea no p e nc o u n t e ra n a l y z e st h e photoelectron yield.101The square root of this photoelectron yield is plotted vs energy of the incident photons. The underground is takenas the average horizontal line through the measurement points for low photon energies. The ionization potential is calculated as intersection of the underground and a fit to the onset of the square root of the pho-toelectron yield. Due to the energy step size and this fitting procedure, which is done manually, an error bar of roughly 60:06 eV must be associated with the obtained ionization energies as we have verified by repeated measurement and evaluation. As compared to UPS measurements of films, PESA has the advantage that the penetration depth of photonsof 7 eV is roughly 10 nm, while for UPS, only the top 0.5 nm of thefilm can contribute, so PESA is measuring bulk properties of the film. Energetic disorder: TSL technique TSL is the phenomenon of luminescent emission after removal of excitation (UV light in our case) under conditions of increasing tem- perature. Our TSL measurements were carried out over a wide tem-perature range from 5 to 330 K using an optical temperature-regulating helium cryostat. When exciting a sample optically with 313 nm light for typically 3 min at 4.2 K, the charge carriers are gener- ated and populate trapping states. Once the sample is heated up,trapped charge carriers are released and then recombine, producing aluminescence emission. TSL measurements were performed in two different regimes: upon uniform heating at the constant heating rate 0.15 K/s and under the so-called fractional heating regime. 102In the latter regime, we apply a temperature-cycling program in which a largenumber of small temperature oscillations are superimposed on a con- stant heating ramp that allows determining the trap depth with high accuracy. This applies when different groups of traps are not well sepa-rated in energy or are continuously distributed, which is of special rele-vance for disordered organic solids where the intrinsic tail states can act as traps at low temperatures. The details of the TSL measurements have been described elsewhere in Refs. 103–105 . Technically, TSL is a relatively simple technique with a straight- forward data analysis. However, it should be noted that the mecha-nism of TSL in amorphous organic semiconductors with a broadenergy distribution of strongly localized states, differs from theChemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 031304 (2021); doi: 10.1063/5.0049513 2, 031304-11 VCAuthor(s) 2021mechanism commonly accepted for crystalline materials with band- type transport where a discrete trapping level model is applicable. Aspecific feature of amorphous solids is that the localized states withinthe lower energy part of the intrinsic DOS distribution can give rise toshallow charge trapping at very low temperatures, and as a conse- quence, TSL can be observed even in materials where the “trap-free limit” has been postulated. Since TSL measurements are normally per-formed after a long dwell time after photoexcitation, the initial energydistribution of localized carriers is formed after low-temperatureenergy relaxation of photogenerated carriers within a Gaussian distri-bution of DOS. Theoretical background for application of TSL for probing the DOS distribution in disordered organic systems has been developed, 105,106using a variable-range hopping formalism and the concept of thermally stimulated carrier random walk within a posi-tionally and energetically random system of hopping sites. The theoryproves that the high temperature wing of the TSL curve in such amor-phous materials should be an exact replica of the deeper portion of theDOS distribution 105,106and yields the effective DOS width. Kadashchuk and co-workers have applied low-temperature fractional TSL to investigate the intrinsic energetic disorder in a variety of impor-tant semiconducting polymers, oligomers, and hybrid metalorganicperovskites (see, for instance Refs. 103–109 ). A clear advantage of TSL is that it is a purely optical and electrode-free technique. This helps toeliminate interface/contact effects and, most importantly, it allows DOS measurements even in materials with large energy disorder where the charge transport is very dispersive. SOFTWARE AND INPUT FILES All simulations were performed with the open-source software package VOTCA. 110Atomistic and polarizable force fields, VOTCA input files, and analysis scripts are available in the Materials Library gitrepository, version 1.0. 111 SUPPLEMENTARY MATERIAL See the supplementary material for additional material proper- ties, including: reorganisation energies, transfer integrals, comparisonof gas phase ionisation energies, and dipole moments. ExperimentalTSL curves are also shown. AUTHORS’ CONTRIBUTIONS A.M. and L.P. contributed equally to the work. ACKNOWLEDGMENTS D.A. acknowledges the BMBF Grant InterPhase (Grant No. FKZ13N13661) and the European Union Horizon 2020 Research and Innovation Programme ‘Widening Materials Models’ under Grant No. 646259 (MOSTOPHOS). This research has beensupported by the King Abdullah University of Science andTechnology (KAUST), via the Competitive Research Grants (CRG)Program. D.A. acknowledges KAUST for hosting his sabbatical.The Ukrainian team acknowledges funding through the EU Marie Skłodowska-Curie ITN TADF lifegrant (Grant No. 812872). This research was also supported by the European Research Councilunder the ERC Grant No. 835133 (ULTRA-LUX), VW Foundation,and by the National Academy of Science of Ukraine (Project No.VC/205) and NRFU 2020.01/0144. K.-H.L. acknowledges thefinancial support from the Swiss NSF Early Postdoc Mobilityfellowship (Grant No. P2ELP2_195156). 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5.0047386.pdf

J. Chem. Phys. 154, 194109 (2021); https://doi.org/10.1063/5.0047386 154, 194109 © 2021 Author(s).Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition Cite as: J. Chem. Phys. 154, 194109 (2021); https://doi.org/10.1063/5.0047386 Submitted: 12 February 2021 . Accepted: 26 April 2021 . Published Online: 20 May 2021 James Shee , Matthias Loipersberger , Diptarka Hait , Joonho Lee , and Martin Head-Gordon COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Efficient propagation of the hierarchical equations of motion using the Tucker and hierarchical Tucker tensors The Journal of Chemical Physics 154, 194104 (2021); https://doi.org/10.1063/5.0050720 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 Spin contamination in MP2 and CC2, a surprising issue The Journal of Chemical Physics 154, 131101 (2021); https://doi.org/10.1063/5.0044362The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition Cite as: J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 Submitted: 12 February 2021 •Accepted: 26 April 2021 • Published Online: 20 May 2021 James Shee,1,a) Matthias Loipersberger,1 Diptarka Hait,1,2 Joonho Lee,3 and Martin Head-Gordon1,2,b) AFFILIATIONS 1Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, USA 2Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3Department of Chemistry, Columbia University, New York, New York 10027, USA a)Author to whom correspondence should be addressed: jshee@berkeley.edu b)Electronic mail: mhg@cchem.berkeley.edu ABSTRACT In this work, we provide a nuanced view of electron correlation in the context of transition metal complexes, reconciling computational characterization via spin and spatial symmetry breaking in single-reference methods with qualitative concepts from ligand-field and molecular orbital theories. These insights provide the tools to reliably diagnose the multi-reference character, and our analysis reveals that while strong (i.e., static) correlation can be found in linear molecules (e.g., diatomics) and weakly bound and antiferromagnetically coupled (monometal- noninnocent ligand or multi-metal) complexes, it is rarely found in the ground-states of mono-transition-metal complexes. This leads to a picture of static correlation that is no more complex for transition metals than it is, e.g., for organic biradicaloids. In contrast, the ability of organometallic species to form more complex interactions, involving both ligand-to-metal σ-donation and metal-to-ligand π-backdonation, places a larger burden on a theory’s treatment of dynamic correlation. We hypothesize that chemical bonds in which inter-electron pair correlation is non-negligible cannot be adequately described by theories using MP2 correlation energies and indeed find large errors vs experiment for carbonyl-dissociation energies from double-hybrid density functionals. A theory’s description of dynamic correlation (and to a less important extent, delocalization error), which affects relative spin-state energetics and thus spin symmetry breaking, is found to govern the efficacy of its use to diagnose static correlation. Published under license by AIP Publishing. https://doi.org/10.1063/5.0047386 I. INTRODUCTION The ab initio modeling of transition metals is a long-sought goal, and to date, the widespread use of Density Functional Theory (DFT) in chemical and biological realms has led to many important discoveries.1–3However, the robust accuracy that modern den- sity functionals4and single-reference (SR) wavefunction meth- ods such as coupled cluster with singles, doubles, and perturba- tive triples (CCSD(T)) have shown for closed-shell organic sys- tems may not necessarily carry over to transition metal com- pounds. Indeed, there are a number of aspects relevant to tran- sition metals that are either less or not at all relevant for typical organic molecules, such as relativistic effects, a balanced treatmentof solvation for redox species, and the likely possibility of con- verging to extrema other than the global minimum depending on the choice of Self-Consistent Field (SCF) algorithm and/or initial guess. Yet, as these are, in principle, well-defined problems with well-defined (albeit often not perfect) solutions,5–8it is often the case that terms such as “strong correlation” and “multi-reference (MR) character” are used as generic explanations in the face of unsystematically erroneous (and thus, apparently mysterious) predictions. Let us start with some accepted working definitions: By multi-reference (MR) character,9we mean that the wavefunction of a chemical system contains more than one determinant with significant weight. Strong correlation is not necessarily implied by J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp MR character as these determinants may only interact very weakly [e.g., unrestricted Hartree–Fock (UHF) solutions for H 2in the dis- sociation limit] but can be said to occur when the MR nature of a system leads to the breakdown or lack of convergence of SR perturbation theory (PT). Herein, we focus only on MR charac- ter, also known as “static correlation.” In the cases of stretched H 2 and biradicaloids such as O 2or benzyne isomers, static correla- tion can be encountered when HOMO–LUMO and singlet–triplet gaps approach (but do not exactly reach) zero. In these situations, the lowest energy singlet wavefunctions have substantial open-shell character and thus require two determinants for a proper qualitative description. Within DFT, Yang and co-workers have likened such a situation to a fractional spin error.10,11Variational single- determinant methods will, in these cases, exhibit spin symmetry breaking (SSB) with the expectation value of the spin operator, ⟨S2⟩, between 0 and 1 (exactly 1 for dissociated H 2and a per- fect biradical). The second aspect that leads to MR character arises from the need of a wavefunction to transform as an irreducible representation of the molecular point group. Variational optimiza- tion of a single-determinant wavefunction, here, can only yield one of, at times, multiple configurations that when superposed yield the correct symmetry, and spatial symmetry breaking (spa- tial SB) then occurs.12Simple examples can be found in HF cal- culations of stretched diatomics such as F+ 2.13While we note that independent-particle theories can also break other intrinsic symme- tries of the electronic Hamiltonian,14,15e.g., those related to complex conjugation and time-reversal, in this work, we will focus on spin and, to a lesser extent, on spatial SB and their relationship to MR character. Transition metal atoms with partially filled valence d shells can exhibit substantial MR character due to the close energetic spacing of many low-lying electronic states in contrast to the spectra of typical closed-shell organic compounds.16Transition metal diatomics have been studied extensively in recent years,16–25and recent precise experiments from Morse26have enabled meaning- ful comparisons between theory and experiment. Errors in the computed bond dissociation energies vs experiment as large as 30 kcal/mol were reported from DFT, and neither the “gold standard” CCSD(T)27nor a MR variant28showed reliable accuracy for all species.22It appears that some diatomics were “easier” to treat than others; however, common MR diagnostics did not corre- late with the accuracy of any method and were shown to point to inconsistent conclusions depending on the particular diagnostics and furthermore typically lack physically interpretive values.22,29 Many of these systems were later investigated with quantum Monte Carlo methods,30which yielded very accurate results vs experiment when consistently employing multi-determinant trial wavefunctions. This level of accuracy, but with respect to near- exact benchmark calculations, could only be attained via high orders of coupled cluster (CC) theory, which are not scalable to larger systems.24Reference 24 implies, as might be expected, that higher orders of CC theory are needed to describe the increasing number of bonds; however, there were many exceptions to this trend. Consider- ing even the simplest bonding motif (i.e., a single bond) as found in metal hydride diatomics, many glaring irregularities could be found [convergence issues or ∼23 kJ/mol inaccuracies at the CCSD(T) level], and no explanation on simple physical grounds has been provided.Metal complexes with a higher coordination number are of relatively greater interest to chemists as the coordination environ- ment resembles that of realistic transition metal catalysts or active sites in biology. 3d-containing complexes with partially filled d shells are of particular interest due to their earth-abundance31and, from the perspectives of electronic structure and reactivity, due to the competing accessibility of low spin (LS) and high spin (HS) configurations. As they are non-linear, Jahn–Teller distortions32 occur to avoid the unequal occupancy of degenerate orbitals, i.e., to break orbital degeneracies due to spatial symmetry. However, the presence of MR character has been suggested in Ref. 33 as a way of rationalizing large errors vs experiment of SR methods. Indeed, within a set of apparently simple and similar metal cations from Ref. 34, B3LYP, B97, and DLPNO-CCSD(T) performed very well for a subset, but very poorly for another. The elucidation of underlying reasons for such aberrant behavior, most notably for complexes such as Fe (NH 3)+ 4and Mn (NH 3)+ 4, is a primary motivation for the present study. We then proceed to highlight the MR character in states that are LS due to antiferromagnetic coupling, as found in mono-metal complexes with redox-noninnocent ligands and oxygen-bridged Mn(III)/Mn(IV) dimers. To provide more details on the suitability of commonly used MR diagnostics for transition metal systems, Wilson and co-workers have explored metrics derived from CC and Configuration Interac- tion (CI), e.g., T1 and D1 (Frobenius and matrix two-norm of the singles amplitude t 1in CCSD, respectively), C2 0(square of the HF configuration in the CISD wavefunction or of the leading configura- tion state function in active-space methods), and %TAE (triples con- tribution to the atomization energy). Those authors concluded that CC diagnostics are insufficient, C2 0from the Complete Active Space Self-Consistent Field (CASSCF35) method is only reliable when a full active space can be considered, and %TAE fails for weakly bound systems such as Zn 2. No linear correlation between any investigated diagnostics and the accuracy of the correlation consistent compos- ite approach36was found.29We note that even if they were reliable, diagnostics based on CCSD amplitudes and the relative contribu- tion of the (T) component would require calculations that scale as the sixth and seventh power of the system size, respectively. Another quantitative approach that can shed light on the presence of MR character essentially aims to quantify the number of unpaired elec- trons in a molecule using the eigenvectors and eigenvalues of the one-particle density matrix, known as natural orbitals (NOs) and NO occupation numbers (NOONs), respectively.37,38More recent extensions supplement a scalar quantity with real-space descriptors identifying the local regions of the molecule responsible for static correlation,39,40and in some cases, canonical Kohn–Sham orbitals are used in combination with a finite-temperature DFT formalism.41 The applicability of CASSCF (e.g., in Ref. 42) is limited because exact CI solvers are feasible only for systems with less than ∼24 active orbitals.43Approximate solvers, e.g., DMRG44or selected CI,45–48enable the use of a larger number of orbitals,49,50yet they are still sensitive to active space specification and nonetheless still scale exponentially. This active-space dependence renders CASSCF calculations, including RASSCF methods,51rather difficult to prop- erly converge. More generally, misleading conclusions can be drawn when active spaces are not sufficiently large.44References 52 and 53 illustrate this difficulty with regard to the MoFe 7S9C catalytic center of nitrogenase. Furthermore, the delicate interplay between J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp MR character and dynamic correlation can sometimes make the choice of sufficiently large active spaces quite unintuitive (e.g., active spaces with double d shells). For the ground-state (GS) of non-linear molecules (with geometries optimized without symmetry constraints), we will explore, in this work, the use of SSB in unrestricted single- determinant wavefunctions formed from orbitals from Kohn–Sham DFT orκ-regularized orbital-optimized MP2 ( κ-UOOMP2)54to detect the presence of MR character in transition metal complexes. It is known55that DFT orbitals, which are optimized in the pres- ence of mean-field electron correlations, do not break spin or spatial symmetries as easily as HF orbitals. With regard to κ-UOOMP2, the inclusion of pairwise additive electron correlation in the orbital optimization has a similar effect and has been used to diag- nose MR character in organic molecules and fullerenes56,57 and in one transition metal containing system (neutral iron porphyrin).58Moreover, given the plethora of MR diagnostics, we take a more appealing route using single-particle orbital theory, drawing on chemical concepts from ligand-field and molecular orbital theories.59–61We note that these models have recently been connected with sophisticated ab initio quantum-chemical methods and successfully applied to corroborate a variety of experimental observations.62,63We take one step further now and seek to explore how ligand-field theoretical arguments can be used to understand and predict the presence of MR character in conjunction with the quantitative use of SSB in SR theories that include some electron cor- relations. Thus, one main goal of this work is to provide, through a series of illustrative and chemically relevant examples, a link between well-established qualitative concepts and the occurrence of a MR wavefunction. Correlation is often partitioned into “static” and “dynamic” contributions. If we assume, as most do, that the former is syn- onymous with MR character, there remains the question of which type of correlation is responsible for the bulk of the errors of the commonly used SR quantum chemical methods. Admittedly, the extent to which static correlation is relevant for transition metal complexes is, at present, unclear. It is often assumed that there is a large degree of dynamic correlation, but can this quantity be connected with the physical properties of the bonding exhibited by transition metal compounds? If so, what order of PT or CC theory is needed? We endeavor to propose a more nuanced description of electron correlation than the strong vs weak distinction, which, at present, permeates the field, and we hope this will guide practition- ers in their selection of appropriate quantum-chemical methods. As an example, we focus on decidedly SR metal-carbonyl complexes and suggest that the pairwise additive correlation energy expres- sion of MP2 is inadequate for the quantitative description of dative interactions consisting of σ-donation and π-backbonding, and thus caution against the use of MP2-based double-hybrid density functionals (DHDFs) for organometallic complexes. II. THEORETICAL OVERVIEW A. Methods A variational theory is the one that minimizes E=⟨Ψ∣ˆH∣Ψ⟩ ⟨Ψ∣Ψ⟩(1)with respect to one or more parameters. In HF theory, Ψis a Slater determinant (antisymmetrized occupied orbitals) and the energy is minimized with respect to the orbital coefficients. In the CASSCF method, Ψis a linear combination of Slater determinants con- structed from all possible electron configurations within a specified set of orbitals called the active space. Both the CI and orbital coeffi- cients are treated as variational parameters. A CASSCF calculation with the full orbital space as the active space is equivalent to full CI (FCI) and is exact within the basis set employed. Approximate FCI calculations are performed with the Adaptive Sampling Con- figuration Interaction (ASCI) method,47,64a selected CI approach that iteratively improves a fixed-size CI wavefunction by select- ing the most important configurations in the Hilbert space (via a first-order PT based selection rule). The convergence of observables such as energy or molecular properties can be gauged by com- paring results from ASCI wavefunctions of increasing size or by considering the second-order perturbative correction65to the ASCI energy. κ-OOMP2 introduces a regularizer to OOMP2 (which opti- mizes orbitals to minimize E0+E(2), where E(2)is not variational) to prevent the MP2 energy from diverging and to ensure a con- tinuous restricted (R) to unrestricted (U) transition during bond breaking and is presented and discussed in detail in Refs. 54 and 56. Theκ-MP2 total energy, which is minimized with respect to orbital rotations in κ-OOMP2, is Eκ−MP2(κ)=E0−1 4∑ ijab∣⟨ij∥ab⟩∣2 Δab ij(1−e−κΔab ij)2 , (2) where i,janda,brepresent the occupied and virtual orbitals, respec- tively. Δab ij=ϵa+ϵb−ϵi−ϵj, and the antisymmetrized two-electron integrals ⟨ij∥ab⟩are defined as ⟨ij∣ab⟩−⟨ij∣ba⟩. Equation (2) reveals that Eκ-MP2(κ→0)=EHF. SSB in the HF solution can occur even for stable molecules at equilibrium bond lengths, which should be well-described by a closed-shell configuration. This situation has been termed “artificial” SSB.56In the limit of κ→∞, unregularized OOMP2 is obtained, which does not have Coulson–Fischer (CF) points66due to divergences accompanying small HOMO–LUMO gaps.67Therefore, in this limit, the theory is strongly though artifi- cially biased in favor of spin symmetry restoration. As the tendency to break spin symmetry clearly depends on the choice of regular- ization parameter, an optimal κhas been chosen in light of two criteria. The first is physically motivated, requiring CF points for single, double, and triple carbon–carbon bonds to occur at increas- ing bond distances. The second is empirically motivated, selecting κ such that errors for reference reaction datasets are minimized: For thermochemistry, this leads to κ=1.45 E−1 h.54 We note that a situation analogous to the latter is relevant to the specification of global hybrid DFT functionals with the exchange part of the correlation energy given as Ex=aEHF x+(1−a)ESL x, (3) where EHF xis the exact HF exchange (EXX) functional and ESL xis the semilocal exchange functional. The SSB behavior of a global hybrid can be modulated by the fraction of EXX employed, i.e., the parameter a. One advantage of κ-UOOMP2 vs DFT is that the J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp former is free of self-interaction error, in which an electron can spuriously interact with itself, and its many-body generalization known as delocalization error (DE).68,69It is well-known that DE tends to (artificially) reduce the extent of SSB (and spatial SB) with pure and global hybrid (i.e., those with a modest amount of EXX, e.g., B3LYP) functionals in cases such as stretched hydrogen fluo- ride, and is also relevant to the description of ligand-to-metal dative bonding. For some of the smaller molecules in this study, we will utilize a variant of CC with double excitations that variationally optimizes a set of orbitals rather than including a singles term in the cluster operator (OOCCD).70 B. Calculation of S2 In 1955, Löwdin derived71that ⟨S2⟩=−N(N−4) 4+∫Γ(r1s1,r2s2∣r1s2,r2s1)dx1dx2, (4) where the two-particle density matrix, Γ, is normalized toN(N−1) 2. The expressions of the two-particle density matrix are known for ROHF- and UHF-based theories along with simple density func- tional theories.72,73The UHF expression is of particular relevance to the present work and can be written as ⟨S2⟩UHF=S(S+1)+Nβ−occ ∑ i¯jS2 i¯j, (5) where S=Nα−Nβ 2and Si¯jis theαβoverlap integral between the i-th spin-up orbital and the j-th spin-down orbital. We emphasize that the ⟨S2⟩value reported for DFT calculations is exact only for the fictitious non-interacting Kohn–Sham system, as the occupied Kohn–Sham orbitals are used instead of those from UHF to compute Eq. (5). Forκ-UOOMP2 and UOOCCD, unless otherwise mentioned, Eq. (5) is used with the orbitals resulting from the minimization procedure. A rigorous expression for ⟨S2⟩UOOCCD can be found in Ref. 74. Regarding κ-UOOMP2, with the first-order MP wavefunc- tionψ1in hand, the expectation value of the spin operator can be rigorously obtained as ⟨S2⟩MP2=⟨ψ0∣S2∣ψ0⟩+2⟨ψ0∣S2∣ψ1⟩, (6) assuming real amplitudes and orbitals. NOONs can be obtained by diagonalizing the one-particle density matrix formed from ψ1. As can be seen from Eq. (2), the amplitudes of the doubly excited states in theκ-OOMP2 wavefunction are, by construction, encouraged to be small (i.e., when Δab ijis small, so is the regularizing term in parentheses and thus the regularized amplitude). Thus, we expect that in most cases, ⟨S2⟩will be similar when computed either by Eq. (5) from κ-UOOMP2 orbitals or by Eq. (6) with the perturbed wavefunction. Similar considerations apply to the NOONs. C. Computational details All calculations were performed with Q-Chem 5.2,75except for CASSCF calculations and geometry optimizations for Fe(II) X 6and the metal carbonyls not in Ref. 33, which were performed with Orca.76These geometry optimizations utilized the DKH Hamiltonian and DK basis sets, for consistency with previous work.33CASSCF calculations were initialized from DFT orbitals (in most cases, obtained with the B3LYP functional). For the ASCI calculations, we use approximate NOs47,64or MCSCF-like orbital optimization within the active space,77which leads to more compact CI wavefunctions and thus lower variational energies for a given wavefunction size. Further details about ASCI can be found in Refs. 47, 64, and 65. ASCI NOONs were computed from the variational CI wavefunction alone without any perturbative corrections. For ASCI calculations of the hydrogen fluoride molecule and transition metal hydrides, the second-order PT correction to the energy ( EPT2) is smaller than 10−3Ha (i.e., 0.6 kcal/mol) in magnitude. This high level of convergence was not possible for para -benzyne, and so we used the full-valence active space of 28 orbitals. The NOONs are converged by 4 ×106ASCI determinants. ASCI calculations were also performed for Fe(H 2O)2+ 6and Fe(CO )2+ 6, with convergence shown in the supplementary material. To obtain ⟨S2⟩values and NOONs, the def2-SV(P) basis set is used78(unless mentioned otherwise). UOOCCD calculations use the frozen-core approximation. While we are well-aware that for ground-state properties, the quantitative accuracy of energetic quantities vs experimental measurements requires much larger basis sets, the def2-SV(P) basis set is found to produce the qualitative descriptions required, and at a much-reduced computational cost that would be expected of a practical diagnostic tool. The GDM algorithm79is used as the default SCF solver for HF, DFT,κ-UOOMP2, and UOOCCD calculations. We confirmed the stability, with respect to orbital rotations, of HF and DFT solutions. For the κ-UOOMP2 calculations, the resolution of the identity approximation was employed with the corresponding auxiliary basis set. The integration grid for DFT calculations consists of 99 radial spheres each with 590 points.80 We employed an energy decomposition analysis (EDA) based on absolutely localized molecular orbitals to decompose the bind- ing energies. The original scheme decomposes the interaction energy into frozen, polarization, and charge transfer contributions.81,82 In order to reveal more insight into the nature of the bidirec- tional charge transfer of some metal–ligand bonds, we augmented the analysis with a further decomposition of the charge transfer energy using the newly developed variational forward–backward decomposition.83 III. RESULTS A. Stretched H2,para -benzyne, and stretched hydrogen fluoride In this section, we investigate three simple systems that can be made to exhibit static correlation and compare the frontier NOONs computed from various approximate unrestricted methods with exact or near-exact reference values from ASCI calculations. Analytical expressions for H 284show that the lowest unoccupied natural orbital (LUNO) occupation number for a single Slater determinant is related to ⟨S2⟩by nLUNO =1−√ 1−⟨S2⟩. (7) J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp While this relation is exact only for H 2, the ability of a SR method to produce, via spin polarization, (fractional) NOONs in agreement with exact values indicates that the presence of SSB reliably reflects the genuine MR character. In Fig. 1, we plot nLUNO vs bond length for three DFT func- tionals with varying amounts of EXX (0%, 20%, and 50% for BLYP, B3LYP, B5050LYP, respectively), HF, and κ-UOOMP2 vs exact results (obtained via CISD). The exact results show the fractional occupation of the antibonding orbital even at short bond lengths, whereas the other methods have zero occupation of the LUNO until beyond the CF point. The CF point is pushed to longer bond lengths going from HF, κ-UOOMP2, to more pure DFT functionals. While all methods provide a satisfactory description of H 2in the dissociation limit, we note that in the region beyond ∼1.5 Å, nLUNO as approximated using DFT orbitals appears to be closer to the exact value than when (rigorously) derived from HF and κ-UOOMP2 theories. As a second test, we consider para -benzyne, a prototypi- cal biradicaloid.85,86Using the cc-pVDZ basis and the geome- try from Ref. 87, we compute reference highest occupied natural orbital (HONO) and LUNO populations with full valence CASSCF (28 electrons in 28 orbitals, using the ASCI solver77), which are compared with values from other methods in Fig. 2. As found above for H 2, adding exact HF exchange systematically shifts the DFT predictions toward the HF result. We note that both B3LYP and κ-UOOMP2 produce NOONs in good agreement with the ASCI estimates. While the above two examples would suggest that B3LYP is an excellent choice to describe SSB and fractional NOONs in these MR situations, its susceptibility to DE warrants caution in polar systems such as stretched hydrogen fluoride. In HF theory, the one-particle self-interaction error is canceled exactly—this is also the case in MP2 theories. Figure 3 shows that the LUNO occupation at the B3LYP FIG. 1. LUNO occupation ( nLUNO) vs internuclear distance for stretched H 2. The exact andκ-UOOMP2 results were obtained with the aug-cc-pVQZ basis, while the DFT results were obtained with the aug-pc-4 basis. FIG. 2. Comparison of LUNO and HONO occupation numbers, computed with various methods, for para-benzyne.κ-UOOMP2-PTwfn represents the values calculated from Eq. (6). level at long bond lengths is significantly lower than the ASCI value because DE leads to contamination of the unrestricted solution with closed-shell, ionic contributions. Adding a higher percentage of EXX into the functional form provides the nonlocal orbital-dependent exchange necessary to describe the derivative discontinuity, and DE can thus be reduced by global hybrids with a larger amount of EXX (e.g., B5050LYP) or range-separated hybrid functionals (e.g., CAM-B3LYP88).89,90B5050LYP provides a substantial improvement over B3LYP, and CAM-B3LYP further improves the predicted NOONs. FIG. 3. LUNO occupation ( nLUNO), as computed with various methods compared to the ASCI benchmark, vs internuclear distance for stretched hydrogen fluoride. Theκ-UOOMP2 results were obtained with the aug-cc-pVQZ basis, and the DFT results were obtained with the aug-pc-4 basis. We show ASCI values with the cc-pVDZ basis, where EPT2could be fully converged ( <10−3Ha). In the cc-pVTZ basis, we checked that nASCI LUNOdeviated by at most 0.05 from the converged value in the cc-pVDZ basis. J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the context of transition metal complexes, DE can have a dramatic effect on the predicted covalency or dative nature of metal–ligand interactions. It has been shown that calculated spin-densities and derived electron-nuclear hyperfine coupling constants (which can be measured by EPR spectroscopy) are sensitive to DE and can lead to extreme variations in predicted paramagnetic nuclear magnetic resonance shifts in the range ofO(1000) ppm.91,92Fortunately, the presence of DE can be diagnosed in straightforward ways, implied, e.g., by deviations from straight-line behavior in plots of energy vs fractional occupation number69,92(see Ref. 93 for the analysis of the fluoride anion relevant to dissociated hydrogen-fluoride). In such cases, methods such asκ-UOOMP2, B5050LYP, and range-separated hybrid functionals are to be preferred over pure or typical global hybrids such as BLYP or B3LYP, respectively. Finally, regarding Fig. 3, we note that using the one-particle density matrix from the κ-UOOMP2 wavefunction to compute nLUNO tracked the value obtained from using the κ-UOOMP2 orbitals in Eq. (5) nearly indistinguishably, except at bond lengths before the CF point. In this region, nLUNO is necessarily zero for a single-determinant wavefunction and can be non-zero due to the doubly excited configurations in the MP1 wavefunction. B. Metal hydride diatomics: Spin and spatial symmetry breaking can imply multireference ground-states In this section, we investigate the 3d transition metal hydrides and will glean insights into the connection between static correlation and symmetry breaking. These systems are small enough such that near-exact wavefunctions can be obtained, and we use ASCI to converge FCI-quality wavefunctions (with a Ne frozen-core for the metal atom). As mentioned in the Introduction, 3d metal diatomics can possess surprisingly complicated electronic structures, resulting in a host of literature showing that SR methods (and some MR ones) have pronounced difficulty in predicting experimental thermochem- istry. As will be discussed below, metal hydrides are linear molecules, point group C ∞v, for which the Jahn–Teller theorem does notapply. Therefore, MR character can arise in the GS due to both spatial and spin symmetry requirements. Figure 4 shows a qualitative schematic of an expected molecu- lar orbital (MO) diagram for metal hydride compounds. The shell of formally non-bonding orbitals (consisting of one σ, twoπ, and twoδorbitals) are, for now, treated as a single degenerate shell on the grounds of very small energetic splittings between the five orbitals. For our analysis of SSB, this will be adequate; however, a closer look is required for our analysis of spatial SB in TiH and CoH. Table I shows the ⟨S2⟩values from UHF, UDFT, κ-UOOMP2, and UOOCCD, compared with the exact spin quantum number. Comparing with the MO diagrams in Fig. 4, the first thing to notice is that a number of diatomics that do not show SSB are HS states (VH, CrH, and MnH). The HS states are known to be of SR nature.94 SSB can only occur when there are paired valence electrons, as in the Fe–Zn hydrides. In addition, a higher spin state must be suf- ficiently close in energy to mix into the unrestricted SR wavefunc- tion of the LS state. In the single-particle/MO picture, this means that the LUMO in the dominant electronic configuration must be FIG. 4. Molecular orbital diagrams for metal hydrides. The orbitals in the red block define the analog of 10Dq for diatomics. low-lying enough to be occupied (via spin-flip excitation) in the HS state. As will be discussed further in Sec. III C, tetrahedral and octa- hedral metal complexes exhibit a splitting between t 2gand e glevels known as 10Dq. Below, we discuss the analogs of this for linear 3d metal hydrides. As shown in the red box of Fig. 4, the analog of 10Dq, which can modulate SSB for FeH, CoH, NiH, and CuH, is the gap between the non-bonding 4s and 3d metal orbitals with the anti- bondingσ∗. The lowest excitation energies, as (roughly) estimated via time-dependent density functional theory (TDDFT) within the Tamm–Dancoff approximation (TDA) with the B3LYP functional and def2-SV(P) basis, for CuH and ZnH are 2.2 and 4.4 eV, respec- tively, suggesting that for these diatomics (though especially the lat- ter), this gap is too large for HS states to compete. In contrast, the spin-contamination found in FeH, CoH, and NiH with all dynami- cally correlated SR theories considered suggests that, for these three species, this gap is small enough such that SSB can occur. Going from CuH to FeH, the fractional population of the σ∗NO, as given by ASCI calculations and shown in Table II, monotonically increases from 0.06 to 0.35. Correspondingly, the gap between the non- bonding manifold and the σ∗orbital can be expected to decrease in this direction due to (i) the decrease in metal atomic electroneg- ativity, which, in the MO picture, increases the energy of the metal AOs and, as a result, the metal non-bonding MOs relative to the σ∗ orbital (which has partial ligand character), and (ii) the increase in metal atomic radius, which, according to ligand-field theory, should decrease the splitting of bonding and antibonding orbitals upon J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I. Diatomic species, term symbols, and exact and calculated ⟨S2⟩values. S2 exact S2 UHF S2 UB3LYP S2 UCAM−B3LYP S2 UB5050LYP S2 κUOOMP2 S2 UOOCCD ScH(1Σ+) 0 1.00 0.60 0.58 0.54 1.00 1.00 TiH(4Φ) 3.75 3.75 3.76 3.76 3.75 3.75 3.75 VH(5Δ) 6.00 6.01 6.01 6.01 6.01 6.01 6.01 CrH(6Σ+) 8.75 8.87 8.79 8.79 8.80 8.82 8.79 MnH(7Σ+) 12.00 12.00 12.00 12.00 12.00 12.00 12.00 FeH(4Δ) 3.75 4.72 4.05 4.10 4.39 4.44 4.48 CoH(3Φ) 2.00 2.95 2.16 2.19 2.48 2.23 2.43 NiH(2Δ) 0.75 1.68 0.81 0.82 0.98 0.79 0.92 CuH(1Σ+) 0 0 0 0 0 0 0 ZnH(2Σ+) 0.75 0.76 0.75 0.76 0.76 0.76 0.76 complexation with the hydride. As a result, the HS state formed by populating σ∗becomes increasingly more energetically competitive with the LS state, resulting in proportional SSB. For ScH, which is the only diatomic in the left-half of the row with significant post-HF SSB, the analog of the 10Dq parameter corresponds to the gap between the 4s and 3d MOs (0.4 eV with TDDFT/TDA). In other words, the intruding HS state responsi- ble for spin-contamination is formed when the spin-down electron in the doubly occupied 4s orbital undergoes a spin-flip excitation into the energetically proximate non-bonding 3d shell. For ScH at the UOOCCD level of theory, we note that computing ⟨S2⟩via Eq. (5) with the optimized orbitals led to a value of 1, whereas using the full UOOCCD wavefunction yielded ⟨S2⟩=1.75. The RooCCD energy is found to be lower than that of UOOCCD (but slightly higher than the ASCI energy), which reflects the fact that CC wave- functions are highly sensitive to (and can be adversely affected by) spin-contamination in the reference state. In light of the four half-occupied ASCI-computed NOONs shown in Table III, the lack of SSB in TiH is rather provoca- tive. Indeed, due to the quartet multiplicity, there are no paired electrons in the 4s-3d MO manifold, which would imply no SSB, TABLE II. Occupation number corresponding to the σ∗natural orbital from ASCI/def2-SV(P) calculations along with other MR diagnostics based on CCSD(T)/cc- pVTZ-DK from Ref. 29. σ∗NOONaT1 D1∣tmax 1∣ % TAE ScH(1Σ+) 0.02 0.04 0.05 1.3 TiH(4Φ) 0.06 0.12 0.13 0.8 VH(5Δ) 0.09 0.21 0.22 0.2 CrH(6Σ+) 0.17 0.43 0.48 1.6 MnH(7Σ+) 0.02 0.05 0.05 −1.6 FeH(4Δ) 0.35 0.10 0.29 0.47 10.6 CoH(3Φ) 0.22 0.07 0.21 0.22 12.4 NiH(2Δ) 0.14 0.06 0.17 0.17 10.6 CuH(1Σ+) 0.06 0.04 0.13 0.09 3.8 ZnH(2Σ+) 0.03 0.08 0.11 −1.7 aThis work.which until now has implied a SR wavefunction. TiH’s four NOONs of 0.5 can be explained in the context of spatial , rather than spin, SB as follows. The4Φterm symbol denotes a doubly degen- erate ( E) irreducible representation (4Φxand4Φy) where, e.g., 4Φx=1√ 2(∣σδ+πx⟩−∣σδ−πy⟩).16The corresponding electron con- figurations are illustrated in Fig. 5, where the wavefunction is a linear combination of two configurations varying in their occupation of the πandδorbitals. The term symbol of the wavefunction for CoH leads to an analogous situation but for two holes rather than two electrons, e.g.,3Φx=1√ 2(∣σ2π2 yδ2 −πxδ+⟩−∣σ2π2 xδ2 +πyδ−⟩),16also illustrated in Fig. 5. Thus, for TiH and CoH, the four 0.5 and 1.5 NOONs, respec- tively, imply the MR character due to spatial symmetry. This can occur in the absence of SSB (TiH) or in addition to it (CoH). TABLE III. Frontier NOONs from ASCI/def2-SV(P) calculations for TiH and CoH. The four NOONs in the central box correspond to πandδorbitals shown in Fig. 5. TiH 1.95 0.99 0.51 0.51 0.50 0.50 0.04 CoH 1.93 1.78 1.50 1.50 1.47 1.47 0.22 FIG. 5. Spatially symmetric4Φxand3Φxwavefunction schematics for TiH and CoH, respectively. J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6. High-spin (HS) and low-spin (LS) electron configurations corresponding to d6Fe(II) X 6complexes. C. Spin symmetry breaking in metal complexes modulated by ligand position in the spectrochemical series For the rest of this paper, we turn to 3d metal complexes with higher coordination numbers, for which Jahn–Teller distortions readily lift any degenerate electron configurations due to spatial symmetry. In this section, we introduce the correspondence between SSB and the magnitude of the ligand-field paramater, 10Dq, which denotes the splitting between t 2gand e gorbitals for tetrahedral (T d) and octahedral (O h) complexes. Whether or not this can be inter- preted as a marker for MR character will be postponed to Sec. III D. Our discussion will center around O hcomplexes, and in particular Fe(II) L 6, which implies a d6configuration. In the O hfield, which yields a three below two d orbital ligand-field splitting, the LS state is a singlet with each of the three t 2gorbitals doubly occupied, whereas the HS state is a quintet formed by unpairing and spin-flipping two electrons from the t 2gto the e gorbitals. These states along with the definition of 10Dq are shown in Fig. 6. A small (large) 10Dq value generally results in the GS being HS (LS), respectively, though a complete analysis involves examining the delicate balance between (i) the energetic cost to promote from t 2gto e g(10Dq), (ii) the ener- getic cost to pair two opposite-spin electrons in the same t 2gorbital, and (iii) the stabilizing exchange interaction between the same-spin electrons. It is well-known that the 10Dq parameter can be modulated by a ligand’s position on the spectrochemical series, which reflects whether the metal (M)–ligand (L) interaction is characterized by L-to-Mσdonation only or, in addition, L-to-M πdonation or M-to-Lπbackbonding. MO theory provides an intuitive model,which corroborates the experimentally determined trend that 10Dq gets smaller going from π-accepting to σ-donating-only to π-donating ligands.95 Table IV shows the calculated ⟨S2⟩values at the UHF, UB3LYP, UCAM-B3LYP, UB5050LYP, and κ-UOOMP2 levels of theory for O hFe(II) complexes in the LS state representing a range of ligand-field strengths. The CO ligand is perhaps the strongest-field π-acceptor in the spectrochemical series, and there is no SSB at any level of theory (not even UHF). We attribute this to 10Dq being sufficiently large such that the HS state is energetically inaccessi- ble and therefore unable to mix into the LS wavefunction. This implies that the LS state is SR with very strong-field/ π-accepting ligands (indeed, as will be argued in Sec. III F, the inaccuracy of MP2-based methods for organometallic thermochemistry is due to other reasons). As the ligand-field strength is attenuated (NH 3and H2O areσ-donation only ligands and the halides are π-donor lig- ands), the magnitude of 10Dq decreases, and the increasing accessi- bility of the HS configuration manifests as deviations from exact ⟨S2⟩ values in the LS state. The trend of increasing SSB going toward weak-field ligands along the spectrochemical series is present in all methods investi- gated. In fact, all DFT and κ-UOOMP2 methods yield qualitatively similar values of ⟨S2⟩[with the exception of the artificial SSB from the B5050LYP functional for Fe(NH 3)2+ 6, which, as might be expected, is nearly half the value found from UHF]. We have used a polarizable continuum model ( ϵ=78.4) for the DFT calculations of all anionic species, as motivated by the interesting case of the com- plex with fluoride (F−) ligands: In the gas phase, the ⟨S2⟩value from B3LYP (1.74) exceeds that from UHF (1.49). The fluorine anion is known to have a positive HOMO eigenvalue when employing func- tionals such as B3LYP in all but exceedingly large basis sets,97and the curvature in a plot of the energy as a function of fractional occupation number is a hallmark of DE, which further encourages long tails in the radial charge density.93,98Indeed, for FeF4− 6in the gas phase with B3LYP, 21 of the 42 occupied Kohn–Sham orbital eigenvalues were greater than zero. When improving (slightly) the description of a continuum orbital via the def2-SVPD basis set, ⟨S2⟩ is further increased to 1.94, which reflects the expected narrowing of the singlet–triplet gap as continuum orbitals become more involved. Using the dielectric characteristic of water solvent, only the HOMO TABLE IV. Calculated ⟨S2⟩values of the LS state of Fe(II) X2+/4− 6. All DFT calculations of anions (i.e., complexes with halide ligands) use the C-PCM polarizable continuum model, with a dielectric constant of 78.4 and a van der Waals radius of 2 ˚A for Fe. 10Dq estimates are obtained from Ref. 96 via the difference in CASPT2 energies of the5T(t4 2ge∗2 g)and5E(t3 2ge∗3 g) states. S2 exact S2 UHF S2 UB3LYP S2 UCAM−B3LYP S2 UB5050LYP S2 κUOOMP2 10Dq (eV) LS Fe(CO)2+ 6 0 0 0 0 0 0 4.90 Fe(NCH )2+ 6 0 0.44 0 0 0 0 3.21 Fe(NH 3)2+ 6 0 1.01 0 0.05 0.45 0 2.69 Fe(H 2OO)2+ 6 0 1.36 0.86 0.90 1.08 0.68 1.64 FeF4− 6 0 1.49 1.03 1.07 1.23 1.04 FeCl4− 6 0 1.71 1.40 1.42 1.53 1.41 FeBr4− 6 0 1.74 1.43 1.46 1.58 1.41 J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp eigenvalue remained positive with a value of 0.01, and the resulting ⟨S2⟩decreased from 1.74 to 1.03, now below the UHF value and in agreement with functionals with more EXX (and thus less DE) and κ-UOOMP2, which is free of DE. We note that increasing ϵand the flexibility of the basis set can make all occupied B3LYP eigenvalues negative, but we did not find any large effect on the SSB behavior (e.g., a HOMO eigenvalue of −0.11 is obtained with ϵ=1000 and the def2-SVPD basis, and ⟨S2⟩=1.20). D. Does spin symmetry breaking imply static correlation or variational collapse? Many of these O hFe(II) compounds are, in fact, prototypical and well-studied spin-crossover complexes. Under appropriate external conditions, e.g., pressure or protein environment, spin- crossover complexes can exhibit transitions between LS and HS states, enabling the precise control of interesting magnetic phenomena.99–102Four of the Fe(II) complexes—[Fe(H 2O)6]2+, [Fe(NH 3)6]2+, [Fe(NCH )6]2+, and [Fe(CO )6]2+—have received significant attention from theoreticians utilizing an array of sophisticated ab initio methods (in the absence of gas-phase experimental measurements).103Spin gaps pose a difficult prob- lem for DFT methods since a range of splittings can be obtained depending on the functional employed (and in particular, for global hybrids, the amount of EXX incorporated).103,104 A notable study of these four Fe(II) complexes compared the results of a host of DFT functionals and wavefunction methods with those from diffusion Monte Carlo (DMC) calculations (employing pseudopotentials and within the fixed-node approximation).105It is claimed therein that the CO species is markedly multi-configurational. In another study, all-electron DMC calculations have been performed. For the complex with the CO ligand, the calculated spin gaps with single-determinant and multi-determinant trial wavefunctions agreed to within the 0.005 Ha error bar, suggesting that this complex is not strongly correlated.104Neese and co-workers did not consider the CO species but carried out a detailed investigation of the remaining three Fe(II) complexes with DLPNO-CCSD(T) methods employing large basis sets.106 Now, we will add to the data in Table IV the finding that all HS states of the four Fe(II) complexes presently under con- sideration did not exhibit any SSB. This observation, found also for the diatomic species mentioned above, is quite general and provides the point of departure for the development of spin-flip approaches to, e.g., TDDFT and wavefunction methods (CI, CC, and more). This can be understood in the context of the quantum theory of angular momentum, where a high spin quantum num- ber can have numerous, e.g., msstates, at least one of which can typically be well-described by a single determinant. For the LS singlet species, going from CO, NCH, NH 3, to H 2O (i.e., going from strong- field/π-accepting toward weak-field/ σ-donation-only), we find that the deviation of ⟨S2⟩UHF increases consistently. UB3LYP completely restores the SSB found in UHF, except in the case of Fe(H 2O)2+ 6. κ-UOOMP2 yields similar conclusions. In disagreement with the aforementioned claims from other groups in the literature, our results strongly suggest that the CO species is notMR, given that no SSB occurs even at the UHF level. As detailed in the supplementary material, the NOONs from an 18e32o ASCI calculation are all eithergreater than 1.93 or less than 0.06, clearly indicating a closed-shell, SR singlet state. On the other hand, all dynamically correlated meth- ods here suggest that the LS H 2O species has HS character mixed in. This SSB behavior is consistent with calculated 10Dq values from Ref. 96, reproduced in Table IV. The adiabatic LS/HS gaps, i.e., derived from separately opti- mized LS and HS geometries, have been calculated in Refs. 106 and 96. Both CASPT2 and CCSD(T) methods predict that only Fe(CO)2+ 6has a LS GS (due to the large 10Dq of 4.9 eV), while the GSs of the three other molecules with weaker-field ligands are HS. Evidently, the NCH ligand reduces 10Dq such that the cost of promoting two electrons to the e gorbitals is more than compen- sated by the stabilization provided by exchange (with four spin-up unpaired electrons) and by unpairing two pairs that had been in the t2gmanifold. NH 3and H 2O ligands continue the trend of reducing 10Dq, and of the three factors mentioned previously, the HS state drops lower in energy due to the increasing ease of promoting from t2gto e g. Yet while the adiabatic energy difference is relevant for deter- mining the proper GS multiplicity when calculating thermochemical properties, it is the vertical gap—i.e., the difference between the LS state and the HS state in the LS geometry—which is directly related to the SSB behavior: At a fixed geometry, when the HS state is the GS, SSB in the LS state (assuming it is a saddle point in the energy sur- face in orbital space) is to be expected from unrestricted variational methods, wherein constraining a molecule’s multiplicity is done by constraining the projection Sz(equal to Nα-Nβ) rather than the value of S2. As a consequence, such methods will include suitable HS contributions into the single-determinant reference in order to minimize its energy. However, the presence of HS contributions at the single determinant level for such species does not necessarily imply that the LS state has significant MR character. Indeed, with a large-enough LS–HS gap, it is quite possible that the lowest LS state is dominated by a single determinant, while being above the HS state in energy. UHF/UKS calculations for such LS states can still exhibit SSB due to inclusion of HS character, despite the actual state being fairly SR (as indicated by NOONs). A classic main-group example of this is CH 2, where the triplet state is the GS, while the lowest singlet state is predominantly closed-shell (HONO occupa- tion of 1.89 and LUNO occupation of 0.09). UHF/UKS calculations onSz=0 CH 2would generate SSB, as it is energetically favorable to contaminate the closed-shell singlet with the lower energy triplet. Several species in Table V provide additional examples, as described later. TABLE V. Vertical energy difference (eV) between the LS state and the HS state in LS geometry. A negative value means that the LS state is more stable. UHF, UB3LYP, andκ-UOOMP2 predictions are compared with reference values from CCSD(T), all in the def2-SV(P) basis set. CCSD(T)aUHF UB3LYP κ-UOOMP2 Fe(CO)2+ 6 −3.73−1.15−4.48 −6.79 Fe(NCH )2+ 6−0.26 1.34 −1.20 −0.83 Fe(NH 3)2+ 6 0.87 1.98 −0.11 0.70 Fe(H 2O)2+ 6 1.83 2.41 0.78 1.78 aR for singlet and U for quintet (the latter, in all cases, is essentially spin-pure). J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Table V reveals that, at the LS geometry, UHF puts the HS state energetically below the LS state in all cases except for the carbonyl complex, which may explain why only the carbonyl complex was spin-pure at this level of theory (Table IV). UB3LYP andκ-UOOMP2 predictions of the relative spin-state ordering for the NCH complex are in agreement with the CCSD(T) reference, which suggests that UHF has overstabilized the HS state (a con- sequence of HF’s neglect of all correlation except for same-spin exchange stabilization) such that its predicted state ordering is incor- rect. Going toward weak-field ligands along the spectrochemical series, UB3LYP predicts a LS-below-HS ordering for the CO, NCH, and NH 3complexes, but the ordering switches for Fe(H 2O)2+ 6. This can explain the onset of SSB in the UB3LYP level of theory at this molecule. Interestingly, the spin-state ordering predicted by κ-UOOMP2 switches from a LS to HS GS at the NH 3complex; however there is no SSB in Table IV; furthermore, we note that the vertical spin gap at this level of theory deviates by 3 eV from the CCSD(T) benchmark for the CO species—a clue that will become relevant in our later discussion of the possible inappropriateness of MP2 in describing interactions such as metal-carbonyl bonds. On the whole, it appears that with a HS GS, SSB in the LS excited state need not imply MR character, but rather what we will refer to as “variational collapse.” Indeed, this is the reason that some sort of spin-projection57,107–109is mandatory in such cases. Consider Fe(H 2O)2+ 6, for which SSB persists not only at the UHF level but at all DFT and κ-UOOMP2 theories. A CASSCF cal- culation with six electrons in five metal d orbitals yields NOONs of 1.960, 1.954, 1.954, 0.066, and 0.066. As shown in the supplementary material, both the 12e14o active space selected following the protocol from Ref. 110 and an 18e20o calculation with the ASCI solver yielded NOONs either greater than 1.95 or less than 0.06. The implied SR character suggests that, in this case, SSB simply reflects the HS-below-LS relative energetics rather than the pres- ence of static correlation. For FeX4− 6, with X =F, Cl, and Br pro- ceeding toward weaker-field/ π-donating ligands, the expected 10Dq decrease is a small effect, with LUNO populations resulting from minimal (e.g., 6e5o) CASSCF calculations of 0.076, 0.100, and 0.105, respectively. For FeF4− 6, we verified that using a larger active space of 6e15o to include a second d shell yielded a similar LUNO of 0.070 (vs 0.076 from 6e5o). The LS state of Fe(II)Br4− 6is less MR than NiH, CoH, and FeH (with LUNO occupations of 0.14, 0.22, and 0.35, respectively), and its LUNO value is strikingly similar to that of the lowest singlet state of CH 2(0.09), which is also predominately closed shell. This analysis suggests that neither the CO nor H 2O LS Fe(II) complexes are MR, in agreement with pre- viously reported D1 diagnostic values of 0.14 and 0.06, which are below the 0.15 threshold suggested by Jiang and co-workers for tran- sition metals.29Rather, the SSB observed from theories that include dynamic correlation is a manifestation of variational collapse. We thus reiterate that for excited states, SSB should be used together with NOONs from a MR theory in order to probe for the presence of MR character. To summarize, this set of Fe(II) complexes suggests that strong- field ligands such as CO (and, e.g., CN) yield single-configurational, LS GSs (which is also the case for 4d and 5d metal complexes, for the same reason—i.e., due to the large 10Dq that results from the strong M–L interaction). Weaker-field ligands on the spectrochem- ical series (e.g., NH 3and H 2O) favor HS GSs due to the attenuationof 10Dq and the stabilizing exchange interactions among same- spin electrons (indeed, as the 10Dq parameter of T dcompounds is roughly half of that of O hcompounds, the former, in general, have HS GSs). Practically speaking, for the calculation of thermochemical properties (for which only GSs are relevant) of these types of coor- dination complexes, SSB will therefore be the exception rather than the rule, and encountering MR character need not be much of a con- cern (though appropriate SR methods must still be chosen with care, vide infra ). E. Identifying multi-reference character from spin symmetry breaking We now seek to uncover chemical themes or circumstances, in addition to the diatomics analyzed above, in which static correlation can arise in the GS of transition metal complexes. Such conditions, while admittedly quite rare, are found in LS metal compounds, which have very weak M–L bonding (encountered, e.g., in the gas phase in particular for σ-donating only ligands with low metal oxidation states), or antiferromagnetically coupled spins occupying either separate metal centers or a metal center and a low-lying π∗ orbital of a redox-noninnocent ligand. 1. Very weak metal–ligand bonding In Ref. 33, the ligand dissociation energies of 34 metal com- plexes, formed via ligand coordination of neutral or cationic 3d metals in the gas phase, were investigated. We have computed ⟨S2⟩ with respect to UHF, UB3LYP, and κ-UOOMP2 orbitals for all GSs involved. Table VI shows the cases for which SSB at the UHF level was restored in both UB3LYP and κ-UOOMP2 cases; Table VII shows those cases for which SSB persisted at the UB3LYP and κ-UOOMP2 levels. TABLE VI. Cases in which spin symmetry breaking at the UHF level is restored with UB3LYP and κ-UOOMP2. S2 exact S2 UHF S2 UB3LYP S2 κUOOMP2 Co(NH 3)+ 3 2 2.78 2.03 2.03 Cr(CO)5(H2) 0 0.70 0 0 Cr(CO)5 0 0.74 0 0 V(CO)+ 5 2 2.35 2.03 2.02 V(CO)+ 6 2 2.32 2.02 2.02 TABLE VII. Cases in which spin symmetry breaking at the UHF level persists in UB3LYP and κ-UOOMP2. S2 exact S2 UHF S2 UB3LYP S2 κUOOMP2 Fe(CH 2O)+ 3 3.75 4.74 3.96 3.86 Fe(CH 2O)+ 4 3.75 4.75 4.11 3.95 Fe(H 2O)+ 3 3.75 4.55 3.89 3.95 Fe(H 2O)+ 4 3.75 4.59 3.86 3.89 Fe(NH 3)+ 3 3.75 4.54 3.86 3.90 Fe(NH 3)+ 4 3.75 3.78 3.94 3.92 Mn(NH 3)+ 3 6 6.86 6.30 6.62 Mn(NH 3)+ 4 6 7.00 6.71 7.00 J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We observe that the molecules in the former group, for which UHF SSB is restored with UB3LYP and κ-UOOMP2, consist of metals with CO ligands, except in one case (which will be discussed after). CO ligands can participate in both σ-donation and π-backbonding. The latter substantially strengthens the metal–ligand bond, increasing 10Dq which decreases the chances of finding a truly strongly correlated state; hence, the spin symmetry restoration when dynamic correlation is included via DFT or κ-UOOMP2. The molecules in the second group (Table VII), for which SSB persists with UB3LYP and κ-UOOMP2, all have weak-field ligands, which predominately participate in σ-donation, i.e., without πback- donation. This is obvious for H 2O and NH 3ligands. The CH 2O ligand has a double bond between C and O atoms and coordinates at the O end, with sp2hybridization that is, in principle, capable of πinteractions with the metal. We performed an EDA calculation, which shows that while the formaldehyde ligand can π-accept, the backbonding charge-transfer component is less than half that found for the CO complex (see Table S2). The weaker π-backbonding to CH 2O vs carbonyl appears to create a 10Dq comparable to σ-only ligands, which enables SSB. We also note that this second group of complexes contains only Fe and Mn ions. Mn+and Fe+are unique, in that they have 4s13dn−1 electronic configurations. When a Lewis base donates two electrons, one must go into a bonding orbital, while the other must go into an antibonding orbital, which lowers the bond order and weakens the covalent bond more so than if the metal atom was 3dn. As implied by bond strength trends with H 2and CO ligands, Fe+and Mn+require 24 and 113 kJ/mol, respectively, to promote the 4s electron to attain the 3dnconfiguration.34This trend is consistent with our finding that the SSB is more severe in the Mn vs the Fe ammonia species; indeed, the Mn molecule is the only species, which has SSB with UB3LYP orbitals in excess of 10%. A parallel can be drawn between the chemical situation that occurs in these very weakly bound metal complexes and that which occurs while stretching bonds. Taking H 2as an example, the gap between the bonding and antibonding MOs decreases as the bond length is stretched; after a certain distance, the near-degeneracy of the orbitals (or equivalently, of the many-body singlet and triplet states) results in an unpairing of the closed-shell singlet state into a two-configurational, biradicaloid singlet wavefunction. When the wavefunction is constrained to a single-determinant with unre- stricted orbitals, beyond the CF point, the broken-symmetry deter- minant, ∣↑↓⟩, acquires partial triplet character since ∣S2=2,ms=0⟩ =1√ 2(∣↑↓⟩+∣↓↑⟩). In these transition metal complexes, we general- ize the two involved multiplicities to LS and HS (quartet and sextet for the Fe complexes and quintet and septet for the Mn complex), and the accessibility of the HS state due to the factors mentioned above can be visualized as a vertical compression of the entire MO diagram. For septet Mn(NH 3)+ 4, the spin-up HOMO has predominately s character (as given by partial Mulliken populations from the B3LYP calculation). This suggests that, at least for this case, the fractional closed-shell unpairing corresponds to population of an orbital not in the t 2gor e gmanifolds, but rather one derived from the 4s metal AO. Indeed, the NOONs from a CASSCF(6e6o) calculation of the quintet have occupation numbers 1.06, 1, 1, 1, 1, and 0.94; the orbitals cor- responding to the 1.06 and 0.94 NOONs have equal contributionsfrom Mn s and d orbitals. The weak bonding apparently compresses the MOs such that the 4s orbital, which is in typical cases well- separated and energetically above the 3d MOs, is close enough to the 3d levels to stabilize the septet (via extra exchange). Indeed, CASSCF(6e6o) (and also κ-UOOMP2) predicts a septet GS with the quintet ∼5 mHa higher in energy, and thus, the quintet NOONs reflecting six unpaired electrons suggest that, rather than diagnos- ing an MR quintet GS, what we have witnessed instead is a varia- tionally driven collapse toward the HS state. This underscores the danger of using small active spaces in CASSCF methods, namely, that the spin-state ordering can be in error without dynamic corre- lation, which can lead to a false diagnosis of MR character. However, even theories that do formally include dynamic correlation lead to inconclusive results: In the def2-TZVPP basis and with the DKH Hamiltonian, BP86 (pure) and B3LYP (global hybrid with 20% EXX) functionals put the quintet below the septet state, while PBE0 (global hybrid with 25% EXX) and HF reverse the ordering. These mul- tiplicities are clearly very close in energy, and the possibility of a septet GS—at odds with conventional chemical intuition—is cur- rently being re-investigated carefully with a number of ab initio methods. Finally, we comment that in contrast to the SSB found in Mn and Fe complexes with ammonia ligands discussed above, in Co(NH 3)+ 3, UHF SSB is artificial (i.e., restored via UB3LYP and κ-UOOMP2) presumably because the 3d8configuration of Co, with- out s occupancy, enables relatively stronger bonding/ligand-field splitting even in the absence of π-backdonation. While we were unable to perfectly correlate deviations in the calculated ligand-dissociation energies using SR methods such as DLPNO-CCSD(T) or DFT vs experiments with SSB at the UB3LYP orκ-UOOMP2 levels of theory, we note that the largest error from DLPNO-CCSD(T) vs experiment among the 34 molecules investi- gated in Ref. 33 was for the Fe(NH 3)+ 4species (9.15 kcal/mol). For the Mn(NH 3)+ 4complex, the B3LYP, B97, M06, PBE0, and ωB97X-V functionals, along with DLPNO-CCSD(T), consistently underesti- mated the experimentally measured ligand-dissociation energy in the range of 4.74–6.88 kcal/mol (though the use of the septet multi- plicity is being explored). We must point out, here, that such weakly bound ions are unlikely to be found in solution. The binding ener- gies are very weak, e.g., 10.0 ±1.7 and 8.6 ±1.4 kcal/mol for Fe and Mn tetra-ammonium species, respectively, and in polar sol- vents such as water, these complexes may readily dissociate. It is also likely that the Mn(I) and Fe(I) states will undergo redox events with the solvent to become the more stable Mn(II) and Fe(II) oxidation states. 2. Redox-active complexes with noninnocent ligands Theab initio prediction of electrochemical redox potentials of transition metal catalysts is a long-sought goal. Encouragingly, pre- vious analyses of the deviations of DFT predictions from experimen- tal measurements have revealed some systematic trends,111and there are indications that computing the potential of the reference redox couple can encourage favorable error cancellation.112However, one can easily find a sizable number of large outliers, and as the pres- ence of MR character in the highly reduced species is one factor that would lead to erroneous predictions, the ability to quickly diagnose these situations is a prerequisite if computational predictions are to be reliable and predictive. J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp When an electrode reduces a homogeneous metal catalyst one or more times, making it redox-active toward substrates such as CO 2 and O 2, the reduction is most often metal-centered.113–117Indeed, a unique property of transition metals is that they can accom- modate multiple oxidation states, which can enable remarkable reactivity. However, large delocalized ligands can have low-lying π∗orbitals, which can be preferentially reduced vs a virtual metal d orbital.118–127Such reduced species can have a LS GS due to the antiferromagnetic (AFM) interaction which can arise between opposite spins localized separately in metal and ligand orbitals. When this AFM stabilization outweighs the potential exchange sta- bilization if all unpaired spins were oriented in the same direction, the GS will be LS with a wavefunction requiring more than one determinant.51,122,125,126,128 From the perspective of SSB, we first investigate a set of metal complexes that was the focus of a recent work by Batista et al. , in which redox potentials in non-aqueous solution were computed and compared to experimental measurements.7This set included MCp 2 and MCp∗ 2where M =Fe, Co, Ni; and M′bpy 3where M′=Fe, Co. Cp, Cp ∗, and bpy denote cyclopentadienyl, pentamethylCp, and bipyridine ligands, respectively.129As shown in Table VIII, SSB in the UHF wavefunction occurs in nearly all species (with the exception of NiCp∗ 2). For the Cp and Cp ∗complexes, UB3LYP and κ-UOOMP2 restore spin-purity in all cases, implying SR character. For the tri-bpy complexes, the notably large UHF SSB is approximately restored in all cases with UB3LYP and κ-UOOMP2. Looking more closely at Co(bpy )2+ 3, for which ⟨S2⟩κUOOMP2 calcu- lated from the Slater determinant of optimized orbitals deviates by 6.6% from the exact value, we find that this deviation increases to 10.4% when evaluated with respect to the first-order wavefunction associated with κ-UOOMP2 [i.e., using Eq. (6)]. To investigate the possibility that DE in the B3LYP functional has led to a bias toward spin symmetry restoration for this Co(II) species, as seen previ- ously for the stretched hydrogen-fluoride molecule in Fig. 3, we find TABLE VIII. ⟨S2⟩values calculated for the complexes in Ref. 7. S2 exact S2 UHF S2 UB3LYP S2 κUOOMP2 FeCp 2 0 1.21 0 0 FeCp+ 2 0.75 1.37 0.78 0.76 CoCp 2 0.75 1.54 0.77 0.76 CoCp+ 2 0 1.32 0 0 NiCp 2 2 2.01 2.01 2.01 NiCp+ 2 0.75 1.65 0.77 0.76 FeCp∗2 0 1.20 0 0 FeCp∗+ 2 0.75 1.47 0.79 0.77 CoCp∗2 0.75 1.53 0.78 0.76 CoCp∗+ 2 0 1.36 0 0 NiCp∗2 2 2.01 2.01 2.01 NiCp∗+ 2 0.75 1.67 0.77 0.75 Fe(bpy)2+ 3 0 3.71 0 0 Fe(bpy)3+ 3 0.75 3.43 0.77 0.79 Co(bpy)2+ 3 0.75 4.42 0.76 0.80 Co(bpy)3+ 3 0 2.66 0 0⟨S2⟩CAM-B3LYP =0.83, identical to the value from the κ-UOOMP2 wavefunction. Figure 7 also shows that SSB can be modulated by the amount of EXX included in global hybrid functionals, reveal- ing that PBE with 20% EXX yields SSB comparable to CAM-B3LYP, which appears sensible given that CAM-B3LYP has 19% short-range EXX (and 65% long-range EXX). Increasing %EXX still further increases ⟨S2⟩toward the HF value. Recall that this arbitrariness is not unique to hybrid DFT functionals as the tendency toward SSB in κ-UOOMP2 is dependent on the value of the κregularizer: Scanning κgives a full view of the symmetry breaking landscape.56 Admittedly, it is difficult to draw conclusions regarding the MR character of the Co(bpy )2+ 3species. On the one hand, x-ray absorp- tion spectra have been well-reproduced by linear combinations of simulated spectra from the LS (doublet) and HS (quartet) states, leading to the claim that Co(bpy )2+ 3is 57% HS and 43% LS, whereas Co(bpy)3+ 3is relatively more monoconfigurational ( ∼80% LS).130 Taking these distributions and assuming the spectra were taken at room temperature imply spin gaps of roughly 0.01 and 0.03 eV for the Co(II) and Co(III) species, respectively, which indeed would imply SC. On the other hand, the bpy ligand is between NH 3and CO on the spectrochemical series suggests that 10Dq should be rel- atively large due to π-backbonding, and therefore, SSB is likely to be restored with a suitable level of dynamic correlation (and indeed, it is with B3LYP). It is also true that UB3LYP was used in Ref. 7 and the calculated Co(III) →Co(II) reduction potential matched the experimental value to a very high degree of accuracy (within a tenth of an eV, when triple- ζbasis sets were used). Thus, negligi- ble SSB from B3LYP and worst case deviations of around 10% from CAM-B3LYP and the κ-UOOMP1 wavefunction suggest that this complex is predominately of SR character with redox-innocent bpy ligands. We now turn to Fe complexes with terpyridine (tpy) or porphyrin ligands, which due to the more delocalized ligand frameworks are expected to have a relatively lower-lying π∗orbital FIG. 7. ⟨S2⟩for the PBE-based hybrid functional as a function of exact HF exchange (EXX) fraction for Co(bpy )2+ 3, compared with values from UHF, κ-UOOMP2, UB3LYP, and UCAM-B3LYP. J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8. Schematic view of the reduction process of Fe tpy and porphyrin com- plexes, as part of their CO2RR catalytic cycles. The red moiety indicates the location of the excess electrons, and L =CH3CN. than bpy. Both complexes are efficient electrocatalysts for the CO 2 reduction reaction (CO2RR) to CO. An important feature of CO2RR electrocatalysts is the substrate selectivity of CO 2against the hydrogen evolution reaction because proton-coupled CO 2reduc- tion and direct proton reduction occur at similar potentials. The incorporation of a redox-active ligand, which when reduced yields a relatively Lewis-acidic metal center, results in the metal favoring CO 2binding over protonation (the formation of a metal hydride is the first step in the hydrogen evolution mechanism). This origin of product-selectivity has been established for the Re (bpy)(CO)3Cl complex123,131and also for iron polypyridine125,126,132and porphyrin122catalysts. Furthermore, the placement of the extra electron in a π∗orbital that is lower-lying than a virtual d orbital results in milder reduction potentials—another desirable feature of an electrocatalyst—that are made milder still due to the additional stabilization from the M–L AFM coupling. Combined spectroscopic and computational work conducted by Neese and co-workers122suggests that the active species of the popular FeTPP catalyst (TPP =tetraphenylporphyrin) is an intermediate-spin Fe(II) center that is antiferromagnetically coupled to a porphyrin diradical anion. The FeTPP catalyst and its deriva- tives are among the most active CO 2reduction catalysts.118,121,133 Recent work by Derrick et al. shows that a tpy-based ligand frame- work (tpyPY2Me) in combination with an iron center is an efficient CO 2reduction catalyst at a low overpotential.126The doubly reduced active form of the catalyst is a singlet with a doubly reduced tpy ligand strongly coupled to an intermediate spin Fe(II) center. This electronic structure was established based on both computationaland spectroscopical evidence. Both complexes and their reduction reactions are depicted in Fig. 8. Table IX shows the SSB behavior for the Fe(II)-tpy species with net charge n =2+, 1+, and 0. The first reduction is known to occupy the noninnocent tpy π∗orbital, leaving the d6Fe(II) center closed- shell and thus forming an overall doublet. Consistent with this picture, both n =2+and singly reduced complexes, while severely spin-contaminated at the UHF level, do not exhibit significant SSB when accounting for dynamic correlation via B3LYP, CAM-B3LYP, and B5050LYP functionals. In contrast, the second reduction is accompanied by the loss of a ligand and a spin transition of the iron center from LS to intermediate spin (i.e., S Fe=1), while the twice- reduced tpy (S tpy=1) couples to the metal center to form an overall singlet. This AFM coupling of opposite spins separately localized on metal and ligand requires a MR description (for fundamentally the same reason that open-shell singlet biradicals require two deter- minants) and manifests as SSB, which persists upon inclusion of dynamic correlation with the three DFT functionals investigated. As expected, the calculated ⟨S2⟩value increases with %EXX in the hybrid functional. In the case of the iron porphyrin, here modeled without the phenyl groups (denoted FeP), the neutral complex has a triplet GS.134As indicated by the complete restoration of the UHF SSB with all DFT functionals and κ-UOOMP2, shown in Table X, this species is predicted to have a SR electronic structure. Both the first and second reductions are ligand-centered and result in M–L AFM cou- pling, which has been observed experimentally. The SSB that persists from HF through all DFT methods (increasing, as in the Fe tpy sys- tems above, with %EXX), for both FeP−and FeP2−corroborate the presence of M–L AFM coupling involving the SFe=1 center and the reduced non-innocent porphyrin. As first encountered in Ref. 58, κ-UOOMP2 predicts unphys- ical GSs for the FeP species. In particular, while the neutral species exhibits very minor spin-contamination, the Mulliken spin density on the Fe atom is found to be 1, as opposed to the expected value of 2 for an Fe-centered triplet. In addition, we found that ⟨S2⟩κUOOMP2 for FeP1−is 0.87, roughly twice as small as the values found with the three hybrid DFT functionals used, and the spin-density on the Fe atom is 0.4 (vs the expected value of 2 for a ligand-centered reduc- tion stabilized by AFM coupling to the metal; UB3LYP gives 1.96). Similarly, for the twice-reduced Fe tpy and FeP species, κ-UOOMP2 predicts spin-pure, closed-shell GSs with a net spin of 0 on the metal, inconsistent with the AFM-coupled states deduced from experimen- tal measurements and predicted from B3LYP, CAM-B3LYP, and B5050LYP functionals. Finally, we remark that MR character arises also when cat- alysts such as metalloporphyrins bind small-molecule substrates such as O 2and NO.135–138These are of great biological relevance TABLE IX. ⟨S2⟩values of dication and singly and doubly reduced Fe-tpy complexes. ωB97X-D geometries are used, taken from Ref. 126. [Fe(II) (tpyPY2Me2−)]nS2 exact S2 UHF S2 UB3LYP S2 UCAM−B3LYP S2 UB5050LYP S2 κUOOMP2 n=2+ 0 2.85 0 0 0 0 n=1+ 0.75 2.94 0.76 0.77 0.78 0.86 n=0 0 4.03 1.48 1.76 1.94 0 J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE X. ⟨S2⟩values of neutral and singly and doubly reduced iron porphyrin (FeP) complexes. LRC- ωPBEh/def2-SV(P) geometries are used. [FeP]nS2 exact S2 UHF S2 UB3LYP S2 UCAM−B3LYP S2 UB5050LYP S2 κUOOMP2 n=0 2 3.98 2.01 2.01 2.01 2.07 n=1−0.75 3.24 1.61 1.69 1.74 0.87 n=2−0 3.03 1.76 2.00 2.26 0 and deserve further investigation alongside catalytic systems for CO2RR and other desirable reactions. The present results show that SSB from these hybrid DFT functionals is a computationally inex- pensive hallmark, which can be used in future in silico catalyst design projects and can inform, if not replace, chemical intuition regarding the possible non-innocence of novel ligand frameworks. 3. Antiferromagnetic coupling in metal–metal dimers The quintessential example of AFM coupling is multi-metal compounds.139–141As a simple example system, we consider the four smallest Mn(III)–Mn(IV) compounds from Ref. 142, which have two bridging oxygens directly connecting the Mn centers. These molecules are shown in Fig. 9. The ⟨S2⟩UB3LYP values for compounds numbered 5, 6, 7, and 11 are all 3.8, which is far from the exact value of 0.75 for these experimentally assigned doublet species. We confirmed that the spin-densities on the Mn atoms are 4 and −3 in all cases (we note that a HS, octet cal- culation needed to be performed first, and the resulting den- sity was used to initialize a subsequent calculation of the doublet state), indicating strong AFM coupling. As expected, for these types of multi-metal states, MR methods with large active spaces are required to obtain exchange-coupling parameters that agree with FIG. 9. Mn(III)–Mn(IV) dimers investigated in this work, with ⟨S2⟩values shown. The purple, red, blue, gray, and white colors indicate Mn, O, N, C, and H atoms, respectively.experiment.44Rather unexpectedly, in many cases, broken- symmetry DFT (i.e., utilizing some form of spin projection) appears capable of producing accurate coupling constants as well, though relative spin-state energetics are nevertheless sensitive to the func- tional employed.143–146 F. A cautionary coda on the use of MP2 and double-hybrid functionals on single-reference organometallics In this section, we emphasize that even in the absence of per- sistent SSB (i.e., for species well-described by a single determinant), computational methods should be chosen with care. In other words, in our view, categorizing a system as either MR or not is only the first step to quantitatively accurate predictions. As an illustra- tive example, we focus on metal complexes with strong-field CO ligands, which are of primary importance in organometallic chem- istry. As discussed above, the MO diagrams of these complexes are characterized by large 10Dq values, which result in spin-pure, LS GSs. However, Fig. 10 shows that DHDFs, which incorporate MP2 correlation energies, consistently overestimate experimental ligand dissociation energies34,147for M–CO complexes. While one might expect that the highest rung of Jacob’s ladder4,148would provide the most accurate results for these organometallic com- pounds, DHDFs drastically underperform simpler functional forms, e.g., the global hybrids B3LYP and B97 and the range-separated hybridωB97X-V. Next, we consider the 3d metal complexes from Ref. 33 with gas-phase experimental ligand-dissociation measurements and select only those for which DLPNO-CCDS(T)/CBS yields accu- rate results. The six suitable compounds are V (H2)+ 4, Co(H2)+ 4, Ti(H 2O)+ 4, Cu(NH 3)+ 4, Cu(CO )+ 4, and Fe(N 2)+ 4. H 2O is aπ-donor, and NH 3ligands can only engage in σ-bonding with the metal. FIG. 10. Comparison of calculated ligand dissociation energies, extrapolated to the CBS limit, with experimental values. Geometries optimized at the UB3LYP- DKH/cc-pVTZ-DKH level; single-points with the indicated functional in the def2- QZVPP basis without scalar relativistic effects. These relativistic effects are very small for 3 dmetal carbonyls,149and the present calculations without scalar rel- ativity agree quite well with DKH calculations (albeit in a triple- ζbasis) from Ref. 33. J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp H2, N 2, and CO ligands are π-acceptors, in the order of increasing backbonding strength (see Table S2). Figure 11 shows, in the cc-pVTZ basis, the deviations of the ligand-dissociation energies calculated with UMP2, κ-UOOMP2, and UCCSD from UCCSD(T). Evidently, while MP2 methods yield reasonable accuracy for transition metal chemistry ( ∼2–3 kcal/mol36) for the complexes with ligands that can only σ-donate or weakly π-accept, large errors are found for Cu(CO )+ 4(interestingly orbital optimization makes the accuracy worse). These results imply that the overes- timation of ligand-dissociation energies with DHDFs vs experi- ment (Fig. 10) stems from an overestimation at the MP2 level. Neese et al. also found that MP2 and OOMP2 produce large (4.8–60 kcal/mol) errors vs CCSD(T) for CO dissociation of four Cr and Ni complexes;150similarly, Hyla-Kryspin and Grimme found for MP2-calculated CO-dissociation enthalpies an average overesti- mation of 19.3 kcal/mol over a set of seven 3d metal carbonyl species (interestingly, while MP3 led to underestimations of even larger average magnitude, the performance of spin-component- scaled MP2 and MP3 was more promising, with MAEs of 9.6 and 3.7 kcal/mol, respectively).151 We hypothesize that while the pairwise additive approximation for the correlation energy of MP2 theory can adequately describe σ-bond only ligands (in the absence of static correlation), it cannot be expected to produce quantitative results for four- and six-electron-like interactions involving both σ-donation and π-backbonding, regardless of the orbital set employed. σ-donation involves one electron pair, while π-backbonding can additionally involve either one or two more electron pairs, depending on the number of available donor–acceptor orbitals. In the case of a metal- carbonyl bond, π-backbonding can involve two pairs of electrons backdonated into the degenerate π∗ xandπ∗ yorbitals of CO, result- ing in a bond that effectively involves six correlated electrons in total. This reasoning begs the following question: For metal carbonyl complexes, or more generally for metal complexes with strong-field FIG. 11. Deviations of the ligand-dissociation energies (kcal/mol) calculated with various methods from UCCSD(T), in the cc-pVTZ basis. A negative deviation denotes underbinding.ligands which can significantly π-accept, will the SSB behavior of κ-UOOMP2 be compromised due to its inability to describe the inter-pair correlations involved in dative bonds characterized by simultaneous σ-donation and π-backdonation? If so, how can SSB due to this be distinguished from SSB due to genuine MR character? The first point is that these inter-pair effects are, in general, smaller than single pair correlation energies. Moreover, when inter-pair correlations are physically significant (as we argue to be the case in organometallic compounds), the gap between LS and HS states is typically already large, as in the Fe(II)(CO )2+ 6system discussed earlier, rendering spin-state mixing an irrelevant concern in most situations. However, for smaller-gap systems such as complexes with redox-noninnocent ligands, e.g., twice-reduced Fe tpy and por- phyrin, we have shown above that κ-UOOMP2 yields qualitatively incorrect predictions regarding the presence of MR character, and thus, the use of SSB from DFT orbitals is recommended. Work is ongoing to disentangle errors due to κ-regularization and the MP2 ansatz itself. IV. CONCLUSIONS While transition metal chemistry has a reputation for being challenging for computational quantum chemistry models, one has to look hard for static correlation in the GSs of mono-metal transi- tion metal complexes. This is because for small 10Dq values, the HS state (which is, in general, SR) is favored due to the exchange sta- bilization of same-spin electrons and the energy “saving” from not having to pair electrons; for large 10Dq values, the LS is energetically far-below the HS state, which prevents the latter from contaminating the total spin of the former, leading again to a SR state. The latter sit- uation occurs when 4d and 5d transition metals are involved. One of the goals of this work, though, is to find relevant situations in which static correlation is present in molecules. We have encountered MR character in the following: 1. Metal hydride diatomics courtesy of the inapplicability of the Jahn–Teller theorem to linear systems, in which spatial symmetry mandates two important determinants for both TiH and CoH. For less electronegative 3d metals, i.e., going from Ni to Co to, mostly strikingly, Fe, we find increasingly non-negligible LUNOs due to the narrowing gap between the non-bonding metal 4s-3d shell and the antibonding orbital (involving metal d2 zand hydrogen 1s), which opens the door to SSB due to the intrusion of higher-spin-state character. 2. Metal complexes with higher coordination numbers and low [i.e., Fe(I) or Mn(I)] oxidation states, which exhibit very weak bond energies (tens of kJ/mol) due to weak-field ( σ-donation only) ligands and metals with singly occupied 4s orbitals pre- venting favorable dative bonding. While less likely to exist in solution, such weak bonding can be encountered in the gas phase (in, e.g., metal-organic frameworks or atmospheric chemistry). 3. Molecules that exhibit AFM coupling resulting in LS GSs. We have demonstrated that this can occur in reduced states of homogeneous catalysts containing redox-noninnocent lig- ands such as terpyridine and porphyrin and also in oxygen- bridged bimetallic Mn(III)–Mn(IV) dimers. J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Forpara -benzyne along with stretched H 2and hydrogen flu- oride, the SSB behavior and NOON predictions from DFT orbitals agree quite well with those of ASCI (a near-exact FCI approxima- tion). As expected, increasing the fraction of EXX in hybrid func- tionals pushed CF points to shorter bond lengths and increased the amount of artificial SSB (i.e., LUNOs in excess of the exact value) approaching that of UHF (i.e., 100% EXX). It is remarkable that computing ⟨S2⟩from the KS orbitals (which represent the solution of a non-interacting problem) yields results similar to those obtained from approximate (e.g., κ-UOOMP2) wavefunction methods and NOONs in close agreement with those from ASCI. However, as the case of stretched hydrogen fluoride reveals, when pure and common hybrid DFT orbitals are used to compute ⟨S2⟩and NOONs, the static correlation error is intertwined with DE, which leads to erroneous results. In such cases, κ-UOOMP2, which is free of DE, is formally more reliable (in addition to the fact that ⟨S2⟩and NOONs are well- defined). We note that range-separated hybrids and functionals with a high percentage of EXX are lower-scaling alternatives in which DE is much-reduced, and the use of orbitals from optimally tuned range-separated hybrid functionals may be a potentially interesting way forward in certain cases. Regarding the diagnosis of static correlation, SSB at the UB3LYP and κ-UOOMP2 levels is shown to be insufficient in case 1 from above as it does not account for a wavefunction that must be multi-configurational to respect spatial symmetry, and generally in LS states which are not the GS due to variational collapse (practical implications of the latter for the calculation of HS–LS spin split- tings will be discussed below). In these special cases, SSB should be used in conjunction with NOONs from a MR method, ideally one with an appropriate treatment of dynamic correlation. Methods that use a HS reference and apply spin-flip excitations to generate the Hilbert space of lower spin multiplicities appear to be good candi- dates to, e.g., probe the possible multi-configurational nature of a LS excited state and have the advantage of a black-box selection of the active space orbitals involved in spin-unpairing from the target LS state. Another important takeaway suggested by this work is that strong correlation is a term that demands clarification, especially in the context of transition metals. The term is frequently used as a synonym of static correlation and MR character, and we have presented SSB as an intuitive and meaningful diagnostic for the GS when spatial symmetry is not an issue. Having established a connection with chemically revealing models such as ligand-field and molecular orbital theories, a picture is painted which suggests that, fundamentally, static correlation in transition metal systems involves the same phenomenon as in organic molecules, e.g., biradicaloids.56,152 In contrast, evidently, it is the dynamic correlation that presents additional difficulties compared to the case of typical organic molecules. For organometallics, four- and six-electron-like inter- actions (or at least inter-electron pair correlations) as relevant to π-donation or π-backdonation on top of σ-donation appear to be important, as illustrated by the failure of MP2 and MP2-based DHDFs in predicting experimental ligand dissociation energies for metal carbonyl complexes. The second idea that supports this conclusion, while admittedly less concrete, is that large errors in DLPNO-CCSD(T) calculations were found when the UHF solution exhibited significant spin-contamination. It is our expectation thatwhen orbitals obtained from theories that include (even approx- imately) dynamic correlation, such as DFT and κ-UOOMP2, are employed, subsequent CC predictions will improve in accuracy, as has already been seen in main group molecules.153,154Thus, in our view, the common notion that transition metals are difficult for traditional electronic structure methods is due to both static correlation and dynamic correlation, yet we stress our finding that truly MR situations are encountered only in the special cases enumerated above (and perhaps a few others), making the proper treatment of dynamic correlation more important in most com- monly encountered cases. There is also an important connection between a theoreti- cal method’s ability to capture dynamic correlation and the ability of SSB to imply MR character, which depends on the accurate prediction of relative energies between two (or possibly more) spin states. In the limit of no correlation other than the exchange interaction between same-spin electrons (required by Fermi statistics/Pauli-exclusion), as is the case in HF theory, HS states are artificially favored relative to LS states. For a system with a LS GS (determined experimentally or by an exact theoreti- cal method), HF (and often CASSCF with small active spaces) will significantly underestimate the LS–HS gap or even incorrectly predict a HS GS. In such cases, the SSB implied by the HF LS wavefunction will be artificially large due to the unphysical inclu- sion of the HS state. The idea behind using orbitals optimized from Kohn–Sham DFT or κ-UOOMP2 is essentially an attempt to more accurately describe relative spin-state energetics via the (approx- imate) inclusion of dynamic correlation, which enables true MR character to be reliably diagnosed. Yet the difficulty involved in accurately predicting LS–HS gaps (in a reasonable amount of time) cannot be overestimated. With regard to DFT, the issue stems from the systematic overstabiliza- tion of HS states as the fraction of EXX is increased. Pure DFT functionals, e.g., PBE, BLYP, or BP86, are justifiably advantageous when investigating transition metal systems in the sense that the resulting orbitals are likely to be free of “artificial” SSB; subsequent PT or CC will converge more quickly, and there is no arbitrari- ness in, e.g., how much EXX to include. However, pure functionals are plagued by high levels of DE and, as a result, are known to yield unphysical charge and spin densities, underestimate barrier heights, and so on. With regard to κ-UOOMP2, we recall (i) the large 3 eV error in the vertical spin gap of the iron hexacarbonyl species [vs CCSD(T) in the same basis] and (ii) the spurious closed-shell GSs implied for the twice-reduced Fe tpy and porphyrin species, which are known experimentally to be open-shell singlet states with AFM coupling between the metal and the ligand. Both (i) and (ii) involve the overstabilization of closed-shell LS states in complexes with strong-field ligands, which form bonds to metal ions involving simultaneous σ-donation and π-backdonation. Evi- dently, for these types of organometallic complexes, the fact that dynamic correlations among multiple electron pairs are entirely missing in MP2 approaches (and thus to some extent in DHDFs incorporating MP2 correlation) has a detrimental impact on the accuracy of predictions regarding thermochemistry (Fig. 10), rela- tive spin-state energetics, and the presence of static correlation via SSB. Further development and assessment of SR methods, in addi- tion to local correlation functionals, capable of describing this type J. Chem. Phys. 154, 194109 (2021); doi: 10.1063/5.0047386 154, 194109-16 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp of dynamic correlation relevant to transition metal bonding is clearly necessary. CCSD and CCSD(T) are limited by their high scaling. The direct variant of the Random Phase Approximation (dRPA),155 which can be implemented to scale as the fourth power of system size,156has been shown to be equivalent to ring-CCD157and there- fore includes terms approximately correlating excitations involving more than one electron pair. dRPA has shown promising accuracy for bond energies of metal carbonyl complexes (among others).158 On the other hand, the improved description of the dissociation limit within the RPA formalism comes at the expense of DE.159 OORPA approaches have been developed,160which do not seem to require regularization, and may be promising in the context of a DHDF.161,162OOCCD and MP3 also approximately include inter- pair excitations, and the scaling of these methods might be reduced by localized approximations or tensor decompositions of the two- electron integrals, enabling their use in DHDFs. Our data suggest, though it remains to be seen, that the performance of DHDFs incorporating such correlation energies163,164may yield improved accuracy for organometallic thermochemistry. The implications of our findings for practical quantum- chemical calculations on TM complexes and catalysis modeling are broadly as follows: a. The MR character in the GS, while relatively rare, can be diag- nosed in a black-box and active-space-free manner through SSB in hybrid DFT calculations (even employing small basis sets). We suggest B3LYP or B5050LYP and range-separated hybrid functionals if substantial DE is detected. Such SSB is typically found to coincide with fractional NOONs from multi-configurational wavefunctions, which together signal that the relative energetics of frontier orbitals are such that multiple spin states are nearly degenerate. b. In the absence of MR character, we can state the following: (i) When the LS state of interest is the GS, then SR wavefunc- tion methods (with an appropriate treatment of dynamic correlation) and DFT are, in principle, capable of providing robust predictions. While the use of hybrid function- als is necessary to diagnose MR character, the choice of functional used to model, e.g., catalytic cycles should be based on a judicious mixture of system-specific cri- teria (e.g., agreement with available experimental mea- surements) and general benchmark performance across TM compounds.30,33,165–167The performance of hybrids is typically superior to that of semi-local functionals. For organometallic complexes with πdonating or accepting lig- ands, such as metal carbonyls, the use of DHDFs based on MP2 correlation energies may not yield improved results beyond hybrids such as B97 or ωB97X-V, as demonstrated here. (ii) When the LS state of interest is notthe GS, HF and hybrid functionals can still exhibit artificial SSB. Care must be taken when evaluating spin gaps and related properties, and we recommend the use of either spin-restricted orbitals or spin- projection of spin-unrestricted solutions. Such approaches may also reduce (though not completely eliminate) the sen- sitivity of DFT-predicted spin-splittings on the choice of functional.c. In the presence of MR character, methods based on a single- determinant reference (e.g., MPn and CC theories) should generally be avoided. DFT, especially when using hybrid func- tionals with a high fraction of EXX, also appears to be unsuit- able (though it is SR only in the context of the fictitious KS system). One could attempt to proceed (semi-quantitatively) using broken-symmetry DFT, where spin-projection can help if there is one leading contaminant107or if Heisenberg physics is operative.109,168When possible (in light of relatively high computational costs), electronic structure approaches that explicitly account for a multi-configurational wavefunction should be used. SUPPLEMENTARY MATERIAL See the supplementary material for NOONs and convergence details for ASCI-SCF calculations of larger molecules ( para -benzyne, [Fe(CO)6]2+, and [Fe(H 2O)6]2+), NOONs and active space selec- tion for a selected CASSCF calculation ([Fe(H 2O)6]2+), and com- plete ALMO-EDA results. ACKNOWLEDGMENTS J.S. thanks Romit Chakraborty for insightful discussions. This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. 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5.0055260.pdf

J. Appl. Phys. 130, 034301 (2021); https://doi.org/10.1063/5.0055260 130, 034301 © 2021 Author(s).Electromagnetic field emitted by core–shell semiconductor nanowires driven by an alternating current Cite as: J. Appl. Phys. 130, 034301 (2021); https://doi.org/10.1063/5.0055260 Submitted: 27 April 2021 . Accepted: 24 June 2021 . Published Online: 15 July 2021 Miguel Urbaneja Torres , Kristjan Ottar Klausen , Anna Sitek , Sigurdur I. Erlingsson , Vidar Gudmundsson , and Andrei Manolescu ARTICLES YOU MAY BE INTERESTED IN Performance prediction of InP/GaAsSb double heterojunction bipolar transistors for THz applications Journal of Applied Physics 130, 034502 (2021); https://doi.org/10.1063/5.0054197 Adaptively reverse design of terahertz metamaterial for electromagnetically induced transparency with generative adversarial network Journal of Applied Physics 130, 033101 (2021); https://doi.org/10.1063/5.0054080 Atomistic origin of compositional pulling effect in wurtzite (B, Al, In) xGa1−xN: A first-principles study Journal of Applied Physics 130, 035704 (2021); https://doi.org/10.1063/5.0050102Electromagnetic field emitted by core –shell semiconductor nanowires driven by an alternating current Cite as: J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 View Online Export Citation CrossMar k Submitted: 27 April 2021 · Accepted: 24 June 2021 · Published Online: 15 July 2021 Miguel Urbaneja Torres,1 Kristjan Ottar Klausen,1 Anna Sitek,1,2 Sigurdur I. Erlingsson,1 Vidar Gudmundsson,3 and Andrei Manolescu1,a) AFFILIATIONS 1Department of Engineering, Reykjavik University, Menntavegur, 1, IS-102 Reykjavik, Iceland 2Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wroclaw University of Science and Technology, Wybrze że Wyspia ńskiego 27, 50-370 Wroclaw, Poland 3Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland a)Author to whom correspondence should be addressed: manoles@ru.is ABSTRACT We consider tubular nanowires with a polygonal cross section. In this geometry, the lowest energy states are separated into two sets, one the corner and the other side-localized states. The presence of an external magnetic field transverse to the nanowire imposes an additionallocalization mechanism: the electrons being pushed sideways relatively to the direction of the field. This effect has important implicationson the current density as it creates current loops induced by the Lorentz force. We calculate numerically the electromagnetic field radiatedby hexagonal, square, and triangular nanowires. We demonstrate that because of the aforementioned localization properties, the radiated field can have a complex distribution determined by the internal geometry of the nanowire. We suggest that measuring the field in the neighborhood of the nanowire could be the basic idea of the tomography of the electron distribution inside it if a smaller receiver antennacould be placed in that zone. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0055260 I. INTRODUCTION Nanowires based on semiconduc tor materials have emerged as an advantageous platform for the realization of diverse applications in multiple areas, such as photonics, 1–4electronics,5–8photovoltaics,9–11 or topological Majorana physics.12,13With diameters of tens to hun- dreds of nanometers, the microscopic crystal structure of the materialsused becomes evident, resulting in polygonal cross sections.Nanowires based on III –V semiconductors fabricated by bottom-up growing techniques are commonly hexagonal, 14,15but other geome- tries also have been obtained, like triangular,16,17square,18and dodecagonal.19Nanowires of core –shell type can also have distinct core and shell polygonal shapes, li ke a triangular shell with a hexago- nal core20,21or the other way around.22 By using various semiconductor materials, growing conditions or controlling the thickness of the core or of the shell, the nanowire properties can be fine-tuned for desired or potential physicalbehavior.23In particular, with an insulating core surrounded by a conductive (doped) shell, the nanowire functions as a tubular con-ductor. Such a conductive nanotube may also be obtained byappropriate band alignment at the core –shell interface 24–26or via the Fermi level pinning at the outer surface of the nanowire.27 In the case of a polygonal shell, a group of low energy elec- trons, including those in the ground-state, tends to localize at thecorners of the shell. 28–31This corner localization can be seen as the effect of particle accumulation at the bending of a two-dimensionalquantum wire, 32representing a transverse polygonal cross section of the shell, or at the regions with the greatest surface curvature along the prismatic tubular shell.33For a polygon with Ncorners, and including the electron spin, there are 2 Nstates localized in the corners. Depending on the shell thickness and sharpness of thecorners, these states are separated by an energy interval varying from a few meV to tens of meV from the next 2 Nstates, which are localized on the sides of the polygon. 34These corner and side statesJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-1 Published under an exclusive license by AIP Publishingcorrespond to low-energy transverse modes of a prismatic shell where, due to the longitudinal motion, they generate states local- ized along edges or facets, respectively. The controlled fabrication of core –shell nanowires and their internal structure motivated significant theoretical research on theelectronic states in tubular prismatic geometry, i.e., in the shell, rel- evant for the optical response, 35–37electrical,38and thermoelectrical conductivity,39or topological physics.40–42At the same time, apart from the remarkable efforts for fabrication, relatively less experi-mental research has been devoted to the electrons localized in theshell. Notably, transport experiments showed oscillations of the longitudinal conductance in the presence of a magnetic field related to flux periodicity. 27,43More recently, the transverse con- ductance and Coulomb blocking effects have been investigated intriangular nanowires. 21Also, optical emission excited by highly focused x rays indicated a nonuniform electron distribution in hex- agonal multishells.44In addition to the localization of electrons on facets of different thicknesses, a single hexagonal shell has alsobeen probed via photoluminescence experiments. 45However, more systematic experimental evidence of the quantum localization ofelectrons in the tubular prismatic geometry, and especially the pres- ence of corner and side localized states with an energy gap separat- ing them, has not been achieved yet. In this paper we exploretheoretically the possibility to relate the quantum localization inthe shell to the electromagnetic field radiated by the nanowire. An interesting application of semiconductor based nanowires is as elements of optoelectronic circuits, where they could functionas nanoantennas, with emitter and/or receiver function. It wasalready demonstrated that individual nanowires made of InAs wereable to detect electromagnetic radiation in the terahertz domain. 46 Later on, nanowires built from InP have been studied by Grzelaet al. in the near infrared domain (850 nm) and shown to have capablity of directional emission and absorption via the internalMie modes. 47,48More recently, a multishell design of cylindrical nanowires has been proposed to achieve a superdirective emission in the optical domain.49 This interest in nanoantennas based on nanowires also moti- vates us to consider the electromagnetic fields radiated by a polygo-nal core –shell nanowires due to an alternating current driven along the nanowire. We focus on core –shell nanowires made of semicon- ductors, where the charge transport occurs only within the shell, which acts as a tubular conductor, whereas the core is an insulator.This situation has been obtained by doping only the material of theshell but not that of the core. 21,43If the radiated field of our nano- wire, seen as an emitter nanoantenna, can be explored by a separate receiver nanoantenna, then the current and charge distributionwithin the nanowire could be obtained by solving an inverseproblem. We emphasize again that the quantum localization of theelectrons in a tubular prismatic geometry is not trivial and offers a variety of application and manipulation possibilities (including contactless 50), which have not been investigated experimentally yet. First, we compute numerically the electronic quantum states for selected geometries and then the current along an infinite pris-matic shell driven by a time-dependent harmonic voltage bias. In this work, we intend to emphasize the consequences of the internal geometry of such a nanowire, i.e., with a tubular prismatic shape,and of the associated localization properties, on the radiatedelectromagnetic field. In principle, such effects can be observed in the neighborhood of the nanowire, i.e., close enough to the lateral surface, where the anisotropic distribution of currents should leadto complex field distribution. Whereas with increasing the distancebetween the observation point and the nanowire, the internalcurrent distribution becomes less and less important, eventually until the radiated field looks similar to that of a simple, unstruc- tured wire of finite length. For these reasons, we shall considernanowires of an infinite length, and we shall calculate the radiatedfield in the proximity of the nanowire. We describe our numerical methodology in Sec. II. Then, in Sec. III, we qualitatively analyze the signature of the prismatic tubular geometry of the nanowire on the structure of the radiatedelectromagnetic field. We discuss the implications of the prismedge and facet localization and, in particular, the anisotropy of theradiated field. Furthermore, we explore the variation of the radiated field when applying an external magnetic field perpendicular to the nanowire. The anisotropy of the radiated field is limited to the dis-tances much smaller than the nanowire length, but depending onthe geometry of the nanowire and on other parameters, this zonecan be considerably large compared to the nanowire radius. Finally, in Sec. IV, we collect our conclusions and comments on the tomog- raphy idea. II. MODEL AND METHODS We start by considering a system of non-interacting electrons confined in a polygonal ring. The model begins with a circular disk situated in the plane ( x,y), which is discretized in polar coordi- nates. 51On this grid, we superimpose polygonal constraints and we retain the points that lie inside the resulting boundaries, excludingthe rest. With this method, we define hexagonal, square, and trian- gular cross sections. Furthermore, we consider the nanowires to be infinite, assuming free particle propagation along their length, i.e.,in the zdirection. The Hamiltonian of the system is then expressed as follows: H¼ (/C0i/C22h∇þeA)2 2meff/C0geffμBσB, (1) where B¼(Bx,By, 0) is an external magnetic field, transversal to the nanowire length and Ais the corresponding vector potential. The vector r¼(x,y,z) defines the positions inside the shell; eis the electron charge; meffandgeffare the effective electron mass and bulk g-factor of the material considered, respectively, μBis the Bohr ’s magneton; and σ¼(σx,σy,σz) are the spin Pauli matrices. The Hilbert space associated with the polar grid is spanned by the position vectors jqi¼j rq,fqi,w h e r e rqandfqare the radial and angular coordinates of site q, respectively, from which the spin is excluded. In order to obtain the eigenstates of the Hamiltonian(1), we solve the problem in two steps: we first obtain the transverse eigenstates jai¼P qψ(q,a)jqiand eigenvalues Ea(a¼1, 2, 3 ;/C1/C1/C1), forB¼0, i.e., the states of an electron in the polygonal ring repre- senting the cross section of the nanowire. Then, we retain the trans-verse states with the lowest energies and together with the plane wave vectors in the zdirection, jki¼exp(ikz)=ffiffiffi Lp ,w h e r e Lis the length of the nanowire (considered infinite in our model), and withJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-2 Published under an exclusive license by AIP Publishingthe spin states jsi¼+1, we form the basis jaksi. Finally, for B=0, we diagonalize numerically the total Hamiltonian (1)in this basis using a discretized series of kvalues and obtain the eigenvalues Emks (m¼1, 2, 3, ...) and eigenstates jmksiof the Hamiltonian (1), expanded in the basis jaksi. The number of transverse states jai needed to reach numerical convergence was typically 4 N(excluding spin). The convergence was checked in several cases for an even larger basis set, and no change of the final results was observed. Thenumerical diagonalizations were performed with the Lapack library. With this approach, we obtain the first and the second lowest-energy groups of states for the infinite prismatic tubular nano- wire, which are localized along the edges or along the facets of the prism, respectively. The charge density associated with these states is ρ(r)¼/C0eX mksFEmks/C0μ kBT/C18/C19 jhrjmksij2/C0nd/C0/C1 , (2) where /C0eis the electron charge and ndrepresents the density of the ionized donors. The next step is to compute the current density J inside the shell carried by the edge- and facet-localized states, J(r)¼X mksFEmks/C0μ kBT/C18/C19 hmksjj(r/C0r0)jmks, (3) withF(u)¼1=[exp( u)þ1] being the Fermi function and u¼(Emks/C0μ)=kBT,w h e r e μis the chemical potential, Tis the temperature, and kBis the Boltzmann constant. The operator j(r,r0)¼e[δ(r/C0r0)vþvδ(r/C0r0)]=2 describes the contribution to the total current density at the position rfrom an electron situated at r0and moving with a velocity v. The velocity is defined by the opera- torv¼i[H,r0]=/C22h.52 If no longitudinal voltage bias is applied, the system is in equi- librium and the contributions to the total current of electronsmoving with opposite velocities in the zdirection compensate each other. When a voltage bias is considered, these contributions no longer cancel out, and the nanowire carries a non-zero total current. In order to simulate the effect of a voltage bias, we create in oursystem an imbalance between electrons in states correspondingto positive and negative velocities, i.e., @E mks=@k.0, and @Emks=@k,0, respectively.53Thus, we consider two different values, harmonically time-dependent, for the chemical potentials μþandμ/C0 for carriers moving in opposite directions, with μ+¼μ+Vsin(ωt), where ωis the frequency, 2 Vis the bias amplitude, and μis the static chemical potential, which is determined by the carrier density at equilibrium. Once the current density distribution is obtained, we calculate the time dependent scalar and vector potentials outside the nano-wire, V(r,t)¼1 4πε0ðρ(r0,t) jr/C0r0jdr0, (4) A(r,t)¼μ0 4πðJ(r0,t) jr/C0r0jdr0, (5) where ε0andμ0are the vacuum electric permittivity and magneticpermeability, respectively, and the integration is carried out inside the shell. To properly obtain the vector potential we consider the approx- imation r/C28L. Having the vector potential, it is straightforward to obtain the electromagnetic radiated field using the relations E¼/C0∇V/C0@A @tandB¼∇/C2A: (6) We thus neglect the retardation effects and displacement currents since we will consider relatively low frequencies and quasi-stationarycurrents. 54In all our calculations, the contribution from the scalar potential was also very small. In fact, in Eq. (4), we neglected the polarization effects inside the nanowire, both due to the lattice dielectric response and due to the electron –electron Coulomb inter- actions. In principle, such screening corrections lead to a dielectricconstant ε.ε 0inside the nanowire and reduce the scalar potential everywhere for the electron densities and frequencies considered in our work. Hence, with a more elaborated model replacing Eq. (4), the contribution of Vto our results will remain negligible. The next step is to estimate the frequency domain we can con- sider with our method. Typical nanowires made of semiconductor materials such as InAs, InP, or GaAS have electron mobilities μein the range of 400 –6000 cm2=(V s), with scattering times in the range of tens of femtoseconds.55,56Thus, for frequencies in the RF domain, from kHz to GHz, achievable within an electric circuitconnected to the nanowire, we can reduce the electron damping and losses in the semiconductor nanowire to the static resistivity and ignore the dynamic corrections. For higher frequencies, towardthe optical domain, the effects of the anisotropic and dynamic per-mittivity must be taken into account both inside and outside thenanowire, for example, as in Ref. 37. In our present RF regime, the current can be injected into the nanowire through source –drain contacts covered with top-gate electrodes. This method has been used in a recent experiment tostudy the current output in core/shell GaSb/InAsSb NWs as a func- tion of AC gate voltage and frequency using a four-probe method. 57In our work, we assume the contacts are far from each other, at several micrometers distance, possibly at the two ends ofthe nanowire such that the length of the resulting antenna is muchlarger than the radius. As we mentioned in the Introduction, we are interested in the field distribution in the vicinity of the nano- wire resulting from the prismatic geometry. At large enough dis-tances from the nanowire its internal prismatic geometry becomesirrelevant, but instead, the geometry of the contacts may play arole. In addition, in practice, the contacts contribute to the imped- ance of the nanowire and, therefore, to the current strength and also possibly to the current distribution in the contact region.Nevertheless, including these details in our model is beyond thescope of our present work, where we want to concentrate on theprimary implications of current carrying tubular prismatic shell. We consider InAs bulk parameters for the nanowire: m eff¼0:023meandgeff¼/C014:9. We use a voltage bias correspond- ing to V¼2:5 meV, and we consider a test frequency of 1 MHz. We are mostly interested in the angular distribution of the radiated field around the nanowire, and for this purpose, the frequency plays no qualitative role. It quantitatively affects not only theJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-3 Published under an exclusive license by AIP Publishingconductivity that drops down in the GHz domain due to damping, but also the power density that, in principle, increases proportion- ally to the frequency squared. For our purpose, we evaluate thepower density of the radiated field with the time-averaged magni-tude of the Poynting vector S(r)¼1 TðT 01 μ0(E(r,t)/C2B(r,t))dt: (7) In all the examples shown in this article, the external radius of the shell (measured from the center of the polygon to one corner) and the thickness of the facets are Rext¼30 and t¼6 nm, respectively. III. RESULTS For sufficiently thin shells with polygonal cross section, like ours, the electrons with the lowest energies are localized in thecorners of the polygon and the electrons in the next layer of energy states are localized on the sides. 34The corner and side states are energetically separated by a gap interval that depends on the shapeand on the aspect ratio of the polygon, i.e., the ratio between thethickness and external radius of the cross section. It increases withdecreasing the shell thickness or the number of corners, and hence, in such a structure, the subspace of corner states is potentially robust to many types of perturbations. 58 A. No external magnetic field InFigs. 1(a) –1(c), we show the energy spectra of nanowires with hexagonal, square, and triangular-hexagonal cross sectionsassuming no external field. The blue-dashed horizontal lines indicate the chemical potential level. The energy intervals between corner and side states are Δ h¼21:2 meV for the hexagon, Δs¼34:4m e V for the square, and Δt¼171:1 meV for the triangle. The chemical potential (blue-dashed line) corresponds to an electron density of n¼1017cm/C03, a regime which can be achieved experimentally.27 For this carrier concentration, the plasma frequency is of the order of 1013Hz, far enough from our frequency domain such that our quasi-static description of the electromagnetic field is reliable.Because of the spin and rotational symmetries of these geometries, the states can be twofold or fourfold degenerate such that multiple energy levels overlap in each spectra. Each group of corner and sidestates consists of 12 levels for the hexagon, eight for the square, andsix for the triangle, and the corresponding degeneracy patterns are2442/2442, 242/242, and 24/42, respectively. 59 When the time-dependent harmonic voltage is applied to the nanowire, the total current is zero at those instants of times whenμ +¼μ, i.e., when sin( ωt)¼0. However, at these instants, the amplitude of the radiated electric field is maximal as it depends onthe time derivative of the current density through the vector poten- tial calculated with Eq. (5). On the contrary, when sin( ωt)¼1, and the applied voltage bias is maximum, i.e., μ þ/C0μ/C0¼2V,t h e current reaches its maximum, but its time derivative vanishes, andso does the radiated electric field. It is important to note that, when μ +=μor sin( ωt)=0, depending on the amplitude of the har- monic voltage applied, electrons may move between different energylevels, which translates into changes of localization and hence of the current distribution and radiated field, within one period of time. InFigs. 1(d) –1(f), we plot the time derivative of the current density, dJ=dt, inside the nanowire cross section. For simplicity, we only show it in the case of μ+¼μ, when the radiated field inten- sity is maximal. We use a carrier density sufficiently low in order to obtain a chemical potential at equilibrium ( μ) within the low energy bands. In the absence of an external magnetic field, the distributions shown for dJ=dtare qualitatively similar to the electron concentra- tions and current density at equilibrium.38,60Obviously, the imposed voltage bias creates an imbalance between carriers moving in opposite directions along the length of the nanowire. In ourcase, the total current is driven along the nanowire ranges betweena few nA to a few μA for voltage bias amplitudes of 1 –10 meV. 38 InFigs. 1(g) –1(i), we show the angular variation of the power density of the radiated electromagnetic field at the distance d¼2Rextfrom the center of the nanowires. The reference minimal values are Smin¼3:01, 3 :06, 1 :76/C210/C04W=m2for the hexagonal, square, and triangular nanowires, respectively. As expected fromthe current density distribution, the resulting field captures the internal geometry of the nanowires manifested by a number of peaks which is equal to the number of corners. The electrons local-ized in each corner form quasi-independent current channels canbe considered as individual nanowires. In classical terms, one can attribute to the radiated field a sin- gular behavior due to the quasi-one or two-dimensional currentdistributions within the prismatic shell with sharp corners and thinsides. 61However, here, we take into account properly the transverse degree of freedom consistently with the quantum localization of the electrons along parallel current filaments with transverse geom- etry and different strengths. This special current distribution leadsto multiple variations in the radiated field, of which the lower onewill have the largest strength. We, thus, expect weakly imposedquadrupole, or hexapole, or higher order pattern in the radiation field, or more unusual patterns, as we will see in Subsection B due to the presence of a magnetic field. Before that, in Fig. 2 , we show how the differences between the maximal and the minimal value of the power density curves,shown in Figs. 1(g) –1(i), decay with the distance from the center of the nanowire, within the near field zone, between d¼2R extand d¼10Rext. The fastest decay is observed for the hexagonal nano- wires, which have the least anisotropic geometry, closest to circular.As expected, the decay is less pronounced with decreasing the number of corners of the nanowire. The signals originating from the square and the triangular nanowires have similar intensity atthe position d¼2R ext, but they decay much slower around the tri- angular than around the square nanowire. The difference atd¼10R extis around 105,1 03, and 102times smaller for the hexag- onal, square, and triangular nanowires, respectively. B. Effects of a transverse external magnetic field In this subsection, we consider a uniform magnetic field per- pendicular to the nanowire axis. Such a magnetic field may drasti- cally perturb the current distribution within the tubular shell, mainly because the field component normal to the nanowire surface, whichJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-4 Published under an exclusive license by AIP Publishingis responsible for the Lorentz force, varies with the angular coordi- nate. In the case of a circular cross section and for strong enough fields, the electrons situated on the sides of the tubular shell, rela-tively to the direction of the magnetic field, experience a nearly zerolocal magnetic field and move on snaking trajectories along the wire in opposite directions. On the contrary, the electrons situated on the top or bottom regions tend to perform local cyclotron loops. 62–66The situation becomes more complex for a polygonal shell, where the localization resulting from the geometry competes with the one induced by the magnetic field.28,38As we pointed out pre- viously, when no longitudinal voltage is applied over the nanowire,the total current vanishes due to the equal number of electrons moving to the left and right. This holds for both zero and non-zero transverse magnetic fields. In the case of the current density J, FIG. 1. (a)–(c) Energy spectra obtained for the three polygonal shapes. The blue-dashed line marks the chemical potential μin equilibrium for a carrier density of n¼1017cm/C03, the arrows indicate the energy gap between the groups of corner and side states. (d) –(f) Time derivative of the current density ( dJ=dt) inside the nano- wire when its amplitude is maximal, i.e, at an instant in time when μ+¼μ. The color scale units are A =(s nm2). (g) –(i) Relative difference of the time-averaged magni- tude of the Poynting vector (radiated power density) as a function of the angle at a distance of 2 Rextfrom the center of the nanowire. The initial 0/C14is set at the rightmost corner of the shell in the figures and increases counterclockwise.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-5 Published under an exclusive license by AIP Publishingwhich reflects the underlying charge and velocity distributions, the presence of a non-zero magnetic field leads to splitting due to theLorentz force. This can be seen in Fig. 3 , where the equilibrium current density is split into channels consisting of electrons with positive and negative velocities, the former ones pushed to the right side of the sample, while the latter to the left side. The currentdensity takes the form of loops along the zaxis of the nanowire which close up at the ends of it (i.e., at infinity in our model).Then, even if the total integrated current is still zero, the current density is not compensated locally. However, in equilibrium, the channels that form each loop are compensated, i.e., the same current flows in both directions, witheach channel paired with the one on the opposite side of the geomet- ric symmetry axis relative to the magnetic field direction. Depending on the localization strength, the pairing can happen within the samecorner or side or on opposite ones. Thus, we observe that in the caseof the hexagon [ Fig. 3(a) ], for which the localization is the weakest, there is one main loop formed on the two sides parallel to the mag- netic field direction, where the snaking states are formed, and two additional loops originate from the split maxima in the corners situ-ated at angles 0 /C14and 180/C14with respect to the magnetic field. In the case of the square, Fig. 3(b) , the situation is similar, with one main loop that captures most of the current and two additional ones on the corners perpendicular to the field direction. For the triangularcase, Fig. 3(c) , where the localization is the strongest, there are three loops and a pair of opposite channels in each corner. If the magnetic field is strong enough, it can mix the two groups of corner and side states, leading to complex changes in theenergy spectra and in the current distributions. The dispersion withrespect to the wave vector kwhen a magnetic field is applied perpen- dicular to one of the edges is shown in Figs. 4(a) –4(c). We use the same carrier density n¼10 17cm/C03for the hexagon and the square cases so that the position of the chemical potential would be at thelevel of the corner states if there was no magnetic field applied, as itwas shown in Figs. 1(a) –1(c), whereas for the triangular case, we now use n¼10 18cm/C03in order to reach the side states. The time derivative of the current density ( dJ=dt), shown in Figs. 4(d) –4(f), contrary to the current distribution, has only either positive or negative sign, as the electrons are oppositely acceleratedor slowed down by the voltage bias, depending on its sign. For clarity, the time derivative of the current distribution over the whole time period is shown in Fig. 7 of the Appendix . InFigs. 4(g) –4(i), we show how the magnetic field affects the relative differences of the power density. The minimum values ofthe power density are now Smin¼1:74, 2 :23, 15 :45/C210/C04W=m2 for the hexagonal, square, and triangular nanowires, respectively. For the hexagonal nanowire, we observe a clear difference withrespect to the case without an external magnetic field. Instead ofthe six corner-related peaks, we now observe only two main ones with some shoulders. The reason is that now the Lorentz force pushes the current density to the sides where the snaking states areformed, which carry most of the current. We note, however, that the time-average Poynting vector is an integration over a complete time period. When the chemical poten- tials are imbalanced, the nanowire quickly carries net current only in one direction and thus, over half of a period the contribution of oneof the two sides is much smaller. On the contrary, the contributionof the two other regions where the additional loops are formedremains relevant during the complete period. Thus, on average, the two peaks of the maximum radiated power density appear on the angles corresponding to the two corners pierced by the field. Forclarity, to illustrate this effect, we also show the current density dis-tribution over a complete period in Fig. 8 of the Appendix . The situation is, to some extent, similar in the case of the square, but the difference in this case lies in the fact that each corner hosts current in both directions. Thus, integrating over aperiod, we still observe four peaks in the power density but nowwith different intensities. In the triangular case, the corner localization is stronger and the current distribution is locked in each corner if the carrier con-centration is too low. With n¼10 17cm/C03as before, the effects of a magnetic field of the order of 1 T are negligible, and the radiatedfield profile is similar to the one in Fig. 1(i) . For this reason, we increase the carrier density to n¼10 18cm/C03so that the chemical potential crosses the next group of states, which are side localized,for which the localization is weaker. The Lorentz force then spreadsthe current loops laterally with respect to the magnetic field. Forthe field orientation shown they are symmetrically enhanced in two corners. This pattern is reflected in the power density curve, where we now observe only two clear peaks at angles corresponding tothose corners, i.e., 120 /C14and 240/C14. FIG. 2. Decay of the relative difference in the log-scale between the minimum and maximum values of the radiated power density as a function of the distance from the center of the nanowire. FIG. 3. Current distribution in equilibrium inside the polygonal shells for n¼1017cm/C03when a magnetic field of B¼1 T is applied perpendicular to one of the edges for the hexagonal (a), square (b), and triangular (c) shells.The units of the color scale are nA =nm /C02.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-6 Published under an exclusive license by AIP PublishingTaking advantage of the localization mechanism induced by the external magnetic field, it is possible to tune the directivity of the nano- wires. In Fig. 5 , we consider the same parameters as in the previous examples, but we change the direction of the magnetic field. As can beseen in Fig. 5 , a change in the direction of the magnetic field does not only shifts the power density curves, as it happens for the hexagonal case but also creates a different localization that leads to different current density distributions. Thus, this change has also implicationson the power density curves, i.e., it mainly affects the overlapping of the contributions between the main loop and the weaker ones for the h e x a g o na n dt h er e l a t i v ed i f f e r e n c ei nt h e i ra m p l i t u d ef o rt h es q u a r e . For the triangular nanowire, the ro tation of the field, now parallel to one of the sides, leads to the accumulation of current in the corneropposite to its direction, which translates in the formation of a domi- nant peak in the power density curve as seen in Fig. 5(i) . The minimal values of the power density are in this later case Smin¼1:74, FIG. 4. As in Fig. 1 , the energy spectra (a) –(c), but now in the presence of an external magnetic field of B¼1 T perpendicular to one of the edges of the nanowire indi- cated by the arrows in (d) –(f). The power distribution outside the nanowire no longer reproduce the geometry of the cross section (g) –(i). The additional dashed green curve in (g) corresponds to the case of B¼0:5 T , shown in order to illustrate the evolution of the curve. The chemical potentials correspond to a carrier density of n¼1017cm/C03in panels (a) and (b) while for the triangular nanowire in (c) n¼1018cm/C03. The color scale units are A =(s nm2).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-7 Published under an exclusive license by AIP Publishing2:37, 15 :62/C210/C04W=m2for the hexagonal, sq uare, and triangular nanowires, respectively. Additionally, as can be seen in Fig. 5(c) , because of the absence of an inversion center for the triangle, the energy curves are no longer symmetric with respect to the wave vector kif the magnetic field points parallel to one of the sides. Finally, for comparison, in Fig. 6 , we show the relative differ- ence of the radiated power density for different orientations of the external magnetic field. For a hexagonal nanowire, where the geo- metric localization is the weakest, a change in the magnetic fieldorientation results in an angular shift of the power density curve. The change of shape of the power distribution can be attributed tothe polygonal geometry, as the electrons are being pushed to differ- ent areas of the shell and different loops are formed [ Fig. 6(a) ]. For the square nanowire, instead, the change of orientation of the fielddoes not shift the curves, as the geometry-induced localization isstronger than for the hexagonal case, but leads to a change of the contribution from each corner [ Fig. 6(b) ]. For the triangular nano- wire, for which we used a higher carrier density ( n¼10 18cm/C03), a FIG. 5. As in Figs. 1 and 4, panels (a) –(c) show the energy spectra, (d) –(f ) the time derivative of the current density, and (g) –(i) the power density around the nanowire, now with the external magnetic field perpendicular to one of the facets of the nanowires in (d) and (e) and parallel to one side for the triangular case (f ). The color scale units are A =(s nm2).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-8 Published under an exclusive license by AIP Publishingrotation of the magnetic field leads to a gradual decrease of the power density emitted by the current hosted by the corner perpen-dicular to its direction as the Lorentz force pushes the current tothe opposite side, where the main loop is formed [ Fig. 6(c) ]. IV. CONCLUSIONS Polygonal semiconductor core –shell nanowires exposed to a transverse external magnetic field incorporate complex electron localization mechanisms, yielding a rich phenomenology. The direction and magnitude of the magnetic field allow for the tunabil-ity of the electron localization and create current loops or channelsof current where electrons travel in opposite directions. The electro- magnetic field emitted by these nanowires subjected to an alternate current along their length depends on the underlying electron andcurrent distribution inside the nanowire. In order to study (some) consequences of the electron localiza- tion inside a prismatic core –shell nanowire, we obtained the radiated power density as a function of the angle in the neighborhood of it. We should mention that we on purpose avoided the term “near field, ”which is more familiar to the electrical engineers and implies a comparison between the wavelength of the radiated field and the dis-tance from the antenna. Still, our nanowires should be considered electrically small antennas, i.e., with the length much smaller than the wavelength. However, for our goal, the frequency and thus the wave-lengths are not essential parameters, the relevant domain for the fieldanisotropy being given by the geometry of the prismatic shell. In a recent work, a nonuniform optical field emitted by hexag- onal core –shell structures with imperfect internal geometry, via space resolved photoluminescence, has been detected and associ-ated with electron localization within the shell. 45In the present paper, we instead considered the radio frequency domain. We showed that the radiated electromagnetic field can capture the elec- tron and current density localization, and we studied how it isaffected by a magnetic field transverse to the nanowire, with differ- ent angular orientations. The magnitude of the power emitted by the nanowires highly depends on the parameters used in the model, i.e., carrier density,amplitude of the voltage bias, and, especially, frequency, and it can vary by orders of magnitudes. Rather than an in-depth study of the magnitude of the emitted power in different situations, our goal hasbeen to demonstrate, more qualitatively than quantitatively, that theresulting field follows the anisotropy of the charge and current distri-bution within the shell of the nanowire and has a tunable directivity, first, via the geometry and internal structure of the nanowire, and second, with an external magnetic field. Although we restricted ourstudy to individual nanowires, the extension of the ideas to arrays ofparallel nanowires, as nanowires are often grown, to achieve a com- bined effect and increased power is straightforward. If the electromagnetic field radiated by our core –shell nanowire could be measured with a receiver nanoantenna, at a sufficientlyshort distance, the data could be used for a tomography of thecharge distribution in the nanowire. The receiver may be another nanowire, shorter than the emitter, made either of a semiconductor 46 or a metal.67The other solution could possibly follow the examples of the near-field optical spectroscopy of a Yagi –Uda nanoantenna,68 or using a nanoantenna array as a receiver.69Another interesting example is a tandem of receiving –transmitting nanoantennas situ- ated 500 nm apart.70Indeed, technical challenges are inevitable, like approaching the two emitter and the receiver at a distance within theemitter near field pattern, adjusting the relative distance and angularposition of the two elements, etc., for example, like in the atomicforce or scanning tunneling microscopy. In addition, depending on the specific structure of the receiver, interaction with the emitter nanowire is possible and that should be included in the reconstruc-tion of the current distribution. The design of a proper setup forsuch an experiment, where all these issues are taken into account, is nevertheless beyond the scope of our present work, which we hope will stimulate an experimental attempt. FIG. 6. Relative difference of the time-averaged Poynting vector as a function of the angle for different orientations of the magnetic field for the three dif ferent cross sections: hexagonal (a), square (b), and triangular (c). The angle of the magnetic field is varied from being perpendicular to one of the corners to bei ng perpendicular to one of the sides in steps of 10/C14for the hexagonal and triangular nanowires and steps of 15/C14for the square. As in the previous cases, the power density is measured at a distance of 2 Rextfrom the center of the nanowire.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-9 Published under an exclusive license by AIP PublishingACKNOWLEDGMENTS This work was supported by the Icelandic Research Fund, Project No. 163438.APPENDIX: SUPPLEMENTARY FIGURES Time derivative of the current density distribution over a com- plete period when a magnetic field of B¼1 T is applied perpendicular FIG. 7. Time derivative of the current density distribution over a complete period when a magnetic field of B¼1 T is applied perpendicular to one of the edges of the nanowire. The carrier concentration is n¼1016cm/C03.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-10 Published under an exclusive license by AIP Publishingto one of the edges of the nanowire ( Fig. 7 ). Current density distribu- tion over a complete period when a magnetic field of B¼1Ti s applied perpendicular to one of the edges of the nanowire ( Fig. 8 ).DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. FIG. 8. Current density distribution over a complete period when a magnetic field of B¼1 T is applied perpendicular to one of the edges of the nanowire. The carrier concentration is n¼1016cm/C03.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 130, 034301 (2021); doi: 10.1063/5.0055260 130, 034301-11 Published under an exclusive license by AIP PublishingREFERENCES 1M. S. Gudiksen, L. J. Lauhon, J. Wang, D. C. Smith, and C. M. Lieber, “Growth of nanowire superlattice structures for nanoscale photonics and electronics, ” Nature 415, 617 –620 (2002). 2P. J. Pauzauskie and P. Yang, “Nanowire photonics, ”Mater. Today 9,3 6–45 (2006). 3D. You, C. Xu, J. Zhao, F. Qin, W. 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5.0051446.pdf

AIP Advances ARTICLE scitation.org/journal/adv Noise investigation in a spin-based four-qubit GaAs block of self-assembled quantum dots Cite as: AIP Advances 11, 065126 (2021); doi: 10.1063/5.0051446 Submitted: 25 March 2021 •Accepted: 4 June 2021 • Published Online: 17 June 2021 Konstantinos Prousalis,1,a) Agis A. Iliadis,2 Evangelos K. Evangelou,3 and Nikos Konofaos1 AFFILIATIONS 1School of Informatics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece 2Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA 3Department of Physics, University of Ioannina, Ioannina 45110, Greece a)Author to whom correspondence should be addressed: kprousalis@csd.auth.gr ABSTRACT Optically controlled self-assembled quantum dots have received substantial attention in the quantum computing area, as techniques for initializing, manipulating, and reading out single spin qubits have been demonstrated in essence. The electron-spin coherence and hole-spin coherence are limited due to noisy quantum effects, and there is a significant need for further evaluation and investigation studies. In this work, the behavior of charge noise and spin noise for a fundamental logic unit of four qubit embedded in an AlAs/GaAs heterostructure is reported based on the modeling and simulation approach in the atomic level to provide a more in-depth analysis and evaluation of quantum noise. The numerical calculations are based on reliable simulation methods, which are consistent with experimental results. The approach presented here can become the basis for scaled-up advanced simulations expanding to larger logical blocks of qubits. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0051446 I. INTRODUCTION When a self-assembled nanostructure is directed to bind one or more electrons (or holes) within solid-state materials such as semi- conductors, then a quantum dot (QD) is formed. Bringing QDs close enough in space to achieve the overlap of the wave functions is necessary for some applications such as the implementation of the quantum memory. Based on emerging fabrication techniques,1–4the coupling of these QDs is possible to be realized in a heterostructered semiconductor system. The spin degrees of the electrons (and/or holes) can then be examined as a two-state quantum system, the qubit.5,6In a magnetic field, the energy levels of spin-up and spin- down are split by the Zeeman effect energy in a non-degenerative way, distinguishing the spin-1/2 qubit model from other spin qubits such as the singlet–triplet or the exchange-only models. In this work, we restrict our scope to the singly confined electron as a spin-1/2 qubit in a self-assembled dot, as it represents a simple yet fundamental model for investigation. The ubiquitous physical phenomenon of quantum noise dis- turbs the ideal concept of a pure two-state spin qubit. The dissipative part of the spin dynamics can be attributed to spin-relaxation anddecoherence mechanisms, both restricting qubit’s lifetime to T1and T2, respectively. In many systems, the relaxation is the leading mech- anism that inevitably contributes to dephasing, which has a much shorter lifetime T∗ 2and constitutes a serious problem. Both mecha- nisms are related to different processes. For a single spin system with an isotropic g-factor, spin relaxation is associated with the decay of spin polarization parallel to the external magnetic field, while spin dephasing is associated with the decay of spin polarization transverse to the magnetic field.7 Various experiments on self-assembled InGaAs quantum dots set the ground for further research in quantum computation. Some cornerstone findings encouraged to meet DiVincenzo’s criteria.8So far, spin lifetime T1is higher than 20 ms,9while decoherence time T2is estimated to be about 3 μs.10,11Dephasing time T∗ 2has reached 0.1μs.12The operation time of Pauli’s gates Zand Xcan achieve the extremely low durations of 8.111,13and 4 ps,11,13,14respectively. The two-qubit gate, such as swap gate, does not take longer than 17 ps.15Yet, initialization and readout are feasible reaching fidelities at about 99%16and 96% or higher,17–20respectively. However, scal- ability continuous to be a serious problem for almost two decades, although the efforts with group IV host materials, such as Si or Ge, AIP Advances 11, 065126 (2021); doi: 10.1063/5.0051446 11, 065126-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 1. (a) Vertical Schottky diode’s heterostructure with the adopted hard- ware parameterization. (b) 3D represen- tation of the adopted device with a hypo- thetically accurate QD lattice. which eliminate the random nuclear spins.21Group III–V quantum dots, such as GaAs, have much better optical properties due to their indirect bandgap. Moreover, some example applications of dynamic decoupling schemes on GaAs qubit systems show a reduction in the surrounding nuclear spin noise,22–24which is an advantage over their group IV counterparts. More specific, these decoupling techniques use complex echo-like sequences to avoid surrounding fluctuations. The present investigation is conducted on noise interactions among four self-assembled QDs. Each of the four InGaAs dots is embedded in the ultra-pure GaAs layer having an AlAs superlat- tice blocking layer and a Schottky diode gate, as shown in Fig. 1. In this system, electron trapping and de-trapping are controlled by the gate voltage and the optical excitation. The major interactions that dissipate the coherent state of a singly confined electron-spin are assumed, and the noise amplitude at the location of a single QD is calculated under the dynamics developed among the four QDs. Noise power can be attributed to noise mechanisms, which result in enhancing the local electric and magnetic fields. The total space charge noise and spin noise are numerically calculated by adopting the simulation approach of Kuhlmann et al. ,25which is success- fully re-examined by a spectroscopic experimental approach based on resonance fluorescence. The simulation method is adjusted for the four-QD system examined here. An analysis of the dynamics for different parameterization is provided, while the Monte Carlo method is employed. The method can also be applied to evaluate the noise amplitude for the scaled-up extended problem of the QD array. Section II develops the adopted noise model with the relevant equations concerning space charge and spin noise. Then, the Monte Carlo simulation method is described, and the results are presented. II. MODELLING SYSTEM INTERACTIONS A quantum device of four self-assembled InGaAs QDs is exam- ined using the Schottky diode’s structure.26,27The parameters of this structure are detailed in Fig. 1 together with the adopted thickness of each layer. QDs in the wetting layer are separated from an n+-backcontact by a dtumthick GaAs tunnel barrier, while directly above the dots a GaAs capping layer of thickness dcapis deposited. The cap layer follows a blocking barrier AlAs/GaAs superlattice of thick- ness dsps, and another capping layer of dcthickness follows. On the surface, a semitransparent gate electrode for the Schottky diode is deposited. A negative voltage Vgis applied between the Schottky gate and the ohmic contact in the back of the structure. The position of self-assembled QDs is assumed to be symmet- rical with precision on a 2 ×2 grid within lattice constant D. Current fabrication techniques may not be able to achieve spatial and spectral precision yet, but modern techniques seem to be close to resolving this issue.1–4 It is known that the experimental procedure of resonance flu- orescence can be used to estimate noise within a QD. The sub- tle nature of a QD can be used by itself as a sensor nano-device to quantify noise interactions. A coherent laser system with a photon detector can be employed to carry out such an experiment by measuring the emitted photons caused from the applied detuned laser. The photon energies of the linearly polarized laser should be far below the energy gap of the host semiconductor avoiding inducement of extra noise. In many cases, the low noise state can be reached by temporary illuminating with a second non-resonant laser light, at very low frequencies. After a few hours, the noise is gradually reduced. Non-resonant optical excitation targets a single InGaAs QD in the monolayer (wetting layer), and an electron–hole pair is created. Almost instantly, the electrons relax rapidly to the back contact and the holes end up in the capping layer/blocking bar- rier interface. The hypothesis that holes are trapped at the interface is strongly supported by Houel et al.28and justifies the occurring space charges of the holes as it is estimated. Particularly, trapped holes are viewed as charge localization centers causing local electric fields that dissipate with time. A. Noise mechanisms The most prominent interactions between the QD electron- spin system and the noise sources are the hyperfine coupling with the surrounding nuclei and the collected excitation of the host semiconductor’s phonons together with the spin–orbit interaction AIP Advances 11, 065126 (2021); doi: 10.1063/5.0051446 11, 065126-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv of the system. Moreover, the existence of possible defects in the cap- ping layer above the dots may be trapped by holes and change the noise dynamics. Theg-factor fluctuations could be neglected since their contri- bution to the trapped electrons in the GaAs material is very weak.29 Other remaining noise sources may come from extra interactions between the phonons and the hyperfine interactions or other device structural factors, but these are considered secondary effects. The main noise mechanisms are due to the trapped holes and the hyperfine interactions. Standalone or combinations of these two noise mechanisms affect variably the resulting electromagnetic field at the location of the QD. The magnetic field is determined mainly by the Overhauser field of the nearby nuclei in the absence of any exter- nal field, while the electric field is determined mainly by the adjacent defects that trap holes. Moreover, phonon scattering and spin–orbit interactions are processes that may end up contributing to the relax- ation and decoherence of the spin states in an accumulated way and at different degrees. In order to obtain a less noisy environment, the resonant fluorescence is assumed to be recorded on a QD at wavelength 950.61 nm, power 0.55 μeV, and temperature 4.2 K, without exter- nal magnetic field. The weak optical excitation suppresses phonon scattering, thus increasing quantum confinement. The properties of the applied laser provide a minimal invasion on the QD, allowing an approximate evaluation of the charge and spin noise due to its high sensitivity. In order to obtain a less noisy environment, the resonant fluorescence is assumed to be recorded on a QD at wavelength 950.61 nm, power 0.55 μeV, and temperature 4.2 K, without exter- nal magnetic field. The weak optical excitation suppresses phonon scattering, thus increasing quantum confinement. The properties of the applied laser provide a minimal invasion on the QD, allowing an approximate evaluation of the charge and spin noise due to its high sensitivity. B. Nuclear spin noise The surrounding nuclear spin ensemble of the localized elec- tron spin provides the development of a spin bath, which is created by about Nequals 104–105nuclear spins located at the area of the lens-shaped QD. On itself, the dot’s spin bath of nuclei has lifetimes of aboutτ=0.1 ms due to the long dipole–dipole correlation time among nuclear spins in GaAs.30,31The isotropic contact hyperfine interaction among electrons and nuclear spins creates the Over- hauser field, and the interaction energy is expressed in the form of the Fermi contact interaction,32–34 Hhyp=v0 gμB∑N i=1Ai∣ψ(ri)∣2ˆIi, (1) where v0is the volume of an InGaAs unit cell, Aiis the hyperfine coupling strength determined by the value of the electron Bloch wave function at the site of each nucleus, ψ(ri)is the electron enve- lope wave function at the i-th nucleus, ˆIiis the spin operator of nuclear spins, gis the electron g-factor constant, and μBis the spin magneton moment. However, the effect of hyperfine interactions can be equivalently described with an effective magnetic field seen by the QD spin, which is commonly referred to as Overhauser field BN,BN=A gμBNeff∑Neff i=1ˆIi, (2) where Neffdenotes the number of nuclear spins inside the top hat envelope function ψ(r). In fact, the electron wave function is approx- imated with a top hat function. The parameter Anow is the average spin nuclei coupling constant. Due to the arbitrary direction of the Overhauser field, the spin ground states become admixed. In our simulation parameterization in Table I, the parameter Neffis∼375, merely reduced, in order to balance the fact that the real spins are larger than 1/2. Moreover, the isotropic part of the electron–hole exchange interaction leaves almost intact X0from the in-plane fluctuations of the nuclear magnetic field. As a result, the exciton energy does not depend on the in-plane Overhauser fields.25,35 According to experiments, the neutral exciton (without the contribution of extra charges) has smaller nuclear spin correlation time than the charged types of excitons.13The nuclear spin bath by itself has a large lifetime of the order of̷hN A, which in the case of GaAs is about ∼0.1 ms.36–39 For the nuclear spin dipole–dipole interaction, the correlation duration is estimated by τ=̷h Edd=̷h μI1BI2, (3) where Eddis the energy of a nuclear dipole with moment μI1in the magnetic field BI2of another dipole at distance dand gyromagnetic ratioγ. The magnetic field of dipole I2can be approximated as TABLE I. Simulation parameters. System condition @T Temperature 4.2 K @⟨δ⟩ Detuning 0 @Bext External magnetic field 0 @T Integration time 10 min @tbin Sampling rate 10 μs @Γ0 Linewidth 0.93 ±0.1μeV @D Lattice distance 350–450 nm Charge noise simulation a Stark shiftα 31.7μeV cm/kV Nh Hole density 0.2 ×1010cm−2 τ0 Center lifetime (unoccupied) 3 s τ1 Center lifetime (occupied) 50 μs p Transition probability 1.6% Spin noise simulation g g-electron factor 0.5 Δ Fine structure 17.3 μeV Neff Number of spins in the top hat 375 envelope function ψ A Hyperfine interaction constant 90 μeV τ0 Center lifetime (unoccupied) 8 μs τ1 Center lifetime (occupied) 8 μs AIP Advances 11, 065126 (2021); doi: 10.1063/5.0051446 11, 065126-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. (a) Random dispersion of defects (red crosses) having tendency to trap holes above four QDs (black dots) with a density of N c=32×106/cm2. (b) Elec- tric field as a function of D for the four-QD system. BI2∼μ0 2πμI2 d3, (4) whereμIi=̷hγiIi. The constant μ0is the vacuum magnetic permeability. C. Coulomb noise Electrostatic noise mainly arises from an ensemble of localized charges, in this case holes, at the interface of the active layer and blocking layer. These are charge localization centers, which may be occupied intermittently or not by a carrier, which for this setting is a hole. Each time a center is occupied by a single hole, an energy shift occurs in the QD. The centers occur with a density Ncand may be occupied with probability por unoccupied with probability 1- p. The electric field Fhiis produced by a trapped hole in a defect at distance dfrom the QD. The intensity is calculated by the algebraic sum of the following Coulomb equations: ˆFhi=ˆFh+ˆFIm1+ˆFIm2, (5) where Fhis the electric field due to the positive charge and, FIm1 andFIm2, due to the induced image force which lowers the potential. These two image carriers are the result of the coulomb interaction between the two positive carriers and their mirror image of two neg- ative charges. The first one in the top gate and a second one in the back contact. The overall electric field Fcover a single QD in the 2×2 lattice is again the algebraic sum of all the interactions caused by the occurring trapped holes, ˆFc=ˆFh1+ˆFh2+ ⋅ ⋅ ⋅ + ˆFhM, (6) where each Fhi(i=1, 2, . . ., M) is the electric field formation due to its respective hole. M is the maximum number of holes that may appear. All these electrostatic fields are estimated by the standard equations of Coulomb, Fhi,x(x,y,z)=±ex 4πϵ0ϵr1 d3, (7) Fhi,y(x,y,z)=±ey 4πϵ0ϵr1 d3, (8)Fhi,z(x,y,z)=±ez 4πϵ0ϵr1 d3. (9) Every QD undergoes 12 interactions due to surrounding charge carriers with each of their image charges being at distance d =√ x2+y2+z2. The distance zis always equal to dcap, as this is the hypothesis in the simulation of Kuhlmann’s work. In Fig. 2, a vivid representation of the random activation of the charged localization centers is demonstrated by the red crosses. In this simulation, about one localization center is occupied in 400 ×400 nm2for each QD. III. SIMULATION METHOD Both electric and magnetic fields are simulated as an ensem- ble of on/off fluctuating point charges and atomic nuclear spins, respectively. In each time step Δt, the overall configuration of the two ensembles determines their current power. In the theory of current noise in semiconductors, the power spectrum of a two- level fluctuator is Lorentzian, while the rates of transition for occu- pied and unoccupied lifetimes are additive.40The same strategy can be adopted for the magnetic field with a reasonable approxima- tion. When the lifetimes of a non-activated (state 0) or activated (state 1) fluctuator are τ0andτ1respectively, then their occur- ring probability is τ0/(τ0+τ1)andτ1/(τ0+τ1), respectively. Hence, given the Lorentzian distribution of both power spectra, the transi- tion probability between the two flipping states can be estimated by p0→1(δt)=1−τ0+τ1e−δt/T τ0+τ1, (10) p1→0(δt)=1−τ0e−δt/T τ0+τ1, (11) where 1/T=1/τ0+1/τ1. The transition probability and a random number generator will set the configuration for the two-level fluctu- ator ensemble for each time step. The localization centers are treated independently. Figure 3 demonstrates how the lifetimes of 0 and 1 states change in time according to Eqs. (10) and (11). The localization centers of the charge carrier’s ensemble can be modeled having density Ncon a 2D array. Holes may be trapped AIP Advances 11, 065126 (2021); doi: 10.1063/5.0051446 11, 065126-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. Probability evolution of localization centers remaining at states 0 and 1. with probability Nh=pNc. Centers directly above and around the dot-Z-axis proximity would have a strong impact on noise, but they are rarely observed about the Z axis allowing for their neglection. All the trapped holes in the four-qubit division will generate an electric field Fhi, and the total overlapping field at the location of a single QD isFc. In a similar and more simplistic way, the localization centers that constitute the effective spin nuclei ensemble can be modeled again as a 2D array. Each nucleus is treated as an on/off fluctuator and forms a magnetic field Bi. The total momentary electromagnetic field will be BN. Both transition rates are assumed to be equal to 1/ τ according to the estimated correlation time. Typically, the power spectrum of a specific signal is estimated by an averaged periodogram method using the relevant recorded experimental data. Recording QD’s emitted resonance fluorescence signal S(t) can accurately lead to the estimation of the noise spec- trum of the QD. The resonance fluorescence signal is the summation of the quantum dot and laser noise: NRF(f)=NQD(f)+Nexp(f). More specifically, Nexparises from the intensity fluctuations in the laser (shot noise) and has a 1/ f2-behavior at low frequencies (<10 Hz). At higher frequencies, it becomes larger than NQD. According to the averaged periodogram method, we apply the fast Fourier transform ( FFT ) to the normalized resonance fluores- cence signal S(t)/⟨S(t)⟩and then calculate the square magnitude of the result, as in the following equation: NRF(f)=∣FFT[S(t) S(t)]∣2 t2 bin/T, (12) where⟨S(t)⟩is the average number of photons per sampling dura- tionΔt=tbinandTis the overall measurement time. The sampling rate is considered as the way to group a specific amount of data within a typical value of 1 μs.The Lorentzian distribution of the emitting resonance fluo- rescence signal could be formulated by the following equation: S(t)=(Γ0/2)2 (αFc(t)+δ0(t)+δ)2+(Γ0/2)2, (13) whereΓ0is the transform-limited linewidth of the hypothetical applied laser, δis the laser detuning, and αis the Stark shift coeffi- cient of the exciton. Detuning δ0represents the blue Zeeman branch due to the magnetic field BN, δ0(t)=±1 2√ Δ2+(gμBBN(t)/2)2, (14) whereΔis the fine structure splitting, gis the electron g-factor, and μBis the Bohr magneton. The positive sign is considered when the laser is tuned above (blue-transition) the resonance frequency, while the negative is true when the resonance fluorescence laser is tuned below (red-transition) the resonance frequency. The Monte Carlo method is used to simulate the modified dynamics of the localized fluctuations under the four-qubit regime, giving reliable results as the generalization of the simulation protocol is feasible. Basic parameters for such a simulation can be the occupa- tion probability p, the density of the charge fluctuators Nc, lifetimes of fluctuating centers, and distance d. IV. SIMULATION RESULTS The simulation results focus on the case of the neutral exci- tonX0complex. The noise spectrum is numerically calculated at the location of one QD interacting with the rest of the three and under the variation of parameter D. A. Noise spectrum in QD Due to the assumed manufacturing symmetry of the four-QD lattice, it would be sensible to consider that approximately the same amount of noise power is aggregated at the location of each QD. Small and non-uniformly dispersed imperfections in the structure may not allow for a tantamount aggregation of noise, but generally it can be controlled by using ultra-pure materials and careful design methods. The basic parameters in our simulator concern the evolution probability pof either electric or spin centers, the QD lattice con- stant D, and the adopted sampling rate with the overall time Tof the simulated experiment. The total noise amplitude NQDfor a QD is presented in Fig. 4 in the frequency range of 10–104Hz with a sampling rate of 10 μs. Smaller sampling time units would offer a somewhat better resolu- tion as it considers the noise amplitude, but it would be relatively demanding in computational power. Even higher frequencies would be demanding, too. Quantum dot noise is just a small portion of the known shot noise, which stems from the optical beam in a typical resonance fluo- rescence experiment. The amplitude of shot noise is usually revealed at low frequencies ( <1 Hz) where it is higher than dot’s noise and becomes approximate to dot’s noise at ∼10 Hz. The unveiling por- tion has a 1/ fαformation, and its slope determines α(typically 0.5<α<2). Noise amplitude fluctuations are reduced for higher AIP Advances 11, 065126 (2021); doi: 10.1063/5.0051446 11, 065126-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 4. Noise spectra on a neutral exciton X0is simulated within a four-QD embedded lattice system at a distance of D =400 nm. The full parameterization is described in Table I. frequencies as the sampling rate is higher and the average noise level is stabilized. Comparing the noise amplitude throughout the frequency spectra 10–104Hz in Fig. 4 with the respective Kuhlmann’s sim- ulation result [Fig. 4(a) in Ref. 25], a relatively small increment is observed. Both simulation results are conducted under the same hosting conditions and simulation parameters (see Table I) for the neutral exciton. Such an augmentation would be expected at D=400 nm for the four QDs, but that is below the threshold of one order of magnitude and such a quantification, mainly due to charge carriers, seems to be interesting. B. Lattice dependence It is important to find the balance between lattice constant D and noise level NQD. Even in the order of a few tens of nanometers, smaller distances may have a large impact on quantum computa- tional speed, so much, as in noise. Both the electric and magnetic fields are examined in the distance Drange between 100 and 900 nm. The continuous and dynamical formation and relaxation of the four occurring excitons within each QD lead to a system of four trapped holes with high probability, under the specific parameterization in Table I. In each of the four ensembles of defects, the position of each hole is random, maintaining a relatively neighboring proxim- ity. The random placement of the emerged holes is similar to this one in Fig. 2 where about five defects are populated per 400 ×400 nm2 surface. Each QD has two QDs at distance Dand one diagonally at distance D√ 2. The Monte Carlo method is employed again to esti- mate the fluctuating electric and magnetic fields during a simulation period T. In Fig. 2(b), the power of the electric field in QD 1is cal- culated under different values of lattice constant Din the range of 100–900 nm. The amplitude of the five discrete (see color- ing) plots in Fig. 5 is reduced as D is increased with a rate that reflects the reciprocal of the cubic Fc. For higher values of D (>500 nm), the plots are calculated almost on the same region. FIG. 5. The electric field is calculated at the location of QD 1for five gradually increasing D values. The full parameterization is given in Table I. When D>500, it is obvious that the power of the electric field is stabilized. Fluctuations in the magnetic field are dominated by the local hyperfine coupling within the dots, making the magnetic noise con- tribution independent of the lattice spacing. On the other hand, the electric power is highly dependent on the lattice constant. The plot in Fig. 2(b) confirms that Fchas a 1/ x3behavior for increasing D. It is supposed that each QD is empty—not charged with captured electrons—and has only one trapped hole above. The main result of this study is summarized in Fig. 6. The aggregated amount of noise is estimated at the center of QD 1while interacting with the rest three QDs. The four ensembles of defects are treated as a common ensemble, which affect undividedly the elec- tromagnetic intensity at the center of the dots maintaining the same parameterization of Table I. The Monte Carlo method was conducted with a sampling rate of 10μs and an integration time of 10 s. It is also supposed that there were no spectral resonances during the simulation. The comparison in Fig. 6 is carried out for a lattice spacing between 300 and 450 nm. The best regime to minimize noise levels without occupying redun- dant square nanometers would be in the region of 400 ±20 nm. Below 100 Hz, we observe higher fluctuations in the amplitude since the QD is driven at a slower rate by the laser. As a result, the sam- pling rate for the Monte Carlo simulation is lower allowing for more scattering of the points. The relationship between the lattice constant Dand the noise levels is examined by calculating the noise spectra for three critical and different distances 300, 350, and 450 nm, in Fig. 6. For D=400 or 450 nm, the noise amplitude is almost the same. It means that we should not expect a significant reduction in noise for higher Dsince the electrostatic interaction in this system among QDs becomes neg- ligible. At D=300 nm, a small amount of noise increases the surface area of the blue plot, which indicates that the electric field affects the noise in, e.g., QD1. For lower D, there is no much increment in noise since the contribution of the trapped holes is small. AIP Advances 11, 065126 (2021); doi: 10.1063/5.0051446 11, 065126-6 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 6. Noise spectra comparison between 300, 400, and 450 nm. The full parameterization is given in Table I. V. CONCLUSIONS In this study, we extend the simulation method of Kuhlmann et al. for a 2 ×2 QD lattice embedded in a GaAs quantum well in a Schottky diode, in order to provide a faithful simulation and investigate the noise amplitude. The role of the distance among QDs is examined under the formation of defects, which trap holes. The optimal position of the four dots was found at about D=450 nm. In the quantum computer architecture engineering context, looking for the optimal interdot spacing in a lattice via the reso- nance fluorescence method may be proved accurate and efficient. Various experiments are based on spectroscopy on an individual self-assembled quantum dot studying the emitted photons, and this method can be exploited for a system of more quantum dots. This simulation sets the base for the next simulation step supposing to be a 3 ×3 lattice of nine QDs, allowing for eight dipole–dipole interactions, which is the maximum affordable con- gestion in a 2D surface. Then, it would be challenging to incorporate the noise overhead of possible probing systems that would act in a high proximity to such a lattice. DATA AVAILABILITY The data that support the findings (simulation program) of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. Schneider, M. Strauß, T. 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5.0048323.pdf

Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi Preparation of individual magnetic sub-levels of4He(23S1) in a supersonic beam using laser optical pumping and magnetic hexapole focusing Cite as: Rev. Sci. Instrum. 92, 073203 (2021); doi: 10.1063/5.0048323 Submitted: 22 February 2021 •Accepted: 14 June 2021 • Published Online: 2 July 2021 Tobias Sixt, Jiwen Guan,a) Alexandra Tsoukala, Simon Hofsäss,b) Thilina Muthu-Arachchige,c) Frank Stienkemeier, and Katrin Dulitzd) AFFILIATIONS Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany a)Current address: National Synchrotron Radiation Laboratory, University of Science and Technology of China, 230029 Hefei, People’s Republic of China. b)Current address: Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany. c)Current address: Institute of Applied Physics, University of Bonn, Wegelerstr. 8, 53115 Bonn, Germany. d)Author to whom correspondence should be addressed: katrin.dulitz@physik.uni-freiburg.de ABSTRACT We compare two different experimental techniques for the magnetic-sub-level preparation of metastable4He in the 23S1level in a supersonic beam, namely, magnetic hexapole focusing and optical pumping by laser radiation. At a beam velocity of v=830 m/s, we deduce from a comparison with a particle trajectory simulation that up to 99% of the metastable atoms are in the MJ′′=+1 sub-level after magnetic hexapole focusing. Using laser optical pumping via the 23P2–23S1transition, we achieve a maximum efficiency of 94% ±3% for the population of the MJ′′=+1 sub-level. For the first time, we show that laser optical pumping via the 23P1–23S1transition can be used to selectively populate each of the three MJ′′sub-levels ( MJ′′=−1, 0,+1). We also find that laser optical pumping leads to higher absolute atom numbers in specific MJ′′ sub-levels than magnetic hexapole focusing. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0048323 I. INTRODUCTION In a He gas discharge, two long-lived, excited (“metastable”) atomic levels are formed by electron-impact excitation from the 11S0electronic ground level: the 23S1level (electronic energy E=19.8 eV1and natural lifetime τ=7870 s2) and the 21S0level (E=20.6 eV1andτ=19.7 ms3). In the following, the metastable He atoms are referred to as spin-polarized when only a single magnetic sub-level of He(23S1) is populated. Such spin-polarized metastable He (HeSP) is used for a wide range of applications. Special interest is currently devoted to He magnetometry for the quantum sensing of very small magnetic fields, e.g., see Refs. 4 and 5. In metastable atom electron spectroscopy,6also referred to as metastable de-excitation spectroscopy, HeSPhas, for example, been used for probingsurface magnetism.7In atom optics, HeSPhas found applications in nanolithography, as well as in atomic waveguides and beam splitters for atom interferometry.8,9HeSPalso serves as a source of polarized electrons10,11and ions,12e.g., for atomic and high-energy nuclear scattering experiments. Besides that, spin-polarized samples of3He(23S1) are used for biomedical imaging, e.g., to visualize the human lung.13–15 Supersonic beams of HeSPare typically produced by opti- cal pumping,16as well as by magnetic (de-)focusing and mag- netic deflection. Optical pumping of4He(23S1) via the 23PJ′–23S1 transition (where J′=0, 1, 2) at a wavelength of λ=1083 nm has first been achieved by Franken, Colegrove, and Schaerer using a helium lamp.17–19A few years later, the optical pumping of the 23S1level of the3He isotope has also been demonstrated using Rev. Sci. Instrum. 92, 073203 (2021); doi: 10.1063/5.0048323 92, 073203-1 Published under an exclusive license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi a similar setup.20,21The more recent use of narrowband laser radia- tion has proven to be particularly efficient for the optical pumping of He in the 23S1state.22–25 Apart from that, the interaction of a spin with an inhomo- geneous magnetic field has also been used for producing spin- polarized atomic beams of3He(23S1) and4He(23S1). These level preparation techniques include Stern–Gerlach deflection,26–31mag- netic hexapole focusing,32–38and Zeeman deceleration.39,40 A comparison between the different techniques for HeSPprepa- ration in a supersonic beam is of paramount importance for experi- mental design considerations. In this article, we describe the results of a comparative study aimed at the laser optical pumping of 4He(23S1) into a single MJ′′sub-level (where MJ′′=−1, 0,+1) and at the magnetic hexapole focusing (defocusing) of the MJ′′=+1 (MJ′′=−1) sub-level of4He(23S1) using an array of two mag- netic hexapoles. We have determined the efficiency for MJ′′-sub- level selection using low-cost fluorescence and surface-ionization detectors, respectively, which can easily be implemented in other experiments. II. EXPERIMENTS Major parts of the experimental setup have already been described elsewhere.41,42Briefly, a pulsed4He beam is produced by a supersonic expansion of4He gas from a high-pressure reservoir (30–40 bar) into the vacuum using a home-built CRUCS valve43 (30μs pulse duration). An electron-seeded plate discharge (attached to the front plate of the valve) is used to excite an ≈4⋅10−5frac- tion of He atoms from the 11S0electronic ground level to the twometastable levels, 21S0and 23S1, referred to as He∗hereafter.41After passing through a 1 mm-diameter skimmer at a distance of 130 mm from the valve exit, the supersonic beam enters a second vacuum chamber, in which a specific magnetic sub-level of the 23S1level is prepared using laser optical pumping or selected using magnetic hexapole focusing (see below). The distance between the skimmer tip and the center of the optical pumping region (hexapole magnets) is 228 mm (331 mm). The He∗flux and the He∗beam velocity are determined using Faraday-cup detection at well-known positions along the supersonic beam axis y. For the experiments on optical pumping, the pulsed valve is operated at room temperature, resulting in a supersonic beam of He∗with a mean longitudinal velocity of v=(1844±6)m/s (250 m/s full width at half maximum, FWHM). For the experi- ments on magnetic hexapole focusing, the pulsed valve is cooled by using a cryocooler (CTI Cryogenics, 350CP), and the valve temperature is actively stabilized to 42 K using PID-controlled resistive heating. This results in a supersonic beam of He∗with a mean longitudinal velocity of v=(830±17)m/s (≈130 m/s FWHM). A. Laser optical pumping The energy-level schemes and the experimental setup used for laser optical pumping are shown in Figs. 1(a) and 1(b), respectively. Optical pumping is achieved by laser excitation via the 23P1–23S1 transition or via the 23P2–23S1transition at λ=1083 nm, respec- tively. The laser light for optical pumping is generated by a combi- nation of a fiber laser and a fiber amplifier (NKT Photonics, Koheras BOOSTIK Y10 PM FM, 2.2 W output power, 10 kHz linewidth). FIG. 1. (a) Left: He energy levels relevant for the experiments described in the main text. The level energies are taken from Ref. 1. Right: Transitions relevant for the laser optical pumping of He(23S1) via the 23P0(top), the 23P1(middle), and the 23P2(bottom) levels, respectively. The relative transition strengths for σ+,π, andσ−excitation are labeled in pink, green, and blue, respectively. (b) Schematic illustration of the experimental setup used for optical pumping and fluorescence detection. HHC =Helmholtz coil, PBS =polarizing beam splitter, λ/4=quarter wave plate, PD =photodiode, P =polarizer, and L =aspheric lens. (c) Schematic drawing of a magnetic hexapole array in Halbach configuration. (d) Sketch of the detection system for magnetic hexapole focusing, including the two Halbach arrays (HAs), the wire (W) detector, and a Faraday-cup (FC) detector. All dimensions in (b)–(d) are given in units of mm, and they are not to scale. Rev. Sci. Instrum. 92, 073203 (2021); doi: 10.1063/5.0048323 92, 073203-2 Published under an exclusive license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi The laser frequency is stabilized using frequency-modulated, satu- rated absorption spectroscopy in a He gas discharge cell. Since the frequency difference between the 23P1and 23P2spin–orbit levels is onlyΔf≈2 GHz,1the laser frequency can be changed between the different transitions without effort. The laser light is guided to the vacuum chamber using a polarization-maintaining single-mode fiber, where it is collimated to a beam diameter of 2 w0≈14 mm ( w0is the Gaussian beam waist). Before the laser beam enters the vacuum chamber, it passes a polarizing beam splitter for polarization clean-up and a quarter wave plate for the production of circularly polarized light. Inside the vacuum chamber, the laser beam crosses the supersonic beam at right angles and parallel to the direction of the magnetic field produced by two coils in near-Helmholtz configuration (radius R=55 cm). The thus-produced homogeneous magnetic field of Bz≤4.5 G provides a uniform quantization axis for all He atoms in the supersonic beam. The level-preparation efficiency is determined by measuring the laser-induced-fluorescence (LIF) of the He atoms in the direc- tion perpendicular to the supersonic beam and the laser beam. The fluorescence light is collected and focused onto an InGaAs photo- diode (Hamamatsu, 1 mm active area diameter, photosensitivity of RPD=0.63 A/W at λ=1080 nm) using two anti-reflection-coated, aspheric lenses (Thorlabs, 75 mm diameter, 60 mm focal length). Due to the symmetric lens configuration [as shown in Fig. 1(b)], the fluorescence collection region in the yzplane is expected to be of the same size as the detection region, which is given by the active area of the photodiode. The output of the photodiode is ampli- fied using a home-built transimpedance amplifier with a gain of GPD≈5⋅105V/A. A rotatable linear polarizer (Thorlabs, extinc- tion ratio>400 : 1 atλ=1083 nm) is mounted in between the lenses in order to analyze the polarization of the fluorescence light. All the optical components of the fluorescence detector are placed into a single lens tube system to ensure the correct alignment of the opti- cal parts. The entire detector assembly is positioned on a transla- tional stage outside the vacuum chamber, which can be moved along theyaxis. Under normal operating conditions, the number of He atoms in the 23S1level is≈109/pulse, as inferred from the signal on the Faraday-cup detector.41For excitation via the 23P2–23S1transition, the time-dependent signal of the photodiode has a peak voltage of UPD,max≈41 mV and an FWHM of 27 μs. The peak flux of detected photons is then inferred from UPD,max using ˙Nph,max=UPD,max hνRPDGPD≈7⋅1011photons s, (1) where his Planck’s constant and νis the corresponding transition frequency. From these measurements, we infer a root-mean-square noise amplitude of Unoise=6.4 mV for a single measurement, which improves to Unoise=0.4 mV by averaging over 300 gas pulses. This results in a signal-to-noise ratio of SNR=U2 PD,max U2 noise=⎧⎪⎪⎨⎪⎪⎩16.2 dB, single measurement 40.3 dB, 300 averages.(2)AtSNR=0 dB, we thus infer a detection limit of ˙Nph, lim≈1⋅1011 photons/s ( ˙Nph, lim≈7⋅109photons/s) for a single measurement (300 averages). B. Magnetic hexapole focusing For the magnetic focusing of He(23S1,MJ′′=+1), we use a set of two Halbach arrays44,45in hexapole configuration, sketched in Fig. 1(c), whose design has already been described previously.46,47Each hexapole array consists of 12 magnetized seg- ments (Arnold Magnetic Technologies, NdFeB, N42SH; a rema- nence of B0=1.3 T), which are glued into an aluminum housing and placed on a position-adjustable rail at a center-to-center distance of 14.6 mm. To determine the focusing properties of the magnet assem- bly, a thin stainless-steel wire [diameter dwire=0.2 mm, labeled “W” in Fig. 1(d)] is used as a position-sensitive Faraday-cup-type detector. Its position along the yand xaxes can be varied by a maximum of 180 and 50 mm, respectively, using a set of two pre- cision linear translators. A second Faraday-cup detector [labeled “FC” in Fig. 1(d)], i.e., a stainless-steel plate of 30 mm diame- ter, is placed behind the wire detector to determine the He∗beam velocity. III. RESULTS AND DISCUSSION A. Laser optical pumping In general, the level preparation efficiency relies on the polar- ization state of the laser radiation, on the energy-level structure of the involved electronic levels, and on the transition strengths for photon absorption and emission. If atoms are excited with σ+(−)-polarized light, the change in angular momentum between the upper and lower levels is ΔMJ′′=MJ′−MJ′′=+1(−1)for every photon-scattering event, where MJ′and MJ′′are the magnetic projection quantum numbers for the upper and the lower mag- netic sub-levels, respectively. For excitation with π-polarized light, ΔMJ′′=0. The transition strengths for the 23PJ′–23S1transitions (where J′=0, 1, 2) in He are shown in Fig. 1(a). As can be seen from Fig. 1(a), all MJ′−MJ′′transition strengths for the 23P2–23S1tran- sition are non-zero. Hence, in this case, multiple excitation cycles withσ+(−)-polarized light lead to equal populations in the 23S1,MJ′′ =+1(−1)and 23P2,MJ′=+2(−2)sub-levels, respectively. In this case, photon emission via this transition continues to occur as long as the atoms are subject to laser excitation. In contrast to that, the emission of photons ceases after a few pumping cycles for σ+(−)exci- tation of the 23P1–23S1transition since all population is trapped in the 23S1,MJ′′=+1(−1)sub-level. Likewise, the excitation of the 23P1–23S1transition using π-polarized light leads to a transfer of population into the 23S1,MJ′′=0 sub-level and photon emission stops as a result of the zero transition strength for the 23P1,MJ′ =0−23S1,MJ′′=0 transition. As can be inferred from Fig. 1(a), the selective population of a single MJ′′sub-level in He(23S1) via the 23P0–23S1transition is more complicated, as it requires two different laser polarization states. Therefore, we have focused our experimental work on the Rev. Sci. Instrum. 92, 073203 (2021); doi: 10.1063/5.0048323 92, 073203-3 Published under an exclusive license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi optical pumping of He(23S1) via the 23P2–23S1and 23P1–23S1tran- sitions, respectively. In the following, we provide a brief descrip- tion of the different optical pumping schemes and the meth- ods used for analyzing and optimizing the sub-level preparation efficiencies. Furthermore, we compare our results with literature values. 1. Optical pumping via the 23P2–23S1transition As stated above, the excitation of the 23P2–23S1transition with σ+(−)-polarized light leads (after a few excitation cycles) to the opti- cal cycling between the 23S1,MJ′′=+1(−1)sub-level and the 23P2, MJ′=+2(−2)sub-level. In this case, the atomic fluorescence only consists ofσ+(−)-polarized light because ΔM=MJ′−MJ′′=+1(−1). During the first optical pumping cycles, and for a non-perfect cir- cular polarization of the input light, sub-levels with MJ′≠+2(−2) are also populated and can decay back to the 23S1level while emit- tingσ−(+)- andπ-polarized photons as well. Therefore, the polariza- tion purity of the emitted fluorescence provides information about the sub-level preparation efficiency. Since the polarization of flu- orescence photons is given with respect to the quantization axis, which is the pointing of the external magnetic field vector, fluo- rescence photons of σ±polarization that are detected along an axis perpendicular to the quantization axis are projected onto a linear polarization. The direction of this projected linear polarization is again perpendicular to the quantization axis. Furthermore, fluores- cence photons of πpolarization are linearly polarized parallel to the quantization axis. Therefore, we analyze the purity of the emitted light using a linear polarizer plate, and the fluorescence intensity can be ascribed to σ±polarization ( πpolarization) if the transmis- sion axis of the polarizer Pis perpendicular (parallel) to the mag- netic field component Bz, respectively.48For example, the results of fluorescence measurements at different polarizer angles are shown in Fig. 2. FIG. 2. Fluorescence intensity as a function of polarizer angle for excitation with σ+-polarized light via the 23P2–23S1transition. The experimental data points are shown as red circles. The black curve is a sine fit to the data. The dashed vertical lines represent the two angles at which the transmission axis of the polarizer Pis perpendicular or parallel to the magnetic field component Bz, respectively. Here, an efficiency of η+1≈93% is determined from the fit to the data.In order to compare the sub-level preparation efficiencies, we define ηi=ρ(MJ′′ i) ∑iρ(MJ′′ i)(3) for producing a specific magnetic sub-level population ρ(MJ′′ i)of He(23S1) (where i=−1, 0,+1). For the 23P2–23S1transition, the effi- ciency for optical pumping into the 23S1,MJ′′=+1(−1)sub-level is thus obtained using η+1(−1)=1−IF(P∥Bz) IF(P∥Bz)+IF(P/⊙◇⊞Bz), (4) where IF(P∥Bz)andIF(P/⊙◇⊞Bz)are the fluorescence intensities for emission at polarizer axes P∥BzandP/⊙◇⊞Bz, respectively. 2. Optical pumping via the 23S1–23P1transition Excitation via the 23P1–23S1transition allows for the selective optical pumping into each of the MJ′′sub-levels in He(23S1). When the atoms are excited with pure σ+(−)-polarized light, all popula- tion is pumped into the 23S1,MJ′′=+1(−1)sub-level. Since this is a dark sub-level, fluorescence emission should only occur in the first few pumping cycles. However, by using a mixture of σ+- and σ−-polarized excitation light, the dark sub-level is remixed so that optical cycling (and thus fluorescence emission) continues to occur. In the present configuration, the input polarization is changed by varying the angle Φof the quarter wave plate compared to the axis of the incident linear laser polarization. The observed change in the flu- orescence intensity as a function of quarter wave plate angle is shown in Fig. 3. The efficiency for pumping into the 23S1,MJ′′=+1(−1) FIG. 3. Red circles: Measured fluorescence intensity for excitation via the 23P1–23S1transition as a function of quarter wave plate angle Φ. The quarter wave plate angles for excitation with pure σ+-polarized light and with an equal mixture ofσ+- andσ−-polarized light are indicated as dashed vertical lines. The inset shows the results of a measurement in a region around Φ=45○taken under different experimental conditions. Rev. Sci. Instrum. 92, 073203 (2021); doi: 10.1063/5.0048323 92, 073203-4 Published under an exclusive license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi sub-level is determined using η+1(−1)=1−IF(σ+(−)) IF(σ++σ−), (5) where IF(σ+(−))andIF(σ++σ−)are the fluorescence intensities for excitation with pure σ+(−)polarization and with a mixture of σ+and σ−polarization, respectively. For optical pumping into the 23S1(MJ′′=0)sub-level, we have used an additional coil pair in near-Helmholtz configuration (a radius of 76 mm and a coil distance of 255 mm), placed at right angles to the other Helmholtz coil pair, to generate a well-defined quantization axis along the xdirection. As a result, the laser beam direction is perpendicular to the magnetic field component Bx. In addition to that, the quarter wave plate is replaced by a half wave plate. By rotating the half wave plate, the angle of polarization is adjusted to be either parallel or perpendicular to Bx. In the latter case, the excitation light is projected onto an equal mixture of σ+ andσ−input polarization, which again causes a remixing of the oth- erwise dark sub-level 23S1,MJ′′=0. Figure 4 shows the change in the fluorescence intensity as a function of half wave plate angle. For pumping into the 23S1,MJ′′=0 sub-level, the sub-level preparation efficiency is thus obtained using η0=1−IF(π∥Bx) IF(π/⊙◇⊞Bx), (6) where IF(π∥Bx)and IF(π/⊙◇⊞Bx)are the fluorescence intensities for excitation using π-polarized light in a direction parallel and perpendicular to the magnetic field component Bx, respectively. 3. Optimization of the sub-level preparation efficiency We have identified several parameters, which strongly affect the sub-level preparation efficiency: the interaction time between the excitation laser light and the sample, the laser intensity, the magnetic field strength, and the purity of the input polarization. FIG. 4. Red circles: Measured fluorescence intensity for excitation via the 23P1–23S1transition as a function of half wave plate angle Φ. The half wave plate angles for excitation with π-polarized light parallel and perpendicular to the magnetic field direction Bxare indicated as dashed vertical lines.During the excitation process, an atom typically scatters sev- eral photons before it is pumped to the designated magnetic sub- level. Since the radiative lifetime of the 23PJ′levels in He is long compared to typical optical pumping transitions in other atoms (τ=1/Γ=97.89 ns1), a comparably long interaction time between the laser beam and the sample has to be achieved. In our case, we have found that a large 1/ e2laser beam diameter of 2 w0≈14 mm is most practical for this purpose. For a supersonic beam with a mean velocity of 1844 m/s, this beam diameter translates into an interaction time of Δtint=7.6μs≫τ. We have studied the influence of the interaction time on the sub-level preparation efficiency ηiby monitoring the fluorescence intensity at different fluorescence detector positions along the yaxis. As can be seen from the colored markers in Fig. 5, the efficiency η+1 forσ+excitation of the 23P2–23S1and 23P1–23S1transitions, respec- tively, increases to a nearly constant value as the detector is moved toward the midpoint of the excitation laser beam. This confirms that, in our experiment, the interaction time does not limit the sub-level preparation efficiency. We have simulated the population transfer process using rate-equation calculations. A detailed description of the rate- equation model can be found in the Appendix. The best fit to our experimental data for excitation via the 23P2–23S1tran- sition and via the 23P1–23S1transition, respectively, is found by assuming that the excitation light is a mixture of 95 % σ+- polarized light and 5 % σ−-polarized light. The admixture of wrongly polarized light also explains why the observed sub-level preparation efficiency is below 100 %. In addition to that, as can be seen from Fig. 1(a), the relative transition strengths for optical pumping with wrongly polarized light are 1 /6 for the 23P1–23S1transition, while it is only 1 /30 for the 23P2–23S1tran- sition. Thus, optical pumping via the 23P1–23S1transition is more sensitive to wrongly polarized excitation light, which explains the observed difference in the sub-level preparation efficiency. FIG. 5. Sub-level preparation efficiencies η+1forσ+excitation of the 23P2–23S1 and 23P1–23S1transitions (see legend), respectively, at different positions of the fluorescence detector along the yaxis. The origin of the position axis denotes the midpoint of the laser beam. Experimental values are shown as markers, and the results of the rate-equation calculations are shown as solid lines. In the calcula- tions, a mixture of 95% σ+-polarized light and 5% σ−-polarized light is assumed for both excitation schemes. Rev. Sci. Instrum. 92, 073203 (2021); doi: 10.1063/5.0048323 92, 073203-5 Published under an exclusive license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi In our setup, such an admixture of wrong input polarization might be caused by imperfections of the quarter wave plate or by the birefringence of the vacuum window. Second, the laser intensity has to be high enough so that the laser-induced power broadening compensates for a detuning of the laser frequency from the atomic resonance. This detuning is caused by the Doppler broadening due to the transverse velocity of the atoms ( ΔDoppler≈12 MHz FWHM) and by the Zeeman shift of the atomic levels ( ΔZeeman<14 MHz). The FWHM of the power broadening can be expressed as Δpower=Γ 2π⋅√ 1+I Isat, (7) where Iis the intensity of the laser light and Isat≈0.16 mW/cm2 (assuming a two-level system) is the saturation intensity of the tran- sition. Therefore, in order to compensate for the Doppler broaden- ing and for the Zeeman shift, the laser intensity has to be I≥12mW cm2, corresponding to a laser power of ≥9 mW for our experiments. From Fig. 6, we can see that the sub-level preparation efficiency forσ+excitation of the 23P2–23S1transition is constant for laser powers P>50 mW. Unfortunately, measurements of the sub-level preparation efficiency at lower laser powers suffer from low sig- nal intensities and are thus less representative. For σ+excitation of the 23P1–23S1transition, we observe that more than 300 mW of laser power are required to reach a constant sub-level preparation efficiency. This power difference might be attributed to a weaker power broadening of the 23P1–23S1line compared to the 23P2–23S1 line as a result of a higher saturation intensity for this transition. As both transitions have the same linewidth, the same initial level, and approximately the same transition frequency, we can see from Eq. (A4) that the squared dipole matrix elements ∣μJ′∣2are propor- tional to the degeneracy factors 2 J′+1. As Isat∝1/∣μJ′∣2, it follows thatIsat(23P1−23S1)/Isat(23P2−23S1)=∣μ2∣2/∣μ1∣2=5/3. FIG. 6. Sub-level preparation efficiency η+1forσ+excitation of the 23P2–23S1 and 23P1–23S1transitions (see legend), respectively, at different laser powers. The data are taken at a 6 mm distance downstream from the midpoint of the laser beam in order to represent the efficiencies at equilibrium.Third, the magnetic bias field has to be large enough to ensure a uniform quantization axis within the optical pumping region so that the contributions of stray fields along other spatial directions are small. In Fig. 7, a scan of the sub-level preparation efficiency η+1for σ+excitation of the 23P2–23S1and 23P1–23S1transitions, respec- tively, is shown as a function of the magnetic field component Bz. The highest efficiency is achieved at field strengths between 2 G ≤ Bz≤3 G for both transitions. This magnetic field range is in line with previous observations reported in the literature.24,49,50At magnetic field strengths Bz>3 G, the sub-level preparation efficiency for exci- tation via the 23P1–23S1(23P2–23S1) transition is decreased (remains constant) compared to the optimum Bzfield range. This is consistent with a decreased scattering rate at higher magnetic fields caused by the increased Zeeman detuning. We have also analyzed the influence of stray magnetic fields along the xandydirections. Using a high-accuracy, three-axis Gauss probe (Stefan Mayer Instruments, ≤1 G, 0.05 mG resolution), we obtain Bx≈0.2 G and By≈0.1 G. At Bz=3 G, this results in an angle ofθ=√ B2x+B2y/Bz≈80 mrad between the magnetic field and thezaxis (cf. the work of Gillot et al.49). We have observed that a further compensation of the magnetic stray fields using additional coils along the xaxis (resulting in θ<40 mrad) does not result in an improved sub-level preparation efficiency. In addition to that, a non-perfect alignment of the laser propagation direction parallel to the quantization axis can induce a similar limit to the achievable sub-level preparation efficiency as the presence of magnetic stray fields. Furthermore, small magnetic-field inhomogeneities within the interaction region, resulting from, e.g., a not perfect Helmholtz coil arrangement or electronic devices in the laboratory, may also limit the sub-level preparation efficiency. In summary, we conclude that the imperfect polarization of the laser light (see the discussion above) is the main limiting factor for the sub-level preparation efficiency. FIG. 7. Sub-level preparation efficiency η+1forσ+excitation of the 23P2–23S1and 23P1–23S1transitions (see legend), respectively, at different bias magnetic field strengths. The data are taken at a 6 mm distance downstream from the midpoint of the laser beam in order to represent the efficiencies at equilibrium. Rev. Sci. Instrum. 92, 073203 (2021); doi: 10.1063/5.0048323 92, 073203-6 Published under an exclusive license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi 4. Comparison with literature values In Table I, we present a summary of the maximum sub-level preparation efficiencies ηi,maxobtained from our experimental data and a comparison with literature values. As can be seen from the table, ourηi,maxvalues are in good agreement with previous results for the laser optical pumping of He(23S1). To the best of our knowl- edge, we are the first to obtain a maximum efficiency >90% for opti- cal pumping into MJ′′=0. The only previous attempt to selectively populate MJ′′=0 was made by Giberson et al.22using linearly polar- ized light resonant with the 23P0–23S1transition and propagating along the quantization axis. For optical pumping into the spin-stretched sub-levels ( MJ′′ =±1), we report a somewhat lower maximum efficiency than pre- vious groups, which we attribute to the aforementioned imperfect laser polarization in our experiments. In addition, we see a devia- tion ofηi,maxfor optical pumping with σ+andσ−-polarized light, especially while exciting via the 23P1–23S1transition. This might be induced by a systematic asymmetry in our setup resulting from, e.g., small magnetic-field inhomogeneities as discussed above. B. Magnetic hexapole focusing The red circles in Fig. 8 show the results of a series of mea- surements, which were obtained using the setup for the mag- netic hexapole focusing of He(23S1,MJ′′=+1) [cf. Figs. 1(c) and 1(d)]. In order to interpret these results, we performed numeri- cal three-dimensional particle trajectory simulations in MATLAB. For these simulations, we use random number distributions for the particle positions and velocities (deduced from the TABLE I. Summary of maximum efficiencies ηi,maxobtained for the laser optical pumping of He(23S1) into selected MJ′′sub-levels in our experiment and a com- parison with literature values. The given uncertainties (two standard deviations) of our experimental results are statistical only. ηi,max(in %) MJ′′=+1 MJ′′=0 MJ′′=−1 23P2–23S1transition This work 94 ±3⋅⋅⋅ 90±3 Granitza et al. (1995)2598.5 ⋅⋅⋅ 98.5 Lynn et al. (1990)2396 ⋅⋅⋅ 96 Giberson et al. (1982)22≈66 ⋅⋅⋅ ≈66 23P1–23S1transition This work 87 ±5 93±4 75±5 Granitza et al. (1995)25<98.5a⋅⋅⋅ <98.5a Wallace et al. (1995)2497 ⋅⋅⋅ 97 23P0–23S1transition Kato et al. (2012)26>99 ⋅⋅⋅ >99 Schearer and Tin (1990)50⋅⋅⋅ ⋅⋅⋅ 96 Giberson et al. (1982)22⋅⋅⋅ 56 ⋅⋅⋅ aNo specific values given. FIG. 8. Red circles: Measured He∗signal intensities on the wire detector at dif- ferent positions yalong the supersonic beam axis and at different transverse positions xafter magnetic hexapole focusing. The yaxis scale is given relative to the center of the two Halbach arrays. Black lines: He∗signal intensities obtained from a numerical particle trajectory simulation. experimental data obtained at the Faraday-cup detector) and a velocity-Verlet algorithm. An initial number of 5 ⋅106particles in each Zeeman sub-level of He(23S1) and He(21S0) is propagated at a time. The magnetic field by the two Halbach arrays is imple- mented using the analytical expressions given in Ref. 47. Particles are removed from the simulation if their transverse position inside a Halbach array exceeds the 3.0 mm inner radius of the assembly [cf. Fig. 1(c)]. In each xydetection plane, the output of the trajectory simu- lation (black lines in Fig. 8) is analyzed over a certain interval of x positions corresponding to the diameter of the wire detector. The experimental results are matched to the simulated data by compar- ing the ratio of areas beneath two Gaussian distributions fitted to the datasets (not shown). Very good agreement between the exper- imental and simulated datasets is achieved by using an effective remanence of B0,eff=1.0 T<B0and an effective wire diameter of dwire,eff=5.0 mm>dwirein the simulations. The decreased rema- nence compared to B0could be due to the demagnetization of the material as a result of the prolonged storage time of the magnets. Likewise, deviations from the ideal Halbach configuration may also be possible as a result of manufacturing defects. The analysis of the simulated results suggests that the strong increase in the He∗signal intensity around x=0 is due to the transverse focusing of the MJ′′=+1 sub-level of the 23S1level. The strongest signal increase, corresponding to the focal point of the device, is at a distance of ≈110 mm from the center of the two Halbach arrays. The remaining signal intensity is mainly due to a mixture of He atoms in the 23S1,MJ′′=0 and 21S0,MJ′′=0 sub- levels. This is also consistent with previous observations.34At time t0=0, we assume a He(21S0)/He(23S1) ratio of 66%, which is in line with the results of previous measurements in our laboratory.42At the focal point, the signal contribution by He atoms in the 23S1, MJ′′=−1 sub-level is decreased by more than a factor of 7 compared to the signal intensity by atoms in the 23S1,MJ′′=0 sub-level. This is a result of the strong transverse magnetic defocusing forces, which are exerted onto the atoms in the MJ′′=−1 sub-level. Rev. Sci. Instrum. 92, 073203 (2021); doi: 10.1063/5.0048323 92, 073203-7 Published under an exclusive license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi The output of the simulation also provides an estimate of the sub-level-selection efficiency for He(23S1,MJ′′=+1),η+1. Under the conditions of our experiment, η+1is nearly constant over a region of Δy≈20 mm around the focal point. However, the efficiency strongly depends on the He beam diameter considered for the analysis. If we assume that the supersonic beam is collimated to the diameter of the wire detector (i.e., 0.2 mm) just in front of this device, we obtain a maximum efficiency η+1,max=99% at the focal point. If we assume the same He beam diameter as in the optical pumping experiments described above (i.e., 2.9 mm), the maximum efficiency η+1,maxat the focal point is reduced to 84%. To further improve the sub-level selectivity, we suggest the use of a bent magnetic guide47,51–54or the use of a central stop behind the Halbach arrays.35–38 IV. CONCLUSION We conclude that both laser optical pumping and magnetic hexapole focusing are very efficient methods for the selective prepa- ration of magnetic sub-levels of He(23S1) in a supersonic beam. We find that optical pumping into the spin-stretched sub-levels of He(23S1) via the 23P2–23S1transition is more efficient than excita- tion via the 23P1–23S1transition. The best performance is achieved forσ+(−)excitation via the 23P2–23S1transition, yielding a maxi- mum efficiency of 94% ±3% (90%±3%) for optical pumping into MJ′′=+1 (MJ′′=−1). Magnetic hexapole focusing is observed to be highly sub-level selective at low forward velocities of the supersonic beam. At v=830 m/s and at the focal point of the hexapole lens system, we infer that up to 99% of the metastable atoms are in the MJ′′=+1 sub-level if an 0.2 mm-diameter region around the center of the supersonic beam axis is considered. The magnetic-hexapole-sub-level-selection technique is attractive because it allows for the quantum-state manipulation of all atomic and molecular species with non-zero spin. Compared to optical pumping, the mechanical setup for mag- netic focusing is rather simple, especially when commercial magnets are used.53 However, optical pumping has several advantages compared to magnetic hexapole focusing. While magnetic focusing is limited to the preparation of sub-level-selected samples in low-field-seeking sub-levels only, optical pumping allows for the selective population of all MJ′′sub-levels, as shown here for the 23P1–23S1transition in He. For optical excitation with π-polarized light, we obtain an efficiency of 93% ±4% for population transfer into MJ′′=0. The creation of a pure MJ′′=0 sub-level might be possible by using mag- netic focusing as well but would require a strong overfocusing of the low-field-seeking quantum states. In our experiments, this may be realized by further reducing the forward velocity of the He∗atoms or by using a longer hexapole magnet array. However, we observe that the number of metastable helium atoms decreases by a factor of ≈2 when the valve temperature is decreased from 300 to 50 K. At the same time, the peak He∗flux within the gas pulse decreases by a factor of≈50, because the longer flight time to the detection region leads to a larger longitudinal spreading of the beam. Optical pump- ing can be applied independently of the velocity of the atoms as long as the discussed requirements for reaching the equilibrium sub-level efficiency are fulfilled. Thus, this technique results in a greater flex- ibility in choosing the valve temperature, and as mentioned above, running the valve at higher temperatures leads to much higher peakfluxes of MJ′′-sub-level-selected atoms. These high peak fluxes are particularly important for applications that benefit from high local densities, such as collision experiments. Besides that, optical pump- ing relies on a transfer of population from a statistical mixture of MJ′′sub-levels into a single sub-level, whereas magnetic hexapole focusing relies on the spatial focusing (defocusing) of the desired (unwanted) MJ′′-sub-level population. Further transmission losses are due to an aperture, which has to be inserted into the beam path behind the magnet assembly in order to eliminate contributions by the 23S1,MJ′′=0 and 21S0,MJ′′=0 sub-levels, whose motion is not influenced by a magnetic field. In the future, we will use the presented sources of MJ′′-sub-level-selected He(23S1) for quantum-state-controlled Penning-ionization studies.41Furthermore, magnetic-sub-level- selected beams of He(23S1) are useful as a starting point for the generation of coherent superposition states. The coherent control of Penning and associative ionization cross sections with such superposition states, for instance, involving the MJ′′=0 sub-level of He(23S1), has been predicted.55In addition to that, helium is of particular interest for high-precision tests of few-electron quantum electrodynamics theory, because it is the simplest two-electron atom.56,57Accurate transition frequency measurements have been performed on ultracold trapped samples58–60and on atomic beams28,61,62of He(23S1) atoms. The measurement of transitions with a zero first-order Zeeman shift (i.e., MJ′=0←23S1,MJ′′=0 transitions) would greatly reduce the experimental uncertainty. ACKNOWLEDGMENTS We thank J. Toscano (JILA) and B. Heazlewood (University of Oxford) for the loan of the magnetic hexapole arrays and for fruitful discussions. This work was financially supported by the German Research Foundation (Project No. DU1804/1-1), by the Fonds der Chemischen Industrie (Liebig Fellowship to K.D.), and by the University of Freiburg (Research Innovation Fund). APPENDIX: RATE-EQUATION CALCULATIONS The equations used for the characterization of the optical pumping process are of the form ˙N′′ i=−N′′ i∑ jWij+∑ jWijN′ j+Γ∑ jξijN′ j, (A1) ˙N′ j=∑ iWijN′′ i−N′ j∑ iWij−ΓN′ j∑ iξij, (A2) where N′′ iandN′ jdenote the populations in the ith and jth magnetic sub-levels of He(23SJ′′=1) and He(23PJ′), respectively, and Γ=1/τ is the spontaneous decay rate of the excited sub-levels according to their natural lifetime τ. The matrix elements for the excitation rate and for the branching ratio between the ith and jth magnetic sub-level are denoted as Wijandξij, respectively. The former are expressed as Wij=2∣μij∣2I ̵h2Γcε0(2J′+1)⋅V(ΔZeeman ,ΔDoppler ,Γ), (A3) Rev. Sci. Instrum. 92, 073203 (2021); doi: 10.1063/5.0048323 92, 073203-8 Published under an exclusive license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi with the laser intensity I, the reduced Planck constant h, the speed of light c, and the vacuum permittivity ε0. The line-broadening fac- torV(ΔZeeman ,ΔDoppler ,Γ)results from a Voigt profile taking into account the Doppler broadening ΔDoppler , the natural linewidth Γ, and the detuning of the transition from resonance due to the Zeeman shift, ΔZeeman . The squared dipole matrix element ∣μij∣2is calculated using the Wigner–Eckart theorem ∣μij∣2=∣μL∣2⋅(2J′+1)(2L′′+1)(2J′′+1)⋅{L′′L′1 J′J′′S}2 ⋅(J′1 J′′ MJ′,jq−MJ′′,i)2 , (A4) where L′′=0 and L′=1 are the quantum numbers for the orbital angular momenta of the lower and the upper levels, S=1 is the quantum number for the total spin, and q=MJ′′,i−MJ′,j=0,±1 denoteπandσ∓polarization, respectively. The spontaneous decay rateΓis used to calculate the squared reduced dipole matrix element ∣μL∣2, Γ=ω3 0 3πε0̵hc32L′′+1 2L′+1∣μL∣2. (A5) Here,ω0is the zero-field transition frequency. In addition to that, we consider a Gaussian distribution of the laser intensity along the yaxis, I=I(y)=I0exp(−2y2 w2 0), (A6) where w0is the beam radius and I0=f⋅2Plaser/(πw2 0)is the peak intensity calculated from the laser power Plaser. The factor f=0.1341 is used to correct for the limited spatial overlap between the laser beam and the supersonic beam. We use the mean forward velocity of the He∗beam in order to transform from the time frame of the rate equations to the position frame of the intensity distribution and to the detector position along the yaxis. The matrix elements for the branching ratio are calculated using the 3- jsymbol ξij=(2J′+1)⋅(J′′1 J′ MJ′′,i−q−MJ′,j)2 . 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5.0049882.pdf

APL Mater. 9, 060904 (2021); https://doi.org/10.1063/5.0049882 9, 060904 © 2021 Author(s).Spin–charge conversion and current vortex in spin–orbit coupled systems Cite as: APL Mater. 9, 060904 (2021); https://doi.org/10.1063/5.0049882 Submitted: 10 March 2021 . Accepted: 14 May 2021 . Published Online: 03 June 2021 Junji Fujimoto , Florian Lange , Satoshi Ejima , Tomonori Shirakawa , Holger Fehske , Seiji Yunoki , and Sadamichi Maekawa COLLECTIONS Paper published as part of the special topic on Emerging Materials for Spin-Charge Interconversion ARTICLES YOU MAY BE INTERESTED IN Recent progress on measurement of spin–charge interconversion in topological insulators using ferromagnetic resonance APL Materials 9, 060702 (2021); https://doi.org/10.1063/5.0049887 Topological insulators for efficient spin–orbit torques APL Materials 9, 060901 (2021); https://doi.org/10.1063/5.0048619 Field-free magnetization switching induced by the unconventional spin–orbit torque from WTe 2 APL Materials 9, 051114 (2021); https://doi.org/10.1063/5.0048926APL Materials PERSPECTIVE scitation.org/journal/apm Spin–charge conversion and current vortex in spin–orbit coupled systems Cite as: APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 Submitted: 10 March 2021 •Accepted: 14 May 2021 • Published Online: 3 June 2021 Junji Fujimoto,1,a) Florian Lange,2 Satoshi Ejima,2,3 Tomonori Shirakawa,4,5 Holger Fehske,2 Seiji Yunoki,3,4,5,6and Sadamichi Maekawa1,6 AFFILIATIONS 1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 2Institut für Physik, Universität Greifswald, 17489 Greifswald, Germany 3RIKEN Cluster for Pioneering Research (CPR), Wako, Saitama 351-0198, Japan 4RIKEN Center for Computational Science (R-CCS), Kobe, Hyogo 650-0047, Japan 5RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan 6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan Note: This paper is part of the Special Topic on Emerging Materials for Spin-Charge Interconversion. a)Present address: Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan. Author to whom correspondence should be addressed: fujimoto.junji@gmail.com ABSTRACT Using response theory, we calculate the charge current vortex generated by spin pumping at a point-like contact in a system with Rashba spin–orbit coupling (SOC). We discuss the spatial profile of the current density for finite temperature and for the zero-temperature limit. The main observation is that the Rashba spin precession leads to a charge current that oscillates as a function of distance from the spin pumping source, which is confirmed by numerical simulations. In our calculations, we consider a Rashba model on a square lattice, for which we first review the basic properties related to charge and spin transport. In particular, we define the charge current and spin current operators for the tight-binding Hamiltonian as the currents coupled linearly with the U (1)and SU(2)gauge potentials, respectively. By analogy to the continuum model, the SOC Hamiltonian on the lattice is then introduced as the generator of the spin current. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0049882 I. INTRODUCTION Generation and detection of non-equilibrium spin angular momentum are of crucial importance in spintronics. In recent decades, research has been extended to the interconversion of spin and other physical quantities, such as charge,1–3heat,4and mechan- ical angular momentum.5–7The spin Hall effect (SHE) is commonly used in experiments to electrically generate spin currents, while the inverse spin Hall effect (ISHE) is used as a detector. The SHE and the ISHE originate from spin–orbit couplings (SOCs) through intrinsic, side-jump scattering and skew-scattering mechanisms, which each contribute to the effects in a different way.8 To experimentally gain more information about the transport behavior of a system, it may be useful to consider different geome- tries for the spin–charge conversion besides the usual configuration for ISHE experiments [Fig. 1(a)]. Here, we propose to investigatethe nonuniform charge current response to a local spin injection [Fig. 1(b)]. For the two-dimensional (2D) Rashba model, which is an important example for the (I)SHE, we show that the charge current in such a configuration forms a vortex whose spatial profile indicates the strength of the SOC9. Specifically, we consider a Rashba system on a square lattice with one site coupled to a classical spin (Fig. 2). The spin oscillates in thexy-plane and, thereby, locally induces a z-polarized spin current. We calculate the induced charge current density analytically in the framework of response theory, obtaining a vortical structure for the dc component that can be regarded as the effect of spin precession and a spin-dependent force due to the Rashba SOC. The results are in agreement with a semiclassical wave-packet analysis (WPA)9as well as with numerical simulations. As in the field of spintronics, the lattice Rashba model has been less commonly used than the continuum one;10we first derive APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 9, 060904-1 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 1. (a) Spin–charge conversion in a typical configuration composed of a fer- romagnet and a heavy metal with a strong spin–orbit coupling (SOC). The spin injection into the heavy metal from the ferromagnet by spin pumping induces an electric voltage via the inverse spin Hall effect (ISHE). (b) A different configuration, in which the spin is injected locally, inducing a nonuniform charge current. For the Rashba SOC with the spin injection, whose polarization is perpendicular to the plane, the charge current forms a vortex. FIG. 2. Schematic figure of the local spin pumping in a Rashba model on a square lattice. The time-dependent classical spin S(t)at the origin is coupled to the elec- tron spin through the exchange interaction. The dynamics of the classical spin induces a charge current vortex. the underlying tight-binding Hamiltonian in Sec. II. Considering the model on a square lattice, we define the charge and spin cur- rents in Secs. III and IV, respectively, by using the local U (1)and SU(2)gauge transformations. These definitions are consistent with the ordinary ones, in which the currents are introduced through the corresponding polarizations and their time derivatives. In a next step, we construct the SOC as the generator of the spin current in Sec. V. Although this is different from the approach in Ref. 11, it results in an equivalent Hamiltonian. In Sec. VI, we finally consider the Rashba system with the classical spin and present the charge current vortex generation by spin pumping. Section VII summa- rizes our work. The Appendix discusses the lattice versions of the continuity equations for charge and spin. II. TIGHT-BINDING MODEL We begin with the ordinary tight-binding model on the square lattice, which is given by ℋt=−t 2∑ n,m(c† n,mcn+1,m+c† n,mcn−1,m +c† n,mcn,m+1+c† n,mcn,m−1+h.c.), (1) where t>0 is the nearest-neighbor hopping parameter and cn,m =t(cn,m,↑,cn,m,↓)is the spinor form of the electron operator for the site at Rn,m=naˆx+maˆy(nandmare integers and ais the lattice constant). The factor 1 /2 is included to avoid double counting.By the Fourier transformation cn,m=1√ N∑ kckeik⋅Rn,m, (2) where Nis the total site number, the Hamiltonian for periodic boundary conditions can be written as ℋt=∑ kTkc† kck, (3) with Tk=−2t(coskxa+coskya). For k≪2π/a, we can approxi- mate the energy dispersion as Tk=−4t+̵h2k2/2mewith the Dirac constant̵h,k=∣k∣, and̵h2 2me≡ta2. In the long-wavelength limit, the system, thus, behaves like a free-electron model with effective elec- tron mass me. Below, we introduce the charge current and spin cur- rent operators for the lattice model and show explicitly that in the long-wavelength limit, they become equivalent to their counterparts for the free-electron model. III. CHARGE CURRENT OPERATOR The charge current operator for the tight-binding model on the square lattice can be derived by exploiting the local U (1)gauge transformation cn,m=Un,m¯cn,m, Un,m=eiϕn,m,ϕn,m≪1. (4) Here,ϕn,m=ϕ(Rn,m)andϕ(r)is a smooth real-valued function slowly varying on the lattice constant scale. When using the transfor- mation (4) on the Hamiltonian (1), combinations of unitary matri- ces for two different positions, such as U† n,mUn+1,m, arise. Because ϕn,m≪1, one can approximate U† n,mUn+1,m≃(1−iϕn,m)(1+iϕn+1,m) ≃1+i(ϕn+1,m−ϕn,m). (5) Expanding ϕn+1,m≃ϕn,m+a∂ ∂xϕ(r)∣r=Rn,mthen yields U† n,mUn+1,m=1−iea ̵hAem x(Rn,m), (6) where eis the elementary charge,30and we have introduced the electromagnetic vector potential as Aem x(r)≡−̵h e∂ ∂xϕ(r). (7) Similarly, we find U† n,mUn,m+1=1−iea ̵hAem y(Rn,m), (8) where Aem y(r)≡−̵h e∂ ∂yϕ(r). (9) APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 9, 060904-2 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm The Hamiltonian can, thus, be written as ℋt=¯ℋt+ℋem, where ¯ℋt is obtained by replacing c(†) n,mwith ¯c(†) n,minℋtandℋemis composed of terms linear in the vector potential, ℋem=ieta ̵h∑ n,m(¯c† n,m¯cn+1,m−¯cn−1,m 2Aem x(Rn,m)−h.c. +¯c† n,m¯cn,m+1−¯cn,m−1 2Aem y(Rn,m)−h.c.). (10) We define the charge current operator as the current coupled lin- early with the vector potential, i.e., jx(Rn,m)=δℋt δAemx(Rn,m)∣ ϕ=0, (11) jy(Rn,m)=δℋt δAemy(Rn,m)∣ ϕ=0, (12) which leads to the following expressions: jx(Rn,m)=ieta2 ̵h(c† n,mcn+1,m−cn−1,m 2a−h.c.), (13) jy(Rn,m)=ieta2 ̵h(c† n,mcn,m+1−cn,m−1 2a−h.c.). (14) In the continuum limit c(†) n,m→c(†)(r), we have cn+1,m−cn−1,m 2a→∂c(r) ∂x,cn,m+1−cn,m−1 2a→∂c(r) ∂y, (15) which yields j(r)=ie̵h 2me(c†∇c−(∇c†)c). (16) Equation (16) is consistent with the paramagnetic charge current operator in the free-electron model. We note that the conventional definition of the total charge current is12 J=∂P ∂t=−i ̵h[P,ℋt], (17) where P=−e∑n,mRn,mc† n,mcn,mis the (total) polarization. By consid- ering that the total charge current is the sum of the local charge currents, J=∑n,mj(Rn,m), expressions equivalent to Eqs. (13) and (14) are obtained. IV. SPIN CURRENT OPERATOR In this section, we derive the spin current operator for the square lattice, by using the local SU (2)gauge transformation, cn,m=Vn,m˜cn,m, Vn,m=eiθωn,m⋅σ,θ≪1, (18) whereσ=(σx,σy,σz)is the vector of Pauli matrices and ∣ωn,m∣=1. Furthermore, ωn,m=ω(Rn,m), with a smooth real-valued vector field ω(r)=(ωx(r),ωy(r),ωz(r))slowly varying on the lattice-constantscale. Equation (18) describes a spin rotation around the vector ωn,m with the angle θ. We can approximate Vn,m≃1+iθωn,m⋅σ, since θ≪1. The gauge transformation of the Hamiltonian (1) then fol- lows in a similar way to the U (1)gauge transformation in Sec. III. To first order in θ, V† n,mVn+1,m=1+iaAx(Rn,m)⋅σ 2, (19) V† n,mVn,m+1=1+iaAy(Rn,m)⋅σ 2, (20) where we have introduced the SU (2)gauge potential, Ax(r)=2θω(r)×∂ ∂xω(r), (21) Ay(r)=2θω(r)×∂ ∂yω(r). (22) The Hamiltonian can now be rewritten as ℋt=˜ℋt+ℋA, where ˜ℋt is the Hamiltonian obtained by replacing c(†) n,mwith ˜c(†) n,minℋtand ℋA=−ita 4∑ n,m(˜c† n,mσ˜cn+1,m−˜cn−1,m 2⋅Ax(Rn,m)−h.c. +˜c† n,mσ˜cn,m+1−σ˜cn,m−1 2⋅Ay(Rn,m)−h.c.). (23) We define the spin current operator as the current coupled linearly with the SU (2)gauge potential, i.e., jα s,x(Rn,m)=δℋt δAαx(Rn,m)∣ θ=0, (24) jα s,y(Rn,m)=δℋt δAαy(Rn,m)∣ θ=0, (25) withα∈{x,y,z}being the spin index, which yields jα s,x(Rn,m)=−ita2 2(c† n,mσαcn+1,m−cn−1,m 2a−h.c.), (26) jα s,y(Rn,m)=−ita2 2(c† n,mσαcn,m+1−cn,m−1 2a−h.c.). (27) In the continuum limit c(†) n,m→c(†)(r), the spin current operator becomes jα s,x(r)=̵h 2̵h 2mei(c†σα∂c ∂x−∂c† ∂xσαc), (28) jα s,y(r)=̵h 2̵h 2mei(c†σα∂c ∂y−∂c† ∂yσαc), (29) which is again consistent with the expression for the free-electron model. APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 9, 060904-3 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm Let us briefly comment on the definition of the spin current operator proposed by Shi et al.13They introduce the spin current operator as the time derivative of the spin-displacement operator rsz, jz s(r)=d(rsz) dt. (30) This definition can be regarded as an extension of Eq. (17). To see this, we define the total spin polarization as Pz =̵h 2∑n,mRn,mc† n,mσzcn,mand the total spin current as its time derivative, Jz s=∂Pz ∂t=−i ̵h[Pz,ℋt]. (31) Comparing with Jz s=∑n,mjz s(Rn,m)then leads to Eqs. (26) and (27) forα=z. V. SPIN–ORBIT COUPLING Although the Rashba SOC in the square lattice has been derived from the atomic SOC with inversion asymmetry,11we here intro- duce it in a different way and obtain the same result. In the contin- uum model, the SOC is the generator of the intrinsic spin current, i.e.,ℋsoc=(2me/̵h2)∑i,αλα i∫drjα s,i(r)with the coefficient λα ispeci- fying the SOC amplitude.14We extend this definition of SOC to the lattice model as ℋsoc=∑ i,αλα i ta2∑ n,mjα s,i(Rn,m), (32) where jα s,i(Rn,m)is the spin current operator with the direction i∈{x,y}and spin index α∈{x,y,z}. The SOC amplitude λα iin the case of the Rashba SOC is λα R,i=αRεiαz=αR⎛ ⎝0 1 0 −1 0 0 0 0 0⎞ ⎠ iα, (33) whereεijkis the antisymmetric tensor and αRis the Rashba parame- ter. Substituting Eq. (33) into Eq. (32) results in ℋR=−iαR 2∑ n,m(c† n,mσycn+1,m−cn−1,m 2a−h.c. −c† n,mσxcn,m+1−cn,m−1 2a+h.c.) (34) or, in the momentum representation, ℋR=αR a∑ kc† k(σysinkxa−σxsinkya)ck =∑ kc† k(λk⋅σ)ck, (35) withλk=(αR/a)(−sinkya, sin kxa, 0). Considering the long- wavelength limit k≪2π/a,(sinkxa)/a≃kx, and(sinkya)/a≃ky, we obtain ℋRashba=αR∑ kc† k(k×σ)zck, (36) which is the well-known Rashba Hamiltonian in the continuum.While we focus here on the Rashba model, the above proce- dure can be used for other types of SOCs too. From the knowledge of the continuum model, the Dresselhaus SOC amplitude may be shown to be λα D,i=βD⎛ ⎝1 0 0 0−1 0 0 0 0⎞ ⎠ iα, (37) whereβDis the Dresselhaus SOC strength. In the case of the Weyl SOC for the square lattice, we have λα W,i=γW⎛ ⎝1 0 0 0 1 0 0 0 0⎞ ⎠ iα, (38) whereγWis the strength of the Weyl SOC. Next, we derive the charge current and spin current operators in the presence of Rashba SOC. By applying the local U (1)gauge transformation (4) to the Rashba Hamiltonian (34), we get ℋR =¯ℋR+ℋ′ em, where ¯ℋRis the Hamiltonian obtained by replacing c(†) n,mwith ¯c(†) n,minℋRand ℋ′ em=−eαR 4̵h∑ n,m(¯c† n,mσy¯cn+1,mAem x(Rn,m)+h.c. +¯c† n,mσy¯cn−1,mAem x(Rn,m)+h.c. −¯c† n,mσx¯cn,m+1Aem y(Rn,m)−h.c. −¯c† n,mσx¯cn,m−1Aem y(Rn,m)−h.c.). (39) By defining the charge current operator analogously to Eqs. (11) and (12), we get the following expressions: j′ x(Rn,m)=−eαR 2̵h(c† n,mσycn+1,m+cn−1,m 2+h.c.), (40) j′ y(Rn,m)=+eαR 2̵h(c† n,mσxcn,m+1+cn,m−1 2+h.c.). (41) In the continuum limit, where (cn+1,m+cn−1,m)/2→c(r), we confirm the correspondence to the anomalous charge current oper- ators in the free-electron model with the Rashba SOC, j′(r)=eαR ̵hc†(ˆz×σ)c, (42) where c(†)=c(†)(r). Note that the anomalous charge current opera- tor (42) is proportional to the spin-density operator s(r)=c†σcas j′(r)=(eαR/̵h)ˆz×s(r). The spin current is derived by using the local SU (2)gauge transformation (18) on the Rashba Hamiltonian (34). We get ℋR =˜ℋR+ℋ′ A, where ˜ℋRis the Hamiltonian obtained by replacing c(†) n,m with ˜c(†) n,mandσα(α=x,y) with ˜σα≡V† n,mσαVn,minℋRand APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 9, 060904-4 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm ℋ′ A=αR 2∑ n,m(˜c† n,m˜σyσα˜cn+1,mAα x(Rn,m)+h.c. +˜c† n,m˜σyσα˜cn−1,mAα x(Rn,m)+h.c. −˜c† n,m˜σxσα˜cn,m+1Aα y(Rn,m)−h.c. −˜c† n,m˜σxσα˜cn,m−1Aα y(Rn,m)−h.c.). (43) The spin current operator for the Rashba Hamiltonian is then defined as j′α s,x(Rn,m)=1 2δℋR δAαx(Rn,m)∣ θ=0, (44) j′α s,y(Rn,m)=1 2δℋR δAαy(Rn,m)∣ θ=0, (45) so that we have j′α s,x(Rn,m)=j(1),α s,x(Rn,m)+j(2),β s,x(Rn,m)εyαβ, (46) j′α s,y(Rn,m)=j(1),α s,y(Rn,m)+j(2),β s,y(Rn,m)εxαβ, (47) where j(1),y s,x(Rn,m)=αR 4(c† n,mcn+1,m+cn−1,m 2+h.c.), (48) j(1),x s,y(Rn,m)=−αR 4(c† n,mcn,m+1+cn,m−1 2+h.c.), (49) and the other components of j(1),α s,iare zero, and j(2),β s,x(Rn,m)=iαR 4(c† n,mσβcn+1,m+cn−1,m 2−h.c.), (50) j(2),β s,y(Rn,m)=−iαR 4(c† n,mσβcn,m+1+cn,m−1 2−h.c.). (51) In the continuum limit, we obtain j(1),y s,x(r)=−j(1),x s,y(r)=1 2αRc†c, (52) j(1),α s,i(r)=εiαz(αR/2)c†c, and j(2),β s,i(r)=0, which means j′α s(r)=j(1),α s(r)=αR 2(ˆα׈z)c†c. (53) This anomalous spin current operator is proportional to the electron density operator n=c†c. Note that in our definition of the spin current operator for the Rashba Hamiltonian [see Eqs. (44) and (45)], an additional factor 1/2 is included. The factor originates from the fact that the Rashba SOC is the generator of the intrinsic spin current. Since the intrinsic spin current is also generated by the SU (2)gauge potential as shown in Eq. (23), the SOC amplitude λα iacts as an SU (2)gauge potential in the continuum model,ℋ′=∑ i,α∫dr˜jα s,i(r)(Ai−ℛ−1λi)α+1 2∑ i,α∫dr1 2˜n(r) ×(Ai−ℛ−1λi)α(Ai−ℛ−1λi)α+𝒪({λα i}2). (54) Here, ˜jα s,i(r)and ˜n(r)are the spin current and electron den- sity operators described by the field operator ˜c(†). Moreover, we have introduced the vector forms of the SU (2)gauge potential Ai=(Ax i,Ay i,Az i)and the SOC amplitude λi=(λx i,λy i,λz i)with the rotational matrix ℛdefined by V†σV=ℛσ. We need to count the order of such a gauge potential including the SOC amplitude, which means that the terms proportional to both the gauge potential and the SOC amplitude correspond to the second order. The fac- tor 1/2 difference in the definitions will be clarified in the Appendix discussing the continuity equations. We now comment on the connection between the definition in Ref. 13 and Eqs. (46) and (47). Calculating Eq. (31) for the Rashba Hamiltonian (34), we find that a torque term Tαarises in addition to the expressions (46) and (47), Tα=αR ta2∑ n,m∑ i,j,βRn,mεijzεαjβjβ s,i(Rn,m). (55) In particular, α=z, Tz=−αR ta2∑ n,mRn,m(jx s,x(Rn,m)+jy s,y(Rn,m)). (56) We emphasize that we have defined the spin current operator that corresponds to the spin operator sαand not to the spin displace- ment operator rsαso that, as shown in the Appendix, the continuity equation for spin is not fulfilled. Instead, a continuity equation holds for the total spin displacement operator. These definitions are equiv- alent in the absence of SOCs, but they generally lead to different expressions in spin–orbit coupled systems. For a meaningful discussion of spin transport, it is important that the spin current is defined properly. One possible proper way to define the spin current operator is based on the local SU (2)gauge transformation and another is based on the continuity equation for spin, both of which should be equivalent. Sometimes, a spin current operator is naively introduced15as¯jα s,i(r)=(sαvi+visα)/2, where vi is the velocity operator. However, the operator ¯jα s,i(r)in the presence of the Rashba SOC is not the same as our expression (53), where the factor 1 /2 is different. Moreover, when one uses the continuity equation for spin to discuss spin accumulation in finite systems, the boundary conditions need to be taken into account.13,16,17 VI. CHARGE CURRENT VORTEX GENERATION BY SPIN PUMPING After deriving the SOC Hamiltonian, we are now able to dis- cuss the charge current vortex generation by spin injection via spin pumping. We consider a tight-binding model on the square lattice with a Rashba SOC, which is coupled to a classical spin at the origin R0,0through the exchange coupling ℋS(t)=−JsS(t)⋅(c† 0,0σc0,0). (57) APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 9, 060904-5 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm We assume that the classical spin has the magnitude Sand precesses with the angular frequency ωin the xy-plane (see Fig. 2), S(t)=S⎛ ⎝cosωt sinωt 0⎞ ⎠, ˙S(t)=Sω⎛ ⎝−sinωt cosωt 0⎞ ⎠. (58) While we can do the following calculation also for Sz≠0, the dc component of the charge current density does not change qualita- tively from the Sz=0 case so that we set Sz=0 for simplicity. The classical spin dynamics generates a charge current through the Rashba SOC. We now evaluate the charge current density within response theory (see Appendix A in Ref. 18 for details), Ji(r,ω)=χα i(r,ω)S(ω)+∫∞ −∞dω′ 2πχαβ i(r,ω,ω′)Sα(ω−ω′) ×Sβ(ω′)+⋅⋅⋅ . (59) In the following, we focus on the dc component of the response in the lowest order with respect to S(ω), which is ϑαβ i(r)=lim ω′→0χαβ i(r, 0,ω′)−χαβ i(r, 0, 0) −iω′. (60) The response coefficient χαβ i(r,ω,ω′)is obtained from the correla- tion function in the Matsubara representation,Xαβ i(r, iωλ, iωλ′)=J2 s 2N∑ qeiq⋅r∬β 0dτdτ′eiωλ(τ−τ′)+iωλ′τ′ ×⟨Tτ,τ′{Ji(q,τ)sα(0,τ′)sβ(0, 0)}⟩, (61) by taking the analytic continuations i ωλ→hω+2i0 and iωλ′ →hω′+i0, Xαβ i(r,̵hω+2i0,̵hω′+i0)=χαβ i(r,ω,ω′). (62) Here, Ji(q,τ)andsα(0,τ)are the charge current operator and the spin operator in the Heisenberg picture, T τ,τ′is the time-ordering operator,β=1/kBTis the inverse temperature, and ⟨⋅⋅⋅⟩is the thermal average. The corresponding operators in the Schrödinger picture are Ji(q)=∑ kc† k−q 2(vk,i+v′ k,i)ck+q 2, (63) with the normal and anomalous velocities vk,i=−2eat̵hsinkiaand v′ k,i=−eαR̵hεziβσβcoskia, and sα(0)=c† 0,0σαc0,0=1 N∑ k,k′c† k′σαck. (64) After some straightforward calculations, we find ϑαβ i(r)=J2 s N3∑ qeiq⋅r∑ k,k′i̵h 2π∫∞ −∞dε(−∂f ∂ε)tr[(vk,i+v′ k,i)GR k+q 2(ε)σαIm[GR k′(ε)]σβGA k−q 2(ε)−(vk,i+v′ k,i)GR k+q 2(ε)σβIm[GR k′(ε)]σαGA k−q 2(ε)] −J2 s N3∑ qeiq⋅r∑ k,k′̵h 4π∫∞ −∞dεf(ε)tr[(vk,i+v′ k,i)GR k+q 2(ε)σα(∂εGR k′(ε))σβGR k−q 2(ε)−(vk,i+v′ k,i)GR k+q 2(ε)σβ ×(∂εGR k′(ε))σαGR k−q 2(ε)−(vk,i+v′ k,i)GA k+q 2(ε)σα(∂εGA k′(ε))σβGA k−q 2(ε)+(vk,i+v′ k,i)GA k+q 2(ε)σβ(∂εGA k′(ε))σαGA k−q 2(ε)], (65) where f(ε)is the Fermi–Dirac distribution function, GR/A kis the retarded/advanced Green function defined by GR/A k(ε)=Gk(ε±iγ), Gk(z)=z+μ−Tk+λk⋅σ Dk(z), (66) Dk(z)=(z+μ−Tk)2−∣λk∣2. (67) Here,μis the chemical potential and γis the level broadening. The terms proportional to f(ε)are expected to be negligible, since Re[1/Dk(ε+iγ)]≈0. We, therefore, may simplify Eq. (65) to ϑαβ i(r)=−̵hJ2 s 4N2εαβz∫∞ −∞dε(−∂f ∂ε)ν(ε)∑ p,qei(p−q)⋅r ×tr[(vp+q 2,i+v′ p+q 2,i)GR p(ε)σzGA q(ε)], (68)whereν(ε)is the density of states given by ν(ε)=−2 πN∑ pIm[GR p(ε)]=1 N∑ pFp(ε;γ) (69) with the Cauchy distribution given by Fp(ε;γ)=1 πγ (ε+μ−Tp+η∣λp∣)2+γ2. (70) It should be noted that Eq. (68) also gives the linear response of the charge current to ˙Sz. Taking the trace, we obtain ϑαβ i(r)=−̵hJ2 s 2εαβz∫∞ −∞dε(−∂f ∂ε)ν(ε) ×[−8ieat ̵h(−diy(r,ε)bix(r,ε)+biy(r,ε)dix(r,ε)) −4ieαR ̵hεziγεγzδ(−ci(r,ε)biδ(r,ε)−ai(r,ε)diδ(r,ε))], (71) APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 9, 060904-6 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm where ai(r,ε)=−π 2N∑ η=±∑ peip⋅rsinpia 2Fp(ε;γ), (72) biα(r,ε)=−π 2N∑ η=±∑ peip⋅rλα p ε+μ−Tpcospia 2Fp(ε;γ), (73) ci(r,ε)=−π 2N∑ η=±∑ peip⋅rcospia 2Fp(ε;γ), (74) diα(r,ε)=−π 2N∑ η=±∑ peip⋅rλα p ε+μ−Tpsinpia 2Fp(ε;γ). (75) Assuming that the chemical potential is low enough to approx- imate the dispersion as parabolic and setting μ+4tasμ, we get the charge current response Jdc(r,t)=D(r)(S(t)×˙S(t))zˆeϕ, (76) where ˆeϕ=(−sinϕ, cosϕ)withϕ=tan−1y/x, and the coefficient is given as D(r)=D1(r)+D2(r)with D1(r)=eJ2 s 8∫∞ −∞dε(−∂f ∂ε)ν(ε)̵h2 2mer{ℬ(r,ε)}2, (77) D2(r)=eJ2 s 8∫∞ −∞dε(−∂f ∂ε)ν(ε)αR𝒞(r,ε)ℬ(r,ε) (78) and ℬ(r,ε)=∑ ηη∫∞ 0dpp2J1(pr)αR ε+μ−̵h2p2 2meFp(ε;γ), (79) 𝒞(r,ε)=∑ η∫∞ 0dppJ 0(pr)Fp(ε;γ). (80) Here, Jn(x)is the Bessel function of order n. By taking the zero-temperature limit and the no-level- broadening limit, which means that −∂f/∂εandFp(ε;γ)are reduced to the delta functions δ(ε)andδ(ε+μ−̵h2p2/2me+ηαRp), respec- tively, we get D1(r)=eJ2 sm2 e 8̵h4νF p2 0̵h2 2mer⎛ ⎝∑ η=±ηpηJ1(∣pη∣r)⎞ ⎠2 , (81) D2(r)=−αReJ2 sm2 e 8̵h4νF p2 0∑ η′=±∣pη′∣J0(∣pη′∣r)∑ η=±ηpηJ1(∣pη∣r) (82) forμ≥−meα2 R/2̵h2≡−ERand D1(r)=D2(r)=0 forμ<−ER. In these expressions, νF=ν(0)=1/2πtforμ≥0 andνF=meαR/̵h2p0t for−ER≤μ<0 are the densities of states at the Fermi level, and pη=p0+ηme ̵h2αR, (83) FIG. 3. The spatial profile of the charge current induced by the classical spin precession. The unit of the current is I0=et/8πah, the Rashba parameter ˜αR =αR/ta=0.1, and the exchange interaction strength Js/t=1. The vortical struc- ture is clearly shown. The clockwise current is represented by the red area, and the counterclockwise current is described by the blue area. A more rapid oscillation is seen, which has the same origin as the Friedel oscillation19and Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction.20–22 p0=√ 2me ̵h2(μ+ER). (84) The charge current density given by Eq. (76) clearly forms a vortical structure. Figure 3 depicts the spatial profile of the current for the Rashba parameter ˜αR=αR/ta=0.1 and chemical potential μ/t=0.5 withγ=0. The changes between clockwise and counter- clockwise current directions originate from the Rashba spin pre- cession as shown by the WPA in Ref. 9. For more details, we plot the radial dependence of the coefficients D1(r),D2(r), and D(r)in Fig. 4. We find that the envelopes of D1(r)andD2(r)agree with the corresponding WPA results [Eqs. (9) and (10) of Ref. 9], DWPA 1(x)=A2 sin2(˜αRx/2) ˜αRx2, (85) DWPA 2(x)=−Asin(˜αRx) x, (86) where x=r/aand A=J2 s˜αR/2πp0at2. The oscillations in D1(r) and D2(r)with shorter wavelengths have the wavenumber 2 p0 =p++p−, which comes from the interference of the two electron bands. They have the same origin as Friedel oscillations19and the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction.20–22Figure 5 shows the temperature dependence of the current. For finite T, the rapid oscillations become weaker as the radius increases and eventually disappears. We can see from Eqs. (85) and (86) that the oscillation period is described by the Rashba parameter. This dependence allows us to estimate the strength of the Rashba SOC. The Rashba SOC ampli- tude may also be determined from the magnetic field induced by the current vortex.9 APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 9, 060904-7 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 4. (a) and (b) The radial dependence of the response coefficients D1(r)and D2(r)atT=0 for parameters μ/t=0.5,˜αR=0.1,Js/t=1, andγ=0. The wave packet analysis (WPA) results are also plotted. The envelopes of D1andD2are in good agreement with the WPA results. (c) The radial dependence of the total response coefficient. To check the response-theory results, we performed numerical calculations. We use a Lanczos transformation to map the Hamil- tonian to a chain representation and employ the infinite boundary conditions described in Ref. 23. The time-dependent currents are then obtained by straightforward numerical diagonalization of the one-electron Hamiltonian. Details of the numerical method can be found in the supplementary material of Ref. 9. Figure 6 displays the results for an exchange-coupling strength of Js/t=0.1 and a FIG. 5. The current profile at finite temperatures. We set the level-broadening parameter as γ/t=5×10−4, and the other model parameters are as those in Figs. 3 and 4. FIG. 6. Numerically calculated charge current due to spin pumping at the origin at T=0. Model parameters are Js/t=0.1,ω=0.1t/h,μ/t=0.5, and ˜αR=0.1. The color of the arrows indicates the charge current direction, and their lengths indicate the magnitude. (a) Snapshot at time 100 h/t. (b) Time average over a period 1 /ω around 100 h/t. The scale in (a) is larger than that in (b) by a factor of 20. (c) Radial dependence of the current in (b) compared with response theory. The currents J1 andJ2correspond to the response coefficients D1(r)andD2(r), respectively. As before, the current is given in units of I0=et/8πah. precession frequency of ω=0.1t/̵hatT=0. The charge current at some fixed time is not symmetric under rotation around the origin and deviates strongly from the analytical expression (76) for the dc component [Fig. 6(a)]. After averaging over a period 1 /ω, however, we recover the vortical structure predicted by the response theory [Fig. 6(b)]. While oscillations with wavenumber 2 p0occur at small r, they are not nearly as strong as those in the analytical result. This may be due to the finite frequency ω, which could have a similar effect to a finite temperature by inducing excitations that are not exactly at the Fermi energy. We would like to make a comment concerning the connection to our previous work. In Ref. 9, we considered a Rashba system cou- pled to a quantum spin chain. The spin injection was achieved by driving a spinon spin current through the spin chain. In this paper, we instead investigate the spin injection by local spin pumping. Both ways of spin injection lead to the same result of the current vor- tex generation in the Rashba system. Hence, we conclude that it is a APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 9, 060904-8 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm universal phenomenon that the local spin injection induces the cur- rent vortex generation in the Rashba system. It should be noted that we discuss the temperature dependence of the current vortex generation in the present work. We also comment on related materials for studying the current vortex generation. Since we consider a Rashba system that is coupled to a classical spin, 2D oxides,24,25Ag/Bi surfaces,26and some van der Waals materials27,28are examples of our model, if a nano-scale magnet could be attached. Moreover, we give a comment concerning the inverse effect of current vortex generation by local spin injection, i.e., local spin extraction induced by a charge current vortex in the Rashba system. In the Rashba system, charge-to-spin conversion occurs because of the Edelstein effect.29Although such charge-to-spin conversion is interesting, it would be difficult to prepare the current vortex around a nanomagnet. VII. SUMMARY Spin–orbit coupling phenomena, such as the (inverse) spin Hall effect, are of central importance in spintronics, as they facilitate the conversion between spin and charge degrees of freedom. A mini- mal model to investigate these effects theoretically is the free two- dimensional electron gas with Rashba spin–orbit coupling. Meso- scopic systems and interface effects are more easily studied using the lattice version of the Rashba model, however. In this paper, we demonstrated how the spin current and charge current opera- tors on a lattice and in the presence of spin–orbit coupling can be derived from appropriate gauge transformations. We, furthermore, showed that, like in the continuum model, the spin–orbit coupling Hamiltonian can be defined as the generator of the intrinsic spin current. As a specific application of the lattice description, we then ana- lyzed the charge current response of the Rashba model to spin injec- tion via local spin pumping, which we modeled by coupling a single site of the lattice to a classical precessing spin. The dc component of the charge current was found to form a vortical structure, where the current direction depends on the distance from the classical spin. This dependence is explained by the Rashba spin precession. We also found additional spatial oscillations with twice the Fermi wavenum- ber, which have a similar origin to Friedel oscillations and the RKKY interaction. At finite temperature, these oscillations disappear for sufficiently long distances. Finally, we performed numerical simu- lations, which include effects beyond the contributions considered in the analytic calculation. Although the snapshot of the spatial pro- file of the current at a certain time has a complicated structure, the time average is consistent with the dc component from the analytic calculation. While we focused here on the Rashba system, the local spin injection will generally induce the nonuniform charge currents in spin–orbit coupled systems, where the spatial profile of the cur- rent should reflect the electronic structure of the system. Hence, the spin–charge conversion by local spin injection has a great potential to investigate the electronic structure such as the SOC strength. ACKNOWLEDGMENTS J.F. was partially supported by the Priority Program of Chi- nese Academy of Sciences (Grant No. XDB28000000). S.E. and F.L.were supported by the Deutsche Forschungsgemeinschaft through Project Nos. EJ7/2-1 and FE398/8-1, respectively. S.M. was sup- ported by the JST CREST Grant (Nos. JPMJCR19J4, JPMJCR1874, and JPMJCR20C1) and the JSPS KAKENHI (Grant Nos. 17H02927 and 20H01865) from the MEXT, Japan. J.F. thanks Y. Ominato for valuable discussion. APPENDIX: CONTINUITY EQUATIONS In this section, we derive the charge and spin continuity equations for the tight-binding Hamiltonian ℋ=ℋt+ℋRwith the Rashba SOC. The continuity equation of the charge Qn,m =−ec† n,mcn,mis evaluated from ˙Qn,m=−e(˙c† n,mcn,m+c† n,m˙cn,m), (A1) with the time derivative of the field operators cn,mgiven by the Heisenberg equation of motion, ˙cn,m=1 i̵h[cn,m,ℋ] =−t i̵h(cn+1,m+cn−1,m+cn,m+1+cn,m−1) −αR 2̵h(σycn+1,m−cn−1,m a−σxcn,m+1−cn,m−1 a). (A2) Substituting Eq. (A2) in Eq. (A1) results in ˙Qn,m=Γt,x+Γt,y+ΓR,x+ΓR,y, (A3) where we have suppressed the indices nandmon the right-hand side, and Γt,x=iet ̵h{(c† n+1,m+c† n−1,m)cn,m−h.c.}, (A4) Γt,y=iet ̵h{(c† n,m+1+c† n,m−1)cn,m−h.c.}, (A5) ΓR,x=eαR 2̵h⎛ ⎝c† n+1,m−c† n−1,m aσycn,m+h.c.⎞ ⎠, (A6) ΓR,y=−eαR 2̵h⎛ ⎝c† n,m+1−c† n,m−1 aσxcn,m+h.c.⎞ ⎠. (A7) In the continuum limit, the first term becomes Γt,x=ieta2 ̵h⎛ ⎝c† n+1,m−2c† n,m+c† n−1,m a2cn,m−h.c.⎞ ⎠ →ie̵h 2me(∂2c† ∂x2c−c†∂2c ∂x2) =−∂jx(r) ∂x, (A8) where we have used Eq. (16). Similarly, Γt,y→∂jy(r)/∂yso that Γt,x+Γt,y→−∇⋅j. (A9) APL Mater. 9, 060904 (2021); doi: 10.1063/5.0049882 9, 060904-9 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm The continuum limits of the Rashba terms are [see Eq. (42)] ΓR,x →(eαR/̵h)∂(c†σyc)/∂xandΓR,y→−(eαR/̵h)∂(c†σxc)/∂y, which may be summarized to ΓR,x+ΓR,y→−∇⋅j′. (A10) From the above, we finally obtain the continuity equation ˙Q=−∇⋅(j+j′) (A11) that expresses the conservation of charge. We now derive the continuity equation for spin in the same manner. The time derivative of the spin operators is ˙sα n,m=̵h 2(˙c† n,mσαcn,m+c† n,mσα˙cn,m). (A12) Substituting Eq. (A2) in this expression, we have ˙sα n,m=Σα t,x+Σα t,y+Σα R,x+Σα R,y, (A13) with Σα t,x=t 2i{(c† n+1,m+c† n−1,m)σαcn,m−h.c.}, (A14) Σα t,y=t 2i{(c† n,m+1+c† n,m−1)σαcn,m−h.c.}, (A15) Σα R,x=−αR 4⎛ ⎝c† n+1,m−c† n−1,m aσyσαcn,m+h.c.⎞ ⎠, (A16) Σα R,y=αR 4⎛ ⎝c† n,m+1−c† n,m−1 aσxσαcn,m+h.c.⎞ ⎠, (A17) where the indices nand mon the right-hand side are again sup- pressed. In the continuum limit, Σα t,x=ta2 2i⎛ ⎝c† n+1,m−2c† n,m+c† n−1,m a2cn,m−h.c.⎞ ⎠ →̵h 2̵h 2mei(∂2c† ∂x2σαc−c†σα∂2c ∂x2) =−∂jα s,x(r) ∂x, (A18) and similarly, Σα t,y→∂jα s,y(r)/∂yso that Σα t,x+Σα t,y→−∇⋅jα s(r). (A19) For the Rashba part, we obtain Σα R,x+Σα R,y→−∇⋅j(1),α s+τα(A20) with the torque term τα=αR ta2∑ i,j,βεijzεαjβjβ s,i(r). (A21)Therefore, the continuity equation for spin in the continuum limit is ˙sα=−∇⋅(jα s+j′α s)+τα. (A22) As indicated by the torque term τα, the spin is not conserved. The underlying reason is that the Rashba SOC breaks the SU (2) rotational symmetry in spin space. It should be noted that the spin current for the lattice Rashba Hamiltonian is defined by Eqs. (44) and (45), in which the additional factor 1/2 is included. The continuity equation (A22) holds for the spin current j′α sas given by Eq. (53). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. E. Hirsch, “Spin Hall effect,” Phys. Rev. Lett. 83, 1834–1837 (1999). 2S. Murakami, N. Nagaosa, and S.-C. Zhang, “Dissipationless quantum spin current at room temperature,” Science 301, 1348–1351 (2003). 3J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, “Universal intrinsic spin Hall effect,” Phys. Rev. Lett. 92, 126603 (2004). 4K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. 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5.0047088.pdf

Defect-free interface between amorphous (Al2O3)1/C0x(SiO 2)xand GaN(0001) revealed by first-principles simulated annealing technique Cite as: Appl. Phys. Lett. 119, 011602 (2021); doi: 10.1063/5.0047088 Submitted: 10 February 2021 .Accepted: 17 June 2021 . Published Online: 6 July 2021 Kenta Chokawa,1,a) Kenji Shiraishi,1,2 and Atsushi Oshiyama1 AFFILIATIONS 1Institute of Materials and Systems for Sustainability, Nagoya University, Nagoya 464-8601, Japan 2Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan a)Author to whom correspondence should be addressed: chokawa@nagoya-u.jp ABSTRACT We report first-principles molecular dynamics (MD) simulations that unveil the interface structures of amorphous mixed oxide (Al2O3)1/C0x(SiO 2)xand GaN polar surfaces. The MD allows us to perform the melt and quench (simulated annealing) simulations to forge distinct amorphous samples. We find that the dangling bonds are completely absent at all the obtained interfaces. This annihilation is due tothe diffusion of appropriate species, O for (Al 2O3)1/C0x(SiO 2)x/GaN(0001) and Al and Si for (Al 2O3)1/C0x(SiO 2)x/GaN(000 /C221), from the amor- phous to the interface and the subsequent formation of strong bonds with both ionicity and covalency at the interface. This absence of thedangling bond indicates the superiority of (Al 2O3)1/C0x(SiO 2)xfilms to Al 2O3or SiO 2as a gate oxide for the GaN–metal–oxide–semiconductor field effect transistor. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0047088 There has been a global trend in recent years toward a zero-carbon society. As important factors to achieve this, power devices with a lowenergy consumption are highly desired in power electronics. GaN hashigh potential as a material for such power devices due to its widebandgap, relatively high carrier mobility, and robustness in harsh envi- ronments, possibly replacing current Si power devices in the future: for instance, the superiority of physical properties of GaN renders the GaN-based metal–oxide–semiconductor field effect transistors (MOSFETs)with the high switching speed and the low energy consumption toreplace the Si-based insulated-gate-bipolar transistors (IGBTs). 1–4 To utilize the advantages of the superior physical properties in MOS devices, however, the exploration and development of gate insu-lators are indispensable. This is due to the fact that the quality of theinterface of the insulator and GaN decisively affects the carrier mobil-ity and the quality of the insulating films is strongly related to the deg- radation during the device operations. In Si devices, SiO 2,w h i c hi s usually formed by the oxidation of Si, plays an excellent role as a gateinsulator. For GaN, Ga 2O3is formed by the oxidation but shows the small conduction band offset (CBO) and valence band offset (VBO) atthe interface with GaN, 5thus being unsuitable for the gate insulator. In this context, deposited amorphous Al 2O3and SiO 2have been examined as gate insulators for GaN in the past.6–13Three important requirements for gate insulators are (i) suffi- ciently large band offsets with the semiconductor, (ii) sufficiently highdielectric property, and (iii) sufficiently low trap density at the inter-face. From this viewpoint, SiO 2has an advantage with its large CBO of 3.2 eV and VBO of 2.4 eV compared with the corresponding CBO and VBO of Al 2O3of 1.7 and 1.6 eV, respectively.14,15Al2O3has an advan- tage with its dielectric constant of 8.5 compared with the value of SiO 2, 3.8. An additional disadvantage of Al 2O3is the crystallization at low temperature, which leads to the formation of the grain boundary, pos-sibly inducing the leakage current in the devices. Hence, the mixed oxide of Al 2O3and SiO 2may be a plausible candidate as the gate insulator for GaN. This mixed amorphous oxide(Al 2O3)1/C0x(SiO 2)x, which we call AlSiO here, and also other ceramics containing Si, O, and Al atoms16,17have been studied in the field of geoscience and materials science for some time.18–24These materials have also attracted attention for their potential in semiconductor tech- nology. The AlSiO dielectric thin film is indeed formed by the alter-nate deposition of SiO 2and Al 2O3followed by the annealing. The fabricated MOS devices with an AlSiO gate oxide show better per-formances than those with SiO 2or Al 2O3:15,25theC–Vmeasurements of the GaN-based MOS capacitors with an (Al 2O3)1/C0x(SiO 2)xgate insulator show the low interfacial defect density and the higher Appl. Phys. Lett. 119, 011602 (2021); doi: 10.1063/5.0047088 119, 011602-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldurability for the dielectric breakdown. It has also been reported that the crystallization temperature of AlSiO becomes higher and the for- mation of grain boundaries during the fabrication process is substan- tially suppressed than Al 2O3.25In addition, from a theoretical perspective, our previous work has revealed that the oxygen vacancy in AlSiO is electrically inactive.26 The superiority of (Al 2O3)1/C0x(SiO 2)xover SiO 2and Al 2O3as observed in the above experiments is somewhat expected from their band offsets and dielectric constants. However, the most important factor for the superiority, i.e., the lower trap density at the interface, is mysterious and its microscopic reason is unknown. There have been theoretical works regarding the interfacial characteristics of ideal crys- talline oxides on the GaN(0001) surface:27–29it has been reported that Ga–O bonds are dominant at the GaN/Al 2O3interfaces without gap states appearing.27–29Yet, it is unclarified whether the existence of those Ga–O bonds is the principal reason for the lower trap density atthe interface of amorphous (Al 2O3)1/C0x(SiO 2)xwith GaN(0001) and even with GaN(000 /C221). In this Letter, we tackle this issue by performing the first-principles molecular dynamics (MD) simulations for the (Al 2O3)1/C0x(SiO 2)x/GaN interface in which a melt-quench technique is employed. This technique is regarded as dynamical simulated annealing30,31and enables us to f o r g ear e a l i s t i ca m o r p h o u s( A l 2O3)1/C0x(SiO 2)x/GaN interface, in con- trast with the previous calculations for the interface with ideal crystalline oxides, such as Al 2O3,S c 2O3,a n dG a 2O3.27–29The present simulations unequivocally show the presence of the defect-free interface of amor- phous (Al 2O3)1/C0x(SiO 2)xand GaN. A similar simulated annealing tech- nique has been successfully applied to amorphous26or heterogeneous interface systems.32–34 The calculations are performed using density functional theory (DFT)35with the Kohn–Sham (KS) scheme36using the Vienna ab-initio simulation package (VASP).37,38We use the generalized gradient approximation of Perdew, Burke, and Ernzerhof to the exchange–correlation energy functional39in our MD simulations and structural-optimization calculations. The hybrid functional by Hyde, Scuseria, and Ernzerhof (HSE)40in which the semilocal approximation in DFT is hybridized with the Hartree–Fock approximation is also used to calculate the density of states (DOS) accurately. We here use the fraction of the Fock exchange and the range-separation parameter as 0.35 and 0.20 A ˚/C01, respectively, in the HSE. By this set up, the bandgap becomes 3.3 eV for wurtzite–GaN, which is 2.9% smaller than the experimental value of 3.4 eV. Interactions between the ionic cores and the valence electrons are described by the projector-aug-mented-wave potential. 41T h ee l e c t r o nw a v ef u n c t i o n sa r ee x p a n d e d by the plane waves up to a kinetic energy of 500 eV. We have obtained the lattice constants a,b,a n d c, and the bulk modulus B0of wurtzite–GaN as a¼b¼3:247 A˚,c=a¼1:626, and B0¼182 GPa, respectively. These agree with experimental values within an error of 1.8% for the lattice constants and 9.9% for the bulk modulus.42The effects of the spin polarization are also examined, but no spin- polarized solutions appear through the calculations. The AlSiO/GaN interface is simulated by a superlattice model of AlSiO layers and GaN layers in which a supercell is of the hexagonal symmetry with the dimensions of 9.74 /C29.74/C234.46 A ˚3. For the Brillouin-zone integra- tion, we use the Cpoint for the MD calculations and structural optimi- zations and the 4 /C24/C21 points for the DOS calculations. Structural optimization is performed until all the atomic forces are less than1m e V / A ˚. To obtain the site-projected local density of states (LDOS), we use the energy spectrum of the Kohn–Sham (KS) states, i.e., the effective single-electron states, in DFT. It is noted that the electronic levels in the bandgap is rigorously the Fermi-level position at whichthe total energies of the different charge states become equal.However, the KS levels correspond to such electronic levels at least qualitatively. The computations are performed using the LOBSTER package 43,44in which the Kohn–Sham orbitals expressed in term of the plane waves are projected to the atomic orbitals. The atomic con-figurations are drawn by using VESTA. 45 We adopt the composition xin (Al 2O3)1/C0x(SiO 2)xasx¼0.38 in the present supercell calculation since this is the possible closest value to the experimental value 0.34–0.36, which shows the best perfor-mance in MOS devices. 15,25The supercell of our superlattice model consists of six bilayers of GaN with the lateral periodicity of 3 /C23, and of 32[(Al 2O3)1/C0x(SiO 2)x]r e g i o n( Fig. 1 ). Hence, our model contains two interfaces, (Al 2O3)1/C0x(SiO 2)x/GaN(0001) and (Al 2O3)1/C0x(SiO 2)x/ GaN(000 /C221) interfaces: the Ga-face (0001) and the N-face (000 /C221) are examined simultaneously. The dimension of the supercell along the[0001] direction is set to be 34.46 A ˚to reproduce the appropriate den- sity of (Al 2O3)1/C0x(SiO 2)xregion (see below). To prepare the initial structure of the (Al 2O3)1/C0x(SiO 2)xregion, we first introduce the a- Al2O3(Al64O96)s l a bw i t h( 0 0 0 1 )a n d( 0 0 0 /C221) surfaces, randomly remove 24 Al atoms and 12 O atoms from there, and then put 12 Siatoms at the Al sites. This procedure renders the mass density ofAlSiO layers to be 2.7 g/cm 3. This is the interpolated value between the mass densities of amorphous Al 2O3(3.05–3.40 g/cm3)46,47and amor- phous SiO 2(2.20 g/cm3) for the present composition x¼0.38. In order to obtain the interface of crystalline GaN and amor- phous AlSiO, the system is first heated up to 6000 K and equilibrated for 5 ps with fixing the GaN layers (layer 1 and layer 2 in Fig. 1 )a tt h e bulk positions. Then, the system is quenched to 2000 K. The quench-ing rate is an important factor to obtain the plausible amorphousstructures. Since we have succeeded in obtaining the bulk amorphous(Al 2O3)1/C0x(SiO 2)xwith clear gaps at a quenching rate of 200 K/ps in a previous work,26we adopt the same rate in the present study. Then, the GaN two bilayers near each interface (layer 1 in Fig. 1 ) are relaxed from the bulk positions and system is equilibrated at 2000 K for 10 ps.Finally, the system is quenched to 0 K at a rate of 200 K/ps followed bythe structural optimization. The amorphization of the AlSiO layer obtained by this heat treatment in the MD simulations is evidenced in the pair correlation functions shown in the supplementary material .I n the present work, we have applied the different initial velocities foreach atom at the beginning of the MD simulations and thus obtainedfour different amorphous AlSiO samples. This is to extract common FIG. 1. Initial structure of the (Al 2O3)1/C0x(SiO 2)x/GaN interface with x¼0.38 in our superlattice model. The bright area surrounded by the lines represents the unit cell.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 011602 (2021); doi: 10.1063/5.0047088 119, 011602-2 Published under an exclusive license by AIP Publishingfeatures appearing at AlSiO/GaN interfaces, irrespective of the struc- tural difference of the amorphous. The local structures of an interface of amorphous AlSiO and Ga- faced GaN(0001), i.e., AlSiO/GaN(0001), obtained in the present dynamical simulated annealing are shown in Fig. 2(a) . In the initial structure of the MD simulations, there are nine interface Ga atoms and each of them is accompanied by a single dangling bond (threefold coordinated Ga) toward the [0001] direction. (See the supplementary material for the pair correlation function in the present system from which we deduce the coordination numbers.) After the simulatedannealing, those Ga dangling bonds are completely annihilated by the O atoms, which have diffused from the AlSiO layer to the interface. We have indeed observed the O diffusion to the interface during snap- shots in the simulated annealing. Each of those O atoms forms a strong bond with the interface Ga atom and the back bonds with eitherAl or Si atoms. It is of note that the formation of this dangling-bond- free interface is common to all the four samples in the present work. This is examined from the distance between the Ga atoms at the inter- face and the nearest neighbor atoms. We have found that the nearest atoms are oxygen, and the interatomic distance is 1.78–2.07 A ˚.T h i s value agrees with the Ga–O bond lengths in crystalline b-Ga 2O3of 1.80–2.08 A ˚,48indicating the formation of the interfacial Ga–O bonds. The absence of the dangling bonds is also evidenced in the partial pair correlation functions and the resulting coordination numbers shown in the supplementary material . The oxygen density at the interface plane is calculated to be about 0.11 atoms/A ˚2. This density is obviously larger than the plane-averaged O density in the AlSiO region, which is computed as about 0.05 atoms/A ˚2, although the latter value depends on the thickness of the AlSiO region in our simulation cell. This indi-cates the oxygen diffusion from the AlSiO region toward the interface during the simulated annealing procedure, which corresponds to the annealing process in the experiments. In addition to the absence of the Ga dangling bonds, we have found none of the Ga–O–Ga bridge struc- ture, the Ga–Al bond, the Ga–Si bond, and the O–O bond near the interface of our four interface models: the AlSiO/GaN(0001) interface consists solely of the Ga–O bonds. N e x t ,w eh a v ep e r f o r m e dB a d e rc h a r g ea n a l y s i s 49–53to examine the partial charge Dqof each atom. The partial charge here is defined as the electron depletion (plus sign) or the accumulation (minus sign) compared with the neutral atom. It is of note that quantitative validityof the Bader analysis is limited to the cases in which the interatomic distances are close to their equilibrium distances. Figure 3 shows thepartial charge distribution for each atomic element as a function of the position zalong the [0001] direction. In Fig. 3 , the position is discre- tized with the width of Dz¼0.8 A˚and the partial charges of the atoms, which are located between zandzþDzin the width, are added, i.e., the total partial charge DQ(z). We obviously observe that Si, Al, and Ga are positively charged, while N and O are negatively charged inthis AlSiO/GaN system. At the AlSiO/GaN(0001) interface ( z/C247A˚), the total partial charge of the Ga atoms is more positive, being larger than that of Ga atoms in the GaN bulk, whereas the total partial chargeof O atoms is more negative than in the amorphous (Al 2O3)1/C0x(SiO 2)x layer. It is of note that the total partial charge of N atoms shows no prominent variation at the interface. The increase in the DQof Ga at the interface reflects the increase in the partial charge Dqof each Ga at the interface: the averaged Dqat the interface is þ1.50, while it is þ1.42 in GaN bulk. On the other hand, the averaged Dqof O at the interface is /C01.46, which is less negative than the averaged Dq¼/C01.61 in the (Al 2O3)1/C0x(SiO 2)xlayer. The more negative DQof O at the interface is, therefore, due to the diffusion of O atoms toward the inter-face as described above. In any case at the AlSiO/GaN(0001) interface, electrons are transferred from the interface Ga to the O, resulting in the formation of stable Ga–O bonds. The strength of the Ga–O bondcompared with the Ga–Al bond has been also found at the Al 2O3/GaN interface by our previous work27using the crystal-orbital-Hamilton- population analysis.43,44 We have found that the interface of amorphous AlSiO with the N-faced GaN(000 /C221), i.e., AlSiO/GaN(000 /C221), is also dangling-bond free.Figure 2(b) shows a local structure of AlSiO/GaN(000 /C221) obtained in the present simulated annealing technique. The nine N atoms at the interface with dangling bonds in the initial structure of the MD simu-lations are completely terminated by either Al or Si atoms, which dif-fuse from the AlSiO region to the interface. We have found that O atoms do not appear at the AlSiO/GaN(000 /C221) interface. Instead, Al FIG. 2. Local interface structures of the (a) amorphous (Al 2O3)1/C0x(SiO 2)x/ GaN(0001) and (b) amorphous (Al 2O3)1/C0x(SiO 2)x/GaN(000 /C221) interfaces with x¼0.38 obtained by our MD simulations. The representative structure among the four samples is shown. Green, blue, and light blue larger balls depict Ga, Si, and Alatoms, respectively, whereas red and blue–gray smaller balls depict O and Natoms, respectively. FIG. 3. The total partial charge DQðzÞdistribution for each atomic element as a function of the position zalong the [0001] direction obtained by the Bader charge analysis. The total partial charge is derived by adding the partial charge Dqof each atom between zandzþDz. In this figure, we take Dz¼0:8A˚, and contributions from the four samples are added. The plus and minus signs represent the electrondepletion and the accumulation compared with the neutral atom, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 011602 (2021); doi: 10.1063/5.0047088 119, 011602-3 Published under an exclusive license by AIP Publishingand Si atoms in this case appear to form Al–N and Si–N bonds, result- ing in the annihilation of the dangling bonds at the interfaces [Fig. 2(b) ]. Again, this feature is common to all the four amorphous s a m p l e s .T h eb o n dl e n g t h so fA l – Na n dS i – Nf o r m e da tt h ei n t e r f a c eare 1.82–2.23 A ˚with averaged value of 1.94 A ˚and 1.73–1.81 A ˚with averaged value of 1.77 A ˚, respectively. These values are slightly larger than the corresponding lengths in crystalline AlN of 1.89 A ˚and in b-Si 3N4of 1.73 A ˚, respectively,42,54but obviously show the bond forma- tion of the interface N with either the Si or the Al. The number of theterminating Al and Si atoms at the interface of our superlattice model isnine, and each atom makes a bond with the corresponding N atom of the GaN side. The bond angles of the interface Ga–N–Al or Ga–N–Si structures range from 92 /C14to 132/C14, and the averaged value is 109/C14with its standard deviation of 6/C14. This averaged value is almost same as the bond angle of Ga–N–Ga in GaN, which is 108/C14. This indicates that the terminating Al and Si atoms are located near the positions that follow the stacking sequence of the GaN region, and the interface is con- structed by an additional atomic layer with a wurtzite structure consist-ing of Al and Si atoms. All the remaining bonds of the terminating Aland Si atoms are formed with the O atoms in the (Al 2O3)1/C0x(SiO 2)x region at the interface [ Fig. 2(b) ]. The coordination number of each atom near the interface is shown in the supplementary material . The right part of Fig. 3 (z/C2427 A˚) shows the distribution of the total partial charge DQðzÞnear the AlSiO/GaN(000 /C221) interface. The DQðzÞof Al and Si clearly shows the accumulation of the positive charge at the interface Al and Si atoms, whereas the DQðzÞof N and O near the interface shows the negative-charge accumulation. The aver- aged partial charge Dqof Al and Si atoms is þ2.45 and þ3.13 at the interface, which are almost same as the values in the (Al 2O3)1/C0x(SiO 2)x layer of þ2.43 and þ3.06, respectively. Thus, the accumulation ofpositive charge is due to the higher Al and Si atomic density than in the (Al2O3)1/C0x(SiO 2)xvia the diffusion toward the interface. The averaged partial charge Dqof N at the interface is /C01.63, which is smaller than the value of /C01.42 in the GaN bulk, resulting in the large total partial charge DQof N at the interface. The averaged partial charge Dqof O near the AlSiO/GaN(000 /C221) interface is /C01.63, which is almost same as in the (Al 2O3)1/C0x(SiO 2)xlayer. On the other hand, the density of O atom in the interface region is 0.13–0.15 atoms/A ˚2, which is higher than the plane-averaged O density of 0.05 atoms/A ˚2in the AlSiO region. The a c c u m u l a t i o no ft h en e g a t i v ec h a r g es h o w ni n Fig. 3DQðz/C2425 A˚)i s due to the higher plane-averaged O density. Namely, the more positive charge DQof Al and Si is compensated by the more negative N atoms and higher density of O atoms near the interface. The N dangling bondsat the unrelaxed interface attract Al and Si atoms, and those Al and Siare, in turn, accompanied by the negatively charged O atoms. This fea-ture along with the quantitative values of the partial charges of each atom at each position found in the present calculations is indicative of the importance of both the ionic and the covalent characters in the(Al 2O3)1/C0x(SiO 2)x/GaN interface. Figure 4 shows the calculated local density of states (LDOS) of our four distinct samples generated by the present simulations [from Figs. 4(a)–4(d) ]. The LDOS integrated over each (0001) plane is plot- ted as a function of the position along the [0001] direction z.T h em o s t important character is the absence of the gap states in whole regions ofthe GaN(0001)/AlSiO/GaN(000 /C221) system. In the previous work, we have found that oxygen vacancies in amorphous AlSiO induce several KS states in the lower half of the energy gap. 26In the present interface system, we have found no O vacancy in spite that the O atoms diffuseto the interface: the oxygen density becomes inhomogeneous and thensuppresses the generation of the O vacancy, leading to the absence of FIG. 4. Calculated local density of states (LDOS) of the amorphous (Al 2O3)1/C0x(SiO 2)x/ GaN interfaces with x¼0.38 obtained by our MD simulations. The results for the four distinct amorphous (Al 2O3)1/C0x(SiO 2)xare shown from (a) to (d). For each model, (i)side view of the atomic structure of AlSiO/GaN interface (ii) and the LDOS are shown. The color code of the atoms is same as in Fig. 1 . The energy origins of the LDOS are set to be the valence band maximum ofGaN.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 011602 (2021); doi: 10.1063/5.0047088 119, 011602-4 Published under an exclusive license by AIP Publishingthe gap state in the amorphous region. Our previous calculations also show that Ga–Al bonds at the crystalline Al 2O3/GaN(0001) interfaces induce KS states in the bandgap of GaN.27In the present AlSiO/ GaN(0001) interfaces, such structures inducing the gap states are absent due to the interface stabilization by the O atoms. This stabi-lization comes from structural flexibility of the mixed amorphous containing both group III and IV elements. The electron counting rule 55may be satisfied near the interface with different kinds of the elements. At the AlSiO/GaN(000 /C221) interface, substantial number of Al–N and Si–N bonds appear. Yet, these bonds induce no gap states presumably due to their similarity to the Ga–N bond. Reducing the carrier traps in MOSFETs is crucial for the device performance. Such traps are usually induced by structural imperfec- tion near the interface, and the dangling bond is one of the majorresources. The present calculations unequivocally reveal that the dan- gling bonds at the AlSiO/GaN interfaces are annihilated by the dynamical simulated annealing, which correspond to the appropriateannealing in experiments. This finding infers the superiority of amor- phous (Al 2O3)1/C0x(SiO 2)xfilms to either Al 2O3or SiO 2films for the gate oxide of GaN MOSFETs, which has been experimentallyobserved. 15The annihilation of the dangling bonds at the interface is due to the diffusion of appropriate species in the amorphous AlSiO toward the interface and the formation of the stable cation–anion bonds. It is noteworthy that such bond formation requires chargetransfer from the AlSiO film to the interface to satisfy the electron counting rule. 55The atomic diffusion combined with the charge trans- fer is mostly possible for the mixed oxides AlSiO. In summary, we have performed first-principles molecular- dynamics simulations that unveil the interface structures of the amor-phous mixed oxide (Al 2O3)1/C0x(SiO 2)xand GaN polar surfaces. The simulated annealing technique in the molecular dynamics simulations enables us to forge four distinct amorphous (Al 2O3)1/C0x(SiO 2)xand then realistic (Al 2O3)1/C0x(SiO 2)x/GaN(0001) and (Al 2O3)1/C0x(SiO 2)x/ GaN(000 /C221) interfaces. We have found that the dangling bonds are completely absent in all the resulting interfaces for both Ga-face and N-face GaN. This annihilation is due to the diffusion of appropriatespecies, O for (Al 2O3)1/C0x(SiO 2)x/GaN(0001) and Al and Si for (Al2O3)1/C0x(SiO 2)x/GaN(000 /C221), from the amorphous to the interface and the subsequent formation of strong bonds with both ionicity andcovalency at the interface. This absence of the dangling bond infers the superiority of (Al 2O3)1/C0x(SiO 2)xfilms to Al 2O3or SiO 2as a gate oxide for GaN-MOSFETs. See the supplementary material for the recipe of the dynamical simulated annealing procedure and for the pair correlation functionsof the amorphous (Al 2O3)1/C0x(SiO 2)x/GaN systems obtained by our MD simulations and the coordination numbers of constituting atoms. This work was supported by the MEXT-Japan research programs, “Program for Research and Development of Next-Generation Semiconductors to Realize an Energy-Saving Society (Contract No. JPJ005357),” “Program for Promoting Research on the SupercomputerFugaku” (Quantum-Theory-Based Multiscale Simulations towardDevelopment of Next-Generation Energy-Saving Semiconductor Devices), and “Program for Creation of Innovative Core Technology for Power Electronics (Contract No. JPJ009777),” and also by the grants-in-aid under Contract No. 18H03873. 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J. Appl. Phys. 129, 094301 (2021); https://doi.org/10.1063/5.0038612 129, 094301 © 2021 Author(s).Majorana and non-Majorana modes in a nanowire in partially proximity to a superconductor Cite as: J. Appl. Phys. 129, 094301 (2021); https://doi.org/10.1063/5.0038612 Submitted: 02 December 2020 . Accepted: 03 February 2021 . Published Online: 02 March 2021 Ze-Gang Liu , Yue-Xin Huang , Guang-Can Guo , and Ming Gong ARTICLES YOU MAY BE INTERESTED IN Majorana qubits for topological quantum computing Physics Today 73, 44 (2020); https://doi.org/10.1063/PT.3.4499 Quantum-spin-Hall phases and 2D topological insulating states in atomically thin layers Journal of Applied Physics 129, 090902 (2021); https://doi.org/10.1063/5.0029326 Observation of 2D transport in Sn- and In-doped Bi 2−xSbxTe3−ySey topological insulator Journal of Applied Physics 129, 095702 (2021); https://doi.org/10.1063/5.0035692Majorana and non-Majorana modes in a nanowire in partially proximity to a superconductor Cite as: J. Appl. Phys. 129, 094301 (2021); doi: 10.1063/5.0038612 View Online Export Citation CrossMar k Submitted: 2 December 2020 · Accepted: 3 February 2021 · Published Online: 2 March 2021 Ze-Gang Liu,1Yue-Xin Huang,1Guang-Can Guo,1,2,3and Ming Gong1,2,3 ,a) AFFILIATIONS 1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People ’s Republic of China 2Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 3CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China a)Author to whom correspondence should be addressed: gongm@ustc.edu.cn ABSTRACT We investigate the Majorana and non-Majorana modes in a nanowire in partial proximity to a superconductor, in which the gapped superconductor will play different roles in different topological regimes. In the gapped topological superconducting phase, it plays the roleof a topological barrier, which confines some localized edge modes in the quantum dot (QD) region. The probability for the wave function in this region can approach unity by tuning the system parameters. These low-lying localized modes exhibit linear spectra with equal energy level spacing, with eigenvalues ε n¼vFnπ=(2L), where vFis the Fermi velocity, Lis the size of the QD region, and n[Z. We demonstrate these features using a spinless nanowire in proximity to a p-wave superconductor and a spin –orbit coupled semiconductor nanowire in proximity to a s-wave superconductor. A simple picture is proposed to understand the behavior of these results. However, in the trivial superconducting phase when both bands are occupied in the spin –orbit coupled mode, we observe some non-Majorana modes, with complicated low-lying excited spectra, which resembles that reported in experiments. These differences are rooted deeply in the bulk-edge correspondence. These observations may be able to facilitate the identification of Majorana zero modes in experiments. Published under license by AIP Publishing. https://doi.org/10.1063/5.0038612 I. INTRODUCTION Topological superconducting phases and the associated self-Hermitian Majorana zero modes (MZMs) have received inten- sive attention in recent years for their non-Abelian exchange statis-tics and their essential role as building blocks for topologicalquantum computation. 1–3Originally, this exotic state was proposed to be realized in some rare p-wave superconducting materials, such as Sr 2RuO 4, which has not yet been convincingly proven to be a p-wave superconductor.3–6In the past decade, several new plat- forms have been proposed to realize this exotic quasi-particle, inwhich the most widely explored platform is the semiconductorInAs and InSb nanowires with strong spin –orbit coupling strengths and large Landé gfactors in proximity to a conventional s-wave superconductor (such as Nb or Al). This platform follows theoriginal proposal by Das Sarma ’s group, 7–9using the fact thatspin–orbit coupling can induce coupling between the spin and momentum. Thus, when the Zeeman field opens a gap in the twobands, the pairing at the same band is allowed even with s-wave interaction, which yields an effective p-wave pairing in the dressed basis (eigenvectors of the single particle Hamiltonian withoutpairing). The large Landé gfactor ensures that even a weak external Zeeman field can induce sizable Zeeman splitting (of the order ofseveral meV), which allows the tuning of chemical potentialbetween the two bands. This platform has been realized in experi-ments based on InSb-NbTiN, InSb-Nb and InAs-Al hybridstructures. 10–12In Ref. 13, the authors have even realized a ferro- magnetic iron chain on the surface of s-wave superconducting lead (Pb), which may also be categorized to this platform although theeffective Hamiltonian is somewhat more complicated. In thissystem, the localization of wave functions at the two open ends canJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 094301 (2021); doi: 10.1063/5.0038612 129, 094301-1 Published under license by AIP Publishing.be directly mapped out using high-resolution spectroscopic imaging technique. Researchers are also trying to explore these MZMs in the heterostructure based on topological insulators (suchas Bi 2Se3thin films) and the s-wave superconductors, in which the surface states of the topological insulator materials play the samerole as spin –orbit coupling, following the proposal by Fu and Kane 14and Fu.15The similar zero-bias conductance peaks in the vortex cores have been reported in Refs. 16–19. Recently, there is also great interest in the searching of the MZMs based on iron-based superconductors with spin fluctuation induced s +-wave pairing.20–26The quantized zero-bias conductance peaks in this new platform have recently been reported in Refs. 27–32. The unambiguous identification of the MZMs is still a great challenge in experiments. The most convincing evidence may comefrom the 4 πperiodicity from the braiding of these zero modes, 2,5 which has not yet been observed in experiments. This statistics has only been simulated using photonic qubits in Ref. 33. From the point of view of theoretical understanding, the evidence from thezero-bias conductance is not unique for MZMs because the similarfeatures can also be induced by some other mechanisms, includingthe Kondo effect, 34–36weak antilocalization,37Andreev bound states,38–45and disorder effect,45–48all of which can lead to trivial zero-bias conductance peaks mimicking the Majorana transportsignatures. 10–13,49–51In Refs. 52and53, the authors have improved the quality of the nanowires using the molecular beam epitaxy method, which is expected to greatly reduce the disorder effect, and the quantized conductance 2 e2=/C22his approached. However, it does not rule out the possible origins by the other mechanisms, whichhave been a subject of intensive theoretical investigation. 38–45 Researchers are also trying to understand the quantized conduc- tances from some trivial interpretation by examining the difference between the MZMs and the Andreev bound states. The under-standing of these modes may also need a more accurate modelingof the nanowire in proximity to the s-wave superconductor. For instance, in Ref. 54, Cao et al. try to understand the decay of the subgap energy levels by assuming a steplike spin –orbit coupling strength in the nanowire. For a finite size nanowire, the lateral con-fined energy levels may also be important, giving rise to the multibandtopological superconducting phase. 55To take the electron –electron interaction into account, the Schrödinger –Poisson equation is also adopted, which can slightly renorm alize the parameters of the nano- wire.56,57Furthermore, the disorder effect is also an intensively studied topic in theory,45–48while a detailed comparison between theory and e x p e r i m e n t si ss t i l lc h a l l e n g i n g . This work is motivated by the experiments of semiconductor nanowires,10–13,49–53,58–61which are partially covered by the s-wave superconductors. The unpaired region in one of the open ends isfrequently termed as a quantum dot (QD), which may confinesome low-lying energy levels. While the physics in this platform has been intensively studied, a picture for the localization of the edge states in this region is still lacking and their analytical proper-ties are rarely discussed in the literature. Here, we focus on thephysics of these edge states, aiming to provide some analyticalexpressions to describe these states based on a simple picture. We find that the topological gapped phase can serve as a barrier for the localized modes in the QD region. These localized modes exhibitlinear spectra with equal energy level spacing, that is,ε n¼vFnπ=2L, where n[Z,vFis the Fermi velocity, and Lis the length of the QD region. A simple mechanism to this result is also given. These confined states are mainly determined by the param-eters of the QD region when the QD length is long enough. Incontrast, the trivial barrier will exhibit totally different behaviors.For example, in the spinless model, no confined states can be observed in the QD region; however, in the spin –orbit coupled model as used in experiments, some non-Majorana modes local-ized in the QD region can be observed. The differences betweenthem are rooted deeply in the bulk-edge correspondence. Theseresults are useful to facilitate the identification of the MZMs in experiments. This paper is organized as follows. In Sec. II, we discuss the properties of the low-energy excited states in a nanowire based on atoy model with p-wave pairing. We will show that the localized wave functions and the level spacing of these states are determined by some simple parameters of the nanowire in the QD region. The general mechanism to ε nin the QD region is presented in Sec.II A. In Sec. III, the similar features are confirmed in a semi- conductor nanowire in proximity with a s-wave superconductor using the setup in experiments. Parameters to quality of these states in this spinful model are also presented. We discuss the relevance of our results to experiments in Subsection III B . Finally, we summary these results in Sec. IV. II. A TOY MODEL WITH p-WAVE PAIRING We first consider the following toy model (see Fig. 1 ) for a spinless nanowire in proximity to a p-wave superconductor, which was first considered by Kitaev for the realization of the MZMs.3 The difference is that in our model the region x[[0,L] is not proximity to the p-wave superconductor. This assumption makes sense even in realistic materials when the proximity length ξ/C28L (ξcan be a few to tens of nanometers62). The corresponding effec- tive Hamiltonian is written as H¼/C0@2 x 2m/C0μ/C18/C19 τz/C0iθ(x/C0L)Δτx@x, (1) where θ(x/C0L) is the Heaviside step function and the Pauli matri- cesτz,xacting on the particle and hole subspaces. We assume an infinity wall in the region where x,0. This model follows inti- mately the setup used in experiments. In Ref. 53, Zhang et al. have considered an InSb nanowire partially covered by the aluminum coating. The QD region is about L¼0:3μm; while the whole length of the nanowire is about 1.2 μm. In the experiment, the profile of the potential of the quantum dot can be tuned by theback gate. In Ref. 61, the QD region in the InSb nanowire is much longer than the above value (roughly about L/difference2/C03μm). In the first experiment for the MZMs by Mourik et al. , 10this region has the size about L¼0:2μm; and in Ref. 49, the QD region has the size about L¼0:15μm. For this reason, we see that this is a common structure used in experiments. This model can be solved analytically, which is one of the most important advantages of this toy model. In the QD region (0,x,L), the eigenvalues can be written as ε(1) k¼+jk2 2m/C0μj,Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 094301 (2021); doi: 10.1063/5.0038612 129, 094301-2 Published under license by AIP Publishing.which is gapless and trivial [see Fig. 1(c) ]. In the paired region (x.L), the eigenvalues are ε(2) k¼+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (k2 2m/C0μ)2þΔ2k2q , which is gapped when μ=0 [see Fig. 1(c) ], with excitation gap given by Eg¼min(jμj,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mΔ2(2μ/C0mΔ2)q ): (2) The paired nanowire becomes topological nontrivial when μ.0. When a state is confined in the QD region with energy smallerthan the bandgap width E g, the paired nanowire plays the role of a confinement potential, similar to the prototype models in textbooks of quantum mechanics. Throughout this work, Egwill be termed as barrier width. In the following, we will investigate the confined spectra in the QD region. We write the wave function in region0,x,Las ψ1(x)¼c1eikexþ~c1e/C0ikex c2eikhxþ~c2e/C0ikhx/C18/C19 , (3) where c1,c2,~c1, and ~c2need to be determined by the boundary conditions. The expressions for kiare determined by the Schrödinger equation Hψ1¼εψ1, with ke¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m(μþε)pand kh¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m(μ/C0ε)p.63With the vanished boundary condition ψ1(0)¼0, we have c1¼/C0~c1,c2¼/C0~c2. In the paired region (x.L), we have ψ2(x)¼X ibiai 1/C18/C19 e/C0qix, (4) where<qi.0 is used to ensure that the wave function must decay to zero when x!1,63indicating localization on the left end. By plugging ψ2(x) into the Schrödinger equation, we have det/C0q2 i=2m/C0μ/C0ε iqiΔ iqiΔ q2 i=2mþμ/C0ε/C18/C19 ¼0, (5) which is used to determine qi. This expression yields q2 i¼2Δ2m2/C02μm+2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε2þΔ4m2/C02Δ2μmq : (6) We find four roots, two of which satisfy Re qi.0. These two solu- tions are denoted as q1and q2. Thus, the wave function in the paired region has the form ψ2¼b1a1 1/C18/C19 e/C0q1xþb2a2 1/C18/C19 e/C0q2x: (7) By the Schrödinger equation Hψ2¼εψ2, we find a1¼iq1Δ q2 1=2mþμþε,a2¼iq2Δ q2 2=2mþμþε: (8) When ε/difference0, we expect q2 i=2mþμ¼iqiΔ, thus ai¼i, which is the origin why the electron and hole sectors have a phase difference by π=2. This feature will be important in the Andreev reflection [see Fig. 1(b) ]. Matching these two wave functions at x¼Lwill give two new boundary conditions ψ1(L)¼ψ2(L),ψ0 1(L)¼ψ0 2(L): (9) We neglect the possible resistance potential at the interface because the nanowires in the QD region and paired region are essentiallythe same. This resistance will not be important when it is lowenough or when the wave functions of the localized modes are fully localized in the QD region (see discussion below). From the above continuous conditions, we determine the values of c 2and b1,2as FIG. 1. (a) The nanowire in partially proximity to a superconductor (SC). (b) Andreev reflection. The gapped topological superconductor (Topo. SC) servesas a barrier for the confined states in the QD region, in which the particle (solid circle) and hole (open circle) will be almost fully localized in this region. (c) The energy bands in the QD region and paired region in momentum space forAndreev reflection near the two Fermi points +k F. The arrows in the QD region represent the moving direction of electrons and holes. (d) A general mechanism forδk¼nπ=(2L) applicable to both models considered in this work. The phase accumulated by this trajectory is given by 2 nπ, with n[Z; see Eq. (23).reand rhare the reflection coefficients of electrons and holes, respectively, in Andreev reflection.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 094301 (2021); doi: 10.1063/5.0038612 129, 094301-3 Published under license by AIP Publishing.follows: c2 c1¼csc(khL)(a1A1/C0a2A2) a1a2(q1/C0q2),b1,2 c1¼2iM1,2 q2/C0q1, (10) where A1,2¼q1,2sin (keL)þkecos (keL), B1,2¼khcos (khL) þq1,2sin (khL),M1¼A2 a1eq1L, and M2¼/C0A1 a2eq2L. In these equa- tions, the constant c1is used for normalization of the whole wave function. We find that the solution of εfrom Eq. (5)is given by the following nonlinear equation: F(ε)¼a1A1B2/C0a2A2B1¼0: (11) When ε/difference0 and a1¼a2¼i, the above solution is reduced to F(ε)¼khcos(khL)sin( keL)/C0kecos(keL)sin( khL)¼0: (12) This simplified expression is independent of q1andq2, which enter the equation only through the coefficients a1and a2. As a result, the solution of εonly depends on the parametrs in the QD region. When ε/difference0, its solution can be solved approximately. Although this toy model is not as practical as the spin –orbit coupled nano- wire in proximity to the s-wave superconductor in the realization of the MZMs, it can give some important insight into the physicsin it. In the following, we discuss the fate of these confined states in two different cases. (1)In the topological superconductor regime . In this regime μ.0,k eand khare real numbers and the wave function in the QD region can be written as ψ1(x)¼2c1sinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m(μþε)px 2c2sinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m(μ/C0ε)px/C18/C19 : (13) In the paired region, ψ2decays in the form of e/C0q1,2x. For the MZMs, the probability that the wave function confined in the QD region is computed as P¼ðL 0jψ1j2dx¼p2/C0psinp p2þsþw/C0wcospþpsinp, (14) where p¼2ffiffiffiffiffiffiffiffiffi2mμpL,w¼4LmΔ, and s¼4Lμ=Δ. The two compo- nents of ψ1have the same probability Pin the confined region. When p¼2nπwith n[Zþ, we find P¼1 1þq=p2¼2mLΔ 1þ2mLΔ/difference1/C01 2mLΔ, (15) which can reach P.0:99 with mLΔ.49:5. We demonstrate this behavior by investigating the oscillation of this probability as afunction of chemical potential in Fig. 2 . For this reason, the proper- ties of the MZMs are purely determined by the parameters of the QD region only if the QD length is long enough, while the gapped topological superconductor merely plays a role of a topologicalbarrier. In this limit, a simplified picture for the low-lying energylevels can be found; however, when the length is small, the leaking of the wavefunction to the superconducting region is also impor- tant, and the low-lying energy levels also depend strongly on thepairing strength in the superconducting region. 65This result is dif- ferent from the case when the whole nanowire is covered by the superconductors, in which the properties of edge modes are alsodetermined by the parameters of superconductors. With thisfeature, the properties of the edge modes can be tailored in experi-ments due to the well-developed semiconductor technology. Next, we consider the properties of the low-lying excitations. When ε/C250, we find k e,h/C25+εffiffiffiffiffim 2μr þffiffiffiffiffiffiffiffiffi 2μmp ,a1,2/C25iΔq1,2 q2 1,2=2mþμ¼i, q1,2/C25mΔ+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ2/C02μ mr ! ,Δ.0,(16) which satisfy <q1,2.0. An important point needs to be empha- sized is that while εcan be treated as a small number during series expansion, εLffiffiffiffiffiffiffiffiffiffiffi 2m=μp should be treated as a large value. Using the above approximate solutions, we can obtain a simplified expressionforF(ε)¼0a s ffiffiffiffiffiffiffiffiffi 2mμp sin εLffiffiffiffiffiffiffi 2m μr/C18/C19 /C0εffiffiffiffiffim 2μr sin(2 Lffiffiffiffiffiffiffiffiffi 2mμp )¼0: (17) To the leading term by assuming jεffiffiffiffi m 2μq j/C28jffiffiffiffiffiffiffiffiffi2mμpj, the solution of the above equation is given by εn¼nπ 2Lffiffiffiffiffi 2μ mr ,n[Z, (18) which means that the properties of the confined states are only determined by the parameters of the nanowire, such as m,μ, and L. FIG. 2. The probability Pfor the MZMs in the QD region as a function of chemical potential μfor different Δ(meV nm) and L(nm). We set m¼23me (see the effective mass of electron in Sr 2RuO 4in Ref. 64,meis the electron rest mass), and the dashed lines indicate the values based on Eq. (15).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 094301 (2021); doi: 10.1063/5.0038612 129, 094301-4 Published under license by AIP Publishing.The nonlinear correction to this solution is small (see Fig. 3 ), thus is neglected. This linear spectrum is also quite different from the confined levels in an infinite square well with width L, which scales asεn/(n=L)2forn[Zþ. The corresponding numerical results for jεnj,Egare pre- sented in Fig. 4(a) .I nFig. 4(b) , we present the computed eigenval- ues as a function of n, which completely agree well with the prediction by Eq. (18). To show that these modes are indeed fully localized in the QD region, we present the wave functions for thesemodes in Fig. 4(c) forn¼0–3. In this plot, we have purposely chosen parameters such thatffiffiffiffiffiffiffiffiffi2mμpL=πis roughly an integer number, thus at the boundary the wave functions will take the maximal value or minimal value alternatively with respect to n.A similar alternation can be observed for other parameters, thoughnot so obviously. (2)In the trivial superconductor regime . In this regime, μ,0 and no confined state can be observed in the QD region. The gap width is determined by E g¼/C0μ, and ke,khare purely imaginary numbers. In the QD region, the wave function can be written as ψ1(x)¼c1(e/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m(jμj/C0ε)p x/C0effiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m(jμj/C0ε)p x) c2(e/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m(jμjþε)p x/C0effiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m(jμjþε)p x) ! , (19) forjεj,Eg. This solution means that the confined state, if existed, will all be localized at the interface x¼L. We find that while the continuous condition ψ1(L)¼ψ2(L) is fulfilled, the condition ψ0 1(L)¼ψ0 2(L) cannot be satisfied simultaneously, leading to the above claim. The absence of confined modes in this region can be understood in the following way. Let us assume that there is arobust zero mode localized at the interface between the QD regionand the paired region near x¼L. One may imagine to introduce an infinitesimal pairing to the QD region, which is still in the topo- logical trivial phase. As a result, we see that both the QD region and the paired region ( x.L) are topological trivial, but the inter- face supports a robust edge mode near x¼L. This conclusion vio- lates the bulk-edge correspondence widely studied for topologicalphase transitions. 66This picture based on proof by contradiction is useful for us to understand the essential difference between topo- logical barrier and trivial barrier, which is deeply rooted in the well-established correspondence principle. A. General mechanism for Eq. (18) This observation points to a more general interpretation to Eq.(18). To this end, let us recall the basic results in Ref. 63by Blonder et al. , in which when the contact resistance of the interface FIG. 3. The real part and imaginary part of Eq. (11), and the approximate expression of Eq. (12). Parameters used are m¼23me,L¼170 nm, μ¼1:0 meV , Δ¼0:4 meV n m. FIG. 4. (a) Confined states below the barrier width Eg(red lines) as a function of chemical potential [see Eq. (2)]. We take Δ¼0:4 and L¼100. (b) Energy levels εnas a function of n=L. Parameters for the symbols are red big squares: L¼100,Δ¼1:7,μ¼0:2; red small squares: L¼100,Δ¼0:3,μ¼0:2; red open circles: L¼100,Δ¼1:7,μ¼1:5; blue open circles: L¼170, Δ¼1:7,μ¼1:5; red solid circles: L¼100,Δ¼1:7,μ¼0:6; blue solid circles: L¼170,Δ¼1:7,μ¼0:6. (c) Wave functions for the confined states with n¼0–3. We have used L¼197:6 (red vertical dashed lines) with Δ¼1:7,μ¼1:5. In all plots, m¼23me. The units for L,Δ,μare nm, meV n m, meV, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 094301 (2021); doi: 10.1063/5.0038612 129, 094301-5 Published under license by AIP Publishing.is absent or it is small enough, the normal reflection of electrons and holes is negligible or strictly forbidden. Then for the in-gap localized modes, only the Andreev reflection is allowed, which isaccompanied by a phase difference π=2. Thus, we have a picture as shown in Fig. 1(d) for the solutions in the QD region. To this end, let us assume the solution that an electron with ε/difference0 is injected into the topological barrier. We can rewrite the wave function in the QD region as ψ 1(x)¼eikexþrhae/C0ikex ae/C0ikhxþreeikhx/C18/C19 , (20) where reandrhare the reflection coefficients of electrons and holes (see Eq. A11 in Ref. 63), with re¼+iand rh¼+iwhen ε¼0. The vanished boundary condition at x¼0 is given by 1þrha¼0, aþre¼0: (21) The above wave function can be understood using the following steps based on only the Andreev reflection [ Fig. 1(b) ]. For an elec- tron inject to the barrier with wave function eikex, it will reach the barrier and is reflected as a hole with wave function as reeikhL. This hole at x¼0 will be reflected as a hole by a normal reflection with wave function as ae/C0ikhx. After that, the hole will be reflected as an electron with wave function arhe/C0ikex. Finally, destructive interfer- ence happens at x¼0 to satisfy the vanished boundary condition. From the phase accumulated in this process, we have [see the tra-jectory in Fig. 1(d) ] e ikeLree/C0ikhLeiπe/C0ikhLrheikeLþ1¼0: (22) From Eq. (21), we find a¼/C0re(the minus sign comes from the normal reflection of hole accompanied by a πphase) and rerh¼1, thus the above condition is given by 2(keL/C0khL)¼2nπ,n[Z: (23) This is one of the central relation in this work. We can assume ke¼kFþδkandkh¼kF/C0δk[seeFig. 1(c) ], then δk/C12L¼nπ,n[Z: (24) This relation has been derived from a different way in Eq. (12), hence εkFþδk/C0εkF¼@εk @kjk¼kF/C18/C19 δk¼vFδk: (25) To the leading term, the above solution yields Eq. (18), which can be further simplified by defining a Fermi velocity vF¼kF=m¼ffiffiffiffiffiffiffiffiffiffiffi 2μ=mp . This picture will be used to study the con- fined states in the second model in Sec. III[see Eq. (28)]. Notice thatδkin this relation is half of that by the normal reflection, such as that in the deep square well, in which δk¼π=L, from the doubled trajectory in Fig. 1(d) .III. A REALISTIC MODEL FOR THE MZMs WITH s-WAVE PAIRING With these results in the p-wave superconducting model in mind, we next try to see whether the similar features can also be found in realistic materials based on a spin –orbit coupled nanowire in proximity to the s-wave superconductor.7,26,67,67–71This model has been used for searching the MZMs in realisticexperiments. 10–13,49–53,58–61If it is true, then this mechanism is also applicable to the other platforms, which are mentioned in Sec. I. We consider the following tight-binding model:71 H¼/C0X i(μcy iciþtcy iciþ1)/C0iαX icy iσyciþ1þh:c: þhX icy iσzciþX iθ(x/C0L)(Δcy i"cy i#þΔ*ci#ci"), (26) with ci¼(ci",ci#)T. In this expression, t,α,h, and Δare hopping amplitude between nearest-neighbor sites, spin –orbit coupling coefficient, Zeeman field, and s-wave pairing strength, respectively, andcy iσ(ciσ) is the creation (annihilation) operator for the fermion. In the topological regime, we have (2t/C0jμj)2þjΔj2,h2,(2tþjμj)2þjΔj2: (27) The single particle spectra can be written as εk¼+(μþ2tcosk)+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2þ4α2sin2kp . In this model, t/C25/C22h2=(2m*a2)¼10 meV for lattice constant a/difference15 nm and m*/difference0:015me(using effective mass of electron in InSb from Ref. 72). We choose this tight-binding model only for the sake of numerical convenience. We only focus on the region when thechemical potential is close to the bottom of the single particlebands [see Fig. 5(c) ]. In Fig. 5(a) , we plot the possible confined states by the superconducting barrier. The corresponding single particle spectra are presented in Fig. 5(c) . We find that when μ/C20/C02t/C0h, the momenta of k iare purely imaginary numbers, thus no confined states can be observed (see proof by contradiction in Sec. II A). In the topological regime, we can find robust confined edge modes with equal level spacing. Using the previous simplepicture, we find the eigenvalues can be written as ε n¼εkFþδk/C0εkF¼nπ 2LvF,n[Z, (28) where vF¼2tsink/C04α2sinkcoskffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2þ4α2sin2kp , (29) with kto be determined by 2 tcoskþμþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2þ4α2sin2kp ¼0. This Fermi velocity may be tuned in a wide range by controlling the chemical potential and Zeeman field, which can directly influ- ence the energy level spacings for these confined states. This linearspectrum is the same as Eq. (18), except for a different Fermi veloc- ityv F. As before, these levels are only determined by the parame- ters of the nanowire [see Fig. 5(b) ] and P/difference1. However, a notable difference is that in the trivial phase regime when both bands areJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 094301 (2021); doi: 10.1063/5.0038612 129, 094301-6 Published under license by AIP Publishing.occupied with kito be real numbers, plane wave states can be con- fined in the QD region, which give rise to ubiquitousnon-Majorana modes with almost zero energy as reported by Chenet al. in Ref. 60[see our data in Fig. 5(a) for~μ.2:0 meV]. In this regime, the eigenvalues of these localized modes are no longer equally spaced, instead, closing and reopening of the energies nearthe Fermi energy without sticking is expected. This bound stateeven with zero energy, which can happen at some magic point, isnot robust against perturbation. The similar behaviors in the excited energy levels are also shown in Fig. 5(a) due to the coupling between the four Fermi points. In Fig. 6 , we show some of the localized modes confined by the superconducting gap in thismodel, which are almost localized in the QD region as expectedfrom the toy model. The wave functions may exhibit different behaviors in these two cases. In the topological regime, the density of wave function oscillates almost periodically; while in the trivialregime, strong interference between the four Fermi points is pre-sented, thus the density of wave function oscillates irregularly in the QD region. In Ref. 60, it has been found that the non-Majorana zero-bias conductance peaks are ubiquitous, the feature of theselocalized modes, if observed in experiments, may provide some further evidence for the identification of the MZMs in experiments. 13 A. Estimation of εnin semiconductor nanowires The low-lying energy levels in the QD region can be estimated using the following way. In a typical nanowire with carrier densityρand Fermi momentum k F¼πρ, the eigenvalues according to Eq.(24) can be obtained using εn¼n/C22h2πρπ 2m*L,n[Z: (30) For the typical values in InSb, we use L¼1μm,m*¼0:015me, and ρ¼105cm/C01, we find εn¼0:251nmeV. This level spacing can be much larger than the dissipation induced broadening from FIG. 6. Wave functions of the confined states in the QD region of the spin –orbit coupled semiconductor nanowire with length L¼1:35μm, where modes (I) to (VI) correspond to the eigenvalues marked in Fig. 5 . The colors show the wave functions of the electron part with spin up (red) and spin down (green). Wehave used μ¼/C0 20 meV (in the topological regime with ~μ¼μþ2t/difference0 meV) andμ¼/C0 16 meV (in the trivial regime when two bands are occupied with ~μ/difference4 meV). The other parameters are the same as that used in Fig. 5(a) . The size of the QD region Lis also comparable to that used in experiments. FIG. 5. Confined states in the s-wave superconducting model. (a) Confined states with jεnj,Eg(red lines). We use ~μ¼μþ2t,t¼10,α¼2:5, Δ¼1, and h¼2,L¼1:35. Confined states can be found in the trivial regime only when kiare real values. (b) Energy levels in the topological regime as a function of n=L, which has the same feature as Fig. 1(b) . Parameters for the symbols are blue open circles L¼6:75,Δ¼0:25,α¼2,h¼0:5; red open circles: L¼3:75,Δ¼0:25,α¼2,h¼0:5; blue big squares: L¼6:75, Δ¼1,α¼2,h¼2; blue small squares: L¼6:75,Δ¼0:25,α¼2,h¼2; blue solid circles: L¼6:75,Δ¼1,α¼8,h¼2; red solid circles: L¼3:75,Δ¼1,α¼8,h¼2. The black solid lines are given by Eq. (28).I n all plots, ~μ¼0. (c) The corresponding single particle band structure. The units forΔ,μ,α,h,tare meV , and the unit of Lisμm.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 094301 (2021); doi: 10.1063/5.0038612 129, 094301-7 Published under license by AIP Publishing.the averaging effect, which is of the order of 10 μeV at low tempera- ture (20 mK).53We expect this value to be tuned in a wide range in experiments by Landρ. This relation is useful because in experi- ments when the mass m*and length Lare known, this energy level spacing can even be used as a tool to determine the carrier densityρ(hence the Fermi velocity v F) in the nanowire as a function of external gate voltage. B. Relevance to experiments We finally discuss the relevance of the above results to experiments for the observation of zero-bias conductance peaks. In Secs. IIandIII, we have assumed that the QD region and the paired region have the same chemical potential, while in experi-ments, the local potential in the QD region can be tuned by theexternal gate voltage. This will not influence our major conclu- sion for the reason that the edge modes are fully localized in the QD region. In this condition, when a finite chemical potentialdifference δμis applied in the QD region with a perturbed Hamiltonian H 0¼Hþδμ, the confined spectra take the follow- ing form: ε0 n¼εnþδμ: (31) For this reason, the zero-bias peaks can be realized for δμ¼/C0εn for arbitrary n[Z. This result may explain why in experiments ubiquitous zero-bias conductance peaks can be observed.60 Despite all that, we expect that the trivial barrier and the topologi- cal barrier may have totally different energy level spacings as afunction of gate voltage. In experiments, a lot of tools such asconductance (and differential conductance) and current can be used to determine these low-lying localized edge modes in the QDs. Since the wave functions are almost localized in the QDregion, we expect the transmission coefficients for all the energylevels are quantitative similar, then these measurements are only determined by the density of states given in Eq. (31) with some finite broadening of the spectra. Thus, the spectra may be directlyextracted from the peaks in these measurements. IV. CONCLUSION To conclude, we discuss the role of the topological barrier for the low-lying localized edge modes, including the MZMs and theAndreev bound states, in the unpaired QD region. These results arerelevant to the experiments in the realization of the MZMs for topological quantum computation. We determine these features analytically in two different models, in which the results can beunderstood based on a simple picture. Our results may provide asimple model for the ubiquitous zero-bias conductance peaksobserved in experiments. 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5.0054025.pdf

J. Appl. Phys. 129, 210902 (2021); https://doi.org/10.1063/5.0054025 129, 210902 © 2021 Author(s).Magnetism in curved geometries Cite as: J. Appl. Phys. 129, 210902 (2021); https://doi.org/10.1063/5.0054025 Submitted: 13 April 2021 . Accepted: 11 May 2021 . Published Online: 01 June 2021 Robert Streubel , Evgeny Y. Tsymbal , and Peter Fischer COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Special optical performance from single upconverting micro/nanoparticles Journal of Applied Physics 129, 210901 (2021); https://doi.org/10.1063/5.0052876 Acoustic nonreciprocity Journal of Applied Physics 129, 210903 (2021); https://doi.org/10.1063/5.0050775 All-optical switch based on novel physics effects Journal of Applied Physics 129, 210906 (2021); https://doi.org/10.1063/5.0048878Magnetism in curved geometries Cite as: J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 View Online Export Citation CrossMar k Submitted: 13 April 2021 · Accepted: 11 May 2021 · Published Online: 1 June 2021 Robert Streubel,1,2,3 ,a) Evgeny Y. Tsymbal,1,2 and Peter Fischer3,4 AFFILIATIONS 1Department of Physics and Astronomy, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA 2Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA 3Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 4Physics Department, UC Santa Cruz, Santa Cruz, California 95064, USA a)Author to whom correspondence should be addressed: streubel@unl.edu ABSTRACT Curvature impacts physical properties across multiple length scales, ranging from the macroscopic scale, where the shape and size vary drastically with the curvature, to the nanoscale at interfaces and inhomogeneities in materials with structural, chemical, electronic, and magnetic short-range order. In quantum materials, where correlations, entanglement, and topology dominate, the curvature opens the path to novel characteristics and phenomena that have recently emerged and could have a dramatic impact on future fundamental and appliedstudies of materials. Particularly, magnetic systems hosting non-collinear and topological states and 3D magnetic nanostructures stronglybenefit from treating curvature as a new design parameter to explore prospective applications in the magnetic field and stress sensing, micro- robotics, and information processing and storage. This Perspective gives an overview of recent progress in synthesis, theory, and characteri- zation studies and discusses future directions, challenges, and application potential of the harnessing curvature for 3D nanomagnetism. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054025 I. INTRODUCTION Understanding the relationship between electronic and mag- netic properties and structural and chemical quantities is one ofthe overarching themes in condensed matter and applied physicsand key to the discovery of novel quantum materials. Correlated electron systems and magnetic materials are particularly interesting because microscopic characteristics of entangled and topologicalstates are heavily determined by local atomic and nanoscale fea-tures. To date, research efforts have largely been focusing on syn-thesizing planar single-crystals, epitaxial films, and multilayer stacks with tailored functionalities originating from the nearly perfect long-range order and symmetry. The existence or absenceof symmetry is essential to many phenomena emergent in topologi-cal insulators, ferroelectric, multiferroic, and magnetic materialswhose physical properties are described by vector order parameters relying on, e.g., spin –orbit coupling. 1–3In fact, current information processing and storage architectures as well as concepts for novelmicroelectronics, including the evolving field of spintronics, 4rely on low-dimensional systems with well-defined symmetry andspecial types of spin –orbit coupling. However, structural and chem- ical inhomogeneities and disorder emerge even in the most perfect materials and at interfaces. A new way of describing thoseimperfections is to assign them a curvature in real, reciprocal, or spin space ( Fig. 1 ). A local curvature can be employed to design systems with spontaneous or inhomogeneous inversion symmetrybreaking and to stabilize 3D magnetization vector fields or to tailor topology and magneto-transport properties in amorphous corre- lated electron systems. 5–7Sculpting 3D curved nanostructures provides means to tailor the curvature on the nanoscale while simultaneously expanding 1D and 2D nanostructures into the third dimension8and is heavily used in microrobotics.9–11 Topological vector fields, such as vortices,12skyrmions13and topological knots,14–17possess a curvature in the vector order parameter space, e.g., spin space. Compared with uniformly polar-ized or topologically trivial configurations, topological vector fields span the Bloch sphere Ntimes with Nreferring to the topological charge. The representation in terms of a Bloch sphere is convenientto describe electromagnetism in solids 18–20and to link topological properties to electronic transport phenomena.21The latter has stimulated a multitude of theoretical and experimental studies ofmagnetic 22–27and polar28,29skyrmions in a large variety of materi- als systems in view of both basic sciences and novel information storage and processing units, such as the racetrack memory.30–32 Alternative concepts propose to use topological states as 3D curvedJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-1 Published under an exclusive license by AIP Publishingmagnonic waveguides33for spin wave-based spintronics,34for neuromorphic35–38and probabilistic39computing, or for topologi- cal magnonics,40,41taking advantage of their quasi-particle charac- ter. The vast majority of magnetic topological vector fields hasbeen stabilized in systems with inversion symmetry breaking, pro- vided either by virtue of their crystal structure 42,43or through the presence of planar interfaces,44–46causing an asymmetric vector spin exchange known as the Dzyaloshinskii –Moriya interaction (DMI).42,43However, those concepts and governing mechanisms are universally applicable to ferroelectric,28,29multiferroic,47and 2D van der Waals materials,48–51as well as to amorphous materials52–54with local inversion symmetry breaking. In fact, systems with a locally varying DMI55,56or a spontaneous symmetry breaking with respect to spin chirality have been proposed for sta- bilizing twisted and anisotropic magnetic solitons,16,57including topological spin knots referred to as hopfions.14–17,58The inherentdilemma of mutually exclusive small topological states and high magnetic ordering temperature, essential to spintronics applica-tions, may be addressed using targeted synthesis of magneticallyordered alloys. 59The prerequisite non-planar arrangement of atoms of the same element can be interpreted as a curved interface within the solid-state material, opening a completely new direction of exploring curvature as a new design parameter. A complementary route to break inversion symmetry without impairing intrinsic properties relies on engineering curved nano-structures and tailoring magnetic exchange interactions. 60 Curvature has been employed to design tubular architectures withvirtually unlimited magnetic domain wall velocity and unidirec-tional spin wave propagation owing to curvature-driven magneto-chirality. 61Since curvature-driven inversion symmetry breaking is conceptually analogous to an emergent DMI, topological states can be created and manipulated solely by curvature62without the need FIG. 1. Magnetism in curved geometries in real, reciprocal, and spin space. Magnetic properties and novel functionalities are governed by the curvature and short-range order alongside elements and composition. Local inversion symmetry breaking by the curvature, strain, and short-range order can promote the format ion of 3D topological spin textures owing to an emergent local vector exchange interaction (DMI) with prospective applications to microelectronics while offering great er flexibility in materials syn- thesis. Geometrically confined structures, such as nanorods, nanotubes, and nanohelices, induce a curvature-driven DMI that discriminates betwe en spin chirality and sup- ports the nucleation of chiral and topological states with unprecedented stability upon current excitation. 3D nanostructures are synthesized by s elf-assembly of nanoparticles, nanoprinting, or etching enable microrobotics in gaseous and liquid phases, fundamental studies on 3D spin frustration, and 3D magn etic logic and storage systems.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-2 Published under an exclusive license by AIP Publishingfor intrinsic inversion symmetry breaking. To this extent, we envi- sion that curvature will be used as a scientific design principle in the form of rough, curved structural, chemical, and magneticinterfaces, gradients, and inhomogeneities/disorder in solid-statematerials. This includes, in particular, artificial magneto-electricmaterials, 63local DMI to stabilize anisotropic topological states, room-temperature skyrmions spanning a few nanometers, and spin waves emanating from and along non-collinear spin textures, suchas chiral domain walls 64,65and 3D topological states,33enabling configurable 3D magnonic crystals. The advantage of designingand implementing curved vector fields over structurally predefined curvature opens a new way to tune on-demand the spin wave dis- persion, i.e., band structure, through twisting and deforming oraltering the topology of the magnetization configuration. Similar to the recent success of expanding low-dimensional magnetism into 3D nanomagnetism, 66–68implementing curvature as a design concept into future magnetic materials requires an inte- grated approach of advanced modeling, synthesis, and characteriza-tion to validate the properties and behavior of curved magneticstructures. Recent developments of analytical and numerical frame-works have allowed for quantifying curvature-induced magneto- chirality, 61,69curvature-driven formation of topological states,62,70 and vector spin exchange on the atomic scale.71Advances in elec- trochemical deposition72and 3D nanoprinting73–76enabled the synthesis of tubular, helical, and more complex nanostructures with ever-growing quality of magnetic and structural properties. Magnetic properties of planar films have been tailored by interfaceengineering taking advantage of improved growth capabilities andab initio guided synthesis. 3Magnetic microscopy, tomography, and scattering77–81at coherent x-ray light sources and aberration- corrected transmission electron microscopy centers have become essential to characterize chemical and structural inhomogeneitieswithin the magnetic material and near interfaces/surfaces, and to vis-ualize 3D magnetization vector fields. Great progress has been madein pushing limits of optical and scanning probe microscopies relying on, e.g., the Kerr effect, superconducting quantum interference device magnetometry, 82and nitrogen-vacancy magnetometry.83 Given the enormous scientific opportunities and challenges with adding curvature as a critical parameter to magnetic materials,this Perspective provides an overview of recent progress in synthesis, theory, and experimental studies and discusses potential future direc- tions of harnessing curvature for 3D nanomagnetism. In particular,we summarize the current state of 3D nanostructures, curvatureeffects, and their relation to topological magnetic states in Sec. II. Current and future technological advances in numerical modeling, synthesis, and characterization, enabling these scientific break-throughs, are discussed in Sec. III. Sections IVandVgive a scientific and technological perspective of the harnessing curvature for basicsciences and prospective applications of 3D nanomagnetism. II. STATE-OF-THE-ART OF CURVATURE-INDUCED EFFECTS Curved geometries are characterized by the spatial distribution of the local inverse radius, i.e., curvature, which can span a wide range from 1/ μm down to 10/nm. Generally, the upper and lower boundaries are governed by extrinsic properties, including theshape and size of 3D nanostructures and structural deformation, and intrinsic properties, such as interfaces, heterogeneity, and dis- order, respectively ( Fig. 1 ). The unique feature of the curvature is its inherent local inversion symmetry breaking which, dependingon its origin, leads to a constant or gradually/randomly changingmodification to magnetic properties. 60,84,85The former refers to the special case of a constant curvature and magnetization orientation with respect to the curvature; the latter to the general case of avarying microscopic or nanoscopic curvature. Note that this appliesto real, reciprocal, and spin space; curved spin geometries in recip-rocal and spin space affect mainly spin excitations and electronic transport due to different spin –orbit coupling phenomena. The effect of a locally varying curvature in the form of structural, chem-ical, electronic, and magnetic inhomogeneities and disorder scaleswith its ratio of magnitude to spatial variation. A sufficiently largeratio can affect magnetic properties and, for instance, stabilize topological spin textures on the corresponding length scale; other- wise, curvature-induced modifications to magnetic interactions willmostly compensate each other. On the other hand, engineeringcurved nanostructures allow for tailoring magnetic exchange inter-actions without impairing intrinsic properties. This approach is fundamentally different from traditionally tuning the shape and size to modify magnetic dipole energies of nanostructures. A. 0D and 1D nanostructures The most prominent properties of nanostructures are the shape and size that alter or even completely suppress magnetism when approaching tens of nanometers. These modifications stem from anincreased surface-to-volume ratio that boosts unfavorable magneticdipole contributions, triggering a high sensitivity to short-rangeorder and location/orientation of the magnetization of adjacent nano- particles. The latter can be employed to design complex 3D nano- structure assemblies of core-shell and solid magnetic nanoparticlespossessing a centered magnetic moment, as well as Janus particleswith an off-center magnetic moment in the form of a well-defined in-plane or perpendicular 86–88magnetization, vortices,89–92or topo- logical states.93Magnetic short-range and even long-range orders manifest in clusters with spin frustration [ Fig. 2(a) ],942D hetero- structured colloidal crystals [ Fig. 2(b) ],95and straight tubular chains with variable diameters.96Theoretical studies revealed novel assemblies beyond straight chains97and flakes, such as meander- ing chains,98rings with different sizes, shapes and topology [Fig. 2(c) ],99–101and shells [ Fig. 2(d) ].102 Physically expanding a spherical nanoparticle along one axis results in cylindrical nanorods with a uniaxial structural and mag- netic symmetry. These structures typically stabilize a longitudinal magnetization similar to planar nanowires lacking a magneto-crystalline anisotropy; all other magnetic properties, such asdomain wall nucleation and motion, magnetization reversal, andspin wave propagation, are fundamentally different due to constant local curvature (circular cross section). Early theoretical works on domain wall nucleation and propagation in nanorods providedquantitative proof for a suppressed Walker breakdown 103for trans- verse walls.104The unprecedented large domain wall velocities have only recently been contested by synthetic antiferromagnets and angular moment-compensated ferrimagnets105,106with inversionJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-3 Published under an exclusive license by AIP PublishingFIG. 2. Magnetization configurations in 0D and 1D nanostructures. (a) –(d) Self-assembly of magnetic Janus particles into (a) and (b) 2D arrays with specific symmetry (static), (c) straight chains, closed loops, and helices (static, length, temperature, and field dependent), and (d) 3D shells (static, electric ch arge-stabilized). (a) Reproduced with permission from Baraban et al. , Phys. Rev. E 77, 031407 (2008). Copyright 2008 American Physical Society. (b) Reproduced with permission from Tsyrenova et al. , Langmuir 35, 6106 –6111 (2019). Copyright 2019 American Chemical Society. (c) Reproduced with permission from Hernández-Rojas and Calvo, Phys. Rev. E 97, 022601 (2018). Copyright 2018 American Physical Society. (d) Reproduced with permission from D. Morphew and D. Chakrabarti, Nanoscale 10, 13875 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (e) 3D non-collinear spin textures forming in FeGe nanorods with DMI revea ling dependence on spatial confinement (diameter). Reproduced with permission from Charilaou and Löfller, Phys. Rev. B 95, 024409 (2017). Copyright 2017 American Physical Society. (f) Magnetization in Co-rich CoNi nanorods with face-centered cubic (fcc) and hexagonal close packed (hcp) crystal structures visualized with elec tron holography. Reproduced with permission from Andersen et al. , ACS Nano 14, 1399 (2020). Copyright 2020 American Chemical Society. (a), (b), and (f) and (c) –(e) are experimental and numerical data, respectively.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-4 Published under an exclusive license by AIP Publishingsymmetry breaking. The combination of uniaxial symmetry and shape of the nanorods causes, depending on the diameter,107either deterministic nucleation of transverse domain walls or Bloch pointsat the center of vortex walls 108–110(Fig. 1 ). More complex non- collinear spin textures emerge in nanorods with an intrinsic inver-sion symmetry breaking and resulting in Dzyaloshinskii –Moriya interaction (DMI), 42,43which reveal a periodic switching between skyrmionic and helical spins for matching and non-matching roddiameter, respectively [ Fig. 2(e) ]. 111 In recent years, tremendous progress has been made in syn- thesis and visualizing the magnetization configuration,112including magnetization reversal process and correlation with local structural and chemical properties. Adopting x-ray photon emission electronmicroscopy 113to conduct transmission experiments and analyzing both direct and shadow XMCD contrast enabled the visualizationof helical spins in curved nanomembranes 114and nanorods115and studying 3D printed nanohelices.116The correlation between local structural and chemical properties and the magnetization configu-ration has been addressed with electron holography in nanorods interms of imperfections, 117–119such as grains and surface roughness, and engineered chemical/structural segmentation [ Fig. 2(f) ].120–122 The latter approach allowed for transforming a simple longitudinal magnetization prevailing in elongated nanorods into helical, vortex,or transverse configurations, which are strong contenders for novelspin torque nanooscillators. Recently, experimental studies of the current-driven domain wall motion in nanorods corroborated the theoretically predicted high velocities. 123 B. 2D curved geometries and curvature effects Nanoparticles (0D) and nanorods (1D) without a magnetic core resemble shell (spherical shell) or ring (nanotube) structures with distinct magnetic properties are governed by topology and curvature. Hollow tubular architectures with longitudinal magneti-zation and vortex domain walls lack, in contrast to nanorods, aBloch point in the center ( Fig. 1 ). These nanostructures promise virtually unlimited magnetic domain wall velocity [ Fig. 3(a) ], 124 unidirectional spin wave propagation [ Fig. 3(b) ],125–128and vortex chirality-dependent standing spin wave spectra,127,129owing to curvature-driven magneto-chirality.61,69The latter refers to the spin chirality selection in nanotubes due to lifted degeneracy between moving vortex walls with opposite circulation. The spin transfer torque tilts the magnetization within the domain wall inward(outward), depending on its circulation, and enhances (impairs)the stability of the vortex wall by reducing (increasing) its magneticstray field. 125In addition to these dynamic modifications, tube diameter or strength of magnetic dipole interactions can be varied to switch between longitudinal, helical and vortex configurations[Fig. 3(c) ]. 130Engineering systems with unidirectional spin wave propagation are appealing for energy efficient magnonics34and cre- ating unidirectional magnetoacoustic waves.131The former has just recently been demonstrated in planar architectures by resonantly exciting non-collinear spin textures, such as vortices,65Bloch points,64and domain walls,132in ferromagnetic and synthetic anti- ferromagnets. Particularly interesting is the spin wave propagation along curved domain walls64and skyrmion tubes33to design 3D reconfigurable magnonic waveguides.Mathematically, curved geometries can be treated as planar systems following a coordinate transformation. The generalized theory of curvilinear micromagnetism60illustrates how local and non-local interactions emerge from the curvature, including a mag-netic exchange interaction similar to DMI ( Fig. 4 ), 66,84,85an easy- surface anisotropy,133and, for rough interfaces and surfaces and heterogeneous materials, i.e., local curvature, a spatial distribution of easy-axis, easy-cone, or easy-plane anisotropy. The vector spinexchange originates from the local inversion symmetry breakingand causes a local preference for spin chirality if the magnetizationand normal vector of the curved surface are not aligned ( Fig. 4 ). For instance, spheres and tubes with radial magnetization do not show a preference. The curvature-driven DMI puts magneto-chirality 69in a broader context and explains the emergence of chiral and topological spin textures in curved surfaces with cylindrical sym-metry, 134cones,84twisted bands,135Möbius bands [ Fig. 3(d) ],136 tori,137bent nanotubes138and rods,139,140nanohelices,63,141,142 shells,143–146and indentations [ Fig. 3(e) ].62,70Antiferromagnetic nanohelices support the formation of coherent magnon condensatesin the momentum space. 142Geometrically tailoring the curvature of nanohelices allows for stabilizing topologically distinct chiral spin textures, such as cycloidal and helicoidal configurations, as well as collinear single-domain and multi-domain states [ Fig. 3(f) ].63,141 These states can be transformed into each other by stretching or squeezing the nanohelix, offering a new approach to design magneto-electric materials without external magnetic fields.63 Similarly, the vortex ground state in ring-shaped nanowires transi- tions upon deformation into the trivial onion state.147These trans- formations represent an unwinding of chiral, topological spintextures into trivial states, triggered by the curvature-induced DMI. Nanoscale indentations enable the stabilization and manipula- tion of topological spin textures, such as skyrmions and skyrmio-niums also known as target skyrmions [ Fig. 3(e) ], 62,70in magnetic materials with otherwise absent inversion symmetry breaking.Relying on exchange instead of magneto-static energies, this mech- anism is fundamentally distinct from using thickness gradients to nucleate vortex lattices in percolated non-planar films. 148The shape, size, and topology of the magnetic state can be tailored byadjusting the local curvature. In a broader sense, these theoreticalstudies infer that local inversion symmetry breaking, due to struc- tural and chemical inhomogeneity, and rough, curved interfaces causes an inhomogeneous, local DMI in real materials. The chal-lenge is to engineer these curved interfaces to promote the forma-tion of topological spin textures instead of randomly canted spins occurring in frustrated spin systems. To date, experimental studies of curvature effects are still rare. Magnetic switching in tubular architectures with radial magnetiza-tion and nanotubes with longitudinal magnetization were visual-ized with magnetic force microscopy 149and superconducting quantum interference device microscopy.82These tabletop tools, providing spatial resolution on the sub-100 nm scale, represent asignificant advancement compared with earlier works using cantile-ver magnetometry. 150,151Azimuthal soft-magnetization configura- tions,152inaccessible in planar geometry, have proven essential to giant magneto-impedance field sensors with unprecedented sensi- tivity.153A first glimpse of the potential of curvature for spintronics was recently given by disentangling spin and charge resistance inJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-5 Published under an exclusive license by AIP PublishingFIG. 3. Numerically predicted curvature effects in 2D curved geometries. (a) Ultra-fast domain wall velocities in nanotubes due to delayed Walker breakdow n associated with spin chirality selection. Reproduced with permission from Yan et al. , Appl. Phys. Lett. 99, 122505 (2011). Copyright 2011 AIP Publishing LLC. (b) Asymmetric magnon dispersion in nanotubes originating from spin chirality selection similar to interfacial DMI in planar systems. Reproduced with permission from Ot álora et al. , Phys. Rev. Lett. 117, 227203 (2016). Copyright 2016 American Physical Society. (c) Transformation of magnetic states in nanotubes with magnetic dipole coupling streng th. From Salinas et al. , Sci. Rep. 8, 10275 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (d) Spin chirality selection in Möbius bands governed by perpendicular magnetic anisotropy. From Pylypovskyi et al. , Phys. Rev. Lett. 114, 197204 (2015). Copyright 2015 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (e) Formation of skyrmions in nanoindentations and spherical surfaces due to emergent DMI associated with loca l inversion symme- try breaking. Reproduced with permission from Kravchuk et al. , Phys. Rev. Lett. 120, 067201 (2018). Copyright 2018 American Physical Society and Phys. Rev. B 94, 144402 (2016). Copyright 2016 American Physical Society. (f) Helicoidal spin textures in nanohelices reversibly transforming into homogeneous an d periodical states upon stretching/compression. From Volkov et al. , Sci. Rep. 8, 866 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY) license.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-6 Published under an exclusive license by AIP PublishingFIG. 4. Theory of curvilinear micromagnetism. Non-local magnetic interactions emerge from a curvature-driven DMI in systems where the magnetization is no t aligned along the normal vector of the curved surface. From Sheka et al. , Commun. Phys. 3, 128 (2020). Copyright 2020 Author(s), licensed under a Creative Commons Attribution (CC BY) license.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-7 Published under an exclusive license by AIP Publishingaluminum nanowires deposited above a groove.154The experiment showed a higher efficiency compared with planar structures, which is essential to low-power spin current electronics and 3D microelec-tronics architectures with curved interconnects. C. Topological states Topological magnetic states are 3D inherently curved magneti- zation vector fields ( Fig. 1 ) that behave like quasi-particles upon magnetic and electric field excitation. 155The magnetic properties and electronic transport phenomena21are linked via solid-state electromagnetism,18–20which sets them apart from chiral domain walls and vortices12and makes them both fundamentally intriguing and relevant to low-power spin-based microelectronics. Since the theoretical prediction of skyrmions13and their experimental observa- tion in single-crystals156,157and thin films,158magnetic skyrmions have been extensively studied in view of current manipulation,159 current creation,160,161and electric detection of individual skyrmions via the topological Hall effect162,163associated with the perpendicular deflection of skyrmions164or, more recently, the Nernst effect.165,166 These investigations had mainly been driven by engineering planar interfaces3to tailor spin –orbit coupling, essential to DMI42,43 (formation), topological Hall effect (detection), and spin –orbit torque (manipulation). Synthesizing ultra-thin multilayer stacks withtailored interfacial DMI 71enabled the stabilization of room- temperature skyrmions in ferromagnets167and ferrimagnets.106,168,169 The latter benefit from significantly enhanced current-driven veloc- ities near angular moment compensation. Using complex oxide materials to grow epitaxial interface heterostructures with brokeninversion symmetry and a large gradient of the electrostatic potentialpromoted the formation of skyrmions at low temperature 170,171 whose size can be controlled by the ferroelectric polarization.172 The smallest room-temperature skyrmionic spin structures (,20 nm) were stabilized by pseudo-random substitution of Si atoms with Co excess atoms in polycrystalline B20 Co –Si materials.59Disorder also exists in multicomponent B20 single- crystals stabilizing topological phases [ Fig. 5(a) ],173such as Fe1/C0y(Co, Mn) y(Ge, Si),59,173–176which crystallize as a chiral lattice with an inherent chemical disorder that becomes chiral to theatomic building blocks. 177A recent theoretical work178showed the necessity of spin frustration to explain the experimentally observed transition between different topological phases in B20 structures [Fig. 5(b) ],173,179including magnetic monopoles on the order of 1 nm,179–181and rebuked the commonly accepted requirement of large DMI to stabilize small topological states. Atomistic simula-tions 182and experimental studies183,184of amorphous ferrimagnets confirmed further the persistence of DMI in structurally and chem- ically disordered materials. In fact, chemical and structural disordercan cause bulk DMI 52,184and stabilize topological states in amor- phous compounds [ Fig. 5(c) ].54This is attributed to an increased Anderson localization185,186and the suppression of electron trans- fer between transition metal atoms that enlarge local density of states and spin –orbit coupling,187local DMI, magneto-resistance, and Hall effects.188However, observing topological knots in poly- crystalline soft-magnetic bulk materials lacking inversion symmetry breaking demonstrated a certain degree of randomness in the occurrence of topological objects similar to magnetic vortices inextended soft-magnetic films.58The coordination number of the amorphous structure can be tuned by the deposition temperature from a high-coordination-phase at low temperatures189to a lower-coordination-phase at room temperature190with a short- range order resembling that of B20 structures. In this context, dis-order refers to locally varying DMI due to atomic short-range order and not to randomly distributed pinning sites, which have theoretically been investigated in view of current-driven skyrmiondynamics. 191–195 Systems with a locally varying DMI55,56or a spontaneous symmetry breaking with respect to spin chirality, triggered by spin frustration, have been proposed for stabilizing twisted and aniso- tropic magnetic solitons,56,57including topological spin knots referred to as hopfions16,17,196[Fig. 5(e) ], and, in part, experimen- tally been observed [ Fig. 5(d) ].54,58,197,198The lowest-order hopfion can be pictured as a spin torus which is twisted along its circumfer- ence continuously transforming between vortex and antivortex. The increased complexity of hopfions generally hinders a determin-istic formation; a recent numerical study proposed to combinespatial confinement with DMI and perpendicular magnetic anisot-ropy to stabilize hopfion-like spin textures [ Fig. 5(f) ]. 196These higher-order, anisotropic topological states possess a vanishing gyro-vector and intrinsically compensate the perpendicular deflec-tion of quasi-particles due to Magnus force promising a straighttrajectory at increased velocities [ Figs. 11(b) and11(c) ]. Examples range from biskyrmions (bound pair of skyrmions with opposite chirality) 199–201and bilayer skyrmions202to antiskyrmions,203,204 skyrmioniums (biaxial skyrmions)205–207and antiskyrmioniums208 to skyrmion bags,209,210and hopfions,211,212as well as antiferro- magnetic topological states.213–217Moreover, the Magnus force can be suppressed by nanoscale modifications to structural and mag- netic properties in the form of tracks and pinning sites,23,218or switching to tubular systems with a corresponding helical skyrmiontrajectory. 219Considering topological states in 2D and disordered materials further benefits novel concepts for manipulating magnetic exchange and topological states via curvature [ Fig. 3(e) ],62,70 voltage,220–225strain,226–228or pressure,229–231which are less effec- tive or even destructive in (poly-)crystalline metallic systems. Thesealternate routes provide a convenient way to twist and deform oreven alter the topology of 3D curved magnetization vector fields needed to design configurable 3D magnonic crystals 33or tunable topological magnonics.40,41 D. Curved spin geometries in reciprocal space In addition to magnetic exchange manifesting collinear, non- collinear, and topological magnetism, spin –orbit coupling enables an efficient charge-to-spin conversion and current-induced spin – orbit torques, mediated by non-trivial spin textures in reciprocalspace. The latter originate from inversion symmetry-breakingDresselhaus 232and Rashba233fields, which impose a spin chirality on the electronic bands ( Fig. 6 ) and generate a non-equilibrium spin polarization. The conversion between charge and spin currentrelies on the (inverse) Edelstein or (inverse) spin Hall effect andhas experimentally been observed at, e.g., non-magnetic metal interfaces 234,235and insulating oxide interfaces,236,237and in ferro- electric materials238and 2D van der Waals heterostructures.239Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-8 Published under an exclusive license by AIP PublishingFIG. 5. T opological states stabilized by inversion symmetry breaking and spin frustration. (a) Topological phase transitions between trigonal and cubic s kyrmion phases in B20 Mn 1/C0xFexGe with xvisualized with Lorentz microscopy (electron intensity shown). From Kanazawa et al. , New J. Phys. 18, 045006 (2016). Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (b) Ab initio calculations of MnGe based on Heisenberg spin frustration without DMI revealing cubic lattice of skyrmions and magnetic monopoles. Reproduced with permission from Mendive-Tapia et al. , Phys. Rev. B 103, 024410 (2021). Copyright 2021 American Physical Society. (c) Coexistence of helical spins and skyrmions in amorphous Fe –Ge films visualized with Lorentz microscopy (electron phase and magnetization depicted). Reproduced with permission from Streubel et al. , Adv. Mater. 33, 2004830 (2021). Copyright 2021 John Wiley and Sons. (d) Anisotropic skyrmions in (left) La1/C0xSrxMnO 3, transforming into each other via field –driven motion of Bloch lines and (right) amorphous Fe –Ge retrieved from Lorentz microscopy. Reproduced with per- mission from Yu et al. , Adv. Mater. 29, 1603958 (2017). Copyright 2017 John Wiley and Sons and from Streubel et al. , Adv. Mater. 33, 2004830 (2021). Copyright 2021 John Wiley and Sons. (e) Formation of anisotropic skyrmions and topological knots (hopfions, preimages, and cross section shown) by Heisenberg spin frustration retrieved from micromagnetic simulations. From Zhang et al. , Nat. Commun. 8, 1717 (2017). Copyright 2017 Author(s), licensed under a Creative Commons Attribution (CC BY) license and Sutcliffe, Phys. Rev. Lett. 118, 247203 (2017). Copyright 2020 American Physical Society. (f) Metastable hopfion relaxed in system with DMI and uniaxial sym- metry by micromagnetic simulations (preimages shown). Reproduced with permission from Balasubramanian et al. , Phys. Rev. Lett. 125, 057201 (2020). Copyright 2020 American Physical Society. (a), (c), and (d) and (b), (e), and (f) are experimental and numerical data, respectively.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-9 Published under an exclusive license by AIP PublishingTopological materials possess a gapless surface state protected by time-reversal symmetry,240whose band dispersion features a Dirac cone decorated with electron spins pointing tangential to the surface.241This spin texture is dramatically changed upon magnetic doping due to local time-reversal symmetry breaking that opens agap at the Dirac point and causes a magnetically inducedhedgehog-like spin configuration ( Fig. 1 ). 242The latter allows for generating spin currents243and producing spin-transfer torques on adjacent ferromagnets.242Chiral crystals, which lack inversion, mirror, or other rotation-inversion symmetries, stabilize topologi-cally non-trivial spin textures whose spin components parallel tothe electron momenta appear around highly symmetric k-points. 244 Both existence and inverted topology in right- and left-handed crystals were recently observed in chiral tellurium crystals,241prom- ising pure spin current generation. In ferroelectric materials, thespin texture is coupled to and can be controlled by the ferroelectricpolarization providing a promising platform to explore the cou- pling between spin, orbital, valley, and lattice degrees of freedom in solids. 245One particular benefit of Rashba spin –orbit coupling is its control by a gate voltage across an interface supporting a 2D elec-tron gas in the form of a spin field-effect transistor. Its practical realization is challenging since non-collinear spin textures possess a reduced spin diffusion length owing to an enhanced magneticimpurity and defect scattering of electrons changing their momen-tum and randomizing the spin. 246This effect can be circumvented by engineering structures where the magnitudes of Rashba and Dresselhaus spin –orbit coupling are equal, resulting in a unidirec- tional spin –orbit field and a momentum-independent spin configu- ration, known as the persistent spin texture ( Fig. 6 ). Under these conditions, the electron motion is accompanied by spin precessionaround the unidirectional spin –orbit field, leading to a spatially periodic mode referred to as a persistent spin helix. 247The latter is robust against spin-independent disorder and offers an infinitespin lifetime. It has experimentally been demonstrated in a 2Delectron gas semiconductor quantum-well structure by tuning quantum-well width and doping, 247,248and theoretically been pre- dicted in bulk oxide materials with a non-symmorphic space group FIG. 6. Spin–orbit coupling phenomena with corresponding spin orientation of the two spin-split electronic sub-bands for systems without inversion symmetry. T he Edelstein effect causes a spin accumulation due to shifted Fermi surfaces with an external electric field. An electric current Jdisplaces the Fermi surface along its flow direction, thereby tilting their spins up (down) for ky.0(ky,0) and creating a spin current in the y-direction (spin Hall effect). L. L. T ao and E. Y . Tsymbal, Nat. Commun. 9, 2763 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY) license and Sinova et al. , Rev. Mod. Phys. 87, 1213 (2015). Copyright 2015 American Physical Society.392Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-10 Published under an exclusive license by AIP Publishingsymmetry.249Examining the role of local curvature, structural, and chemical disorder in view of real-space inversion symmetry break- ing and their potential to enhance charge-to-spin conversion andincrease critical temperatures is critical particularly to the emergentfield of topological amorphous materials. 5 III. TECHNOLOGICAL ADVANCES Scientific advances and future directions are heavily inter- twined with technological advances in numerical modeling, synthe- sis and characterization. Their development throughout the last decade has diversified both the original research perspective andscientific community. A. Numerical modeling Accurate modeling of magnetic systems and magnetic interactions in solid-state materials is more important than ever before to accelerate materials discovery, predict magnetic phase transitions, static, and dynamic properties, and refute or corroborate analytical and experi-mental data. This is mainly because of an increased complexity com-pared to, e.g., planar ferromagnetic Permalloy (Ni 80Fe20)f i l m s ,a n d driven by the use of multiple elements, including transition metal, metalloids, heavy-element, and rare-earth materials, curved nanostruc- tures, disorder, and link to spin –orbit coupling and magneto-transport phenomena. Affordable parallel computing with multi-core centralprocessing units (CPU) and graphics processing units (GPU), as wellas inexpensive memory have helped establish numerical simulations as a mainstream technique to address curvature phenomena. Numerous public-domain finite element/difference method software packages are available to model magnetic materials withDMI, including legacy OOMMF, 250MuMax3,251and Fidimag.252 Atomistic solvers, such as Vampire,253Spirit,254and Fidimag,252 enable a more accurate modeling of singular magnetic spin textures,e.g., Bloch points/lines and skyrmions, antiferro- and ferrimagnetism,helimagnets, and frustrated systems, as well as 3D curved nanostruc-tures, structural and chemical disorder, and temperature effects. Micromagnetic simulations of 3D curved geometries with arbitrary shape can be carried out with Nmag, 255which is a powerful frame- work in combination with the HLib library.256Future developments will accommodate computational intense calculations of elastic prop-erties and magnetostriction, and time-dependent deformation and motion of realistic multifunctional materials. One leap in this direc- tion has been done by Boris Computational Spintronics, 257am u l t i - physics software with incorporated heat flow solver, electronic trans-port solver, temperature-dependent material parameters, andmechanical stress –strain solver. While these micromagnetic platforms offer insight into magnetic states, magnetization reversal processes, size effects, and current- and field-driven spin excitations, they dorely on physical parameters such as saturation magnetization, mag-netic exchange interactions, magnetic anisotropy, etc., typicallyretrieved from experiments or ab initio calculations. In the wake of interface and curvature engineering, density functional theory is essential to determining the dependence of inter-face and curvature effects, including DMI and spin –orbit torque, on used elements, and structural and chemical order. Three of the most popular ab initio frameworks are FLEUR, 258VASP,259and Quantum ESPRESSO,260which provide means to model band structures andquantify atomic DMI values. The numerical results of exchange- coupled systems strongly depend on the atomic coordinates, which are typically approximated according to their crystalline structures.However, this presumption is invalid for inhomogeneous and disor-dered materials. Arguably, the actual coordinates of each individualatom are virtually impossible to determine; the systems can however be approximated according to their short-range order that can be quantified with molecular dynamics simulations using, e.g.,LAMMPS. 261The latter simulates dynamic processes of assembly, nucleation and diffusion during synthesis or upon external stimula-tion on the atomic scale. This hierarchical approach of modeling will become more important to future studies of real materials with imperfections, disorder and highly inhomogeneous regions, includ-ing amorphous materials and interfaces. B. Synthesis Engineering interfaces has been a focus of recent research on nanomagnetic materials, primarily due to the possibility to harnessthe spin –orbit coupling induced by symmetry breaking effects at such interfaces. 2,3These efforts have been guided by ab initio calcu- lations to identify the best pairing of heavy-element material or oxide and magnetic element in view of largest DMI values71to sta- bilize chiral spin textures and topological states. Magneto-transportproperties, such as the spin Hall effect and spin –orbit torque, essential to current manipulation of chiral spin textures have typi-cally been phenomenologically optimized and correlated to ab initio calculations. In conjunction with exploring different classes of materials, including atomic monolayers, epitaxial, polycrystalline andamorphous films as well as 2D materials, this approach has flour-ished owing to employing both intrinsic and extrinsic (interface)properties, which offers new functionalities and greater flexibility in materials synthesis. The next decade will show to which extent struc- tural and chemical inhomogeneity, disorder, and curved interfacescan be harnessed to tailor magnetic exchange interactions andmanipulate topological states in solid-state materials. The synthesis of 3D nanostructures utilizing electrochemi- cal deposition [ Fig. 7(a) ], 72,262,263two-photon lithography [Fig. 7(b) ],75,264–267and focused electron beam-induced deposi- tion [ Fig. 7(c) ]73,74h a ss e e nt r e m e n d o u sp r o g r e s sp a r t i c u l a r l y with respect to controlling sh ape, roughness, morphology, homogeneity and purity. Electrochemical deposition using porous alumina or gyroid polymer268templates enabled the syn- thesis of nanorods, nanotubes nanohelices, multi-segmentedspecimens, 115,117–122,269and 3D networks270? –272with variable diameter ( ,50 nm), length ( /difference1μm), and metallic materials [Fig. 7(a) ]. The default trigonal symmetry of the porous template was circumvented by focused ion beam guided anodization.270 Focused electron beam-induced deposition has taken the leadin synthesizing 3D nanostructures with virtually any shapeand curvature, including nanowires, 116,274,275networks [Fig. 7(c) ],276–278and topological structures.279Relying on the dissociation of adsorbed metal-cabonyl precursor molecules bythe electron beam, the printed metallic nanostructures typicallyi n c o r p o r a t ec a r b o no ro x y g e ni m p u r i t i e so f *10%, which can be reduced using reactive gases during synthesis or post-growth. In-depth studies of process parameters, such as growth rate,Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-11 Published under an exclusive license by AIP PublishingFIG. 7. Synthesis of 3D nanostructures using bottom-up techniques. (a) Electrochemical deposition into porous alumina templates for fabrication of magne tic nanorod arrays, 3D magnetic networks, and nanohelices. Reproduced with permission from Chen et al. , Langmuir 27, 800 (2011). Copyright 2011 American Chemical Society; from Wagner et al. , Adv. Electron. Mater. 7, 2001069 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution (CC BY) license; and reproduced with permission from Sattayasamitsathit et al. , Nanoscale 6, 9415 (2014). Copyright 2014 Royal Society of Chemistry. (b) Polymer templates obtained by two-photon lithography (left) and additional etching and pyrolysis (right) revealing significant shrinkage. Reproduced with permission from Seniutinas et al. , Microelectron. Eng. 191, 25 (2018). Copyright 2018 Elsevier. (c) Nanoprinting of 3D nanostructures utilizing electron beam-induced deposition through dissociation of metal-cabony l precursor molecules. From Janbaz et al. , Sci. Adv. 3, eaao1595 (2017). Copyright 2017 Author(s), licensed under a Creative Commons Attribution (CC BY) license.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-12 Published under an exclusive license by AIP Publishingprecursor depletion/diffusion and heat load,277,280and computer-aided nanofabrication,278,281including Monte Carlo simulations of reaction –diffusion processes, have been essential to advancing 3D nanoprinting and investigating curvature andtopology effects in 3D nanostructures. Alternate techniques for 3D nanostructuring are printing polymeric 3D nanotemplates with two-photon lithography and subsequent metal deposition, 264implosion fabrication harnessing shrinkage and dehydration of hydrogel scaffolds,282and self- assembly of nanoparticles on curved liquid –liquid interfaces to structure liquids283–285that can be endowed with a remanent mag- netization ( Fig. 10 ).76,286,287Nanoindentations with engineered cur- vature can be carved out via dry etching with ion irradiation priorto non-epitaxial film deposition. Post-growth nanoscale modifica-tions to magnetic exchange, anisotropy and saturation magnetiza-tion may be performed with low-current ion irradiation. 218 Another versatile technique with respect to tailored magneticproperties is strain engineering rolled-up nanotech 288–290that facilitates internal strain gradients to manufacture tubular magneticgeometries with variable diameter/curvature and thick-ness. 152,291,292Subsequently, strain engineering has been general- ized to synthesize shape-morphing micromachines ( Fig. 13 ),293–296 reconfigurable actuators,297,298and shape memory polymers299 with magnetic functionality. C. Characterization Whether 3D nanostructures or topological magnetic states, the challenge with characterizing 3D magnetization vector fields is the complexity and ambiguity of many characterization techniques due to the lack of knowledge about all three magnetization compo-nents and their spatial distribution at a sufficient spatial resolution.Joint studies harnessing multimodal techniques and subsequentdetection of remaining components have provided means to identify stable magnetic states. The most advanced tools with respect to resolu- tion and sensitivity are x-ray and electron techniques, complementedby tabletop instruments, such as sc anning probe and optical micros- copy, magneto-transport, electro n spin resonance spectroscopy, and magnetic neutron scattering revealing internal spin structures of nano- particles. 300,301Choosing state-of-the-art inst rumentation is typically a compromise between high sensitivity, high spatial resolution, temporalresolution, and accessibility. Additi onal constraints are element specif- icity, interaction between probe an d magnetization, and environment, e.g., applying current/voltage, strain/pressure and magnetic fields or changing temperature and gas/solutions, to create, manipulate anddetect magnetic states. 1. Advanced electron and x-ray characterization Electron microscopy80,81combines subatomic spatial resolution and beam coherence. One prime example harnessing both quantitiesis atomic scalar tomography to examine atomic order, internal defects, and strain of nanoparticles in vacuum 302–305and liquid cells.306–308An adequate technique to visualize the magnetization on the atomic scale would tremendously benefit the study of antiferro-magnets, disordered materials and topological states spanning only a few atoms. Future demonstrations might be accomplished by record- ing the diffraction pattern using 4D scanning transmission electronmicroscopy similar to current approaches for strain mapping. 304For now, 3D magnetization vector fields can be reconstructed on the nanoscale using vector field tomography based on off-axis or in-lineelectron holography [ Fig. 8(a) ]. 122,309,310Electron holography allows for studying the interaction of electromagnetic waves with 3Dnanostructures 117–119,311or thin films54,157,183,197,205,312–315on the nanoscale [ Fig. 8(b) ]. In-line holography, also known as Fresnel mode Lorentz microscopy, retrieves the electron phase from a focalplane series using transport-of-intensity equation 316or Gerchberg – Saxton algorithm317without the need for a biprism and reference beam. The latter can also be avoided by leveraging differential phase contrast.318–320Magnetic (vector) and electrostatic/structural (scalar) contributions to the electron phase can be separated by subtractingthe phase of the magnetically saturated state or of the flipped sample.A third option unique to the Gerchberg-Saxton algorithm takesinto account different length scales and phase amplitudes 54,183as well as the slow convergence of low-frequency components of non-electrostatic features during the iterative phase retrieval.321 The simultaneous detection of both in-plane components of themagnetic induction (two-dimensional gradient of the electronphase) sets electron microscopy apart from x-ray techniques and is essential to time-resolved studies of the vast majority of mag- netic systems. X-ray spectromicroscopies, 79harnessing x-ray magnetic circular dichroism or x-ray magnetic linear dichroism as element- specific absorption contrast mechanism, have been a workhorse for quantifying orbital and spin moments322,323and visualizing magne- tization configurations and spin excitations on the tens of nanome-ter scale. Time-resolved measurements concerned thermal spinfluctuations, and current- and magnetic field-driven nucleation and manipulation of chiral domain walls, topological states and magnons. Orbital moments offer insight into the local electronorbital alignment. The limitation to one magnetization componentcan be overcome, for stable magnetic states, by tilting the sample toaccess another component or performing vector field tomography. Within the last few years, magnetic x-ray tomography has matured from prototypical demonstration with soft x rays [ Fig. 8(c) ] 292to full-scale soft and hard x-ray tomography [ Fig. 8(d) ]58,324–326to stroboscopic tomographic imaging of driven magnetization dynam-ics. 327This incredible progress has been possible by algorithm and hardware development at numerous synchrotron facilities. Alternatively, resonant x-ray scattering328can be employed to determine periodicity, spin chirality, and depth profile of periodicstructures, such as skyrmion lattices in magnetic 329,330and ferro- electric331materials. The interference pattern under the magnetic diffraction peak created by coherent x rays was used to reconstructaperiodic magnetization vector fields on the nanoscale harnessingptychography 77,78and to study thermal spin fluctuations near, e.g., topological phase transitions on the nanosecond time scale with x-ray photon correlation spectroscopy at free electron laser facilities.332 Phase contrast imaging, such as x ray and electron ptychogra- phy, holography, and tomography techniques, is based on wavepropagation and offers superior sensitivity and contrast compared with conventional microscopy. At synchrotron facilities, a coherent x-ray beam is currently generated by a pinhole smaller than 10 μm that clips more than 90% of the beam. Free electron lasers andJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-13 Published under an exclusive license by AIP PublishingFIG. 8. Advanced electron and x-ray characterization of 3D curved geometries. (a) Magnetization in Co/Cu multilayered nanorods obtained from holographic vector field electron tomography. From Wolf et al. , Commun. Phys. 2, 87 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (b) Electron phase contrast imaging of (left) chiral ferrimagnetism in amorphous structures and (right) spin frustration in 3D printed nanostructu res visualized with Lorentz microscopy and electron holography, respectively. Reproduced with permission from Streubel et al. , Adv. Mater. 30, 1800199 (2018). Copyright 2018 John Wiley and Sons and Llandro et al. , Nano Lett. 20, 3642 (2020). Copyright 2020 American Chemical Society. (c) 3D imaging of radial magnetization in tubular Co/Pd microstructures using soft x rays. From Streubel et al. , Nat. Commun. 6, 7612 (2015). Copyright 2015 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (d) The 3D reconstruction of magnetic singularities in striped domain patterns in Ni 80Fe20=NdCo 5=Ni80Fe20films. From Hierro-Rodriguez et al. , Nat. Commun. 11, 6382 (2020). Copyright 2020 Author(s), licensed under a Creative Commons Attribution (CC BY) license.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-14 Published under an exclusive license by AIP Publishingfield-emission aberration-corrected transmission electron micro- scopes already provide a coherent x ray and electron beam, respec- tively. Ongoing developments of faster, more sensitive detectors,and better optics and sources, e.g., monochromatic, brilliant,smaller, and coherent beams, offered by aberration-corrected trans-mission electron microscopes and next-generation diffraction- limited light sources, will significantly lower data acquisition time and improve illumination conditions and accessibility to a limitednumber of high-end instruments. Answering scientific questionsconcerning the relation between magnetization, structural andchemical order, topology and electric response will strongly rely on advancing operandi and time-resolved capabilities ranging from millisecond (pump-free) 54,333to picosecond (pump –probe)334,335 time scales. The former is limited by the detector, the latter by the pulse width and pulse separation. Nanofabrication advancesinclude developing platforms for current, voltage and piezoelectric strain manipulation, and correlating magneto-transport properties, such as the topological Hall effect, with the magnetization configu-ration of individual topological states. The latter will benefit fromincreasing the number of aberration-corrected transmission elec-tron microscopes and x-ray beamline endstations with liquid helium cryostat holders and allow for studying topological states in topological insulators and magnetic systems with possible quantumfluctuations. Ambient experiments using, e.g., oxygen and hydro-gen gas provide means to modify interface/surface chemistry (spin – orbit coupling) or exchange interaction by reversible hydrogen intercalation. Liquid cells based on amorphous silicon nitridenanomembranes or graphene can be used to control magnetism viachemical means or study self-assembly of nanoparticles. 2. Tabletop instrumentation Magneto-optical Kerr effect m agnetometry and microscopy,336 offering sensitivity to normal, transverse, or longitudinal magnetiza- tion components of the outer 20 nm provided means to study magne-tization reversal processes and current-driven manipulation of micrometer-sized topological states 161and 3D nanostructures by ana- lyzing reflections from different surface regions [ Fig. 9(a) ].291,337–339 These measurements can be combined with magneto-transport exper- iments to retrieve magneto-resistance and (topological) Hall coeffi-cient for non-collinear spin textures [ Fig. 9(b) ], 181or with micro-Hall probes to determine the magnetic hysteresis loops for the entire 3D nanostructures [ Fig. 9(c) ].281,340Magnetic imaging on the tens of nanometer scale is commonly carried out with magnetic force micros-copy 341by probing the second derivative of the normal stray field. Recent advances in magnetic tip customization have enabled simulta- neous measurements of multiple components,342a configurable tip magnetization to track normal and in-plane components,343 nanotube-based monopole sensors,344and a significantly enhanced sensitivity.345These steps are essential to quantitative magnetic force microscopy and reconstructing the 3D magnetization vector field. The challenge with magnetic force microscopy is the magnetic dipole inter- action between tip and sample that alters the states in soft-magneticsystems or drags magnetic domain walls in relatively hard-magneticmaterials. This limitation has been addressed with non-invasive scan- ning probe microscopies. Superco nducting quantum interference device microscopy 82,346measures the magnetic induction on thenanoscale, which was demonstrated with ferromagnetic nanotubes and nanocubes [ Fig. 9(d) ]. Nitrogen-vacancy scanning probe microscopy83,347emerged as a highly sensitive technique to image non-collinear spin textures at the nanoscale in 2D van der Waalsmaterials 348and antiferromagnets349and reconstruct the full 3D mag- netization vector field.350Extended magnetic phases were classified with respect to topology and chirality of the 3D spin textures using electron spin resonance spectroscopy.231,351,352The latter also allows for quantifying magnetic exchange stiffness and damping. Leveragingt h em a g n o nd i s p e r s i o ni ni n v e r s i o ns y m m e t r y - b r o k e ns y s t e m s ,Brillouin light scattering provided means to quantify DMI near the interface, including spin chira lity inversion in ferrimagnets. 184 IV. SCIENTIFIC PERSPECTIVE Growing expertise and capabilities in modeling, synthesis and characterization will enable researchers to explore rich sciences not only in magnetism and condensed matter physics, but also in close conjunction with engineering, biology, and chemistry where themagnetization is either of central importance or a mean to improvefunctionality. In 3D nanomagnetism, this is reflected in an increas- ing number of theoretical and experimental works that harness cur- vature in 3D nanostructures and real, disordered materials tomanipulate topological states. A. 3D nanostructures We anticipate three major research directions concerning assemblies of nanoparticles, spin frustration in 3D nanostructures,and magnetization vector fields and spin excitations in curvedgeometries. 1. Nanoparticle assembly Experimental investigation of theoretically predicted nanopar- ticle assemblies with non-trivial geometries, such as shells, rings,helices, and nanopatterns with different sizes, shapes, and topology will face challenges with the inherent size and shape distribution of nanoparticles that cause disorder and deformation of the assembly.Numerical modeling will need to account for these experimentallimitations to provide better insight into the self-assembly and itsapplication potential. Disordered particle crystals and curved dipole-coupled systems can be used to explore disorder and curva- ture effects and to determine to which extent dipole systems resem-ble exchange-coupled materials. While the vast majority of experimental and theoretical studies of nanoparticle assembly in solution 353has relied on mag- netic dipole interactions, there is ample opportunity to employ mechanical (gravity, surface tension) and chemical (pH, ligands)means. One prominent example is the assembly and jamming ofnanoparticles at inversion symmetry broken liquid –liquid interfaces, which provide a reversible structural transformation between liquid and glassy states [ Figs. 10(a) and10(d) ]. 76,354The glassy state can be pictured as a skeleton enclosing the liquid core with potentially highanisotropic, non-equilibrium shape. In combination with superpara-magnetic nanoparticles, this approach enables a transition between paramagnetic ferrofluid and ferromagnetic liquid housing 2D ferro- magnetism on the curved liquid interface. 76,286,287,355With eachJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-15 Published under an exclusive license by AIP PublishingFIG. 9. T abletop characterization tools for 3D nanomagnetism. (a) Surface-sensitive magneto-optical Kerr effect magnetometry of 3D conduit nanoprinted by focused elec- tron beam-induced deposition of Permalloy and tubular Permalloy cap structure. Reproduced with permission from Sanz-Hernández et al. , ACS Nano 11, 11066 (2017). Copyright 2017 American Chemical Society and Streubel et al. , Nano Lett. 12, 3961 (2012). Copyright 2012 American Chemical Society. (b) Magneto-transport in B20 MnGe single-crystal showing temperature-dependent topological Hall effect due to emergence of topological states, including magnetic monopoles shown on the right. Reproduced with permission from Kanazawa et al. , Phys. Rev. Lett. 125, 137202 (2020). Copyright 2020 American Physical Society. (c) Micro-Hall effect measurements on CoFe nanocube frames printed by focused electron beam-induced deposition. From Al Mamoori et al. , Materials 11, 289 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (d) Superconducting quantum interference device microscopy visualizing the magnetization r eversal process in ferrmagnetic nanotubes. Reproduced with permission from Vasyukov et al. , Nano Lett. 18, 964 (2018). Copyright 2018 American Chemical Society. All techniques can be applied to 3D curved geometries.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-16 Published under an exclusive license by AIP PublishingFIG. 10. Self-assembly of nanoparticle building blocks into 3D hierarchical systems in liquid environment. (a) Interfacial jamming of superparamagnetic nanoparticles at liquid –liquid interfaces forming ferromagnetic liquid droplets with reconfigurable shape and preserved magnetization observed in hydrodynamics experi ments. Reproduced with permission from Liu et al. , Science 365, 264 (2019). Copyright 2019 American Association for the Advancement of Science. (b) Isotropic elastic and magnetic proper- ties stemming from structural short-range order of jammed nanoparticles imaged with transmission electron microscopy in their dried state. Reprod uced with permission from Wu et al. , Proc. Natl. Acad. Sci. U.S.A. 118, e2017355118 (2021). Copyright 2021 National Academy of Sciences. (c) In-field and zero-field assembly of nanoparticles from a mixture of dispersed superparamagnetic and non-magnetic nanoparticles revealing distinct heterostructures with enhanced magnetic anisot ropy and remanent mag- netization. The depicted states are modeled with molecular dynamics and micromagnetic simulations and leave out non-magnetic nanoparticles. (d) S elf-assembly of Au/C0/C0 Fe3O4dumbbell-like nanoparticles with packing parameter of (left) 0.63 and (right) 0.84 visualized with transmission electron microscopy. Reproduced with permis- sion from Liu et al. , Nano Lett. 20, 8773 (2020). Copyright 2020 American Chemical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-17 Published under an exclusive license by AIP Publishingnanoparticle acting as a uniformly magnetized macrospin, thermal spin excitations, magnetic short-range, and long-range order depend on the structural short-range order of adjacent nanoparticles[Fig. 10(b) ], 286similar to macrospins in planar systems.356,357These systems have the potential to become a versatile platform to investi-gate spin liquid to spin glass transitions in planar and curved geome- tries which can be controlled by chemical properties, such as pH and ligands, stress and strain, electric and magnetic fields. The pH affectsthe screening of electrostatic charges and thus the separationbetween negatively charged nanoparticles jammed at the interface.Jamming mixtures of non-magnetic and superparamagnetic nano- particles in the presence of an external magnetic field provides a route to adaptive reconfigurable 3D printing and heterostructuring[Fig. 10(c) ], as known from 2D ferrofluids. 358Long chains359–361and lattices362form at liquid –liquid and liquid –air interfaces in alternat- ing in-plane magnetic fields where the balance between viscous and magnetic torque, and magnetic attraction and hydrodynamic repul- sion govern the stability and mechanical response of the dynamicallystable, ordered structures. Jamming these assemblies will inhibit themobility of individual particles, producing a locked remanent magne-tization of the entire droplet. This novel approach stimulates to reim- agine magnetism and microrobotics from the perspective of liquids with solid-state functionalities 355and prospective applications to vis- cosity engineering, magnetically functionalized liquid –crystalline and plastic –crystalline phases,363–365organic synthesis in living cells,366 and encapsulation and triggered release of cargo.354,361 2. 3D spin frustration The expansion of frustrated dipole systems from planar artifi- cial spin ice structures into 3D space will be essential to the resem-blance of inherently 3D, frustrated exchange-coupled materials. 367 Although providing valuable insight into spin frustration, thecurrent approach is restricted to 2D planar systems and improperlyscaled nearest and next-nearest neighbor interactions due to thefinite size of nanoislands. Advances in focused electron beam induced deposition, two-photon lithography and electrochemical deposition will allow for synthesizing single- and multi-componentnanostructures where magnetic moments are confined to verticesor connecting segments. This bottom up approach enables apathway to design 3D geometrically frustrated heterostructures with various symmetry and geometry of isotropic lattices, 2D layered structures, and deliberately disordered systems. Tailoring shape andsize (extrinsic) or anisotropy, exchange and Curie temperature(intrinsic) of multi-component heterostructures translates to anensemble of magnetically harder and softer materials. While single- component nanostructures are simpler to manufacture, they are less realistic since exchange interactions affect thermal spin fluctuationsand local ground states. Multi-component materials with differentthermal expansion coefficients will open a path to reversibly trans-form isotropic into anisotropic spin frustrated systems by changing dipole interactions which is particularly impactful for isotropic mac- rospins with negligible energy barrier, such as XY, Kitaev spins, andspin glasses. The realization of these complex transitions is unique tomacrospin incorporated into a nanostructure matrix and not feasible to achieve in a conventional way, e.g., changing temperature and studying systems with various lattice parameters. While the primaryinterest in 3D frustrated heterostructures relies on basic science, they may also find application in random number generation or magnetic dipole logic. 368 3. Non-collinear spin textures in 3D nanostructures The accumulated knowledge of and advances in synthesis, characterization, and numerical modeling of 3D nanostructures will help boost research efforts to manufacture 3D networks for spintronics (race track memory), magnonics (spin wave excitation),and neuromorphic computing (spin oscillator). These prospectiveapplications require high metal purity and minimal interfaceroughness to guarantee adequate magnetic exchange interactions, and spin-transfer or spin –orbit torque. The current manipulation of spin textures is not only essential with respect to applicationbut also a critical need as structural complexity prevents aneffective control by external magnetic fields. We envision two routes going forward addressing domain wall manipulation in 3D networks, and more complex non-collinear spin textures andtopological states in nanostructures. There are numerous chal-lenges to overcome before realizing microelectronics based ontopological states in 3D networks. The coherent, synchronized motion of domain walls via spin- transfer or spin –orbit torque necessitates replica of domain walls with the same chirality, type, magnetic moment, width, and thick-ness. Since domain wall shape and type are governed by extrinsic(shape, thickness, curvature) and intrinsic (exchange, saturation magnetization, anisotropy) properties, particular attention will be devoted to architectures of multiple components, i.e., vertices andconnecting segments, bent and modulated nanostructures thataffect both kinetics and dynamics of domain walls. To some extent,slightly different domain wall velocities can be compensated using shift registration in the form of periodic pinning sites. Similarly important is to ensure compatibility with planar systems, whichdemands a mean to effectively couple chiral spin textures into andout of the 3D networks. 337Alternatively, new ways to nucleate and manipulate chiral domain walls in 3D networks are needed, taking advantage of, e.g., local stray fields predefining domain wall chiral-ity and varying cross section to alter current density beyond thenucleation threshold. A combination of strain and temperaturemanipulation of local magnetic exchange and anisotropy could enable logical operations in the form of gates and splitters. Even considering the most ideal case of an amorphous metallic networkwithout grain boundary pinning, questions remain about contactresistance, heat dissipation, pinning at corners, and curvatureeffects. The challenge will be to harness rather than to compensate these effects. The multitude of physical parameters influencing magnetization vector fields and spin excitations in 3D networks makes the studyand optimization of individual components essential, particularly, inview of more complex non-collinear spin textures and topological states. A major milestone is the synthesis of 3D nanostructures with inversion symmetry breaking to stabilize and retain chiral, non-collinear spin textures. The common approach of single-crystals withinversion symmetry-broken unit cells is impossible to achieve with the vast majority of bottom-up nanofabrication techniques, and impractical in view of 3D networks and application. This leavesJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-18 Published under an exclusive license by AIP Publishingcurvature, interfaces, and short-range order in amorphous materials. Biaxial 3D networks provide interfacial and curvature-induced inver- sion symmetry breaking, while enabling spin –orbit torque manipula- tion. It is unclear, though, how non-collinear spin textures wrappingthe magnetic shell would transition at vertices and intersections dueto changes of topology. Switching from magnetic shell to magnetic core allows free navigation through the 3D network at the expense of compensated inversion symmetry breaking. These symmetry argu-ments underline the essence of local variations in curvature, inter-faces and short-range order to consolidate a sizable DMI. Current efforts to synthesize and investigate cylindrical and tubular nanostructures in view of magnetic states and magnetiza- tion reversal process will expand to include domain wall dynamics,spin excitations, and 3D imaging of the magnetization vector fieldas well as its correlation with local structural and chemical proper-ties in terms of imperfections and engineered chemical/structural segmentation. The latter allows for stabilizing non-collinear spin textures, such as helices, skyrmions and skyrmioniums, which mayserve as novel 3D spin oscillators or magnonic crystals for, e.g.,speech and pattern recognition. Structural transformation by virtueof thinning (conical shape) or bending (curvature) will further provide means to localize non-collinear spin textures and transition regions between states with distinct topology. This includes reversi-ble switching between collinear and non-collinear spin textures inbent nanostructures, such as helices and rings, harnessing strain- mediated curvature modifications to the magnetic exchange without magnetostriction, magnetic fields, and current flow. Whilea first demonstration can be given by mechanical stretching andcompression, designing artificial magneto-electric materials 63will involve incorporating nanostructures in, e.g., piezoelectric solgel lead zirconate titanate matrices.3693D magnonic crystals will likely be realized using cylindrical or tubular nanostructures with longitu-dinal and azimuthal magnetization configurations owing to a pro-found theoretical understanding and significant advances insynthesis capabilities. The emanation of magnetic spin waves from domain walls separating uniformly magnetized domains simplifies analysis, and addresses the fundamental question of curvature-driven magneto-chirality selection of vortex domain walls and aunidirectional spin wave propagation in tubular geometries thatcan be tuned by magnetic fields, strain and curvature. A more chal- lenging subject is the spin wave emanating from non-collinear, topological spin textures in 3D nanostructures, and their depen-dence on both structural and magnetic properties, including, inparticular, chirality, topology, and periodicity. This close relation- ship makes them highly appealing from the perspective of quantum materials for non-volatile, analog information processing. B. Topological states stabilized by curvature and short-range order The research on topological solitons in condensed matter will diversify with a strong emphasis on expanding the zoo of topologi- cal magnetization vector fields in homogeneous and inhomoge-neous materials and harnessing curvature, disorder, strain, andvoltage to tailor type, strength, and inhomogeneity of magnetic exchange interactions on the nano- and atomic scales. Overcoming physical and technological limitations of ferromagnetic Néel andBloch type skyrmions will be addressed by exploring ferrimagnetic, antiferromagnetic, and multiferroic isotropic and anisotropic topo- logical states, such as higher-order skyrmions and hopfions. Thesestudies will thrive on multimodal investigations combiningmagneto-resistance measurements with magnetic imaging of indi-vidual and ensembles of topological states with different chirality, topological charge and dimensions. We anticipate similar proce- dures for dynamic experiments of spin excitations, such ascurrent-, voltage-, and strain-driven motion, nucleation, and spinwave propagation ( Fig. 11 ). Reconstructing thermally stable 3D magnetization vector field with magnetic tomography will provide unambiguous evidence of its topology. Synthesis and experiment will be guided by numerical modeling of the most realistic possibleconfigurations relying on molecular dynamics, ab initio , micromag- netic, and Monte Carlo simulations. A more detailed discussion oftechnological advances is given in Sec. III. Symmetry and order are regarded essential to quantum materials ranging from supercon- ductors to topological insulators to topological magnetic and polarstates. Vector spin exchange, i.e., DMI, is an indirect magneticexchange interaction mediated by conduction electrons of adjacentatoms that reveals, similar to RKKY exchange coupling, 370–372a spatial oscillatory behavior of both sign and magnitude and is highly sensitive to structural and chemical order.373The correspond- ing local DMI can be homogeneous, inhomogeneous, or random onthe microscale and ideally requires a sub-atomically accurate place- ment of elements and atoms to tailor topological objects and their current-induced motion [ Figs. 11(a) and11(b) ]. 56,212However, this is experimentally impractical in view of both efforts and materials syn-thesis. Instead, we envision that research will focus on structural andchemical disorder in the form of random substitution/intercalation of atoms, rough interfaces, and amorphicity to tailor interatomic exchange on the atomic and nanoscale. Recent theoretical and experi-mental works have shown promise of t hese alternate, unconventional means and reinforced the need for a profound understanding of fun-damental mechanisms. This ranges from probing and understanding to engineering and harnessing curv ature, structural and chemical short-range order in exchange-coupled systems to stabilize topologi-cally non-trivial states with unprecedented small feature sizes andtailored symmetry. Room-temperature skyrmions spanning a fewnanometers are currently futuristic but may be realized by enlarging the mean distance between spin-pol arized atoms via pseudo-random atom substitution in inversion symmetry-broken systems that leavesthe exchange stiffness and DMI exchange constant unaffected. 59 Amorphous magnetic materials exhibit, due to suppressed electron transfer between transition metal atoms and enlarged local density of states and spin –orbit coupling,187increased magneto- resistance and Hall effects and provide greater flexibility in materi-als synthesis and manipulations via current, voltage, strain, andcurvature. In contrast to single-crystals and epitaxial films, they can be grown on virtually any planar, curved or modulated substrate, and will allow for exploring compositions and phases inaccessiblein crystalline form due to, e.g., phase segregation. Minimalmagneto-crystalline anisotropy and sensitivity to short-range ordermake amorphous materials ideal prototypical systems to examine curvature-driven DMI, local inversion symmetry breaking and spin chirality selection in terms of stabilized topological magnetic statesand magneto-transport phenomena ( Fig. 3 ). This includes, inJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-19 Published under an exclusive license by AIP PublishingFIG. 11. Manipulation strategies for topological magnetic states, excluding curvature-induced effects discussed in Fig. 3 . (a) Spin Hall effect-induced motion of anisotropic skyrmions.Reproduced with permission from Jin et al. , Appl. Phys. Lett. 114,1 9 2 4 0 1 (2019). Copyright 2019 AIP PublishingLLC. (b) Current-driven motion of hop-fions by spin-transfer torque revealing rotation and expansion or shrinking depending on spin damping αand non- adiabatic coefficient β.B o t t o mi m a g e shows cross-sectional view of emergent magnetic field with velocities for vortex and antivortex configuration.Reproduced with permission from Liuet al. , Phys. Rev. Lett. 124,1 2 7 2 0 4 (2020). Copyright 2020 American Physical Society. (c) Optically inducedcurrent pulse nucleation and current-driven displacement by spin –orbit torque of skyrmionium as compared with sky- rmion. From Göbel et al. ,S c i .R e p . 9, 12119 (2019). Copyright 2019 Author(s),licensed under a Creative Commons Attribution (CC BY) license. (d) Voltage control of magnetic anisotropy mediatedby adjacent dielectric layer for directionalmotion of topological states. Reproduced with permission from X .W a n g ,W .L .G a n ,J .C .M a r t i n e z ,F .N. T an, M. B. A. Jalil, and W . S. Lew,Nanoscale 10, 733 (2018). Copyright 2018 Royal Society of Chemistry. (e) Magnetic trapping of magnetic sky-rmion by vertical magnons with largeorbital angular momentum. Reproduced with permission from Jiang et al. ,P h y s . Rev. Lett. 124, 217204 (2020). Copyright 2020 American PhysicalSociety. (f) Strain manipulation of mag- netic exchange interaction in Co/Pt mul- tilayers. Reproduced with permissionfrom Gusev et al. , Phys. Rev. Lett. 124, 157202 (2020). Copyright 2020 American Physical Society. (g) Formation of skyrmions by helium ionirradiation-induced modifications to mag-netic anisotropy and exchange in Pt/Co/ MgO. Reproduced with permission from Juge et al. ,N a n oL e t t . 21,2 9 8 9( 2 0 2 1 ) . Copyright 2021 American ChemicalSociety. (a) –(e) and (f ) and (g) are numerical and experimental data, respectively.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-20 Published under an exclusive license by AIP Publishingparticular, the visualization of topological states with different shape, size, and topology, stabilized by isotropic and anisotropic nanoscale indentations and sculptures, and quantification of aniso-tropic exchange interactions and magneto-transport propertiesalong, e.g., grooves. Selective release from the substrate andmechanical manipulation of free-standing films in the form of local curvature, bending, tensile and compressive strain and pressure are additional intriguing routes to alter magnetic exchange and nucle-ate and switch topological states. Electric voltage can be used tomanipulate magnetic exchange and anisotropy via modifications tothe electron density of states near the Fermi level [ Fig. 11(d) ] 221 and strain-mediated coupling [ Fig. 11(f) ],228altering both degree of inversion symmetry breaking and magneto-resistance. The highsensitivity to short-range order of amorphous materials on theverge between conductors and insulators will enable strain tuningof DMI far exceeding the values for polycrystalline films of one order of magnitude using 0.1% in-plane deformation. 228The spatial variations in voltage consolidate a reconfigurable curvedinterface with respect to magnetic exchange that, in principle, canresemble mechanical curvature and induce a local DMI. Theseefforts can be combined with helium ion irradiation to tailor mag- netic exchange, anisotropy, and magnetic moment on the nano- scale [ Fig. 11(g) ]. 218The latter allows for writing tracks for skyrmion nucleation and motion preventing deflection due to theMagnus force [ Fig. 11(c) ]. 207Since voltage manipulation of the electronic structures and magnetic exchange is typically limited to insulators and ultra-thin materials due to screening effects in con-ductors, it will be highly interesting to see experimental studies onthe potential of amorphous materials. Initial investigations will concern extended and nanopatterned homogeneous systems whose compositions are chosen according to their crystalline B20 counterparts with sizable DMI. In long-term,heterogeneous and individually optimized layered structures with con-tinuously varying composition, el ements, morphology, and magnetic properties may emerge, enabling, e.g., the formation of topological knots with the unique current-driven motion [ Fig. 11(b) ]. 212This includes different types and strengths of exchange interactions, mag-netic anisotropy, and transition temperatures, as well as multifunc-tional films with, e.g., magnetic and ferro-/piezoelectric properties.These materials have been synthesized as layered heterostructures or single-crystals and been subject to co mpatibility limitations. The latter are lifted in amorphous media requiring only short-range order.Multifunctional materials promis e voltage, strain, and curvature control of topological magnetism, an tiferromagnetism, and multiferro- ism, magnetic control of polar topological states, and amorphous topological insulators and superconductors 6,374whose properties can be locally configured by topological magnetic states. An alternateroute will focus on atomic layered structures to host topological statesspanning a few atoms while taking advantage of negligible damping/ pinning due to hybridization of ordered p-orbitals. Both characteristics make 2D van der Waals materials exceptionally susceptible to disorderand voltage and enable a selectiv er e l e a s ef r o mt h es u b s t r a t et o examine curvature and strain effects all-electrically and via magneticimaging. 48–51 Discovering new materials and means to manipulate individ- ual topological states can, in long-term, be accompanied by investi-gations of collective behavior and spin excitations. This pertains toboth magnetization dynamics and phase transitions between topo- logically distinct states, and their relation to structural and chemical short-range order, and local DMI. Spin waves are interesting withrespect to disorder-induced topological magnonics, 7and their lateral and particularly vertical propagation along, e.g., skyrmiontubes, which is critical to envisioning configurable 3D magnonic crystals. Vertical magnons with large orbital angular momentum provide further means to manipulate topological states[Fig. 11(e) ]. 375The advantage of designing and implementing curved vector fields over structurally predefined curvature is anew way to tune on-demand the spin wave dispersion, i.e., band structure and topology, through twisting and deforming, or alter- ing the topology of the magnetization configuration. In contrastto topological states confined to 3D nanostructures, extendedfilms offer further collective behavior and potentially a route todesign 3D networks of topological states and (topological) spin wave guides. V. TECHNOLOGICAL PERSPECTIVE Despite a strong focus on basic sciences and the early stage of research and development, numerous technological applications ofcurved magnetic geometries have been proposed and, in part, beenrealized. They differentiate themselves from planar technologies by enhanced performance, novel functionality, and/or higher effi- ciency (lower power consumption). Similar to scientific advances,structural properties have taken the lead in both sensing and micro-robotics applications. We anticipate a growing interest in structural,chemical, magnetic, and electronic curved geometries in quantum materials to manipulate chiral and topological states for novel sensing and microelectronics based on spintronics. The fundamen-tal aspects of adding curvature as a critical parameter to magneticmaterials were discussed in detail in Secs. IIandIV. A. Sensing One of the earliest and most tangible beneficiaries of curved magnetic geometries are flexible and stretchable magneto-resistive sensors ( Fig. 12 ), 376which can be synthesized by conventional thin film deposition directly onto a flexible and/or stretchable polymericsubstrate or via selective release and subsequent transfer onto virtu-ally any surface. The high-quality structural, chemical and magnetic properties enable sensing capabilities, comparable with rigid speci- mens on, e.g., silicon wafers, relying on giant magnetic resistanceand impedance, 153,377,378Hall effect, and giant stress resistance and impedance.379While functional multilayer stacks placed either in the neutral plane or onto micrometer-thick foils experience minimal impact from strain and stress, films on thicker substrates suffer magnetostriction which benefits giant stress impedance mea-surements in the form of an altered magnetic susceptibility andanisotropy. These characteristics provide the foundation for sensingfluid and gas flow (bending), thermal and mechanical expansions of planar and curved geometries (interfacial strain), and the physi- cal orientation within a constant magnetic field or of a variablemagnetic field as an inexpensive and thin alternative to semicon-ductor Hall probes. The latter will empower contactless position sensing for magnetic bearings, 3D “touch ”screen and wearable navigation devices (position and movement in 3D), and on-skinJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-21 Published under an exclusive license by AIP Publishinginteractive electronics for, e.g., augmented and virtual reality applications.376,380–383Switching from extended films to 3D nano- structures, such as nanopillars, tubes, and helices, will offer spatially resolved sensing capabilities384–386on the submicrometer scale and improved sensitivity to both magnetic field and stress.Magneto-resistive vector field sensing could be realized with 3Dnetworks of nanostructures. Given that the magnetic anisotropy and susceptibility of superparamagnetic nanoparticle coatings are highly dependent on the assemblies ’short-range order, they can be used to monitor and detect thermal and mechanical expansions aswell as cracks in and twists of conducting wires based on imped- ance changes. Similarly, local magnetic fields in liquids can be probed by analyzing self-assembly of superparamagneticnanoparticles. The magnetic phase transition from paramagnetism to ferromagnetism of superparamagnetic nanoparticles upon jamming in liquid environment could be adapted to sensing pH with bio-compatible hydrogels. 387,388The latter are highly sensitive to small changes in pH leading to a hysteresis-free shrinkage andexpansion, which will affect the mean distance between embeddednanoparticles and their remanent magnetization and boost sensitivity. 389,390 B. Microrobotics Magnetic nano- and microstructures dispersed in liquid and gaseous environments are strong contenders for microrobotics FIG. 12. Development of magnetosensitive e-skins based on magneto-resistive sensing. From G. S. Canon Bermudez and D. Makarov, Adv. Funct. Mater. 2007788 (2021). Copyright 2021 Author(s), licensed under a Creative Commons Attribution (CC BY) license.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-22 Published under an exclusive license by AIP Publishingbecause of a high susceptibility to external magnetic fields, which enables the remote control of the translational, and rotational motion, orientation, and direction, and the selection of differentmode of operation. Mechanical actuation and selective transforma-tion of shape-morphing micromachines with distinct local mag-netic and elastic properties 294–296can be realized by chemically, temperature or magnetic field-driven structural deformation, and adapted to complex origami [ Figs. 13(a) and13(b) ],295,297,391cargo delivery in liquid293,294,298,354,361and gaseous294environments, vis- cosity/turbulence engineering (microfluidics),359,361and surface roughness/modulation (optics). The magnetic functionality is typi- cally given by ferromagnetic nanoparticles, embedded in elastic films or magnetic films with tailored magnetic properties, and themechanical response to a rotating or constant magnetic field. Thelatter offers quasi-static 297and dynamic [millisecond time scale, Fig. 13(c) ]298actuation, including microrobots that walk, crawl, roll, and climb in air and liquid environment,294bio-inspired fla- gellated micromachines [ Fig. 13(d) ],293and printing of complex 3D structures that emerge from 2D planar films.295These develop- ments pave the way toward prospective applications in life sciencesand engineering, such as drug and cargo delivery, directional tissue growth, microsurgery, and artificial fertilization, as well as func- tional and reconfigurable microfluidic channels, adaptive opticalelements, viscosity engineering, local magnetic field sensing andgeneration, and actuation. 9–11The reconfigurable magnetization of ferromagnetic liquid droplets provides a mean to promote magneti- cally aligned cell differentiation and proliferation, which is particu-larly appealing for blood vessels, cartilage, and nerve tissueregeneration in living cells. 366In-field assembly and jamming of mixed phases of non-magnetic and superparamagnetic nanoparti- cles on liquid –liquid interfaces will enable reversible magnetic field- sensitive nanopatterning and designing birefringent, refractive, dif-fractive and potentially chiral liquid optical components. Choosingligands with the potential to significantly reduce interfacial tensionwill provide means to generate micrometer, potentially, sub- micrometer, droplets from parent specimens owing to spontaneous emulsification. A magnetic field promoting the assembly andagglomeration of nanoparticles at the interface will enable astimulated emulsification in the form of an explosive release offerromagnetic microdroplets. The viscosity of lubricating liquids can be enhanced by DC magnetic field-induced assembly of superparamagnetic nanoparticles into chains, tubes, flakes, orrings, which decelerates translational and rotational motion ofmotors; disassembling may occur naturally at remanence or within an AC magnetic field. C. Microelectronics Compared with sensing and microrobotics, realizing low- power microelectronics by harnessing curved geometries is a long-term effort that requires substantial scientific, technological and, to some extent, conceptual advances regarding, in particular, imple- mentation and integration. Examples range from more conven-tional mechanisms, such as dipole spin frustration and currentmanipulation, to voltage, curvature, and topology control. Spin frustration in 3D heterogeneous nanostructures may find applica- tion in random number generation and magnetic dipole logic.Current-driven domain wall manipulation and propagation in 3D networks provide a path toward 3D racetrack memory devices anddomain wall logic, as recently demonstrated in 2D, 368with superior storage density owing to minimal footprint. Incorporating curved nanorods in piezoelectric matrices enables voltage-induced switch-ing between collinear and non-collinear spin textures with distinctmagneto-resistance; this approach allows for designing artificialnon-volatile magneto-electric materials based on strain-mediated curvature modifications to the magnetic exchange without magne- tostriction, magnetic fields, and current flow. Thickness-modulatedlow-damping materials, such as yttrium iron garnet, or 3D net-works thereof can be explored in reference to their capability toserve as tunable ferromagnetic oscillators with multiple indepen- dent narrow resonances. These efforts will help launch 5G cellular communication services in the originally intended high-frequencyband of (24 /difference28) GHz to bolster future demand in bandwidth and rate, which is prevented by the currently employed complementary metal –oxide –semiconductor (CMOS) voltage-controlled oscillators. Novel 3D spin oscillators and magnonic crystals for, e.g., speech and pattern recognition can be realized with non-collinear spin tex-tures, such as chiral domain walls, vortices, helices, and topologicalstates, formed in segments of 3D networks or individual cylindrical nanostructures. The close relation between spin wave excitation and spin chirality, topology, and periodicity as well as structural proper-ties offers a significantly larger parameter space than conventionalnanospin oscillators and magnonics based on a uniform magnetiza-tion. These devices will take advantage of spatial confinement and directionality of the spin excitation (higher efficiency), and a distinct dispersion relation with potential topological features. Prototypicalsystems may be based on reconfigurable vortices in non-planarantidot arrays, domain walls pinned at corners of bent planar and3D nanostructures, and domain walls separating uniformly magne- tized domains in cylindrical or tubular nanostructures with longitu- dinal and azimuthal magnetization configurations. Harnessing chiralspin textures in 3D nanostructures and extended films have theadvantage of designing and implementing curved magnetization vector fields over structurally predefined curvature and opens a new way to tune on-demand the spin wave dispersion through twistingand deforming, or altering the topology of the magnetization config-uration. Extended films offer further collective behavior and a routeto design 3D networks of topological states and topological spin wave guides, which is intriguing from the perspective of novel quantum materials. A potential commercialization demands an all-electric characterization of the spin oscillators and magnonic materi-als probing the transmitted current signal as a fingerprint of thermaland radio frequency spin wave excitations. Greater flexibility in materials synthesis of quantum materials and correlated electron systems can be accomplished in the form ofamorphous and polycrystalline materials with local inversion sym-metry breaking. Focusing on the short-range order instead of globalsymmetry will allow for designing multifunctional materials which are typically incompatible due to mismatching lattice constants, sym- metry and phase segregation. These materials offer voltage, strainand curvature control of topological magnetism, antiferromagnetismand multiferroism, magnetic control of polar topological states, and amorphous topological insulators and superconductors whose prop- erties can be tuned by local decoration with topological magneticJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-23 Published under an exclusive license by AIP PublishingFIG. 13. Shape-morphing magnetic materials in gaseous and liquid environment. (a) Magnetic nanoparticles incorporated into elastomer matrix whose short- range order and magnetic anisotropy can be tuned at elevated temperatures in a magnetic field. Depicted shapes emerge from planar sheet in the presence of a magnet ic field. Reproduced with permission from Song et al. , Nano Lett. 20, 5185 (2020). Copyright 2020 American Chemical Society. (b) Programmed self-assembly of frames equipped with multipole permanent magnets. From Niu et al. , Proc. Natl. Acad. Sci. U.S.A. 116, 24402 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (c) Millisecond actuation based on perpendicular magnetized films exposed to linear AC magnetic field. From Wang et al. , Commun. Mater. 1, 67 (2020). Copyright 2020 Author(s),licensed under a Creative Commons Attribution (CC BY) license.298(d) Bio-inspired flagellated micromachines with magnetization stemming from embedded aligned ferromagnetic nanoparticles driven by a rotating magnetic field in liquid environment. From Huang et al. , Nat. Commun. 7, 12263 (2016). Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license.293Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210902 (2021); doi: 10.1063/5.0054025 129, 210902-24 Published under an exclusive license by AIP Publishingstates. Local voltage applications will provide a mean to consolidate a reconfigurable curved interface with respect to magnetic exchange, and to alter local DMI. The latter is envisioned to promote the for-mation of complex 3D magnetization vector fields all-electrically,e.g., creating and deleting skyrmioniums and antiskyrmioniums, aswell as anisotropic solitons like topological knots, which is essential to post-CMOS microelectronics. VI. CONCLUSION Harnessing the curvature as a design parameter to tailor and manipulate magnetic properties of non-collinear and topological states as well as of 3D magnetic nanostructures is a vital emergent field with ample opportunity for basic and applied sciences.Despite its primary focus on basic sciences and early stage ofresearch and development, a multitude of prospective applicationshave emerged, including magnetic field and stress sensing, microro- botics, and information processing and storage. Their realization requires an integrated approach of modeling, synthesis, and charac-terization across multiple length scales. This Perspective presentedrecent advances in basic and applied sciences and technology in the context of ongoing research efforts that we hope will guide and stimulate future directions. AUTHORS ’CONTRIBUTIONS R.S. wrote the manuscript with suggestions from E.Y.T. and P.F. ACKNOWLEDGMENTS This work was supported by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under U.S. Department of Energy Contract No. DE-AC02-05-CH11231 and Nebraska EPSCoR under the FIRSTAward No. OIA-1557417. E.Y.T. acknowledges the support of theNational Science Foundation (NSF) through MRSEC (NSF AwardDMR-1420645) and EPSCoR RII Track-1 (NSF Award OIA-2044049) programs. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Manchon, H. C. 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5.0048884.pdf

Band energy landscapes in twisted homobilayers of transition metal dichalcogenides Cite as: Appl. Phys. Lett. 118, 241602 (2021); doi: 10.1063/5.0048884 Submitted: 27 February 2021 .Accepted: 31 May 2021 . Published Online: 17 June 2021 F.Ferreira,1,2,a) S. J.Magorrian,1,2 V. V. Enaldiev,1,2,3 D. A. Ruiz-Tijerina,4 and V. I. Fal’ko1,2,5,a) AFFILIATIONS 1Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom 2National Graphene Institute, University of Manchester, Booth St E, Manchester M13 9PL, United Kingdom 3Kotel’nikov Institute of Radio-Engineering and Electronics of the Russian Academy of Sciences, 11-7 Mokhovaya St, Moscow 125009, Russia 4Secretar /C19ıa Acad /C19emica, Instituto de F /C19ısica, Universidad Nacional Aut /C19onoma de M /C19exico, Cd. de M /C19exico C.P. 04510, Mexico 5Henry Royce Institute for Advanced Materials, University of Manchester, Manchester M13 9PL, United Kingdom Note: This paper is part of the APL Special Collection on Twisted 2D Electronic and Photonic Materials and Devices. a)Authors to whom correspondence should be addressed: fabio.ferreira@postgrad.manchester.ac.uk and vladimir.falko@manchester.ac.uk ABSTRACT Twistronic assembly of 2D materials employs the twist angle between adjacent layers as a tuning parameter for designing the electronic and optical properties of van der Waals heterostructures. Here, we study how interlayer hybridization, weak ferroelectric charge transfer betweenlayers, and a piezoelectric response to deformations set the valence and conduction band edges across the moir /C19e supercell in twistronic homobilayers of MoS 2, MoSe 2,W S 2, and WSe 2. We show that, due to the lack of inversion symmetry in the monolayer crystals, bilayers with parallel (P) and antiparallel (AP) unit cell orientations display contrasting behaviors. For P-bilayers at small twist angles, we find band edges in the middle of triangular domains of preferential stacking. In AP-bilayers at marginal twist angles ( hAP<1/C14), the band edges are located in small regions around the intersections of domain walls, giving highly localized quantum dot states. VC2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http:// creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/5.0048884 The assembly of van der Waals heterostructures has potential for tailoring the properties of 2D materials.1Recently, it has been shown that twisted bilayer graphene exhibits intrinsic unconventional super- conductivity2and a Mott insulating phase.3These effects have been related to the localization of electrons, in particular, stacking areas ofthe moir /C19e superlattice (mSL), potentially enhanced by lattice recon- struction, promoting energetically favorable Bernal stacking. 4–6 L a t t i c er e c o n s t r u c t i o na l s ot a k e sp l a c ei nb i l a y e r so ft w i s t e dt r a n - sition metal dichalcogenides7–16(TMDs), where it gives rise to the for- mation of preferential stacking domains separated by a network ofdomain walls (DWs). The shape and size of these domains depend on the mutual orientation of the unit cells of the individual crystals, which can be parallel (P) or antiparallel (AP), whereas the superlattice perioddepends on the misalignment angle between the crystallographic axesof the layers. These two orientations fundamentally differ in that AP- bilayers possess inversion symmetry for all local stacking configura- tions, whereas for P-bilayers, both inversion and mirror reflectionsymmetries are generally broken. Below we discuss how this symmetrybreaking in P-bilayers leads to: (a) interlayer charge transfer due to hybridization of the conduction bands (CBs) in one layer with thevalence bands (VBs) of the other and (b) layer-asymmetric piezoelec- tric charges caused by lattice reconstruction concentrated around the network of domain walls and corners. Some of these effects wererecently discussed in relation to the structural characterization of TMD homobilayers and heterobilayers, 7–16and in twisted hexagonal boron nitride (hBN).17–19In both P- and AP-bilayers, the band ener- gies depend on the interplay between the above effects and the inter-layer hybridization of band edge states, which is most prominent for the states around the C- and Q-valleys and marginally relevant for the K-valley band edges. To model twisted TMD homobilayers, we use a multiscale approach 10for describing lattice reconstruction and computing piezo- electric charge and potential distributions, complemented by density functional theory (DFT)-parametrized Hamiltonians for interlayer hybridization of band edge states developed to describe an arbitrarylocal stacking. The first step is to use an earlier parametrized 10 Appl. Phys. Lett. 118, 241602 (2021); doi: 10.1063/5.0048884 118, 241602-1 VCAuthor(s) 2021Applied Physics Letters ARTICLE scitation.org/journal/aplinterlayer adhesion potential to compute atomic reconstruction of twisted TMDs for various twist-angles using an elasticity theory. We take into account both lateral and vertical deformations of the layers, see S1 in the supplementary material . This gives us the pattern of local interlayer distance dand lateral offset r0, which determine the stacking of the layers, as well as the piezoelectric potential distribution across the mSL. The second step is to compute the band structures of bilayers with various offsets (using DFT implemented in Q UANTUM ESPRESSO package20) in order to parametrize r0-dependent Hamiltonians describing interlayer hybridization (see S2 and S3 in the supplemen- tary material ). Diagonalizing the Hamiltonians, which take into account piezopotentials, we build maps for the local band edge ener- gies across the mSL. We analyze the band edge properties in the valence band (VB), near both the C- and K-valleys, and the conduction band (CB), near both the K- and Q-valleys. We consider both options ( Cand K) for the location of the VB edge in the Brillouin zone, as their relative ener- gies may depend on the encapsulating environment. In the presence of hBN, the energies of chalcogen orbitals, which are strongly represented at theC-valley, may be shifted with respect to the metal dx2/C0y2and dxy orbitals that determine the K-valley energy,21with the magnitude and direction of the shift determined by the WSe 2/hBN band alignment and the properties of their hybridization. Local stacking configurations, distinguished by a relative lateral offset of the two lattices in the bilayer (see Fig. S2 in the supplementary material ), vary across the moir /C19e supercell, with lattice reconstruction promoting energetically favorable configurations. For P-stacking, these are MX0/XM0, in which a metal atom M(M0) is below(above) a chalco- gen atom X(X0) in the opposite layer (X and X0refer to chalcogen atoms in bottom and top layer and M and M0refer to transition metal atoms in bottom and top layer). For AP-stacking, the favorable config- uration is 2H, as found in bulk TMD crystals, and features two interlayer pairs of vertically aligned metal/chalcogen atoms. Other important high-symmetry stackings include XX0(both P and AP), for which chalcogen atoms are vertically aligned, and for AP stacking MM0, which has aligned metal atoms (Fig. S2 in the supplementary material ). P-bilayer superlattices are comprised of two triangular domains with MX0and XM0stacking separated by domain walls with XX0stacking near hexagonal domain wall lattice sites, whereas in AP-bilayers there are large 2H domains separated by a honeycomb network of domain walls with XX0and MM0stacking at their intersec- tions.10Below, we aim to determine which stacking areas in the moir /C19e supercell host the band edges, which is essential for determining and predicting the formation of quantum dots in marginally twisted struc- tures. We separately discuss features of the K-, Q-, and C-valley band edges for P- and AP-bilayers. For AP-bilayers, the position of the VB at Cis determined by the interplay between the piezopotential and interlayer hybridization of resonant band edges. Due to symmetry, the piezopotential induced by lateral lattice reconstruction is the same on both layers;10interlayer hybridization is also sensitive to lattice deformations but in the vertical direction. This is because the wavefunctions at Ccarry a substantial weight of pzorbitals of chalcogens, which strongly overlap between the layers with a pronounced dependence on the interlayer distance.21 2H domains, which provide energetically more favorable stack- ing, also feature the closest interlayer distance, promoting the higher position of the top VB due to interlayer hybridization of the VB edges.At the same time, piezocharges are largest and negative (hence, the electron piezopotential is highest) at XX0regions,10attracting holes from the centers of 2H domains toward XX0corners. The trigonal symmetry of the honeycomb domain structure suggests that the result-ing VB edges appear as three maxima labeled in Fig. 1 as 2H c.T h i s behavior is corroborated by the plots shown on the left hand side of Fig. 1 , where band edge energies in areas with local 2H, MM0,a n dX X0 stacking configurations are compared with each other and with the energy of those local maxima at 2H c. The data shown in Fig. 1 for all TMDs indicate that they all display the same VB behavior at C. For large twist angles, the piezopotential magnitude decreases, whereas the contribution of interlayer hybridization remains the same. This promotes the maximum value for the VB energy at the center of2H domains, while the minimum value switches from MM 0-t oX X0- stacking regions. This is shown in the left-hand side panels of Fig. 1 , where the crossover between the two regimes takes place at hAP/C250:6/C14. For AP-bilayers, the VB and CB edge energies at K are domi- nated by the piezopotential contribution. This is because interlayer hybridization between K-valley states in AP-bilayers is weak: the band edge states are dominated by the orbitals of metals, buried inside the layer,21and their spin-valley locking due to spin–orbit splitting makes monolayer bands with the same spin off-resonant. A domain wall structure is fully developed for twist angles hAP/H113510:6/C14, giving rise to large piezopotentials around domain wall intersections. For such twist angles, the piezopotential forms quan-tum-dot-like wells of depth /C24150 meV for electrons and holes in the MM 0and XX0intersections of the domain wall structure, respectively, shown in Fig. 1 . In a periodic moir /C19e supercell, quantized states local- ized in each of these quantum dots should give rise to narrow bands located at the CB and VB edges of marginally twisted bilayers. Uponincreasing the twist angle, the magnitude of the piezopotential decreases and the location of its minimum in the supercell shifts to 2H a r e a s ,f o l l o w e db yt h ep o s i t i o no ft h eV Bb a n de d g e .T h eC Be d g e remains in the MM 0-stacked regions due to a residual attractive piezo- potential for electrons even at larger angles. For AP-bilayers, the position of the CB edge at the six Q-valleys is determined by the piezopotential and by interlayer coupling varia- tions across the mSL. For hAP/C201/C14, the piezopotential overcomes hybridization, forming quantum dots for electrons at MM0corners of 2H domains. For hAP>3/C14, the Q-valley CB edge shifts to the center of 2H regions. For P-bilayers, the VB edge at Cis dominated by interlayer hybridization of chalcogen orbitals, which is an order of magnitudestronger than both the piezoelectric and ferroelectric potentials (Table S4 in the supplementary material ). A weak ferroelectric potential arises due to the interlayer charge transfer caused by off-resonant hybridiza- tion of conduction bands in one layer with valence bands in the oppo- site layer. Such a spontaneous out-of-plane charge polarization isallowed due to the lack of inversion symmetry in P-bilayers. 22,23For the full range of twist angles, the VB at Clies in the MX0/XM0 domains, where hybridization is largest because the interlayer distance is smallest (see Fig. 2 ). For P-bilayers, the VB and CB edges at K are determined by a competition among interlayer charge transfer, the piezopotential, and resonant interlayer hybridization at the band edges. All three are com-parable for the K-valley states but vary differently across the mSL.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 241602 (2021); doi: 10.1063/5.0048884 118, 241602-2 VCAuthor(s) 2021Interlayer hybridization vanishes for MX0and XM0stackings due to the symmetry of the K-valley Bloch states in the honeycomb lattice,24 whereas the potential step due to ferroelectric charge transfer is largest (Tables S4 and S5 in the supplementary material ). The piezopotential, which has opposite signs in the two layers, contributes to the splittingof the resonantly coupled bands rather than to the modulation of theiraverage position, as in AP-bilayers. 25Splitting in XX0areas is only determined by interlayer hybridization, as the ferroelectric potential and piezopotential are absent there. The piezopotential distributionstrongly depends on twist angle. Also, piezo- and ferroelectric poten-tials have opposite signs, and fully or partially cancel each other out inareas where both are present. At marginal twist angles h P/H113511/C14,t h i s leads to flat CB and VB edges across MX0and XM0domains, see Fig. 2 , forming shallow ( /C2530 meV) triangular traps for electrons and holes, respectively. As the twist angle is increased, the piezopotential expands into the MX0/XM0domains, with similar magnitude and opposite sign to the potential caused by the charge transfer effect, reducing the splitting between the bands ( supplementary material Fig. 13). For the VB, this is sufficient to move the band edge to the XX0areas, where the splitting due to weak interlayer hybridization persists, and for the CB, the MX0/ XM0and XX0regions become very close in energy, see right hand pan- els of Fig. 2 .For P-bilayers, the Q-valley CB edge is mainly affected by the var- iation of resonant interlayer hybridization of monolayer Q-valleystates across the mSL, with only weak effects from the layer- asymmetric piezo- and ferroelectric potentials. For h P/C202/C14,t h eC B edge appears at one-dimensional channels zigzagging across the mSL.Such channels are /C2425 meV deep for P-WSe 2but shallower for the other TMDs. The asymmetry seen in the band edge maps is due to the underlying asymmetry of the Q-valley wave functions in each layer, interplaying with the lateral interlayer offset, which is differently ori-ented at different segments of the DW web. In this regime, XX 0areas play the role of potential barriers, acting as scatterers between thechannels for CB electrons. For h P/C212/C14, the channels broaden to form anisotropic landscapes. Despite their small amplitude, “potential” var- iations in such landscapes would determine anisotropic moir /C19em i n i - bands for Q-valley electrons. Finally, we have established how the interplay among interlayer hybridization, piezoelectric potential, and ferroelectric charge transfer affects the band edges for electrons in the K- and Q-valleys in the CB, and K- and C-valleys in the VB of TMD bilayers. We have analyzed all these possibilities because the influence of the encapsulating mate-rial or a substrate may affect their mutual alignment. For example, theDFT modeling of isolated bilayers points toward Q-valley CB edge for all four TMDs studied here, whereas recent experimental studies 26of FIG. 1. Left panels: variation of the VB energy with twist angle hAPfor different stacking configurations in AP-bilayers at the C- and K-valleys. The VB edge at the C-valley is located at the corners of 2H domains (labeled as 2H c) for marginal twist angles. These corners can be seen in the C-valley VB edge map for hAP¼0:2/C14. The arrows in the left panels mark the difference between the C-valley VB energy at the corners and in the center of 2H domains. Right panels: variation of the CB energy with hAPfor different stacking configurations in AP-bilayers at the K- and Q-valleys. The middle panels show maps for the VB edge at the C- and K-valleys and for the CB edge at the K- and 6Q1- valleys in AP-MoS 2forhAP¼0:2/C14andhAP¼3/C14. The zigzag orientations of DWs make the Q-valley CB edge maps C3-symmetric, leading to the same CB edge landscapes for6Q1,2,3. The vacuum level is set to 0 eV. See the supplementary material , Figs. S4–S7 for maps of the other TMDs.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 241602 (2021); doi: 10.1063/5.0048884 118, 241602-3 VCAuthor(s) 20212H MoS 2bi- and tri-layers indicate the persistence of K-valley band edges. Then, the crossovers between different misalignment angleregimes presented in this paper and highlighted in Figs. 1 and2repre- sent the variety of scenarios for the formation of quantum dot arrays that determines Hubbard physics for narrow minibands of electrons and holes in twistronic bilayers with different encapsulation environments. See the supplementary material for more details about the multi- scale approach for describing lattice reconstruction and computing piezoelectric charge and potential distributions. More details about DFT band structure calculations and effective Hamiltonians used todescribe band edges at C- ,K ,a n dQ - v a l l e y sc a na l s ob es e e ni nt h e supplementary material . We thank C. Yelgel, N. Walet, Q. Tong, M. Chen, F. Xiao, H. Yu, and W. Yao for discussions. This work has been supported by EPSRC Grant Nos. EP/S019367/1, EP/V007033/1, EP/S030719/1, andEP/N010345/1, ERC Synergy Grant Hetero2D, Lloyd’s Register Foundation Nanotechnology Grant, European Graphene Flagship Core 3 Project, and EU Quantum Technology Flagship Project 2D-SIPC. Computational resources were provided by the Computational Shared Facility of the University of Manchester andthe ARCHER2 UK National Supercomputing Service ( https:// www.archer2.ac.uk ) through EPSRC Access to HPC Project No. e672.DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1A. K. Geim and I. V. 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Tadmor, “Structural and electron diffraction scaling oftwisted graphene bilayers,” J. Mech. Phys. Solids 112, 225–238 (2018). 6Y. Wang, Z. Wang, W. Yao, G.-B. Liu, and H. Yu, “Interlayer coupling in com- mensurate and incommensurate bilayer structures of transition-metaldichalcogenides,” Phys. Rev. B 95, 115429 (2017). 7S. Carr, D. Massatt, S. B. Torrisi, P. Cazeaux, M. Luskin, and E. Kaxiras, “Relaxation and domain formation in incommensurate two-dimensional heter- ostructures,” Phys. Rev. B 98, 224102 (2018). FIG. 2. Left panels: variation of the VB energy with twist angle hPfor XX0- and MX0-stacking configurations in P-bilayers at the C- and K-valleys. Right panels: variation of the CB energy with hPfor XX0- and MX0-stacking configurations in P-bilayers at the K- and Q-valleys. The middle panels show maps for the VB edge at the C- and K-valleys, and for the CB edge at the K- and 6Q1-valleys in P-MoS 2forhP¼0:2/C14andhP¼3/C14. Note that for all angles, the Q-valley maps are anisotropic, with the anisotropy axis rotated by6120/C14when going from Q 1to Q 2and Q 3. For hP¼3/C14, the low contrast of the CB and VB K-valley maps reflect a negligible variation of the band edge energy ( <20 meV). The vacuum level is set to 0 eV. See the supplementary material , Figs. S8–S11 for maps of the other TMDs.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 241602 (2021); doi: 10.1063/5.0048884 118, 241602-4 VCAuthor(s) 20218M. H. Naik and M. Jain, “Ultraflatbands and shear solitons in moir /C19e patterns of twisted bilayer transition metal dichalcogenides,” Phys. Rev. Lett. 121, 266401 (2018). 9M. R. Rosenberger, H.-J. Chuang, M. Phillips, V. P. Oleshko, K. M. McCreary,S. V. Sivaram, C. S. Hellberg, and B. T. Jonker, “Twist angle-dependent atomic reconstruction and moir /C19e patterns in transition metal dichalcogenide hetero- structures,” ACS Nano 14, 4550–4558 (2020). 10V. V. Enaldiev, V. Z /C19olyomi, C. Yelgel, S. J. Magorrian, and V. I. Fal’ko, “Stacking domains and dislocation networks in marginally twisted bilayers of transition metal dichalcogenides,” Phys. Rev. Lett. 124, 206101 (2020). 11A. Weston, Y. Zou, V. Enaldiev, A. Summerfield, N. Clark, V. Z /C19olyomi, A. Graham, C. Yelgel, S. Magorrian, M. Zhou, J. Zultak, D. Hopkinson, A. Barinov,T. H. Bointon, A. Kretinin, N. R. Wilson, P. H. Beton, V. I. Fal’ko, S. J. Haigh, and R. Gorbachev, “Atomic reconstruction in twisted bilayers of transition metal dichalcogenides,” Nat. Nanotechnol. 15, 592 (2020). 12L. J. McGilly, A. Kerelsky, N. R. Finney, K. Shapovalov, E.-M. Shih, A. Ghiotto, Y. Zeng, S. L. Moore, W. Wu, Y. Bai, K. Watanabe, T. Taniguchi, M. Stengel, L. Zhou, J. Hone, X. Zhu, D. N. Basov, C. Dean, C. E. Dreyer, and A. N. Pasupathy, “Visualization of moir /C19e superlattices,” Nat. Nanotechnol. 15, 580–584 (2020). 13D. Edelberg, H. Kumar, V. Shenoy, H. Ochoa, and A. N. Pasupathy, “Tunable strain soliton networks confine electrons in van der Waals materials,” Nat. Phys. 16, 1097 (2020). 14L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes, C. Tan, M. Claassen, D. M. Kennes, Y. Bai, B. Kim, K. Watanabe, T. Taniguchi, X. Zhu, J. Hone, A. Rubio,A. N. Pasupathy, and C. R. Dean, “Correlated electronic phases in twisted bilayer transition metal dichalcogenides,” Nat. Mater. 19, 861–866 (2020). 15V. V. Enaldiev, F. Ferreira, S. J. Magorrian, and V. I. Fal’ko, “Piezoelectric net- works and ferroelectric domains in twistronic superlattices in WS 2/MoS 2and WSe 2/MoSe 2bilayers,” 2D Mater. 8, 025030 (2021). 16V. Vitale, K. Atalar, A. A. Mostofi, and J. Lischner, “Flat band properties of twisted transition metal dichalcogenide homo- and heterobilayers of MoS 2, MoSe 2,W S 2and WSe 2,”arXiv:2102.03259 (2021).17C. R. Woods, P. Ares, H. Nevison-Andrews, M. J. Holwill, R. Fabregas, F. Guinea, A. K. Geim, K. S. Novoselov, N. R. Walet, and L. Fumagalli, “Charge- polarized interfacial superlattices in marginally twisted hexagonal boronnitride,” Nat. Commun. 12, 1 (2021). 18K. Yasuda, X. Wang, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, “Stacking-engineered ferroelectricity in bilayer boron nitride,” arXiv:2010.06600 (2020). 19N. R. Walet and F. Guinea, “Flat bands, strains, and charge distribution in twisted-bilayer hBN,” arXiv:2011.14237 (2020). 20P. Giannozzi, O. Baseggio, P. Bonf /C18a, D. Brunato, R. Car, I. Carnimeo, C. Cavazzoni, S. de Gironcoli, P. Delugas, F. Ferrari Ruffino, A. Ferretti, N.Marzari, I. Timrov, A. Urru, and S. Baroni, “Q UANTUM ESPRESSO toward the exascale,” J. Chem. Phys. 152, 154105 (2020). 21A. Korm /C19anyos, G. Burkard, M. Gmitra, J. Fabian, V. Z /C19olyomi, N. D. Drummond, and V. Fal’ko, “k /C1p theory for two-dimensional transition metal dichalcogenide semiconductors,” 2D Mater. 2, 022001 (2015). 22L. Li and M. Wu, “Binary compound bilayer and multilayer with vertical polar- izations: Two-dimensional ferroelectrics, multiferroics, and nanogenerators,” ACS Nano 11, 6382–6388 (2017). 23Q. Tong, M. Chen, F. Xiao, H. Yu, and W. Yao, “Interferences of electrostatic moir /C19e potentials and bichromatic superlattices of electrons and excitons in transition metal dichalcogenides,” 2D Mater. 8, 025007 (2020). 24D. A. Ruiz-Tijerina and V. I. Fal’ko, “Interlayer hybridization and moir /C19e super- lattice minibands for electrons and excitons in heterobilayers of transition-metal dichalcogenides,” Phys. Rev. B 99, 125424 (2019). 25Unlike the case of AP-bilayers, in P-bilayers, the top and bottom layers have piezocharges of opposite sign. This is due to the signs of the piezocoefficientsin the expression for piezocharge density. See the supplementary material ofRef.10. 26M. Masseroni, T. Davatz, R. Pisoni, F. K. de Vries, P. Rickhaus, T. Taniguchi, K. Watanabe, V. Fal’ko, T. Ihn, and K. Ensslin, “Electron transport in dual-gated three-layer MoS 2,”Phys. Rev. Res. 3, 023047 (2021).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 241602 (2021); doi: 10.1063/5.0048884 118, 241602-5 VCAuthor(s) 2021

5.0048042.pdf

APL Mater. 9, 050904 (2021); https://doi.org/10.1063/5.0048042 9, 050904 © 2021 Author(s).Negligible spin–charge conversion in Bi films and Bi/Ag(Cu) bilayers Cite as: APL Mater. 9, 050904 (2021); https://doi.org/10.1063/5.0048042 Submitted: 18 February 2021 . Accepted: 05 May 2021 . Published Online: 18 May 2021 Di Yue , Weiwei Lin , and C. L. Chien COLLECTIONS Paper published as part of the special topic on Emerging Materials for Spin-Charge Interconversion ARTICLES YOU MAY BE INTERESTED IN Strongly heat-assisted spin–orbit torque switching of a ferrimagnetic insulator APL Materials 9, 051117 (2021); https://doi.org/10.1063/5.0049103 Pure spin current phenomena Applied Physics Letters 117, 190501 (2020); https://doi.org/10.1063/5.0032368 Field-free magnetization switching induced by the unconventional spin–orbit torque from WTe 2 APL Materials 9, 051114 (2021); https://doi.org/10.1063/5.0048926APL Materials PERSPECTIVE scitation.org/journal/apm Negligible spin–charge conversion in Bi films and Bi/Ag(Cu) bilayers Cite as: APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 Submitted: 18 February 2021 •Accepted: 5 May 2021 • Published Online: 18 May 2021 Di Yue,1,2 Weiwei Lin,1 and C. L. Chien1,a) AFFILIATIONS 1Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA 2State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China Note: This paper is part of the Special Topic on Emerging Materials for Spin-Charge Interconversion. a)Author to whom correspondence should be addressed: clchien@jhu.edu ABSTRACT Spin pumping experiments using ferromagnetic metals have reported highly efficient spin–charge conversion in Bi and at the Bi/Ag interface, possibly due to the inverse Rashba–Edelstein effect. However, longitudinal spin Seebeck effect experiments using the yttrium iron garnet ferrimagnetic insulator in Bi films and Bi/Ag bilayers do not show evidence of appreciable spin-to-charge conversion except the large Nernst signal inherent to Bi. These contrasting conclusions highlight the differences between magnetic metals and magnetic insulators as spin current injectors. Only the detected voltages that adhere to the inverse spin Hall effect of jC=(2e/̵h)θSHjS×σare due to spin currents. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0048042 INTRODUCTION The spin Hall effect (SHE) converts a charge current jCinto a pure spin current jSin a conductor with substantial spin–orbit coupling (SOC).1,2Conversely, the inverse spin Hall effect (ISHE)2,3 converts a pure spin current into a charge current as described by jC=(2e/̵h)θSHjS×σ, where eis the electronic charge,̵his the reduced Planck constant, θSHis the spin Hall angle, and σis the spin index. Conversion between spin current and charge current is essential for the exploration of pure spin current phenomena and cutting- edge spintronics. The efficiency of spin–charge conversion is usually quantified by the spin Hall angle θSHwith values ∣θSH∣≤1. When a pure spin current flows in a metal, the spin-to-charge conver- sion efficiency lies between two limits. If none of the charge carriers are deflected sideway, θSH=0, whereas if all the charge carriers are deflected sideway, ∣θSH∣=1. The sinusoidal angular dependence was experimentally well established.4Since the strength of the SOC scales with the atomic number Z, many (but not all) heavy metals (e.g., Ta, W, and Pt) have large θSHvalues.5,6Likewise, some 3 dmetals (e.g., Cr and Ni) also have substantial θSH.7Since the best heavy metals have θSHvalues of the order of only 0.1, it is always attractive to identify materials with even larger θSH, including exploiting alloys, doping, and interfaces.Bi, being the heaviest non-radioactive element with Z=83, has attracted much attention. Theories suggested that Bi may exhibit the quantum spin Hall effect (QSHE) due to its strong diamagnetism.8 A close relationship between the QSHE and the diamagnetism of Dirac electrons has been suggested in Bi.9,10Due to nearly cross- ing bands with strong SOC, the calculated spin Hall conductivity for Bi is more than twice that of Pt.11A ratio of spin Hall conduc- tivity to electrical conductivity ( θSH) of 0.008 was reported for Bi (200 nm) via spin accumulation measured in an attached permal- loy (Py≈Ni80Fe20) at the temperature of 3 K.12On the other hand, spin pumping (SP) experiments in Bi(110)/Py bilayers have reported much larger θSH=−0.071 and+0.019 near the Bi/Py interface and in the Bi bulk, respectively, illustrating the substantial θSHand the benefits of interfaces.13 The θSHvalues of several common metals, such as Cu and Ag, are negligibly small.5However, it has been reported that a small amount of heavy element Bi impurities ( <0.5%) in Cu induces a huge θSHof−0.24 as reported in nonlocal lateral spin valves (LSVs) with Py, where the large θSHhas been attributed to the domination of skew scattering.14When the Cu host is replaced with Ag, the θSH is reduced by a factor of 10.15 Remarkably, inserting a thin Ag layer as in NiFe/Ag/Bi greatly enhances the current and voltage signals in the SP experiments than APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 9, 050904-1 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm those in NiFe/Bi.16The observed large signals in NiFe/Ag/Bi, if attributed to ISHE, would imply the unphysical result of θSH>1, thus leading to the proposal of a novel mechanism of spin-to-charge conversion at the Bi/Ag Rashba interface,17namely, the inverse Rashba–Edelstein effect (IREE).18The efficiency of spin-to-charge conversion at the Ag/Bi interface is larger than that in Pt.16The IREE has been reported in NiFe/Ag/Sb whose IREE length λIREE is a fac- tor of 10 smaller than that in NiFe/Ag/Bi via SP.19Moreover, the SP signals of Ag/Bi/Fe and Bi/Fe have opposite signs.20The spin-to- charge conversion efficiency measured in Ni 20Fe80/Ag/Bi by SP has been shown to be nearly independent of temperature.21However, in Py/Cu/Bi(111) LSV, the spin-current-induced charge current, which is attributed to the IREE at the Cu/Bi interface, is found to change the direction with increasing temperature.22 The broadband terahertz emission observed in CoFeB/Ag/Bi under incident femtosecond laser pulses has been reported to be larger than those in CoFeB/Ag and CoFeB/Bi and likewise attributed to the IREE at the Ag/Bi interface.23The sign change of the THz signal has been reported for Fe(2 nm)/Ag(2 nm)/Bi(3 nm) and Fe(2 nm)/Bi(3 nm)/Ag(2 nm), in support of the IREE mechanism at the Bi/Ag interface.24Recently, THz emission has been observed in single Bi thin films (without an adjacent magnetic layer) when irradi- ated by circularly polarized femtosecond laser pulses and attributed to the photoinduced ISHE in Bi.25 Using a spin-polarized positron beam, opposite surface spin polarizations have been found between Bi/Ag/Al 2O3and Ag/Bi/Al 2O3with the application of a charge current in the same direction, suggesting charge-to-spin conversion induced by the Rashba–Edelstein effect.26Charge-to-spin conversion at the Ag/Bi interface has also been reported in spin torque ferromagnetic resonance (ST-FMR) experiments in Py/Ag/Bi.27Recently, size- able ST-FMR has been investigated in Bi/Py.28Angle-dependent magnetoresistance observed in CoFeB/Ag/Bi trilayers has been attributed to the simultaneous action of the direct and inverse Rashba–Edelstein effects at the Bi/Ag interface.29,30Combining the electrical measurements in a mesoscopic H-shaped structure and current induced magnetization switching observed in a Fe/Cu/ Bi(0.15 nm)/Cu/MgO(111), a large value of θSH=−0.12 has been deduced at T=5 K for the Cu film delta doped with 0.15 nm Bi.31 Notably, by using the magneto-optical Kerr effect, optical spin orientation, and spin pumping, the spin-to-charge conversion gen- erated by the IREE was observed in 1–3-nm-thick Bi films corre- sponding to the presence of Bi nanocrystals at the surface of Ge(111), whose efficiency is comparable to that of the Ag/Bi interface.32As the film size increases, the Bi film becomes continuous and semimetal- lic, leading to the cancellation of spin–charge conversion occurring at opposite surfaces, resulting in an average spin–charge conversion that progressively decreases and disappears.32Thus, many publica- tions, as listed in Table I, have reported evidence of very substan- tial spin-to-charge conversion and IREE in Bi, Bi/Ag, and related materials. One notes that most of the reports on the substantial spin–charge conversion in these systems employ ferromagnetic (FM) metals (e.g., Fe and Py) using high frequency SP spin cur- rent injection.13,16,19–21,33It is known that in SP experiments with FM metals, the FMR microwave also induces heat and other par- asitic effects, including spin rectification via the anisotropic mag- netoresistance effect, planar Hall effect, anomalous Hall effect, andanomalous Nernst effect.34–36Some of these accompanying effects, especially those that manifest the same symmetry as that of spin- to-charge conversion, are difficult to eliminate.35,36The detected voltages are taken entirely as that of spin-to-charge conversion, whose efficiency may be grossly overestimated.35,36Furthermore, FM metals also have their own inherent spin-polarized charge current effects, which further complicate the measurement of the spin–charge conversion.37In contrast, magnetic insulators, with no charge carriers, enable the generation of pure spin current without the charge carrier contributions, thus being more advantageous for the quantification of spin–charge conversion.37–39 When the FM metal (e.g., NiFe) is replaced by a ferrimagnetic insulator , e.g., yttrium iron garnet (YIG), the deduced values of θSH of Bi are orders of magnitude smaller . Using SP in Bi(111)/YIG bilayers, the deduced θSHvalue was only 0.000 12 for a 60 nm Bi layer, two orders of magnitude smaller.40The sign of the spin–charge conversion current for Bi/Ag/YIG and Ag/Bi/YIG also remains the same, and not opposite as in some earlier IREE results.41Some studies show that supposedly the Seebeck voltage due to the spin- wave-induced lateral temperature gradient can still be detected in Sb/SiO 2/YIG, when the spin current from YIG has been completely blocked by an inserted SiO 2layer.42These conflicting experimental results question whether one can attribute the measured voltage in SP experiments40,41to spin–charge conversion. Recently, by utilizing the spin Peltier effect (SPE) in Cu 1−xBix/YIG (x=0–1), the Cu–Bi binary alloys do not show remarkable SHE,43in contrast to the giant θSHclaimed previously.14,15,31 In addition to SP, the longitudinal spin Seebeck effect (LSSE) is another method to inject spin current using an out-of-plane tem- perature gradient without the high frequency excitations.39,44The LSSE experiments in Pt/YIG and Ta/YIG reveal only the spin current effect with opposite voltages.44It is highly desirable to perform LSSE experiments in Bi/YIG and Bi/Ag/YIG to assess the spin current conversion. Only the voltage that adheres to jC=(2e/̵h)θSHjS×σis due to spin-to-charge conversion. Previously, the LSSE experiments in Bi/YIG and Bi/Ag/YIG show that although pure spin current has been injected into the Bi layer and the Bi/Ag bilayer, but there is no detectable signal of spin-to-charge conversion, except the distinc- tive Nernst signal from the Bi layer.45In this Perspective, we show additional experimental results and provide more discussions on the negligibly small spin-to-charge conversion in Bi films and Bi/Ag(Cu) bilayers on YIG. SAMPLE FABRICATION AND STRUCTURAL CHARACTERIZATION We used DC magnetron sputtering to deposit Bi and Ag lay- ers onto YIG substrates in a chamber with a base pressure of 5×10−8Torr. Yttrium iron garnet (YIG =Y3Fe5O12) is a highly stable ferrimagnetic oxide with T C=560 K. The melting point of Bi is rather low at 271○C, which posts some difficulties in sputtering. The Bi deposition was conducted at about 3 W at a rate of 0.18 nm/s onto YIG substrates maintained at liquid nitrogen temperature. We used two types of YIG substrates: 0.5-mm-thick polycrystalline YIG slab (YIG slab) and 5- μm-thick single crystalline YIG film grown by liquid-phase-epitaxy on GGG(111) substrates (YIG/GGG). The crystal structures, thickness, and roughness of the thin film samples were characterized by x-ray diffraction (XRD) and x-ray reflectivity APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 9, 050904-2 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm TABLE I. Bi layer samples and thickness in nm, deposition method, layer orientation in rhombohedral notation, method of measurement, spin Hall angle ( θSH), spin diffusion length ( λSD), IREE length ( λIREE), and references. Bi deposition, orientation Sample (thickness in nm) (rhombohedral) Measurement θSH λSD(nm) λIREE (nm) Reference Bi(200)/Py(25) LSV 0.008 12 Bi(5–100)/Py(10) e-beam evap. SP −0.071 (Bi/Py) 2.4 (Bi/Py) 13 0.019 (bulk Bi) 50 (Bi) Bi(110) Bi(5–250)/Py(16) e-beam evap. SP 0.02 8 33 Amorphous Bi Bi(7–40)/FeO x(2)/Fe(10) e-beam evap. SP 0.0158 15.3 20 Bi(111) textured Bi(1.2–10)/Py(5) MBE ST-FMR 0.03–0.06 28 Bi(5–60)/YIG Bi(111) SP 0.00012 20 40 Bi(5–60)/YIG Low temperature LSSE <2×10−50.43–0.97 45 DC sputtering Bi(110) textured Bi(15–60)/YIG MBE LSSE <2×10−545 Single crystal Bi(111) Bi(8)/YIG Thermal evap. LSSE <2×10−545 Bi(111), Bi(110) MgO/Bi(30–100) RF sputtering THz 30 ±5 25 Bi(111),(110) NiFe(15)/Ag(0–20)/Bi(8) Evaporation SP 0.2–0.33 16 Bi(111) NiFe(15)/Ag(5)/Bi(8) SP 0.11 ±0.02 19 Ag(20)/Bi(8)/FeO x(2)/Fe(10) e-beam evap. SP −0.33 20 Ni20Fe80(20)/Ag(10)/Bi(10) Evaporation SP 0.03 (300 K) 21 0.04 (40 K) Py(35)/Cu(100)/Bi(20) e-beam evap. LSV 0.009 (300 K) 15 Bi(111) textured −0.001 (10 K) Fe(2)/Bi(0–4)/Ag(2) Polycrystalline Bi THz 17 22 Fe(2)/Ag(2)/Bi(0–4) 0.08–0.54 Bi(0–5)/Ag(25)/Al 2O3 Thermal Positron 2.1 26 Ag(25–500)/Bi(8)/Al 2O3 Deposition Py(9)/Ag(2–15)/Bi(4) Sputtering ST-FMR 0.18 0.1 27 Bi(20)/Ag(30)/YIG Bi(110) SP 0.001–0.07 41 Ag(30)/Bi(20)/YIG Bi(111) Bi(8–15)/Ag(2)/YIG Low-T DC LSSE <2×10−545 Sputtering Bi(110) Evaporation Bi(111), Bi(110) Bi(0–10)/Ge MBE Optical spin 0.05 32 Bi nanocrystals orientation, SP Py(30)/Cu/Cu 99.5Bi0.5(20) Sputtering LSV −0.24 (10 K) 14 Py(30)/Cu/Ag 99Bi1(20) Sputtering LSV −0.023 (10 K) 15 Fe(0.8)/Cu(3)/Bi(0.15)/Cu(7)/MgO MBE H-bar, SOT −0.12±0.01(5 K) 170 ±12(5 K) 31 switching Cu 1−xBix/YIG (x=0–1) Sputtering SP No clear SHE 43 APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 9, 050904-3 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 1. XRD and XRR results of Bi/YIG grown by low-temperature sputtering. (a) XRD results (log scale) of Bi(60 nm)/YIG/GGG. (b) XRD results (linear scale) of the Bi(400 nm)/YIG slab. The inset shows the log-scale plot of the data. (c) XRR results of the Bi/YIG slab (scaled by a factor of 103) and Ag/SiO x/Si. The red curves are fits to XRR results, indicating that the thickness of Bi(Ag) is 31.6 nm (30.5 nm) and the surface roughness of Bi(Ag) is 0.4 nm (0.5 nm). (d) XRR of superlattice [Bi(8 nm)/Ag(1.5 nm)] 5/SiO x/Si g. Data in (b)–(d) are obtained on the same x-ray facility, while data in (a) are obtained on a different x-ray facility. Panels (c) and (d) are reproduced with permission from Yue et al. , Phys. Rev. Lett. 121, 037201 (2018). Copyright 2018 American Physical Society. APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 9, 050904-4 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm (XRR) with the Cu Kαradiation. As shown in Figs. 1(a) and 1(b), on both YIG/GGG and YIG slab substrates, the Bi layer fabricated by low-temperature sputtering shows a preferred orientation along the rhombohedral-(110) direction, as indicated by the prominent Bi(110) and Bi(220) XRD peaks, same as the texture of Bi films in Ref. 13. The XRD results of Bi/Ag/YIG/GGG samples, as reported previously,45indicate a Bi(110) texture on top of the Ag(111) texture structure. From the XRR measurements, we can determine the film thick- ness and roughness, as shown in Fig. 1(c). In particular, the values of roughness of the Bi and Ag layers are 0.4 and 0.5 nm respectively, much smaller than the roughness of 3.5 nm in both Bi and Ag lay- ers reported in previous SP works with NiFe/Ag/Bi/SiO 2/Si.16The interfacial roughness of Bi/Ag in our work is also small, as indi- cated by the XRR of the superlattice [Bi(8 nm)/Ag(1.5 nm)] 5/SiO x/Si [Fig. 1(d)], where several characteristic features revealing the five repeats of bilayer are also evident.Several other layers, including Pt and W for spin-to-charge conversion via the ISHE and, in some cases, a thin MgO layer for blocking the pure spin current, were also deposited by sputtering at room temperature without breaking the vacuum. In addition to the sputtered sample, we have also studied epitaxial Bi(111) films grown by molecular beam epitaxy (MBE) on YIG/GGG substrates and polycrystalline Bi films and Bi/Ag bilayers grown by thermal evap- oration with considerable roughness.45The LSSE results of these samples fabricated by MBE and thermal evaporation are consistent with those made by sputtering.45In this Perspective, we mainly focus on the sputtered samples. SPIN-TO-CHARGE CONVERSION IN Bi The scheme of LSSE with the thermal injection of spin cur- rent from YIG is shown in Fig. 2(a). The out-of-plane temperature gradient (∇zT) causes YIG to inject a pure spin current jSinto the FIG. 2. LSSE measurements in thin film/YIG. (a) Schematic diagram for LSSE, where the measurements were performed by placing the sample between a Cu block heat sink and a resistance heater to generate an out-of-plane temperature gradient along the z axis. The heat flux was kept constant at 0.06 W/mm2with a temperature gradient of about 10 K/mm across the YIG substrates. The transverse thermovoltages along the y axis were measured with an in-plane magnetic field along the x axis. The distance between the two voltage leads is about 6 mm. Transverse thermovoltages as a function of the in-plane applied magnetic field measured in (b) Pt(3 nm)/YIG/GGG and Pt(3 nm)/MgO(3 nm)/YIG/GGG, where the inset shows the results in W(3 nm)/YIG/GGG, (c) Bi(15 nm)/YIG/GGG and Bi(15 nm)/MgO(3 nm)/YIG/GGG, and (d) Bi(5 nm)/YIG/GGG, Bi(8 nm)/YIG/GGG, Bi(15 nm)/YIG/GGG, and Bi(60 nm)/YIG/GGG, where the inset shows an enlarged plot of the thermovoltages in Bi(5 nm)/YIG/GGG. Panel (a) is reproduced with permission from Yue et al. , Phys. Rev. Lett. 121, 037201 (2018). Copyright 2018 American Physical Society. APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 9, 050904-5 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm adjacent metallic layer, where spin-to-charge conversion occurs via the ISHE giving rise to an electric current, often detected as a DC voltage associated with an electric field EISHE∝jS×σ, where jSis the spin current parallel to the temperature gradient ∇zTandσis the spin index set by the direction of the surface magnetization M of YIG. The thermal injection of spin current by LSSE with a mag- netic insulator enjoys the advantages of a simple injection and detec- tion scheme without the complexities due to radio-frequency elec- tromagnetic induction in SP. The anomalous Nernst effect, which dominates in FM metals, is also absent in magnetic insulators. We have rigorously evaluated the LSSE, where metals with opposite θSHgive rise to ISHE voltages of opposite signs, and that the ISHE voltage vanishes when the spin current has been blocked by a nonmagnetic insulating layer such as AlO xand MgO.44,45As shown in Fig. 2(b), the transverse thermovoltages in the Pt film with a posi- tiveθSHand the W film with a negative θSHare readily observed with opposite signs, and the field dependence follows that of the mag- netic hysteresis loop of the YIG/GGG substrates. After a 3-nm-thick MgO layer has been inserted between the Pt layer and the YIG/GGG substrate, the thermovoltage vanishes in Pt/MgO/YIG/GGG, demonstrating that the nonmagnetic insulating MgO(3 nm) layer has completely blocked the spin current. These LSSE experiments conclusively establish pure spin current phenomena in Pt/YIG and W/YIG. In short, these experiments establish the validity of jC=(2e/̵h)θSHjS×σ, that the field dependence reflects that of σand hence that of the YIG magnetization, and that the ISHE must vanish when the spin current jShas been blocked. Volt- ages that do not adhere to this description are unrelated to spin currents. However, dramatically different behavior has been observed in Bi/YIG. As shown in Fig. 2(c), the transverse thermovoltages in Bi(15 nm)/YIG/GGG show no correlation with the magnetiza- tion of YIG. Instead, these thermovoltages are linearly dependent on the applied magnetic field without saturation. After a 3-nm-thick MgO layer has been inserted between the Bi layer and the YIG/GGG substrate, the thermovoltages in Bi(15 nm)/MgO(3 nm)/YIG/GGG remain essentially unchanged from those in Bi(15 nm)/YIG/GGG. These experiments show rather succinctly that, especially since the MgO(3 nm) layer should have completely blocked the spin current, the measured voltages in Bi/YIG/GGG are unrelated to spin current phenomena. We studied Bi films with different thicknesses from 5 to 60 nm on the YIG/GGG substrates, as shown in Fig. 2(d). The thermovolt- ages in all these Bi/YIG/GGG samples are linearly dependent on the applied magnetic field and show no correlation with the magneti- zation switching of YIG, hence unrelated to spin current. Instead, these are the characteristics of the ordinary Nernst effect, the ther- mal counterpart of the ordinary Hall effect. In the ordinary Nernst effect, the charge current driven by a thermal gradient gives rise to a transverse electric field ENE∝∇ T×Bdue to the Lorentz force. The linear field dependence and the sinusoidal angular dependence of the Nernst effect are well known. The ordinary Nernst effect scales with the Bfield, whereas the LSSE in Pt/YIG and W/YIG scales with the surface magnetization Mof YIG. The Nernst effect is negligible in many common metals, such as Ag and Pt, but Bi exhibits one of the largest Nernst effect due to the small Fermi energy and large elec- tron mobility. In addition to the very different field dependence, the Nernst signal in Bi films increases with increasing thickness, whereasthe LSSE signal in Pt/YIG decreases with increasing Pt thickness limited by the spin diffusion length. The Nernst effect in BiSb/Co bilayers may generate spurious signals in the spin–orbit torque har- monic voltage measurements, leading to an overestimation of θSHof BiSb.46 Besides the Bi films grown on the 5- μm-thick single crystalline YIG film on GGG(111) substrates (YIG/GGG), we have also stud- ied Bi films grown on the 0.5-mm-thick polycrystalline YIG slab (YIG slab). As shown in Fig. 3(a), the transverse thermovoltages in the Bi(8 nm)/YIG slab are again proportional to the external mag- netic field without saturation at high magnetic fields even after the magnetization of the YIG slab has been saturated. There is a small plateau at low magnetic fields. This plateau is a common feature in all bulk YIG substrates due to the switching of the surface domains and demagnetizing effects of YIG substrates.47,48After subtracting the linear-field-dependent part from the thermovoltages, the resid- ual part of the thermovoltages may appear to show a hysteretic-like behavior, as shown in Fig. 3(b). However, these are also notrelated to the pure spin current phenomena because, as shown in Fig. 3(c), the thermovoltages remain intact after a MgO(3 nm) layer has been inserted between Bi and YIG slab, totally blocking the spin current. These thermovoltages are from the Nernst effect of the Bi films, and the small plateau at low magnetic fields just reflects the rever- sal of the magnetization in the YIG slabs and changes in the stray field,45as shown in the inset of Fig. 3(c). To confirm this argu- ment, we intentionally introduce the stray field to the pure Nernst effect in Bi/SiO x/Si. As shown in Fig. 3(d), the transverse thermo- voltages in Bi films on nonmagnetic SiO x/Si, which linearly depend on the external magnetic field, are purely from the Nernst effect in Bi. We intentionally placed a YIG slab onto the Bi/SiO x/Si sample and performed the thermovoltages measurements and reproduced the plateau feature at low magnetic fields, as shown in Fig. 3(e). The YIG slab and the Bi films were placed in close proximity with no pos- sibility of spin current injection into Bi films from the YIG slab, but the stray field of the bulk YIG slab can act on the transport proper- ties in Bi films. Again, the plateau is unrelated to pure spin current phenomena. Contrary to many previous conclusions from SP experiments using FM metals, we found nomeasurable evidence of spin current contribution to the thermovoltages from LSSE experiment in the Bi/YIG samples. The key question is whether the spin current has actually been injected into the Bi layer at all. To address this issue, we have measured Pt(3 nm)/Bi( tBi)/YIG and W(3 nm)/Bi( tBi)/YIG in LSSE, designed to detect the emergent spin current through the Bi layer by Pt and W, respectively. As shown in Fig. 3(f), we have indeed observed the spin-to-charge conversion signals in the Pt(3 nm)/Bi (4 nm)/YIG slab and W(3 nm)/Bi(4 nm)/YIG slab with opposite signs due to the different signs of θSHin Pt and W. The signals in W(3 nm)/Bi( tBi)/YIG gradually decreases with increasing Bi layer thickness tBiand becomes barely detectable in W(3 nm)/Bi (8 nm)/YIG. These signals are those of ISHE from spin current that has passed through the Bi layer and converted in the Pt and W layers. Thus, we have demonstrated that the pure spin current has been injected via LSSE from YIG into the Bi layer and has passed through the Bi layer but with negligibly small spin-to-charge con- version within Bi. We have estimated an upper limit of θSHin Bi as 2×10−5.45Besides the Bi films fabricated by low temperature sput- tering, we have also studied the thermovoltages in single-crystalline APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 9, 050904-6 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 3. LSSE results of Bi layers. (a) Transverse thermovoltages as a function of the in-plane applied magnetic field measured in Bi(8 nm)/YIG slab with the blue dashed line as the linear fit to the high-field data. (b) The residual transverse thermovoltages in Bi(8 nm)/YIG slab after the blue straight line in (a) has been subtracted. Transverse thermovoltages as a function of the in-plane applied magnetic field measured (c) in Bi(8 nm)/YIG slab and Bi(8 nm)/MgO(3 nm)/YIG slab, where the inset shows the schematic diagram of stray fields caused by the YIG slab, (d) in Bi(30 nm)/SiO x/Si grown by thermal evaporation, or (e) with a YIG slab spacer between the heater and the Bi(30 nm)/SiO x/Si sample. (f) Transverse thermovoltages as a function of the in-plane applied magnetic field measured in W(3 nm)/Bi(2 nm)/YIG slab, W(3 nm)/Bi(4 nm)/YIG slab, W(3 nm)/Bi(8 nm)/YIG slab, and Pt(3 nm)/Bi(4 nm)/YIG slab (scaled by a factor of 20). Results in other samples with Pt(3 nm)/Bi( tBi)/YIG slab and W(3 nm)/Bi( tBi)/YIG slab structures have been shown in Ref. 45. Bi(111) films grown by MBE and polycrystalline Bi films grown by thermal evaporation on YIG substrates, where we also observed only the Nernst signals in the Bi films and found no evidence of spin-to-charge conversion.45 SPIN-TO-CHARGE CONVERSION IN Bi/Ag AND Ag/Bi Both SP and THz experiments have reported greatly enhanced spin-to-charge conversion in Bi/Ag bilayers. We next describe spin injection into Bi/Ag bilayers by LSSE in YIG. As shown in Fig. 4(a), the transverse thermovoltages in Bi(5, 15, 60 nm)/Ag(2 nm) /YIG/GGG are all linearly dependent on the applied magnetic field and show no correlation with the magnetization of YIG. As shown in Fig. 4(b), the MgO(3 nm) spin current blocking layer also has no effect. These thermovoltages are those resulted from the ordi- nary Nernst effect in Bi and not the pure spin current. Furthermore, we have detected the emergent spin current passing through the Bi/Ag bilayers by measuring the thermovoltages in the Pt (3 nm) /Bi(tBi)/Ag( tAg)/YIG slab structures. As shown in Fig. 4(c), the spin-to-charge conversion signals in Pt(3 nm)/Bi(1 nm)/Ag(1 nm) /YIG slab and Pt(3 nm)/Bi(2 nm)/Ag(1 nm)/YIG slab have been clearly observed and become barely detectable in Pt(3 nm)/Bi(2 nm) /Ag(2 nm)/YIG slab. Thus, spin current has been injected into the Bi/Ag bilayers but with no detectable spin-to-charge conver- sion signals. We have found no evidence of the large IREE at theBi/Ag interface, which has been claimed with even larger spin-to- charge conversion efficiency than that of Pt by SP experiments with NiFe/Ag/Bi.16 There have been suggestions of opposite signs of spin-to- charge conversion at Bi/Ag and Ag/Bi interfaces when the spin current is injected from different sides of the interface by assum- ing the existence of the IREE at the Bi/Ag interfaces.24,26In our experiments, we have also performed the LSSE experiments on the Ag(5 nm)/Bi(10 nm)/YIG slab and Bi(10 nm)/Ag(5 nm)/YIG slab, as shown in Figs. 4(d) and 4(e). We find that the thermovoltages in both the Ag/Bi/YIG slab and Bi/Ag/YIG slab increase with the external magnetic field without saturation at high magnetic fields where the magnetization of the YIG slab has been saturated. At low fields, the thermovoltages have a plateau feature that reflects the reversal of the YIG magnetization and change of the stray field. Thus, these thermovoltages are not from the spin current but from the Nernst effect in the Bi layer. We found no evidence of spin- to-charge conversion in Bi/Ag and Ag/Bi bilayers on YIG, not to mention the sign change of spin-to-charge conversion in Bi/Ag and Ag/Bi. One of the key reasons for the IREE at Bi/Ag interfaces is that in the angle-resolved photoemission spectroscopy (ARPES) measure- ments, a large Rashba coefficient ( αR=3.05 eV Å) has been found in the Ag(111) film covered by 1/3 of a monolayer of Bi atoms, which forms a Bi/Ag(111) surface alloy system.17However, in most APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 9, 050904-7 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 4. Transverse thermovoltages as a function of the in-plane applied magnetic field measured in (a) Bi(5 nm)/Ag(2 nm)/YIG/GGG, Bi(15 nm)/Ag(2 nm)/YIG/GGG, and Bi(60 nm)/Ag(2 nm)/YIG/GGG, where the inset shows an enlarged plot of the thermovoltages in Bi(5 nm)/Ag(2 nm)/YIG/GGG, (b) Bi(8 nm)/Ag(3 nm)/YIG slab and Bi(8 nm)/Ag(3 nm)/MgO(3 nm)/YIG slab, (c) Pt(3 nm)/Bi(1 nm)/Ag(1 nm)/YIG slab, Pt(3 nm)/Bi(2 nm)/Ag(1 nm)/YIG slab, and Pt(3 nm)/Bi(2 nm)/Ag(2 nm)/YIG slab, (d) Ag(5 nm)/Bi(10 nm)/YIG slab, (e) Bi(10 nm)/Ag(5 nm)/YIG slab, and (f) Bi(0.18 nm)/Ag(3 nm)/YIG slab. of the spin current transport experiments, Bi/Ag bilayer structures with much thicker Bi layers were used. In our experiments, we attempted to fabricate a Bi/Ag surface alloy system by sputtering a 0.18±0.09 nm-thick Bi layer on a Ag(3 nm) layer, which cor- responds to a coverage of Bi atoms about (5.1±2.5)×1014 atoms/cm2, close to the theoretical value 4.6 ×1014atoms/cm2in the 1/3 of a monolayer of Bi on Ag(111). In the LSSE experiments on the Bi(0.18 nm)/Ag(3 nm)/YIG slab, as shown in Fig. 4(f), the thermovoltages are negligibly small.SPIN-TO-CHARGE CONVERSION IN Bi/Cu Besides the Bi/Ag system, very large values of the Rashba coef- ficient have also been reported at the interface of Bi with other non- magnetic materials by ARPES measurements, such as the Bi/Cu(111) surface alloy system with αR=0.82 eV Å,49which may lead to a large IREE in the Bi/Cu bilayer systems. In our experiments, we fabricated Bi/Cu bilayers by low temperature sputtering on the YIG substrates and conducted the LSSE experiments. As shown in Fig. 5(a), the FIG. 5. Transverse thermovoltages as a function of the in-plane applied magnetic field measured in (a) Bi(8 nm)/Cu(2 nm)/YIG/GGG and Bi(15 nm)/Cu(2 nm)/YIG/GGG and (b) Bi(15 nm)/Cu(3 nm)/YIG slab and Bi(15 nm)/Cu(3 nm)/MgO(3 nm)/YIG slab. APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 9, 050904-8 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm transverse thermovoltages in Bi(8, 15 nm)/Cu(2 nm)/YIG/GGG are all linearly dependent on the applied magnetic field and with no cor- relation to the magnetization of YIG from the ordinary Nernst effect in Bi and not from the pure spin current. Besides, we find the the transverse thermovoltages in the Bi(15 nm)/Cu(3 nm)/YIG slab and in Bi(15 nm)/Cu(3 nm)/MgO(3 nm)/YIG slab are almost the same [Fig. 5(b)], which indicates conclusively that the thermovoltages in Bi/Cu/YIG are not from the spin current. Very large spin-to-charge conversion effects have been reported for various layers with Bi, including Bi, Bi/Ag, and Bi/Cu by spin pumping experiments using ferromagnetic metals as spin injec- tors. We have investigated Bi, Bi/Ag, Bi/Cu, and additional struc- tures using LSSE with a ferromagnetic insulator YIG. We include only voltages that follow jC=(2e/̵h)θSHjS×σas due to spin-to- charge conversion with voltage V ISHE shown in Fig. 6. We have found negligible V ISHE in Bi, Bi/Ag, and Bi/Cu despite substantial voltage due to the Nernst effect in Bi. Although the injected spin current undergoes negligible spin-to-charge conversion within Bi, it can readily be detected afterward by Pt or W, resulting in a large VISHE of positive or negative value, respectively. ANISOTROPIC SPIN CURRENT EFFECTS Dependence of spin–charge conversion on the crystalline ori- entation of heavy metals has attracted much interest in recent years. By using ST-FMR in epitaxial Pt(6 nm)/Fe(6 nm)/MgO[001] grown by MBE, θSHin Pt is found to be the same along different in-plane crystalline directions.50In contrast, the damping-like spin–orbit torque is 1.3 times smaller in the epitaxial Co/Pt(111) bilayers com- pared to the polycrystalline films, whereas the field-like spin–orbit torque values are of comparable magnitude.51The spin Hall magne- toresistance (SMR) of epitaxial Pt/Co bilayers on MgO(110) single crystal substrates is strongly anisotropic and depends on the applied FIG. 6. The extracted inverse spin Hall voltage V ISHEfrom various metallic layer structures of Pt, Pt/Bi, Pt/Bi/Ag, Bi, Bi/Ag, Ag/Bi, Bi/Cu, W/Bi, and W on YIG using the longitudinal spin Seebeck effect spin injection scheme.current direction with respect to the two primary in-plane crystal directions [001] and [1 10] in the Pt layer, suggesting the invariance of the spin diffusion length and an anisotropic Rashba–Edelstein effect.52Very recently, it has been found that the spin Hall angle and the damping-like torque in epitaxial Pt(110) grown on MgO(110) single-crystal substrates are 20% larger when current is applied along the [001] crystallographic direction as compared to [1 10], a difference attributed to the bulk contributions of the SHE in the Pt layer through its anisotropic resistance in this specific orientation.53 Bi is one of the most unusual materials, a semimetal with unique transport properties ranging from very small effective mass to extremely anisotropic Fermi surfaces of both electrons and holes. As such, the spin current effects are likely to be highly anisotropic with marked differences for thin films of different orientations. In Table I, we note the orientations of the Bi layers in various reports if available. To date, spin current effects have mostly been observed in Bi(111) and Bi(110) without notable differences from those of polycrystalline Bi. However, it may be beneficial to fabri- cate Bi of other orientations to reveal anisotropic spin current effects, if any. DISCREPANCIES BETWEEN MAGNETIC METALS AND MAGNETIC INSULATORS AS SPIN INJECTORS SP and LSSE can inject spin current from a FM material into Bi, in which the ISHE gives rise to a voltage. Numerous SP and THz measurements in Bi and related materials have reported very large voltages and concluded very substantial spin-to-charge conversion. In contrast, the LSSE measurements on Bi have also observed size- able voltage but concluded that they are notrelated to spin current. Instead, there is negligible spin-to-charge conversion in Bi. We stress that only the voltage that adheres to jC=(2e/̵h)θSHjS×σis due to spin-to-charge conversion. In all the LSSE experiments on Bi layers, large voltages have been detected but fail to survive two stringent tests for spin currents: the detected voltage must display a magnetic field dependence as that of the spin index σ, the same as that of the magnetization of the FM injector, and the detected voltage must vanish when the spin current jShas been deliberately blocked. The voltages observed in LSSE experiments in Bi are not related to spin current but due to the large Nernst effect inherent to Bi. Spin-to- charge conversion in Bi and Bi/Ag has been concluded to be the strongest by SP using FM metals and yet negligibly small by LSSE using magnetic insulators. SP and LSSE are two most widely used methods to inject a spin current from FM materials, either a metal or insulator, into a metal where a resultant voltage from ISHE is detected. How- ever, in addition to spin current phenomena, SP and LSSE are susceptible to other unintended parasitic effects that also gener- ate voltages. These extra voltages, unrelated to spin-to-charge con- version, must be excluded. In LSSE, the most notable parasitic effect is the anomalous Nernst effect but only in FM metals. Con- sequently, the LSSE measurements have avoided FM metals alto- gether and use exclusively magnetic insulators, such as YIG. Fur- ther scrutiny is still required to assure that the detected voltages are those due to spin currents and not from other parasitic effects, such as the Nernst effect, which is usually negligible but extremely large in Bi. APL Mater. 9, 050904 (2021); doi: 10.1063/5.0048042 9, 050904-9 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm In contrast, in both SP and ST-FMR measurements, FM metals have been the most common spin current injectors. FMR not only injects spin current but also generates a slew of parasitic effects.35,36 When an FM metal (Py) is replaced by an insulator (YIG), SP exper- iment in Bi/YIG shows voltages that are orders of magnitude smaller than those in Bi/Py. It is astonishing that they could be so dia- metrically different in Bi materials. When FM metals are used as spin injectors in SP, the parasitic effects are likely to be resistiv- ity dependent, more acute in spin-to-charge detecting layers with high resistivities. The resistivity of Ag thin films (6–10 μΩcm) is far smaller than that in Bi thin films (100–200 μΩcm), which is a semimetal. The microwave induced charge current in NiFe/Ag/Bi can be larger than that of NiFe/Bi.45The rectification effects may be further enhanced due to the different Schottky barriers at the Ag/Bi and the NiFe/Bi interfaces.45The microwave absorption may increase due to the impedance change with the insertion of the Ag layer.45These are likely origins of the greatly enhanced voltages in NiFe/Ag/Bi. Since the early experiments of SHE, although some experi- ments have included p-orbital based semiconductors, semimetals, and topological insulators, to date, most accepted materials with strong spin-to-charge capabilities are based on elements with delec- trons, 5 d(e.g., Pt), 4 d, and some 3 d(e.g., Cr). Elements with p electrons have not been well established, and in the case of Bi, it is controversial. SUMMARY SP experiments using FM metals have reported very large spin current effects in Bi, Bi/Ag, and Bi/Cu, far larger than those in Pt and W, the heavy metals with some of the largest efficiency of spin-to-charge conversion. The voltage signals, believed to be due to spin current, have been so large that other mechanisms, such as the IREE, need to be invoked. We used LSSE, another established spin injection method, in conjunction with the ferrimagnetic insu- lator YIG and observed sizeable voltages but negligibly small spin- to-charge conversion in Bi and Bi/Ag, and Bi/Cu, similar to those in metals Cu and Ag. We stress that only the voltage that adheres to jC=(2e/̵h)θSHjS×σis due to spin-to-charge conversion. We show that pure spin current has been injected from YIG into the Bi layer, which accommodates negligible spin–charge conversion. While SP with FM metals reveals Bi/Ag as one of the largest, the LSSE effect with insulating YIG shows one of the smallest, spin–charge con- version effect. FM metals with various charge and spin-polarized current effects generate a slew of parasitic effects in SP and ST-FMR that may complicate the extraction of those from pure spin currents. Magnetic insulators, without the complexities of charge carriers and their parasitic effect, are advantageous for assessing spin-charge conversion in various materials. ACKNOWLEDGMENTS The work at Johns Hopkins University was supported by the U.S. Department of Energy, Basic Energy Science (Award No. DE- SC0009390). W.L. was supported, in part, by U.S. National Science Foundation DMREF (Grant No. 1729555). D.Y. was supported, in part, by the National Nature Science Foundation of China (Grant No. 12004075).DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. 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5.0051816.pdf

J. Appl. Phys. 129, 223907 (2021); https://doi.org/10.1063/5.0051816 129, 223907 © 2021 Author(s).Computational investigation of half-Heusler/ MgO magnetic tunnel junctions with (001) orientation Cite as: J. Appl. Phys. 129, 223907 (2021); https://doi.org/10.1063/5.0051816 Submitted: 28 March 2021 . Accepted: 24 May 2021 . Published Online: 10 June 2021 Jianhua Ma , Yunkun Xie , Kamaram Munira , Avik W. Ghosh , and William H. Butler ARTICLES YOU MAY BE INTERESTED IN Optical and electronic properties of SiTe x (x = 1, 2) from first-principles Journal of Applied Physics 129, 224305 (2021); https://doi.org/10.1063/5.0054391 On the temperature-dependent characteristics of perpendicular shape anisotropy-spin transfer torque-magnetic random access memories Journal of Applied Physics 129, 223903 (2021); https://doi.org/10.1063/5.0054356 Optical phonon modes, static and high-frequency dielectric constants, and effective electron mass parameter in cubic In 2O3 Journal of Applied Physics 129, 225102 (2021); https://doi.org/10.1063/5.0052848Computational investigation of half-Heusler/MgO magnetic tunnel junctions with (001) orientation Cite as: J. Appl. Phys. 129, 223907 (2021); doi: 10.1063/5.0051816 View Online Export Citation CrossMar k Submitted: 28 March 2021 · Accepted: 24 May 2021 · Published Online: 10 June 2021 Jianhua Ma,1,a) Yunkun Xie,1Kamaram Munira,2Avik W. Ghosh,1,3,b)and William H. Butler2,4,c) AFFILIATIONS 1Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, Virginia 22904, USA 2Center for Materials for Information Technology, University of Alabama, Tuscaloosa, Alabama 35401, USA 3Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA 4Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35401, USA a)Author to whom correspondence should be addressed: jm9yq@virginia.edu b)Electronic mail: ag7rq@virginia.edu c)Electronic mail: wbutler@mint.ua.edu ABSTRACT A series of half-metallic XYZ half-Heusler alloys is combined with MgO to create Heusler –MgO junctions. The electronic and magnetic properties of these junctions are investigated. The strong oxidation between metal and oxygen atoms causes the systems with pure YY inter-faces to be the most stable cases. We conclude that uniaxial anisotropy can be induced in Heusler layers adjacent to MgO. The type of inter- face layers determines the half-metallicity and anisotropy (in-plane or perpendicular) in the Heusler –MgO junctions. The capacity to retain both half-metallicity and perpendicular magnetic anisotropy in NiMnSb/MgO and CoTiSn/MgO junctions with a MnMn interface layermakes these structures potential candidates as electrode layers in spin transfer torque random access memory devices. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0051816 I. INTRODUCTION Spin transfer torque random access memory (STT-RAM) has attracted a great deal of attention and has now been commercial-ized based on its potential to combine the speed of static RAM (SRAM), the density of dynamic RAM (DRAM), and the nonvo- latility of flash while providing excellent scalability and outstand-ing endurance. 1The magnetic tunneling junction (MTJ) inside an STT-RAM cell stores information in the relative magnetic orienta-tions of the electrode layers, with STT switching resulting in mag- netization reversal of a softer free layer. In order to fabricate a practical STT-RAM, a robust magnetic tunnel junction (MTJ)with optimized and matched interfaces between barriers and elec-trodes is necessary. 2This requires a magnetic material with high spin polarization, low saturation magnetization, low magnetic damping, and perpendicular magnetic anisotropy (PMA).3If the materials are compatible with MgO at crystal interfaces, thatwould make them particularly promising for spintronic applica-tions due to the known ability of MgO to symmetry filter the various tunneling bands.Half-metals (HMs) have been widely studied as one of the most promising electrode materials in the field of spintronics. These materials have a gap at the Fermi level in one spin channel for the bulk while showing metallic behavior in another spin channel. The resultant highly spin-polarized current is critical to a low energyread –write current drive. The first half-metal, the half-Heusler alloy NiMnSb, was recognized in a calculation by de Groot and collabora-tors in 1983. 4Since then, a series of half-metal and near-half-metal Heusler alloys has been identified theoretically.5–10In order to do a systematic search within the Heusler alloy family, we created aHeusler alloy database including electronic structures, magnetism, and structure stability studies of 378 half-Heusler alloys. 11From the half-Heusler database, we have identified 45 half-metals and 34near-half-metals with negative formation energy that follow theSlater –Pauling rule of three electrons per atom. Unfortunately, most of the above half-metals and near-half-metals do not have compati-ble lattice constants with MgO. Furthermore, their C1 bcubic sym- metry prevents any uniaxial magnetocrystalline anisotropy (MCA)needed for data retention as a magnetic layer.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223907 (2021); doi: 10.1063/5.0051816 129, 223907-1 Published under an exclusive license by AIP PublishingIn this paper, we identify six half-metals and near-half-metals with less than 1% lattice mismatch with MgO, and we calculate first principles electronic bands of their heterostructures. Section II describes the supercell structures simulated, while Sec. IIIpresents the electronic structure results of various Heusler –MgO systems. We show that a large ( .105) ballistic tunnel magnetoresistance (TMR) is possible in these systems as the half-metallicity persists with MgO. II. STRUCTURE AND COMPUTATIONAL METHOD Calculations were performed for periodic Heusler –MgO junc- tion superlattices consisting of five layers of MgO (B1 unit cell,lattice constant a¼5:95 Å) and five layers of the half-Heusler alloys using density-functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP) 12and using a plane wave basis set and the projector augmented wave (PAW) method.13 A uniform cut-off energy of 520 eV was implemented for all calcula-tions. The Perdew –Burke –Ernzerhof (PBE) version of the general- ized gradient approximation (GGA) to the exchange-correlation functional of DFT was adopted. 14In order to take into account all degrees of freedom for the Heusler –MgO interfaces, we performed full structural optimizations for the superlattice parameters and internal coordinates. The convergence criterion of the change in total energy was set to 10/C05eV. We took a 12 /C212/C22k-point mesh for the (100) transverse interface between the alloy and MgO. The half-Heusler alloy of the form XYZ crystallizes in the face-centered cubic C1 bstructure ( F/C2243 m) with one formula unit per unit cell. It consists of four sublattices: X at the (1 4,14,14), Y at the (1 2,12,12), Z at the (0,0,0), and the vacancies at the (34,34,34) sites. The six half-Heusler alloys we studied here are CrScAs, MnVSb, FeTiSb, CoTiSn, NiTiIn, and NiMnSb. To the best of our knowledge, noneof these have been synthesized except for NiMnSb. 15The calculated lattice constant of NiMnSb (5.91 Å) is within 1% of the experimen- tal result (5.92 Å).15The [100] direction of these alloys and the [110] direction of MgO present a nearly perfect lattice match. We studied four Heusler terminations as presented in Fig. 1 : (a) X (three layers of X and two layers of YZ in total across the Heusler slab), (b) XX (two layers of XX, one layer of X, and two layers of YZ in total), (c) YZ (two layers of X and three layers of YZin total), and (d) YY (two layers of YY, one layer of YZ, and twolayers of X in total) terminals. The XX termination means that thevacancies in the bulk are occupied by X atoms at the interfaces, but the X layers inside the bulk still retain their vacancies. The YY termi- nations mean that the Z atoms in the YZ termination are replacedby Y atoms. More structural details are discussed in Sec. III. III. RESULTS AND DISCUSSION We performed full ionic relaxations for the six Heusler/MgO supercells to obtain their equilibrium structures. There are four ter- minations for each Heusler/MgO supercell as shown in Fig. 1 . The bond types and lengths at the different interfaces are listed inTable I . For example, the Cr/OMg interface is composed of a layer containing only Cr atoms and a layer containing both O and Mg atoms with direct Cr –O bonds. We notice that the bond length is the shortest at the X/OMg-terminated interfaces in all sixHeusler –MgO junctions, while the optimized Y/Z –O bonds are typically the longest at the YZ/OMg interfaces. We suppose thatthe longer bond results from the bigger atomic size of Ti, Sc, Sb,and Sn. We also analyzed the bonding strength of the interfaces by calculating their binding energies W.Wis the binding energy of the two interfaces on both sides of the Heusler layer, W¼E XYZ =MgO/C0EXYZ/C0EMgO, (1) where EXYZ =MgOis the total energy of the optimized Heusler –MgO junctions, while EXYZ(EMgO) is the total energy of the Heusler (MgO) slab surrounded by vacuum in a supercell. The lattice parameters of the slab supercells are taken to be the same as those FIG. 1. Side views of the superlattices of the half-Heusler(110)/MgO junctions: (a) X/OMg-terminated interface, (b) XX/OMg-terminated interface, (c) YZ/OMg-terminated interface, and (d) YY/OMg-terminated interface. Color note: X (red), Y (blue), Z (green), Mg (orange), and O (pink).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223907 (2021); doi: 10.1063/5.0051816 129, 223907-2 Published under an exclusive license by AIP Publishingof the optimized Heusler –MgO junctions, and no further relaxation is performed for the slab supercells. We need to point out that relative stability of the interfaces of MTJs with Heusler electrodescould also be studied employing more elaborated ab initio atom- istic thermodynamics. 16,17These calculations can be considered in the future work. We found that the YY/OMg interfaces present the lowest binding energies for most Heusler –MgO junctions except CrScAs – MgO. The mechanism behind the binding energies is complicatedfor different interface termination. First, the X-rich (XX) interface has more X –O bonds than the X interface with vacancies per unit cell; therefore, the X-rich interface has lower binding energy thanthe X interface. Second, the Y element is the most electropositiveelement in XYZ; therefore, we propose that the oxidation andbonding between Y –O atoms is stronger than X –O atoms. Previous investigation also predicted that full-Heusler alloys tend to stabilize in pure XX or YY layers 18due to strong oxidation. Third, the bigger atomic size at the interface may increase the bond length.The size of Sc is the largest among the Y elements in this work;therefore, the bond length (2.32 Å) at the Sc-rich (ScSc) interface is larger than the Cr-rich (CrCr) interface (2.10 Å). The Cr-rich inter- face has lower binding energy than the Sc-rich interface. Table II lists the geometric and magnetic properties of Heusler –MgO junctions. a ris the lattice parameter obtained from the layer parallel with the interface after relaxation, while cris the average lattice parameter of the four “internal ”spacings in fiveHeusler layers. If the cr/arvalue is larger than 1.01, we interpret that this Heusler layer has been distorted to a tetragonal structure. The in-plane lattice variation is calculated compared to the bulklattice constants. The largest variation is 3.8% for the ScSc interface.The possible reason for a large variation for this case is that thebigger atomic size of Sc distorts the interface layer more seriously. However, most of the in-plane lattice variation is less than 1%. As the thickness of Heusler layers increases, the in-plane lattice varia-tion and distortion will be reduced. Table II also includes the saturation magnetization ( M S) and the induced anisotropy on one interface side ( ?K) . The induced anisotropy is calculated by comparing the differences between ground state energies, whose magnetic moments are aligned along(001) and (100) directions. A positive anisotropy Ksuggests that the magnetization prefers to be perpendicular to the Heusler –MgO interfacial plane, while a negative one implies that the magnetization tends to lie in-plane. We also consider the competition between induced anisotropy and shape anisotropy because shape anisotropyin thin films usually dominates over magnetocrystalline anisotropy.We define the critical thickness tas the threshold value for induced anisotropy to overcome shape anisotropy when thickness is less than t.T h e tis introduced in a shape anisotropy equation, E induced ¼Eshape¼1 2tμ0M2 S, (2) where Einduced andEshape are the induced anisotropy energy and the shape anisotropy energy through the unit area in-plane on one inter- face side, respectively. The critical thickness is listed in Table II .I f the thickness crof the Heusler layers is less than critical thickness t, the induced anisotropy can overcome the in-plane shape anisotropy. Although the uniaxial anisotropy is induced by the tetragonal distortion in the Heusler layers, its value and easy axis direction are not determined by cr/ar. The value of uniaxial anisotropy is signifi- cantly conditioned by the atoms at the Heusler layer interface. Forexample, for TiSb interface layers at the FeTiSb/MgO junction, the anisotropy is in-plane, while Fe and FeFe interface layers generate sig- nificant perpendicular anisotropy. Also, if the interface layer thicknessc ris less than the corresponding critical thickness t, its induced per- pendicular anisotropy dominates over the shape anisotropy. The final column in Table II is whether the supercell still retains half-metallicity (HM) or near-half-metallicity (NHM). From the final column, it is obvious that locally modified stoichi-ometry at the interface changes the metallicity. For example, theVV interface is near-half-metallic for the MnVSb/MgO structure,while the VSb interface is metallic. We confirm that all the layers including the middle VSb layer in the VSb termination structure are metallic. Therefore, combination of the X- or Y-rich interfaceterminations may drastically change its physical properties at thecenter of the Heusler layer. As discussed before, an ideal Heusler magnetic electrode needs to have not only enough PMA but also adequate half- metallicity of the Heusler layers. The latter attribute is very difficultto retain in supercells and is in fact governed by the interface type.Interestingly, we find that only systems with YY interface layers sometimes remain half-metallic or near-half-metallic, as seen in Figs. 2 and 3, while all other interfaces lose their half-metallicity.TABLE I. Bond lengths and binding energy Wfor the half-Heusler(100)/MgO interfaces. Interface terminal Bond type Bond length (Å) W(eV) Cr Cr –O 2.06 −2.69 CrCr Cr –O 2.10 −3.83 ScAs Sc –O/As –O 2.30/3.11 −1.74 ScSc Sc –O 2.32 −3.74 Mn Mn –O 2.03 −2.34 MnMn Mn –O 2.13 −2.92 VSb V –O/Sb –O 2.32/2.88 −1.51 VV V –O 2.10 −4.26 Fe Fe –O 1.98 −2.47 FeFe Fe –O 2.05 −3.99 TiSb Ti –O/Sb –O 2.34/2.92 −1.64 TiTi Ti –O 2.18 −4.35 Co Co –O 1.96 −2.47 CoCo Co –O 2.02 −4.21 TiSn Ti –O/Sn –O 2.29/2.74 −2.00 TiTi Ti –O 2.18 −4.23 Ni Ni –O 2.00 −2.05 NiNi Ni –O 2.07 −3.21 TiIn Ti –O/In –O 2.26/2.57 −2.39 TiTi Ti –O 2.18 −4.35 Ni Ni –O 2.01 −2.06 NiNi Ni –O 2.08 −3.22 MnSb Mn –O/Sb –O 2.46/3.01 −1.84 MnMn Mn –O 2.18 −3.65Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223907 (2021); doi: 10.1063/5.0051816 129, 223907-3 Published under an exclusive license by AIP PublishingOne might argue that the in-plane distortion of aralso influences half-metallicity so that the MnVSb/MgO junction with the VVinterface layer resembles the bulk system with near-half-metallicity,while other systems become metallic. However, in the case of NiMnSb, the amount of in-plane distortion is the same for both NiNi and MnMn interface layers, but the Heusler layer loses itshalf-metallicity only for the NiNi interface layer. It has been veri-fied experimentally by spin-resolved x-ray photoemission spectro-scopy (SR-XPS) of an NiMnSb/MgO(100) junction that an Mn interface layer can be achieved, and some oxidization of Mn exists when an MgO(001) epitaxial barrier is grown on top of NiMnSbusing molecular beam epitaxy (MBE). 19,20However, the half- metallicity and the high TMR ratio of the NiMnSb –MgO junction have not yet been observed in experiments due to a decrease of the Mn magnetization at the interface with MgO.19,21Much like anisot- ropy, the interface layer seems to play a dominant role in determin-ing half-metallicity over in-plane distortion. We also point out thatalthough induced PMA is predicted for an NiMnSb/MgO junction with a pure MnMn interface, the large saturation magnetization M Sof NiMnSb will force the magnetization to lie in-plane becauseits thickness cr(5.56 Å) is very close to its corresponding critical thickness t(3.68 Å). This is in fact the same as other near-half-metallic Heusler –MgO junctions except CoTiSn. IV. BALLISTIC I–VCHARACTERISTICS OF HALF-HEUSLER –MgO MTJs While the tunnel magnetoresistance (TMR) of the superlattice will depend on defect states, spin depolarization, and incoherent scattering at room temperature, such effects tend to be geometry specific and hard to predict. Accordingly, a computational measureof the impact of half-metallicity is the ballistic current, whichwould at the least include the impact of a finite bias and thevarious effective masses across the tunnel junction. The transport calculations are performed in SMEAGOL, 22which employs the nonequilibrium Green ’s function (NEGF) method combined with density-functional theory (DFT) in the SIESTA package23on a pseudo-atomic orbital basis set.24Double- ζ25–27is used for all orbitals, and the generalized gradient approximation (GGA) with Perdew –Burke –Ernzerhof (PBE) functional is adopted throughoutTABLE II. Summary of the geometric and magnetic properties of various half-Heusler –MgO junctions. We list the calculated lattice constant in bulk a, interface type, relaxed lattice constants arandcr, relaxed ratio between cr/ar, saturation magnetization MS, induced anisotropy ( ⊥K), critical thickness t, and half-metallicity (bulk CrScAs is in the tet- ragonal phase and it has two lattice constants in the second column). XYZa(Å) in bulkInterface (Å)Relaxed ar,cr(Å)% change inarfor HeuslerRelaxed cr/arMS (emu/cm3)⊥K×1 06 (erg/cm3)Critical thickness, t(Å) Half-metallic? CrScAs (HM) 5.92, 6.54 ScAs 5.87, 6.86 −0.79 1.1684 618 −0.89 No Cr 5.92, 6.92 0.01 1.1693 896 −2.40 No CrCr 6.03, 6.23 0.19 1.0428 747 −0.31 No ScSc 6.15, 6.08 3.8 0.9889 475 0.79 7.25 No MnVSb (HM) 5.92 VSb 5.90, 6.13 −0.27 1.0371 59 6.75 207.55 No Mn 5.96, 6.03 0.64 1.0116 372 −1.21 No MnMn 6.00, 6.08 1.30 1.0156 737 0.66 20.14 No VV 5.92, 5.52 0 0.9345 381 −1.18 NHM FeTiSb (NHM) 5.95 TiSb 5.89, 5.93 −0.94 1.0063 72 −1.68 No Fe 5.92, 5.96 −0.44 1.0055 208 5.27 177.56 No FeFe 5.98, 5.87 0.45 0.9882 421 5.09 168.90 No TiTi 5.98, 5.64 0.55 0.9431 0 0 No CoTiSn (NHM) 5.93 TiSn 5.91, 5.93 −0.30 1.0028 0 0 No Co 5.90, 5.93 −0.43 1.0049 14 1.19 38.44 No CoCo 5.96, 5.80 0.51 0.9805 0.90 −0.00017 No TiTi 5.98, 5.66 0.89 0.9461 160 1.28 39.64 NHM NiTiIn (NHM) 5.99 TiIn 5.97, 5.88 −0.26 0.9846 70 0.238 7.53 No Ni 5.92, 6.09 −1.19 1.0295 0.01 0 No NiNi 5.94, 6.16 −0.85 1.0375 13 −1.79 No TiTi 6.00, 5.74 0.17 0.96 248 0.056 1.71 NHM NiMnSb (HM) 5.91 MnSb 5.89, 5.76 −0.42 0.9802 1130 8.10 14.40 No Ni 5.93, 5.78 0.27 0.9757 811 0.018 0.03 No NiNi 5.96, 5.83 0.79 0.9791 796 −2.83 No MnMn 5.96, 5.56 0.79 0.9327 1881 2.15 3.68 HMJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223907 (2021); doi: 10.1063/5.0051816 129, 223907-4 Published under an exclusive license by AIP Publishingall calculations. The electronic temperature is set at 300 K. The spin-dependent current ( σ¼",#) is calculated from Iσ(V)¼e hð Tσ(E;V)(fL/C0fR)dE, (3) Tσ(E;V)¼TrΓσ LGσΓσRGσy/C2/C3 , (4) where Tσ(E;V) is the bias-dependent transmission and fL,Rare the Fermi –Dirac distribution for the left/right Heusler contact, respec- tively. The transmission is calculated using NEGF,28,29where Gσisthe retarded Green ’sf u n c t i o na n d Γσ L,Rare energy-dependent broad- ening matrices obtained from the anti-Hermitian parts of the recur- sively computed22left/right contact self-energies. At equilibrium, a 5/C25k?-point mesh is used to converge the charge density, and then a2 0/C220k?-point mesh is used to calculate the spin-dependent current. At a finite bias, a constant potential is added to the Heusler contacts and a linear Laplace potential is added in the MgO. This approximation has been verified to be appropriate for magnetictunnel junctions where the voltage drops mostly in the high resistantMgO. 30Such an approximation considerably reduces the computa- tional cost for self-consistent calculations in SMEAGOL. Figure 4 shows the ballistic I–Vcharacteristics of the six MTJs. Among them, NiMnSb shows higher current density ( /difference108A=cm2) than the other ( /difference105A=cm2). The reason for the big difference in current amplitude is because for NiMnSb, the Fermi energy in theconducting spin channel lies in the middle of the s band, which has the lowest decay rate inside MgO due to the same orbital sym- metry as the Δ 1band in MgO. Other systems are symmetry fil- tered as their conducting spin channel Fermi energies lie in thenon-Δ 1symmetry bands,31potentially due to the different work functions of the electrode materials. Therefore, the energy barrier heights between MgO and the electrodes are different, resulting in different tunneling probabilities. MTJs are widely used in read units where the tunnel magneto- resistance (TMR) ratio is a key metric. The TMR can be calculated from the currents in parallel and anti-parallel configuration, TMR¼Ip/C0Iap Iap: (5) Figure 5 shows the TMR ratio as a function of the voltage for the six MTJs. Out of the six junctions, four of them show high TMR ata low bias. It is worth mentioning that the specific value of theTMR ratio for a single junction is not important because ourcurrent calculations assume full b allistic transport without any FIG. 2. Local density of states (DOS) for an MnVSb –MgO periodic supercell with a VV/OMg interface. The supercell stays NHM (the bandgap is opened in the spin-down channel). The maximum valence band edge is /C00:1549 eV , and the minimum conduction band edge is /C00:0349 eV . The bandgap is 0.12 eV . The bandgaps in all the layers are the same. FIG. 3. Local density of states for a NiMnSb –MgO periodic supercell with an MnMn/OMg interface. The supercell stays HM (the bandgap is opened in the spin-down channel). The maximum valence band edge is /C00:0927 eV , and the minimum conduction band edge is 0.1773 eV . The bandgap is 0.27 eV . Thebandgaps in all the layers are the same. FIG. 4. I–Vs of the half-Heusler/MgO/half-Heusler junction.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223907 (2021); doi: 10.1063/5.0051816 129, 223907-5 Published under an exclusive license by AIP Publishingspin scattering mechanism. Since the denominator Iapis a very small number, any spin scattering can alter the TMR greatly.Depending on the specific mechanism, the TMR can vary with tem-perature 32or magnetic impurity concentration. Still, comparing theballistic TMRs across different junctions provides a convenient reference point for which electrode materials might be better suited to low energy read operation in MgO-based MTJs. Some of the systems show a negative differential resistance (NDR) at a moderate bias. Such an NDR is not hard to see in ide-alized geometries with sharp features in their density of states (e.g., Fig. 9 in Ref. 33). We use CoTiSn to illustrate the underlying mechanism. NDR here arises due to a competition between anever-expanding Fermi window under a bias vs the progressivealignment then misalignment between sharp peaks in the contactdensities of states sweeping past each other. The increasing Fermi window dominates the increase of current at a low bias, between 0 and 0 :22 V. After that, the transmission starts to shrink with growing misalignment between contact densities of states awayfrom zero bias. In the bottom plot of Fig. 6 ,w ec l e a r l ys e et h a t within 0.22 –0.35 V, the product of the contact DOS in the bias window keeps dropping for both spin channels. The competing effects lead to the observed NDR in the I –V. In reality, the NDR is probably hard to observe as the sharp DOS features get washedout by non-idealities, defects, and imperfections. V. CONCLUSION AND FUTURE WORK In this paper, we establish that it is possible to obtain uniaxial perpendicular magnetic anisotropy (PMA) and to retain half- metallicity (HM) in a Heusler –MgO junction, leading to an enor- mous ballistic finite temperature TMR between 10 3and 105at a low bias. Our results motivate further search and discovery ofhalf-Heusler alloys as potential electrode materials in STT-RAMdevices. Further theoretical works including chemical and group theoretical analyses are needed to explain the observed relation between the interface layer vs PMA and HM. ACKNOWLEDGMENTS The authors acknowledge funding support from the National Science Foundation (NSF) DMREF-1235230 and NSF-SHF-1514219.The authors also acknowledge Advanced Research ComputingServices at the University of Virginia and high-performance comput-ing staff from the Center for Materials for Information Technology at the University of Alabama for providing technical support that has contributed to the results in this paper. The computational work wasdone using the high-performance computing cluster at the Center forMaterials for Information Technology, University of Alabama, andthe Rivanna high-performance cluster at the University of Virginia. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J.-G. Zhu, Proc. IEEE 96, 1786 (2008). 2K. Munira, W. H. Butler, and A. W. Ghosh, IEEE Trans. Electron Dev. 59, 2221 (2012). 3E. Chen, D. Apalkov, Z. Diao, A. Driskill-Smith, D. Druist, D. Lottis, V. Nikitin, X. Tang, S. Watts, S. Wang, S. Wolf, A. Ghosh, J. Lu, S. Poon, FIG. 5. The tunnel magnetoresistance of half-Heusler/MgO magnetic tunnel junctions. FIG. 6. (a)I–Vof CoTiSn/MgO/CoTiSn in a parallel configuration. (b) The product of the DOS from left and right electrodes at a finite bias.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223907 (2021); doi: 10.1063/5.0051816 129, 223907-6 Published under an exclusive license by AIP PublishingM. Stan, W. Butler, S. Gupta, C. Mewes, T. Mewes, and P. Visscher, IEEE Trans. Magn. 46, 1873 (2010). 4R. A. de Groot, F. M. Mueller, P. G. v. Engen, and K. H. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983). 5I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 134428 (2002). 6I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 174429 (2002). 7H. C. Kandpal, C. Felser, and R. Seshadri, J. Phys. D: Appl. Phys. 39, 776 (2006). 8I. Galanakis, P. Mavropoulos, and P. H. Dederichs, J. Phys. D: Appl. Phys. 39, 765 (2006). 9I. Galanakis, J. Phys.: Condens. Matter 16, 3089 (2004). 10S. Skaftouros, K. Özdo ğan, E. Şaşıoğlu, and I. Galanakis, Phys. Rev. B 87, 024420 (2013). 11W. H. Butler, A. W. Ghosh et al. , see http://heusleralloys.mint.ua.edu/ for “Heuslers Home ”(2016). 12G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996). 13P. E. Blöchl, O. Jepsen, and O. K. Andersen, Phys. Rev. B 49, 16223 (1994). 14J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 15K. Buschow, P. van Engen, and R. Jongebreur, J. Magn. Magn. Mater. 38, 1 (1983). 16B. Pradines, L. Calmels, and R. Arras, Phys. Rev. Appl. 15, 034009 (2021). 17B. Hülsen, M. Scheffler, and P. Kratzer, Phys. Rev. Lett. 103, 046802 (2009). 18K. Munira, J. Romero, and W. H. Butler, J. Appl. Phys. 115, 17B731 (2014). 19M. Sicot, P. Turban, S. Andrieu, A. Tagliaferri, C. D. Nadai, N. Brookes, F. Bertran, and F. Fortuna, J. Magn. Magn. Mater. 303, 54 (2006).20P. Turban, S. Andrieu, B. Kierren, E. Snoeck, C. Teodorescu, and A. Traverse, Phys. Rev. B 65, 134417 (2002). 21P. Turban, S. Andrieu, E. Snoeck, B. Kierren, and C. Teodorescu, J. Magn. Magn. Mater. 240, 427 (2002). 22A. R. Rocha, V. M. García-Suárez, S. Bailey, C. Lambert, J. Ferrer, and S. Sanvito, Phys. Rev. B 73, 085414 (2006). 23J. Soler, E. Artacho, J. Gale, A. García, J. Junquera, P. Ordejón, and D. Sanchez-Portal, J. Phys.: Condens. Matter 14, 2745 (2002). 24J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón, and D. Sánchez-Portal, J. Phys.: Condens. Matter 14, 2745 (2002). 25S. Huzinaga, Comput. Phys. Rep. 2, 281 (1985). 26R. Poirier, R. Kari, and I. G. Csizmadia, Handbook of Gaussian Basis Sets (Netherlands, Elsevier, 1985). 27J. Junquera, O. Paz, D. Sánchez-Portal, and E. Artacho, Phys. Rev. B 64, 235111 (2001). 28S. Datta, Quantum Transport: Atom to Transistor (Cambridge University Press, 2005). 29A. Ghosh, Nanoelectronics —A Molecular View (World Scientific, 2016). 30Y. Xie, I. Rungger, K. Munira, M. Stamenova, S. Sanvito, and A. W. Ghosh, Nanomagnetic and Spintronic Devices for Energy-Efficient Memory and Computing (John Wiley & Sons, Ltd, 2016), Vol. 91. 31W. H. Butler, Sci. Technol. Adv. Mater. 9(1), 014106 (2008). 32A. MacDonald, T. Jungwirth, and M. Kasner, Phys. Rev. Lett. 81, 705 (1998). 33P. Damle, A. Ghosh, and S. Datta, Chem. Phys. 281, 171 (2002).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 223907 (2021); doi: 10.1063/5.0051816 129, 223907-7 Published under an exclusive license by AIP Publishing

5.0048619.pdf

APL Mater. 9, 060901 (2021); https://doi.org/10.1063/5.0048619 9, 060901 © 2021 Author(s).Topological insulators for efficient spin– orbit torques Cite as: APL Mater. 9, 060901 (2021); https://doi.org/10.1063/5.0048619 Submitted: 24 February 2021 . Accepted: 13 May 2021 . Published Online: 25 May 2021 Jiahao Han , and Luqiao Liu COLLECTIONS Paper published as part of the special topic on Emerging Materials for Spin-Charge Interconversion ARTICLES YOU MAY BE INTERESTED IN Spin–orbit torque characterization in a nutshell APL Materials 9, 030902 (2021); https://doi.org/10.1063/5.0041123 Field-free magnetization switching induced by the unconventional spin–orbit torque from WTe 2 APL Materials 9, 051114 (2021); https://doi.org/10.1063/5.0048926 Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147APL Materials PERSPECTIVE scitation.org/journal/apm Topological insulators for efficient spin–orbit torques Cite as: APL Mater. 9, 060901 (2021); doi: 10.1063/5.0048619 Submitted: 24 February 2021 •Accepted: 13 May 2021 • Published Online: 25 May 2021 Jiahao Han and Luqiao Liua) AFFILIATIONS Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Note: This paper is part of the Special Topic on Emerging Materials for Spin-Charge Interconversion. a)Author to whom correspondence should be addressed: luqiao@mit.edu ABSTRACT Current-induced magnetic switching via spin–orbit torques has been extensively pursued for memory and logic applications with promising energy efficiency. Topological insulators are a group of materials with spin-momentum locked electronic states at the surface due to spin–orbit coupling, which can be harnessed to reach strong spin–orbit torques. In this paper, we summarize and compare the methods for calibrating the charge-spin conversion efficiency in topological insulators, with which topological insulators are identified as outstanding spin–orbit torque generators compared with the well-studied heavy metals. We then review the results of magnetic switching under reduced current density in topological insulator/ferromagnet heterostructures. Finally, we provide insights on current challenges as well as possible exploration directions in the emerging field of topological spintronics. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0048619 I. INTRODUCTION In the past decade of spintronic studies, current-induced mag- netic switching via spin–orbit torques (SOTs) has been extensively pursued as a promising technique for writing information in the magnetic random-access memory with low power consumption, unlimited endurance, and fast operation time.1The key to gener- ate strong SOTs is to find materials with large spin–orbit coupling that can convert charge current into non-equilibrium spins to inter- act with magnetic moments. Heavy metals such as Pt,2,3Ta,4and W5were among the first that have been utilized to generate non- equilibrium spins via the spin Hall effect or the Rashba–Edelstein effect. The spin Hall effect6converts a three-dimensional charge current density Jc(in the unit of A/m2) to a transverse spin cur- rent density (2e/̵h)Js(in the unit of A/m2;eis the electron charge and̵his the reduced Planck’s constant) with spin polarization σ. The spin current is expressed as (2e/̵h)Js=θSHσ×Jc, with θSHbeing the unitless spin Hall angle. Besides the spin Hall effect in the bulk, the Rashba–Edelstein effect of the two-dimensional electron states at the surface can be another mechanism to generate spin polarization.7The injected two-dimensional charge current density(jc, in the unit of A/m) can be converted to a three-dimensional spin current density (2e/̵h)Jsvia the Rashba–Edelstein-effect coeffi- cient ( q, in the unit of nm−1) expressed as8(2e/̵h)Js=qσ×jc. When neighbored by a ferromagnetic material with magnetic moment m, the non-equilibrium spins σcan exert SOTs to modulate the magnetic dynamics or even switch the magnetic orientation, which enables the core function of magnetic memory and logic devices. In general, the SOTs can be decomposed to a field-like term τFL ∼σ×mand a damping-like term τDL∼m×(σ×m), the latter of which usually accounts for magnetic switching.1It is worth not- ing that the generation of damping-like torques is not restricted to the categories of conventional spin-transfer torques or spin Hall effect-induced SOTs. The non-equilibrium spin σfrom both bulk and interfacial spin–orbit coupling can give rise to τDLwhen σ is absorbed by the adjacent magnetic material via spin flipping scattering.1,9,10 The recent discovery of topological insulators (TIs) provides tremendous opportunities to further improve the efficiency of charge-spin conversions. TIs are a class of materials with spin–orbit coupling that is strong enough to invert the sequence of bands at certain high symmetry points and leads to topologically protected APL Mater. 9, 060901 (2021); doi: 10.1063/5.0048619 9, 060901-1 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm conducing electronic states at the surfaces11–14[Fig. 1(a)]. The topo- logical surface states exhibit a helical Dirac cone character in the dispersion relation, where the spin orientation of electrons is locked with their wavevector [Figs. 1(b) and 1(c)]. When a charge current flows through the surface, which can be described in the wavevec- tor space as a shift in the electron distribution at the Fermi surface, the number of forward-going electrons dominates over backward- going ones, leading to a net spin accumulation because of the spin-momentum locking15,16[Fig. 1(d)]. Compared to heavy metals, TIs can, in principle, provide much higher non-equilibrium spin accumulation with the same amount of charge current. The bulk band topology in TIs gives rise to topological surface states with one-to-one locking for spin and momentum orientations.8With an applied charge current, this mechanism provides higher spin accumulation than the Rashba spin–orbit coupling that causes two surface bands with opposite spin-momentum locking chirality. Here, we note that because of the bulk–surface correspondence, the surface states of TI can also be viewed as a holographic manifestation of the bulk topology.17 We treat the contributions from the intrinsic topologically deter- mined bulk band as the same with the one from the topological surface state. Besides the surface spin-momentum locking, exotic FIG. 1. Topological surface states and spin-momentum locking in TIs. (a) Real space picture of the conducting surface states in an ideal TI. (b) Dirac cone of the topological surface states in the wavevector space, where the spin and momentum of electrons are one-to-one locked to each other at the Fermi level. (c) Angle- resolved photoemission spectrum that indicates the bulk and surface bands of a six-quintuple-layer-thick Bi 2Se3film. (d) Top view of the Dirac cone crossed by the Fermi level with spin-momentum locking. With a flow of charge current, the shift in the electron distribution in the wavevector space induces non-equilibrium spins. Panel (c) is adapted with permission from Zhang et al. , Nat. Phys. 6, 584 (2010). Copyright 2010 Springer Nature Limited.spin–orbit effects in TIs may lead to large spin accumulation from the bulk states18–20through extrinsic non-topological mechanisms as well. Thanks to the development of advanced material growth and characterization techniques in the past decades, various TI thin films have been synthesized and have been demonstrated to own non-trivial topological electronic structures via the angle-resolved photoemission spectroscopy (ARPES).21–23These efforts provide the foundation for exploiting TI thin films in spintronic devices as effi- cient SOT sources. So far, the most extensively studied group of TIs is the bismuth-based compounds including Bi xSb1−x, Bi 2Se3, Bi 2Te3, (BixSb1−x)2Te3, and so on,13,14,21,24grown by molecular beam epi- taxy, which allows one to precisely control the layer-by-layer growth of the epitaxial structure. Whereas the Fermi surface in Bi xSb1−x, Bi2Se3, and Bi 2Te3usually intersects with the bulk bands, the tun- able Bi:Sb ratio in (Bi xSb1−x)2Te3allows one to adjust the Fermi level to within the bandgap of the bulk states21,24so that only the topological surface states are conductive. Other TI materials such asα-Sn25and SmB 626–29have also demonstrated topological bands that enable charge-spin conversion. Very recently, magnetron sput- tering, a technique that has been widely adapted in semiconductor manufacturing, has been utilized to grow TI compounds with giant charge-spin conversion.30 In Secs. II and III, we will review the major progresses toward the goal of utilizing TIs for efficient magnetic switching, which involves determinations of SOT efficiency generated by TIs, experimental demonstrations of magnetic switching with TI het- erostructures, and optimization of the magnetic switching at room temperature to make TIs compatible with practical application considerations. II. QUANTIFICATION OF THE SPIN–ORBIT TORQUE EFFICIENCY Two pioneering works that explore the SOT generated by TIs were reported in 2014 by Fan et al.31and Mellnik et al.32indepen- dently. Fan et al. utilized the second harmonic magnetometry, which was first applied to bilayers of heavy metal/ferromagnetic films with perpendicular magnetic anisotropy (PMA),33,34to calibrate the SOT effective field in the Cr-(Bi xSb1−x)2Te3/(Bi xSb1−x)2Te3(magnetically doped TI/intrinsic TI) heterostructure at a temperature of 1.9 K.31In this method, a low-frequency alternating current is injected to gen- erate the SOT and induces low-frequency oscillations of magnetic moments. The periodic change in the anomalous Hall resistance further accounts for the generation of the second harmonic compo- nent of the Hall voltage [Fig. 2(a)]. The voltage detection can also be replaced by the magneto-optic Kerr signal that is sensitive to mag- netic moment tiltings,35where the effective spin Hall angle is deter- mined to be 90 at a temperature of 2.5 K and decreases dramatically as the temperature increases. In the room-temperature experiment done by Mellnik et al. , an electrical current with microwave frequency is injected into the Bi2Se3/NiFe bilayer.32The SOT generated by the oscillating cur- rent in Bi 2Se3causes resonant precession of the NiFe magnetization, yielding resistance oscillations due to the anisotropic magnetore- sistance effect.36The mixing between the applied alternating cur- rent and the oscillating resistance leads to a direct voltage, from which the magnitude of the damping-like (in-plane) and field-like APL Mater. 9, 060901 (2021); doi: 10.1063/5.0048619 9, 060901-2 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 2. Experimental proof of the SOT generated from TIs. (a) Second-harmonic anomalous Hall resistance as a function of in-plane external magnetic field, measured in the Cr-(Bi xSb1−x)2Te3/(Bi xSb1−x)2Te3sample. The shaded regions I, II, and III represent a single-domain state pointing in the −ydirection, magnetization reversal, and a single-domain state pointing in the +ydirection, respectively. The insets in regions I and III show the tilted magnetization around its equilibrium position under a low-frequency alternating current. (b) Top panel: schematic of the spin-torque ferromagnetic resonance setup for a Bi 2Se3/NiFe sample. The radio frequency current IRFgenereates net spin accumulation and thus the spin torques τon the NiFe moments. Bottom panel: mixing voltage measured in the spin-torque ferromagnetic resonance experiment. Panel (a) is adapted with permission from Fan et al. , Nat. Mater. 13, 699 (2014). Copyright 2014 Springer Nature Limited. Panel (b) is adapted with permission from Mellnik et al. , Nature 511, 449 (2014). Copyright 2014 Springer Nature Limited. (out-of-plane) SOT can be extracted [Fig. 2(b)]. The determined effective spin Hall angle, which represents the damping-like SOT strength per unit current density, is an order of magnitude higher than the heavy metals, pointing toward the strategy of realizing low- power magnetic memory devices using TIs as room-temperature SOT sources. A temperature-dependent study further reveals that the topological surface states are mostly responsible for the SOT in this TI at low temperature.37Here, we note that although the spin Hall angle was originally defined for the spin Hall effect, it is often used as a general figure of merit to reflect the SOT effi- ciency even when the exact mechanism is not the conventional three-dimensional spin Hall effect. This effective term of the spin Hall angle is defined as the ratio between the spin current density, calculated from the detected SOT, and the averaged charge current density across the thickness of the spin–orbit material. The third method usually used for quantifying SOT is the domain wall magnetometry approach, where a direct current and a constant biasing magnetic field are applied simultaneously. Within chiral domain walls, the SOT acts as an effective field in the out-of- plane direction for the magnetic thin film with PMA. The SOT effec- tive field can assist or impede the domain wall movement and thus the magnetic switching, which is reflected by a shift in the coercive field in anomalous Hall resistance measurement when sweeping an out-of-plane field38[Figs. 3(a) and 3(b)]. In this method, the SOT is directly compared with applied magnetic fields rather than changes in Hall resistance, which prevents the parasitic contributions from complicated transport phenomena in TIs.39,40As expected, the SOT efficiency still exceeds that of heavy metals detected by the same protocol.41 Being the reciprocal effect of the SOT-induced ferromagnetic resonance, spin pumping is a representative method to verify the spin-to-charge conversion in TIs. By exciting magnetic resonancevia the inductive method in a ferromagnet, spin current is injected to the neighbored TI layer, which is converted to a charge current through the surface spin-momentum locking or the bulk inverse spin Hall effect19,20,42,43[Figs. 3(c) and 3(d)]. Among the experiments in bismuth-based TIs, although the reported spin pumping phe- nomena are qualitatively consistent, the spin-to-charge conversion ratio, characterized by the effective spin Hall angle (ratio between the averaged charge current density and the spin current density), varies by several orders of magnitude (10−4to 101). The origin of the large discrepancy is still inconclusive but possibly comes from the following aspects: (i) differences in measurement temperature and element doping, which modulates the proportion of surface and bulk conductivities; (ii) parasitic effects with the spin pumping when TI is accompanied with a metallic ferromagnet, in particular, the spin rectification effect originating from microwave irradiation or thermal effects;44–46and (iii) variation of film thickness and inter- face quality during the sample preparation. In addition to giving rise to a spin pumping voltage, it has been shown that the spin-charge interconversion and the exchange interaction from the topologi- cal surface states can lead to phenomena such as enhanced Gilbert damping and additional magnetic anisotropy on magnetic materi- als, which modulates their ferromagnetic resonance behaviors.47–50 Besides the experiments discussed above that utilize SOTs to mod- ulate magnetic orientation or oscillation, other magneto-transport methods such as spin Seebeck effect,51lateral spin valve,52–59tunnel- ing spectroscopy,60and bilinear magnetoelectric resistance61have been utilized to confirm the robust charge-spin interconversion in TIs as well. The surface spin-momentum locking mechanism can be stud- ied from the dependence of the spin-charge conversion on the Fermi level position. For the purpose of emphasizing the two-dimensional nature of the surface states, the ratio between the two-dimensional APL Mater. 9, 060901 (2021); doi: 10.1063/5.0048619 9, 060901-3 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 3. Magnetotransport and dynamic measurements of the charge-spin conversions in TIs. (a) SOT-driven domain wall motion in a PMA layer under an in-plane magnetic field Hx, which fixes the domain wall chirality. The domain wall moments experience an effective SOT field Heff zalong the out-of-plane direction and move along the opposite directions, as indicated by the domain wall velocity vDW. (b) Hall resistance vs applied out-of-plane field under positive and negative direct currents and an in-plane bias field in the Bi 2Se3/CoTb sample. The center shift corresponds to the SOT effective field ( Heff z). (c) Schematic of the spin pumping from NiFe to a TI layer. (d) Spin pumping voltage in the Bi 1.5Sb0.5Te1.7Se1.3/NiFe sample. (e) Interface charge-to-spin conversion efficiency qICSas a function of Sb composition xin (Bi xSb1−x)2Te3, measured by spin-torque ferromagnetic resonance. The inset shows the band structure and Fermi level position for samples with different Sb concentrations. (Bi xSb1−x)2Te3with conductive surface states and insulating bulk states are expected to exist with 0.5 <x<0.9. (f) Spin pumping voltage as a function of Bi ratio xin Cr 0.08(BixSb1−x)1.92Te3/Y3Fe5O12. Panel (b) is adapted with permission from Han et al. , Phys. Rev. Lett. 119, 077702 (2017). Copyright 2017 American Physical Society. Panels (c) and (d) are adapted with permission from Shiomi et al. , Phys. Rev. Lett. 113, 196601 (2014). Copyright 2014 American Physical Society. Panel (e) is adapted with permission from Kondou et al. , Nat. Phys. 12, 1027 (2016). Copyright 2016 Springer Nature Limited. Panel (f) is adapted with permission from Wang et al. , Phys. Rev. Res. 1, 012014(R) (2019). Copyright 2019 American Physical Society. charge current density jcand the three-dimensional spin current density Jsin the Rashba–Edelstein configuration is more appropriate to describe the conversion efficiency. Via the spin-torque ferromag- netic resonance in (Bi xSb1−x)2Te3, Kondou et al. found that for x =0.5, 0.7, and 0.9, where the Fermi level falls into the bulk bandgap but is apart from the Dirac point, the charge-to-spin conversion ratio q=(2eJs)/(̵hjc)(in the unit of nm−1) is almost constant.62This is because in this range, only the topological surface states are conduc- tive and contribute to the charge-to-spin conversion [Fig. 3(e)]. An additional feature in this experiment is the small value of qwhen the Fermi level is very close to the Dirac point. As pointed out in Ref. 62, this effect may come from (i) the inhomogeneities in the surface states around the Dirac point, which deviates the charge flow from the electric field direction and weaken the net spin accumula- tion, and (ii) the spacer layer may induce an additional Rashba spin splitting and partially cancel the spin-momentum locking effect. On the other hand, using second harmonic measurement and current- induced switching, Wu et al.63showed that the SOT effective field reaches maximum when the Fermi level lies in the bulk bandgap. This is consistent with another Fermi level dependent SOT measure- ment, where the Fermi level position is tuned by a gate voltage.64 It is shown that when the Fermi level only intersects with the sur- face band, the SOT efficiency is larger than the bulk conducting case. The application of gate voltage provides an additional degree of free- dom to control SOT. Moreover, in a spin pumping experiment withCr0.08(BixSb1−x)1.92Te3/Y3Fe5O12bilayers, Wang et al.17observed a constant spin pumping signal when the Fermi level is tuned within the bulk bandgap [Fig. 3(f)]. This is consistent with the unchanged charge-to-spin conversion efficiency in the direct measurements on SOT when the conductivity only comes from the surface states. We note that the usage of the ferromagnetic insulator as the spin cur- rent source17,65excludes possible artifacts brought by conductive ferromagnets. The charge-spin conversion efficiency is also influenced by the TI thickness t, which can reflect the conversion mechanism of the interfacial spin-momentum locking or the bulk spin Hall effect. First, for an ideal TI, the spin-to-charge conversion is expected to be solely originated from surface states (with thickness ts). It is also assumed that only the surface conducts current and the bulk is insu- lating. Under a fixed spin injection current density ( Js), the spin pumping charge current Ispwill first increase and then saturate with t. The saturation point corresponds to a complete separation of the top and bottom surfaces ( t>2ts) [Fig. 4(a)]. Because increasing the bulk thickness does not contribute to the conductivity of an ideal TI, the spin pumping voltage Vspis expected to have a platform as well for t>2ts[Fig. 4(b)]. Meanwhile, for the three-dimensional spin Hall effect, Ispincreases and gradually saturates with tafter exceed- ing 2 λsf, where λsfdenotes the spin diffusion length [Fig. 4(c)]. As t increases, Vspshould have a decreasing behavior, which is from the reduced resistance [Fig. 4(d)]. APL Mater. 9, 060901 (2021); doi: 10.1063/5.0048619 9, 060901-4 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 4. Thickness tdependence of the spin pumping signals. [(a) and (b)] Spin pumping charge current Ispand voltage Vspin an ideal TI, where the surface states with a thickness of tssolely provide the conductivity and the spin-to-charge conver- sion through spin-momentum locking. [(c) and (d)] IspandVspin a system with a conducting bulk and the (inverse) spin Hall effect. λsfrepresents the spin diffusion length. Now, by comparing Fig. 4(a) with Fig. 4(c), and Fig. 4(b) with Fig. 4(d), one can see that the biggest difference between the ideal TI and spin Hall effect is that Vspremains constant in the large thick- ness limit for the TI case due to the insulating bulk state. However, in real TIs studied so far (with thickness dependence), the bulk state is often not fully insulating. Therefore, Vspalso decreases with t, as reported in Ref. 65, because Bi 2Se3is bulk conducting. On the other hand, Ispsaturates at a small thickness of ∼2 nm in Ref. 65, but at a much larger thickness of ∼30 nm in Ref. 19. These values may be correlated with the surface thickness in TI and a possibly larger spin diffusion length in a metal-like system, respectively. Fur- thermore, in polycrystalline films, the thickness dependence can be influenced by size effects, such as the charge-spin conversion fromquantum confinement in granular Bi xSe1−xfilms (see more details in Sec. III). III. SPIN–ORBIT TORQUE SWITCHING INDUCED BY TOPOLOGICAL INSULATORS Current-induced magnetic switching via the giant SOT from TIs was realized for the first time in the Cr- (BixSb1−x)2Te3/(Bi xSb1−x)2Te3heterostructures31and later in Cr-(Bi xSb1−x)2Te3sandwiched by different insulators,64where the Cr-doped TI becomes magnetic with PMA at cryogenic temperatures [Fig. 5(a)]. The magnetic reversal is reflected by the sign change of the anomalous Hall resistance when sweeping the direct current, which has been extensively used in the SOT switching experiments in heavy/metal/ferromagnet structures. At an experimental temperature of 1.9 K, the critical switching current below 105A/cm2(two orders of magnitude smaller than the typical values of heavy metals) stimulated significant interest in pursuing ultralow power dissipation memory and logic devices using TIs. In later studies, to suppress possible extra electrical contributions from TI heterostructures,39Yasuda et al.39and Che et al.35developed another method of using pulsed current to induce magnetic switching and direct current with a much weaker amplitude to sense the Hall resistance. The critical current density with a slightly larger magnitude is still far below that of heavy metals. Due to the low Curie temperature of the employed ferro- magnetic materials, magnetic switching was only realized at a few Kelvin in the magnetically doped TI heterostructures. Room tem- perature SOT switching, which is critical for most applications, requires the development of ferromagnetic thin films with appropri- ate anisotropy (PMA or uniaxial in-plane anisotropy) when accom- panied with TI. So far, progresses have been made using different strategies. The first is to utilize rare earth-transition metal alloys, in which two magnetic sublattices are antiferromagnetically cou- pled. The PMA is realized through the growth-induced strain in the bulk and is less sensitive to the underlayer structures.66,67In the CoTb/Bi 2Se3bilayer, Han et al. demonstrated room-temperature SOT switching41[Fig. 5(b)], with the SOT efficiency several times higher than heavy metals Pt and Ta. This approach was also verified in other similar structures68such as Bi 2Se3/GdFeCo and (BixSb1−x)2Te3/GdFeCo. We note that the SOT efficiencies obtained FIG. 5. Current-induced magnetic switching in TI based heterostructures with PMA. (a) Switching in the Cr- (BixSb1−x)2Te3/(Bi xSb1−x)2Te3sample at 1.9 K. (b) Switching in the Bi2Se3/CoTb sample at room tem- perature. Panel (a) is adapted with permission from Fan et al. , Nat. Mater. 13, 699 (2014). Copyright 2014 Springer Nature Limited. Panel (b) is adapted with permission from Han et al. , Phys. Rev. Lett. 119, 077702 (2017). Copyright 2017 American Physical Society. APL Mater. 9, 060901 (2021); doi: 10.1063/5.0048619 9, 060901-5 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm from the rare earth-transition metal alloys are usually smaller than those obtained from the traditional ferromagnetic electrodes, which is potentially due to a lower interfacial spin transparency in these stacks. In these experiments, it is shown that (Bi xSb1−x)2Te3with a more insulating bulk provides higher efficiency for magnetic switch- ing than Bi 2Se3, suggesting that the SOT generation benefits from a more concentrated charge current in topological surface states.41,68 The second way is to grow an ultrathin non-magnetic seeding layer above TI, which gives rise to the needed texture for reaching PMA in the ferromagnetic layer grown on top. PMA multilayers such as Ti (or Mo)/CoFeB/MgO and Ta/CoFeB/Gd/CoFeB have been grown on TIs for magnetic switching.30,63,69In these multilayers, because of the small value or the opposite sign of the spin Hall angle, contributions from the non-magnetic seeding metals can be excluded and the dominant source of the SOT is identified to be the TI layer. In these stacks, the SOT efficiency is usually enhanced when the bulk of the TI is tuned to be less conductive, suggest- ing the important role of the topological surface states. The third method is to develop ferromagnetic films that are epitaxial on top of TIs, which provides correct crystal orientation and stain to gener- ate PMA. For example, the epitaxial structure of Bi 0.9Sb0.1/MaGa70 shows a giant effective spin Hall angle of 52 and a low switching current density of 105A/cm2. The fourth method is to develop in- plane magnets whose easy-axis is collinear with the injected spin so that the magnetic moment can be switched by the SOT that over- comes the Gilbert damping. This configuration has been verified in the Bi 2Se3/NiFe bilayer by the magneto-optic Kerr imaging.71It is also possible to electrically probe the in-plane magnetic reversal via the unidirectional magnetoresistance effects.72–74 In parallel with the strategy that utilizes topological surface states in epitaxial TI films, exploring novel TI compounds with exotic grain effects provide another approach for generating giant SOT. Using the SOT from the Bi xSe1−xthin films prepared by mag- netron sputtering, Dc et al. observed switching of the CoFeB-based PMA multilayers at room temperature,30with a critical switching current density in the order of 105A/cm2. In the polycrystalline BixSe1−xfilm, the band structure of grains with reduced dimensions reveals additional states that have surface character and are local- ized mainly on the grain sidewalls and corner. Such a quantum confinement effect leads to a charge-to-spin conversion efficiency that enhances with reduced size and dimensionality of the nanoscale grains. The high spin-to-charge conversion was verified by the spin pumping experiment later,75in which the spin pumping signal decreases in thicker granular Bi xSe1−xfilms. This trend suggests that the enhanced quantum confinement in thinner films is critical to the large charge-spin conversion. Besides Bi xSe1−x, high SOT efficiency was also obtained in sputtered Bi 2Te3films.76These works highlight the potential of fabricating TI-based spintronic devices on silicon substrates using industry friendly techniques. In TI/metallic ferromagnet stacks, the high resistivity of TIs usually causes significant current shunting through the ferromag- netic layer. One strategy to fully exploit the efficiency of TI is to utilize magnetic insulators as the free layer for switching. Potential candidates include rare earth iron garnets or barium ferrites with PMA.77,78Liet al.78reported switching of an insulating ferromagnet BaFe 12O19using the SOT from Bi 2Se3. The switching efficiency at 3 K was found to be 300 times higher than that at room temperature and 30 times higher than that in Pt/BaFe 12O19. The decreasing trendof the SOT efficiency with temperature is consistent with a number of reports in TI-based systems and suggests the presence of more pronounced topological surface states at low temperatures. Another approach to avoid shunting is to separate the TI and the conductive ferromagnet by inserting an insulating magnetic layer that blocks charge current, which, on the other hand, allows the transmission of spin angular momentum in the form of spin wave or spin fluc- tuations. In the Bi 2Se3/NiO/NiFe trilayer structure, Wang et al.79 demonstrated that the non-equilibrium spins generated from Bi 2Se3 can transmit through the antiferromagnetic insulator NiO up to 25 nm and switch the NiFe moments. Finally, we summarize from the literature the key features and parameters in the SOT switching experiments using TIs and heavy metals (Table I). Besides the effective spin Hall angle θSHand the critical current for switching Jcr, we list the power consumption for switching ferromagnetic electrodes in unit magnetic volume, expressed as J2 cr/σ, where σis the electrical conductivity of the spin–orbit materials. We can see that TIs stand out as the favorable materials for magnetic switching because they benefit from reduced critical current and low power consumption. However, Jcrnot only depends on the properties of the TI layer but is also affected by the thermal stability of the ferromagnetic layer as well as the applied in-plane bias field in experiments with PMA materials. To reveal the intrinsic SOT generation, one can replace Jcrby 1/θSHbecause in spin–orbit materials with the same geometrical dimensions, the critical current density for switching scales with the inverse of the spin Hall angle. We find that TIs still represent an energy efficient candidate when 1 /θ2 SHσ, or equivalently σ/σ2 SH, is adopted as the comparison metric ( σSHis the spin Hall conductivity). These fea- tures suggest that the strategy of utilizing conductive surface states with strong spin-momentum locking is desired for maximizing the SOT and power efficiency. In reality, it may not be very rigorous to directly compare the absolute values of SOT efficiencies from differ- ent works because the actual values depend heavily on the details of the sample quality and experimental technique in each study. Never- theless, some presentations of SOT and power efficiencies may still be useful to capture the main features. For a fairer comparison, it is helpful to measure on different materials using the same technique within one particular study.41,63,68 We would like to emphasize here that in the majority of exist- ing TI/ferromagnet devices, a large portion of the applied current flows in the conductive ferromagnetic layer, which has not been taken into account in the power calculation above. Considering the “wasted” power in ferromagnetic metals, the comparison on power consumption in Table I can look differently, where some of the recently studied heavy metal alloys exhibit very competitive power performances.80,81The unnecessary dissipation from magnetic met- als can be potentially avoided by using an insulator as the magnetic free layer, developing more conductive TIs, or inserting an insula- tor between the TI and the ferromagnet, which has been discussed above. We note that in Table I, the spin Hall angle of many TI films is larger than unity. In the original definition of the three-dimensional spin Hall effect in the diffusive region, θSH=(2e/̵h)Js/Jcrepresents the ratio between the transverse spin current density and longitu- dinal charge current density. Therefore, θSHshould be interpreted as the tangent of the angle formed by the electrons’ trajectory and the longitudinal direction. In the case of a small deflection angle, the APL Mater. 9, 060901 (2021); doi: 10.1063/5.0048619 9, 060901-6 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apmTABLE I. Summary of the key parameters in the SOT switching experiments using TIs and heavy metals. Parameters Reference 31 Reference 39 Reference 41 Reference 71 Reference 70 Reference 30 Reference 68 Reference 78 Reference 3 Spin–orbit material (Bi,Sb) 2Te3 (Bi,Sb) 2Te3 Bi2Se3 Bi2Se3 Bi0.9Sb0.1 BixSe1−x (Bi,Sb) 2Te3 Bi2Se3 Pt Ferromagnet Cr-(Bi,Sb) 2Te3Cr-(Bi,Sb) 2Te3 CoTb NiFeaMnGa CoFeBbGdFeCo BaFe 12O19 Co Substrate GaAs (111) InP (111) GaAs (111) Al 2O3(0001) GaAs (001) SiO 2 Al2O3(0001) Al 2O3(0001) SiO 2 Temperature (K) 1.9 2 RTcRT RT RT RT 3–200 RT Spin Hall angle θSH 140 ⋅ ⋅ ⋅c0.16 1 52 18.6 3 267 (3 K) 0.03 Conductivity σ(S/cm) 220 ⋅ ⋅ ⋅ 943 400 2.5 ×10378 182 660–760 2 ×104 Critical current Jcr (A/cm2)8.9×1042.5×1062.8×1066×1051.5×1064.3×1051.2×1051–2×1062.8×107 Power Jcr2/σ(W/cm3) 3.6 ×107⋅ ⋅ ⋅ 8.3×1099×1089×1081.3×1097.9×1092–6×1091.4×1010 1/(θSH2σ) normalized with Pt4×10−6⋅ ⋅ ⋅ 0.7 0.05 3×10−67×10−40.01 3×10−71 aNiFe has in-plane magnetic anisotropy. Other ferromagnets in this table have PMA. bCoFeB here represents a multilayer containing Ta/CoFeB/Gd/CoFeB. c“⋅ ⋅ ⋅” means information not shown in the reference. RT means room temperature.so-defined spin Hall angle is close to the angle of the trajectory. In the extreme case that the trajectory angle gets close to 90○, the spin Hall angle goes toward infinity. In this sense, the larger-than-unity spin Hall angle is a natural reflection of the “quantum spin Hall state” that is expected from high quality topological insulators. IV. CONCLUSIONS AND OUTLOOK Thanks to the tremendous efforts from the TI and the spin- tronic communities in developing high quality TI films and charac- terizing the outstanding SOT efficiency, TIs have been recognized as a promising candidate for building magnetic memory devices with ultralow dissipation. However, there are still a few bottlenecks to be overcome before TIs can be readily used as the spin current source in real applications. In most of the existing studies, the synthesis of TI films requires the costly technique of molecular beam epitaxy. Deposition techniques that are more compatible with the CMOS manufacture process should be developed to produce TI films with large scale. As discussed earlier, the sputtered Bi xSe1−xfilm pro- vides an industry-friendly approach that explores the giant SOT.30 Furthermore, for practical magnetic random-access memory tech- nology, TIs need to be integrated with the state-of-the-art magnetic tunnel junctions, for example, CoFeB/MgO/CoFeB stacks with PMA and tunneling magnetoresistance (TMR) bigger than 100%.82Engi- neering the TI surface that can support the PMA of magnetic lay- ers and avoid atomic diffusion during high-temperature fabrication would be a critical step. Possible ways include developing TI films or adding insertion layers with good thermal stability and appropriate crystal orientation, which helps in forming (001)-oriented CoFeB.83 As a final goal, a demonstration of magnetic tunnel junctions inte- grated with TI layers that have high TMR and efficient SOT switch- ing are highly desirable for the practical applications of TI-based spintronics.84 Besides the potential improvements in practical devices, there remain many open questions on the fundamental interpretation of the TI induced SOT phenomena, even on the basic effect in the simplest sample structure, i.e., the switching and efficiency in the TI/ferromagnet bilayer. First of all, it is not fully understood why the experimentally determined charge-spin conversion efficiency varies significantly among different TI systems. Besides variances in the sample quality and measurement technique, there exists a more fundamental question on how to determine the contributions from the bulk and surface. As is discussed earlier in this review, besides the intrinsic topological bulk bands that give rise to the surface states, it seems that in experiments the bulk states can make additional con- tributions through extrinsic non-topological effects. How to separate the topological trivial and non-trivial effects in SOT remains a chal- lenging task. Moreover, regarding the topological non-trivial origin, it will be helpful to correlate the SOT with the structure of surface bands in the same material stack. As a good starting point, people have combined spin pumping with ARPES in a TI material α-Sn, where the spin-to-charge conversion coexists with the topological surface states.25 There exist many other interesting topics beyond the traditional scheme of switching ferromagnetic materials. So far, TIs have mostly been used to generate in-plane spins to switch PMA materials, which provides better thermal stability than in-plane magnets.82Since the APL Mater. 9, 060901 (2021); doi: 10.1063/5.0048619 9, 060901-7 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm spins are perpendicular to the easy axis of the magnet, the damping- like SOT plays the role as an effective magnetic field, which needs to overcome the anisotropy field in the order of 0.1–1 T for reaching magnetic switching in a thermally stable nanomagnet, thus requir- ing a huge current density. However, when the spins are collinear with the magnetic easy axis, the switching current can be largely reduced because the SOT is used to overcome the Gilbert damp- ing.4,85Therefore, it is highly desired to switch PMA materials with out-of-plane spins.86–90Designing TI structures with low symmetry along the certain crystal direction is a promising way to generate out-of-plane spins87,91while benefiting from the demonstrated giant charge-to-spin conversion. Finally, we highlight the rich physics and applications when combining TIs with antiferromagnets. On one hand, the antiferromagnetic exchange coupling92–94with the Dirac fermions at the interface can give rise to emergent phenom- ena such as topologically nontrivial spin textures.95On the other hand, the giant SOT from TIs can be potentially used to displace the spin texture and switch the Néel vector. 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5.0052845.pdf

Valley polarized conductance quantization in bilayer graphene narrow quantum point contact Cite as: Appl. Phys. Lett. 118, 263102 (2021); doi: 10.1063/5.0052845 Submitted: 1 April 2021 .Accepted: 30 May 2021 . Published Online: 30 June 2021 Kohei Sakanashi,1,a) Naoto Wada,1Kentaro Murase,1Kenichi Oto,2Gil-Ho Kim,3 Kenji Watanabe,4 Takashi Taniguchi,5Jonathan P. Bird,6 David K. Ferry,7 and Nobuyuki Aoki1,a) AFFILIATIONS 1Department of Materials Science, Chiba University, Chiba 263-8522, Japan 2Department of Physics, Chiba University, Chiba 263-8522, Japan 3School of Electronic and Electrical Engineering and Sungkyunkwan Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University (SKKU), Suwon 16419, South Korea 4International Centre for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan 5Research Centre for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan 6Department of Electrical Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260, USA 7School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287-5706, USA a)Authors to whom correspondence should be addressed: k.sakanashi@chiba-u.jp andn-aoki@faculty.chiba-u.jp ABSTRACT In this study, we fabricated quantum point contacts narrower than 100 nm by using an electrostatic potential to open the finite bandgap by applying a perpendicular electric field to bilayer graphene encapsulated between hexagonal boron nitride sheets. The conductance across the quantum point contact was quantized at a high perpendicular-displacement field as high as 1 V/nm at low temperature, and the quantization unit was 2 e2/hinstead of mixed spin and valley degeneracy of 4 e2/h. This lifted degeneracy state in the quantum point contact indicates the presence of valley polarized state coming from potential profile or effective displacement field in one-dimensional channel. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0052845 A short one-dimensional channel, namely, a quantum point con- tact (QPC), defined in two-dimensional materials has been receiving much attention for future spin-valley device applications such as a spin-valley filter or spin-valve effect.1,2Several QPC devices have already been realized in two-dimensional materials such as electrostat- ically induced bilayer graphene (BLG),3–10etching-defined monolayer graphene,11–14or transitional metal dichalcogenide.15–17However, the phenomena attributed to degrees of freedom, such as valley and spin in the QPC, are still unclear and controversial. Monolayer graphene, which has Dirac dispersion, is not applica- ble for electrostatically confined nanostructures such as the QPC due to its lack of an energy gap;18therefore, an etching-defined QPC struc- ture was used, which is also used in conventional high-electron mobil- ity transistor systems in several ways.11,14Such graphene QPC devices show strong disorder and inhomogeneity effects, consequently causing inter-valley scattering,19localized state effects,20or low transmission probability.19Even bilayer graphene has no energy gap in the intrinsic situation; however, a bandgap can open with a perpendicular electricfield.21,22Then, electrically confined nanostructures can be realized in BLG by fabricating sandwiched structure with top and bottom metallic gates.3–10However, in order to form a bandgap sufficient for defining the QPC in the BLG, a relatively high displacement field ( >0.3 V/nm) is necessary by applying opposite-polarity voltages between the top(bottom) split gates and the global back (top) gate, 21–23and then the carrier density in the two-dimensional electron (hole) gas region(2DEG or 2DHG), except underneath the split gates, becomes high—up to the order of a few 10 12cm/C02causing lower mobility. Forming a well-defined QPC channel electrostatically requires an adequate high displacement field inside the QPC channel that runs between the splitgates. However, BLG has a relatively short Fermi wavelength com-pared to a conventional HEMT system, 18and an even shorter channel width between the split gates ( <50 nm), so it is difficult to pinch-off the current channel because of the fringing effect of the electric field atthe edge of the split gates. 9To achieve the pinch-off properties and well tunability of Fermi wavelength in the QPC channel, a triple-gate (a channel-gate) structure, which has an additional high- kdielectric, Appl. Phys. Lett. 118, 263102 (2021); doi: 10.1063/5.0052845 118, 263102-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apland a top gate were utilized, and then clear quantized conductance steps were observed in units of 4 e2/h.3,4,7,8In these structures, the QPC channel width is typically 150–250 nm, and the pinch-off can be achieved with depleting the carriers in the channel by applying an appropriate voltage to the channel gate without squeezing the channel width. Then, the conductance steps usually appear at every 4 e2/h. However, the value of the steps sometimes appears at 2 e2/hdepending on samples. So the unit of the conductance quantization is not a unique value and is still unclear.3In this paper, we realized split-gate structures less than 100 nm and studied the transport properties bychanging the confinement potential profile by changing the voltage applying to the split gates. In this paper, we mainly measured two types of devices, named device A [ Figs. 1(a) and1(b)] and device B [ Figs. 1(c) and1(d)]. All BLG flakes and hexagonal boron nitride (hBN) crystals used for the two devices were exfoliated onto SiO 2/Si chips from bulk crystals and carefully chosen by means of optical microscopy, Raman spectroscopy, and atomic force microscopy (AFM) to ensure the thickness and cleanness. A stack of a hBN/BLG/hBN heterostructure was assembled by PC (6% Poly bisphenol A Carbonate dissolved in chloroform)/PDMS based dry-transfer method 24a n dr e l e a s e do nt h et o po ft h e cleaned 300 nm-thick SiO 2layer on the pþþ-Si substrate working as a global back gate (BG) for device A. The split gate electrodes (SG) for QPC1 were fabricated on the stack at the final step as schematically shown in Fig. 1(b) . For device B, prior to attaching the stack, the thin metallic SG electrodes for the QPC2 were fabricated on the SiO 2sur- face, and the top gate covering the QPC2 and 2DEG (2DHG) region was fabricated on the stack as shown in Fig. 1(d) . The Si substrate was also used to control the perpendicular electric field. All the electrodes were fabricated by means of electron beam lithography (EBL) and electron beam deposition (EBD) of Cr 1/AuPd 9 nm. The channel region was patterned by standard EBL, followed by reactive ion etching (RIE) in a gas mixture of CHF 3(10) and O 2(4.5 sccm) in ahome-built etcher at 60 W. Source-drain electrodes were formed by the one dimensional edge contact technique24through the EBL, RIE, and EBD (Cr 3/Pd 15/Au 80 nm) with the same PMMA mask to avoid contamination and any self-bias effect.25We record typical series resis- tance of a few hundreds to k Xin our devices. All of the SG have the same dimension of 100 nm-wide and 100-nm-long at the top with a triangular shape, and the coarse part has a 1- lm-width as shown in the inset of Fig. 1(a) . All electrical measurements were measured by standard lock-in techniques at the frequency of the 17.7 Hz with a small excitation voltage, below 100 lV, in four terminal configurations for device A and two terminals for device B at low temperature in a JANIS3He refrigerator. During the measurement, QPC1 in device A was defined by applying positive and negative gate voltages to the SG and the BG, respectively, in order to apply a perpendicular electric field. In the device B, the QPC2 was defined in the same manner; how- ever, the global BG was biased in the same polarity to the SG in order to assist the perpendicular electric field within the conducting 1D channel and to decrease the carrier density in the 2DEG (2DHG) region for increasing the mobility. Figure 1(e) shows a resistance curve of device A as a function of the back-gate voltage ( VBG) at 2 K. The highest field-effect carrier mobility was 57 000 cm2/V s. Figure 1(f) shows a resistance curve of device B as a function of the global top-gate voltage ( VTG)a t2K .T h e residual impurity density in device A is below 1011cm/C02(see the sup- plementary material Fig. S1). This high-quality BLG device has a lon- ger mean free path than the width and the length of the split gates. The resistance map of the VBGand the top split gate voltage ( VSG) sweeps shows higher resistance peeks at around VBG¼/C03V f o r t h e Dirac point of the BLG and at along the slope upward to the left related to the higher displacement field, D, comparable with earlier studies,3,9where D¼(DtopþDbottom )/2 where Dtop¼/C0etop(Vsg /C0VCNP)/dtopandDbottom¼ebottom (Vbg/C0VCNP)/dbottom )21,22(see the supplementary material Fig. S2). Each eanddare determined by using FIG. 1. (a) Optical microscope image and (b) schematic view of the device A. Red dotted line indicates the position of the BLG flake embedded in the hBN layers. Th e inset in (a) shows AFM image of the tip of the SG. The scale bar is 500 nm. (c) Optical microscope image and (d) schematic view of the device B. White dotted lines ind icates the posi- tion the QPC2 underneath the hBN/BLG/hBN stack. (e) VBGdependence of the two-terminal resistance of device A at 2 K. (f) VTGdependence of the two-terminal resistance of the device B at 2 K. All other gates were grounded during the mesurements.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 263102 (2021); doi: 10.1063/5.0052845 118, 263102-2 Published under an exclusive license by AIP Publishingthe dielectric constant and the thickness of the dielectric layers, respec- tively. When the jDjexceeds 1 V/nm, the BLG underneath the SGs can open a certain level of bandgap at the Dirac point, and the charge is depleted enough to define the QPC.21Figure 2(a) shows a typical transfer curve by sweeping the VSGtoward the positive voltage by applying fixed VBG¼/C080 V. The conductance decreased slowly at 0 <VSG<4 V due to the formation of pþppþjunction (regime I) since high hole density in 2DHG was induced by negatively biasing theglobal back gate electrode and the carrier density underneath the SGs was slightly decreased. In the middle displacement field regime, the 2DHG under SGs was getting depleted by approaching the Diracpoint, and the conductance starts to decrease drastically (regime II). Under a high displacement field region, jDj/C241V / n m , a s u f fi c i e n t bandgap was formed at the Dirac point in the BLG underneath the SGelectrodes (see the supplementary material Fig. S3). The 2DHG in the region was depleted since the Fermi level of BLG was tuned to the middle of bandgap, and then a QPC structure was electrostaticallydefined. The conducting channel was getting squeezed, and quantized plateaus appeared in the conductance curve (regime III) are magnified inFig. 2(b) . After exceeding the critical V SGvalue of aroundVSG¼9.3 V, the conductance increases again (regime IV) by the Fermi level reaching the conduction band edge underneath the SG and conduction begins to revive by forming a pþnpþjunction due to Klein tunneling.26Note that both carrier density and displacement field strength in the constriction are changed during the single VSG sweeping. The quantization of the conductance is unclear compared to such QPCs in a conventional semiconductor heterostructure due to the shorter mean free path in this system and the geometrical shape ofthe SG electrodes. Nevertheless, the derivative of conductance with respect to V SGshows well-developed maxima corresponding to the position of the plateaus at steps of 2 e2/hcovering a wide range of con- ductance, where the derivative of conductance is usually negative due t ot h ep - t y p eQ P Ca ss h o w ni n Fig. 2(b) . However, the conductance did not go below 2 /C22e2/hin this sample even by increasing the nega- tive back gate voltage up to /C0110 V (see the supplementary material Fig. S2). The reasons why the QPC did not reach a pinch-off could beconsidered as follows: The QPC can be defined only during the V SG range when the Fermi level under the SG is in the bandgap (see the supplementary material Fig. S3); therefore, only a finite number of c o n d u c t a n c es t e p s( fi v es t e p si nt h i sc a s e )c a nb eo b s e r v e dw i t h i nt h e energy (gate voltage) window. Since the carrier density in the 2DHGregion is rather high, up to 5 /C210 12cm/C02, due to the large back gate voltage ( >/C080 V) to induce sufficient displacement field to open the bandgap under the SGs, the stray field from the tip of the SG into theQPC channel is not sufficient to deplete the carrier density completely in the channel. Moreover, in order to achieve the pinch-off condition in the QPC, a perpendicular electric field is necessary to open a suffi- cient bandgap in the channel region. However, it is not sufficient only from the tip of the SGs even using a narrow separation less than100 nm. Another set of gate electrodes provide such a channel gate 3–5,7,8to assist the perpendicular electric field to achieve the pinch-off. Nevertheless, our results exhibit a feature of the level degeneracy of quantized conductance with a unit of 2 e2/hafter subtraction of series resistance ( Rseries/C241kX)o ft h e2 D H Gr e g i o nf r o mt h em e a - sured 4-t resistance ( R4t),GQPC¼1/(R4t/C0Rseries)i ne2/hunit formula shown in Fig. 2(b) . Due to zero magnetic field and relatively small spin–orbit coupling in the BLG system, such a BLG-QPC should show the quantization units of g¼4 for the valley (2) and the spin degener- acy (2). Our results suggest a possibility of breaking the valley degener-acy in the narrow constriction. The quantization unit can be modified by transmission probability Twhen the constriction geometry is not uniform or quantum wire case instead of QPC geometry. However, the transmission probability would not be restricted so much from T/C241 since the length of the QPC (100) is sufficiently shorter than the mean free path ( /C24500 nm), estimated from the field-effect mobility (57 000 cm 2/V s) near the Dirac point. Another possibility is interval- ley scattering due to the scattering at atomic defects, sample edge roughness, or variations in the etching defined hard-wall potential. However, our electrostatically induced soft-wall and smooth confine-ment potential, which has no dissipative edge or roughness, should show the transparency T/C241. Similar studies have been performed using electrostatically induced QPCs and quantum wires in BLG,where they have reported quantized conductance steps of 4 e 2/h.T h e main differences of these former studies from our device are the use of a wide channel more than 200 nm and an additional channel gate structure in order to assist the perpendicular electric field to achieve FIG. 2. (a) Conductance (red) and its derivative (blue) with respect to VSG(dG/ dVSG) of QPC1 in device A, obtained by sweeping VSGatVBG¼/C0 80 V. The back- ground color indicates the different transport regimes (I–IV). (b) Left panel: conduc-tance curve at regime III in (a) after subtraction of series resistance. Right panel:differential conductance ( dG/dV SG) of the curve in the left panel.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 263102 (2021); doi: 10.1063/5.0052845 118, 263102-3 Published under an exclusive license by AIP Publishingthe pinch-off condition.3–5,7On the other hand, such QPC structures having a narrow constriction width less than 100 nm (similar to our devices) tend to show g¼2,9,10whose QPCs are used for defining a quantum dot. Considering the differences of the channel width, the landscapes of the confinement potential would be different; a rectan- gular potential profile could be considered for the wide channel and then the additional channel gate decreased carrier density and increased the Fermi wavelength to achieve the mode at the discrete energy level. On the other hand, the narrow constriction, less than 100 nm, used in QPC1, provides a parabolic-soft-wall potential and then the degeneracy may be removed by the confinement effect. By applying an out-of-plane magnetic field, the one-dimensional discrete energy levels split and cross each other due to the formation and theevolution of Landau levels at low magnetic field; 3,5,7therefore, the con- ductance steps do not appear regularly compared to the results of in- plain magnetic field application.10However, at higher magnetic field B¼9 T, the conductance steps are still slightly complicated as the unit of conductance quantization becomes e2/hdue to the Zeeman splitting over the full gate voltage range in regime III as shown in Fig. 3 (see the supplementary material Fig. S4 for the Landau fan plot from B¼0t o 9 T). Different from the former BLG-QPC studies using a channel gate structure,3–5,7,8no significant enlargement of the width of the conduc- tance plateau is observed even at 9 T due to the channel width (<8 0n m )n a r r o w e rt h a nt h ec y c l o t r o nr a d i o si no u rQ P Cd e v i c e . Such a g¼2 quantization behavior has also been confirmed in another BLG-QPC device, QPC2 in device B, which has a triple-gate structure shown in Figs. 1(c) and1(d) and higher mobility. Clear con- ductance steps with a unit of 2 e2/hcan be observed at 3–6 /C22e2/hafter subtracting Rseries as shown in Fig. 4 . In this experiment, the jDj between the top gate and the SG was 0.7 in addition to the jDjof 0.7 V/nm between the top gate and the BG for the assistance of the perpendicular electric field in the 2DHG region including the channel. Although the minimum conductance decreased down to 2 e2/h,i tcould not achieve the pinch-off due to the small energy window of the depletion at the SG region. Some additional small conductance stepscan be observed in Fig. 4 . These features may originate from quantum intereference due to disorder in the QPC region, or from Fabry–P /C19erot interference in the split-gate region, resulting from parallel conduction due to the presence of insufficient gate voltage. 26–30These additional steps can be observed also due to hole-hole intereaction, although themobility of our devices may not be suifficiently large to allow suchmany-body effects. 31–33 In summary, we have demonstrated electrostatically confined QPC structures narrower than 100 nm without the assistance of addi-tional channel gates and observed the well-developed quantized con-ductance at steps of 2 e 2/hinstead of 4 e2/halthough the conductance is not fully pinched-off. The step of 2 e2/hsuggests the breaking of at least one degeneracy such as valley polarization by perpendicular electric fields, and these results indicate the narrow confinement potential pro-file may be responsible for lifting the valley degeneracy suggesting forthe application as such a valley filter device. See the supplementary material for the device quality estimation, resistance evolution in dual-gate sweep, and how to define the BLG- QPC. The authors thank Yu Saito for fruitful discussion on device fabrication. This work was supported by the JSPS KAKENHI Grant Nos. 18H0812 and 20J20052, JSPS Bilateral Joint Research withNRF, Chiba University VBL project, and Iketani Science andTechnology Foundation. The work was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) with file No. 2021K2A9A2A08000168. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. FIG. 4. Conductance of QPC2 in device B as a function of bottom split gate voltage VSGatVTG¼/C0 4 Vand VBG¼45 V. FIG. 3. Conductance of QPC1 in device A as a function of VSGatVBG¼/C0 110 V andB¼9T .Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 263102 (2021); doi: 10.1063/5.0052845 118, 263102-4 Published under an exclusive license by AIP PublishingREFERENCES 1A. Rycerz, J. Tworzydło, and C. W. J. Beenakker, Nat. Phys. 3, 172 (2007). 2D. Xiao, W. Yao, and Q. Niu, Phys. Rev. Lett. 99, 236809 (2007). 3H. Overweg, H. Eggimann, X. Chen, S. Slizovskiy, M. Eich, R. Pisoni, Y. Lee, P. Rickhaus, K. Watanabe, T. Taniguchi, V. Fal’Ko, T. Ihn, and K. Ensslin, Nano Lett. 18, 553 (2018). 4H. Overweg, A. Knothe, T. Fabian, L. Linhart, P. Rickhaus, L. Wernli, K. Watanabe, T. Taniguchi, D. S /C19anchez, J. Burgd €orfer, F. 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5.0052301.pdf

Appl. Phys. Lett. 118, 232402 (2021); https://doi.org/10.1063/5.0052301 118, 232402 © 2021 Author(s).Proximity-induced anisotropic magnetoresistance in magnetized topological insulators Cite as: Appl. Phys. Lett. 118, 232402 (2021); https://doi.org/10.1063/5.0052301 Submitted: 30 March 2021 . Accepted: 20 May 2021 . Published Online: 11 June 2021 Joseph Sklenar , Yingjie Zhang , Matthias Benjamin Jungfleisch , Youngseok Kim , Yiran Xiao , Gregory J. MacDougall , Matthew J. Gilbert , Axel Hoffmann , Peter Schiffer , and Nadya Mason COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Enhancement of photon detection in superconducting nanowire single photon detector exposed to oscillating magnetic field Applied Physics Letters 118, 232603 (2021); https://doi.org/10.1063/5.0046262 Coupling of a single tin-vacancy center to a photonic crystal cavity in diamond Applied Physics Letters 118, 230601 (2021); https://doi.org/10.1063/5.0051675 Unidirectional transport of superparamagnetic beads and biological cells along oval magnetic elements Applied Physics Letters 118, 232405 (2021); https://doi.org/10.1063/5.0044310Proximity-induced anisotropic magnetoresistance in magnetized topological insulators Cite as: Appl. Phys. Lett. 118, 232402 (2021); doi: 10.1063/5.0052301 Submitted: 30 March 2021 .Accepted: 20 May 2021 . Published Online: 11 June 2021 Joseph Sklenar,1,2,a) Yingjie Zhang,3 Matthias Benjamin Jungfleisch,4,5 Youngseok Kim,6Yiran Xiao,1 Gregory J. MacDougall,1 Matthew J. Gilbert,6,7Axel Hoffmann,1,3,4,6 Peter Schiffer,1,8,9 and Nadya Mason1 AFFILIATIONS 1Department of Physics Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801, USA 2Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48202, USA 3Department of Materials Science and Engineering, University of Illinois, Urbana, Illinois 61801, USA 4Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA 5Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA 6Department of Electrical and Computer Engineering, University of Illinois, Urbana, Illinois 61801, USA 7Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA 8Department of Physics, Yale University, New Haven, Connecticut 06520, USA 9Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA a)Author to whom correspondence should be addressed: jnsklenar@wayne.edu ABSTRACT Topological insulators (TIs) host spin-momentum locked surface states that are inherently susceptible to magnetic proximity modulations, making them promising for nano-electronic, spintronic, and quantum computing applications. While much effort has been devoted to study- ing (quantum) anomalous Hall effects in magnetic magnetically doped TIs, the inherent magnetoresistance (MR) properties in magnetic proximity-coupled surface states remain largely unexplored. Here, we directly exfoliate Bi 2Se3TI flakes onto a magnetic insulator, yttrium iron garnet, and measure the MR at various temperatures. We experimentally observe an anisotropic magnetoresistance that is consistent with a magnetized surface state. Our results indicate that the TI has magnetic anisotropy out of the sample plane, which opens an energy gapbetween the surface states. By applying a magnetic field along any in-plane orientation, the magnetization of the TI rotates toward the plane and the gap closes. Consequently, we observe a large ( /C246.5%) MR signal that is attributed to an interplay between coherent rotation of magnetization within a topological insulator and abrupt switching of magnetization in the underlying magnetic insulator. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0052301 In metallic ferromagnetic films, anisotropic magnetoresistance (AMR) 1refers to the dependence of a film’s resistivity on the orienta- tion of magnetization with respect to the current passing through it. In spintronic research, AMR plays a central role in a wide variety of experimental contexts. For example, the AMR of a ferromagnetic layer can be used to detect magnetization dynamics2–4within magnetic het- erostructures designed so that spin torques drive the magnetization. Often, TIs are involved in these experiments because they are capable of generating large spin torques5–7and easily facilitate spin-to-charge conversion.8,9On the other hand, the use of TIs as a source of AMR in these spintronic experiments is, relative to torque effects, under- explored. Because of the strong spin–orbit coupling intrinsic to TIs, it is anticipated that AMR effects in TIs should be large.10We report a magnetoresistance effect that is phenomenologically consistent with a proximity-AMR predicted to exist in undoped, mag- netic surface states of a TI that are adjacent to a ferromagnetic insula- tor.10In this report, we will refer to this effect as a proximty-AMR, owing to the fact that the dependence of this AMR on magnetization orientation differs from conventional AMR behavior. The proximity- AMR of the TI does not arise from magnetic impurities; instead, this AMR originates from an energy gap opening at the conducting surfacestates. Similar effects are expected in antiferromagnetic Dirac semime- tals, where the N /C19eel vector orientation determines high and low resis- tance states. 11The proximty-AMR we measure has a high resistance when magnetization is perpendicular to the surface and a low resis- tance when magnetization lies in the plane of the surface. In Appl. Phys. Lett. 118, 232402 (2021); doi: 10.1063/5.0052301 118, 232402-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplcomparison, conventional AMR of metallic ferromagnets has a high resistance when the magnetization is parallel to the current and a low resistance when perpendicular to the current.1The maximum percent change we find in resistance due to proximity-AMR is roughly 6.5% at1.8 K. This is significantly larger than spin Hall magnetoresistance(0.01%), an AMR-like effect induced in metals with strong spin–orbit coupling by magnetic insulators. 12 In order to study the proximity-AMR effect of the TI, we use bilayers consisting of a TI on a ferromagnetic insulator, which pro-vides an exchange interaction at the interface to “proximity”-magnetize the TI surface state. 13–16We note that the other strategy to magnetize a TI is to intentionally dope the material with magnetic ele- ments.17That approach has been key to the observation of the quan- tum anomalous Hall effect within a magnetic TI.18,19More recently, in the magnetically doped TI, AMR behavior has been observed20and used to characterize an asymmetric magnetoresistance effect due to magnon scattering. To rule out additional AMR effects that can arisefrom magnetic impurities, we chose to employ the magnetic proximity effect so that the AMR is intrinsic to the surface-state.10 We mechanically exfoliated Bi 2Se3flakes21–24directly onto the magnetic insulator yttrium iron garnet (YIG), as illustrated and shownthrough atomic force microscopy in Figs. 1(a) and1(b).T h eY I Gi s 25 nm thick and grown on a gadolinium gallium garnet (GGG) sub- strate via sputtering, 25–27and the flake is 30 nm thick. The YIG behaves as an in-plane ferromagnet with a low coercive field ( <2O e ) from 1.8 to 300 K.25–27Magnetometry over the relevant experimental temperature range is shown in Fig. 1(c) . Exfoliation of the Bi 2Se3leads to variable flake sizes, thicknesses (ranging from 25 to 60 nm), and resistivities. The most resistive flakes in this thickness range show proximity-induced ferromagnetic magnetotransport effects, asdescribed in this manuscript. More conductive flakes often show noproximity-induced ferromagnetic behavior; instead, these flakesreproduce previously reported weak anti-localization effects observed in TIs. 28–30 FIG. 1. (a) An illustration of a 30 nm Bi 2Se3flake exfoliated on YIG and contacted by Ti/Au electrodes. The coordinate system depicts that the current flows in the x-direction and an external magnetic field is applied in the xyplane at an angle /. The magnetization of the TI cants toward the in-plane direction of the external field as described by the polar angle w. (b) Atomic force microscopy image of a TI flake. False shaded yellow rectangles represent the patterned Ti/Au electrodes. (c) SQUID magnetometry of t he bare YIG film shown over the temperature range where we observed ferromagnetic effects in the magnetoresistance. Note that the coercive field of the YIG is le ss than 2 Oe over the relevant temperature range. (d) Resistance vs temperature for a resistive Bi 2Se3flake (red) compared with a conductive flake (blue) that are exfoliated on the same YIG substrate.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 232402 (2021); doi: 10.1063/5.0052301 118, 232402-2 Published under an exclusive license by AIP PublishingThe measurement coordinate system and relevant field and mag- netization vectors are illustrated in Fig. 1(a) . The magnetic field, H,i s applied at an in-plane angle /relative to the current. Because of per- pendicular magnetic anisotropy (PMA) in the TI,31the magnetization of the TI ( MTI) is not generally co-linear to the applied field. A polar angle, w, is used to describe the orientation of MTI.W em a k et w o - point measurements of the resistance across flakes as a function of applied magnetic field. In Fig. 1(d) ,w ep l o tt h et e m p e r a t u r e - dependent resistance of the flake shown in Fig. 1(b) ,a sw e l la st h a to fa similarly sized more conductive flake. The more resistive flake (red) d i s p l a y sas i g n i fi c a n tr i s ei nr e s i s t a n c ea st h et e m p e r a t u r ei sl o w e r e d ,while the more conductive flake (blue) shows a decrease in resistance. These are characteristic behaviors of TIs with Fermi level within the bulk gap (semiconducting behavior) and within the bulk conductionband (metallic behavior). 32Magnetoresistance of the conductive flake i ss h o w ni nt h e supplementary material Fig. S8, and no ferromagnetic behavior is observed. We note that due to the insulating nature of thesubstrate, electrostatic charging effects were problematic. The most robust samples were those measured in a two-point configuration, i.e., electrodes patterned transverse to the current often failed. Thus, we donot have Hall resistivity data as a companion to our longitudinal data. Below 4 K, the magnetoresistance of the resistive Bi 2Se3/YIG bilayers exhibits a strong 6.5% change in the MR as a function of field. In this low-temperature range, there are also pronounced ferromag-netic hysteresis effects in the MR signal. In Fig. 2(a) ,w ep l o tt h eM Rcurves for different temperatures at /¼90 /C14. Here, current and field are perpendicular to one another and the field is swept from negativeto positive values. At 4 K, there are negative dips in the magnetoresis- tance that are hysteretic; a similar effect has been previously observed and attributed to enhanced conduction from domain walls in the adja- cent magnetic insulator 13(see supplementary material Section 1.2). Below 4 K, a pronounced magnetoresistance peak emerges, whichcauses the zero-field resistance to considerably increase. In Figs. 2(b) and2(c), we show the magnetoresistance at 1.8 K for data taken at both /¼0 /C14and/¼90/C14. Unlike conventional AMR, we see a large signal when the field is applied in-plane-parallel and perpendicular to the current. Furthermore, when we apply a field of 5 kOe and measure the MR as the field is rotated in the xyplane, we observe no change in the resistance as a function of /; this is markedly different behavior from conventional AMR in metallic ferromagnets (see supplementary material Section 3). A near equal amplitude and ferromagnetic signal at both /¼0/C14and/¼90/C14, combined with no variation in the resis- tance as a function of /at large fields, is consistent with the TI having PMA where a high resistance state occurs at remanence from the mag-netization pointing out-of-plane. By applying an in-plane magnetic field at any orientation, we rotate the magnetization from a high resis- tance state to a low resistance state. We note that there is some weak anisotropy in the shape of the AMR signal when comparing /¼0 /C14 and/¼90/C14.I nsupplementary material Section 4, we discuss possible origins of this anisotropy. We now describe theoretical modeling and FIG. 2. (a) The temperature-dependent magnetoresistance for an in-plane appliedfield at /¼90/C14is shown. The field is swept from positive to negative, and a resistance offset is manually added for visi-bility purposes. A magnetoresistance peakemerges below 4.0 K, which saturates at fields above a few hundred Oe. (b) and (c) We show experimental data at 1.8 K for/¼0 /C14and /¼90/C14, respectively. Magnetic hysteresis is clearly observed for both configurations, and the presence of a magnetoresistance peak at both orienta-tions is inconsistent with the conventionalAMR of metallic ferromagnets.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 232402 (2021); doi: 10.1063/5.0052301 118, 232402-3 Published under an exclusive license by AIP Publishingmicromagnetic simulations, which support the assertion of PMA in the magnetized TI layer. We attribute the low-temperature MR peak to AMR in the TI arising from proximity-induced, perpendicular magnetization. Theoretically, if the Fermi energy of a TI is close to the Dirac point, a magnetized TI should have PMA.31In fact, neutron scat- tering measurements have provided evidence for magnetizationwith PMA induced in Bi 2Se3by the magnetic insulator, EuS, which has in-plane anisotropy.15InFig. 3(a) ,w ei l l u s t r a t eh o w proximity-AMR arises in a TI with PMA. At zero-field, the mag-netization of the TI is nearly parallel to the z-direction (out-of- plane) opening a gap in the surface states; canting from the z- direction is expected if the underlying magnetic insulator has in-plane anisotropy. When the magnetization has a large out-of- plane component, the resistance is then expected to be (relatively) large. As a field is applied in the plane of the sample, the magneti-zation rotates toward the plane and the gap will close. Throughout this process, the resistance continuously decreases as the out-of- plane component of the magnetization is reduced. We emphasizethat the angular dependence of the AMR in the TI only relies on the polar angle, wðHÞ, which describe the orientation of the out- of-plane magnetization as a function of the applied field. This is incontrast with conventional AMR, 1spin Hall magnetoresistance,12 and AMR induced by Rashba spin–orbit coupling,33where, quitegenerally, the magnetoresistance depends on the orientation of the magnetization relative to the current. To theoretically model the data, we need to obtain the field evolu- tion of wðHÞ. Previously, a diffusive transport model was formulated to explain how the resistance of a TI changes as the surface state becomes gapped by the magnetization.10With wðHÞ,t h i st r a n s p o r t model can be utilized, and the angular-dependence of proximity-AMRcan be written as 10 qxx¼/C22hh e2EFse1þn2cosw2 1/C0n2cosw2; (1) n¼fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þf2cosw2p : (2) In the above expression f¼D=EF,Dis the proximity-induced exchange energy, while EFis the Fermi energy relative to the Dirac p o i n tw h e nt h e r ei sn om a g n e t i z a t i o ni nt h e z-direction. Additionally, seis the electronic elastic scattering rate, eis the charge of the electron, his Planck’s constant, and /C22his the reduced Planck’s constant. A simple way to obtain wðHÞis to minimize a free energy expres- sion that treats the TI as a magnetic thin film with perpendicular anisotropy,34 F¼/C0 HM TIsinwþ2pM2 TIcos2w/C0Kucos2w; (3) FIG. 3. (a) Illustration of the AMR mecha- nism in a proximity magnetized TI. Perpendicular magnetization induced inthe TI by the YIG will open a gap (2 D) between the surface states. Applying an in-plane field ( H) rotates the magnetization (M TI) toward the in-plane direction, which leads to a decrease in the bandgap. (b)We plot the remnant magnetic moment configuration of our micromagnetic simula- tions after a field is applied along the þx- direction. Note how there is appreciablespin canting out-of-plane near the YIG-TI interface due to exchange coupling between competing anisotropies. (c) Weplot the field evolution of the z-componentof the TI magnetization and the x- component of the YIG magnetization obtained from micromagnetic simulations.The red and blue arrows indicate thesweep direction of the external field. The z-component of the magnetization in the TI is greatest right at the switching field ofthe YIG (offset from zero), which leads toa kink in the magnetoresistance. The cir- cled pairs of curves and the correspond- ing black arrows indicate which curvescorrespond to the x(z)-component of themagnetization. (d) We fit the magnetore- sistance, informed by micromagnetic sim- ulations, to the resistance expression inEq. (1). Red and blue arrows indicate sweep direction.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 232402 (2021); doi: 10.1063/5.0052301 118, 232402-4 Published under an exclusive license by AIP Publishingwhere Kuis a uniaxial anisotropy coefficient. With wðHÞacquired, the resistance line shape can, in principle, be fit with just two parameters, D=EFandKu. However, this approach does not capture the hysteretic features in the line shape; it only describes the smoothly varying com- ponent of the resistance. This is not surprising, given that this treat- ment of the data is phenomenologically similar to that observed withconventional AMR behavior when an applied field is along a hard axis. 35Still, the free energy model has utility as a simple approach to estimate D=EFand Ku, and we employ this model to do so. In Fig. 4(a),w ep l o t D=EFas a function of temperature as extracted from our free energy analysis for /¼90/C14. We find that D=EFincreases in a lin- ear manner between 3.5 and 2.0 K from approximately 5% to 14%.From the same analysis, we can estimate K uand we find that, within experimental error, it was independent of temperature corresponding to an effective uniaxial field of nearly 14 kOe. A more detailed discus- sion of this analysis is found in supplementary material Section 4. We note that similar results are obtained by performing the same analysisfor/¼0 /C14. A more faithful representation of the magnetic bilayer system can be constructed using micromagnetic simulations, where a ferro- magnet with in-plane anisotropy (the YIG) is exchange coupled to an out-of-plane ferromagnet (the TI surface). Using Object Oriented MicroMagnetic framework (OOMMF),36we built a 1D discretized lat- tice of 13 layers (YIG) with in-plane anisotropy coupled with 2 layers (TI) having perpendicular anisotropy. An illustration of a micromag- netic remnant moment configuration is shown in Fig. 3(b) .M a t e r i a lparameters were informed by our data as well as the literature, and a detailed discussion of simulations is found in supplementary material Section 5. We simulated the application of an external field and recorded the magnetization as a function of field. At every field step in the simulation, we averaged and normalized the magnetization to a unit vector. The z-component of the magnetization unit vector, as a function of external field, is wðHÞ; used in conjunction with Eq. (1), the magnetoresistance is modeled. InFig. 3(c) , we plot micromagnetic simulations of the aver- aged x-a n d z-components of the magnetization in the YIG and the TI, respectively. In Fig. 3(d) , we plot magnetoresistance generated from micromagnetic simulations as black lines on top of the exper- imental data. Taken together, these simulations reveal that the large, smoothly varying component of the signal originates from the tendency of the TI-magnetization to coherently rotate as an in- plane field is applied. However, because the YIG has in-plane anisotropy, the YIG-magnetization will suddenly switch as the field is reversed. Because the two materials have orthogonal anisotro- pies, exchange coupling causes the remanent magnetization state of TI to be slightly canted toward the sample plane. When the magnetic field is reversed, the magnetization of the TI begins to rotate further out of the sample plane causing the resistance to fur- ther increase and leading to a “spike” in the resistance offset from remanence. The resistance is largest just before the switching fieldof the YIG, after which the magnetization of the TI further rotates toward the plane lowering the resistance. FIG. 4. (a)D=EFis plotted as a function of temperature as red squares for / ¼90/C14. The micromagnetically obtained, temporally and spatially averaged, z-com- ponent of the magnetization in the TI is plotted as a function of temperature as blue squares. (b) An illustration of themagnetic ordering within the TI. From leftto right, the diagrams indicate the effective field that the TI (drawn in blue) experien- ces along with a potential magnetizationconfiguration. The magnetization in the TItends to order along an axis canted from the out-of-plane axis due to competing anisotropies between YIG and the TI, andthis direction is indicated as ^h. The aver- aged, effective magnetization of the TI is illustrated by the quantity ^M eff.(c) and (d) We plot the time trajectories of the spa-tially averaged magnetization of the TI asa unit vector on the unit sphere at 1 K and 4 K, based on micromagnetic simulations.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 232402 (2021); doi: 10.1063/5.0052301 118, 232402-5 Published under an exclusive license by AIP PublishingOur experimental data are phenomenologically consistent with theory, which only takes into account surface-state contributions tothe AMR. However, we need to consider whether bulk conduction in the TI could also contribute to AMR-like behaviors. 12Recent studies have shown that even for cases where there is significant bulk conduc-tion, the magnetotransport is sensitive to the surface states. 37,38 Further, in a metal or TI adjacent to a ferromagnet, proximity effects are usually observed to extend less than 3 nm into the non-magnetic metal.15,39This implies that ferromagnetic proximity effects, observed in a TI adjacent to a magnet, only exist at and near the surface of a TI. We also note some indirect evidence, which suggests that, for the resis- tive samples we measure, the Fermi level lies within the bulk bandgap.Hall measurements of our bulk BiSe 3single crystal (which is grown with excess Se40) reveal a carrier density40of 5/C21017cm/C03,av a l u e consistent with carrier densities measured in ARPES studies, which observe the Fermi energy within or near the bulk bandgap.41This is consistent with the observed increase in sample resistance as the tem- perature is lowered in our resistive samples. We posit that the temperature dependence of D=EFarises from the magnetic ordering of the TI near 4 K, as illustrated in Fig. 4(b) . Alternatively, a change in magnetic anisotropy as a function of tem-perature 31could describe the temperature dependence, but this effect is extremely small over the temperature range we measure (see supple- mentary material Section 6). To investigate thermal fluctuations over the temperature range, we experimentally measured and performedtemperature-dependent micromagnetic simulations. We precondition the micromagnetic calculations using material parameters obtained from modeling the line shape of the experimental data. In Fig. 4(a) ,w e plot the time and spatially averaged z-component of the magnetization i nt h eT Ia sb l u es q u a r e s .F r o m1 Kt o5 K ,t h eo u t - o f - p l a n em a g n e t i c moment decreases monotonically due to thermal fluctuations. In Figs. 4(c)and4(d), we plot the time-dependent trajectories of the spatially averaged TI magnetization on a unit sphere at 1 K and 4 K at zero- field. At 1 K, the magnetization tends to be aligned with PMA ( ^z), slightly canted toward the YIG magnetization ( ^x). At 4 K, the magneti- zation frequently switches between the two canted static equilibrium directions. All of these data taken together indicate that a low- temperature magnetic ordering effect is responsible for the emergenceof proximity-AMR below 4 K. In conclusion, we have reported an anisotropic magnetoresis- tance effect, which arises from a proximity-magnetized surface state ofthe TI. The change in the resistance is controlled by the component of the net magnetization, which is normal to the sample surface. This perpendicular magnetization opens a gap in the TI surface state, whichleads to an increase in resistance. The resistance of the TI changes by 6.5%, which is significantly larger than similar effects induced by mag- netic insulators into spin Hall metals. 12By electrically gating the TI, one can tune the Fermi energy closer to the Dirac point and enhancethis effect. See the supplementary material section that houses additional magnetotransport data and analysis. Furthermore, simulation details are provided and other mechanisms for the temperature dependenceobserved in the data are discussed and ruled out. AUTHORS’ CONTRIBUTIONS J.S and Y. Z. contributed equally to this work.Work at The University of Illinois at Urbana-Champaign was supported by the National Science Foundation MRSEC programunder NSF Award No. DMR-1720633. NM acknowledges support from the Army under No. W911NF-20–1-0024. G.J.M and Y.X. acknowledge support from the U.S. Department of Energy underGrant No. DE-SC0012368. Y.Z. acknowledges support from the University of Illinois. Work at Argonne, including YIG thin film growth, was supported by the U.S. Department of Energy, Office ofScience, Materials Science and Engineering Division. All authorscontributed to the data analysis and manuscript preparation. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1T. McGuire and R. 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5.0053633.pdf

AIP Advances ARTICLE scitation.org/journal/adv Performance improvement of Cu 2ZnSn(S,Se) 4 thin-film solar cells by optimizing the selenization temperature Cite as: AIP Advances 11, 075205 (2021); doi: 10.1063/5.0053633 Submitted: 9 April 2021 •Accepted: 10 June 2021 • Published Online: 1 July 2021 Xiaogong Lv,1,2 Chengjun Zhu,1,a) Yanchun Yang,1Ruijian Liu,1Wenliang Fan,2and Yiming Wang1 AFFILIATIONS 1Key Laboratory of Semiconductor Photovoltaic Technology of Inner Mongolia Autonomous Region, School of Physical Science and Technology, Inner Mongolia University, 235 West Daxue Street, Hohhot 010021, China 2Ordos Institute of Technology, Ordos 017000, Inner Mongolia, China a)Author to whom correspondence should be addressed: cjzhu@imu.edu.cn ABSTRACT In this work, Cu 2ZnSnS 4(CZTS) precursor films were deposited using a water-based solution approach. Subsequently, selenization was performed at different temperatures in the range of 480–610○C to prepare Cu 2ZnSn(S,Se) 4(CZTSSe) absorber-layer films. The effects of the selenization temperature on the crystallinity, structure, morphology, and photoelectric properties of CZTSSe thin films, as well as the performance of solar cells constructed using these films, were systematically studied. The absorber-layer films selenized at different temperatures all formed pure-phase CZTSSe and had basically the same film thickness. It was found that application of an optimal sel- enization temperature can enhance the crystallinity, crystal grain size, and mobility and reduce the resistivity of CZTSSe films. Selenization at 550○C resulted in the largest grain size ( ∼μm), the highest crystallinity, the highest mobility (4.29 cm2V−1s−1), the lowest resistivity (3.13 ×102Ωcm), the thinner fine-grained layer, a bandgap value of 1.21 eV, and a Cu-poor, Zn-rich elemental composition [Cu/(Zn +Sn) =0.85 and Zn/Sn =1.16]. The power-conversion efficiency was improved from 3.04% in a CZTSSe cell device with an absorber layer sel- enized at 480○C to 4.69% in a film selenized at 550○C. This was mainly due to the improvement of the crystallinity, crystal grain growth, and reduction of the fine-grained layer of the CZTSSe film. These results show that optimizing the selenization temperature is essen- tial for enhancing the performance and the ultimate device efficiency of CZTSSe absorber layers prepared using a water-based solution approach. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0053633 I. INTRODUCTION The last several decades have seen remarkable development of Cu(In,Ga)Se 2(CIGS) solar cells, achieving a 22.6% power- conversion efficiency (PCE).1However, the rarity and high cost of In and Ga have increased the need for alternative materials. It is possible to obtain Cu 2ZnSn(S,Se) 4(CZTSSe) with a crystal architec- ture similar to that of CIGS by substituting Zn and Sn for In and Ga in CIGS. Kesterite CZTSSe p-type semiconductor materials have been regarded as some of the most suitable choices for photovoltaic applications because they are nontoxic, are composed of elements that are abundant, have an excellent visible-light absorption coeffi- cient (>104cm−1), have an adjustable direct bandgap (1.0–1.5 eV),and—most importantly—have the highest theoretical efficiency of 32.2% according to the Shockley–Queisser limit.2–5 Several approaches have been adopted for fabrication of CZTSSe solar-cell devices. The molecular-precursor solution approach is preferred because of its simplicity, low cost, controllable element ratio, low energy consumption, high output, and easy large- area production. Using a hydrazine-based processing technique, Kim et al. improved the efficiency of CZTSSe devices to a record- breaking 12.7%.6However, the highly toxic and explosive nature of hydrazine limits its industrial applications. Therefore, there is an urgent need to find an environmentally friendly and less toxic sol- vent to replace hydrazine in the process of preparing CZTSSe solar cells. To date, a variety of low-toxicity and safe substitute organic AIP Advances 11, 075205 (2021); doi: 10.1063/5.0053633 11, 075205-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv solvents have been found and developed. For example, Xin et al. used dimethyl sulfoxide, achieving an efficiency of 11.8%.7Ethanol and 1- butylamine,8CS2and thioglycolic acid,9ethanolamine and thiogly- colic acid,10thioglycolic acid and methylamine,11ethanedithiol and ethylenediamine,122-methoxyethanol,13and other solvents have also shown good performance in the preparation of CZTSSe solar cells. However, these solvents are also toxic. It is well known that water is the cheapest and most environmentally friendly solvent, and it does not react with air. Therefore, water-based solution approaches for preparing CZTSSe solar cells have drawn consider- able attention. Tian et al. fabricated a CZTSSe cell having a record 6.62% efficiency,14confirming the potential of water-based solution approaches for producing highly efficient CZTSSe solar cells. The post-selenization temperature is a critical parameter for controlling the structure, crystallinity, elemental composition, and electrical and optical properties of CZTSSe absorber films, as well as the ultimate device efficiency.15,16Therefore, determining the opti- mal selenization temperature is crucial for enhancing the perfor- mance of these devices. In this work, CZTS precursor films were synthesized using a water-based solution approach combined with a spin-coating pro- cess, and CZTSSe films were then prepared by selenization. Solar-cell devices were then fabricated as per the classical structure (soda-lime glass [SLG]/Mo/CZTSSe/CdS/i-ZnO/ITO/Al). The influences of the selenization temperature on the crystallinity, electrical and optical properties, morphology, and architecture of CZTSSe films and the main parameters of the solar cells were investigated. The results show that optimizing the selenization temperature enables improve- ment of the CZTSSe film quality and the solar-cell performance. This work establishes an experimental foundation for the preparation of high-efficiency CZTSSe cell devices using water-based solution methods. II. EXPERIMENTAL METHODS A. Materials and synthesis of the CZTS precursor solution The precursor materials SnO (99.9% purity), ZnO (99.99% purity), CuO (99.9% purity), HSCH 2COO–NH 4+(70 wt. %), and Se (99.9% purity) were procured from Shanghai Aladdin Bio- Chem Technology Co. Additionally, NH 2CSNH 2(99% purity), CdSO 4⋅8/3H 2O (99% purity), and NH 4OH (25 wt. %) were pro- cured from Sinopharm Chemical Reagent Co. All these chemicals and reagents can be directly used in commercial production without further purification. As shown in Fig. 1, SnO (2.0 mmol), ZnO (2.4 mmol), and CuO (3.52 mmol) were added to a solution mixture ofammonium thioglycolate (3 ml) and deionized water (3 ml). The mixture was then magnetically agitated at room temperature for about 1 h until it became light yellow and transparent. The CZTS precursor solution was obtained after centrifuging the mixture at 13 000 rpm for 3 min. B. Formation of the CZTSSe absorber layer An SLG (18 ×18×1.1) mm3sputtered with a Mo layer (∼400 nm) was spin-coated using a Cu 2ZnSnS 4(CZTS) precursor liquid at 3000 rpm for 30 s. Then, a 5-min pre-annealing of the SLG was performed at 300○C on a hot plate. A CZTS precursor film with an ideal thickness ( ∼1.5μm) was fabricated through six repetitions of pre-annealing–spin coating. All the steps were carried out in a glove chamber filled with argon. Finally, selenization of the CZTS precursor film was performed for 10 min at different temperatures (480–610○C) in a circular graphite box including Se powder, which was placed in a fast thermoprocessing furnace with flowing nitrogen to synthesize the CZTSSe film. C. Fabrication of CZTSSe solar cells Fabrication of CZTSSe devices was accomplished as per the conventional architecture, namely, SLG/Mo/CZTSSe/CdS/i- ZnO/ITO/Al. As a first step, the CZTSSe film ( ∼1.5μm) was deposited with a CdS buffer layer ( ∼60 nm) using the chemical-bath approach. One layer of each of i-ZnO ( ∼70 nm) and ITO ( ∼250 nm) was then successively deposited via RF and DC sputtering processes. Through thermal evaporation, an Al mesh electrode ( ∼2.0μm) was deposited on top of the ITO layer. The last step was mechanically isolating the sample into nine CZTSSe cell devices (active area: 0.22 cm2). D. Characterization An x-ray diffractometer (MiniFlex 600, Rigaku) was used to acquire XRD spectra, and an x-ray photoelectron spectrom- eter (Amicus, Kratos) was used to perform XPS analysis. For collection of scanning electron microscopy (SEM) micrographs, an SEM system (S-4800, Hitachi) with an energy-dispersive x- ray (EDAX) spectrometer (QUANTAX 200, Bruker) was used. A UV–visible spectrophotometer (LAMBDA 750, PerkinElmer) was employed for determination of UV–visible spectra. A Hall mea- surement system (HMS-3000, Ecopia) was used to determine the Hall effect in the samples. A solar simulator (7-SCSpec, Sofn) was used to detect the J–Vcurves under simulated AM 1.5 G illumination. FIG. 1. Schematic diagram of the CZTSSe solar-cell fabrication process. AIP Advances 11, 075205 (2021); doi: 10.1063/5.0053633 11, 075205-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv III. RESULTS AND DISCUSSION Figure 2 shows XRD patterns from the CZTS precursor film and the CZTSSe films prepared at different selenization tempera- tures (480–610○C). The three dominant XRD peaks appearing at 27.10○, 28.40○, and 30.66○were ascribed to the (100), (002), and (101) crystalline planes, respectively, in the wurtzite CZTS archi- tecture [Fig. 2(a)]. The broad and weak diffraction peaks indicate that the CZTS crystal grains were small; the average grain size of the CZTS precursor film estimated via the Debye–Scherrer formula was about 18 nm. In Fig. 2(b), it can be seen that major diffraction peaks appear for all the selenized CZTSSe films at 27.27○, 45.24○, and 53.72○. These can be, respectively, ascribed to the (112), (204)/(220), and (312)/(116) crystalline planes in the kesterite CZTSSe archi- tecture.17–19There were no other diffraction peaks in the XRD patterns, indicating that pure-phase CZTSSe was obtained, which is a prerequisite for fabricating CZTSSe devices with high perfor- mance. After selenization, the films’ structures were transformed from metastable wurtzite to kesterite, in line with the results of Tian et al.14When selenized at 550○C, the (112) peaks were clearly inten- sified compared to selenization at a temperature of 480○C, and this was accompanied by a reduction in the full width at half maximum, indicating larger crystal grains and higher crystallinity. In contrast, the intensity of the (112) peaks decreased at 610○C, revealing that the crystallinity of the film decreased, perhaps because CZTSSe decomposed at this high temperature. Overall, the CZTSSe film sel- enized at 550○C had the best crystallinity and the largest crystal grains. The XPS-acquired chemical valence states of Cu, Zn, Sn, S, and Se for the 550○C-selenized CZTSSe film are displayed in Fig. 3. The Cu 2p binding energies peaked at 932.0 and 951.9 eV, showing a 19.9 eV difference, indicating Cu+ions. The Zn 2p peaks appeared at 1021.8 eV (2p 3/2) and 1044.8 eV (2p 1/2). The distance of 23.0 eV between these two peaks indicates the presence of Zn2+. The splitting peaks of Sn 3d were located at 485.9 and 494.3 eV, showing an 8.4 eV difference in energy, which indicates Sn4+ions. The S 2p peaks appeared at 161.9 eV (2p 3/2) and 163.2 eV(2p 1/2), and the distance of 1.3 eV between these two peaks indicates the presence of S2−. The peaks of Se 3d at 53.8 eV indicate Se2−ions. These results verified that the synthesized CZTSSe compounds are consistent with previous studies.20–22 Figure 4 shows the top-view and cross-sectional SEM micro- graphs of the as-deposited CZTS precursor film, as well as simi- lar images of the CZTSSe films selenized at different temperatures. The CZTS precursor film was very smooth, dense, and uniform in thickness ( ∼1.5μm), without pinholes or cracks, indicating a high- quality CZTS nanocrystalline film [Fig. 4(a)]. Larger crystal grains emerged on the surface of the film, but they were discontinuous because the lower selenization temperature resulted in poor crys- tallinity [Fig. 4(b)]. Although the CZTSSe crystal grains covered the surface, there were still many small grains owing to insuffi- cient crystallization [Fig. 4(c)]. The crystal grains further grew to the micrometer level and were evenly distributed, which reduced the carrier recombination and improved both the conductivity and photoconductivity of the absorber layer [Fig. 4(d)].23Moreover, the growth of crystal grains also can enhance the diffusion length and lifetime of minority carriers, leading to an increase in short-circuit current ( Jsc).24,25As the selenization temperature was increased, the grain size decreased significantly, and small holes appeared on the surface, which may have been caused by the decomposition of the CZTSSe film at high selenization temperatures [Figs. 4(e) and 4(f)]. These small holes formed inside the CZTSSe absorber layer and at the interface between the absorber layer (CZTSSe) and the buffer layer (CdS) will form a recombination center of carriers and led to a reduction in Jsc.26After selenization at different temperatures, the thicknesses of the CZTSSe films were about the same ( ∼1.5μm), and these formed a bilayer architecture comprising an upper layer of large grains and a lower layer of fine grains [Figs. 4(b)–4(f)]. This was attributed to the difficulty of selenium vapor diffusion to the bottom of the CZTS precursor film during the selenization process. The pre- vious literature has reported that the fine-grained layer can cause high series resistance and limit the transport of photogenerated car- riers, thereby reducing device performance.17It can be seen from FIG. 2. XRD spectra from (a) the CZTS precursor film and (b) CZTSSe films selenized at 480, 510, 550, 580, and 610○C. AIP Advances 11, 075205 (2021); doi: 10.1063/5.0053633 11, 075205-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. XPS results for (a) Cu 2p, (b) Zn 2p, (c) Sn 3d, (d) S 2p, and (e) Se 3d in the 550○C-selenized CZTSSe film. Fig. 4 that during the process of increasing the selenization tem- perature from 480 to 550○C, the thickness of the fine-grained layer decreased, while the thickness of the large-grained layer increased. The sample selenized at 550○C had the largest thickness of the large- grained layer, and it was flat and dense with no holes, which was conducive to reducing the series resistance and increasing Jsc. The elemental compositions of the CZTS precursor film and the 550○C-selenized CZTSSe film, as revealed by EDAX spectroscopy, are shown in Table I. The results reveal the successful synthesis of a copper-poor and zinc-rich CZTSSe film, which is highly desir- able. Fine regulation of the element ratio, and especially obtaining acopper-poor and zinc-rich ratio, is extremely important for fabricat- ing high-performance CZTSSe solar cells. For CZTSSe devices, both the open-circuit voltage ( Voc) and Jscare enhanced since a copper- poor and zinc-rich ratio can inhibit deep Cu Znantisite defects in the film, as well as related defect clusters including Cu Zn+ZnCu, Cu Zn +SnZn, and 2Cu Zn+SnZn.27The elemental ratios of the CZTS pre- cursor film [Cu/(Zn +Sn)=0.79 and Zn/Sn =1.17] and the CZTSSe film selenized at 550○C [Cu/(Zn +Sn)=0.85 and Zn/Sn =1.16] were basically the same as that of the precursor liquid, suggesting that these ratios can be effectively controlled with a water-based solution approach. FIG. 4. Top-view (upper panels) and cross-sectional (lower panels) SEM micrographs of (a) the CZTS precursor film and CZTSSe films selenized at (b) 480○C, (c) 510○C, (d) 550○C, (e) 580○C, and (f) 610○C. AIP Advances 11, 075205 (2021); doi: 10.1063/5.0053633 11, 075205-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv TABLE I. Chemical compositions of the CZTS precursor film and the CZTSSe film selenized at 550○C. Composition (at. %) Compositional ratio Film Cu Zn Sn S Se Cu/(Zn +Sn) Zn/Sn (Cu +Zn+Sn)/(S +Se) Precursor 19.33 13.19 11.32 56.15 0 0.79 1.17 0.78 Selenized at 550○C 20.56 12.96 11.17 15.08 40.23 0.85 1.16 0.74 FIG. 5. UV–visible spectra of (a) the CZTS precursor film and (b) the 550○C-selenized CZTSSe film. The insets show plots of ( αhυ)2vshυ. Figure 5 shows UV–visible spectra from the CZTS precursor film and the 550○C-selenized CZTSSe film. As presented in the insets of Figs. 5(a) and 5(b), bandgap calculations were accomplished by extrapolating the linear part of the curve for both films. Com- pared to the CZTS layer (1.50 eV), the CZTSSe film (1.21 eV) exhib- ited a smaller bandgap. The valence band maximum (VBM) is the anti-bonding state of the Cu d and S p orbitals, and the conduc- tion band minimum (CBM) is the anti-bonding state of the Sn s and S s orbitals of Cu 2ZnSnS 4. When large Se atoms (1.98 Å) par- tially replace small S atoms (1.84 Å) along with alloying into the host lattice, the VBM moves up and the CBM moves down, result- ing in a decrease in the bandgap value ( Eg=CBM −VBM).28–30 The replacement of elements was also confirmed by the above ele- mental composition analysis results (Table I). After selenization, the bandgap of the absorber-layer film was reduced from 1.5 to 1.21 eV, which expanded the light-absorption range and increased the carrier concentration. Figure 6 shows the variations in resistivity, carrier mobility, and carrier concentration of CZTSSe films with the selenization tem- perature. Based on these results, the CZTSSe films can be classi- fied as p-type semiconductor materials. The carrier concentration depends on the defect density in the film induced by the elemen- tal ratio.31As previously noted, the elemental ratios of the selenized CZTSSe films are fundamentally identical to that of the precursor liquid, which results in insignificant variation of their carrier con- centrations. The microstructure is an important factor influencing a film’s carrier mobility. A large-grained and dense microstructure weakens the grain-boundary scattering effect and promotes carriermobility,32whereas a poor microstructure reduces mobility. The high carrier mobility (0.11–4.29 cm2V−1s−1) within a 480–550○C selenization temperature range was due to the growth of grains in the films. Carrier mobility was highest when the films were selenized at 550○C. With higher selenization temperatures, the mobility reduces as a result of the decrease in crystal grain size and the appearance of holes in the film. When the carrier concentration is similar, the microstructure plays a critical role in the resistivity of the CZTSSe film. Because crystal-grain development reduces the number of grain FIG. 6. Resistivity, carrier mobility, and carrier concentration revealed by Hall measurements of CZTSSe films selenized at different temperatures. AIP Advances 11, 075205 (2021); doi: 10.1063/5.0053633 11, 075205-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv boundaries, the carriers can move more freely along their paths,33 resulting in lower resistivity. The CZTSSe film selenized at 550○C had the lowest resistivity (3.13 ×102Ωcm) due to its dense and large-grained microstructure. The I–Vcurve analysis showed that optimizing the seleniza- tion temperature enables enhancement of the CZTSSe film’s con- ductivity. Figure 7 shows the I–Vcurves of CZTSSe films selenized at various temperatures. As the selenization temperature increases from 480 to 550○C, the current density also increases (1.6 ×10−3to 18.35 ×10−3mA) at 2.0 V, and it reaches its maximum at 550○C. As the selenization temperature is increased further, the current density begins to decrease again. These results are consistent with the Hall measurements. Figure 8 shows J–Vcurves of CZTSSe cell devices with films selenized at 480, 550, and 610○C under standard AM 1.5G illumi- nation. Table II illustrates the fill factor (FF), Voc,Jsc, and PCE of the cell devices. The cell device with films selenized at 550○C had the highest PCE (4.69%), with an FF value of 36.58%, a Vocvalue of 0.36 V, and a Jscvalue of 31.20 mA/cm2. Compared to the device with films selenized at 480○C, its Jscand PCE were 36% and 54% higher, respectively; they were also higher than those of the device with films selenized at 610○C. The FF and Vocvalues of the solar-cell devices with films selenized at 480, 550, and 610○C were basically the same. Therefore, the higher PCE of the device with films selenized at 550○C was mainly due to its higher Jscvalue. The thickness of the absorber-layer film, the grain size, the thickness of the fine-grained layer, and the interface defects between the absorber layer and the buffer layer will all affect the Jscvalue of the solar cell. From the above characterization and analysis, it can be said that the thickness of CZTSSe absorber layers selenized at different tem- peratures is basically the same, and a pure CZTSSe phase is formed without any secondary phase. In addition, the manufacturing pro- cess for the device is also the same. Therefore, the improvement of the crystallinity, the growth of the crystal grains, and the reduction of the fine-grained layer were the main contributors to the increase in the Jscvalue of the cell device with films selenized at 550○C compared to that of the cell device with films selenized at 480○C. FIG. 7. I–Vcurves of CZTSSe films selenized at different temperatures. (Inset) Schematic illustration of the equipment used for the I–Vmeasurements. FIG. 8. J–Vcurves of CZTSSe solar cells with films selenized at 480, 550, and 610○C under AM 1.5G illumination. (Inset) Schematic structure of the cell devices. TABLE II. Photovoltaic parameters of CZTSSe cell devices with films selenized at 480, 550, and 610○C. Selenization Jsc Active temperature (○C)Voc(V) (mA/cm2) FF (%) PCE (%) area (cm2) 480 0.34 22.95 33.97 3.04 0.22 550 0.36 31.20 36.58 4.69 0.22 610 0.36 26.43 37.55 4.09 0.22 As the grain size decreased and the holes appeared in the CZTSSe film selenized at 610○C,Jscof the cell device with films selenized at 610○C was less than that of the cell device with films selenized at 550○C. IV. CONCLUSIONS In summary, a simple, environmentally friendly, and low-cost water-based solution method was adopted to fabricate kesterite CZTSSe absorber layers and solar cells. The effects of different sel- enization temperatures (480–610○C) on the structure, morphology, composition, and photoelectric properties of the CZTSSe absorber- layer films were studied. The results show that the properties of these CZTSSe films depend greatly on the selenization temperature. Applying an optimal selenization temperature can improve the qual- ities of the films. The CZTSSe film selenized at 550○C was found to have the best crystallinity, the largest grain size ( ∼μm), the thin- ner fine-grained layer, the highest mobility (4.29 cm2V−1s−1), and the lowest resistivity (3.13 ×102Ωcm). Importantly, an increase in the value of Jscin the solar-cell device with a film selenized at 550○C led to a higher PCE (4.69%) compared to that of the device with films selenized at 480○C (3.04%). These results show that the optimization of the selenization temperature is critical for fabricat- ing high-efficiency solar cells using a water-based solution approach. In the future, we will continue to optimize the selenization process to reduce or even eliminate the fine-grained layer in the absorber film, and we will use cation doping to further suppress the body defect AIP Advances 11, 075205 (2021); doi: 10.1063/5.0053633 11, 075205-6 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv concentration of the absorber layer for improvement of the Jscand Vocvalues of these solar cells. ACKNOWLEDGMENTS This study was financially supported by the National Natu- ral Science Foundation of China (Grant No. 62064010), the Nat- ural Science Foundation of Inner Mongolia Autonomous Region (Grant Nos. 2020MS05004 and 2019MS01022), the Scientific Research Foundation of Ordos Institute of Technology (Grant Nos. KYZD2019004 and KYYB2018010), and the Scientific Research Foundation of the Higher Education Institutions of Inner Mongolia Autonomous Region of China (Grant No. NJZY16379). 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10.0005181.pdf

Low Temp. Phys. 47, 533 (2021); https://doi.org/10.1063/10.0005181 47, 533 © 2021 Author(s).Superlattice on the surface of a nanotube Cite as: Low Temp. Phys. 47, 533 (2021); https://doi.org/10.1063/10.0005181 Submitted: 21 May 2021 . Published Online: 27 July 2021 A. M. Ermolaev , and G. I. Rashba ARTICLES YOU MAY BE INTERESTED IN Dynamical Green’s function for elastic half-space, and energy losses due to collision Low Temperature Physics 47, 555 (2021); https://doi.org/10.1063/10.0005183 Temperature gradient and transport of heat and charge in a semiconductor structure Low Temperature Physics 47, 550 (2021); https://doi.org/10.1063/10.0005182 Bose systems in linear traps: Exact calculations versus effective space dimensionality Low Temperature Physics 47, 577 (2021); https://doi.org/10.1063/10.0005185Superlattice on the surface of a nanotube Cite as: Fiz. Nizk. Temp. 47,5 7 7 –595 (July 2021); doi: 10.1063/10.0005181 View Online Export Citation CrossMar k Submitted: 21 May 2021 A. M. Ermolaev1and G. I. Rashba2,a) AFFILIATIONS 1Academician I. M. Lifshitz Theoretical Physics Department 2V. N. Karazin Kharkiv National University, Kharkiv 61022, Ukraine a)Author to whom correspondence should be addressed: georgiy.i.rashba@gmail.com ABSTRACT The results of theoretical studies of the thermodynamic, kinetic, and high-frequency properties of the electron gas on the surface of a nanotube in a magnetic field in the presence of a longitudinal s uperlattice are presented. Nano-dimensions of the motion area lead to energy quantization. Its multiply connected structure in the presence of a magnetic field leads to effects that are derived from the Aharonov –Bohm effect. It is shown that the curvature of a nanotube, even in the absence of a magnetic field, causes new macroscopic oscillation effects such as de Haas –van Alphen oscillations, which are associat ed with the quantization of the transverse electron motion energy and with the root peculiarities of the den sity of electron states on the nanotube surface. Thermodynamic potentials and heat capacity of the electron gas on the tube are calculated in the gas approximation. The Kubo formula for the con- ductivity tensor of the electron gas on the nanotube surface is obt ained. The Landau damping regions of electromagnetic waves on a tube are determined and the beats are theoretically predicted on the graph of the dependence of conductivity on tube parameters.In the hydrodynamic approximation, the plasma waves on the surface of a semiconductor nanotube with a superlattice are consid-ered. It is shown that optical and acoustic plasmons can propagate along a tube with one kind of carrier. Electron spin waves on the surface of a semiconductor nanotube with a superlattice in a ma gnetic field are studied. The spectra and areas of collisionless damping of these waves are found. We have shown that the spin wave damping is absent in these areas if the tubes with a degener-ate electron gas have small radius. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/10.0005181 Contents 1. Introduction 1 2. Nanotubes with superlattices: classification and research development2 2.1. Classification of nanotubes 2 2.2. Thermodynamics of nanotubes 2 2.3. Response function 22.4. Collective excitations 33. Electron energy spectrum and density of states on the nanotube surface with superlattice3 3.1. Electron energy spectrum 3 3.2. Density of electron states 44. Thermodynamic quantities of a nanotube with a superlattice 54.1. Degenerate electron gas 5 4.2. Non-degenerate electron gas 6 5. Response function of electron gas on a tube with superlattice 85.1. Conductivity tensor 85.2. Magnetic susceptibility 136. Collective excitations on a tube with a superlattice 13 6.1. Plasmons 13 6.2. Spin waves 15 7. Conclusion 16 1. INTRODUCTION In the sixties of the last century, Moisey Isaakovich Kaganov, working on the book,1could not ignore the promising direction in solid state physics –semiconductors with superlattices, which was rapidly developing. Superlattice is an additional translational sym-metry artificially created on the surface and in the sample volume. The physical properties of materials with an additional artifi- cially created periodic structure (superlattice) differ significantly from the corresponding properties of homogeneous bodies. Theadditional potential of the superlattice modifies the band structureof the base material. Since the period of the quantum superlattice d is much larger than the lattice constant, the Brillouin zone is divided into a number of minibands. This creates narrow subbandsLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,533 Published under an exclusive license by AIP Publishing(minibands) separated by forbidden regions in the conduction band and the valence band of the initial crystal. However, for such periodicity to significantly affect the behavior of quasiparticles (electrons, phonons, magnons), certain conditions must be met. First, the average energy of a quasiparticle should be comparable with the miniband width, and second, the superlattice period should be much less than the quasiparticle mean free path l:d/C28l. This inequality can be rewritten as Δ/C29/C22h/τ, where Δis the half- width of the miniband, τis the relaxation time. Superlattices, in which the above conditions are satisfied, are called quantumsuperlattices. The idea of creating a quantum superlattice was first expressed by L. V. Keldysh, 2who proposed to use a powerful ultrasonic wave to obtain additional periodic potential. The first samples of quantum superlattices were synthesized as early as 1971 by themolecular epitaxy method. 3,4Continuous progress of methods of molecular beam epitaxy from metallic-organic compounds made it possible to create high-quality heterostructures based on the GaAs-Ga 1−xAlxAs system.5Currently superlattices are the main ele- ments of a new technology called band-gap engineering. The main results obtained by Moisey Isaakovich in this area are published in Refs. 6–10. In Ref. 6the theory of acoustoelec- tronic interaction in crystals with superlattices is developed. Reference 7contains the calculation of the thermodynamic quanti- ties of superlattices in a magnetic field: chemical potential, mag-netic moment, and sample temperature for a nondegenerate anddegenerate electron gas. Oscillations in temperature are also found in this article. They are used as a method for adiabatic cooling of the sample. Reference 8is devoted to the theory of electrical con- ductivity of semiconductors with superlattices in a quantizing mag-netic field. This paper shows that a change in the quantizingmagnetic field applied along the axis of a one-dimensional super- lattice induces metal-dielectric phase transitions. Also in this paper, the characteristics of the photoconductivity of the superlattice in aquantizing magnetic field are calculated. In Ref. 9, the spectrum of low-frequency electromagnetic oscillations in a superlattice was cal-culated under the conditions of the quantum Hall effect and an assumption was made about the possibility of experimental detec- tion of these waves in Hall dielectrics. A popular presentation ofthe basic ideas of the physics of semiconductors with superlatticesis contained in the book. 10 The properties of massive semiconductors with superlattices are considered in many publications by domestic and foreignauthors. Extensive reviews and monographs have been written, theydescribe a number of important aspects of superlattice physics. 11–17 The advances in modern nanoelectronics are associated both with the development of technology and with advances in funda- mental physics, describing the thermodynamic and high-frequencyproperties of nanostructures. Advances in physics and the technol-ogy of solid-state nanostructures have led to the creation of a scien-tific foundation for their widespread use in nanoelectronics. 18In this regard, the techniques and methods well known in the theory of massive semiconductors with superlattices, being applied to nanoob-jects, are filled with new content. Solid-state nanostructures arenano-sized objects characterized by the presence of inhomogeneitiesof various nature and configurations within semiconductor and dielectric media. The range of these nanostructures is quite wide:quantum wires, 18,19quantum dots,18,20fullerenes and nanotubes.21 Although these objects differ in their physical nature, they are united by their very small size in one or several directions. These dimen-sions are only one or two orders of magnitude larger than the char-acteristic interatomic distance. Under these conditions, the quantumnature of the motion of current carriers manifests itself in an essen- tial way. It is well known that the reception, transmission and pro- cessing of information in monomolecular structures are based onquantum processes of charge transfer in them. This circumstanceleads to the widespread usage in scientific literature of such terms, asquantum computers, quantum information systems. The physical mechanisms, used in these structures, are the tunneling effect and the interaction of charged quasiparticles (conduction electrons) witha periodic potential, they have also been encountered earlier in solidstate physics and the theory of massive superlattices. However, asapplied to nanotubes, the nature of these effects significantly changes due to the quasi-one-dimensional type of conductivity and tube cur- vature. Therefore, the known results, related to 3D macroscopicsamples, cannot be transferred to nanotubes. Thus, there is a needfor new fundamental studies of thermodynamic and electromagneticprocesses in quasi-1D macromolecules in general and in nanotubes in particular. The purpose of this review article is to summarize the results of the authors ’investigations in the field of the physical properties of nanotubes with superlattices. These studies cover the thermody- namics of nanotubes, 22dynamic conductivity,23,24collective excita- tions (plasmons and Landau –Silin spin waves) of the electron gas of nanotubes with a longitudinal superlattice25–34and are based on a unified approach. The approach developed here uses the analogybetween physical processes in nanotubes with superlattices and processes in massive superlattices, which were studied by M. I. Kaganov. 6–10The review material is presented using a number of important problems in the physics of nanotubes as examples. Itshould be emphasized that the problems considered as a complex,in addition to theoretical and applied interest, serve as bright illus- tration of the general trends in the development of modern physics of nanostructures. 2. NANOTUBES WITH SUPERLATTICES: CLASSIFICATION AND RESEARCH DEVELOPMENT 2.1. Classification of nanotubes Thirty years have passed since the discovery of carbon nano- tubes by Iijima. 35However, the interest in these nanosystems is so great that in recent years a new direction in physics and technologyhas emerged –carbon nanomaterial science. Nanotubes are pre- pared by rolling up a graphene sheet (or two-dimensional hetero- structure) into a tube. Depending on the rolling-up manner, thetube has metallic, semiconductor, or dielectric properties. Manyarticles and reviews have appeared in the world of scientific litera-ture (see, for example, Refs. 36–38), in which the properties of nanotubes are studied. They are interesting to physicists because nanotubes are dielectric, semiconductor, metal, so the methodsdeveloped to study these systems are transferred to nanotubes andother electron nanosystems on curved surfaces. 38To study their properties, it is necessary to synthesize the methods of quantum mechanics, statistics and kinetics and Riemannian geometry. A newLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,534 Published under an exclusive license by AIP Publishingparameter appearing in theory (the curvature of the structure) con- tributes to enriching the picture of phenomena in nanosystems increasing the ways to control their properties. In electronicsystems on curved surfaces, effects have already been discoveredthat have no analogue in systems with flat geometry. These includeeffects of hybridization of size and magnetic quantization of the motion of conduction electrons, modification of the electron Hamiltonian, 38specific resonances in the scattering of electrons in carbon nanotubes39and quantum wires19by impurity atoms. The logic behind the development of solid state physics is such that currently the objects of investigations are not only three dimensional systems with superlattices,6–17but also low- dimensional systems. Modern technologies allow creating not onlynanotubes but nanotubes with superlattices. Along with flatsuperlattices, 2,4,40–45also ones with cylindrical symmetry exist.18 They are of radial and longitudinal types.18,46The radial superlat- tice is a set of coaxial cylinders, while the longitudinal one looks like a set of coaxial rings of the same radius. The tubes with longi-tudinal superlattice are prepared by lithographic methods. It can beobtained by embedding fullerenes or other additives to the nano-tube or when the nanotube is attached to a substrate for charge exchange. 47In such a system, there exists the periodic potential acting upon electrons moving along the tube. 2.2. Thermodynamics of nanotubes The thermodynamic functions of electron gas on a nanotube surface have been studied in literature.48–54In Ref. 51the geometri- cal effects in ideal quantum gases of electrons, photons and phonons in confined space were considered. In Ref. 52a thermody- namic analysis of the boron-nitride nanotubes nucleation on thecatalysts surface was performed. In Refs. 53and 54the chemical potential, energy, pressure and the work function of an electronicgas on a conducting carbon nanotube surface under zero tempera- ture are calculated. Within the framework of the Hartree –Fock approximation the contact electron-electron interaction is takeninto account. Analytical form of the work function of carbon nano-tubes was derived in the paper. 54At large radii of nanotubes the limit to the work function of graphene was done. Superlattice at the surface of a carbon nanotube has been pre- viously studied.55,56In Ref. 56authors estimated orbital magnetiza- tion of the electron gas at the surface of the nanotube with thesuperlattice in a magnetic field parallel to the axes of the tube and the superlattice. Using the model suggested by the authors, 56we calculated the heat capacity of the degenerate electron gas.22In Subsections 4.1, 4.2 we present the results of calculations of suchthermodynamic functions of the degenerate and nondegenerateelectron gas on the semiconductor cylindrical nanotube surface in a longitudinal magnetic field as chemical potential, internal energy and heat capacity. We employ the effective mass approximationand Poisson summation formula for the calculation of the densityof states. 22 2.3. Response function Simplified conductivity models are usually used for studying electromagnetic waves propagation in the cylindrical geometry systems, for example, in nanotubes. The metal cylinderconductivity is often believed to be endless, and the dielectric per- mittivity of the matter in which cylinder is dipped is considered to be constant or only frequency dependent. Conductivity ’s spatial dispersion is usually not taken into account. Nonetheless, the elec-tromagnetic field ’s nature in the tube, its waveguide characteristics are sensitive to the surface currents. Therefore, the electron gas conductivity tensor components calculation problem with allow- ance for the spatial and time dispersion is worth consideration. Inconnection with increased interest in currents within the cylindricalconductors, the authors of Ref. 57have calculated the longitudinal conductivity for solid and hollow cylinders without superlattice in magnetic field and considered quantum electromagnetic waves in such systems. Exact expressions for all the components of the con-ductivity tensor for degenerate and nondegenerate electron gas onthe nanotube surface without superlattice are presented in Ref. 58. It is worth to be clarified how the superlattice affects this tensor. In Sec. 5.1, the components of the dynamic conductivity tensor are calculated based on the effective mass model for a nanotube with alongitudinal superlattice in a magnetic field. The superlattice axisand the magnetic field strength vector are assumed to be parallel tothe tube axis. The reaction of the electron gas of a nanotube to a weak alter- nating magnetic field is characterized by the tensor of dynamicmagnetic susceptibility. The components of the magnetic suscepti-bility tensor presented in Subsection 5.2 will be used to solve the dispersion equation to determine the spectrum of Landau –Silin spin waves on a tube with a superlattice in Subsection 6.2. 27 2.4 Collective excitations Plasma waves on the surface of carbon35and semiconductor nanotubes59,60were studied in Refs. 61–67. Plasmon in the nanotubes are studied mainly in approximation of randomphases 34,62,66,67and in the hydrodynamic approximation.65,68In the framework of the hydrodynamic approach, the plasma waves on the surface of a nanotube with a longitudinal superlattice inSubsection 6.1 are considered. Not only longitudinal, but alsotransverse electron currents are taken into account. It was shownthat both optical and acoustic plasmons can propagate through a tube with one type of carriers. 32 Electron spin waves on the surface of a semiconductor nano- tube with a superlattice in a magnetic field have been consideredin the Subsection 6.2. These waves in bulk conductors were pre-dicted by Landau 69and Silin.70Their properties in bulk conduc- tors were considered in Refs. 71–73. Subsection 6.2 discusses spin-wave spectra on the surface of a nanotube with a superlatticeand regions of collisionless damping of waves. It is shown thatspin waves are not damped in small-radius tubes with a degener- ate electron gas. 31 3. ELECTRON ENERGY SPECTRUM AND DENSITY OF STATES ON THE NANOTUBE SURFACE WITHSUPERLATTICE 3.1. Electron energy spectrum The conduction electron energy spectrum in the carbon and semiconductor nanotubes has a band nature. A small electronLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,535 Published under an exclusive license by AIP Publishingdensity near the band edge permits to use the effective mass approximation. This approximation allows describing the proper- ties of such systems qualitatively and often also quantitatively. Energy of electron with effective mass m*on the surface of a cylindrical nanotube with radius awith longitudinal superlattice consists of the energy of rotational motion, l2/2m*a2, and that of longitudinal motion Δ(1−coskd), where l= 0,± 1, …and kare projections of electron angular moment and momentum, respec-tively, onto the axis of the tube. The expression Δ(1−coskd)i s usually used in the tight-binding model of electrons in a crystallattice. 1,7,8,10,11,13–15Here Δand dare, respectively, amplitude and period of modulating potential on the tube surface. If kd << 1, this expression becomes k2/2m, where m*¼1/Δd2. Here-inafter, the Planck ’s constant is set to unity. In the magnetic field B, parallel to the tube axis Z, the energy of electron rotational motion becomes equal to ε0(l+η)2,74where ε0=( 2m,a2)−1is the rotational quantum, η=Φ/Φ0is the ratio of magnetic flux Φ=πa2Bthrough the tube crosssection to the flux quantum, Φ0=2πc/e( eis the elec- tron charge, cis the velocity of light). Taking into consideration the spin splitting of levels, we obtain the electron energy εikσ¼ε0(1þη)2þΔ(1/C0coskd)þσμBB,( 3 :1) where μβis the magnetic moment of an electron, σ= ±1 corre- sponds to two spin orientations. The longitudinal effective mass of an electron is supposed to be equal to its transversal one. Flux ratio η=Φ/Φ0is included in Eq. (3.1) in the form of l+η. This allows limiting ηto 0≤η≤1. The order of miniband location depends on η.I fη< 1/2, we have ε0η2<ε−1<ε+1<ε−2<…Ifη>1/2 then ε−1<ε0η2<ε−2<…Here the spin level splitting will not be taken into consideration. At η< 1/2 lower miniband is within [ε0η2,ε0η2+2Δ], and the next is within [ ε−1,ε−2+2Δ]. Energy gap between them is equal to ε0(1−2η)−2Δ. Width of the kth gap between ( k+ 1)th and kth minibands ( k= 1,3, …) is equal to ε0k(1−2η)−2Δ. Usually in experiments with nanotubes of radius a∼10−7−10−6cmε0/C29Δ, and the relationship between fluxes in different fields is far less than unity, therefore minibands don ’t overlap. However, with an increase in tube radius their overlap isinevitable. The effect of Coulomb interaction of electrons on the tube onto the energy spectrum to Hartree –Fock approximation was dis- cussed in Ref. 75. Screening the electron-electron interactions was studied in Refs. 76and 77. The Hartree-Fock correction to the spectrum (3.1) in the model of contact interaction of electrons has a usual form gn /C0σ,78where nσis the surface density of electrons with spin projection σ,gis the Fourier component of electron short-range interaction energy. In this case, the electron energy onthe tube with a longitudinal superlattice is 27 εσ ik¼ε0(1þη)2þΔ(1/C0coskd)þgn/C0σþσμBB: (3:2) The first term in Eq. (3.2) refers to the quantized levels of the cir- cular motion of electrons on the tube in the magnetic field, thesecond term is the energy of the longitudinal motion of the elec-trons, and the third and fourth terms are the exchange shift and the spin splitting of the levels, respectively. The energy spectrum of the longitudinal motion of electrons consists of narrow minibandswith the widths 2 Δseparated by energy gaps. The minibands can overlap. Small-radius tubes correspond to the case with a smallnumber of occupied lower minibands. Figure 1 shows schematically spectrum (3.1) in the first Brillouin zone ( −π/d<k < π/d, when two lower spin-split minibands 0 ±(l=0 ,σ= ±1) overlap. We consider the case of η< 1/ 2 when the positions of the lower boundaries ε+ lof the minibands satisfy the inequalities ε/C0 0,εþ 0 ,ε/C0 /C01,εþ /C01,...The miniband overlapping region [ εþ 0,ε/C0 0þ2Δ] inFig. 1 has the width 2 Δ−Ωwith Ω¼gδnþ2μB, where δn=n −−n+. 3.2. Density of electron states Electron density of states with the spectrum (3.1) is calculated according to the formula ν(ε)¼X ikσδ(ε/C0εikσ): This equals ν(ε)¼L πdX iσΘ(ε/C0εσ l)θ(εσ lþ2Δ/C0ε)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(ε/C0εσ l)(εσ lþ2Δ/C0ε)p : (3:3) Here εσ l¼εσ l0,Lis the tube length, Θis the Heaviside function. In the absence of a superlattice, Eq. (3.1) represents a system of one-dimensional subbands with root singularities of state density at their boundaries εσ l. Modulating potential converts this spectrum FIG. 1. Electron energy (3.1) in two overlapping minibands 0±.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,536 Published under an exclusive license by AIP Publishingto minibands 2 Λwide with boundaries εσ landεσ l+2Δ.Figure 2 shows the dimensionless density of states A¼πνdε0/2L(3.3) in the two lower minibands of the spectrum (3.1) as a function of ε/ε0for parameters η= 0.1, Δ/ε0= 0.1, usually used in experiments.56When ε/C29ε0, the sum of lincluded in (3.3) can be substi- tuted with an integral expression. As a result, the spectrum of the nanotubes becomes continuous, and density of states is nowequal to ν(ε)¼4L πdffiffiffiffiffiffiffiffiffiffi2Δε0p Kffiffiffiffiffiffiε 2Δr/C18/C19 , ε,2Δ, 4L πdffiffiffiffiffiffiffiffiffiffiffiffi2Δεε 0p Kffiffiffiffiffiffi 2Δ εr ! ,ε.2Δ,8 >>>>< >>>>:(3:4) where K(k) is the complete elliptic integral of the first kind with modulus k. 79Considering the abovementioned relationship between m*withΔandd, we are reassured that (3.4) represents the density of states of a two-dimensional electron gas with a one-dimensional superlattice in the absence of a magnetic field, occupy-ing a band with area S=2παL. This system can be obtained by cutting the tube along its length and turning it inside out to form a surface. If ε/C282Δfrom (3.4) the density of states of a two- dimensional electron gas in the absence of a superlattice isobtained: ν 0¼m*S/π. Poisson formula is used for the calculation of l, included in Eq. (3.3),a t ε/C29ε0. Then ν¼νmonþνosc, where νmon is the monotonic component of the density of states of Eq. (3.3), and νosc is the oscillating component. The latter contains Fourier integral with a finite limits, where integrand has a root singularity at the limits of integration. Asymptote of the far Fourier component of this integral is known.80From it we obtained νosc(ε)¼4L πdε0ε0 ε/C16/C173/4ε 2Δ/C16/C171/2X1 l¼11ffiffi lpcos 2πlΦ Φ0cos 2 πffiffiffiffiffiε ε0r /C0π 4/C18/C19 ,ε0/C28ε,2Δ, νosc(ε)¼4L πdε0ε0 ε/C16/C173/4ε 2Δ/C16/C171/2X1 l¼11ffiffi lpcos 2πlΦ Φ0cos 2 πffiffiffiffiffiε ε0r /C0π 4/C18/C19 þ1/C02Δ ε/C18/C19/C01/4 cos 2 πlffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε ε01/C02Δ ε/C18/C19s þπ 4 ! "# , ε/C29ε0,ε.2Δ:(3:5) Function (3.5) oscillates with the change in electron energy and magnetic flux Φ. The amplitude of oscillations decreases with increase in energy proportionally to ε−1/4. 4. THERMODYNAMIC QUANTITIES OF A NANOTUBE WITH A SUPERLATTICE 4.1. Degenerate electron gas Using density of states (3.4) and (3.5) let us calculate the number of electrons N, their energy E, chemical potential μand heat capacity C. Let us consider degenerate gas at the surface of the nanotube with a longitudinal superlattice. In the case appropriate for nanotubes with a small radius, when at zero temperature electrons partially fill only the lowerminiband, we obtained N¼4L πdarcsinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμ0/C0ε 2Δ,r E¼4L πdΔ1þε Δ/C16/C17 arcsinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμ0/C0ε/C0 2Δr /C01 2Δffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (μ0/C0ε/C0)(εþ2Δ/C0μ0)p/C20/C21 : (4:1) Here ε−=ε0η2is the lower limit of spectrum (3.1),μ0is the Fermi energy. From Eq. (4.1) Fermi energy is found μ0¼ε/C0þ2Δsin2πdnlin 4: FIG. 2. Density of states (3.3) at the two lowest minibands of the spectrum (3.1) for parameter values given in the text.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,537 Published under an exclusive license by AIP PublishingThe energy of a completely filled miniband is equal E¼2LΔ d1þε/C0 Δ/C16/C17 , where nlin=N/Lis linear electron density. In order to obtain heat capacity of electron gases one must perform Sommerfeld expansion81,82of the functions Nand Eof powers of T/μ, where Tis the temperature (Boltzmann constant equal unity is assumed). This is possible if the chemical potential is located far from the features of state densities, i.e., the following inequalities must be met T/C28μ/C0ε/C0,T/C28εþ/C0μ,( 4 :2) where ε±are the upper and lower boundaries of the last partially filled miniband. Corrections on the order of T2in expansion of N andEare equal NT¼πLT2 3d(μ0/C0ε/C0þΔ)[(μ0/C0ε/C0)(εþ/C0μ0)]/C03/2, ET¼πLT2 3d[μ0Δ/C0ε/C0(εþ/C0μ0)][(μ0/C0ε/C0)(εþ/C0μ0)]/C03/2: IfT/C28μ0/C0ε/C0/C282Δthen corrections in chemical potential and energy due to temperature are equal δμ¼π2T2 12(μ0/C0ε/C0), δE¼πLΔT2 3d(2Δ)3/2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμ0/C0ε/C0p :(4:3) δΕtakes into account a term present due to the dependence of chemical potential on temperature. From (4.3) we obtained mono- tonic component of the nanotube ’s heat capacity Cmon¼πLT 3dffiffiffiffiffiffi 2Δpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμ0/C0ε/C0p : (4:4) Using Eq. (3.4) heat capacity of an electron gas with a super- lattice in the absence of a magnetic field at low temperatures can beobtained. If μ 0<2Δheat capacity equals C¼TS 3dffiffiffiffiffiffim* Δr 1/C0μ0 2Δ/C16/C17/C01 Effiffiffiffiffiffiμ0 2Δr/C18/C19 þ1/C0μ0 2Δ/C16/C17 Kffiffiffiffiffiffiμ0 2Δr/C18/C19 /C20/C21 ,( 4 :5) where E(k) is the complete elliptic integral of the second kind.79 Coefficient at Tin this formula is calculated precisely. If μ0/C282Δ, from Eq. (4.5) standard expression for the heat capacity of an elec- tron gas without a superlattice is obtained: C¼πm*TS/3, where density of states ν0is used. In accord with the Pauli principle heat capacity (4.4) and (4.5) is proportional to the temperature. However, proportionality coefficient is a complex function of the μ0/Δparameter.Oscillating components Nand Eat conditions (4.2) and ε0/C28μ0,2Δare equal NOSC EOSC/C18/C19 ¼4(ε0μ0)1/4L π2dffiffiffi 2p1 μ0/C18/C19 /C2X1 l¼11 l3/2cos 2 πlΦ Φ0/C18/C19 sin 2 πlffiffiffiffiffiμ0 ε0r /C0π 4/C18/C19 /C2λl sinhλl, (4:6) where λl=π2lT/(ε0μ0)1/2. Functions (4.6) experience oscillations similar to de Haas –van Alphen and AharonovBohm type oscilla- tions with changes in μ1/2 0related to electron density and magnetic fluxΦ. The first are due to passage of root singularity of state density (3.3) at miniband boundaries through Fermi energy. This brings oscillations in consideration closer to de Haas –van Alphen type oscillations in a magnetic field.81,82However nonequidistance of energy levels of cross-sectional movement of electrons in thetube brings about ( μ 0/ε0)1/2in phase with oscillations (4.6). These oscillations exist in absence of a magnetic field. Their period is equal to τ¼1/ffiffiffiffiffiffiffiffiffiffi2m*ap.A measurement of the period allows one to obtain effective mass of an electron. Amplitude of oscillationsdecreases with an increase in temperature, as it does in the usual case of de Haas –van Alphen effect in a quantizing magnetic field. 81,82 From Eq. (4.6) let us obtain the oscillating term of heat capac- ity of a nanotube: COSC¼4μ0L dffiffiffiffiffiffi 2Δp (ε0μ0)1/4X1 l¼11ffiffi lpcos 2 πlΦ Φ0/C18/C19 /C2sin 2 πlffiffiffiffiffiμ0 ε0r /C0π 4/C18/C191 sinhλl(1/C0λlcothλl): (4:7) With an increase in temperature monotonic component of heat capacity (4.4) exceeds the oscillating component (4.7) if T.μ0(μ0/ε0)1/4 Figure 3 illustrates dependence of the amplitude of the main harmonic of the oscillating component of heat capacity (4.7) B¼4ffiffiffi 2pL dε0 Δ/C16/C171/2μ0 ε0/C18/C193/4λlcothλl/C01 sinhλl on the temperature when Φ/Φ0= 0.1 for the values of GaAs parameters that are usually used in experiments:83m*= 0.07 m0 (where m0is the mass of a free electron), a=1 0−6cm,μ0/ε0= 10, L=1 0 μm,Δ= 1 meV, d= 3500 Å. Amplitude of Breaches its maximum value at temperature Tm/(ε0μ0)1/2(αdenotes proportionality). 4.2. Non-degenerate electron gas At a fixed number of electrons chemical potential of a nonde- generate electron gas can be determined from equation N¼X mkσexp[β(μ/C0εmkσ)], (4 :8)Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,538 Published under an exclusive license by AIP Publishingwhere βis the reverse temperature. Sums included in this expres- sion are determined precisely. For estimating the sum by mthe fol- lowing formula is used84 X1 m¼/C01exp[/C0x(mþυ)2]¼ffiffiffiπ xrX1 l¼/C01exp/C0π2l2 x/C18/C19 cos 2π/υ,x.0 The sum by kis reduced to Bessel ’s modified function of the first kind85 I0(x)¼1 πðπ 0df/C1ex/C0cosf: As a result, solution of Eq. (4.8) has the form μ¼1 βInNd 2Lffiffiffiffiffiffiffiffiffiffiffiffiffiffi βε0 πeβΔr( /C2coshβμBB/C1I0(βΔ)1þ2X1 l¼1exp/C0π2l2 βε0/C18/C19 cos 2πlΦ Φ0 ! "#/C01) : (4:9) This shows that chemical potential undergoes Aharonov –Bohm type oscillations with a change in magnetic field crossing the tube.De Haas –van Alphen type oscillations are not present in this case. In the absence of a superlattice they were considered in article. 86 Energy of an electron gas can be calculated by equation81,82 E¼/C0N@ @βInX mkσexp(/C0βεmkσ):It equals E¼N 2β2βε0 mþΦ Φ0/C18/C192*+ þ1/C02βΔ1/C0I0 0(βΔ) I0(βΔ)/C20/C21 /C02βμBBthβμBB() , (4:10) where Pmhi¼P mPmexp(/C0βεm) P mexp(/C0βεm),εm¼ε0mþΦ Φ0/C18/C192 : Derivative with respect to the argument of the Bessel function is marked with a prime ( ‘). Heat capacity of an electron gas equals C¼N 22(βε0)2mþΦ Φ0/C18/C194*+ /C0 mþΦ Φ0/C18/C192*+2 "#( þ1þ2(βΔ)2[I00 0(βΔ)I0(βΔ)/C0(I0 0(βΔ))2] /C2[I0(βΔ)]/C02þ2βμBB coshβμBB/C18/C192"# ) : (4:11) Separate terms in Eqs. (4.10) and(4.11) agree with the energy term in(3.1). The first term on the right side of expression (4.10) repre- sents the average energy of centripetal motion of electrons at the surface of the nanotube, the second and third terms are due to lon-gitudinal motion of electrons along the tube, and the last term isdue to spin splitting of energy levels of an electron in a magneticfield. It coincides with the energy of a two-level system with dis- tance 2μ BBbetween the levels. Expression (4.11) shows that the presence of a magnetic field does not affect the heat capacity termpresent due to electron motion. At the same time, modulation doesnot affect heat capacity related to centripetal motion of electronsand spin splitting of levels. Using the presentation of a Bessel func- tion as a row and its asymptote, “longitudinal ”component of heat capacity (4.11) is confirmed to be equal to C jj¼N 2{1þ2(βΔ)2[I00 0(βΔ)I0(βΔ)/C0(I0 0(βΔ))2]/C2[I0(βΔ)]/C02} ¼N 2,βΔ/C281, N,βΔ/C281:( This result agrees with the classical theory on equipartition of energy about degrees of freedom.81Its physical meaning is obvious. If energy of thermal motion of electrons β−1is small compared to the modulating potential amplitude, the electrons oscillate slightlyin the modulating potential gaps. These oscillations make a contri-bution to heat capacity in the amount of N.I fβ −1exceeds modula- tion amplitude Δ, the electrons move freely along the tube. Contribution of this motion to heat capacity is equal to N/2. Thus, term C||changes from NtoN/2 as temperature increases. “Transverse ”part of heat capacity depends on the magnetic flux. In weak magnetic fields, the inequality holds Φ/C28Φ0. This allows the dismissal of magnetic field influence on the “transverse ” FIG. 3. T emperature dependence of the amplitude of the oscillating component of heat capacity (4.7) for parameter values given in the text.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,539 Published under an exclusive license by AIP Publishingcomponent of heat capacity C?Then the following limiting expres- sions can be obtained: C?¼N/2, βε0/C281 N(βε0)2exp(/C0βε0),βε0/C291:/C26 As expected, high temperature limit of C?is in accord with the theorem on equipartition of energy about degrees of freedom. 5. RESPONSE FUNCTION OF ELECTRON GAS ON A TUBE WITH SUPERLATTICE 5.1. Conductivity tensor For the nanotube with superlattice in magnetic field, the surface electron gas linear response to an electromagnetic wave E=E0expi(mw+qz−ωt) is characterized by conductivity two- dimensional tensor σαβ(m,q,ω). Here Eis the electric field of wave, mis the integer number, qandωare the wave vector and fre- quency of the wave, wand zare cylindrical coordinates. The density of surface current on the tube is jα(m,q,ω)¼X βσαβ(m,q,ω)Eβ(m,q,ω), (5 :1) where jα(m,q,ω) and Eβ(m,q,ω) are cylindrical harmonics of j and Evectors. Kubo ’s formula for the conductivity tensor of elec- tron gas on the surface of the nanotube with superlattice is58 ααβ(m,q,ω)¼ie2n m*ωδαβþ1 ωð1 0dteiωt[^Jα(m,q,t),^Jβ(/C0m,/C0q, 0)]/C10/C11 , (5:2) where nis surface density of electrons, J(m,q,t) is the cylindrical harmonics of current density operator in the external magneticfield B. The angle brackets denote the average value of the operator commutator. The quantum constant was assumed as unity. The components of J(m,q) vector are ^J w¼/C02e m*affiffiffi SpX lklþηþm 2/C16/C17 ^aþ lk^a(lþm)(kþq), ^Jz¼/C02e m*ffiffiffi SpX lkkþq 2/C16/C17 ^aþ lk^a(lþm)(kþq),(5:3) where ^alkand ^aþ lkare operators of annihilation and creation of elec- trons in jlkistate, S=2πaLis the surface area for the tube with length L. Spin splitting of levels is not considered in Eq. (5.3). From Eqs. (5.2) and (5.3) we obtain the components of con- ductivity tensor: σww¼ie2n m*ωþi2e2 m2 *a2ωSX lkf(εlk) /C2lþηþm 2/C16/C172 εlk/C0ε(lþm)(kþq)þωþi0/C0lþηþm 2/C16/C172 ε(l/C0m)(k/C0q)/C0εlkþωþi02 643 75, (5:4)σwz¼σzw¼i2e2 m2 *aωSX lkf(εlk) /C2lþηþm 2/C16/C17 kþq 2/C16/C17 εlk/C0ε(lþm)(kþq)þωþi0/C0lþη/C0m 2/C16/C17 k/C0q 2/C16/C17 ε(l/C0m)(k/C0q)/C0εlkþωþi02 43 5, (5:5) σzz¼ie2n m*ωþi2e2 m2 *ωSX lkf(εlk) /C2kþq 2/C16/C172 εlk/C0ε(lþm)(kþq)þωþi0/C0k/C0q 2/C16/C172 ε(l/C0m)(k/C0q)/C0εlkþωþi02 643 75: (5:6) Here fis Fermi function, εlk¼ε0(lþη)2þΔ(1/C0coskd): (5:7) The second term addend in the right part of Eq. (5.7) is often used in the theory of semiconductor superlattices.7,8,10,11–15The real parts of the components σwwandσzzare even functions of mand ω, while imaginary parts are odd ones. At zero temperature in sum- mation Σkthe values kin the formulas (5.4) –(5.6) are limited to gap−kl≤k≤kl,where kl¼1 darccosεlþΔ/C0μ0 Δ is the maximum momentum of the electrons in the miniband l,εl=ε0(l+η)2is the miniband boundaries. If q= 0, at zero temper- ature from the formulas (5.4) –(5.6) we calculate the components of dynamical conductivity tensor: Reσww(m,ω)¼e2 πm2 *a3ωX lkllþηþm 2/C16/C172/C20 /C2δ(ω/C0Ωþ)/C0lþη/C0m 2/C16/C172 δ(ω/C0Ω/C0)/C21 , Imσww(m,ω)¼e2n m*ωþe2 π2m2 *a3ωX lkllþηþm 2/C16/C172 ω/C0Ωþ/C0lþη/C0m 2/C16/C172 ω/C0Ω/C02 643 75, σwz(m,ω)¼0, (5 :8) Reσzz(m,ω)¼e2 3πm2 *aωX lk3 l[δ(ω/C0Ωþ)/C0δ(ω/C0Ω/C0)], Imσzz(m,ω)¼e2n m*ωþe2 3π2m2 *aωX lk3 l1 ω/C0Ωþ/C01 ω/C0Ω/C0/C20/C21 :(5:9) Here Ω±=ε0m[2(l+η)±m] are frequencies of direct transitions of electrons between the miniband boundaries s lin the field of elec- tromagnetic wave. During the transitions, conservation laws for longitudinal components of angular moment, momentum, andLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,540 Published under an exclusive license by AIP Publishingenergy are satisfied. At zero temperature, the summation over lin Eqs. (5.8) and (5.9) is limited by the condition jεlþΔ/C0μ0j/C20Δ. This means that Fermi energy is concentrated within the miniband.The minibands are positioned in the intervals [ ε l,εlþ2Δ] and have the width 2 Δ. Generally, the semiconductor nanotubes with radius a∼(10−7−10−6) cm in magnetic field B∼105G are used.In this case, the electrons of the semiconductor nanotube occupy little quantity of bottom minibands, which boundaries at η< 1/2 satisfy the inequality ε0η2,ε/C01,ε1,ε/C02,... In the quantum limit where n,1/πad, Fermi energy is concentrated in the bottom miniband l=0 [ε0η2,ε0η2+2Δ]. In this case, in the absence of spatial dispersion, from Eqs. (5.8) and(5.9) we obtain Reσww¼e2k0 πm2 *a3ω/C2 ηþm 2/C16/C172 δ(ω/C0ε0m(2ηþm))/C0η/C0m 2/C16/C172 δ(ω/C0ε0m(2η/C0m))η/C0m 2/C16/C172 δ(ω/C0ε0m(2η/C0m))/C20 /C21 , Imσww¼e2n m*ωþe2k0 π2m2 *a3ωηþm 2/C16/C172 ω/C0ε0m(2ηþm)/C0η/C0m 2/C16/C172 ω/C0ε0m(2η/C0m)2 643 75,(5:10) Reσzz¼e2k3 0 3πm2 *aω[δ(ω/C0ε0m(2ηþm))/C0δ(ω/C0ε0m(2η/C0m))], Imσzz¼e2n m*ωe2k3 0 3π2m2 *aω1 ω/C0ε0m(2ηþm)/C01 ω/C0ε0m(2η/C0m)/C20/C21 : (5:11) Here Ω±=ε0m(2η±m). The superlattice parameters Δand dare included in Eqs. (5.10) and(5.11) only via the maximum momen- tum k0of electrons in the bottom miniband. In the absence of superlattice: Δ→∞,d→∞,d2Δ→m/C01 *.T h e n kl=[ 2m*(μ−εl)]1/2 and the Eqs. (5.10) and(5.11) agree with ones obtained in Ref. 58.A t m= 0 ,o n l yt h ei m a g i n a r yp a r t e2n/m *ωremains in Eqs. (5.10) and (5.11) , while the real part is zero. This determines the electromagnetic wave energy absorbed by electrons. In the absence of direct and indi- rect transitions of electrons, the absorption is zero. As the electron density grows, the number of addends in Eqs. (5.8) and(5.9) increases. If Fermi energy is concentrated in the second miniband, the oscillatorforces of electron resonance transitions in Eqs. (5.8) and (5.9) are determined by values k 0andk−1.T h e s ea r ei n c l u d e di nE q s . (5.8) and (5.9) if the minibands are over-lapped, i.e. ε0η2þ2Δ.ε/C01,a n d Fermi energy is concentrated in the overlap area [ ε−1,ε0η2+2Δ]. Otherwise, the overlapping of minibands is absent. Then themaximum momentum of electrons k 0in the completely occupied bottom miniband corresponds to Brillouin zone boundary π/d. In the quantum limit, taking into account the spatial dispersion, the real part of conductivity depends on Fermi level position in thebottom miniband. If μ 0is positioned in the bottom half of the mini- band ( ε0η2<μ0<ε0η2+Δ,q<π/2d)f r o mE q . (5.6) we obtain k/C0¼1 darcsinjω/C0Ωþj 2Δsinqd 2,ω/C0,ω,ωþ, ω+¼Ωþ+2Δsinqd 2α0sinqd 2þffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C0α2 0q cosqd 2/C18/C19 , α0¼ε0η2þΔ/C0μ0 Δ:The real parts of other compone nts of conductivity tensor are obtained from Re σzzusing substitution of ( k−)2by (ηþm/2)2/a2in Reσwwand by k/C0(ηþm/2)/ain Reσwz If the Fermi level is positioned in the upper half of the mini- band ( ε0η2þΔ,μ0,ε0η2þ2Δ,π/2d,q,π/d), we obtain Reσzz¼e2(kþ)2 2πm2 *adω4Δ2sin2qd 2/C0(ω/C0Ωþ)2/C20/C21/C01/2 , where kþ¼π d/C0k/C0,ω/C0,ω,ωþ, ω+¼Ωþ+2Δsinqd 2jα0jsinqd 2þffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C0α2 0q cosqd 2/C18/C19 : The real part of conductivity is nonzero in the area of Landau damping [ ω−,ω+] of electromagnetic waves in the tube. In the quasi-classical case, the quantization of electron circular rotation can be neglected. That is possible under condition ofε 0/C28μ0. Substituting the lsummation by integrals in Eqs. (5.8) and(5.9), we obtain Reσww¼e2ω 8πm2 *(aε0jmj)3(kþIþ/C0k/C0I/C0), Reσzz¼e2 6πm2 *aε0jmjω(k3 þIþ/C0k3 /C0I/C0),(5:12) where k+¼1 darccosε+þΔ/C0μ0 Δ,ε+¼ε0ω 2mε0+m 2/C18/C192 , I+(m,ω)¼Θω 2mε0+m 2þffiffiffiffiffiμ0 ε0r /C18/C19 Θffiffiffiffiffiμ0 ε0r /C0ω 2mε0+m 2/C18/C19Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,541 Published under an exclusive license by AIP PublishingandΘis the Heaviside function. If ω> 0 and jmj,2ffiffiffiffiffiffiffiffiffiffiffiμ0/ε0p, the Eq. (5.12) become as follows Reσww¼e2ω 8πm2 *(aε0jmj)3(kþ/C0k/C0), 0,ω,ω/C0, kþ, ω/C0,ω,ωþ, 0, ω.ωþ,8 < :(5:13) Reσzz¼e2 6πm2 *aε0jmjω(k3 þ/C0k3 /C0), 0,ω,ω/C0, k3 þ, ω/C0,ω,ωþ, 0, ω.ωþ,8 < :(5:14) where ω+¼2ε0jmjffiffiffiffiffiffiffiffiffiffiffiμ0/ε0p+ε0m2. Equation (5.13) and(5.14) are Reσww¼e2 8πm2 *ε2 0(ajmj)3dGa, Reσzz¼e2 6πm2 *ε2 0ajmjd3Gb, where Ga(x)xarccosε0(x/C0(μ0/ε0))2 4m2Δ/C0arccosε0(xþμ0/ε0))2 4m2Δ/C20/C21 ,0,x,x/C0, xarccosε0(x/C0μ0/ε0))2 4m2Δ, x/C0,x,xþ, 0, x.xþ,8 >>>>< >>>>: G a(x)1 xarccosε0(x/C0(μ0/ε0))2 4m2Δ/C18/C19 3 /C0arccosε0(xþμ0/ε0))2 4m2Δ/C18/C19 3"# ,0,x,x/C0, 1 xarccosε0(x/C0(μ0/ε0))2 4m2Δ/C18/C19 3 , x/C0,x,xþ, 0, x.xþ,8 >>>>>>< >>>>>>: x=ω/ε 0,x±=ω±/ε0. The value ( μ0/ε0)12inω±is equal to the classical angular moment akFof Fermi electron in the circular orbit ( kFis Fermi momentum). In Fig. 4 the dependences of Ga [Fig. 4(a) ], and Gb[Fig. 4(b) ] functions on xare shown for parame- ters m= 10, μ0/ε0=Δ/ε0= 100 typical for semiconductor superlattices. Atjmj.2ffiffiffiffiffiffiffiffiffiffiffiμ0/ε0pwe obtained Reσww¼e2ω 8πm2 *(aε0jmj)30, 0,ω,ω, kþ,ω/C0,ω,ωþ, 0, ω.ωþ,8 < :(5:15) Reσzz¼e2 6πm2 *aε0jmj0, 0,ω,ω, k3 þ,ω/C0,ω,ωþ, 0, ω.ωþ,8 < :(5:16) where now ω+¼+2ε0jmjffiffiffiffiffiffiffiffiffiffiffiμ0/ε0pþε0m2. Functions (5.15) and (5.16) are represented as Reσww¼e2 8πm2 *ε2 0(ajmj)3dFa,Reσzz¼e2 6πm2 *ε2 0ajmjd3Fb, where Fa(x)¼arccosε0(x/C0m2)2 4m2Δ,x/C0,x,xþ, Fb(x)¼1 xarccosε0(x/C0m2)2 4m2Δ"#3 ,x/C0,x,xþ, x=ω/ε0,x±=ω±/ε0.I nFig. 5 the dependences of Fa[Fig. 5 (a)], and [ Fig. 5 (b)] Fbfunctions on xare shown for parameters m=5 , μ0/ε0=Δ/ε0= 4 under condition of jmj.2ffiffiffiffiffiffiffiffiffiffiffiμ0/ε0p. As the real part of conductivity is connected with electromag- netic field energy absorbed by electrons, the Eqs. (5.13) –(5.16) determine the boundaries of Landau damping for waves withpositive and negative helicity. These boundaries are parabolas inthe“angular moment-frequency ”plane. The real part of the con- ductivity determines the damping decrement of electromagnetic waves on the tube. High-frequency asymptotics of the conductivityLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,542 Published under an exclusive license by AIP Publishingimaginary part at ω/C29Ω+are Imσww¼e2n m*ωþ2e2m π2m2 *a3ω2X lkl(lþη), (5 :17) Imσzz¼e2n m*ωþ2e2ε0m2 3π2m2 *aω3X lk3 l: (5:18) Ifε0/C28μ0, the sums by lincluded in Eqs. (5.17) and(5.18) are cal- culated by Poisson formula. Then the Eqs. (5.17) and (5.18) for components contain monotonic ( σmon) addends and oscillating (σosc) ones. These depend on the ratio of Fermi energy to the mini- band width. When ω/C29Ω+,ε0/C28μ0andμ0<2Δafter integration by parts and replacing the integration variable, the oscillating part of the sum J¼P lkl(lþη) in formula (5.17) is equal toJosc¼/C04 πdffiffiffiffiffiμ ε0rX1 r¼11 rsin 2πrηð1 0dyy2cos 2 πrffiffiffiffiffiμ ε0r y/C18/C19 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/C0y2)(y2þα2)p , α¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2Δ/C0μ)/μp : Asymptotic of this integral under μ0/C29ε0as is known.80As a result, the imaginary part of the transverse conductivity (5.17) is equal Imσww¼e2n m*ω/C023/2e2m π3m2 *a3dω2μ ε0/C18/C191/4μ Δ/C16/C171/2 /C2X1 r¼1sin 2πrη r3/2cos 2 πrffiffiffiffiffiμ ε0r /C0π 4/C18/C19 : (5:19) FIG. 5. The real part of conductivity from Eqs. (5.15) and (5.16) depending on frequency for values of parameters referred in the text under condition of jmj.2ffiffiffiffiffiffiffiffiffiffi μ0/ε0p . FIG. 4. The real part of conductivity (5.13) and (5.14) depending on frequency for values of parameters referred in the text under condition of jmj,2ffiffiffiffiffiffiffiffiffiffi μ0/ε0p .Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,543 Published under an exclusive license by AIP PublishingI nt h ec a s eo f μ0>2Δwe obtain, Imσww¼e2n m*ω/C023/2e2m π3m2 *a3dω2μ ε0/C18/C191/4μ Δ/C16/C171/2X1 r¼1sin 2πrη r3/2 /C2cos 2 πrffiffiffiffiffiμ ε0r /C0π 4/C18/C19 þμ/C02Δ μ/C18/C193/4 cos 2 πrffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ/C02Δ ε0r þπ 4/C18/C19"# : (5:20) The Eqs. (5.19) and (5.20) undergoes Aharonov-Bohm oscillations under variation of magnetic flux th rough the tube cross-section. The oscillation period is e qual to the flux quantum Φ0. Also the oscilla- tions looking like de Haas –van Alphen ones exist. They are caused by transition of root singularities of electron density of states at the mini- band boundaries through Fermi boundary due to the tube radius var- iation or changing the electron density. The latter is related withFermi energy as follows μ 0¼π 2dnffiffiffiffiffiffi Δ m*r ,μ/C282Δ, 1 8m*(πdn)2,μ/C292Δ:8 >>< >>: Analyzing the dependence of oscillations (5.19) on (adn)1/2we obtain the period τ¼πaffiffiffiffiffiffiffiffiffim*Δp/C0/C1/C01/2. Ifμ0<2Δ, only the miniband bottom boundaries εlpass through Fermi boundary when the tube parameters change. As a result, in Eq. (5.19) the base frequency of oscillations is present only. The second addend in Eq. (5.20) exists because at μ0>2Δnot only miniband εlbottom boundaries transverse Fermi level but the upper ones εl+2Δas well. Existence of two oscillation frequencies in Eq. (5.20) causes the beats in the plot of conductivity versus the tube parameters. They are similar to the beats of plasma and spinwaves spectra in the tube. 26,27IfΔ/C28μ0the relative difference of conductivity oscillation frequencies and amplitudes in Eq. (5.20) is of the order of Δ/μ0. As this parameter increases, the beats turn into weak modulations and disappear at μ0<2Δ. The sum by lin longitudinal conductivity (5.18) is calculated by Poisson formula as well. Consequently, the conductivity contains monotonic and oscil-lating components. At μ 0>2Δthey are equals Imσmon zz¼e2n m*ωþ2e2ε0m2 3π2m2 *aω3d3Jmon,( 5 :21) Imσosc zz¼2e2ε0m2 3π2m2 *aω3d3X1 r¼1cos 2πrη/C1Jr osc,( 5 :22) where Jmon(b)¼2ðb ffiffiffiffiffiffiffiffiffiffi b2/C02c2pdxarccosx2/C0b2þc2 c2/C18/C19 3 ,( 5 :23)Jr osc(b)¼4ðb ffiffiffiffiffiffiffiffiffiffi b2/C02c2pdxcos(2 πrx) arccosx2/C0b2þc2 c2/C18/C19 3 , b¼(μ/ε0)1/2,c2¼Δ/ε0:(5:24) The integrals (5.23) and (5.24) are not calculated exactly. InFig. 6(a) , the dependence Jmon (b) calculated numerically is shown. Solid-, dotted-, and chain-line curves correspond to c2= 10, 20, 30, respectively. The Josc(b) dependence at r= 1 and c2=1 5 i s shown in Fig. 6(b) . In accordance with formulas (5.21) –(5.24) the monotonic part of the conductivity and the amplitude of the oscillating partdecrease with frequency ω n, increases. Figure 6(b) shows the weak modulations are caused by the beats. FIG. 6. Monotonic (a) and oscillating (b) components of longitudinal conductivity (5.21) and (5.22) atμ0>2Δas the functions of ( μ0/ε0)1/2for parameter values referred in the text.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,544 Published under an exclusive license by AIP PublishingThe imaginary part of the transverse conductivity (5.19) and(5.20) behaves similarly. It is includes into the dispersion equa- tion for the electromagnetic waves spectrum. The beats and oscilla-tions obtained here there exist only in the quasiclassical case.Their reasons were described above. The ratio of the oscillationamplitude of the transverse conductivity (5.19) to the amplitude of the oscillation in the absence of the superlattice 58is of order a/d(ε0/Δ)1/2. The longitudinal conductivity oscillations of de Haas –van Alphen type exist also in the absence of magnetic field. 5.2. Magnetic susceptibility Within the framework of random phase approximation,78,87,88 circular components of the tensor of dynamical spin susceptibility of an electron gas with the spectrum (3.2) on the tube surface are89–91 χ+(m,q,ω)¼χ0 +(m,q,ω)1/C0g 2μ2 Bχ0 +(m,q,ω)/C20/C21/C01 ,( 5 :25) where χ0 +(m,q,ω)¼2μ2 B SX lkf(ε(lþm)(kþq)+)/C0f(εlk+) ωþεlk+/C0ε(lþm)(kþq)+þi0(5:26) is the susceptibility of the ideal electron gas with spectrum (3.2), χ±=χxx±iχyx,m,qand ωare, respectively, angular moment, moment and frequency of the spin wave. The real and imaginary parts of the component χ0 /C0(5.26) of electron degenerated gas are Reχ0 /C0(m,q,ω)¼μ2 B 4π2adΔsinqd 2X lPðkþ l/C0q 2ðÞ d /C0kþ l/C0q 2ðÞ ddx/C1(C/C0/C0sinx)/C012 664 /C0Pð/C0k/C0 lq 2ðÞ d /C0k/C0 lq 2ðÞ ddx/C1(Cþ/C0sinx)/C013 775,( 5 :27) Imχ0 /C0(m,q,ω)¼μ2 B 4π2adΔsinqd 2/C12/C12/C12/C12/C12/C12/C12/C12 /C2P lPÐk/C0 lþq 2ðÞ d /C0k/C0 lþq 2ðÞ ddx/C1δ(sinx/C0Cþ)2 4 /C0Ðkþ l/C0q 2ðÞ d /C0kþ l/C0q 2ðÞ ddx/C1δ(sinx/C0C/C0)3 5, (5:28) where k+ l¼1 darccosε+ lþΔ/C0μ0 Δis the maximum moment of electrons in the miniband with (l, σ=+) number, C+¼ω/C0Ω+ 2Δsinqd 2,Ω+¼ε0m[2(lþη)+m]þΩ (5:29) are frequencies of vertical transitions of electrons between miniband boundaries [ ε+ l,ε+ lþ2Δ]( w h e r e ε+ l¼ε0(lþη)2þσμBBþgn/C0σ) with spin-flip transition /C0!þ under action of an alternating field. Sum over lin(5.27) and (5.28) is limited by the condition jε+ lþΔ/C0μ0j,Δwhich means that Fermi energy is in the miniband (l,σ=± ) . 6. COLLECTIVE EXCITATIONS ON A TUBE WITH A SUPERLATTICE 6.1 Plasmons In the framework of the hydrodynamic approach, using the continuity equation for electron liquid and Poisson equation for electrical potential, the authors65,66have obtained the dispersion equation for the spectrum of surface plasma waves on the tube: ω¼4πam2 a2Imσww(m,ω)þq2Imσzz(m,ω)/C20/C21 Im(qa)Km(qa)(6:1) where mis the projection of the plasmon angular moment on the tube axis z;σwwandσzzare components of electron gas dynamical conductivity in absentia of spatial dispersion ( qυ0/C28ω,υ0is the Fermi velocity) in cylindrical coordinates w,z;ImandKmare modi- fied Bessel functions. The Eq. (6.1) is true also for the tube with a superlattice. Substituting Drude expression for conductivity ie2n/m*ωinto Eq.(6.1), we obtained the known spectrum for intraband ( m=0 ) and interband ( m≠0) plasmons:65–67 Ω2 mq¼4πe2an m*m2 a2þq2/C20/C21 /C2Im(qa)Km(qa): (6:2) The Eq. (6.2) does not take into account the interband current caused by quantum transitions of electrons in the wave fieldbetween the minibands. Taking that into account, the transverse component of dynamical conductivity tensor for electron gas on the tube is σ ww¼ie2n m*ωþi2e2 m1 *a2ωSX lkf(εlk) /C2 lþm 2/C16/C172 (ω/C0Ωþþi0)/C01/C0lþm 2/C16/C172 (ω/C0Ω/C0þi0)/C01/C20/C21 ,( 6 :3) where Ω±=ε0m(2l±m) are frequencies of direct transitions of electrons between the minibands. The longitudinal conductivityσ zzis obtained from (6.3) using substitution of a/C02lþm 2/C0/C12byk2 andσwz=σzw= 0. In Eq. (6.3) we apply the electron energy on the surface of the semiconductor nanotube with a superlattice:26,74 εlk¼εol2þΔ(1/C0coskd). The imaginary part of the interbandLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,545 Published under an exclusive license by AIP Publishingconductivity has resonance singularities at frequencies Ω±. The Landau attenuation is concentrated in narrow bands δω∼Δqd near these frequencies.27 In formulas for conductivity, we restrict ourselves to the quantum limit where electrons in the degenerated gas occupy par-tially only the lower miniband l= 0 with width 2 Δand their density does not exceed 1/ πad. In this case, the solution of Eq. (6.1) is defined by the parameter α m¼3m2/4(ak0)2. That is connected with forces of oscillators for resonance transitions 0 !mof elec- trons between the minibands. Here k0¼1 darccosΔ/C0μ0 Δ is the maximum momentum of an electron in the miniband with l=0 . Ifαm< 1, there exists a series of branches in the plas-mon spectrum ω2 mw¼1 2{ω2 mþΩ2 mqþ[(ω2 mþΩ2 mq)2 þ16 3a m/C16/C174 ω2 mqΩ2mq(1/C0αm)(qk0)21þq2a2 m2/C18/C19 /C01#1/2) , (6:4) where ωm=ε0m2are frequencies of electron single-particle transitions 0 →m.Figure 7 shows the frequency of the wave (6.4) ω0 1q¼ω1q/Ω10(solid line) and wave (6.2) Ω0 1q¼Ω1q/Ω10(dashed curve) as a function of x=qaform= 1 and α1= 0.75. Here Ω10 ¼(2πe2n/m*a)1/2is the limiting frequency for the wave with the spectrum (6.2). Parameter values m*¼0:64/C110/C028g (GaAs), a=1 0−7cm,k0a= 1 are used. Under the condition α1< 1 the Fermilevel lies in the upper half of the miniband. If αm.1, then two branches are connected with each 0 →mtransition: ω2 +(m,q)¼1 2{ω2 mþΩ2 mq+[(ω2 mþΩ2 mq)2 /C016 3a m/C16/C174 ω2 mΩ2mq(αm/C01)(qk0)21þq2a2 m2/C18/C19 /C01#1/2) : (6:5) Figure 8 shows the dependence of the wave frequencies (6.5) ω0 +(1,q)¼ω+(1,q)/Ω10(solid and dash-dotted curves) and wave (6.2) Ω0 1q¼Ω1q/Ω10(dashed curve) as a function of x=qa under m= 1 and α1= 3. The above mentioned values of m*,a, and k0a= 0.5 were used. In this case the Fermi level lies in the lower half of the miniband. The branches (6.4) andω+(6.5) are posi- tioned above ωm, and the branches ω_(6.5) are below ωm. In the limit of long waves ( qa/C281) and at αm< 1 from Eq.(6.4) we obtain ω2 1q¼ω2 101þ1 2Ω2 10 ω2 1þΩ2 10lnqaþ4 3ω2 1Ω2 10 (ω2 1Ω2 10)2(1/C0α1)(k0a)2(qa)2"# , (6:6) ω2 mq¼ω2 m01þΩ2 m0 ω2 mþΩ2 10m2/C02 2m2(m2/C01)(qa)2/C20 þ4 3m2ω2 mΩ2 m0 (ω2 mΩ2 m0)2(1/C0αm)(k0a)2(qa)2# ,m¼+2,+3,... (6:7) FIG. 7. The dispersion curves of waves with the spectrum (6.4) (solid line) and with the spectrum (6.2) (dashed line) under m=1 ,α1< 1 are shown. Parameter values are given in the text. FIG. 8. The dispersion curves of waves with the spectrum (6.5) (solid and dash-dotted curves) and with the spectrum (6.2) (dashed line) under m=1 , α1> 1 are shown. The parameters values are given in the text.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,546 Published under an exclusive license by AIP PublishingThe critical frequencies of waves with spectra (6.6) and (6.7) are ω2 m0¼ω2 mþΩ2 m0¼ε2 0m4þ2e2k0jmj πm*a2: (6:8) The frequency depolarization shift in Eq. (6.8) contains the period and the amplitude of the superlattice modulating potential.Atα m> 1 the expressions (6.6) and (6.7) are true for the upper branch ω+. The bottom branch ω−has the sound spectrum ω_(m, q)=cmq, where c2 m¼4a2 3m4(k0a)2ω2 mΩ2 m0 ω2 m0(αm/C01): (6:9) Optical ω+and acoustic ω−branches are connected with in-phase and anti-phase density oscillations of electrons which participate in longitudinal and transversal motion on the tube. 6.2. Spin waves In the random-phase approximation, the dispersion equation for the spectrum of transverse spin waves on a tube with a superlat- tice in the magnetic field has the form34 1/C0g 2μ2 Bχ0 +(m,q,ω)¼0: (6:10) Components χ±correspond to spin transversal waves with positive (−) and negative (+) helicity. The plus (minus) sign in χ±corre- sponds to transverse Landau-Silin spin waves69–72with a negative (positive) chirality. The solution of Eq. (6.10) for a degenerate electron gas depends on the position of the Fermi level μ0. If the electron density nsatisfies the inequality n,1 2π2a(k/C0 0þkþ 0), (6 :11) the Fermi level occurs in the miniband overlapping region in Fig. 1 . Inequality (6.11) involves k+ 0¼1 darccosε+ 0þΔ/C0μ0 Δ, which is the maximum electron momentum in the miniband 0±.I f the minibands do not overlap and the level μ0is situated in the second miniband then k/C0 0in Eq. (6.11) should be replaced by π/d. The graphical analysis of Eq. (6.10) in the case εþ 0,μ0, ε/C0 0þ2Δindicates that each mvalue, i. e., each spin-flip /C0!þ electron transition 0/C0!mþbetween the minibands l= 0 and l=mcorresponds to two branches of the magnon spectrum with a positive chirality. These branches are situated between the frequen-cies of single-electron transitions between the minibands Ω +¼ε0m[2η+m]þΩ: In the long-wavelength limit 2 Δsinqd 2/C12/C12/C12/C12/C12/C12/C28jω/C0Ω +j/C16/C17 , themagnon spectrum with a positive chirality reads ω+(q)¼ω0 +þα+sin2qd 2,( 6 :12) where ω0 +¼1 2[ΩþþΩ/C0/C0υ(k/C0 0/C0kþ 0)]+1 2[(Ωþ/C0Ω/C0)2 /C02υ(k/C0 0/C0kþ 0)(Ωþ/C0Ω/C0)þυ2(k/C0 0/C0kþ 0)/C04υ(kþ 0Ωþ/C0k/C0 0Ω/C0)]1/2 (6:13) are the limiting frequencies of the modes, υ¼g 2π2a, α+¼2Δsinkþ 0d(ω0 +/C0Ωþ)2þsink/C0 0d(ω0 +/C0Ω/C0)2 kþ 0d(ω0 +/C0Ωþ)2/C0k/C0 0d(ω0 +/C0Ω/C0)2:(6:14) If the minibands 0−and 0+do not overlap, k/C0 0in Eqs. (6.13) and (6.14) must be replaced by π/d. The spectrum of negative- chirality spin waves can be found from Eqs. (6.13) and (6.14) by exchanging the spin indices /C0$þ and a sign change of Ω. In the case of weak electron-electron interaction υ/C28dΩ+, we find from Eqs. (6.13) and(6.14) ω0 +¼Ω+/C0υk+ 0,( 6 :15) α+¼+2Δsink+ 0d k+ 0d,( 6 :16) where υk+ 0is the depolarization frequency shift. The upper (lower) branch ω+(ω−) of the magnon spectrum has a negative (positive) chirality. Collisionless damping of spin waves is given by the imagi-nary part of susceptibility (5.26) . In the case of a degenerate elec- tron gas, it is Imχ /C0¼μ2 B 2πadX l4Δ2sin2qd 2/C0(ω/C0Ωþ)2/C20/C21/C01/2 ,( 6 :17) where frequency is in the interval Ωþ/C02Δsinqd 2,ω,Ωþþ2Δsinqd 2, Imχ/C0¼μ2 B 2πadX l4Δ2sin2qd 2/C0(ω/C0Ω/C0)2/C20/C21/C01/2 ,(6:18) where frequency is in the interval Ω/C0/C02Δsinqd 2,ω,Ω/C0þ2Δsinqd 2: The Landau damping of spin waves is nonzero in the Stoner sectors of the q−ωplane bounded by the curves ω+¼Ω++2Δsinqd 2.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,533 (2021); doi: 10.1063/10.0005181 47,547 Published under an exclusive license by AIP PublishingDispersion curves (6.12) are situated outside the Stoner sectors, i. e., the spin waves considered in this Subsection are undamped. To observe the effects associated with these modes the distances betweenthe edges υk + 0¼2π2aυn+of the Stoner sector and the limiting fre- quencies must exceed both the thermal and impurity broadening ofthe electron energy levels. 7. CONCLUSION Superlattice at the surface of a nanotube has a significant impact on its properties. It can be obtained by embedding fuller-enes or other additives to the nanotube or when the nanotube isattached to a substrate for charge exchange. 47In the absence of a superlattice, the tube spectrum in a longitudinal magnetic field is a collection of one-dimensional subbands located next to each otherand having nonequidistant boundaries. 22,26,27,34,86,89Periodic mod- ulating potential artificially created at the surface of the tube con- verts the spectrum into a system of minibands, the widths of which are determined by the amplitude of the modulating potential.22 In a longitudinal magnetic field its amplitude and period dependon the magnetic field strength. Energy gaps separating the mini-bands have widths defined by the ratio of the miniband width to the magnitude of the rotational quantum and depend on magnetic field strength. 26,27Density of electronic states has a root singularity at miniband borders.22As the radius of the tube increases, the minibands overlap resulting in a continuous spectrum.22 In this review, the density of states, chemical potential, energy, and heat capacity of a degenerate and non-degenerate electron gas at the surface of a nanotube with metallic conductivity character ina longitudinal magnetic field have been presented. 22We show that abovementioned thermodynamic values include monotonic andoscillating components. In agreement with Pauli principle mono- tonic heat capacity component of a degenerate electron gas is pro- portional to the temperature. 22Heat capacity displays de Haas –van Alphen type oscillations due to the passage of state density rootsingularity through the Fermi boundary with a change in electron density. 22These oscillations persist in the absence of a magnetic field. Heat capacity also displays Aharonov –Bohm type oscillations when magnetic field flux through cross-sectional of the tube isvaried. Heat capacity studies allow observation of the transition ofmodulating potential from the localized gaps mode to the mode of free motion along the tube. 22 The obtained in the present review formulas for the conduc- tivity tensor components may be applied for studying electromag-netic wave propagation in nanotubes with superlattices based onAl xGa1−x/GaAS, InGaAs/GaAs, InAs/GaAs, GeSi/Si heterojunc- tions and in carbon nano-tubes in the regime of metallic conduc- tivity. The real part of conductivity determines the wave energyabsorbed by electrons. 24In the degenerated electron gas, this is non-zero in the areas of Landau collisionless damping.24Knowing the positions of transparency windows for the waves, it is possible to improve the waveguide characteristics of nanotubes.21The imag- inary part of conductivity is included into the dispersion equationfor electromagnetic wave spectrum. 32,65–67This has the resonance singularities at frequencies of electron direct transitions between minibands. Usually, near these frequencies there exist new branches in the wave spectrum and related band transparency. Observationof conductivity oscillations of de Haas –van Alphen type allows determining the electron effective mass, Fermi momentum, rota- tional quantum and superlattice parameters dand Δ.24These values are included in the oscillation amplitude and period expres-sions. Revealing the instant of appearing the beats under variationof the nanotube parameters gives the opportunity to obtain the ratio of Fermi energy to miniband width. In the framework of the hydrodynamic approach, the plasma waves on the surface of a nanotube with a longitudinal superlatticewere discussed. 32Not only longitudinal electron current but also transversal one has been taken into consideration. It has been shown that both optical and acoustical plasmons could propagate along the tube with one sort of carrier.32The results of this review can be used in studying the magnetic scattering of neutrons by thespin magnetization current of conduction band electrons on a tube.The cross-sections of scattering by spin waves and Stoner excita- tions are of interest. This problem was solved earlier for a two- dimensional electron gas on a plane. 73The curvature of a cylinder should manifest itself in additional features of the scattering cross-section. 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5.0050277.pdf

J. Chem. Phys. 154, 214106 (2021); https://doi.org/10.1063/5.0050277 154, 214106 © 2021 Author(s).Correlation-driven phenomena in periodic molecular systems from variational two- electron reduced density matrix theory Cite as: J. Chem. Phys. 154, 214106 (2021); https://doi.org/10.1063/5.0050277 Submitted: 13 March 2021 . Accepted: 17 May 2021 . Published Online: 02 June 2021 Simon Ewing , and David A. Mazziotti ARTICLES YOU MAY BE INTERESTED IN Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). II. The full rank NEVPT2 (FR-NEVPT2) formulation The Journal of Chemical Physics 154, 214113 (2021); https://doi.org/10.1063/5.0051218 Ultrafast x-ray pump x-ray probe transient absorption spectroscopy: A computational study and proposed experiment probing core-valence electronic correlations in solvated complexes The Journal of Chemical Physics 154, 214107 (2021); https://doi.org/10.1063/5.0047381 Model protein excited states: MRCI calculations with large active spaces vs CC2 method The Journal of Chemical Physics 154, 214105 (2021); https://doi.org/10.1063/5.0048146The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Correlation-driven phenomena in periodic molecular systems from variational two-electron reduced density matrix theory Cite as: J. Chem. Phys. 154, 214106 (2021); doi: 10.1063/5.0050277 Submitted: 13 March 2021 •Accepted: 17 May 2021 • Published Online: 2 June 2021 Simon Ewing and David A. Mazziottia) AFFILIATIONS Department of Chemistry and The James Frank Institute, The University of Chicago, Chicago, Illinois 60637, USA a)Author to whom correspondence should be addressed: damazz@uchicago.edu ABSTRACT Correlation-driven phenomena in molecular periodic systems are challenging to predict computationally not only because such systems are periodically infinite but also because they are typically strongly correlated. Here, we generalize the variational two-electron reduced density matrix (2-RDM) theory to compute the energies and properties of strongly correlated periodic systems. The 2-RDM of the unit cell is directly computed subject to necessary N-representability conditions such that the unit-cell 2-RDM represents at least one N-electron density matrix. Two canonical but non-trivial systems, periodic metallic hydrogen chains and periodic acenes, are treated to demonstrate the methodology. We show that while single-reference correlation theories do not capture the strong (static) correla- tion effects in either of these molecular systems, the periodic variational 2-RDM theory predicts the Mott metal-to-insulator transition in the hydrogen chains and the length-dependent polyradical formation in acenes. For both hydrogen chains and acenes, the peri- odic calculations are compared with previous non-periodic calculations with the results showing a significant change in energies and increase in the electron correlation from the periodic boundary conditions. The 2-RDM theory, which allows for much larger active spaces than are traditionally possible, is applicable to studying correlation-driven phenomena in general periodic molecular solids and materials. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0050277 I. INTRODUCTION Computing the electronic structure of extended molecules and materials can reveal important information, including the bandgap, reactivity, and bulk properties, such as conductivity or polarizabil- ity. As such, accurate methods for computing the electronic struc- ture of materials is of interest to many fields, including the study of inorganic polymers, organic electronics, semiconductors, and super- conductors.1–12Electronic structure calculations on extended mate- rials are computationally tractable in cases where periodic bound- ary conditions can be imposed, which separate the electrons and orbitals into smaller, periodically repeating, unit cells. Commonly used methods for such calculations include density functional the- ory (DFT),13–16GW approximations,17quantum Monte Carlo meth- ods,18and coupled cluster (CC) theory.19Many of these meth- ods, however, have difficulty computing strongly correlated periodic materials with high accuracy at an efficient computational cost, and hence, there is a need for further advances in methods and theories.Here, we present an approach to the calculation of electronic structures for periodic materials in the gamma-point approxima- tion based on the variational calculation of the two-electron reduced density matrix (2-RDM). In the variational 2-RDM (v2RDM) method,20–33the 2-RDM is constrained by N-representability con- ditions that are necessary for the 2-RDM to represent at least one N-electron density matrix. Because these conditions are necessary, the minimization of the energy with respect to the 2-RDM gener- ates a lower bound on the ground-state energy in the given basis set. Furthermore, because the constraints known as p-positivity condi- tions restrict the metric matrices of qparticles and (p−q)holes to be positive semidefinite,34,35the variational 2-RDM method has a well-defined, physical solution even in the presence of strong electron correlation. The two-positivity conditions, which include the non-negativity of the particle–particle2D, hole–hole2Q, and particle–hole2Gmatrices, have been shown to be capable of describ- ing strongly correlated phenomena, including polyradical charac- ter in extended conjugated systems,27,36ligand non-innocence in J. Chem. Phys. 154, 214106 (2021); doi: 10.1063/5.0050277 154, 214106-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1. The acene chain systems used for calculations have various numbers of repetitions of the unit cell inside one periodic box. Every unit cell contains four carbon atoms and two H atoms for a total of 26 electrons and 40 orbitals in the 6-31G basis set. Of those orbitals, four electrons and four orbitals ([4,4] active space) were used from each unit cell in the active space calculations. transition-metal complexes,37,38non-superexchange mechanisms in bridged transition-metal dimers,39and exciton condensation in elec- tron double layers.12 Extension of the v2RDM theory to treat periodic boundary con- ditions allows us to compute the electronic structure of strongly correlated periodic molecules without performing multiple molec- ular (open boundary condition) calculations and extrapolating to the infinite-length limit. Because the periodic v2RDM method with the two-positivity conditions only scales as O(r6), where ris the number of active orbitals in the unit cell, we can perform cal- culations with much larger active spaces than possible with tra- ditional methods.37,40Through the N-representability conditions, the v2RDM theory includes multi-body correlation that is diffi- cult for many traditional periodic methods to capture. Although we present the periodic v2RDM method in the gamma-point approx- imation here, the formalism can be extended to include k-point sampling. To demonstrate the periodic v2RDM theory, we treat two extended systems, known for exhibiting strong electron corre- lation: hydrogen chains and acene chains. Both systems are difficult to treat accurately with traditional methods, such as second-order many-body perturbation theory, configuration interaction with sin- gle and double excitations, or coupled cluster theory with single and double excitations extended to periodic boundary conditions. The hydrogen chains undergo a Mott metal-to-insulator transi- tion41,42upon dissociation with the insulator phase being strongly correlated, whereas the acene chains (Fig. 1) become strongly cor- related with polyradical character as the lengths of the chains increase. II. THEORY We discuss the v2RDM theory, periodic boundary conditions, and their combination into a periodic v2RDM method. A. Variational two-electron reduced density matrix methods The 1-RDM and 2-RDM are defined by integrating the full N-electron density matrix over the spatial and spin coordinates of all but one or two electrons, respectively. Using second quantization, we can represent the elements of the 1- and 2-RDMs as1Di j=⟨Ψ∣ˆa† iˆaj∣Ψ⟩, (1) 2Di,j k,l=1 2⟨Ψ∣ˆa† iˆa† jˆalˆak∣Ψ⟩, (2) where ˆa† mand ˆamare the second-quantized creation and annihila- tion operators for spin orbital ∣ψm⟩.20–23,26,43,44Notably, the 1-RDM can be derived from the 2-RDM by integrating over the spatial and spin coordinates for one of the two electrons. These spin RDMs can be converted into spatial RDMs by tracing over the spin of the elec- trons. Eigenvalues of spatial 1-RDMs, also known as spatial natural orbital occupation numbers,45represent the numbers of electrons in the spatial orbitals, and by the Pauli exclusion principle, they are constrained to lie between 0 and 2. A signature of strong correlation, or contributions from multiple Slater determinants, is the presence of fractionally filled orbitals, which are characterized by eigenvalues of the spatial 1-RDM near 1.46Similarly, eigenvalues of 2-RDMs rep- resent the number of electrons in each two-electron function, known as a geminal, and partially filled geminals contribute to correlation. For systems that have at most pairwise interactions, the molec- ular energy can be written as a functional of the two-electron reduced density matrix, E=Tr(1H1D)+Tr(2V2D), (3) where1Hand2Vare the reduced Hamiltonian matrices of the one-electron and two-electron integrals. The variational 2-RDM (v2RDM) method minimizes the ground-state energy in Eq. (3), with the 2-RDM as the fundamental variable rather than the wave- function.22,24,47Often, the v2RDM method is combined with the notion of an active space, a set of orbitals that are correlated in a mean field of the remaining orbitals.27,48A system where nelectrons are allowed to fill rspatial orbitals is denoted as an [n,r]active space. Ifnis equal to the total number of electrons and ris equal to the size of the basis set, then the calculation approximates the energies from full configuration interaction (FCI). Central to RDM calculations is the concept of N-represent- ability, which requires that an RDM represents a physical N-electron density matrix.35,43,49The simplest and most familiar N- representability constraint is the Pauli exclusion principle, which states that the eigenvalues of the spatial 1-RDM must lie between 0 and 2. Similar constraints exist for the two-electron RDM, the particle–hole RDM (2G), and the two-hole RDM (2Q), namely, 2D⪰0,2G⪰0, and2Q⪰0, restricting the eigenvalues of all three matrices to be non-negative. All three of these constraints restrict the space of valid 2-RDMs because2Gand2Qare related by linear map- pings to the 2-RDM.22Minimizing the energy from a 2-RDM subject to these constraints generates an optimization problem known as a semidefinite program.47,50–52The solution of the semidefinite pro- gram yields a lower bound to the ground-state energy from a com- plete active space configuration interaction (CASCI) calculation48 but with a computational scaling that is polynomial in the size of the active space. B. Periodic boundary conditions For molecules with extended structures, additional symme- tries must be exploited to make accurate electronic structure J. Chem. Phys. 154, 214106 (2021); doi: 10.1063/5.0050277 154, 214106-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp calculations tractable. Calculations on periodic molecules can be altered to account for the periodic boundary conditions (PBCs) of the molecule by using periodic basis functions. Bloch waves are the simplest way to construct a complete basis set of periodic functions, but the underlying structure for these basis functions can vary and are usually chosen to be either Gaussian53,54or plane waves.55–58In general, Bloch waves can be defined as ∣Ψk(r)⟩=eik⋅ru(r), (4) where u(r)is a function for which the periodic boundary condi- tions are satisfied and kis a momentum vector. For many cases, the gamma point, where only k=0 wave vectors are included in the basis set, gives an accurate approximation to the complete basis while greatly simplifying the computational complexity. Although plane waves automatically satisfy the periodic boundary conditions, they do not resolve atomic details as readily as Gaussian basis functions. In the case of Gaussian functions, periodic boundary conditions can be imposed through a local basis approximation54 ∣Ψk(r)⟩=∑ Teik⋅Tu(r−T), (5) where Tis a lattice translational vector and u(r−T)is a local Gaus- sian atomic basis function. This approximation accounts for period- icity by summing over images of each basis function in neighbor- ing cells. As the number of translational vectors in the sum over T increases, the wavefunction will more closely approach the periodic boundary conditions because the closest cells are being explicitly included. Once the basis set is chosen to satisfy the boundary con- ditions, the one-electron and two-electron integral matrices can be computed and used in the v2RDM method. We use the PySCF built- in integration methods to evaluate these integrals using the local basis approximation.54 Without the use of periodic boundary conditions, extended systems are typically approximated by computing molecular sys- tems of varying sizes and extrapolating to the infinite limit. In this form of analysis, several assumptions are made about the energy density of the system so that the energies of the different systems can be subtracted, leaving an “effectively edgeless” system. Specifically, it is assumed that the system is long enough that the system can be broken into several classes of edge subsystems and edgeless (quasi- periodic) subsystems. By subtracting the energy of smaller systems that contain no (or fewer) edgeless subsystems, the energy of the central systems can be approximated. However, gamma point cal- culations compute these systems directly due to the PBCs used in the unit cell, and these calculations are often much cheaper because there is no need to compute the electronic structure of a super- cell. Additionally, we posit that gamma point calculations better represent the extended systems than molecular calculations of the same size for two main reasons: first, molecular calculations typi- cally require several calculations to get an energy so that the analysis above can be performed. Second, periodic calculations eliminate the computational time and power needed to compute the electronic structure of the edge subsystems, which are ultimately discarded. Additionally, this discarding of data unnecessarily complicates the analysis of these computations for observables other than the energy.In both forms of analysis, the assumption that the (quasi) peri- odic unit cells are identical—no phase change between cells—can be relaxed to get a more accurate representation of the extended structure, since long-range order can affect the electronic struc- ture in extended systems. For molecular calculations, the analysis above is simply extended further to larger systems, which reduces the inaccuracy due to each approximation. Using PBCs, there are two main methods for relaxing this assumption: including k-points or including more repeating units in the unit cell. Including k- points explicitly adds basis functions that allow for phase changes between unit cells. Similarly, including more repeating units in the unit cell allows the wavefunction to change phase between repeat- ing units within the cell. The results of a gamma point calcula- tion with Nrepeated units in the unit cell can be shown to be equivalent to a k-point calculation on a single unit cell with N k- points because the volume of the Brillouin zone is inversely pro- portional to the volume of the unit cell. The main motivation for k-point calculations is to exploit translational symmetry to decrease computation time, which is the subject of future work for this method. III. RESULTS A. Methods Hydrogen chain symmetric dissociation curves were com- puted with H 10in the unit cell using the correlation-consistent polarized valence double-zeta (cc-pVDZ)59basis set. Both molecular and periodic calculations were completed in a [10,20] active space. Additional calculations were performed using DFT (B3LYP functional), configuration interaction with single and double excitations (CISD), and Møller–Plesset second-order perturbation theory (MP2)60methods. Each of these calculations was performed using the implementations in PySCF.54,61 Additionally, the electronic structure of acene chains were computed using the periodic v2RDM method and the 6-31G basis set, with crystal structure coordinates obtained from the American Mineralogist Crystal Structure Database62and explic- itly shown in supplementary material, Table S2. A series of cal- culations were performed, with 1–10 repetitions of the unit cell inside the periodic box. For each calculation, we used a [4,4] active space per repeated unit, which accounts for the complete π- space. Therefore, the ten-unit calculation had an active space of [40,40]. This procedure ensured that the energy per repeated unit and occupation numbers were converged and revealed informa- tion about the parity of the periodic orbitals. The occupation num- bers from periodic v2RDM were then compared to those obtained via gamma-point calculations using PySCF-built-in periodic cou- pled cluster methods.54Finally, the energy, natural orbital (NO) occupation numbers, and several forms of entropy were computed for molecular acene chains with 2, 4, 6, and 8 rings using the v2RDM method. The molecular calculations had active spaces of [10,10], [18,18], [26,26], and [34,34] respectively, corresponding once again to the full π-space. Geometries for these molecular chains were obtained from the supplementary data from Ref. 2. These results are compared to the calculations using the periodic method. J. Chem. Phys. 154, 214106 (2021); doi: 10.1063/5.0050277 154, 214106-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp B. Hydrogen chain Previous work on the extended hydrogen chain has focused on obtaining the dissociation curve through various extrapolation techniques, starting from molecular calculations.64,65Molecular energies for multiple chain lengths are calculated and then com- pared with each other to extrapolate to the infinite chain length, or the thermodynamic, limit. We use the PBC approach to compute the ground-state energy of the hydrogen chain, which removes the edge effects entirely without extrapolation. To ver- ify the periodic-system-to-molecule limit, we present ground-state energies for one-, two-, and three-dimensional hydrogen clus- ters with varying periodic box sizes in supplementary material, Fig. S1. Examining the dissociation curve for H 10in Fig. 2, we observe that the molecular and periodic systems converge to the same energy, around 5 hartree, as the spacing between hydrogen atoms increases. In this regime, the hydrogen atoms begin to behave inde- pendently, so the total energy is approximately ten times the energy of a single hydrogen atom. However, interesting differences develop as the separation distance decreases. The equilibrium bond length lies at 0.95–1.05 Å, and the periodic system has a deeper well by 6.6 mhartree per atom. This agrees with the previous work by Motta et al.64that indicates through extrapolating molecular calcu- lations to the infinite limit that the periodic system has a deeper well by about 4 mhartree per atom. One metric that has been used66–68to discuss the Mott metal- to-insulator transition is the sum of the magnitudes of the off- diagonal elements of the 1-RDM in the atomic orbital basis, denoted γ. Because the metal-to-insulator transition is a long-range effect, we ignore elements corresponding to orbitals on a single atom, instead summing over the off-diagonal elements that correspond to elec- tron density shared between atoms. We add these elements of the 1-RDM in quadrature as a parallel to the Frobenius norm of the matrix. See the supplementary material for additional details. When the system has long-range order, this metric will be large, indicating strong metallic behavior, and when the system has no long-range order, this metric will be small, indicating strong insulating behav- ior. Figure 3 shows the metric as a function of the atomic spacing FIG. 2. Dissociation curves for H 10using molecular v2RDM and periodic v2RDM with a [10,20] active space and the cc-pVDZ basis set. Results agree with previous data,63,64 showing the equilibrium bond length at 0.95–1.05 Å. A comparison to a periodic CASCI method is presented in supplementary material, Table S1.in H 10. The Mott transition metric for periodic Hartree–Fock calcu- lations is nearly constant for all bond lengths. A similar absence of the transition to an insulator is seen from the DFT with the B3LYP functional, MP2, and CISD (as observed in previous work,51the cou- pled cluster calculations with single and double excitations do not converge far beyond the equilibrium bond length, and hence, it is not included in the reported data). By contrast, the periodic v2RDM cal- culations exhibit the expected transition from metallic to insulating properties as the bond length increases. C. Acene chain NO occupations for periodic calculations are shown in Table I and Fig. 4, which show that the known polyradical nature69–71 of acene chains is only recovered by v2RDM for periodic boxes with an even number of unit cells included. The coupled clus- ter singles-doubles (CCSD) method with periodic boundary con- ditions only recovers static correlation in the two unit cell case. Notably, the CCSD calculation does not exhibit significant static correlation in the four unit cell case. Multi-reference wavefunc- tions are needed to accurately capture static correlation, and CCSD, a single-reference method, fails to accurately capture static cor- relation in larger systems. This result is consistent with previ- ous limitations with CCSD observed in the computation of finite acene chains.27The highest-occupied natural orbitals (HONOs) and lowest-unoccupied natural orbitals (LUNOs) that display biradical character have a 180○phase change between unit cells (see Fig. 5), which explains the requirement for an even number of unit cells to achieve the biradical character. This also agrees with previous molecular (non-periodic) calculations suggesting this same parity.36 Although one unit cell contains only four carbon atoms and two hydrogen atoms, gamma-point calculations should be performed on a box containing two unit cells to accurately reflect molecular properties. Electron densities for eight ring molecular, eight-unit peri- odic, and nine-unit periodic calculations are included in Fig. 5. These images show clearly that the HONO −1 and LUNO +1 FIG. 3. Mott metal-to-insulator metric as a function of hydrogen atom spacing for H 10for various periodic methods. Hartree–Fock and other post-Hartree-Fock calculations produce a metric that is relatively constant across all bond lengths, whereas both molecular66and periodic v2RDM calculations have an appropriate transition from the metal to insulator as the bond length increases. J. Chem. Phys. 154, 214106 (2021); doi: 10.1063/5.0050277 154, 214106-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I. Natural orbital occupations for acene chains with 1–4 unit cells in the periodic box. v2RDM calculations show strong correlation for even numbers of unit cells, whereas CCSD fails to recover strong correlation except in the two-unit cell calculation. CCSD calculations were only performed up to four unit cells due to the expensive memory requirements for longer chains. Natural orbital occupation numbers One unit cell Two unit cells Three unit cells Four unit cells CCSD v2RDM CCSD v2RDM CCSD v2RDM CCSD v2RDM HONO −2 1.9594 2.0000 1.9082 1.9036 1.9361 1.9321 1.8992 1.8662 HONO −1 1.9580 1.9842 1.9055 1.8918 1.9032 1.8515 1.8992 1.8662 HONO 1.9440 1.9743 1.4740 1.3558 1.9032 1.8515 1.7194 1.3479 LUNO 0.0475 0.0263 0.5215 0.6457 0.0902 0.1511 0.2782 0.6597 LUNO +1 0.0391 0.0152 0.0845 0.1109 0.0902 0.1511 0.0922 0.1309 LUNO +2 0.0301 0.0000 0.0841 0.0943 0.0548 0.0688 0.0922 0.1309 orbitals from the molecular calculation have even parity, while the HONO and LUNO have odd parity. Additionally, the elec- tron density of the even parity orbitals remain unchanged in the eight-unit periodic calculation, and the odd parity orbitals remain unchanged in the nine-unit periodic calculation (after translation along the periodic axis). As a result, the occupations of the HONO and LUNO closely match those in the nine-unit periodic calcula- tion, whereas the HONO −1 and LUNO +1 orbital occupations closely match those in the eight-unit periodic calculation (shown in Fig. 4). Notably, each orbital in the nine-unit periodic calculation has a single “defect” that looks like the electron density along the edge of the molecular calculation. Because there are an odd num- ber of repeating units in the unit cell and gamma point calculations can only recover orbitals with an even number of antinodes, the defects effectively stretch an even number of antinodes across the unit cell. Finally, the symmetry of the eight-unit periodic calculation of HONO and LUNO explains the early convergence of the occupa- tion. Since these orbitals are just four copies of the two-antinode pat- tern fully contained in two repeated units, these orbitals, in particu- lar, stay the same for any even number of repeated units in the unit cell. Earlier results using molecular DMRG2also indicate the polyradical behavior of acene chains, but significant staticcorrelation is only recovered for chain lengths longer than six rings, and the partial occupation levels only reach the values obtained here (1.35 HONO and 0.66 LUNO occupations for six unit cells) for chain lengths of ten rings. Thus, we conclude that this periodic v2RDM method accurately captures strong correlation efficiently compared to DMRG methods, in that fewer atoms and orbitals are needed to observe strong correlation effects. Additionally, this method is gen- eralizable to two- or three- dimensional systems [by summing over neighboring cells in two or three axes in Eq. (5)], whereas DMRG methods are typically restricted to one-dimensional systems. For both the periodic and molecular acene calculations, we include the one-electron Von Neumann Entropy (1-VNE) and a two-electron form of entropy of quantum entropy (we call the connected entropy) originally created by Prezhdo72in Fig. 6. The connected entropy is defined as 2Sc=NS(1D)−S(2D), (6) where Nis the number of electrons and S(M)=−Tr(MlnM)is the Von Neumann entropy. Both the 1-VNE and the connected entropy are extensive, and the 1-VNE quantifies the amount of bipartite correlation, whereas the connected entropy quantifies the amount of tripartite correlation in the system. To compare these FIG. 4. NO occupations for periodic (a) and molecular (b) acene chain calcula- tions. In (a), the even-unit calculations are colored blue and the odd-unit calcu- lations are colored red to emphasize the differences in occupation trends between the two sets of calculations. Notably, the odd-unit calculations have a simi- lar HONO and LUNO occupation trend as the molecular calculations, whereas the even-unit calculations have the same HONO and LUNO occupations across all calculations. J. Chem. Phys. 154, 214106 (2021); doi: 10.1063/5.0050277 154, 214106-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5. Images of electron density for HONO −1, HONO, LUNO, and LUNO +1 for molecular [8-acene (a)] and peri- odic [eight-unit (b) and nine-unit (c)] cal- culations. The dashed lines in the struc- tures for the periodic calculations repre- sent the lattice boundary. quantities between calculations, we divide each by the num- ber of orbitals in the calculation to compare the intensive forms of the entropy. Because all the calculations performed use the complete π-space, these intensive quantities are directly comparable. The intensive VNE is relatively constant for both molecular and periodic calculations, indicating that bipartite correlation is not increasing, despite the increase in correlation as longer chains are considered. In contrast, the connected entropy increases steadily as the chain length increases, indicating an increase in tripartite FIG. 6. One-electron Von Neumann entropy and connected entropy (2Sc) of molec- ular and periodic acene chains of various lengths. Periodic calculations have active spaces of [4 n, 4n], where nis the number of repeated units in the unit cell, and molecular calculations have active spaces of [4 n+2, 4n+2], where nis the num- ber of rings in the calculation. The entropy of each calculation has been divided by the number of orbitals to get an intensive entropy measure that can be compared directly between calculations.correlation. Additionally, the periodic calculations have more of each type of entropy than the molecular calculations for all chain lengths. As a result, the two-unit periodic calculation recovers more correlation than any molecular calculation performed, which emphasizes the importance of including periodic boundary condi- tions to capture static correlation in extended systems. IV. DISCUSSION AND CONCLUSIONS We have generalized the variational two-electron reduced den- sity matrix (v2RDM) electronic structure theory, which incorporates periodic boundary conditions for extended structures. Using peri- odic boundary conditions can greatly simplify the computational complexity and accuracy of calculations by removing edge effects. Even gamma-point calculations, for which the electronic structure repeats with no phase change between unit cells and, therefore, does not account for long-range or low-frequency contributions to the wavefunction, can recover a large proportion of the correlation due to periodicity. The proposed periodic v2RDM method was shown to account for a large degree of static correlation due to the multirefer- ence nature of 2-RDMs and to correlate electrons between periodic cells. The ability to capture strong correlation in extended structures will prove beneficial in the study of many systems, including systems with extended π-systems, inorganic polymers, and materials, among others. Additionally, we have confirmed previous work regarding the equilibrium bond length of the extended hydrogen chain. This is significant because we can efficiently recover an accurate estimate for the ground-state energy of the hydrogen chain system without performing expensive calculations with different chain lengths. We also showed that the periodic v2RDM method can capture the metal- to-insulator transition for the hydrogen chain, which Hartree–Fock, MP2, CISD, and CCSD calculations are unable to do. The success of the v2RDM method is due to its accurate treatment of the strong correlation upon dissociation and its correct description of the peri- odic nature of the wavefunction. With the application of this method to the acene chain, we have shown that parity effects require that two crystallographic unit cells are used for gamma-point electronic structure calculations. Due to the ability to represent multireference systems, the 2-RDM methods can accurately capture strongly correlated phe- nomena in materials without the exponential scaling of full config- uration interaction, allowing the use of much larger active spaces J. Chem. Phys. 154, 214106 (2021); doi: 10.1063/5.0050277 154, 214106-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp than traditionally possible for such systems. The combination of these attributes has the potential for more realistic descriptions of correlation-driven phenomena in extended π-systems, semiconduc- tors, superconductors, organometallic polymers, as well as other materials. 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5.0053806.pdf

Spin dynamics in GaN/Al 0.1Ga0.9N quantum well with complex band edge structure Cite as: Appl. Phys. Lett. 118, 252107 (2021); doi: 10.1063/5.0053806 Submitted: 11 April 2021 .Accepted: 6 June 2021 . Published Online: 21 June 2021 Shixiong Zhang,1 Ning Tang,1,2,3,a) Xingchen Liu,1 Xiaoyue Zhang,1LeiFu,1 Yunfan Zhang,1Teng Fan,1 Zhenhao Sun,1Weikun Ge,1and Bo Shen1,2,3 AFFILIATIONS 1State Key Laboratory of Artificial Microstructure and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China 2Frontiers Science Center for Nano-Optoelectronics and Collaboration Innovation Center of Quantum Matter, Peking University, Beijing 100871, China 3Peking University Yangtze Delta Institute of Optoelectronics, Nantong 226010, Jiangsu, China a)Author to whom correspondence should be addressed: ntang@pku.edu.cn ABSTRACT Spectrally distinguished spin relaxation dynamics in a single GaN/Al 0.1Ga0.9N quantum well was investigated by a time-resolved Kerr rotation spectrum at room temperature. Three spin relaxation processes were well distinguished by a photon energy upon the excitation energy being resonated with the bandgap of various layers. It is observed that the electron spin relaxation time of 7 ps in a GaN quantumwell is much shorter than that of 140 ps in an Al 0.1Ga0.9N barrier layer due to the considerable polarization electric field at a GaN/ Al0.1Ga0.9N heterointerface. For electrons in bulk GaN and Al 0.1Ga0.9N, the dominant role of electron–photon scattering and alloy disorder scattering in the anisotropic D’yakonov–Perel’ (DP) relaxation was revealed by the photoexcited electron density and magnetic field depen- dence of the spin relaxation time. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0053806 Due to an electric-field controllable spin–orbit coupling (SOC) and potential device applications at room temperature, GaN-based semiconductors have attracted a lot of interest in the community of semiconductor spintronics.1–5For bulk GaN, the influence of temperature,6doping density,7and external magnetic field8,9on the electron spin relaxation well elucidates the D’yakonov–Perel’ (DP) spin relaxation mechanism in wurtzite semiconductors. With the help of suppression of the DP scattering in GaN nanowires10and other low dimensional structures,11the electron spin relaxation time could be enhanced greatly. A two-dimensional electron gas (2DEG) in GaN- based heterostructures or a quantum well (QW) is proposed as a potential candidate for making spintronic devices because of its excel- lent electrical properties such as high electron concentration and mobility.12,13Furthermore, the intrinsic polarization electric field induces a strong and tunable Rashba SOC14,15that can compensate the Dresselhaus SOC to manipulate spin relaxation.16Therefore, it is imperative to study the fundamental spin properties of 2DEG forimplementing the novel spintronic device functions. The optical pump and probe technique is a simple and powerful approach that can detect the spin relaxation process with ultrahigh time resolution and give the spin relaxation time directly. However, almost the same gfactor ofelectrons in the GaN/AlGaN QW 17and a barrier layer18leads to a difficulty for extracting respective optical signals.19The spin relax- ation dynamics has, therefore, rarely been investigated in the GaN/AlGaN QW. In this work, the spin relaxation dynamics in a single GaN/ Al 0.1Ga0.9N QW was studied by a photon-energy dependent time- resolved Kerr rotation (TRKR). Electron spin relaxation processes in the GaN/Al 0.1Ga0.9NQ W ,A l 0.1Ga0.9N barrier, and bulk GaN are clearly distinguished by choosing an appropriate excitation energy. The enormous difference of spin relaxation time in the three regions is well clarified based on the DP theory, which is gov- erned by the Rashba contribution. Moreover, in bulk GaN and Al0.1Ga0.9N, the external magnetic field strength and carrier den- sity dependence of the spin relaxation time further reveal the dom- inant role of the DP mechanism. As shown in Fig. 1(a) ,t h es i n g l eG a N / A l 0.1Ga0.9NQ Ww a s grown by metal organic chemical vapor deposition (MOCVD). It con- sists of a 3 lm undoped GaN buffer layer grown on a silicon substrate, a3 0 0 n mA l 0.1Ga0.9N barrier, a 5 nm GaN well, and a 20 nm Al0.1Ga0.9N barrier. The TRKR measurements were carried out at room temperature. An external magnetic field was applied in Viogt Appl. Phys. Lett. 118, 252107 (2021); doi: 10.1063/5.0053806 118, 252107-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplgeometry to cause the Larmor precession of the electron spin. The femtosecond pump and probe pulses were supplied by a mode-locked Ti: sapphire laser with a repetition rate of 80 MHz. A secondharmonic generator was used to obtain the pulses with photonenergy tunable in the range of 2.296–3.647 eV. The pump and probe beams were separated by a polarization beam splitter. The probe beam was linearly polarized and delayed by a mechanical time delayline. Meanwhile, the pump beam was modulated periodically to leftcircularly polarized light r þ(or right circularly polarized light r/C0) by a photo-elastic modulator. To extract the weak spin signals fromthe background noise, a modulation frequency 50 kHz was set as thereference frequency of a lock-in amplifier. The pumping light spot has a diameter of about 100 lm. On the surface of a spin-polarized sample, different reflectivity for different circularly polarized lightwould induce Kerr rotation of a linearly polarized (probe) beam.The spin relaxation process could then be exhibited by the time-dependent Kerr rotation. In order to identify the band structure of a single GaN/ Al 0.1Ga0.9N QW, photoluminescence (PL) spectroscopy measure- ments were conducted at room temperature with an excitation energyof 3.815 eV. As shown in Fig. 1(b) , three luminescence peaks emerge from the sample. The peak at 3.421 eV stems from the bulk GaNbuffer layer. At higher photon energy, the double-peak Lorentzian fit-ting for the PL spectrum demonstrates that the photon energies of the peaks’ center are 3.513 and 3.586 eV. According to the Al composition dependent bandgap in an AlGaN alloy, 12,13the dominant peak at 3.586 eV is attributed to the Al 0.1Ga0.9N barrier, while a weak lumines- cence peak at 3.513 eV is due to the recombination of electrons andholes in the GaN QW. 20It is supposed that the considerable polariza- tion electric field would induce the spatial separation between the wave functions of electrons and holes in the GaN QW, thus resulting in the Stark effect and a weak luminescence intensity. Based on the above results from PL spectrum, photon-energy dependent TRKR measurements were used to distinguish the spinrelaxation dynamics of electrons in various layers. As depicted inFig. 2(a) , the spin precession of electrons in the bulk GaN buffer layer could be observed at a photon energy of 3.402 eV, when electrons inthe bulk GaN were excited alone. As the excitation energy was tuned to make the resonant transition in the GaN QW, a fast spin relaxationprocess was observed as shown in Fig. 2(b) . Once the external mag- netic field was applied, the amplitude of a TRKR signal would decreasedue to spin precession. In this case, the photon with an energy of3.513 eV is capable of exciting the electron–hole pairs in the bulk GaNas well, and the spin polarization of photoexcited electrons in the bulkGaN would, however, vanish under the optical transition far above thebandgap. Therefore, it is the spin polarization of 2DEG in the GaNQW that indeed causes the TRKR signals with fast decay in Fig. 2(b) . In contrast, a slow spin relaxation signature was detected at an excita-tion energy of 3.573 eV in Fig. 2(a) , which approaches to the band edge of the Al 0.1Ga0.9N barrier. In general, a part of photoexcited car- riers in the Al 0.1Ga0.9N barrier would transfer to the GaN QW imme- diately, and then theoretically the TRKR signals would be contributedfrom both the Al 0.1Ga0.9N barrier and GaN QW. Thanks to a fast electron spin relaxation in the GaN QW, the slow spin relaxationsignature should describe the electron spin relaxation process in theAl 0.1Ga0.9N barrier alone. To quantify these processes, the spin relaxation time sswas determined by fitting the TRKR signal using a single-exponentialdecay Aexp/C0t=s s ðÞ for Bext¼0 and to a damped cosine Aexp/C0t=ss ðÞ cosðxtÞforBext>0, where Ais the amplitude of the spin polarization, x¼glBBext=/C22hrepresents the Larmor frequency, /C22h is the reduced Planck constant, and lBis the Bohr magneton.9It deserves to be mentioned that the decay of the spin polarization comesfrom both the spin relaxation and recombination of carriers. For thebulk GaN and Al 0.1Ga0.9N, the carriers’ lifetime is about 20 ps6only shorter than that of the spin relaxation time; but it is crucial to remind us that the spin relaxation of holes is even faster, within hundreds of femtoseconds. After that, the remaining holes are unpolarized, andconsequently, there will be polarized electrons left over during andafter the later period of the carrier recombination process, which dom-inate the major spin polarization decay process. In order to obtain thespin relaxation time precisely, the Kerr signals at t>50 ps were used for the single exponential fittings as shown in Fig. 2(a) ,w h e nt h e recombination of carriers has almost finished and the decay of Kerr FIG. 1. (a) Schematic diagram of a sample structure and an experimental configuration. (b) Photoluminescence spectrum of the sample at room temperature and t he double- peak Lorentzian fitting results for raw data. The excitation energy and power are 3.815 eV and 10 mW, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252107 (2021); doi: 10.1063/5.0053806 118, 252107-2 Published under an exclusive license by AIP Publishingsignals purely originates from the spin relaxation. In the GaN QW, the considerable polarization electric field would induce the spatial separa-tion between the wave functions of electrons and holes, thus indeed resulting in a long carriers’ lifetime of hundreds of picoseconds. 21At the initial stage t<50 ps, the slow recombination of carriers can be neglected and the fast spin relaxation mainly contributes to the decay of Kerr signals as shown in Fig. 2(b) . Therefore, the single exponential fittings can give the spin relaxation time directly. It is found that the spin relaxation time of electrons in the GaN QW sQW s¼7p si sm u c h shorter than that in the bulk GaN sGaN s¼40 ps or in the Al 0.1Ga0.9N barrier sAlGaN s ¼140 ps. The enormous difference in spin relaxation time originates from the individual SOC and momentum scattering in different layers, which could be well explained with a DP spin relaxa- tion mechanism. In bulk semiconductors with wurtzite structure, the Hamiltonian for both Rashba and Dresselhaus terms is6 HSO¼HR SOþHD SO¼/C22h 2XkðÞ/C1r; (1) with XkðÞ¼2 /C22haeþcebk2 z/C0k2 k/C16/C17 hi ky /C0aeþcebk2 z/C0k2 k/C16/C17 hi kx 00 BBB@1 CCCA; (2) where zk0001½/C138 ,xk11 20½/C138 ,yk1100½/C138 ,k2 k¼k2 xþk2 y,ris the vector of the Pauli spin matrices, aedescribes the Rashba coefficient, and ceandbare the parameters of Dresselhaus SOC. The DP spin relaxa- tion rate is determined by6 Cij¼1 2hX2kðÞisp¼1 2dijX2hi /C0XiXj/C10/C11/C16/C17 sp; (3) where spis the momentum scattering time and /C1/C1/C1hi denotes the momentum distribution. For bulk wurtzite GaN, it is reported that theRashba contribution is dominant in the nondegenerate regime atroom temperature. 6The electron momentum can be described by the Boltzmann distribution at room temperature, and the DP spin relaxa- tion time in the bulk GaN and Al 0.1Ga0.9N barrier can be expressed as ss¼/C22h4 4a2 em/C3kBTs/C01 p; (4) where m/C3is the electron effective mass and kBis the Boltzmann con- stant. For GaN/AlGaN QW, however, the large polarization electricfield induces giant Rashba SOC, the Dresselhaus term can thus beignored. 22Therefore, the DP spin relaxation time in the GaN QW could also be estimated in Eq. (4).16Furthermore, the DP spin relaxa- tion time in the bulk GaN, Al 0.1Ga0.9N barrier, and GaN QW is calcu- lated using the parameter value listed in Table I . It is worth noting that the large Rashba coefficient23and long momentum scattering time (high electron mobility)12,13at the GaN/AlGaN heterointerface result in the short spin relaxation time. Meanwhile, the fast momen- tum scattering rate due to the additional alloy disorder scattering would lead to a long spin relaxation time in the Al 0.1Ga0.9N alloy. FIG. 2. (a) TRKR signals at an excitation energy of 3.402 and 3.573 eV, corresponding to the electron spin relaxation in the bulk GaN buffer layer and the Al 0.1Ga0.9N barrier. Damped-cosine fitting results give the corresponding spin relaxation time. (b) TRKR signals at an excitation energy of 3.513 eV at Bext¼0 and Bext¼0:5 T. The oscillations after the fast decay might originate from the weak spin signals under the excitation of the region with low Al component in the AlGaN barrier. TABLE I. Rashba coefficient ae, electron effective mass m/C3, momentum scattering time sp, and calculations of the DP spin relaxation time ssin the bulk GaN, Al0.1Ga0.9N/GaN QW, and Al 0.1Ga0.9N barrier. ae(meV A ˚)m/C3(me)sp(fs) ss(ps) Bulk GaN 9a0.2a40a49 Al0.1Ga0.9N/GaN QWb22c0.145d100 5 Al0.1Ga0.9N barrier 8.2e0.22f15g144 aFrom Ref. 6. bEstimated from the parameter values in Al 0.1Ga0.9N/GaN heterostructures. cFrom Ref. 23. dFrom Ref. 27. eLinearly interpolated by the parameters of GaN and AlN in Refs. 6and28. fLinearly interpolated by the parameters of GaN and AlN in Ref. 18. gFrom Ref. 26.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252107 (2021); doi: 10.1063/5.0053806 118, 252107-3 Published under an exclusive license by AIP PublishingComparing Fig. 2 andTable I , it is demonstrated that the calculations are well consistent with our experimental results. In the following, to elucidate the DP spin relaxation mecha- nism, the impacts of an external magnetic field strength and pho-toexcited carrier density on the spin relaxation time were discussed in detail. First, in the bulk GaN layer, the anisotropic SOC depicted in Eq. (2)would lead to anisotropic spin relaxation under the framework of DP mechanism, which has been depicted in Ref. 9. Consequently, s B s¼4 3s0 sis expected as shown in Fig. 3(a) ,w h e r e sB s(s0 s) denotes the spin relaxation time with (without) the external magnetic field. Figure 3(b) illustrates the carrier density depen- dence of the spin relaxation time. The anisotropic spin relaxationhas been evidenced within our observation range. It is found that the spin relaxation time has a weak dependence of electron density. Under our experimental conditions, the photoexcited electrondensity is actually much less than the critical density n c/C251:7 /C21018cm/C03, determined by kBT¼3p2nc/C0/C12=3/C22h2=2m/C3: (5)Then the nondegenerate regime should be assumed. In the nondegen- erate regime, the inhomogeneous broadening X2kðÞ/C10/C11 is nearly inde- pendent of electron density as indicated in Eq. (4),w h e r et h eR a s h b a coefficient aeis considered not to vary with electron density.7The momentum scattering time spremains constant since the electro- n–phonon scattering dominates in undoped GaN at room tempera- ture.24Hence, the spin relaxation time has a weak dependence of electron density. It is concluded that the electron spin relaxation in thebulk GaN is dominated by the DP mechanism, where electron–pho-non scattering plays the key role. Owing to a fast spin relaxation in the GaN QW, it is difficult to explore the change of electron spin relaxation time limited by the timeresolution of the TRKR measurements. Figure 4(a) shows the spin relaxation time of electrons in the Al 0.1Ga0.9N barrier as a function of an external magnetic field, which also satisfies the anisotropic spinrelaxation, similar to that in the case of bulk GaN. However, the spinrelaxation time here shows an increasing trend with the photoexcited electron density, as illustrated in Fig. 4(b) .C o m p a r e dw i t ht h e momentum scattering in the bulk GaN, the alloy disorder scattering FIG. 3. Magnetic field strength (a) and photoexcited electron density (b) dependence of the electron spin relaxation time in the bulk GaN. FIG. 4. Magnetic field strength (a) and photoexcited electron density (b) dependence of the electron spin relaxation time in the Al 0.1Ga0.9N barrier.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 252107 (2021); doi: 10.1063/5.0053806 118, 252107-4 Published under an exclusive license by AIP Publishinghas a profound effect on the electron momentum scattering time in the Al 0.1Ga0.9N barrier. With increasing the carrier density, the Coulomb screen effect becomes more pronounced, and the alloy disor-d e rs c a t t e r i n gi st h u se n h a n c e d . 25,26Whereas once the carrier density goes up to the degenerate regime, the Coulomb screen effect reaches saturation and hence the alloy disorder scattering changes insignifi-cantly. Therefore, according to Eq. (4), the spin relaxation time grows with the carrier density and the growth rate becomes gradually slower.In brief, the DP mechanism assisted with the alloy disorder scatteringexerts a crucial effect on the electron spin relaxation in the Al 0.1Ga0.9N barrier. In conclusion, the electron spin relaxation in the bulk GaN, GaN/Al 0.1Ga0.9N QW and the Al 0.1Ga0.9N barrier was studied system- atically by a photon-energy dependent TRKR spectrum at room tem-perature. Three spin relaxation processes could be well distinguished by selective resonate optical transitions in various layers. The enor- mous difference in the spin relaxation time was well explained withthe DP theory, which is governed by the Rashba contribution. A shortspin relaxation time of subpicosecond in the GaN/Al 0.1Ga0.9NQ W was obtained, which results from the large Rashba coefficient and along momentum scattering time induced by the considerable polariza-tion electric field. For bulk GaN and Al 0.1Ga0.9N, it is observed that the anisotropic SOC induces anisotropic DP spin relaxation. The elec-t r o n – p h o n o ns c a t t e r i n ga n da l l o yd i s o r d e rs c a t t e r i n gp l a yk e yr o l e si nthe DP mechanism, respectively, in the two cases. These findings arehelpful to understand the spin relaxation processes in the nitride quan-tum systems with a conspicuous polarization effect, and hence in favorof possible applications for GaN based spintronic devices. This work was supported by the National Key Research and Development Program of China (Nos. 2018YFB0406603 and2018YFE0125700) and the National Natural Science Foundation ofChina (Nos. 61574006, 61522401, 61927806, 61521004, and11634002). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1H. J. Chang, T. W. Chen, J. W. Chen, W. C. Hong, W. C. Tsai, Y. F. Chen, and G. Y. Guo, Phys. Rev. Lett. 98(13), 136403 (2007). 2T.-E. Park, Y. H. Park, J.-M. Lee, S. W. Kim, H. G. Park, B.-C. Min, H.-J. Kim, H. C. Koo, H.-J. Choi, S. H. Han, M. Johnson, and J. Chang, Nat. Commun. 8, 15722 (2017). 3A. Bhattacharya, M. Z. Baten, and P. Bhattacharya, Appl. Phys. Lett. 108(4), 042406 (2016).4S. 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5.0051461.pdf

J. Appl. Phys. 129, 234302 (2021); https://doi.org/10.1063/5.0051461 129, 234302 © 2021 Author(s).Tuning thermoelectric efficiency of monolayer indium nitride by mechanical strain Cite as: J. Appl. Phys. 129, 234302 (2021); https://doi.org/10.1063/5.0051461 Submitted: 25 March 2021 . Accepted: 28 May 2021 . Published Online: 15 June 2021 M. M. Cicek , M. Demirtas , and E. Durgun ARTICLES YOU MAY BE INTERESTED IN Interfacial engineering for the enhancement of interfacial thermal conductance in GaN/AlN heterostructure Journal of Applied Physics 129, 235102 (2021); https://doi.org/10.1063/5.0052742 Forward and reverse current transport mechanisms in tungsten carbide Schottky contacts on AlGaN/GaN heterostructures Journal of Applied Physics 129, 234501 (2021); https://doi.org/10.1063/5.0052079 Optical cloaking and invisibility: From fiction toward a technological reality Journal of Applied Physics 129, 231101 (2021); https://doi.org/10.1063/5.0048846Tuning thermoelectric efficiency of monolayer indium nitride by mechanical strain Cite as: J. Appl. Phys. 129, 234302 (2021); doi: 10.1063/5.0051461 View Online Export Citation CrossMar k Submitted: 25 March 2021 · Accepted: 28 May 2021 · Published Online: 15 June 2021 M. M. Cicek,1,2 M. Demirtas,1and E. Durgun1,a) AFFILIATIONS 1UNAM—National Nanotechnology Research Center and Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey 2Department of Engineering Physics, Faculty of Engineering, Ankara University, Ankara 06100, Turkey a)Author to whom correspondence should be addressed: durgun@unam.bilkent.edu.tr ABSTRACT Tuning the thermoelectric efficiency of a material is a complicated task as it requires the control of interrelated parameters. In this respect, various methods have been suggested to enhance the figure of merit (ZT), including the utilization of low-dimensional systems. Motivatedby the effect of strain on intrinsic properties of two-dimensional materials, we examine the thermoelectric response of monolayer indium nitride (h-InN) under low biaxial strain ( +1%) by using ab initio methods together with solving Boltzmann transport equations for elec- trons and phonons. Our results indicate that among the critical parameters, while the Seebeck coefficient is not affected prominently, electri-cal conductivity can increase up to three times, and lattice thermal conductivity can decrease to half at /C01% strain where valence band convergence is achieved. This results in significant enhancement of ZT, especially for p-type h-InN, and it reaches 0.50 with achievablecarrier concentrations ( /difference10 13cm/C02) at room temperature. Thermoelectric efficiency further increases with elevated temperatures and rises up to 1.32 at 700 K, where the system remains to be dynamically stable, suggesting h-InN as a promising material for high-temperature ther- moelectric applications. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0051461 I. INTRODUCTION Thermoelectric energy conversion that can convert waste heat directly into electricity (and vice versa) has drawn attention over the past few decades as a promising energy harvesting strategy.1,2The efficiency of thermoelectric materials is evaluated by the dimension-less figure of merit, ZT ¼S 2σT=κ,w h e r e Sis the Seebeck coeffi- cient, σis the electrical conductivity, S2σis the power factor (PF), T is the absolute temperature, and κis the sum of electronic ( κe)a n d lattice ( κl) thermal conductivity.3Thermoelectric materials with high efficiency should simultaneously have a large PF and a low κ,w h i c h is a challenging task due to the interdependence of these parameters.Therefore, several approaches have been suggested to enhance ZT, including nanostructuring, 4,5band engineering,3,6–8and chemical modification.9,10However, the desired efficiency for high technologi- cal devices to be integrated into a vast range of sectors could nothave been achieved yet. 11 An alternative and promising strategy to optimize ZT is utiliz- ing two-dimensional (2D) materials as reduced dimensions, which can lead to better thermoelectric conversion efficiency compared tothe bulk counterparts.12–14For instance, a decrease in the thickness (i.e., confinement length) and the thermal de Broglie wavelength can enhance the PF.13Together with this, electrons and phonons have different mean free paths due to the confinement, whichallows engineering their contributions to thermal conductivity individually. 12Accordingly, superior thermoelectric properties of several 2D materials have been demonstrated14–16and their poten- tial to be used in thermoelectric devices is revealed.17,18Moreover, 2D semiconductors naturally offer high tunability of thermoelectric response with chemical modification, strain, and stacking.18–24 Within the quest for identifying potential 2D materials for ther- moelectric applications, the class of group III-nitrides that possess novel electronic and optical properties25are also taken into consider- ation. Although monolayers of BN, AlN, and GaN have been reported to have low ZT values, the relatively high (low) electrical (thermal) conductivity of monolayer hexagonal indium nitride(h-InN) leads to good thermoelectric efficiency. 26,27Albeit 2D InN has not been realized yet, the stability of h-InN has been revealed and its fundamental properties have been characterized.28,29Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 234302 (2021); doi: 10.1063/5.0051461 129, 234302-1 Published under an exclusive license by AIP PublishingAdditionally, the synthesis of InN in thin film and nanostructure forms has been achieved and their thermoelectric properties have been investigated.30–33Accordingly, h-InN holds the promise to be used in thermoelectric applications subsequent to further enhance-ment of its intrinsic thermoelectric efficiency. With this motivation, in this work, we study the effect of low mechanical strain on the thermoelectric response of h-InN by using first-principle methods and Boltzmann transport theory.First, the optimized geometries and corresponding electronic bandstructures are obtained under compressive/tensile strain ( +1%). Next, the strain-dependent relaxation time ( τ), which is required to include the effect of scattering mechanisms (i.e., electron –phonon coupling), is estimated. The variation of these coefficients [Seebeckcoefficient ( S), electrical conductivity ( σ), power factor ( S 2σ), and electronic thermal conductivity ( κe)] with carrier concentration (electron and hole) are calculated for each strain value. To calculate the phonon transport properties, a primarily accurate phonon spec- trum is obtained and then the alteration of lattice thermal conduc-tivity ( κ l) with strain and temperature is examined. Finally, the figure of merit (ZT) under strain is calculated for varying tempera-ture and carrier concentration, and significant enhancement in thermoelectric efficiency of p-type h-InN under low compressive strain is revealed. II. METHOD The first principles calculations based on density functional theory (DFT) were performed using Vienna ab initio simulation package (VASP). 34–37Perdew –Burke –Ernzerhof (PBE) representa- tion of the generalized gradient approximation (GGA) was employed to describe the exchange-correlation functional.38The electronic band structures were also calculated with the inclusion ofspin –orbit coupling (SOC). 39The projector augmented wave (PAW)40,41potentials (including semi-core d-electrons) with a kinetic energy cutoff of 550 eV was used. The Brillouin zone (BZ) was sampled with 15 /C215/C21 k-point mesh during the structural relaxation until the energy difference between two electronic stepswas smaller than 10 /C06eV and the maximum Hellmann –Feynman force acting on each atom was less than 10/C05eV/Å. The vacuum space of 20 Å was set to eliminate the long-range interactions between the periodic images. The crystal structures were visualized by VESTA software.42 The electronic parts of the thermoelectric transport coeffi- cients were calculated by solving the semiclassical Boltzmann trans- port equation (BTE) by using the BoltzTraP code.43,44In the frame of the Boltzmann transport theory, the electronic transport coeffi-cients can be revealed as S αβ(T,μ)¼1 eTVσαβ(T,μ)ð σαβ(E)(E/C0μ)/C0@fμ(T,E) @E/C20/C21 dE, (1) σαβ(T,μ)¼1 Vð σαβ(E)/C0@fμ(T,E) @E/C20/C21 dE, (2) where αandβare the tensorial indices; E,T,V, and μare the elec- tron band energy, temperature, volume of the unit cell, and chemi- cal potential, respectively; fμis the Fermi distribution function.Theσαβ(E) is the energy projected conductivity tensors and can be calculated from the accurate band structure formula σαβ(E)¼1 NX i,kτ(i,k)vα(i,k)vβ(i,k)δ[E/C0E(i,k)], (3) where Nthe number of k-points used in the calculation, iis the band index, τ(i,k) is the relaxation time, vα(i,k) is the αcompo- nent of the band velocity, and E(i,k) is the energy of the state at k-point in the ith band. A denser k-point mesh (60 /C260/C21) was used to obtain accu- rate Fourier interpolation of the Kohn –Sham eigenvalues, which is used for solving BTE under the constant relaxation time approxi- mation in terms of carrier concentration. The spin –orbit coupling (SOC) was not included in the calculations of transport properties,as the band dispersion energies near the Fermi level were notaltered dramatically when SOC was taken into account (Fig. S1 in thesupplementary material ). The electronic transport coefficients S,σ/τ, and K e/τwere calculated as a function of carrier concentra- tion and temperature for each biaxial strain. To obtain ZT,45the relaxation time ( τ) was calculated by using deformation potential theory.46The theory relies on the coupling between electrons and acoustic phonons (one of the dominant scattering mechanism47) and is widely used in semiconductors,48,49 τ¼(μ/C1m*)=e, (4) with the band effective mass ( m*) defined as50 m*¼/C22h2 d2E(k)=dk2(5) and the carrier mobility ( μ) estimated as μ¼2e/C22h3Y2D 3kBTm*2E2 d, (6) where /C22his the reduced Planck constant, kis the wave vector, E(k)i s the energy of an electron at wavevector kin the corresponding band, eis the electron charge, Y2Dis the in-plane stiffness (Y2D¼c2 11/C0c2 12 c11, where cij’s are the elastic constants), kBis the Boltzmann constant, Tis the temperature, and Edis the deforma- tion potential constant. The calculated Y2Dfor h-InN is 65 N/m and Ed’s for hole and electrons are 5 and 20 eV, respectively. The lattice thermal conductivity ( κl) was obtained iteratively followed by calculation of the harmonic and anharmonic force con- stants based on the zeroth and fully iterative solution of theBoltzmann transport equation considering phonon –phonon interac- tion 51implemented in ShengBTE packages,52which takes dominant phonon scattering mechanisms into account (i.e., three-phonon pro- cesses and isotopic disorder). The strain-dependent phonon spectra were calculated by using the finite displacement method within5/C25/C21 supercells. 53Taking fifth nearest-neighbor interactions into account was tested to be sufficient to obtain converged anhar- monic force constants. The 50 /C250/C21 q-point grid along with the largest van der Walls thickness (3.74 Å) was used. The Born effectiveJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 234302 (2021); doi: 10.1063/5.0051461 129, 234302-2 Published under an exclusive license by AIP Publishingcharges and dielectric constants were derived based on the density functional perturbation theory (DFPT), which was added as a cor- rection to the dynamic matrix to take long-range electrostatic inter-actions into consideration. III. RESULTS AND DISCUSSION A. Crystal structure and electronic properties The optimized crystal structure of h-InN in the absence of strain is shown in Fig. 1(a) . Similar to the other group III-N mono- layers, h-InN has a planar honeycomb structure and belongs to the P-6M2 space group. 25The bond distance (d In/C0N) and the lattice constant ( ja!j) are calculated as 2.07 and 3.59 Å, respectively. The total charge density isosurface [ Fig. 1(b) ] represents the distribution of the electrons over the hexagonal lattice and reveals the strongin-plane σ-bonds upon hybridization of sp 2orbitals. The planarity is sustained by π-bonds formed by pzorbitals perpendicular to the lattice plane.25Bader analysis indicates a charge transfer of 1.39 jej from In to N, which is expected due to electronegativity differencebetween cation and anion atoms. The electronegativity differencealso leads to polarization of In –N bonds as revealed by the electron localization function (ELF) shown in Fig. 1(c) . The electronic band structure of h-InN along the symmetry directions of the hexagonal BZ, as well as the corresponding orbitalprojected density of states (PDOS), is shown in Fig. 2 (the elec- tronic band structures calculated with the inclusion of SOC are shown in Fig. S1 in the supplementary material ). The valence band maximum and the conduction band minimum occur at the K andΓsymmetry points, respectively, and the indirect bandgap ( E g/C0i)i s calculated as 0.56 eV (1.60 eV) within the PBE (HSE06) level. The highest valence band (HVB) is relatively flat, resulting in a high density of states (DOS) near the Fermi level ( EF), and originates mainly from the N- porbital. The lowest conduction band (LCB) is primarily composed of In- pand N- sorbital and more dispersive. To reveal the thermoelectric response of strained h-InN, we begin with the variation of electronic band structure with biaxial strain within +1%. For 2D systems, practically biaxial strain can bemore easily applied by considering the lattice mismatch between 2D materials and substrates.54The electronic structure calculations are kept at the PBE level as the band dispersion captures the trans-port properties and it is more crucial than the bandgap. Althoughthe PBE underestimates the bandgap, it is reliable to calculate thethermoelectric coefficients since the band profile can be obtained with sufficient accuracy. It should be noted that apart from the bandgap, the electronic band structures calculated with PBE andHSE06 have the same profile. In this respect, the shifts of the LCBand HVB under biaxial strain are shown in Fig. 3 . These shifts lead to a monotonic increase (decrease) of indirect ( E g/C0i) and direct bandgap ( Eg/C0d) with compressive (tensile) strain. Furthermore, the difference between them narrows with compressive strain andmerges at /C01%. 55The LCB is composed of a single parabolic valley and undergoes notable shifts in energy as well as becomes less dis- persive as applied strain varies from þ1% to /C01%. On the other FIG. 1. (a) The top and side views of the crystal structure of h-InN. The purple and light blue spheres indicate In and N atoms, respectively. The net charge oneach atom is given and the primitive unit cell is represented by the solid arrows. (b) The isosurface of the total charge density (isosurface level is set to 0.03 eÅ /C03) and (c) in-plane electron localization function (ELF). The increase in electron localization from 0 to 1 is indicated with blue to red color code. FIG. 2. Electronic band structure and orbital projected density of states (PDOS) of monolayer h-InN. The fundamental bandgap is shaded and the indirect (Eg/C0i) and direct bandgap ( Eg/C0d) are shown. FIG. 3. The variation of the valence and conduction bands in the proximity of the Fermi level ( EF) with biaxial strain. The inset illustrates the change of the indirect ( Eg/C0i) and direct bandgap ( Eg/C0d) under the biaxial strain.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 234302 (2021); doi: 10.1063/5.0051461 129, 234302-3 Published under an exclusive license by AIP Publishinghand, the valence band profile in the proximity of EFis more complex. There are twofold degenerate bands at the Γ-point, which are located at a slightly lower energy level than the HVB at theK-point. With compressive strain, while HVB at the K-pointremains almost intact, the bands at the Γpoint move upward and reduce the energy difference with HVB. This leads to a band con- vergence between /C00:5% and /C01% resulting in an increasing band degeneracy (i.e., the band extrema of the multiple bands have thesame or comparable energy within a few k BT).7In addition to band convergence, these bands are flattened, which also affects theelectronic transport features (see below). When compressive strain increases beyond /C01%, energy bands at Γ-point replace the original HVB and band degeneracy gradually disappears. Additionally, wealso calculated the electronic band structure of h-InN under uniax-ial strain ( +1%). The obtained results are reported in Fig. S5 in the supplementary material . It is noticed that the profile of band struc- tures (or evaluation of band dispersions) is similar under both types of strain. On the other hand, the band convergence undercompressive strain is less evident for the uniaxial case. From the strain-dependent electronic band structures, the band effective mass ( m *)50and carrier mobility ( μ) of charge carri- ers are obtained [Eqs. (5)and(6)] and these quantities can be used to estimate the relaxation time ( τ) by using Eq. (4). The m*andμ are calculated separately for hole ( m* h,μh) and electron ( m* e,μe) car- riers at 300 K, and their variation with the biaxial strain is given inFigs. 4(a) and 4(b). For all strain levels, m* his higher than m* e, which can be attributed to more dispersive LCB with respect to HVB. In a similar manner, as μ/1=m*,μhis lower than μewithin these strain levels. The sharp decrease (increase) in m* h(μh)a t /C00:5% [Fig. 4(a) ] is related to the modification of valance bands in the vicinity of EFwith strain as explained above.56For the case of electron carriers, m* e(μe) monotonically increases (decreases) in response to biaxial strain from þ1% to /C01% correlated with the cur- vature alternation of LCB. As the band dispersions are not affectedwith the inclusion of SOC, m *,μ,a n d τare estimated at the level of PBE. The obtained results (strain- and temperature-dependent) are summarized in Tables S1 and S2 in the supplementary material .I t should be also noted that while applying the deformation potentialtheory, only the electron –acoustic phonon coupling, which is one of the dominant scattering mechanisms is taken into account. 27,47,57,58 The other possible scattering mechanisms59,60(i.e., polar optical phonon scattering and intervalley scattering) are not considered due to the challenges regarding the computation of electron –phonon coupling in 2D systems.61,62For these reasons, our discussion is restricted with the intrinsic scattering based on the deformationpotential theory, which yields an upper limit for the mobility. Thus, our reported transport coefficients are within the lower limit, which should increase when the different scattering mechanisms are takeninto account. The variation of temperature-dependent mobility ofh-InN with the inclusion of acoustic and polar phonon scattering is s h o w ni nF i g .S 4i nt h e supplementary material . B. Electronic transport properties On the basis of electronic band structure, strain-dependent electronic transport coefficients ( S,σ, PF) can be evaluated by solving BTE and the results are given in Fig. 5 for both charge car- riers (p- and n-type) as a function of carrier concentration ( ρ)a t 300 K. First, in the absence of strain, the absolute value of Seebeckcoefficient ( jS hjandjSej) decreases for both p- and n-type h-InN with ρas shown in Fig. 5(a) , which is in alignment with the Mahan –Sofo theory.63Additionally, jShjis larger than jSejwithin the studied range. For instance, the maximum value of jShjis /difference700μV=K at room temperature, which is double of jSej (/difference350μV=K). This difference in jSjfor electron and hole carriers originates from the density of states effective mass, which is large for flatbands with high DOS around EF. A less dispersive and degenerate HVB (hole doping) when compared with LCB (electrondoping) leads to larger jSj. The applied biaxial strain has no signifi- cant effect on the Seebeck coefficient, therefore a similar trend isobtained with/without strain. In contrast to jSj, the electronic conductivity ( σ) increases with the increasing ρand for a given ρ,σ his smaller than σe [Fig. 5(b) ]. When strain is applied, σedecreases (increases) mono- tonically with compressive (tensile) strain, following the similartrend with μ ein the range of +1%. In the case of p-type doping, while the effect of tensile strain on σhis minor, it dramatically increases with compressive strain. This increase is also correlatedwith band convergence as it leads to multiple pathways of conduct-ing channels within similar energy and thus enhances σ h. These results suggest that p-type h-InN can demonstrate better perfor- mance than n-type system in the presence of strain. FIG. 4. The band effective mass ( m/C3) and carrier mobility ( μ) of h-InN at 300 K for (a) p-type (hole carriers) and (b) n-type doping (electron carriers) as a func-tion of the applied biaxial strain.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 234302 (2021); doi: 10.1063/5.0051461 129, 234302-4 Published under an exclusive license by AIP PublishingAsjSjdecreases and σincreases with ρ, their collective effect on thermoelectric response can be revealed by calculating thepower factor (PF= σS 2). The variation of PF with ρunder strain is shown in Fig. 5(c) .P F his a significantly larger PF e, resulting in a nearly threefold magnification at the same carrier concentration in the absence of strain. While biaxial strain has minimal effect onPFe,P F his notably enhanced under compressive strain. For instance, the maximum of PF hobtained at 8 /C21013cm/C02hole con- centration is 6.9 mW/(m K2) under /C01% strain and this value is two times higher than the corresponding PF hwithout strain. Therefore, the band convergence resulting from small compressivestrain leads to enlarged PF. C. Thermal transport properties A low thermal conductivity is an essential factor to achieve high thermoelectric performance. The electronic contribution to the thermal conductivity ( κ e) can be estimated by applying the Wiedemann –Franz law,44 κe¼LσT, (7) where Lis the Lorentz number (2 :4/C210/C08WΩK/C02for free elec- trons), and σis the electrical conductivity obtained from BTE. Accordingly, κefollows the same trend as σ, within the range of 0.31 –0.76 W m/C01K/C02as strain applied from þ1% to /C01% at 300 K. Its overall magnitude is much smaller than lattice thermal conduc-tivity ( κ l) and, therefore, does not affect ZT significantly. To obtain κl, an accurate phonon spectrum should be obtained in the first place. The phonon dispersion together withphonon partial density of states (pPDOS) is shown in Fig. 6(a) FIG. 5. (a) Seebeck coefficient ( jSj), (b) electrical conductivity ( σ), and (c) power factor ( PF¼σS2) as a function of carrier concentration ( ρ) at 300 K for both n- and p-type doping. FIG. 6. (a) Phonon band structure and partial phonon density of states (pPDOS) for unstrained h-InN. (b) The variation of lattice thermal conductivity (κl) with temperature and (c) the variation of phonon bandgap with strain. κlat /C01% is shown with dashed lines indicating the onset of imaginary frequencies at this compressive strain level.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 234302 (2021); doi: 10.1063/5.0051461 129, 234302-5 Published under an exclusive license by AIP Publishing(the strain-dependent phonon spectrum is shown in Fig. S2 in the supplementary material ). h-InN has three acoustic and three optical branches associated with two basis atoms in the primitivecell. While the transverse (TA) and longitudinal acoustic (LA)branches have a linear dispersion with the wave vector near the Γ point, the dispersion of flexural acoustic (ZA) branch is quadratic. The flexural optical branch (ZO) is coupled to the LA branch, which is ascribed to the planarity of group III-N monolayers. 64,65 The pPDOS analyses indicate that the optical phonon branches are governed by the vibration of light N atom, whereas the vibration ofmore massive In atom dominates the high frequency of LA branch. In the monoatomic chain model, bonding stiffness (i.e., inter- atomic force constant) and atomic mass in the primitive cell deter-mine the scale of phonon dispersion frequencies. When theinteratomic force constants are weaker, and the mass of the atomsis heavier, the phonon modes shift to lower frequencies. In this regard, the highest frequency of the normal mode vibration at the Γ-point for h-InN is 18 THz, which is lower than that of h-GaN (21 THz) and h-BN (40 THz). 64Ascending the frequency results in a low Debye temperature66(Θd¼/C22hωi, max=kband ωi, max is the maximum phonon frequency at the zone boundary of ith acoustic phonon branch), above which all the phonon modes are excited.67 Thus, a small Θdindicates an intense three-phonon scattering and leads to low κlregarding the Slack model68,69and clarifies the lowering of κlfrom B to In in the group III-N monolayers. The κl is calculated as 6.77 W/mK at room temperature for h-InN and it is lower than those of h-BN (245 W/mK)70and h-GaN (14.9 W/m),64,71in accordance with variation of Θd(1678, 522, and 143 K for h-BN, h-GaN, and h-InN, respectively). The low κlof h-InN can also be associated with the weak phonon harmonic inter- action and strong anharmonic scattering.72It should be noted that while calculating κl,p h o n o n –phonon interactions are involved and electron –phonon coupling is not taken into account. Electron – phonon coupling is substantial in highly doped semiconductors andmetals due to possessing enough electron states around the Fermi surface. 73Accordingly, in h-InN, this interaction is not expected to be strong. The suggested approach is applied to various 2D systemsincluding Group III-N monolayers, 64,74and also tested on prototype materials such as graphene.64,74The obtained results are in agree- ment with experimental results75and theoretical predictions. The variation of κlwith temperature is shown in Fig. 6(b) .I n the absence of strain, κldecreases with increasing temperature fol- lowing a T/C01dependence.76This type of variation suggests that the phonon transport scattering is mainly dominated by the Umklapp process, which modifies the thermal resistance.77When strain is applied, κldecreases (increases) with compressive (tensile) strain. Whereas the tensile strain leads to more isolated N- sbands, the compressive strain increases the overlapping wave functions of N- s lone-pair electrons with bonding electrons of adjacent In atoms. This induces nonlinear electrostatic forces upon thermal agitation, leading to an increase in the phonon anharmonicity and thusreducing the κ l.78–80Therefore, κlis reduced under the compres- sive strain as a result of enhanced lone-pair interaction, which isproposed as the dominant mechanism. The variation of κ lwith strain is similar to other 2D Group III-N systems81and this reduc- tion is also observed in other 2D systems such as silicene82,83and phosphorene.84The phonon gap between the highest frequency ofacoustic and the lowest frequency of optical branches increases (decreases) with compressive (tensile) strain [ Fig. 6(c) , and Fig. S3 in the supplementary material ]. Compressive strain softens the phonon modes and gather them in a smaller frequency range. Thisincreases the probability of phonon scattering and thereby reducesthe phonon relaxation time. 85,86As the acoustic phonon branches primarily contribute to the thermal conductivity, softening of these modes overall results in reduced group velocity and decreasedthermal conductivity. 76By contrast, tensile strain induces phonon stiffening, which often enhances the thermal conductivity. It shouldbe noted that imaginary frequencies start to emerge at higher com- pressive strain levels and thus the mechanical strain should be kept low to obtain realistic results. D. Figure of merit By combining all the obtained results, the figure of merit (ZT) for doped h-InN can be estimated. ZT as a function of ρat room FIG. 7. (a) The variation of figure of merit (ZT) with carrier concentration for n- and p-type doping at 300 K and (b) variation of strain-dependent ZT hwith temperature.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 234302 (2021); doi: 10.1063/5.0051461 129, 234302-6 Published under an exclusive license by AIP Publishingtemperature is shown in Fig. 7(a) . For doping concentrations, a realistic range (1 /C210/C013–8/C210/C013cm/C02), which is attainable in 2D materials, is considered.87The ZT of the n- and p-type doping is labeled as ZT eand ZT h, respectively. In the absence of strain, the maximum of ZT h(0.06) is threefold higher than ZT e(0.02), which is correlated with the variation of PF [ Fig. 5(c) ]. While ZT increases with compressive strain for both charge carriers, it is more promi- nent for ZT hwhere a rise by a factor of five is noticed at a strain level of /C01%. The increase in ZT hunder the compressive strain is aligned with the simultaneous decrease of κland the increase of PF. Such a dramatic enhancement is not obtained for ZT eas shown inFig. 7(a) . To reveal the thermoelectric performance of h-InN at elevated temperatures, we investigate the variation of ZT hwith tem- perature for each strain level and the results are given in Fig. 7(b) . The temperature range is selected according to the moleculardynamics results (Fig. S3 in the supplementary material ), which indicates that the h-InN is dynamically stable up to 700 K. As expected, ZT hincreases with temperature, indicating that h-InN exhibits better performance at high temperatures. ZT hreaches its maximum value (0.50 at 300 K and approaches up to 1.32 at700 K) at /C01% strain level, where band convergence is obtained. 88 The obtained ZT values are larger than the conventional thermo- electric materials, for instance, Bi 2Te3(1.0),89SnSe (0.70),90and PbTe (0.30).91 IV. CONCLUSION In summary, we have examined the strain-dependent thermo- electric properties of monolayer h-InN by using ab initio methods together with solving Boltzmann transport equations. Our resultsshow that the p-type system exhibits better performance than n-type, which is linked to the valence band profile near the Fermi level. Low compressive strain at the level of /C01% induces band con- vergence, which leads to a significant increase in PF together with asubstantial decrease in κ l. This modification results in high ZT, which is calculated as 0.50 at room temperature for the p-type system. Additionally, ZT enhances with elevated temperatures and reaches up to 1.32 at 700 K, at which h-InN maintains its dynami-cal stability. Our results indicate that the thermoelectric perfor-mance of p-type h-InN can be significantly enhanced with applicable (low) compressive strain at realistic doping levels and suggest this system as a promising material for high-temperaturethermoelectric applications. SUPPLEMENTARY MATERIAL See the supplementary material for additional details on elec- tronic band structures calculated with spin –orbit coupling, strain- dependent phonon spectra, molecular dynamics simulationsresults, strain- and temperature-dependent effective mass and relax-ation time values, the effect of polar phonon scattering on mobility, variation of electronic band structures under uniaxial strain and convergence tests regarding electronic, and phononic transport. ACKNOWLEDGMENTS The calculations were performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TR-Gride-Infrastructure) and the National Center for High Performance Computing of Turkey (UHeM) under Grant No. 5007092019. This work was supported by the Scientific and Technological ResearchCouncil of Turkey (TUBITAK) under Project No. 117F241. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1L. Yang, Z.-G. Chen, M. S. Dargusch, and J. Zou, Adv. Energy Mater. 8, 1701797 (2018). 2L. E. Bell, Science 321, 1457 (2008). 3J. He and T. M. Tritt, Science 357, eaak9997 (2017). 4M. S. Dresselhaus, G. Chen, M. Y. Tang, R. Yang, H. Lee, D. Wang, Z. Ren, J.-P. Fleurial, and P. Gogna, Adv. Mater. 19, 1043 (2007). 5M. G. Kanatzidis, Chem. Mater. 22, 648 (2009). 6G. Tan, F. Shi, S. Hao, H. Chi, T. P. Bailey, L.-D. Zhao, C. Uher, C. Wolverton, V. P. Dravid, and M. G. Kanatzidis, J. Am. 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5.0047381.pdf

J. Chem. Phys. 154, 214107 (2021); https://doi.org/10.1063/5.0047381 154, 214107Ultrafast x-ray pump x-ray probe transient absorption spectroscopy: A computational study and proposed experiment probing core-valence electronic correlations in solvated complexes Cite as: J. Chem. Phys. 154, 214107 (2021); https://doi.org/10.1063/5.0047381 Submitted: 12 February 2021 . Accepted: 18 May 2021 . Published Online: 02 June 2021 Chelsea E. Liekhus-Schmaltz , Phay J. Ho , Robert B. Weakly , Andrew Aquila , Robert W. Schoenlein , Munira Khalil , and Niranjan Govind ARTICLES YOU MAY BE INTERESTED IN Model protein excited states: MRCI calculations with large active spaces vs CC2 method The Journal of Chemical Physics 154, 214105 (2021); https://doi.org/10.1063/5.0048146 Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). I. Revisiting the NEVPT2 construction The Journal of Chemical Physics 154, 214111 (2021); https://doi.org/10.1063/5.0051211 Correlation-driven phenomena in periodic molecular systems from variational two-electron reduced density matrix theory The Journal of Chemical Physics 154, 214106 (2021); https://doi.org/10.1063/5.0050277The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ultrafast x-ray pump x-ray probe transient absorption spectroscopy: A computational study and proposed experiment probing core-valence electronic correlations in solvated complexes Cite as: J. Chem. Phys. 154, 214107 (2021); doi: 10.1063/5.0047381 Submitted: 12 February 2021 •Accepted: 18 May 2021 • Published Online: 2 June 2021 Chelsea E. Liekhus-Schmaltz,1,a) Phay J. Ho,2 Robert B. Weakly,1 Andrew Aquila,3 Robert W. Schoenlein,4,3 Munira Khalil,1 and Niranjan Govind5,a) AFFILIATIONS 1Department of Chemistry, University of Washington, Seattle, Washington 98195, USA 2Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 3LCLS, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 4Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 5Physical and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, Washington 99352, USA a)Authors to whom correspondence should be addressed: cliekhus@uw.edu and niri.govind@pnnl.gov ABSTRACT Femtosecond x-ray pump–x-ray probe experiments are currently possible at free electron lasers such as the linac coherent light source, which opens new opportunities for studying solvated transition metal complexes. In order to make the most effective use of these kinds of experiments, it is necessary to determine which chemical properties an x-ray probe pulse will measure. We have combined electron cascade calculations and excited-state time-dependent density functional theory calculations to predict the initial state prepared by an x-ray pump and the subsequent x-ray probe spectra at the Fe K-edge in the solvated model transition metal complex, K 4FeII(CN) 6. We find several key spectral features that report on the ligand-field splitting and the 3p and 3d electron interactions. We then show how these features could be measured in an experiment. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0047381 I. INTRODUCTION The advent of multicolor, ultrafast x-ray pulses at free electron lasers (FELs), such as the linac coherent light source (LCLS), offers the possibility of performing novel x-ray pump–x-ray probe (XPXP) spectroscopy experiments. X-ray pulses can now be separated by ∼30% of their photon energies, and future technologies will allow experimenters to arbitrarily tune the energy of one soft x-ray pulse and one hard x-ray pulse at the Tender X-ray Instrument at LCLS.1–5 XPXP experiments will therefore be able to measure an atomic site-specific electronic excitation with an atomic site-specific probe potentially tuned to a different element to directly measure charge flow in a molecule. For example, with sufficiently short ( ∼100 as) and intense pulses, the x-ray pump can be used in a stimulatedx-ray Raman measurement, creating a localized electronic excitation that can be probed with an x-ray pulse.6–10While pairs of different colored attosecond pulses are not currently demonstrated, longer x-ray pulses can also be used to generate novel, atomic site-specific excitations to study electronic dynamics. Most XPXP spectroscopy experiments have been conducted on gas phase molecules using electron and/or ion coincidence measure- ments.11–16These experiments have successfully studied the molec- ular dynamics of the selected gas phase molecules, and as a result, considerable effort has been made to extend these techniques into the liquid and solid phases.17–21Recent literature suggests that XPXP techniques8,22,23could be useful for studying solvated transition metal systems, which are common models for photocatalysis and photosynthesis. J. Chem. Phys. 154, 214107 (2021); doi: 10.1063/5.0047381 154, 214107-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Optical or IR pump–x-ray probe measurements have already proven that x-ray absorption spectroscopy is a useful tool for study- ing the transient electronic character in solvated transition metal complexes.24–28The limitation of these techniques lies in the pump energy. While optical pulses can be used to initiate charge transfer and measure subsequent changes in the valence electronic structure, they cannot initiate or measure changes in the local charge density around specific atoms. Instead, in an ultrafast, XPXP experiment, we can initiate complex orbital relaxations and electron motions by specifically ionizing the Fe atom, for example, and then monitor the changes via transmissive X-ray Absorption Near-Edge Spectroscopy (XANES). Femtosecond XPXP measurements provide the opportunity to extend these experiments to both shorter timescales by avoiding group velocity mismatch and by exciting novel electronic states to study detailed aspects of the electronic structure. Both benefiting and complicating this approach is the added layer of complexity brought by the x-ray pump pulse. X-ray pulses generate a core hole in the excited atom, which is subsequently filled FIG. 1. Experimental and calculated Fe K-edge XANES of K 4FeII(CN) 6and the complimentary complex K 3FeIII(CN) 6. Both oxidation states have B and C peaks, which correspond to a 1s to e gandπ∗transition. The A peak is only visible in the FeIIIoxidation state where there is a vacancy in the t 2gorbital. The calculated roots are broadened by 1.5 eV and scaled by 4 ×104from the raw oscillator strength. Figure 2 shows these transitions in more detail. The experimental data are given by arbitrary units.by higher lying electrons via fluorescence and Auger–Meitner decay, resulting in the ejection of one or more electrons and a superposition of many electronic states after a few femtoseconds.29Probing this host of new electronic states gives researchers access to previously hidden electronic interactions, answering the following questions: What role do core electrons play in electron transport processes? How do high lying core electrons impact the ligand-field splitting? The diversity of electronic states to be probed requires a knowledge of their distribution resulting from the cascade caused by the x-ray pump pulse. While x-ray absorption and emission spectra of solvated sys- tems have been successfully calculated using time-dependent density functional theory (TDDFT)30–36and electron cascades have been studied in liquid and solid samples,37electron cascades have not yet been studied in solvated transition metal systems to which most state-of-the-art electron cascade calculations are not readily appli- cable. In this paper, we attempt to bridge this knowledge gap by utilizing an atomic electron cascade calculation38,39to guide our TDDFT-based x-ray absorption near-edge spectroscopy (XANES) calculations. The system we have chosen to study is K 4FeII(CN) 6dissolved in water, shown in the inset of Fig. 1. This model complex has been well-studied in table-top experiments, in synchrotrons, and at FELs due to its ultrafast metal–ligand interactions and its role as a model for other metal centered ligand complexes.40Our study shows that the novel electronic states prepared by a femtosecond x-ray pulse provide a unique measurement of key chemical signa- tures, such as the effect of 3p and 3d electron interactions on the ligand-field energy splitting and the effect of different charged states on solute–solvent interactions. II. COMPUTATIONAL DETAILS All DFT- and TDDFT-based XANES calculations were per- formed with the NWChem computational chemistry program.30,41,42 The PBE0 exchange–correlation functional43was used as we have previously shown that this functional is sufficiently accurate to describe the overall structure, dynamics, and spectra over a wide energy range for solvated transition metal complexes.33,34The COSMO (COnductor-like Screening MOdel)44,45implicit solvation model with a dielectric constant of 80.1 was used to represent the water solvent instead of an explicit solvent representation to reduce the complexity of the model. All-electron calculations were performed using the 6-311G∗∗basis set46,47for the light atoms and the Sapporo-TZP-2012 basis for the Fe atom.48Pre- viously converged geometries from Ref. 33 and tabulated in the supplementary material were used and were kept fixed for all calculations. For the electron cascade dynamics, we used our previously developed on-the-fly Monte Carlo rate equation method,49,50which has been used to model nonlinear x-ray multiphoton processes and inner-shell relaxation in atoms and molecules. In this method, all electronic processes are treated as random quantum processes, where the probabilities are weighted by the transition rates. The rates are computed using the Hartree–Fock–Slater (HFS) electronic struc- ture method, which includes relativistic corrections and spin–orbit coupling terms, following the procedure outlined in Ref. 51. We have used this method to study x-ray interaction with bromine J. Chem. Phys. 154, 214107 (2021); doi: 10.1063/5.0047381 154, 214107-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ions.52Briefly, we first calculate the non-relativistic HFS energies and wave functions for all the occupied and unoccupied orbitals. From these wave functions and the HFS potential, the relativis- tic energy corrections (relativistic mass-velocity and Darwin cor- rections) and spin–orbit energies are computed. To compute the inner-shell decay rate, the non-relativistic HFS wave functions and orbital energies with relativistic and spin–orbit corrections are used. We note that these cross sections and rates are calculated for spin-averaged electronic configurations, but they are sensitive to the occupation numbers and energies of the individual spin–orbit orbitals. We used 100 000 trajectories to obtain a converged time pro- file of the cascade dynamics. The following steps are repeated until the transitions cease. In our cascade dynamics simulations, each tra- jectory is started with a 1s hole in the Fe2+ion in the low-spin electronic configuration to approximate the low-spin K 4FeII(CN) 6 complex and the rates of individual Auger and fluorescence decay channels are computed. A random number is then generated to determine if an electronic transition takes place in a given time step. A transition occurs if the calculated probability of the transition is greater than the randomly generated number. If no electronic transition occurs, the time step is advanced and a new random number is generated. If a transition occurs, one decay chan- nel is selected and the Auger and fluorescence rates are computed for the new selected electronic configuration. The time step is then advanced, and another random number is generated to determine if there is a transition. These steps are repeated until there are no more transitions. From the time history of these transitions, the time profile of all participating electronic configurations can be computed. III. GROUND STATE EXPERIMENTAL AND CALCULATED SPECTRA We begin by examining the ground state Fe K-edge XANES spectra of K 4FeII(CN) 6and K 3FeIII(CN) 6to establish the essential XANES features, which will be used as a baseline when describ- ing the excited-state Fe K-edge features in Sec. IV. Given that the x-ray pump will produce a highly ionized compound, we first compare ground state spectra of two analogous compounds, with the oxidation state of Fe varied between them to provide refer- ence points for understanding the spectra to come. We present both experimental and calculated XANES spectra of K 4FeII(CN) 6 and K 3FeIII(CN) 6. Experimental XANES data were taken at the Advanced Light Source (ALS) at beamline 10.3.2 and are reproduced from Ref. 33, where a more detailed experimental account can be found. The experimental and calculated XANES spectra of the FeII and FeIII oxidation states of K 4FeII(CN) 6are shown in Fig. 1. Both K4FeII(CN) 6and K 3FeIII(CN) 6have B peaks, which are quadrupole transitions from the 1s to e gorbitals, and C peaks, which correspond to primarily 1s to π∗orbital transitions (shown in detail in Fig. 2). K3FeIII(CN) 6has an “A” peak, corresponding to a quadrupole tran- sition from the 1s to t 2gorbital, which has a vacancy in this oxi- dation state. Both the A and C peaks are significantly modified by the change in oxidation state with the A peak appearing and the C peak blueshifting as the Fe oxidation state changes from II to III. The energy splitting between the A and B peaks is a key signature of the FIG. 2. Comparison of orbitals. Atomic orbitals of the bare Fe2+ion are compared with the K 4FeII(CN) 6molecular orbitals. The 3p orbitals in the solvated system are well represented by the ion, while the 3d orbitals are strongly modulated by the ligands. Also shown are the electronic transitions involved in the XANES spectrum. We label the 1s to t 2gtransition “A,” the 1s to e gtransition “B,” and the 1s to π∗ transition “C.” The splittings are not drawn to scale. Both sets of orbitals are drawn with the same isovalues. Orbitals of K 4FeII(CN) 6are further scaled in unison to show the molecular frame and ligand contributions to e gand t 2gmolecular orbitals. valency of the Fe atom in Fe(CN) 6and is a direct measure of the ligand-field splitting, which is 3.3 eV in the experimental spectrum and 3.2 eV in the calculated spectrum.33 While the position of the A and B peaks is well repre- sented by the calculation, the C peak is less so. The lowest lying roots of the C peak are 1.4 eV removed from the experimental data, and the C peak maximum is further removed, more than 3 eV from the experiment. This discrepancy is largely due to the strong solvent interactions involved in the π∗orbitals, which are not fully captured by the implicit solvent model. In our earlier work,33we have shown how the C peak is strongly dependent on the solvent. The C peak is therefore an important marker for probing solvent–solute interactions. All three of these peaks pro- vide important insights into the valency, ligand-field splitting, and solvent–solute interactions. In Secs. IV and V, we will show that the XPXP spectrum has similar peaks that provide additional measures of these properties and provide more opportunity to study these interactions. IV. X-RAY PUMP–X-RAY PROBE CALCULATIONS We will consider the following x-ray pump–x-ray probe pro- cedure: a femtosecond x-ray pump with photon energy above the Fe K-edge removes a 1s electron to initiate an electron cascade and an x-ray probe that is below the Fe K-edge observes the XANES J. Chem. Phys. 154, 214107 (2021); doi: 10.1063/5.0047381 154, 214107-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp signal. The calculated electron cascade informs the XANES calcula- tions by reporting the type and distribution of both core and valence holes in the system as a function of time. While we can calculate the Fe K-edge XANES spectrum of K 4FeII(CN) 6, at this time, it is not straightforward to calculate the electron cascade of the solvated sys- tem. Therefore, to simplify our calculations, we have performed the electron cascade calculations on an isolated Fe2+ion, which has the same charge as the Fe oxidation state and low-spin configuration as in K 4FeII(CN) 6as described earlier. We then use the resulting elec- tronic configurations to determine which configurations (holes) to consider as references in our XANES calculations. The accuracy of this simplification is largely determined by the similarities and differences between the atomic and molecular orbitals. In Fig. 2, we show the 3p and 3d atomic and molecular orbitals of both the Fe2+ion and K 4FeII(CN) 6to compare the two. From visual inspection, we see that both 3p orbitals are very simi- lar. This is because the 3p molecular orbital is completely composed of Fe atomic orbitals. The 3d molecular orbital shows obvious dif- ferences between the ion and the molecule due to the octahedral metal–ligand coordination. In contrast to the 3p orbital, the Fe t 2g orbital is hybridized with the cyanide ligand orbitals. It is likely, therefore, that electron cascade processes that involve 3p and lower orbitals are well represented by the Fe2+ion, while those processes that involve electrons in the 3d orbitals are less likely to be accurately modeled. A. Electron cascade: X-ray pump calculation The result of the electron cascade calculation is shown in Fig. 3 where the top ten most probable electronic configurations after 10–100 fs are displayed. These electronic states compose 75% of the total population at the start of 10 fs. We see that the original 1s hole has resulted in configurations with primarily 3p or 3d holes. Some of these configurations have additional lower lying holes in the 3s orbital. The probability of having a vacancy in 2s or 2p after 10 fs is small. The remaining states not shown in Fig. 3 also contain pri- marily 3s, 3p, and 3d holes. Overall, in the 10–100 fs window, more than 99% of the configurations has one or more 3p or 3d holes and about 17% of these configurations also have a 3s hole. For the rest of our analysis, we use the resulting population statistics to motivate studying the effect of several combinations of 3p and 3d holes on the XANES spectrum of K 4FeII(CN) 6. B. Fe K-edge XANES: X-ray probe calculations Having identified 3p and 3d orbitals as the most important electron hole locations after a 1s ionization event, we now cal- culate the resulting XANES spectra of the solvated system with these reference hole configurations. The spectra we generate use the same geometry, basis set, and exchange correlation as those in the ground state K 4FeII(CN) 6calculation. The reference hole con- figurations are generated by converging the Kohn–Sham orbitals with specified molecular orbital occupancies using the “occup” block feature in NWChem, which are then used for XANES cal- culations. By using TDDFT, we acknowledge that the spectra can only be captured within the space of single excitations. However, we believe this approach can provide a first order estimate in combination with the appropriate reference. The final calculated FIG. 3. Electron configurations after electron cascade. The probabilities of the top ten most likely electron configurations after 10–100 fs are shown above. Lines of the same color have the same lowest electron hole, and lines of the same style have the same highest electron hole. To prepare for the XANES portion of the cal- culation, we rely on the Monte Carlo electron cascade calculation to identify which holes are the most relevant to a potential x-ray pump–x-ray probe experiment. We see that 3p and 3d orbitals are the most likely places for an electron hole after 10–100 fs. XANES spectra of K 4FeII(CN) 6in the ground state and five differ- ent excited electronic configurations that involve 3p and t 2gholes are shown in Fig. 4. Each XANES spectrum shows the character- istic B and C peaks that represent 1s to e gand π∗transitions, respectively. The first aspect we focus on is the appearance of new peaks in the spectra shown in Fig. 4. There are two main sets of addi- tional peaks that appear, the A peak and a much stronger lower energy peak around 7060 eV. The A peak appears when there is a vacancy in the t 2gorbital as in the configurations [Ne] 3s23p63t5 2g, [Ne] 3s23p6t4 2g, and [Ne] 3s23p5t5 2g. The strength of the A tran- sition also scales with the t 2gorbital vacancy, which can be seen by comparing [Ne] 3s23p6t5 2gand [Ne] 3s23p63t4 2g. In addition, the splitting between the A and B peaks increases with the orbital vacancy going from ∼3.2 to 5.4 eV as the hole vacancy increases from 1 to 2. Initial states with a hole in the t 2gorbital provide a unique opportunity to examine the key chemical properties of K 4FeII(CN) 6. As in the ground state, the splitting between the A and B peaks provides a direct measurement of the splitting between the t 2g and e gorbitals as well as a measurement of the Fe oxidation state. Measuring this splitting with different electron configura- tions, such as with or without a 3p hole provides the opportunity to examine how the ligand-field splitting is affected by interactions between the 3p and 3d electrons or the oxidation state of the Fe atom. The lower energy peaks around 7060 eV, which we will refer to as the 3p peaks, appear in the configurations [Ne] 3s23p53t6 2g, [Ne] 3s23p53t5 2g, and [Ne] 3s23p43t6 2gwhere there is a 3p vacancy. Here, the transitions are much stronger since they are dipole allowed, com- pared to the A and B peaks, which are dipole forbidden. Again, the strength of the transition scales with the vacancy of the 3p orbital. As with the A peaks, the 3p peaks also shift with t 2gand 3p vacancy, J. Chem. Phys. 154, 214107 (2021); doi: 10.1063/5.0047381 154, 214107-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. Calculated XANES spectra of several electron configurations with 3p and t2gholes. The spectra are calculated using 100 roots, broadened by 1.5 eV, and shifted by 143.8 eV to match the K 4FeII(CN) 6ground state XANES. The top panel shows the spectra of electronic configurations with filled 3p orbitals, while the bot- tom panel shows the spectra of electronic configurations with at least one hole in the 3p orbitals. For easy visualization, we only plot the A, B, and C peak regions in the main figures. The A peaks are indicated by arrows. The inset shows the lower energy peaks caused by a 3p vacancy. once again providing a measure of the interaction between the 3p and t 2gelectrons. The electron configuration also affects the C peak. Removing electrons from the t 2gorbital blueshifts the broad C peak by ∼2.5 eV, while removing a 3p electron causes an ∼7.5 eV shift. As we saw in the ground state calculation, the C peak is sensitive to solvent–solute interactions.33Therefore, the size of the shifts reflects both the rela- tive screening ability of the different orbitals on the excited 1s elec- tron and changes in solvent–solute interactions due to the various charge states. The C peak can therefore measure not only the effect of the overall charge on solvent interactions as noted in earlier stud- ies but also the position of the charge within the total electronic state depending on the hole position. In an XPXP experiment, we expect that some portion of the states we have discussed above will contribute to the initial state since they would appear after ∼10 fs, a timescale short enough that significant nuclear or valence electronic motion would not yet occur. We also see in Fig. 3 that the electron cascade does not significantly evolve after this time, meaning that further changes would then be dominated by metal–ligand or solute–solvent interactions. While we do not consider the evolution of these peaks as a function ofpump–probe delay in this manuscript, the peaks that we identified in this discussion would be good candidates for monitoring in a time evolving system. V. PROPOSED EXPERIMENT In consideration of the proposed experiment, it is prudent to underscore an easily overlooked outcome of the calculated XANES spectra. As seen in Fig. 4, a 3p core hole shifts the valence energies sufficiently so that peaks A of 3p6tn 2gspecies are isolated from those that have a 3p core hole. Thus, we are left with two separable exper- iments. Probing the red edge of the A–C region reports on species without 3p holes. Probing the 3p region reports on those species with a 3p hole. The C and 3p peaks and the splitting between the A and B peaks were identified in Sec. IV B as useful measurements in an XPXP experiment. Given that the A, B, and C peaks are sensitive to the ligand-field splitting and the solute–solvent interactions, an ideal experiment would measure the contributions of all three peaks in the excited electronic state simultaneously. The spectral conges- tion in the B–C region, shown in Fig. 4, limits this global resolution. We therefore explore the feasibility of measuring an individual A peak, spectrally isolated from this congestion, and by considering the strong signal strengths of dipole allowed transitions in the 3p region. Of those electronic configurations containing an A peak, the [Ne] 3s23p6t4 2gconfiguration is both predicted by the electronic cascade and its A peak is generally well separated from the other calculated XANES transitions. Note that in Fig. 4, peaks A–C of the electronic state with 3p holes are blueshifted from those with a full p shell. The A peak of configurations with additional holes in the 3p would be further blueshifted, removing them from the present consideration. Moreover, the splitting resulting from holes in t 2gfurther redshifts the A peaks. These two interactions make the [Ne] 3s23p6t4 2gconfiguration the most likely candidate for an individually resolved electronic state in the region. This provides the best opportunity to calculate the feasibility of the experiment in this region, neglecting more highly ionized species. Resolving 1s to 3p transitions is less dependent on the signal size as they are dipole allowed. Both x-ray pulses would be focused onto a liquid jet of the solvated sample. The first pulse would be tuned to above the Fe K-edge to remove a 1s electron and initiate the cascade and result- ing dynamics. We assume an ∼10 fs delay between the two pulses, which represents the time at which most electron cascade dynam- ics conclude. While, certainly, some valence dynamics occur on this timescale, there is a trade-off between capturing those dynamics and probing stationary configurations. The scale of this delay should be reconsidered with the realization of attosecond pulses but is neces- sarily given the current technology available. It is considered to be a portion of the instrument response function of the experiment. Just as the experiment is limited in the short timescale ( <10 fs) by the resolution of the electron cascade and the available femtosecond pulse durations, it is also limited on the long timescale ( >100 fs) as nuclear dynamics dictate the molecular response. The assumption of fixed molecular geometry breaks down, as would the molecule itself. The second pulse would be below the Fe K-edge and span the appropriate region of the XANES spectrum. Two spectrometers J. Chem. Phys. 154, 214107 (2021); doi: 10.1063/5.0047381 154, 214107-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp would be used to measure the signal. One would be downstream of the interaction region to measure the experimental spectrum, and one would be a transmissive spectrometer upstream to mea- sure the incoming x-ray pulses’ shot to shot spectrum.53While the exact spectrum of the pump pulse is not a source of noise, the spectrum of the probe pulse is critical to extracting the spec- troscopic signal. Such a configuration is best conducted in the CXI (Coherent X-ray Interaction) hutch at LCLS,54where we can make use of their liquid jet capabilities and the tight x-ray focus and place the upstream spectrometer in the x-ray transport tun- nel. We cannot generate a full expected experimental signal since we have not calculated all of the most likely electronic states pre- dicted by the electron cascade. Instead, we can predict the expected signal contribution from electronic states in which we are most interested. To generate the expected experimental signal, we assume an initial excitation rate of 10% and a 100 μm liquid jet with a 500 mM concentration of K 4FeII(CN) 6. We use the tabulated x-ray absorp- tion properties of Fe to predict the absorption edge size.55We can then appropriately scale the calculated XANES spectrum and superimpose it on our scaled experimental ground state absorp- tion edge, appropriately shifted with the B peak position to rep- resent the change in the ionization energy. The resultant signal is shown in the lower panel of Fig. 5 for the [Ne] 3s23p6t4 2g electronic state where we have assumed that this state makes up 10% of the resultant initial electronic state. The corresponding ground state absorption spectrum generated from the same edge and the calculated XANES spectrum along with the absorption due to the water and cyanide ligands is shown in the upper panel of Fig. 5. To observe this signal, we would measure a difference spectrum between the pumped and ground state spectrum. The uncertainty of a difference signal at the A peak energy can be estimated from the definition of the absorption spectrum, A(hν), relative to the mea- surement at the downstream spectrometer, M, and the measurement at the upstream spectrometer, P, at photon energy hν, M(hν)=P(hν)(1−A(hν)). (1) The uncertainty of the difference spectrum is then given by δA(hν)=A⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪(δM M)2 +(δP P)2 . (2) Assuming that shot-noise-limited detection is consistent with pho- ton counting and thatδM M≈δP P, δΔA(hν)=4A(hν)⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪1 Nphotons(hν), (3) where Nphotons(hν)is the number of photons in the x-ray probe at photon energy hνover the bandwidth of one pixel. For the measurement in question, we will estimate that Nphotons(hν)≈11 000 for each pulse, which is the pixel well depth of the ePix10k detector currently installed at the CXI hutch. From the upper panel of Fig. 5, we estimate that the total absorption Abetween 7110 and 7125 eV is 0.22. At 120 Hz, 4.5 h of data FIG. 5. Predicted contributed signal. The expected absorption signal from the [Ne] 3s23p6t4 2gelectronic state, scaled for its expected ∼10% probability and ∼10% overall excitation rate is shown in the lower panel. Included is both the broadened signal and the absorption edge. The inset shows the A and B peaks in more detail. The upper panel shows the ground state absorption using the same edge, the cal- culated spectrum, and the background absorption from the ligands and the solvent for comparison. This figure allows us to estimate the signal size in order to deter- mine the averaging time necessary for a 120 Hz experiment. The absorption edge is the combined absorption of all atoms, including solvent, according to tabulated values.55 collection will approximately provide the necessary number of pho- tons to resolve the change in absorption of 6 ×10−6at the A peak. Averaging over several pixels would reduce this averaging time. To resolve the approximately ten times larger C peak, only 2.7 min of data collection is necessary. With the prospect of LCLS-II coming online, which promises MHz repetition rates, measuring both the A and C peaks is well within feasibility limits. VI. CONCLUSION In order to fully exploit the unique element-specific capabili- ties of x-ray pump–x-ray probe experiments in solvated molecular systems, it is essential to understand the specific electronic states resulting from electron cascades and how these states are mani- fest in x-ray absorption spectra (as measured by the probe). We have proposed combining an atomic electron cascade calculation with a TDDFT-based XANES calculation to accomplish this goal in a solvated model transition metal complex. The electron cascade J. Chem. Phys. 154, 214107 (2021); doi: 10.1063/5.0047381 154, 214107-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp calculation allowed us to identify which molecular orbitals were most likely missing electrons after the cascade. We then calculated the XANES spectra of several electronic configurations with the cor- responding holes. We believe this approach will be necessary for interpreting and predicting XPXP measurements, and the method- ology laid out in this manuscript could be one way to attain these results. We then evaluated how these different electron holes could be useful for measuring chemically relevant properties, such as the ligand-field splitting, and 3p and 3d electron interactions. From this, we identified the A–B peak splitting and the C peak in the [Ne] 3s23p6t4 2gelectron configuration as one potential target for experimental observation. We laid out an experimental plan that will allow us to test our computational approach and prediction and showed that this experiment is currently feasible under present LCLS conditions. Understanding these interactions in model transi- tion metal systems lays the ground work for studies on more com- plex interactions such as those found in solvated cyanide-bridged mixed-valence bimetallic electron transport compounds involving, for example, a combination of 3d and 4d transition metal cen- ters. As XPXP measurements continue to mature, they will provide new insight into not only solvated transition metal complexes but also solvated systems in general, and the computational and exper- imental techniques presented here can serve as a guide for that development. SUPPLEMENTARY MATERIAL See the supplementary material for tables of molecular geome- tries used in this work. ACKNOWLEDGMENTS The authors acknowledge support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division under Award Nos. DE-SC0019277 (C.E.L.-S., R.B.W., and M.K.), KC-030105172685 (N.G.), DE-AC02-76SF00515 (R.W.S.), and DE-AC02-06CH11357 (P.J.H.). R.B.W. acknowledges support from the NSF Graduate Research Fellowship Program under Grant No. DGE-1762114. This research benefited from computational resources provided by EMSL, a DOE Office of Science User Facility sponsored by the Office of Biological and Environmental Research and located at the Pacific Northwest National Laboratory (PNNL). PNNL is operated by Bat- telle Memorial Institute for the United States Department of Energy under DOE Contract No. DE-AC05-76RL1830. 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APL Materials PERSPECTIVE scitation.org/journal/apm Exploiting random phenomena in magnetic materials for data security, logics, and neuromorphic computing: Challenges and prospects Cite as: APL Mater. 9, 070903 (2021); doi: 10.1063/5.0055400 Submitted: 28 April 2021 •Accepted: 10 June 2021 • Published Online: 8 July 2021 C. Navau1,a) and J. Sort1,2,a) AFFILIATIONS 1Departament de Física, Universitat Autònoma de Barcelona, E-08193 Cerdanyola Del Vallès, Spain 2Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluís Companys 23, E-08010 Barcelona, Spain a)Authors to whom correspondence should be addressed: Carles.Navau@uab.cat and jordi.sort@uab.cat ABSTRACT Random phenomena are ubiquitous in magnetism. They include, for example: the random orientation of magnetization in an assembly of non-interacting isotropic magnets; arbitrary maze domain patterns in magnetic multilayers with out-of-plane anisotropy, random polar- ization, and chirality of an array of magnetic vortices; or Brownian skyrmion motion, among others. Usually, for memory applications, randomness needs to be avoided to reduce noise and enhance stability and endurance. However, these uncontrolled magnetic effects, espe- cially when incorporated in magnetic random-access memories, offer a wide range of new opportunities in, e.g., stochastic computing, the generation of true random numbers, or physical unclonable functions for data security. Partial control of randomness leads to tunable prob- abilistic bits, which are of interest for neuromorphic computing and for new logic paradigms, as a first step toward quantum computing. In this Perspective, we present and analyze typical materials that exhibit stochastic magnetic phenomena and we show some examples of emerging applications. The current challenges in terms of material development, as well as new strategies to tune stochasticity, enhance energy efficiency, and improve operation speeds are discussed, aiming to provide new prospects and opportunities in this compelling research field. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0055400 I. INTRODUCTION Magnetic materials exhibit a plethora of stochastic (or random) effects. Such randomness occurs in the magnetic orientation of small magnetic nanoparticles over time,1in the Brownian trajectories of skyrmion quasiparticles,2or in the polarity and chirality of magnetic vortices generated in micrometer-sized soft-magnetic disks,3among others. For some applications (e.g., conventional memory devices), randomness is highly undesirable (since it can generate noise and loss of information), and strategies have been put forward to avoid it. However, randomness can be used to generate true random num- bers at hardware level or in new computing paradigms,4as will be described in this Perspective. Partial control of randomness with external stimuli (voltage, current, particle/dot shape modifications,and structural defects) can find applications in logics and in neuro- morphic computing. Probabilistic bits (i.e., bits with a probability to occur higher or lower than fully random 50%) are considered a first step toward quantum q-bits.5Figure 1 illustrates the main concepts related to stochasticity in magnetic phenomena and the prospective applications. The goal of this Perspective is to analyze the state of the art of materials and phenomena related to magnetic random- ness, with emphasis on the current challenges (control of random- ness, energy efficiency, CMOS integration, or operation speed) and future opportunities in different technological fields, encompassing data security, probabilistic computing, or neuromorphic computing (see Fig. 1). The reported effects can be of interest to deal with data in which uncertainty is inherently present. It has been long recog- nized that probabilistic algorithms can tackle some specific problems APL Mater. 9, 070903 (2021); doi: 10.1063/5.0055400 9, 070903-1 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 1. Illustration of the main phenomena that lead to magnetic random effects and the prospective applications in different research domains. PMA stands for “perpendicular magnetic anisotropy.” much more efficiently than classical algorithms using a deterministic computer.6While the areal density of information in magnetic stor- age media has been drastically increasing over the years, new chal- lenges posed by artificial intelligence and the need to tackle big data require changes in the computing paradigms, in which stochasticity can play a key role. II. STOCHASTIC EFFECTS IN MAGNETIC MATERIALS It is well known that the magnetic dipoles of an assembly of small magnets will orient at random in the as-prepared state or when cooled through the Curie temperature if they are not affected by dipolar interactions. As an example, Fig. 2(b) shows the magnetic-force microscopy (MFM) image of random “up” (white) and “down” (black) single-domain states of an array of Ni pillars with out-of-plane magnetization grown by electrodeposition.7 A single-domain nanomagnet can be characterized by an energy barrier ( ΔE) that separates two possible states, each of them with the magnetic moment oriented in opposite directions. When the magnet is placed in one of these two states, it stays there for a certain amount time, called “retention time” ( τ), which is given byτ=τ0exp(ΔE/k BT), where τ0is a material constant, k Bis the Boltzmann constant, and T is temperature. In a first approximation, FIG. 2. (a) Scanning electron microscopy (SEM) image and (b) magnetic-force microscopy (MFM) micrographs of an array of Ni pillars (90 nm diameter, 220 nm height), showing random “single-domain” behavior with out-of-plane magnetiza- tion. Reproduced from Ross et al. , J. Appl. Phys. 91, 6848 (2002) with the permission of AIP Publishing. (c) Magnetic contrast image acquired by x-ray photoemission electron microscopy (XPEEM) at the Fe L-edge of an array of Ni80Fe20/Ir20Mn80disks (showing dissimilar magnetic configurations) in the frame- work of an exchange bias investigation. Reproduced from Salazar-Alvarez et al. , Appl. Phys. Lett. 95, 012510 (2009) with the permission of AIP Publishing.the energy barrier can be expressed as ΔE=1/2MSVH K, where M S is the saturation magnetization, V is volume, and H Kis the effec- tive anisotropy field. While τis of the order of years in non-volatile memories, the time can be engineered to be much shorter (sub- second) using suitable nanomaterials (with low H K, M S, and V val- ues). When the size of a magnet is sufficiently small (approaching the superparamagnetic limit), its orientation will spontaneously fluctu- ate with time between two binary states due to thermal agitation. The readout from such tiny magnets, once the magnetic signal is trans- duced into a voltage or current, will result in telegraphic noise,1and this can be considered as a simplified version of a stochastic device. This concept has been applied to magnetic tunnel junctions (MTJs) comprising a free layer with small in-plane magnetic anisotropy (superparamagnetic MTJs), where thermal activation effects induce stochastic fluctuations in the magnetoresistance values at time scales below 5 ns.8 In magnetic objects of larger dimensions, the magnetization reversal mode can be tailored to arbitrarily fluctuate between dif- ferent mechanisms. In a previous work, we adjusted the thickness and diameter of circular magnetic disks to critical values close to the boundaries between single-domain and vortex states so that the disks could reverse their magnetization arbitrarily either by coherent rotation or through vortex formation. This is illustrated in Fig. 2(c), which shows the element-specific polarized x-ray photoemission electron microscopy (XPEEM) image at the Fe L-edge, in the absence of magnetic field applied, of an array of Ni 80Fe20(6 nm)/Ir 20Mn 80 (5 nm) disks (2 μm in diameter) deposited onto a naturally oxidized Si wafer. Disks 1 and 4 form vortex states at remanence, while disks 2 and 3 are single domains.9 Remarkably, even when all disks in an array form vortex states (e.g., in 100 nm-thick Ni 80Fe20disks with 1 μm diameter), the chi- rality of the vortex and polarity of the vortex core are random if no symmetry-breaking geometrical constraints are applied to the disk shape (see Fig. 3). The white and black spots at the cen- ter of the disks, imaged using full-field magnetic transmission soft x-ray microscopy (MTXM) [see Fig. 3(b)], reveal positive and nega- tive polarity of the vortex core, respectively.3The chirality is deter- mined from the in-plane magnetic component [Fig. 3(a)], taking into account the vortex core polarity. Another example is con- tinuous multilayers with perpendicular effective anisotropy, which exhibit magnetic domains that form random maze patterns each time the system is demagnetized, provided that there are no struc- tural defects (e.g., grain boundaries) that influence the labyrinthic paths. There are many other examples of random phenomena in mag- netic materials. Steels and other types of ferromagnets exhibit the Barkhausen effect, which is related to noise in the magnetic output signal when the material is subject to an applied magnetic field and the size of the domains fluctuates with time in discrete steps due to structural defects in the crystal lattice.10Concerning magnetic “moving entities,” stochastic domain wall pinning has been pre- dicted11in magnetic nanowire devices due to the influence of ther- mal perturbations on the domain walls’ dynamics. Random ther- mally driven effects also manifest in the trajectories of magnetic skyrmions in unconstrained, defect-free, ultra-thin magnetic films (see Fig. 4).12,13Skyrmions are spin swirling quasiparticles that have generated great expectations in recent years due to their potential to act as small information carriers.14 APL Mater. 9, 070903 (2021); doi: 10.1063/5.0055400 9, 070903-2 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 3. Full-field magnetic transmission soft x-ray microscopy (MTXM) images of the in-plane (a) and out-of-plane (b) magnetic components taken at remanence in an array of 100-nm-thick Ni 80Fe20disks (1 μm diameter). The sense of rota- tion of the in-plane magnetization (i.e., vortex chirality) is indicated with the white arrows in (a). The black and white spots at the center of the disks are upward and downward vortex cores (positive and negative polarity), respectively (b). The vor- tex configuration of each disk is illustrated in (c). Reproduced with permission from Imet al. , Nat. Commun. 3, 983 (2012). Copyright 2012, Nature Publishing Group. FIG. 4. Non-deterministic trajectories, determined by Kerr microscopy imaging, of selected skyrmions formed at 296 K in Ta (5 nm)/Co 20Fe60B20(1 nm)/Ta (0.08 nm)/MgO (2 nm)/Ta (5 nm) stacks. All the skyrmions were set to start at position (0, 0). The timescale of the observation was in the range of seconds to minutes. The inset shows the time-averaged mean squared displacement (MSD) (black line) and the linear fit of the data (red dashed line). Reproduced with per- mission from Zázvorka et al. , Nat. Nanotechnol. 14, 658 (2019) Copyright 2019, Nature Publishing Group. III. IS IT POSSIBLE TO CONTROL RANDOMNESS IN MAGNETIC MATERIALS? While fully random magnetic effects are appealing for stochas- tic computing or true random number generators (RNGs) (see Sec. IV), partial control of randomness can yield other interestingapplications. For example, the retention time in low energy bar- rier nanomagnets, τ, can be manipulated with current or voltage. By doing so, one can generate probabilistic p-bits, which have a prob- ability of being “1” or “0” different than 50% (which would be the fully stochastic case). These p-bits are considered as an intermediate between classical bits (with deterministic “0” and “1” orientations) and quantum q-bits (with coherent superpositions of “0” and “1”).1 Another example is the control of double-shifted loops in [Pt/Co] nmultilayers with out-of-plane magnetization that we could achieve using the coupling with an adjacent antiferromagnetic (AFM) Ir 20Mn 80layer (exchange bias effect).15This is illustrated in Fig. 5. Configuration (a) is the typical maze domain patterns with random shapes obtained after demagnetizing the [Pt/Co] nmul- tilayer. An equal proportion of white and dark stripes (indicat- ing regions with “up” and “down” magnetization, respectively) is observed by MFM. When this pattern is coupled to the adjacent AFM layer, a double-shifted loop (50% shifted to the right and 50% shifted to the left) is obtained. By annealing the system close to the exchange bias blocking temperature and subsequently cooling under different magnetic fields, the degree of randomness is decreased. For larger magnetic fields, the amount of “white” regions progres- sively increases [configuration (b)] while the magnetization ampli- tude of the left-shifted hysteresis loop also increases. If the system is cooled with a sufficiently high magnetic field, an almost determinis- tic single-domain state is generated [configuration (c)] and the loop shape approaches that of a single-shifted hysteresis loop. During the last few years, we have also proposed strategies to manipulate and control the chirality and polarity of magnetic vor- tices by incorporating asymmetrical variations in the thickness of FIG. 5. (Left) Magnetic-hysteresis loops measured along the perpendicular to the plane of a [Pt/Co]n multilayer coupled to IrMn, after field cooling from T =520 K using different magnetic field values (H FC). The various loops show the progressive tuning of the vertical magnetization amplitude depending on H FC. (Right) Depen- dence of the normalized magnetization amplitude of the sub-loop shifted toward the negative fields, ΔM, vs the squareness ratio (M r/Ms). The insets show the cor- responding magnetic-force microscopy images for (a) M r/Ms=0, (b) M r/Ms=0.43, and (c) M r/Ms=0.93. Note that each image has a side length of 20 μm. Repro- duced with permission from Brück et al. , Adv. Mater. 17, 2978 (2005). Copyright 2005, Wiley. APL Mater. 9, 070903 (2021); doi: 10.1063/5.0055400 9, 070903-3 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm the dots.16A similar approach consists in breaking the circular rota- tional symmetry by, e.g., asymmetrical notches.17Although experi- mental realization of these effects requires the use of advanced litho- graphic procedures, these procedures are now quite manageable, and therefore, a variety of works demonstrating geometrical control of the vortex state have been reported.18–22 Very recently, we have also used micromagnetic simulations to study the dynamics of skyrmions in the presence of defects (pinning sites) or borders. The probability that a skyrmion becomes trapped in a pinning center or the probability of its survival along a race- track can be calculated as a function of temperature and the length of the track.23Indeed, inherent or artificially created defects in the crystallographic structure of the films can modify the trajectory of skyrmions in a partly deterministic manner. In spite of the previous examples, a number of different chal- lenges, in terms of materials properties and technological advances, need to be overcome for a successful accomplishment of fully stochastic magnetic effects and the probabilistic (i.e., partial) con- trol of this randomness. Namely, superparamagnetic MTJs require the use of very thin amorphous soft-magnetic layers with very low anisotropy. So far, almost all examples are based on CoFeB. Con- versely, for the coupling between FM and AFM domains, a rela- tively high uniaxial magnetic anisotropy is required in the ferro- magnet, while the anisotropy in the antiferromagnet and the size of AFM domains should be small or at least commensurate to the domain size in the ferromagnet. This needs to be accompa- nied with a relatively high interfacial exchange constant so that the strength of the coupling is sufficient to overcome the magnetostatic energy in the ferromagnet. Finally, formation of skyrmions at room temperature only occurs in a limited number of systems, such as Ta/Co 40Fe40B20/TaO x,24,25in asymmetric Co nanodot arrays with the in-plane magnetic easy axis grown on Co/Pd underlayers with perpendicular anisotropy,22or in Pt/Co/Ta and Pt/CoFeB/MgO stacks with ultrathin Co and CoFeB layers,26among others.27The control of structural defects (e.g., periodic pinning sites) in these layers, with the aim of guiding skyrmions motion, again requires dedicated lithography procedures and is not always straightforward. IV. HIGHLIGHTS OF EMERGING APPLICATIONS BASED ON MAGNETIC STOCHASTIC EFFECTS A. Exploiting magnetic randomness for true random number generators Random numbers play a crucial role in the encryption of secure data for communication and storage purposes. Taxonomically, ran- dom number generators (RNGs) can be classified into two large cat- egories: pseudo-RNGs, which use deterministic software algorithms to generate a sequence of random numbers (and are thus vulnerable to hacking cyberattacks), and true-RNGs, which are implemented at the hardware level and generate sequences of random numbers using non-deterministic, uncontrollable physical events (thus being cryp- tographically more secure).4From a physical viewpoint, true-RNGs can benefit from two types of randomness sources: fluctuations (e.g., thermal noise, shot current noise, and chaotic fluctuation of a semiconductor laser) and stochastic switching events (radioactive decay intervals, magnetization switching in nanomagnets, switching events in resistive-random-acces memories (RAMs), and current- triggered random transitions in spintronic spin-transfer-torqueSTT-magnetoresistive random-acces memories (MRAMs)).28,29In the following, the basic idea of STT-MRAM, in which a pulse gen- erator utilizes the stochastic nature of spin-torque switching in an MTJ to generate true random numbers,28is illustrated (Fig. 6). An MTJ consists of two nanoscale ferromagnetic (FM) layers separated by a spacer layer (typically MgO). While the magneti- zation of one of the layers (“reference” layer) is pinned in a par- ticular direction (usually through the coupling with an antiferro- magnet), the magnetization of the other layer (free layer) can be switched by external stimuli, such as a magnetic field or a spin cur- rent. Depending on the relative orientation of the two FM layers, the device exhibits two resistance states: “high” (when the two lay- ers are antiparallel to each other) or “low” (when the two layers show parallel orientation of their magnetization). These two states are sep- arated by an energy barrier that is determined by the anisotropy and volume of the FM layers.30As mentioned in Sec. II, thermal agitation in nanomagnets brings about a distribution of switching fields. A similar effect occurs when these nanomagnets constitute the free layer of an MTJ. The switching current of a STT-MTJ nanopil- lar also has a distribution due to thermal agitation. The switching probability between the two resistance states depends on the cur- rent amplitude and pulse width. Both parameters can be adjusted so that the switching probability of a given MTJ is close to 0.5. In this way, these switching events are the source of true random num- bers.4,31–33The main challenge of this approach is the Joule heating power dissipation, which is not negligible and hinders large scale integration of these types of devices. As will be described in detail in Sec. V, the energy efficiency challenge can be tackled by the devel- opment of voltage-controlled (i.e., magnetoelectric) MRAMs, where voltage could be used (instead of current) to tailor the anisotropy of the free layer and the magnetic easy axis direction. Implemen- tation of voltage-controlled MRAMs requires precise control of the thickness and microstructure of the magnetic free layer so that its properties can be sensitive to an applied electric field. Another chal- lenge is that the STT-MTJ share the same access path for the read and write operations, thereby causing interference problems and some lack of reliability. This has been overcome using the spin–orbit torque (SOT) effect, instead of STT, which utilizes three terminals instead of two (read/write paths are thus separated).34In this case, the switching between parallel and antiparallel states is induced by FIG. 6. Schematic diagram of STT-MRAM “spin dice.” Current pulses are injected into a MTJ with perpendicular anisotropy to switch the magnetization in the free layer by spin-transfer-torque (STT).28Highly reliable random bits can be obtained by comparing resistance values of an array of MTJs comprised in the STT-MRAM. This is sometimes referred to as “differential concept.”4Reproduced with permis- sion from Fukushima et al. , Appl. Phys. Express 7, 083001 (2014). Copyright 2014, The Japan Society of Applied Physics. APL Mater. 9, 070903 (2021); doi: 10.1063/5.0055400 9, 070903-4 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm an in-plane current that flows along a heavy metal (Pt, Ta, W, and Ir) grown underneath the free layer.35,36In spite of a better robust- ness, energy loss due to heating effects continues to be a challenge in SOT-RAM true-RNGs. Randomness in the trajectories of skyrmions (Fig. 4) can also be used to generate random numbers or random sequences. A particular design combined the stochastic dynamics of syn- thetic AFM skyrmions, driven by spin-polarized currents, with the racetrack memory concept using a topologically bifurcated path with one input branch and two output branches.37The authors demonstrated that under the steady action of current, skyrmions starting from a continuous nucleation in the input branch were stochastically divided into the two output branches of the device. B. Exploiting magnetic randomness for physical unclonable functions (data security) Traditionally, protection of confidential information has relied on the use of passwords generated either by the user or by encryp- tion software. This type of protection is often vulnerable to cyber- attacks (phishing, malware, ransomware, and eavesdropping). To minimize security breaches, a new concept was proposed in 2002 when the utilization of the so-called “physical unclonable functions” (PUFs) was first suggested as a hardware alternative to conven- tional algorithmic “passwords” or encryption procedures.38A PUF is defined as a collection of robust, singular, unclonable, and unpre- dictable physical properties of a device, which make it unique (i.e., “device fingerprint”).39,40Such properties can be the result of non-controllable manufacturing imperfections or can originate from stochastic physical phenomena that occur during device oper- ation. These features can be used to generate a series of queries and device responses technically called “challenge-response pairs” (CRPs). Authentication is based on the physical verification of these CRPs. There are many variables that can be used to generate PUFs: local variations of the capacitance across a film containing ran- dom assemblies of dielectric particles (“coating PUF”), stochastic switching of superparamagnetic MRAMs with time, and variations of the magnetoresistance state (high/low) in the memory units of STT-MRAMs. Some recent PUF schemes have even taken advan- tage of carbon nanotubes41or plasmonic particles and quantum materials.42PUFs can be implemented with small hardware invest- ment, and they are more energy efficient than tamper-proof security packages existing in the market. First magnetic PUF designs date back to 1994.43They were engineered to make unique magnetic swipe cards, such as credit and identity cards, by blending FM particles with random sizes and shapes on a receptor layer. The PUF principle was thus based on the manufacturing variability. More sophisticated mag- netic PUFs include an MTJ operating using the toggle principle (where switching is induced by applied magnetic fields44or the STT-MRAMs).45Toggle-MRAM-PUFs take advantage of geomet- ric variations between different MTJ cells, which are used as a digital signature of the device.44Usually, the free layer is affected by thermal agitation. In the case of superparamagnetic MRAM-PUFs, random switching occurs with time in each cell (Sec. II). The working princi- ple of STT-MRAM PUFs is similar to the one described in Sec. IV A for true-RNGs. However, to establish the set of CRPs, an array ofSTT-MRAM cells (typically in crossbar geometry) is needed and each MTJ is set to be in a random configuration of high and low resistance states when a critical switching current is applied.33,46–50 Generally speaking, PUF architectures utilizing non-volatile memories are quite new, mostly developed during the last five years. These include, for example, PUFs based on resistive switching mate- rials (RE-RAM PUFs), phase change materials (PCM-PUFs), ferro- electric oxides (FE-RAM PUFs), toggle magnetoresistance (MRAM PUFs), or STT-MRAM PUFs. The advantages and disadvantages of each type of these data security primitives depend on the proper- ties of the constituent materials and the involved switching mecha- nisms. STT-RAM PUFs are among the most energy efficient (with 10−2pJ/bit), orders of magnitude more efficient than PCM-PUFs (10 pJ/bit), RE-RAM PUFs, or toggle-MRAM PUFs (10−1pJ/bit). Access speed in STT-RAM PUFs (1–10 ns) is also faster than in FE-RAM PUFs (10–100 ns) or flash PUFs ( >103ns) and comparable to RE-RAM PUFs or toggle MRAM-PUFs.39,40,51–59Thus, magnetic materials exhibiting stochastic transitions can indeed offer unique opportunities in the exciting field of data security. C. Exploiting magnetic randomness for logics and probabilistic computing Probabilistic computing is emerging as an intermediate com- putation paradigm that lies between classical digital computing and quantum computing. At present, quantum computing suffers from some important challenges, such as low-temperature operation and the limited number of many-body interactions that can be imple- mented. These challenges can be tackled to some extent through probabilistic computing. The basic computation units are the prob- abilistic p-bits, which are classical entities spontaneously fluctuating in time between “1” and “0” states.1,5,34,60The full potential of p-bits can be realized only when the probability of occurrence of the two possible states can be modulated by an external input.34Figure 7 shows a schematic comparison between the large energy barrier in the MTJ of a conventional MRAM memory (which would lead to retention times of the order of years) and that of a p-bit based using MTJs with a thinner free layer (leading to retention times of the order of ms).5 FIG. 7. Illustration of the energy profiles between the parallel and antiparallel states of the free and pinned magnetization orientations in (a) a MTJ for typical MRAM technology (with a large energy barrier) and (b) a probabilistic bit (with much lower energy barrier and retention times in the order of ms). Adapted from Borders et al. , Nature 573, 390 (2019). Copyright 2019, Nature Publishing Group. APL Mater. 9, 070903 (2021); doi: 10.1063/5.0055400 9, 070903-5 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm The energy barrier of the p-bit can be tuned by adjust- ing the thickness, saturation magnetization, and effective mag- netic anisotropy of the free layer in the MTJs. The occurrence of dipolar interactions between the free and pinned layers can lead to not completely random parallel and antiparallel (i.e., non- fully isotropic) configurations, resulting in some probabilities for each of the two states that can be modulated to values differ- ent than 0.5.60In STT-MRAMs, such switching probability can be tuned by proper adjustment of the spin-torque currents. In some other cases, the anisotropy of the free layer can be controlled with voltage (by magnetoelectric effects). This offers a wide range of opportunities to use magnetic materials for the implementation of p-bits. In particular, different types of operations (e.g., factoriza- tion, invertible logic, Bayesian inference, and combinatorial opti- mization) can be performed using matrices of correlated MTJ- based p-bit networks that can be read by, e.g., precharge sense amplifiers.61,62 Similarly, by tailoring the design of stochastic nanomagnets (e.g., tuning the effective magnetic anisotropy of Co 60Fe20B20circu- lar disks or combining pairs of coupled nanomagnets with in-plane and perpendicular-to-plane anisotropies), the use of probabilistic computing for spin logic operations has been proposed and, to some extent, experimentally demonstrated.63–66 Within the field of stochastic computing, a promising route using skyrmions is the ability to process random bit-streams, instead of individual bits. A bit-stream formed by N bits with “1” or “0” is characterized by the stochastic number p, which is the probability of observing a “1” in the stream. In this sense, a perfect control of indi- vidual skyrmions is not critical, and the system would be highly tol- erant to soft errors: a collapse of a single skyrmion would not modify significantly the overall pvalue.67,68When two bit-streams are com- puted in an AND logic gate (as example), the pvalue for the output bit-stream is the multiplication of the “p” values of the input bit- streams. The use of skyrmions allows for very energy efficient and fast processing, thus being promising candidates for massive parallel computing.69However, to ensure this multiplication property, it is necessary that the input bit-streams are uncorrelated. The thermal- driven randomness in the skyrmion movement has been proposed as the mechanism for reshuffling one bit-stream, before entering the computation gate, in order to avoid the undesired correlations while maintaining the pvalue.68In Fig. 8, we show a very recent experi- mental validation of the skyrmionic reshuffling using a continuous single layer stack of Ta(5)/Co 20Fe60B20(1)/Ta(0.08)/MgO(2)/Ta(5) (in parentheses, the thickness of the layers in nm).12This reshuf- fling is an off-chip system, although recent developments on in- chip creation, manipulation, and detection of skyrmions70point to the in-chip integration of skyrmionic systems for stochastic computing. Another possibility for taking advantage of randomness in skyrmion movement is to consider the interaction with specially designed defects. Recently, by considering this interaction, we have shown that, due to thermal diffusion, the probability that a skyrmion avoids the defect or is being trapped in it can change continuously from 0 to 1 by varying the position where the skyrmion is generated, as shown in Fig. 9.23Moreover, we also calculated the probability of survival of a skyrmion along a nanotrack, which depends on the driving current and the interaction with the borders of the track. This, together with evaluation of thermal collapse,71opens the path FIG. 8. Reshuffler operation with skyrmion nucleation by a direct current. (a) The input signal is constructed as a time frame in which the skyrmion crosses the blue threshold line. The output is produced on crossing the orange line. (b) The corresponding input signal is depicted in blue (top), and the resulting output signal is depicted in orange (bottom). The radius of the reshuffling chamber is 40 μm. The Pearson correlation factor between the input and the output was evaluated as 0.11±0.14. Reproduced with permission from Zázvorka et al. , Nat. Nanotechnol. 14, 658 (2019). Copyright 2019, Nature Publishing Group. of using finite nanotracks as probabilistic selectors of skyrmionic bits or bit-streams, as well as for designing stochastic logic gates. Thus, defects, either engineered or naturally occurring, can have strong interaction with the skyrmions. Recent works stud- ied the dynamics of skyrmions considering random distribution of defects72or randomly distributed granularity.73Moreover, the dynamics of skyrmions (or other skyrmionic-like structures) in the presence of defects is highly non-linear, which yields the possibility of having critical points (attractors, repeller, and/or saddle points) depending on the input parameters.74This could result in interest- ing chaotic scenarios.75These systems are at an early stage of their development but could be also envisaged to randomize or shuffle information. D. Exploiting magnetic randomness for neuromorphic computing Neuromorphic computing imitates the physical structure of a human brain to perform computational operations. The possible utilization of magnetic randomness for neuromorphic computing applications is still in its infancy.30,36,76–80Yet, stochastic effects are of primary importance in artificial neural networks that are intended to mimic the brain capability to do “pattern recognition” (i.e., object identification by resemblance). It has been proposed that arrays of interconnected magnetic nanowires could trigger further develop- ments in the field of neuromorphic computing.81Artificial synapses should possess sufficiently high fault tolerance in order to be able to identify a given object even if there are some errors or differences between the input signal as the pre-recorded pattern (similar to how the brain works). In recent years, there is growing evidence from the neurologists’ community that the brain performs probabilistic computation based on partially random operations of neurons, synapses, and dendrites. This effect should be emulated in artificial intelligence systems, APL Mater. 9, 070903 (2021); doi: 10.1063/5.0055400 9, 070903-6 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 9. Snapshots of the probability density (color bar in units of 10−3nm−2) of the presence of a skyrmion as time runs (from left to right, time tin ns). vhis proportional to the driving spin-current density. This probability can be tuned from 0 to 1 by changing the initial position of the skyrmion and the temperature (bottom). The circle corresponds to the snapshots shown. The two lobules in the last snapshot show clearly that there is a probability (around 0.45 in this case) of being trapped and a probability of scaping (around 0.55). Reproduced with permission from Castell-Queralt et al. , Phys. Rev. B 101, 140404(R) (2020). Copyright 2020, American Physical Society. particularly for deep learning approaches. The field of magnetoelec- tricity (voltage control of magnetism) has brought some compelling results into this direction.30,79It has been theoretically proposed that, by using voltage to manipulate the relative orientation of the pinned and free layers in MTJs, such structures can lead to low-power spik- ing neurons. Magnetoelectric neurons would be encoded through the control of externally applied voltage signals in each individual MTJ.30The idea is that by applying appropriately shaped voltage pulses, the MTJs can be switched in a probabilistic (rather than fully stochastic) manner. In this way, a probabilistic output stream is gen- erated with a probability that depends on the magnitude of the input stimuli.76 The well-known spike timing-dependent plasticity behavior has also been accomplished recently by means of magneto-ionics.80 This refers to a particular class of magnetoelectric actuation in which an applied voltage triggers the motion of ions from (or toward) a target magnetic material, depending on the voltage polar- ity, in order to modify its magnetic properties in a non-volatile (permanent) manner. So far, spike timing-dependent plasticity and other neuromorphic features have been studied in detail in Co/GdO xbilayers, where GdO xis an ionic conductor able to provide O2−ions to Co (to reduce its magnetization, emulating forgetting processes, or synaptic depression) or to remove O2−ions from CoO (to increase the magnetization, emulating learning processes, or synaptic potentiation). Non-linear voltage-driven magnetization (M) vs time potentiation/depression curves have been observed in a number of other electrolyte-gated magnetoelectric materials.82,83 As an example, Fig. 10(b) shows potentiation–depression mag- netic transitions that we measured in mesoporous FeO xpatterneddisks under application of negative and positive voltage long-term pulses.84While there is a general trend for M to always increase (or decrease) for negative (or positive) voltages, respectively, dif- ferences in the exact M values among dots (and samples) were also encountered. Namely, due to microstructural differences among the patterned dots, the exact O2−ion diffusion channels generated in dif- ferent dots were different. Consequently, stochastic magneto-ionic responses were observed. Further randomness can be induced by interconnecting the dots in several paths and applying different voltage pulses in each of the electrical contacts [Fig. 10(c)]. A further development toward neuromorphic computing using magnetic materials is displayed in Fig. 10(d).85,86The basic structure consists of an elongated ferromagnet/heavy metal heterostructure able to stabilize domain walls, which can be moved with the elec- tric current using a three-terminal device (spin-Hall effect). The FM layer in contact with the heavy metal is actually the free layer of a MTJ where multi-level resistive states can be generated through the motion of the domain wall in the free layer. That is, depend- ing on the position of the domain wall, the fraction of the free layer parallel or antiparallel to the pinned layer varies in an analog man- ner, as required for neuromorphic computing. Simulations on the operation principle of this device suggest that it has the potential to achieve energy consumption as low as 2 pJ per synaptic event, which is comparable to the energy consumption in biological synapses. Skyrmionic systems have also been proposed for neuromor- phic computing.87Again, the tunable current-driven dynamics of skyrmions allows designing systems mimicking the biological neu- rons, where “skyrmionic synapses” can reproduce the synaptic plas- ticity and a gradient of perpendicular magnetic anisotropy along APL Mater. 9, 070903 (2021); doi: 10.1063/5.0055400 9, 070903-7 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 10. (a) Scanning electron microscopy image of an array of magnetoelectrically actuated mesoporous FeO xdisks.84(b) Variation of the normalized magnetization with time for FeO xduring two consecutive cycles, first applying ΔV=−10 V (to induce synaptic potentiation) and then ΔV=+10 V (to induce synaptic depression). (c) A solid- state magnetoelectric device that consists of an ordered array of micro-/nano-scale FeO xinterconnected dots, each of them in direct contact with an oxygen getter/donor layer (e.g., Gd, which will become GdO xupon voltage application). The dots can be electrically connected in a combinatorial fashion, i.e., using different paths, therefore, forming a number of possible circuits. (d) Design of a device able to encode multilevel resistive states through the position of the domain wall in the FM free layer due to the passage of a charge current of appropriate magnitude between terminals T2 and T3. The device state can be “read” between terminals T1 and T3 using the tunneling junction. Note that PL, TB, FL, and HM denote the pinned layer, tunnel barrier, free layer, and heavy metal layer, respectively. Reproduced from Roy et al. , J. Appl. Phys. 123, 210901 (2018) with the permission of AIP Publishing. a track can be tuned to simulate the leaky-integrate-fire function of a neuron.88,89Experimental versions of such artificial synapses have been built using skyrmions on [Pt (3 nm)/Gd 24Fe66.6Co9.4 (9 nm)/MgO (1 nm)] 20ferrimagnetic multilayer stacks, while sim- ulations showed an 89% accuracy in pattern recognition using all-electric skyrmionic artificial synapses.87 V. FUTURE OUTLOOK The utilization of random phenomena in magnetic materials offers formidable challenges for future implementation of innova- tive technologies, such as new data security primitives, probabilis- tic logics (with prospects toward quantum computing), or artificial neural networks. Some of the described materials and proposed lay- outs are compatible with CMOS architectures, and many of them can be readily developed at the chip level (e.g., crossbar arrays for STT-RAMs, arrays of nanomagnets for stochastic computing and logics, and room-temperature skyrmions). However, the current state of the art in these fields faces several challenges, such as (i) need for improved control of randomness for probabilistic com- puting (beyond the approaches described in Sec. III), (ii) need forenhanced energy efficiency (bearing in mind that 20% of the global energy will be spent in information technologies by 203090), (iii) fur- ther compatibility with existing CMOS technologies, (iv) enhanced endurance (cyclability) to increase the lifetime of devices, or (v) faster operation speed (ideally toward sub- μs), among others. Many of newly proposed stochastic magnetic devices are based on STT-MRAMs. The endurance of STT-MRAM PUF systems is sometimes compromised by the dielectric breakdown of the insu- lating barrier layer resulting from repeated applied electric pulses, which cause stress and degradation. Similar to true-RNGs, the use of (SOT)-PUFs has been recently suggested as an alternative to overcome this limitation.35,91Interestingly, new strategies are being proposed to enhance tamper resistance (ability to detect a cyberattack) by means of new STT- and SOT-MRAM PUF archi- tectures,47thereby surpassing the functionalities of other types of non-magnetic PUFs. Recently, STT magnetic tunnel junction archi- tectures have also been proposed for their use as spintronic nano- oscillators, which allow for new neuromorphic functionalities at high frequencies, such as pattern recognition.92,93However, this technology (STT-MRAMs or SOT-MRAMs) utilizes spin-polarized currents, which still involves a significant power dissipation in the APL Mater. 9, 070903 (2021); doi: 10.1063/5.0055400 9, 070903-8 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm form of Joule heating effect. In recent years, voltage-driven switch- ing of nanomagnets has gained interest as an alternative to STT effects. By using voltage instead of current to control the mag- netic properties of materials, important energy savings are expected. Switching energies as low as 10−6pJ/bit have been theoretically pre- dicted for optimized strain-mediated magnetoelectric materials.55 So far, switching energies of the order of 10−3pJ/bit have been experimentally reported in both multiferroic heterostructures and magneto-ionic systems.55,94Yet, the implementation of stochastic devices based on the use of magnetoelectric materials is still at the early stages.29,95,96Aspects such as clamping effects with the sub- strate (in the case of strain-mediated ferroelectric/magnetostrictive bilayers), reduction of the needed threshold voltages to induce the desired effects, or full reversibility upon successive application of repeated voltage pulses still require further optimization and new materials engineering. In turn, magneto-ionic materials based on O2−ion migration often suffer from limited cyclability (endurance) (sometimes not exceeding 100 cycles), and the rates at which ions diffuse can be rather slow ( ∼seconds or, in some cases, ms). These switching rates are too slow for memory applications but can still be convenient for neuromorphic applications. Progress at the mate- rial level has been accomplished by replacing materials based on O2−ion motion by others governed by different ion species, such as N3−or H+, which have been shown to diffuse faster.97,98Remarkably, it has been proposed that the proton-based magneto-ionic approach can also induce new functionalities, such as Dzyaloshinskii–Moriya interactions.99Further materials engineering could boost magneto- ionics. In particular, control of structural defects [e.g., vacancies) and grain boundaries (which often act as fast diffusion paths100) are suitable strategies to improve the performance of these types of materials. In this direction, the creation of vertically aligned nanos- tructures (VANs) with magneto-ionic characteristics can be pro- posed as an alternative to continuous thin films or planar patterned dots.101,102 Concerning skyrmions, although they can be moved with high efficiency (with currents of the order of 106Am−2, i.e., several orders of magnitude lower than the currents needed to move domain walls) and progress in the control of individual skyrmions has been made, there is still a large room for further adjustment of their random- ness.103Issues such as the interaction with defects; thermal influence on realistic devices; and faster and better reproducible ways of gener- ating, moving, and detecting skyrmions need deeper investigations. In addition, other skyrmionic-like structures can also be studied for randomized systems (e.g., AFM skyrmions and antiskyrmions). Concerning operation speeds, true-RNGs and PUFs based on STT-MRAMs are already very fast (of the order of ns). Con- versely, faster rates are needed in magneto-ionic materials. In skyrmions, velocities larger than 10 m/s have been reported,104 but the speed is largely affected by the granularity of the track or the presence of defects. Probabilistic control of the randomness of skyrmion Brownian motion via periodic pinning sites in their propagation tracks faces the challenge of the concomitant reduc- tion in the working speed of the resulting logic device. Hence, fur- ther research is needed to optimize this interplay of effects and to implement stochastic/probabilistic devices operating at competitive speeds. Finally, stochastic effects have also been reported, very recently, in artificial spin networks, a particular type of magneticmetamaterial in which a large number of nanoscale magnetic elements are coupled together to render controllable random responses (i.e., tunable probability distributions) based on stochas- ticity in magnetic domain wall motion (domain wall sinks) at the nanoscale.105It has been claimed that such types of materials can be exploited to perform complex computational tasks, beyond conven- tional Von Neumann computing paradigms. ACKNOWLEDGMENTS This project received funding from the European Research Council (MAGIC-SWITCH 2019-Proof of Concept Grant No. 875018), the Agencia Estatal de Investigación of the Spanish Government (Grant Nos. MAT2017-86357-C3-1-R and PID2019- 104670GB-I00 and associated FEDER), and the Generalitat de Catalunya (Grant Nos. 2017-SGR-292 and 2017-SGR-105). We would like to thank Dr. C. Navarro-Senent, Dr. A. Nicolenco, and J. 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5.0050054.pdf

J. Chem. Phys. 154, 234103 (2021); https://doi.org/10.1063/5.0050054 154, 234103 © 2021 Author(s).Focal-point approach with pair-specific cusp correction for coupled-cluster theory Cite as: J. Chem. Phys. 154, 234103 (2021); https://doi.org/10.1063/5.0050054 Submitted: 11 March 2021 . Accepted: 21 May 2021 . Published Online: 17 June 2021 Andreas Irmler , Alejandro Gallo , and Andreas Grüneis ARTICLES YOU MAY BE INTERESTED IN Requirements for an accurate dispersion-corrected density functional The Journal of Chemical Physics 154, 230902 (2021); https://doi.org/10.1063/5.0050993 Chemical physics software The Journal of Chemical Physics 155, 010401 (2021); https://doi.org/10.1063/5.0059886 Size-consistent explicitly correlated triple excitation correction The Journal of Chemical Physics 155, 034107 (2021); https://doi.org/10.1063/5.0057426The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Focal-point approach with pair-specific cusp correction for coupled-cluster theory Cite as: J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 Submitted: 11 March 2021 •Accepted: 21 May 2021 • Published Online: 17 June 2021 Andreas Irmler,a) Alejandro Gallo, and Andreas Grüneis AFFILIATIONS Institute for Theoretical Physics, TU Wien, Wiedner Hauptstraße 8-10/136, 1040 Vienna, Austria a)Author to whom correspondence should be addressed: andreas.irmler@tuwien.ac.at ABSTRACT We present a basis set correction scheme for the coupled-cluster singles and doubles (CCSD) method. The scheme is based on employing frozen natural orbitals (FNOs) and diagrammatically decomposed contributions to the electronic correlation energy, which dominate the basis set incompleteness error (BSIE). As recently discussed in the work of Irmler et al . [Phys. Rev. Lett. 123, 156401 (2019)], the BSIE of the CCSD correlation energy is dominated by the second-order Møller–Plesset (MP2) perturbation energy and the particle–particle ladder term. Here, we derive a simple approximation to the BSIE of the particle–particle ladder term that effectively corresponds to a rescaled pair-specific MP2 BSIE, where the scaling factor depends on the spatially averaged correlation hole depth of the coupled-cluster and first-order pair wavefunctions. The evaluation of the derived expressions is simple to implement in any existing code. We demonstrate the effectiveness of the method for the uniform electron gas. Furthermore, we apply the method to coupled-cluster theory calculations of atoms and molecules using FNOs. Employing the proposed correction and an increasing number of FNOs per occupied orbital, we demonstrate for a test set that rapidly convergent closed and open-shell reaction energies, atomization energies, electron affinities, and ionization potentials can be obtained. Moreover, we show that a similarly excellent trade-off between required virtual orbital basis set size and remaining BSIEs can be achieved for the perturbative triples contribution to the CCSD(T) energy employing FNOs and the (T∗) approximation. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0050054 I. INTRODUCTION Traditional quantum chemical theories approximate the many- electron wavefunction by a linear combination of Slater determi- nants constructed from one-electron orbitals. Unfortunately, this expansion causes a frustratingly slow convergence of many calcu- lated properties to the complete basis set limit. This means that a significant part of the computational cost in many-electron pertur- bation theory calculations originates from the need to include large numbers of one-electron basis functions to achieve the desired level of precision. Many techniques have been developed to accelerate the convergence to the complete basis set limit including explic- itly correlated methods, transcorrelated methods, and focal-point approaches.1–17More recently, a density-functional theory based approach has also been employed to correct for basis set incomplete- ness errors.18Furthermore, there exist a wide range of techniques that aim at extrapolating the complete basis set limit.19–21 Explicitly correlated methods are undeniably among the most reliable and efficient methods to correct for the basis setincompleteness error (BSIE) in quantum chemical many-electron theories. They account for the first-order cusp condition in the many-electron wavefunction ansatz explicitly and are commonly referred to as F12 theories, where F12 stands for a two-electron cor- relation factor that enables a compact expansion of the wavefunction at short interelectronic distances.1–6Despite the need for additional many-electron integrals such as three- and sometimes even four- electron integrals, explicitly correlated coupled-cluster theories have been implemented in a computationally efficient manner, allowing for numerically stable and precise ab initio calculations of molecular systems.1,2,5,22–28 Focal-point approaches seek to combine low and high level the- ories in a computationally efficient manner to correct for the BSIE in the high-level calculation. In this work, we seek to correct for the BSIE of coupled-cluster theory using corrections based on second- order Møller–Plesset perturbation theory (MP2). Unfortunately, the BSIEs of MP2 and coupled-cluster singles and doubles (CCSD) ener- gies usually differ significantly. This can be attributed to the so-called “interference effect” that leads to a reduction in the BSIE of CCSD J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 154, 234103-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp theory compared to MP2 theory.29,30Peterson et al. account for this effect using complete basis set (CBS) limit model chemistry via an empirical formula.31Similar approaches include rescaled explic- itly correlated MP2 energies.32We stress that the particle–particle ladder (PPL) contribution to the CCSD correlation energy is the leading cause of the interference effect. Therefore, its contribution to the BSIE needs to be treated as accurately as possible to attain well converged CCSD energies.33–35In this context, it is noteworthy that studies on the basis set convergence in third-order perturba- tion theory have also revealed the significance of corresponding PPL contributions.36,37 Recently, we have explored a focal-point approach to cor- rect for the BSIE of CCSD correlation energies that is based on its diagrammatic decomposition into MP2, PPL, and the so-called rest term.34The decomposition of the CCSD correlation energy into contributions that dominate the BSIE (MP2 and PPL) and the rest can be very useful in ab initio calculations for the follow- ing two reasons. First, the MP2 correlation energy in the complete basis set limit can be computed using algorithms with low com- putational complexity, removing the corresponding BSIE almost exactly.38–41Second, constructing approximations to the BSIE in the PPL term is a much simpler challenge than correcting the BSIE in all possible diagrammatic contributions to the CCSD correla- tion energy. Furthermore, the MP2 and PPL terms can be regarded as an independent electron pair approximation to CCSD theory, which simplifies the development of an electron pair-specific BSIE correction. In Ref. 35, we have shown that even a simple cor- rection to the PPL BSIE, which is based on rescaling a corre- sponding MP2 correction, achieves significant reductions in the BSIE of the CCSD correlation energy. However, the attained level of precision did not reach F12-like quality for all investigated properties. Here, we present a computationally efficient and significantly more accurate method to correct for the BSIE in the PPL term of the CCSD correlation energy. Compared to our previous work in Ref. 35, the improvements result from the following two modifica- tions. First, we employ frozen natural orbitals for the virtual orbital manifold calculated from approximate one-particle reduced density matrices.42Second, and most importantly, we introduce approxima- tions to the electron pair wavefunctions appearing in the expression for the PPL term, making it possible to compute BSIE corrections with low computational cost but high accuracy. To this end, we develop a pair-specific mean-field ansatz that exhibits an average correlation hole depth that agrees with the spatially averaged corre- lation hole depth of the coupled-cluster or first-order wavefunction in a finite basis set. Employing this mean-field ansatz in the bra or ket state of the PPL expression leads to considerable simplifications. The derived pair-specific BSIE correction effectively corresponds to the rescaled MP2 BSIE. As already mentioned above, the scaling factor depends on averaged correlation hole depths. In hindsight, our work corroborates the success of CCSD basis set corrections that are based on rescaling MP2 BSIEs. We note that the averaged pair-specific correlation hole depth can be computed from the electronic tran- sition structure factor, which introduces the need for two-electron integrals employing the δ(r12)kernel. However, these additional integrals do not increase significantly the computational complexity of a conventional MP2 calculation in a finite basis set. We demon- strate that the introduced BSIE correction yields CCSD correlationenergies that converge rapidly with respect to the number of frozen natural orbitals. II. THEORY A. The pair-specific decomposed PPL correlation energy in the CBS limit In this work, we introduce a correction to the BSIE of the CCSD correlation energy. The CCSD correlation energy is given by ECCSD c=∑ ij∑ abTab ij(2⟨ij∣V∣ab⟩−⟨ji∣V∣ab⟩). (1) Tcd ijis computed from the CCSD singles ( ta i) and doubles ( tab ij) ampli- tudes as Tab ij=tab ij+ta itb j.ta iandtab ijare obtained by solving the corre- sponding amplitude equations.43,44Here, we employ a diagrammatic decomposition of the CCSD correlation energy that is obtained by substituting the doubles amplitude in Eq. (1) with corresponding contributions from the right-hand side of the converged amplitude equation, which is given by tab ij=⟨ab∣V∣ij⟩/Δij ab⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ driver/mp2+∑ cd⟨ab∣V∣cd⟩Tcd ij/Δij ab ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ ppl+⋅⋅⋅ . (2) Δij ab=εi+εj−εa−εbare the one-electron energy differences in the Hartree–Fock approximation. This yields the following correlation energy contributions:34 ECCSD c=Emp2+Eppl+Erest, (3) where Emp2corresponds to the MP2 correlation energy Emp2=∑ ij∑ abWij ab⟨ab∣V∣ij⟩ (4) and the particle–particle ladder term is defined as Eppl=∑ ij∑ abWij ab∑ cd⟨ab∣V∣cd⟩Tcd ij. (5) We note that Emp2was referred to as Edriverin previous related work.34,45Wij abis given by Wij ab=2⟨ij∣V∣ab⟩−⟨ji∣V∣ab⟩ ϵi+ϵj−ϵa−ϵb. (6) For the sake of brevity, we define Erestsuch that it contains all remaining contributions to the CCSD correlation energy. All equa- tions refer to spin-restricted spatial orbitals with the Coulomb integrals defined by ⟨ij∣V∣ab⟩=∬dr1dr2ϕ∗ i(r1)ϕ∗ j(r2)v(r12)ϕa(r1)ϕb(r2), (7) using the Coulomb kernel v(r12)=1/∣r1−r2∣. The following discussion is based on the premise that the finite virtual orbital manifold is spanned by a set of canonical orbitals, which needs to be augmented with additional virtual orbitals to reach the CBS limit, while the occupied orbitals are fully converged J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 154, 234103-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to the CBS limit regardless of the approximations used in the virtual orbital space. This situation closely resembles ab initio calculations employing re-canonicalized frozen natural orbitals.42We choose the following index labels for occupied and virtual spatial orbitals: i,j,k,...occupied states, a,b,c,...virtual states in finite basis, α,β,γ,...augmented virtual states, A,B,C,...union of all virtual states. The particle–particle ladder term ( Eppl) defined by Eq. (5) can also be expressed as ϵA ij=⟨Ψ(1) ij∣V∣Ψcc ij⟩, (8) where ⟨Ψ(1) ij∣=∑ abWij ab⟨ab∣ (9) and ∣Ψcc ij⟩=∑ cdTcd ij∣cd⟩. (10) Herein, ∣Ψ(1) ij⟩refers to the linearized first-order wavefunction, whereas ∣Ψcc ij⟩resembles a coupled-cluster pair wavefunction. Con- sequently, the PPL term only couples wavefunctions with the same occupied pair index. ∣pq⟩(⟨pq∣) is implicitly defined by Eq. (7) and corresponds to a (complex conjugate) product of the pqorbital pair. We note that the exponential CCSD pair wavefunction ansatz cannot be expressed by ∣Ψcc ij⟩only but also depends explicitly on the HF wavefunction and other polynomials of singles and doubles amplitudes not included in the definition of ∣Ψcc ij⟩. We formally define the CBS limit of the linearized first-order and coupled-cluster-like pair wavefunction by ∣Ψ(1)−cbs ij⟩=∣Ψ(1) ij⟩+∣δ(1) ij⟩ (11) and ∣Ψcc−cbs ij⟩=∣Ψcc ij⟩+∣δcc ij⟩, (12) respectively. ∣δ(1) ij⟩and∣δcc ij⟩are defined such that they correct for the BSIE in the respective parent wavefunctions ∣Ψ(1) ij⟩and∣Ψcc ij⟩. Fol- lowing the notation of this article, the latter are obtained from a vir- tual basis set a. Already in 1985, Kutzelnigg discussed that the con- ventional expansion, using products of one-electron states, cannot represent the wavefunction accurately at regions where the interelec- tronic distance approaches zero.46Thus, for increasing one-electron basis set sizes, the contribution of ∣δ(1) ij⟩and∣δcc ij⟩will largely be localized to the cusp region at small interelectronic distances. Sub- stituting the above BSIE corrections into Eq. (8) yields the following contributions to the PPL energy in the CBS limit: ϵB ij=⟨δ(1) ij∣V∣Ψcc ij⟩, (13) ϵC ij=⟨Ψ(1) ij∣V∣δcc ij⟩, (14) ϵD ij=⟨δ(1) ij∣V∣δcc ij⟩. (15)Consequently, the CBS limit formally reads Eppl−cbs=∑ ij(ϵA ij+ϵB ij+ϵC ij+ϵD ij). (16) This work outlines an efficient approximation to Eppl-cbs. To this end, we analyze ϵB ijandϵC ijand provide suitable approximations for them in Secs. II B–II D. We disregard ϵD ijsince it is of second order in the BSIE of the pair wavefunctions ( δ). We recall that ϵA ijis evaluated in the conventional coupled-cluster calculation using the finite basis set. B. Coupling of ∣δ(1) ij⟩to∣Ψcc ij⟩ We now turn to the expression for ϵB ijand employ the resolution of the identity (RI), ϵB ij=∑ αβ⟨δ(1) ij∣αβ⟩⟨αβ∣V∣Ψcc ij⟩ +∑ aβ⟨δ(1) ij∣aβ⟩⟨aβ∣V∣Ψcc ij⟩ +∑ αb⟨δ(1) ij∣αb⟩⟨αb∣V∣Ψcc ij⟩. (17) The above equation can be interpreted as the coupling of the change of the first-order wavefunction to ∣Ψcc ij⟩. Due to ⟨δ(1) ij∣ab⟩=0, only projectors that involve at least one state from the augmented vir- tual basis have to be included in the RI. The orbitals ϕαandϕβare strongly oscillating in space, i.e., they bear large wave number and/or high angular momentum number. In contrast, ∣Ψcc ij⟩is expected to be much smoother. Following fundamental ideas of scattering theory, we replace the complicated scattering problem with a much simpler one, by means of the following approximation: ∣Ψcc ij⟩=∑ cdTcd ij∣cd⟩≈∣ij⟩gcc ij, (18) where∣ij⟩is a mean-field state constructed from the Hartree–Fock orbitals of the occupied pair iandj. We stress that the left-hand side and right-hand side of Eq. (18) are orthogonal. However, this approximation is only used in the context of evaluation of matrix elements in expressions such as Eq. (13), which include the Coulomb interaction. The scaling factor gcc ijis chosen such that the spatially averaged correlation hole depths of the correlated wavefunction and its mean- field approximation are equated after projection onto the occupied space of the same electron pair, ∑ cdTcd ij⟨ij∣δ(r12)∣cd⟩=⟨ij∣δ(r12)∣ij⟩gcc ij. (19) The appearing integrals are defined in an analogous manner to Eq. (7) but with the Coulomb kernel replaced by the Dirac delta functionδ(r12). When using Gaussian basis functions, this requires only minor modifications of the original integral routines (see J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 154, 234103-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ref. 47). From Eq. (19), we obtain an explicit expression for the pair-specific correlation hole depth scaling factor given by gcc ij=∑cdTcd ij⟨ij∣δ(r12)∣cd⟩ ⟨ij∣δ(r12)∣ij⟩. (20) For the sake of brevity in the following paragraphs, we introduce a projection operator ˆgijthat yields an approximate mean-field state when applied to any correlated electron pair state such that ∣ij⟩gcc ij=ˆgij∣Ψcc ij⟩. (21) To get a better understanding of the above approximation, we now inspect the explicit expression for ϵB ijof a singlet state, which is given by ⟨δij∣V∣ψij⟩=∬ dr12dr12̃δ∗ ij(r12,r12)1 ∣r12∣̃ψij(r12,r12). (22) Here, the electron-pair functions δijandψijhave been transformed to a real-space representation in r12=r1−r2and ¯r12=r1+r2. Because ˜δijis largely localized around the cusp region, it effectively screens the Coulomb kernel at large interelectronic distances ∣r12∣.ϵB ij is therefore dominated by contributions from short interelectronic distances. Moreover, ˜ψijis a smooth function in the cusp region compared to ˜δij, suggesting that ˜ψij(r12,¯r12)≈˜ψij(r12=0,¯r12) (23) is a reasonable approximation. ˜ψij(0,¯r12)is the correlation hole depth as a function of ¯r12. The central approximation of this work is based on employing a mean field ansatz for ˜ψij(0,¯r12)that is obtained by projecting ˜ψij(0,¯r12)onto a corresponding zeroth-order mean-field wavefunction and ensuring that the spatially averaged correlation depths of the mean-field ansatz and ˜ψijagree. This is achieved using the pair-specific projection operator ˆgijdefined in Eq. (21). Using the mean-field approximation described above, ϵB ijcan be approximated as follows: ϵB ij=⟨δ(1) ij∣V∣Ψcc ij⟩≈⟨δ(1) ij∣V∣ij⟩ ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ Δϵ(2) ijgcc ij, (24) where Δϵ(2) ijrefers to the pair-specific BSIE correction of the MP2 correlation energy. Thus, we have shown that the ϵB ijcontribution to the PPL term can be approximated using Δϵ(2) ijtimes a scaling factor that depends on the spatially averaged correlation hole depth of∣Ψcc ij⟩. C. Coupling of ∣δcc ij⟩to∣Ψ(1) ij⟩ We now focus on the coupling between the first-order wave- function and the BSIE correction to Ψcc ij. Using once more the RI, we write Eq. (14) in the following way:ϵC ij=∑ CD⟨Ψ(1) ij∣V∣CD⟩⟨CD∣δcc ij⟩ ≈∑ CD⟨Ψ(1) ij∣ˆg† ijV∣CD⟩⟨CD∣δcc ij⟩ =g(1) ij∑ CD⟨ij∣V∣CD⟩⟨CD∣δcc ij⟩. (25) In the above equation, we have approximated the first-order state by a mean-field state that exhibits an identical spatially averaged corre- lation hole depth. Furthermore, the exact expression for ∣δcc ij⟩is not accessible, as we do not intend to solve the coupled-cluster equations in the large basis set. Moreover, we note that ⟨cd∣δcc ij⟩≠0, which is in contrast to the BSIE of the first-order state, where ⟨cd∣δ(1) ij⟩=0. Therefore, we approximate the orbital representation of ∣δcc ij⟩includ- ing only the dominant contributions (driver and PPL) in the com- plete basis set limit of the amplitude equations, as defined by Eq. (2), ⟨CD∣δcc ij⟩≈⟨CD∣δ(1) ij⟩ ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ (I)⊕⟨γζ∣V∣Ψcc ij⟩ ϵi+ϵj−ϵγ−ϵζ ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ (II) ⊕⟨cζ∣V∣Ψcc ij⟩ ϵi+ϵj−ϵc−ϵζ ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ (III)⊕⟨γd∣V∣Ψcc ij⟩ ϵi+ϵj−ϵγ−ϵd ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ (IV) ⊕⟨CD∣V∣δcc ij⟩ ϵi+ϵj−ϵC−ϵD⊕⋅⋅⋅ . (26) The direct sum notation is used to emphasize the fact that γis a subset of C. In the following, we consider only those terms defined by (I)−(IV)because they are of zeroth-order in δ⋅V, while the rest isO(δ⋅V). We now turn to the contributions of the terms defined by(I)−(IV)toϵC ij. Inserting (I) from Eq. (26) into the last line of Eq. (25) yields ϵC ij(I)=g(1) ijΔϵ(2) ij. (27) To account for the contributions of (II)−(IV)toϵC ij, we again approximate ∣Ψcc ij⟩using the mean-field ansatz defined by Eq. (21). Inserting the resulting approximations into the last line of Eq. (25) yields ϵC ij(II−IV)=g(1) ijΔϵ(2) ijgcc ij. (28) Therefore, our final approximation to Eq. (14) is given by ϵC ij≈ϵC ij(I)+ϵC ij(II−IV)=Δϵ(2) ij(g(1) ij+g(1) ijgcc ij), (29) which corresponds again to Δϵ(2) ijscaled by a factor that depends on the correlation hole depths of ∣Ψcc ij⟩and∣Ψ(1) ij⟩. We note that a corresponding BSIE correction in MP3 the- ory would have to include ϵC ij(I)only. However, in CCSD theory, the BSIE of ∣Ψcc ij⟩is not well approximated using δ(1) ij. Therefore, ϵC ij(II−IV)accounts for the change of δ(1) ijdue to the most impor- tant PPL coupling terms linear in ∣Ψcc ij⟩. The coupling strength of these terms is on the order of gcc ijand needs to be included to attain high accuracy. J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 154, 234103-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp D. The pair-specific PPL basis set correction We now summarize the final approximation to the BSIE cor- rection of the PPL energy, ϵB ij+ϵC ij≈Δϵ(2) ij(gcc ij+g(1) ij+g(1) ijgcc ij). (30) We stress that the contribution of ϵD ijdefined in Eq. (15) has been neglected because it is not of leading order in δ. We arrive at the following approximate CBS limit expression of the PPL energy: Eps−ppl=Eppl+∑ ijΔϵ(2) ij(gcc ij+g(1) ij+g(1) ijgcc ij) ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ Δps−ppl. (31) At this point, we note again that in the above expression the pair- specific correlation hole depth scaling factors gcc ijandg(1) ijare com- puted in a finite basis set, whereas Δϵ(2) ijrefers to the BSIE correction of the pair-specific MP2 correlation energy. III. THE UNIFORM TWO-ELECTRON GAS In order to assess the presented approximations, we first study a particularly simple model system—the three-dimensional uniform electron gas (UEG). The details of this model are described, for instance, in Ref. 45. For the here performed analysis, it is enough to study only two electrons in a homogeneous positive background. The singlet ground state Hartree–Fock (HF) wavefunction is a constant function, and the virtual HF states are plane-waves, ϕa(r)=1√ Ωeikar, (32) with HF eigenvalues ϵa=1 2ka2−4π Ωk2 a. (33) Notice that the eigenenergies are ordered with respect to the length of the corresponding momentum vector. The unit cell volume is given by Ω. This simplifies the four-index integrals to ⟨ii∣V∣ab⟩=4π Ω∣ka∣2δka+kb,0. (34) Consequently, the MP2 energy expression contains only a single sum, Emp2=∑ a⟨ii∣V∣a¯a⟩ ϵi+ϵi−ϵa−ϵ¯a⟨a¯a∣V∣ii⟩. (35) We use the notation ¯afor the virtual orbital with momentum vector −ka. The PPL energy expression reads Eppl=∑ a⟨ii∣V∣a¯a⟩ ϵi+ϵi−ϵa−ϵ¯a∑ c⟨a¯a∣V∣c¯c⟩tc¯c ii. (36) In a finite basis set calculation, the number of virtual basis func- tions Nvhas to be truncated. For the UEG model system, this is typically done by introducing a cutoff wave vector k1and consid- ering only virtual states with ∣ka∣<k1. Following the ideas of Sec. II, we introduce a second cutoff k2specifying the augmented virtualstatesα, with k1≤∣kα∣<k2. Hence, we can write the following four contributions to the total PPL energy: ϵA ii=∑ a⟨ii∣V∣a¯a⟩ ϵi+ϵi−ϵa−ϵ¯a∑ c⟨a¯a∣V∣c¯c⟩tc¯c ii, ϵB ii=∑ α⟨ii∣V∣α¯α⟩ ϵi+ϵi−ϵα−ϵ¯α∑ c⟨α¯α∣V∣c¯c⟩tc¯c ii, ϵC ii=∑ a⟨ii∣V∣a¯a⟩ ϵi+ϵi−ϵa−ϵ¯a∑ γ⟨a¯a∣V∣γ¯γ⟩tγ¯γ ii, ϵD ii=∑ α⟨ii∣V∣α¯α⟩ ϵi+ϵi−ϵα−ϵ¯α∑ γ⟨α¯α∣V∣γ¯γ⟩tγ¯γ ii.(37) We stress that one important feature of the UEG model con- sists in the fact that enlarging the basis set does not alter the occu- pied and virtual orbitals. We now examine the proposed approxi- mations numerically. We choose the union of all virtual states to be a very large number of 30 046 states, which can be considered a good approximation to the CBS limit for the present system. In the following, we gradually increase the number of virtual states in the finite basis and evaluate the approximate expressions for ϵB iiand ϵC iiin Eqs. (24) and (29) and compare them to the exact result in Eq. (37). The results for increasing numbers of virtual states are given in Table I. The contribution of ϵB iiis roughly twice as large asϵC ii. We find that both energy contributions can be approximated with remarkable accuracy using the presented expressions. Although the approximations made for ϵB iiandϵC iidiffer, we cannot observe any significant differences in the accuracy of both terms. The term ϵD ii, for which no approximation was introduced, converges consid- erably faster, when compared to the other two contributions ϵB iiand ϵC ii. Hence, the BSIE of the PPL contribution can be reduced by a large portion successfully. It appears that the remaining deviation is roughly in the same order of magnitude as the rest contribu- tion. Therefore, it is a reasonable approximation to neglect both contributions from ϵD iiandΔErest. For the above analysis, we have employed the fully converged CCSD amplitudes expanded in a basis of 30 046 virtual states. The amplitudes have been partitioned according to the cutoff k1into sets corresponding to tα¯α iiandta¯a ii, which have been used to compute ϵB ii TABLE I. BSIEs for the two-electron UEG with rs=3.5a.u. Reference energies are obtained from a calculation with 30 046 virtuals. Referred to exact (ex.) is the eval- uation of Eq. (37) for the converged amplitudes using 30 046 virtuals and using Nv orbitals in the finite basis. Estimates (est.) are evaluated using Eqs. (24) and (29). The BSIE of the rest term is given in the last column and calculated between results obtained with Nvand 30 046 virtual orbitals. All energies are given in mH. ϵBϵC ϵD Nv ex. est. ex. est. ex. ΔErest 26 0.582 0.560 0.255 0.262 0.178 −0.065 56 0.343 0.332 0.157 0.162 0.076 −0.049 122 0.171 0.166 0.079 0.083 0.027 −0.029 250 0.087 0.084 0.047 0.043 0.010 −0.015 514 0.043 0.043 0.021 0.022 0.004 −0.008 J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 154, 234103-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp andϵC ii. However, in practice and for the following benchmark sys- tems, we employ only CCSD amplitudes that have been calculated using a finite virtual orbital basis set. IV. COMPUTATIONAL DETAILS In Secs. V and VI, we present results obtained for a set of benchmark systems including 107 molecules and atoms. We employ aug-cc-pVXZ basis sets for first-row elements and aug-cc-pV(X +d)Z basis sets for second-row elements.48,49These basis sets will be denoted as AVXZ throughout this work. We obtained the refer- ence energies using the quantum chemistry package PSI4.50We have modified the code such that the Epplcontribution is extracted from the calculation as described in Ref. 35. For the CBS limit estimates, we use AV5Z and AV6Z energies and the extrapolation formula EX=ECBS+a/X3, with the basis set cardinal number X. This for- mula is used to get CBS estimates of all three individual terms: Emp2, Eppl, and Erest. We use unrestricted Hartree–Fock orbital functions and corresponding CCSD implementations for all open-shell sys- tems. All correlation energy calculations in this work used the frozen core approximation. In addition, (F12∗) calculations are performed using TURBO- MOLE22,51–53and the AVDZ, AVTZ, and AVQZ basis sets. We employ default settings; however, we use the RI basis aug-cc-pV5Z developed by Hättig54in all calculations. We note that these large RI basis sets are employed for all types of auxiliary functions in the TURBOMOLE implementation, i.e., $cbas, $jkbas, and $cabs. All results in the main text employ γ=1.0 in the parametrization of the correlation factor. Further results using a different γparameter can be found in the supplementary material. The derived approximate BSIE corrections to the PPL term were computed using our own coupled-cluster code cc4s, LIB- INT2,55and CTF.56In these calculations, the Hartree–Fock ground state wavefunction was obtained with the NWChem package57 and interfaced to cc4s as described in Ref. 58. We stress that the employed basis set correction defined by Eq. (31) depends on the pair specific MP2 BSIE correction Δϵ(2) ijand the correlation hole depth scaling factors gcc ij/g(1) ij. These terms need to be computed using a consistent set of occupied orbitals. In practice, this is compli- cated for states that belong to a degenerate set, which allows for arbi- trary unitary rotations among the degenerate subspace. This is par- ticularly problematic for the way the estimate for Δϵ(2) ijis obtained in this work, as it involves the results from separate MP2 calculations with different basis sets. In this work, we avoid arbitrary unitary rotations among degenerate sets of orbitals by introducing point charges far away from the molecules and atoms that break corre- sponding symmetries, lifting all possibly problematic degeneracies. These point charges are sufficiently far away to ensure that all com- puted correlation energies change by a numerically negligible small amount. For the newly introduced basis set correction scheme, we con- struct frozen natural orbitals (FNOs) on the level of second-order perturbation theory.42,59,60We truncate the virtual space used for the CCSD calculations by choosing only NvFNOs with the largest occupation number, where Nv=Xno×max(No,α,No,β)with Xno ∈[12, 16, 20, 24, 28, 32]. No,αand No,βrefer to the number of occupied spin-up and spin-down orbitals, respectively. We stress FIG. 1. Distribution of the number of virtual orbitals per occupied orbital for all 107 studied systems when employing an atom-centered AVXZ basis set. The same Gaussian function was used to smear the data points. that we use large basis sets (AVQZ and AV5Z) for the construc- tion of FNOs. Therefore, the number of virtual orbitals, Nv, is defined differently than for conventional quantum chemical cal- culations. In conventional calculations with atom-centered basis sets, the total number of orbitals is independent of the number of occupied orbitals but depends only on the atomic species for a chosen basis set. Yet, we seek to compare the BSIEs of correla- tion energies calculated using both approaches. To provide an esti- mate for which the cardinal number in the AVXZ basis set family corresponds on average to which number of virtual orbitals per occupied orbital, Fig. 1 depicts Nv/Nofor all studied atomic and molecular systems employing conventional AVXZ (X =D, T, Q, 5) basis sets. We find that AVDZ and AVTZ roughly correspond toXno=12 and Xno=20, respectively. Later, it will be numeri- cally verified that our choice of a fixed number of virtuals per occupied leads to a well-balanced energy description for different reactants. We have calculated the correlation energies of in total 107 molecules and atoms. Thereupon, we evaluated 26 closed-shell reac- tion energies (REc), 39 open-shell reaction energies (REo), 44 atom- ization energies (AEs), 16 electron affinities (EAs), and 22 ionization potentials (IPs). This benchmark set is a subset of the one studied by Knizia et al.5We had to exclude a number of molecules from their benchmark set as some of the molecules have been too large to be treated with our workflow. For some other molecules, we were not able to converge to a common HF ground state with neither of the three packages NWChem, TURBOMOLE, and PSI4. These molecules have also been excluded from our benchmark. A detailed list of the calculated molecules and corresponding reactions can be found in the supplementary material. V. RESULTS This section presents results for molecular systems and is orga- nized as follows: In Sec. V A, we assess the convergence of the dia- grammatically decomposed CCSD correlation energy contributions, confirming that the BSIEs in the total energy are dominated by the MP2 and PPL terms. In Sec. V B, we show that this behavior per- sists for most quantities computed from the total energies including reaction energies, atomization energies, ionization potentials, and electron attachment energies. In addition, we explore the accuracy of the derived approximate correction to the BSIE of the PPL term for all investigated quantities. In Sec. V C, we assess the accuracy of J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 154, 234103-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the corrected total CCSD energies and related quantities using two practical settings for the introduced focal-point approach and the respective BSIE corrections to the PPL term. The obtained results are compared to conventional CCSD and CCSD(F12∗) approaches. Section V D discusses our findings for the (T) and (T∗) correlation energy contributions using FNOs. A. Total energies We begin the analysis of the molecular systems by present- ing results for the basis set errors of the diagrammatically decom- posed correlation energy contributions for 107 molecules and atoms. Figure 2 depicts the BSIEs of the PPL ( ΔEppl), MP2 ( ΔEmp2), and rest (ΔErest) contributions. Furthermore, the BSIEs of the PPL ener- gies corrected according to Eq. (31) are also depicted ( ΔEps-ppl). The BSIEs are estimated using reference values obtained from a [56] extrapolation. The correlation energies are evaluated using a Hartree–Fock reference wavefunction and 16 FNOs per occupied orbital to approximate the virtual orbital manifold. This basis set size is on average between AVDZ and AVTZ, as can be seen in Fig. 1. For the construction of the FNOs, the one-particle reduced density matrix at the level of MP2 was calculated in an AV5Z basis set. Our findings show that MP2 energies calculated using 16 FNOs per occu- pied orbital exhibit by far the largest BSIEs when compared to the other contributions. In contrast to MP2, Erestis significantly better converged. This analysis reveals that Erestcan already be well approx- imated using a smaller number of natural orbitals than required for EpplandEmp2. However, adding the basis set correction to Epplas defined in Eq. (31) significantly reduces the remaining BSIE such thatΔEps-pplbecomes comparable to ΔErestfor all studied systems. This demonstrates impressively that the approximation derived in Sec. II can transfer its accuracy from the uniform electron gas model system to real atoms and molecules. B. Energy differences More decisive than well converged total energies is the question of how the proposed correction scheme works for energy differ- ences. Therefore, we analyze the BSIEs for the different channels FIG. 2. Distribution of the basis set incompleteness error (BSIE) of various inves- tigated energy channels (MP2, PPL, and rest) including the corrected PPL energy (ΔEps-ppl) for 107 studied systems. The energies were calculated using 16 frozen natural orbitals per occupied orbital and are referenced to [56] values. The same Gaussian function was used to smear the data points.(Eppl,Emp2, and Erest) for REc, REo, and AEs. The results are sum- marized in Table II for increasing numbers of FNOs as well as for the basis sets AVDZ-AV6Z. The MP2 contribution shows the largest BSIE followed by the PPL contribution. This is in accordance with the findings for the total energies, discussed in Sec. V A. We stress that only in the case of REc, the BSIE of ErestandEpplis of compara- ble magnitude. Furthermore, we note that the computed errors using FNOs for some systems become larger again or do not reduce signifi- cantly for Xno>24. We attribute this behavior to not sufficiently well converged FNOs. When approaching Xno>24, one would require even bigger basis sets than the employed AV5Z for the construction of the FNOs. Generally, it is not expected that the errors are signif- icantly smaller than when using all possible virtual orbitals in the AV5Z basis set. In particular, for the large basis sets, the BSIE of the rest con- tribution is remarkably small; the rms deviation for AV5Z is only around 0.3 kJ/mol and lower. A similar high accuracy can be attained when using only a comparably small number of 20 FNOs per occu- pied orbital, which achieves rms deviations of around 0.5 kJ/mol for the reaction energies and 1 kJ/mol for atomization energies. For REo and AEs, the PPL contribution converges signifi- cantly slower with respect to the basis set size compared to Erest. Furthermore, the BSIE cannot be diminished considerably with a finite number of FNOs. This behavior changes when taking the proposed correction into account. Compared to the uncorrected PPL contribution, the BSIE of the corrected PPL contribution is reduced by a factor of four and more, when using only 20 FNOs per occupied or less. For REc, the correction has no significant effect. In summary, rms deviations of the rest contributions ( ΔErest) and corrected PPL ( ΔEps-ppl) contributions are on the scale of 1 kJ/mol when using 20 FNOs per occupied orbital. Reaching a simi- lar accuracy by employing conventional basis set calculations would require a [34] extrapolation. C. Benchmarking a practical focal-point approach Based on the findings in Secs. V A and V B, we now define and assess a practical focal-point approach to compute total CCSD energies. For an even-tempered composition, we combine extrapo- lated MP2 energies with CCSD calculations employing FNOs. We introduce the following two compositions: EFPa=Emp2([34])+Erest(12)+Eppl(12) =Eccsd(12)−Emp2(12)+Emp2([34]) (38) and EFPb=Emp2([45])+Erest(20)+Eppl(20) =Eccsd(20)−Emp2(20)+Emp2([45]). (39) Emp2(Xno),Eccsd(Xno),Erest(Xno), and Eppl(Xno)refer to the cor- responding correlation energy contributions calculated employ- ingXnoFNOs per occupied orbital. Emp2([34])and Emp2([45]) refer to MP2 correlation energies obtained from a [34] and [45] extrapolation, respectively. For the first ansatz ( EFPa), we construct the FNOs using an AVQZ calculation, whereas for the second J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 154, 234103-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. BSIEs of correlation energy contributions to REc, REo, and AEs. Shown are the rms deviations to the [56] reference. The results have been obtained using 12–32 FNOs per occupied orbitals ( Xno), [23]–[45] extrapolations, and for different conventional basis sets ranging from AVDZ up to AV6Z. REc (kJ/mol) REo (kJ/mol) AEs (kJ/mol) ΔEmp2ΔEpplΔErestΔEps-pplΔEmp2ΔEpplΔErestΔEps-pplΔEmp2ΔEpplΔErestΔEps-ppl Xno=12 6.396 1.850 1.212 1.199 28.325 8.811 1.513 1.986 39.350 14.985 2.783 2.257 Xno=16 4.576 1.302 0.822 0.931 20.556 6.509 0.783 1.502 28.230 11.118 1.437 1.321 Xno=20 2.978 0.877 0.437 0.739 15.771 5.224 0.489 1.390 22.094 8.941 1.005 0.949 Xno=24 2.240 0.540 0.338 0.667 12.429 4.130 0.308 1.257 17.231 7.115 0.650 0.723 Xno=28 2.289 0.626 0.299 0.536 9.416 3.114 0.242 1.284 14.024 5.930 0.600 0.664 Xno=32 2.476 0.649 0.516 0.485 7.241 2.398 0.392 1.325 11.494 4.903 0.559 0.627 AVDZ 15.207 2.400 3.468 ⋅⋅⋅ 43.096 10.296 5.200 ⋅⋅⋅ 67.192 21.166 12.269 ⋅⋅⋅ AVTZ 6.981 1.767 2.040 ⋅⋅⋅ 20.598 5.675 2.544 ⋅⋅⋅ 27.203 9.567 4.153 ⋅⋅⋅ AVQZ 3.212 1.077 0.653 ⋅⋅⋅ 8.945 2.666 0.624 ⋅⋅⋅ 11.866 4.321 0.831 ⋅⋅⋅ AV5Z 1.885 0.640 0.288 ⋅⋅⋅ 4.553 1.370 0.300 ⋅⋅⋅ 6.122 2.257 0.274 ⋅⋅⋅ AV6Z 1.092 0.371 0.164 ⋅⋅⋅ 2.636 0.793 0.177 ⋅⋅⋅ 3.541 1.306 0.158 ⋅⋅⋅ [23] 5.420 1.708 2.049 ⋅⋅⋅ 11.576 3.901 2.852 ⋅⋅⋅ 11.131 4.941 2.534 ⋅⋅⋅ [34] 2.007 0.842 0.815 ⋅⋅⋅ 1.393 0.662 1.468 ⋅⋅⋅ 1.237 0.708 1.845 ⋅⋅⋅ [45] 0.730 0.262 0.148 ⋅⋅⋅ 0.719 0.237 0.400 ⋅⋅⋅ 0.458 0.172 0.416 ⋅⋅⋅ ansatz ( EFPb) the AV5Z basis set is used. In this section, we will explore benchmark results obtained using both approaches with and without the introduced Δps−ppl correction that depends on the respective pair-specific extrapolated MP2 energies and corre- lation hole depths. The corresponding BSIEs are summarized in Table III. The uncorrected focal-point approaches EFPaand EFPbyield only satisfying BSIEs for the closed-shell reaction energies. Here, FPa achieves the quality of the [34] result, although the CCSD calcula- tion is performed with a significantly smaller virtual space of only 12 FNOs per occupied orbital. For the open-shell reactions and other properties, the focal-point method performs significantly worse with rms deviations between 3 and 12 kJ/mol and a maximum error of around 25 kJ/mol.The focal-point approaches including the Δps−ppl correction yield significantly more consistent BSIEs for all studied energy dif- ferences. The rms deviations are 1.5–3 kJ/mol and around 1 kJ/mol for FPa and FPb, respectively. For the FPb +Δps−ppl approach, the maximum deviation is below 4 kJ/mol for all considered reactions. We note that for the closed-shell reactions the corrected focal- point results show larger rms deviations and larger maximum errors than the uncorrected variants. We attribute this to fortuitous error cancellation between the individual energy contributions to the CCSD correlation energy. This is only visible when the Δps−ppl correction is insignificant as it is the case for the closed-shell reac- tion energies (see Sec. V B). Correcting for the BSIE in the PPL term reduces this error compensation, causing slightly larger BSIEs TABLE III. CCSD valence correlation energy basis set incompleteness error for closed-shell reaction (REc), open-shell reac- tions (REo), atomization energies (AEs), ionization potentials (IPs), and electron affinities (EAs). The reference is obtained from a [56] extrapolation. Two different variants of the focal-point approximation are used with and without correction. The details are found in the main text. REc (kJ/mol) REo (kJ/mol) AEs (kJ/mol) IPs (kJ/mol) EAs (kJ/mol) max rms max rms max rms max rms max rms FPa 4.823 1.905 15.403 7.130 24.223 11.786 10.244 3.301 15.384 5.414 FPb 2.828 1.029 10.998 5.445 16.660 8.461 6.837 2.652 9.203 3.766 FPa+Δps−ppl 7.248 2.541 8.836 3.024 9.017 2.400 3.359 1.520 5.666 1.603 FPb+Δps−ppl 3.253 1.102 2.424 1.049 3.491 0.848 1.475 0.733 3.062 0.828 [23] 15.637 5.814 33.638 10.460 26.000 8.443 8.814 4.366 6.170 3.033 [34] 6.222 2.031 4.545 1.938 3.965 1.874 1.993 0.828 1.772 0.895 [45] 1.137 0.438 1.864 0.650 1.302 0.532 0.213 0.112 0.181 0.113 (F12∗)@AVDZ 9.859 3.163 14.625 4.274 11.957 5.821 8.375 4.632 7.059 4.720 (F12∗)@AVTZ 3.838 1.434 5.946 1.657 3.834 1.520 2.685 1.393 1.810 1.189 (F12∗)@AVQZ 2.233 0.869 1.727 0.719 1.795 0.532 0.898 0.543 1.015 0.472 J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 154, 234103-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp for closed-shell reaction energies. However, the results for REc obtained including the Δps−ppl correction are of comparable size to open-shell reactions and other properties. The extrapolation using the AVDZ and AVTZ basis sets shows large maximum errors of up to 30 kJ/mol, and the rms deviation ranges from 3 to 10 kJ/mol. The [34] extrapolation yields satisfy- ing results with rms deviations of around 2 kJ/mol, for IPs and EAs already below 1 kJ/mol. Although the [45] extrapolation yields the best statistical results, a CCSD calculation with the large AV5Z basis set is only possible for small systems. We stress that more sophis- ticated extrapolation techniques were already tested for the original version of the employed benchmark set. The results can be found in the supplementary material of Ref. 5. Knizia et al. concluded that “[⋅⋅⋅] in our benchmark set there are only few systems where using either extrapolation scheme makes a noteworthy difference.” Thus, the corrected FPa ansatz is to be preferred over [23] extrapola- tion and the corrected FPb seems to be superior compared with the [34] extrapolation. We stress that in both cases larger HF and MP2 calculations have to be performed in order to obtain the final result. For comparison, CCSD(F12∗) results are also given for three different basis sets. The F12 results obtained using the AVQZ basis set reach almost the quality of the [45] extrapolation, with rms deviations well below 1 kJ/mol. For the (F12∗)@AVTZ results, the rms deviations are only around 1.5 kJ/mol, whereas (F12∗)@AVDZ yields results that show rms deviations with about 3–5 kJ/mol. We note that (F12∗)@AVTZ yields results with smaller rms devi- ations than the [34] extrapolation. We note that the size of the virtual space in the CCSD calculation for (F12∗)@AVDZ and FPa +Δps−ppl is similar. The same is true for (F12∗)@AVTZ and FPb+Δps−ppl. However, the FPa and FPb approaches require HF and MP2 calculations using the AVQZ and AV5Z basis sets, respectively. Therefore, the entire computational cost of the pro- posed focal-point approaches depends strongly on the efficiency of the employed HF and MP2 algorithms. Further statistical analysisof the test set using HF and conventional CCSD is provided in the supplementary material. D. Perturbative triples contribution Having assessed the introduced focal-point approach for the CCSD method, we now turn to the discussion of BSIEs in the perturbative triples contribution to the CCSD(T) energies calculated using FNOs. In addition to the conventional approach of comput- ing the (T) contribution, we will also explore the (T∗) approxi- mation, which approximates the CBS limit of (T) by rescaling the finite basis set result with a factor estimated on the level of MP2 theory as outlined in Ref. 5. In this work, the scaling factor cor- responds to Emp2([45])/Emp2(Xno). The results are summarized in Table IV. The rms deviations for the AVDZ are 2–8 kJ/mol, and even with the corrected values, denoted as (T∗), the errors are within the range of 2–3 kJ/mol. With increasing cardinal numbers, the devia- tions reduce considerably. AVQZ results show deviations of up to 1 kJ/mol, reducing even further for the (T∗) correction where they do not surpass 0.5 kJ/mol. As it is apparent from the presented data, the usage of FNOs together with the (T∗) ansatz seems to be highly effective. Already 12 FNOs per occupied orbital suffice to reduce the rms deviations to 0.7 kJ/mol and lower. When using 20 FNOs instead, this deviation reduces smoothly below 0.35 kJ/mol. We stress that computing the (T∗) scaling factor using a [56] extrapolation instead of [45] extrapolation has almost no effect (0.05 kJ/mol in the rms BSIEs). Considering the findings for the BSIEs listed in Tables III and IV in combination shows that it is possible to obtain CCSD(T) correlation energy estimates of REc, REo, AEs, IPs, and EAs with a root mean square deviation from the CBS limit below 4 kJ/mol using 12 FNOs per occupied orbital only. Employing 20 FNOs per occupied orbital reduces the rms BSIE to around 1 kJ/mol for all computed energy differences. A detailed summary of all computed energies and BSIEs can be found in the supplementary material. TABLE IV. BSIE of the (T) contribution to closed-shell reaction (REc), open-shell reactions (REo), atomization energies (AEs), ionization potentials (IPs), and electron affinities (EAs). Shown are the rms deviations to the [56] reference. REc (kJ/mol) REo (kJ/mol) AEs (kJ/mol) IPs (kJ/mol) EAs (kJ/mol) (T) (T∗) (T) (T∗) (T) (T∗) (T) (T∗) (T) (T∗) Xno=12 0.914 0.522 1.925 0.456 2.881 0.633 0.808 0.387 1.378 0.710 Xno=16 0.583 0.385 1.175 0.316 1.765 0.350 0.522 0.252 0.877 0.415 Xno=20 0.418 0.265 0.808 0.255 1.336 0.259 0.381 0.177 0.679 0.333 Xno=24 0.292 0.203 0.616 0.227 0.989 0.236 0.298 0.122 0.506 0.207 Xno=28 0.268 0.183 0.487 0.241 0.824 0.209 0.260 0.110 0.421 0.166 Xno=32 0.206 0.233 0.397 0.244 0.699 0.192 0.223 0.092 0.364 0.129 AVDZ 1.976 3.231 5.176 3.216 8.176 2.826 3.139 2.364 4.330 2.676 AVTZ 1.055 0.823 1.485 0.997 2.165 0.450 0.862 0.432 1.275 0.566 AVQZ 0.509 0.269 0.667 0.433 0.913 0.153 0.366 0.185 0.571 0.277 AV5Z 0.270 0.135 0.303 0.231 0.425 0.087 0.179 0.088 0.296 0.151 AV6Z 0.157 0.078 0.191 0.134 0.246 0.049 0.103 0.050 0.171 0.087 [23] 0.888 ⋅⋅⋅ 0.705⋅⋅⋅ 0.570⋅⋅⋅ 0.232⋅⋅⋅ 0.170⋅⋅⋅ [34] 0.255 ⋅⋅⋅ 0.188⋅⋅⋅ 0.133⋅⋅⋅ 0.066⋅⋅⋅ 0.121⋅⋅⋅ [45] 0.035 ⋅⋅⋅ 0.088⋅⋅⋅ 0.093⋅⋅⋅ 0.024⋅⋅⋅ 0.040⋅⋅⋅ J. Chem. Phys. 154, 234103 (2021); doi: 10.1063/5.0050054 154, 234103-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp VI. SUMMARY AND CONCLUSION In this work, we introduced a new CCSD basis set correc- tion scheme that employs FNOs and exhibits an excellent trade- off between the virtual orbital basis set size and remaining BSIE. The introduced correction aims at removing the BSIEs of the most important contributions to the CCSD correlation energy in the CBS limit originating from the MP2 and PPL terms. We stress that our approach is electron pair-specific. The presented results for reaction energies, atomization energies, ionization potentials, and electron attachment energies exhibit a rapid convergence with respect to the basis set size. Furthermore, we have shown that the CBS limit of the (T) contribution to the CCSD(T) energy can be approximated with a similar efficiency when employing FNOs and a rescaling procedure previously referred to as (T∗).5 The introduced focal-point approach is an interesting alterna- tive to conventional basis set truncation techniques that are based on cardinal numbers. FNOs give access to much more finely incre- mented basis set sizes while maintaining a stable and rapid conver- gence of many properties to the CBS limit. We did not observe sig- nificant shell filling effects that might cause non-monotonic energy convergence with respect to the FNO basis set size. However, we note that the presented approach relies on the availability of com- putational efficient algorithms to compute MP2 energies and FNOs. We stress that our approach is directly transferable to ab initio calculations employing pseudo-potentials. This will be beneficial for solid state calculations in a plane-wave basis set, which is one of the main motivations for this research. Most solid state calcu- lations using coupled-cluster theory and plane-wave basis sets per- formed so far employ FNOs, and the electronic transition structure factor needed to compute correlation hole depths is readily avail- able.61–63Furthermore, we stress that the outlined basis set correc- tions will also be interesting to other many-electron theories, where similar interference effects play an important role, for instance, auxiliary-field quantum Monte Carlo.64 Finally, we note that the introduced basis set correction scheme is expected to work reliably in systems exhibiting correlation hole depths that vary strongly between different electron pairs. This includes situations where core-valence electron correlation effects play an important role. 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J. Chem. Phys. 154, 214111 (2021); https://doi.org/10.1063/5.0051211 154, 214111 © 2021 Author(s).Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). I. Revisiting the NEVPT2 construction Cite as: J. Chem. Phys. 154, 214111 (2021); https://doi.org/10.1063/5.0051211 Submitted: 23 March 2021 . Accepted: 09 May 2021 . Published Online: 04 June 2021 Yang Guo , Kantharuban Sivalingam , and Frank Neese ARTICLES YOU MAY BE INTERESTED IN Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). II. The full rank NEVPT2 (FR-NEVPT2) formulation The Journal of Chemical Physics 154, 214113 (2021); https://doi.org/10.1063/5.0051218 Spin contamination in MP2 and CC2, a surprising issue The Journal of Chemical Physics 154, 131101 (2021); https://doi.org/10.1063/5.0044362 Model protein excited states: MRCI calculations with large active spaces vs CC2 method The Journal of Chemical Physics 154, 214105 (2021); https://doi.org/10.1063/5.0048146The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). I. Revisiting the NEVPT2 construction Cite as: J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 Submitted: 23 March 2021 •Accepted: 9 May 2021 • Published Online: 4 June 2021 Yang Guo,1Kantharuban Sivalingam,2 and Frank Neese2,a) AFFILIATIONS 1Qingdao Institute for Theoretical and Computational Sciences, Shandong University, Qingdao, Shandong 266237, China 2Max-Planck-Institut für Kohlenforschung, Kaiser-Wilhelm-Platz 1, D-45470 Mülheim an der Ruhr, Germany a)Author to whom correspondence should be addressed: Frank.Neese@kofo.mpg.de ABSTRACT Over the last decade, the second-order N-electron valence state perturbation theory (NEVPT2) has developed into a widely used multiref- erence perturbation method. To apply NEVPT2 to systems with large active spaces, the computational bottleneck is the construction of the fourth-order reduced density matrix. Both its generation and storage become quickly problematic beyond the usual maximum active space of about 15 active orbitals. To reduce the computational cost of handling fourth-order density matrices, the cumulant approximation (CU) has been proposed in several studies. A more conventional strategy to address the higher-order density matrices is the pre-screening approx- imation (PS), which is the default one in the ORCA program package since 2010. In the present work, the performance of the CU, PS, and extended PS (EPS) approximations for the fourth-order density matrices is compared. Following a pedagogical introduction to NEVPT2, contraction schemes, as well as the approximations to density matrices, and the intruder state problem are discussed. The CU approximation, while potentially leading to large computational savings, virtually always leads to intruder states. With the PS approximation, the computa- tional savings are more modest. However, in conjunction with conservative cutoffs, it produces stable results. The EPS approximation to the fourth-order density matrices can reproduce very accurate NEVPT2 results without any intruder states. However, its computational cost is not much lower than that of the canonical algorithm. Moreover, we found that a good indicator of intrude states problems in any approximation to high order density matrices is the eigenspectra of the Koopmans matrices. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0051211 I. INTRODUCTION In the past few decades, complete active space self-consistent field (CASSCF) and complete active space configuration interaction (CASCI) methods have been the most successful methods to recover static correlation of strongly correlated systems.1The CASSCF or CASCI method can be considered as a multi-level method, electrons within the active space are treated at full configuration interaction (FCI) level, whereas the other electrons (core electrons) are treated at the Hartree–Fock level. Because the computational cost of the FCI step grows exponentially with the size of active space, stan- dard CASSCF/CASCI algorithms can only tackle systems with less than about 20 active orbitals. To overcome the active space limita- tion of the CASCI/CASSCF methods, many approximate wave func- tion Ansätze have been employed, including selected CI/CIPSI,2–4density matrix renormalized group (DMRG),5,6and FCI quantum Monte Carlo (FCIQMC) theories.7They all serve as substitutes of the FCI step in canonical CASCI/CASSCF. It has been proven that static correlation can be accurately described by those approx- imate CASCI/CASSCF treatments. Calculations with several dozen active orbitals and active electrons have been reported using these methods.8–11 Despite these impressive achievements, it is well known that CASCI/CASSCF can only deliver qualitatively correct results, where only a small fraction of all electrons in the system are part of the active space. To quantitatively describe larger strongly correlated systems, the dynamic correlation of core–active, core–virtual, and active–virtual orbitals has to be treated as well. Thus, multireference (MR) dynamic correlation calculations, including MR perturbation theories (MRPT), MR configuration interaction methods (MRCI), J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-1 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp or even MR coupled cluster Ansätze (MRCC), have to be performed on top of CASCI/CASSCF references. These MR methods have been applied to describe bond-breaking processes, spin states of transi- tion metal complexes, excited states, and many other multirefer- ence situations. Excellent reviews describing these MR methods are available.12–14 Given its comparatively low computational cost paired with their at least approximate size consistency, MRPT methods are more popular than MRCI (not size consistent) or MRCC (com- putationally often prohibitive) in studying electronic structures of strongly correlated systems.15–17Among the widely available MRPT methods, the complete active space second-order perturbation the- ory (CASPT2)18and second-order N-electron valence state per- turbation theory (NEVPT2)19have been extensively calibrated and widely used.20–22Much progress has been made based on canon- ical CASPT2 and NEVPT2, for example, analytical gradients,23–26 explicit correlation methods,27,28and linear scaling extensions using pair natural orbitals (PNOs).29,30Besides the above topics, one of the most active fields is to extend the application of these methods to large active spaces using approximated CASCI/CASSCF references based on selected CI/CIPSI, DMRG, or FCIQMC. To develop CASPT2 or NEVPT2 algorithms for large active space references, one of the most difficult steps is the construc- tion of higher-order reduced density matrices (RDMs), e.g., the third- or fourth-order RDMs that are required for both CASPT2 and NEVPT2. The third order RDM can still be handled fairly well for systems with a few dozen active orbitals, and it is also required for the construction and normalization of the internally contracted configuration states functions (CSFs) that span the first- order interaction space (FOIS) in CASPT2 and NEVPT2. However, the fourth-order RDM quickly becomes unmanageable. There are generally three ways to overcome this bottleneck: (a) approximat- ing the fourth-order RDMs, (b) directly contracting RDMs with the zeroth-order Hamiltonian during the construction of orthonor- mal IC basis, and (c) using (partially) uncontracted wave functions to avoid higher-order RDMs altogether. To obtain approximate higher-order density matrices, the cumulant approximation is one of the most efficient ways, and it has been extensively explored in the contracted Schrödinger equation method.31–33Later, the spin- free cumulant approximation has been reported by Kutzelnigg, Mukherjee, and Shamasundar.34–36Its performance was investi- gated by Zgid and co-workers using strongly contracted NEVPT2 (SC-NEVPT2).37Based on the spin-free cumulant approximation, Yanai and co-workers developed the first DMRG-CASPT2 algo- rithm.38The cumulant approximation based DMRG-CASPT2 algo- rithm is further pursued by Pierloot and co-workers39as well as Nakatani and Guo.40Noting the “false intruder states” introduced by the cumulant approximation,37Chan and co-workers developed alternative DMRG based SC-NEVPT2 algorithms.41,42Reiher and co-workers reported a DMRG based partially contracted NEVPT2 (PC-NEVPT2) method as well.43By avoiding the construction of the fourth-order RDMs explicitly, Kurashige and Yanai,44and Wouters and co-workers45reported DMRG-CASPT2 implementa- tions. Approximate SC-NEVPT2 algorithms based on the FCIQMC wave function were reported by Sharma and co-workers, and by Booth and co-workers.46,47Alternatively, the fourth-order RDM in CASPT2 or NEVPT2 can be avoided using an uncontracted wave function Ansatz . The pioneering study by Celani and Werner48andrecent works by Chan and co-workers,49Sharma and co-workers,50 and Giner and co-workers51explored this direction. Recently, much attention was devoted to selected CI methods of the configuration interaction from iteratively selected zeroth order space (CIPSI) family, given its clear and simple construction and its ability to converge toward the exact FCI wave function.52–57In ORCA,58different selected CI algorithms [called “iterative config- uration expansion” (ICE)] have been implemented based on the Slater determinants, configuration state function (CSF), or spatial configurations (CFGs), respectively, and have been publicly available since 2016. These implementations will be described in full detail elsewhere.59 Thus, the next important step is to develop MR methods based on the ICE algorithms to compute the remaining dynamic corre- lation. The NEVPT2 method is a reliable candidate to achieve this target. Over the past 15 years, we have gained a host of experience with this method and came to appreciate the attractive features of this method as being computationally efficient, intruder state free, and size consistent. A large number of successful applications, pre- dominantly in the field of molecular magnetism, attest to the virtues of NEVPT2 in computational chemistry.15,16,60However, it has been reported that using approximate density matrices leads to the occur- rence of intruder states in NEVPT2.37,43In the present paper, three approximations of higher-order RDMs have been carefully exam- ined using a test set, consisting of excitation energies of organic molecules, transition metal complexes, and potential energy sur- faces (PESs) of diatomic molecules. The occurrence of intruder states under the three approximations is carefully analyzed. II. METHODOLOGY A. Recapitulation of multireference perturbation theory The zeroth-order wave functions of MRPT theory are usually defined as ∣Ψ(0)⟩=∑ IBI∣I⟩⋅ (1) Here, the set of ∣I⟩is suitable many-particle basis functions, com- monly CSFs or Slater determinants in a complete or restricted active space. The coefficients BIare the expansion coefficients, usually determined variationally by diagonalizing the full Hamiltonian over the set of ∣I⟩, thus providing the zeroth order wave function with energy E0. To define a second-order perturbation theory, the non- relativistic Born–Oppenheimer Hamiltonian is partitioned into the zeroth-order part H0and the perturbation part V such that H=H0+V=PH 0P+QH 0Q+V⋅ (2) In MRPT2, the operators Pand Qare projectors onto the refer- ence space and singly and doubly excited CSFs of the reference wave functions, respectively. The first-order wave functions, consisting of singles and doubles excitations, are also known as FOIS, ∣Ψ(1)⟩=∑ I,pqBI,pqˆEp q∣I⟩+∑ I,pqrsBI,pqrsˆEp qˆEr s∣I⟩⋅ (3) J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-2 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In Eq. (3), the spin-traced excitation operators ˆEq pare defined by second quantized operators, ˆEq p=ˆa+ q↑ˆap↑+ˆa+ q↓ˆap↓. (4) By using real functions to represent zeroth- and first-order wave functions, the coefficients of ∣Ψ(1)⟩can be determined by minimiz- ing the Hylleraas functional Hyl=⟨Ψ(1)∣H0∣Ψ(1)⟩+2⟨Ψ(1)∣V∣Ψ(0)⟩⋅ (5) The definition of H0plays a central role in perturbation theory, and its choice differs in various MRPT methods. In the NEVPT2 theory, the Dyall Hamiltonian is chosen to define the zeroth-order Hamiltonian61 HDyall=Hinact Dyall+Hact Dyall+C, ⎛ ⎝Hinact Dyall=∑ ijFijˆEj i+∑ abFabˆEb a, Hact Dyall=∑ tuhtuˆEt u+1 2∑ tuvw(tv∣uw)(ˆEt vˆEu w−δu vˆEt w), C=∑ i2hii+∑ ij2(ii∣jj)−(ij∣ij)−∑ i2Fii⎞ ⎠. (6) The constant Cin Eq. (6) ensures that the reference CAS energy ECASis recovered when acting HDyall on the CASSCF reference wave function. A brief explanation of various notations used in the present work is given in Table I. In Table I, the spin-free pre-density matrix (PDM) and spin-free RDM can be transformed into each other. For example, the fourth-order RDM can be obtained by normal ordering operators in the fourth-order PDM, γpqrs pqrs=Γpqrs pqrs−δqpΓrps rqs−δrpγpqs rqs−δspγpqr sqr−δrqγpqs prs −δsqγpqr psr−δsrγpqr pqs−δrpδsrγpq sq−δrpδsqγpq rs −δspδrqγpq sr−δrqδsrγpq ps⋅ (7)Thus, in the following discussion, the PDMs and RDMs will be referred to as “density matrices” and will not be further distin- guished unless otherwise specified. B. Introduction to different contraction schemes in NEVPT2 In MR theory, the first-order wave function consists of all the CSFs that can interact directly with the zeroth-order wave function through the Hamiltonian. This space is referred to as the first-order interacting space (FOIS). A variety of contraction schemes to span the FOIS exists in the literature, including the strongly contracted (SC), internally contracted (IC), and uncontracted (UC) construc- tions. In either of these schemes, the FOIS can be grouped into eight distinct excitation subspaces depending on the number of hole and particle labels that occur in the excitation operators. The original NEVPT2 work introduces the SC, the partially contracted (PC), and the UC-NEVPT2,62where a given contraction scheme is applied to the entire first-order wave function. However, it is also possible to choose a mixed representation for the different subspaces.23,63,64A comparison of various contraction schemes has been performed by Angeli and co-workers in the context of NEVPT265and by some of us in the framework of MRCI.66 One advantage of mixed UC and IC schemes is that higher- order density matrices can be avoided. In the Werner–Knowles contraction scheme, the three subspaces involving two virtual indices are internally contracted, whereas the other subspaces are left uncontracted.63Thus, the calculations of fourth-order density matrices is completely bypassed. Such contraction schemes, where some of the FOIS subspaces are internally contracted, should, in our opinion, rightfully be called “partially contracted” (PC).48The con- traction scheme, where all eight subspaces are internally contracted, we prefer is referred to as “fully internally contracted” (FIC). We note that this is not in keeping with the literature, where Angeli and Malrieu refer to what we call FIC as PC, whereas no specific name has been assigned to the mixed contraction schemes. Thus, through- out our work and in keeping with our previous publications,28,66the name FIC-NEVPT2 will be used throughout the present papers in place of PC-NEVPT2. TABLE I. Summary of the general notation. Inactive orbitals i,j,k,l Active orbitals t,u,v,w,x,y Transformed active indices μ,v Virtual orbitals a,b,c,d Arbitrary orbitals p,q,r,s One-electron integrals hpq Two electron repulsion integrals in (11 ∣22) (Mulliken) notation (pq∣rs) Reference wave function ∣Ψ(0)⟩=∑IBI∣I⟩ Reference CSFs ∣I⟩,∣J⟩ Internally contracted CSFs ∣Ψpr qs⟩ Pre-density matrix (PDM) γt⋅⋅⋅v u⋅⋅⋅w=⟨Ψ(0)∣ˆEt u⋅⋅⋅ˆEv w∣Ψ(0)⟩ Spin-free reduced density matrix (RDM) Γt⋅⋅⋅v u⋅⋅⋅w=⟨Ψ0∣ˆEt⋅⋅⋅v u⋅⋅⋅w∣Ψ0⟩ J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-3 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Consistent with other MR theories, the computational cost of UC-NEVPT2 is much larger than SC- or FIC-NEVPT2, which is probably the reason why this method is essentially never used in practice. In both SC- and FIC-NEVPT2, the contraction of FOIS leads to a compact representation of the wave functions as well as low computational costs while introducing a tolerable error in the final error relative to UC-NEVPT2.65 Among the different NEVPT2 algorithms, SC-NEVPT2 is the most efficient one, given that it completely avoids any iterative pro- cedure for the first-order wave function or the storage of amplitudes. However, it is not unitary invariant with respect to the rotations within the doubly occupied and virtual spaces.67This renders the use of this contraction scheme highly problematic in the context of local correlation theories.29We also found that SC-MRCISD is very inaccurate compared to FIC-MRCISD.66Another consequence of this lack of unitary invariance is that the analytical gradient of SC- NEVPT2 suffers from numerical instabilities.25Thus, in the present work, we focus on FIC-NEVPT2 and will drop the “FIC” prefix unless specifically noted. C. Higher-order density matrices in NEVPT2 The working equations of the NEVPT2 method have been reported in detail elsewhere.29,62Since there is no inter subspace interaction in NEVPT2, the total correlation energies of NEVPT2 can be computed subspace by subspace. For NEVPT2 calcula- tions with large active space references, the Siand Sasubspaces become computational bottlenecks, since they require the fourth- order RDM. The working equations of the Sasubspace serve as an illustration. The IC first-order wave function of the Sasubspace reads ∣ΨSa⟩=∑ tuva∣Ψav tu⟩=∑ tuvaTav tuˆEa tˆEv u∣Ψ(0)⟩⋅ (8) The final correlation energy of the Sasubspace can be computed by E(2) Sa=∑ tuv,aTav tu⟨Ψ(0)∣V∣Ψav tu⟩⋅ (9) Due to the fact that the IC wave functions are nonorthogonal and potentially linear dependent, the corresponding amplitude Tav tuin Eqs. (8) and (9) should be computed by Tav tu=∑ μCtuv,μTa μ, (10) where Ta μare the amplitudes computed with transformed active indices and Ctuv,μis the corresponding transformation matrix. To compute Ctuv,μ, a generalized eigenvalue equation restricted within active space must be solved, ∑ tuv⟨Ψav tu∣Hact Dyall∣Ψav tu⟩Ctuv,μ =Eμ∑ tuv⟨Ψav tu∣Ψav tu⟩Ctuv,μ(KC μ=EμMC μ). (11) The left-hand side of Eq. (11) is the augmented Koopmans matrix, abbreviated as K, which can be further simplified to be the usual Koopmans matrix ˜K. As shown in Table I, the transformed active indices in the eigenbasis of the Dyall Hamiltonian are collectivelydenoted with μ. In the eigenbasis, the correlation energy and ampli- tude expressions of Sasubspace in Eq. (9) can be rewritten as E(2) Sa=∑ μ,aTa μ⟨Ψ(0)∣V∣Ψa μ⟩, (12) Ta μ=−⟨Ψ(0)∣V∣Ψa μ⟩ Eμ+Faa. (13) Since the Dyall Hamiltonian includes the two-body terms in the active part, the fifth-order density matrices are needed to compute the augmented Koopmans matrices of the Siand Sasubspaces. The calculation and storage of such density matrices is a major bottle- neck in large active space NEVPT2 calculations, e.g., with 14 active orbitals, the third-order density matrix requires 60 MB storage, while the fourth-order matrix accounts for 10 GB and the fifth-order one much as 2 TB of data. As pointed out by Dyall and later by Angeli and co-workers,61,62 the fifth-order density matrix can be avoided by using the rank- reduction trick. To achieve that, a commutator is introduced into Eq. (11), ⟨Ψav tu∣HDyall∣Ψav tu⟩=⟨Ψ0∣ˆEu vˆEt a[HDyall,ˆEa tˆEv u]∣Ψ0⟩ +⟨Ψ0∣ˆEu vˆEt aˆEa tˆEv uHDyall∣Ψ0⟩ =⟨Ψ0∣ˆEu vˆEt a[HDyall,ˆEa tˆEv u]∣Ψ0⟩ +ECAS⟨Ψav tu∣Ψav tu⟩ =⟨Ψ0∣ˆEu vˆEt a[Hact Dyall,ˆEa tˆEv u]∣Ψ0⟩ +(Faa+ECAS)⟨Ψav tu∣Ψav tu⟩⋅ (14) The explicit expression of the first term in Eq. (14) is the Koopmans matrices ˜Kand presented elsewhere.62To the best of our knowledge, most, if not all, of the reported NEVPT2 implementations are based on the rank-reduction formulation, in which only up to fourth-order density matrices are needed. The rank-reduced NEVPT2 is exact as long as exact CASCI references are used. Hence, this elegant formu- lation avoids a major bottleneck in NEVPT2 that would otherwise arise from the fifth order RDM. Nevertheless, the size of active space in NEVPT2 calculations is still severely limited due to the need to construct the fourth-order density matrices. In Secs. II D–II F, three approximations to reduce the computational cost of fourth-order density matrices are introduced. Approximated fourth-order density matrices only affect Eμin the denominator of Eq. (13). D. Pre-screening (PS) approximation In ORCA, in place of RDMs, working equations are expressed in terms of PDMs (see Table I). The fourth-order PDM is defined as γprtv qsuw=⟨Ψ(0)∣ˆEp qˆEr sˆEt uˆEv w∣Ψ(0)⟩ =∑ I,JBIBJ⟨I∣ˆEp qˆEr sˆEt uˆEv w∣J⟩⋅ (15) In practice, these higher-order PDMs are evaluated by introducing a resolution of identity (RI) CSF or CFG space.68The RI space ∣K⟩is J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-4 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp constructed by exciting and de-exciting an electron from the refer- ence space, which is always complete and does not compromise the accuracy. By inserting ∣K⟩into Eq. (15), the PDM is evaluated as γprtv qsuw=∑ I,J,K,L,MBIBJ⟨I∣ˆEp q∣K⟩⟨K∣ˆEr s∣L⟩⟨L∣ˆEt u∣M⟩⟨M∣ˆEv w∣J⟩⋅(16) Not all of the CSFs in the reference wave function are equally impor- tant. In fact, there are many CSFs that have negligible coefficients or weights in the wave function. Under the pre-screening (PS) approx- imation, the wave function is truncated before the evaluation of the higher-order PDMs. In ORCA, CFGs (a configuration is defined by all CSFs with the same occupation pattern irrespective of their spin coupling) are building blocks to express the CASCI wave functions. Thus, in the current work, the PS approximation truncates the CAS wave functions based on CFGs. If the weight of a CFG is smaller than a given threshold ( Tps), all CSFs in the CFG are eliminated from the zeroth-order wave function. After the truncation, the approximate wave function reads ∣Ψ0⟩≈∣Ψ0⟩=∑ IBI∣I⟩ =∑ CFG _N∑ I∈CFG _NBI∣I⟩⎛ ⎝∑ I∈CFG _NBI2>Tps⎞ ⎠. (17) This approximation is designed to eliminate “dead wood” from the reference space in an attempt to enhance the efficiency of the sub- sequent computational steps. In fact, such truncated wave functions may be considered as “ a posteriori ” selected CI wave functions. With the truncated reference ∣Ψ0⟩, the corresponding fourth-order PDM can be approximated by γprtv qsuw≈∑ I,JBIBj⟨I∣ˆEp qˆEr sˆEt uˆEv w∣J⟩. (18) Note that the reduction of the reference space also implies a smaller RI space. In principle, the PS scheme can be applied to PDMs of arbitrary rank. The ad hoc truncation leads to a small error in the trace, which can be corrected by a renormalization of the truncated wave function ∣Ψ0⟩. In the present work, after the truncation of CAS space, the CI coefficients BIare renormalized. E. Extended PS (EPS) approximation The PS approximation to the fourth-order density matrices is straightforward and reduces the computation cost. However, as shown in Sec. III, it cannot deliver reliable results with loose Tps thresholds. To improve the accuracy of the PS approximation, an extension is proposed. Under the PS approximation, the CAS wave function can be partitioned into two components as follows: ∣Ψ(0)⟩=∣Ψ(0)⟩+∣˜Ψ(0)⟩ =∑ IBI∣I⟩+∑ ˜JB˜J∣˜J⟩(BI2>Tps,B˜J2≤Tps), (19) where the strong part is ∣Ψ0⟩and the weak part is ∣˜Ψ0⟩. The previ- ousTpsthreshold in PS can be used to partition the complete CASspace. In the PS approximation, only the strong part of the wave function is used to compute approximate density matrices, Eq. (18). The contributions involving CSFs in the ∣˜Ψ(0)⟩space are neglected. In the extended PS (EPS) approximation, two additional terms are considered in the approximate fourth-order PDM as well, γtuvw pqrs≈⟨Ψ0∣ˆEt pˆEu qˆEv rˆEw s∣Ψ0⟩+⟨˜Ψ0∣ˆEt pˆEu qˆEv rˆEw s∣Ψ0⟩ +⟨Ψ0∣ˆEt pˆEu qˆEv rˆEw s∣˜Ψ0⟩⋅ (20) A pilot implementation of the EPS approximation of the fourth- order PDM is realized for the Siand Sasubspace in NEVPT2. The EPS approximation has the same scaling as the canonical PDM construction and is thus computationally more expensive than the PS approximation for the fourth-order PDM. As will be shown in Sec. III, the accuracy of the fourth-order PDM approximated with PS(10−14) settings is as exact as that with PS(0.0) settings. To fur- ther speed up the EPS approximation, not all ∣˜J⟩, orthogonal to the strong part of CAS space, are considered in the weak wave function ∣˜Ψ0⟩. The configurations with B˜J2smaller than 10−14are neglected in the weak part as well during the construction of fourth-order PDM in the EPS approximation. F. Cumulant (CU) approximation Reduced density matrices of a given order can be expanded in terms of anti-symmetrized products of lower-ranked reduced den- sity matrices and a residual part of the given order. The latter is called the cumulant, and its neglect allows a fast and factorizable reconstruction of the higher-order density matrices.31In the spin- orbital basis, the cumulant expansion is best represented using the Grassmann product ( ∧) as an anti-symmetrizer, dq1q2⋅⋅⋅qm p1p2⋅⋅⋅pm∧ds1s2⋅⋅⋅sn r1r2⋅⋅⋅rm =[1 (m+n)!]2 ∑ P,Qsgn(P)sgn(Q)P(q1⋅⋅⋅sn) ×Q(p1⋅⋅⋅rn)dq1q2⋅⋅⋅qm p1p2⋅⋅⋅pmds1s2⋅⋅⋅sn r1r2⋅⋅⋅rm, (21) where P(Q)permutes all upper (lower) indices of the two tensors. The parity of the permutations is expressed as sgn (P)and sgn(Q), respectively. Denoting the cumulant as Δ, the fourth-order RDM dtuvw pqrs(in spin-orbital basis) is compactly written as dtuvw pqrs=Δtuvw pqrs+16Δtuv pqr∧dw s+18Δtu pq∧Δvw rs+72dt p∧du q∧dvw rs +24dt p∧du q∧dv r∧dw s⋅ (22) Similarly, the spin-orbital based third-order RDM becomes dtuv pqr=Δtuv pqr+9dt p∧Δuv qr+6dt p∧du q∧dv r⋅ (23) In this approximate scheme, the RDMs of a given order nare approximated by including all contributions from lower-rank den- sity matrices m<nbut neglecting the cumulant of rank n. The NEVPT2, being implemented as a spin-adapted approach, requires spin-free density matrices. Explicit formulas for the spin- free cumulants of third and fourth-order density matrix have been J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-5 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp derived by Shamasundar, Kutzelnigg, and Mukherjee.36The afore- mentioned formulas are also employed in the SC-NEVPT2 for- mulations of the Chan group,37as well as in the DMRG-CASPT2 and DMRG-MRCISD approaches of the Yanai group.44,69In the CU(4) variant, the fourth-order cumulant Δtuvw pqrsis omitted, whereas the CU(3,4) variant neglects both fourth- and third-order cumu- lants. In the context of DMRG-CASPT2, the CU(4) variant has been found to be robust,38which has also been confirmed in a more recent study of Phung et al.39However, the complete neglect of the cumulants is not unproblematic. As demonstrated by Zgid et al. ,37 the CU(4) approximation for SC-NEVPT2 leads to “false intruder states.” “False intruder states” arise from poorly approximated Eμ in the denominator. For example, the cumulant approximated FIC- MRCI, developed by Saitow, Yanai, and co-workers, shows con- vergence problems and is no longer variational.69The cumulant approximated FIC-MRCC, discussed in the Ph.D. thesis of Hanauer, shows similar convergence symptoms.70Consequently, a number of authors have questioned the principle validity of the cumulant omis- sion, as the higher-order cumulants are not necessarily small.71,72 Adapting Yanai’s spin-free formulation of the cumulant approxi- mation, the CU(4) and CU(3,4) approximations for the Siand Sa subspaces in NEVPT2 have been implemented in ORCA, based on the rank-reduced NEVPT2 formulation.G. Intruder states in NEVPT2 The occurrence of intruder states is a potential problem in many MRPT methods. Although there have been many studies about intruder state problems, there appears to be no generally accepted definition of “intruder states.” The energies of QH0Q in Eq. (2) of some MRPT2 formulations, perhaps most promi- nently CASPT2, can become near degenerate with the zeroth-order energy, E0=⟨Ψ(0)∣H0∣Ψ(0)⟩, and result in a singular perturbation expansion. The near-degeneracy is an artifact of the zeroth-order Hamiltonian and not present in the exact Hamiltonian. Various numerical techniques have been developed to cope with this situation. These include imaginary shifts and real shifts to avoid small denominators in various MRPT2 variants.73–78In addition to near-degeneracy, the definition of the zeroth-order Hamiltonian can cause another artifact: FOIS CSFs with lower energies than the reference wave function. These CSFs have positive energy contributions in the perturbation expansion and should also be considered to represent intruder states. Thus, in the context of NEVPT2, the definition of “intruder states” in Sasubspace means Eμ−Faa<TIS, (24) TABLE II. The absolute energies of three lowest roots, S 0, S1, and S 2, of stilbene molecules. The results using EPS, PS, and CU approximations with respect to the exact NEVPT2 energies are given in eV, with mean absolute deviations (MAD) and standard deviations (SD). The bold entries have intruder spectra in the Saand Si subspace. State (dihedral) NEVPT2 (a.u.) EPS (10−8) EPS (10−6) EPS (10−4) PS (10−12) PS (10−10) PS (10−8) PS (10−6) CU(4) S0(15○) −539.492 258 0.001 0.008 0.042 0.000 0.024 0.012 0.250 0.005 S1(15○) −539.333 953 0.002 0.018 0.084 0.000 0.000 0.006 −0.087−0.005 S2(15○) −539.338 184 0.003 0.023 0.111 0.000 0.000 0.001 0.097 0.020 S0(92○) −539.436 354 0.001 −0.007 0.036 0.000 0.000 0.000 0.036 0.004 S1(92○) −539.312 358 0.002 0.013 0.066 0.000 0.000 −0.037 0.041 −0.049 S2(92○) −539.312 259 0.002 0.013 0.075 0.000 0.000 0.013 0.060 0.034 S0(180○) −539.507 956 0.001 0.007 0.036 0.000 0.000 0.003 0.115 0.009 S1(180○) −539.341 803 0.002 0.017 0.081 0.000 0.002 0.015 −0.010 0.002 S2(180○) −539.344 432 0.003 0.021 0.104 0.000 0.000 0.045 −0.354 0.005 MAD ⋅⋅⋅ 0.002 0.014 0.071 0.000 0.003 0.015 0.117 0.015 SD ⋅⋅⋅ 0.002 0.015 0.075 0.000 0.008 0.021 0.158 0.021 TABLE III. Relative energies (in eV) of NEVPT2 between the cis-/trans -stilbene isomers and the transition state. The deviations of various EPS, PS, and CU approximations with respect to the relative energies from the canonical NEVPT2 calculation are given in eV, with MAD and SD. Energy difference NEVPT2 EPS (10−8) EPS (10−6) EPS (10−4) PS (10−12) PS (10−10) PS (10−8) PS (10−6) CU(4) S0(92○)–S 0(15○) 1.521 0.000 −0.015 −0.006 0.000 −0.024 −0.012 −0.214−0.001 S1(92○)–S 1(15○) 0.588 0.000 −0.004 −0.018 0.000 0.000 −0.043 0.128 −0.044 S2(92○)–S 2(15○) 0.705 −0.001 −0.010 −0.036 0.000 0.000 0.011 −0.037 0.014 S0(180○)–S 0(15○)−0.427 0.000 −0.001 −0.006 0.000 −0.024 −0.009 −0.135 0.004 S1(180○)–S 1(15○)−0.214 0.000 −0.001 −0.003 0.000 0.002 0.009 0.077 0.007 S2(180○)–S 2(15○)−0.170 0.000 −0.002 −0.007 0.000 0.000 0.044 −0.452−0.015 MAD ⋅⋅⋅ 0.000 0.006 0.013 0.000 0.008 0.021 0.174 0.014 SD ⋅⋅⋅ 0.001 0.008 0.017 0.000 0.014 0.027 0.221 0.020 J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-6 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV. Absolute energies of the adenine ground (11Ag) and excited states. The deviation of EPS, PS, and CU approximations with respect to the canonical NEVPT2 are given in eV, with MAD and SD. The bold entries have intruder spectra in the SaandSisubspace. State NEVPT2 (a.u.) EPS (10−8) EPS (10−6) EPS (10−4) PS (10−12) PS (10−10) PS (10−8) PS (10−6) CU(4) 11A′−466.637 220 0.001 0.010 0.083 0.000 0.000 0.016 0.052 0.008 21A′−466.436 480 0.001 0.014 0.137 0.000 0.000 −0.297 0.315 0.011 31A′−466.457 945 0.001 0.020 0.196 0.000 0.000 0.052 1.046 0.081 41A′−466.383 849 0.001 0.017 0.173 0.000 0.000 0.051 32.594 0.033 51A′−466.387 096 0.001 0.022 0.232 0.000 0.000 −0.017 0.810 −0.009 61A′−466.402 979 0.002 0.027 0.290 0.000 0.000 −0.053 0.295 −0.109 71A′−466.359 307 0.001 0.023 0.218 0.000 0.000 −0.033 −0.128 0.017 11A′′−466.437 710 0.001 0.015 0.139 0.000 0.000 0.000 −5.047 0.167 21A′′−466.414 178 0.001 0.015 0.132 0.000 0.000 −0.157 0.250 0.019 MAD ⋅⋅⋅ 0.001 0.018 0.178 0.000 0.000 0.075 4.504 0.051 SD ⋅⋅⋅ 0.001 0.019 0.187 0.000 0.000 0.117 11.004 0.073 where Eμhas been defined in Eq. (11). TISis a predefined small energy difference, for example, 0.005 hartree.77Such a definition of intruder states depends on not only the eigenvalues of the Koopman’ matrices but also on the orbital energies of the inactive spaces.The definition given in Eq. (24) might not be very suitable for NEVPT2. As discussed by Angeli and co-workers, in NEVPT2, all Eμare computed by generalized eigenvalue equations involving the active part of the Dyall Hamiltonian. An explanation of the physical meaning of Eμwas given by Angeli and co-workers.79Thus, Eq. (11) TABLE V. Absolute energies of the naphthalene ground state (11Ag) and excited states. The deviation of EPS, PS, and CU approximations with respect to the canonical NEVPT2 are given in eV, with MAD and SD. The bold entries have intruder spectra in the SaandSisubspace. State NEVPT2 (a.u.) EPS (10−8) EPS (10−6) EPS (10−4) PS (10−12) PS (10−10) PS (10−8) PS (10−6) CU(4) 11Ag−385.236 655 0.000 0.001 0.008 0.000 0.000 0.000 0.006 0.000 21Ag−385.007154 0.000 0.001 0.020 0.000 0.000 0.000 −0.011 −0.005 31Ag−384.986 489 0.000 0.002 0.025 0.000 0.000 0.000 −0.002 −0.143 41Ag−384.953 713 0.000 0.002 0.027 0.000 0.000 0.000 0.017 −0.001 11B3u−385.074 046 0.000 0.001 0.013 0.000 0.000 0.000 0.017 −0.026 21B3u−384.945 873 0.000 0.002 0.027 0.000 0.000 0.000 0.006 −0.010 31B3u−385.032 487 0.000 0.003 0.040 0.000 0.000 0.000 0.020 −0.026 11B2u−385.084 192 0.000 0.002 0.026 0.000 0.000 0.000 0.070 −0.072 21B2u−385.023 867 0.000 0.002 0.035 0.000 0.000 0.000 0.034 0.005 31B2u−384.901 406 0.000 0.002 0.025 0.000 0.000 0.000 0.061 0.022 11B1g−385.009 955 0.000 0.002 0.024 0.000 0.000 0.000 −0.005 −0.036 21B1g−384.954 085 0.000 0.002 0.023 0.000 0.000 0.000 0.033 −0.134 13B2u−385.114 957 0.000 0.001 0.012 0.000 0.000 0.000 3.357 0.006 23B2u−385.063 156 0.000 0.001 0.018 0.000 0.000 0.000 0.613 −0.005 13B3u−385.080 985 0.000 0.001 0.018 0.000 0.000 0.000 −0.006 −0.193 23B3u−385.082 862 0.000 0.002 0.024 0.000 0.000 0.000 0.082 0.015 13B1g−385.066 090 0.000 0.001 0.016 0.000 0.000 0.000 0.004 0.009 13Ag−385.028 625 0.000 0.002 0.020 0.000 0.000 0.000 0.004 0.004 23Ag−384.994 713 0.000 0.002 0.024 0.000 0.000 0.000 −0.012 0.003 33Ag−385.019 789 0.000 0.003 0.037 0.000 0.000 0.000 −0.044 −0.069 MAD ⋅⋅⋅ 0.000 0.002 0.023 0.000 0.000 0.000 0.220 0.039 SD ⋅⋅⋅ 0.000 0.002 0.024 0.000 0.000 0.000 0.764 0.067 J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-7 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp should lead to positive Eμ, provided that reasonable reference wave functions are chosen.67Hence, Eμ>0 (25) is considered a requirement for an intruder state free method. Note that to solve the generalized eigenvalue equations for Eμ, there is a pre-defined parameter to remove the linear dependencies of IC wave functions. To the best of our knowledge, there is no unique value for this threshold. If the chosen threshold is too small, numerical insta- bilities will result. If the threshold is chosen too large, major errors are introduced into the results. In ORCA, a threshold of 10−6is cho- sen as the default setting, which produces stable NEVPT2 results in the vast majority of cases we have met, in practice, over the course of more than a decade. However, if approximate density matrices are supplied to the NEVPT2 method, there might be some grossly inaccurate or even negative Eμthat will lead to small or unphysical denominators. The latter are considered “false intruder states” in CU SC-NEVPT2.37In our view, it is sensible to differentiate “false intruders” from thebona fide intruder states that arise from a specific choice of the zeroth order Hamiltonian H 0. By contrast, “false intruders” arise from approximations within the chosen perturbation framework and would be absent without the approximations. The final result from both types of intruders will be similar—a divergent, useless per- turbation energy. However, the underlying causes for this behavior are very different. In the canonical transformation algorithm80as well as CU DMRG-MRCI,69loose thresholds as high as 10−2have been applied in order to avoid “false intruder states” (variational collapses). However, it is not clear to us why these large thresholds necessary should only remove the “false intruders” from the approximated Eμ values. In the present work, 10−6is still chosen as the threshold to remove the linear dependency of IC wave functions throughout the present work. In Tables II–IX, if there is at least a denominator with a value smaller than 0.005, TIS=0.005, the result is considered to be contaminated by “intruder spectra” and highlighted as such. TABLE VI. The deviation of adenine and naphthalene vertical excitation energies using EPS, PS, and CU approximations with respect to the exact NEVPT2 results (in eV), with MAD and SD. Excited state NEVPT2 EPS (10−8) EPS (10−6) EPS (10−4) PS (10−10) PS (10−8) PS (10−6) CU(4) Adenine 21A′5.462 0.000 0.004 0.054 0.000 −0.314 0.263 0.003 31A′4.878 0.001 0.011 0.113 0.000 0.035 0.994 0.072 41A′6.895 0.000 0.007 0.090 0.000 0.035 32.543 0.024 51A′6.806 0.001 0.012 0.149 0.000 −0.034 0.758 −0.018 61A′6.374 0.001 0.017 0.207 0.000 −0.070 0.243 −0.117 71A′7.562 0.001 0.013 0.135 0.000 −0.049 −0.179 0.008 11A′′5.429 0.000 0.005 0.056 0.000 −0.017 −5.098 0.159 21A′′6.069 0.000 0.005 0.049 0.000 −0.173 0.199 0.011 Naphthalene 13B2u 3.312 0.000 0.001 0.005 0.000 0.000 3.351 0.006 23B2u 4.721 0.000 0.001 0.010 0.000 0.000 0.607 −0.005 13B3u 4.236 0.000 0.001 0.011 0.000 0.000 −0.011−0.193 23B3u 4.185 0.000 0.002 0.016 0.000 0.000 0.076 0.014 13B1g 4.641 0.000 0.001 0.008 0.000 0.000 −0.002 0.008 13Ag 5.661 0.000 0.001 0.012 0.000 0.000 −0.001 0.004 23Ag 6.584 0.000 0.002 0.017 0.000 0.000 −0.018 0.003 33Ag 5.901 0.000 0.002 0.029 0.000 0.000 −0.050−0.070 21Ag 6.245 0.000 0.001 0.012 0.000 0.000 −0.017−0.005 31Ag 6.807 0.000 0.001 0.018 0.000 0.000 −0.008−0.143 41Ag 7.699 0.000 0.001 0.020 0.000 0.000 0.011 −0.001 11B3u 4.425 0.000 0.000 0.006 0.000 0.000 0.012 −0.027 21B3u 7.913 0.000 0.001 0.020 0.000 0.000 0.000 −0.011 31B3u 5.556 0.000 0.002 0.032 0.000 0.000 0.015 −0.027 11B2u 4.149 0.000 0.001 0.018 0.000 0.000 0.065 −0.072 21B2u 5.790 0.000 0.002 0.027 0.000 0.000 0.029 0.004 31B2u 9.123 0.000 0.001 0.017 0.000 0.000 0.055 0.021 11B1g 6.169 0.000 0.001 0.017 0.000 0.000 −0.011−0.037 21B1g 7.689 0.000 0.001 0.016 0.000 0.000 0.027 −0.134 MAD ⋅⋅⋅ ⋅⋅⋅ 0.000 0.004 0.043 0.000 0.027 1.653 0.044 SD ⋅⋅⋅ ⋅⋅⋅ 0.000 0.006 0.066 0.000 0.072 6.378 0.071 J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-8 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VII. Absolute energies of [Cr(NH 3)6]3+ground state (GS) and low-lying quartet and doublet states. The deviation of the EPS, PS, and CU approximations with respect to the canonical NEVPT2 are given in eV, with MAD and SD. The bold entries have intruder spectra in the SaandSisubspace. States NEVPT2 (a.u.) EPS (10−8) EPS (10−6) EPS (10−4) PS (10−12) PS (10−10) PS (10−8) PS (10−6) CU(4) 4GS−1381.803 735 0.000 0.000 0.001 0.000 0.000 −0.001 0.004 0.001 14T2g−1381.707 369 0.000 0.000 0.000 0.000 0.000 0.000 0.006 0.001 −1381.707 358 0.000 0.000 0.000 0.000 0.000 −0.002−0.020 0.014 −1381.707 439 0.000 0.000 0.001 0.000 0.000 0.000 0.009 0.000 14T1g−1381.675 925 0.000 0.000 0.004 0.000 0.000 0.001 0.002 0.014 −1381.675 920 0.000 0.000 −0.050 0.000 0.000 −0.001 0.013 0.019 −1381.675 688 0.000 0.000 0.002 0.000 0.000 0.000 0.015 −0.002 24T1g−1381.596 972 0.000 0.000 0.002 0.000 0.000 0.000 0.003 0.001 −1381.596 951 0.000 0.000 0.002 0.000 0.000 −0.006 0.001 −0.044 −1381.596 923 0.000 0.000 0.002 0.000 0.000 0.001 −0.017 0.002 12Eg−1381.728 513 0.000 0.000 0.002 0.000 0.000 −0.001 0.002 0.000 −1381.728 506 0.000 0.000 0.001 0.000 0.000 0.000 −0.003−0.070 12T1g−1381.725 774 0.000 0.000 0.001 0.000 0.000 0.000 0.002 0.005 −1381.725 770 0.000 0.000 0.001 0.000 0.000 0.000 −0.025−0.008 −1381.725 760 0.000 0.000 −0.007 0.000 0.000 −0.001−0.002−0.003 12T2g−1381.691 040 0.000 0.000 0.001 0.000 0.000 0.000 0.005 −0.005 −1381.691 030 0.000 0.000 0.001 0.000 0.000 −0.004 0.016 −0.007 −1381.691 004 0.000 0.000 0.001 0.000 0.000 0.001 0.016 0.013 12A1g−1381.646 823 0.000 0.000 0.003 0.000 0.000 0.000 0.000 0.001 MAD ⋅⋅⋅ 0.000 0.000 0.004 0.000 0.000 0.001 0.008 0.011 SD ⋅⋅⋅ 0.000 0.000 0.012 0.000 0.000 0.002 0.011 0.020 Furthermore, a discussion about “false intruder states” is conducted in Sec. III E, where a comparison between exact Koopmans energies Eμand approximated ones are performed. III. RESULTS AND DISCUSSION In principle, the PS, EPS, and CU approximations to the den- sity matrices discussed in Sec. II can be applied to all subspaces in NEVPT2, except Sijab(no density matrix is needed in Sijab). However,to better understand these approximations, in the present work, only the fourth-order density matrices in the Siand Sasubspaces are approximated, since these are the most “vulnerable” with respect to small denominators. The other six subspaces are left untouched. To test the performance of the three approximations, a test set is proposed, featuring potential energy surfaces, excited states of organic molecules, and transition metal complexes. All geometries of the molecules in the test set can be found in the supplementary material. Unless otherwise stated, the calculations are carried out TABLE VIII. Absolute energies of [Cu(NH 3)4]2+the ground state (GS), d–d, and LMCT states. The deviation of the EPS, PS, and CU approximations with respect to the canonical NEVPT2 results is given in eV. The bold entries have intruder spectra in the SaandSisubspace. States NEVPT2 (a.u.) EPS (10−8) EPS (10−6) EPS (10−4) PS (10−12) PS (10−10) PS (10−8) PS (10−6) CU(4) GS−1864.990 055 0.000 0.001 0.007 0.000 0.000 0.112 −0.004 0.000 12B2g−1864.908 804 0.000 0.000 0.008 0.000 0.000 −0.005 −0.001 0.000 12A1g−1864.897 615 0.000 0.001 0.011 0.000 0.000 −0.008 0.007 0.001 12Eg−1864.896 114 0.000 0.000 0.007 0.000 0.000 0.000 0.039 0.000 −1864.896 069 0.000 0.000 0.005 0.000 0.000 −1.123 0.191 0.000 12Eu−1864.783 248 0.000 0.001 0.011 0.000 0.000 0.071 0.400 −0.001 −1864.783 193 0.000 0.001 0.016 0.000 0.000 −0.271 −0.123−0.001 22B1g−1864.764 283 0.000 0.001 0.016 0.000 0.000 −0.025 0.024 0.001 22A1g−1864.717 391 0.000 0.001 0.019 0.000 −0.001 0.005 0.087 −0.002 MAD ⋅⋅⋅ 0.000 0.001 0.012 0.000 0.000 0.189 0.109 0.001 SD ⋅⋅⋅ 0.000 0.001 0.012 0.000 0.000 0.409 0.166 0.001 J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-9 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IX. Excitation energies (in eV) of [Cr(NH 3)6]3+and [Cu(NH 3)4]2+. The deviation of the EPS, PS, and CU approximations and the statistics of the absolute deviation are given in eV. Excited states NEVPT2 EPS (10−8) EPS (10−6) EPS (10−4) PS (10−12) PS (10−10) PS (10−8) PS (10−6) CU(4) [Cr(NH 3)6]3+14T2g 2.622 0.000 0.000 −0.001 0.000 0.000 0.001 0.002 0.000 2.623 0.000 0.000 −0.001 0.000 0.000 −0.001−0.024 0.012 2.620 0.000 0.000 −0.001 0.000 0.000 0.001 0.005 −0.002 14T1g 3.478 0.000 0.000 0.003 0.000 0.000 0.002 −0.001 0.012 3.478 0.000 0.000 −0.052 0.000 0.000 0.000 0.009 0.018 3.484 0.000 0.000 0.000 0.000 0.000 0.001 0.012 −0.004 24T1g 5.626 0.000 0.000 0.000 0.000 0.000 0.001 −0.001−0.001 5.627 0.000 0.000 0.000 0.000 0.000 −0.005−0.003−0.045 5.628 0.000 0.000 0.001 0.000 0.000 0.002 −0.021 0.001 12Eg 2.047 0.000 0.000 0.001 0.000 0.000 0.000 −0.002−0.001 2.047 0.000 0.000 0.000 0.000 0.000 0.001 −0.007−0.071 12T1g 2.121 0.000 0.000 0.000 0.000 0.000 0.001 −0.002 0.003 2.122 0.000 0.000 0.000 0.000 0.000 0.000 −0.029−0.009 2.122 0.000 0.000 −0.009 0.000 0.000 0.000 −0.005−0.005 12T2g 3.067 0.000 0.000 0.000 0.000 0.000 0.001 0.002 −0.007 3.067 0.000 0.000 0.000 0.000 0.000 −0.003 0.012 −0.008 3.068 0.000 0.000 −0.001 0.000 0.000 0.002 0.012 0.011 12A1g 4.270 0.000 0.000 0.002 0.000 0.000 0.001 −0.004 0.000 [Cu(NH 3)4]2+12B2g 2.211 0.000 0.000 0.001 0.000 0.000 −0.117 0.003 0.000 12A1g 2.515 0.000 0.000 0.004 0.000 0.000 −0.119 0.010 0.001 12Eg 2.556 0.000 0.000 0.000 0.000 0.000 −0.112 0.043 0.001 2.558 0.000 0.000 −0.002 0.000 0.000 −1.235 0.194 0.000 12Eu 5.628 0.000 0.000 0.004 0.000 0.000 −0.041 0.404 −0.001 5.629 0.000 0.000 0.010 0.000 0.000 −0.383−0.120 0.000 22B1g 6.144 0.000 0.001 0.009 0.000 0.000 −0.137 0.028 0.001 22A1g 7.420 0.000 0.001 0.012 0.000 0.000 −0.107 0.091 −0.001 MAD 0.000 0.000 0.004 0.000 0.000 0.087 0.040 0.008 SD 0.000 0.000 0.011 0.000 0.000 0.259 0.094 0.018 with the def2-TZVPP basis81and using the RI-JK approxima- tion together with the Def2/JK auxiliary basis82as implemented in ORCA.58Note that the Koopmans matrices constructed from approximate fourth-order density matrices are not symmetric any- more. To avoid the diagonalization of non-symmetric matrices, a symmetrization step is performed before solving the generalized eigenvalue problems. The same adjustment has been used by Yanai and co-workers.69 A. Stilbene isomerization Stilbene is a prototypical molecule for photoreactions and has been extensively studied by theory and experiment.83In a simpli- fied model, the cis–trans isomerization of the stilbene molecules is described by the torsion of the central carbon double bond. Here, the isomerization is initiated by photoexcitation from the S 0ground state to S 1excited states, which proceeds to the transition state at a dihedral angle of nearly 90○, from where it decays to the ground state leading to cisand trans isomers. The S 2state, with a mini- mum at 90○, plays a key role in the process as well. The structures are reported in the supplementary material and obtained from astate-averaged CASSCF(2,2) relaxed surface scan. An accurate description of the electronic structure, however, requires an active space that comprises the entire conjugated πorbitals, i.e., CAS(14,14). On the singlet PES, this leads to 2.7 ×106CSFs, a cal- culation that is still close to the maximum of what is possible for an exact CASCI using standard hardware and hence a perfect oppor- tunity to study the timings of the various approximate approaches (Fig. 1). State-averaged CASSCF references over three singlet states are used for the subsequent NEVPT2 calculations. From Fig. 1, it is evident that the computational bottleneck is by far the construc- tion of the fourth-order density matrices. The EPS approximation is the most time consuming scheme. With a threshold of 10−6, it nevertheless still halves the computational cost of the fourth-order PDM construction. With the exception of the EPS (10−4) results, the EPS approximation produces the most accurate NEVPT2 results of the three approximations (Table II). The savings offered by the PS scheme are moderate. With the usual conservative threshold of 10−10,the calculation time reduces from nine days to two days for a single processor job. Relaxing the threshold further to 10−8and 10−6leads to further significant savings that amount to 88% and 99% of the total calculation time. The CU(4) scheme offers even J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-10 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1. Breakdown of the total density matrix construction timings (in hours) for the lowest stilbene singlet states S0(92○) using various approximations computed with a single core of Intel(R) Core (TM) i7-6700HQ CPU (2.60 GHz). The CASCI and the construc- tion of first- and second-order RDM (CASCI +1RDM+2RDM) takes 0.4 h, the construction of third-order RDM (3RDM) takes 1.3 h, while the CU(4) approximation takes less than 0.1 h. greater computational savings of almost two orders of magnitude in the processing of the fourth-order density. However, the CU(4) results from Table III are less accurate than the PS(10−10) values with errors up to and exceeding 0.04 eV. They also vary widely in magnitude and size for different states and different geometries. The PS approximation is still somewhat accurate with a threshold of 10−8, but the 10−6results are unusable. In the calculations with EPS(10−4), PS(10−10), PS(10−8), PS(10−6), and CU(4) approxima- tions, intruder states are observed for all states and geometries in theSaand Sisubspaces. However, the intruder states do not always lead to inaccurate results, which is due to the fact that most intruder states interact only weakly with the reference through the respective numerators.74 Whether the errors discussed above are still acceptable may be considered to be a matter of taste. The errors in the excitation energies are summarized in Table III from which it is evident that with the exception of PS(10−6), all errors are still below 0.1 eV. From Table III, the CU(4) approximation emerges as a particu- larly attractive option since the errors are not exceeding 0.05 eV, and most of them are even below 0.02 eV. This might be considered satisfactory, given that our previous benchmark shows that the system error of NEVPT2 is estimated to be 0.24 eV for vertical excitation energies on organic molecules.22All results in Table III, except for PS(10−6) results, are within these accuracy margins. B. Adenine and naphthalene: Vertical excitation energies Schapiro et al. assessed the overall performance of the NEVPT2 approach for vertical excitation energies against the CC3 method, which is considered to be of benchmark quality.22A complete study of the approximated schemes on the full benchmark is outside the scope of this work. However, to further illustrate the effect of the EPS, PS, and CU approximations on vertical excita- tion energies, we instead picked two examples with larger active spaces from Thiel’s benchmark set, namely, adenine and naphtha- lene.20In the former, the CAS(18,13) is chosen such that all π,π∗ and nitrogen lone-pair orbitals are included. In case of naphthalene,the chosen CAS(10,10) consists of the complete π-conjugated system. In the state-averaged CASSCF procedure, the molecular orbitals are averaged over the singlet or triplet states separately. The absolute errors as well as the relative errors are given in Tables IV–VI. In all EPS calculations, there is not a single intruder state observed. The calculations with tight thresholds (10−8) are practi- cally indistinguishable from the reference results, and those with a threshold of 10−6still provide an accuracy of about 0.02 eV. This would be considered sufficient for almost all practical appli- cations. Even with a crude threshold of 10−4, the EPS approxima- tion still predicts reasonably accurate absolute energies for naph- thalene, with deviations of less than 0.04 eV from the exact results. With the EPS approximation, the absolute deviations for adenine are slightly larger, where the largest error is about 0.29 eV. The PS results, from the thresholds as 10−12, are devoid of intruder states. For calculations with PS(10−10) settings, the intruder states (negative denominators) only appear in the Sisubspace of ade- nine, while there are no intruder states for naphthalene. The errors in the absolute energies of adenine (Table IV) and naphthalene (Table V) have strong fluctuations for the PS approximation with a loose threshold 10−6and CU(4) approximations. With PS(10−6) or CU(4) settings, there are obvious intruder states for all states. Inter- estingly, the PS(10−8) produce very accurate results for naphthalene, while for adenine, the maximum absolute deviation is as large as 0.30 eV. The vertical excitation energies have remarkably small MAD and SD for all EPS and PS results with conservative cutoffs (Table VI). There is almost no error for excited state energies from the EPS(10−8) and PS(10−12) approximations. The accuracy of the CU(4) approximation is close to that of EPS(10−4) and PS(10−8) approximations, which, in turn, are more accurate than the PS(10−6) results. C. Excitation energies of [Cr(NH 3)6]3+ and [Cu(NH 3)4]2+ Ligand field splitting and ligand to metal charge transfer (LMCT) in transition metal complexes are often studied using MR J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-11 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp methods. In order to describe target states, besides d–d excitations on metal centers, important sigma bonding/anti-bonding partner orbitals from ligands have to be included in the active space as well. Following the double shell protocol introduced by Pierloot and co- workers,84more accurate results can be obtained by extending the active space with another set of dorbitals. The determination of active spaces for the two complexes follows these rules. As an example, the ligand field splitting of approximately octa- hedral [Cr(III)(NH 3)6]3+is studied using CAS(7,12) as the active space. The CASSCF orbitals are averaged over the nine doublet and ten quartet states, comprising the eight lowest ligand field states. The deviations of NEVPT2 using various approximated fourth-order density matrices with respect to the canonical ones are reported in Table VII. We note in passing that the CASSCF results per- fectly reproduce the state degeneracies that are expected for such an octahedral compound. However, the degeneracies are not retained for the NEVPT2 results due to the construction of the FOIS, a well-known shortcoming of the method.85,86From the canonical NEVPT2 results, the degeneracy is lifted by less than 0.01 eV. The approximated variants add further inaccuracies to the formally degenerate states. In the case of the CU(4), the largest artificial splitting is observed for the2Egstate with about 0.07 eV. Similar to the results given in Secs. III A and III B, the PS approxima- tion with the conservative threshold 10−12does not lead to any intruder states and produces accurate absolute energies. Intruder states are observed in all states for crude EPS, PS as well as the CU(4) approximations. However, the absolute deviations intro- duced by the intruders are very small. The largest error is less than 0.07 eV. A challenging application for MRPT is to compute ligand metal charge transfer (LMCT) states of transition metal complexes, where large active spaces are usually needed to get valid results. A well- studied example is the low-lying LMCT states of [Cu(NH 3)4]2+. To model the four lowest LMCT states and the five d–d excited states, an active space CAS(17,14) is adopted, including the double copper d shells as well as the relevant bonding partners of the dominant ligand character. Table VIII shows the absolute deviation of the EPS, PS, and CU approximations with respect to canonical NEVPT2 results. The EPS(10−8), EPS(10−6), PS(10−12), PS(10−10), and CU(4) results are in excellent agreement with the reference calculations, although there are still intruder states in these calculations. The excitation energies of the two complexes are given in Table IX. The EPS approximation is the most stable one. With a threshold of 10−6, the EPS approximation does reproduce the same results as canonical NEVPT2. For both complexes, the CU(4) approximation is much more accurate than that from PS(10−8) and PS(10−6) approximations for excitation energies (Table IX), especially for [Cu(NH 3)4]2+. The results from PS(10−8) are even worse than that from PS(10−6). This illustrates once more that the accuracy of the PS approximation is not monotonic as the Tps thresholds decrease. The CU(4) approximation delivers satisfactory results that are only slightly worse than that from PS(10−10) and EPS(10−4). D. Potential energy surfaces (PESs) of Cr 2and N 2 Scanning potential energy surfaces are excellent probes to detect intruder states since they manifest themselves in form ofhumps, spikes, and discontinuities. The N 2molecule, with its triple bond, is a typical candidate and was studied by Zgid et al. with cumu- lant approximated SC-NEVPT2. In the following, a CAS(10,8) is chosen for the N 2PES calculations, where the 2 s and 2p orbitals form the active space, with the cc-pVQZ basis set. No RI approxi- mation was used for the calculations reported in this subsection. The absolute and relative errors of N 2PESs with PS, CU(4), and EPS approximations are depicted in Fig. 2. With Tps≤10−10, there is almost no intruder state with the PS approximation, while using CU(4) and PS approximations with relaxed thresholds, intruder states (negative denominators) appear for all geometries. The CU approximate FIC-NEVPT2 results are consistent with the SC-NEVPT2 variant studied by Zgid et al.37Throughout the whole PES, the CU(4) calculations with the NEVPT2 Ansatz are even more erratic compared to the reported CU(4) SC-NEVPT2 results.37 Using the PS approximation with Tps=10−8, the PESs are almost as accurate as the exact results. Apparently, the intruder states, here, do not interact strongly with the reference wave functions. How- ever, with Tps=10−6, the PES produced by the PS approximation also becomes erratic. These unsatisfactory results are mainly due to poorly approximated Eμin the Sasubspace (the energy contribu- tion from the Sisubspace are very small in all results, less than 0.1 milli-hartree). Compared to the PS results, the EPS approximated PESs are much more accurate. Throughout the whole PES of the N 2 molecule, there are no intruder states. Even with the very crude Tps threshold, 10−4, the largest deviation throughout the whole PES is below 1.0 mEh. To qualitatively understand these results, a detailed comparison of Eμin the Sasubspace is performed in Sec. III E. The PES of the chromium dimer is a much-studied guinea pig for multireference methods. The bonding in the chromium dimer formally represents a sextuple bond, and hence, it is much more complex than the triple bond in the nitrogen molecule studied above. The dimer has been studied by Zgid et al. with CU(4) SC- NEVPT2 and by Yanai et al. with the CU(4)/CU(3,4) CASPT2.37,44 Interestingly, in the latter, the cumulant approximated CASPT2 has exceptionally large errors as compared to the N 2, where the errors were still acceptable. In calculations using the PS and CU approx- imated density matrices, the active space is chosen as CAS(12,12), which consists of the 3d and 4s orbitals of both Cr atoms, with the Wachters +f ANO basis set (Fig. 3). Slightly different from the results for N 2, with Tps=10−8, the PS approximation introduces errors along with the PES. The deviations from the PS(10−6) and CU(4) approximations are not acceptable. In the case of the Cr 2 molecule, intruder states are observed even with tight PS(10−12) settings, although both PS(10−10) and PS(10−12) provide accurate PESs. With Tps=10−6settings, the EPS produces a smooth PES. However, given the complex bonding of Cr atoms, the results with the larger Tps=10−4setting deteriorate. Here, the largest error is more than 6.0 mEh. Nevertheless, computing PESs, the results with the EPS approximation remain consistently more stable than the PS and CU approximations. E. Energy spectra of N 2inSasubspace As discussed in Subsection III D, intruder states are observed for the N 2molecule with PS(10−6), PS(10−8) as well as CU(4) approximations. However, the accuracy of PESs of these approxi- mations are different. The large deviations introduced by the CU(4) J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-12 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. N2dissociation curve for the ground state with the EPS, PS, and CU approximations: (a) absolute energies of the PS and CU approximations, (b) relative errors of the PS and CU approximations with respect to the canonical NEVPT2 results, and (c) relative errors of the EPS approximation with respect to the canonical NEVPT2 results. J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-13 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. Cr2dissociation curve for the ground state with the PS and CU approximations: (a) absolute energies of PS and CU approximations, (b) relative errors of PS and CU approximations with respect to the canonical NEVPT2 results, (c) relative errors of the EPS approximation with respect to the canonical NEVPT2 results. J. Chem. Phys. 154, 214111 (2021); doi: 10.1063/5.0051211 154, 214111-14 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and PS(10−6) approximations could not be explained simply by intruder states in the Sasubspace. As shown in Sec. II C, the approx- imate fourth-order density matrices only affect the denominators in Eq. (13), more specifically the Koopmans matrices of the Sa and Sisubspace. To qualitatively evaluate these approximations, it is interesting to study the Koopmans energies Eμthat enter the denominators. The Eμcomputed by Eq. (11) for two representative geometries, 1.1 Å (equilibrium) and 2.2 Å (bond-breaking), is shown in Fig. 4. FIG. 4. The Koopmans energies Eμ(in Hartree) of the Sasubspace using various approximated density matrices for the N 2molecule at (a) N–N bond =1.1 Å and (b) N–N bond =2.2 Å. The exact Koopmans eigenvalues are used as reference. The numbers in parentheses denote the number of negative ones, some of which are out of the plotted region.The consistency of these spectra with respect to the exact ones can be used to evaluate the accuracy of the EPS, PS, and CU approxima- tions. In the canonical NEVPT2 calculations throughout the whole N2PES, there is not a single negative Eμin the Sasubspace. This is also true for the EPS approximation in both calculations. Pro- vided that a rational active space and Tpscutoffs are chosen, the EPS scheme produces accurate density matrices for NEVPT2, which are more stable than the PS and CU approximation. Both PS and CU approximations can lead to negative Eμand consequently to intruder states problems in the approximated NEVPT2 procedure. It is obvi- ous from the inspection of Fig. 4 that the PS approximations produce more accurate Eμthan CU(4) for both cases. Interestingly, the results from PS(10−6) at the bond-breaking (strongly correlated) geometry are more accurate than those for the equilibrium one. In Fig. 4, the number of negative Eμvalues are given in the parentheses as well. In general, the energies of the doubly occupied and virtual MOs enter the NEVPT2 denominator as positive contributions. Thus, singular or negative denominators in Saand Sisubspaces can only occur in the presence of negative Koopmans energies. If approxi- mate density matrices are employed to compute Eμin the NEVPT2 formulation, negative Koopmans energies can, in principle, appear in all the FOIS subspaces. Note that the Sijabsubspace is an excep- tion as it does not involve the Koopmans matrices in the first place. The origin of these negative eigenvalues will be discussed in Paper II of this series. It is clearly seen that in Fig. 4, the results from the CU(4) approximation lead to much more severe problems than those of the PS approximation. Thus, the CU(4) approximation is far less stable than the PS approximation in the context of intruder states. IV. CONCLUSIONS In the present work, three approximations of the fourth-order density matrix have been examined. Our results show that the EPS approximation is the most robust. With a loose threshold Tps=10−6, it can deliver accurate enough results even for the challenging Cr 2 molecule. However, the EPS approximation will not reduce the scal- ing of the fourth-order density matrix constructions, which is dif- ficult to extend to large active space calculations directly. The PS approximation will produce reliable NEVPT2 results if conservative thresholds are used, e.g., Tps≤10−10. However, the computational savings brought about by the approximation are somewhat limited. The CU approximation for the fourth-order density matrix leads to much larger computational savings of up to two orders of magni- tude. However, it does not always lead to stable and occasionally even to disastrous NEVPT2 results. The poor accuracy observed with the PS approximation with crude thresholds and occasionally observed with the CU(4) approximation is not always due to con- ventional intruder states. Instead, the deterioration of the results from approximating the fourth-order RDM can be traced back to grossly inaccurate Koopmans energies Eμ. Our results demonstrate that the approximated Eμmay become very negative. This is unphys- ical and a sure sign of problematic behavior in the calculation. In Paper II of this series, the origin of intruder states associated with approximate RDMs in the NEVPT2 approach will be discussed and improved approximations, which avoid negative eigenvalues, will be reported. J. Chem. 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5.0049418.pdf

J. Chem. Phys. 154, 204304 (2021); https://doi.org/10.1063/5.0049418 154, 204304 © 2021 Author(s).Local vs global approaches to treat two equivalent methyl internal rotations and 14N nuclear quadrupole coupling of 2,5- dimethylpyrrole Cite as: J. Chem. Phys. 154, 204304 (2021); https://doi.org/10.1063/5.0049418 Submitted: 04 March 2021 . Accepted: 30 April 2021 . Published Online: 26 May 2021 Thuy Nguyen , Wolfgang Stahl , Ha Vinh Lam Nguyen , and Isabelle Kleiner ARTICLES YOU MAY BE INTERESTED IN Cubic aromaticity in ligand-stabilized doped Au superatoms The Journal of Chemical Physics 154, 204303 (2021); https://doi.org/10.1063/5.0050127 The effects of proton tunneling, 14N quadrupole coupling, and methyl internal rotations in the microwave spectrum of ethyl methyl amine The Journal of Chemical Physics 153, 184308 (2020); https://doi.org/10.1063/5.0025650 Photo-cycloreversion mechanism in diarylethenes revisited: A multireference quantum- chemical study at the ODM2/MRCI level The Journal of Chemical Physics 154, 204305 (2021); https://doi.org/10.1063/5.0045830The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Local vs global approaches to treat two equivalent methyl internal rotations and14N nuclear quadrupole coupling of 2,5-dimethylpyrrole Cite as: J. Chem. Phys. 154, 204304 (2021); doi: 10.1063/5.0049418 Submitted: 4 March 2021 •Accepted: 30 April 2021 • Published Online: 26 May 2021 Thuy Nguyen,1Wolfgang Stahl,2,a) Ha Vinh Lam Nguyen,1,3,b) and Isabelle Kleiner1 AFFILIATIONS 1Laboratoire Interuniversitaire des Systèmes Atmosphériques (LISA), CNRS UMR 7583, Université Paris-Est Créteil, Université de Paris, Institut Pierre Simon Laplace, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France 2Institute of Physical Chemistry, RWTH Aachen University, Landoltweg 2, 52074 Aachen, Germany 3Institut Universitaire de France (IUF), 1 rue Descartes, 75231 Paris Cedex 05, France a)Deceased. b)Author to whom correspondence should be addressed: lam.nguyen@lisa.ipsl.fr ABSTRACT The microwave spectrum of 2,5-dimethylpyrrole was recorded using a molecular jet Fourier transform microwave spectrometer oper- ating in the frequency range from 2 to 26.5 GHz. Only one stable conformer was observed as expected and confirmed by quantum chemical calculations carried out to complement the experimental analysis. The two equivalent methyl groups cause each rotational transition to split into four torsional species, which is combined with the quadrupole hyperfine splittings in the same order of mag- nitude arising from the14N nucleus. This results in a complicated spectrum feature. The spectral assignment was done separately for each torsional species. Two global fits were carried out using the XIAM code and the BELGI -C2v-2Tops -hyperfine code, a modified ver- sion of the BELGI -C2v-2Tops code, giving satisfactory root-mean-square deviations. The potential barriers to internal rotation of the two methyl groups were determined to be V3=317.208(16) cm−1. The molecular parameters were obtained with high accuracy, pro- viding all necessary ground state information for further investigations in higher frequency ranges and on excited torsional-vibrational states. Published under license by AIP Publishing. https://doi.org/10.1063/5.0049418 I. INTRODUCTION Besides the classic topics in structural chemistry,1large ampli- tude motions (LAMs), e.g., internal rotation,2ring puckering,3 and inversion motion,4form a very active area in microwave spectroscopy. The effect of a methyl top on the rotational spectrum is that each rotational transition exhibits a tor- sional fine structure caused by the interaction of the methyl internal and the overall rotation. This fine structure, consist- ing of A–E doublets, depends on the height of the poten- tial barrier hindering the internal rotation. If the barrier is infinite, the internal motion of the methyl group corresponds to simple harmonic oscillation, whereas if the barrier is very low, the internal motion corresponds to an essentially free rotation.In the presence of two methyl internal rotors, the fine struc- ture consists of quartets (two equivalent tops) or quintets (two inequivalent tops). Analyzing such microwave spectra is a rather complicated task. Fitting the spectrum with sufficient accuracy often requires effective Hamiltonians to be included in the model. Due to challenges in both spectral assignment and fit, only a limited number of two-top molecules have been investigated, as reviewed by Nguyen and Kleiner.2Unlike inequivalent two-top cases where the point group of the frame symmetry can be C 1or C s, in the cases of two equivalent methyl groups, due to the higher molecular symmetry requirement, e.g., C 2, C 2h, or C 2v, the num- ber of studies reported in the literature are even smaller. Some examples of equivalent two-top molecules are dimethylgermane,5 dimethylamine,6dimethylketene,7isobutylene,8propane,92,6- lutidine,10,11difluorodimethylsilane,122-bromopropane,13dimethyl J. Chem. Phys. 154, 204304 (2021); doi: 10.1063/5.0049418 154, 204304-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp sulfate,14acetone,15,16dimethyl ether,17diethyl ketone,18dimethyl sulfide,192,5-dimethylthiophene,202,5-dimethylfuran,21and 2,6- dimethylfluorobenzene.22The spectral analysis can be challenging in some cases like 2,6-lutidine,10,11that understanding the microwave spectrum still remains incomplete. In the present work, we stud- ied the microwave spectrum of 2,5-dimethylpyrrole (25DMP), a nitrogen-containing five-membered ring that contains two equiva- lent methyl groups and a nitrogen atom, with support from quantum chemical calculations. The investigation adds an important contri- bution to the limited number of studies on two equivalent methyl rotors. While high barriers result in rather small torsional splittings in the order of a few tens of kHz to a few MHz as found for 4- and 5-methylthiazole,23,24or 2-chloro-4-fluorotoluene,25in the case of low barriers, the splittings can be up to several GHz,26–28making the spectral assignment difficult. If the molecule contains a14N nucleus that causes a quadrupole hyperfine structure with splittings also from a few tens of kHz to 2 MHz in addition, the spectral anal- ysis is even challenging for high barrier cases, since the hyperfine patterns overlap with the torsional splittings.29To confirm that the assignments are correct, it is often useful to fit each torsional species separately. For 25DMP, we wrote a program called WS18 that takes into account the quadrupole coupling of one14N nucleus for this purpose. The effects of internal rotation arising from two methyl groups in combination with one weakly coupling14N quadrupole nucleus can be treated in a global fit with the program XIAM .30How- ever, XIAM has its weakness in reproducing the experimental data to measurement accuracy if the torsional barrier is low.31–33It is often discussed that a reason might be the neglection of interac- tions between different v tstates, but a study on m-methylanisole has proven this assumption to be false.34Recent studies on 4-methylacetophenone,353-fluorotoluene,36and m-methylanisole37 strongly suggest the limited number of parameters available in XIAM to be the main reason. The initial version of the BELGI -Csprogram38has been extended to a hyperfine version for one-top molecules, BELGI - Cs-hyperfine ,39and recently also for molecules with two inequiva- lent methyl tops, BELGI -Cs-2Tops -hyperfine .31In many cases, the BELGI code shows great advantages over XIAM in reducing the root-mean-square (rms) deviations.26,31,39–41To treat the microwave spectrum of 25DMP, we modified the BELGI -C2v-2Tops code that was used to fit the rotational spectrum of dimethyl sulfide, prop- erly treating molecules containing two equivalent methyl tops with C2vmolecular symmetry,19toBELGI -C2v-2Tops -hyperfine to con- sider the quadrupole coupling of one14N nucleus. The results of BELGI -C2v-2Tops -hyperfine will be compared with those of XIAM . II. THEORETICAL A. Quantum chemical calculations Quantum chemical calculations were performed at the MP2/cc- pVDZ level of theory using the GAMESS program.42The obtained rotational constants, the V3term of the methyl torsional potentials, and the14N nuclear quadrupole coupling constants (NQCCs) were used as initial values for assigning the spectrum of 25DMP. Optimizations of the molecular geometry of 25DMP at the MP2/cc-pVDZ level of theory yielded only one stable conformer FIG. 1. The molecular geometry of 25DMP calculated at the MP2/cc-pVDZ level of theory. The atom numbering and the principal axes of inertia a- and b- are given. with the geometry shown in Fig. 1. The dihedral angles α1=∠(C3, C2, C 6, H 13) and α2=∠(C4, C 5, C 7, H 16) are 0○at equilibrium. The nuclear coordinates in the principal axis orientation are given in Table SI in the supplementary material. In order to calculate the 14N NQCCs, the method of Bailey43was applied as it is known to give very reliable results.31,44–46The electric field gradient at the site of the 14N nucleus in 25DMP was calculated at the B3PW91/6-311 +G(d,p) level using the molecular geometry shown in Fig. 1. The calibration factor eQ/h =4.599 MHz/a.u. recommended for conjugated πsys- tems was used.44The resulting quadrupole coupling constants are χaa=1.3243, χbb=1.665, and χcc=−2.996 MHz. Due to the C 2v symmetry of the molecule, all off-diagonal elements are zero. A two-dimensional potential energy surface (2D-PES) was calculated at the MP2/cc-pVDZ level by varying the dihedral angles α1andα2to predict the barriers to internal rotation and to study the possible potential coupling effects between the two methyl rotors, as given in Fig. 2. The symmetry adapted Fourier coefficients describing the PES are collected in Table SII in the supplementary material. From the PES, we obtain a V3value of 281.9 cm−1for each top with a V6contribution of ∼42.6 cm−1. Moreover, the top–top potential coupling contributes with 10% to the value of the barrier to internal rotation, indicating a weak anti-gearing motion between the two tops, as will be discussed in detail in Sec. IV C. B. Molecular symmetry In 25DMP, two equivalent methyl groups are attached to a C 2v symmetric frame. Its molecular symmetry group is G 36with the well- known irreducible representations A 1, A2, A3, A4, E1, E2, E3, E4, and G and the character table given in Table SIII in the supplementary material.47In this work, we follow the labeling scheme (00), (01), (11), and (12) that naturally arises from the semi-direct product decomposition G 36=(CI 3⊗CI 3)⋉C2vas reported by Ezra48and dis- cussed for 2,5-dimethylthiophene by Van et al.20but with detailed intuitive interpretation. The label ( σ1,σ2)⋅X of the irreducible repre- sentations consists of two parts. The first part arises from the invari- ant subgroup of G 36, which is the direct product CI 3⊗CI 3of the two intrinsic (superscript I) C 3permutation groups of the internal rotors. Their irreducible representations ( σ1,σ2) with σi=0, 1,−1 J. Chem. Phys. 154, 204304 (2021); doi: 10.1063/5.0049418 154, 204304-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. The potential energy surface of 25DMP calculated at the MP2/cc-pVDZ level of theory obtained by varying the dihedral angles α1=(C3, C2, C6, H13) and α2=(C4, C5, C7, H16) in a 10○grid, corresponding to the rotations of the methyl groups. The energies (in percent) are color-coded. The global energy minimum is atE=−287.897 702 hartree (0%). represent the transformation properties of the C 3-adapted planar rotor wave functions ei(3k+σ)α,k∈Z, and the torsional angle α, according to the three species A, E a, and E bof the C 3group, respec- tively. For better readability, we use the number 2 in the symme- try labels for the σi=−1 irreducible representation. Under the C 2v frame symmetry, the direct product CI 3⊗CI 3decomposes into four orbits {(00)}, {(12), (21)}, {(11), (22)}, and {(01), (10), (02), (20)}. One representative of each orbit is chosen for the symmetry label. The second part, the label X, is formed by the irreducible represen- tations of the little co-group of a given ( σ1,σ2) orbit. Here, we will give an intuitive description of this little co-group. The expectation values of the angular momentum of a methyl group are zero for the σi=0 states. For σi=1 and−1 states, the angular momenta are non- zero with opposite signs. This can be imagined as no rotation for σi=0 and a clockwise or a counterclockwise rotation for σi=±1. In the case of the (00) species, none of the tops rotates. There is no displacement of the methyl groups, and in this state, the little co-group is C 2vwith its irreducible representations A 1, B1, A 2, and B2. Therefore, we have the G 36species (00) ⋅A1, (00)⋅B1, (00)⋅A2, and (00)⋅B2corresponding to the traditional A 1, A 2, A 3, and A 4 labels, respectively. For the (12) orbit, one top rotates clockwise, and the other one rotates counterclockwise. Looking at a snapshot at a given time, the point group of the frame symmetry is no longer C 2v but C ssymmetry. There is no longer an abmirror plane but only a bcmirror plane. From the irreducible representations A′and A′′of Cs, we obtain the G 36species (12) ⋅A′and (12)⋅A′′corresponding to E1and E 2, respectively. In the case of the (11) orbit, both tops rotate either clockwise or counterclockwise. Almost all the time, there is no longer any mirror plane in the molecule, but the C 2axis, coincid- ing with the baxis, remains. The little co-group of the (11) orbit is C2with the irreducible representations A and B. We obtain the G 36 irreducible representations (11) ⋅A and (11)⋅B, corresponding to E 3 and E 4. Finally, the (01) orbit describes the case where one top does not rotate and the other one rotates clockwise or counterclockwise. A snapshot of the molecule will not show any non-trivial symmetry elements. The little co-group is C 1with the species A, and the G 36 irreducible representation is therefore (01) ⋅A corresponding to G. A comparison of both labeling schemes shows that direct informationof the degeneracy of a species, i.e., the non-fold, double-fold, and fourfold degeneracy of A, E, and G, respectively, is lost. Instead, we obtain intuitive information on the behavior of the internal rotors for a given G 36irreducible representation. 25DMP has nine protons and one14N nucleus, from which 29⋅3=1536 spin functions arise. The selection rules along with the spin statistical weights are given in Table I. In comparison to 2,5-dimethylthiophene,20a factor of 6 was found because the NH group, replacing the sulfur atom in 2,5-dimethylthiophene, adds a proton with two spin states and a14N nucleus with three spin states, while the32S nucleus has only one spin state. Without taking the spin statistical weights into account, the inten- sities of all torsional components are the same (see, for example, Fig. 4 in Ref. 35). When the spin weight was considered, the calculated intensities of torsional components are in fairly good agreement with the observed intensities as depicted in Fig. 3. Only for the (12) species, which should be about half as intense as the (00) species, we found that the two lines feature almost the same intensity and have no reasonable explanation for this observation. C. The WS18 code The complexity of the spectra of 25DMP caused by two equiv- alent methyl groups and the presence of a14N nucleus makes the spectral assignment a challenging task. The splittings of the torsional species for some rotational transitions are within 1 MHz and over- lap with the hyperfine splittings of the14N nucleus. Using a global fit for assignments was very time-consuming and uncertain. From experiences in the literature, separately fitting the torsional species is a good alternative to check the assignments, as successfully done in the investigations in Refs. 22, 35, and 49. To take into account the quadrupole coupling of the14N nucleus, the WS18 code was written. The effective Hamiltonian used in our study is H=Hr+Hcd+Hop+Hnq, (1) with the pure rotational part Hr=AP2 z+B1 4(P2 ++P2 −+P+P−+P−P+) −C1 4(P2 ++P2 −−P+P−−P−P+), (2) the quartic centrifugal distortion terms Hcd=−ΔJP4−ΔJKP2P2 z−ΔKP4 z−δJP2(P2 ++P2 −) −δK1 2{P2 z,(P2 ++P2 −)}, (3) TABLE I. Spin statistical weights of torsional components of allowed transitions of 25DMP. ee↔oo eo ↔oe (00)⋅A1↔(00)⋅A2 216 (00) ⋅B1↔(00)⋅B2 168 (12)⋅A′↔(12)⋅A′′96 (12) ⋅A′↔(12)⋅A′′96 (11)⋅A↔(11)⋅A 120 (11) ⋅B↔(11)⋅B 72 (01)⋅A↔(01)⋅A 384 (01) ⋅A↔(01)⋅A 384 J. Chem. Phys. 154, 204304 (2021); doi: 10.1063/5.0049418 154, 204304-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3. The broadband scan and the simulated spectrum of the four torsional species (00) in red, (01) in green, (11) in blue, and (12) in purple of the rotational transition 4 22←413of 25DMP. the odd-order angular momentum terms Hop=qPz+r1 2(P++P−)+qJP2Pz+qKP3 z+⋅⋅⋅ , (4) and the nuclear quadrupole coupling Hamiltonian Hnq=V(2)⋅Q(2), χaa=2eQV(2) 0, χbb−χcc=√ 6eQ(V(2) 2+V(2) −2).(5) The Hamiltonian matrix is set up in the symmetric top basis. Only matrix elements diagonal in Jwere considered, leading to a matrix size of (2 J+1)(2J+1). Both real and complex matrix elements are allowed. The code can be used for fitting microwave spectra of molecules with any point group of the frame symmetry. Effective Hamiltonian terms can be added from the input file in WS18, similar to the aixPAM code written for treating the rotational spectrum of one-top molecules.34These terms are given as a sum of products of the fundamental operators P2,Pz, the step-up and step-down oper- ators P+=Px+iPy,P−=Px−iPy, which are coded as P2, Pz, P +, and P−, respectively. As an example, the operator rJ, multiplying 1 2(P2P++P2P−), is coded as rJ0.5P2P+ rJ0.5P2P−.(6) D. The BELGI -C2v-2Tops -hyperfine code In order to treat the LAMs originating from two methyl groups and a weak nuclear quadrupole coupling caused by a14N nucleus in the rotational spectrum of 25DMP, the BELGI -C2v-2Tops code19 was modified to BELGI -C2v-2Tops -hyperfine . The method is similar to that applied for 4,5-dimethylthiazole.31The only difference is the molecular point group that is C sin the case of 4,5-dimethylthiazole and C 2vin 25DMP. The Hamiltonian used in BELGI -C2v-2Tops -hyperfine is the same as in BELGI -C2v-2Tops ,19which is based on the Hamiltonianinitially introduced by Ohashi et al.49It is written in a slightly mod- ified Principal Axis Method (quasi-PAM). In the Hamilton oper- ator of Eq. (6) from Ref. 49, which we also used, the axis system (x, y, z) is chosen in such a way that quadratic cross terms in the angular momentum components Px,Py, and Pzare kept fixed to zero at lowest order, which is different from the traditional PAM Hamiltonian with internal rotation [see Eq. (7) of Ref. 49]. The relations between the parameters of our quasi-PAM and the tradi- tional PAM system are defined in Eqs. (9)–(12) of Ref. 49. Unlike in Ref. 49, we follow a two-step diagonalization of the matrix associated with the Hamiltonian. More detailed descriptions are referred to Refs. 19 and 50. The quadrupole energies for each rotational transition in a given torsional state caused by the14N nucleus are computed in the BELGI -C2v-2Tops -hyperfine code using the perturbation approach where the hyperfine energy expression is Eh f(I,J,F)=2f(I,J,F) J(J+1)[χaa⟨P2 z⟩+χbb⟨P2 x⟩−(χaa+χbb)⟨P2 y⟩ +χab⟨PzPx+PxPz⟩] (7) with the Casimir function f(I,J,F).51 To facilitate the calculations, the numerical expectation values of the quadratic angular momentum components ⟨P2 x⟩,⟨P2 y⟩,⟨P2 z⟩, and⟨PzPx+PxPz⟩are quantified and then transferred into Eq. (7). This method allows us to determine the structure of all hyperfine patterns and take the nuclear quadrupole coupling constants into account in a global fit that already treats the internal rotation of two equivalent methyl groups in BELGI -C2v-2Tops .19 III. EXPERIMENTAL A. Measurements The rotational spectra of 25DMP were measured with a molec- ular jet Fourier transform microwave spectrometer operating in the frequency range from 2.0 to 26.5 GHz.5225DMP was purchased from TCI Europe, Zwijndrecht, Belgium, with a stated purity of over 97% and was used without any further purification. A small piece of a pipe cleaner containing the substance was put inside a steel tube close to the nozzle. Helium was used as carrier gas and J. Chem. Phys. 154, 204304 (2021); doi: 10.1063/5.0049418 154, 204304-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp flowed at a backing pressure of ∼200 kPa over the substance. The mixture of helium and 25DMP was then expanded into the cavity. A broadband scan with a step width of 250 kHz was recorded in the frequency range from 10.6 to 12.5 GHz. Eventually, all spectra were recorded at high resolution where Doppler splittings are observed due to the coaxial arrangement between the molecular beam and the resonators. The measurement accuracy is ∼4 kHz for all lines. B. Spectral analysis Due to the C 2vsymmetry of 25DMP, we only expected one dipole moment component along the baxis, which is in agree- ment with the results from quantum chemical calculations stating that∣μa∣=0.00,∣μb∣=2.10, and ∣μc∣=0.00 D. Using the calculated rotational constants, the V3potential of the methyl groups, and angles between the principal aaxis and the internal rotor axes, a spectrum was predicted using the XIAM code.30We first neglected the nuclear quadrupole hyperfine structure and could identify the four torsional states (00), (01), (11), (12) of some intense transi- tions such as 2 12←101, 404←313, and 4 22←413by comparing the broadband scan and the predicted spectrum. Using those lines, a fit was carried out, and the fitted molecular parameters enabled us to predict and find further lines in the frequency range of the spectrometer. The hyperfine structures arising from the nuclear quadrupole coupling of the14N nucleus were observed for all torsional transitions. The assignment of the hyperfine patterns was first done separately for each torsional species using the WS18 code. Subsequently, all lines are fitted globally with the pro- grams XIAM and BELGI -C2v-2Tops -hyperfine . The frequency list along with the residuals obtained with XIAM ,BELGI -C2v-2Tops - hyperfine , and WS18 is given in Table SIV in the supplementary material.IV. RESULTS AND DISCUSSION A. Separate fits The four torsional species (00), (01), (11), and (12) were fitted separately with the parameters summarized in Table II. All sepa- rate fits yielded reasonable standard deviations that are close to the measurement accuracy, showing that all lines are assigned correctly. All geometry parameters in the separate fits are effective and do not present the real geometry of the molecule. B. Global fits The microwave spectrum of 25DMP was fitted globally with the programs XIAM and BELGI -C2v-2Tops -hyperfine with the results given in Table III. Both fits give satisfactory rms deviations. In Table III, all parameters are referred to the principal axis system. As theBELGI -C2v-2Tops -hyperfine program works in the quasi-PAM system, the rotational and the NQCCs need to be converted into the principal axis system for comparison. The complete set of fitted parameters using the BELGI -C2v-2Tops -hyperfine code is presented in Table IV. C. Discussion The different approaches used in the XIAM and BELGI -C2v- 2Tops -hyperfine codes lead to different values of fitted parameters, which are given in Tables III and IV, respectively. The rotational constants obtained from the BELGI -C2v-2Tops -hyperfine fit, after converting them to the principal axis system, agree well with those obtained from the XIAM program. The small deviations are prob- ably not only due to different terms included in the two fits car- ried out in different coordinate systems but also due to the conver- sion process. The values of the calculated rotational constants A,B, and Cdeviate by−1.6%,−0.86%, and −1.02%, respectively, to the TABLE II. Molecular parameters of all torsional species of 25DMP obtained from fits with WS18. Par.aUnit (00) (01) (11) (12) A MHz 6294.595 98(87) 6291.658 26(19) 6288.723 8(19) 6288.719 17(62) B MHz 2016.972 45(33) 2016.916 15(10) 2016.861 03(37) 2016.860 32(23) C MHz 1557.134 64(36) 1557.134 73(10) 1557.135 85(23) 1557.135 95(14) ΔJ kHz 0.125 8(82) 0.100 4(11) 0.103 4(41) 0.099 1(22) ΔJK kHz −0.152(39) −0.172 3(52) −0.235(23) −0.259(13) ΔK kHz 5.28(20) 3.913(23) 2.74(41) 2.30(12) δJ kHz 0.052 1(79) 0.030 1(61) 0.031 8(50) 0.0285(19) q MHz ⋅⋅⋅ 44.941 77(36) ⋅⋅⋅ 89.843 2(18) r MHz ⋅⋅⋅ 6.086(37) 12.311(76) ⋅⋅⋅ qJ kHz ⋅⋅⋅ − 1.240(13) ⋅⋅⋅ − 2.588(60) qK kHz ⋅⋅⋅ − 41.645(64) ⋅⋅⋅ − 8.452(45) χaa MHz 1.325 3(25) 1.324 3(15) 1.326 2(20) 1.330 1(16) χbb−χcc MHz 4.628 7(49) 4.633 9(25) 4.645 0(35) 4.633 1(29) Nb44 98 55 77 σckHz 4.1 2.5 2.4 2.1 aAll parameters refer to the principal axis system. Watson’s A reduction and Irrepresentation were used. Single standard errors in the unit of the last digits are given in parentheses. bNumber of hyperfine components. cRoot-mean-square deviation of the fit. J. Chem. Phys. 154, 204304 (2021); doi: 10.1063/5.0049418 154, 204304-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III. Molecular parameters of 25DMP in the principal axis system obtained with XIAM (FitXIAM ) and BELGI -C2v-2Tops -hyperfine (FitBELGI ) as well as values calculated at the MP2/cc-pVDZ level of theory. Parameters from the quasi-PAM system are converted into the PAM system using the conversion code introduced in Sec. 2.3 of Ref. 31. Operator Par.aUnit Fit XIAM FitBELGIbAb initioc P2 a A MHz 6290.618 62(82) 6290.69(32) 6192.8 P2 b B MHz 2016.959 64(16) 2016.96(17) 1999.7 P2 c C MHz 1557.073 34(14) 1557.063 40(11) 1541.2 −P4ΔJ kHz 0.101 0(16) 0.0964 −P2P2 a ΔJK kHz −0.205 8(96) −0.2100 −P4 a ΔK kHz 3.294(39) 3.3804 −2P2(P2 a−P2 c) δJ kHz 0.0284 0(99) 0.0289 −{P2 a,(P2 a−P2 c)} δK kHz ⋅⋅⋅ 0.0580 F0 GHz 158d160.238(72)e158 (pαi−ρPa)2F GHz 164.034 0e164.960 5f {(pα1−⃗ρ†⃗Pr1),(pα2−⃗ρ†⃗Pr2)} F12 GHz −5.4211e−5.74(12)e (1/2)[1−cos(3 α)] V3 cm−1317.208(16) 317.920 5(41) 281.9 FPapαiρ 0.0370e0.036 460(18)e {(pαi−⃗ρ†⃗Pri),P2 a} Dpi2K MHz 0.699 6(89) χaa MHz 1.327 2(21) 1.326 7(14) 1.324 3 χbb−χcc MHz 4.639 0(36) 4.638 8(31) 4.661 0 ∠(i1,a)gdeg 157.130(22) 156.955(24)e156.41 ∠(i1,b) deg 67.123(22) 66.955(24)e66.42 ∠(i2,a)gdeg 22.888(22) 23.045(24)e23.59 ∠(i2,b) deg 67.112(22) 113.045(24)e66.42 Nqh274 274 rmsikHz 4.8 3.4 aAll parameters refer to the principal axis system. Watson’s A reduction and Irrepresentation were used. bRotational and quadrupole coupling constants obtained by transformation from the quasi-PAM to the PAM system. cCalculated at the MP2/cc-pVDZ level. The centrifugal distortion constants were obtained by harmonic frequency calculations. dFixed to the calculated value. eDerived parameter in XIAM . For BELGI -C2v-2Tops -hyperfine , the fitted parameters are q1(=q2), which multiplies the operator Pzpα1(see Table IV). The value of the ρparameter can be obtained from q1=q2=−2F1ρ1zand r1=−r2=−2F1ρ1x.f12is a floated parameter, with the relation f12=−2F12(see text). fFixed value. In BELGI -C2v-2Tops -hyperfine ,F1(orf1in the notation of Ref. 49) =F2(orf2), multiplying the operators p2 α1=p2 α2. g∠(i,c)=90○for both rotors due to symmetry. hNumber of hyperfine components. iRoot-mean-square deviation of the fit. experimental ones obtained with XIAM . Since the rotational con- stants obtained from ab initio calculations are evaluated at equilib- rium geometry and the experimental constants refer to the vibra- tional ground state, an agreement better than 1% is hardly feasible. In the XIAM fit, four centrifugal distortion constants (except δK) are accurately determined, while only three of them ( ΔJ,ΔK, and ΔJK) are needed to model the microwave spectra of 25DMP with BELGI . The calculated values of centrifugal distortion constants are in very good agreement with the experimental ones (see Table III). To predict the V3potential and determine the contributions of higher order terms such as V6and V9in the Fourier expan- sion of the potential function, as well as to study the potential coupling between the methyl groups, three different sets of sym- metry adapted terms were used to parameterize the potential func- tions. For comparison, the contour plots drawn with the Fourier coefficients obtained from those three different parameterizations are given in Fig. 4. Due to symmetry, only a portion of each PES with the dihedral angles α1andα2ranging from −60○to 60○is necessary.The PES (a) in Fig. 4 is obtained after fitting the data points with a constant term and the Fourier term [cos(3 α1)+cos(3 α2)], with which all data points can be reproduced with a maximum devi- ation of 10% within the dynamic range. If we included, in addition, the Fourier terms [cos(6 α1)+cos(6 α2)] and [cos(9 α1)+cos(9 α2)], associating with V6and V9, respectively, as illustrated in the PES (b), the maximum deviation decreased to 2.8%, and the minimum in (b) is significantly broader than that in (a). The PES (c) represents one unit cell of the PES given in Fig. 2 where also potential cou- pling terms between the methyl groups are taken into account and the maximum deviation is 1.13%. The potential minimum is slightly distorted along the anti-diagonal, which can be interpreted as an anti-gearing motion of the methyl groups close to the minimum. The V6andV9contributions are ∼15% and 3%, respectively. Experimen- tal information on these contributions cannot be obtained from our spectrum containing only vibrational ground state rotational tran- sitions, which only yields an effective V3barrier for both rotors. The ab initio results show that the coupling between the two methyl groups is rather weak with contributions of 1.8% and 11% for Vcc J. Chem. Phys. 154, 204304 (2021); doi: 10.1063/5.0049418 154, 204304-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV. Spectroscopic constants of 25DMP in quasi-PAM obtained with the program BELGI -C2v-2Tops -hyperfine . OperatoraPar.bUnit Value P2 z A MHz 6736.22(17) P2 x B MHz 2024.68 9(11) P2 y C MHz 1557.063 40(11) −P4ΔJ kHz 0.039 2(17) −P4 z ΔK kHz 3.433(28) −P2P2 z ΔJK kHz −0.156 1(69) p2 1=p2 2 F1=F2 GHz 164.960 5c p1p2 f12 GHz 11.47(12) (1/2)[1−cos(3 α1)] V3,1 cm−1317.920 5(41) Pzp1 q1 GHz −12.332 9(45) Pxp1 r1 GHz −1.569 1(12) Pxp1P2r1J MHz −11.91(28) 2χaa MHz 2.653 3(29) 2χbb MHz 3.312 1(27) 2χcc MHz −5.965 4(29) Nqd274 rmsekHz 3.4 aAll parameters refer to the quasi-PAM system. Px,Py, and Pzare the components of the overall rotation angular momentum, p1andp2are the angular momentum of the first and the second top, and αiis the internal rotation angle. bThe relations between the parameters of top 1 and top 2 are V32=V31,V32J=V31J, q1=q2,r2=−r1, and r2J=−r1J. cFixed values. dNumber of hyperfine components. eRoot-mean-square deviation of the fit. and Vss, respectively. This is in agreement with our experimental finding that potential coupling terms were not needed in the fits. The details of symmetry adapted terms and their values are given in Table SII in the supplementary material. The values of V3are determined to be 317.208(16) and 317.920 5(41) cm−1with XIAM and BELGI , respectively. They are in fairly good agreement but not within their error. Qualitatively speaking, theV3value should not depend on the axis system (PAM or quasi- PAM) in use. Nevertheless, it depends on different sets of parameters floated in the fit, causing the slight difference between XIAM and BELGI . The V3potential calculated at the MP2/cc-pVDZ level of theory is 11.2% lower than the observed value, thus giving only the correct FIG. 5. Comparison of the barriers to internal rotation (in cm−1) of (a) 2-methylthiophene ( 1)53and 2,5-dimethylthiophene ( 2),20(b) 2-methylpyrrole (3)46and 2,5-dimethylpyrrole ( 4, this study), and (c) 2-methylfuran ( 5)54and 2,5-dimethylfuran ( 6).21For 2-methylthiophene, the uncertainty was not given in Ref. 53. order of magnitude. While comparing the value of the V3poten- tial obtained for 25DMP with that of 2-methylpyrrole46and those of other five-membered rings, as shown in Fig. 5, we found that for 2- and 2,5-methylated heterocyclic rings, the barrier to internal rota- tion is lowest in mono-methylated and di-methylated thiophenes (1,2), higher in substituted pyrroles ( 3,4), and highest in substi- tuted furans ( 5,6). The same holds true when comparing mono- methylated thiazoles, imidazoles, and oxazoles.55We also observe that the barrier is always larger in dimethylated heterocycles com- pared to the monomethylated ones. The lower the barrier, the more pronounced this effect. It should be noted that the correlation between the potential barrier V3and∠(i,a) is−1.000, showing that if the barrier of V3 increases, the value of ∠(i,a) will decrease simultaneously. In the BELGI approach, the angle ∠(i,a) is not available as a floatable parameter directly but we observed a correlation between V3and F1=F2. This problem has been solved in previous studies by fixing the values of F1and F2to those derived from the XIAM fit.31How- ever, in the case of 25DMP, the fit did not converge, and a slight FIG. 4. The PES of 25DMP calculated at the MP2/cc-pVDZ level parameter- ized using three different sets of symme- try adapted terms: (a) only the Fourier term [cos(3 α1)+cos(3 α2)] was used; (b) the Fourier terms [cos(6 α1)+cos(6 α2) and cos(9 α1)+cos(9 α2)] are included in addition; and (c) potential coupling terms between the methyl groups are also taken into account. J. Chem. Phys. 154, 204304 (2021); doi: 10.1063/5.0049418 154, 204304-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp change in Fleads to a large different in V3even though the rms remains the same. We then tried several values of Fstarting from the value derived with XIAM in steps of 0.05. The value of F1=F2 finally chosen yielded a nicely converged fit. The XIAM code is based on the rigid top–rigid frame model in which parameters related with the molecular geometry such as ∠(i,a),∠(i,b), and F12are available as derived parameters. F12can be floated, but it is not recommended since it would lead to the dis- tortion of the molecule. In contrast, the BELGI program employed a different approach in which the angles ∠(i,a) and∠(i,b) are not directly involved. Instead, several parameters related to those angles ( q1,r1,f12) can be floated, as for the two-inequivalent- top Hamiltonian introduced by Ohashi et al.49As given in the footnote of Table III, F1=F2was kept fixed, and the constants q1and r1are floated. It is clear in Table IV from the operators they multiply that q1and r 1are essentially −2F1ρ1aand−2F1ρ1b,50 respectively, with similar definitions for top 2. In the present study, for two equivalent tops, the following parameters were set to be equal: F1=F2,V3,1=V3,2,q1=q2. (8) To be consistent with the convention used in Jabri et al. ,19r2was set to be equal but with opposite sign to r1, r2=−r1. (9) The magnitude and the signs of those parameters can be derived from Eq. (11) of Ref. 49. It should be noted that the angle θabout the yaxis relating the a,b,cprincipal axes to the x,y,zrho- axes is zero in the case of two equivalent tops. Due to this choice of sign from r2=−r1, we also have the relation f12=−2F12. In our previous study on 2-methylpyrrole, the methylation effect arising from one methyl group attached to the pyrrole ring on NQCCs was found to be negligible.46In the present study, two methyl groups with similar functionalities are involved, rais- ing an interesting question whether they possess more significant influence than that of one methyl group. Since the coupling con- stants obtained in the principal axes of inertia cannot be used directly for comparison, we attempt to compare their values in the principal axes of the field gradient at the nitrogen nucleus in each molecule. For molecules having a plane of symmetry, the principal axis of inertia and a principal axis of the field gradient are collinear with the axis perpendicular to the plane. Assuming that the caxis of the inertial moment coincides with the princi- pal axis yof the field gradient, the NQCCs in the inertia χaaand χbband those in the field gradient χxxandχzzare related by the expression ⎛ ⎜ ⎝χaa χbb⎞ ⎟ ⎠=⎛ ⎜ ⎝sin2θza cos2θzacos2θza sin2θza⎞ ⎟ ⎠⋅⎛ ⎜ ⎝χxx χzz⎞ ⎟ ⎠, (10) where θzais the angle between the zaxis and the aaxis, as illustrated in Fig. 6. Because 25DMP has C 2vsymmetry, the principal axes of the field gradient at the nitrogen nucleus are the same as the principal axes of inertia. Therefore, θza=0○,χxx=χaa=1.3272 MHz, and χzz=χbb=1.6559 MHz. For 2-methylpyrrole,46using the experi- mentally deduced NQCCs χaa=1.3345 MHz and χbb=1.5127 MHz FIG. 6. The principal axes of inertia of the molecule and the principal axes of the field gradient at the nitrogen atom of 2-methylpyrrole.46 together with the calculated value of χab=0.0908 MHz, the angle θzais determined to be 81.4○. Consequently, χxxand χzzare 1.2964 MHz and 1.5508 MHz, respectively. Since the two molecules both have a plane of symmetry, the χyy=χcc and χyyvalues are −2.9831 and −2.8472 MHz for 25DMP and 2-methylpyrrole,46respectively. The results show that the second methyl group adjoining to the nitrogen atom in 25DMP increases the NQCC values, especially χzz, of the nitrogen atom. Nevertheless, the effect of methylations on the pyrrole ring is still negligible, confirming the results found in Ref. 46. V. CONCLUSION The microwave spectrum of 25DMP with four torsional species arising from two equivalent methyl internal rotations in combina- tion with the presence of quadrupole hyperfine structures originat- ing from a14N nucleus was recorded under molecular jet conditions and successfully assigned with support of quantum chemical calcula- tions. The WS18 code, a local approach where the different torsional species were fitted separately, enables us to verify the correctness of the assignment. The BELGI -C2v-2Tops -hyperfine version of the BELGI code and the program XIAM offer proper tools for a global fit of molecules featuring a C 2vsymmetry and a weakly coupling quadrupole nucleus, yielding highly accurate molecular parameters and NQCCs. Comparing 25DMP with the mono-methylated deriva- tive 2-methylpyrrole, we conclude that the second methyl group adjoining to the nitrogen atom in 25DMP does not significantly change the field gradient around the nitrogen atom but slightly increases the NQCC values. SUPPLEMENTARY MATERIAL See the supplementary material for the Cartesian coordinates, coefficients of the PESs, G 36character table with the adapted spin weight for 25DMP, and frequency list. J. Chem. Phys. 154, 204304 (2021); doi: 10.1063/5.0049418 154, 204304-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ACKNOWLEDGMENTS T.N. thanks the Université de Paris for a Ph.D. fellowship. This work was supported by the Agence Nationale de la Recherche ANR (Project ID ANR-18-CE29-0011) and partly supported by the Programme National Physique et Chimie du Milieu Intestel- laire (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES. 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5.0049577.pdf

AIP Advances 11, 055020 (2021); https://doi.org/10.1063/5.0049577 11, 055020 © 2021 Author(s).Planar Hall effect in c-axis textured films of Bi85Sb15 topological insulator Cite as: AIP Advances 11, 055020 (2021); https://doi.org/10.1063/5.0049577 Submitted: 06 March 2021 . Accepted: 03 May 2021 . Published Online: 14 May 2021 Ramesh C. Budhani , Joshua S. Higgins , Deandre McAlmont , and Johnpierre Paglione ARTICLES YOU MAY BE INTERESTED IN Boosting spintronics with superconductivity APL Materials 9, 050703 (2021); https://doi.org/10.1063/5.0048904 Field-free magnetization switching induced by the unconventional spin–orbit torque from WTe 2 APL Materials 9, 051114 (2021); https://doi.org/10.1063/5.0048926 Strongly heat-assisted spin–orbit torque switching of a ferrimagnetic insulator APL Materials 9, 051117 (2021); https://doi.org/10.1063/5.0049103AIP Advances ARTICLE scitation.org/journal/adv Planar Hall effect in c-axis textured films of Bi 85Sb15topological insulator Cite as: AIP Advances 11, 055020 (2021); doi: 10.1063/5.0049577 Submitted: 6 March 2021 •Accepted: 3 May 2021 • Published Online: 14 May 2021 Ramesh C. Budhani,1,2,a) Joshua S. Higgins,2 Deandre McAlmont,1and Johnpierre Paglione2 AFFILIATIONS 1Department of Physics, Morgan State University, Baltimore, Maryland 21251, USA 2Maryland Quantum Materials Center and Department of Physics, University of Maryland, College Park, Maryland 20742, USA a)Author to whom correspondence should be addressed: ramesh.budhani@morgan.edu ABSTRACT Measurements of the planar Hall effect (PHE) and anisotropic magnetoresistance (AMR) in polycrystalline films of topological insulator Bi85Sb15are reported. The observation of PHE and AMR in these films of carrier density ≈2×1019electrons/cm3is like the behavior of in-plane field transport in thin films of metallic ferromagnets. However, the amplitudes of PHE ( ΔρPHE) and AMR ( Δρxx) are at variance. ΔρPHEandΔρxxalso undergo a sign reversal near ≈160 K. We compare these results with the reported PHE of topological insulators and Weyl semimetals and discuss possible scenarios for anisotropic backscattering of charge carriers in this non-magnetic alloy. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0049577 I. INTRODUCTION Bismuth–antimony alloys (Bi 1−xSbx) are well-known thermo- electric (TE) materials.1,2Their TE characteristics emanate from a tunable electronic band structure achieved by adjusting the Bi/Sb ratio in the alloy. This material has attracted much attention in recent years on the recognition of a strong spin–orbit interaction (SOI) driven band crossing in the composition range of 0.03 <x <0.22.3,4For x≈0.03, it acquires a Dirac-like metallic state, which changes to a 3D Weyl semimetal on application of a magnetic field, with signatures of the chiral anomaly in longitudinal magnetore- sistance (LMR).5For 0.09<x<0.22, (Bi 1−xSbx) is a 3D topolog- ical insulator (TI), as established by angle resolved photoemission measurements on single crystals6and epitaxial thin films.7Elec- tronic transport measurements on such crystals are characterized by a metal-like resistivity at low temperatures and the presence of weak Shubnikov–de Haas oscillations in the magnetic field depen- dence of longitudinal ( ρxx) and Hall ( ρxy) resistivity.8These features of electronic transport have been attributed to spin–momentum locked surface states. However, counter arguments suggesting subtle changes in bulk conduction at lower temperatures due to improved coherence and effectiveness of inadvertent doping have been given as well. The low effective mass of charge carriers and large dielec- tric function of Bi 1−xSbxmake the impurity conduction dominantat low temperatures.9The topological phase of (Bi 1−xSbx) has been identified as an excellent spin–orbit torque (SOT) material for spin- tronic applications.10–13In epitaxial bilayers of BiSb and a ferromag- net (FM) like MnGa, the spin–momentum locked surface states of the former pump a large spin current into the FM layer under the action of a charge current driven Rashba–Edelstein effect (REE).14 The REE torque on the magnetization of the FM layer has been established to be much larger than the spin Hall effect driven torque of a heavy metal like Pt. Interestingly, two recent studies10,15have indicated that the polycrystalline films of Bi 1−xSbxmade by a scalable process like sputtering are quite effective in producing spin currents to torque the magnetization of FeMn, FeCoB, and CoTb thin films. These observations have motivated us to undertake a detailed study of electronic transport in sputter-deposited polycrystalline films of BiSb. Although polycrystalline films of (Bi 1−xSbx) alloys have been studied previously by a number of groups,16–18the focus of those studies has been their applicability as a thermoelectric material. Our objective here is to compare the low temperature (T ≥2 K) magneto- transport in polycrystalline films of Bi 85Sb15with that of epitaxial films and single crystals where the existence of a topological phase has been established. We also seek to find the existence of the pla- nar Hall effect (PHE) and anisotropic magnetoresistance (AMR), which have been seen earlier in several non-magnetic topological insulators19–24and the Dirac/Weyl family of semimetals.25–28 AIP Advances 11, 055020 (2021); doi: 10.1063/5.0049577 11, 055020-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv II. EXPERIMENTAL DETAILS Bi85Sb15films were deposited on thermally oxidized silicon wafers by magnetron sputtering of a stoichiometric 2-in. diameter alloy target in a laser deposition/sputtering hybrid load-lock cham- ber with a base pressure of ∼7×10−8Torr. The Bi 85Sb15alloy has a low melting temperature (T m∼300○C)29and high sputter- ing yield.30The sputter gun was operated at low power ( ≈25 W) to avoid surface melting of the target and to ensure a low growth rate (∼0.15 nm/s). Films were deposited at ambient temperature and at 100 and 150○C. It was noticed that the higher deposition temper- ature and excessive growth rates result in rough films. This is a perennial issue with the growth of thin films of low melting point alloys and elements.30The crystallographic structure of these films has been evaluated with x-ray diffraction. For measurements of magneto-transport, films were deposited through a shadow mask in a Hall bar geometry with the bar dimensions of 300 ×3000 μm2. Transport measurements were carried out in a physical prop- erty measurement system in the temperature and field ranges of 2–300 K and 0 to ±9 T, respectively. The use of a vertical sam- ple rotator allowed in-plane and out-of-plane rotation of the film for measurements of anisotropic magnetoresistance and planar Hall effect. III. RESULTS The binary equilibrium phase diagram of (Bi 1−xSbx) shows complete solubility of the two elements for all values of x, leading to a single-phase material of rhombohedral structure.31However, due to the low melting points of Bi and Sb, the growth or anneal- ing of BiSb films at T >TLiquidus may result in phase separation; therefore, we have deposited the films at only T <150○C. Figure 1 compares the Θ–2Θx-ray diffraction profiles of the two films grown at 35 and 100○C. The x-ray profile of the film deposited at 35○C shows diffraction peaks corresponding to several allowed hkl-indices with (001) reflections being most prominent. The higher intensity of such reflections suggests a predominantly c-axis oriented growth along the c-axis of the rhombohedral cell. This preferential growth FIG. 1. X-ray diffraction profile of the Bi 85Sb15thin film deposited on thermally oxidized silicon at 35 and 100○C. The diffraction peaks have been indexed for the Miller indices of the rhombohedral structure of the BiSb alloy. A preferred orientation of the film along the c-axis is evident in these data.becomes prominent at 100○C as indicated by the suppression of the intensity of reflections corresponding to non-zero values of h and k indices. These observations are consistent with the results of Rochford et al.16on Bi 80Sb20films deposited on thermally oxi- dized silicon by cosputtering of elemental targets. Films of Bi 85Sb15 deposited by radio frequency sputtering of the alloy target on (0001) sapphire reveal a preferential c-axis growth due to a better c-plane lattice match between sapphire and BiSb.31 Here, we focus on electron transport in the BiSb films deposited at≈35○C. Ambient temperature growth of the TI film is preferred when it is deposited on amorphous ferromagnets like FeGaB, FeCoB, and Fe–Gd alloys to avoid their crystallization. The inset of Fig. 2 shows the zero-field longitudinal resistivity of the film between 2 and 300 K. The resistivity first rises on lowering the temperature down to∼180 K, and then, this rise tapers off, leading to a resistivity of ∼2.0 mΩcm at 2 K. This behavior of resistivity is comparable to that reported by Fan et al.31for sputter-deposited films on (0001) sapphire. It is also worth comparing the resistivity of these films with those made by molecular beam epitaxy. The data of Cho et al.2for films grown by MBE on CdTe crystals reveal a resistivity of ∼0.2 mΩ cm at 300 K, which rises to ∼2.5 mΩcm at 2 K. While the resistiv- ity ratioρ(2 K)/ρ(300 K) of the MBE grown films is the same ( ∼1.2) as that of the sputter-deposited films reported here, the temperature dependence of resistivity is strikingly different in the two cases. The resistivity of sputtered polycrystalline films first rises and then tapers off, while for the MBE films,2the rise is faster at lower temperatures. Interestingly, the ρxx(2 K)/ρxx(300 K) ratio for Bi 1−xSbxsingle crys- tals with x =0.09 is≈1.8,8with aρxx(300 K) of ∼0.16 mΩcm. The temperature dependence of ρxxis similar to that of the sputtered film, barring some signatures of a metallic conduction at T <50 K, which is presumably due to well-defined conducting surface states in single crystal samples. The main panel of Fig. 2 shows the Hall resistivity ( ρxy) of the ambient temperature deposited film measured between 2 and 20 K as a function of magnetic field. ρxyis linear in field for μ0H≤3 T and does not show any temperature dependence. In the frame- work of a simple Drude model, it yields a carrier density of 1.97 ×1019electrons/cm3and carrier mobility of ∼203 cm2V−1s−1. The slight upward curvature of ρxyvs H at the higher fields suggests a FIG. 2. Hall resistivity ( ρxy) of the Bi 85Sb15film measured between 2 and 20 K as a function of magnetic field. The linear portion of ρxyvsμ0H has been used to calcu- late the carrier concentration. The Hall resistivity shows a very weak temperature dependence at high fields. The inset of the figure shows the resistivity ( ρxx) of the film in zero-field measured between 2 and 300 K. AIP Advances 11, 055020 (2021); doi: 10.1063/5.0049577 11, 055020-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv two-carrier scenario for electron transport in this system. The high field data also reveal a very weak temperature dependence in the temperature range of 2–20 K. While there is a lack of carrier con- centration and mobility data on epitaxial and polycrystalline films of Bi85Sb15, the carrier concentration in our films is ∼2 orders of mag- nitude higher than that reported by Taskin and Ando8for x=0.09 BiSb single crystals and the carrier mobility of these films is smaller by the same factor. The measurements of the electrical resistivity of 3D topologi- cal insulators and Dirac/Weyl semimetals in a configuration where the current density, magnetic field, and induced electric field are in the same plane have generated considerable interest due to the pres- ence of anisotropic magnetoresistance and planar Hall effect,19–28 which have traditionally been the signatures of electronic transport in a magnetically ordered metal.32We carried out the measure- ments of the electrical resistivity of BiSb thin films in a field-in-plane geometry. Here, the transport current J x(=25 A/cm2) flows in the x direction, and the induced electric fields E xxand E xyin the x and y directions, respectively, are measured as the magnetic field is rotated in the xy plane from −90○to 270○. The angle ϕ=0 corresponds to the situation where H and J xare parallel. The induced electric field in the direction of J xyields magnetoresistivity, whereas the orthogonal field E xyresults in the planar Hall effect. The components of the resistivity tensor are expressed as20,32 ρxx=ρ/⊙◇⊞−Δρcos2ϕ, (1) ρPHE=−Δρsinϕcosϕ, (2) whereΔρ=ρ/⊙◇⊞−ρ†, withρ/⊙◇⊞andρ†corresponding to H perpendic- ular to J xand H parallel to J x, respectively. The anisotropic magne- toresistance ( Δρ) andρPHEare characteristic features of spin–orbit scattering dominated electronic transport in magnetic alloys due tocoexisting s and d bands near the Fermi energy.32Interestingly, how- ever, PHE and anisotropic magnetoresistance have been found in non-magnetic 3D topological insulators like Bi 2Te3and Bi 2Se3,19–24 including the observation of a nonlinear response at higher cur- rent densities in epitaxial films of Bi 2Se3, which depends on the direction of current with respect to the crystal axis of the mono- layer.23,24Appreciable values of AMR and PHE have been observed in Dirac and Weyl semimetals as well.25–28While in the latter class of semimetals, this anisotropic transport has been attributed to break- ing of chiral symmetry, which results in a large negative longitu- dinal magnetoresistance (NLMR), varied interpretations have been proposed for PHE in 3D TIs where no NLMR is seen. Bi1−xSbxis one of the first reported 3D TIs. While the measure- ments of the Hall resistivity and orbital MR in single crystals and epi- taxial films of Bi 1−xSbxhave been reported earlier,6,8,9data on PHE and AMR are lacking. Figure 3(a) shows the variation of ρxxandρxy at 2 K and+9 T as the sample is rotated to change the angle between magnetic field and current density directions from −90○to 270○. A similar measurement of ρxxandρxyat 2 K with the field direction reversed is shown in Fig. 3(b). We note that the positions of extrema inρxxandρxydata of Figs. 3(a) and 3(b) are consistent with Eqs. (1) and (2). However, there is a noticeable asymmetry in the behavior ofρxyfor the two field orientations. There are two factors that con- tribute to this asymmetry. First is a normal Hall voltage that results from a non-zero out-of-plane component of the magnetic field due to a misalignment of the film plane and the plane of rotation. This contribution is antisymmetric in field and can be eliminated on sym- metrization of the ( +H) and (−H) data. Second is a zero-field mis- alignment voltage across the Hall contacts, which will add to the PHE voltage on symmetrization of ρxy[{(ρxy(+H)+ρxy(−H)}/2]. This constant resistance can be subtracted from the measured ρxy(H) provided its value is small such that the AMR induced change in it is insignificant compared to the true ρxy(H). In addition to these two factors, another contaminant of ρxycomes from the orbital FIG. 3. The variation of ρxxandρxyas a function of the angle ( ϕ) between the direction of a 9 T field and the direction of current density J x. (a)+9 T, (b) −9 T, and (c)ρxxandρxyon symmetrization of the data shown in (a) and (b). The solid lines in (c) are fits to the equation of the typeρxx(ϕ)=ρxx(0)−Δρxxcos2ϕand ρxy(ϕ)=ρxy(0)−Δρxysinϕcosϕ+B cos2(ϕ) to theρxxandρxydata, respec- tively. (d) shows the variation of antisym- metricρxyextracted from (a) and (b) and the variation of coefficient B extracted from the fit in (c). AIP Advances 11, 055020 (2021); doi: 10.1063/5.0049577 11, 055020-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv magnetoresistance (OMR) of the misaligned section of the trans- verse contacts when the sample is tilted with respect to the plane of rotation. This would add a cos2ϕterm in Eq. (2). The misalignment of the plane of rotation and the plane of the sample adds an error in the value of AMR as well. If the OMR of the sample is large, then the normal component of the field will add an OMR contribution toρ/⊙◇⊞xxandρ//xx. We address these errors by first symmetrizing the ρxx(ϕ) andρxy(ϕ) data of Figs. 3(a) and 3(b). The result of this pro- cedure is displayed in Fig. 3(c), along with the fits of ρxx(ϕ) to Eq. (1) and ofρxy(ϕ) to a function of the type ρxy(ϕ)=A+B sin(ϕ) cos (ϕ) +C cos2(ϕ). The last term of this equation considers the error in ρxydue to OMR of the misaligned section, as discussed earlier. The magnitude of this error [C cos2(ϕ)] and the antisymmetric contribu- tion toρxyare displayed in Fig. 3(d). Comparing the peak amplitude ofρxyin Fig. 3(c) and the peak amplitude of the C cos2(ϕ) term in Fig. 3(d) shows that the maximum error introduced by this term in the measurement of ρxyis≤±3.5%. Moreover, a comparison of FIG. 4. The angle (ϕ) dependence of ρPHEandρxxhas been measured at 2 K for several values of magnetic field. (a) and (b) show the symmetrized ρxyandρxx data as a function of angle.the peak value of antisymmetric ρxy(≈3μΩcm) at 9 T in Fig. 3(d) with the Hall resistance data shown in Fig. 2 linearized in field (slope ≈70μΩcm/T) yields a tilt angle of ≈0.6○with respect to the plane of the sample. The angular dependencies of the symmetrized ρxy andρxxat 2 K for several values of field are shown in Figs. 4(a) and 4(b), respectively. In the plot of Fig. 4(a), the constant offset value ofΔρxyat zero-field has been subtracted, whereas Fig. 4(b) displays the variation of ρxx(ϕ) with respect to the average value of ρxxat ϕ=±90○. The dominant sin( ϕ)⋅cos(ϕ) and cos2(ϕ) dependencies of these quantities are evident in Figs. 4(a) and 4(b), respectively. The variation of the peak amplitudes of ΔρPHEandΔρxx(ρxx//−ρxx/⊙◇⊞) extracted from these plots as a function of magnetic field is shown in Fig. 5. The two important conclusions that can be drawn from these data are: (1) the longitudinal magnetoresistance is positive with a field dependence of the type Δρxx∼Hα(α=1.5) and (2) the PHE amplitude is larger than Δρxx. In the inset of Fig. 5, we plot the variation of orbital magne- toresistance ( ΔROMR=[{R xx(H)−Rxx(0)}/R xx(0)]×100) with field normal to the film plane, longitudinal magnetoresistance (LMR) (ΔRLMR=[{R//xx(H)−Rxx(0)}/R xx(0)]×100) with field parallel (//) to J x, and anisotropic magnetoresistance ( ΔRAMR=[{R//xx(H) −R/⊙◇⊞xx(H)}/R/⊙◇⊞xx(H)]×100). Two noteworthy conclusions that can be drawn from these data are: (1) the LMR is larger than OMR and (2) the AMR is small but positive, indicating the absence of any chiral anomaly. The resistance tensor of a two-dimensional magnetic film for an in-plane magnetic field predicts that the amplitudes of ΔρPHEand Δρxxshould be the same. The large difference seen in the values ofΔρPHEandΔρxxin this case points toward some subtle differ- ences in the origin of these components in the resistance tensor. One might argue20that in systems with large orbital magnetoresistance, the out of film plane magnetic field due to sample misalignment may significantly change the value of Δρxx. An estimation of this error can be made by looking at the results of the orbital MR mea- surements shown in Fig. 5 and the estimated tilt of 0.6○from the antisymmetric ρxyin Fig. 3(d). This much tilt at 9 T would result FIG. 5. Amplitudes of ΔρPHE(ϕ) andΔρxx(ϕ) at 2 K plotted as a function of magnetic field. The inset shows the variation of out-of-plane and in-plane field magnetoresistance and anisotropic magnetoresistance as a function of applied field. AIP Advances 11, 055020 (2021); doi: 10.1063/5.0049577 11, 055020-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv in a≈95 mT perpendicular field. From Fig. 5, we conclude that the effect of this misalignment of field on the measurement of Δρxxis negligible. We have also considered the possibility of contamination of Δρxxby the large thermoelectric power of BiSb alloys.1,2This effect may get accentuated by the large distance ( ≈2000μm) between the Vxxpads compared to the distance ( ≈300μm) between V xypads if any thermal gradients across the length of the sample are produced by uneven cooling. However, this speculated thermoelectric contri- bution to the longitudinal voltage will lead to an asymmetry in V xxat ϕ=0○andϕ=180○, which we do not see. Clearly, the difference in the amplitudes of ΔρxxandΔρxyPHEdoes not appear to be a spurious effect emerging from any misalignment or thermal gradients. Figures 6(a) and 6(b), respectively, show the angular depen- dence ofΔρPHEandΔρxxat several temperatures in the range of 10–160 K. A significant drop in the amplitude of PHE and AMR is seen on increasing the temperature, followed by a sign change in the temperature window of 150–200 K. The amplitudes of ΔρPHEand FIG. 6. Angle (ϕ) dependence of ΔρPHEandΔρxxat several temperatures from 2 to 160 K is shown in (a) and (b), respectively. The data have been obtained by symmetrization of the plus and minus field response. FIG. 7. Variation of the PHE amplitude [ ΔρPHE(ϕ)] andΔρxx(ϕ) amplitude as a function of temperature. The inset of the figure shows d ρxx/dT vs T in zero-field. Δρxxare plotted in Fig. 7. The change in the sign of these two quan- tities appears to correlate with the inflection point in the temperature dependence of ρxx, as can be seen in the inset where we have plotted dρxx/dT vs T. IV. DISCUSSION The noteworthy features of the in-plane magnetoresistance of these Bi 85Sb15thin films are: (1) observation of a planar Hall effect, (2) a difference in the amplitudes of ΔρPHEandΔρxx, and (3) a sign reversal of these two quantities in the vicinity of 150 K. Although BiSb is non-magnetic, this first observation of PHE in BiSb is con- sistent with the recent reports of PHE in non-magnetic semimet- als like (Bi 1−xSbx)2Te3,20bismuth,31and MoTe 225,26with non-trivial band topology. The observation of PHE in (Bi 1−xSbx)2Te3has been attributed to scattering by magnetic impurities present in the sam- ple.20The anisotropy of the Fermi surface and the resulting large dif- ference in the magnetic field dependence of ρ/⊙◇⊞xxandρ//xxhave also been argued to be the source of PHE in some systems.26,33We have analyzed the field dependence of ρ/⊙◇⊞xxandρ//xxat 2 K. The resistance rises as∼Hα, withαas 1.46 and 1.51 for the /⊙◇⊞and // measurements, respectively. A treatment of electron transport in the framework of a semiclassical Boltzmann transport equation attributes PHE in topo- logical insulators to orbital magnetism of Bloch electrons, which is non-zero because of the symmetry breaking in-plane magnetic field.22,27This model, however, does not predict a difference in the value ofΔρPHEandΔρxx. The experimental data of Taskin et al.20 on the (Bi 1−xSbx)2Te3crystal show a large difference in ΔRPHEand ΔRxx, which the authors have attributed to a contamination of the signal by an OMR contribution arising from the misalignment of the sample plane and the plane of rotation. However, this scenario does not apply in the present case as we have shown that the difference inΔρPHEandΔρxxcannot be attributed to a tilt of the sample plane. Similarly, a change in the sign of planar Hall and anisotropic mag- netoresistance at higher temperatures is difficult to explain based on misalignment and/or impurity scattering. A plausible description of this difference in the amplitudes of ΔρPHEandΔρxxas well as of the sign change has been given by AIP Advances 11, 055020 (2021); doi: 10.1063/5.0049577 11, 055020-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv Zheng et al.28where they consider the tilt of Dirac cones in the TI induced by the in-plane magnetic field. This tilt contributes to anisotropic backscattering, which is enhanced further by impurity resonant states and may lead to a sign change in AMR. The change in the sign of AMR of our BiSb films is consistent with this picture. V. CONCLUSIONS In summary, we have addressed the behavior of longitudinal and Hall resistivities of highly oriented thin films of the Bi 85Sb15 topological insulator, which has been established as a superior spin torque material for spintronic applications.11–15The overall features of the out-of-plane magnetic field transport are comparable to ear- lier reports on MBE grown films. The in-plane field transport reveals a striking planar Hall effect whose magnitude is larger by a factor of≈2 as compared to the magnitude of AMR. Moreover, both PHE and AMR undergo a sign change on raising the sample temperature beyond∼150 K. These new features of the in-plane magnetic field transport presumably arise due to anisotropic scattering of Dirac electrons in a planar magnetic field. ACKNOWLEDGMENTS This research was supported by the Air Force Office of Sci- entific Research, Grant No. FA9550-19-1-0082 awarded to Mor- gan State University. J.P. acknowledges support from the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant No. GBMF9071, and J.S.H. acknowledges support from the National Institute of Standards and Technology Cooperative Agreement No. 70NANB17H301. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1W. M. Yim and A. Amith, “Bi ⋅Sb alloys for magneto-thermoelectric and thermo- magnetic cooling,” Solid. State. Electron. 15, 1141 (1972). 2S. Cho, A. DiVenere, G. K. Wong, J. B. Ketterson, and J. R. Meyer, Phys. Rev. B 59, 10691 (1999). 3J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B 78, 045426 (2008). 4H. Guo, K. Sugawara, A. Takayama, S. Souma, T. Sato, N. Satoh, A. Ohnishi, M. Kitaura, M. Sasaki, Q.-K. Xue, and T. Takahashi, Phys. Rev. B 83, 201104(R) (2011). 5D. Shin, Y. Lee, M. Sasaki, Y. H. Jeong, F. Weickert, J. B. Betts, H.-J. Kim, K.-S. Kim, and J. Kim, Nat. Mater. 16, 1096 (2017).6A. Nishide, A. A. Taskin, T. Yasuo, T. Okuda, A. Kakizaki, T. Hirahara, N. Kan, F. Komori, Y. Ando, and I. Matsuda, Phys. Rev. B 81, 041309(R) (2010). 7T. Hirahara, Y. Sakamoto, Y. Saisyu, H. Miyazaki, S. Kimura, T. Okuda, I. Matsuda, S. Murakami, and S. Hasegawa, Phys. Rev. B 81, 165422 (2010). 8A. A. Taskin and Y. Ando, Phys. Rev. B 80, 085303 (2009). 9D. Minh Vu, W. Shon, J.-S. Rhyee, M. Sasaki, A. Ohnishi, K.-S. Kim, and H.-J. Kim, Phys. Rev. B 100, 125162 (2019). 10Z. Chi, Y.-C. Lau, X. Xu, T. Ohkubo, K. Hono, and M. Hayashi, Sci. Adv. 6, 2324 (2020). 11N. H. Duy Khang, Y. Ueda, K. Yao, and P. N. Hai, J. Appl. Phys. 122, 143903 (2017). 12N. H. Duy Khang, Y. Ueda, and P. Nam Hai, Nat. Mater. 17, 808 (2018). 13N. Roschewsky, E. S. Walker, P. Gowtham, S. Muschinske, F. Hellman, R. B. Seth, and S. Salahuddin, Phys. Rev. B 99, 195103 (2019). 14F. Hellman et al. , Rev. Mod. Phys. 89, 025006 (2017). 15N. H. Duy Khang, S. Nakano, T. Shirokura, Y. Miyamoto, and P. Nam Hai, Sci. Rep. 10, 12185 (2020). 16C. Rochford, D. L. Medlin, K. J. Erickson, and M. P. Siegal, APL Mater. 3, 126106 (2015). 17R. C. Mallik and V. D. Das, J. Appl. Phys. 98, 023710 (2005). 18T. Massana, C. N. Afonso, A. K. Petford-long, and R. C. Doole, Thin Solid Films 288, 186 (1996). 19S. Wiedmann, A. Jost, B. Fauque, J. van Dijk, M. J. Meijer, T. Khouri, P. Oezzini, S. Grauer, S. Schreyeck, C. Brune, H. Buhmann, L. W. Molenkamp, and N. E. Hussey, Phys. Rev. B 94, 081302 (2016). 20A. A. Taskin, H. F. Legg, F. Yang, S. Sasaki, Y. Kanai, K. Matsumoto, A. Rosch, and Y. Ando, Nat. Commun. 8, 1340 (2017). 21A. A. Burkov, Phys. Rev. B 96, 041110 (2017). 22S. Nandy, A. Taraphder, and S. Tewari, Sci. Rep. 8, 14983 (2018). 23P. He, S. S.-L. Zhang, D. Zhu, Y. Liu, Y. Wang, J. Yu, G. Vignale, and H. Yang, Nat. Phys. 14, 495 (2018). 24P. He, S. S.-L. Zhang, D. Zhu, S. Shi, O. G. Heinonen, G. Vignale, and H. Yang, Phys. Rev. Lett. 123, 016801 (2019). 25F. C. Chen, X. Luo, J. Yan, Y. Sun, H. Y. Lv, W. J. Lu, C. Y. Xi, P. Tong, Z. G. Sheng, X. B. Zhu, W. H. Song, and Y. P. Sun, Phys. Rev. B 98, 041114(R) (2018). 26D. D. Liang, Y. J. Wang, W. L. Zhen, J. Yang, S. R. Weng, X. Yan, Y. Y. Han, W. Tong, W. K. Zhu, L. Pi, and C. J. Zhang, AIP Adv. 9, 055015 (2019). 27S. Nandy, G. Sharma, A. Taraphder, and S. Tewari, Phys. Rev. Lett. 119, 176804 (2017). 28S.-H. Zheng, H.-J. Duan, J.-K. Wang, J.-Y. Li, M.-X. Deng, and R.-Q. Wang, Phys. Rev. B 101, 041408(R) (2020). 29H. Okamoto, J. Phase Equilib. Diffus. 33, 493 (2012). 30Handbook of Sputter Deposition Technology , 2nd ed., edited by K. Wasa, I. Kanno, and H. Kotera (Elsevier, 2012), p. 42. 31T. Fan, M. Tobah, T. Shirokura, N. Huynh Duy Khang, and P. Nam Hai, Jpn. J. Appl. Phys., Part 1 59, 063001 (2020). 32I. A. Campbell, A. Fert, and O. Jaoul, J. Phys. C: Solid State Phys. 3, S95 (1970). 33S.-Y. Yang, K. Chang, S. Stuart, and P. Parkin, Phys. Rev. Res. 2, 022029(R) (2020). AIP Advances 11, 055020 (2021); doi: 10.1063/5.0049577 11, 055020-6 © Author(s) 2021

5.0054222.pdf

J. Chem. Phys. 154, 200901 (2021); https://doi.org/10.1063/5.0054222 154, 200901 © 2021 Author(s).More than little fragments of matter: Electronic and molecular structures of clusters Cite as: J. Chem. Phys. 154, 200901 (2021); https://doi.org/10.1063/5.0054222 Submitted: 15 April 2021 . Accepted: 09 May 2021 . Published Online: 27 May 2021 Jarrett L. Mason , Carley N. Folluo , and Caroline Chick Jarrold ARTICLES YOU MAY BE INTERESTED IN Rhodium chemistry: A gas phase cluster study The Journal of Chemical Physics 154, 180901 (2021); https://doi.org/10.1063/5.0046529 From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597 Building intuition for binding free energy calculations: Bound state definition, restraints, and symmetry The Journal of Chemical Physics 154, 204101 (2021); https://doi.org/10.1063/5.0046853The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp More than little fragments of matter: Electronic and molecular structures of clusters Cite as: J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 Submitted: 15 April 2021 •Accepted: 9 May 2021 • Published Online: 27 May 2021 Jarrett L. Mason, Carley N. Folluo, and Caroline Chick Jarrolda) AFFILIATIONS Department of Chemistry, Indiana University, 800 East Kirkwood Avenue, Bloomington, Indiana 47405, USA a)Author to whom correspondence should be addressed: cjarrold@indiana.edu ABSTRACT Small clusters have captured the imaginations of experimentalists and theorists alike for decades. In addition to providing insight into the evolution of properties between the atomic or molecular limits and the bulk, small clusters have revealed a myriad of fascinating properties that make them interesting in their own right. This perspective reviews how the application of anion photoelectron (PE) spectroscopy, typically coupled with supporting calculations, is particularly well-suited to probing the molecular and electronic structure of small clusters. Clusters provide a powerful platform for the study of the properties of local phenomena (e.g., dopants or defect sites in heterogeneous catalysts), the evolution of the band structure and the transition from semiconductor to metallic behavior in metal clusters, control of electronic structures of clusters through electron donating or withdrawing ligands, and the control of magnetic properties by interactions between the photoelectron and remnant neutral states, among other important topics of fundamental interest. This perspective revisits historical, groundbreaking anion PE spectroscopic finding and details more recent advances and insight gleaned from the PE spectra of small covalently or ionically bound clusters. The properties of the broad range of systems studied are uniquely small-cluster like in that incremental size differences are associated with striking changes in stability, electronic structures, and symmetry, but they can also be readily related to larger or bulk species in a broader range of materials and applications. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054222 I. INTRODUCTION With the development of increasingly sophisticated physical chemistry tools, both experimental and theoretical, that arose in the later part of the 1900s, the study of small fragments of matter with sizes of 2–10 atoms or molecules flourished. Nearly three and a half decades ago, Kaldor, Cox and Zakin, several of the earlier investiga- tors of these species referred to as clusters, asked “How large does the cluster have to be before solid state theoretical description applies? What are the magnetic properties of “naked” and “dressed” clus- ters? Is there catalytic chemistry possible on such clusters, etc.?”1 This exciting eruption of research was built on the seminal work by Moskovits and Hulse,2who co-deposited metal vapor and inert gases to create small fragments of matter. After many conferences, literature reviews, and compilations,3–9the motivation for cluster research goes beyond the discovery of enticing properties that might arise along the growth of matter from the atomic or single molecule limit to the bulk. To set the stage, elemental and molecular clusters are typi- cally separated in categories both by their size and by their bindingforces. Small clusters (1 <n<100 atoms) are commonly built atom-by-atom, or molecule by molecule, in a bottom-up approach and exhibit significant changes in properties such as ionization or bond dissociation energies with incremental increases in size. This punctuated variation in small cluster properties with size lies in contrast with nanoclusters, or nanoparticles, which are com- monly formed by either synthetic bottom-up or mechanical top- down approaches. Nanocluster properties are affected by quantum confinement but exhibit a smooth evolution toward bulk properties upon addition of one atom or molecular unit, often with n−1/3- or volume-dependence. Within the small cluster category, another distinction lies in the bonding between constituent atoms or molecules. For exam- ple, water clusters, (H 2O)n, or clusters carrying an extra proton,10 electron,11or some charged species12,13represent one of the more actively studied clusters14–16among the class of molecular clusters or ion–molecule complexes.17These and weakly bound systems such as helium clusters, He n, are distinct from carbon clusters, C n, or more ionic cluster such as metal oxides, MxOy, in which all con- stituent atoms are connected by a network of covalent and/or ionic J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp bonds. The latter of these has been commonly invoked as models for heterogeneous catalysts.18–63 As noted above, there have been several reviews to date on cluster studies. Examples include overviews on the use of vibrational spectroscopies for structure determination,64,65matrix- isolated metal clusters as models for heterogeneous catalysis,66gen- eral surveys of transition metal oxides,67transition metal cluster magnetic properties,68and metal cluster reactivity.69,70This perspec- tive will focus on the application of anion photoelectron (PE) spec- troscopy toward probing the electronic and molecular structures of “hard” clusters71and the interesting properties that arise in a range of systems. In simplest terms, anion PE spectroscopy involves pho- todetachment of a negatively charged cluster, M− n, and analysis of the energy of the resulting photoelectrons, M− n+hv→Mn+e−KE. (1) As will be detailed further below, the electron kinetic energies, e−KE, reflect the energy between the initial anion state(s) and the final neutral state(s). We note here that some clusters are known to sup- port doubly charged anionic states, despite the destabilization from intramolecular coulomb repulsion.72We will not consider dianions in this perspective, but they are a separate, interesting set of species. Anion PE spectroscopy is particularly well-suited to cluster study. Figure 1 shows a schematic of the (a) simple direct detach- ment process in which a m/z-selected negative ion is photodetached, resulting in a neutral in a distribution of vibrational levels, with the FIG. 1. Relative energy levels associated with direct detachment (a) Mx+e− ←M− x+hvand an indirect/autodetachment pathway (b) Mx+e−←[M− x]∗ ←M− x+hv. The representative spectra below indicate the expected transi- tion intensities via direct detachment (Franck–Condon progression) and indirect detachment (high intensity at low e−KE) pathways.total energy balanced by the e−KEof the photoelectrons. The bottom panel shows the hypothetical distribution of e−KEfor a simple sys- tem in which a vibrational progression is resolved. As will be noted below, “hard” clusters can have complex electronic structures and therefore a high density of rovibronic levels of the anion that lie above the detachment limit, which can lead to indirect electron ejec- tion processes. One such example is shown in Fig. 1(b): The incident photon is resonant with such an excited level of the anion, which can lead to internal conversion to a very high vibrational level of the anionic ground state and electron loss through thermionic emission, which favors low e−KEstatistically. An additional powerful feature of anion PE spectroscopy is the ability to map the low-lying electronic structures of neutral clus- ters, which provides a view of the evolution of electronic structure as a function of cluster size and composition. Figure 2 schemati- cally illustrates how the evolution of the electronic structure of small FIG. 2. Illustration of the evolution of the electronic structure of a small group Mo- oxide clusters (MoO 3bulk stoichiometry, bandgap ∼3.0 eV) as oxidation increases from Mo 3O3to Mo 3O6and to Mo 3O9. The red horizontal lines represent relative energies of the fully occupied orbitals described as predominantly O 2p orbitals (correlating to the bulk valence band). Blue horizontal lines represent Mo-local 4d and 5s orbitals (correlating to the bulk conduction band). Arrows indicate electronic occupancy; “red” electrons indicate the excess electron in the anion associated with the lowest energy detachment transition, with “green” electrons indicating other accessible detachment transitions. J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp metal oxide clusters, such as Mo 3Oy−, from undercoordinated (sub- oxide) to fully oxidized (bulk stoichiometric) can be probed using anion PE spectroscopy. As with bulk MoO 3, the clusters have mixed ionic/covalent character. The red horizontal lines represent orbitals predominantly described as O 2p orbitals, which have some cova- lent overlap with the Mo 4d orbitals; these orbitals correlate with the valence band of bulk MoO 3. The turquoise lines represent the metal-local 5s and 4d orbitals that correlate with the bulk conduction band. Anion PE spectroscopy accesses neutral states via detachment of electrons from the highest lying valence orbitals. In this simple illustration, the “red” electron is the excess electron in the anion, which is in the highest occupied orbital and “green” electrons can additionally be detached with a ultraviolet (UV) or visible photon. In suboxides, multiple valence electrons occupy the Mo-local orbitals. Because they are energetically close-lying, the PE spectrum will show multiple overlapping transitions associated with, for example, the detachment of the “red” and “green” electrons, with differences in thee−KEvalues associated with the different final states equaling the energy difference between the close-lying states. For example, the PE spectrum of Mo 3O3−exhibits at least five overlapping tran- sitions within a narrow energy range. The spectrum becomes less congested with transitions as the number of electrons occupying the crush of metal orbitals decreases to the point of the stoichiomet- ric cluster. Note that in the stoichiometric Mo 3O9−cluster, the PE spectrum of which is shown hypothetically based on the spectrum of the W 3O9−congener,73the energy difference between the two lowest energy detachment transitions reflects the bulk analog of the HOMO–LUMO gap! The electron affinity represents the relative stability of the anion and neutral, which reflects a particular stability of open or closed shell electronic structures, physical stability of the structure, etc. Therefore, neutral clusters with particularly stable structures can have relatively low electron affinities. As noted above, charged clusters are readily m/z-selected prior to photodetachment measurements. However, mass selectivity is not a unique advantage to anion PE spectral studies. Charged clusters are common targets of reactivity studies that probe the size and composition-dependent chemical properties, and many of the studies on cluster models for catalysis noted above imple- ment mass selection of cluster reactants, products, or both. Col- lision induced dissociation in guided ion beams,74,75which can measure the bond energies as a function of size and composition, can identify clusters with particular stabilities and can be extended to measure the melting temperatures of clusters as a function of size.76Vibrational and electronic action spectra also leverage the mass selectability of chromophore and mass analysis of daughter ions.77,78 II. METHODS The general experimental strategy for measuring anion PE spectra of negative ions is to couple a cluster ion source to a mass spectrometer or mass filter. The photodetachment laser is then inter- sected with the ion beam of a selected m/z, and the photoelec- tron kinetic energy distribution is measured. The success of this general approach to cluster study is in no small part reliant on the computational characterization of potential molecular and elec- tronic structure characterization, which shall be discussed briefly inSec. II C. Following is a survey of several approaches taken to anion PE spectroscopic interrogations of “hard” clusters. A. Cluster sources Different strategies for cluster production have been formu- lated over the years. Metal cluster production techniques, including seeded oven sources, sputtering sources, liquid–metal ion sources, and laser vaporization sources, were reviewed by de Heer in 1993.7 As the PE spectrometers have used both continuous wave (CW) and pulsed lasers to photodetach the negative ions, both CW and pulsed cluster sources have been used in the studies surveyed in this perspective. Lineberger, whose seminal PE spectroscopy studies imple- mented CW lasers, developed a flowing afterglow ion source by cathodic sputtering from a DC discharge.79,80In a pulsed variation, the pulsed arc cluster ion source (PACIS) introduced by Meiwes- Broer and co-workers used high voltage discharge coinciding with a pulse of high-pressure buffer gas to produce a wide range of clusters of the material of the metal electrodes.81,82 Smalley83and Bondybey84simultaneously developed laser abla- tion sources using bulk targets made of the desired cluster material. The Smalley-style source coupled a pulsed molecular beam valve in close proximity to the ablation spot on the target, with a high- pressure pulse of helium issued from the valve entraining the metal vapor generated by ablation, which allowed for the production of larger, internally cold clusters. Laser ablation targets include bare metal rods, pressed pellet targets, liquid metal targets, and powder- coated rods.85–88A schematic of a Smalley-type source used in our laboratory89is shown in Fig. 3(a). Several variations of this cluster source and their applications are included in a review by Dun- can,36and cluster sources featuring two separate ablation targets for controlled production bimetallic clusters have been detailed.90,91 A variety of ns pulsed lasers have been used in ablation sources, although femtosecond lasers may increase the cluster sizes pro- duced.92–94Ablation sources generate a mixture of cation, anion, and neutral cluster species with varying compositions for different charged species, making m/z selection essential.37 Production of ligated clusters has been achieved by several groups using a two-valve scheme in which one valve introduces a high-pressure buffer gas that sweeps over the plasma generated by ablation of a target to produce the clusters, while a second valve injects a ligating compound downstream of the point of ablation, giving clusters time to coalesce and cool.95 Wet-synthetized ligated clusters can also be introduced into the gas phase. For example, a high flux cluster beam source developed by Palmer and co-workers is operated by sputtering metal cluster ions from a matrix.67,68More recently, Wang and co-workers brought cryogenically cooled ion traps into common use in anion PE spec- troscopy studies and beyond.96These traps can couple CW sources such as electrospray ionization (ESI) sources into a pulsed mass spectrometer [e.g., Fig. 3(b)] by using ion traps that accumulate ions, injecting them then into a pulsed mass spectrometer at the exper- imental repetition rate. This approach can also be applied to laser ablation sources, typically to cryogenically cool ions in the trap prior to spectroscopic investigation. The clear advantage of coupling an ESI source is the ability to introduce clusters generated via benchtop synthetic methods. J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 3. Representative cluster sources: (a) the pulsed “Smalley-style” source that uses laser ablation of a target material coupled with a pulsed molecular beam valve described in Ref. 89 and (b) a CW style ESI/ion trap source for introducing bench-top synthesized clusters into the gas phase. Reproduced with permission from Wang, J. Chem. Phys. 143, 040901 (2015). Copyright 2015 AIP Publishing LLC. B. Photodetachment and e−KEanalysis Depending on whether the source is pulsed or CW, the mass selection component is a time-of-flight or quadrupole mass spec- trometer or a Wein filter.97In either case, with a mass-filtered CW or pulsed anion beam, the anions are intersected and selectively photodetached with a fixed-frequency laser as per Eq. (1), yielding neutrals in a distribution of electronic, vibrational, and rotational states and photoelectrons in a distribution of kinetic energies, e−KE, per e−KE=hν−EA−Eneutral e,vib,rot+Eanion e,vib,rot. (2) If the anionic precursors are internally cold (a valid assumption for ions that have been cryogenically trapped, vide supra , is that Eanion e,vib,rot ≈0), the e−KEdistribution will reflect the energies of rovibronic states of the neutral relative to a single level of the anion. In gen- eral, one-electron transitions are the most intense, and Δs=±1/2. There are notable exceptions to this rule. The e−KEvalues for any given transition are photon energy dependent. However, PE spectra are typically reported in terms of the photon energy-independent binding energy, e−BE, related viathe simple relationship e−BE=hν−e−KE, (3) which allows for direct comparison of spectra collected with differ- ent detachment photon energies. Photoelectrons generated from a randomly oriented ensemble of anions have angular distribution relative to the electric field vec- tor of the laser described by the following equation formulated by Cooper and Zare:98 ∂σ ∂Ω=σtotal 4π[1+β(E)(3 2cos2θ−1 2)], (4) where θis the angle between the direction of the ejected electron and incident photon polarization, σtotalrefers to the total cross section, andβis an asymmetry parameter, ranging in value from 2 to −1. The asymmetry parameter in atomic systems is a function of the angular momentum, ℓ, of the outgoing photoelectron wave, which relates to the atomic electronic orbital of origin via conservation of angular momentum. That is, detachment of an electron from an s- orbital yields photoelectrons with ℓ=1 and p-orbitals yield ℓ=0, 2. For molecules with less-than-spherical symmetry, the relationship is less straightforward; an elegant and physically insightful description of the relationship between the outgoing photoelectron waves and the molecular orbitals of origin has been provided by Sanov.99 The methods of electron kinetic energy analysis in CW exper- iments implement tools such as hemispherical energy analysis,100 while in pulsed experiments, photoelectron time-of-flight, either in a magnetic bottle or in a field-free drift tube, has been in common use along with photoelectron imaging.101A comparison of these three types of electron detection methods is shown in Fig. 4. Both the hemispherical analyzer and field-free time-of-flight have low collection efficiency. The magnetic bottle time-of-flight approach, which has 100% collection efficiency, typically has a lower duty cycle (e.g., 10 Hz) although recent developments in electron trapping offer higher duty cycles, e.g., with synchrotron radiation facilities.102 A particularly creative and powerful combination of ion source and anion PE spectroscopy was demonstrated by Kappes and co- workers, who coupled an electrospray source to an ion mobility spectrometry setup, which allowed structural isomer separation of a particular m/z prior to interrogation using a magnetic bottle anion PE spectrometer.103As will be noted below, competitive structural isomers of clusters frequently co-exist in small- to mid-sized ele- mental clusters and metal oxide clusters in lower-than-traditional oxidation states. Photoelectron imaging, which was brought into broader use in the anion PE spectroscopy community by Sanov,104leverages a technique developed in pioneering work by Chandler and Houston based on a design of Eppink and Parker.105A great advantage of this technique is the simultaneous measurement of electron kinetic energy (via velocity) and angular distributions as the 3D electron cloud is projected onto a 2D surface. Attending this capability is the added ability to probe indirect detachment pathways that affect photoelectron angular distributions, such as autodetachment.106,107 The push to higher resolution photoelectron imaging has made it a powerful tool for studying a broader range of anionic sys- tems.108,109Slow electron velocity imaging (SEVI)110,111has emerged as a more reliable and broadly applicable version of high-resolution J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 4. Representative electron detec- tion schemes: (a) the field-free time-of- flight drift tube used to measure drift times of the fraction of photoelectrons that reach the detector (introducing a magnetic bottle scheme allows 100% collection efficiency), (b) the velocity map imaging setup that uses extractors and repellers to project the 3D elec- tron cloud onto a 2D phosphor screen, and (c) the hemispherical analyzer in which only electrons of a specific kinetic energy are allowed to pass and reach the detector. Modes (a) and (b) are used in pulsed experiments, and mode (c) is typically used in CW experiments. (c) Reproduced with permission from Travers et al. , J. Chem. Phys. 111, 5349 (1999). Copyright 1999 AIP Publishing LLC. PE spectroscopy than zero electron kinetic energy (ZEKE) spec- troscopy of anions, the latter of which involved tuning through the detachment continuum and discriminating against the detection of photoelectrons with greater than near-zero kinetic energy.112Anion ZEKE was limited to the study of detachment transitions yielding ℓ=0 photoelectrons, i.e., with non-zero detachment cross sections per the Wigner threshold law,113 σ∝σ0(Ehv−Ethreshold)ℓ+1 2. (5) In SEVI experiments, similarly, the photon energy is held anywhere between 0.01 and 0.5 eV above the detachment threshold; at each wavelength, a high-resolution photoelectron spectrum is obtained for a limited e−KErange and a complete spectrum may be stitched together.114–116While ZEKE generally suffers from grueling collec- tion times, SEVI has a high collection efficiency and maintains the measurement of photoelectron angular distributions.117 Ultrafast anion PE spectroscopy118–120has been conducted on a number of cluster systems to follow the evolution of anionic states with time. As a one-photon detachment tool, ultrafast lasers do notoffer an advantage over CW or nanosecond lasers. However, in a time-resolved pump–probe setup, dynamics of excited electronic states can be measured. In this technique, one photon drives an exci- tation, and the second photon detaches the time-evolving cluster anion. This technique has been used to explore the intermolecular dynamics in soft clusters, e.g., dihalide anions solvated by a finite number of CO 2molecules, or electronic relaxation in metal clusters. Excellent reviews of this now mature technique were presented by Stolow et al.121and Fielding and co-workers.122 C. Computational spectroscopy A thorough survey of computational methods that have sup- ported anion PE spectroscopic studies of clusters is beyond the scope of this perspective. However, it is important to note that a more detailed and insightful picture of the properties of clusters can be gained from reconciling experimental spectra with computational results on the anionic and neutral species. Without a general start- ing point in terms of which cluster electronic and molecular struc- tures are viable for consideration, assignment of spectra is limited to J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp very low-order interpretation, as small covalently or ionically bound clusters can potentially form numerous different structures. Identifying viable stable structures and wavefunctions without a priori knowledge of the structures can be achieved using a vari- ety of approaches to global minimum energy searches, a number of which have been summarized by Jian et al.123Density functional theory (DFT) calculations with or without single-point calculations conducted at more accurate (and expensive) levels are commonly used for structure optimization using a range of quantum chemistry packages.124Calculations readily provide zero-order spectroscopic parameters to compare the spectra: The difference in the zero-point corrected total energies of the global minimum energy neutral and anion corresponds to the adiabatic electron affinity, which often, but not always, corresponds to the lowest e−BEorigin of the experimen- tally observed transitions. This does not apply to situations in which the lowest energy isomer of the anion has zero Franck–Condon overlap with the neutral (e.g., the anion and neutral favor very different molecular structures) or if the transition between similar structures is not one-electron allowed. Similarly, it is straightforward to calculate the energy of the neutral cluster confined to the structure of the anion. The energy dif- ference corresponds to the vertical detachment energy, which is the e−BEat which the intensity of an electronic transition reaches a max- imum (highest Franck–Condon overlap). Time-dependent DFT can help in assigning excited state transitions observed in the spectra. More definitive structural assignments can be made with spec- tral simulations based on more detailed spectroscopic parameters determined from the differences between the initial anion and final neutral states, the respective vibrational frequencies, and nor- mal coordinates. Simulations that calculate Franck–Condon over- laps between the anion and neutral vibronic levels provide a more detailed vibronic profile of detachment transitions. Home-written codes, such as the one developed in our own group,125can con- volute stick spectra with the e−KE-dependent experimental res- olution and allow for different “temperatures” associated with different vibrational modes. This control takes into account the non-equilibrium conditions in some cluster sources in which lower- frequency modes can cool more efficiently than higher-frequency modes. Of course, simulations are most helpful when vibrational fea- tures are resolved in the associated anion PE spectra, which is not always the case, particularly with larger clusters. Other open access spectral simulations programs such as ezFCF (formerly ezSpectrum) and ezDyson are also widely used for simulating experimental PE spectra.126 Useful constructs from the computational studies of anion and neutral states are Dyson orbitals,127,128which allow visualization of the difference between the N-electron state and the N −1 electron state, and Natural Ionization Orbitals (NIOs), which add insight into electronic relaxation associated with the final N −1 state.129 The latter can further be used to calculate the photodetachment cross sections and PADs of transitions using standard DFT model chemistries.130For systems with multiple close-lying photodetach- ment transitions, this additional information can enrich the spec- tral assignments and the overall picture of the cluster electronic structure. In very simple terms, the NIO picture can provide a con- trast between electron detachment transitions that can be described as purely one-electron and more complex detachment processes. Examples of both are shown in Fig. 5. In the case of the Ce-doped FIG. 5. NIO visualizations of the orbital vacated by a photoelectron (a) for the case of a purely one-electron transition predicted for the Ce (cream)-doped B 6 (pink) cluster anion [Ref. 131] and (b) in the3A′→4A′′transition in MoVO 4− better characterized as a two-electron process involving detachment of an electron localized from an orbital localized on the Mo (turquoise) center and relaxation of an electron from the V center (gray) to the Mo center, shown in the dashed box (Ref. 132). O atoms are indicated in red. Adapted with the permission from (a) Mason et al. , J. Phys. Chem. A 123, 2040 (2019). Copyright (2019) American Chemical Society, and (b) Thompson et al. , J. Chem. Phys. 146, 104301 (2017). Copyright 2017 AIP Publishing LLC. B6cluster anion [Fig. 5(a)],131the detachment process can be visu- alized as creating a hole in the orbital situated on the Ce center, while in the case of the heterometallic MoVO 4−cluster [Fig. 5(b)],132 the transition between from the3A′anionic ground state to the 4A′′neutral ground state, which was relatively very low in inten- sity in the experimental spectrum, can be described as detachment accompanied by significant relaxation of a remaining metal-local electron. III. ANION PE SPECTRA OF CLUSTERS ACROSS THE PERIODIC TABLE While it would be nearly impossible to fully account for the pro- lific output of anion PE spectroscopic studies conducted on “hard” clusters, this perspective will survey several noteworthy and repre- sentative studies, both foundational and more recent. The studies are organized in terms of the type of clusters, classified as pure ele- mental, mixed (e.g., ionic, metal oxides, otherwise mixed elemental clusters), doped (clusters of predominantly one element with a single atom of a different element), and ligated clusters. A. Elemental clusters Anion PE photodetachment techniques have been applied toward the study of elemental clusters across the Periodic Table from the alkali metals133to I 3−(thought this molecule is considered J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp a reaction intermediate rather than a “hard” elemental cluster)134 and down to the f-block elements.135Starting this particular section in the p-block, early work on carbon clusters was driven by their importance in astrochemistry and combustion.136A review dedi- cated to carbon clusters, their size evolution, and the discovery of C60and carbon nanotubes would be encyclopedic in size. In addition to anion PE and ZEKE spectral studies of small carbon clusters, studies on heavier group IV elemental clusters of silicon,137–139germanium,140,141tin,142,143and lead144,145have revealed the fundamental connection between cluster and bulk properties. In particular, the persistent low-dimensionality of car- bon clusters is related to the unique stability of πbonding in carbon, a property behind the stability of graphite, while the energetic acces- sibility of unoccupied nd orbitals of the group IV elements leads to the formation of three-dimensional clusters in the very small size regime. This effect is reflected very plainly in the spectra. Figure 6 shows the anion PE spectra of odd-numbered carbon146and tin clusters for comparison. The near-vertical appearance and system- atic increase in electron binding energy with the size for the carbon clusters reflect the nearly identical linear structures for both the anion and neutral, and stabilization from charge delocalization with increasing chain length leads to higher binding energies. In contrast, the more compact 2D (in the case of Sn 3−) and 3D structures of small tin clusters result in multiple, close-lying, and vibrationally broadened electronic transitions. Relativistic effects become more important with the heavier elements, with spin–orbit splitting com- peting with Jahn–Teller effects in governing cluster structures, such as Sn 3.143While Si and Ge clusters were not found to form cage structures known for several magic C n−/Cnclusters, Sn 12and Pb 12 were found to form stable, icosahedral structures.142,144Related to this high-symmetry molecular character, Pb 12−, studied by VUVPE spectroscopy,147showed non-metallic behavior, as evidenced by the low screening of a core hole. Metallic behavior emerged forn>20. In contrast to group IV elements, the p-block neighbors, nitro- gen and oxygen, are in gas phase at standard temperature and pres- sure. N 2does not bind an electron, but O 2does. O 4−is a very com- plex, thoroughly investigated molecular anion in which the charge is shared between two O 21/2−molecules.148–150Larger oxygen clus- ters tend to be described as “soft” [O 4−]⋅(O2)nclusters. The heav- ier group V151,152and VI153elements have been among the earlier clusters that were studied by this community. Boron has inspired interest because of applications in high energy density storage materials and graphite/graphene alternatives. Boron’s one-electron deficiency compared to neighboring carbon imbues these clusters with interesting electronic properties, par- ticularly with the boron cluster propensity for establishing aro- maticity. Bowen reported early foundational work on boron clus- ters;165since then, prolific boron cluster studies154by Wang and co-workers have been at the forefront of boron cluster research, exploring exciting planar, vacancy-punctuated hexagonal struc- tures, puckered structures, borospherenes with interesting electronic structures,155and dopant atom effects,156with cluster sizes well into the nanomaterial regime.157 Situated below boron, aluminum clusters are also worth a com- ment, having been studied by anion PE spectroscopy by several groups over the past few decades.158–161Early studies were partly motivated by the seminal findings on apparent spherical shell- closing patterns of alkali metal clusters from abundance spectra, modeled within the jellium picture of a uniformly positive sphere supporting electrons.162The question of whether aluminum clusters could similarly be modeled by the jellium picture arose in the early FIG. 6. Comparison of the anion PE spectra of two series of group IV clus- ters obtained using 4.66 eV photon ener- gies (except in the case of Sn 3−, which was obtained with 3.495 eV, but the e−KEvalues are set to 4.66 eV −e−BE for direct comparison). Linear carbon cluster anions (Ref. 146) exhibit near- vertical electronic transitions to linear neutral clusters (odd-numbered clusters shown; even-numbered cluster spec- tra are similar but have systematically higher EAs), while tin cluster anions form two- or three-dimensional structures with numerous close-lying electronic states (Ref. 143). C n−spectra reproduced from Arnold et al. , J. Chem. Phys. 95, 8753 (1991). Copyright 1991 AIP Publishing LLC and Sn n−spectra adapted with the permission from Moravec et al. , J. Chem. Phys. 110, 5079 (1999). Copyright 1999 AIP Publishing LLC. J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp 1990s.163The combination of anion PE spectroscopy and higher- level calculations did show the jellium electronic shell structure for several of the clusters, highlighting the punctuated evolution of clus- ter properties in the small cluster size regime and underscoring the enhanced fundamental understanding that can be gleaned by inti- mate coupling between experimental and computational efforts.164 In a number of studies, Al 13−was found to be a particularly stable species and exemplary of a superatom.165–171Superatoms are defined as small, stable, and compact clusters with quantum states resem- bling the electronic orbitals in atoms. Examples of mixed-elemental clusters exhibiting superatom character are included below. Transition metal clusters, beyond the heavier p-block metal elemental clusters, present a different level of complexity in terms of electronic structures due to partially filled nd orbitals.4They were also among the first elemental clusters studied by anion PE spectroscopy.172–175Of particular interest in earlier studies was the evolution of band structure and metallic behavior,176but they also offered an enticing window into the evolution of magnetic proper- ties. As noted above, anion PE spectroscopy can directly map the high density of electronic states evocative of metallic behavior.177 Because of the fundamental inseparability of electronic and molecu- lar structures, using the combination of theory and PE spectroscopy to determine molecular structures has also been a productive approach. While inert as bulk, small supported gold nanoparticles have demonstrated a range of catalytic activity,178which had augmented the motivation for exploring the electronic and molecular structures of small gold clusters. Wang and co-workers explored a range of gold clusters,179the smallest of which assume planar (2D) structures. They demonstrated a transition from 2D to 3D structures at n=12, with Au 12−found to coexist in both 2D and 3D structures.180–183 Planar structures, rather than 3D structures for the smaller gold clusters, were rationalized by 5d–6s hybridization enabled by rela- tivistic effects.184Larger gold clusters, e.g., Au 26−, also studied by Wang and co-workers have been shown to have several energetically competitive molecular architectures that coexist experimentally, including core–shell, cage, tubular, and hexagonal motifs, as illus- trated in Fig. 7, which shows four structures identified compu- tationally that appear to contribute to all the observed PE spec- trum of Au 26−.182The structural diversity has been similarly con- firmed by the studies of Kappes and co-workers, who have employed electron diffraction in their studies of trapped and size-selected gold cluster anions185in addition to numerous other metallic systems.186,187 Direct detachment transitions can generally be treated as instantaneous one-electron processes in simple molecules, but this description is not necessarily appropriate when considering more complex metal clusters, and the high density of electronic states in the neutral suggests the same of the precursor anions. Early evidence of this was seen in the form of thermionic emission in the PE spec- tra of small tungsten188and other189cluster anions. As illustrated in Fig. 1(b), thermionic emission occurs when the anion is photoex- cited to another higher-lying anionic state, followed by intercon- version to a high vibrational level of the ground state of the anion, whereupon the electron is ejected. Because electron ejection fol- lows the initial excitation event, the photoelectron angular distribu- tion of thermionically emitted electrons is isotropic.190Thermionic emission from bulk emitters is statistical, and W n−clusters exhibit FIG. 7. Energetically competitive structures of Au 26−(Ref. 182). The variation in structure underscores the fluxionality of the metallic gold cluster anions. Reprinted with permission from Schaefer et al. , ACS Nano. 8, 7413 (2014). Copyright 2014 American Chemical Society. statistical thermionic emission in clusters as small as n=4. A dif- ferent indirect detachment process frequently encountered in anion photodetachment spectroscopies is autodetachment in which ejec- tion of an electron by a long-lived ( ca.ps) rovibrationally excited quasibound state of the anion is observed to follow propensity rules associated with electronic, vibrational, or rotational relaxation to the final neutral state, with the ejected electron having a discrete kinetic energy associated with the relaxation. This process, which has been observed in a number of soft cluster studies, will not be described further here. The high density of electronic states in transition metal clus- ters also opens the door to the study of the dynamics of excited electronic states of cluster anions, which have been measured using time-resolved PE spectroscopic techniques. In simple terms, a pump laser excites the cluster anion to an excited, bound, or quasibound electronic state, and the evolution of that excited state is probed by a delayed second pulse, which detaches the electron allowing the measurement of the PE spectrum of the clusters from the evolving excited state. The appearance of the PE spectrum changes with the delay between the excitation and detachment pulses. Using ultrafast anion PE spectroscopy, evidence of non- metallic behavior in small Pb cluster anions was observed: Despite the density of electronic states at larger cluster sizes, picosecond relaxation times suggested non-metallic behavior,191up to and including Pb 38−, which showed a remarkable polarizability differ- ence than the elemental bulk. Similarly, intra- and inter-band exci- tations in Hg n−clusters have been explored by Neumark and co- workers.119,120These clusters are, on the one hand, very simple, with each atom contributing a 5d106s2 1S0electronic structure, resulting in a fully occupied 6s “band,” with a single electron in the 6p “band” in the cluster anion. They were able to discern three different time- dependent processes, including 6s →6p interband excitation with Auger ejection, 6s →6s intraband excitation, with time-evolving detachment of the excited state, and 6s →6p excitation with time- evolving detachment of the directly excited electronic state. Life- times were dependent on the cluster size. In sum, these experiments J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp offer deep insight into the evolution of band structures with cluster size.192 Koop et al. employed ultrafast anion PE spectroscopic stud- ies of a range of elemental clusters, including p-block and transi- tion metal clusters, in the search of long-lived excited electronic states.193They made the curious observation that only cluster anions accompanied by spectra with narrow features had lifetimes on the ps timescale. The rationale was that narrow features reflected low coupling between the molecular and electronic structures, which suggests non-bonding character of the orbitals associated with the excitation. Time resolved PE spectroscopy of “hard” clusters such as those described above has not been as commonly adopted as it has been for “soft” clusters, many of which are relevant in biological charge transfer or radiation damage processes.194 B. Ionic clusters, metal oxide clusters, and other binary and higher-order clusters If one of the original curiosities surrounding elemental clusters was the size at which bulk properties began to emerge, one of the questions arising in metal oxides or other ionic clusters was how the localized bonding in the bulk materials would be reflected in small clusters. In addition, small clusters offer a platform for studying non-stoichiometric species. For example, our laboratory has exten- sively characterized numerous metal oxide clusters, both homo- 195–198and heterometallic,199–203in lower-than-traditional oxidation states (suboxides). One of the earliest studies on ionic clusters was reported by the Bowen group, who demonstrated the slow evolution toward bulk F-center energy in (CsI) n−clusters.204These stoichiometric clusters have low electron affinities, unlike the incrementally non- stoichiometric alkali halide ( MX)nX−clusters, which have electron binding energies higher than the bare X−halide anion. The work by Bowen on the stoichiometric clusters showed a smooth n−1/3depen- dence on binding energy, which is what is expected in a simple quan- tum confinement picture, suggesting that the sizes sampled in this study ( n≥13) were in the range that can be described as “confined” rather than molecular. Wang and co-workers reported a detailed determination of the structures of smaller Na xClx+1−(x=1–4) clusters.205The clusters in this very small size range cannot assume any structure that resembles the cubic bulk structure, but they did feature connectivity with alter- nating Cl–Na–Cl atoms. An interesting feature of the clusters was that the excess negative charge was largely evenly shared between the halide atoms, with the electron binding energy increasing with x. Bonding in metal oxide clusters can be described as a mixture of ionic and covalent bonding. Metal oxide clusters offer a platform for determining the evolution of cluster properties with sequential oxidation, as illustrated in Fig. 1. Unsaturated, suboxide clusters offer an enticing atomic-scale view of the oxidation process, a non- smooth process because of their small size. The value addition of computational studies provides greater detail. For example, after the discovery of the first Mn 12single molecule magnet, Bowen and co- workers used a combination of anion PE spectroscopy and theory to explore the magnetic properties in small, profoundly reduced Mn 5O−and Mn 6O−clusters, revealing the existence of multipleisomags, including non-magnetic species that would be undetected using other methods.206 While the electronic and molecular structures of small metal oxide clusters do reflect the bulk properties to a certain extent, particularly in fully oxidized clusters (e.g., Mo 3O9−in Fig. 2), the combination of small size below which any proto-bulk structure is energetically favorable and lower-than-traditional oxidation states can lead to different structures favored for the anion and the neu- tral.207In such a case, the anion PE spectrum does not provide information on the adiabatic EA or structure of the lowest energy neutral structure. In the event of close-lying structures for both the anion and neutral, cluster anions tend to favor more extended structures or structures that can otherwise support a more delo- calized charge distribution. In the case of Al 3O3−,49MoVO 3−,199 and several other clusters studied in our laboratory, at least two structural isomers of the anions were found to be computation- ally very close in energy and co-populated the ion beam, with one of the structures emerging definitively more favorable for the neutral. An anion beam “hole-burning” strategy was adopted to par- tially deplete the anion with the lower binding energy (i.e., the structure associated with the more stable neutral) using a lower- photon energy laser prior to the ion packet entering the main laser interaction region. Figure 8 shows the spectra of (a) Al 3O3− and (b) MoVO 3−obtained using 3.495 eV photon energy, both with and without the partial “bleaching” of one isomer, along with the deconvoluted spectra. Figure 8(c) shows how spectral features associated with the lower-energy neutral structure can be partially bleached if the anions are close in energy. Indeed, a series of studies on transition metal suboxide structures showed a relationship between the number M–O–M bridge bonds vs M =O terminal bonds and the electron binding energy, since neu- trals favored the former, which make the cluster more com- pact, and anions favor the latter, which accommodates charge delocalization. Anion PE spectra of lanthanide oxides have posed interest- ing challenges to the currently available low-cost electronic struc- ture calculations.208–214With partially filled, core-like 4f subshells, a crush of close-lying electronic states with nearly identical orbital occupancies, and also with close-lying ferromagnetically and anti- ferromagnetically coupled states, their relative energies appear to be dependent on the charge state. Recently, we demonstrated that lan- thanide suboxide clusters near the center of the row, such as Sm (4f5or 4f6) or Gd (4 f7), exhibit the effects of strong photoelectron- valence electron (PEVE) interactions,215hallmarked by anti-Wigner threshold law behavior and the prevalence of shake-up transitions. Figure 9(a) shows one such example: the PE spectrum of Sm 2O− measured using 3.495 eV photon energy exhibits an intense band Xand two lower–intensity transitions or manifolds of transitions, A andB, at higher e−BE(i.e., lower e−KE). With lower photon energy, thee−KEvalues of all transitions drop and approach zero for Aand B, yet the intensity of A(also less obviously, B) increases relative to band X. This behavior was rationalized as due to longer PEVE inter- action times with lower e−KE, leading more higher probabilities of mapping the perturbed [neutral +e−] system onto excited neutral final states, e.g., AandB. The [neutral +e−] interaction time is on the fs timescale, regardless of the photon energy used in the detachment. However, J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 8. Raw anion PE spectra of (a) Al 3O3−(Ref. 49) and (b) MoVO 3−(Ref. 199) obtained using 3.495 eV photon energy (top, solid trace) and of the ion packets after the lower EA isomer was partially depleted with hv1(2.330 eV for Al 3O3−and 2.016 eV for MoVO 3−, dotted line at top). Bottom spectra show the deconvolution of the spectral contribution associated with the two different structural isomers for each. Turquoise represents Mo, gray represents V, and red represents O. (c) Schematic of relative energies of the anions and neutrals of isomers A and B in which isomer B is much lower in energy for the neutral. If A−and B−are close in energy, then B−can be partially depleted by a hv1, which does not detach A−prior to the photoelectron spectrum being obtained using hv2. (a) Adapted with permission from Akin and Jarrold, J. Chem. Phys. 118, 1773 (2003). Copyright 2003 AIP Publishing LLC, and (b) Mann et al. , J. Phys. Chem. A. 114, 11312 (2010). Copyright 2010 American Chemical Society. electronic excitations are on the same timescale. Figure 9(b) shows one point of comparison: the electric field from the ejected elec- tron felt by the neutral as a function of time for three different photon energies, assuming the momentum of the ejected electron remains constant, and a binding energy of ∼1 eV. An externally applied electric field is a time-dependent perturbation on the elec- tronic structure of the neutral, particularly with highly polarizable outer valence orbitals. During the (evolving) perturbation, orbitals can be described as a time-dependent linear combination of the unperturbed orbitals, resulting in a wider range of final neutral states beyond the one-electron picture being accessed. In addition, fields on the order of 0.4 V Å−1are predicted to switch the relative sta- bilities of ferromagnetically coupled and antiferromagnetically cou- pled states.216The color-coded arrows in the bottom panel of Fig. 9 show the duration of fields over 0.4 V Å−1for three different photon energies. For systems in which the high density of low-lying states, FIG. 9. (a) PE spectra of Sm 2O−obtained over a range of photon energies (Ref. 215), showing the increase in the relative intensities of transitions to excited states with the decrease in photon energy and photoelectron e−KE.The excited state populations increase with the increase in photoelectron–valence electron interaction time. (b) Field from the photoelectron as a function of time for tran- sitions to excited states, driven by three photon energies. The color-coded arrows indicate the time at which the field is 0.4 V Å−1, which is sufficient to change the relative energies of FM and AFM states in computational studies (Ref. 216) on similar systems. (a) Reprinted with permission from Mason et al. , J. Phys. Chem. Lett. 10, 144 (2019). Copyright 2019 American Chemical Society. including close-lying ferromagnetic and antiferromagnetic states, create rich time-evolving [neutral +e−] interactions, the detach- ment process cannot be viewed as instantaneous, and transitions that would traditionally be described as shake-up or spin forbidden are apparently common. Preceding was just a sampling of some of the interesting find- ings in mixed clusters, including purely ionic stoichiometric, incre- mentally non-stoichiometric, and several different types of metal oxides, ranging from profoundly reduced to stoichiometric. Beyond these classes of clusters studied by anion PE spectroscopy are metal sulfides,217,218mixed III–V, and II–VI cluster models for semicon- ductors and elemental combinations selected to model interfaces such as metal–semiconductor junctions or supported metal catalysts J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp at an atomic level. As an example, particularly stable metal–carbon clusters, or met-cars,219,220were discovered and explored in the gas phase, including via anion PE spectroscopy.221The met-cars pos- sess alluring stoichiometries (e.g., Ti 8C12) and high-symmetry struc- tures, and appeared as magic numbers in mass spectra of ionic species. While these systems continue to be explored computation- ally as model catalysts, realization of their applications has proven elusive. Mixed metal oxides, i.e., ternary clusters, add another dimension for exploring more complex electronic structures. Near-neighbor combinations such as the MoVO y−,199MoNbO y−,201 Mo xWx′Oy−,196,203Mn xMox′Oy−,87and Ce xSmx′Oy−131,214studied in our group revealed that modest differences in metal oxophilic- ity can result in very pronounced differences in oxidation states of the individual metal centers for any suboxide species, which may underlie the particular activity of doped metal oxides in a reduced environment. Trans-periodic combinations, such as Al MOy− (M=Mo, W)200,202or Ce xOyPtn−,212are described as ionic com- plexes, Al+[MOy]2−and [Ce xOy]+Ptn2−, which have implications for supported metal and metal oxide catalysts, and the electrostatic environment at the interface between the catalyst and support mate- rial. While cluster models of catalysts have obvious limitations in accounting for all the chemical and physical features at play in applied systems, they do provide important insight into important atomic and molecular scale features that govern catalyst–substrate interactions. C. “Doped” clusters As a variation on the theme of mixed-elemental clusters, doped clusters can be distinguished as being prevalently one element (or one compound, such as a monometallic oxide) to which a dis- parate atom is added. Doped clusters have a rich history in clus- ter studies. The notion and appeal of clusters that are air stable were in full flower following the discovery of C 60and were further fueled by the production of air-stable single-metal atom encapsu- lated fullerenes.222Endohedrally doped cage clusters, an idea built on fullerene encapsulated metals,223have continued to be actively explored. These high-symmetry species, and understanding their growth, have motivated the study of smaller metal-doped group IV clusters using anion PE spectroscopy,224–227which, when com- bined with computational studies, show the evolution in electronic and structural properties as the clusters grow toward the more sym- metric and stable cage-like structures. Nuanced information such as charge distribution between the dopant and the main cluster can also be gleaned from calculations, provided some reconciliation between the calculated structures and observed spectra.228These studies show that within the small (n <20) cluster size regime, a sin- gle dopant atom can have a profound effect on the cluster structure and stability. In a similar vein to magic number met-cars and other metal-doped group IV clusters, the concept of “designer magnetic superatoms” has been explored by Zhang et al.229They proposed that simple elemental superatomic clusters doped with a single mag- netic atom, e.g., VNa 7, would provide stability with a high mag- netic moment. Rather than relying on the traditional approach of a Stern–Gerlach setup to measure the magnetic moments of the V-doped Na clusters, they relied on comparing experimental andcomputational spectra, with the calculations on structures and spin states in agreement with the observed spectra providing the mag- netic moment. In an elegant extension of their work on elemental boron clusters, Wang and co-workers have conducted a series of stud- ies on doped boron clusters, MBn−,123,230–232with dopants rang- ing from Li to transition metals to lanthanides. The strength of B–B bonding results in structures that could aptly be described as B x−cluster-ligated metal centers. For example, the high-spin metalloboron cluster, MnB 6−, studied with high resolution PEI spectroscopy was shown to have a planar teardrop structure with Mn at its “tip”233and a stable aromatic electronic structure. Other transition metal doped boron clusters assume beautiful flower-like structures, with the metal center as the pistil and the ring- or drum- like boron clusters structure encircling the metal center, as shown in Fig. 10.230 Lanthanide hexaborides are cluster materials used in electron emitters and have exotic magnetic properties. In the bulk, B 6octa- hedral anions form a cubic lattice with the lanthanide cations, but in small clusters, the negatively charged B 6unit is planar or near- planar. Bowen and co-workers234and our group131have probed SmB 6−and CeB 6−, respectively, where both have the lowest energy structures evocative of a teardrop with the metal center at the tip. CeB 6was determined to be highly ionic, with Ce2+(B6)3−aptly describing the anion, while the spectrum of SmB 6−coupled with ab initio calculations suggested some covalent character involving the 4f orbitals of Sm overlapping with B 6-local orbitals. Wang and co-workers recently reported a bent triple-decker inverse-sandwich structure of La 3B14−, which gives hints of the more compact boride cluster structures found in the bulk material.235 D. Ligated clusters As noted in the Introduction, a large body of metal and metal oxide cluster studies implement clusters as models for catalysts. A particularly useful feature of cluster models is the ability to emulate defect sites in a controlled, albeit somewhat reductionist, manner. However, there have been numerous experimental studies, outside of anion PE spectroscopy, on metal and metal oxide clusters ligated with CO 2, CO, H 2, ethylene, and other molecules that have shed FIG. 10. PE spectrum and structure of the Co-doped B 16−cluster (Ref. 230). Popov et al. , Nat. Commun. 6, 8654 (2015). Copyright 2015 licensed under a Cre- ative Commons Attribution (CC BY 4.0) license. The authors edited the figures from Ref. 230 by superimposing the molecular structure on the spectrum. J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-11 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 11. From Ref. 250: photodetach- ment difference mass spectra of [Au 25(SC 2H4Ph)18]−at (a) 355 nm and (b) 266 nm. The asterisks point to the depletion of parent ions. PE spectra of [Au 25(SC 2H4Ph) 18]−at (c) 355 nm and (d) 266 nm. Red solid and black dotted lines correspond to the experimental data and simulated curves for thermionic emission from the Au13core, respectively. Reprinted with permission from Hirata et al. , J. Phys. Chem. C 123, 13174 (2019). Copyright 2019 American Chemical Society. light on the interactions between catalytically active sites and sub- strates. One of the more effective experimental tools in this arena has proven to be IR predissociation spectra of ions.15,22–25,77,78The added value of anion photodetachment studies on cluster com- plexes with molecules such as H 2,236CO,237–240C2H,241and CO 2 and larger molecules such as amines242and benzene243is a direct insight into how the charge state impacts the cluster–ligand inter- actions. For example, in metal cluster carbonyl complexes, the excess charge on the anion leads to enhanced π∗back-donation, and detachment spectra exhibit progressions in the C–O stretch, resulting from preparing the neutral cluster complex with elongated carbonyl bonds.244Studies on smaller, ligated heterometallic clus- ters can give insight into the nature of doping sites or cooperative catalyst.245,246 More recent studies on ligated clusters leverage advances in coupling ESI sources and cryogenic ion traps to pulsed anion photo- electron spectrometers,247which have facilitated studies on solution- phase synthesized ligated clusters. As anionic ligated clusters can also shed ligands while being introduced into the gas phase by ESI, probing the progression, or tuning, of properties of the bare clus- ters to saturated, ligated clusters in the absence of solvent molecules is achieved. An illustration of this effect can be found in work of Bowen and co-workers on ligated cobalt sulfide clusters.248,249By sequentially ligating the superatomic Co 6S8cluster with electron- donating triethyl phosphine ligands, which form dative bonds with the metal centers, they demonstrated a sequential decrease in EA of the neutral.248Furthermore, sequential substitution of CO lig- ands for the triethyl phosphine ligands, i.e., Co 6S8(PEt 3)6−x(CO) x (x=0–3), resulted in a systematic increase in EA. Additional insight from the computational component of this study included the demagnetization of the Co 6S8core with ligation. An example of anion PE studies on ligated gold and silver clusters synthesized via benchtop methods for making molecularly homogeneous species is the work by Hirata et al. , who introduced [Au 25(SR) 18]−and [Ag 25(SR) 18]−thiolate complexes into a magnetic bottle instrument.250In the case of these ligated metallic species, photodissociation through loss of thiolate (anionic) ligands com- petes with photodetachment, as shown in Figs. 11(a) and 11(b). Direct detachment spectra were observed with 3.495 eV photonenergy [Fig. 11(c)], with the spectrum exhibiting two broad tran- sitions, allowing a determination of the EA. However, with 4.661 eV photon energy, thermionic emission was observed exclusively, sug- gesting a large oscillator strength direct absorption transition com- pletely usurping any direct detachment. These findings underscore the need to take into account indirect processes as a matter of course for more complex systems, and indeed, direct detachment may not be observed in studies on certain anions. IV. OUTLOOK Anion PE spectroscopies on the wide array of “hard” clusters have brought a wealth of insight into the evolution of properties with size and composition. With advances in the techniques for gen- erating, mass-selecting, and probing cluster anions using different flavors of anion detachment, it has become evident that additional, exciting information can be gleaned from these types of studies. A. Mining additional information embedded in anion PE spectra As described in Sec. III D, the anion PE spectra of [Au 25(SR) 18]−and [Ag 25(SR) 18]−clusters exhibited primarily thermionic emission, rather than detachment, under ultraviolet radiation. At some size or cluster composition, does anion pho- todetachment no longer provide useful information? Can more be gleaned from indirect processes? Given that this effect was clearly photon energy-dependent, measuring PE spectra over a range of photon energies and plotting the ratio of direct detachment to thermionic emission signal could provide the gas phase electronic absorption spectrum of the anions while concurrently mapping the neutral electronic structures. If mass analysis is conducted in paral- lel, photodissociation vs photoexcitation/thermionic emission path- ways can be explored, which would provide more insight into the nature of the excitations. Similarly, more can be gleaned from anion PE spectra of clus- ters with exceptionally complex electronic structures, such as the lanthanide suboxide species, with a more rigorous theoretical treat- ment of the detachment process. Computational results are essential to a large body of cluster study, yet the photodetachment process is J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-12 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp not fully understood. Detachment transitions have generally been treated as “sudden” or “instantaneous,” while more recent exper- imental studies on strongly correlated systems demonstrate that they are not and that strong photoelectron–valence electron inter- actions can add a dimension to the analysis of the detachment spec- tra, including mapping the energies and character of excited states that are dark with respect to direct detachment. Additional detailed information on the multi-stable magnetic states and the effects of the time-dependent inhomogeneous electric field generated by the photoelectron, or other interesting phenomena such as the multifer- roic states of Rh nclusters,251can be better understood with more detailed theoretical support. Furthermore, a more rigorous treat- ment of the time-dependent nature of detachment transitions would deeply enrich our understanding of increasingly complex systems, including a number of large non-cluster systems (e.g., extended conjugated biological molecules) with high densities of states, and electron–neutral interactions involved in electron-driven chemistry more generally. B. Dynamics of the electronic states of neutral clusters With the ever-improving spectroscopic “shutter speed,” the ultrafast dynamics of electronic excitation and relaxation can be probed. The work by Neumark on Hg nanions119,120reveals the range of size-dependent relaxation processes associated with the proto- band structure of small semiconductor clusters; unusually fast relax- ation times coupled with nuclear wave packet motion with excita- tion were also observed. In principle, these types of measurements can be extended to neutral clusters. Most of the anion photode- tachment spectra of clusters described in this perspective exhibit transitions to excited neutral states (including multi-electron tran- sitions), the temporal evolution of which can further be explored either by a second ionizing pulse or through sensitive fluorescence measurements. These measurements will be complicated if there are multiple excited states accessed in the detachment and fur- ther complicated with vibrationally broadened transitions. However, the fact that different spin states are accessed via anion detach- ment will create an interesting landscape, wherein spin-allowed vs spin-forbidden relaxations can be inferred. Exploration into the use of x-ray free electron lasers (XFELs) as a route for core- level photoelectron spectroscopy and ultrafast dynamics of size- selected anionic clusters is also currently under way.252Although challenging, these studies offer promising insight into the relax- ation dynamics of the neutral cluster and photoelectron-neutral interactions. In the case of time-resolved detachment-ionization measure- ments, ionization energies of cluster are generally significantly higher than their associated electron affinities, although certain classes of clusters, most notably, alkali, lanthanide, and lanthanide suboxide clusters, have very low ionization energies, ∼4 eV,253so ionizing excited states of the neutral clusters would be in the visi- ble to ultraviolet range. Using the lanthanide oxide clusters with the spectra exhibiting shake-up (multi-electron) transitions to excited states, the fates of these states can be followed by mapping the total cation yield as a function of delay time. Furthermore, the detach- ment energy has been shown to change the relative abundances of excited states. Measuring the e−KEof the whole system, which willinclude detachment and ionization transitions, as a function of the detachment energy will provide a tidy tie line through anion detach- ment spectroscopies and PFI-ZEKE measurements while providing dynamical information on intermediate neutral states. C. Stepping toward more realistic models for catalysts The prospect of building new bulk catalytic materials from clus- ter deposition underscores the overall importance of cluster studies while simultaneously motivating the development of instrumenta- tion for cluster species selectivity. An important question is how do the electronic properties of isolated clusters change upon deposi- tion? Anderson and co-workers have devoted significant effort into studying the size-selected deposition of metal atoms onto surfaces of support materials such as TiO 2.254,255When deposited on a sur- face, the catalytic properties of metal clusters have unique, discrete energy levels from the respective individual atoms, gas phase clus- ters, and bulk material. Howard-Fabretto and Andersson discussed this phenomenon in a recent review, which focuses primarily on the photocatalytic properties of Au and Ru clusters deposited on semiconductor surfaces.256Current cluster beam methods are capa- ble of building bulk catalytic materials in a laboratory setting, but preparative scale cluster deposition techniques are still needed.257 Ideally, the integrity (or enhancement) of the clusters’ catalytic activity is maintained by deposition onto a weakly interacting sub- strate, such as highly ordered graphite,258which also motivates fur- ther study leading toward developing predictability of the prop- erties of clusters measured in the gas phase, after deposition on surfaces. Cluster models also are useful in shedding light on defect sites in bulk material. These sites often present in the form of edge sites, basal planes, and vacancies and sometimes contribute signif- icantly to the overall catalytic activity of the material. In regard to the oxygen reduction reaction (ORR) and hydrogen evolution reaction (HER), these defect sites are widely considered to be the most active catalytic sites in the material due to their tuned electronic properties.19–22Studies for the past two decades have focused on tuning the electronic properties of these sites by dop- ing. Since these sites are localized, clusters studies are commonly utilized to gain information on catalytic behaviors of these materi- als. For example, metal sulfides have localized bonding and defect sites, making cluster models an excellent representation of molec- ular catalytic processes.23In addition, graphene point and line defects have been shown to have ORR electrocatalytic capability with energy barriers comparable to platinum(111).20These cluster studies have a primary interest in understanding molecular reac- tion pathways that lead to kinetic macroscopic properties of the material. D. Underexplored techniques and questions The more common application of cryogenic traps coupled to cluster sources in the past decade has allowed us to look at internally cold systems, which decongest spectra by suppressing hot bands and rotational broadening, allowing for more definitive and simplified analyses. Clusters generally do not reach the temperature of the cryo- genic trap, however. Helium nanodroplets, on the other hand, are an ultracold vehicle for spectroscopic measurements. Metal clusters J. Chem. Phys. 154, 200901 (2021); doi: 10.1063/5.0054222 154, 200901-13 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp have been synthesized in He droplets,259and neutral photoioniza- tion spectra of Mg clusters in He nanodroplets have been mea- sured,260offering the tantalizing possibility of anion PE spectroscopy of ultracold clusters in He nanodroplets. One potential pitfall would be broadening by He “boiling” off the neutral cluster due to the large change in cluster–He interaction with the charge state. In a reversal of using photodetachment to probe strong photoelectron-valence electron interactions near detachment threshold, strong electron–neutral interactions in certain systems may lead to interesting electron attachment processes. Using older electron transmission spectroscopy techniques on these more exotic species, the anions of which can be m/z selected and photodetached prior to interrogation by a monochromatic electron, unbound anionic states that are strongly coupled to bound states can be explored. Such studies may yield more insight into “doorway” states for anion formation and electron momentum control of anion or neutral state populations. Certainly, the range of cluster systems studied by anion pho- todetachment can continue to augment our understanding of the evolution of the band structure from the atomic or molecular scale to the bulk, the nature of defect sites in catalysts, the catalytic activities of small clusters of inert bulk, and strong electron neutral interac- tions. Beyond these areas that have already been visited to varying degrees, entanglement, quantum information, single molecule mag- nets, and spintronics offer areas into which anion PE spectroscopy studies of small clusters could continue to expand and flourish. ACKNOWLEDGMENTS C.C.J. is grateful for enlightening conversations with Professor Hrant P. Hratchian (UC Merced) and for additional graphics from Professor K.H. Bowen (JHU), Professor L.-S. Wang (Brown), and Professor G.B. Ellison (CU Boulder). This work was supported by the Indiana University College of Arts and Sciences. 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5.0047710.pdf

Appl. Phys. Lett. 118, 261601 (2021); https://doi.org/10.1063/5.0047710 118, 261601 © 2021 Author(s).In situ heteroepitaxial construction and transport properties of lattice-matched α- Ir2O3/ α-Ga2O3 p-n heterojunction Cite as: Appl. Phys. Lett. 118, 261601 (2021); https://doi.org/10.1063/5.0047710 Submitted: 16 February 2021 . Accepted: 12 June 2021 . Published Online: 28 June 2021 J. G. Hao , H. H. Gong , X. H. Chen , Y. Xu , F.-F. Ren , S. L. Gu , R. Zhang , Y. D. Zheng , and J. D. Ye ARTICLES YOU MAY BE INTERESTED IN Ultra-wide bandgap corundum-structured p-type α-(Ir,Ga) 2O3 alloys for α-Ga2O3 electronics Applied Physics Letters 118, 102104 (2021); https://doi.org/10.1063/5.0027297 Toward the predictive discovery of ambipolarly dopable ultra-wide-band-gap semiconductors: The case of rutile GeO 2 Applied Physics Letters 118, 260501 (2021); https://doi.org/10.1063/5.0056674 A review of Ga 2O3 materials, processing, and devices Applied Physics Reviews 5, 011301 (2018); https://doi.org/10.1063/1.5006941In situ heteroepitaxial construction and transport properties of lattice-matched a-Ir2O3/a-Ga 2O3p-n heterojunction Cite as: Appl. Phys. Lett. 118, 261601 (2021); doi: 10.1063/5.0047710 Submitted: 16 February 2021 .Accepted: 12 June 2021 . Published Online: 28 June 2021 J. G. Hao,1H. H. Gong,1X. H. Chen,1 Y.Xu,1F.-F. Ren,1,2 S. L.Gu,1R.Zhang,1Y. D. Zheng,1and J. D. Ye1,2,a) AFFILIATIONS 1School of Electronic Science and Engineering, Nanjing University, Nanjing 210023, China 2Research Institute of Shenzhen, Nanjing University, Shenzhen 518000, China a)Author to whom correspondence should be addressed: yejd@nju.edu.cn ABSTRACT The construction of Ga 2O3-based p-n heterojunction offers an alternative strategy to realize bipolar power devices; however, lattice mismatch usually leads to undesirable device performance and makes interface engineering more challenging. In this work, we demonstrated the con- struction of lattice-matched p-n heterojunctions by the in situ hetero-epitaxy of p-type a-Ir2O3on n-type Si-doped a-Ga 2O3using the mist- chemical vapor deposition technique. The a-Ga 2O3/a-Ir2O3p-n heterojunction shows single-crystalline corundum structures and well- defined rectifying characteristics. The transport mechanism has been identified to be space-charge-limited current conduction, which isinduced by interfacial traps in an ultrathin disordered layer at the a-Ga 2O3/a-Ir2O3interface. Through thermal treatment in oxygen ambient, interfacial trapping states are suppressed, and more shallow acceptors of Ir vacancies are activated, both of which lead to the profound reduction of reverse leakage current, thus the improved current rectification ratio. The p-type a-Ir2O3with advantages of lattice matching to a-Ga 2O3provides a promising strategy to realize high-performance bipolar power devices. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0047710 Recent advances based on the metastable a-phase gallium oxide (Ga 2O3) have triggered increasing research interest due to its growth compatibility with sapphire substrates and the enhanced breakdownfield endowed by its large bandgap (5.1–5.3 eV). 1To date, high-quality a-Ga 2O3epilayers and a-(In)Al 2–2xGa2xO3ternary alloying films with a large tunable bandgap have been achieved by mist-chemical vapordeposition (mist-CVD), 2molecular beam epitaxy (MBE),3and halide vapor phase epitaxy (HVPE) techniques.4,5Device-oriented imple- mentations based on a-Ga 2O3, such as a vertical Schottky diode with a low on-resistance of 0.4 m Xcm2and a breakdown voltage of 855 V,6 have been also demonstrated. It is expected to deliver higher output power, lower energy consumption, larger Baliga’s figure of merit, andthe enhanced heat dissipation capability. To allow the design of bipolardevices, the construction of p-n heterojunction is a favorable strategyfor achieving high-performance photodetectors, solar cells, and powerelectronic devices. 7–13Several reports demonstrated b-Ga 2O3hetero- junctions integrated with p-type NiO,14–16Cu2O,17CuI,18Cr2O3,19 and SnO,20while lattice mismatch makes interface engineering more challenging. Kan et al. have first reported the band alignment of a-Ga 2O3/a-Ir2O3p-n heterojunction.21,22However, charge transportmechanisms and the interface engineering have not been fully explored yet. In this work, we demonstrated the in situ epitaxy of lattice-matched a-Ga 2O3/a-Ir2O3heterojunction using the mist-CVD technique. Both Si-doped a-Ga 2O3and unintentionally doped a-Ir2O3 epilayers exhibit single crystalline in pure corundum phases. The well- defined rectification characteristic is obtained for the as-grown p-nheterojunction and further improved by the post-annealing. Thedetailed transport mechanisms mediated by interfacial traps have beendiscussed. Thea-Ga 2O3anda-Ir2O3epilayers were grown using a home- made hot-wall atmosphere mist-CVD system. Sapphire substrateswere cleaned consecutively with acetone, alcohol, and de-ionizedwater. Gallium acetylacetonate and chloro-(3-cyanopropyl)-dimethyl-silane were used as Ga and Si precursors, respectively. These precur-sors were solved in the de-ionized water (0.05 mol/l) with 1.5%hydrochloric acid and atomized into micrometer-sized droplets by a2.4 MHz ultrasonic transducer. Then high-purity O 2carrier gas was used to carry the precursors into the growth chamber with a flow rateof 650 sccm. The growth was optimized at the temperature of 470 /C14C under the atmospheric pressure.23Fora-Ir2O3, iridium chloride Appl. Phys. Lett. 118, 261601 (2021); doi: 10.1063/5.0047710 118, 261601-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apl(IrCl 3) was employed as the Ir precursor, which was solved in the de-ionized water (0.005 mol/l) with 1% HCl. Other growth param- eters are the same as the Si-doped a-Ga 2O3growth. High- resolution x-ray diffraction (HRXRD) measurements were per-formed using a D8 diffractometer equipped with a Cu Ka1(1.540 56 A ˚) line and a Ge (004) channel-cut crystal. For the reciprocal space mapping (RSM) measurement, a four-bounce dual channel Si (022) analyzer (DCA) mounted on the detectorstage was used. High-resolution transmission electron microscopy(HRTEM) and energy-dispersive x-ray spectroscopy (EDS) wereperformed using a monochromated FEI Titan STEM operated at 200 kV. Hall measurements were performed at room temperature by using Ti/Au as Ohmic contacts. Transmittance spectra wererecorded using a UV-vis-NIR spectrophotometer. For the fabrica-tion of the p-n heterojunction diodes, the a-Ir 2O3top layer was selectively etched by the ion-coupled plasma process down to the Si-doped a-Ga 2O3, and subsequently, Ti/Au (50/100 nm) metal electrodes with a diameter of 200 lm were deposited using electron beam evaporation to form the good Ohmic contactson both a-Ga 2O3and a-Ir2O3. The current–voltage (I–V) and capacitance–voltage (C–V) characteristics were measured by Keithley 2635A and Agilent 4980A source meters, respectively.Figure 1(a) shows the XRD 2 h-xscan patterns of a-Ga 2O3, a-Ir2O3epilayers on sapphire, and the as-grown a-Ga 2O3/a-Ir2O3het- erojunction. Dominant diffraction peaks at 39.92/C14, 40.27/C14, and 41.68/C14 are corresponding to (0006) planes of a-Ir2O3,a-Ga 2O3,a n d a-Al2O3, respectively.21No other diffraction peaks are observable, indicating the pure corundum phase of a-Ga 2O3anda-Ir2O3epilayers. Figure 1(b) shows the normalized x-ray x-scan rocking curves (XRC) of a-Ir2O3 anda-Ga 2O3epilayers. The full-width-at-half maximum (FWHM) of a-Ga 2O3(0006) is only 0.019/C14, and thus its corresponding screw dislo- cation density is estimated to be 1.34 /C2106cm/C02.21In comparison, the broadening FWHM of 0.761/C14fora-Ir2O3indicates the growth is not coherent due to the large in-plane lattice mismatch of 5.0% with an a-Al2O3substrate. For the a-Ir2O3epitaxy on lattice-matched a-Ga 2O3, the relatively large surface roughness of the underneath thick a-Ga 2O3epilayer leads to the imperfect initial nucleation and the suc- cessive non-coherent growth of a-Ir2O3. To investigate the residual strain and lattice relaxation, the XRD reciprocal space mapping (RSM) for (104) plane of the a-Ga 2O3/a- Ir2O3heterojunction is characterized and shown in Fig. 2(c) . Diffraction peaks corresponding to the a-Ir2O3,a-Ga 2O3,a n d a-Al2O3are distinct with different reciprocal space coordinate values (Qxand Q z). The in-plane relationship is apparently to be a-Ir2O3 FIG. 1. (a) XRD 2 h-xscan patterns of a-Ga 2O3,a-Ir2O3epilayers directly grown on sapphire, and the as-grown a-Ga 2O3/a-Ir2O3heterojunction; (b) XRD x-scan rocking cure of a-Ga 2O3anda-Ir2O3(0006) plane; (c) reciprocal space mapping of x-ray ð1014Þdiffraction for the as-grown a-Ga 2O3/a-Ir2O3heterojunction. (d) Absorption spectra for a-Ir2O3anda-Ga 2O3epilayers directly grown on sapphire.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 261601 (2021); doi: 10.1063/5.0047710 118, 261601-2 Published under an exclusive license by AIP Publishing[100]// a-Ga 2O3[100]// a-Al2O3[100]. As summarized in Table S1 of thesupplementary material , the measured Q xand Q zdetermined from RSM are almost same as the calculated values of free-standingmaterials, which suggest that the lattice of both a-Ir 2O3anda-Ga 2O3 epilayers is fully relaxed. The Q xdifference between a-Ga 2O3and a-Ir2O3shows a very small in-plane lattice mismatch of about 0.3%, as schematically shown in Fig. 2(a) . Such lattice matching is a prerequi- site for the construction of good heterojunctions. Note that an unde-finable X peak appears between the diffraction patterns of a-Al 2O3 anda-Ga 2O3, which has been reported to be resulted from an interfa- cial layer. It may be related to the cation intermixing and the solid-state reaction between Ga 2O3and Al 2O3.24 Figure 1(d) shows the absorption coefficient ( a) spectra for a-Ir2O3anda-Ga 2O3epilayers, which are obtained from the transmit- tance spectra in Fig. S1. By fitting the absorption spectra by the Taucrelation of ahv/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihv/C0E gp,25optical bandgaps ( Eg)o fa-Ir2O3and a-Ga 2O3are determined to be 2.93 and 5.14 eV, respectively. A negli- gible band tail for a-Ga 2O3epilayer verifies its high crystalline quality, while a broadband tail near the band edge of a-Ir2O3is observed. It is well described by the Urbach rule of a¼a0exp½ðhv/C0EgÞ=Eu/C138,26where the characteristic energy Euis estimated to be 0.43 eV. These sub-gap states are induced by the interfacial lattice disorder in thevicinity of phase transition and consequently play a critical role todetermine the carrier transport and device performance. Figures 2(b) and2(d)are cross-sectional HRTEM images to eval- uate interfacial microstructures and lattice alignments, while Fig. 2(c) shows the EDS element mapping across the a-Ir 2O3/a-Ga 2O3hetero- interface. An obvious periodic distortion at the a-Ga 2O3/a-Al2O3 interface is observed in Fig. 2(b) with a clear contrast along the [100] direction. The average spacing of the misfit dislocation confined at thea-Ga 2O3/a-Al2O3interface is estimated to be 8.6 nm, which corre- sponds to the matching of 20 a-Ga 2O3cells and 21 a-Al2O3cells, as schematically shown in Fig. 2(a) . Such semi-coherent growth of a- Ga2O3has been reported in the frame of multiple domain matching.27 InFig. 2(d) , the uniform interface brightness contrast indicates the presence of an ultrathin disordered layer (about 2 nm) at the a-Ir2O3/ a-Ga 2O3interface, which still exhibits the corundum structure. As dis- cussed above, the imperfect initial nucleation of a-Ir2O3upon the rela- tively rough surface of a-Ga 2O3leads to the high lattice disorder at the interface, which is eventually developed into inversion domains with FIG. 2. (a) The schematic of lattice alignment of a-Ir2O3,a-Ga 2O3, and a-Al2O3; (b) the cross-sectional HRTEM image of a-Ga 2O3/a-Al2O3; (c) EDS chemical maps and (d) HRTEM image across a-Ir2O3/a-Ga 2O3interface; SAED patterns for (e) the a-Ir2O3bulk, (f) a-Ir2O3/a-Ga 2O3interface, and (g) the a-Ga 2O3bulk region.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 261601 (2021); doi: 10.1063/5.0047710 118, 261601-3 Published under an exclusive license by AIP Publishingdistinct antiphase boundaries of a-Ir2O3epilayer, well supported by the XRD analysis. Furthermore, accompanied by strain relaxation, the cation intermixing effect of Ga and Ir and even solid-state reaction occurs during the in situ epitaxial process, as evidenced by the EDS mapping in Fig. 2(c) . The oxygen distribution is relatively rich at the interface with respect to the bulk regions, which implies that cation vacancies (V Irand V Ga) are more abundant at the interface region. As the shallow acceptor in a-Ir2O3,VIrwould contribute more holes and enhance the interfacial conductivity, and on the other hand, more car- rier trapping centers are inevitably introduced. Figs. 2(e)–2(g) show the selective-area electron diffraction (SAED) patterns of the bulk a- Ir2O3,a-Ir2O3/a-Ga 2O3heterointerface, and the bulk a-Ga 2O3,r e s p e c - tively, viewed along the [010] zone axis. The observation of sharp and discrete bright spots suggests that both a-Ir2O3anda-Ga 2O3are single crystalline with pure corundum phase. In Fig. 2(f) , the splitting of dif- fraction spots for the high-index reflections of a-Ir2O3anda-Ga 2O3 indicates the relaxation of misfit strain. The in-plane reciprocal space splitting is determined to be about 0.3%, verifying that the a-Ir2O3epi- l a y e ri sf u l l yr e l a x e d . The disordered interfacial microstructures are expected to modify electronic properties and carrier transport in the resultant device. The hole concentration and mobility of the a-Ir2O3layer directly grown on sapphire were measured to be 2.16 /C21020cm/C03and 0.58 cm2/V s, respectively. Such degenerated p-type characteristics is verified by the valence band X-ray photoemission spectroscopy (VB-XPS) in Fig. S2. It has been reported that the spinel ZnM 2O4(M¼Ir, Rh, Co) oxides, composed of zinc and transition metals with a d6closed shell configuration, unanimously exhibit p-type conductivity at room tem- perature.28–31An octahedrally coordinated Ir ion normally has a large t2g-egcrystal-field splitting, and the low-spin state with only a t 2goccu- pation makes an open shell Ir4þion, which behaves effectively like p electrons and dominates the valence band maximum (VBM).28 Similar to the spinel ZnIr 2O4,t h eV B - X P Ss p e c t r u mo f a-Ir2O3shows that the top of valence band is dominated by low-spin Ir 5d t 2gstates hybridized with O 2p states, and the Fermi level is located near the Ir 5d t 2gorbital band. Theoretical calculations predict that the octahedral-coordinated V Irin ZnIr 2O4exhibits negative formation energies and behaves as shallow acceptors to induce unoccupied states near the VBM.28As all Ir cation ions have the same octahedral occu- pancy in both a-Ir2O3and ZnIr 2O4, their p-type conductivity may be resulted from the same intrinsic defects, most likely, V Irshallow acceptors. Note that, as both water precursor and oxygen carrier gas are strong oxidants, the mist-CVD growth can be considered in the O-rich condition, which is confirmed by the nonstoichiometric Ir/O atomic ratio of about 0.4 for a-Ir2O3determined by the XPS measure- ment. Upon annealing in oxygen ambient at 300 and 400/C14C, more V Ir are activated, resulting in the slight increase in the hole concentrationto be 2.26 /C210 20and 2.38 /C21020cm/C03, and the hole mobility is improved to be 0.80 and 0.97 cm2/V s, respectively. In comparison, the Si-doped a-Ga 2O3layer has an electron concentration of 1.5/C21019cm/C03and a mobility of 43.2 cm2/V s, which has no obvious changes upon annealing. Figure 3(a) shows the cross-sectional schematic of the hetero- junction, which is composed of a 1.2 lm-thick p-type a-Ir2O3and a 1.4lm-thick Si-doped a-Ga 2O3on sapphire. It was reported that the band alignment of the a-Ga 2O3/a-Ir2O3heterojunction is a staggered- gap (type-II) with the valence- and conduction-band offsets of 3.34and 1.04 eV, respectively, as shown in Fig. 3(b) .21Figures 3(c) and3(d) show the linear and semi-logarithmic scaled I–V characteristics of the as-grown a-Ga 2O3/a-Ir2O3heterojunction diode (denoted as As- grown HJ) and those annealed at 300/C14C( A @ 3 0 0/C14CH J )a n d4 0 0/C14C (A@400/C14C HJ), respectively. The as-grown diode exhibited an obvious rectification characteristic but with a low rectifying ratio of 43.2 at 63 V. Upon annealing in oxygen ambient, the leakage current is remarkably reduced with the improved rectification ratio to 312 at 63 V. Furthermore, the as-grown heterojunction exhibits double- barrier characteristics with a low turn-on voltage (V ON)o f0 . 9 7V , while the V ONvalues for other two annealed diodes are 2.48 and 1.21 V, respectively. Considering the built-in potential and band offsets, a forward voltage of 2.4 V is necessary to overcome the total barrier height for electrons across the a-Ir2O3/a-Ga 2O3inter- face.21However, except for the 300/C14C annealed diode, the experi- mental V ONvalues are lower than the barrier height, indicating the domination of other transport mechanisms in the presence ofinterfacial traps. Quantitative analysis of I–V characteristics is performed in terms of the thermionic emission (TE) and the series resistance effect with the differential equation of @V=@lnðIÞ¼nkT=qþIR S,w h e r e RSis the series resistance and nis the ideality factor.32Figure 4(a) shows the dependence of @V=@lnðIÞonI, in which the slope and intercept rep- resent RSandn, respectively. Overall, all plots exhibit linear dependen- ces on the injection current, yielding nvalues of 34.04, 18.49, and 4.41 and RSof 1.82, 1.67, and 3.89 k X, for the as-grown and annealed diodes. The significant deviation of electrical performance from the ideal TE, such as high ideality factors, is mainly originated from the carrier transport mediated by interfacial traps.32It is also noted that an obvious deviation at low inject current was observed for the as-grown diode, which indicates that the charge transport is not governed by the thermionic emission (TE). Figure 4(b) shows the I–V curves in double-logarithmic scale. The charge transport mainly follows the trap-controlled space-charge-limited conduction (SCLC) mechanismwith three distinguished regimes. At the low bias, the current exhibits the linear bias-dependence in terms of J/ðkTÞ lV,w h e r e l¼Ech=kT andEchare the characteristic energy of localized trapping states.33At the moderate bias, the current is governed by the Mark–Helfrich rela- tion of J/V1þlowing to the activation of trapped carriers.34At high bias, all localized states are filled by the injected carriers, and thus the charge transport obeys the trap-free Mott–Gurney square law ofJ/V 2.33The differential power is calculated and depicted in Fig. 4(c) bymðVÞ¼@logI=@logV. It is found that mexhibits a gradual incre- ment between 1.0 and 2.0 over the scanned reverse bias range, while three regimes of the mvariation are identified at forward bias. At low forward bias, mis lower than 2.0. A sharp rise at the bias of 0.2 V indicates the onset of the trap-filling process. At moderate voltage, two distinct peaks appear with the maximum mvalues of 3.51, 4.45, and 6.39 for the investigated three diodes, respectively. It lies in the Mark–Helfrich regime, suggesting that carriers are activated and filled into traps with different energy levels.29In terms of Ech¼ðm/C01Þ=kT, the corresponding Echof band tail states is esti- mated to be 0.065, 0.089, and 0.14 eV. Note that the characteristic energy ( Ech¼0.065 eV) of the band tail states for the as-grown HJ is much lower than the Urbach characteristic energy (0.43 eV) of the a-Ir2O3epilayer. The discrepancy could be originated from that only shallow acceptors near the VBM of p-Ir 2O3are activated, and otherApplied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 261601 (2021); doi: 10.1063/5.0047710 118, 261601-4 Published under an exclusive license by AIP Publishingtraps with deeper energy levels cannot contribute to the carrier trans- port. The increased Echfor the annealed heterojunction is a result of the increased p-type conductivity because more V Irare activated dur- ing the annealing process, as evidenced by the increased current at the same forward bias and higher hole concentrations. At high bias, the m values of all diodes decrease down to 2.0, entering the trap-freeMott–Gurney regime. Trap-filled limit voltage (V TFL)i sd e fi n e dt h e bias with the onset of m¼2. In this circumstance, all localized traps are filled. Further applied bias results in the injection of free carriers that cannot be trapped.29Figure 4(c) shows that the V TFLvalue for the as-grown heterojunction is about 2.10 V, and thus, the average trapdensity ( N bt) is estimated to be 4.03 /C21018cm/C03in terms of Nbt ¼ee0VTFL=qd2(d¼1.2lma n d e/C255 are the thickness and dielectric constant of a-Ir2O3, respectively).33The carrier transport mediated by interfacial traps has been also verified by the increased reverse leakagecurrent in the pulsed I–V sweeping mode (Fig. S3). The trapping anddetrapping processes of carriers occur alternately during the pulsepumping and intervals, which lead to capacitive current superimposed on the DC component.The carrier trapping and emitter processes involved in sub-gap states can be also evaluated by the capacitance–frequency (C– f)c h a r a c - teristics. It is well known that the interfacial traps result in a superim- posed capacitance ( C it) parallel to the depletion capacitance (C D).35 Figure 4(d) shows the C– fplots for the as-grown HJ at different reverse biases. At low frequencies, the capacitance shows an obvious change inresponse to the reverse bias, indicating that more traps are distributedaround junction interface. As the frequency increases from 1 kHz to 1 MHz, the capacitance decreases monotonously from 1.49 to 0.042 nF. As the capture and escape of carriers in traps are slow processes, thetrapped carriers cannot respond quickly at high frequency, and thus, thecapacitance at high frequency (C HF) should be only contributed by C D. In this regard, the interfacial trap density (D it) can be estimated to be 2.88/C21013eV/C01cm/C02byDit¼Cit=ðq2AÞ¼ð CLF/C0CHFÞ=ðq2AÞ,35 where Ais the junction area, C LFand C HFare the capacitances at low and high frequencies, respectively. In terms of Nt¼Dit/C22kT=t (t¼2 nm is the interfacial layer thickness), the corresponding bulk con- centration of interfacial traps ( Nt)i s7 . 4 9 /C21018cm/C03,w h i c hi sa l m o s t t w i c ea sh i g ha st h a to ft h et r a pd e n s i t y( Nbt)o f4 . 0 3 /C21018cm/C03in the FIG. 3. (a) The schematic structure and (b) energy band diagram of the a-Ir2O3/a-Ga 2O3heterojunction; (c) linear and (b) semi-logarithmic scaled I–V characteristics of the heterojunction diodes.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 261601 (2021); doi: 10.1063/5.0047710 118, 261601-5 Published under an exclusive license by AIP Publishinga-Ir2O3bulk region. Thus, it is safe to conclude that the reverse leakage is dominated by carrier transport mediated by deep-level traps in thelattice distorted interfacial layer. It also supports that the reduced reverse leakage upon annealing is a result of the remarkable reduction of trap densities. In summary, the in situ epitaxy of lattice-matched a-Ir 2O3/ a-Ga 2O3p-n heterojunctions has been demonstrated, which exhibits single-crystalline corundum structures and well-defined rectifying electronic characteristics. In the presence of ultrathin disordered inter- facial layer, the transport mechanism was identified to be dominatedby space-charge-limited current conduction. The reverse leakage cur-rent was remarkably reduced through the thermal treatment in oxygenambient, owing to the suppression of interfacial traps and the activa- tion of V Irshallow acceptors. These results indicate that the latticematching a-Ir2O3/a-Ga 2O3controlling the interface engineering is an alternative strategy to realize high-performance bipolar power devices. See the supplementary material for the optical transmittance spectra of a-Ir2O3anda-Ga 2O3epilayers directly grown on sapphire ( F i g .S 1 ) ,t h eV B - X P Ss p e c t r ao f a-Ir2O3epilayer (Fig. S2), and the pulsed I–V results of the diode annealed at 400/C14C( F i g .S 3 ) .T h e experimental reciprocal lattice vectors derived from the RSM of Fig. 2(c)and the calculated ones are summarized in Table S1. This work was supported by the National Nature Science Foundation of China (Nos. 61774081 and 91850112), the State Key R&D project of Jiangsu (Nos. BE2018115 and BE2019103), theState Key R&D project of Guangdong (No. 2020B010174002), theFIG. 4. (a) Dependence of @V=@lnðIÞonI; (b) log-log J–V curves; (c) differential power index as a function of bias; (d) capacitance as a function of frequency (C– f) for the as-grown heterojunction at different reverse biases.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 261601 (2021); doi: 10.1063/5.0047710 118, 261601-6 Published under an exclusive license by AIP PublishingJoint Youth Fund of Ministry of Education for Equipment Pre- research (No. 6141A02033237), and the Shenzhen FundamentalResearch Project (Nos. JCYJ20180307154632609 andJCYJ20180307163240991). DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1S. J. Pearton, J. Yang, P. H. Cary, F. Ren, J. Kim, M. J. Tadjer, and M. A. Mastro, Appl. Phys. Rev. 5, 011301 (2018). 2G. T. Dang, S. Sato, Y. Tagashira, T. Yasuoka, L. Liu, and T. Kawaharamura, APL Mater. 8, 101101 (2020). 3M. Kracht, A. Karg, M. Feneberg, J. Bl €asing, J. Sch €ormann, R. Goldhahn, and M. Eickhoff, Phys. Rev. Appl. 10, 024047 (2018). 4D.-W. Jeon, H. Son, J. Hwang, A. Y. Polyakov, N. B. Smirnov, I. V. Shchemerov, A. V. Chernykh, A. I. Kochkova, S. J. Pearton, and I.-H. Lee, APL Mater. 6, 121110 (2018). 5R. Jinno, C. S. 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5.0053309.pdf

Appl. Phys. Lett. 119, 013101 (2021); https://doi.org/10.1063/5.0053309 119, 013101 © 2021 Author(s).Growth of PtSe2 few-layer films on NbN superconducting substrate Cite as: Appl. Phys. Lett. 119, 013101 (2021); https://doi.org/10.1063/5.0053309 Submitted: 06 April 2021 . Accepted: 17 June 2021 . Published Online: 06 July 2021 Michaela Sojková , Jana Hrdá , Serhii Volkov , Karol Vegso , Ashin Shaji , Tatiana Vojteková , Lenka Pribusová Slušná , Norbert Gál , Edmund Dobročka , Peter Siffalovic , Tomáš Roch , Maroš Gregor , and Martin Hulman ARTICLES YOU MAY BE INTERESTED IN Electrical tuning of the spin–orbit interaction in nanowire by transparent ZnO gate grown by atomic layer deposition Applied Physics Letters 119, 013102 (2021); https://doi.org/10.1063/5.0051281 Magnetoelectric coupling in micropatterned BaTiO 3/CoFe 2O4 epitaxial thin film structures: Augmentation and site-dependency Applied Physics Letters 119, 012901 (2021); https://doi.org/10.1063/5.0056038 Defect-free interface between amorphous (Al 2O3)1−x(SiO 2)x and GaN(0001) revealed by first- principles simulated annealing technique Applied Physics Letters 119, 011602 (2021); https://doi.org/10.1063/5.0047088Growth of PtSe 2few-layer films on NbN superconducting substrate Cite as: Appl. Phys. Lett. 119, 013101 (2021); doi: 10.1063/5.0053309 Submitted: 6 April 2021 .Accepted: 17 June 2021 . Published Online: 6 July 2021 Michaela Sojkov /C19a,1,a) Jana Hrd /C19a,1 Serhii Volkov,2Karol Vegso,3,4Ashin Shaji,3 Tatiana Vojtekov /C19a,1 Lenka Pribusov /C19aSlu/C20sn/C19a,1Norbert G/C19al,1Edmund Dobroc ˇka,1Peter Siffalovic,3,4 Tom /C19a/C20sRoch,2 Maro /C20sGregor,2 and Martin Hulman1 AFFILIATIONS 1Institute of Electrical Engineering, Slovak Academy of Sciences, D /C19ubravsk /C19a cesta 9, 84104 Bratislava, Slovakia 2Department of Experimental Physics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynska dolina F2, 842 48 Bratislava, Slovakia 3Institute of Physics, Slovak Academy of Sciences, D /C19ubravsk /C19a cesta 9, 84511 Bratislava, Slovakia 4Centre for Advanced Materials Application, D /C19ubravk /C19a cesta 9, 84511 Bratislava, Slovakia a)Author to whom correspondence should be addressed: michaela.sojkova@savba.sk ABSTRACT Few-layer films of transition metal dichalcogenides have emerged as promising candidates for applications in electronics. Within this group of 2D materials, platinum diselenide (PtSe 2) was predicted to be a compound with one of the highest charge carrier mobility. Recently, the successful integration of group III–V nitride semiconductors with NbN x-based superconductors was reported with a semiconductor transistor grown directly on a crystalline superconductor. This opens up the possibility of combining the macroscopic quantum effects ofsuperconductors with the electronic, photonic, and piezoelectric properties of the semiconducting material. Here, we report on thefabrication of a few-layer PtSe 2film on top of an NbN substrate layer by selenization of pre-deposited 3 nm thick Pt layers. We found the selenization parameters preserving the chemical and structural integrity of both the PtSe 2and NbN films. The PtSe 2film alignment can be tuned by varying the nitrogen flow rate through the reaction chamber. The superconducting critical temperature of NbN is only slightly reduced in the optimized samples compared to pristine NbN. The carrier mobility in PtSe 2layers determined from Hall measurements is below 1 cm2/V s. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0053309 In recent years, 2D transition metal dichalcogenides (TMDs) have become one of the most studied research topics due to their con-siderable potential for future nanoelectronics, 1–3thanks to several advantages. The weak van der Waals (vdW) interaction between thelayers and the absence of surface dangling bonds allows the facile fab-rication of heterostructures. The broken inversion symmetry and spin–orbit coupling in some monolayer TMDs provide us with a new platform for studying valley electronics and fabricating spintronicdevices. 4,5The transition from indirect to direct bandgap with the decreasing layer number down to monolayer, which is observed insome TMDs (e.g., 2H MoS 2,2 HW S e 2), stimulates the development of various optoelectronic applications—photodetectors, photovoltaicdevices, and light-emitting devices. 3PtSe 2b e l o n g st oac l a s so fT M D materials intensively studied due to its widely tunable bandgap, highcarrier mobility, and excellent air stability. 6Thermodynamically favored 1T-phase of the PtSe 2structure crystallizes in the D3 3d(P3m1)space group of the trigonal system.7Single-layer and few-layer PtSe 2is a p-type semiconductor, whereas thicker films exhibit typical semi-metallic characteristics. 8Charge carrier mobility values increase with the decreasing number of layers.6,9 Several methods, including molecular beam epitaxy,10chemical vapor deposition,9,11and the selenization of thin platinum films9,12 [so-called thermally assisted conversion (TAC)], have been reportedfor the growth of monolayer and few-layer PtSe 2films. Similar to MoS 2, PtSe 2can grow aligned horizontally or vertically to the substrate plane. Han et al.13observed that the crystallographic orientation of PtSe 2layers transits from “horizontal-to-vertical” with the increasing layer thickness, which happens even at the low growthtemperature of 400 /C14C. Interestingly, this structural transition is accompanied by the drastic change in the electrical transport proper-ties as clarified by the FET characterization. While vertically aligned 2D PtSe 2layers exhibit metallic characteristics, strong FET gate Appl. Phys. Lett. 119, 013101 (2021); doi: 10.1063/5.0053309 119, 013101-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplresponses were observed with horizontally aligned 2D PtSe 2layers pre- pared from very thin Pt films ( /C240.75 nm).13,14 Niobium nitride (NbN) is a classical superconductor15with a rela- tively high zero transition temperature (near 16 K), large superconduct- ing energy gap, and thermal cyclability. Easy fabrication makes this superconductor attractive for many applications, such as superconduct- ing electronics circuits, quantum computing, or high-frequency devi-ces. 16,17NbN occurs in three crystal structures—hexagonal, tetragonal, and cubic.18However, NbN polymorphism and a high affinity to oxy- g e nm a k et h ef a b r i c a t i o no ft h i nfi l m sb e l o w5 0 n mw i t hg o o ds u p e r - conducting properties and high Tc challenging. Reactive magnetron sputtering,19high-temperature chemical vapor deposition (HTCVD),20 and atomic layer deposition (ALD)21are typically used methods to fab- ricate the thin NbN films. Recently, Roch et al.22fabricated high-quality 50 nm thick NbN films grown on the sapphire substrate by pulsed laser deposition (PLD). They observed a superconducting transition temper- ature of 16.6 K, the highest value reported to date for an NbN film of that thickness. Several authors reported on the role of nitrogen during the fabrication process. Farha et al.23prepared NbN films by PLD. They found that the nitrogen pressure affects the film composition and phase. Increased N 2pressure is necessary to obtain the cubic NbN phase; how- ever, further increase led to a phase transition from the mixed (hexago- nal Nb 2N and cubic NbN) to single hexagonal Nb 2N phase. Polakovic ˇ et al.24showed that the ion beam sputtered NbN exhibits lower sensitiv- ity to nitrogen concentrations during deposition than that of the films prepared by reactive sputtering. Since semiconductor transistors are approaching their limits, quantum technologies are expected to deliver faster computation and guaranteed secure communication.25This requires the fabrication of new devices, such as superconducting single-photon detectors26or Josephson junction flux qubits,27which exploit the macroscopic mani- festation of quantum properties in superconductors. Combining a semiconductor with a superconductor on a single platform is expected to provide devices with improved performance. Yan et al.25used a molecular beam epitaxy to grow and integrate niobium nitride (NbNx)-based superconductors with a wide-bandgap family of semi- conductors—silicon carbide, gallium nitride (GaN), and aluminumgallium nitride (AlGaN). However, the fabrication of III–V nitride semiconductors usually requires such high temperatures (more than 700 /C14C) that the superconducting NbN layer is destroyed. Thus, semi- conductors that can be grown at lower temperatures are crucial for the fabrication of semiconductor-on-NbN devices. In this work, few-layer PtSe 2films were prepared on top of a superconducting NbN substrate by selenizing thin Pt films. We studied the influence of the growth parameters on the structural and electrical properties of the PtSe 2layer and the NbN substrate superconductivity. We found that the nitrogen flow rate through the reaction chamber during selenization is the critical parameter to avoid NbN substrate degradation. Higher nitrogen pressure suppressed undesirable incorpo- ration of oxygen caused by the higher chemical binding energy of Nb- Oi nc o m p a r i s o nw i t hN b - N .21More importantly, PtSe 2growth causes only a minor decrease in the NbN zero transition temperature. High-quality NbN films (50 nm thick) were prepared on a c-plane sapphire substrate using the pulsed laser deposition as described in our previous work.28X-ray diffraction (XRD) analyses confirmed the fcc-NbN cubic structure (ICDD 01-071-0162). Only 111 and 222 diffractions were visible in the XRD pattern, pointing to(111) preferential orientation of NbN on sapphire. Resistance vs tem- perature [R(T)] characteristics were measured using a standard DC four-probe measurements. The zero resistivity transition temperature (TC0) is around 15.5 K for a set of NbN films. Subsequently, a thin (3 nm) platinum film was magnetron sputtered directly on an NbNfilm and annealed in the presence of the selenium powder in a custom- designed CVD chamber. 9,29We chose thicker Pt layers (3 nm thick) to ensure the visibility of platinum diselenide in the XRD spectra. A heat-ing rate of 25 /C14C/min was used in all cases. The samples were annealed for 30 and 120 minutes at two different terminal temperatures. Thelower one of 400 /C14C is typical for thermally assisted conversion of Pt layers. On the other hand, PtSe 2films with improved crystallinity were obtained at an annealing temperature of 550/C14C.9We also varied a nitrogen flow rate to study its influence on PtSe 2formation and NbN stability. PtSe 2layer is about four times thicker than the parent Pt film, s ot h et h i c k n e s so ft h efi n a lP t S e 2samples is 12–14 nm.9 A confocal Raman microscope (Alpha 300 R, WiTec, Germany) equipped with a laser with an excitation wavelength of 532 nm wasused to record Raman spectra. All measurements were performed at FIG. 1. (a) Normalized Raman spectra of PtSe 2films prepared at (a) 400/C14C and (b) at 600/C14C for 30 and 120 min with different N 2flow rates. The spectra were nor- malized to the intensity of the E gpeak.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 013101 (2021); doi: 10.1063/5.0053309 119, 013101-2 Published under an exclusive license by AIP Publishingambient conditions. Figure 1 shows the Raman spectra of PtSe 2films on NbN prepared at 400 [ Fig. 1(a) ] and 550/C14C[Fig. 1(b) ] with differ- ent N 2flows. Only the part of the Raman spectrum is shown in Fig. 1 that reveals the lines belonging to PtSe 2,n a m e l y ,t h eE gmode at around 175 cm/C01and the A 1gmode at /C24207 cm/C01.T h es p e c t r an o r - malized to the E gline intensity show a changing ratio of the two Raman peaks. The intensity of the A 1gline increases with the decreas- ing nitrogen flow rate. The increase in the A 1g/Egratio is considered a signature of the increasing thickness of the PtSe 2layer.30,31This expla- nation, however, is not applicable here as the thicknesses of all samplesare comparable. Instead, the alternating intensity ratio can be identi- fied with the transition of the layer’s alignment from the horizontal to the vertical configuration, as can be seen from the XRD measurementsdiscussed below. The thin films’ crystal structure was studied by XRD in a sym- metrical h/2hconfiguration. We used a Bruker D8 DISCOVERdiffractometer equipped with a rotating anode (Cu-K a) and operating at a power of 12 kW. Figure 2(a) shows XRD patterns of a reference NbN substrate and PtSe 2films grown on NbN at 400/C14C and different growth conditions. As shown in the figure, the presence of 111 diffrac-tions of NbN proves the NbN layers’ stability at this growth tempera-ture. On the other hand, the PtSe 2layers gradually change their orientation, depending on the nitrogen flow rate. At the lowest rate of 25 sccm, the PtSe 2220 diffraction indicates a vertical alignment (VA) of the layer with respect to the NbN substrate. This diffraction gradu-ally disappears, and a new one corresponding to 001 appears as the N 2 flow rate increases. The PtSe 2layer alignment changes from vertical to horizontal with the increasing flow rate, analogous to the situation encountered in MoS 2.29These observations are further supported by the results obtained from grazing-incidence wide-angle x-ray scatter-ing (GIWAXS) measurements. GIWAXS measurements were per-formed using a home-built system based on the micro-focus x-ray FIG. 2. (a) XRD pattern of the NbN sub- strate layer and PtSe 2films prepared by the selenization of a 3 nm thick platinum layer at 400/C14C on the NbN substrate using different nitrogen flow rates andannealing times. (b) GIWAXS patterns ofthe same samples as in (a). Violet squares label PtSe 2diffractions.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 013101 (2021); doi: 10.1063/5.0053309 119, 013101-3 Published under an exclusive license by AIP Publishingsource (CuK a,IlS, Incoatec) and two-dimensional x-ray detector (Pilatus 200 K, Dectris). Figure 2(b) shows that the orientation of PtSe 2 indeed depends on the nitrogen flow rate used during layer seleniza- tion. Two strong 001 diffractions of PtSe 2atqxy/C2461.16 A ˚/C01are visi- ble in the GIWAXS pattern when the flow rate was the lowest (25 sccm N 2). The 001 diffraction vector is perpendicular to the PtSe 2 basal planes, and the “lying-down” orientation of the vector seen inFig. 2(b) reflects the vertical alignment of the basal planes. These dif- fractions became weaker when the flow rate increased (to 250 sccm). When the nitrogen flow rate was higher than 350 sccm, the 001 dif- fraction is observed as a pronounced arc at q z/C241.16 A ˚/C01,c o r r e s p o n d - ing to the “ standing up ” orientation. In this case, the 001 diffraction spot is masked by the missing wedge due to geometrical constraints imposed by the GIWAXS measurement mode. Now, the PtSe 2basal planes are parallel with the underlying NbN, i.e., they are aligned hori- zontally. This observation is consistent with the results from XRD measurements in the symmetrical configuration shown in Fig. 2(a) . Here, the 001 diffraction at 2 h/C2416.5/C14starts to be visible for the filmsprepared with a higher flow rate and becomes even more pronounced and narrowed after applying a longer annealing time [see inset in Fig. 1(a)]. Note that the NbN peaks are outside the range of the GIWAXS experimental set-up used. An increase in the selenization temperature to 550/C14C led to the destruction of the NbN substrate layer when the N 2flow rate was low [Fig. 3(a) ]. At the lowest flow rate, NbN completely converted to nio- bium and niobium oxide Nb 2O5. Only 220 and 100 diffraction of PtSe 2is visible in the XRD pattern suggesting its vertical alignment (VA). At 250 sccm, Nb, NbN, and Nb 2O5peaks are present together with that of VA PtSe 2. The oxide formation using a lower nitrogen flow rate is also evident from the GIWAXS pattern [see Fig. 3(c) ]. In addition to the 001 diffractions of VA PtSe 2atqxy/C2461.16 A ˚/C01,t h e niobium oxide diffractions indexed as 001, 108, 180, 181, and 200 are also visible. The nitrogen flow rate higher than 350 sccm suppressed both the decomposition of NbN and oxide formation. Only diffrac-tions belonging to PtSe 2and NbN were identified in the XRD pattern. Moreover, the appearance of a single 001 diffraction along the qz FIG. 3. (a) XRD pattern of NbN substrate layer and PtSe 2films prepared by seleni- zation of 3 nm thick platinum layer at 550/C14C on the NbN substrate using differ- ent nitrogen flow rates and annealingtimes. (b) Schematic sketch showing the arrangement of a PtSe 2films on an NbN/ sapphire substrate. (c) GIWAXS patternsof the same samples as in (a). Violetsquares and red circles label PtSe 2and Nb2O5diffractions, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 013101 (2021); doi: 10.1063/5.0053309 119, 013101-4 Published under an exclusive license by AIP Publishingdirection means that the PtSe 2layers are aligned horizontally. The position of this diffraction shifts toward a lower 2 hangle compared to data reported for a single crystal (ICDD 00-018-0970). This suggestschanges in the unit cell parameters—an increase in c-parameter and a decrease in a-parameter. We suppose that the matching of PtSe 2lattice to that of the NbN substrate caused this change. Therefore, we per-formed a u-scan of 101 PtSe 2diffraction [see inset in Fig. 3(a) ]. An indication of six peaks 60/C14apart is visible in the spectrum,, implying a weak ordering of PtSe 2layers in the a-bplane. The possible coinci- dence of NbN and PtSe 2lattices is schematically shown in Fig. 3(b) . The contraction of the PtSe 2aparameter helps to overcome the signif- icant difference of more than 10% between the in-plane lattice param-eters of the NbN substrate and the PtSe 2layer. GIWAXS measurements [ Fig. 3(c) ] confirmed the vertical PtSe 2alignment for low N 2flow rates (diffraction spots at qxy/C2461.16 A ˚/C01)a n dt h eh o r i - zontal alignment (diffraction at qz/C241.16 A ˚/C01)f o rt h eh i g h e s to n e . Evidently, higher flow rates are necessary during the selenization pro-cess to suppress the NbN decomposition. Naturally, an interesting question to be answered in this study is whether NbN retains its superconductivity even after the growth ofthe PtSe 2layer. R(T) measurements were performed using the stan- dard DC four-probe method (see Fig. 4 ). As expected, the NbN sub- strates lost superconductivity when lower N 2flow rates (25, 250 sccm) and the higher selenization temperature were used. On the otherhand, increasing the flow rate caused the NbN layers to stay supercon-ducting for both selenization temperatures. We observed zero transi-tion temperature T C0decreasing while the R(T) curves remain essentially unchanged. Although the longer annealing time resulted in a more significant reduction in zero transition temperature, the latter remains above 13 K for all samples examined. In any case, the zerotransition temperature observed in our experiments is higher than thatin GaN/NbN heterostructures. 25 It is known that the crystal structure quality of NbN influences the superconducting transport properties. Therefore, the micro-structure and texture of the film play an essential role. 28PtSe 2films were prepared at temperatures close to the growth temperature of NbN. As seen from Fig. 4 , not only the values of criticaltemperatures were changed but also the values of DTC(T at 90% RN—T at 10% R N, where R Nis resistance in the normal state close to the transition to superconducting state). It increased from 0.04 K for NbN up to 0.58 K for the sample selenized at 550/C14C. We sup- pose that during selenization annealing, some recrystallization pro-cess in NbN film occurs and the number of grain boundariesincreases. This leads to the decrease in zero transition temperatureand an increase in DT C. Finally, we performed Hall effect measurements to determine the charge carrier mobility in the PtSe 2films. All measurements were done at room temperature. We used indium contacts pressed into thecorners of the sample and measured the conductivity ( r)i nt h ev a n der Pauw configuration. Hall coefficient R Hand the charge carrier concentration were estimated from the measurement in the magneticfield. From these quantities, the charge carrier mobility l Hwas calcu- lated as lH¼(RH/C1r). The sign of the Hall coefficient determines the type of charge carriers. In all our samples, n-type charge carriers domi- nate. Based on these observations and the fact that the final thicknessof PtSe 2films is 12–14 nm, we suppose that the films are rather semi- metallic than semiconducting, in accordance with Ciarrocchi et al.32 The charge carrier mobility ranged from 0.2 to 0.8 cm2/V s for the samples with the horizontal alignment. These values seem to be at leastone order of magnitude smaller than those for PtSe 2layers grown on sapphire.9On the other hand, the values of charge carrier mobility reported in the literature range from few tenths to tens of cm2/V s even for samples prepared on the same type of substrate.6,9For that reason, lower mobility values of PtSe 2films prepared on NbN sub- strate are not unexpected. However, an exact explanation for theseparticular values is currently lacking. We could not estimate the mobil-ity in the vertically aligned PtSe 2samples because of the samples’ very high resistance and missing Hall effect. This can be understood as the electrons have to jump between weakly interacting PtSe 2atomic planes and nanosized grains forming the thin layers. In conclusion, we reported on the deposition of few-layer PtSe 2 films on a thin NbN substrate by selenizing a Pt layer at relatively low temperatures (up to 550/C14C). As-prepared heterostructures were char- acterized by Raman spectroscopy, XRD and GIWAXS measurements, electrical transport, and Hall measurements. We have optimized the fabrication conditions so that the NbN substrate is structurally andchemically unchanged after the PtSe 2growth. It has been discovered that the parameter controlling the properties of NbN and the align-ment of the PtSe 2layer is the nitrogen flow rate used during seleniza- tion. An NbN layer within the NbN/PtSe 2heterostructure remains superconducting; the zero transition temperature is lower than in pris- tine NbN by only 2–3 K. In optimized samples, T C0is still above 13 K, depending on the details of PtSe 2growth. The charge carrier mobility ranges from 0.2 to 0.8 cm2/V s in the samples with horizontally aligned PtSe 2. The integration of semiconductors and transistor gain elements with NbN-based superconductors opens up several new opportunities,such as single-photon detectors integrated with HEMT amplifiers forsecure quantum communications or HEMT microwave amplifiers with superconducting Josephson junctions, providing a platform for superconducting qubits. This work was supported by the Slovak Research and Development Agency (Nos. APVV-15-0693, APVV-17-0352, APVV-17-0560, and APVV-19-0365) and Slovak Grant Agency for FIG. 4. R(T) dependences of the PtSe 2/NbN heterostructures prepared at different growth conditions.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 013101 (2021); doi: 10.1063/5.0053309 119, 013101-5 Published under an exclusive license by AIP PublishingScience (Nos. VEGA 2/0149/17 and VEGA 2/0059/21). This study was performed during the implementation of the project Building-up Centre for advanced material application of the Slovak Academyof Sciences, ITMS project code 313021T081 supported by Research and Innovation Operational Programme funded by the ERDF. We acknowledge J /C19an D /C19erer for the deposition of platinum films. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1Y. Gong, Z. Lin, Y.-X. Chen, Q. Khan, C. Wang, B. Zhang, G. Nie, N. Xie, and D. Li, Nano-Micro Lett. 12, 174 (2020). 2Y. Liu, X. Duan, Y. Huang, and X. Duan, Chem. Soc. 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Armstrong, K. Hafidi, G. Karapetrov, and V. Novosad, APL Mater. 6, 076107 (2018). 25R. Yan, G. Khalsa, S. Vishwanath, Y. Han, J. Wright, S. Rouvimov, D. S. Katzer, N. Nepal, B. P. Downey, D. A. Muller, H. G. Xing, D. J. Meyer, and D.Jena, Nature 555, 183 (2018). 26G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, Appl. Phys. Lett. 79, 705 (2001). 27J. E. Mooij, Science 285, 1036 (1999). 28S. Volkov, M. Gregor, T. Roch, L. Satrapinskyy, B. Granc ˇicˇ, T. Fiantok, and A. Plecenik, J. Electr. Eng. 70, 89 (2019). 29M. Sojkov /C19a, K. Vegso, N. Mrkyvkova, J. Hagara, P. Hut /C19ar, A. Rosov /C19a, M. /C20Caplovic ˇov/C19a, U. Ludacka, V. Sk /C19akalov /C19a, E. Majkov /C19a, P. Siffalovic, and M. Hulman, RSC Adv. 9, 29645 (2019). 30M. O’Brien, N. McEvoy, C. Motta, J.-Y. Zheng, N. C. Berner, J. Kotakoski, K. Elibol, T. J. Pennycook, J. C. Meyer, C. Yim, M. Abid, T. Hallam, J. F.Donegan, S. Sanvito, and G. S. Duesberg, 2D Mater. 3, 021004 (2016). 31W. Jiang, X. Wang, Y. Chen, G. Wu, K. Ba, N. Xuan, Y. Sun, P. Gong, J. Bao, H. Shen, T. Lin, X. Meng, J. Wang, and Z. Sun, InfoMat 2, 12013 (2019). 32A. Ciarrocchi, A. Avsar, D. Ovchinnikov, and A. Kis, Nat. Commun. 9, 919 (2018).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 013101 (2021); doi: 10.1063/5.0053309 119, 013101-6 Published under an exclusive license by AIP Publishing

5.0043914.pdf

Sequential tunability of red and white light emissions in Sm-activated ZnO phosphors by up- and downconversion mechanisms Cite as: J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 View Online Export Citation CrossMar k Submitted: 12 January 2021 · Accepted: 5 June 2021 · Published Online: 28 June 2021 Puneet Kaur,1,a) Kriti,1Simranpreet Kaur,1Rahul,2 Pargam Vashishtha,3Govind Gupta,3Chung-Li Dong,4 Chi-Liang Chen,5 Asokan Kandasami,6 and Davinder Paul Singh1,a) AFFILIATIONS 1Department of Physics, Guru Nanak Dev University, Amritsar, Punjab 143001, India 2Centre for Nanoscience and Nanotechnology, Block II, Sector 25, Panjab University, Chandigarh 160014, India 3CSIR-National Physical Laboratory, New Delhi 110012, India 4Department of Physics, Tamkang University, Tamsui 25137, Taiwan 5National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan 6Inter-University Accelerator Centre, New Delhi 110067, India Note: This paper is part of the Special Topic on Emerging Materials and Devices for Efficient Light Generation. a)Authors to whom correspondence should be addressed: puneetphy.rsh@gndu.ac.in anddpsingh.phy@gndu.ac.in ABSTRACT Inorganic ZnO modified using rare earth (RE) ions is proposed as an alternative source of energy harvesting over the whole solar spectrum by utilizing the down- and upconversion excitation mechanisms. The present investigation reports the tunability of white/red light possess-ing excellent color rendering index and color quality scale by employing down/upconversions from Sm-activated ZnO phosphors. Theoccurrence of intra-4f transitions of Sm 3+ions in both up- and downconversion signifies the energy transfer from defect centers of the host lattice to the dopant sites (Sm3+). A mechanism is explicated with the help of an energy level diagram for down/upconversion to provide a clear understanding of the host –guest energy transfer and the involvement of various defect states. As a proof-of-concept, these findings demonstrate an inexpensive and clean approach to solid-state lighting and solar cell industries by extending the spectral range from theultraviolet to infrared region. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0043914 I. INTRODUCTION Alarming concerns about climate change and global warming demand humans to focus on alternative resources of energy toreduce fossil fuel consumption and to control environmental pollu-tion. Energy-saving illumination technologies utilizing solar energyare effective methods to reduce power utilization, which at present accounts for a total of 19% of global electricity consumption and 1.9GT CO 2release annually.1Therefore, the requirement of highly efficient lighting technology to significantly reduce worldwidepower demand and CO 2release becomes critical. Solid-state light- ing (SSL) based on light-emitting diodes (LEDs) have significant advantages in terms of low power consumption, high efficiency, flexibility, low cost, easy maintenance, long lifetime, improved lightquality, etc., and hence these are promising alternatives to tradi- tional incandescent and fluorescent bulbs.2–5However, convention- ally used omnipresent light sources comprise the RGB approach, where the generation of white light is controlled by mixing of red, green, and blue LED chips.6Another common strategy involves a combination of InGaN chips and Y 3Al5O12:Ce3+phosphors to yield cool yellow-white light.5It is quite difficult to maintain the color balance of these conventional incandescent and fluorescence sources. To overcome these prevailing issues, single-phase inor- ganic visible-light-emitting phosphors synthesized via a simple chemical approach are drawing attention. These are environment friendly and are potential candidates for the SSL industry owing to their extensive applications in the field of plasma display panels,Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-1 Published under an exclusive license by AIP Publishingwhite LEDs, liquid crystal display, biomedical imaging and therapy, fingerprint detection, sensors, solar power generation, etc.7,8Such materials may also provide a long-term solution to energy demand,global warming, and climate change. Among these, ZnO exhibits“full-spectrum lighting ”(FSL), with a broad absorption band in the UV region of the electromagnetic (EM) spectrum and emerges as a potential candidate for SSL. 6,9However, to achieve white light illu- mination and color tunability, certain modifications in the defectstructure demand chemical doping. Theoretically, the FSL can be achieved via down- and upcon- versions in ZnO by transferring excitation energy to the longer wavelength emitting activator ions such as rare earth (RE) ions. Inorganic phosphor with an activator is an efficient tool to explorewhite light for FSL. 10Activation through RE ions enhances the emission efficiency by controlling the nonradiative (NR) paths andenergy loss when energy is transferred from the host to RE ions. 11 By altering the composition of RE ions in host materials, one cantailor the lattice parameters and energy transfer (ET) since origi-nally forbidden transitions (of RE ions) are partially allowed whenthese are incorporated in oxide hosts. 12–14This strategy will lead to the development of novel white light phosphors with tunable emis- sion properties such as color temperature, stability, purity, etc. Previous studies reported the downconverted host –guest ET resulting in green and red luminescence owing to the5D4→7Fj and5D0→7Fjtransitions for Tb3+and Eu3+doped ZnO nanorods, respectively. Various RE ions such as Tb, Dy, Eu, Ce, etc. were incorporated in ZnO and investigated by many research groups tounderstand the multi-color emissions originating from bothdefects as well as f–ftransitions of RE ions. 15,16They observed the concentration quenching phenomenon at higher concentrations of RE ions based on the Judd-O-Felt theory that imposes the con- straints on the practical feasibility of using them in various opto-electronic applications. 17However, attaining white light from single-phase inorganic semiconductors like ZnO, by carefullycontrolling the dilute concentration of RE ions as well as ET among host and activator ions is still an open challenge for the scientific community. Hence, it requires in-depth investigations tounderstand and address these open questions regarding thedynamics of ET by carefully controlling the host –guest and defect interactions. Apart from an alternative lighting source under UV excitation (downconversion), RE-doped ZnO exhibits remarkable applicationsin the field of energy conversion under near-infrared (NIR) radia-tion excitations through upconversion luminescence (UCL). 18 Currently used solar energy based devices utilize around 40% −50% of the whole solar spectra comprising of the visible region(400−800 nm) only. The rest of the spectral distribution compris- ing of UV and IR regions is not absorbed due to the insensitivity ofsolar devices. 19Hence, this leads to the significant wastage of solar energy as well as the low conversion efficiency of these devices. Therefore, photon conversion from UV/NIR excitations into visibleradiations can efficiently compensate for the nonabsorbable regionof EM spectra for dye sensitized (DSSC) and organic solar cells. 20 Previously, these phosphors have been investigated under synchro- tron based vacuum ultraviolet (VUV) light and cool white light emissions have been achieved under these excitations.21To achieve FSL herein, we propose that the RE-doped ZnO phosphors can beefficiently used to convert photons from UV/NIR by exploiting down and upconversion mechanisms. In the present study, Sm3+-activated ZnO compounds pre- pared via the facile coprecipitation method were employed to studythe extended spectral response in UV and IR regions by employingdown- and upconversion mechanisms. The mechanism of energy transfer is responsible for overall white (red) light emissions by employing the photon up (down) conversion processes are illus-trated with the help of energy level diagram. Furthermore, it alsoevaluates the various spectroscopic parameters of the emitted lightsuch as color chromaticity (CIE), correlated color temperature (CCT), color rendering index (CRI), and color quality scale (CQS) as a function of Sm doping concentrations for the development ofSSL devices. II. EXPERIMENTAL SECTION/METHODS The undoped (pristine) and Sm-activated ZnO samples were synthesized via the facile chemical coprecipitation route, 22in which the respective acetates of each precursor were taken according to the chemical formula Zn 1−xSmxO (x = 0, 0.005, 0.01, 0.015, and 0.02) and annealed at 600 C for 3 h in the air. Detailed informationrelated to the synthesis process is reported elsewhere. 23These Sm-doped ZnO samples with the formula Zn 1−xSmxO( x = 0 , 0.005, 0.01, 0.015, and 0.02) are referred hereafter as ZnO:S0, ZnO: S1, ZnO:S2, ZnO:S3, and ZnO:S4, respectively. The doping concen-trations are further verified using energy dispersive x-ray measure-ments with the help of a (Supra ZIESS) scanning electronmicroscope (SEM) at 15.0 kV, which are listed in Table S1 in the supplementary material . X-ray powder diffraction (XRD) studies were performed with the help of Shimadzu XRD (MAXIMAXRD-7000) equipped with a CuK αanode ( λ= 1.5408 Å). The diffuse reflectance spectra at room temperature were recorded usinga (UV –Vis–NIR) Perkin Elmer Lambda 35 spectrometer in the spectral range of 200 –600 nm with a resolution of ± 1.0 nm. The emission spectra were measured by the Edinburgh FLS-980 PLsystem. He-Cd laser and a Xe lamp (200 −1800 nm) were used as the excitation sources along with the optical detectors in the range of 230 −1700 nm. The time-resolved PL (TRPL) measurements were carried out using an Edinburg instruments FLS 980 equippedwith Crylas TCSPC pulse laser. X-ray absorption measurements atroom temperature were performed at BL-17C1 at the NationalSynchrotron Radiation Research Center (NSRRC), Taiwan. III. RESULTS AND DISCUSSION All the samples investigated in this study (ZnO:S0, ZnO:S1, ZnO:S2, ZnO:S3, and ZnO:S4) were characterized by XRD (see Fig. S1 in the supplementary material ) and found to be free from impurities with the crystal structure of hexagonal wurtzite,and the details related to sample preparation and characterizationsare published elsewhere. 21Figure 1(a) shows the diffuse reflectance spectra (DRS) of ZnO:S0, ZnO:S1, ZnO:S2, ZnO:S3, and ZnO:S4 recorded in the UV –Vis wavelength range of 200 −600 nm. The direct bandgap energies (E g) calculated using the Kulbeka –Munk (K–M) function24indicate a decrease in the bandgap values with dilute doping of Sm ions (see Fig. S3 in the supplementary material ). Such doping results in sp–dspin-exchange interactions between theJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-2 Published under an exclusive license by AIP Publishingband electrons and localized 3delectrons.25A strong mixing of these orbitals leads to the formation of defect bands within the for-bidden bandgap. The contraction of the bandgap indicates that new levels are formed near the edge of the conduction band (CB) and valence band (VB). The significant lowering of the values of thebandgap indicates an increase in the deformities after doping andthe enhancement of defects. 22 Electrons excited to the edge of CB experience disorder due to the presence of various defects and the thermal vibrations.26,27The local electric field originating from the defects perturb the densityof state of electrons along with CB and VB edges leading to bandtailing called “Urbach tail. ”The presence of donor impurity levels that merge with the CB also leads to band tailing. 28The energy associated with the Urbach tail known as Urbach energy26is calcu- lated using the relation α¼α0exphυ Eu/C18/C19 , (1) and it is listed in Table I . It reveals information about the optical transition between occupied VB tails to unoccupied states at theedge of CB. It possesses dimensions of energy with quantitative information on possible defect states. The calculated Urbach energyvalues are found to be increasing with Sm concentrations as shown inFig. 1(b) . This indicates that the lattice or structural disorder is induced due to the replacement of Zn 2+with Sm3+ions.29Thus, the addition of Sm ions leads to more disorder in the system with adecrement in crystalline quality as observed from bandgap calcula-tions. 28These results are consistent with XRD results, which have been explained in our previous work.21Sm doping facilitates the formation of localized defect states within the bandgap leading tothe narrowing as well as band tailing. 30With the increase in the doping concentrations, the electron concentration increases that could induce band tailing in the forbidden region and also lead to the possibility of e−-e−and e−-impurity scatterings. Thus, an increase in the Urbach energy along with a decrement in thebandgap confirms the formation and enhancement of variousdefect bands in between the VB and CB. Figure 2(a) shows the downconversion (DC) emission spectra of undoped and Sm-activated ZnO under 325 nm (3.81 eV) excita-tion. The emission comprises of two sub-bands: (i) the near bandedge (NBE) emission in the UV region that arises due to the recom-bination of free electrons and (ii) the defect level emission (DLE) peaks in the visible region (400 −700 nm) arising due to the various defect levels. The emission spectra of ZnO:S0 and ZnO:S1 consist ofthree significant peaks: NBE (A), orange (F), and red (G) emissions.The NBE is observed at 381 nm and broad visible emission of twobands (F and G) centred at around 605 and 696 nm in orange and red regions, respectively. On increasing the Sm concentration, the NBE and red emissions remain the same. However, the orange (F)emission peak shows the blue shift resulting in the emission bandaround 500 nm. The emission band centered around 500 nm for ZnO:S2 is mathematically deconvoluted by using three Gaussian components centred at 465 (B), 500 (C), and 558 (D) nm and FIG. 1. (a) Diffuse reflectance spectra of undoped and Sm-doped ZnO, (b) variation of bandgap and Urbach energy as a function of Sm concentrations. TABLE I. Bandgap energies, Urbach energy, and average decay lifetime ( τavg)o f various emissions under the 325 nm excitation. SampleEg (eV)Eu (eV)τavg(s) (490−520 nm)τavg(s) (590−635 nm)τavg(s) (635−700 nm) ZnO:S0 3.250 0.133 … 4.54 × 10−63.80 × 10−6 ZnO:S1 3.241 0.155 … 4.22 × 10−86.48 × 10−8 ZnO:S2 3.223 0.166 6.64 × 10−8… 5.44 × 10−8 ZnO:S3 3.216 0.179 3.36 × 10−8… 5.82 × 10−8 ZnO:S4 3.203 0.180 7.75 × 10−86.60 × 10−87.32 × 10−8Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-3 Published under an exclusive license by AIP Publishingattributed to blue, cyan, and green emissions, respectively [ Fig. 2(b) ]. The deconvoluted emission spectrum of ZnO:S3 indicates a redshiftin blue and cyan emission peaks. Furthermore, the green emission (D) band shifts into the yellow (E) region (574 nm) leading to emis- sion in the yellow region. On further increasing Sm concentration tox = 0.025, this band further shifts and results in an orange emission(F) centered at ∼590 nm [as shown in the deconvoluted spectrum in Fig. 2(c) ] along with the small red shift in blue and cyan emission bands. The broad visible PL spanning from violet to red signifies thepresence and contribution of various intrinsic defect states such as zinc interstitials (Zn i), extended zinc interstitials [(Zn i)ext], zinc vacancies (V Zn), oxygen vacancies (V o), etc. It is worth noted here that these defect states arise during the growth process of ZnO par- ticularly in the low-temperature regime.31 The NBE emission of ZnO is attributed to the recombination of photon-induced electrons and holes through an exciton –exciton collision process.32The visible emission can be attributed to various surface and deep level defects. It is observed that the defect FIG. 2. (a) Downconversion emission spectra, (b) –(d) deconvoluted emission spectra of undoped and Sm-activated ZnO showing various defect related emissions.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-4 Published under an exclusive license by AIP Publishingstates are more active and nonradiative decay paths reduce with the Sm doping. The blue emission observed in the range of (2.63−2.67 eV) is attributed to the presence of (Zn i)ext. It results due to the transition of an electron from (Zn i)extto (V Zn). (Zn i)ext lies 0.54 −0.63 eV below the CB whereas V Znlies 0.1 eV above the VB. The electron from the bottom of CB relaxes nonradiatively to (Zn i)extthat further recombines with the hole at the acceptor level. The orange band centered ∼600 nm is correlated with the presence of interstitial defects of oxygen (O i) and zinc (Zn i). The full- potential muffin-tin orbital method predicts that Zn ilies at 0.22 eV below the CB and O iat 1.06 eV above the VB. The electron relaxes to Zn ilevel nonradiatively, recombines with the holes present at the acceptor level radiatively, and leads to an orange emission band.The red emission band at 696 nm arises due to the presence ofoxygen vacancies in the system. V ois theoretically predicted to lie at 1.57 eV above the VB. The electron that excites into the CB by 325 nm excitation recombines with the hole present at V oleading to the red emission. This emission peak remains almost constantfor all the Sm concentrations investigated in this study. The cyan and green emission bands are observed in the range of 500 −560 nm and are attributed to the presence of oxygen vacancies. The oxygen vacancies exist in three different states: V o (neutral, having 2 electrons lying at 1.57 eV above the VB), V o+ (charged, having 1 electron or unoccupied lying at 0.84 eV below CB), V o++(charged, unoccupied lying near the edge of CB).33 These cyan and green emissions originate due to the presence of singly charged oxygen vacancies (V o+). The trapped electron in the CB due to the V o+results in the formation of neutral oxygen vacancy (V o*). Then, the radiative recombination of electrons from V o* with the holes present at the edge of VB results in the subsequent emission band.34The yellow emission centered ∼570 nm originates due to the presence of oxygen interstitials. The electron from CB radiatively recombines with the photo-excited hole at O ilevel and results in yellow emission. Lifetime measurements of ZnO and Sm-doped ZnO at cyan, red, and orange emissions as shown in Fig. 3 were carried out to further understand the contribution of defect states and energytransfer mechanism. The decay curve of the pristine sample isobtained with 325 nm excitation, and it exhibits triexponentialdecay behaviour governed by Eq. (2), I(t)¼A 1exp/C0t τ1/C18/C19 þA2exp/C0t τ2/C18/C19 þA3exp/C0t τ3/C18/C19 :(2) However, the Sm-activated ZnO exhibits a bi-exponential decay curve as in Eq. (3), I(t)¼A1exp/C0t τ1/C18/C19 þA2exp/C0t τ2/C18/C19 , (3) where τi’s are components of decay time and A i’sa r eap r e - exponential factor (relative amplitude) of the fitting functions.35,36 The effective average decay lifetime is calculated as37 τavg¼ΣAiτ2 i ΣAiτi(4)and are given in Table I . The nonexponential, multi-decay component-time profile with a decay time of several microseconds indicates the presence of various defect states.38The third order decay of the pristine sample, ZnO:S0, indicates more nonradiativecontributions in the overall emission. 39However, after doping of Sm the decay profile shifts toward more radiative decay involving only two decay components. For both the orange and red emission bands, τavgdecreases after the Sm doping into the host lattice. The reduction in τavgindi- cates the enhancement of radiative recombination. Thus, the Smions facilitate the recombination channels present in the host –RE system over the nonradiative as compared to the pristine sample. To explain the sudden decrease in the average decay time of orange and red emissions for Sm-doped ZnO, it is important toquantify the valency state of Sm ions. Figure 4 shows normalized L III-edge XANES spectra of Sm-doped ZnO phosphors. The pres- ence of a strong intense single white line (6723 eV) corresponds to the 2p −5d electronic transitions. The spectra of Sm-doped ZnO phosphors are identical to that of the Sm 2O3powder and conveys a 3+ vacancy state of Sm ions.40Also, the absence of a double peak rules out the existence of mixed valency Sm ions.41Furthermore, the spectral features of all the samples are fairly identical to each other, hence nullifying any kind of change in the valence state ofSm ions as a function of doping concentration. 42 For the red emission band, a sudden decrease in τavgwith the addition of Sm can be explained on the basis that Sm3+ions intro- duce extra pathways in addition to existing decay channels. Theabove findings suggest that in the doped samples apart fromoxygen vacancies the transition from 4G5/2to6H11/2levels of Sm3+ also contributes toward the red emission band. Similarly for orange emission, the extra decay path involved due to the addition of Sm involves4G5/2to4H9/2transition of Sm3+as shown in the energy level diagram ( Fig. 5 ).43,44However, due to electron –phonon cou- pling, the broad emission bands are observed instead of sharpspikes raising due to Sm 3+.45,46For ZnO:S2 and ZnO:S3, the orange band vanishes and the new emissions originate in blue, cyan, green, and yellow regions indicating the ET among variousdefects of ZnO due to Sm doping. τ avgof the ZnO:S3 sample for cyan emission is lower as compared to the orange emission. Samahet al. reported ET from host defect states to 5D0energy level of Eu2+is responsible for orange and red emissions in Eu-doped ZnO thin films.47Kumar et al. ascribed the white light emission from 5 mol. % Tb3+due to cross-relaxation among the host and dopant ions.48The phenomenon stated above demonstrates that the ET mechanism within the various defect levels as well as to Sm3+ results in white light emission as a function of Sm concentration. Enhancement in the DLE after the doping of Sm indicates an increase in the defect levels of the host lattice due to the strain andthe formation of oxygen and zinc vacancies and interstitial defects. The peak position provides information about the depth and type of defects present in the host. The energy level diagram, shown inFig. 5 , is based on the theoretically predicted defect levels and energy positions corresponding to the emission bands obtained with/without deconvolution of PL emission spectra representing the most probable defect states involved in the Sm-doped ZnO system. Figure 6 shows the calculated values of color coordinates in the CIE 1931 chromaticity diagram. The values of CIE coordinatesJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-5 Published under an exclusive license by AIP Publishingalong with CCT values are listed in Table II . The undoped ZnO lies i nt h eo r a n g er e g i o no ft h ev i s i b l es p e c t r a .F o rS m>0 . 0 1 ,t h ee m i s s i o ncolor shifts toward the white region of the spectrum indicating the effect of Sm concentration on the optical behavior of the ZnO phos- phors. The CIE (x,y) coordinates for the samples: ZnO:S2, ZnO: S3, and ZnO:S4 are at (0.286, 0.385); (0.238, 0.353); and (0.287, 0.370),respectively. These are in close vicinity of the ideal white light illumi-nate (0.333, 0.333) as shown in Fig. 6 . The calculated CCT values show variation from the warm to cool region as the Sm concentration increases. The psychology and phys i o l o g yo fh u m a nb e i n g si ss i g n i f i - cantly influenced by color temperature. 1It is well known that white light with high CCT or high blue content values is ideal for lighting in schools, offices, hospitals, etc. wher e highly visual activity and concen- trations are required.49These findings suggest that the coexistence ofblue, cyan, green, and yellow emissions triggered due to Sm doping with concentration >0.01 and lead to the cool white light emissionunder the UV excitation, thus, making them an ideal candidate for white LEDs with the transition from the warm to cool region as a func- tion of Sm concentration. Furthermore, the room temperature upconversion (UC) lumi- nescence spectra of the Sm-activated ZnO phosphors with a differ-ent doping concentration of Sm 3+ions were recorded by using an IR excitation of 980 nm in the wavelength range of 600 –800 nm. The optimized UC emission spectrum of ZnO:S3 is shown inFigs. 7(a) and its enlarged spectrum in 7(b). The resulting spectrum consists of two emission peaks centred at 655 nm in the visible red region and 734 nm in the NIR region. The concentration effect of Sm doping on the observed red and NIR bands is shown in FIG. 3. Time-resolved photoluminescence decay curves of undoped and ZnO:Sm for downconverted cyan, orange, and red emissions.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-6 Published under an exclusive license by AIP PublishingFig. 7(c) . The maximum UC emission is observed in ZnO:S3. These optical emissions in the visible region are assigned tointra-4f transitions of the Sm 3+ions.50The radiative emission band at 655 nm results because of direct transition from the 4G5/2→6H9/2level of Sm3+51,52as an outcome of host –guest FIG. 4. XANES spectrum of Sm L III-edge spectra of (a) ZnO:S1, (b) ZnO:S2, (c) ZnO:S3 and (d) ZnO:S4 and (e) Sm 2O3. FIG. 5. Energy level diagram showing various radiative (solid color lines) and nonradiative (dashed color lines) transitions indicating host –guest energy transfer in case of down conversion. FIG. 6. Energy level diagram showing various radiative (solid color lines) and non- radiative (dashed color lines) transitions indicating host –guest energy transfer.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-7 Published under an exclusive license by AIP Publishingresonance energy transfer (RET) shown in the energy level diagram (Fig. 9 ). The NIR emission observed at 734 nm is ascribed to the spectral overlap of the radiative recombinations resulting due tothe involvement of interstitial of Zn and O and the transition of 4G5/2→6H11/2. At the lower growth temperatures, deep donor Zn i are attributed for the NIR emission upon above bandgap excita-tions. A similar kind of NIR emission in the case of ZnO has beenreported by Nguyen et al. and Shih et al. 53,54According to the full- potential muffin tin orbital theory (Zn i)extlies at (0.54 −0.63) eV below the CB minima. The plausible mechanism for NIR emission comprises of transition of the excited electron by the edge of CB tothe (Zn i)extfrom where it recombines with the holes present at O i which lies at 1.06 eV above the VB and the4G5/2→6H11/2intra-4f transition of Sm3+ions as shown in Fig. 9 . To understand the UC mechanism, the power dependence of red and NIR emissions for Sm-doped ZnO phosphors werestudied. The UC luminescence intensity (I) is directly proportionalto the nth power of the pump laser power (P), i.e., I /P n, where “n”is the number of pump photons absorbed per UC emission.55 The value “n”can be obtained from the slope of the double loga- rithmic plot of I vs P. As shown in Fig. 8 , the slope of red emission is∼2, indicating that two photons are involved in the UC process. However, in the case of 735 nm emission, the calculated value of n is lower than the predicted value. The decrement in the slope values is regulated by the competi- tion among the linear decay and the upconversion processes. Itfurther signifies that the resonance energy transfer and cross relaxa- tions play a vital role in the NIR emission.56–58This behaviour has been also theoretically proven by Chao et al. and Pollnau et al.59,60 The possible mechanism for visible UC is due to the absorp- tion of two or more photons in the IR region involving various processes such as ground state absorption (GSA), two-photonabsorption (TPA), and two-step two-photon absorption (TS-TPA)through some intermediate virtual states as detailed in the energylevel diagram (see Fig. 9 ). 61In this case, the UC takes place either via host or dopant ion. The TS-TPA involving virtual states which are replaced by real states (intrinsic or extrinsic defects or both)present in the host lattice excites electron. 62Such real states can be excited using low excitation density available from lamp sources instead of laser sources via TS-TPA.63Similar UC emission was reported by Stehr et al . in the case of bulk and nanostructured ZnO. It was confirmed from the photo-EPR studies that UC takesplace via the TS-TPA process. 63Another possible mechanism for the excitation of electrons from 980 nm involves an intraband of Sm3+ions. The GSA pumping photons populate the6F11/2and6F9/2 levels of Sm3+ions. The electron excited via TS-TPA in host ZnO partly shares its energy with the electrons present in the6F11/2and 4M15/2state of Sm3+ions via resonance energy transfer (RET) as shown in the energy level diagram. These excited electrons might transfer their energy to the Zn idefect states from where it decays to shallow acceptor levels resulting in strong NIR emission. It is alsopossible that the excited electrons at 4M15/2will decay nonradiativelyTABLE II. CIE 1931 chromaticity colur coordinates, CCT , CRI, CQS, and color purity as a function of excitation wavelength as well as Sm concentration. Sample325 nm 980 nm (x, y) CCT CRI CQS Color purity (x, y) CCT CRI CQS Color purity ZnO:S0 (0.467,0.446) 2856 78.70 71.11 76.83 …… … …… ZnO:S1 (0.503, 0.456) 2503 73.71 64.67 87.54 (0.725,0.270) 5855 88.83 81.47 98.23ZnO:S2 (0.286, 0.385) 7215 73.84 65.86 12.35 (0.725,0.270) 5851 88.78 81.30 98.27ZnO:S3 (0.238,0.353) 10 666 63.20 52.12 30.64 (0.725,0.270) 5849 88.95 81.71 98.29ZnO:S4 (0.287, 0.370) 7373 78.44 72.61 11.91 (0.725,0.270) 5851 89.16 82.35 98.26 FIG. 7. (a) Upconversion emission spectrum of ZnO:S3 under 980 nm excitations, (b) enlarged view of Spectrum in the range of 630 −670 nm, and (c) variation of visible and NIR upconverted emission intensities as a function of Sm concentration.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-8 Published under an exclusive license by AIP Publishing(NR) or via cross relaxations (CR) to4G5/2level of Sm3+ion from where it relaxes to6H11/2and6H9/2by emitting the photons centred around 655 and 735 nm.64 The CIE coordinates estimated using the commission of Internationale de l ’Eclairage (CIE) 1931 color matching functions, confirms red emission with small variation as doping concentration changes as shown in Fig. 6 . In the visible region, at Sm concentra- tion, x = 0.015 (i.e., ZnO:S3), the ideal red color is achieved with98.29% color purity ( Table II ). From the above studies, it is evident that multicolour, as well as cool white luminescence, can be achieved under UV-NIR excita- tions from RE-activated ZnO. It is observed that under UV excita- tion characteristic emissions arise from the host as well as dopantions. UV excitations are initially absorbed via the host throughbandgap excitations and the energy is then transferred to variousdefect states via nonradiative decays. From the various donor levels, multicolor radiative transitions, as well as ET to Sm 3+ions, takesplace resulting in the intra-4f transitions as shown in Fig. 10 . Various colored emissions combine and result in a white hue visible to the human eye as a function of Sm concentrations. Uponexcitation in the IR region, the energy is absorbed via TS-TPAalong with the direct excitation of Sm 3+ions ( Fig. 9 ). This causes two emission bands comprising the red and NIR emissions that are identified due to both host and dopant ions. All the above results reveal that these phosphors under UV-NIR excitations can betuned to be used as a potential phosphor in the visible lightingindustry. The average CRI under UV excitation is found to be ahigh value ∼68 to 78, indicating that reasonable rendering quality cool white light in these phosphors as listed in Table II . Under IR excitation, the red emission corresponds to CRI ∼88. It is indicative that these sources are best suited for natural and accurate light forthe indoor environment. Although, CRI is used to appraise the per-formance of light emitters still it lacks provide complete affirmation of the good color saturation. Therefore, the CQS values were FIG. 8. Logarithmic plots of pump power vs emission intensity for red and NIR emission under the excitation of 980 nm for Sm-doped ZnO phosphors.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-9 Published under an exclusive license by AIP PublishingFIG. 10. The schematic energy level diagram for both DC and UC showing the possible excitation –emission mechanisms. FIG. 9. Energy level diagram depicting the possible transition mechanism responsible for emission due to the upconversion.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 243106 (2021); doi: 10.1063/5.0043914 129, 243106-10 Published under an exclusive license by AIP Publishingcalculated that includes recommended 14 test color samples instead of 8 as in the case of CRI. Under UV excitation, the CQS value is close to the ideal white daylight ( ∼80). These perorations provide a shortcut for the evolution of ideal CRI and CQS cool white/ mono-chromatic red-emitting phosphors ideal for SSL. IV. CONCLUSION The DC emission spectra reveal tunability from the warm orange to cool white region of the visible spectrum with CRI for ideal white daylight as a function of Sm 3+ions concentration. This study unearths the significant ET from hosts to Sm3+ions, whereas, the UC spectra indicate cool red visible emission with 98% colorpurity and excellent CRI resulting due to intra-4f transition 4G5/2→6H9/2of the Sm3+ions. XANES spectra confirm the 3+ oxi- dation state of Sm ions. The occurrence of Sm3+intra-4f transitions under both UC and DC signify ET from host to guest and throughthe defects in ZnO. The TRPL measurements of both pristine andSm-activated ZnO in the current study reveal significantly lowerdecay rates. The emergence of slow decay rates upon Sm doping further assures ET to the activated RE ions for which the host defects act as a trap center. The energy level diagram based on thetheoretically predicted defect levels and energy corresponding tothe emission bands obtained with/without deconvolution of PL emission spectra further provides a clear understanding of the ET from host to guest and the involvement of various defect statesevident from UV –Vis spectroscopy. The above results reveal that activation of ZnO host via Sm 3+ion provides an attractive path to attain multicolor as well as white light emissions under UC/DC and paves the way for such materials to be efficiently used in SSL and solar cell industries. SUPPLEMENTARY MATERIAL See the supplementary material for information about the structure and bandgap variation as a function of Sm concentration. ACKNOWLEDGMENTS This work was supported by the Department of Science and Technology, New Delhi (Grant No. IF160408). The authors declarethat they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1P. Shen, M.-S. Lin, and C. Lin, Sci. Rep. 4, 5307 (2015). 2E. Taylor-Shaw, E. Angioni, N. J. Findlay, B. Breig, A. R. Inigo, J. Bruckbauer, D. J. Wallis, P. J. 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5.0045936.pdf

J. Chem. Phys. 154, 184703 (2021); https://doi.org/10.1063/5.0045936 154, 184703 © 2021 Author(s).Isolated Pd atoms in a silver matrix: Spectroscopic and chemical properties Cite as: J. Chem. Phys. 154, 184703 (2021); https://doi.org/10.1063/5.0045936 Submitted: 30 January 2021 . Accepted: 21 April 2021 . Published Online: 11 May 2021 Caroline Hartwig , Kevin Schweinar , Travis E. Jones , Sebastian Beeg , Franz-Philipp Schmidt , Robert Schlögl , and Mark Greiner COLLECTIONS Paper published as part of the special topic on Heterogeneous Single-Atom Catalysis ARTICLES YOU MAY BE INTERESTED IN Surface composition of AgPd single-atom alloy catalyst in an oxidative environment The Journal of Chemical Physics 154, 174708 (2021); https://doi.org/10.1063/5.0045999 The influence of palladium on the hydrogenation of acetylene on Ag(111) The Journal of Chemical Physics 154, 184701 (2021); https://doi.org/10.1063/5.0050587 Non-destructive detection of large molecules without mass limitation The Journal of Chemical Physics 154, 184203 (2021); https://doi.org/10.1063/5.0046693The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Isolated Pd atoms in a silver matrix: Spectroscopic and chemical properties Cite as: J. Chem. Phys. 154, 184703 (2021); doi: 10.1063/5.0045936 Submitted: 30 January 2021 •Accepted: 21 April 2021 • Published Online: 11 May 2021 Caroline Hartwig,1 Kevin Schweinar,2 Travis E. Jones,3 Sebastian Beeg,1Franz-Philipp Schmidt,1,3 Robert Schlögl,1,3and Mark Greiner1,a) AFFILIATIONS 1Max Planck Institute for Chemical Energy Conversion, Mülheim an der Ruhr, Germany 2Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany 3Fritz Haber Institute of the Max Planck Society, Berlin, Germany Note: This paper is part of the JCP Special Topic on Heterogeneous Single-Atom Catalysis. a)Author to whom correspondence should be addressed: mark.greiner@cec.mpg.de ABSTRACT Over the past decade, single-atom alloys (SAAs) have been a lively topic of research due to their potential for achieving novel catalytic proper- ties and circumventing some known limitations of heterogeneous catalysts, such as scaling relationships. In researching SAAs, it is important to recognize experimental evidence of peculiarities in their electronic structure. When an isolated atom is embedded in a matrix of foreign atoms, it exhibits spectroscopic signatures that reflect its surrounding chemical environment. In the present work, using photoemission spec- troscopy and computational chemistry, we discuss the experimental evidence from Ag 0.98Pd0.02SAAs that show free-atom-like characteristics in their electronic structure. In particular, the broad Pd4d valence band states of the bulk Pd metal become a narrow band in the alloy. The measured photoemission spectra were compared with the calculated photoemission signal of a free Pd atom in the gas phase with very good agreement, suggesting that the Pd4d states in the alloy exhibit very weak hybridization with their surroundings and are therefore electronically isolated. Since AgPd alloys are known for their superior performance in the industrially relevant semi-hydrogenation of acetylene, we consid- ered whether it is worthwhile to drive the dilution of Pd in the inert Ag host to the single-atom level. We conclude that although site-isolation provides beneficial electronic structure changes to the Pd centers due to the difficulty in activating H 2on Ag, utilizing such SAAs in acety- lene semi-hydrogenation would require either a higher Pd concentration to bring isolated sites sufficiently close together or an H 2-activating support. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0045936 .,s I. INTRODUCTION During the last decade, single-atom alloys (SAAs) have gained considerable attention in heterogeneous catalysis research,1espe- cially in selective hydrogenation reactions.2–9Usually, in a SAA, a dilute active metal of group 10 (Ni, Pd, Pt) is substituted in a noble metal host of group 11 (Cu, Ag, Au).1–11By this approach, only small amounts of the expensive active metal are used.10 For selective hydrogenation reactions, the ability of SAAs to dissociate H 211–13and the spillover of hydrides from the active sin- gle sites to host metal atoms11,14,15are often discussed, which both influence the catalytic selectivity compared to the bulk active metal.One selective hydrogenation reaction of considerable interest is the semi-hydrogenation of acetylene toward ethylene. Ethylene, the monomer for the large-scale industrial product polyethylene, is mainly produced by thermal cracking of hydrocarbons such as naph- tha. During this process, small amounts (0.5%–2%) of acetylene are produced. Acetylene has to be diminished to the ppm level since it poisons the catalyst in the ethylene polymerization reaction.16 Bimetallic AgPd alloys are industrially applied in this reaction17–19 since they show increased selectivity toward ethylene compared to monometallic Pd catalysts, which also form side products, such as fully hydrogenated ethane and polymers, known as green oil, which poison the catalyst.20–22The superior performance of AgPd alloys in J. Chem. Phys. 154, 184703 (2021); doi: 10.1063/5.0045936 154, 184703-1 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the semi-hydrogenation of acetylene can be referred to their elec- tronic structure, and the question arises how the catalytic properties of AgPd change when the dilution is pushed to the Pd site-isolation limit in a SAA. The changes to catalytic properties upon alloying are gener- ally classified as ensemble or ligand effects. Ensemble effects refer to the change in the coordination environment that occurs upon alloying, while ligand effects refer to the change in the electronic structure of metals when they are alloyed together.23Since ligand effects alter a metal atom’s electronic structure, they can, in princi- ple, be observed using electronic-structure sensitive spectroscopies, such as photoemission spectroscopy. In many alloys, differences in electro-negativity between the constituent elements give rise to a partial charge transfer.24,25Charge transfer can, in some cases, be observed by shifts in core-level XPS peaks (as long as the shift is not counter-acted by some other process, such as final-state effects).26,27 Additionally, the electronic structure of the surrounding matrix can alter the screening properties of the photoemission process, giving rise to additional binding energy shifts28and possible changes in peak line shape.29Changes to the electronic structure might also be observed in the valence band spectra, which, to a first approxima- tion, represent the cross section weighted valence projected density of states (PDOS) of the alloy and have an impact on the line shape and the core level shifts (CLSs).26 In some SAAs, the solute exhibits a very weak interaction with the matrix element, resulting in an electronically isolated metal site, where the solute’s valence states resemble a free-atom state. This behavior can be seen in the d-band of the solute. While most tran- sition metals in their pure form have d-bands that are several eV in width, in certain alloys, the solute’s d-bands become very narrow.5 This phenomenon has been shown recently for Ag 0.995Cu 0.00530 and can also be found in older literature about transition metals containing impurities of another transition metal.31–33 In the present work, we demonstrate the spectroscopic char- acteristics of an Ag 0.98Pd0.02SAA and discuss how they are a con- sequence of its electronic structure and coordination environment by comparing with the computed photoemission signal (PES) of a free Pd atom. We also discuss whether the geometric and electronic site-isolation of Pd in the Ag 0.98Pd0.02model catalyst can mani- fest themselves in the catalytic behavior in the semi-hydrogenation of acetylene. This discussion contributes to an improved under- standing of the nature of active sites in SAAs and single-atom catalysts. II. EXPERIMENTAL SECTION A. Synthesis of the Ag 0.98Pd0.02alloy Quantitative amounts of Ag (slugs 3 ×3 mm2, 99.99%) and Pd (granules <7 mm, 99.95%) purchased from EvoChem were melted in a light oven, with a four-time re-melting process for homogeniza- tion. Afterward, the alloy was cold-rolled to a 1.6 mm thick foil and then annealed for 6 h at 800○C to promote grain growth. The foil was cut into smaller pieces and mechanically polished step by step until a roughness of 1 μm was archived by using a diamond suspen- sion as an abrasive. Subsequent annealing in 0.5 mbar O 2at 500○C led to the surface segregation of impurities such as Cu, K, S, Si, and Cl, which could then be removed by Ar+sputtering. This procedurewas repeated several times to clean the sample surface. The last step was heating in UHV at 500○C. Additionally, an Ag 0.95Pd0.05alloy was prepared using the same procedure. B. Polycrystalline Pd and Ag foil The 0.1 mm thick Pd foil was purchased from Alfa Aesar (99.9%), and the Ag foil from Sigma-Aldrich (99.99%) is 0.5 mm thick. Both samples were cleaned by several Ar+sputtering and annealing cycles (in O 2, H 2and vacuum). C. XPS experiments For the x-ray photoemission spectroscopy (XPS) measure- ments, two different near ambient pressure (NAP) XPS setups were used: (i) a lab source (NAP) XPS using monochromatic Al K α (1487 eV) radiation and a Phoibos NAP-150 hemispherical ana- lyzer from SPECS GmbH, and (ii) the (NAP) XPS setup at the UE56-2_PGM1 beamline at Bessy II, which is also equipped with a hemispherical analyzer from SPECS GmbH. For the experiments performed in this work, they were both operated in ultrahigh vacuum. Before performing XPS measurements, the samples were Ar+ sputter cleaned for 15 min. All spectra of the Ag 0.98Pd0.02alloy [except those of Figs. 3(c) and 3(d)] were measured using syn- chrotron radiation; thereby, excitation energies of 520 eV for Ag3d, 485 eV for Pd3d, and 200 eV for the valence band spectra were used. Reference spectra of the polycrystalline Pd and Ag foil were collected using the laboratory XPS (with Al K αexcitation). All peaks were ana- lyzed using the CasaXPS software and fitted using a U2 Tougaard background. D. STEM–EDX measurements Scanning transmission electron microscopy (STEM) in com- bination with energy-dispersive x-ray spectroscopy (EDX) was per- formed using a Thermo Fisher Talos F200X at 200 kV. The focused electron beam was raster scanned across the region of interest— 222×620 nm3large—and EDX spectra were collected by a four- quadrant detector (Super-X detection system, Thermo Fisher) from each scanning point (696 ×1940 spectra). The scanning step size was 320 pm, and the acquisition time was 10 μs per pixel. 70 frames were acquired, and the collected EDX spectra of each frame were summed up, resulting in an improved signal-to-noise ratio. A beam current of 3.2 nA was used to resolve the low Pd signal within the sample. For quantification, back- ground subtracted Pd–K and Ag–K lines were considered (using an empirical power law fitting). The peak areas were weighted by Brown–Powell ionization cross sections, as given within the analysis software (Velox 2.13, Thermo Fisher Scientific). E. Calculations The 4d PES spectra of the free Pd atom were computed using Quanty using Slater integrals and the spin–orbit coupling parameter computed for the neutral free atom at the Hartree–Fock level and empirical spectral broadening.34Density functional theory (DFT) calculations of the solids were performed using the Quantum ESPRESSO package version 6.4.135at the PBE level using pseudopo- tentials from the PSlibrary36with a kinetic energy (charge density) cutoff of 60 Ry (600 Ry) for scalar relativistic and fully relativistic J. Chem. Phys. 154, 184703 (2021); doi: 10.1063/5.0045936 154, 184703-2 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp simulations. AgPd SAAs were computed by substituting a single Ag atom with a Pd atom in (2 ×2×2) and (3 ×3×3) crystallographic supercells of face-centered-cubic (fcc) Ag, and relaxing all atom positions and lattice vectors until forces dropped below 10−3a.u., the cell-pressure dropped below 0.5 kbar, and the change in total energy was below 10−4Ry at the scalar relativistic level. Fixed geometry cal- culations were performed including spin–orbit coupling for Pd. The PDOS was generated with a broadening of 0.17 eV. Core level shifts were computed using the ΔSCF method. III. RESULTS AND DISCUSSION A. Homogeneity and electronic structure of Ag 0.98Pd0.02SAA in comparison with its plain constituents Before going into the details of the valence states of the Ag0.98Pd0.02alloy, it is necessary to verify that the Pd atoms are indeed present as geometrically isolated single sites in the Ag host. This is a challenging but inevitable part of all SAA studies. For the case of AgPd alloys, it is especially difficult since the constituents are neighboring atoms in the Periodic Table, and therefore, it is not pos- sible to distinguish them by their contrast using high-angle annular dark-field (HAADF)-STEM. However, it is reasonable to expect that in low concentrations, Pd is present statistically as isolated atoms, since the AgPd phase diagram shows them to be completely misci- ble,37and that the heat of formation is always negative, indicating an attraction between Ag and Pd.38Additionally, DFT calculations reveal a positive aggregation energy for the formation of Pd dimers and trimers in an Ag host. From these studies, it can be concluded that Pd atoms prefer to be isolated.39–41 Further evidence for the Pd atom isolation was found in stud- ies on AgPd alloys using a variety of methods including STM,42CO adsorption,3and a simulation method based on machine learning.43 To confirm these previous findings, we have used EDX mapping in a TEM on a thin piece of the alloy. Figure 1(a) showsa dark field (DF) image of the analyzed sample position. The simul- taneously collected EDX signal results in Ag and Pd maps as given in Fig. 1(b). The uniform distribution of the Pd signal in the EDX map indicates that the Pd is homogeneously distributed. The low Pd signal, and therefore high signal-to-noise ratio, hinders a clear inter- pretation on the very local scale (for a more detailed discussion on the homogeneity of Pd, see Fig. S1 in the supplementary material). However, integrating the EDX signal over the whole region shown in panels (a) and (b) gives a clear Pd peak [yellow arrows in Fig. 1(c)], which is identified by 2 ±0.5 at. % [see Sec. II (the Experimental section) for details on the quantification]. Stronger evidence of the isolated nature of the Pd atoms can be seen in the XPS core level and valence band spectra, as explained here, with a discussion of the differences in the electronic struc- ture of the Pd3d core level and the valence band states of the Ag0.98Pd0.02alloy and pure Pd. Figure 2(a) shows the Pd3d spec- trum of bulk metallic Pd. The peaks are asymmetric (asymmetry factor at 10% peak height is 0.4), with a tail extending to the high binding energy side of the main peak. The asymmetry is a result of intrinsic energy losses caused by interactions of the core-level electrons with the valence band electrons.29,44The degree of asym- metry depends on the local DOS at the Fermi level.45The DOS at the Fermi edge for bulk Pd is very high [Fig. 2(b)] because the Pd4d states are not completely filled, having an electron configuration of 4d9.55s0.5.45,46 In contrast, the Pd3d line shape in the Ag 0.98Pd0.02alloy is quite symmetric (asymmetry factor of 0.95). The reason for the more sym- metric shape is the low DOS around the Fermi level in the alloy. In the alloy, the Pd4d band becomes filled so that the states around the Fermi level in AgPd are primarily Ag5s [Fig. 2(b)]. Thus, few valence excitations are available to cause the energy loss that gives rise to peak asymmetry. Additionally, it can be seen in the valence band spectra that the Pd4d valence states are well separated from the Ag4d states [Fig. 2(b)], which can be explained by weak wave-function mixing of the metal 4d states, as was previously also shown for a Ag 0.995Cu 0.005 FIG. 1 . STEM–EDX analysis: (a) dark field image of the Ag/Pd sample. (b) EDX maps showing the elemental distribution of Ag (left) and Pd (right) over the sample position as shown in (a). The color-coded signal corresponds to the net counts of the corresponding EDX signal (i.e., after background subtraction). (c) EDX spectrum extracted from the whole region shown in (a) and (b) (top). The Ag–K peaks are clearly resolved, while the Pd signal is hardly visible (yellow arrows) due to its low content of ∼2 at. % relative to the Ag signal. Therefore, a magnified view of the same spectrum is shown in the lower panel, now clearly displaying the Pd-K αbut also Pd-K βpeak. The blue lines are Gaussian fits to the experimental data (gray dots). J. Chem. Phys. 154, 184703 (2021); doi: 10.1063/5.0045936 154, 184703-3 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Comparison of (a) the Pd3d and (b) the valence states of the Ag 0.98Pd0.02 alloy with the polycrystalline Pd foil. SAA,30and by DFT calculations of the DOS of multiple SAA combi- nations.5The weak wave-function mixing results from the fact that the filled Ag4d valence states are located below the Fermi level. For this reason, almost no hybridization of the Pd4d and Ag4d states is possible; hence, the Pd electrons are located at the Pd atom forming a narrow, electronically isolated virtual bound state.26 Spectroscopic evidence of the homogeneity of the AgPd SAA can be found in the XPS line shape of the Ag3d 5/2and the Pd3d 5/2 signals. As mentioned above, the line shape depends on the DOS at the Fermi level. In a homogeneous alloy, the components share the same DOS; therefore, they are expected to have similar line shapes(in the absence of satellites on the solute atom). Figures 3(a) and 3(b) show a comparison of the line shape of Pd3d in Ag 0.98Pd0.02with that of Ag3d. Here, we fit the Ag3d 5/2signal using the generalized Voigt line shape (LF in CasaXPS 2.3.23) on a Tougaard background. The exact same line shape was then used to fit the Pd3d 5/2signal. As one can see in Figs. 3(a) and 3(b), an identical line shape can be used to fit both signals reasonably well, suggesting that they share a similar electronic structure, both in terms of initial-state and final-state. Further indication for successful alloying is the core level shift (CLS) of the Ag3d 5/2peak from 368.20 eV for pure Ag to 368.18 eV for Ag 0.98Pd0.02and 368.10 eV in the case of an Ag 0.95Pd0.05alloy. The peak shift goes hand-in-hand with an increase in the FWHM of Ag3d 5/2from 0.70 eV for the Ag reference sample to 0.78 eV for the Ag 0.95Pd0.05sample [Fig. 3(c)]. Calculations of the CLS of Ag, as a function of the Pd neighbors, revealed a linear correlation toward lower binding energies of Ag.26,47The CLS of Ag3d toward lower binding energies, as the concentration of Pd increases, can be explained by charge transfer of Ag5s states to Pd4d states, as it can also be observed for the metal-to-oxygen charge transfer in oxidized Ag.48This negative shift is attributed to final-state effects, where the screening charge has bonding 5s character.48The impact of charge transfer is very weak for the highly diluted Ag 0.98Pd0.02alloy, where the CLS accounts only to 0.02 eV. As the concentration of Pd in Ag increases, additional hybridization effects lead also to a more nega- tive shift of the Ag3d states.26The broadening of the Ag core level signal suggests that there are distinct CLSs for Ag atoms adjacent to Pd atoms and Ag atoms not in the direct neighborhood of Pd atoms.47 Additionally, slight differences in the binding energy of the Pd3d 5/2core level could be observed from 335.04 eV for the pure Pd metal, 335.12 eV for Ag 0.98Pd0.02, to 335.06 eV for Ag 0.95Pd0.05 [Fig. 3(d)]. It should be mentioned that the CLS does not show a lin- ear trend in the case of Pd, and it is important to distinguish between initial- and final-state effects. Due to the higher density of states FIG 3 . Fitting of the Ag3d 5/2(a) and Pd3d 5/2(b) signals of Ag 0.98Pd0.02using the same line shape. An identical line shape is a sign for a homogeneous alloy because it means that the constituents share the same DOS. CLS and peak broadening of Ag3d 5/2(c) and Pd3d 5/2(d) due to alloying in an Ag 0.98Pd0.02and AgPd 0.58Pd0.05alloy compared to pure Ag and Pd. The colored points represent the FWHM. J. Chem. Phys. 154, 184703 (2021); doi: 10.1063/5.0045936 154, 184703-4 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp near the Fermi level in the bulk Pd metal, compared to Ag 0.98Pd0.02, Doniach–Sunjic core-hole screening,44which leads to asymmetry, plays a more dominant role in the Pd metal than in Ag 0.98Pd0.02, where there is almost no screening charge at the Fermi level. Since bulk Pd and Pd in the AgPd SAA have very different DOS, it is not straightforward to directly compare and discuss the Pd3d peak positions of the Pd metal and Ag 0.98Pd0.02. In contrast, Ag 0.98Pd0.02 and Ag 0.95Pd0.05 have very similar valence electronic structures, and the Pd3d peak positions can be more directly compared. In this case, we see that the peak position shifts slightly toward lower binding energy, as the Pd concentration increases. As mentioned above, the Pd4d virtual bound state in AgPd SAA shifts below the Fermi level and becomes very narrow due to the localization of the Pd electrons around the atom because of the weak wave function mixing of the Ag4d and Pd4d states. One would expect that this effect would lead to a decrease in the bind- ing energy for the Pd3d core levels in the initial state since the Pd4d band becomes more populated and more negatively charged due to the charge transfer of Ag5s states toward Pd4d states. However, in an AgPd SAA, the screening charge at the Fermi level changes its character from 4d orbitals with a high DOS in pure Pd to sp char- acter from the neighboring Ag atoms with low DOS at the Fermi level. The low DOS at the Fermi edge leads to very little core-hole relaxation and hence to a pronounced final-state effect. As a conse- quence of the reduction in screening charge, the shift toward higher binding energies becomes more pronounced as the Pd concentration decreases toward infinite dilution.26Therefore, the positive shift is higher for Ag 0.98Pd0.02than for Ag 0.95Pd0.05. In order to validate the measured CLS, the CLS of AgPd SAAs in comparison to Pd bulk was computed by substituting a single Ag atom with a Pd atom in (2 ×2×2) and (3 ×3×3) crystallographic super-cells of fcc Ag, revealing Ag 31Pd1(Ag 0.97Pd0.03) and Ag 107Pd1(Ag 0.99Pd0.01), respectively. For both supercells, the computed Pd3d 5/2CLS is 0.08 eV, which is precisely the shift we measured for the Ag 0.98Pd0.02alloy [Fig. 3(d)]. This result demonstrates that no further CLS and elec- tronic structure changes are expected when diluting Pd below 3 at. % in Ag. It should be noted that the discussed CLSs for Ag- and Pd3d 5/2states are very small, but previous researchers performed experimental and theoretical calculations of the CLS in Ag–Pd alloys over the whole concentration range.26,49Thereby, they found that the effect of inter-atomic d-electron charge transfer, which is often discussed as origin for CLSs, is negligible for Ag–Pd alloys.26 B. Ag 0.98Pd0.02valence band In this section, we analyze in more detail the valence band of Ag0.98Pd0.02measured with synchrotron radiation and discuss the findings by considering the literature from the 1960s to 1980s, where electronic structure changes induced by transition metal impuri- ties in another transition metal were intensively studied.31–33The materials previously studied are, in principle, the same as the mate- rials now referred to as SAAs. As shown in Fig. 2(b), the Pd4d states are well separated from the Ag4d states, but the FWHM of the Pd4d virtual bound states is 0.85 eV broader than it was for the Ag 0.995Cu 0.005 SAA (0.5 eV).30This observation opens the question: Which parameters have an impact on the width of thevalence states? Parameters giving rise to broadening are crystal field (d–d) coupling, s–d coupling, and spin–orbit coupling. In general, the d-band width increases along a row of the Periodic Table and reaches its maximum in group 5, and then it decreases again as the d-band becomes filled. Along the column from 3d to 5d elements, the d-band width increases due to more interacting electrons.50Con- sequently, the fact that Pd is a 4d and Cu is a 3d element and that Pd is in group 10, whereas Cu is in group 11 leads to broader Pd valence states. Norris and Meyers claimed already in 1971 that even at infi- nite dilution, there is broadening of the Pd4d bound states due to the spin–orbit splitting and the s–d coupling. The d–d interactions (crystal field effects) can be neglected in the case of transition metal impurities in the host metal.31In 1985, van der Marel et al. inves- tigated experimentally (UPS) and theoretically Pd and Pt induced changes in noble-metal density of states.30From the UPS results of a Ag 0.97Pd0.03alloy, it became clear that spin–orbit splitting plays a role for the broadening of the Pd4d valence states. Consequently, we fitted the Pd4d states of the Ag 98Pd0.02sample using two sig- nals with the typical area ratio for 4d states of 2:3. In Fig. 4(a), it can be seen that this approach results in a good fit for the Pd4d signal shape. The Pd4d 5/2and the Pd4d 3/2peak in the fit are cen- tered at a binding energy of between 1.8 and 2.2 eV. The posi- tions are identical with the one observed in the literature for the Ag0.97Pd0.03alloy.30In that publication, the authors found an identi- cal FWHM of 0.8 ±0.1 eV for the entire Pd4d signal. Furthermore, they theoretically calculated the valence band using a spin–orbit cou- pling and an s–d coupling parameter but no crystal field coupling parameter. This agrees with the predictions of Norris and Meyers that only spin–orbit coupling and s–d interactions have an influence on the broadening.33 To estimate, if the Pd atoms in the Ag 0.98Pd0.02alloy really behave like electronically isolated free-atoms sites, as it was pre- dicted for the free-atom like Ag 0.995Cu 0.005 SAA,30one can compare the impurity valence d-states of the alloy with the calculated states of a single metal atom in the gas phase. The result of the calcu- lations of the photoemission signal (PES) of such an isolated gas phase atom is shown in Fig. 4(b), which demonstrates that spin– orbit splitting plays a significant role in the valence states of a free Pd atom. The calculated splitting is 0.4 eV and is very close to the result obtained by fitting the Pd4d valence states in the XPS spectra [Fig. 4(a)]. Since spin–orbit splitting can account for the Pd4d band width, there might be very little hybridization between the Pd and Ag valence d-states, and therefore a free-atom like behavior of Pd.51–53To verify this proposition, the Pd4d PDOS of Ag31Pd1was computed by fully relativistic DFT and a broadening of 0.17 eV [Fig. 4(d)]. It can be observed that there is also a weak Pd4d electron density between −6 and −3 eV, the range of the Ag4d states, and consequently, there is a small degree of hybridization between Ag and Pd. For comparison, the PDOS of bulk Pd is also plotted [Fig. 4(c)], which shows the width and the high DOS at the Fermi edge of Pd4d. Our Ag 0.98Pd0.02 SAA provides a reference for the often- discussed phenomenon of site-isolation, whereby in this case, geo- metric site-isolation is accompanied by electronic site-isolation. In Sec. III C, we will discuss the effect of the geometric and electronic site-isolation on the catalytic performance of AgPd SAAs in the selective hydrogenation of acetylene. J. Chem. Phys. 154, 184703 (2021); doi: 10.1063/5.0045936 154, 184703-5 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . (a) The Pd4d valence states are fitted using two peaks due to spin–orbit splitting, and thereby, the typical area ratio of 2:3 is used. (b) Calculated PES of a free Pd atom with spin–orbit splitting. The energy scale is adjusted to the XPS experiment. The calculated splitting is 0.4 eV, closely matching the splitting in the measured signal. (c) PDOS of bulk Pd. (d) Pd4d PDOS of Ag 31Pd1including the spin–orbit splitting of Pd4d 3/2and Pd4d 5/2. C. Discussion of the catalytic properties of AgPd SAA in selective acetylene hydrogenation Previous work has shown that bimetallic AgPd alloys can exhibit increased selectivity toward ethylene, compared to monometallic Pd catalysts, which also form side products, due to its high activity.20,21,54,55Those side products are fully hydrogenated ethane and polymers, known as green oil, which poison the cata- lyst.20–22It is believed that alloying Pd with Ag increases ethylene selectivity for two reasons: (i) it hinders the formation of sub-surface hydrogen, which is responsible for the full hydrogenation path- way toward ethane,21,55,56and (ii) it weakens the binding strengths toward all surface intermediates, leading to increased desorption of the desired product, ethylene.57–59The latter effect is because the d-band of Pd becomes filled due to alloying,26,60,61which weakens the adsorption of π-electron donors such as acetylene and ethylene, since they can donate fewer electrons into the Pd d-band.20In the present work, the valence band spectra (Fig. 4) indicate a full d-band for Pd, strongly supporting this model. Due to the filled d-band, the ethylene desorption barrier becomes lower than the hydrogenation barrier,57and also C–C coupling, which leads to the formation of green oil, is unlikely to occur.62 On the other hand, one can say that Ag is inactive in acetylene hydrogenation since (defect free) Ag ensembles can neither adsorb acetylene63,64nor activate H 2.65Therefore, although the selectivity increases when Pd is alloyed with Ag, the overall activity of the selec- tive hydrogenation reaction decreases with increasing Ag content. Consequently, the question arises: Which Ag:Pd ratio is ideal to cat- alyze the reaction. Khan and co-workers thoroughly summarized the requirements of an ideal AgPd catalyst:21“the Pd–Ag system should have, on the one hand, the Pd-rich surface to dissociate hydrogen and catalyze reaction with acetylene and, on the other hand, an Ag-rich core to prevent hydrogen migration into the particle .” In their work, they showed that an AgPd model catalyst became inactive at veryhigh Ag surface coverages. In essence, if H 2is activated at Pd sites and if acetylene adsorbs at Pd sites, then in the case of site-isolation, the adsorbates must be able to diffuse to each other for the reaction to occur. Tierney and co-workers demonstrated theoretically13and by STM experiments14that H 2dissociation can occur on isolated Pd sites of CuPd and AuPd SAAs; however, they found that only for CuPd SAAs, a subsequent spillover of hydrides to the host atoms (Cu) was possible.11,14,15Previous DFT calculations of the formation energy of adsorbed H atoms by Darby et al. revealed formation ener- gies of 0.16 eV for a Ag(111) surface and −0.12 eV for a Pd-doped Ag(111) surface, whereas the energies for a Pd-doped Cu(111) and plain Cu(111) surface were between −0.27 and −0.26 eV, respec- tively.66These values verify why spillover of H atoms is likely to occur on CuPd SAAs but not on AgPd. Thus, we expect that without H spillover, isolated Pd sites in AgPd SAAs must be sufficiently close to one another for semi-hydrogenation to occur. The work of Armbrüster and co-workers on Ga–Pd inter- metallics demonstrated such an effect.67–69In that example, Pd and Ga are covalently bonded in a ordered crystal structure dif- ferent from their constituents. Pd atoms are isolated in that no neighboring Pd atoms are present in the first coordination sphere.68 The isolation of Pd sites hinders the formation of bulk hydrides and therefore avoids the formation of fully hydrogenated ethane,69while the close proximity of neighboring Pd atoms allows for adsorbates to encounter one another. The experiments show that GaPd 2com- pared to GaPd has a 30-fold higher activity but the same selectivity (75%). The reason for the lower activity of GaPd is explained by the differences in the valence band structure. GaPd has a lower DOS at the Fermi edge (0.2 states eV−1atom−1), a lower lying d-band cen- ter (0.4 eV), and a narrower d-band width (0.9 eV) than GaPd 2,67,68 and similar effects were observed for the Pd valence structure of AgPd in the present work. As mentioned above for the AgPd alloys, this electronic effect (i.e., filling of the d-states) enhances the J. Chem. Phys. 154, 184703 (2021); doi: 10.1063/5.0045936 154, 184703-6 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp selectivity toward ethylene. In the case of AgPd, site-isolation could, in principle, still be obtained at Pd concentrations as high as 7 at. %— that is, one Pd atom for every 12 Ag atoms—such that in the fcc unit cell, the 12 atoms in the coordination sphere of a Pd atom are all Ag. While close proximity of isolated sites can enable the semi- hydrogenation reaction to occur, the hydrogen uptake can also be increased by the support. In a previous study by Pei et al. , it was found that nanoparticle-sized, SiO 2supported AgPd SAA catalysts were efficient and highly selective in the hydrogenation of acetylene in an excess of ethylene.3In their study, H 2activation was likely enhanced by the presence of the support. Claus and co-workers’ work on H–D exchange experiments on Ag/SiO 2, investigated by FTIR spectroscopy, revealed that even the bare SiO 2support is able to activate H 2.70,71Other groups found that the H 2dissociation on such non-reducible supports can be attributed to defect sites in the support.72,73It was experimentally shown that the H 2activation increases for Ag nanoparticles on the SiO 2support,70,71which could be explained by H–H exchange at the metal-support interface, where a H atom on the metal exchanges with a proton of the OH (silanol) group on the SiO 2support.73,74Thereby, residual water molecules accelerate the H–H exchange.73,75Such metal–support interactions are possible for the supported AgPd/SiO 2SAA catalysts, but not for the unsupported AgPd SAA foil examined in the present work. In addition, the AgPd nanoparticles in AgPd/SiO 2might contain additional electron-deficient sites (defect sites and step and corner atoms), which lead to valence d-band vacancies and the facilita- tion of metal–H bond formation. Hence, Ag atoms in nanoparticles might be able activate H 2.71,76 In summary, a hydrogenation reaction requires two initial steps, the dissociation of hydrogen and the adsorption of the hydro- carbon (here acetylene). The alloying of Pd with another metal such as Ga or Ag hinders the formation of sub-surface hydrogen and also weakens the binding toward the desired product (ethylene) due to d-band filling, and both effects increase the selectivity.51,52,69The dis- sociation of hydrogen is possible at the single-atom Pd sites in SAAs, and for CuPd SAAs, a subsequent spillover of H atoms to the Cu host occurs, increasing the hydrogen uptake.14,15Additionally, the spillover enables the reaction of the hydrides with acetylene, which is adsorbed on the Pd sites. AuPd and AgPd SAAs are not expected to exhibit H 2spillover to the host due to weak binding between H atoms and Ag or Au.11,13,14,66Hence, the H atoms cannot approach and react with acetylene when the isolated Pd sites are far away from each other. In intermetallics such as GaPd, the Pd sites are also geometrically isolated but close enough for the adsorbates (H and acetylene) to react with each other.67,69In supported SAA catalysts, such as AgPd/SiO 2, H 2can be activated at the defect sites of the sup- port, and also electron-deficient Ag sites might activate H 2.71Those two effects increase the H 2uptake and enable the reaction of H 2and acetylene. IV. CONCLUSION Here, we have investigated an unsupported AgPd single-atom alloy foil using electron spectroscopy and computational chemistry. This Ag 0.98Pd0.02alloy is one of the few documented cases of elec- tronic site-isolation in heterogeneous catalysis, and we demonstrateand discuss the ways in which its unusual free-atom like electronic structure is manifested in the measured spectra. In particular, a comparison of the core level Pd3d line shapes of Ag0.98Pd0.02and bulk Pd revealed a symmetric line shape for Pd3d in AgPd. The change in the line shape can be referred to the change in the local density of states at the Fermi level. Bulk Pd has very high density of states due to unfilled Pd4d states, which lead to asymmet- ric core-level line shapes. In contrast, the Pd4d states in the AgPd single-atom alloy are very narrow and filled and are hence shifted below the Fermi level. Furthermore, it was found that the Ag3d and Pd3d core-level states can be fitted using exactly the same line shape, which indicates that both metals share the same density of states and that the Pd atoms are homogeneously distributed in the alloy. Addi- tionally, the Pd4d valence states in Ag 0.98Pd0.02are well separated from the Ag4d valence states. A comparison of the Pd4d valence states in the single-atom alloy with a calculated photoemission signal of a free Pd atom in gas phase showed good agreement. Conse- quently, there is a weak wave-function mixing of the Pd4d states with their surroundings, which verifies the electronic site-isolation of the Pd atoms. We also discussed the effect of the site-isolation on the catalytic activity in acetylene hydrogenation. From the discussion, we con- cluded that the geometric and electronic Pd site-isolation in the inert Ag host should reduce the catalytic performance of the Pd atoms, as adsorbates (H 2and acetylene) on isolated Pd sites of the Ag 0.98Pd0.02 alloy would be too far away from each other to react. This issue can be avoided when the host material participates in the reaction, e.g., in CuPd SAAs, H 2will be dissociated by the Pd sites followed by spillover to the Cu host.15In addition, defect sites in the support are able to activate H 2and hence increase the H 2uptake and the catalytic performance.3,71,73 SUPPLEMENTARY MATERIAL Analysis of the homogeneity of Pd in Ag by STEM–EDX, cat- alytic investigation of AgPd SAA and Pd in acetylene hydrogenation, and information of the Ag3d to Pd3d ratio by heating the AgPd SAA in vacuum and in H 2can be found in the supplementary material. ACKNOWLEDGMENTS We acknowledge the Max Planck Society and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)— Grant No. 388390466—TRR 247 for funding. We also acknowledge the Helmholtz-Zentrum Berlin for the use of their infrastructure. Furthermore, we thank Detre Teschner and Frederic Sulzmann from the Fritz Haber Institute for the scientific advice and support during the synchrotron measurements. DATA AVAILABILITY The XPS data that support the findings of this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.4481984, https://doi.org/10.5281/zenodo.4482000, and https://doi.org/10.5281 /zenodo.4482138. J. Chem. Phys. 154, 184703 (2021); doi: 10.1063/5.0045936 154, 184703-7 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp REFERENCES 1R. T. Hannagan, G. Giannakakis, M. Flytzani-Stephanopoulos, and E. C. H. Sykes, Chem. Rev. 120, 12044 (2020). 2M. B. Boucher, B. Zugic, G. Cladaras, J. Kammert, M. D. Marcinkowski, T. 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5.0046080.pdf

Cavity quantum electrodynamics design with single photon emitters in hexagonal boron nitride Cite as: Appl. Phys. Lett. 118, 244003 (2021); doi: 10.1063/5.0046080 Submitted: 31 January 2021 .Accepted: 27 May 2021 . Published Online: 16 June 2021 Yanan Wang,1 Jaesung Lee,1 Jesse Berezovsky,2 and Philip X.-L. Feng1,a) AFFILIATIONS 1Department of Electrical and Computer Engineering, Herbert Wertheim College of Engineering, University of Florida, Gainesville, Florida 32611, USA 2Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106, USA Note: This paper is part of the APL Special Collection on Non-Classical Light Emitters and Single-Photon Detectors. a)Author to whom correspondence should be addressed :philip.feng@ufl.edu ABSTRACT Hexagonal boron nitride (h-BN), a prevalent insulating crystal for dielectric and encapsulation layers in two-dimensional (2D) nanoelec- tronics and a structural material in 2D nanoelectromechanical systems, has also rapidly emerged as a promising platform for quantum pho- tonics with the recent discovery of optically active defect centers and associated spin states. Combined with measured emissioncharacteristics, here we propose and numerically investigate the cavity quantum electrodynamics scheme, incorporating these defect-enabledsingle photon emitters (SPEs) in h-BN microdisk resonators. The whispering-gallery nature of microdisks can support multiple families ofcavity resonances with different radial and azimuthal mode indices simultaneously, overcoming the challenges in coinciding a single point defect with the maximum electric field of an optical mode both spatially and spectrally. The excellent characteristics of h-BN SPEs, including exceptional emission rate, considerably high Debye–Waller factor, and Fourier transform limited linewidth at room temperature, renderstrong coupling with the ratio of coupling to decay rates g/max( c,j) predicated as high as 500. This study not only provides insight into the emitter–cavity interaction, but also contributes toward realizing h-BN photonic components, such as low-threshold microcavity lasers andhigh-purity single photon sources, critical for linear optics quantum computing and quantum networking applications. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0046080 Single photon emitters (SPEs) in solid-state platforms are among the most crucial ingredients for developing integrated quantum pho- tonic circuits, 1,2which are envisaged to revolutionize information processing, communication, and sensing technologies in the future.Thanks to the advances in materials science and nanotechnology, aproliferation of optically active defect centers has been achieved inwide-bandgap (WBG) materials, 3–7which can be considered as “inverted atoms”—atomic impurities in otherwise perfect crystals associated with quantized optical transitions. The wide-bandgapattributes encompassing excellent electronic isolation and thermal andchemical stability promise robust single photon emission and excep- tional spin coherence even at room temperature. 8–10 Analogous to the SPEs in the conventional WBG materials (e.g., diamond and silicon carbide), the existence of defect-related SPEs in a hallmark van der Waals (vdW) WBG crystal, namely, hexagonalboron nitride (h-BN), was reported in late 2015 and sparked growingresearch interest. 11Extensive photophysical studies have led to encour- aging milestones, including control of single photon emission viaelectrical/magnetic/strain fields,12–14optically detected magnetic reso- nance,15,16and Rabi oscillation under resonant excitation.17Even though the physical nature of these defect centers is still under investi- gation, h-BN has proven to be a promising platform to explore light–matter interaction and enable on-demand single photon sources,endowed with large emission rate ( >10 6counts/s),11strong zero- phonon emission (Debye–Waller factor, FDW/C240.8),18high quantum efficiency ( /C2487%),19Fourier transform (FT) limited linewidth at room temperature,18and single photon purity even at 800 K.20 Toward the realization of quantum functionalities based on SPEs, efficient coupling to high-quality optical devices that can direct emis-sion into a single spatial/spectral mode and enhance the emission ratewith unit efficiency is requisite. Initial experiments have demonstrated coupling of h-BN defect centers to linear photonic crystal cavities, 21,22 silicon nitride (Si 3N4) microdisk resonators,23and dielectric Bragg microcavities,24while all in weak coupling regime with Purcell factors (FP) below 10. This is because in the emitter–cavity systems relying on heterogeneous integration,22,23the emission dipole is coupled to the Appl. Phys. Lett. 118, 244003 (2021); doi: 10.1063/5.0046080 118, 244003-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplevanescent field but not the maximum field confined inside the optical cavity. Moreover, a wide spectral range of emission has been observed from h-BN, with zero-phonon lines (ZPLs) spanning from ultraviolet (UV) to near-infrared (NIR) wavelengths.25–27Most current cavity designs work at a specific wavelength, and the ones comprising pho-tonic crystal lattices 21,22and fixed Fabry–P /C19erot cavities24are incompe- tent to afford tunability over a broad wavelength range. It can be seen that important challenges lie in matching the frequency of cavity reso- nance with the ZPL and aligning the SPE dipole with the maximum electric field of the optical mode simultaneously. In this Letter, we propose a coupled emitter–cavity system built upon defect center embedded within h-BN microdisk ( Fig. 1 ). The dif- ficulty of spatially and spectrally aligning the SPE with the cavity mode can be resolved in such a whispering-gallery geometry, of which cavity resonance can be flexibly engineered by tuning the radial and azi- muthal mode indices. Critical analysis of the ZPL energies of h-BN SPEs and the resonance wavelength, quality ( Q) factor, mode volume as functions of microdisk geometry have been performed. We further relate these parameters to the coupling and decay rates in cavity quan- tum electrodynamics (cavity-QED), leading to a quantitative and com- prehensive understanding of the emitter–photon interaction in strong coupling regimes. To address the frequency matching issue raised by the broad range of emission, we start with surveying the spectral distribution of emission from 560 nm to 750 nm. It is noticeable that we exclude the emitters with high photon energies (4–6 eV) in this study, because they may originate from localized exciton transitions close to the h-BN bandgap ( /C245.9 eV).28We evaluate the emission spectra from dozens of h-BN flakes mechanically exfoliated and then transferred onto patterned silicon dioxide on silicon (290 nm SiO 2on Si) sub- strates, by utilizing custom-built confocal microscope systems andspectroscopic techniques. 29The systems consist of high-magnification optical objectives (50 /C2and 100 /C2) and spectrometers (Horiba iHR550and Princeton Instruments 2500) with charge-coupled device (CCD) cameras sufficiently sensitive to record weak emission signals. Flakes possessing optically active defect centers produce bright fluorescence that can be directly visualized in fluorescence images with the below- bandgap excitation from a sub-milliwatt 532 nm laser. In our experi- ments, individual spectra are collected from isolated bright spots in fluorescence images with spatial resolution below 1 lm. Correlation measurements are performed by using the Hanbury Brown and Twiss (HBT) interferometry with single photon counting avalanche photodi-odes (Excelitas Technologies Inc.) and a timing module (PicoQuant Inc.), to further verify the single photon nature of the emission process. Figure 2(a) displays four typical spectra from our measurements at room temperature, most of which can be fitted into two Lorentz peaks with a relatively sharp line and a broader feature in lower energy, assigned as zero-phonon line (ZPL) and phonon sideband(PSB), respectively. It can be seen that the energies of different emitters spread over a considerably broad spectral range, while the ZPL wave- lengths cluster in four distinct positions, with peaks around 575 nm, 625 nm, 680 nm, and 725 nm and linewidth below 10 nm. We sum- marize in Fig. 2(b) the data from our observations, into a histogram of occurrences, combined with data from measurements at cryogenic temperatures. 25,27Each cluster of emission could originate from a spe- cific defect composition and involve at least one defect level lying deep within the semiconductor bandgap. However, the variations in local environment, such as strain and electric field from trapped charges, cause shifts or splitting of defect states and inhomogeneous broaden- ing of emission, accounting for a nearly continuous distribution in ZPL energies. An in-depth understanding of SPEs could be gained by compar- ing them with ab initio calculations.30–32Distinct from the well- studied defect-related SPEs in three-dimensional (3D) WBG crystals, such as nitrogen vacancy (NV) centers in diamond,3silicon vacancy (VSi) and divacancy (DV) centers in silicon carbide,4,5the atomic ori- gin of emission in h-BN, is still under debate. Here, we consider two types of defect sites, negatively charged boron vacancy (V B/C0)a n d carbon-related center (V NCB), which have been proposed to interpret the recent observation of optically detected magnetic resonance (ODMR).15,16,31,32As predicted in Ref. 31,VB/C0center could give rise to six intrinsic orbital levels and the triplet transition (1)3E0!(1)3A20 with dominant (1)3A2!(1)3A20component, of which transition energy DE¼1.78–1.83 eV agrees well with the emission detected around 680 nm [ Fig. 2(c) ]. Similarly, the transition (1)3A100!(1)3A20is calculated to possess an energy DE¼2.00–2.13 eV, corresponding to the ZPLs between 580 nm and 620 nm. Spin-triplet (2)3B1to (1)3B1 transition of the V NCBcenter, consisting of one nitrogen vacancy and one neighboring carbon atom substitution,32associates with ZPL energy of /C241.75 eV and wavelength at /C24710 nm [ Fig. 2(d) ]. In accordance with the spectral distribution of ZPLs discussed above, we explore the resonance behaviors of microdisk at 585 nm, 635 nm, and 715 nm, via finite-element eigenfrequency simulations. Taking advantage of the azimuthal symmetry, only a two-dimensional cross section of microdisk is simulated by employing the 2D axisym- metric space dimension in COMSOL MultiphysicsTM.33It is worth- while noting that the wavelength-dependent refractive indices and birefringence of h-BN are applied in our numerical simulations. As demonstrated in a recent report,34h-BN flakes exhibit significant FIG. 1. Schematic illustration of a coupled emitter–cavity system consisting of h-BN microdisk cavity and embedded defect center, which can serve as an intriguing plat-form for exploring the quantum light–matter interactions and facilitating functionalquantum photonic devices.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 244003 (2021); doi: 10.1063/5.0046080 118, 244003-2 Published under an exclusive license by AIP Publishingbirefringence due to the peculiar vdW layered structure, and in-plane refractive index nkand out-of-plane index n?can be calculated using Sellmeier’s equations n2 kkðÞ¼1þ3:336k2 k2/C026 322;n2 ?kðÞ¼1þ2:2631k2 k2/C026 981;(1) where the wavelength kis in nanometers. Relatively small out-of-plane refractive index ( /C241.8) compared to the in-plane index ( /C242.1) over the wavelengths of interest results in comparatively weak confinementof modes along the direction perpendicular to the h-BN layers, which can be directly visualized in the electric-field distribution as shown in Figs. 3(a) and3(b). Whispering-gallery modes of microdisk resonator follow the rela- tion mk¼2pr diskneff,w h e r e mis the azimuthal index, kis the reso- nance wavelength, rdiskis the microdisk radius, and neffis the effective refractive index. For device design, the resonance wavelength can beprecisely engineered by tuning the disk radius and azimuthal index.Figure 3(c) depicts the capability of matching the fundamental quasi-transverse-electric (quasi-TE 00) and quasi-transverse-magnetic (quasi-TM 00) modes with different ZPL wavelengths at the same time. For instance, a microdisk with radius rdisk¼2.1lm and thickness hdisk¼400 nm can support quasi-TE 00mode around 585 nm, 635 nm, and 715 nm with azimuthal index m¼40, 36, and 31, respectively. FIG. 2. Spectral distribution of emission and proposed defect centers in h-BN. (a) Typical emission spectra of defect-related SPEs in mechanically exfolia ted h-BN flakes. The raw spectra (hollow symbols) are processed with baseline subtraction, normalized to unity intensity, and fitted to Lorentz peak functions (solid lin es). (b) Histogram plot of ZPL energies for h-BN SPEs based on data in this work (purple), Ref. 25(light green), and Ref. 27(orange). The occurrence is normalized by dividing the total number of emitters. Illustrations of (c) negatively charged boron vacancy V B/C0and (d) carbon-related center V NCB, and their corresponding defect states within the bandgap (not in scale, including data from Refs. 31and32). ABS: absorption; PL: photoluminescence. FIG. 3. Whispering-gallery modes of h-BN microdisk cavity at different ZPL wave- lengths. Electric-field distribution of (a) quasi-TE 00and (b) quasi-TM 00modes for a microdisk ( rdisk¼2.5lm,hdisk¼400 nm) around 585 nm. (c) FEM-simulated cavity resonances as functions of disk radius and azimuthal mode number around585 nm, 635 nm, 715 nm, respectively. Red solid lines: quasi-TE 00mode; blue solid lines: quasi-TM 00mode.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 244003 (2021); doi: 10.1063/5.0046080 118, 244003-3 Published under an exclusive license by AIP PublishingThe free spectral range (FSR) of such a microdisk at 635 nm is calcu- lated as 14.3 nm. Although the predicated FSR is smaller than the wavelength difference between ZPL and PSB reported in Ref. 18,t h e large Debye–Waller factor of 0.8 and high quantum efficiency of/C2487% can ensure that the coupling between the cavity mode and ZPL is dominant. The PSB can be further filtered out by designing themicrodisk-waveguide or microdisk-fiber coupling schemes in realmeasurements. Further characterization of microdisk cavity is focused on reso- nances around 635 nm, at which FT limited lines from h-BN SPEshave been reported under resonant excitation at room temperature. 18 To determine the cutoff thickness of microdisk, the mechanical prop-erties of h-BN are also taken into consideration. A minimum thickness of 100 nm is set based on our previous studies to ensure that the h-BN flake is rigid enough to facilitate a pedestal-supported structure. 35,36 From the mode analysis performed with a fixed microdisk radiusr disk¼2.5lm, we can notice that the radiation-limited quality factor (Qrad) merely reaches 102for quasi-TM 00mode at /C24635 nm, when the microdisk thickness hdiskis below 150 nm [ Fig. 4(a) ]. As hdiskincreases to 500 nm, Qradescalates and saturates around 1016and 109for quasi- TE00and quasi-TM 00modes, respectively. Significantly lower Qradof quasi-TM 00mode confirms weak confinement resulting from smaller refractive index along out-of-plane direction. Effective mode volume(V eff)b e l o w lm3can be achieved for microdisk with hdisk>150 nm. In parallel, the influence of microdisk radius rdiskis investigated with thickness hdiskfixed at 400 nm [ Fig. 4(b) ]. Both quasi-TE 00and quasi-TM 00modes exhibit exponential dependence of Qradand approximately linear dependence of Veffonrdiskin the study range.For quasi-TE 00mode, Veffcan be as small as 0.5 lm3(equal to /C242k3), while maintaining Qradover 1010, at which level radiation losses are not expected to be the dominant loss mechanism. To date, the measured values from the h-BN microdisk with similar dimensions are in order ofQ/C2410 3,37,38presumably limited by the fabrication imperfections. In a coupled emitter–cavity system, the spatial mode density inside the cavity can be altered substantially; hence, the spontaneous emission rate can be either enhanced ( FP>1) or inhibited ( FP<1), determined by the Purcell factor Fp/C173 4p2k n0/C18/C193Q Veff; (2) where n0is the refractive index at the location of the emitter. Purcell factor as a function of quality factor Qa n de f f e c t i v em o d ev o l u m e Veffis plotted as Fig. 4(c) .W i t hs u b - lm3mode volume, enhancement (FP>1) can be realized even when Q¼103,a n d FPcan be as large as 300 if Qis improved to 105. Since the probability of spontaneous emis- sion placing a photon into the cavity is given by b¼FP/(FPþ1), deter- ministic photon emission into a single field mode is possible, if FPis sizable to make b/C251. The potential coupling strength can be described by the cooperativity parameter C/C173 4p2k n0/C18/C193Q Veffczpl ctotal/C12/C12/C12/C12~Eð~rdÞ ~Eð~rmÞ/C12/C12/C12/C122 ¼FPczpl ctotal/C12/C12/C12/C12~Eð~rdÞ ~Eð~rmÞ/C12/C12/C12/C122 ;(3) where czplis the rate of zero-phonon emission at wavelength kand ctotalis the total rate of spontaneous emission, czpl/ctotal¼FDW./C12/C12/C12/C12~Eð~rdÞ ~Eð~rmÞ/C12/C12/C12/C122 represents the overlapping of the emitter dipole and the maxi- mum electric field inside the cavity. Applying the reported Debye–Waller factor FDW/C240.8 and assuming a perfect alignment between the emitter and cavity mode, a theoretical expectation of the cooperativity parameter is over 200 for a microdisk with Q¼105. It is known that QandVeffdescribe the cavity decay rate ( j)a n d peak electric field strength within the cavity, respectively. To further translate the cavity mode analysis performed above to the standard parameters studied in cavity-QED, we evaluate the cavity decay rate j/2p¼x/4pQand coherent coupling rate g=2p¼1 2sspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ck2ssp 2pn3 0Veffs ; (4) where ssp¼2.65 ns is spontaneous emission lifetime and the value is adopted from the FT limited lines with emitter dephasing rate c/2p ¼63 MHz.18We also assume Q¼Qrad, and the defect center is opti- mally positioned within the cavity field. As shown in Figs. 5(a) and 5(b), the cavity decay rate decreases as the thickness and radius of the microdisk increase for both quasi-TE 00and quasi-TM 00modes. The coupling gradually strengthens with increasing thickness, while becomes weaker with enlarging radius. By comparing the coupling rate g/2pto the cavity decay rate j/2pand the emitter dephasing rate c/2p,w ec a nf u r t h e rd e t e r m i n e whether a system is in strong coupling or not [ Fig. 5(c) ]. It is notice- able that for the proposed system, strong coupling regime with g>max(j,c) can be reached as thickness hdiskexceeds 200 nm [Figs. 5(d) and 5(e)]. Because dissipation in a strongly coupled emitter–cavity system results from either cavity decay or emitter FIG. 4. Geometry dependence of mode confinement in h-BN microdisk cavity. Radiation-limited quality factor Qradand effective mode volume Veffas functions of microdisk (a) thickness and (b) radius around 635 nm. (c) Purcell factor scaling asa function of QandV effaround 635 nm. Red solid lines and red squares: quasi- TE00mode; blue solid lines and blue triangles: quasi-TM 00mode.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 244003 (2021); doi: 10.1063/5.0046080 118, 244003-4 Published under an exclusive license by AIP Publishingdephasing, the ratio of gto the maximum decay rate in the system g/max( j,c) is calculated, representing the number of coherent exchanges of energy (Rabi oscillations) between the emitter and cavity. Except for the microdisk with hdisk<250 nm (or rdisk <1.7lm), the loss is dominated by emitter dephasing and g/max( j,c) plateaus at a value above 500. In summary, we have synergized the experimental and analytical efforts to understand the defect-related single photon emission in h-BN. Utilizing the spectral configurability benefited from whispering- gallery geometry, we have proposed a coupled emitter–cavity systembased on h-BN SPE embedded within h-BN microdisk, numerically explored the cavity characteristics, and provided the design and theo- retical evaluation of light–matter interaction via cavity-QED analysis.The methodology and quantitative metrics developed in this study can serve as practical guidelines for designing and facilitating h-BN pho- tonic devices and integrated systems toward quantum science and engineering applications. The authors are thankful for the support from the National Science Foundation (NSF) via EFRI ACQUIRE program (Grant No. EFMA 1641099) and its Supplemental Funding through the Research Experience and Mentoring (REM) program. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. L. O’Brien, A. Furusawa, and J. 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5.0048146.pdf

J. Chem. Phys. 154, 214105 (2021); https://doi.org/10.1063/5.0048146 154, 214105 © 2021 Author(s).Model protein excited states: MRCI calculations with large active spaces vs CC2 method Cite as: J. Chem. Phys. 154, 214105 (2021); https://doi.org/10.1063/5.0048146 Submitted: 19 February 2021 . Accepted: 02 May 2021 . Published Online: 02 June 2021 Valérie Brenner , Thibaut Véry , Michael W. Schmidt , Mark S. Gordon , Sophie Hoyau , and Nadia Ben Amor COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical Chemistry ARTICLES YOU MAY BE INTERESTED IN Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). I. Revisiting the NEVPT2 construction The Journal of Chemical Physics 154, 214111 (2021); https://doi.org/10.1063/5.0051211 Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). II. The full rank NEVPT2 (FR-NEVPT2) formulation The Journal of Chemical Physics 154, 214113 (2021); https://doi.org/10.1063/5.0051218 Spin contamination in MP2 and CC2, a surprising issue The Journal of Chemical Physics 154, 131101 (2021); https://doi.org/10.1063/5.0044362The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Model protein excited states: MRCI calculations with large active spaces vs CC2 method Cite as: J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 Submitted: 19 February 2021 •Accepted: 2 May 2021 • Published Online: 2 June 2021 Valérie Brenner,1 Thibaut Véry,1,a)Michael W. Schmidt,2Mark S. Gordon,2 Sophie Hoyau,3 and Nadia Ben Amor3,4,b) AFFILIATIONS 1LIDYL, CEA, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette, France 2Department of Chemistry, Iowa State University, Ames, Iowa 5001, USA 3Université de Toulouse, UPS, LCPQ (Laboratoire de Chimie et Physique Quantiques), IRSAMC, 118, rte de Narbonne, F-31062 Toulouse Cedex, France 4CNRS, UPS, LCPQ (Laboratoire de Chimie et Physique Quantiques), IRSAMC, 118, rte de Narbonne, F-31062 Toulouse Cedex, France Note: This paper is part of the JCP Special Collection in Honor of Women in Chemical Physics and Physical Chemistry. a)Present address: IDRIS–CNRS, Orsay Cedex F-91403, France. b)Author to whom correspondence should be addressed: nadia.benamor@irsamc.ups-tlse.fr ABSTRACT Benchmarking calculations on excited states of models of phenylalanine protein chains are presented to assess the ability of alternative meth- ods to the standard and most commonly used multiconfigurational wave function-based method, the complete active space self-consistent field (CASSCF), in recovering the non-dynamical correlation for systems that become not affordable by the CASSCF. The exploration of larger active spaces beyond the CASSCF limit is benchmarked through three strategies based on the reduction in the number of determinants: the restricted active space self-consistent field, the generalized active space self-consistent field (GASSCF), and the occupation-restricted mul- tiple active space (ORMAS) schemes. The remaining dynamic correlation effects are then added by the complete active space second-order perturbation theory and by the multireference difference dedicated configuration interaction methods. In parallel, the approximate second- order coupled cluster (CC2), already proven to be successful for small building blocks of model proteins in one of our previous works [Ben Amor et al. , J. Chem. Phys. 148, 184105 (2018)], is investigated to assess its performances for larger systems. Among the different alternative strategies to CASSCF, our results highlight the greatest efficiency of the GASSCF and ORMAS schemes in the systematic reduction of the configuration interaction expansion without loss of accuracy in both nature and excitation energies of both singlet ππ∗and n π∗ COexcited states with respect to the equivalent CASSCF calculations. Guidelines for an optimum applicability of this scheme to systems requiring active spaces beyond the complete active space limit are then proposed. Finally, the extension of the CC2 method to such large systems without loss of accuracy is demonstrated, highlighting the great potential of this method to treat accurately excited states, mainly single reference, of very large systems. Published under license by AIP Publishing. https://doi.org/10.1063/5.0048146 I. INTRODUCTION Quantum chemistry now provides a large panel of tools to tackle the excited state calculation of molecular systems.1–4How- ever, still today, highly accurate methods, because of their high computational demands (consuming time as well as memory and disk resources), remain limited to small systems and even very small systems if their dynamics is addressed. One of the challengesis then to benchmark approximate methods against highly accu- rate ones, the objective being to define computational protocols using more efficient but less reliable methods. Once such bench- marks have been carried out and once such a protocol has been defined and validated, excited state potential energy surfaces can be investigated, and then, their photoinduced chemical dynamics can be explored. In this spirit, we recently developed a protocol5–8 combining three levels of theory to study both the excited state J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp spectroscopy and dynamics of models of phenylalanine pro- tein chains. First, the time-dependent density functional theory (TDDFT)9is used in non-adiabatic dynamics simulations in order to qualitatively investigate the deactivation mechanisms, and then, two higher levels of theory, the standard approximate coupled clus- ter singles and doubles method (CC2)10–14and a multireference configuration interaction (MRCI) method,15–17are used in order to address them quantitatively. One key point in the development of this computational pro- tocol was to demonstrate the validity of the CC2 method for models of phenylalanine protein chains by comparison with MRCI calcula- tions, the method that allows one to access the properties of excited states of such systems, the first and the second derivatives of the energy being affordable. The comparison between CC2 and MRCI calculations was already done in our previous work on a building block of proteins, a capped peptide containing one residue.5We now evaluate if these reliable performances of the CC2 method can be extrapolated to larger systems such as capped peptides contain- ing more residues and/or containing more extended side chains. However, the active space size that can be affordable in a com- plete active space self-consistent field (CASSCF)18calculation is reached with the capped peptide containing one residue, and alter- natives to the CASSCF must, therefore, be found to tackle larger systems. Two theoretical challenges arise for the multireference (MR) methods when expanding the size of the systems: the ability to obtain accurate zeroth-order reference wave functions (WFs) with the optimization of the orbitals and configuration interaction (CI) coefficients, on one hand, in order to obtain the major part of the non-dynamical or static correlation and, on the other hand, the introduction of the dynamical electronic correlation. Multiconfig- uration self-consistent field (MCSCF)19–21or CASSCF are usually the methods used in the first step where, at least, all the orbitals for which the occupation numbers vary significantly are defined as active. However, these methods are currently limited to 18 elec- trons in 18 orbitals as the number of determinants becomes too large in the configuration interaction part. To overcome this limit, a solution is to restrict the number of determinants, for exam- ple, by partitioning the active space into groups of orbitals as has been proposed in different methods such as the configuration inter- action spaces with restrictions on the orbital occupancies,22,23the macroconfiguration approach,24the Restricted Active Space (RAS) SCF (RASSCF)25,26method, the Generalized Active Space (GAS) SCF (GASSCF)27–30method, or the Occupation-Restricted Multi- ple Active Space (ORMAS)31,32method. The RASSCF scheme splits the active space into three subspaces, while the restriction acts on the degree of excitation of the determinants. In the GASSCF and in the ORMAS methods, the number of subspaces is not lim- ited. The difference between them consists essentially in how the electron excitation between subspaces is managed. Starting from these zeroth-order wave functions, the dynamical correlation can be taken into account by multireference second-order Pertubation Theory (PT): multireference Møller–Plesset (MRMP) perturbation theory,33,34multiconfigurational quasi-degenerate perturbation the- ory (MCQDPT),35,36Complete Active Space with Perturbation at the Second Order (CASPT2),37or second-order N-electron valence state perturbation theory (NEVPT2)38,39methods on the CASSCF wave functions, RASPT240on the RASSCF ones, and ORMAS-PT41on theORMAS ones. The GASPT242method has also been developed but is not yet available in the standard version of MOLCAS.43–46Multiref- erence Configuration Interaction (MRCI) can also be used to intro- duce dynamical correlation on top of these zeroth-order wave func- tions. However, the computational cost of these methods is much greater than that of perturbative ones, and without further approxi- mations, the MRCI methods are restricted to rather small systems. The reduction in the CI-matrix size can be obtained by selecting only the most important determinants or configurations. The most evident way is to restrict the excitation degree, for example, from the full CI to the single and double CI or Difference Dedicated CI (DDCI),47,48where all two hole–two particle excitations external to the active space are excluded, which is suitable for the calculation of vertical excitation energies. As dynamic electron correlation is a local phenomenon, long-range interactions can also be neglected, and linear-scaling CASPT2,49NEVPT2,50or MRCI15–17,51–54methods have been developed. We report here first an evaluation of the performances of differ- ent alternatives to CASSCF on a series of capped peptides of increas- ing size, the NAPA Bconformer of the N-acetylphenylalaninylamide and conformers of larger systems, the capped Ac–Gly–Phe–NH 2A and Ac–Gln–Phe–NH 2Cconformers, two dipeptides with differ- ent side chains. The model proteins, the CC2 method as well as the multireference approaches used, and, in particular, the princi- ple and parameters of three different alternatives to CASSCF among the most recent ones are described in the Sec. II. In the Sec. III, the CC2 and multireference results are presented and discussed on the series of systems, the criteria of the selection of the most effi- cient alternative to CASSCF being detailed. Finally, the validity of the CC2 methods for larger systems is demonstrated in Sec. IV by comparison with the multireference calculations. II. METHODS AND COMPUTATIONAL DETAILS A. Model proteins The model proteins first consisted of a conformer of our reference system, a capped peptide with one residue, the N- acetylphenylalaninylamide (NAPA B), and second consisted of con- formers of two larger systems, the Aconformer of the capped Ac–Gly–Phe–NH 2dipeptide, which contained one glycine (Gly) and one phenylalanine (Phe), and the Cconformer of the capped Ac–Gln–Phe–NH 2dipeptide, which contained one phenylalanine (Phe) and one glutamine (Gln) residue, a residue which also bores an amide group in the side chain. Low-lying excited states, i.e., the lowest ππ∗excited state localized on the phenyl ring and the low- est n π∗ COexcited states localized on the peptide bonds (one state per amide group) were investigated at the CC2/cc-pVDZ optimized geometry of the lowest ππ∗excited state.6All these excited state conformers adopted prototypical secondary structural features of proteins, these global structures being preserved from their ground state (Fig. 1 and Ref. 6 for details). The NAPA Bconformer corre- sponded to a γ-turn folded conformation stabilized by a C 7H-bond and an NH ⋅ ⋅ ⋅πbond. The Aconformer of Ac–Gly–Phe–NH 2cor- responded to the 2 7ribbon extended conformation stabilized by two successive C 7H-bonds (double γ-turn). The Cconformer of Ac–Gln–Phe–NH 2corresponded to a type I β-turn backbone, sta- bilized by a C 10H-bond combined to a side chain/main chain C 7 H-bond bridging the NH site of the first peptide bond to the oxygen J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1. NAPA (Ac–Phe–NH 2)B, Ac–Gly–Phe–NH 2A, and Ac–Gly–Phe–NH 2C conformers. The numbering of the peptide bonds starts from the CH 3CO group and continues along the backbone, taking into account the amide groups of the side chains. atom of the Gln residue side chain CO–NH 2group labeled 7δ. In addition, it contained a NH ⋅ ⋅ ⋅πbond, which implied the hydrogen atom of the second peptide bond. B. CC2 method CC210–14calculations were carried out with the TURBOMOLE package.55,56All the CC2 calculations were performed by using the resolution-of-identity (RI)57approximation for the electron repul- sion integrals in the correlation treatment and the description of the excitation processes. The Dunning cc-pVDZ correlation consistent basis sets58were employed in connection with optimized auxiliary basis sets57for the RI approximation. Frozen core for the 1s elec- trons was employed, and all calculations were carried out in the C1 point-group symmetry. Ten singlet states were considered, and D 1, D2diagnostics and % ⟨E1∣E1⟩biorthogonal norm were calculated in order to evaluate the capability of the CC2 method to properly describe the ground and excited states of such systems.5,6,14,59,60The convergence criterion used in single point energy calculations was 10−8on the density for the HF calculation, 10−9for the RI-CC2 ground state energy in the iterative coupled-cluster methods, and 10−6for the convergence threshold for the norm of residual vectors in eigenvalue problems for the RI-CC2 excited state calculations. In the geometry optimization, the convergence criterion used corre- sponds to a norm of the Cartesian gradient lower than 10−4a.u. The harmonic frequencies were calculated by numerical differentiationof the analytic gradients using central differences and a step length of 0.02 a.u. This also allowed verifying that the optimized geometries correspond to true minima. Orbital-relaxed first-order properties were determined; in par- ticular, the density and then CC2 differences between the den- sity of excited states and that of the ground state were per- formed. In addition, a post-processing tool interfaced to TURBO- MOLE, Nancy_EX-2.0,61was used in order to analyze the den- sity and character of the excited states and obtain, at the CC2 level, the so-called natural transition orbitals (NTOs)62,63of each excited state. Instead of describing one excitation with multi- ple canonical spin orbital couples, all the physical information on the nature of the electronic transition was gathered in one (sometimes two) couple(s) (from one occupied to one virtual) of NTOs allowing an unambiguous characterization of the nature of excited states. Moreover, the contributions of the NTOs to the wave function were more directly and accurately comparable to the MRCI weights of the determinants in the total wave function than the contributions of the canonical occupied–unoccupied HF orbitals. C. Multireference approaches Multireference (MR) wave function (WF) approaches were used to investigate the electronic transitions in a two step pro- cedure, the first one including the non-dynamical correlation led to zeroth-order reference wave functions and was followed by the introduction of dynamical correlation. Taking into account non- dynamical correlation was done—as a reference calculation—for the NAPA Band Ac–Gly–Phe–NH 2Acomplexes by means of the CASSCF18method using the MOLCAS package. In order to reduce the bottleneck of the active space, three different methods were also used: RASSCF,25,26GASSCF,27–29,64and the quasi-equivalent ORMAS.31The RASSCF and GASSCF methods are implemented in the MOLCAS package, while ORMAS is part of the GAMESS (General Atomic and Molecular Electronic Structure System) pack- age.65–67As the orbitals were assigned to different subspaces, local- ized orbitals were necessary. The SCF orbitals were localized with the DoLo code68,69as the starting point of the MOLCAS calcula- tions, or using Pipek–Mezey localization70in the GAMESS pack- age, for the ORMAS ones. All the calculations were based on the Dunning correlation consistent basis sets cc-pVDZ.58For NAPA B, some CASSCF/CASPT2, Q +DDCI, and CC26calculations were also performed with the enlarged cc-pVTZ basis sets, but as the effects were weak with a very important computational cost (see supple- mentary material, Appendix S4), only the cc-pVDZ basis set has been kept for the other systems. The Cholesky decomposition tech- nique71,72has been used (threshold of 10−8a.u.) in the MOLCAS calculations. To provide a good description of all considered states, it was necessary to include the involved orbitals into the active space. In the studied complexes, the low-lying states corresponded to local excitations: (i) π→π∗centered on the phenyl group or (ii) n→π∗ COcentered on peptide bonds [where n is N or O lone pairs (pure-p lone pair)]. To accurately describe these states, the active space should include all the πand π∗orbitals of the phenyl and of the carbonyl groups as well as lone pairs of the nitrogen and oxy- gen atoms (the highest in energy, the pure-p lone pair). This active J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp space corresponded to 18 electrons in 14 orbitals [CAS(18,14)] for the NAPA Bcomplex. For the larger systems (Ac–Gly–Phe–NH 2A and Ac–Gln–Phe–NH 2C), with one or two more peptide bonds, the active space contained 24 electrons in 18 orbitals or 30 elec- trons in 22 orbitals, respectively. The former led to a cumbersome but still tractable active space, while the latter one was unfeasible and required the use of RASSCF, GASSCF, and ORMAS alternative methods. In the RASSCF, the active space was separated into three sub- spaces: (i) RAS1 concerned occupied orbitals in which a limited number of holes was allowed, (ii) RAS2 corresponded to the com- plete active space, and (iii) RAS3 was the subspace containing the virtual orbitals, and the number of particles that could be created in this subspace had to be defined. Although this method is not com- monly used, compared to the CASSCF one, in the same way, the orbitals that have notable varying occupation numbers have to be included in the RAS2. The notation used to define the RASSCF cal- culation was RAS (number of active electrons, number of holes in RAS1, number of particles in RAS3; number of orbitals in RAS1, in RAS2, and in RAS3), the number of active electrons referring to the total number of electrons in the three subspaces. In the present case, the partition of the RAS subspaces can be done in a state specific way: the aromatic cycle orbitals in the RAS2 (or the orbitals of each peptide bond) plus the complementary orbitals in RAS1 and RAS3 to reach the complete set of active orbitals. The obtained results are presented and discussed in Appendix S5 of the supplementary material. Indeed, this led to different sets of orbitals, while the phi- losophy of the DDCI method is based on a common set of state averaged orbitals. This is why we chose in the following a common set of subspaces to describe all the states at the RASSCF/RASPT2 lev- els. For this, the RASSCF code was used as a Configuration Interac- tion (CI) one, coupled with the optimization of the orbitals. Indeed, the RAS2 subspace was empty; all the occupied active orbitals in the main determinant were placed in the RAS1 subspace and the virtual active ones in the RAS3 subspace, and the number of holes and particles defined the level of the CI (SD, SDT, SDTQ, etc.). For NAPA B, this led to a RAS(18,n,m; 9,0,5), where n was the number of holes, identical to the number m of particles, and took the val- ues 3, 4, or 5. This partition is illustrated in Fig. 2. For the larger systems, we used for Ac–Gly–Phe–NH 2Aa RAS(24,n,n; 12,0,6), n=3, 4, 5, and for Ac–Gln–Phe–NH 2Ca RAS(30,n,n; 15,0,7), n=3. The goal was to determine, for the NAPA Band Ac–Gly–Phe–NH 2Asystems, how many holes and particles are necessary to reach the CASSCF results. In the ORMAS/GASSCF methods, the total active space was divided into several subspaces. One subspace was dedicated to the π FIG. 2. NAPA B. Definition of the three active subspaces for the RASSCF. FIG. 3. Definition of the multiple active spaces for the ORMAS and GASSCF for NAPA B. orbitals of the cycle (phenyl group), while the other ones concerned each peptide bond (Fig. 3). In each of these subspaces, all determi- nants were generated. Electron excitation between subspaces could be allowed by defining the minimum and maximum number of elec- trons in each subspace for ORMAS, while it was a cumulative num- ber of electrons that was used in the GASSCF. In some cases, the two methods are not equivalent29even though in the present study they are. Indeed, as the targeted states were supposed to be local states, ππ∗centered on the phenyl group or n π∗ COon each peptide bond, electron excitation between subspaces was not allowed. However, this partition was not equivalent to multiple CASSCF calculations on each active space. In the GASSCF or ORMAS, all the states could be calculated in the same time and state averaged orbitals are obtained, necessary for DDCI calculations. Furthermore, simultaneous excita- tions inside the different sub-systems were allowed. Then, if n π∗ CO states involved excitations on different peptide bonds, they should be properly described. The generated determinants corresponded to all possible combinations of alpha and beta strings that kept the defined numbers of electrons in each subspace, leading, for exam- ple, to combinations of two triplets on the peptide subspaces on one side and a quintet on the aromatic cycle on the other side. There were many possibilities, and the maximal total excitation degree of a determinant could be large, 10 for NAPA B, corresponding to a di- excited determinant on each peptide subspace and a hexa-excitation on the aromatic cycle. This maximal total excitation degree was increased to 12 for Ac–Gly–Phe–NH 2A(3 peptide bonds) and 14 for Ac–Gln–Phe–NH 2C(3 peptide bonds and an amide group). How- ever, the ORMAS/GASSCF allowed a drastic elimination of the less important determinants, compared to the CASSCF. One can note that ORMAS and GASSCF can also be used to perform RASSCF calculations. The orbitals of the singlet states are averaged in all cases. CASPT237calculations were performed on RASSCF/CASSCF reference wave functions to introduce dynamical correlation. GASPT2 is not yet available in the MOLCAS package. A level shift73,74of 0.2 a.u. was used as well as the standard Ioniza- tion Potential Electron Affinity (IPEA) shift75in the MOLCAS J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp calculations. The quasi-degenerate second-order perturbation the- ory for the ORMAS, ORMAS-PT method, implemented in the GAMESS package, was also applied. In that case, level shifting was not necessary. The DDCI method developed by Caballol et al. ,47,48a configu- ration interaction method, was also used to determine the correlated energies and wave functions, starting from the optimized and then localized RASSCF, CASSCF, or GASSCF orbitals. This method was used in order to reduce the number of determinants by neglecting those coming from two hole–two particle excitations external to the active space and was relevant to calculate the correlation energy that contributed to the energy difference between ground and excited states for which average starting orbitals were mandatory. In the DDCI calculations, the complete active space was reduced accord- ing to the targeted states, and several separate CI calculations were performed. The active spaces were the same as the subspaces defined in the GASSCF: (i) πand π∗orbitals for the local excitations cen- tered on the phenyl group ( π→π∗), corresponding to six electrons in six orbitals, and (ii) for each peptide group, the nitrogen and oxygen pure-p lone pairs (n), πcoand π∗ COorbitals for the local excitations on each peptide bond (n →π∗ CO), which led to six elec- trons in four orbitals. These small active spaces were denoted as CAS iwith i=1 to the number of peptide bonds. For NAPA B, a larger active space was used with all the orbitals of the peptide bonds, i.e., eight electrons in 12 orbitals. This active space allowed to check the relevance of the reduced (6,4) active space. However, it could not be used for larger systems due to increasing number of peptide bonds. Size-extensivity errors were accounted by the a posteriori Davidson correction,76and the corrected excitation ener- gies were denoted as Q +DDCI. In our previous study on NAPA B and in the present work, the DDCI method using localized orbitals was used in the quasi-linear-scaling version15–17performed with the EXSCI program77interfaced with MOLCAS. To neglect long-range interactions, thresholds on the exchange integrals were applied. This quasi-linear-scaling CI has been validated by comparison with cal- culations performed with CASPT2 in our previous work on NAPA B.5The same division of the molecular orbitals (MOs) in four zones was used in the present work. The first zone, zone 0, contained the active space of the targeted states completed by the remaining active orbitals of the full CAS. Zone 1 contained the remaining σorbitals of the whole molecule. Zone 2 was defined by the non-valence vir- tual orbitals of the atoms involved in the active space, while zone 3 contained the rest of these non-valence orbitals. The same parame- ters as in our previous study on NAPA Bwere applied, i.e., a very small 0.0001 a.u. threshold on the zone 0 and largest ones on the other ones as defined in the supplementary material (Table S2) and detailed in our previous work.5 III. RESULTS A. CC2 results The ground state calculations of the conformers exhibited D1/D2values in the 0.083–0.087/0.17–0.27 ranges, respectively, while the excited state calculations exhibited a D 2value equal to 0.25–0.27 with a biorthogonal norm % ⟨E1∣E1⟩≥89%. The D1 and D2 diagnostics computed from the single and double substitution amplitudes in the CC2 wave function were reliable indicators when static or dynamic correlation effects are not adequately treated at theCC2 level: their magnitude is correlated with the performances of the CC2 method. The obtained values confirmed the reliability of the CC2 calculations on these systems even if some of them cor- responded to the upper limit of the recommended values (D 1/D2 up to 0.15/0.25).14Moreover, in the excited states, the contributions of the canonical occupied–unoccupied HF orbitals to the total wave function change were larger or equal to 96% for the ππ∗states and between 72% and 94% for the other states. The CC2/cc-pVDZ exci- tation energies of the first low-lying excited states ( a priori one ππ∗ and one n π∗ COper amide group) of NAPA B(three excited states), Ac–Gly–Phe–NH 2A(four states), and Ac–Gln–Phe–NH 2C(five states) conformers in their ππ∗CC2/cc-pVDZ optimized geometry are reported in Table I. In addition, Table I contains the couple(s) of NTOs (occupied and virtual) of each state for which we obtained a contribution to the wave function greater than 10%. Moreover, the contours of the difference between the CC2 density of the differ- ent low-lying excited states and that of the ground state are shown in Fig. S1-1-3. Whatever the conformers, the first excited state is a locally ππ∗exited state centered on the phenyl ring. For NAPA B, the second and third excited states are locally n π∗ COstates, each one centered on a peptide bond, the second peptide bond, that of the C-terminal side, for the second state and the first peptide bond, that of the N-terminal side, for the third one. For Ac–Gly–Phe–NH 2 A, the next three states are locally n π∗ COexcited states centered on two peptide bonds or one peptide bond. The second and the third excited states are localized on the two same peptide bonds, the sec- ond and the third, but with opposite weights ( ∼80% and∼20%), whereas the fourth excited state is localized on the first peptide bond. For Ac–Gln–Phe–NH 2C, the third, fourth, and fifth states are locally n π∗ COexcited states centered on one peptide bond (3, 1, and 2, respectively). The second state results from a combination of excitations of different nature, ππ∗and n π∗(CT), resulting in an n[ π∗ CO(4) π∗] couple of NTOs involving the same lone pair, the oxygen pure-p lone pair of the fourth peptide bond, and a virtual NTO showing πlocalized contributions both on the peptide bond 4 [π∗ CO(4) ] and on the phenyl group ( π∗). B. Selected reference approaches: Non-dynamical correlation In NAPA B, with all methods, the lowest n π∗ COstate is located on the second peptide bond (C-terminal side) and the second one on the first peptide bond (N-terminal side), following the number- ing shown in Fig. 1. The RASSCF calculations were done for 3, 4, and 5 holes/particles as well as at the CASSCF and GASSCF/ORMAS lev- els. The results are presented in Table II. For n =3, the difference to the CASSCF result for the ππ∗state is +0.25 eV, reduced to +0.10 eV with n=4, while for the n π∗ CO, the overestimation is 0.17 eV for n=3 and 0.25 eV for n =4. With n=5, the largest difference between the RASSCF and the CASSCF is again reduced to reach an overesti- mation of only 0.02 eV. The ORMAS and GASSCF reached almost the same accuracy (overestimation of 0.03 eV). The number of determinants (Table V) is very different between each type of calculation as well as the number of iterations to obtain the convergence and the computational time of each iter- ation (Table II). The GASSCF number of determinants corresponds to 5.5% of the CASSCF one, while, for equivalent accuracy, the RASSCF n =5 generated 22.6% of the total number of the CASSCF J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I. CC2/cc-pVDZ excitation energies of the first low-lying excited states (one ππ∗and one n π∗COper amide group) of NAPA B, Ac–Gly–Phe–NH 2A, and Ac–Gln–Phe–NH 2Cconformers and the couple(s) of NTOs with contribution to the wave function larger than 10%. Evert(eV) NTOs (%, occupied →virtual) NAPA Bππ∗4.82 55% 44% nπ∗ CO(2) 5.63 98% nπ∗ CO(1) 5.78 99% Ac–Gly–Phe–NH 2Aππ∗4.77 58% 42% nπ∗ CO(2,3) 5.60 77% 22% nπ∗ CO(3,2) 5.64 78% 21% nπ∗ CO(1) 5.69 99% Ac–Gln–Phe–NH 2Cππ∗4.76 56% 44% n[π∗ CO(4) π∗] 5.54 97% nπ∗ CO(3) 5.65 97% nπ∗ CO(1) 5.67 97% nπ∗ CO(2) 5.82 99% J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II. NAPA B. RASSCF, GASSCF, ORMAS, and CASSCF energies. ΔE (eV)RASSCF (3h3p) RASSCF (4h4p) RASSCF (5h5p) GASSCF (0h0p) ORMAS (0h0p) CASSCF RAS(18,3,3; 9,0,5) RAS(18,4,4; 9,0,5) RAS(18,5,5; 9,0,5) GS (a.u.) −683.510 968 −683.522 861 −683.523 073 −683.523 403 −683.523 405 −683.523 796 ππ∗4.78 4.63 4.53 4.54 4.54 4.53 nπ∗ CO5.93 6.01 5.79 5.81 5.81 5.78 6.06 6.14 5.91 5.91 5.91 5.89 No. of iterationsa411 1.39254 433125 Time/iteration (min) 0.49 2.69 0.46 9.74 Total time 11 h 11 min 12 h 21 min 38 min 4 h 15 min aThe different calculations were performed on a bi-(4c) Intel Xeon E5-2637 v3 machine, using one processor and 19 GB of memory. determinants. The convergence of the RASSCF was very slow. For convergence thresholds on the energy and on the rotation of the molecular orbitals of 10−8a.u. and 10−4, respectively, the RASSCF n=3 time per iteration is similar to that of the GASSCF but required many more iterations (411 vs 43) and led to a significant overesti- mation compared to the CASSCF values as for the RASSCF n =4. One can note some difficulties in the convergence of the RASSCF n=4, depending on the starting point. Indeed, starting—as in the other cases—from the SCF orbitals, the convergence was not only extremely slow but it also led to states whose nature was not totally in agreement with the CASSCF solutions. However, it was not the case when the starting orbitals were those of the CASSCF. Only the time per iteration was then relevant for a comparison with other meth- ods (Table II). For the RASSCF n =5, the number of iterations was reduced and the results gained accuracy. Finally, to reach an accu- racy similar to that of the CASSCF, the RASSCF converged in 12 h (n=5), whereas the GASSCF converged in only 21 min, with a time per iteration and a number of iterations ∼5 times smaller. The CASSCF converged in fewer iterations than all the other methods, but the time per iteration was the greatest. In conclusion, the total time of the RASSCF calculations n =3–5 was much greater than that of the CASSCF, and only the GASSCF was more efficient, 21 min instead of 4 h 15 min, while allowing equivalent accuracy. The performances of ORMAS calculations (31 iterations in 38 min) were similar to those of the GASSCF (43 iterations in 31 min), slightly slowed down by the need to calculate all the spin multiplicities of theintermediate states, for a total of 14 triplet and singlet states instead of the five singlets. For Ac–Gly–Phe–NH 2A(Table III), one can note that all the n π∗ COstates involve the same peptide bonds regardless of the method used; the first state is located on the third peptide bond, the second one on the first peptide bond, and the last state on the second peptide bond (Fig. 1). Concerning the energetics, overestimations of 0.2 eV ( ππ∗state) and 0.3 eV (n π∗ COstates) were obtained with the RASSCF n =3 reduced to 0.10 and 0.24 eV for n =4. One more time, the RASSCF n =5 gave comparable results to the GASSCF/ORMAS ones, with an overestimation of 0.03 eV relative to the CASSCF results. As in the case of NAPA B, the convergence of the RASSCF was very slow: 197 iterations for n =3, 86 for n =5, while the calcu- lation for n =4 did not converge after more than 400 iterations. The convergence thresholds of the RASSCF n =4 calculation were then raised to 10−7a.u. on the energy and 10−3on the MO rotation to converge in 153 iterations, but a larger error on the n →π∗ COexci- tation energies was observed. The CASSCF calculation, 172 ×106 determinants (Table V), was at the limits of what could be done. The ratio of the number of determinants between the RASSCF n =5 and the GASSCF was still to the advantage of the latter (1.6% instead of 3.4% of the CASSCF space). However, it was to a lesser extent than for NAPA B(5.5% vs 22.6%). The CASSCF for Ac–Gln–Phe–NH 2Cwas intractable. Due to the near-degeneracies of the first and second n π∗ COstates as well as the third and fourth ones, the nature of the different states differed TABLE III. Ac–Gly–Phe–NH 2A. RASSCF, GASSCF, ORMAS, and CASSCF energies. ΔE (eV)RASSCF (3h3p) RASSCFa(4h4p) RASSCF (5h5p)GASSCF (0h0p) ORMAS (0h0p) CASSCFRAS(24,3,3; 12,0,6) RAS(24,4,4; 12,0,6) RAS(24,5,5; 12,0,6) GS (a.u.) −890.354 353 −890.369 503 −890.372 827 −890.373 389 −890.373 389 −890.374 093 ππ 4.82 4.68 4.54 4.53 4.53 4.52 nπ∗ CO(3)6.02 6.06 5.86 5.86 5.86 5.83 nπ∗ CO(1)6.12 6.16 5.95 5.95 5.95 5.92 nπ∗ CO(2)6.13 6.17 5.96 5.97 5.97 5.93 aRASSCF n =4 convergence thresholds of 10−7a.u. on the energy and 10−3on the MO rotation instead of 10−8and 10−4, respectively. J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV. Ac–Gln–Phe–NH 2C. RASSCF and GASSCF energies. RASSCF (3h3p) ≪RASSCF≫(5h5p) GASSCF ΔE (eV) RAS(30,3,3; 15,0,7) RAS(30,5,5; 15,0,7) (0h0p) GS (a.u.) −1136.243 231 −1136.266 795 −1136.268 378 ππ∗4.89 4.57 4.56 nπ∗ CO6.22(3)6.07(1,3)6.04(1) 6.25(1)6.07(3,1)6.06(3) 6.32(4)6.15(4)6.15(2) 6.35(2)6.18(2)6.15(4) according to the method used. However, the first and the second nπ∗ COstates involved the first and third peptide bonds, while the two last states involved the second and the fourth peptide bonds (Fig. 1). Even the RASSCF calculations were problematic due to the slow convergence, and only the RASSCF n =3 was obtained. The “RASSCF” n =5 results were obtained thanks to the GASSCF code, equivalent to the RASSCF one, with a better convergence. The RASSCF results are compared to the GASSCF ones in Table IV. As for NAPA B, the error is larger for the ππ∗state (0.33 eV) than for n π∗ COstates (0.18–0.20 eV) for the RASSCF n =3, while for n=5, this discrepancy in accuracy according to the nature of the states disappeared, and the results were found to be similar to the GASSCF ones. One can note that the total number of determinants is now smaller for the RASSCF n =5 than for the GASSCF one (Table V), but the calculation time is largely in favor of the latter. The ORMAS method was not applied to this very large compound as the calculation required considerable computational time and huge memory resources and was not necessary for the subsequent CI calculations. In conclusion, for the RASSCF strategies, it was necessary to go until an excitation degree of 5 to obtain results in good agreement with the CASSCF ones. The convergence was slow for all excita- tion degrees and even problematic for n =4. The RASSCF code used as a configuration interaction coupled with the optimization of theorbitals was not very efficient, and as mentioned previously, the par- tition of the RAS subspaces in a state specific way (Appendix S5 of the supplementary material) gave relevant results. However, the par- tition used was suitable for using the DDCI method, based on a com- mon set of state-averaged orbitals. The number of determinants of the GASSCF grew more rapidly with the number of subspaces than that in the RASSCF case for which the number of subspaces was, in the present case, always limited to 2 whatever the number of pep- tide bonds. Indeed, for Ac–Gln–Phe–NH 2C, the number of deter- minants became larger for the GASSCF compared to the RASSCF n=5. Actually, the excitation degree of the determinants in the GASSCF could be larger than that in the RASSCF due to the multiplication of the subspaces. For example, in the case of Ac–Gln–Phe–NH 2C, the combination of a mono-excitation on each peptide subspace coupled to a di-excitation on the cycle cor- responded to hexa-excited determinants that were not present in the RASSCF n =5. As mentioned in Sec. II C, the maximal total excitation degree is 14 for this system. Nevertheless, the GASSCF or ORMAS methods using small active subspaces seemed then to be the most relevant strategy to calculate non-dynamical energy for a very reasonable computational cost. The comparison with CASSCF results when possible also gave good agreement, with a maximal overestimation of 0.03 eV on the excitation energies whatever the nature of state. C. Selected reference approaches: Dynamical correlation Dynamical correlation was taken into account using the CASPT2 method on RASSCF/CASSCF reference wave functions and also thanks to the Davidson corrected DDCI configuration interaction method. The optimized RASSCF, GASSCF, or CASSCF orbitals were used as the starting point of the DDCI. The results are presented in Tables VI–VIII. As presented in the computational details, the small active spaces used in the DDCI calculations were the subspaces defined in the GASSCF: (i) CAS(6,6) for the phenyl group, (ii) CAS(6,4) for each peptide bond denoted as CAS iwith i =1 to the number of peptide bonds (Fig. 1), and (iii) for NAPA B, CAS(8,12) with all the orbitals of the peptide bonds. TABLE V. Dimensions of the RASSCF, GASSCF, and CASSCF methods (in GAMESS). The percentages in brackets are the ratio of the number of determinants of the different methods compared to the number of determinants obtained in the CASSCF. Subspaces NAPA B Ac–Gly–Phe–NH 2A Ac–Gln–Phe–NH 2C RASSCF (18,n,n; 9,0,5) (24,n,n; 12,0,6) (30,n,n; 15,0,7) Number of det (n =3) 36 916 (0.9%) 158 669 (0.05%) 510 546 (0.002%) Number of det (n =4) 243 376 (6.1%) 1 787 219 (0.5%) 8 812 371 (0.03%) Number of det (n =5) 905 128 (22.6%) 11 577 923 (3.4%) 89 200 497 (0.3%) Subspaces (number of e, number of Mos) GASSCF/ORMAS (6,6)/(6,4)/(6,4) (6,6)/(6,4)/(6,4)/(6,4) (6,6)/(6,4)/(6,4)/(6,4)/(6,4) Number of det 220 192 (5.5%) 5 510 848 (1.6%) 140 843 008 (0.5%) CASSCF (18,14) (24,18) (30,22) Number of det 4 008 004 (100%) 344 622 096 (100%) 29 085 255 936 (100%) J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI. NAPA B. RASPT2, CASPT2, ORMAS-PT, and Q +DDCI corresponding to the different sets of optimized MOs. Ref. WF RASSCF n =3 MOs RASSCF n =4 MOs RASSCF n =5 MOs ORMAS MOs GASSCF MOs CASSCF MOs ΔE (eV) RASPT2 Q +DDCI RASPT2 Q +DDCI RASPT2 Q +DDCI ORMAS-PT Q +DDCI CASPT2 Q +DDCI ππ∗ 4.64CAS(6,6) 4.54CAS(6,6) 4.54CAS(6,6) 4.42CAS(6,6) 4.57CAS(6,6) 4.73 4.72 4.71 4.72 4.72 nπ∗ CO5.72CAS(12,8) 5.61CAS(12,8) 5.76CAS(12,8) 5.67CAS(12,8) 5.68CAS(12,8) 5.82 5.82 5.84 5.83 5.82 6.05 6.02 6.05 6.03 6.02 5.89 CAS1/CAS2 5.77 CAS1/CAS2 5.92 CAS1/CAS2 5.85 CAS1/CAS2 5.83 CAS1/CAS2 5.83 5.83 5.84 5.84 5.83 5.97 5.96 5.98 5.98 5.98 TABLE VII. Ac–Gly–Phe–NH 2A. RASPT2, CASPT2, and Q +DDCI corresponding to the different sets of optimized MOs. Superscripts (1), (2), and (3) refer to the peptide bond numbering (see Fig. 1). RASSCF n =3 MOs RASSCF n =4 MOs RASSCF n =5 MOs GASSCF MOs CASSCF MOs ΔE (eV) RASPT2 Q +DDCI RASPT2 Q +DDCI RASPT2 Q +DDCI Q +DDCI CASPT2 Q +DDCI CAS(6,6) CAS(6,6) CAS(6,6) CAS(6,6) CAS(6,6) ππ∗4.62 4.66 4.45 4.64 4.46 4.65 4.66 4.57 4.65 CAS1/2/3 CAS1/2/3 CAS1/2/3 CAS1/2/3 CAS1/2/3 nπ∗ CO 5.63(3)5.79(3)5.76(3)5.80(3)5.62(3)5.78(3)5.79(3)5.65(3)5.77(3) 5.71(2)5.88(2)5.78(2)5.85(1)5.67(2)5.87(2)5.88(1)5.66(2)5.87(1) 5.76(1)5.89(1)5.85(1)5.87(2)5.73(1)5.87(1)5.90(2)5.71(1)5.88(2) For NAPA B, the analysis of the CASSCF wave function (expressed in local orbitals) of the n π∗ COexcited states shows a major contribution of the n O→π∗ COmono-excitation and the nNnO→(π∗ CO)2di-excitation in the same subspace counted for 5%. There is also a non-negligible contribution (8%–9%) of di- excitations, implying the second peptide bond (n N→π∗ CO) cou- pled with n O→π∗ COin the considered state. These simultaneous excitations are also present in the DDCI wave functions with a (12,8) active space and even in larger proportions: 8.5% of n NnO →(π∗ CO)2intra-di-excitation and 16% (n O→π∗ CO)2(nN→π∗ CO)1inter-di-excitation. For the second n Oπ∗ COstate, they are almost equivalent, 9%–14%, respectively. Of course, the wave functions of the DDCI with reduced (6,4) active spaces, with a separate treat- ment of the peptide bonds, were different. Indeed, the weight of the (nO→π∗ CO)i(nN→π∗ CO)jintra-excitation was reduced (11%–12% vs 14%–16%) as it was only introduced by a mono-excitation on a reference determinant and then less correlated by the other ones. The weight of the main determinant is larger (61%) than that of the Q+DDCI on the CAS(12,8) active space (48%), while the weight of the inter-di-excitation is reduced (5% vs 9%). TABLE VIII. Ac–Gln–Phe–NH 2C. RASPT2 (n =3) and Q +DDCI with different sets of optimized MOs. Superscripts (1), (2), (3), and (4) refer to the peptide bond numbering (see Fig. 1). RASSCF n =3 MOs “RASSCF”an=5 MOs GASSCF MOs ΔE (eV) RASPT2 Q +DDCI CAS1/2/3/4 Q +DDCI CAS1/2/3/4 Q +DDCI CAS1/2/3/4 ππ∗4.64 4.63 4.62 4.63 nπ∗ CO5.70(3)5.98(3)5.97(3)5.98(3) 5.70(1)5.99(4)5.99(4)6.00(4) 5.71(4)6.03(2)6.03(2)6.04(2) 5.85(2)6.05(1)6.05(1)6.05(1) a“RASSCF” n =5 was obtained thanks to the GASSCF. J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IX. CC2/cc-pVDZ and GASSCF-Q +DDCI excitation energies of the first low-lying excited states (one ππ∗and one nπ∗COper amide group) of NAPA B, Ac–Gly–Phe–NH 2A, and Ac–Gln–Phe–NH 2Cconformers. GASSCF-Q +DDCI CC2 GASSCF-Q +DDCI CC2 GASSCF-Q +DDCI CC2 ΔE (eV) NAPA B Ac–Gly–Phe–NH 2A Ac–Gln–Phe–NH 2C ππ∗4.72 4.82 4.66 4.77 4.63 4.76 nπ∗ CO 5.84 5.63 5.79 5.64 5.98 5.65 5.98 5.78 5.88 5.69 6.00a5.54a 5.90 5.60 6.04 5.82 6.05 5.67 aValues in italics indicate that the nature of this state differs from CC2 to GASSCF-Q +DDCI (see the text). The RASPT2 for n =3 presented a difference with the CASPT2 of+0.04 to +0.07 eV (Table VI). For n =4, this difference was of −0.03 eV for the ππ∗state and −0.06 eV for the n π∗ COstates. For n=5, the deviation to the CASPT2 is −0.03 eV for the ππ∗state and+0.08/+0.09 eV for n π∗ COstates. The ORMAS-PT results were found very similar to those of the CASPT2 ones for the n π∗ COstates, while the ππ∗state presented a difference of 0.13 eV. The two per- turbative treatments are not equivalent, and additionally, the use of the IPEA shift in the CASPT2 method corrects the underestimation of the excitation energies. Furthermore, a level shift of 0.2 a.u. was necessary to avoid intruder states in the CASPT2 contrary to the ORMAS-PT for which no level shift was necessary. The use of level shift tends to increase excitation energies. Whatever the MO set used, i.e., RASSCF, GASSCF, or CASSCF MOs, the Q +DDCI results were not very sensitive and gave almost the same excitation energies. Compared to the CASPT2 results, the difference was of +0.15 eV for the ππ∗state and +0.15/+0.19 eV for the n π∗ COstates. The excitation energies were then shifted by about the same amount. By comparing the two active spaces used to describe the n π∗ CO states—CAS(12,8) or the two CAS i(6,4)—at the Q +DDCI level, one can note that there is a small difference: around 0.01 eV for the first excited state and 0.04 eV on the second one. To make possible the calculation of these excitation energies in the larger systems, this strategy could then be used, provided that the contribution of the other peptides in each n π∗ COremained small. For Ac–Gly–Phe–NH 2A, the analysis of the wave functions of the n π∗ COstates between the different methods presents some differences. The lowest n π∗ COstate is located on the third peptide bond (see Fig. 1) in all cases. The second and third states calcu- lated at the CASPT2 level are located on the second and first peptide bonds, respectively. At the RASPT2 levels, the nature of the different nπ∗ COstates is the same. The Q +DDCI method found the second and the third n π∗ COstates almost degenerate, and inversion of these two states was encountered for the Q +DDCI calculations using the RASSCF n =4, GASSCF, and CASSCF MOs. The first and second states were found close in energy at the CASPT2 level but not at the RASPT2 ones, except for the n =4 which was badly converged at the RASSCF level. The analysis of the DDCI wave functions of the n π∗ COstates showed that the excitation n O→π∗ COhas a weight of 60% for the two first states and 58% for the last one, and the intra di-excitation nNnO→(π∗ CO)2counts for 10%–12%, comparable to the equiva- lent weights in NAPA B. There were also contributions of the interdi-excitations (8%), which were enhanced compared to NAPA Bas there was one peptide bond more. Concerning the energetics (Table VII), the difference between the CASPT2 and Q +DDCI excitation energies is smaller than that for NAPA B:+0.08 eV compared to +0.15 eV for the ππ∗state and similar for the n π∗ COstates: +0.12 to 0.21 eV compared to +0.14 to 0.19 eV with the small CAS(6,4) active spaces. As for NAPA B, the set of molecular orbitals used did not have a notable effect on the Q +DDCI results (0.03 eV). The RASPT2 for n =3 presents an overestimation of +0.05 eV compared to the CASPT2 for the ππ∗ state, while it is an underestimation of −0.12 eV for n =4. For n π∗ CO states, the RASPT2 n =3 shows small differences with the CASPT2 (−0.02/+0.05 eV), while they are larger for n =4 (+0.11/+0.16 eV). For RASPT2 n =5, the agreement with the CASPT2 is similar to that obtained for n =4 for the ππ∗state but strongly improved for the nπ∗ COstates ( −0.03/+0.02 eV). For Ac–Gln–Phe–NH 2C(Table VIII), in the DDCI wave func- tions of the n π∗ COstates, the weight of the intra-di-excitation n NnO →(π∗ CO)2is similar for all the states (9.7%–10.2%) and compara- ble with that the smaller systems, with the exception of the second state for which it is 11.4%. The weight of the inter-di-excitations is 12.3%–12.4% for three of the states, the second one is again slightly different with a weight of 11.7%. The weight of the main refer- ence is slightly lower as the number of peptides increases, between 57% and 58% instead of 58%–60% for Ac–Gly–Phe–NH 2Aand 61% for NAPA B. Only the RASSCF n =3 was affordable with MOLCAS in computing time and quality of convergence; then, the comparison was here between the RASPT2 n =3 and the Q +DDCI on the same set of MOs and on the GASSCF one. The “RASSCF” n=5 was obtained thanks to the GASSCF code, and the pertur- bation treatment is not currently possible on these zeroth-order wave functions. For RASPT2 n =3, there is a difference of +0.01 eV for the ππ∗state and −0.19/−0.35 eV for the n π∗ COstates com- pared to the Q +DDCI calculations. As for the smaller systems, the MOs has almost no influence on the Q +DDCI excitation energies. The three lowest n π∗ COstates are almost degenerate at the RASPT2 level, and the last one was found to be +0.14 eV higher. At the Q +DDCI level, the largest gap between these states is 0.07 eV. This small gap between the states explains the inver- sion of the n π∗ COstates between RASPT2 n =3 and Q +DDCI calculations. In conclusion, the Q +DDCI calculations are not very sensitive to the starting MOs, which suggests that they are all sufficiently sat- isfactory. The RASPT2 is much more sensitive to the zeroth-order J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp wave functions, and only the n =5 results are comparable to the CASPT2 ones. The active spaces used in the DDCI calculations contain the same number of electrons and orbitals for the three systems; then, the dimensions of the quasi-linear-scaling CI calculations were almost the same: between 119 ×106and 155 ×106of determinants with that CAS(6,6) describing the ππ∗state, while the CAS i(6,4) generate less than 7 ×106determinants for the n π∗ COstates. The large active space also used to describe these states in the case of NAPA B, CAS(12,8), generated about 580 ×106of determinants. It was then impossible to enlarge this active space with the additional peptide groups of the larger systems. All the dimensions of the CI calculations can be found in the supplementary material (S2). IV. PERFORMANCES OF THE CC2 METHOD Along the series of the capped peptides, both the nature and the energetics obtained at the CC2 level for the lowest ππ∗excited states, the state for which the geometry has been optimized at this level, are in very good agreement with those obtained at the GASSCF- Q+DDCI level (Table IX). Indeed, the excitation energies were over- estimated for all the capped peptides of around 0.11 eV, a systematic overestimation independent of the size and inferior to the standard error of the CASPT2 method ( ±0.2 eV). Moreover, this overestima- tion is of the same order as that observed between the MR meth- ods themselves, i.e., between CASPT2 and GASSCF-Q +DDCI, for example. The nature of n π∗ COstates is also well described at the CC2 level, and only one excepted difference was observed com- pared to the GASSCF-DDCI. This difference concerns the lowest nπ∗ COstate of Ac–Gln–Phe–NH 2C, whose CC2 wave function exhibits a non-negligible contribution of electronic charge transfer from the backbone to the phenyl ring, a CT contribution (see the NTO couple in Table I and the difference between the density of the ground state and this state on Fig. S1-3), whereas the wave func- tion obtained at the GASSCF-DDCI level does not present such a contribution as electron excitation between spaces was not allowed. The CC2 excitation energies of the n π∗ COstates are well repro- duced compared to the GASSCF-Q +DDCI level, but unlike the ππ∗ states, they are underestimated for all the capped peptides, and this underestimation was a little larger for the Ac–Gln–Phe–NH 2Cand larger than the overestimation observed for the ππ∗states. Indeed, if a similar underestimation has been observed for NAPA Band Ac–Gly–Phe–NH 2A(in average −0.21 eV), this underestimation was in average equal to 0.35 eV for Ac–Gln–Phe–NH 2with a max- imum for the n π∗ CO, which exhibits a CT contribution ( −0.46 eV) and a minimum for the n π∗ COstate localized on the second peptide bond ( −0.22 eV). In conclusion, the CC2 method reproduces well the nature of the excited states independent of the size of the systems. In addi- tion, even if the discrepancies of the excitation energies can depend on the nature of the excited states [in average +0.11 eV for the ππ∗ excited states and −0.21 eV (NAPA Band Ac–Gly–Phe–NH 2A) /−0.35 eV (Ac–Gln–Phe–NH 2C) for the n π∗ COstates], the CC2 excitation energies exhibit a systematic discrepancy compared to the GASSCF-Q +DDCI ones, an overestimation or an underestimation, for each type of excited state independent of the size of the systems. Moreover, these discrepancies are only a little higher than those obtained between the more sophisticated MRCI methodsinvestigated in this work and of the same order of the standard error obtained for these MRCI methods, i.e., ±0.2 eV. V. CONCLUSIONS All three alternatives to CASSCF, namely, RASSCF, ORMAS, and GASSCF, yield results in good agreement (wave function and energetics) with the CASSCF, where that can be performed. Of these, the ORMAS and the GASSCF are the most efficient, especially for larger systems such as capped peptides containing at least two residues. Indeed, the RASSCF requires going up to n =5 to obtain the CASSCF accuracy and quickly becomes not tractable such as in the case of Ac–Gln–Phe–NH 2C. When adding dynamical correlation, both the nature and the energetics of the excited states are also well reproduced at the GASSCF-Q +DDCI level compared with the CASPT2 level. Even if the CASPT2 and the CI methods are state-of-the-art methods, they also both present some drawbacks: (1) for the CI calculations, the use of the smallest active spaces and the size-consistency error, par- tially corrected by the Davidson procedure, and (2) for the CASPT2 calculations, the use of IPEA and level shifts or simply the fact that energies are not upper bounds of the real ones. The basis set also has a significant effect on the IPEA shift.78In the present study, the zeroth-order wave function plays an important role in the CASPT2 calculations, while the CI method seems not to be very sensitive to it. The CASPT2 method is currently used and gives very reason- able results on UV–visible spectroscopies, while the DDCI method is the most accurate method to study magnetic problems and hardly used for spectroscopic ones. The two methods can then be seen as complementary. In addition, the question of which method has to be used no longer arises for very large systems since the CASPT2 calculations become intractable. In order to partition the full CAS into subspaces, the use of the RAS/GAS/ORMAS methods implies a good understanding of the electronic structure of the different states. Furthermore, in the present case, as the considered excitations are local, interspace exci- tations between different GAS spaces can be ignored. To go further, investigating the charge transfer states is a challenging task as elec- tron excitation between subspaces will generate a large amount of determinants and as these states are also found higher in energy at the non-dynamical correlated level, with the necessity to average the orbitals on about 20 states.5 Finally, the discrepancies obtained by the CC2 method for the excitation energies exhibit a systematic overestimation or under- estimation according to the nature of the excited states compared to those obtained by the MRCI methods, these discrepancies being independent of the size of the systems. The extension of the CC2 method to such large systems without loss of accuracy was then demonstrated, highlighting the great potential of this method to treat accurately excited states, mainly single reference, of very large systems with the crucial advantage of having access not only to the energy but also to both the gradient and the Hessian. DEDICATION The authors dedicate this article to Rosa Caballol and Ria Broer for their contributions to the development of theoretical methods J. Chem. Phys. 154, 214105 (2021); doi: 10.1063/5.0048146 154, 214105-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to accurately describe electron correlation effects. In particular, the difference dedicated configuration interaction method is an accurate and versatile strategy to calculate excitation energies as illustrated in this paper. SUPPLEMENTARY MATERIAL See the supplementary material for Appendix S1: contours of the difference between the CC2 density of the different low- lying excited states [ ππ∗(±0.0015 a.u.) and others ( ±0.03 a.u.)] and that of the ground state of NAPA B, Ac–Gly–Phe–NH 2A, and Ac–Gln–Phe–NH 2C; Appendix S2: dimensions of the CI cal- culations and thresholds for the quasi-linear-scaling CI; Appendix S3: geometries; Appendix S4: NAPA B: effect of the basis set; and Appendix S5: NAPA B: different choices of RAS subspaces. ACKNOWLEDGMENTS This work received financial support from the Agence Nationale de la Recherche (ANR) (Grant No. ANR-14-CE06-0019- 01-ESBODYR). This work was granted access to the HPC facil- ity of TGCC/CINES/IDRIS under Grant Nos. 2016-t2016087540, 2017-A0010807540, and 2020-A0070807540 awarded by GENCI (Grand Équipement National de Calcul Intensif) and to the CCRT High Performance Computing (HPC) facility at CEA under Grant No. CCRT2016/CCRT2017/CCRT2020-p606bren. N.B.A. thanks the computing facility CALMIP for the allocation of comput- ing resources (Project No. P16009) at the University of Toulouse (UPS). M.S.G. and M.W.S. acknowledge support from a U.S. National Science Foundation Software Infrastructure (SI2) (Grant No. OCI-1047772). DATA AVAILABILITY The data that support the findings of this study are available within this article and its supplementary material. REFERENCES 1L. González, D. Escudero, and L. Serrano-Andrés, ChemPhysChem 13, 28 (2012). 2M. E. Casida and M. Huix-Rotllant, Annu. Rev. Phys. Chem. 63, 287 (2012). 3K. Sneskov and O. Christiansen, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2, 566 (2012). 4D. Roca-Sanjuán, F. Aquilante, and R. Lindh, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2, 585 (2012). 5N. Ben Amor, S. Hoyau, D. Maynau, and V. Brenner, J. Chem. Phys. 148, 184105 (2018). 6M.-S. Dupuy, E. Gloaguen, B. Tardivel, M. Mons, and V. Brenner, J. Chem. Theory Comput. 16, 601 (2020). 7M. Mali ˇs, Y. Loquais, E. 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5.0051631.pdf

Appl. Phys. Lett. 118, 222107 (2021); https://doi.org/10.1063/5.0051631 118, 222107 © 2021 Author(s).Solution-processed amorphous p-type Cu- Sn-I thin films for transparent Cu-Sn-I/IGZO p–n junctions Cite as: Appl. Phys. Lett. 118, 222107 (2021); https://doi.org/10.1063/5.0051631 Submitted: 26 March 2021 . Accepted: 24 May 2021 . Published Online: 03 June 2021 Haijuan Wu , Lingyan Liang , Xiaolong Wang , Hengbo Zhang , Jinbiao Bao , and Hongtao Cao ARTICLES YOU MAY BE INTERESTED IN Fully epitaxial ferroelectric ScAlN grown by molecular beam epitaxy Applied Physics Letters 118, 223504 (2021); https://doi.org/10.1063/5.0054539 Perspective on the future of silicon photonics and electronics Applied Physics Letters 118, 220501 (2021); https://doi.org/10.1063/5.0050117 2DEGs formed in AlN/GaN HEMT structures with AlN grown at low temperature Applied Physics Letters 118, 222103 (2021); https://doi.org/10.1063/5.0050584Solution-processed amorphous p-type Cu-Sn-I thin films for transparent Cu-Sn-I/IGZO p–n junctions Cite as: Appl. Phys. Lett. 118, 222107 (2021); doi: 10.1063/5.0051631 Submitted: 26 March 2021 .Accepted: 24 May 2021 . Published Online: 3 June 2021 Haijuan Wu,1,2Lingyan Liang,2,a) Xiaolong Wang,2Hengbo Zhang,2,3Jinbiao Bao,1and Hongtao Cao2,3,a) AFFILIATIONS 1Department of Polymer Science and Engineering, School of Material Science and Chemical Engineering, Ningbo University, Ningbo 315211, Zhejiang, People’s Republic of China 2Laboratory of Advanced Nano Materials and Devices, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, People’s Republic of China 3Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049,People’s Republic of China a)Authors to whom correspondence should be addressed: lly@nimte.ac.cn andh_cao@nimte.ac.cn ABSTRACT P-type Cu-Sn-I thin films with different Sn contents ( CSn) were fabricated in air via a simple and low-cost spin-coating method. Sn additive facilitates the amorphization of CuI, and a complete amorphous phase of Cu-Sn-I film is achieved at CSn¼15%. With increasing CSn, the optical bandgap increases and refractive index decreases, probably due to the influence of Sn-additive on both the electronic structure and phase state of the films. The air-processed Sn-free CuI films show p-type conduction with hole mobility and a concentration of 17.3 cm2/V/C01 s/C01and 1.1 /C21019cm/C03, and an increasing trend of resistivity is observed along with a large drop in hole concentration during the Sn- inspired amorphization process. Moreover, transparent Cu-Sn-I/IGZO p–n junctions were constructed, exhibiting the optimum rectifyingcharacteristic at C Sn¼15% with a forward-to-reverse ratio of 6.2 /C2103. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0051631 Portable and/or “invisible” electronics/optoelectronics have attracted more and more attention.1–3Transparent amorphous oxide semiconductors (TAOSs) are intrinsically appropriate for the develop-ment of newly emerging electronic/optoelectronic devices becausethey have many merits such as no grain boundary, ultra-smooth surface/interface, arbitrariness in substrate selection, and low- temperature/large-area preparation. 4–7However, current TAOS mate- rials (such as In-Ga-Zn-O) with high carrier mobility are n-typeconductive, limiting their applications to unipolar n-type devices. 4,8 Hence, it is a big challenge to obtain p-type inorganic semiconductormaterials comparable with TAOS, which are both amorphous andtransparent. Copper iodide (CuI) is an intrinsically p-type semiconductor material with good transparency (bandgap /C243.1 eV) as well as the advantages of nontoxic and abundant reserves. 9,10Up to now, CuI has been used as the active material of thin-film transistors, pyroelectricsand p–n junctions, the hole transport layer of solar cells, and soon. 11–14The valence band maximum (VBM) of CuI is mainly contrib- uted by Cuþ3d and I/C05p orbitals.9The Cuþ3d orbital determines the VBM’s energy level in favor of stable hole doping.15The largespatial spread ( >200 pm) of I/C05p orbital promises a good hole path- way that facilitates high hole mobility even in an amorphous phase.15 Consequently, amorphization of CuI has been an attractive topic.However, previous investigations demonstrated that CuI exhibitspoly-crystallization regardless of preparation methods. 13,16,17Addition of heterovalent elements is a common strategy to suppress the crystal- lization. Jun et al. selected Sn as the additive and fabricated amorphous Cu-Sn-I thin films with Hall mobility of 8.9 cm2V/C01s/C01,w h i c hi s comparable to that of polycrystalline CuI thin films (8 cm2V/C01s/C01).15 Nevertheless, Li et al. reported that Sn additive did not act as an inhibi- tor of fast crystallization of CuI.18Hence, it is necessary to further investigate the detailed evolutions of Cu-Sn-I’s physical properties andthe applications in devices. At present, CuI thin films can be synthesized by various methods. It is more common that copper undergoes a chemical reaction in anenvironment full of iodine vapor, and the Cu film quickly transformsi n t oap o l y c r y s t a l l i n e c-CuI (with a zinc blende structure) film. 9,12The c-CuI film can also be synthesized by vacuum deposition technology such as thermal evaporation, pulsed laser deposition, and sputter-ing. 9,10,16Compared to vacuum technology, a solution method has Appl. Phys. Lett. 118, 222107 (2021); doi: 10.1063/5.0051631 118, 222107-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplattracted increasing attention due to various advantages such as cost reduction, low-temperature fabrication, air process, and precise con- trol of stoichiometric ratios.18,19In this paper, we discuss the prepara- tion and characteristics of solution-processed Cu-Sn-I thin films via systematically investigating the evolutions of their microstructural, morphological, optical, and electrical properties on the Sn content. Furthermore, combining p-type amorphous Cu-Sn-I with n-type In- Ga-Zn-O (IGZO) thin-films, we construct transparent p–n junctions with a significant rectifying effect. The CuI:SnI 4precursor solutions were prepared by mixing CuI (aladdin, 99.95%) and SnI 4(aladdin, 99.99%) in 2-methoxyethanol (Sigma-Aldrich, 99%) and ethanolamine (aladdin, 99%). The molar ratio CSn¼NSn/(NSnþNCu) was set between 0% and 20% by fixing NSnþNCu¼0.4 M. The CuI:SnI 4precursor solutions were spin- coated onto the oxygen-plasma-treated substrates at 5000 rpm for 40 s and then thermally treated at 140/C14C for 4 h in atmospheric environ- ment (with a relative humidity of 30%–40%). Substrates include glasses with patterned ITO stripes of 200/200 lm in width/space for p – nj u n c t i o nc o n s t r u c t i o na n dS i O 2(150 nm)/Si wafers for film char- acterizations. The annealed Cu-Sn-I films are 40–50 nm in thickness. After that, n-type IGZO thin-films of 45 nm were prepared by magne- tron sputtering from an IGZO (In:Ga:Zn ¼1:1:1) ceramic target with a DC power of 80 W, a Ar/O 2flux ratio of 14/6, and a working pres- sure of 0.11 Pa. Finally, 20-nm-thick Ti and 30-nm-thick Au were evaporated as the top electrodes of p–n junctions by e-beam evapora- tion through a shadow mask, with width/space of 200/200 lm. 100 nm-thick ITO transparent electrodes were deposited as the top electrodes for full-transparent devices by magnetron sputtering with a DC power of 60 W, a Ar/O 2flux ratio of 30/0.3, and a working pres- sure of 0.11 Pa. The fabricated films were characterized by x-ray dif- fraction (XRD, Bruker D8 advance x-ray diffractometer, Cu Ka¼1.5418 A ˚radiation), atomic force microscopy (AFM, Veeco Dimension 3100) measurement, a variable angle spectroscopicellipsometry (M-2000DI, J. A. Woollam, Inc.), and Hall measurement (ACCENT, HL5500). The electrical performance of p–n junctions was measured by a semiconductor parameter analyzer (Keithley 4200) at room temperature in the dark and air atmosphere. Figure 1(a) shows the XRD patterns in h–2hscans of the Cu-Sn-I thin films with different CSn. The pure CuI films have a preferred dif- fraction peak at 25.6/C14assigned to the (111) diffraction plane of c-CuI. With increasing CSn, the (111) diffraction peak gets weaker and disap- pears for the films at CSn/C2110%. AFM images in Fig. 1(b) indicate CSn-depended surface morphology. Wheatear-like particles sprawl on the surface of the Sn-free CuI film with some micro-pores in it. For the films with 5%-Sn additive, there are many particles of different sizes scattered in a plane matrix. The particles get larger but fewer at CSn¼10% and completely disappear at CSn¼15%. Smooth surface f a c i l i t a t e sas m o o t hi n t e r f a c e ,w h i c hi sb e n e fi c i a lf o rt h ec o n s t r u c t i o n of high-performance devices with a multi-film structure. The averagedroot-mean-square (rms) roughness also calculated and plotted as a function of C sninFig. 1(b) . The rms roughness of CSn-0%, CSn-5%, CSn-10%, CSn-15%, and CSn-20% thin films are 3.51, 0.228, 0.339, 0.2, and 0.314 nm, respectively. It illustrates that the significant decrease in the surface roughness of Cu-Sn-I films with increasing CSncan be closely coupled with the amorphization enhancement, as observed from XRD results. Taking both the XRD and AFM results into consid- eration, it seems that the amorphization of CuI begins since the addi- tive of Sn, and an amorphous Cu-Sn-I film without any CuI particles is obtained at CSn¼15% on the film surface. However, many small particles reappear on the Cu-Sn-I film of CSn¼20%. These results imply that the CSnis a crucial factor to achieve a complete amorphous phase of Cu-Sn-I film. The absorption coefficient ( a) and refractive index ( n)c a nb e obtained by the spectroscopic ellipsometry measurement and analysis. The ellipsometric angle wand phase difference Dwere recorded in the photon energy ( h/C23) range of 0.73–6.2 eV at incidence angles of 55/C14, FIG. 1. (a) XRD patterns and (b) AFM patterns of polycrystalline c-CuI and Cu-Sn-I films. Average rms roughness as a function of CSnis also plotted.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 222107 (2021); doi: 10.1063/5.0051631 118, 222107-2 Published under an exclusive license by AIP Publishing65/C14,a n d7 5/C14. The samples are simulated using a three-phase model consisting of Si substrate/SiO 2dielectric film/Cu-Sn-I film. One or two Tauc–Lorentz dispersion functions were adopted to represent thedielectric function of the Cu-Sn-I films. Figure 2(a) plots the ( ah/C23) 2vs h/C23curves. Direct optical bandgaps ( Eg) of all the films are obtained from the energy intercept by extrapolating the linear portion of ( aht)2 vshttoa¼0. The extracted Egis plotted as a function of CSninFig. 2(b).T h e Egof the pure CuI film is /C242.92 eV, very close to those of previously reported c-CuI.13Upon increasing CSn,t h e Egcontinuously increases, /C243.07 eV at CSn¼20%. The wide bandgap will promote visible-light transparency. Figure 2(c) illustrates the refractive index (n)–wavelength ( k)c u r v e s .T h e nvalues over the whole measured k range show a large drop after 5%-Sn additive and continuouslydecrease slowly with increasing C Sn(from 2.17 to 1.94). The nvaria- tion on CSnis reverse to the Eg,a sd e p i c t e di n Fig. 2(b) , where the n values at 550 nm are summarized as a function of CSn. (In general, the refractive index is associated with the absorption coefficient, so it is better to compare the refractive index at a wavelength, where the absorption coefficient of materials is zero. It can be found that therefractive index always shows little change with the wavelength when the corresponding photon energy is below the material bandgap. Here in, the 550 nm is chosen as an example.) To make clear the variationsofE gand nonCSn, a pure SnI 4film was also fabricated as a control under the same conditions as the Cu-Sn-I films. The SnI 4film has larger Eg(/C243.27 eV) and smaller nvalues (1.73 at 550 nm) compared to the c-CuI, agreeing well with the variations of both Egandnof the Cu-Sn-I films. This implies that the Sn additive might change the elec- t r o n i cs t r u c t u r eo fC u Ia n di n d u c et h ec o v a r i a t i o no f Egandn.O nt h e other hand, the film packing density (or the volume fraction of poros- ity) also has a big impact on the n values in theory.20The transition from polycrystalline to amorphous phase may also contribute to thevariations of E gand n, since the transition commonly associates with changes in the defect amount and/or film packing density.21 The electrical properties of the Cu-Sn-I films were repetitively tested by the Hall-effect measurement, with the statistical results listedinTable I . The Sn-free CuI film exhibits p-type conductive with a hole concentration up to 1.1 /C210 19cm/C03and a hole mobility of 17.3 cm2V/C01s/C01similar to previous reports (10–45 cm2V/C01s/C01).13,16,22 Carrier concentration close to 1019cm/C03is difficult to be tuned by applied biases when the film is used as the active layer of devices such as thin-film transistors and p–n junctions.12,23With Sn additive, the film resistivity vastly increases with distinct drops in both hole mobil- ity and concentration. The same effect was also reported by adding other dopant elements such as Zn.15T h ed e c r e a s ei nh o l em o b i l i t y may be related to the decrease in hole concentration according to the percolation theory.15The films with CSnof 15% and 20% show a high resistivity beyond the measurement limit of our Hall system. In brief, the tunable electrical properties, in particular, the carrier concentra- tion, would pave a way to construct electronic or optoelectronic devi- ces based on amorphous Cu-Sn-I films. Cu-Sn-I/IGZO p–n junctions with crossbar-structure electrodes were fabricated, as schematically shown in the inset of Fig. 3(b) .T h e current–voltage ( I–V) curves of all the devices are shown in Fig. 3(a) in logarithmic y coordinate. The Sn-free CuI devices show forward and reverse currents of near mA without any rectification behavior, probably due to the rough CuI/IGZO interface and/or high conductiv- ity of the CuI films. With increasing CSn, the device rectification effect becomes more and more evident, accompanied by the decrease in both the conductivity and surface roughness of the Cu-Sn-I films. The forward-to-reverse current ratio at 2.5 V reaches 3.0 /C2103at CSn¼15% but decreases to 2.5 /C2102atCSn¼20%. The decrease is very probably due to the declined quality of both the Cu-Sn-I film and Cu-Sn-I/IGZO interface. The surface morphology characterization (Fig. 1 ) indicates that many small grain particles reappear on the Sn- 20% films, which implies that the chemical component is no longer well-distributed and the Cu-Sn-I/IGZO interface is not so smooth as the Sn-15% case. In short, the device at CSn¼15% shows the best performance. First, the electrical contact properties of the Cu-Sn-I film/ITO electrode and IGZO film/Ti-Au electrode are examined, respec- tively, and linear I–V relationships are observed confirming the Ohmic like contact (Fig. S1). A representative I–V curve in linear coordinates is depicted in Fig. 3(b) with a forward turn-on voltage of/C241.75 V and a low leakage current under reverse bias. In addi- tion, the ideality factor ( nide) of the diode is an important parameter that ranges from 1.0 to 2.0 for an ideal diode. According to the Sah–Noyce–Shockley (SNS) theory,24,25the ideality factor can be extracted and is /C245.5 for the device at CSn¼15%, as demonstrated inFig. 3(c) . The larger ideality factor may be ascribed to inter- diffusion process, poor interface quality, high defect density at the hetero-interface, high series resistance or parasitic rectifying junc- tions within the device, etc.24,26 FIG. 2. (a) Plots of ( ah/C23)2vsh/C23of Cu-Sn-I thin films. (b) Band gap and refractive index as a function of CSn. (c) Refractive index spectra of Cu-Sn-I thin films.TABLE I. Hall-effect measurement results of the fabricated Cu-Sn-I films. CSn(%)Resistivity (Xcm)Mobility (cm2V/C01s/C01)Hole concentration (cm/C03) 0 0.040 17.3 1.1 /C21019 5 1.28 0.795 7.9 /C21018 10 35.53 0.995 1.6 /C21016 15 >100 /C1/C1/C1 /C1/C1/C1 20 >100 /C1/C1/C1 /C1/C1/C1Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 222107 (2021); doi: 10.1063/5.0051631 118, 222107-3 Published under an exclusive license by AIP PublishingFurthermore, full-transparent p–n junctions were prepared by using ITO instead of Ti/Au top electrodes. The transmittance spectra of the ITO glass, ITO/IGZO/Cu-Sn-I/ITO glass, are shown in Fig. S2. The p–n junction arrays are highly transparent, with marginaltransmittance ( /C2472.3% when averaged between 400 and 800 nm) loss in comparison with the ITO glass substrate ( /C2473.8%). As seen in the inset of Fig. 3(d) , the red petals of the flower can be clearly distin- guished through our full-transparent devices. The full-transparentp–n junctions have shown a forward turn-on voltage of 1.09 V, a for-ward-to-reverse current ratio of 6.2 /C210 3, and an ideality factor of 4.2 (Fig. S3). The full-transparent devices demonstrate overall perfor- mance improvement compared to the Ti/Au-electrode devices, veryprobably due to the decrease in the contact resistance between theIGZO film and ITO electrode (Fig. S4). The investigation is still ongoing to improve the overall performance of the p–n junctions by optimizing the properties of both p-type CuI-based films and n-typeoxide films. In summary, amorphous p-type Cu-Sn-I thin films and their p–n heterojunctions were prepared by the solution method, and the microstructural, morphological, optical, and electrical properties of the Cu-Sn-I thin films were systematically discussed as a function ofC Sn. It is found that the CSnplays an important role in the amorph- ization of CuI, and a complete amorphous phase of Cu-Sn-I film was achieved at CSn¼15% with an ultra-smooth surface, an optical bandgap of 3.05 eV, and a relatively high resistivity. This optimizedCu-Sn-I films facilitated the realization of fully transparent p–n junc-tion with IGZO, showing a forward-to-reverse ratio up to 6.2 /C210 3. The achievement of Cu-Sn-I-based p–n junctions makes an opening step forward to realize the practical application of the amorphousCuI-based material.See the supplementary material for contact properties between semiconductors and electrodes, device transmission curves, and ln(I)–Vcurves of the fully transparent devices. This project was supported by the International Cooperation Key Project, Bureau of International Cooperation, the Chinese Academy of Sciences (No. 174433KYSB20180050), the NationalNatural Science Foundation of China (Grant No. U20A20209), and the Natural Science Foundation of Zhejiang Province (Grant No. LD21F040002). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1L. Liu, Z. Niu, and J. Chen, Chem. Soc. Rev. 45(15), 4340 (2016). 2B. Anasori, M. R. Lukatskaya, and Y. Gogotsi, Nat. Rev. Mater. 2(2), 16098 (2017). 3D. Yu, Q. Qian, L. Wei, W. Jiang, K. Goh, J. Wei, J. Zhang, and Y. Chen,Chem. Soc. Rev. 44(3), 647 (2015). 4S. Kono, Y. Magari, M. Mori, S. G. Mehadi Aman, N. Fruehauf, H. Furuta, and M. Furuta, Jpn. J. Appl. Phys., Part 1 60(SB), SBBM05 (2021). 5A. Suresh, P. Wellenius, A. Dhawan, and J. Muth, Appl. Phys. Lett. 90(12), 123512 (2007). 6S. R. Thomas, P. Pattanasattayavong, and T. D. Anthopoulos, Chem. Soc. Rev. 42(16), 6910 (2013). 7L. Petti, N. M €unzenrieder, C. Vogt, H. Faber, L. B €uthe, G. Cantarella, F. Bottacchi, T. D. Anthopoulos, and G. Tr €oster, Appl. Phys. Rev. 3(2), 021303 (2016). 8T. Kamiya, K. Nomura, and H. Hosono, Sci. Technol. Adv. Mater. 11(4), 044305 (2010). FIG. 3. (a) Typical I–Vcurve of the Cu- Sn-I/IGZO p–n junctions with different CSn. (b) I–Vcurve in linear coordinates of the Cu-Sn-I/IGZO p–n junction atC Sn¼15%. The inset plots the device schematic diagram. (c) The experimental and fitting ln( I)–Vcurves. (d) I–Vcurve of Cu-Sn-I/IGZO fully transparent p–n junc-tion, the inset shows the optical photo onour devices with a flower behind.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 222107 (2021); doi: 10.1063/5.0051631 118, 222107-4 Published under an exclusive license by AIP Publishing9M. Grundmann, F. L. Schein, M. Lorenz, T. B €ontgen, J. Lenzner, and H. von Wenckstern, Phys. Status Solidi A 210(9), 1671 (2013). 10C. Yang, M. Kneibeta, M. Lorenz, and M. Grundmann, Proc. Natl. Acad. Sci. U. S. A. 113(46), 12929 (2016). 11C. Yang, D. Souchay, M. Kneiss, M. Bogner, H. M. Wei, M. Lorenz, O. Oeckler, G. Benstetter, Y. Q. Fu, and M. Grundmann, Nat. Commun. 8, 16076 (2017). 12A. Liu, H. Zhu, W. T. Park, S. J. Kim, H. Kim, M. G. Kim, and Y. Y. Noh, Nat. Commun. 11(1), 4309 (2020). 13C. H. Choi, J. Y. Gorecki, Z. Fang, M. Allen, S. J. Li, L. Y. Lin, C. C. Cheng, and C. H. Chang, J. Mater. Chem. C 4(43), 10309 (2016). 14Y. Q. Chen, J. Y. Chu, L. W. Li, A. S. Yerramilli, Y. H. He, H. X. Yang, Y. X. Shen, and T. L. Alford, J. Mater. Sci.: Mater. Electron. 32, 12929 (2021). 15T. Jun, J. Kim, M. Sasase, and H. Hosono, Adv. Mater. 30(12), e1706573 (2018). 16N. Yamada, R. Ino, and Y. Ninomiya, Chem. Mater. 28(14), 4971 (2016). 17H. J. Lee, S. Lee, Y. Ji, K. G. Cho, K. S. Choi, C. Jeon, K. H. Lee, and K. Hong, ACS Appl. Mater. Interfaces 11(43), 40243 (2019).18S. Y. Li, Y. Zhang, W. Yang, and X. S. Fang, Adv. Mater. Interfaces 6(13), 1900669 (2019). 19S. Choi, S. Song, T. Kim, J. C. Shin, J. W. Jo, S. K. Park, and Y. H. Kim, Micromachines 11(12), 1035 (2020). 20L. Y. Liang, H. T. Cao, Q. Liu, K. M. Jiang, Z. M. Liu, F. Zhuge, and F. L. Deng, ACS Appl. Mater. Interfaces 6(4), 2255 (2014). 21D. Saha, R. S. Ajimsha, K. Rajiv, C. Mukherjee, M. Gupta, P. Misra, and L. M. Kukreja, Appl. Surf. Sci. 315, 116 (2014). 22S. Lee, H. J. Lee, Y. N. Ji, S. M. Choi, K. H. Lee, and K. Hong, J. Mater. Chem. C8(28), 9608 (2020). 23A. Liu, H. Zhu, W. T. Park, S. J. Kang, Y. Xu, M. G. Kim, and Y. Y. Noh, Adv. Mater. 30, e1802379 (2018). 24X. X. Li, L. Y. Liang, H. T. Cao, R. F. Qin, H. L. Zhang, J. H. Gao, and F. Zhuge, Appl. Phys. Lett. 106(13), 132102 (2015). 25R. F. Qin, H. T. Cao, L. Y. Liang, Y. F. Xie, F. Zhuge, H. L. Zhang, J. H. Gao, K. Javaid, C. Liu, and W. Z. Sun, Appl. Phys. Lett. 108(14), 142104 (2016). 26Y. S. Lee, D. Chua, R. E. Brandt, S. C. Siah, J. V. Li, J. P. Mailoa, S. W. Lee, R. G. Gordon, and T. Buonassisi, Adv. Mater. 26(27), 4704 (2014).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 222107 (2021); doi: 10.1063/5.0051631 118, 222107-5 Published under an exclusive license by AIP Publishing

5.0047053.pdf

AIP Advances ARTICLE scitation.org/journal/adv Electronic and relating behavior of Mn-doped ZnO nanostructures: An x-ray absorption spectroscopy study Cite as: AIP Advances 11, 065027 (2021); doi: 10.1063/5.0047053 Submitted: 12 February 2021 •Accepted: 5 June 2021 • Published Online: 17 June 2021 Michael W. Murphy,1Laura Bovo,2 Gregorio Bottaro,3Lidia Armelao,1,2,4,a) and Tsun-Kong Sham1,a) AFFILIATIONS 1Department of Chemistry, University of Western Ontario, London, Ontario N6A 5B7, Canada 2Department of Chemistry, University of Padova, via F. Marzolo 1, Padova 35131, Italy 3Institute of Condensed Matter Chemistry and Technologies for Energy (ICMATE), National Research Council (CNR), Department of Chemistry, University of Padova, via F. Marzolo 1, Padova 35131, Italy 4Department of Chemical Sciences and Materials Technologies (DSCTM), National Research Council (CNR), Piazzale A. Moro 7, Roma 00185, Italy a)Authors to whom correspondence should be addressed: lidia.armelao@unipd.it and tsham@uwo.ca ABSTRACT Controlled synthesis of Mn-doped ZnO nanostructures with Mn concentrations of 1%, 3%, and 10% at. has been carried out using sol–gel methods and temperature treatments at 400, 600, and 800○C. It is found that Mn is successfully introduced into the hcp oxide lattice of ZnO nanoparticles of a range of sizes from a few nm to 102nm, depending on temperature conditions. It is also found that a secondary phase appears as the Mn concentration and processing temperature increase, most probably in the form appropriately described as MnO x clusters on the surface, although the dominant component remains hcp ZnO. The x-ray absorption near edge structure at all edges of interest reveals that the Mn2+ion substitutes Zn2+at the tetrahedral site and that the secondary phase exhibits a clear signature of the octahedral local environment at the Mn L 3,2and O K-edge. X-ray excited optical luminescence excited at 1085 eV (just above the Zn L 3,2edge) shows that the characteristic bandgap emission is slightly blue shifted and the luminescence from both the bandgap and defect emission is quenched somewhat with the latter significantly shifted to longer wavelengths in the region observed for surface and near surface defects. The Mn- doped samples processed at low temperature are poor light emitters due to the high degree of disorder and improve markedly with annealing at higher temperature. The magnetic properties of these systems were also investigated. The results suggest that Mn doping impedes radiative recombination, which is in favor of improved photocatalytic behavior. The implication of these findings is discussed. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0047053 I. INTRODUCTION Modern electronic components are steadily decreasing in size to increase efficiency; however, there is a quantum mechanical limit in size as components near nanoscale dimensions. Spintronics (spin +electronics) has emerged as an alternative technology that uses the electron spin rather than its charge as a new way to read and write information during storage and processing. Dilute magnetic semi- conductors (DMSs), which favorably combine both ferromagnetic and semiconducting properties, have attracted considerable inter- est for their potential applications in spintronic devices ever since Dietl et al. theorized that room temperature ferromagnetism (FM) in wide-bandgap semiconductors might exist.1Much of the earliereffort toward DMS spintronic devices has focused on the synthesis and characterization of transition metal (TM)-doped II–VI materi- als, with prominent interest in Mn-doped ZnO, henceforth denoted as Mn:ZnO, due to a proposed and measured Curie temperature above room temperature. However, much controversy surrounds this system as some researchers have observed both paramagnetism (PM) and anti-FM in Mn:ZnO2in addition to both the low temper- ature (∼45 K) and the room temperature FM previously observed by others.3,4Furthermore, it is still unclear whether the FM is intrin- sic (carrier-induced) or due to Mn-related secondary phases.5,6The disagreements arise, at least in part, from incomplete characteriza- tion of the system in terms of morphology and crystallinity. Mn:ZnO systems with the same composition are likely to show different AIP Advances 11, 065027 (2021); doi: 10.1063/5.0047053 11, 065027-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv behavior if their morphology and crystallinity vary. While the mag- netic behavior of Mn-doped ZnO is continually being pursued,7,8 interest in Mn-doped ZnO has shifted in recent years to its pho- tocatalytic and relating functionalities pertaining to electron–hole separation following photoexcitation, such as photocatalysis,9–15 photodegradation of organic molecules,16and anti-bacterial17prop- erties, among others. Doping may facilitate electron–hole sepa- ration and, hence, their reactivity for catalytic processes. Due to vast interest, it is of great importance to explore the relationship between the local environment around Mn and the electronic struc- ture, optical properties, and magnetic properties of the Mn:ZnO system. In this work, we report the preparation of a series of Mn:ZnO nanostructures using a sol–gel method followed by annealing; their structure and morphology, the x-ray absorption near-edge structure (XANES) at the Mn and O K-edge and the Zn and Mn L 3,2-edges, as well as x-ray excited optical luminescence (XEOL), and magnetic properties were investigated. These experiments will provide con- siderable information on the electronic structure of the system, such as the oxidation state and coordination of the Mn dopant, as well as how Mn doping affects the overall electronic, optical, and magnetic behavior of the system. XANES and XEOL are synchrotron techniques. XANES tracks the modulation of the x-ray absorption coefficient at an absorp- tion edge of an element in a chemical environment relative to that of a free atom, of which the photon energy dependent absorption coefficient is monotonic. The modulation in the absorption coeffi- cient arises from core to unoccupied state dipole transitions. The photoelectron probes the neighboring atoms via backscattering; the interference of the outgoing and backscattered electron wave at the absorbing atom produces oscillatory modulations in the absorp- tion coefficient unique to the local structure (identity of the neigh- boring atom, interatomic distance, local symmetry, and dynamics). Therefore, XANES is sensitive to the local structure of the atom of interest. XEOL monitors the energy transfer by materials that absorb x rays and convert them to optical photons (near IR to ultravi- olet).18–20The process is largely based on the inelastic scattering (energy loss) of the photoelectron and Auger electrons resulting from x-ray absorption. The cascade is known as the thermaliza- tion of energetic electrons, resulting in electrons at the bottom of the conduction band and holes at the top of the valence band. The recombination of the electrons and holes radiatively via excitons or defect energy levels in the bandgap will produce optical photons associated with the energy of the optical bandgap and defect states, respectively. The branching ratio of the bandgap vs defect emission can often be used to evaluate the crystallinity of the materials of interest.18,19 II. EXPERIMENTAL A. Synthesis Zn1−xMn xO nanostructures were prepared by sol–gel synthesis, and a schematic of the process is illustrated in Fig. S1. Stoichiometric amounts (x =0, 0.01, 0.03, and 0.10) of zinc acetate and manganese acetate (TMs totaling 0.009 mol) along with citric acid (0.009 mol) were dissolved in 30 ml of water and 80 ml of 30% ammonium hydroxide. The mixture was stirred for 6 h to make a sol, which wasthen placed in a drying oven for 16 h at 90○C to form the gel. The gel was placed in a high temperature furnace at 400○C for 4 h where it was cracked to produce the nanopowder as observed under an electron microscope (see below). The nanopowder was then divided into three portions: one portion was heated for an additional 2 h at 600○C, the second was heated for 2 h at 800○C, and the third was left as is. ZnO nanoparticles with 0%, 1%, 3%, and 10% Mn were prepared. B. Morphology from SEM The SEM images are displayed in Fig. S2, which show nanopar- ticles with a clear variation in the particle size ranging from ∼5 to 100 nm. This variation in the particle size is shown to be tempera- ture dependent rather than dopant dependent. The Mn:ZnO sam- ples processed at 400○C have a very small particle size of ∼5–10 nm, whereas the samples processed at 800○C are 25–100 nm in size. Upon annealing at 800○C, it is evident that agglomeration and ripening of the particles occur, resulting in larger particle sizes. There is no distinguishable difference between the 1% and 10% Mn samples processed at the same temperature, which indicates that the dopant concentration does not affect the particle size (Fig. S1). C. Crystalline structure from XRD The x-ray powder diffraction patterns are shown in Fig. S3 where we see that hcp ZnO is the only phase at low Mn concen- trations, from 0% to 3%, and at low temperature annealing. Con- versely, there are at least two phases present in the Zn 0.97Mn 0.03O samples at 800○C as well as in the Zn 0.90Mn 0.10O samples at 600 and 800○C. All the nanoparticles contain a wurtzite ZnO (a=3.25 Å and c =5.21 Å) phase (JCPDS PDF file no. 01-074-9939) and the secondary phase, when it is present, is of cubic ZnMnO 3 (a=8.35 Å) (JCPDS PDF file no. 00-019-1461) denoted by the black square. The intensity of the peaks associated with the sec- ondary phase increases with increasing processing temperature and dopant concentration although it remains a minor component. The ZnMnO 3impurities are a common issue in the synthesis of Mn:ZnO and have been detected by other studies.21,22A comparison of the XRD spectra of Mn:ZnO processed at 400, 600, and 800○C reveals a drastic broadening in the peaks as the processing temperature is decreased. This peak broadening is a common effect in XRD due to a decrease in the particle size.23This is tracked with the Scher- rer equation,24showing that the trend is consistent with the SEM observation. D. XANES and XEOL Mn L 3,2-edge, O K-edge, and Zn L 3,2-edge measurements were made on the undulator-based SGM beamline 11ID-1 of the Cana- dian Light Source (CLS). XANES spectra were recorded in total electron yield (TEY) and fluorescence yield (FLY).25XEOL spec- tra were collected using a QE65000 Scientific grade spectrometer (Ocean Optics) and a network CCD camera (Axis 2120), allowing for the collection of the entire optical range (200–900 nm). Mn K-edge and Zn K-edge measurements were conducted at the CLS@APS, sec- tor 20 of the Advanced Photon Source (APS). Spectra were obtained in transmission, total electron yield (TEY), or x-ray fluorescence yield (FLY) as appropriate. AIP Advances 11, 065027 (2021); doi: 10.1063/5.0047053 11, 065027-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv E. Magnetic measurements Magnetic measurements, such as the hysteresis loops of selected specimens, were conducted using a SQUID (superconducting quan- tum interference device) at the Institute for Research in Materials (IRM) at Dalhousie University, Halifax, Canada. III. X-RAY ABSORPTION NEAR EDGE STRUCTURE (XANES) A. Mn and Zn K-edge XANES Figure 1 shows the Mn and Zn K-edge XANES of the Mn:ZnO system of different compositions and phases. The Mn K-edge XANES arising from dipole transition from Mn 1 s→4pis shown in Fig. 1(a). A comparison of the Mn K-edge XANES with the ZnO K-edge XANES relative to the threshold (E o=0 eV) aligns the XANES features to the same energy scale above the threshold [Fig. 1(b)]. The threshold of Mn:ZnO is similar to that of MnO with Mn in a 2+oxidation state [Fig. 1(c)]. The peak locations A, B, and C in Fig. 1(a) are identical for all the doped samples, indicating that they have a very similar environment albeit with some noticeablebroadening at low temperature anneal due to disorder. With increasing processing temperature, the peaks sharpen, indicating an increase in crystallinity.26The Mn K-edge XANES of samples pro- cessed at 800○C resembles that of the Zn K-edge of ZnO [Fig. 1(b)], showing that Mn is in a similar tetrahedral local environment and substitutes Zn in the ZnO lattice as (Mn) ZnO [Fig. 1(d)]. Previous studies of the Mn K-edge have suggested that Mn substitutes Zn and no secondary phases are present.26–29The small pre-edge feature at 6540 eV (peak A) is a result of a weak s→d quadrupole transition associated with p-dhybridization. This, in turn, is only possible if the Mn site does not contain an inversion center, as is present in a tetrahedral configuration such as (Mn)ZnO, or when there is sizable local distortion.30The pre-edge peak is present in all nanoparticle samples, but close examination shows a decrease in intensity at a processing temperature of 800○C. This is an indication that a secondary phase, such as cubic ZnMnO 3, possibly forms as some of the Mn converts from a tetrahedral geom- etry to an octahedral geometry at a high processing temperature (800○C). Since this new phase is a minor component, the XANES features are still dictated by the Mn presence in the substitutional site. FIG. 1. (a) Mn K-edge XANES of Mn:ZnO nanoparticles. (b) Mn K-edge XANES of Mn:ZnO compared to the Zn K-edge of ZnO relative to threshold (E o=0 eV); top spectrum: Zn K-edge; the rest: Mn K-edge. (c) Mn K-edge of the reference of Mn oxides where the vertical line at 6550 eV is the E oof Mn:ZnO. (d) Local structure of ZnO showing Zn tetrahedrally bonded to four oxygens. AIP Advances 11, 065027 (2021); doi: 10.1063/5.0047053 11, 065027-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. Zn L 3,2-edge XANES spectra of Mn:ZnO nanoparticles recorded in TEY. (a) Comparison of Mn:ZnO with different Mn concentrations and annealing temperatures. (b) Comparison of ZnO with Mn:ZnO with 1% and 10% Mn and annealed at 800○C. B. Zn L 3,2edge and O K-edge XANES The Zn L 3,2edge XANES probes the unoccupied s character in the conduction band since the Zn 3d is full, and the O K-edge probes the unoccupied state of O 2p character in the conduction band. The Zn L 3,2-edge XANES spectra for Zn 1−xMn xO nanoparticles are shown in Fig. 2 where we see that there is no change in the absorp- tion edge threshold or the location of features in the XANES spectra for systems with 1%–3% Mn at all annealing temperatures. This indi- cates that the ZnO crystal structure remains essentially intact upon doping. Again, the spectral features sharpen at higher temperature as the crystallinity improves. However, while the fine structures at the Zn L 3,2-edge remain intact for the 1% sample, there is a noticeable deviation when the Mn concentration reaches 10% [Fig. 2(b)]. This change reveals anincrease in the overall disorder of the system resulting from the formation of a secondary phase. The secondary phase contains Zn in the form of a ZnMnO complex, most likely ZnMnO 3with an octahedral O environment as indicated by the XRD data (Fig. S3). This changes the local environment of some Zn sites in the system from a tetrahedral coordination in ZnO to an octahedral coordina- tion in ZnMnO 3, resulting in the combinatorial Zn L 3,2-edge XANES spectra in Fig. 2(b). Secondary phase formation is evident at the O K-edge XANES in Fig. 3. From Fig. 3(a), it is apparent that systems with 1%–3% Mn exhibit identical features to ZnO. The XANES spectra of 1% Mn-doped samples processed at 400 and 600○C and the 3% Mn- doped sample at 400○C [Fig. 3(a)] show virtually identical features to those of pure ZnO. They resemble the model spectrum obtained FIG. 3. (a) O K-edge XANES spectra of the Mn:ZnO nanoparticles and (b) O K-edge XANES spectra of Mn:ZnO and MnO (cubic—O hMn), Mn 2O3(spinel—O hMn), and MnO 2(rutile—O hMn) standards (b). AIP Advances 11, 065027 (2021); doi: 10.1063/5.0047053 11, 065027-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv by FEFF8 calculations containing tetrahedrally substituting Mn2+ in Zn vacancies with additional Zn vacancies in the crystal lattice (MnZn +VZn) described in the literature.28 A closer look reveals a small pre-edge shoulder at ∼532 eV for samples with the Mn concentration as low as 1% at 800○C and 3% at 600○C. This is a signature of Mn in the O henvironment.31This feature grows in the 10% sample under high temperature treatment. Thus, the pre-edge shoulder at 532 eV is present in systems where Mn no longer has a purely substitutional (Zn vacancy) role and forms an oxide complex with a lower binding energy. From the loca- tion and shape of the pre-edge feature (Fig. 3), the O in the secondary phase is bonded to either Mn3+or Mn4+in the O henvironment via p-dhybridization with the metal d bands. This observation is con- sistent with the recently compiled system of O K-edge XANES of 3d metal oxide in an O henvironment exhibiting a broad or a split reso- nance (pending the occupancy of d hole counts) via transition to O 2p states hybridized with the e gand t 2g3d orbitals of the transition metal.31The intensity of this pre-edge feature can be used as an indi- cation for secondary phase formation. In this case, the ZnO wurtzite structure remains the dominant component. C. Mn L 3,2-edge XANES The Mn L 3,2edge is the most sensitive edge in exploring the local properties of Mn since dipole transition allows the probing of the unoccupied states of Mn 3d 5/2,3/2 character in the conduction band. Experimentally, the L-edge is also more chemically sensitive partly because of the high energy resolution of the beamline optic, E/ΔE∼10 000 (5000 routinely, 0.13 at 640 eV), and partly because of the core–hole lifetime broadening of the Mn L 3-edge (0.2 eV), which is significantly narrower than that of the K-edge (1.12 eV).Figure 4(a) shows the comparison of the Mn L 3,2-edge XANES of all samples of interest, including MnO, Mn 2O3, and MnO 2stan- dards with d5, d4and d5mixed valence, and d3configurations, respectively. Looking at the Mn L 3,2-edge XANES [Fig. 4(a)], it is apparent that while the 1% sample is Mn2+like after 400 and 600○C anneal, signs of secondary phase formation at 800○C are evident as are in higher Mn concentrations at lower annealing temperature. Given the 1% Mn:ZnO Mn K-edge XANES as the representative of the MnO 4tetrahedral moiety of Mn2+in a ZnO matrix [Fig. 1(b)], spectral features differ from this, which is evidence for a secondary phase. Mn XANES in MnO is representative of a Mn2+in an octa- hedral crystal field surrounded by oxygen [Fig. 4(b)], as is the 1% sample at 400○C for the tetrahedral environment. It should be noted that the Mn 2p to 3d dipole transition is complicated by the presence of the crystal field and Jahn–Teller distortion, not to mention pos- sible exchange interaction with the surrounding and the proximity of the 2p 3/2and 2p 1/2states in energy; thus, the number of peaks observed in the 2p to 3d transition is not always intuitive. The secondary phase formation in the low Mn concentration systems appears to be temperature dependent as noted above: the 1% Mn samples processed at 400 and 600○C do not show signs of sec- ondary phases, which are, instead, only present at 800○C [Fig. 4(a)]. A similar trend can be seen for the 3% Mn samples with secondary phase formation evident in the 600 and 800○C samples but not in the 400○C sample. The onset of an impurity phase seen in the Mn L3,2-edge XANES is the appearance of Mn 2O3like features [Fig. 4(c)] although the Mn2+signal is likely buried under a stronger Mn3+sig- nal. The Mn L 3,2-edge XANES trend correlates well with the increase in the pre-edge peak seen in the O K-edge XANES (Fig. 3). From these results, we propose that Mn2+is first incorporated into the ZnO substitutionally, replacing the Zn2+ion as evident FIG. 4. (a) Mn L 3,2-edge XANES spectra of Mn:ZnO nanoparticles and MnO, Mn 2O3, and MnO 2powder standards. (b) and (c) Comparison of representative 1% and 10% systems with MnO and Mn 2O3, respectively. The energy scale is aligned to show the chemical shift. AIP Advances 11, 065027 (2021); doi: 10.1063/5.0047053 11, 065027-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv from Figs. 1(b), 3, and 4. However, its local structure is different from that of MnO (Mn2+, high spin, coordination number of 6, Ohsymmetry) as seen in the relative peak intensities and its resem- blance to (Mn)ZnO (high spin). In (Mn)ZnO, the O coordination number is 4 with T dsymmetry, as described in the literature.32 With increasing Mn concentration and processing temperature, the Mn3+state appears, resulting in spectral features similar to those of Mn 2O3as seen in Fig. 4(c) and the corresponding O K-edge in Fig. 3(b). The change in Mn valency from 2 +to 3+and 4+is fre- quently seen in the shift to higher photon energy and a decrease in the L 3to L 2peak ratios.4,32–34Although it is difficult to confirm the presence of the Mn4+, which is required for ZnMnO 3forma- tion, ZnMnO 3is most often found in the form of a mixed phase along with ZnO and spinel ZnMn 2O4.32While it is possible that both Mn3+containing ZnMnO 3and Mn4+containing ZnMn 2O4impu- rities exist in the nanopowder and more likely in the near surface region of the nanoparticles. XRD might miss ZnMn 2O4, which is either too weak to be detected due to its small cluster size or obscured by the dominant ZnO signal. However, it is more probable that mixed valence oxide clusters, MnO x, are formed upon high temper- ature annealing. The spectral results are entirely consistent with this notion. IV. X-RAY EXCITED OPTICAL LUMINESCENCE (XEOL) The XEOL spectra are recorded at 1085 eV excitation energy, which is just above the Zn L 3,2-edge. At this energy, the SGMbeamline has excellent flux and Zn will be excited preferentially (Fig. S4). Based on the absorption coefficient of Zn, Mn, and O at this energy, Zn absorbs most of the flux and the penetration depth is on the order of 102nm. More details are displayed in Fig. S4 where the attenuation of the photon for 1% and 10% Mn:ZnO samples is shown. The XEOL spectra of the Mn:ZnO nanoparticles processed at 800○C with concentrations from 0% to 10% are shown in Fig. 5(a). They all show a sharp peak at ∼3.2 eV (380 nm), which is the emis- sion from the optical bandgap, and a broad band, and the defect emission, which varies with the dopant concentration in the visi- ble from green to red. The small variation in optical bandgap energy [inset of Fig. 5(a)] is due to the variation in size distribution and quantum confinement. The overall luminescence intensity decreases with the increasing dopant concentration. By normalizing the XEOL spectra to the intensity of the bandgap emission [Fig. 5(b)], one can see that there is a significant change in the overall luminescence and the branching ratio between the defect emission and the bandgap emission upon doping. The addition of Mn decreases the overall luminescence and the defect emission relative to the bandgap emis- sion. A close examination of the bandgap emission [Fig. 5(a) inset] reveals a blue shift of the optical bandgap as the Mn concentration increases (by 40 meV from undoped to 10% doped). This could be due to a change in the binding energy of the exciton as well as quan- tum confinement noted above. The major difference between the Mn-doped and undoped ZnO systems can be seen in the defect emis- sion; the Mn:ZnO nanoparticles contain defect levels lower in energy FIG. 5. XEOL spectra of ZnO:Mn nanoparticles at 1085 eV: (a) Comparison of the XEOL from 0% to 10% Mn:ZnO samples annealed at 800○C; the inset shows the expansion of the bandgap emission. (b) Same set of spectra as (a) with the bandgap intensity normalized. (c) XEOL from the 3% Mn:ZnO sample processed from 400 to 800○C. AIP Advances 11, 065027 (2021); doi: 10.1063/5.0047053 11, 065027-6 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 6. Hysteresis loop of 10% (top) Mn:ZnO processed at 800○C and 1% (bottom) Mn:ZnO processed at 400○C. Measurements were made at 5 K. than those in the ZnO. The 550 nm defect emission is commonly observed in ZnO nanostructures.35,36Incidentally, 550 nm also cor- responds to the d–d transition from the4T1to6A1state of Mn2+.37 Thus, energy transfer to Mn2+will quench the ZnO defect emis- sion. The∼1.8 eV (680) emission most likely corresponds to surface defects on the Mn:ZnO nanoparticles, such as O vacancies near Mn sites that are lower in energy than the O vacancies near Zn sites, resulting in the red shift of the defect band compared to ZnO. Similar surface state emission has been observed in Eu-doped ZnO.36Fur- thermore, the luminescence at all dopant concentrations increases in intensity with increasing annealing temperature. An example is shown in Fig. 5(c) for 3% Mn:ZnO [Fig. 5(c)]. A similar enhance- ment of the luminescence intensity of Mn:ZnO systems by anneal- ing at 800○C has been previously reported.38It is almost certainly due to improved crystallinity and the trapping of electrons in the O vacancies. The overall reduction of optical luminescence intensity upon Mn doping shows that Mn doping inevitably reduces radia- tive electron–hole recombination via the optical channel, which, inturn, enhances electron–hole separation and, hence, its performance in photocatalysis and photodegradation.9–16 V. MAGNETISM We now return to magnetism, which triggered the initial flurry of the study of magnetic semiconductors. Figure 6 (top and bottom panels) shows the magnetic hysteresis for the 10% Mn:ZnO and 1% Mn:ZnO processed at 800 and 400○C, respectively. A list of the mag- netic behavior and coercivity, Hc, for all samples is summarized in Table I. The 10% Mn:ZnO (800○C) sample displays ferromagnetic ordering at 5 K (Fig. 6, top panel), but no such signature is observed at room temperature. In fact, all samples display a purely paramag- netic state at room temperature. Additionally, the linear shape of the hysteresis loop at low temperature does not rule out the presence of some paramagnetic contribution to the sample. The 1% Mn:ZnO (400○C) sample displays some degree of ferromagnetic ordering TABLE I. Magnetic behavior at 5 K for Mn:ZnO nanoparticles. Temp (○C) 1 (%) Hc (Oe) 3 (%) Mn Hc (Oe) 10 (%) Hc (Oe) 400 SPM +FM 10 SPM +FM 15 SPM +FM 85 600 FM 4 FM 70 FM +PM 275 800 FM +PM 110 FM 580 FM +PM 225 AIP Advances 11, 065027 (2021); doi: 10.1063/5.0047053 11, 065027-7 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv although a superparamagnetic contribution cannot be discounted. Super-paramagnetism (SPM) is common in ferromagnetic materials once the (domain) particle size is <10 nm. The saturation magneti- zation occurs when His∼70 000 Oe for all samples. The coercivity at 5 K for the 1% and 10% Mn:ZnO is ∼110 and∼225 Oe, respec- tively. In general, the coercivity increases with increasing processing temperature and Mn concentration. The increase in coercivity is linked to an increase in the domain size in large particles processed at higher temperatures. The samples that contain higher amounts of Mn secondary phases have hysteresis loops with more paramagnetic character (more linear). Comparing the coercivity in the 3% Mn 800○C sample with the 10% Mn 800○C sample, for example, there is a noticeable decrease with increasing Mn secondary phase formation. The formation of secondary phases causes a shift from ferromagnetic ordering toward paramagnetic ordering, decreasing the long-range magnetic order of the system. These results indicate that samples with the major- ity of Mn as (Mn)ZnO show ferromagnetic behavior, while those containing a higher amount of Mn secondary phase display both ferromagnetic and paramagnetic behaviors at 5 K. VI. SUMMARY AND CONCLUSION Mn:ZnO nanoparticles prepared via the sol–gel method at var- ious concentrations (1%, 3%, and 10%) and annealing temperatures (400, 600, and 800○C) have been analyzed. It is found that the as- prepared samples are poor crystalline materials and the crystallinity improves markedly upon annealing as the annealing temperature increases. The improved crystallinity, however, is accompanied by the precipitation of a secondary phase; the higher the Mn concen- tration, the lower the temperature of the secondary phase, most likely containing MnO xclusters on the surface. Regardless, the dom- inant phase remains the Mn-doped substitutional phase. The Mn L3,2-edge and O K-edge are found to be very sensitive to secondary phase formation, and the pre-edge feature in the O K-edge provides semi-quantitative information as to the amount of secondary phase formed. The XEOL result clearly shows that radiative electron–hole recombination is noticeably quenched in Mn:ZnO systems com- pared to the ZnO nanostructure. This observation strongly suggests that Mn doping will enhance electron–hole separation and, hence, the performance of the system in photocatalysis and photodegrada- tion. Secondary phase formation also leads to a decrease in the coer- civity and a shift from FM toward PM. These results can help explain why different Mn concentrations of Zn 1−xMn xO systems can have different magnetic behaviors. Furthermore, these results help shed light on the issue of why Mn:ZnO systems with identical Mn concen- trations show discrepancies in their FM measurements based on the processing temperature. Although further tests are needed to deter- mine the composition of the secondary phase, these results clearly show that both low Mn concentration and low temperature process- ing are essential if secondary phase formation is to be avoided. These considerations will prove vital to the use of Mn-doped ZnO DMS systems for spintronics and photocatalytic applications. SUPPLEMENTARY MATERIAL See the supplementary material for a schematic of the synthe- sis and SEM and XRD data as well as penetration depth of relevant elements.AUTHORS’ CONTRIBUTIONS M.W.M. participated in the synthesis and did all synchrotron measurements and analyses. L.B. and G.B. participated in the syn- thesis and laboratory characterization. L.A. and T.-K.S. supervised this work. All authors contributed to the preparation and read the manuscript. ACKNOWLEDGMENTS Research at the University of Western Ontario (UWO) was supported by NSERC (Grant No. RGPIN-2019-05926), CFI, OIT, and CRC (TKS). The SEM measurement was conducted at Nanofab- rication Lab, UWO. M.W.M. acknowledges the warm hospitality of the University of Padova during his visit. Magnetic measure- ments were made in IMR of Dalhousie University. This research was, in part, conducted at the Canadian Light Source, which was supported by CIHR, NRC, NSERC, and University of Saskatchewan. This research also used resources of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Depart- ment of Energy (DOE) Office of Science by Argonne National Lab- oratory, and was supported by the US DOE under Contract No. DE-AC02-06CH11357. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Ferrand, Science 287, 1019 (2000). 2A. Tiwari, C. Jin, A. Kvit, D. Kumar, J. F. Muth, and J. Narayan, Solid State Commun. 121, 371 (2002). 3S. W. Jung, S.-J. An, G.-C. Yi, C. U. Jung, S.-I. Lee, and S. Cho, Appl. Phys. Lett. 80, 4561 (2002). 4P. Sharma, A. Gupta, K. V. Rao, F. J. Owens, R. Sharma, R. Ahuja, J. M. O. Guillen, B. Johansson, and G. A. Gehring, Nat. Mater. 2, 673 (2003). 5X. 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5.0056480.pdf

J. Chem. Phys. 154, 244103 (2021); https://doi.org/10.1063/5.0056480 154, 244103 © 2021 Author(s).Quantum chemistry for molecules at extreme pressure on graphical processing units: Implementation of extreme-pressure polarizable continuum model Cite as: J. Chem. Phys. 154, 244103 (2021); https://doi.org/10.1063/5.0056480 Submitted: 11 May 2021 . Accepted: 07 June 2021 . Published Online: 23 June 2021 Ariel Gale , Eugen Hruska , and Fang Liu ARTICLES YOU MAY BE INTERESTED IN Orbital optimization in nonorthogonal multiconfigurational self-consistent field applied to the study of conical intersections and avoided crossings The Journal of Chemical Physics 154, 244101 (2021); https://doi.org/10.1063/5.0053615 Chemical physics software The Journal of Chemical Physics 155, 010401 (2021); https://doi.org/10.1063/5.0059886 Machine learned Hückel theory: Interfacing physics and deep neural networks The Journal of Chemical Physics 154, 244108 (2021); https://doi.org/10.1063/5.0052857The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Quantum chemistry for molecules at extreme pressure on graphical processing units: Implementation of extreme-pressure polarizable continuum model Cite as: J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 Submitted: 11 May 2021 •Accepted: 7 June 2021 • Published Online: 23 June 2021 Ariel Gale, Eugen Hruska, and Fang Liua) AFFILIATIONS Department of Chemistry, Emory University, Atlanta, Georgia 30322, USA a)Author to whom correspondence should be addressed: fang.liu@emory.edu ABSTRACT Pressure plays essential roles in chemistry by altering structures and controlling chemical reactions. The extreme-pressure polarizable con- tinuum model (XP-PCM) is an emerging method with an efficient quantum mechanical description of small- and medium-sized molecules at high pressure (on the order of GPa). However, its application to large molecular systems was previously hampered by a CPU computation bottleneck: the Pauli repulsion potential unique to XP-PCM requires the evaluation of a large number of electric field integrals, resulting in significant computational overhead compared to the gas-phase or standard-pressure polarizable continuum model calculations. Here, we exploit advances in graphical processing units (GPUs) to accelerate the XP-PCM-integral evaluations. This enables high-pressure quantum chemistry simulation of proteins that used to be computationally intractable. We benchmarked the performance using 18 small proteins in aqueous solutions. Using a single GPU, our method evaluates the XP-PCM free energy of a protein with over 500 atoms and 4000 basis functions within half an hour. The time taken by the XP-PCM-integral evaluation is typically 1% of the time taken for a gas-phase density functional theory (DFT) on the same system. The overall XP-PCM calculations require less computational effort than that for their gas-phase counterpart due to the improved convergence of self-consistent field iterations. Therefore, the description of the high-pressure effects with our GPU-accelerated XP-PCM is feasible for any molecule tractable for gas-phase DFT calculation. We have also validated the accuracy of our method on small molecules whose properties under high pressure are known from experiments or previous theoretical studies. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0056480 I. INTRODUCTION Pressure plays important roles in chemistry by inducing phase transitions of molecule crystals,1–3altering chemical bonds,4,5con- trolling chemical reaction rates,6–8and tuning photochemical reac- tions.9,10Modeling the influence of pressure in quantum chemi- cal calculations is of great importance to reveal the mechanism of exotic phenomena under high pressure, including pressure- induced α-helix stabilization,11–13formation mechanism of amino acids in deep space,14,15and metallic behavior of hydrogen under extreme pressure.16–18Highly efficient quantum chemistry simula- tion under high pressure will also enable virtual high-throughput screening19using pressure as a tuning parameter for chemical discovery.Periodic density functional theory (DFT) and Hartree–Fock have been used to investigate high-pressure effects in materials and molecule crystals in the past few decades.20–23However, pressure effects on large molecules, such as proteins, can hardly be simulated with a periodic DFT approach due to the high computational cost for systems with hundreds or thousands of atoms per unit cell. Clas- sical molecular dynamics, as a low-cost alternative, lacks the descrip- tion of electronic structure changes under pressure. Its accuracy in describing high-pressure phenomena is highly dependent on the force field parameterization24and could lead to results contradicting experimental findings.25–27 The extreme pressure polarizable continuum model (XP- PCM) by Cammi and co-workers28–31emerges as a computation- ally efficient approach to incorporate pressure effects (on the order J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-1 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp of GPa29) into quantum chemistry calculations on single molecules. It has been applied to the study of small- to medium-sized molecules from single atoms,30,32,33small organic molecules,29,34to crystals35,36 and fullerenes.37While XP-PCM DFT calculations are much more efficient than their periodic DFT counterparts,36their applications in quantum chemistry calculation of large molecular and biomolec- ular systems are still hampered by the high computational over- head for evaluating large numbers of related electron integrals. Unlike the DFT calculations in the gas phase or in standard- pressure conductor-like screening models,38,39XP-PCM DFT cal- culations involve the unique Pauli repulsion potential, whose eval- uation requires O(MN2)of electric field integrals. Here, Nis the number of basis functions and Mis the number of XP-PCM cav- ity grid points typically at O(10N)orO(100N). This large number of electric field integrals can result in computational overhead many times greater than the total runtime of the corresponding gas-phase DFT calculation. Because of this prohibitively high computational overhead, no XP-PCM calculation has been reported for proteins. Graphical Processing Units (GPUs) are especially suitable for par- allel computing involving massive data40and have been success- fully applied in accelerating various types of electron integrals by numerous groups.41–44Speedups of more than two orders of magni- tude have been observed in the GPU-accelerated quantum chemistry methods at different levels of theory, including Hartree–Fock,45,81 density functional theory,46second-order Moller–Plesset perturba- tion theory,47coupled-cluster theory,48and multireference meth- ods.49,50Here, we exploit the advances of GPUs to accelerate the elec- tric field integrals uniquely required in XP-PCM calculations and enable quantum chemistry simulation of large molecular systems under high pressure. II. THEORY The XP-PCM theory is an extension of conductor-like screen- ing models (COSMO,38C-PCM,39GCOSMO,51and IEF-PCM52–54), which are introduced to describe the free energy of solvated molecules. In these models, the solute molecule is embedded in a dielectric continuum with permittivity ε, forming a cavity with unit permittivity. The solute polarizes the continuum, whose elec- tric field is described by a set of polarization charges on the cavity surface. Then, the free energy of a solvated system in C-PCM can be expressed as GC−PCM=E0+Gpol, (1) where E0is the energy of the solute and Gpolis the electrostatic component of the solvation free energy represented by the interac- tion between the polarization charges and the solute, in addition to the self-energy of the surface charges. Numerous publications have described the detailed formalism of Gpol38,39,51,55,56and algorithms for large molecular systems,44,57–60so we will not elaborate on them in this work. To describe molecules at extreme pressure, Cammi and co- workers28–31proposed the XP-PCM method, where the free energy of the system at the given pressure pis GXP−PCM(p)=E0(p)+Gpol(p)+Gr(p)+Gcav(p). (2)Compared to the free energy formula of C-PCM [Eq. (1)], the XP-PCM free energy [Eq. (2)] introduces the Pauli repulsion con- tribution, Gr, and the cavitation energy term, Gcav. The cavitation energy Gcavis the isotherm–isobar reversible work required for the formation of a void cavity to host the molec- ular solute in the pure solvent at the given pressure and temper- ature.34,61,62The Gcavterm does not contribute to the electronic Hamiltonian of the solute and is sometimes omitted in XP-PCM implementations.30The XP-PCM free energy excluding Gcavis denoted as Ger=E0(p)+Gpol(p)+Gr(p). (3) In this work, we only focus on Gerwithout considering Gcav. The Pauli repulsion term, Gr, describes the exchange–repulsion term of the interaction energy of the solute electrons and the solvent elec- trons. Evaluation of Grmodifies the electronic Hamiltonian and is essential for implementing XP-PCM in self-consistent field (SCF) calculations. In Subsections II A–II D, we describe the essential equations for XP-PCM, focusing on the evaluation of Gr. A. Basic formula of Pauli repulsion potential In the XP-PCM model, the electronic Schrödinger equation for the solute molecule is given by (ˆH0+ˆVpol+ˆVr)Ψ=EΨ, (4) where ˆH0is the Hamiltonian of the solute molecule in vacuum, Ψ is the solute wave function, ˆVpolis the electrostatic solute–solvent interaction covered in C-PCM, ˆVris the Pauli repulsion operator, andEis the energy eigenvalue. The Pauli repulsion operator corresponds to a repulsive poten- tial located at the boundary of the solute cavity, ˆVr=N ∑ i∫ˆρ(r)Γ(r)dr. (5) Here, ˆρ(r)=∑N iδ(r−ri)is the electron density operator over the N electrons of the solute molecule and the repulsion potential Γ(r)is a step barrier potential at the boundary of the cavity, Γ(r)=ZΘC(r), ΘC(r)={0,r⊆DC 1,r⊈DC,(6) whereDCdenotes the domain of the physical space inside the cavity, and the height of the step barrier, Z, is determined by the extent to which the cavity is compressed and can be expressed as a function of the cavity scaling parameter f, Z(f)=Z0(Vc(f) Vc(f0))−(3+η) 3 . (7) Here, f0=1.2 is the value of fat the standard condition of pres- sure, Vc(f)is the volume of the molecular cavity obtained with the J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-2 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp cavity scaling parameter f,ηis a semi-empirical parameter that gauges how strong the Pauli repulsive barrier of the external medium is, and Z0is the barrier at standard pressure calculated from the following equation [adapted from Eq. (13) of Ref. 63]: Z0=(4π/ξ)ρBnB pair. (8) Here, ρBis the number density of the solvent molecule B, nB pairis the number of valence electron pairs of the solvent, and ξ=0.7 is the exponent of the Gaussian representation of localized orbitals.63 In practical implementations, an empirical scaling coefficient is applied,63given by Z0=0.063 ρBnB val MB, (9) where ρBis redefined as the density of the solvent relative to the den- sity of water at 298 K, nB valis the number of valence electrons of the solvent, and MBis the molecular weight of the solvent. In HF/DFT, ∣Ψ⟩is the ground state Slater determinant, so we can apply the rules for the integral of Slater determinants with one- electron operators64and get Gr=⟨Ψ∣ˆVr∣Ψ⟩=∑ μν∑ σ=α,βPσ μν⟨μ∣Γ(r)∣ν⟩, (10) where Pσis the density matrix of the solute electrons with spin σ and μand νare atomic basis functions. The contribution of Pauli repulsion to the Fock matrix is hr μν=∂Gr ∂Pσμν=⟨μ∣Γ(r)∣ν⟩. (11) Combining Eqs. (6), (7), and (11), hr μνcan be rewritten as63 hr μν=Z(f)(Sμν−S(in) μν), (12) where Sμνis an element of the overlap matrix and S(in) μν=−1 4π∯ S(C)(Eμν⋅ˆn)dS (13) is the electric flux of the electric field ( Eμν) contributed by the electron density μ(r)ν(r)through the cavity surface S(C). Here, ˆnis the surface normal vector. Gauss’s law is used to convert the volume integral inside the cavity [Eqs. (5) and (6)] to the surface integral in Eqs. (12) and (13). A detailed derivation is provided in the supplementary material, Text S1. Therefore, in practical XP-PCM implementations, the essential task is to evaluate the Pauli repulsion matrix hr[Eq. (11)] and, hence, the one-electron integrals S(in) μν. We will elaborate on the details of the numerical evaluation of S(in) μνin Sec. II C after introducing the discretization scheme in Sec. II B. B. Discretization of the molecular cavity surface in switching-Gaussian approach In conductor-like screening models, the electrostatic interac- tion ˆVpolis evaluated numerically by discretizing the cavity surfaceinto “tesserae.”38To evaluate the S(in) μν integrals in XP-PCM, we use the readily built solvent accessible surface (SAS) discretized by the switching-Gaussian (SWIG) approach55to be consistent with our implementation of ˆVpol(Fig. 1). In this discretization scheme, the molecular surface is formed from inter-locking van der Waals (VDW) spheres centered around the composing atoms, and the surface of each sphere is discretized by Lebedev grid points.55 The geometry of the solvent molecule is represented by expanding the VDW radii with a scaling factor fand adding an optional effec- tive solvent radius Rsolv. The radius of each VDW sphere, RJ, is then expressed as RJ=f R0 J+Rsolvf/f0, (14) where R0 Jis the Bondi radius65,66of atom J and Rsolvis the sol- vent radius, and the scaling factor f/f0forRsolvis adapted from the scaling of the solvent radius in the solvent exclusion surface (SES).29 To avoid singularities in the evaluation of ˆVpol, surface polar- ization charges are presented as spherical Gaussian functions cen- tered at the grid point, and the Gaussian exponent for the kth point charge belonging to the Ith nucleus is given as ζk=ζ RI√wk, (15) where ζis an optimized exponent for the specific Lebedev quadra- ture level being used (as tabulated by York and Karplus),55wkis the FIG. 1. Schematic presentation of the discretization of the SAS in the SWIG approach for a water molecule as an example. The atomic spheres have centers RAand radii { f⋅R0 A+Rsolvf/f0}, where { R0 A} are the VDW radii, fis the cavity scaling factor, and Rsolvis the solvent radius. The space inside the cavity (white background) and outside the cavity (blue background) is denoted by r⊆DCand r⊈DC, respectively. Cavity surface grid points (located at { rk} with norm vectors {ˆnk}) are presented by dots colored by the element of the center atom (red for O and gray for H). Transparency of the grid points indicates the value of the switch- ing function [opaque for exposed points with Sk≈1 and transparent for partially “buried” points with Sk≪1;Sis defined in Eq. (16)]. An example pair density μ(r)ν(r)contributed by the atomic basis functions μ(r)(centered around R1) andν(r)(centered around R3) is presented by yellow volume, with its electric field presented by Eμν. J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-3 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Lebedev quadrature67weight for the kth point, and RIis the atomic radius of the Ith nucleus that the kth tessera belongs to. To get a smooth change in cavity surface area during geom- etry optimization, the switching function is introduced to indicate how much a grid point is either buried inside the molecular surface or exposed and accessible to solvent molecules. For the improved SWIG (ISWIG) scheme56used by default in our implementation, the switching function is defined as Sk=atoms ∏ J,k∉JSwf(rk,RJ), (16) Swf(rk,RJ)=1−1 2{erf[ζk(RJ−∣rk−RJ∣)] +erf[ζk(RJ+∣rk−RJ∣)]}, (17) where rkis the location of the kth Lebedev point, RJis the location of the Jth nucleus with atomic radius RJ, and erf is the Gauss error function. The area of the kth tessara can then be calculated as ak=wkR2 ISk. (18) C. Numerical evaluation of Pauli repulsion integrals With the discretization scheme of Sec. II B (see also Fig. 1), the S(in) μνintegral [Eq. (13)] can be rewritten as S(in) μν=−1 4πM ∑ k(Ek μν⋅ˆnk)ak. (19) Here, ˆnkis the norm vector of the kth tessera pointing outward from the cavity, akis the area of the kth tessera [Eq. (18)], and Ek μνis the electric field caused by the electron distribution of basis function pair μ(r)ν(r)at the kth tessera center, Ek μν=−∫μ(r)ν(r)rk−r ∣rk−r∣3dr, (20) where rkis the position of the kth Lebedev grid point (Fig. 1). It is worth noting that this type of electric field integral does not exist in C-PCM, where only the electric potential integrals are evaluated to obtain ˆVpol. As usual, the atom-centered basis functions are con- tractions over a set of primitive atom-centered Gaussian functions, μ(r)=lμ ∑ i=1cμiχi(r). (21) Thus, the one-electron integral S(in) μν[Eq. (19)] can be expressed as S(in) μν=1 4πM ∑ k(μ(r)μ(r)∣(rk−r)⋅ˆnk ∣rk−r∣3)ak =1 4πlμ ∑ i=1lν ∑ j=1M ∑ kcμicνj[χi(r)χj(r)∣(rk−r)⋅ˆnk ∣rk−r∣3]ak, (22)where we use brackets to denote one-electron integrals over prim- itive basis functions and parentheses to denote such integrals for contracted basis functions. In the following, we use the indices μ,ν for contracted basis functions and the indices i,jfor primitive Gaus- sian basis functions. We discuss the GPU algorithm for evaluating S(in) μνin Sec. III. D. Numerical calculation of pressure We use the numerical fitting approach proposed by Fukuda et al.68to calculate the pressure passociated with each value of the cavity scaling factor f. For a given molecule with fixed structure, multiple XP-PCM calculations with different fvalues are performed to obtain a series of Gervalues. These Gervalues are fitted as a non- linear function of the associated cavity volumes, Vc, based on the following expression:68 Ger(Vc)=Ger(V0 c)+aVc⎡⎢⎢⎢⎢⎣1 b−1(V0 c Vc)b +1⎤⎥⎥⎥⎥⎦+cVc, (23) where a,b, and care fitting parameters. Then, the pressure p can be computed by differentiation (derivation is available in the supplementary material, Text S2), p(f)=−∂Ger(Vc) ∂Vc=a⎡⎢⎢⎢⎢⎣(V0 c Vc)b −1⎤⎥⎥⎥⎥⎦−c. (24) III. IMPLEMENTATION ON GPUs Unlike the ˆVpolrelated integrals that need to be re-evaluated in each SCF iteration, S(in) μνis evaluated only once before the SCF calcu- lation starts and is directly added to the core Hamiltonian. However, this does not mean that the evaluation of S(in) μν is computation- ally trivial. In Sec. V A, we will demonstrate that XP-PCM usually requires a significantly denser grid than regular C-PCM to ensure numerical integration accuracy, which increases computational cost significantly if no acceleration strategy is applied. Building S(in) μν requires one-electron integral evaluations and involves a significant amount of data parallelism, making it well suited for GPU acceleration. We elaborate the GPU-based acceler- ation strategies in Subsections III A and III B. A. Fine-grained parallelism Analogous to our GPU-based implementation of ˆVpolrelated integrals in C-PCM,44,57we wrote six separate GPU kernels for eval- uating S(in) μνof the following angular momentum classes: ss,sp,sd, pp,pd, and dd. Each individual GPU thread calculates integrals cor- responding to a batch of primitive pairs sharing the same set of pair quantities, similar to the one thread ↔one batch mapping43origi- nally proposed for the evaluation of Coulomb integrals. For instance, in the sskernel, each GPU thread calculates a single integral, [χs1χs2], in each loop, whereas in the spkernel, each GPU thread calculates three primitive pairs, [χsχx p],[χsχy p], and[χsχz p]. The algorithm for evaluating S(in) μν forsppairs is shown schematically in Fig. 2 for a system with one sshell and two pshells J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-4 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2. Algorithm for calculating S(in)forspintegrals of a system composed of one sshell and two pshells (the sshell contains two primitive Gaussian functions each; the first and second pshells have two and three primitive Gaussian functions, respectively). On top of the graph, the pale green array represents primitive pairs belonging to sp shell pairs. The GPU cores are represented by orange squares (threads) embedded in pale yellow rectangles (one-dimensional blocks with six threads/block). The 1 ×6 block is used for illustrative purposes only, and a 1 ×128 block is used in actual implementation. The output is a 3 ×Nthreads array where each GPU thread generates three integrals for primitive pairs [χsχx p],[χsχy p], and[χsχz p]. Primitive pair integrals are finally added to the Fock matrix entry of the corresponding contracted function pair. All red lines and text indicate contracted Gaussian integrals. Blue arrows and text indicate memory operations. and a GPU block size of 1 ×6 threads. The sshell contains two primitive Gaussian functions; the first and second pshells have two and three primitive Gaussian functions, respectively. A block of size 1×6 is used for illustrative purposes. In practice, a 1 ×128 block is used for optimal occupancy and memory coalescence. Primitive pairs, χiχj, that make negligible contributions are not calculated, and these are determined by using a Schwartz-like bound69with a cutoff εscreen=10−14atomic units, [ij∣Schwartz=[χiχj∣χiχj]1/2<εscreen. (25) Here, we use a tighter threshold than the default 10−12threshold for ˆVpol44because Grusually has a smaller magnitude than Gpoland thus is more sensitive to the integral threshold. The surviving pair quantities are preloaded to the GPU global memory, and each GPU thread fetches a batch of 3 spprimitive pairs sharing the same set of pair quantities at the beginning of the integral kernel. Quantities related to each Lebedev grid point (area ak, coor- dinates rk, and norm vector ˆnkpointing toward outside of the cavity) are also preloaded in global memory. Each GPU thread loops over all Lebedev grid points to accumulate the electric flux of the electric field contributed by its primitive pair [χiχj]through all tesserae ofthe cavity surface as follows: S(in) ij=1 4πM ∑ k[χi(r)χj(r)∣(rk−r)⋅ˆnk ∣rk−r∣3]ak. (26) It is worth noting that for the GPU kernel shown in Fig. 2, which evaluates the spangular momentum class, three integrals in the form of Eq. (26) are evaluated by each GPU thread for the primitive pairs [χsχx p],[χsχy p], and[χsχz p]. Evaluation of the primitive integral of Eq. (26) is discussed in Subsection III B. The result is stored to an output array in GPU global memory, which is later copied to the central processing unit (CPU) memory after the accumulation of S(in) ijis done. The last step is to form S(in) μν, S(in) μν=lμ ∑ i=1lν ∑ j=1cμicνjS(in) ij, (27) on the CPU by adding each entry of the output array (primitive pair) to its corresponding atomic orbital pair entry. All algorithms discussed above can be easily generalized to other angular momentum classes other than sp. The numbers of J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-5 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp primitive pairs evaluated by each GPU thread in momentum classes ss,sp,sd,pp,pd, and ddare 1, 3, 6, 9, 18, and 36, respectively, since our implementation uses the Cartesian format basis function and each dorbital has six components. These kernels are launched sequentially. B. Evaluation of primitive integrals The primitive electric field integral of Eq. (26) for different angular momentum classes are evaluated analytically based on the algorithm of McMurchie and Davidson.70For a pair of Cartesian Gaussian basis functions χi=xn Ayl Azm Aexp(−αAr2 A), (28) χj=x¯n By¯l Bz¯m Bexp(−αBr2 B) (29) centered at A=(xA,yA,zA)andB=(xB,yB,zB), the product χiχjcan be expanded as combinations of Hermite polynomial Gaussians, [χiχj∣=n+¯n ∑ Nm+¯m ∑ Ml+¯l ∑ LDNLMΛN(xP)ΛL(yP)ΛM(zP) ×exp(−αPr2 P) =∑ NLMDNLM[NLM∣. (30) Here, the Hermite polynomial Gaussian Λjis related to the Hermite polynomial HjbyΓj(xP;αP)=αj/2 PHj(α1/2 PxP),P=(xp,yp,zp)is the center of the product Gaussian function formed from the overlap of the two Gaussian functions in Eqs. (28) and (29), and αPis the Gaussian exponent of the product Gaussian; indices (N,L,M)run over all possible combinations within the appropriate range. The electric field integral of χiχjcan then be calculated as combinations of the electric field integral of Hermite polynomial Gaussians ( N,M,L), S(in) ij=1 4πM ∑ kak∑ NLMDNLM([NLM∣xkr−3 k]ˆnk,x +[NLM∣ykr−3 k]ˆnk,y+[NLM∣zkr−3 k]ˆnk,z), (31) where ˆnk,x,ˆnk,y, and ˆnk,zare the components of ˆnkin x, y, and z directions, and the integrals of [NML∣are given by [NLM∣xkr−3 k]=−(2π/αP)RN+1,L,M, [NLM∣ykr−3 k]=−(2π/αP)RN,L+1,M, [NLM∣zkr−3 k]=−(2π/αP)RN,L,M+1.(32) Here, the auxiliary functions RNLM can be calculated from recursive relations to tabulated Boys functions.71 We wrote separate GPU kernel functions for evaluating the primitive integrals of each momentum class. Each kernel function generates expansions of primitive pairs [Eq. (30)] based on recursive relations of DNLM71and then evaluates the electric field integrals of Eq. (32) in terms of Boys functions.IV. COMPUTATIONAL DETAILS We have implemented a GPU-accelerated XP-PCM formula- tion in a development version of the TERACHEM72,73package. All XP-PCM calculations use parameters stated as follows unless oth- erwise specified. An ISWIG screening threshold of 10−8is used, meaning that molecular surface (MS) points with a switching func- tion value less than this threshold are ignored. The Pauli repulsion gauge parameter ηis set to 6 as recommended by the literature.30 This choice of ηis known to give a dependence of the computed pressure on the cavity volume Vcin reasonable agreement with the dependence of the experimental pressure on the molar volume in molecular solids.34 To study the computational performance of our GPU imple- mentation on large biomolecules, we select a test set of 18 experi- mental protein structures74obtained with aqueous solution nuclear magnetic resonance spectroscopy (NMR) where inclusion of a sol- vent environment was essential to find optimized structures in good agreement with experimental results (Fig. 3). The proteins range in size from around 70 to 500 atoms, and their detailed proper- ties (PDB ID, number of residues and atoms, charge, and secondary structure) are summarized in the supplementary material, Table S1. For these test molecules, we conduct a number of XP-PCM single- point energy evaluations at the B3LYP75–77/6-31G∗78level of theory, with the cavity scaling factor fvalues ranging from 0.85 to 1.20 (pressure values on the order 100 to 1 GPa; see the supplementary material, Figs. S1 and S2). The environment dielectric constant cor- responds to aqueous solvation ( ε=78.39, ρB=1.0,nB val=8,MB=18). The default revised Bondi radii66are used ( R0 H=1.1 Å, R0 C=1.7 Å, FIG. 3. Structures for the benchmark proteins for the XP-PCM performance test. For each protein, the PDB ID, the number of atoms, and the number of orbitals with 6-31G∗basis set are listed. J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-6 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp R0 N=1.55 Å, R0 O=1.52 Å, and R0 S=1.8 Å). Similar calculations are carried out in the gas phase and in C-PCM implicit solvent to test the computational cost of adding in the XP-PCM high-pressure envi- ronment. All timings have been obtained using a single core of the Intel Xeon Gold 6248 “Cascade Lake” CPU clocked at 2.50 GHz and one NVIDIA Tesla V100 GPU. To validate our XP-PCM implementation, we carry out XP-PCM calculations on an argon (Ar) atom and an acetylene molecule. To simplify the comparison with previous XP-PCM works by Cammi et al. on these systems,30we have used the same simula- tion parameters wherever possible. Therefore, we fix the geometry at the equilibrium geometry optimized in the gas phase without XP-PCM. Due to the lack of f-type basis functions in TERACHEM , we are not able to use the aug-cc-pVTZ basis set used in the work of Cammi,30and the single-point calculations are carried out at the B3LYP/aug-cc-pVDZ level of theory. The environment dielectric constant corresponds to cyclohexane ( ε=2.0165, ρB=0.779, nB val =36,MB=84.16). We use the same set of Bondi atomic radii ( R0 H =1.2 Å, R0 C=1.7 Å, and R0 Ar=1.88 Å) as in the work of Cammi30 to facilitate the comparison of results. The hydrogen atom radius is slightly different from the default revised Bondi radius66used in TERACHEM (R0 H=1.1). The cavity uses an ISWIG56discretization density of 1202 Lebedev points/atom and cavity radii that are varied by applying a scaling factor fon the Bondi radii.65 V. RESULTS AND DISCUSSION The GPU implementation of XP-PCM could be a computa- tionally efficient approach to investigate the electronic structure of molecular and biomolecular systems. In this section, we investigate the efficiency and accuracy of our XP-PCM implementations. We first look for the appropriate discretization level to obtain numeri- cally converged XP-PCM results for a small protein. With the opti- mal discretization parameter, we benchmark the performance of the XP-PCM calculation of a set of proteins varying in sizes to estimate the time scaling and the applicability of this to large biomolecules. Then, we compare the computational performance of the XP-PCM, C-PCM, and the gas phase counterpart to evaluate the extra compu- tational cost for describing the pressure and solvent effects. Finally, we assess the quality of our implementation for describing high- pressure effects by applying the method on an argon atom and an acetylene molecule. A. Convergence with respect to discretization level In this subsection, we look for the optimal XP-PCM cav- ity discretization level that balances the numerical accuracy and computational cost. We examine the convergence of the XP-PCM calculation with respect to the discretization level of the cavity surface, which is defined as the density of Lebedev grid points per sphere (Fig. 4). For the tested peptide (PDB ID: 3FTR), both the Pauli repulsion energy ( Gr) and the electrostatic solvation free energy ( Gpol) values vary with the discretization levels and, hence, determine the conver- gence behavior of the total free energy ( Ger). At low, medium, high, and very high discretization levels (26–50, 110–302, 434–770, and 974–1202 points/atom), the relative error in Gpolis typically less than 5%, 1%, 0.3%, and 0.04%, whereas the relative error for Grrapidly FIG. 4. Convergence of the free energies ( Gr,Gpol,GerinEh) and runtime (in seconds) with respect to the discretization level (points/atom) for the XP-PCM B3LYP/6-31G∗calculation of a peptide (PDB ID: 3FTR). A constant cavity scal- ing factor f=1.0 is used for all calculations. The main chain of 3FTR is shown in the inset, with carbon, nitrogen, oxygen, and hydrogen colored gray, blue, red, and white, respectively. All XP-PCM calculations are conducted with terachem using a single core of Intel Xeon Gold 6248 “Cascade Lake” CPU clocked at 2.50 GHz and one NVIDIA Tesla V100 GPU. changes from 111% to 44%, 7%, and 2% (Fig. 4 and Table I). Since GpolandGrhave similar magnitudes and opposite signs, it is essen- tial to reach sufficient numerical accuracy for both terms to ensure the accuracy of the total free energy. Ideally, a very high discretization level ( ⩾974 points/atom) is preferred, but the total runtime is twice the runtime of the medium discretization level (Fig. 4). As the grid density increases, the number of effective (non-buried) Lebedev points increases linearly, and the number of XP-PCM related primitive integrals (for both hrand ˆVpol) increases linearly (supplementary material, Table S2). Since ˆVpolis re-evaluated for each SCF iteration, the increase in computational cost with grid density is magnified, resulting in very high compu- tational overhead ( <50% of total runtime) at a very high discretiza- tion level (supplementary material, Table S3). To balance accuracy and efficiency, a high discretization level of 434 points/atom is cho- sen for performance tests in Secs. V B and V C unless otherwise specified. It is worth noting that this choice of discretization level for XP-PCM is significantly higher than the recommended discretiza- tion level for C-PCM,55which only has free energy contribution from Gpol. As shown in our test for 3FTR (Fig. 4) as well as pre- vious works in this field,55,56sufficient accuracy for Gpol(error≈1 kcal/mol) can already be reached at a medium discretization level (∼110 points/atom). Apart from the convergence of free energies, the convergence of cavity volume is also crucial for XP-PCM because the volume is explicitly needed for deriving the pressure [Eq. (5)]. Our choice of 434 points/atom has less than 0.2% error in cavity volume for the tested system, significantly lower than the 2% J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-7 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I. Free energies ( GrandGpolinEh) obtained at different discretization levels compared to the results obtained with the highest grid density (1202 points/atom) for the XP-PCM B3LYP/6-31G∗calculation with a cavity scaling factor f=1.0 of a protein fibril (PDB ID: 3FTR). Grid densityRelative error (%) Discretization level (pts./atom) Gr(Eh) Gpol(Eh) Gr Gpol Low26 0.451 286 −0.392 746 111.32 4.51 50 0.451 692 −0.401 015 111.51 2.50 Medium110 0.305 759 −0.406 418 43.18 1.18 194 0.264 476 −0.408 335 23.84 0.72 302 0.246 270 −0.409 623 15.32 0.40 High434 0.229 174 −0.410 099 7.31 0.29 590 0.223 057 −0.410 603 4.45 0.16 770 0.218 007 −0.410 924 2.08 0.09 Very high974 0.218 098 −0.411 119 2.13 0.04 1202 0.213 555 −0.411 281 0.00 0.00 error at low discretization levels (supplementary material, Table S2). Because of the higher discretization level required in XP- PCM, a GPU-accelerated implementation is even more critical for XP-PCM than C-PCM to ensure its applicability to large molecules where the electric field integral evaluations may dominate the total runtime. B. Performance for large molecules Two primary concerns about applying XP-PCM to large molecules are the time scaling of the algorithm and the efficiency compared to its gas phase or normal-pressure solution-phase coun- terparts. To test these, we collected the timings of XP-PCM calcu- lation of a set of proteins at different pressures presented by differ- ent cavity scaling factors f(Fig. 5). The observed empirical scaling of the evaluation of the Pauli repulsion matrix hrunique to XP-PCM is O(N1.9) regardless of the cavity scaling factor f. Here, the normal ( f=1.2), moderately compressed ( f=1.0), and highly compressed ( f=0.85) cavities correspond to pressure values on the order of 1, 10, and 100 GPa, respectively (supplementary material, Figs. S1 and S2). However, the prefactor of the scaling increases as f decreases, meaning that the XP-PCM calculations at higher pressure have higher computational costs for evaluating hr. This is a natu- ral result of the fact that the number of effective molecule surface grid points (not “buried” in the cavity) roughly increases linearly as fincreases (supplementary material, Fig. S3). As the scaling factor fdecreases, the radius of the atom-centered spheres decreases, and there is less overlap between the spheres and more exposure of grid points. The total runtime of XP-PCM follows a similar trend. The run- time has O(N2.0) scaling at all fvalues, but larger prefactors are observed at lower fvalues (Fig. 5). Here, the total runtime includes the evaluation of hr, the electrostatic solvent effect term ˆVpol, and other terms in regular gas-phase SCF. It is worth noting that the ˆVpolevaluation also needs more computational cost at lower fval- ues because of the increase in grid points. For the largest protein(PDB ID: 2KJM) in the benchmark set, the time for evaluating ˆVpol doubles as the cavity is compressed from f=1.2 to f=0.8, while the time for evaluating hralso increases by 1.75 times (supplementary material, Figs. S4 and S5). In summary, our XP-PCM implementation demonstrates a sub-quadratic scaling based on tests of molecules with up to 5000 basis functions, which is similar to the performance of our GPU- accelerated implementation of C-PCM. C. Performance comparison to gas phase To obtain a comprehensive comparison between XP-PCM and its gas phase or normal-pressure solution-phase counterparts, we decompose the timings for XP-PCM at f=1.0 (moderately com- pressed, pat about 10 GPa) into different contributions (Fig. 6). The total runtime of XP-PCM can be partitioned into the XP-PCM- specific part and other parts in common with gas-phase SCF. The former includes three major components: building the cavity sur- face, constructing the Pauli repulsion matrix hr, and evaluating elec- trostatic solvent effects related terms ˆVpol. Among these terms, ˆVpol takes the majority of the time because the related integrals depend on solute electron density and are re-evaluated in each SCF iteration. The molecular cavity and the hrmatrix only need to be built once before SCF starts and take less than 0.2% and 1.4% of the time for any tested protein. In total, the percentage of time taken by the XP-PCM- specific terms fluctuates between 12% and 26% with an average value of 17%. Based on these timings, and assuming that SCF converges similarly in different environments, we expect that the runtime of an XP-PCM calculation is about 1.2X of the gas-phase runtime for the same system. However, direct comparison of the runtime for XP-PCM, C-PCM, and gas-phase calculations shows that XP-PCM and C-PCM require less runtime than the gas-phase counterpart for all tested proteins (Fig. 7). This is caused by the fact that DFT calculations of large molecules such as proteins tend to converge much faster in XP-PCM J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-8 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5. Timings (in seconds) for the XP-PCM B3LYP/6-31G∗single-point calcula- tions for a set of 18 benchmark proteins in normal ( f=1.2,pat∼1 GPa), mod- erately compressed ( f=1.0,pat∼10 GPa), and highly compressed ( f=0.85, pat∼100 GPa) cavities. An ISWIG discretization scheme is used with 434 Lebe- dev points/atom. Timing data are presented with dots, with curves with respective colors showing the empirical scaling fitted by the power function. Timings for the evaluation of the Pauli repulsion integral ( hr) (upper) and for the total single-point calculation (lower), with representative proteins of different sizes shown in the inset structure. All XP-PCM calculations are conducted with terachem using a single core of Intel Xeon Gold 6248 “Cascade Lake” CPU clocked at 2.50 GHz and one NVIDIA Tesla V100 GPU. FIG. 6. Percentage of the runtime spent on different components of XP-PCM calculations of 18 small proteins using B3LYP/6-31G∗. The constructions of the molecular cavity, Pauli repulsion matrix hr, and solvation electrostatic interactions ˆVpoland their sum are represented by blue, red, and green dots, respectively. The sum of these XP-PCM-specific components is denoted with gray dots. For each component, the average value over the 18 protein set is indicated by a dotted line with the corresponding color. All XP-PCM calculations are conducted with terachem using a single core of Intel Xeon Gold 6248 “Cascade Lake” CPU clocked at 2.50 GHz and one NVIDIA Tesla V100 GPU. than in the gas phase, as was observed for the comparison between C-PCM and gas phase in our previous works.44,57The number of SCF iterations taken by the gas-phase calculation is 1.4X to 22.8X of that taken by XP-PCM (supplementary material, Table S4). Two proteins (PDB IDs: 1ODP and 2KJM) that failed convergence in 2000 iterations in the gas phase also successfully converged in XP- PCM within 17 steps. Hence, we expect that in practical applications, XP-PCM is computationally feasible for any large molecular system that is computationally tractable in the gas phase. D. Compressed argon atom To assess the accuracy of our implementation for describing high-pressure effects, we compare our XP-PCM calculation of a FIG. 7. Relative runtime (in seconds) taken by XP-PCM ( f=1.0) and C-PCM B3LYP/6-31G∗single-point energy evaluations compared to their gas-phase counterpart for benchmark proteins ordered by increasing size. Two proteins (PDB ID: 1ODP and 2KJM) in the benchmark set are not included because of unconverged gas-phase calculations. An ISWIG discretization scheme is used with 434 Lebedev points/atom. Runtimes for XP-PCM and C-PCM are visually the same because the extra time for building hrin XP-PCM is negligible due to our efficient implementation. Timings were obtained with a terachem single core of Intel Xeon Gold 6248 “Cascade Lake” CPU clocked at 2.50 GHz and one NVIDIA Tesla V100 GPU. J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-9 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8. (Left) XP-PCM free energy of a compressed argon atom in cyclohexane solvent as a function of cavity volume (cavity scaling factor f=0.85–1.20). Geris calculated relative to its value obtained with f0=1.2. (Right) Comparison of the volume compression of argon ( Vc/V1) as a function of pressure obtained with XP-PCM calculation (this work and the work of Cammi et al.30) and experiments.79V1is the reference volume of the cavity corresponding to the pressure p=1.1 GPa. Schematic illustration of the argon atom in a compressed cavity is shown in the insets. The length of arrows indicates the strength of pressure. compressed argon atom with previous theoretical and experimen- tal results. The free energy Gerat different cavity volumes ( Vc) is obtained with XP-PCM calculations at different cavity scaling factor values ( f=0.85–1.20, Fig. 8 and supplementary material, Table S5). To obtain the corresponding pressures, we used Eq. (23) to fit Geras a function of Vc(Fig. 8 and supplementary material, Table S5). The fitting parameters for compressed argon are a=1.3048 ×10−4Eh/Å3, b=5.4057, and c =−1.7593 ×10−4Eh/Å3. Using these parameters with Eq. (24), we determined the pressure as a function of the cavity volume (Fig. 8 and supplementary material, Table S6). We see excel- lent agreement with the previous XP-PCM work of Cammi for the free energy and the pressure functions due to the usage of a slightly different basis set (see details in Sec. IV). We have also compared the volume compression as a function of pressure to the experimental values for solid argon compression and found very good agreement (Fig. 8). E. Compressed acetylene molecule We further assessed the XP-PCM description of pressure effects in an acetylene molecule, where the cavity has an irregular shape in contrast to argon’s spherical cavity. With a similar numerical fit- ting approach, we obtained the pressure as a function of the cavity volume (Fig. 9 and supplementary material, Table S7). The fitting parameters for compressed acetylene are a =1.5457 ×10−4Eh/Å3, b=5.9367, and c =−1.8521 ×10−4Eh/Å3. Although there are some differences in the free energy values, the p–Vccurve gives a good agreement with the results of Cammi.30 It is not surprising that the energies are not the same, as we have noticed some differences in the cavity definition. Although we obtained the same cavity volumes as the results of Cammi for the argon atom, we noticed that our cavity volumes for acetylene are systematically lower at all fvalues, even when the same set of atomic radii are used (supplementary material, Table S8). The differ- ence is likely caused by the fact that we use the switching-Gaussian approach to smooth the cavity surface, which can influence the cal- culation of Vc. To facilitate direct comparison with the work of Cammi, we added a small solvent radius (0.135 Å) to our cavity to FIG. 9. Relative electronic energy Gerand pressure with respect to cavity volume from XP-PCM B3LYP/aug-cc-pVDZ single-point calculations ( f=0.9–1.2) for a compressed acetylene molecule in cyclohexane. XP-PCM free energy Geras a function of cavity volume for a compressed acetylene molecule in cyclohexane solvent (upper). Geris calculated relative to its value obtained with a cavity scaling factor f0=1.2. Pressure–volume relationship derived from the numerical fitting approach of Eq. (23) (lower). A schematic illustration of the acetylene molecule in a compressed cavity is shown in the insets. The length of arrows indicates the strength of pressure. J. Chem. Phys. 154, 244103 (2021); doi: 10.1063/5.0056480 154, 244103-10 Published under an exclusive license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp expand the volumes to be more comparable to the results of Cammi, and the final results are reported in Fig. 9. However, we can still see subtle differences in volumes between this work and the work of Cammi at the same fvalue. This difference in cavity construction also influences hr, leading to the difference in Ger(Fig. 9). VI. CONCLUSIONS In this work, we demonstrated that by implementing the Pauli repulsion integrals of primitive basis functions with fine-grained parallelism, the free energy of the XP-PCM method can be efficiently evaluated on GPUs and can be applied to the simulation of large molecules under high pressure. The performance was tested by calculating the XP-PCM free energy of 18 proteins with a size range of 70–500 atoms at high pres- sure (on the order of 1–100 GPa). The benchmark calculations are used to demonstrate the feasibility of applying the method on large molecules under high pressure with up to 5000 orbitals. We achieve the same scaling as the C-PCM method, showing that the evaluation of Pauli repulsion integrals in XP-PCM only introduces a minimal increase in computational cost. For all tested proteins, the XP-PCM calculation took less runtime than its gas-phase counterpart due to improved SCF convergence. We showed that our XP-PCM imple- mentation is feasible for any system that can be calculated in the gas phase. We also validated the accuracy of our implementation by comparing the XP-PCM calculated pressure–volume relation- ship with previous XP-PCM and experimental results. Very good agreement is obtained for an argon atom and an acetylene molecule. In the future, we will extend our acceleration strategies to the evaluation of analytical energy gradients29and analytical pressure30 of the XP-PCM method. These efforts will enable efficient geometry optimization and ab initio molecular dynamics of large molecular systems under pressure. SUPPLEMENTARY MATERIAL See the supplementary material for the characteristics of the benchmark protein set; the convergence of cavity volume as a func- tion of grid density for 3FTR; detailed timings as a function of grid density for protein 3FTR; the number of grid points, runtime, and percentage runtime as functions of the cavity scaling factor ffor protein 2KJM; the comparison of XP-PCM and gas-phase timings and SCF iterations for the benchmark protein dataset; Gerand pres- sure as functions of cavity volume for an argon atom and acety- lene; the calculated and experimental volume compression Vc/V1 as a function of pressure for an argon atom; and cavity volume as a function of ffor acetylene obtained with difference cavity radii (PDF) and also for geometries of all proteins in the benchmark set and geometries of an argon atom and an acetylene molecule (ZIP). ACKNOWLEDGMENTS This work was supported by start-up funds provided by Emory University. This work used the Extreme Science and Engineer- ing Discovery Environment80(XSEDE) Bridges-2 at the Pitts- burgh Supercomputing Center through Allocation No. CHE200099,which is supported by National Science Foundation Grant No. ACI-1548562. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1R. J. Hemley, “Effects of high pressure on molecules,” Annu. Rev. Phys. Chem. 51, 763 (2000). 2O. Mishima and H. E. Stanley, “The relationship between liquid, supercooled and glassy water,” Nature 396, 329 (1998). 3Y. Fujii, K. Hase, N. Hamaya, Y. Ohishi, A. Onodera, O. Shimomura, and K. Takemura, “Pressure-induced face-centered-cubic phase of monatomic metal- lic iodine,” Phys. Rev. Lett. 58, 796 (1987). 4L. Pauling, The Nature of the Chemical Bond (Cornell University Press, Ithaca, NY, 1960), Vol. 260. 5S. Duwal, Y.-J. Ryu, M. Kim, C.-S. Yoo, S. Bang, K. Kim, and N. H. 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5.0050340.pdf

AIP Advances 11, 065113 (2021); https://doi.org/10.1063/5.0050340 11, 065113 © 2021 Author(s).Comparison of the magnetic properties of bismuth substituted thulium iron garnet and yttrium iron garnet films Cite as: AIP Advances 11, 065113 (2021); https://doi.org/10.1063/5.0050340 Submitted: 14 March 2021 . Accepted: 26 April 2021 . Published Online: 04 June 2021 Yuanjing Zhang , Qinghui Yang , Xiuting Liu , Ding Zhang , Yiheng Rao , and Huaiwu Zhang COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN 3C-SiC-induced peak emission intensity in photoluminescence spectrum of SiC/SiO 2 core– shell nanowires using first-principles calculations AIP Advances 11, 065214 (2021); https://doi.org/10.1063/5.0050501 Nanowaveguide-illuminated fluorescence correlation spectroscopy for single molecule studies AIP Advances 11, 065112 (2021); https://doi.org/10.1063/5.0051679 Spin wave propagation in a ferrimagnetic thin film with perpendicular magnetic anisotropy Applied Physics Letters 117, 232407 (2020); https://doi.org/10.1063/5.0024424AIP Advances ARTICLE scitation.org/journal/adv Comparison of the magnetic properties of bismuth substituted thulium iron garnet and yttrium iron garnet films Cite as: AIP Advances 11, 065113 (2021); doi: 10.1063/5.0050340 Submitted: 14 March 2021 •Accepted: 26 April 2021 • Published Online: 4 June 2021 Yuanjing Zhang, Qinghui Yang,a) Xiuting Liu, Ding Zhang, Yiheng Rao, and Huaiwu Zhanga) AFFILIATIONS State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China a)Authors to whom correspondence should be addressed: yangqinghui@uestc.edu.cn and hwzhang@uestc.edu.cn ABSTRACT Rare-earth iron garnet thin films with perpendicular magnetic anisotropy (PMA) have recently attracted a great deal of attention for spintron- ics applications. Bismuth substituted iron garnets are particularly popular among these various films because Bi3+with a larger ion radius can adjust the lattice constant, strain state, and PMA of the films. In this paper, Tm 2BiFe 5O12(TmBiIG) and Y 2BiFe 5O12(YBiIG) garnet films with a series of thicknesses are prepared by radio frequency magnetron sputtering, and these films exhibit robust PMA. The microstructural proper- ties, magnetic properties, and the anomalous Hall effect of these two kinds of films are discussed in detail. Due to their larger magnetostriction coefficient and proper tensile strain, TmBiIG films exhibit better PMA than YBiIG films, which have lower damping. As the thickness of TmBiIG and YBiIG films increases, the PMA becomes weaker, and the 40 nm YBiIG turns back to in-plane easy magnetization, but PMA is still obvious for the 64 nm TmBiIG. The ferromagnetic resonance linewidth of the 32 nm TmBiIG film is 249.08 Oe @ 13 GHz, and the damp- ing factor is 1.49 ×10−2, which is close to that of Tm 3Fe5O12. YBiIG films have better damping characteristics than TmBiIG films; however, the value is larger than that of yttrium iron garnet because of surface roughness and defects caused by larger lattice mismatch. In addition, on account of fewer defects and smaller surface roughness, the Hall voltage and Hall resistivity in TmBiIG/Pt heterostructures are larger than in YBiIG/Pt. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0050340 I. INTRODUCTION Rare-earth iron garnet (RIG) films, especially yttrium iron garnet (Y 3Fe5O12, YIG) films, have played a significant role in microwave devices1,2and magneto-optical devices3,4because of the narrow ferromagnetic resonance (FMR) linewidth and large specific Faraday rotation angle. In recent years, Kajiwara and co-workers successfully demonstrated spin pumping and spin- transfer-torque in a YIG/Pt ferromagnetic insulator/heavy metal (FMI/HM) system.5This provoked intensive studies of magnon- ics and spintronics,6–8and a lot of research had focused on FMIs by the spin wave and spin current in FMI/HM het- erostructures.9–12The property of the spintronics device is closely related to FMIs’ magnetization direction,13–15so it is necessary to prepare garnet films with perpendicular magnetic anisotropy (PMA).With the development of film growth technologies, high- quality RIG films with low damping were prepared by radio fre- quency (RF) magnetron sputtering,16,17pulsed laser deposition (PLD),15,18and liquid phase epitaxy (LPE).19,20Typically, YIG films display an in-plane easy magnetization axis since the uniaxial anisotropy is not strong enough to overcome the shape anisotropy. However, stress-induced anisotropy and growth-induced anisotropy could be introduced by changing the growth parameters to sur- pass shape anisotropy, and then, PMA would arise in the garnet films.15,21–26The stress state needed is dependent on the magne- tostriction coefficient of the films. Ortiz et al.21obtained PMA with compressive strain in EuIG and TbIG, with positive (111) magne- tostriction coefficients. The (111) magnetostriction coefficients of most RIG films are negative,14so tensile strain is helpful to PMA and the lattice constant of films is expected to be smaller than that of substrates. For instance, Quindeau and co-workers found AIP Advances 11, 065113 (2021); doi: 10.1063/5.0050340 11, 065113-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv that Tm 3Fe5O12(TmIG) films can have robust PMA,15which was mainly contributed by stress-induced anisotropy influenced by large magnetostriction coefficients. Tang and co-workers further enlarged stress-induced anisotropy by adopting a substituted gadolinium gal- lium garnet (SGGG) with a larger lattice constant as the substrate of TmIG.24However, Fu and co-workers suggested that the extra- large strain would relax, so they prepared YIG films by adding a thin buffer layer of Sm 3Ga5O12on the top of the substrate, and the YIG films successfully obtained PMA.25Bi3+, with a larger ion radius (1.03 Å) than Y3+(0.9 Å) and Tm3+(0.869 Å), can expand the lat- tice constant. In addition, Bi3+occupies the dodecahedral position, which has a significant effect on the growth-induced anisotropy of films grown by LPE.26,27 In this work, Tm 2BiFe 5O12(TmBiIG) films and Y 2BiFe 5O12 (YBiIG) films are deposited by RF magnetron sputtering. The lat- tice constants of TmIG and YIG are 12.324 and 12.376 Å, respec- tively, and the substitution of Bi3+is destined to expand the lat- tice. The lattice constant of the prepared YBiIG films is 12.459 Å on an average, which exceeds that of the Gd 3Ga5O12(GGG) sub- strate,17with a value of 12.383 Å. The lattice constant of the SGGG (Gd 2.6Ca0.4Ga4.1Mg 0.25Zr0.65O12) substrate is larger than that of YBiIG, with a value of 12.497 Å, so the SGGG is chosen to prepare YBiIG films. Therefore, by adjusting growth parameters and choos- ing different kinds of substrates, all TmBiIG films are prepared on GGG and all YBiIG films are prepared on SGGG substrates under proper conditions, respectively. The XRD spectrum measurements demonstrate tensile strain in both kinds of films, and the magnetic hysteresis loops indicate that the films successfully obtained PMA. Moreover, the differentiation of magnetic properties and magnetic anisotropy of the two films is discussed, and the anomalous Hall effect (AHE) of the heterostructures is analyzed. II. EXPERIMENTS The TmBiIG and YBiIG films are prepared by an ACS-4000- C4 four-target RF magnetron sputtering system manufactured by the ULVAC company. All of the TmBiIG films are deposited on GGG substrates, and YBiIG films are deposited on SGGG substrates; both the films and substrates are in the (111) crystal orientation. The targets are obtained by solid-state sintering with high purity Tm 2O3(99.999%) or Y 2O3(99.999%), Fe 2O3(99.999%), and Bi 2O3 (99.999%); in addition, an extra 5 wt. % of Bi 2O3is added to compen- sate for evaporation. The growth temperature is 400○C and the gas pressure of argon is about 1 Pa during sputtering. TmBiIG films with thicknesses of 16, 32, 48, and 64 nm and YBiIG films with thicknesses of 10, 20, 30, and 40 nm are obtained. Afterward, TmBiIG films areannealed at 900○C and YBiIG films are annealed at 850○C in air. The annealing temperatures are experimentally relative to their best PMA, respectively. The morphological characterization and crystal structure anal- ysis are performed by atomic force microscopy (AFM, SEIKO SPA-300HV, Japan) and high-resolution x-ray diffraction (HRXRD, D1 Evolution, JVS, Germany). The magnetic properties are ana- lyzed through a vibration sample magnetometer (VSM, BHV525, IWATSH, Japan) and magneto-optic Kerr apparatus (NanoMOKE3, Durham Magneto Optics, Britain). The FMR property is measured using a vector-network-analyzer ferromagnetic resonance (VNA- FMR) system at room temperature. The AHE measurement is ana- lyzed utilizing FMI/HM heterostructures. III. RESULTS AND DISCUSSION A. Morphological and structural characterization The XRD spectra of these films are obtained by scanning the ω/2θof the (444) diffraction peaks, the intensity spectra of TmBiIG films are shown in Fig. 1(a), while that of YBiIG is shown in our past paper.17There are two diffraction peaks in each XRD spectrum, and the higher one is the diffraction peak of the substrate. The high- resolution transmission electron microscopy (TEM) measurement has proved that these films were mono-crystalline.17The peaks of both TmBiIG and YBiIG films are on the right side of the substrates, which indicates that the lattice constants of both kinds of films are smaller than those of the respective substrates. These two kinds of films are in the state of tensile strain. Then, the distortion δand the lattice mismatch Δacan be calculated from the diffraction peaks by the following formulas and are listed in Table I: 2dsinθ=λ, (1) dhkl=a√ h2+k2+l2, (2) Δa=as−af, (3) δ=Δa as. (4) Here,λis the x-ray wavelength of 1.5406 Å; θis the diffraction angle of substrates and films; [hkl] is the [444] diffraction peak; and as(af) represent the lattice parameters of substrates (films), respectively. FIG. 1. The XRD spectrum and AFM image of the TmBiIG films; (a) the XRD spectrum of 16 and 64 nm TmBiIG films; (b) the relationship between the rms value and film thickness; and (c) the sur- face morphology of the 16 nm TmBiIG film. AIP Advances 11, 065113 (2021); doi: 10.1063/5.0050340 11, 065113-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv TABLE I. The growth parameters and properties of the TmBiIG and YBiIG films. No. t(nm) δ(%) Δa(10−3nm) Kλ u(103erg/cm3) 4πMs(Gs) rms (nm) H∥−H/⊙◇⊞(Oe) TmBiIG-1 16 0.517 6.41 −13.998 1403.2 0.7 952 TmBiIG-2 32 0.623 7.72 −16.859 1289.4 0.3 920 TmBiIG-3 48 0.735 9.12 −19.916 1287.5 0.5 814 TmBiIG-4 64 0.718 8.90 −19.436 1297.9 0.4 650 YBiIG-1 10 1.32 16.54 −18.567 1688.8 5.4 440 YBiIG-2 20 1.71 21.45 −24.078 1583.9 8.4 230 YBiIG-3 30 1.68 21.07 −23.652 1582.3 9.3 80 YBiIG-4 40 1.63 20.34 −22.832 1679.5 7.8 −70 The root-mean-square (rms) surface roughness is determined by three-dimensional AFM for each sample over a 2 ×2μm2range. The surface morphology of the 16 nm TmBiIG is shown in Fig. 1(c), and the relationship of the rms value with the film thickness is shown in Fig. 1(b). The rms values of YBiIG films,17also shown in Table I, are on the order of nanometers. However, the rms values of TmBiIG films only range from 0.2 to 0.7 nm, almost one tenth of YBiIG films. The larger rms of YBiIG films is mainly caused by the larger lattice mismatch with the SGGG. B. Magnetic characterization The effective magnetic anisotropy Keffof RIG films is mainly influenced by magnetocrystalline anisotropy Ka, shape anisotropy Ks, stress-induced anisotropy Kλ u, and growth-induced anisotropy Kg u. For most RIG films, with a negative magnetocrystalline anisotropy constant, a negative Keffmeans that the film has obtained PMA. Typically, YIG films display an in-plane easy axis due to the existence of Ksowing to their cubic structure. Therefore, it is essential to introduce uniaxial anisotropy to overcome Ksand, thus, to achieve the goal to manipulate RIG films performing PMA. The values of Keff,Kλ u, and Kscan be calculated by the following formulas:27,28 Ke f f=Ka+Kλ u+Kg u+Ks, (5) Ks=2πM2 S, (6) Kλ u=−3 2λ111E 1−νΔa/⊙◇⊞ a, (7) K1(x)=K1(0)(1+0.08x), (8)λ111(x)=λ111(0)(1+0.23x), (9) whereλ111,E,ν,Δa/⊙◇⊞,a, and xstand for the (111) magnetostriction coefficient, Young’s modulus ( ∼2.0×1012dyne/cm2), Poisson ratio (∼0.29), lattice mismatch of (111) orientation (approximately equal to negative Δa), lattice constant of substrates, and element con- tent of Bi3+, respectively. Kais mostly determined by the first cubic magnetocrystalline anisotropy constant K1,−5.8×103erg/cm3for TmIG15and−5.7×103erg/cm3for YIG.25According to formula (8), the K1of TmBiIG and YBiIG can be calculated. The saturation magnetization 4 πMSis listed in Table I, and the average value for TmBiIG films is 1319.47 Gs and that for YBiIG films is 1633.63 Gs. The normalized hysteresis loops of TmBiIG films with the field par- allel and perpendicular to the film plane are shown in Fig. 2(a). Conspicuously, TmBiIG films need to overcome a smaller shape anisotropy according to formula (6). In addition, the incorpora- tion of Bi3+would introduce Kg uto offset Ks.27The value of λ111 is−5.2×10−6for TmIG and −2.7×10−6for YIG,24and then, the λ111of TmBiIG and YBiIG could be obtained by formula (9); the calculated Kλ uof the TmBiIG and YBiIG films is listed in Table I. The Kλ uof YBiIG films is larger than that of TmBiIG, but 4 πMSis larger too, so the difference in the absolute value of Kλ uandKsof YBiIG films would be smaller. In addition, previous reports revealed that an extra-large strain would lead to a strong in-plane relaxation, which would thus weaken PMA.25All in all, considering Ka,Ks, Kg u, and Kλ u, the TmBiIG films are easier to obtain PMA than YBiIG films. Obviously, from Fig. 2(a), it is observed that the TmBiIG films successfully reveal PMA, and the difference between the in-plane saturation field H∥and out-of-plane saturation field H/⊙◇⊞is listed in Table I. It can also be found that H/⊙◇⊞would increase as the films FIG. 2. The magnetic hysteresis loops and Kerr signal of TmBiIG films; (a) the normalized hysteresis loops of TmBiIG films magnetized by an out-of-plane magnetic field; (b) the normalized hys- teresis loops of 64 nm TmBiIG by in- plane and out-of-plane magnetic fields; (c) the out-of-plane normalized magneto- optical MOKE signal of TmBiIG films; and (d) the relationship between the sat- uration field and film thickness. AIP Advances 11, 065113 (2021); doi: 10.1063/5.0050340 11, 065113-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. The FMR property of the 32 nm TmBiIG films: (a) the FMR linewidth with frequency from 4 to 17 GHz; (b) the FMR intensity of the 32 nm TmBiIG film at 13 GHz; and (c) the relationship between the FMR frequency and magnetic field. became thicker, which suggests the PMA becomes weaker, but even the 64 nm TmBiIG film shows great PMA. The normal- ized magneto-optical Kerr signal loops of the TmBiIG films with the field perpendicular to the film plane are shown in Fig. 2(c), and the relationship of the magnetic field HXSneeded for saturation depending on the thickness of films is shown in Fig. 2(d). Similarly, as the thickness increases, HXSbecomes larger. The damping factor αis an important parameter for spin-wave devices, so the FMR property is analyzed by a VNA-FMR system applying a magnetic field parallel to the film at room tempera- ture. Figure 3(a) shows the relationship between the FMR linewidth and frequency, and Fig. 3(b) is the FMR curve of the 32-nm-thick TmBiIG film excited at 13 GHz. The FMR linewidth ΔHcalculated by the Lorentzian fit is 249.08 Oe @ 13 GHz, which is similar to the value of the TmIG bulk.29The relationship between FMR fre- quency fand FMR magnetic field Hris shown in Fig. 3(c). Then, α is calculated by the following formulas:30 f=γμ0 2π(Hr−4πMe f f), (10) ΔH=ΔH0+4πα γf. (11)Here,γμ0/2πis the gyromagnetic ratio, Meffis the effective satu- ration magnetization, and ΔH0is the non-uniform linewidth. The calculatedγμ0/2πis 2.26 MHz/Oe and αis 1.49 ×10−2for the 32-nm-thick TmBiIG, 1.23 ×10−2for the 48-nm-thick TmBiIG, and 1.67×10−2for 64-nm-thick TmBiIG film. The measured damping factor of YBiIG films17is smaller than that of TmBiIG films, but the value is much larger than that of YIG films.31Because of the incorporation of Bi3+, there would appear more dislocation, vacancy, and defects, which further increase the FMR linewidth of YBiIG films.32However, the linear relation- ship between the linewidth and resonance frequency indicates that there is no obvious two magnon scattering,33whereas the prepared TmBiIG films have less defect, less lattice mismatch, and smaller rms, so the damping factor of these films is close to that of TmIG films.29 C. Anomalous Hall effect Spin Hall resistance measurements are carried out on a Hall bar made by TmBiIG/Pt and YBiIG/Pt heterostructures,34as shown in Fig. 4(a). The relationship of Hall resistivity and magnetization could be expressed by the following formula:35 ρxy=R0Hz+RsMz. (12) FIG. 4. The schematic of the spin Hall resistance measurement and the cal- culated Hall resistance of TmBiIG/Pt and YBiIG/Pt heterostructures; (a) the schematic of the spin Hall resistance measurement with the out-of-plane mag- netic field; (b) the thickness of Pt was 10 nm, and the thicknesses of FMIs were 10/20 nm YBiIG and 16 nm TmBiIG; and (c) the thickness of TmBiIG was 16 nm and the thickness of Pt was 5/8/10 nm. AIP Advances 11, 065113 (2021); doi: 10.1063/5.0050340 11, 065113-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv TABLE II. The AHE property of FMI/HM heterostructures. TFMI(nm) THM(nm) I(mA) VH(μV) RH(mΩ)ρ(Ωm) 16 nm TmBiIG 5 3 5.9 1.96 2.35 ×10−6 16 nm TmBiIG 8 3 3.2 1.06 0.80 ×10−6 16 nm TmBiIG 10 3 1.9 0.63 0.38 ×10−6 10 nm YBiIG 10 4 0.3 0.075 0.45 ×10−7 20 nm YBiIG 10 1 0.5 0.5 3 ×10−7 Here, R0is the ordinary Hall resistant part and Rsis the anoma- lous Hall resistant part. A magnetic field is applied perpendic- ular to the structure, a fixed current is exerted along the long side of the Hall bar, and then a nanovoltmeter is used to mea- sure the Hall voltage on the short side. The curves of calculated Hall resistance vary with the magnetic field, shown in Figs. 4(b) and 4(c), whose shape and tendency are similar to their hystere- sis loops, respectively. In each curve, the linear background with a small slope is the ordinary Hall resistant part, while the dra- matically changing part is the anomalous Hall resistant part. The applied current I, tested voltage (VH), calculated Hall resistance (RH), and resistivity (ρ)are listed in Table II, where TFMIis the parameter of the TmBiIG or YBiIG film and THMis the thickness of Pt. The Hall resistivity of TmBiIG/Pt is larger than that of YBiIG/Pt, which is mainly caused by a smaller rms value and less defect in TmBiIG films, and the signal-to-noise ratio is large for the curves of TmBiIG/Pt. Furthermore, when the current is fixed in TmBiIG/Pt heterostructures, VH,RHandρwill decrease as the thickness of Pt changes from 5 to 10 nm. This is mainly caused by the following two reasons. On the one hand, Pt is coupled by the TmBiIG film due to the magnetic proximity effect, so the AHE would be diluted by the paramagnetic component Pt as its thickness increases. On the other hand, due to the Spin Hall-AHE, spin pre- cession emerges around the exchange field.24These two mechanisms lead to an increase in VH,RH, andρas the thickness of Pt decreases. IV. CONCLUSION A series of mono-crystalline TmBiIG and YBiIG films with PMA prepared by RF magnetron sputtering are studied in this paper. The ion radius of Tm3+is smaller than that of Y3+, the ion radius of Y3+is smaller than that of Bi3+, and the lattice mismatch of TmBiIG on GGG is less than that of YBiIG on the SGGG, resulting in larger roughness of YBiIG films. A larger rms value of YBiIG films leads to a larger damping factor than that of YIG; however, the damping factor of TmBiIG films is close to that of TmIG. TmBiIG films, with lower saturation magnetization than YBiIG films, need to overcome smaller shape anisotropy. Overall, considering the larger magne- tostriction coefficient, smaller shape anisotropy, and proper stress- induced anisotropy, TmBiIG films exhibit better PMA than YBiIG films. VH,RH, andρincrease in TmBiIG/Pt heterostructures as the thickness of Pt decreases, resulting from the proximity effect and spin Hall-AHE. The Hall resistivity in TmBiIG/Pt heterostructures is larger than in YBiIG/Pt on account of fewer defects and smaller surface roughness, which could be beneficial to the implementation of low-power spin logic devices.ACKNOWLEDGMENTS This work was supported by the International Coop- eration Project under Grant No. SQ2018YFE0205600; the National Key Research and Development Plan under Grant No. 2016YFA0300801; the National Natural Science Foundation of China under Grant Nos. 51472046, 51272036, 51002021, 61131005, 61831012, and 51572042; the National Key Scientific Instrument and Equipment Development Project under Grant No. 51827802; and the Science Challenge Project under Grant No. TZ2018003. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. H. Collins, J. D. Adam, and Z. M. Bardai, “One-port magnetostatic wave resonator,” Proc. IEEE 65(7), 1090–1092 (1977). 2S. A. Manuilov, R. 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AIP Advances 11, 065113 (2021); doi: 10.1063/5.0050340 11, 065113-6 © Author(s) 2021

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J. Appl. Phys. 129, 214302 (2021); https://doi.org/10.1063/5.0050672 129, 214302 © 2021 Author(s).Topological Josephson bifurcation amplifier: Semiclassical theory Cite as: J. Appl. Phys. 129, 214302 (2021); https://doi.org/10.1063/5.0050672 Submitted: 17 March 2021 . Accepted: 16 May 2021 . Published Online: 03 June 2021 Samuel Boutin , Pedro L. S. Lopes , Anqi Mu , Udson C. Mendes , and Ion Garate COLLECTIONS Paper published as part of the special topic on Topological Materials and Devices ARTICLES YOU MAY BE INTERESTED IN Acoustic nonreciprocity Journal of Applied Physics 129, 210903 (2021); https://doi.org/10.1063/5.0050775 Magnetism in curved geometries Journal of Applied Physics 129, 210902 (2021); https://doi.org/10.1063/5.0054025 All-optical switch based on novel physics effects Journal of Applied Physics 129, 210906 (2021); https://doi.org/10.1063/5.0048878Topological Josephson bifurcation amplifier: Semiclassical theory Cite as: J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 View Online Export Citation CrossMar k Submitted: 17 March 2021 · Accepted: 16 May 2021 · Published Online: 3 June 2021 Samuel Boutin,1,2,a) Pedro L. S. Lopes,3Anqi Mu,4 Udson C. Mendes,5and Ion Garate1 AFFILIATIONS 1Département de physique, Institut quantique and Regroupement Québécois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada 2Station Q, Microsoft Quantum, Santa Barbara, California 93106-6105, USA 3Department of Physics and Stewart Blusson Institute for Quantum Matter, University of British Columbia, Vancouver V6T 1Z1, Canada 4Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027, USA 5Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia-Go, Brazil Note: This paper is part of the Special Topic on Topological Materials and Devices. a)Author to whom correspondence should be addressed: Sam.Boutin@gmail.com ABSTRACT Amplifiers based on Josephson junctions allow for a fast and noninvasive readout of superconducting qubits. Motivated by the ongoing progress toward the realization of fault-tolerant qubits based on Majorana bound states, we investigate the topological counterpart of the Josephson bifurcation amplifier. We predict that the bifurcation dynamics of a topological Josephson junction driven in the appropriate parameter regime may be used as an additional tool to detect the emergence of Majorana bound states. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0050672 I. INTRODUCTION In the past decade, the discovery and characterization of Majorana bound states (MBSs) in solid-state devices has become a landmark of the field of topological materials and devices.1One promising implementation of MBS relies on semiconducting nano-wires with strong spin –orbit coupling and proximity-induced superconductivity. 2In this setup, a magnetic field can induce a topological phase transition, where MBSs emerge as localized states at the ends of the nanowire. Because of their spatial localization and the non-Abelian exchange statistics, these MBSs are regardedas potential building blocks for fault-tolerant qubits. 3 In anticipation of the realization of MBS-based qubits, there has been a strong interest on the theoretical front to integrate MBS in circuit quantum electrodynamics (cQED) architectures,4–19the latter of which are widely employed in the readout and control ofsolid-state qubits. 20This theoretical effort has been accompanied by experimental progress toward the realization of superconductingcircuits that are compatible with sizeable magnetic fields. 21–26 Motivated by the aforementioned developments, in the present work, we introduce a topological version of the Josephsonbifurcation amplifier (JBA), which is simply a JBA made from a topological Josephson junction. The original JBA is an amplifierdesigned to read out the state of superconducting qubits in topo-logically trivial Josephson junctions. 27,28It is based on the transi- tion of an RF-driven Josephson junction between two distinctoscillation states near a dynamical bifurcation point. The mainadvantages of the JBA are high speed, high sensitivity, and nonin-vasiveness. These advantages have led to some of the first single-shot qubit readouts of superconducting qubits. 29–31Yet, the diffi- culty of calibration for different devices has resulted in JBAs givingway to simpler designs for qubit measurement, such as theJosephson parametric amplifier. 32–34In our theoretical study, we show that the sensitive binary features and the high signal-to-noiseratio of the bifurcation dynamics offer new ways to detect the emer- gence of MBS in topological Josephson junctions. The remainder of this paper is organized as follows. Section II provides a short review of the main concepts of the JBA and intro- duces its topological version. Section IIIhighlights the signatures of the topological phase transition in key parameters governing thetopological Josephson bifurcation amplifier (TJBA). In Sec. IV,w eJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-1 Published under an exclusive license by AIP Publishingpropose to use the TJBA in order to detect the coexisting 2 πand 4πperiodicities in a topological Josephson junction. Conclusions and final remarks are presented in Sec. V. II. SEMICLASSICAL THEORY OF THE TJBA A. Josephson bifurcation amplifier: A review We describe the topological Josephson bifurcation amplifier using the resistively and capacitively shunted (RCSJ) model of aJosephson junction. 35,36In this model ( Fig. 1 ), the Kirchhoff ’sl a w for the electric current traversing the junction reads Cw0@2δ @t2þw0 R@δ @tþ1 w0@U @δ¼Id(t), (1) where δis the gauge-invariant superconducting phase difference between the two superconducting electrodes, CandRare the effec- tive capacitance of the junction and the characteristic impedance of the microwave source generator, w0¼/C22h=2eis the reduced flux quantum, Idis the bias current, and Uis the grand potential of the junction at temperature T. The first, second, and third terms in the left hand side of Eq. (1)correspond to the displacement current, the dissipative ohmic current and the dissipationless (Josephson and Majorana) current, respectively. The critical current of thejunction is given by w/C01 0max( @U=@δ). In the RCSJ model, δis treated as a classical variable, whereas U(δ) is computed by diagonalizing a quantum mechanical Hamiltonian for each value of δ. Charging energy is not included in the Hamiltonian; instead, the phase fluctuations produced by thecharging energy are incorporated via the displacement current inEq.(1). This semiclassical approach is justified in the “transmon regime ”that is relevant to the operation of the JBA. 28In such a regime, the Josephson energy of the junction [approximately given by (U(π)/C0U(0))=2] greatly exceeds the charging energy EC¼e2=(2C). In thermodynamic equilibrium, fermion parity constraints are absent for timescales that are long compared to quasiparticle poi- soning times. Then, the grand potential37takes the form U(δ)¼/C0 kBTX n.0ln 2 coshϵn(δ) 2kBT/C18/C19 /C20/C21 , (2)where ϵn(δ) are the single-particle energies of the system, with n¼ +1,+2,...andϵn¼/C0ϵ/C0ndue to particle-hole symmetry (by convention, we take ϵn.0 for n.0). For T!0, Eq. (2)reduces to the ground state energy U(δ)¼/C01 2X n.0ϵn(δ): (3) The eigenvalues ϵn(δ) can be obtained by diagonalizing a tight- binding Hamiltonian describing a single-channel, one-dimensionalsuperconducting/normal/superconducting heterostructure of finitelength, where two long superconducting electrodes are separated by a weak link in the normal state. More details of the model are pre- sented in Sec. II C. Equation (1)describes the dynamics of a fictitious particle of mass Cand position w0δmoving in a potential U(δ) under the action of an external force Id. We consider a noisy sinusoidal AC-bias current, Id(t)¼idsin(ωdt)þIN(t), (4) where ωdand idare, respectively, the drive frequency and ampli- tude, and IN(t) is a white-noise current produced by thermal fluc- tuations. We will focus in the case in which idis small compared to the critical current of the junction. Then, the dynamics of the parti-cle is akin to that of a nonlinear harmonic oscillator centered inone of the minima of U(δ), which are located at δ m¼2πm (m[Z). Because U(δ)¼U(δþ2π) in the absence of parity con- straints, we can concentrate on oscillations near δ¼0 without loss of generality. Following the small oscillation approximation, Eq. (1)can be rewritten as @2δ @t2þκ@δ @tþω2 pδþλδ3¼Id(t) Cw0, (5) where the damping rate κ¼1=RC, the plasma frequency ωpin the harmonic approximation and the leading anharmonicity parameterλplay a central role in the functioning of the JBA. They can be extracted from Uvia ω p¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8U(2)(0)ECp =/C22h, λ¼4U(4)(0)EC 3/C22h2,(6) where U(n)(0)¼(@nU=@δn)jδ¼0. Figure 2 illustrates its solution in the absence of current noise. The various curves in this figure are well-known in the literature;28 here, we highlight the most important points for completeness andlater reference. In a frame rotating at the drive frequency, i.e.,taking δ(t)¼Re[δ ze/C0iωdt], we solve for the steady-state solution of Eq.(5). Ignoring rapidly oscillating terms (rotating-wave approxi- mation), Fig. 2(a) displays the modulus of δzas a function of ωd, for different values of id. For small id, the response of the oscillator is a Lorentzian peaked at ωd¼ωp. As the driving amplitude increases, the response of the oscillator becomes peaked at frequen- cies lower than ωp[this is due to the fact that the anharmonicity FIG. 1. Circuit diagram of the RCSJ model for the dynamics of a topological Josephson junction. The cross stands for the dissipationless Cooper pair and single-particle tunneling (the latter present only in the topological phase). The AC bias current is Id, while CandRstand for the capacitance of the junction and the resistance.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-2 Published under an exclusive license by AIP Publishingparameter in Eq. (6)is negative]. Beyond a critical value of id(but still well below the critical current of the junction), there emerges afinite interval of ω dfor which the response of the oscillator is multi-valued, with three values of jδzjfor each ωd. In such an inter- val, the intermediate value of jδzjis not stable and the oscillator is thus said to be in a bistable regime, with a low- and a large-amplitude oscillation steady state. Figure 2(b) shows the dependence of the bistable regime on the drive frequency. Bistability takes place when ω d/C20ωc; ωp/C0ffiffiffi 3p κ=2 and id[(i/C0,iþ), where i/C0and iþare the lower and upper bifurcation currents, respectively. These currents obey therelations i 2 +¼/C08w2 0C2 81λ2ωδωΣ+ffiffiffiffiffiffiffiffiffiffi ffi r(ωd)p hi /C23ω2 dκ2þω2 δω2Σ+ωδωΣffiffiffiffiffiffiffiffiffiffi ffi r(ωd)p hi , (7) where r(ωd)¼(ω2 d/C0ω2 p)2/C03ω2 dκ2, ωδ¼ωp/C0ωd, and ωΣ¼ωdþωp. At the bifurcation currents, the response of the oscillator switches between single-valued and two-valued. Thus, fori d,i/C0(id.iþ), the oscillation amplitude is single-valued and low (high). The fact that bifurcation currents depend on ωpandλ will be important below. Figure 2(c) presents the dependence of the steady-state solu- tionjδzjon the drive amplitude, for a fixed value of the drive fre- quency [ ωd,ωc, red dashed vertical line in Fig. 2(b) ]. In the bistable regime (blue shaded central region), which of the two stable steady states is reached depends on the initial state of the system and how the drive is turned on. In the case where the driveis turned on sufficiently slowly (compared to the timescale 1 =κ) and in the absence of current noise, the steady state follows an hys- teresis curve. For an initial state where jδ zj/difference0, the low-amplitude steady-state is realized for any id,iþ, and the system suddenlyjumps to the high-amplitude solution at id¼iþ. Driving the the Josephson junction close to iþthus allows to detect small changes in the upper bifurcation current, independently from the width of the bistable region jiþ/C0i/C0j. Thus far, we have reviewed the phenomena of bifurcation and bistability in the absence of current noise ( IN¼0). In the presence of noise, due to either thermal or quantum fluctuations, the steadystate reached by the system is not deterministic. We define the bifur- cation probability Pas the probability of the system reaching the high-amplitude state after evolution of the system for a time T 0.28,30 To estimate P, we integrate Eq. (5)for a time T0.1=κandM/C291 realizations of the noise IN(t). We consider uncorrelated white noise by taking IN(t) from a zero-mean Gaussian distribution of variance σ2 N¼4kBT=R. We then estimate the bifurcation probability by P/C25nH=M,w h e r e nHis the number of high amplitude steady-state solutions. This probability Pis experimentally measurable. Figure 3 illustrates the dependence of Pon the drive amplitude in the presence of current noise. When id,i/C0and id.iþ, the oscillator is with certainty in the low and high amplitude states,respectively. Due to the hysteretic nature of the bifurcation dynam-ics,Pis a sharp step function at i d¼iþin the absence of noise (blue curve). Current noise increases the width of the step function in the region id&iþ. As we consider a low amplitude initial state and a smooth turn on of the drive, Pdoes not depend on the lower bifurcation current ( i/C0/C252:25 nA in Fig. 3 ). Because the bifurca- tion currents are sensitive to the intrinsic parameters of the junc-tion (such as ω pand λ), small variations of the latter lead to significant changes in Pwhen idlies in the vicinity of iþ.A sw e explain below, this high sensitivity may allow for new ways tomeasure MBS properties in topological Josephson junctions. B. Limiting forms of U(δ) Before turning to a more microscopic description of a Josephson junction by calculating the grand potential U(δ)f r o ma FIG. 2. (a) Response of a nonlinear oscillator (a Josephson junction) for different drive amplitudes idin a frame rotating at the drive frequency ωd. Here, δ(t)¼Re[δze/C0iωdt] is the superconducting phase difference across the weak link, ωpis the Josephson plasmon frequency, and icis the critical drive current for the onset of the bistable region (not to be confused with the critical current of the junction, which is parametrically higher). The vertical dotted line indicates the critical frequ ency of the pump, ωc¼ωp/C0ffiffi ffi 3p κ=2( w i t h κ/C01¼RC), below which bistability emerges. The shaded area indicates the bistable regime of the id¼1:5ic(purple) curve. (b) Phase diagram for bistability. The colored regions correspond to the low amplitude state (orange), the bistable regime (blue) and the high amplitude state (green). The red dashed line corresponds to panel (c). (c) Amplitude of δas a function of the amplitude of the drive. From left to right, the curve is separated into three regions: (i) low amplitude state (orange), bistable regime (blue), and high amplitude state (green). The lower and upper bifurcation currents are i/C0andiþ, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-3 Published under an exclusive license by AIP Publishingtight-binding model, it is helpful to consider two approximate lim- iting forms of U(δ) in ideal tunnel junctions. These limiting cases will be useful to interpret some of the numerical results of the fol- lowing sections. In the case of conventional tunnel junction at low tempera- tures, the ground state energy is U(δ)¼/C0 EJcosδþconstant, where EJis the Josephson energy associated to the tunneling of Cooper pairs and “constant ”is a δ-independent number. From Eq.(6), the plasmon frequency and anharmonicity parameter are then /C22hωp¼ffiffiffiffiffiffiffiffiffiffiffiffi8EJECpandλ¼/C0ω2 p=6. In this case, λ=ω2 pis a cons- tant because U(δ) has a single harmonic (cos δ). In the case of a topological tunnel junction, one instead finds (disregarding parity constraints) U(δ)≃/C0EMjcos(δ=2)j /C0EJcosδþconstant, where EMis the Majorana energy associated to the tunneling of single electrons. In particular, if the junctiontransparency is very low, E J/C28EMand U(δ)≃/C0EMjcos(δ=2)j.I n this tunnel regime where Cooper pair tunneling across the junction is negligible compared to single-particle tunneling, we haveω p/differenceffiffiffiffiffiffiffiffiffiffiffiffiffiffi2EMECp=/C22handλ/difference/C0ω2 p=24. Once again, λ=ω2 pis a negative constant but of smaller amplitudes. In general, when the δ-dependence of U(δ) has more than one harmonic (e.g., in topological junctions with comparable EJ and EM, or even in trivial junctions that are neither in the tunnel- ing nor in the perfectly transparent regime38),λ=ω2 pwill not be a constant. Finally, for later reference, we note that in the literature of superconducting qubits,20anharmonicity in the transmon regime is often defined as λ0¼(E2/C0E1)/C0(E1/C0E0), where Enare the energy levels for the quantum anharmonic oscillator evaluated tofirst order in the strength of the quartic potential. The relation between λ 0and our anharmonicity parameter is λ0¼6λEC=ω2 p. Thus, λ0¼/C0 ECin a conventional tunnel junction.C. Microscopic model of a topological junction In order to account for topological effects, we calculate the grand potential U(δ) defined in Eq. (2)by diagonalizing a tight- binding model of a 1D nanowire junction (see Fig. 4 for a sketch of the model). As this model has been extensively reviewed in the literature, see, e.g., Ref. 39, we limit ourselves to providing a brief description. We consider spinful fermions with a nnihilation (creation) operator c(y) j,σ(σ[",#fg ). Introducing the spinor ψy j¼(cy j,",cy j,#, /C0cj,#,cj,"), the Hamiltonian reads H¼X jψy jhiψjþψy jþ1ujψjþh:c:/C16/C17 , (8) where hjand ujare (respectively) the onsite and hopping matrices, with site-dependent parameters in order to create a junction. We definethese matrices in terms of two sets of Pauli matrices, σ αand τα (α[x,y,zfg ), acting, respectively, on the spin and the particle-hole sector of the spinor. Electrons can hop between lattice sites with a spin-independent nearest-neighbor tunneling amplitude tand, due to spin –orbit coupling, a spin-flip tunneling amplitude α.T h e n ,t h e hopping matrix is uj¼/C0 tjτz/C0iασyτz, (9) where tj¼taway from the junction interface (see Fig. 4 and below). T h eo n s i t em a t r i xr e a d s hj¼(2t/C0μ)τzþBσxþΔjτx,( 1 0 ) where the local mean-field superconducting gap ΔjisΔin the left superconducting lead, 0 in the weak link and Δeiδin the right super- conducting lead. The Zeeman energy Bassociated to the applied FIG. 3. Probability of finding the Josephson oscillator in the high-amplitude state for different noise strengths after evolution time T0. In order to extract the probability, Eq. (5)is solved in the rotating frame for M¼500 noise realizations. In all cases, we take the initial condition δ(0)¼@tδ(0)¼0, and the parameters κ¼ωp=25, ωd¼ωp/C03κ=2,ωp=2π¼5 GHz, EC=h¼0:1 GHz, and κT0/difference250. Due to the hysteretic nature of the JBA, in the absence of noise, the transition of Pfrom 0 to 1 takes place at id¼iþ(dashed black line) for a smooth switch-on of the drive on a timescale τwithκτ/C291 (here κτ/difference30). FIG. 4. Pictorial representation of the tight-binding model for the topological Josephson junction. Green dots correspond to the weak link. The spin-independent and spin-flip tunneling amplitudes are tandα, respectively. The parameter for the interfaces between the superconducting electrodes andthe weak link is γ/C201. The proximity-induced s-wave superconducting gap on the electrodes has an amplitude Δ, with a phase difference δbetween the two electrodes. The Zeeman splitting caused by the external magnetic field is B. The chemical potential in the electrodes is μ, while that of the weak link is μ 0. Unless otherwise stated, we take μ¼μ0.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-4 Published under an exclusive license by AIP Publishingmagnetic field is assumed to be spatially homogeneous. Likewise, the chemical potential μis assumed to be uniform and identical in the two superconducting electrodes, though we allow for a mismatch inthe chemical potentials of the electrodes and the weak link. The valueμ¼0 corresponds to the chemical potential being in the middle of the Zeeman gap of the normal state electronic bands. Using parameters relevant to InAs nanowires, 2we take t¼25 meV and α¼t=10/difference2:5 meV (corresponding to a lattice constant a¼10 nm and an electronic effective mass m*¼0:016me, with methe bare electron mass). We take through- out the proximity-induced s-wave superconducting gap to be Δ¼0:9 meV. We also neglect the dependence of ΔonBand on μ. In reality, this dependence is smooth across a topological phasetransition, and thus clearly distinguishable from the sharp featureswe will be discussing below. It is well-known that variations of B andμcan drive the superconducting electrodes across a topological phase transition. 40The superconducting electrodes are in the topo- logical phase (with a MBS in each extremity of the superconduc-tors) for B.B c;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ2þμ2p , and in the trivial phase otherwise. To model the effect of a potential barrier at the superconduc- tor/normal interface, we use a phenomenological transparency parameter γ[[0, 1]. Both the hopping amplitude tand the spin –orbit coupling parameter αare scaled by γat the interface. Throughout the text, we consider the cases γ¼0:6 and γ¼1t o study, respectively, the effect of a large potential barrier at the junc- tion interface and the case of a clean junction with no potential barrier. We denote the former as the “tunnel ”regime and the latter as the “transparent ”regime. However, we note that the junction transparency also depends on the normal state Fermi velocityv F41,42and is hence a function of B,αand most importantly μ.F o r example, even when γ¼1, the junction ’s characteristics (e.g., ωp) can resemble those of a true tunnel junction if μis sufficiently negative.In the numerical simulations, unless mentioned otherwise, the weak link is two sites long ( “short junction ”regime), whereas each superconducting electrode contains 200 sites. The main qualitativepoints are relatively insensitive to the strength of spin –orbit cou- pling, though larger values of αmake it easier to observe the rele- vant features. Likewise, longer junction lengths do not change the main features discussed below, though they host additional proper- ties that complicate the JBA-based detection of the topologicalphase transition; we elaborate on this issue below. III. DETECTION OF THE TOPOLOGICAL PHASE TRANSITION In Sec. II, we have demonstrated that the JBA can be highly sensitive to changes in the current-phase characteristics of aJosephson junction. It is therefore natural to wonder whether a JBA could detect the emergence of Majorana bound states. Here, we investigate the signatures of the topological phase transition in keyJBA observables. We begin by discussing the plasmon frequencyand the anharmonicity parameter at zero (Sec. III A ) and finite (Sec. III B ) temperature. This analysis sets the stage for the subse- quent computation (Sec. III C ) of bifurcation currents across the topological phase transition. In this section, we will focus on ther-modynamic equilibrium, where Eq. (2)applies. In Sec. IV, we will depart from thermodynamic equilibrium in order to calculate ω p andλfor different many-body states. A. Plasmon frequency and anharmonicity parameter at zero temperature Figure 5 shows the dependence of the Josephson plasmon fre- quency ωpand the anharmonicity parameter λon the magnetic field, for small oscillations of the superconducting phase differencein the vicinity of δ¼0, in the many-body ground state. We show FIG. 5. Dependence of the Josephson plasmon frequency ωpand the anharmonicity parameter λon the Zeeman energy B, for a short junction at zero temperature. We take a small charging energy EC=h¼100 MHz; see main text for other parameters of the model. The derivatives of ωpandλwith respect to Bare also shown as a func- tion of B. In the thermodynamic limit, the topological phase transition takes place at Bc¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ2þμ2p ,B.Bccorresponding to the topological regime. (a) –(d) “Tunneling ” regime ( γ¼0:6), (e) –(h) “Transparent ”regime ( γ¼1). (c) Dashed lines indicate the expected value of λ=ω2 p¼/C0 1=6 for a ground state energy of the form U(δ)/cosδ, and λ=ω2 p¼/C0 1=24 for U(δ)/cos(δ=2).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-5 Published under an exclusive license by AIP Publishingthe results both in the “tunneling ”regime [ γ¼0:6,Figs. 5(a) –5(d)] and in the “transparent ”regime [ γ¼1,Figs. 5(e) –5(h)]. In all cases, the most salient feature is the pronounced peak in @ωp=@Band @λ=@Bat the topological phase transition. As we explain next, these peaks can be attributed to the emergence of MBS. In the tunneling regime,43Fig. 5(a) evidences an abrupt growth of the plasmon frequency at the onset of the topological phase ( ωpgrows by more than 0 :1 GHz as B=Bcchanges by less than 10%). This behavior can be understood by analyzing ϵn(δ)a s a function of δ. In our model for a topologically trivial tunnel junc- tion, all single-particle “bands ”are rather flat in δ(they would be completely independent of δifγ¼0). Accordingly, U(δ) depends weakly on δand hence ωpis relatively small. In the topological phase, the single-particle bands remain flat, except for those thatresult from the hybridization of MBS. The latter disperse paramet-rically more strongly with δ, because their “bandwidth ”scales with the square root of the transparency, rather than with the transpar- ency itself (as is the case for the non-Majorana bands). Hence, thisexplains why ω pincreases strongly at the onset of the topological phase, thus producing a strong peak in @ωp=@B[Fig. 5(b) ]. The aforementioned argument, derived in the context of a single-channel nanowire model, applies also to a less idealized tunnel junction that may host multiple subbands in the normalstate. The simplest way to see this is by adding a term of the form/C0E J0cosδto the ground state energy, where EJ0models the contri- bution from the multiple subbands to the conventional Josephson energy of the junction. As a result of the additional term in theenergy, ω pis shifted upwards in both the trivial and the topological regimes, but the ramp up of ωpand the peak of @ωp=@Bat the onset of the topological phase remain unchanged.44 A related behavior has been identified for the DC critical current in earlier works, with a marked increase at the onset of the topologicalphase. 45–47At first glance, a similarity in the behavior of ωpand the critical current is not surprising, the latter being proportional to ω2 pin conventional Josephson junctions. Yet, for the unconventional junc- tions we are interested in, this relation of proportionality does not apply. Instead, all one can assert is that ωpdepends on the characteris- tics of the single-particle bands near δ¼0, while the critical current depends on the slope of the single-particle bands near large ( ≃π) values of δ. Moreover, from an experimental point of view, the mea- surement of ωpis qualitatively different from the measurement of the critical current; it can therefore offer an alternative way to probe theinterpretation of a recent experiment reporting an enhanced criticalc u r r e n ta tt h eo n s e to ft h et o p o l o g i c a lp h a s e . 48 Let us now discuss the anharmonicity parameter. Still in the tunneling regime, Fig. 5(c) reflects a peculiar evolution of λ=ω2 p across the topological transition, which can be understood analyti- cally. Deep in the trivial regime, we find λ=ω2 p¼/C01=6, as expected for a conventional Josephson junction with U(δ)¼/C0EJcosδ. Deep in the topological regime, λ=ω2 ptends asymptotically to /C01=24, as expected for U(δ)¼/C0 EMjcos(δ=2)j(cf. Sec. II B). In other words, in a tunnel junction, the evolution of λ=ω2 pacross the topological phase transition reveals the emergence of MBS. Figure 5 also displays the results for several values of μ=Δ.E v e n though we consider μto be homogeneous throughout the system [i.e., μ¼μ0inFig. 1(b) ], we have verified that the results do not change significantly when we vary μin the superconductingelectrodes while keeping μ0pinned to zero in the weak link. We note thatωpincreases with μ=ΔinFig. 5 . This effect can be understood by observing that, for fixed γ, the junction transparency is a monotoni- cally decreasing function of the ratio of the barrier amplitude and thenormal-state Fermi velocity v F.41Asμ=t/C281 for all the cases consid- ered (Fermi level near the bottom of the band), increasing μmeans a higher vFand hence an increase of the junction transparency. Although Fig. 5 shows only positive values of μ, we have checked that, for μ,0, the junction transparency effectively decreases. For μ=Δ&/C04, the behavior expected for a tunnel junction is found irre- spective of the value of γas the Fermi level in the normal state then lies below the bottom of the band (insulating regime). Another effect of increasing μ=Δis the appearance of oscilla- tions in ωpandλas a function of B; their amplitude grows with B. These oscillations are the microwave counterpart of the ones pre-dicted in the DC critical current by Cayao et al. 45The oscillations take place only in the topological phase and arise from the hybridi- zation between MBS localized at opposite extremities of each super-conducting electrode, as well as from the hybridization between theMBS localized at the two opposite extremities of the entire system.Accordingly, the oscillations are washed out when the length of the superconducting electrodes becomes long enough (we find no evi- dence of them when doubling the electrodes size to 400 sites). Let us now discuss the transparent junction regime γ¼1[ s e e Figs. 5(e) –5(h)]. In this regime, the behavior of ω pat the topological phase transition is also reminiscent of that predicted in earlier works for the DC critical current.45,49In those works, a kink of the critical current at the topological phase transition was identified and attrib-uted to a band inversion taking place at the topological phase transi-tion. A similar mechanism is at play in our case, as we explain next. The underlying explanation begins by recognizing that a long superconducting electrode in the trivial phase hosts two differentand quasi-independent p-wave superconducting gaps in the energy spectrum: 45,49,50one at the inner Fermi points (the “Δ/C0gap”)a n d the other at the outer Fermi points (the “Δþgap”). When B/C28α, Δþ≃Δ.A sBincreases, Δþdecreases gradually and gets significantly suppressed when B/C29α.T h e Δ/C0gap is far more sensitive to B,v a n - ishing when B¼Bc. This gap-closing point marks the topological phase transition. Near the transition, we have Δ/C0≃jB/C0Bcj. The next part of the explanation is to recall that in a short, transparent and topologically trivial junction, there is a single Andreev bound (or quasi-bound) state associated to each of the twop-wave gaps of the electrodes. 51In this regime, the rest of the single- particle states (the so-called scattering states) are largely dispersion- less in δ, and therefore the main contribution to ωpandλoriginates from the two Andreev states bound by ΔþandΔ/C0. Although there is no simple analytical expression for the Andreev bound state disper-sions in the presence of a generic magnetic field, 42qualitative insight can be gained by neglecting the coupling between the two p-wave gaps, adopting the Andreev approximation, and assuming perfect transparency. Then, we posit ϵ+(δ)≃Δ+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C0sin2(δ=2)p for the two ABS in the trivial phase, and we get ωp≃ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (Δ/C0þΔþ)ECp =/C22h, λ≃/C0(Δ/C0þΔþ)EC=(24/C22h2):(11)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-6 Published under an exclusive license by AIP PublishingAcross a topological phase transition from the trivial to the topological phase, Δ/C0crosses zero and inverts its sign, thereby changing its character from a superconducting gap to a magneticgap. Accordingly, in the topological phase, Δ /C0no longer binds an Andreev state. Therefore, the contribution from Δ/C0toωpandλin Eq.(11) is turned off for B/C21Bc. This results in a kink for ωpand λas a function of BatB¼Bc(and corresponding discontinuities in@ωp=@Band @λ=@B). Of course, such reasoning of “suddenly turning off ”the contribution from Δ/C0atB¼Bcmakes sense only at zero temperature; finite-temperature effects will be discussed inSec.III B. Some of the features in Fig. 5 match qualitatively with the behavior expected from Eq. (11). First, ω pis decreasing as we approach the topological phase transition. Second, a kink is visibleat the transition, especially for larger values of μ=Δ, and ω pcontin- ues to decrease as we get deeper in the topological phase (because Δþkeeps decreasing as Bis made larger). On the other hand, the results from Fig. 5(g) do not match qualitatively with Eq. (11).F o r one thing, λ=ω2 pis not a constant. Also, a clear kink in ωpat the topological phase transition is present only if μ=Δis sufficiently large. There are various possible reasons for these discrepancies. First, the analytical expressions for ϵ+(δ) are approximately valid at B¼0, but may fail qualitatively at larger B.E v e na t B¼0, γ¼1 does not guarantee a perfect transparency of the junction; the actual transparency depends on the value of μas well. If the transparency is not unity, one expects various competing harmon- ics in ϵ+(δ) near δ¼0 (rather than just cos( δ=2)), which in turn leads to λ=ω2 pnot being constant. Second, the Andreev approxima- tion upon which Eq. (11) is based is not satisfied in the low elec- tron density regime required to have a topological Josephson junction; this is evidenced by the single-particle energy spectra we have calculated (not shown), wherein the ABS in the trivial phasehave significant gaps at δ¼π. Third, near δ¼0, the Andreev state bound by Δ þis buried in the continuum of scattering states;49,50 this is due to the fact that Δþ/C29Δ/C0in the vicinity of the phase transition. In other words, the ABS bound by Δþundergoes multi- ple avoided crossings with scattering states near δ¼0. Consequently, it is not accurate to neglect the δ-dependence of those scattering states, nor is it to ignore their contribution to ωp andλ. Next, we compare our results to relevant works in the litera- ture. Our finding that @ωp=@Band @λ=@Bdisplay prominent peaks atB¼Bcis seemingly more optimistic than that of a recent numerical study by Keselman et al. ,16who conclude that there are “no strong signatures of the topological transition itself on the simulated frequency spectra of the junction. ”This discrepancy in viewpoint might be explained by the different parameter regimesconsidered. In particular, the exact numerical treatment of chargingenergy in Ref. 16comes at the expense of studying smaller systems (/difference40 sites), where finite-size effects smoothen the phase transition into a crossover. Below, we show that finite-temperature effects canlikewise broaden and eventually erase the sharp signatures in@ω p=@B. In another recent work,18Avila et al. have, to the contrary, reported an abrupt dip in the anharmonicity parameter λ0at B¼Bc, stating it to be a “precise smoking gun ”for the topological phase transition. We recall (see discussion at the end of Sec. II B) thatλ0¼6ECλ=ω2 p.The theoretical treatments of Avila et al. and Keselman et al. differ from ours in that they treat the charging energy quantum mechanically, while we do so semiclassically. Yet, both approachesought to yield similar results in the transmon regime. In Fig. 5(g) , we do find that λ=ω 2 pcan have a dip at B¼Bcin the transparent regime, though it is nowhere as pronounced as in Fig. 9(b) of Ref. 18. Moreover, the dip is not a generic feature when we vary parameters such as γandμ[e.g., it is absent in Fig. 5(c) ]. What appears more generic in our case is the peak in @λ=@BatB¼Bc, whose origin we have explained above. On a related note, while Avila et al. investigate the behavior of the nonlinear Josephson inductance in the topological and trivial phases, they do not point out specific signatures at the phase transi-tion (at least Fig. 8 of Ref. 18does not show enough values of Bto ascertain any signatures). Yet, the inverse of this nonlinear induc-tance, close to δ¼0, is proportional to ω 2 p. It would be interesting to find out whether the theory of Ref. 18agrees with our prediction for@ωp=@Bnear Bc. Thus far, we have concentrated on short junctions either in the tunneling or in the transparent regimes. In junctions of inter-mediate transparency, ω pdoes not display a kink or a sharp upturn at the topological phase transition; the peak of @ωp=@BatB¼Bc can then be less pronounced. Even in those cases, we still observe a clear feature in @λ=@BatB¼Bc. Longer junctions show the same phenomenology as the short junctions when it comes to the behavior of ωpandλat the topolog- ical phase transition (we have simulated weak links with up to 50sites). However, one additional aspect of the longer junctions isthat they may host accidental (imposed neither by electronic topol-ogy nor by symmetry), parameter-sensitive, zero-energy touchings atδ¼0 in the topologically trivial phase. 52Because of particle- hole symmetry inherent to the single-particle spectrum, thepositive-energy and negative-energy Andreev states that cross atzero energy have opposite curvatures and anharmonicities. As aresult, the curvature of the occupied (negative-energy) single- particle states changes abruptly as a function of Bat the accidental zero-energy touchings. A corresponding discontinuity takes placeinω pandλ(seeFig. 6 for an example). The magnitude of the dis- continuities is diminished when the δ-dependence of the ABS at the zero-energy touching is smaller; hence, the effect is generally less pronounced in tunnel junctions. We note that no such discon- tinuities occur at the true topological phase transition, wherepositive- and negative-energy Andreev bound states coalesce atzero energy (as opposed to crossing it) as a function of B. Hence, measuring ω pandλas a function of Bmay allow to distinguish an accidental gap closing in the trivial phase (with a discontinuity inω p) from a gap closing associated to a topological phase transition (with no discontinuity in ωpbut a peak in @ωp=@B). Unfortunately, the aforementioned distinction is blurred at finite temperatures, where the discontinuities are transformed into rapid but continuous variations. Hence, at nonzero temperature,the behavior of ω pandλat accidental zero-energy touchings in the trivial phase can mimic the signatures of the topological phasetransition. In this case, one way to disentangle the true MBS signa- tures is to analyze the dependence of the peaks in μ; experiments of this type are feasible. 53The MBS-related peak at the phase transi- tion will be displaced in BasBc¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ2þΔ2p , while the peaks ofJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-7 Published under an exclusive license by AIP Publishingtrivial origin will not generically show such dependence, and can even disappear completely as μis varied (see Fig. 6 ). To conclude this subsection, we remark that our main results do not rely on the fact that a magnetic field has been used to induce a topological phase transition. Similar conclusions arereached when Bis held fixed and the topological phase transition is driven by variations in μ. In that case, we find that @ω p=@μand @λ=@μshow strong peaks at the topological phase transition, for the same reasons as the ones alluded to above.B. Plasmon frequency and anharmonicity parameter at finite temperature In the preceding subsection, we have investigated ωpandλat zero temperature, and have identified signatures of the topological phase transition therein. In this subsection, we study the effect of finite temperature on those features. We concentrate on tempera-tures that are low compared to the proximity-induced s-wave superconducting gap Δ. We find that the peaks of @ω p=@Band @λ=@Bat the topological phase transition are thermally broadened, though they remain noticeable at experimentally attainable temper- atures. We also find a nontrivial temperature dependence of ωpin the topological phase of tunnel junctions, which can be ascribed tothe MBS localized on the opposite extremities of the weak link. Our results are summarized in Fig. 7 . The main aspects of this figure can be understood from the finite temperature expressions [derived from Eqs. (2)and(6)] for the plasmon frequency and the anharmonicity parameter near δ¼0, ω 2 p¼/C04EC /C22h2X n.0tanhϵn(0) 2kBT/C20/C21 ϵ(2) n(0), λ¼/C02EC 3/C22h2X n.0tanhϵn(0) 2kBT/C20/C21 ϵ(4) n(0)/C20 þ3 2kBTsech2ϵn(0) 2kBT/C20/C21 ϵ(2) n(0)/C0/C1 2/C21 , (12) where ϵn(0) is the energy of the nth single particle state at δ¼0 [recall that ϵn(δ)/C210 for n.0], and ϵ(k) n(0) denotes the kth deriv- ative of ϵnwith respect to δ(evaluated at δ¼0). It follows from Eq. (12) that, whenever the energy of the lowest-lying n.0 state at δ¼0 largely exceeds kBT,ωpandλwill have a negligible temperature-dependence. In other words, the FIG. 6. Behavior of the plasmon frequency ωpas a function of the Zeeman energy B, for 20 sites in the weak link, high transparency ( γ¼1) and zero tem- perature. A discontinuity is present in the trivial phase ( B,Bc) for the case μ¼0. This discontinuity emerges from the accidental zero-energy crossing of low-lying Andreev bound states at δ¼0. No such crossing is present at the topological phase transition ( B¼Bc), where low-lying Andreev-bound states instead coalesce to zero energy. Thus, there are no discontinuities in ωpat B¼Bc. FIG. 7. Dependence of the Josephson plasmon frequency ωpand the anharmonicity parameter λon the Zeeman energy B, for a short junction at finite temperature. Parameter values are the same as in Fig. 5 , with μ¼3Δ. The derivatives of ωpandλwith respect to Bare also shown as a function of B. In the thermodynamic limit and at zero temperature, the topological phase transition takes place at Bc¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ2þμ2p ,B.Bccorresponding to the topological regime. (a) –(d) “Tunneling ”regime (γ¼0:6), (e) –(h) “Transparent ”regime ( γ¼1).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-8 Published under an exclusive license by AIP Publishingcondition for a significant temperature-dependence of ωpandλis that kBT*ϵ1(0) [with ϵ1(0) being the smallest single-particle eigenenergy]. Another useful piece of information in Eq. (12) is that, for a fixed n, the contribution of the given single-particle level toωpis suppressed when kBT/C29ϵn(0). Let us now analyze in detail the content of Fig. 7 . We divide the discussion in three parts, according to three regimes: (i) the trivial phase, (ii) the onset of the topological phase, and (iii) deepin the topological phase. In the trivial phase of short junctions, there are no subgap states in our model at δ¼0 (regardless of the transparency). 52 Therefore, ϵ1(0)¼Δ/C0and the temperature-dependence of ωpandλ is strong only if kBT*Δ/C0. At low temperatures, such a condition is satisfied only close to the topological phase transition (becauseΔ /C0!0w h e n B!Bc). This explains why, in Figs. 7(a) ,7(c),7(e), and 7(g), the curves for different temperatures are superimposed deep in the trivial phase, but begin to diverge closer to the topologi- cal phase transition. The fact that kBT/C29Δ/C0is easily attained at the topological phase transition also explains why the zero-temperaturekinks in ω pandλ, which take place at the topological phase transi- tion in highly transparent junctions [ Figs. 5(e) –5(h)], are washed out by thermal effects. Still, remnants of the kinks survive at low (kBT/C28Δ) but finite temperatures, in the form of broadened peaks for@ωp=@Band @λ=@B. These peaks are evident in Figs. 7(b) ,7(d), 7(f),a n d 7(h). At the onset of the topological phase ( B≃Bc), four MBS emerge. Two of them, localized on the outer extremities of thesuperconducting electrodes, have near-zero energies that are essen-tially independent of δin the vicinity of δ¼0; these dispersionless states do not contribute to ω pandλ. For this reason, we will hereaf- ter omit the two “outer ”MBS from the discussion. In contrast, the two “inner ”MBS localized on the opposite extremities of the weak link have a non-negligible bandwidth in δ, given by the hybridization energy EM; these dispersive states can contribute appreciably to ωpand λ. In a junction of high transparency, ϵ1(0)¼Δ/C0/C28EM≃Δþ. Hence, in such a junction, the leading temperature-dependence of ωpandλatkBT*Δ/C0originates mainly from the non-Majorana states. In contrast, in a junction of low trans-parency, ϵ 1(0)¼EM,Δ/C0/C28Δþand the leading temperature- dependence at kBT*EMoriginates from the inner MBS. As the system is driven deeper into the topological phase, Fig. 7 reflects two possible scenarios. In the scenario of Figs. 7(e) –7(h), the junction is highly transparent and there are no dispersive subgap states at δ¼0. In this case, ϵ1(0)¼min(Δ/C0,Δþ), where we once again ignore the outer MBS. IfΔ/C0,Δþ(which is common not far from the topological phase transition), ϵ1(0)¼Δ/C0grows linearly with the magnetic field. Therefore, the temperature-dependence of ωpand λweakens gradually as Bgrows, and becomes once again negligible when min( Δ/C0,Δþ)/C29kBT. This is why, in Figs. 7(e) and 7(g), the curves for different temperatures tend to converge deep in thetopological phase. Contrastingly, in the scenario of Figs. 7(a) –7(d), the junction has low transparency. In this case, there are dispersive subgap MBS of energy E Matδ¼0, so that ϵ1(0)¼EM. Moreover, the low junc- tion transparency implies EM/C28Δ/C0. Then, as temperature is raised well above EM, the contribution of MBS to ωpis suppressed via thetanh ( EM=2kBT)/C281 factor in Eq. (12). At the same time, the ener- gies of the higher-energy (non-Majorana) scattering states are rela- tively dispersionless in δand therefore make a relatively modest contribution to ωp, much like in the trivial phase. As a result, ωpis strongly reduced in the topological phase when kBT/C29EM. This explains the traits of Fig. 7(a) in the topological phase. One intriguing outcome from the foregoing discussion is that it appears to be possible to thermally switch on and off the contri-bution from MBS to ω pin short tunnel junctions, by going from kBT/C28EMtokBT/C29EM. This statement applies to longer junc- tions of low transparency as well. Although such junctions host low-energy Andreev-bound states in the trivial phase, their δ-dependence in the tunneling regime is relatively small; therefore, changing their thermal occupancy does not make a substantial dif-ference to ω p. We have verified this point for junctions with 20 and 50 sites in the weak link, and jμj/C28Δ. C. Bifurcation currents Thus far, we have been concerned with the prediction and understanding of peaks in the derivatives of ωpandλacross a topo- logical phase transition. The connection between ωpandλwith the operation of a JBA is made by Eq. (7). From this equation, it is clear that the bifurcation currents of the TJBA will display signa- tures of a topological phase transition. The results are summarized in Fig. 8 , which shows the lower and upper bifurcation currents as a function of the magnetic field, forshort junctions of low [ Figs. 8(a) –8(d)]a n dh i g h[ Figs. 8(e) –8(h)] transparency. The drive frequency ω dis chosen to be below ωp,a s otherwise there is no bifurcation (cf. Sec. II). The bifurcation currents are much smaller in the tunneling regime than in the transparent regime, because the critical current ofthe junction is much lower in the former. At the topological phasetransition, the behaviour of bifurcation currents as function of mag- netic field parallels that of ω p, and thus their derivatives display peaks at B¼Bc. These peaks are broadened at finite temperature. IV. DETECTION OF THE 4 πJOSEPHSON EFFECT One of the characteristic signatures of the presence of MBS in a Josephson junction is the 4 π-periodic Josephson effect.54,55The detection of this effect has been an active research topic in recent years. The leading experimental reports include missing Shapiro steps56,57and the halving of the Josephson radiation frequency.58 In this section, we describe an alternative (albeit indirect) micro- wave signature of the 4 πJosephson effect in short tunnel junctions, which is amenable to detection by a TJBA. We begin by showing ωpandλin a short tunnel junction, as a function of the magnetic field, for several of the lowest-energymany-body states ( Fig. 9 ). In order to obtain these results, U(δ)i n Eq.(6)has been replaced by the energy of individual many-body states. This is unlike in Sec. III, where we concentrated on ω pandλ in thermodynamic equilibrium. In the lowest-energy many-body state, all single-particle energy levels nwith ϵn(δ),0 are occupied; its corresponding energy is given by Eq. (3). The excited many- body states are constructed by creating particle-hole excitations on top of the lowest-energy state (one such excitation involves empty- ing a single-particle state /C0nand populating a single-particle stateJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-9 Published under an exclusive license by AIP Publishingn, for n.0); their corresponding energies are given by adding to Eq.(3)the excitation energies of the particle-hole pairs. According to Figs. 9(a) and9(b), in the trivial phase of a tunnel junction, ωpandλare very similar for all the low-lying many-body states. Then, at the onset of the topological phase, both ωpandλsplit into two branches, which diverge gradually from one another as themagnetic field increases. Deeper in the topological phase, the lower branch of ω pcan reach zero at a critical value of the magnetic field and subsequently disappear. The magn itude of this upper critical field depends on microscopic details and can be partly controlled by shunt-ing the topological Josephson junction with an ordinary Josephsonjunction of Josephson energy E J0(with EJ0.0). As mentioned in Sec.III A ,EJ0may also model the contribution of multiple subbands to the Josephson energy in a quasi one-dimensional nanowire. The larger the value of EJ0, the higher the critical field at which the lower plasmon branch disappears in the topological phase. For large enoughE J0, the frequency of the lower plasmon branch saturates before reach- ing zero and the branch never disappears. The preceding observations from Figs. 9(a) and 9(b) are closely related to the emergence of the 4 π-periodic Josephson effect in the topological phase. Next, we explain this connection by ana-lyzing the energy spectrum of the junction ( Fig. 10 ). In the trivial regime of the tunnel junction, the dispersion of the single-particle states as a function of δis approximately flat [Fig. 10(a) ]. Hence, when E J0is sizeable, the energies of the many- body states [ Fig. 10(c) ] are of the form /C0EJcosδþconst, where EJ≃EJ0and “const ”is a term independent of δ(though it takes different values for different many-body states). Note that EJis FIG. 9. Representative examples of the plasmon frequency ωpand the anhar- monicity parameter λnear δ¼0 for the four lowest-lying many-body states in a short, tunnel ( γ¼0:6) junction. The junction is shunted with an ordinary Josephson junction with Josephson energy EJ0=h¼25 GHz. The charging energy is EC=h¼100 MHz. The two rows correspond to two different values of the chemical potential with (a) and (b) μ¼0, (c) and (d) μ¼2Δ. The trans- parency of the junction is larger for μ¼2Δ. In the trivial phase of a junction with low transparency [panels (a) and (b)], ωpandλare all similar to one another. At the onset of the topological phase, ωpandλsplit into two different values. For sufficiently small values of EJ0, the lower branch of ωpandλdisap- pears when Bexceeds a critical magnetic field (this critical field is larger than Bc). FIG. 8. (a) and (b) Behavior of lower and upper bifurcation currents, respectively, i/C0in panel (a) and iþin panel (b) in a short driven Josephson junction with γ¼0:6, as a function of the Zeeman energy Band drive frequency ωd. The dashed black line indicates the plasmon frequency in GHz ( ωp=2π). The white regions on the top panels denote areas of parameter space where no bifurcation arises. (c) Line cuts of panels (a) and (b) at fixed drive frequency ωd=2π¼2:6 GHz. The corresponding cuts are indicated in panel (a) (dotted orange horizontal line) and panel (b) (dashed-dotted green horizontal line). (d) The derivatives @i+=@Bdisplay peaks at the topological phase transition ( B¼Bc). These peaks are gradually broadened as temperature is increased (data not shown). (e) –(h) Same as panels (a) –(d) but for the case of a trans- parent junction where γ¼1. Line cuts in panel (g) correspond to ωd=2π/C253:94 GHz. In all panels, the parameters are T¼0:02Δ=kB,μ¼Δ,EC=h¼0:1 GHz, and we add an external Josephson energy EJ0=h¼10 GHz.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-10 Published under an exclusive license by AIP Publishingapproximately the same for all the many-body states. This explains why ωpandλare very similar for different many-body states in the topologically trivial regime. The situation is different in the topologically nontrivial regime of the tunnel junction, where the subgap single-particle states asso-ciated to the inner MBS display increased dispersion [ Fig. 10(b) ]; this dispersion is qualitatively of the form +E Mcos(δ=2), where EM.0 is the hybridization energy of the inner MBS. The occupation of these dispersive Majorana bands determines the parity pof the inner MBS. Many-body states have a well-defined value of p,w i t h p¼0 (1) if the localized single-particle level due to the hybridized MBS is empty (filled). Then, the energies of the many- body states take the approximate form /C0EJcosδ/C0(/C01)p EMcos(δ=2)þconst, where EJ≃EJ0.O n c ea g a i n ,t h em a g n i t u d eo f EJand EMdoes not change from one many-body state to another.The cos( δ=2) term in the energy is responsible for the 4 π-periodic Josephson effect; indeed, if pis conserved, the many-body energies (and thus the supercurrent) are 4 π-periodic in δ. We now come to the key observation. In the topological phase, the many-body states with p¼0 have an absolute minimum atδ¼0 (mod 4 π) [see Fig. 10(d) ]. In contrast, the many-body states with p¼1 show a local (metastable) minimum at δ¼0 (mod 4 π), if and only if EM[(0, 4EJ). This local minimum exists because of the coexistence of /C0EJcos(δ) andþEMcos(δ=2) terms in the energy. When it exists, the local minimum of p¼1 states at δ¼0 differs in curvature and anharmonicity from the absolute minima of the p¼0 states at δ¼0. As a result, from Eq. (6), the plasmon frequencies and anharmonicities associated to small oscil-lations around δ¼0a r e p-dependent, /C22hω p(p)¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8ECEJþ(/C01)pEM 4/C18/C19s , λ(p)¼/C04EC 3/C22h2EJþ(/C01)pEM 16/C18/C19 :(13) When EM¼0,ωpandλare identical for all many-body states. As the system enters the topological phase, EMgrows together with Δ/C0 and two distinct values of ωpand λemerge for p¼0, 1. This explains the main trends observed in Figs. 9(a) and9(b). IfEM¼4EJ, the curvature of the local minimum for p¼1 states at δ¼0 vanishes, and thus ωp(1)¼0. For EM.4EJ, the many-body states with p¼1 have a maximum (rather than a local mimimum) at δ¼0; small oscillations can only take place in the vicinity of δ¼2π. But, because the δ¼2πminimum for p¼1 states is identical to the δ¼0 minimum for p¼0 states, it follows that there is only one branch of ωpandλwhen EJ.4EM. When EJ0≃0, a magnetic field slightly higher than the one required to attain the topological phase is sufficient to lead toE M.4EJ. The reason for this is that EM=EJ, being inversely pro- portional to γ, can be large in the tunneling regime (at least for the single-channel nanowire we consider). The interval of the magnetic field in which the metastable minimum is present is enhanced byincreasing E J0. For a sufficiently large EJ0,EMnever exceeds 4 EJ0, no matter how high Bis; an example of this situation is illustrated inFig. 9 , where the lower branch of ωpsaturates at a finite value at high B. This saturation has to do with the fact that Δþ, decreasing with B, eventually becomes smaller than Δ/C0deep in the topological phase, thereby stunting and reversing the growth of EMas the mag- netic field increases. The aforementioned features are echoed in the bifurcation currents of the TJBA. Figure 11(a) shows iþand i/C0for the lowest many-body states, in a short tunnel junction. In the trivial phase,the bifurcation currents for the low-lying many-body states are allsimilar. Beyond the topological phase transition, i þand i/C0split into two branches corresponding to p¼0 and p¼1 states. Accordingly, the switching probabilities between low- and high- amplitude oscillating states near δ¼0 become p-dependent [see Figs. 11(b) –11(c) ]. Roughly, p¼0, 1 can be regarded as two quantum states with distinct JBA switching probabilities. Let us now discuss the experimental observability of the 4π-periodic Josephson effect in a TJBA. The underlying idea is to FIG. 10. (a) A representative example of the single-particle spectrum in a short tunnel junction, in the topologically trivial phase ( B¼4Bc=5,μ¼0). The single-particle states are dispersionless and there are no subgap states. (b) A representative example of the single-particle spectrum in a short tunnel junction,in the topological phase ( B¼6B c=5,μ¼0). The Majorana bound states are visible inside the bulk superconducting gap. The dispersion of the inner MBS is strong when compared to those of the scattering states. (c) and (d) Low-lying many-body states built from (a) and (b), respectively. In the case of many-bodystates, we have added a term E J0(1/C0cosδ) to all states, with EJ0=Δ¼0:05; this energy originates from an ordinary Josephson junction connected in paral- lel. In the trivial phase, all many-body states have similar shapes, dictated by /C0EJ0cosδ. In the topological phase, a Majorana term /C0(1)pEMcos(δ=2) emerges, where ( /C01)pis the parity of the occupation of the inner MBS. As a consequence, the first many-body excited state, corresponding to a parity-flip of the inner MBS, has a metastable minimum at δ¼0 (thick solid gray and dotted black curves). The solid blue (dashed orange) curves corresponds toeven (odd) global fermionic parity states.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-11 Published under an exclusive license by AIP Publishinginduce transitions between the absolute and local minima of the energy spectrum, and to measure the frequency of small oscillations ofδaround those minima quickly enough (before energy relaxation to the absolute minimum takes place). We begin by noting that thevertical-in- δtransition between the p¼0 and p¼1 minima cannot be induced by microwave pulses, because electron –photon interactions preserve the MBS parity. 17This is unlike the case of the two quantum states in a superconducting qubit, the transitionbetween which is routinely induced by microwave pulses. Nonetheless, there are still two ways one can probe the pres- ence of the metastable minimum with a JBA scheme. One is by letting the system spontaneously switch between the stable andmetastable minima. This will occur due to quasiparticle poisoning or other vertical-in- δprocesses that violate the conservation of p. By repeating a JBA measurement many times for a fixed Bandμ, one will find two very different switching probabilities on the topo-logical phase (provided that E M/C29kBT), but not on the trivial phase. In addition, this type of measurement may allow to extract the lifetime τpof the inner MBS parity pexperimentally (provided that the escape time of δaway from the local minimum through thermal and quantum fluctuations is longer than τp). A more deterministic way of switching between stable and metastable minima is by applying current pulses.35A current pulse of short duration and amplitude comparable to the critical current of the junction can be applied to vary δaway from one minimum of the energy. For fixed p, changes in δthat are odd multiples of 2 π will connect a local minimum with an absolute minimum [seeFig. 10(d) ]. The precise control of δwith the current pulse is not required, however; what is important is that, after the current pulse stops, δwill begin to oscillate around the nearest minimum. If E M[(0, 4EJ), sometimes this minimum will be a local minimum, and other times it will be an absolute minimum. By measuring thestate of the oscillator (low-amplitude vs high-amplitude) immedi- ately after the current pulse is switched off, and repeating the whole process multiple times, two distinct switching probabilities will beapparent in the topological phase, but not in the trivial phase. Thismeasurement protocol is analogous to the JBA-based measurement of Rabi oscillations and relaxation times in conventional supercon- ducting qubits. 28 For the preceding measurement protocol to function, it is nec- essary that the lifetime of the metastable minimum be long com-pared to the JBA measurement time, the latter being of the order of 0.1–1μs. 28As mentioned above, the metastable minimum has a lifetime that is in part determined by the quantum/thermal fluctua-tions over the local potential barrier, as well as vertical-in- δinelas- tic processes that flip p. When 4 E J/C0EM/C29kBT, thermal fluctuations over the barrier are exponentially suppressed. Large quantum fluctuations of the phase (tunneling across the energy barrier around the metastable minimum) are likewise exponentiallysuppressed if the charging energy of the junction is small comparedto the Josephson energy. Under these conditions, the lifetime of themetastable minimum is limited by quasiparticle poisoning pro- cesses, which are indeed slow compared to the timescale in which a JBA experiment can be realized. 59 So far, we have restricted our discussion to short tunnel junc- tions. The reason for this is that, in more transparent junctions of any length, the situation is less favorable for the TJBA-based detec- tion of the 4 π-periodic Josephson effect. This is so for two main reasons. First, the scattering states display significant dispersion inmore transparent junctions. Therefore, there is a wide spread of ω p andλfor different many-body states, even in the trivial regime; this behavior is already incipient in Figs. 9(c) and 9(d), for junctions with γ¼0:6. Second, for B*Bc, the Majorana bands near δ¼0 are buried within the scattering states (because EM/C29Δ/C0in a transparent junction). As a result, the many-body states with p¼1 are part of a continuum of states, which drastically decreases their lifetime and observability. Concerning longer tunnel junctions (up to 50 sites in the weak link were studied), they do show the desired effect, but it is FIG. 11. (a) Lower and upper bifurcation currents for the first four many-body states, in a short tunnel junction ( γ¼0:6,μ¼0; see Fig. 9 for other parame- ters). The bifurcation currents for oscillations about δ¼0 are split into two branches in the topological phase, depending on the parity ( /C01)pof the inner Majorana bound state ( p¼0 for orange and blue curves, p¼1 for green and red curves). (b) and (c) Bifurcation probability Pas a function of the RF-drive amplitude for the four lowest energy many-body states. (b) B/difference0:9Bcand (c) B/difference1:25Bc. The variance of the current noise is chosen as σN¼2 nA.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-12 Published under an exclusive license by AIP Publishingmasked by the fact that there abound non-Majorana subgap states that disperse significantly. This leads to an array of different ωpand λfor the low-lying many-body states, in the topological phase. Yet, the fact that one of the plasmon frequencies drops appreciably inthe vicinity of the topological phase transition (even vanishing at acritical magnetic field if E J0is sufficiently small) still holds and is related to the emergence of MBS. We conclude this section by comparing our results to earlier works in the literature. First, the TJBA-based detection scheme forthe 4 π-periodic Josephson effect relies on the dynamics of the superconducting phase difference near a minimum of the energy (e.g., δ¼0). As such, it is robust under energy gaps that can occur atδ¼πdue to parity violating processes. It also differs from exist- ing proposals and experiments to measure the 4 πJosephson effect. The latter rely on swiping δover multiple full periods along a single many-body state of fixed p; the swipe must be fast enough to overcome any putative energy gap at δ¼π(through Landau – Zener tunneling). Second, the splitting of ω pand λthat we have reported is related to features previously predicted for microwave spectro-scopy. 8,16,18In Ref. 8, it was found that energy levels of a usual transmon qubit are split into doublets when EM=0. From Fig. 2 in that paper, it is evident that the plasmon frequencies would alsobe split into two for different sectors of the inner MBS parity. Yet,the authors restricted themselves to the regime E M/C28EJ, in which the splitting in ωpandλis very small, and also no systematic study ofωpand λwas carried out as a function of the microscopic parameters of the system. Similarly, in Refs. 16and18, a splitting of a spectral line in the microwave spectrum was noted at the topo-logical phase transition in a way that is reminiscent of Fig. 9(c) . These studies focused on the charge qubit parameter regime where the observed splitting was dominated by the charging energy, whileour work focus on the complementary transmon parameter regimewhere the splitting is dominated by E M. The possibility that the lower branch of ωpandλmay disappear in the topological phase of tunnel junctions (for EM.4EJ) was not recognized in earlier works. V. CONCLUSIONS In summary, we have proposed that a bifurcation amplifier built from a topological Josephson junction (a topological Josephson bifurca- tion amplifier, or TJBA) can be an interesting device to study variousaspects related to Majorana bound states and their emergence in semi-conductor nanowires with proximity-induced superconductivity. Much of our effort has been devoted toward understanding the behavior of the Josephson plasmon frequency ω pand the anharmonicity parameter λas a function of the externally applied Zeeman field B, in the transmon regime, at zero and finite tempera- ture, as well as for different many-body states. The reason for thisfocus is that ω pandλare key variables that govern the operation of the bifurcation amplifier; small changes in those quantities can be sensed when the TJBA is driven in the vicinity of the bifurcationpoint. Though our analysis is based on a minimal model of a nano-wire (single-channel, free from disorder), we have attempted to extract physical statements and results whose relevance should transcend the model ’s simplicity.Our work contains two main results. First, in the regimes of low and high transparency, @ω p=@Band @λ=@Bdisplay pro- nounced peaks at the topological phase transition, which arereflected in the bifurcation currents of the TJBA. These peaks areassociated to the emergence of MBS. Finite temperature broadensthe peaks, which nonetheless remain significant provided that the thermal energy is well below ( ,10%) the proximity-induced s-wave superconducting gap. Remarkably, in tunnel junctions, the contri-bution from MBS to ω pcan be turned on and off by varying the temperature. Another remarkable aspect is the evolution of theratio λ=ω 2 pacross a topological phase transition in a junction of low transparency, at zero temperature. In the trivial phase, λ=ω2 ptends to a plateau of /C01=6, which corresponds to the value found in a conventional Josephson junction. In contrast, in the topologicalphase, λ=ω 2 ptends to a plateau of /C01=24. The 1 =4 factor between the two plateaux originates from a fundamental change in the dependence of the ground state energy on the superconducting phase difference δ: it is cos δin the trivial regime and jcos(δ=2)jin the topological regime (the absolute value applies in the absence offermion parity constraints). The change is again due to the appear-ance of MBS. Second, the TJBA can spot the emergence of unconventional periodicities in the current-phase characteristics of the junction,such as the 4 π-periodic Josephson effect that is induced by Bwhen the junction undergoes a topological phase transition. The TJBA-based approach for measuring the 4 π-periodic Josephson effect is centered on the dynamics of δnear an energy minimum; it therefore differs from proposals and experiments that rely onswiping δover multiple full periods. The underlying concept relies on monitoring ω pandλfor different many-body states near δ¼0. Because of the coexistence between cos δand cos( δ=2) terms in the ground state energy, ωpandλsplit into two branches each in the topological phase; the same behavior is present in the bifurcationcurrents of the TJBA. This splitting originates from MBS. The above predictions and results are easier to validate in junctions that contain no subgap states other than the MBS. In our model, such is the case when the junction is short. In longer junc-tions, zero-energy Andreev bound states arise even in the trivialphase, thereby mimicking the signatures of MBS. In their presence,additional analysis is necessary in order to discern topological sig- natures from trivial ones. Similar issues complicate most detection schemes of MBS in semiconductor nanowires. Although a TJBA has not yet been built, the ongoing progress in combining nanowire-based superconducting qubit technology with the magnetic fields required to realize MBS suggests that its fabrication is within reach. Ideally, the present work may providean incentive for the experimental development of a TJBA, whichwill in turn encourage further theoretical studies on the device.Obvious tasks on the theory side include the adoption of more real- istic models of nanowires, as well as a fully quantum mechanical treatment of the TJBA dynamics. Note added in proof: In the final stages of this work, a preprint appeared 60in which an analytical theory for the dynamics of a topological Josephson junction coupled to a cavity is presented. The findings of Ref. 60, which include a fork-like feature of the cavity frequency pull at the topological phase transition, areconsistent with our results.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-13 Published under an exclusive license by AIP PublishingACKNOWLEDGMENTS This research was financed in part by the Canada First Research Excellence Fund and the Natural Science and EngineeringCouncil of Canada. Numerical calculations were partly done with computer resources from Calcul Québec and Compute Canada. U.C.M. acknowledges the support from CNPq-Brazil (Project No.309171/2019-9). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1For current perspectives of the field, see, e.g., S. M. Frolov, M. J. Manfra, and J. D. Sau, Nat. Phys. 16, 718 (2020); E. Prada, P. San-Jose, M. W. A. de Moor, A. Geresdi, E. J. H. Lee, J. Klinovaja, D. Loss, J. Nygard, R. Aguado and L. P. Kouwenhoven, Nat. Rev. Phys. 2, 575 (2020). 2For a review of the theory and the experiments, see, e.g., R. 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B 101, 224501 (2020). 51This discussion does not apply to junctions of low transparency, because scat- tering at the interface between the electrodes and the weak link invalidates the picture of two quasi-independent p-wave gaps.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214302 (2021); doi: 10.1063/5.0050672 129, 214302-14 Published under an exclusive license by AIP Publishing52Andreev bound states with near-zero energy (dubbed “quasi-Majorana states ”) have been predicted to be ubiquitous in the trivial phase of Josephson junctions with smooth potential barriers, and to mimic the behavior of Majorana bound states; see, e.g., A. Vuik, B. Nijholt, A. R. Akhmerov, and M. Wimmer, SciPost Phys. 7, 061 (2019); The quasi-Majorana bound states are absent if the barrier potential is steep. 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5.0053400.pdf

AIP Advances 11, 065313 (2021); https://doi.org/10.1063/5.0053400 11, 065313 © 2021 Author(s).Magnetic transitions and structural characteristics of Mn-doped α-Fe2O3/silica nanocomposites Cite as: AIP Advances 11, 065313 (2021); https://doi.org/10.1063/5.0053400 Submitted: 07 April 2021 . Accepted: 17 May 2021 . Published Online: 08 June 2021 Hyon-Min Song , Ivo Atanasov , and Jeffrey I. Zink ARTICLES YOU MAY BE INTERESTED IN Enhanced magnetic moment with cobalt dopant in SnS 2 semiconductor APL Materials 9, 051106 (2021); https://doi.org/10.1063/5.0048885 Curie–Weiss behavior of the low-temperature paramagnetic susceptibility of semiconductors doped and compensated with hydrogen-like impurities AIP Advances 11, 055016 (2021); https://doi.org/10.1063/5.0048886 Ultrawide bandgap semiconductors Applied Physics Letters 118, 200401 (2021); https://doi.org/10.1063/5.0055292AIP Advances ARTICLE scitation.org/journal/adv Magnetic transitions and structural characteristics of Mn-doped α-Fe 2O3/silica nanocomposites Cite as: AIP Advances 11, 065313 (2021); doi: 10.1063/5.0053400 Submitted: 7 April 2021 •Accepted: 17 May 2021 • Published Online: 8 June 2021 Hyon-Min Song,1,a) Ivo Atanasov,2 and Jeffrey I. Zink1 AFFILIATIONS 1Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, California 90095-1569, USA 2Electron Imaging Center for Nanomachines, California NanoSystems Institute, University of California, Los Angeles, Los Angeles, California 90095-1569, USA a)Present address: Department of Chemistry, Dong-A University, Busan 604-714, South Korea. Author to whom correspondence should be addressed: hyonmin1@dau.ac.kr ABSTRACT Hematite ( α-Fe 2O3) has become popular these days for their photocatalytic activities of water splitting. Metal-doped hematite materials are interesting as well for the bandgap engineering and for resolving fast charge–hole recombination. In this study, magnetism and ionic behaviors of rare manganese-doped α-Fe 2O3/silica nanocomposites are investigated. These nanocomposites are prepared by the impregnation method with a mixture of metal halides, followed by rapid heating (30○C/min) under air condition. When the molar ratio between FeCl 3⋅6H 2O and MnCl 2⋅4H 2O is 2.97, wasp-waisted hysteresis and ferromagnetism with the Curie temperatures of 56.1 and 58.0 K are observed for the nanocomposites annealed at 600○C for the duration of 3 and 7 h, respectively, while dominant spin glass states are observed for the nanocomposites annealed at 500○C. In x-ray diffraction patterns, mixed phases of α-Fe 2O3are identified, whereas crystalline metallic Mn or Mn oxides are hardly found. Electron energy-loss spectroscopy study indicates that Mn2+is severely oxidized, and with this oxidation of Mn2+, Si becomes more metallic. When the molar ratio between Fe and Mn halides is 7.32, magnetism is affected by a small amount of γ-Fe 2O3, and spin glass states and the competition between ferromagnetism and antiferromagnetism are observed in the long temperature range. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0053400 INTRODUCTION Among many transition metal oxides, iron oxides are proba- bly of most interest for their complex structures and rich magnetism as well as their technological importance. Along with goethite, hematite ( α-Fe 2O3) is thermodynamically the most stable iron oxide. It exists in trigonal form with a space group of R-3c. One difference from the other iron oxides, in particular, from cubic spinel γ-Fe 2O3 and Fe 3O4, is that it prefers to exist as a perfect lattice with few metal vacancies. Hematite recently gained attention for its photocatalytic effect with the bandgap energy of 2.1 eV. This belongs to the visible range, and it strongly absorbs visible light, which is used as the pho- tocatalysts under visible light irradiation.1Drawbacks of rather fast charge recombination and the poor efficiency of water splitting have been investigated by using oxygen vacancies near the interfaces.2 There are two transition temperatures in α-Fe 2O3: one as the Morin temperature around 263 K3and the other as the Néeltemperature ( TN) around 960 K, under which antiferromagnetic (AFM) alignment is maintained. The high TNof hematite is note- worthy, and if TNis reduced to room temperature, it has a technological importance to be applied for AFM/FM interfaces in an exceedingly simpler approach. In regard to the effect of size, transi- tion temperatures decrease with the decrease in particle size,4and the Morin temperature is not typically observed in the particles of less than 10 nm in size5or below 20 nm due to the loss of the AFM axis on the surface spins.6 When hematite nanoparticles (NPs) are supported on a matrix, transition temperatures decrease and magnetism is affected by the matrix. Silica-supported hematites are unique materials for their cat- alytic activities and their heterogeneous nature and have been stud- ied as early as 1966.7In Si-modified hematites, Si can prolong the lifetime of charge carriers and enhance the photocatalytic activity of hematites.8Mn-doped α-Fe 2O3/silica materials are rare, and in nanosize, the magnetism of these materials has not been reported. AIP Advances 11, 065313 (2021); doi: 10.1063/5.0053400 11, 065313-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv In this study, magnetism and structural characteristics of Mn-added α-Fe 2O3/silica nanocomposites are investigated. These nanocom- posites are prepared by rapid thermal annealing under air condition (30○C/min).9Mixture of ferromagnetism (FM) and superparam- agnetism is observed from the wasp-waisted hysteresis regardless of the annealing temperature, and the transition between FM and superparamagnetism is observed when excess Mn (molar ratio of 1:2.97) is added. Electron energy-loss spectroscopy (EELS) mea- surements reveal that Mn(II) is severely oxidized, with Si being more metallic character in the nanocomposites. With the oxida- tion of Mn(II), nanocomposites become more ionic, and crystalline Mn oxides are hardly found in x-ray diffraction (XRD) patterns. Despite this poor crystallinity of Mn oxides, Mn dopants domi- nate magnetic properties, with the coercivity of as much as 1978 Oe and with the reduction of transition temperatures to as low as 56.1 K. EXPERIMENTAL SECTION Commercial chemicals and solvents with analytical grades were purchased from Sigma-Aldrich and were used as received. X-ray diffraction (XRD) was measured with the Panalytical X’Pert Pro x-ray powder diffractometer using CuK αradiation ( λ=1.54 056 Å) in a θ–θmode. Transmission electron microscopy (TEM) images were obtained with the FEI Titan 80–300 kV S/TEM using a field effect gun operating at 300 kV. Scanning transmission electron microscopy (STEM) images were taken with a high-angle annu- lar dark field detector. TEM samples were prepared by sonicat- ing the mixture of silica nanocomposites (2.5 mg) and absolute methanol (0.1 ml), followed by dropcasting onto carbon-coated Cu grids. EELS measurements were conducted as follows: Si–L edge, O–K edge, Mn–L edge, and Fe–L edge were acquired from bulk electron energy loss spectra with the energy dispersion of 0.03 eV channel−1. Using the energy filter (Gatan, Inc., GIF Quan- tum ERS), the diffraction mode was adopted and the spectra were obtained in the area with about 200 nm diameter, while the diffrac- tion aperture is inserted. The Superconducting Quantum Inter- ference Device (SQUID, Quantum Design, Inc., San Diego, CA, USA) was used to measure the magnetic properties. MCM-41 type mesoporous silica was prepared according to the literature method.10 Synthesis of magnetized silica nanocomposites The mixture of iron(III) chloride hexahydrate [FeCl 3⋅6H 2O, Chemical Abstracts Service (CAS) number: 10025-77-1, 15.0 g, 55.5 mmol] and manganese(II) chloride tetrahydrate (MnCl 2⋅4H 2O, CAS number: 13446-34-9, 3.70 g, 18.7 mmol) was dissolved in deionized water (10 ml). This aqueous solution was filtered with a 25 mm GD/X non-sterile syringe filter (with a pore size of 0.2 μm and a diameter of 13 mm by Whatman). Mesoporous silica (150 mg) was dried and finely grinded before being placed on the glass filter (a disc diameter of 40 mm and a capacity of 80 ml). With vacuum applied, the aqueous mixture of metal halides was dropped on mesoporous silica. After repeated infiltration ( ×10), yellowish mesoporous silica was dried under vacuum for 2.5 h. The dried sample was trans- ferred to the annealing crucible. Thermal annealing was performed under air condition at the heating rate of 30○C/min. Rod-shapenanocomposites were annealed at 500○C for 7 h or at 600○C for 3 h. Spherical shape nanocomposites were annealed at 600○C for 7 h. Cooling was conducted at a rate of 20○C/min. For preparing the mixture of metal halides with a molar ratio of 7.32, FeCl 3⋅6H 2O (15.0 g, 55.5 mmol) and MnCl 2⋅4H 2O (1.50 g, 7.58 mmol) were mixed with deionized water (10 ml). Spherical shape mesoporous sil- ica was used as a template for impregnation, and the annealing was conducted at 500○C for 12 h. Other experimental conditions are the same as in the case of the 2.97 mol ratio. RESULTS AND DISCUSSION Mesoporous silica templates before impregnation of metal halides were prepared by base-catalyzed hydrolysis.10Raw meso- porous silica of spherical shape [Fig. 1(a)] and rod shape [Fig. 1(e)] before impregnation are shown in STEM images. The framed rods in Figs. 1(b) and 1(f) were obtained by annealing at 500○C for 7 h. The spheres in Figs. 1(c) and 1(g) were obtained by the annealing at 600○C for 7 h, and the rods in Figs. 1(d) and 1(h) were obtained by the annealing at 600○C for 3 h. The precursors for the impreg- nation are FeCl 3⋅6H 2O and MnCl 2⋅4H 2O, and the molar ratio is 2.97:1. XRD patterns (Fig. 2) indicate that the phase of hematite is dominant, but there are mixtures of more complicated crys- talline structures [Fig. 2(b)]. Popular Mn oxides such as MnO (face- centered cubic, Fm-3m), Mn 2O3(orthorhombic, Pbca , or body- centered cubic, Ia-3), and Mn 3O4(tetragonal, I41/amd ) were not identified. In the rods annealed at 500○C, the weak phase of γ-Fe 2O3 is observed. There are cation vacancies in γ-Fe 2O3with the complete oxygen lattice surrounding these vacancies.11γ-Fe 2O3is metastable and undergoes phase change to α-Fe 2O3at high temperature, i.e., 450○C for 30 nm NPs12and 470○C for 12 nm NPs.13In the pres- ence of foreign metal cations, this transition of γ-Fe 2O3toα-Fe 2O3 is retarded. It is known that γ-Fe 2O3with Mn3+doping (8.5%) does not transform γ-Fe 2O3toα-Fe 2O3at 500○C, while it completely transforms to α-Fe 2O3at 600○C with the same amount of Mn doping.14Not only Mn, trace amount of metals (Co, Ni, Zn, Cu, Al, V, and Cr, <0.01 mol. %) slows the transition of γ-Fe 2O3toslows the transition o α-Fe 2O3.15When γ-Fe 2O3is inside silica, it is thermally stable and the transition temperature to α-Fe 2O3is higher.16It is also noticed that 3–4 nm γ-Fe 2O3NPs in the silica matrix undergo transition to α-Fe 2O3at 700○C.17 At the annealing temperature of 600○C, the rhombohedral phase ( R-3c) ofα-Fe 2O3is observed without cubic spinels, and two different hematite structures in each nanocomposite were identified (phase 1 and phase 2, Table I). Cell parameters of the rods are smaller than those of spheres. In spheres, one hematite structure with the cell parameters of a=5.016 Å and c=13.689 Å and the other hematite structure with a=4.967 Å and c=13.518 Å are found, while in the rods, one with the cell parameters of a=5.003 Å and c=13.656 Å and the other with a=4.966 Å and c=13.534 Å are identified (Table I). Compared to the cell parameters of hematite in other studies, such as a=5.0356 Å and c=13.7489 Å (JCPDS 33-0664), cell parameters in this study are quite small, which might be affected by the incorpora- tion of foreign metal cations, preferably Si4+rather than Mn2+in the sites of Fe3+due to the smaller size of Si4+. It is commonly observed that in the presence of a large amount of Si such that the molar ratio of Si/(Si +Fe) is over 0.270, Si incorporation into α-Fe 2O3is found.18 AIP Advances 11, 065313 (2021); doi: 10.1063/5.0053400 11, 065313-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 1. STEM images of (a) spherical and (e) rod-shape mesoporous silica before impregnation. (b) TEM and (f) STEM images of rod-shape nanocomposites annealed at 500○C for 7 h. (c) TEM and (g) STEM images of spherical nanocomposites annealed at 600○C for 7 h. (d) TEM and (h) STEM images of rod-shape nanocomposites annealed at 600○C for 3 h. FIG. 2. XRD patterns of nanocomposites in the range of (a) 25○–100○and (b) 30○–40○with a fitting of (104) and (110) planes. The magnetism of α-Fe 2O3/SiO 2nanocomposites was exam- ined in both field-dependent ( M–H) and temperature-dependent magnetization ( M–T) measurements. M–Hcurves in Fig. 3 show two characteristics. One is the wasp-waisted hysteresis loop with a narrowing hysteresis near zero applied magnetic fields and the other is the overall hard magnetism, which is exhibited in the reluctance TABLE I. Cell parameters of Mn-doped α-Fe2O3/SiO 2nanocomposites obtained from the XRD measurement. Phase 1 (Å) Phase 2 (Å) Rods 500 (7 h) a=5.033 c=13.789 a=5.021 c=13.719 Spheres 600 (7 h) a=5.016 c=13.689 a=4.967 c=13.517 Rods 600 (3 h) a=5.003 c=13.656 a=4.966 c=13.534to be saturated. In the M–Hcurve of rod shape nanocomposites annealed at 500○C for 7 h [Fig. 3(a)], wasp-waisted loops imply a mixture of magnetic phases with different coercivities.19Those in Fig. 3 all show similar patterns of mixed phases of FM and super- paramagnetism.20There is soft FM both at 300 K and at 5 K, and this is thought to be contributed by γ-Fe 2O3, as the peaks of γ-Fe 2O3 were observed in the XRD pattern, and the transition of γ-Fe 2O3to α-Fe 2O3is retarded in the presence of Mn ions.14The coercivity is 108 Oe at 300 K and 779 Oe at 5 K (Table II), and a sharp drop near zero applied fields implies the presence of superparamagnetism or from the effect of the canted spins of AFM, as they undergo sudden drop of magnetization when the applied field approaches zero. In spherical nanocomposites annealed at 600○C for 7 h [Fig. 3(b)], relatively large coercivity of 1978 Oe at 5 K is observed. There are both AFM and FM, and the large coercivity is due to the hard magnetic behaviors from FM materials. Nanocomposites that AIP Advances 11, 065313 (2021); doi: 10.1063/5.0053400 11, 065313-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. M–Hcurves of (a) rod-shape nanocomposites annealed at 500○C for 7 h. (b) Spherical nanocomposites annealed at 600○C for 7 h. (c) Rod-shape nanocomposites annealed at 600○C for 3 h. are composed of Fe and Mn transition metals can hardly achieve such a large coercivity. One of the iron oxides, ε-Fe 2O3phase, which is commonly synthesized within the silica matrix due to their struc- tural instability, displays a coercivity of over 7000 Oe. However, the formation of the ε-Fe 2O3phase starts at 900○C,21which is higher than the annealing temperature in this study. In general, the increase in coercivity implies the increase in anisotropy. There are two pos- sible mechanisms for the increase in anisotropy. One is the long range magnetostatic interactions with each magnetic moment being ordered,22and the other is the interparticle dipolar interactions. The dipole is of naturally anisotropic character, and the dipole–dipole interaction in the long range is also manifested, for example, by cal- culations in magnetic microwires.23Wasp-waisted hysteresis is also observed in the spherical nanocomposites, and it implies a mixture of FM and superparamagnetism. The difference from the other two materials is that there is a significant contribution from AFM. The interaction between FM and AFM is reflected on the asymmetric hysteresis loop shift along the field direction ( −2027 Oe and 1930 Oe, 5 K; see Table II), which suggests exchange anisotropy between AFM and FM. At 300 K, the hysteresis loop is characteristic of AFM and much smaller saturation magnetization (0.976 emu/g) is observed compared to that measured at 5 K (3.64 emu/g). These spheres were annealed at 600○C for 7 h, and for this duration, AFM is thought to develop significantly. In the M–Hcurve of the rods annealed at 600○C for 3 h [Fig. 3(c)], mixed phases of FM and superparamagnetism are exhib- ited at 5 K, which is similar to the rods annealed at 500○C for 7 h.The difference is that wasp waist-like hysteresis is also seen at 300 K. The transition of γ-Fe 2O3toα-Fe 2O3is more complete, but it is believed that γ-Fe 2O3is not completely oxidized to α-Fe 2O3for 3 h of annealing at 600○C. There is a loop shift along the field direction (−594 Oe, 503 Oe, 5 K, Table II), and similar exchange anisotropy as the spheres annealed at 600○C for 7 h (Table II) implies the presence of both FM and AFM. While the conversion to α-Fe 2O3is retarded, γ-Fe 2O3is known to be poorly crystallized with one sixth of the octahedral Fe position, or one ninth of the total Fe position in cubic spinel is vacant. Although in small amount, the satura- tion magnetization of γ-Fe 2O3is higher than that of α-Fe 2O3. Due to the incomplete transition of γ-Fe 2O3toα-Fe 2O3, an AFM/FM interface is created with the core γ-Fe 2O3phase surrounded by the outer shells of α-Fe 2O3. It is believed that the shape of the silica sup- port, whether spheres or rods, does not influence the magnetism of the supported particles. In addition, it is also believed that the dia- magnetic silica matrix hinders dipolar attractions between the par- ticles and thus more hard magnetic behavior develops under silica support.24 InM–Tcurves of the rods annealed at 500○C for 7 h [Fig. 4(a)], the magnetization is small and the value of zero-field-cooled (ZFC) measurement is larger than that of FC. This is not a typical M–T curve, but similar behavior is found in hematite nanorods,25Ni fer- rite NPs,26doped Mn oxides,27and even Co ferrite NPs with mag- netostriction.28It suggests mixtures of FM, AFM, and canted spin structures with little interaction between them. The monotonous decrease in magnetization with the correlation in ZFC and FC TABLE II. M–Hmagnetization data of Mn-doped α-Fe2O3/SiO 2nanocomposites. MS(emu/g)aM0(emu/g)bHC(Oe) 5 K 300 K 5 K 300 K 5 K 300 K Rods 500 (7 h) 3.81 1.65 1.070 −1.080 0.405 −0.415 −783 775 −108 108 Spheres 600 (7 h) 3.64 0.976 0.809 −0.801 0.170 −0.162 −2027 1930 −249 239 Rods 600 (3 h) 5.82 2.77 1.566 −1.777 0.495 −0.689 −594 503 −106 93.6 aMagnetization measured at the applied magnetic field of 5 T. bMagnetization at the applied magnetic field of 0 T on the hysteresis curves. AIP Advances 11, 065313 (2021); doi: 10.1063/5.0053400 11, 065313-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 4. M–Tcurves of (a) rod-shape nanocomposites annealed at 500○C for 7 h, (b) spherical nanocomposites annealed at 600○C for 7 h, and (c) rod- shape nanocomposites annealed at 600 ○C for 3 h. (d) The graph of 1/ Mvs temperature of the spherical nanocom- posites annealed at 600○C for 7 h (blue circle) and rod-shape nanocomposites annealed at 600○C for 3 h (red circle). indicates spin glass-like surface disorder behavior.29Magnetization of both ZFC and FC decreases without significant magnetic tran- sitions in the long temperature range, which is contributed by the disordered surface spins. Spin glass states and the lack of interaction between FM and AFM are also shown in M–Hcurves in Fig. 3(a), with little loop shift along the field direction. The magnitude of the differentiation in FC measurement from 5 to 300 K (0.073–0.064 emu) is small. This is seen in the frustrated spin systems and also implies that the spins of FM and AFM compete each other in the long temperature range. InM–Tcurves of the rods and spheres annealed at 600○C, the peaks in ZFC and the increase in magnetization at low temperature in FC measurements are observed. This sharp increase in FC mea- surements is characteristic of FM materials, and it indicates signifi- cant change in magnetic ordering. The Curie temperature ( Tc) of the spheres annealed at 600○C is 58.0 K [Fig. 4(b)] and that of the rods annealed at 600○C is 56.1 K, which are obtained by fitting FC curves with the equation of power law fit with M(T)=M0(1−T/Tc)β. In the curves of 1/ Mvs temperature in Fig. 4(d), linear patterns over the transition temperature implies paramagnetism. Excess Mn ions are those that contribute to this paramagnetism. In the strong interaction between FM and AFM, FM suppresses AFM during cool- ing under the applied magnetic field (FC condition), and the spins of FM are spontaneously aligned. In the rods that were annealed at 600○C for 3 h, magnetization increases after the appearance of a peak in the ZFC measurement. During the cooling under zero applied fields, followed by the increase in temperature, dis- ordered spins are aligned by the FM component with the increas- ing thermal energy. The peaks of Figs. 4(b) and 4(c) in the ZFC measurement are similar to superparamagnetic blocking tempera- ture after which paramagnetism follows. However, magnetization in ZFC decreases monotonously at the lower temperature, which is different from the usual ferromagnetic–paramagnetic transition.In addition, bifurcation of ZFC and FC curves exists in the entire temperature range and it implies other phases of magnetism in addi- tion to superparamagnetism. The small magnitude of magnetiza- tion in ZFC measurements is also noteworthy. Under zero applied magnetic fields after demagnetization, nanocomposites were cooled from 300 to 5 K, and then the magnetic fields of 50 or 100 Oe were applied. The spins of ferromagnetism are blocked at low tempera- ture in the frozen state and respond slowly with the increase in the temperature, and the magnetization increases. The small magneti- zation under ZFC conditions implies that ferromagnetism is sup- pressed under ZFC measurements. In the FC condition, magnetiza- tion was measured from 300 to 5 K under the magnetic fields of 50 or 100 Oe. The magnetization increases sharply when the temperature approaches Tc. The spins align quickly and the anisotropy increases. This is hardly achievable or observable in hematite or the mixture ofγ-Fe 2O3and α-Fe 2O3. Long range FM ordering is induced by the dipolar interaction between particles, which is assumed to be affected by the spins of Mn ions or oxygenated Mn ions such as Mn3+–O or Mn4+–O bondings. In order to study the behaviors of metal cations in amorphous silica, EELS were measured in STEM mode. Without measuring the absolute edge energies, the spectra were obtained with the main peaks being aligned at 110.8 eV for the Si–L 2,3edge, at 540.0 eV for the O–K edge, at 640.0 eV for the Mn2+L3edge, and at 710.3 eV for the Fe2+L3edge. The exponential decay (AE−r) was adjusted accord- ing to the pre-edge, and this function was used as a background for the subtraction. The Si–L 2,3edge and O–K edge of the raw amorphous silica before the impregnation of metal halides were used as a reference [Figs. 5(a) and 5(b)]. In the Si–L 2,3edge of amorphous silica, the shoulder peak at 108.8 eV and the main peak at 110.8 eV (L 3edge) along with the peak at 117.5 eV (L 2edge) are characteristic of the amorphous structure.30Compared to the spectrum of amorphous AIP Advances 11, 065313 (2021); doi: 10.1063/5.0053400 11, 065313-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 5. EEL spectra of Mn-doped α-Fe 2O3/SiO 2nanocomposites with (a) Si–L, (b) O–K, (c) Fe–L, and (d) Mn–L ionization edges after background subtraction. silica, the annealed nanocomposites show two characteristics. One is the lower onset energy of the Si–L 2,3edge, and this is related to the lower oxidation number of Si and more ionic bonding by the distortion of Si–O–Si bonding.31Tetrahedral SiO 4is distorted and the covalent nature has changed accordingly due to the change in Si–O–Si bonding to Si–O–M bonding (M: Mn or Fe).32The other characteristics are that with the increase in annealing tem- perature, the peak of the Si–L 2,3edge is broader, and this reflects many different types of bond lengths and angles involved in Si–O–Si bonding.33 In the O–K edge, the peaks at 540.7 eV, which arise due to the crystal field splitting of Si 3d orbital, are sharper and the splitting is narrower with the increase in annealing temperature [Fig. 5(b)]. Sharp and intense peaks, particularly of spherical nanocomposites annealed at 600○C, imply oxygen ordering of SiO 44−tetrahedra within the silica matrix.34The absence of the splitting of O–K edge means that the splitting of crystal field is small and the nanocom- posites are more ionic. The pre-edge also changes to lower energy loss as the annealing temperature increases, which implies additional interaction of O with metal cations and the generation of complex transition metal–O bondings. Typically, the onset energy of O–K edge decreases with the increase in the oxidation state of metals. Diffusion of Mn2+and Fe3+into SiO 4tetrahedra and the oxidation of Mn(II) and the reduction of Si4+are thought to occur. Overall, the change in oxidation states of Mn is more dynamic and liable in the amorphous silica matrix, and with the change in the oxidation states of Mn, the broad feature of the Si–L 2,3edge and the shift of theonset energy of the O–K edge imply that Si has also many different oxidation states. In the spectra of the Mn L- 2,3edge, the low L 3/L2intensity ratio is noteworthy [Fig. 5(d), Table III]. It is generally regarded that the higher the oxidation state of Mn, the lower the L 3/L2intensity ratio is.35Two nanocomposites have lower L 3/L2intensity ratios than MnO (4.8). It suggests that Mn is severely oxidized during the rapid annealing, and the shape of the Mn L- 2,3edge is similar to that of Mn oxides with Mn having a high oxidation number.36The annealing at 500○C makes Mn oxidized more severely than the annealing of that at 600○C, which suggests that with the smaller size of Mn(IV), the diffusion of Mn ions into SiO 4tetrahedra is more active at 500○C, and the diffusion is more difficult at 600○C because of the ordering of O and Si at 600○C. When Mn is in high oxidation state, they exist as oxygenated ions instead of being present as free metal cations. Therefore, the bondings of Mn3+–O and Mn4+–O become more plausible. The Fe L- 2,3edge in Fig. 5(c) shows more complicated TABLE III. EELS data of Mn-doped α-Fe2O3/SiO 2nanocomposites. Rods at 500○C Spheres at 600○C ΔE Mn(L 2-L3) (eV) 10.65 10.55 Intensity ratio: Mn(L 3/L2) 2.04 2.52 ΔE Fe(L 2-L3) (eV) 12.90 13.09 Intensity ratio: Fe(L 3/L2) 5.29 5.18 AIP Advances 11, 065313 (2021); doi: 10.1063/5.0053400 11, 065313-6 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 6. (a) TEM image, (b) XRD pat- tern, (c) M–H, and (d) M–Tmagneti- zation curves of spherical nanocompos- ites annealed at 500○C for 12 h using metal halide mixtures of Fe and Mn with a molar ratio of 7.32. structures, which has significant difference from Fe3+of hematites.37 The intensity ratio of the Fe L- 2,3edge implies the presence of Fe2+, while the L 3edge differs from the general shape of hematites and possibly involves Fe2+. For the comparison of the effect of the Mn amount on the magnetism of nanocomposites, the mixture of FeCl 3⋅6H 2O and MnCl 2⋅4H 2O with a molar ratio of 7.32:1 is impregnated in the spherical silica substrate. The target temperature is 500○C (30○C/min), and the nanocomposites were annealed for 12 h under air condition. A similar shape of framed silica was observed in the TEM image [Fig. 6(a)], and the XRD pattern in Fig. 6(b) indicates a dominant α-Fe 2O3phase with the little contribution from γ-Fe 2O3. During 12 h of annealing at 500○C, incomplete transition from γ-Fe 2O3toα-Fe 2O3is observed. The smooth turn at zero applied magnetic fields in the M–Hcurve [Fig. 6(c)] indicates the exis- tence of superparamagnetism along with AFM and FM at 5 K, while both FM and AFM are observed at 300 K. Soft FM from the small amount of γ-Fe 2O3governs the hysteresis loop with small coercivity, although AFM is not suppressed at 5 K by showing hard magnetism. InM–Tcurves [Fig. 6(d)], the decrease in magnetization at low tem- perature from 5 to 43 K in the FC measurement is the paramagnetic behavior, presumably due to the contribution from the high mag- netic moment of Mn2+. This paramagnetism is reflected in relatively large magnetization of ZFC at 5 K (0.176 emu/g) and the small differ- ence of magnetization between ZFC and FC at 5 K. Over 43 K, there are both FM and AFM, and also, there is a competition between FM and AFM, yet with little interaction between them. It indicates the presence of γ-Fe 2O3and α-Fe 2O3. The long plateau with little change in magnetization also implies this competition. Ga-addedα-Fe 2O338and Ti-added α-Fe 2O339show low temperature param- agnetism and spin frustration from the competition between FM and AFM. CONCLUSION Mn-doped hematite/SiO 2nanocomposites were prepared by the impregnation method using mesoporous silica as the tem- plate. At the annealing temperature of 500○C, a mixture of mag- netic phases with different coercivities exists with little interaction between them, as the hysteresis loop shift is not observed and the spin glass-like states with the spins of AFM and FM competing in the long temperature range are noticed. AFM develops signifi- cantly at the annealing temperature of 600○C for 7 h of annealing, and exchange coupling is observed with the hysteresis loop shift and with a coercivity of 1978 Oe. The transition between ferro- magnetism and superparamagnetism is observed with Tcof 58.0 K. EELS measurements indicate that the nanocomposites become more ionic with the increase in annealing temperature. Mn2+is severely oxidized during the thermal annealing, but Si becomes more metallic and the presence of both Fe3+and Fe2+is observed. It is thought that Mn is actively involved in the magnetic activities in the nanocom- posites, although the crystalline forms of any Mn oxides were not identified. There are few studies on nanosize Mn-doped iron oxide/SiO 2materials, and this study suggests that large coercivity as much as 1978 Oe can be achieved under a certain condition in the nanocomposites that are composed of only Fe and Mn transition metals. AIP Advances 11, 065313 (2021); doi: 10.1063/5.0053400 11, 065313-7 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. J. Cowan, C. J. Barnett, S. R. Pendlebury, M. Barroso, K. Sivula, M. Grätzel, J. R. Durrant, and D. R. Klug, “Activation energies for the rate-limiting step in water photooxidation by nanostructured α-Fe 2O3and TiO 2,” J. Am. Chem. Soc. 133, 10134–10140 (2011). 2Z. Zhang, I. Karimata, H. Nagashima, S. Muto, K. Ohara, K. Sugimoto, and T. Tachikawa, “Interfacial oxygen vacancies yielding long-lived holes in hematite mesocrystal-based photoanodes,” Nat. 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5.0036285.pdf

Appl. Phys. Lett. 118, 052408 (2021); https://doi.org/10.1063/5.0036285 118, 052408 © 2021 Author(s).Spin–orbit torque rectifier for weak RF energy harvesting Cite as: Appl. Phys. Lett. 118, 052408 (2021); https://doi.org/10.1063/5.0036285 Submitted: 03 November 2020 . Accepted: 13 January 2021 . Published Online: 02 February 2021 Shehrin Sayed , Sayeef Salahuddin , and Eli Yablonovitch ARTICLES YOU MAY BE INTERESTED IN Enhancement of spin–orbit torque in WTe 2/perpendicular magnetic anisotropy heterostructures Applied Physics Letters 118, 052406 (2021); https://doi.org/10.1063/5.0039069 Micromagnetic understanding of switching and self-oscillations in ferrimagnetic materials Applied Physics Letters 118, 052403 (2021); https://doi.org/10.1063/5.0038635 Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147Spin–orbit torque rectifier for weak RF energy harvesting Cite as: Appl. Phys. Lett. 118, 052408 (2021); doi: 10.1063/5.0036285 Submitted: 3 November 2020 .Accepted: 13 January 2021 . Published Online: 2 February 2021 Shehrin Sayed,a) Sayeef Salahuddin, and Eli Yablonovitch AFFILIATIONS Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720, USA a)Author to whom correspondence should be addressed: ssayed@berkeley.edu ABSTRACT We propose a rectifier concept, simultaneously utilizing the Hall effect and the spin–orbit-torque, that is well matched to the low impedance of antennas. This rectifier is promising for general radio detection and, particularly, for harvesting ambient weak radio signals, where conven-tional rectification fails to operate. The Hall effect and spin–orbit-torque are both proportional to current density, which improves inverselywith the device cross-sectional area, providing a large signal at the nanoscale. A single device made using existing materials can provide 200lV DC from 500 nW of radio frequency (RF) power. A series array of such devices can efficiently enhance the DC voltage to 300 mV while matching the receiver antenna impedance. Such magnetic devices can convert weak RF power into DC power with substantial effi-ciency at low voltage and low impedance where conventional semiconductor rectifiers fail. Published under license by AIP Publishing. https://doi.org/10.1063/5.0036285 Harvesting of the ambient radio frequency (RF) energy is of great current interest 1,2for self-powered devices and circuits, especially to power the emerging applications, e.g., the internet of things, wearabledevices, various sensors and implants, and 3D integrated circuits. Touse the weak ambient RF power density for such applications, we need rectifiers capable of operating much below 1 lWR Fp o w e rw i t hh i g h efficiencies. 2,3However, the development of such technologies is severely limited by the conventional semiconductor rectifiers, whichfail to operate in the weak RF limit. 4There is much on-going effort to achieve efficient semiconductor rectifier technologies in the weak RF limit4–7as well as to explore different mechanisms for rectification.8–12 In this Letter, we propose a rectifier concept, simultaneously uti- lizing the Hall effect and the spin–orbit torque (SOT). Both phenom- ena scale with the current density and improve inversely with thedevice cross-sectional area, providing a large signal at the nanoscale.The basic idea is to inject an RF current in a Hall material to generatea Hall voltage and use the same RF current in a spin–orbit material tocontrol a magnet with current-induced SOT. The magnet then applies a magnetic field to the Hall material, leading to a rectification of the Hall voltage. We use a magnet with low anisotropy energy to make thedevice highly sensitive to weak RF currents. Materials exhibiting SOT 13–16are being extensively studied for switching stable ferromagnets (FMs) in memory17,18and logic19,20 applications. Here, we point out a different application of SOT in therectification of weak RF by coupling it to conventional Hall devices.Using material parameters for InAs as the Hall layer, Bi 2Se3as the spin–orbit (SO) layer, and a soft ferrite as the ferromagnet (FM), wecalculate the DC voltage in a 100-nm-thin device to be /C24200lVf r o m an/C24500 nW RF power. An array of such devices with series connec- tions can efficiently enhance the DC voltage to /C24300 mV from the same RF power while matching the low impedance of the receiver antenna. We estimate a high RF-to-DC power conversion efficiency of/C2471% from such a weak RF power, where conventional technologies exhibit low efficiencies. 4 T h em e c h a n i s mo ft h eS O Tr e c t i fi e rc a nb ee x p l a i n e du s i n ga Hall bar and a solenoid connected in parallel, as shown in Fig. 1(a) .I t is well known that a current-carrying conductor placed in a magnetic field ( B) exhibits a voltage drop in the direction orthogonal to both the current and the B-field due to the Hall effect. If a fraction of the current flows through a solenoid underneath the Hall bar to generatetheB-field, the Hall voltage will be unidirectional irrespective of the current direction, leading to a rectification. However, we need a sole-noid with a large number of turns to produce a substantial B-field from a weak current, which will lead to a large device size and will limit the operation to low frequencies due to the large resultantinductance. W ep r o p o s eas t r u c t u r es h o w ni n Fig. 1(b) , where the solenoid is replaced with a bilayer consisting of a SO material and a FM with alow anisotropy energy barrier. We design the device such that an RFcurrent ( I RF) equally divides between the Hall and the SO layers. The Appl. Phys. Lett. 118, 052408 (2021); doi: 10.1063/5.0036285 118, 052408-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplFM magnetization, on average, follows the fraction of IRFflowing in t h eS Ol a y e r ,p r o v i d e dt h a tt h ec u r r e n ti ss u f fi c i e n t l yh i g h e rt h a na minimum value Imin. The FM applies a B-field on the Hall material along the M-direction, leading to a rectified Hall voltage Vþ/C0V/C0¼Vout¼qH 2tHM/C18/C19 IRF; (1) where qHis the Hall resistivity, tHis the Hall layer thickness, and Mis the normalized magnetization along the easy-axis. The minimum current ( Imin) for rectification is related to the spin-torque driven switching current, which is given by the following expression for a single domain magnet:21,22 Imin¼1 hSHwso L4q /C22hEba; (2) where hSHis the spin Hall angle, wsois the SO layer width, Lis the device length, ais the Gilbert damping, qis the electron charge, and /C22h is the reduced Planck’s constant. Here, Ebis the anisotropy energy of t h eF M ,w h i c hd e p e n d so nt h ec o e r c i v efi e l d( Hc), the saturation mag- netization ( Ms), and the FM volume. To lower the minimum current,we need a SO with high hSH,a nF Mw i t hl o w Eb, and an FM–SO inter- face with low a. We assume that the effect of the demagnetizing field is negligible. Note that the magnetization dynamics is analyzed with thestochastic Landau–Lifshitz–Gilbert (s-LLG) equation and not limitedto Eq. (2). We use Eq. (2)only for an analytical understanding of the parameters. Experimentally, E bhas been reduced by lowering the total mag- netic moment ( Ms/C2volume)23or by tuning the FM thickness to opti- mize near the transition point between in-plane and perpendicularanisotropies 24or by using isotropic geometries.25We consider a soft ferrite that exhibits a low Hca n das m a l lF Mv o l u m et oa c h i e v eas u b - stantially low Eb. Such a FM with very low Ebcan, in principle, switch stochastically between þ1a n d /C01 due to the thermal noise, which is taken into account in our s-LLG equation-based simulations. A strongSOT can pin the magnetization to one of the states. A stochastic FM,on average, follows the current-induced SOT, and the average Mfol- lows the following relation: 11 hMi¼tanhIRF Isat:/C18/C19 ; (3)FIG. 1. The mechanism of the spin–orbit torque (SOT) rectifier. (a) The basic principle explained with a Hall bar and a solenoid connected in parallel. A curren t divides equally between them and the magnetic field ( B) in the solenoid follows the current, leading to a unidirectional Hall voltage. (b) The proposed SOT rectifier structure, where the sole- noid is replaced with a bilayer consisting of a spin–orbit material and a soft magnet. The magnetization, M, follows the current in the spin–orbit layer due to the SOT and the magnet applies a B-field to the Hall bar, giving a similar mechanism to (a).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052408 (2021); doi: 10.1063/5.0036285 118, 052408-2 Published under license by AIP Publishingwhere Isat:is the current required to fully saturate the FM. In a completely stochastic FM driven by high thermal noise, Isat:¼Imin with Eb¼3 2kBT.11Low anisotropy energy magnets can achieve a wide frequency bandwidth of operation, depending on the total magnetic moment in the FM and the angular momentum conservation.11 We design the device such that the length, width, and thickness of each layer (Hall, SO, and FM layers) are 100 nm, 100 nm, and50 nm, respectively. We consider Bi 2Se3with resistivity /C242mXcm and hSH/C253:527as the SO layer. We consider a soft ferrite with Eb/C252kBT, calculated from Hc¼0:4O ea n d l0Ms¼0:5T .28Such a lowEbalong with a¼0:01 is expected to provide Imin/C250:1lA, calculated using Eq. (2). However, various nonidealities can increase Iminand our s-LLG simulation captures a general picture for a single domain magnet. The amplitude of the fraction of IRFin the SO layer is set to 5 /C2Imin¼0:5lA so that the FM can easily follow it. The total IRFis set to 1 lA. We use parameters for InAs as the Hall material and set the doping concentration ( ne)t o1 017cm/C03to get a resistivity q¼1=qneln /C252mXcm, matched with the SO layer. Here, mobility29ln/C253 /C2104cm2V/C01s/C01. This ensures an equal division of the RF current in the Hall and the SO layers. The current shunting in the FM is neg- ligible because ferrites are orders of magnitude higher resistive30than the Hall and the SO layers. The total device resistance Rdev¼200X. The Hall coefficient is RH¼1=qne/C2562:5c m3/C. The magnetic layer applies a B-field along its easy axis31,32and perpendicular to the Hall layer. Given that a soft ferrite can provide a saturation field of l0Ms¼0:5T28and assuming that the field lines in a thin Hall layer do not degrade much, we calculate an expected maximum value of qH¼RHl0Ms/C253:75 mXcm. The maximum rectified Hall voltage is 375 lV calculated from Eq. (1)with IRF¼1lA. Previously, a large Hall voltage induced by the remnant B-field of a permanent nano- magnet has been demonstrated.33Here, we saturate a nano-magnet having low anisotropy energy using current-induced SOT and apply the saturation field to the Hall layer. We have analyzed the SOT rectifier using experimentally bench- marked34SPICE models for the Hall,26the SO,35and the FM36layers, as shown in Fig. 2(a) , which considers both charge and spin transport phenomena within physics-based circuit models. Our SPICE simula- tions consider thermal noise in both the electronic circuit and within the low anisotropy energy magnet. We apply IRF¼I0sinð2pftÞalong the AC leg of the Hall bar with the amplitude I0¼1lA and frequency f¼2.4 GHz, as shown in Fig. 2(b) . The operation of the proposed recti- fier does not depend on the shape of the signal and will efficiently con- vert an alternating signal to a DC signal, as long as the current amplitude is sufficiently greater than Imin. The current generates non- equilibrium spins in the SO material, which applies SOT to the FM. The magnetization dynamics under the SOT was calculated using the s-LLG equation implemented as a SPICE model.34Both the field-like and damping-like torques can be present; however, damping-like torque generated by the nonequilibrium spin current is the dominant compo- nent in the material considered here. Field-like torque in our discussion arises from a current-induced Oersted field, which is very small, but taken into account within our s-LLG simulations. The magnetization of the FM with low-anisotropy energy nicely follows the IRF, due to the strong SOT, as shown in Fig. 2(c) .T h eF M applies a B-field on the Hall layer, which in conjunction with the frac- tion of the IRFflowing in the Hall layer yields a Hall voltage response(Vout) as a function of time ( t), similar to a full-wave rectifier, as shown inFig. 2(d) .N o t et h a t Voutexhibits negative peaks when IRFchanges the sign, which arises due to Iminand switching/response time of the FM. This causes Voutto deviate from the expected ideal case where Vout/jIRFj,s e e Fig. 2(d) , leading to a lowering of the DC voltage from the ideal case. Also, note that the size of the negative peaks varies in our s-LLG simulations due to the stochastic nature of the FM drivenby thermal noise. However, the area of the positive regions is muchlarger compared to the negative region, indicating an average DC,which can charge up a capacitor to /C24210lV, as shown in Fig. 2(e) . V outinFig. 2(d) is noisy due to the presence of thermal noise in the circuit. Such noisy behavior is not visible when a capacitor isconnected at the output, see Fig. 2(e) . We can further enhance the DC voltage strength for a given RF power by connecting multiple devices in series in an array while matching the array impedance with the antenna. For calculation, we consider a WiFi router positioned 5 m from the array as the RF source,seeFig. 3(a) , which transmits P WiFi¼100 mW at f¼2.4 GHz (i.e., wavelength k/C2412:5 cm). The received RF power by the antenna at R¼5 m is calculated using the Friis equation37asPRF¼ðk=4pRÞ2 PWiFi/C25500 nW, which can provide 50 lAr m s ,i ft h ea r r a yFIG. 2. Characteristics of the spin–orbit torque rectifier. (a) Setup and models for simulation. (b) Input RF current ( IRF). (c) The response of the magnetization ( M)t o IRF. (d) Hall voltage ( Vout) showing a response similar to a full-wave rectifier. (e) The rectified Hall voltage can charge up a capacitor. Hall model parameters:26 Rv¼qW=2LtH;Rh¼qL=2WtH;Iv¼VabA=ðqHWÞ, and Ih¼/C0VcdA=ðqHLÞ. SO model parameters: R0¼qL=A;Ic0¼wsohSHVef=q;Is0¼wsohSHVab=q; Rs¼qkssinhðtso=ksÞ=A, and Rsh¼qkscothðtso=2ksÞ=A. Here, ksis the spin diffusion length in the SO.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052408 (2021); doi: 10.1063/5.0036285 118, 052408-3 Published under license by AIP Publishingimpedance is matched to the antenna at 50 X.H e r e ,w eh a v ea s s u m e d isotropic antennas with unity gain. The number of parallel branches in the array can be K¼ffiffiffi 2p /C250lA=IRF/C2571. To match the array impedance to the antenna impedance (50 X), each of the parallel branches can have Ndevices with resistance Rdevin series, where N/C2Rdev=K¼50X. Here, N/C2518 using Rdev¼200X. We can enhance the DC voltage by /C24N/C2Ktimes by adding the DC paths of all the devices in series. In this example, if we can add all the devices in series as shown in Fig. 3(a) , the maximum rectified Hall voltage will be /C24480 mV from the same RF power of 500 nW, which considering the nonidealities can provide a DC voltage of /C24300 mV. We can connect two consecu- tive devices using a capacitor ( /C2410 pF) to reduce the leakage of thegenerated DC within the series-connected devices. To avoid an AC leakage through the DC path between two consecutive parallel branches, we can connect the DC paths using an inductor, which willact as a short circuit for the DC signal. The capacitors and inductors will make the area of the array larger, roughly on the order of /C242m m 2 for the present example. The inductors and capacitors are such that their reactances cancel out, and the SOT rectifiers in the array receive the maximum power from the antenna. The AC path of the SOT recti-fi e rb e h a v e sl i k eal i n e a rr e s i s t o ra n dt h ed e v i c ed o e sn o tc o n t a i na n y internal space charge regions like conventional semiconductor devices. However, parasitics arising from interconnects and contacts can make the impedance matching challenging. An optimized layout to achieve minimal parasitics and analysis with a distributed model 38we leave for as e p a r a t ew o r k . The open circuit output DC voltage for a given input RF power (i.e., S¼Vout=PRF) is defined as the sensitivity of the RF detector. Various semiconductor diodes can offer high zero-bias sensitiv- ity,6,39–41on the order of /C24106lV/lW from an input RF power of /C241 lW.6Recently, magnetic tunnel junction (MTJ)-based diodes reported very high sensitivity,9,10,42on the order of /C24105lV/lWf r o m1 0 0 nW.42However, either an external magnetic or an electric bias is used to enhance the sensitivity of MTJ diodes and zero-bias sensitivity is on the order of 102/C24103. A single SOT rectifier can provide a zero-bias sensitivity of 750 lV/lW and an optimized array can provide 4:8/C2105lV/lW, from an input RF power in the range of 500 nW, seeFig. 3(b) . Such a high zero-bias sensitivity can result in a low noise equivalent power (NEP) given by NEP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4kBTRdevp=S,39,40where kB is the Boltzmann constant and Tis the temperature. For a single SOT rectifier, the expected noise-equivalent power is approximately2.4 pW/ffiffiffiffiffiffi Hzp (calculated using R dev¼400X). The array is matched at Rdev¼50Xand the expected NEP is approximately 1.9 fW/ffiffiffiffiffiffi Hzp . The curvature coefficient of a detector is defined as43 c¼d2I dV2/C18/C19/C30dI dV/C18/C19 : (4) For a conventional diode with I¼IS½expðqV=ðmkBTÞÞ /C0 1/C138,w eh a v e c¼q=ðmkBTÞ,w h e r e mis the diode nonideality factor. For Schottky diodes, the theoretical limit is c¼q=kBT,w h i c hi s3 8 . 6 5 V–1at T¼300 K. Backward tunnel diodes39,43have exhibited a chigher than this theoretical limit, on the order of 50 to 70 V–1at zero bias. The proposed SOT rectifier can exhibit high zero bias c, on the order of 104V–1,s e e Fig. 4(a) .F o rv e r yl a r g ei n p u tc u r r e n t( IRF/C29Isat:), the DC voltage of a SOT rectifier is proportional to the current, i.e.,V cd/jIRFj. For a smaller input current, the current–voltage relation exhibits high curvature leading to a high c. This feature is promising for general radio detection applications from weak signals. For a SOT rectifier, I¼Vout=Rcdis the output DC current and V¼IRFRabis the input AC voltage in Eq. (4),w h e r e RabandRcdare the resistances between nodes a and b and c and d, respectively, see Fig. 2(a) . The efficiency of a rectifier is determined by the maximum DC power, PDC ;max, produced from a given RF power, PRF,a s g¼PDC ;max PRF¼1 4V2 cd=Rcd I2 abRab; (5) where VcdandRcdare the open circuit voltage and source resistance between nodes c and d [see Fig. 2(a) ].Iab¼I0=ffiffiffi 2p is the rms value of FIG. 3. (a) An array of SOT rectifiers designed to provide a large DC voltage from an RF power, while matching the antenna impedance. (b) Zero-bias sensitivity and comparison with existing devices.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052408 (2021); doi: 10.1063/5.0036285 118, 052408-4 Published under license by AIP PublishingIRFand Rabis the resistance between a and b nodes. Here, Rab ¼qL=Wtdis the longitudinal resistance, Land Ware the device length and width, and td¼tsoþtHis the total device thickness. We extract equivalent Rcd¼2Rvþ2R2 v=Rhfrom the model in Fig. 2(a) , using wye-delta transformation when a and b nodes are short circuited. Here, Rv¼qW=2LtHandRh¼qL=2WtHare the vertical and horizontal resistors in the model. We use Eq. (1)to estimate Vcd¼ÐT 0ðqH=tHÞjIRFjdt¼2 pðqH=tHÞI0.E q u a t i o n (5)becomes g¼ffiffiffi 2p qH pq !2 td tH1 1þW2=L2: In this example, we design the device such that it provides a high effi- ciency while matching the antenna impedance, and we have chosent d/C252tHandW/C25L,w h i c hi nE q . (5)gives g¼ðffiffi 2p pqH qÞ2.T h u s ,w e get an efficiency value of /C2471% for the materials and device parame- ters under consideration, which is observed for /C21500 nW RF power. The efficiency degrades for lower input RF power. We compare the efficiency of the SOT rectifier with the conventional semiconductor4,5,7 and magnetic8technologies in Fig. 4(b) . In conclusion, we propose a nanoscale rectifier concept that is promising for general radio detection and, particularly, for harvesting ambient weak radio signals, where conventional rectification fails to operate. We analyze a single device in SPICE using existing materials parameters and show that it can provide 200 lV DC from 500 nW of RF power. A series array of such devices can efficiently enhance theDC voltage to 300 mV while matching the receiver antenna impedance. The expected efficiency is /C2471% at such a low RF power, which makes this nanoscale device promising for powering of the emerging applica- tions such as wearable electronics, self-powered sensors, and implants. This work was supported by the Center for Energy Efficient Electronics Science (E3S), NSF Award No. 0939514.DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1S. Hemour and K. Wu, “Radio-frequency rectifier for electromagnetic energy harvesting: Development path and future outlook,” Proc. IEEE 102, 1667–1691 (2014). 2S. Kim, R. Vyas, J. Bito, K. Niotaki, A. Collado, A. Georgiadis, and M. M.Tentzeris, “Ambient RF energy-harvesting technologies for self-sustainable standalone wireless sensor platforms,” Proc. IEEE 102, 1649–1666 (2014). 3F. Giuppi, K. Niotaki, A. Collado, and A. Georgiadis, “Challenges in energy harvesting techniques for autonomous self-powered wireless sensors,” in 2013 European Microwave Conference (2013), pp. 854–857. 4S. Hemour, Y. Zhao, C. H. P. Lorenz, D. Houssameddine, Y. Gui, C. Hu, andK. Wu, “Towards low-power high-efficiency RF and microwave energy har-vesting,” IEEE Trans. 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J. Chem. Phys. 154, 204102 (2021); https://doi.org/10.1063/5.0050680 154, 204102 © 2021 Author(s).Chasing unphysical TD-DFT excited states in transition metal complexes with a simple diagnostic tool Cite as: J. Chem. Phys. 154, 204102 (2021); https://doi.org/10.1063/5.0050680 Submitted: 17 March 2021 . Accepted: 05 May 2021 . Published Online: 25 May 2021 Federica Maschietto , Marco Campetella , Juan Sanz García , Carlo Adamo , and Ilaria Ciofini COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical Chemistry ARTICLES YOU MAY BE INTERESTED IN More than little fragments of matter: Electronic and molecular structures of clusters The Journal of Chemical Physics 154, 200901 (2021); https://doi.org/10.1063/5.0054222 Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition The Journal of Chemical Physics 154, 194109 (2021); https://doi.org/10.1063/5.0047386 Spin contamination in MP2 and CC2, a surprising issue The Journal of Chemical Physics 154, 131101 (2021); https://doi.org/10.1063/5.0044362The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Chasing unphysical TD-DFT excited states in transition metal complexes with a simple diagnostic tool Cite as: J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 Submitted: 17 March 2021 •Accepted: 5 May 2021 • Published Online: 25 May 2021 Federica Maschietto,1,2,a) Marco Campetella,1,3 Juan Sanz García,1,4 Carlo Adamo,1,5 and Ilaria Ciofini1,a) AFFILIATIONS 1Chimie ParisTech, PSL University, CNRS, Institute of Chemistry for Life and Health Sciences, Theoretical Chemistry and Modelling, 75005 Paris, France 2Department of Chemistry, Yale University, New Haven, Connecticut 06520-8107, USA 3Consiglio Nazionale Delle Ricerche (CNR) SPIN, Area di Ricerca di Tor Vergata, Via del Fosso del Cavaliere 100, 00133 Roma, Italy 4MSME, Univ. Gustave Eiffel, CNRS UMR 8208, F-77454 Marne-la-Vallée, France 5Institut Universitaire de France, 103 Bd Saint-Michel, F-75005 Paris, France Note: This paper is part of the JCP Special Collection in Honor of Women in Chemical Physics and Physical Chemistry. a)Authors to whom correspondence should be addressed: federica.maschietto@yale.edu and ilaria.ciofini@chimieparistech.psl.eu ABSTRACT Transition Metal Complexes (TMCs) are known for the rich variety of their excited states showing different nature and degrees of locality. Describing the energies of these excited states with the same degree of accuracy is still problematic when using time-dependent density functional theory in conjunction with the most current density functional approximations. In particular, the presence of unphysically low lying excited states possessing a relevant Charge Transfer (CT) character may significantly affect the spectra computed at such a level of theory and, more relevantly, the interpretation of their photophysical behavior. In this work, we propose an improved version of the MAC index, recently proposed by the authors and collaborators, as a simple and computationally inexpensive diagnostic tool that can be used for the detection and correction of the unphysically predicted low lying excited states. The analysis, performed on five prototype TMCs, shows that spurious and ghost states can appear in a wide spectral range and that it is difficult to detect them only on the basis of their CT extent. Indeed, both delocalization of the excited state and CT extent are criteria that must be combined, as in the MACindex, to detect unphysical states. Published under license by AIP Publishing. https://doi.org/10.1063/5.0050680 I. INTRODUCTION The prediction of the nature and properties of excited states of transition metal complexes (TMCs) is a fertile research area.1–9Due to their peculiar photochemical and photophysical properties,10–12 organometallic complexes of several metals are nowadays exploited in many different fields, spanning from photovoltaics13–17to medical and biological applications.18–23 However, photophysical and photochemical processes occur- ring in TMC are intrinsically complex and extremely interesting to study, yet difficult to understand based on sole experimentalinvestigations. Indeed, the use of theoretical approaches is crucial to correctly model the behavior of such complexes upon irradiation and to acquire a realistic description of their excited state manifold.1–9,16–18TMCs, though, cumulate most of the complexities inherent to electronic structure investiga- tions such as a high density of low lying electronic states of a different character [from charge transfer (CT) to locally excited (LE)], the presence of nearly degenerate states, and/or that of relevant relativistic effects. A proper description of these compounds is thus a challenging task for any quantum method. J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 154, 204102-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp On the one hand, post-Hartree–Fock methods, such as complete-active-space self-consistent field (CASSCF)24and restricted-active-space SCF (RASSCF) variants25often improved by adding a perturbative correction (CASPT2),26have proven to yield high accuracy in describing both vertical absorption and photochemical behavior of metal complexes,27although somehow limited in their domain of applicability by their computational burden. Moreover, these methods require a selection of a relevant and system-dependent active space, which makes them impractical and of heavy usage for routine applications. On the other hand, density functional theory (DFT) and its time-dependent extension (TD-DFT) benefit from their favorable scaling and cost to accuracy ratio, which has determined their widespread diffusion for the description of metal-based complexes.1 In addition, density rooted approaches limit the user dependency to the choice of exchange–correlation (XC) functional to be used, which, in practice, renders these methods easy to use, although impressively accurate in the description of structural and spectro- scopic properties of such class of compounds, at least for what concerns the electronic ground state and the lowest excited states. Not surprisingly, TD-DFT calculations and the corresponding sim- ulated spectra are increasingly used to support and interpret exper- imental findings, providing useful insights into the nature of the observed transitions and thus providing a precious help to interpret the photophysics of such compounds.5,7,8,16,17,22,23 In most cases, the reliability of the theoretical method is assessed based on the agreement between the computed and the experimentally observed spectra. However, thisexperimental/theoretical spectral matching may result from an accidental compensation of errors or hide some methodological issues.28,29In this respect, it is well known that TD-DFT may fail to describe excitations with a significant through-space charge-transfer (CT) character depending on the exchange–correlation functional used.28,29Indeed, this failure could be related to the self-interaction error,32,33the incorrect asymptotic behavior,34,35and the missing derivative discontinuity36of the exchange–correlation (XC) func- tional used. These errors particularly affect functionals resting on the generalized gradient approximation (GGA)37while being partially cured by the introduction of Hartree–Fock exchange, including global (GH)31and range separated (RSH) hybrids, double hybrids,38,39or purposely tuned hybrids.40 These known limitations of density functional approxima- tions30,31,41,42(DFAs) may lead de-facto to an erroneous descrip- tion of CT excited states and to the prediction of unphysically low lying excited states. This applies with no exception not only to organic push–pull dyes on intramolecular CT excitations, as already largely documented in the literature,29but also to metal- containing complexes.1In the latter case, the errors are even more relevant for the interpretation of their photophysical prop- erties since both CT and LE states very close in energy may be present but unequally affected by the limitations of the DFA used. To diagnose these problems, a number of descriptors have been developed over the years, with the double role of identifying the nature of the excited electronic states and, possibly, evaluating the TD-DFT error, which could be intrinsically related to it. An FIG. 1. Structures of the Ru(II) complexes investigated in this study together with the representation of the bidentate and tridentate ligands involved. J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 154, 204102-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp overview of these indices is outside the scope of the present paper; here, we only recall that they have been successfully and widely applied in the literature and that they are nowadays available in several codes.43–51 A less explored field concerns the validation of such indices for TMCs. Indeed, in this class of compounds, a rich variety of excited states showing a different degree of locality exists, includ- ing, for instance, metal-centered (MC), ligand-centered (LC), metal- to-ligand-charge transfer (MLCT), ligand-to-metal-charge-transfer (LMCT), intra-ligand-charge-transfer (ILCT), and ligand-to-ligand- charge-transfer (LLCT) states. Most often, these excited states are identified by inspection of the associated natural transition orbitals (NTOs)52or by the use of CT descriptors53–57always assuming that their relative energy is correctly predicted. Nonetheless, this latter point is of crucial importance when aiming to accurately interpret the photophysics and photochemistry of TMC. In order to (partially) fulfill this gap, in the present contribu- tion, we examine the lowest excited-state levels of five octahedrally substituted Ru(II) complexes, shown in Fig. 1. The [Ru(bpy) 3]2+(1) and [Ru(tpy) 2]2+(2) complexes are prototype homoleptic systems, where the Ru atom is coordi- nated with either three bpy (2,2′-bipyridine) or two tpy (2,2:2′, 6′-terpyridine) ligands. The photophysics of these systems has extensively been studied in the literature,58,59and they constitute the skeleton of many of the Ru complexes developed for various applications.20,60,61Three heteroleptic ([Ru(bpy) 2(dppz)]2+(3), [Ru(bpy) 2(tpphz)]2+(4), and [(pbbzim)Ru(tpy-HImzPh 3)]2+(5) are derivatives of the two homoleptic scaffolds, substituted with the dppz (dipyrido[3,2-a:2′,3′-c]phenazine), tpphz (tetrapyrido[3,2-a:2′,3′-c:3′′,2′′-h:2′′′,3′′′-j]phenazine), pbbzim (2,6-bis(benzimidazole-2-yl)pyridine), and tpy-HImzPh3 (4′-[4- (4,5-diphenyl-1H-imidazol-2-yl)-phenyl]-[2,2′:6′,2′′]terpyridine) ligands, respectively, expected to promote the formation of low lying excited states of MLCT character.59,62–64 All these compounds have been previously experimentally characterized,58,59,62–64and they will be used to assess the quality of a diagnostic tool enabling to evaluate the quality of the predicted electronic transitions using three largely applied DFAs. The purpose of this study is, indeed, to propose a diagnos- tic index enabling to spot when TD-DFT fails in the description of excited states and, more generally, to provide a simple strat- egy to assess on the fly the validity of different DFAs, in the con- text of prediction of the spectral properties of TMCs. This index should enable us to disclose the presence of unphysical states and to derive a possible correction of the wrongly predicted transi- tion energies. The present analysis, besides its theoretical interest in evaluating XC functionals performances, may be worthwhile to be considered while designing and optimizing metal complexes based on the outcomes of TD-DFT predictions, especially when targeting charge-transfer or charge-separated excited states for experimental applications. II. ANALYSIS OF VERTICAL EXCITATIONS USING THEMACINDEX As mentioned above, a direct consequence of the use of approximated XC functionals in the framework of TD-DFT is that excitations with substantial through-space CT character may bepoorly described with the consequence that their excitation ener- gies can be severely underestimated. These excited states are the so-called “ghost” states. They mostly correspond to through-space electronic transitions with a significant CT character and negligi- ble hole–particle overlap, which appear at very low energies.30,31 The presence of such spurious states, while not necessarily affect- ing the spectral shape, can affect the interpretation and prediction of the photophysical behavior of molecular systems, implying, for instance, that an energetically higher bright state could decay non- radiatively into the lower CT states, leading to an electron-transfer quenching of the excited state fluorescence. In this respect, the lim- itations of a selected DFA have an impact that is much larger than its numerical performances (i.e., the error in the computed transi- tion energies with respect to a given reference), leading to a wrong interpretation of the photophysical behavior of the system under investigation. These ghost states generally have vanishing oscillator strength. Although energy underestimation and negligible oscillator strength can be used as criteria to detect such unphysical states, these indicators may be not sufficient to unambiguously identify them as “ghosts.” Furthermore, as the CT error strongly depends on the cho- sen DFA, the degree by which these are mistaken varies significantly: a given DFA for a given system may provide a correct description and energy estimation of excited states with limited CT character while incorrectly underestimate the energy of other, longer range, CT states. To address the difficulty in identifying and eventually correct excited states of different nature in the TD-DFT framework, we have recently introduced a diagnostic index, the so-called MAC.65The MACindex is based on the formula originally proposed by Mul- liken66to estimate a lower bound to the energy associated with an intermolecular CT state ( ωCT) in the case of donor to acceptor dyad and lately discussed in the framework of TD-DFT by Head-Gordon and collaborators,30 ωCT=IPD−EA A−1 R. (1) Here, R represents the distance between the electron donor/acceptor moieties, IPDis the ionization potential of the donor, and EA Ais the electron affinity of the acceptor. 1 /Ris thus the electrostatic inter- action between the hole and the electron (in atomic units) after a donor-to-acceptor CT excitation. In the original MACindex,65these quantities were computed from Kohn–Sham (KS) orbital eigenvalues ( IPDand EA D) and the hole–electron distance (R) was estimated by the DCTindex.45This index provides a measure of the hole–electron distance from the barycenters of the charge densities corresponding to an increase and decrease in the electron density upon electronic excitation. A more extensive description of this index can be found in the supplementary material. As such, it represents the effective (aver- age) charge/hole distance produced in an electronic excitation and, therefore, can be used, as in the original MACformulation, to provide a realistic estimate of the electrostatic contribution to the electronic transition energy [the last term of Eq. (1)] both in the case of intra- and intermolecular electron transfer. In order to apply the MACindex to TMCs and aiming at ame- liorating the estimate of the first two terms of Eq. (1) with respect to our previous formulation,65the following expression will be used J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 154, 204102-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp here: MAC=∑iac2 ia(εDFA−HF a−εDFA−HF i ) ∑iac2 ia−1 R. (2) Indeed, to improve the original MACindex, for each computed excited state, IPDand EA Aare estimated by a weighted average of the eigenvalues of the contributing KS molecular orbitals being computed at the Hartree–Fock level, on top of the converged DFT orbitals. This allows one to correct the underestimation of TD-DFT- virtual orbital eigenvalues. Indeed, by this single HF cycle, we miti- gate the known KS-bias, which compromises the energy of the empty orbitals with excessively negative correlation energy values (in an absolute sense), artificially lowering their values. The improved hole and particle orbital energies relative to the transition considered are denoted here as εDFA−HF i andεDFA−HF a , respectively. The weights, cia, are the Configuration Interaction (CI) coefficients obtained as solutions of TD-DFT equations.67The procedure is easy and fast as it merely requires to perform an estimation at the HF level of the molecular orbital energies on top of the converged Kohn–Sham orbitals. As in the case of the previous formulation,65by construction, theMACindex depends on the quality of ground and excited state densities used to compute DCT. Concerning the excited states elec- tronic density, recent publications have highlighted the importance of density relaxation effects in TD-DFT,68,69which can be estimated by employing the so-called Z-vector formalism.70With this type of computation, one performs a post-linear response treatment of the excited state calculation, resulting in the addition of an off-diagonal contribution to the transition-density, which accounts for the redis- tribution of the electron density in the orbital space. The resulting quantity is the so-called “relaxed” difference density matrix, which unlike the “unrelaxed” density matrix accounts, at least partially, for the orbital relaxation following the hole/particle generation. The projection in real space of the relaxed and unrelaxed difference den- sity matrices defines what we usually refer to as relaxed and unre- laxed densities.28,71Both density definitions can be used to evaluate DCTand yield the corresponding indicesRDCTandUDCT. The for- mer reflects the spatial extent associated with a given transition, where the electronic density is allowed to gradually change and adapt to the final configuration, while the latter reproduces the CT dis- tance, measured directly upon vertical excitation. In other words, the relaxed (RDCT) counterpart, unlike its unrelaxed (UDCT) coun- terpart, accounts for the redistribution of the electronic charge due to the excitation. Clearly, this double definition of DCTleads to two distinct MACindices:UMACandRMAC. The effect of the use of these two variants is detailed below. The MACindex, as defined in Eq. (2), defines a lower bound to the excitation energy associated with a given charge transfer transi- tion and can be used to diagnose the presence of unphysically pre- dicted low-lying excited states due to artifacts of the XC functional applied. A given TD-DFT transition will be, therefore, identified as a ghost if its energy is lower than the corresponding MACindex, while proper CT excitations will have an energy greater than the MACvalue. Thus, for each electronic transition, two cases can be distinguished, ETD−DFT<MAC→unphysical CT state, (3) ETD−DFT>MAC→real CT state. (4)Clearly, it follows from Eq. (1)—and from its equivalent formula- tion in Eq. (2)—that the condition in Eq. (4) is always fulfilled in the case of transitions of local character that are possessing a negligible charge-transfer character and for which the R (or the DCT) value is close to zero. Indeed, the Mulliken formula is relevant only for tran- sitions of CT-type, and in the following, we will focus on such kinds of excitations. In the case of states predicted at unphysical low energy, that is, states that fulfill the criteria of Eq. (3), an additional distinction is made here: those with zero oscillator strength will be classified as “ghost” states and labeled “G.” By contrast, CT excitations that have a non-vanishing oscillator strength, but still appear too low in energy, will be referred to as “spurious” states and labeled “S.” III. COMPUTATIONAL DETAILS All electronic structure calculations were performed using the Gaussian 16 quantum package.72Starting from the crystallographic x-ray structures, complexes 1–5 were optimized in acetonitrile using the Polarizable Continuum Model (PCM)73following the same protocol, as described in Ref. 74. We employed DFT using the standard hybrid functional B3LYP75,76with the 6-31G(d,p) basis set with one set of d polarization functions for the second-row elements and a set of p polarization functions for the hydrogen atoms.77For the ruthenium atom, we used the LANL08 effective core potential (including 28 core electrons) and associated basis set for valence electrons.78Additionally, we performed vibrational fre- quency calculations—at the same level of theory—to ensure that all structures correspond to minima. Although the assumption of con- sidering in all cases B3LYP structures can induce bias and errors when aiming at comparing with experimental data, in the present case, it has the advantage of allowing to directly compare the behav- ior of the different functionals in predicting vertical excited state energies and nature, which is indeed the main goal of the present paper. The UV–Vis absorption spectra of all complexes were then computed using the TD-DFT approach and three different function- als, namely, the hybrid functional B3LYP,75,76the long-range cor- rected CAM-B3LYP79functional, and the hybrid functional PBE0,80 using the previously defined basis set. In TDDFT calculations, only spin-allowed (singlet–singlet) transitions were considered. Excited states were also computed using the Configuration Interaction with Singles (CIS) excitation approach since as it includes 100% HF exchange, it follows that no unphysical states should be found. As shown by the tables reported in the supplementary material, indeed, no ghost states are detected by the MACanalysis at the CIS level, thus providing a proof of the reliability and consistency of this index. Nonetheless, due to the incomplete treatment of correlation effects, CIS is normally overestimating transition energies and the corre- sponding transition energy values are here only considered as a ghost-free reference more than reference to assess the accuracy of TD-DFT excitation energy values and will thus not be discussed in detail in this text. The natural transition orbitals (NTOs) relative to the 30 low- est transitions were also computed to inspect the character of the transitions. These are collected in Figs. SI.3–SI.7. Finally, to investi- gate the presence of methodological artifacts, related to the choice of the exchange–correlation functional, we computed the MAC index J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 154, 204102-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp on the lowest 30 transitions of each complex, using both unrelaxed and relaxed electron densities. Accordingly, the U and R labels, in the following, will denote that the corresponding MAC value is obtained using either of the two density definitions. This analysis was performed using an in-house developed tool. For complexes 3,4,and 5,difference density plots, computed using relaxed densities for each transition, are collected in Figs. SI.8–SI.10. IV. RESULTS AND DISCUSSION In this work, we will show how the diagnostic index defined as in Eq. (2) can provide relevant insights to detect ghost and spu- rious CT states in TMCs focusing on two global hybrid (B3LYP and PBE0) and one range-separated (CAM-B3LYP) functionals, all widely employed in recent literature for TMC, and analyzing the lowest lying excited states of the five selected metal complexes depicted in Fig. 1. Figure 2 shows the computed and experimental absorption pro- files of the five selected Ru(II) compounds in acetonitrile. When π-accepting ligands, such as polypyridyl, are coordinated to Ru(II), the complex exhibits intense singlet–singlet MLCT transitions in the visible region.60This behavior is typical for both tris-bidentate and bis-tridentate complexes, although a slight red-shift of the max- imum absorption wavelength is often observed for the more rod- like bis-tridentate ones.18The intense charge transfer MLCT tran- sitions are also more easily formed in the presence of ligands ableto delocalize density at the excited state far from the metal cen- ter, that is, to give rise to excited states with larger CT distance. Thus, in Fig. 2, the complexes bearing ligands with higher electron- accepting capabilities show more intense and red-shifted MLCT bands. Clearly, metal-centered and ligand-centered transitions can also be identified in the electronic spectrum of each of the com- pounds. A list of all computed transitions and associated assign- ments is provided in the supplementary material in Tables SI.1–SI.5, together with a representation of all NTOs involved reported in Figs. SI.3–SI.7. The absorption profiles of 1, calculated with different methods, are shown in the upper left panel of Fig. 2. In agreement with pre- vious works,12,58the broadband around 400–500 nm corresponds to multiple MLCT transitions—d6(Ru) toπ∗(bpy). Indeed, in this region of the spectrum, the most intense transitions are ascribed to the population of degenerate excited states ES7 and ES8 both in the case of B3LYP (at 424 nm) and PBE0 (at 405 nm), these states being of MLCT character. The same transitions appear higher in energy at 352 nm and correspond to excited states ES5 and ES6—in the case of CAM-B3LYP, in agreement with previous reports, show- ing an overestimation of transition energies at the range separated level in the case of Ru polypyridine complexes.81The natural tran- sition orbitals (NTOs) reported in the supplementary material, Fig. SI.3, and the corresponding computed energy values and oscillator strengths, reported in Table SI.1, support these assignments. Of note, and as expected due to the presence of unsubstituted bpy ligands, although being of MLCT character, these transitions are rather local, FIG. 2. Absorption spectrum of 1,2,3,4, and5computed using different function- als, along with the corresponding experi- mental spectra, retrieved from Refs. 47 and 49–52. The simulated spectra are computed by a Gaussian convolution of the computed vertical transitions using a full-width half maximum value of 0.4 eV. Computational details are provided in the supplementary material. J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 154, 204102-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the charge transfer extent being quite limited as evident from the associated DCTvalues. A qualitatively similar description can be made for complex 2, the first intense transition, independently of the method con- sidered, being predicted to be related to the population of excited state ES5, an MLCT state characterized by a relatively low CT dis- tance and computed at 447, 427, and 382 nm at B3LYP, PBE0, and CAM-B3LYP levels, respectively (see Fig. SI.2). In the case of GHs only, a second transition of comparable intensity also contributes to the first absorption band at 420 and 398 nm, for B3LYP and PBE0, respectively. In agreement with the experimental data, overall, the spec- tral features of complex 3are not significantly different from those of complex 1, the associated bands being calculated slightly red- shifted using both GH functionals and CAM-B3LYP. In this case, for both GHs, three electronic transitions mostly contribute to the first absorption band. These transitions involve the population of excited states ES4, ES9, and ES11 in the case of B3LYP (computed at 455, 423, and 416 nm, respectively) and excited states ES4, ES9, and ES11 (computed at 429, 402, and 393 nm, respectively) at the PBE0 level. In the case of CAM-B3LYP, on the other hand, a very intense tran- sition at 355 nm determines the absorption band maxima, while a shoulder at higher energies appears due to other less intense tran- sitions. All these transitions correspond to MLCT bands although not localized on the same moieties of the molecule (see, for instance, Fig. SI.5). In the case of complex 4, the situation is similar: several tran- sitions of different MLCT types (see Fig. SI.5) but of comparable intensities are computed to contribute to the first absorption bandat 445, 421, and 414 nm at the B3LYP level and 417, 401, and 353 nm at the PBE0 level (see Table SI.4). On the other hand, using CAM- B3LYP, a single dominant transition at 357 nm gives rise to the first absorption band. This leads to redshifted absorption bands in the case of GHs with respect to the RSH. The calculated absorption profile of 5is qualitatively different from the one of complexes 1–4, independently of the DFA consid- ered. The first intense band calculated with both GHs is due to the convolution of several bright transitions of different MLCT charac- ters, the most intense being in both cases related to be population of excited state ES11 [computed at 548 and 530 nm at B3LYP and PBE0, respectively (see Table SI.5)]. Contrary to 1–4, the first intense transition is computed at lower energy at the CAM-B3LYP level, at 576 nm, though showing the same M to the tpy MLCT character. In order to evaluate if the computed spectra may be influ- enced by the presence of spurious states and if the interpretation of the photophysical properties of these compounds may be mis- leading due to the prediction of low-lying ghost states, a diagnos- tic analysis based on the MACindex was performed. Its results are graphically reported in Figs. 3 and 4 and listed in Tables SI.1–SI.5 together with the previously described transition energies, oscilla- tor strength, and character. BothUMACandRMACwere calculated at different levels of theory. The use of unrelaxed densities, rather than the costlier relaxed ones, is indeed particularly convenient in the case of metal complexes, where the number of excited states to analyze is often large. Unrelaxed densities can be computed all at once, in a single TD-DFT calculation, for all desired vertical states at a given geometry, with significant savings in terms of time and resources. Relaxation effects, by contrast, need to be computed FIG. 3. (Left)UDCTand (right)RDCTvalues of the lowest 30 states of compounds 1and2, with respect to their energy computed with B3LYP (orange), PBE0 (blue), and CAM-B3LYP (green). Each state is assigned a symbol: according to the MACdiagnostic, dots for ghost states (G), diamonds for spurious states (S), squares for charge-transfer (CT) states, and pentagons for local excitations (L). J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 154, 204102-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4. (Left)UDCTvalues of the lowest 30 states of compounds 3, 4, and5, with respect to their energy computed with B3LYP (orange), PBE0 (blue), and CAM-B3LYP (green). Each state is assigned a symbol: according to the MACdiagnostic, dots for ghost states (G), diamonds for spurious states (S), squares for charge-transfer (CT) states, and pentagons for local excitations (L). (Right) Histograms displaying the number of excited states of each kind of transition at different levels of theory. separately for each excited state, thus being more computationally expensive but with the advantage of delivering a refined descrip- tion of the density distribution upon transition. As shown in our previous work68where we investigated the impact of the relaxation on CT states for different DFAs, the two approaches are equivalent only when aiming for a qualitative interpretation of the nature of electronic excitations. Indeed, if both relaxed and unrelaxed den- sities can, in principle, be used to compute the MACindex, the latter—unrelaxed densities—will, in general, result in an overesti- mation of the CT character and consequent exaggeration on the number of ghost and spurious states. Therefore, an inexpensive diag- nostic analysis based on unrelaxed densities can be interesting as a preliminary screening but shall be confirmed by computingrelaxation effects for those states that are classified as unphysical in the first place. This becomes clear analyzing Fig. 3 where diagnostic analy- sis based on bothUMACandRMACof the lowest excitations of complexes 1and 2is depicted. Similar energy vs DCTplots for complexes 3,4,and 5can be found in the supplementary material (Fig. SI.13). Each excited state is assigned a label according to its MACvalue and represented using markers of different sizes and types. Transitions possessingUDCTvalues below 2.0 Å are denoted as local (L), while those having DCTvalues CT greater or equal to 2.0 Å are defined as charge-transfer (CT). Using this classification, most MLCT transitions are actually classified as local since they involve density displacement on the ligand atoms closest to the metal J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 154, 204102-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp center. In addition, transitions with excitation energy values lower than the corresponding MACare classified either as ghost (G) states, if their oscillator strength value is below 0.001, or as spurious (S), otherwise. Ghost states, which correspond to unphysical low-lying through-space transitions, are expected to appear mostly in the low energy/high DCTrange. For complexes 1and 2,one would actu- ally expect to find no ghost state even at the GH level. Indeed, the CT character of their excited states is rather limited (with values of UDCTbetween 0.0 and 3.1 Å). As expected, only one state, namely, the third one computed with the B3LYP functional—see Table SI.1—is classified as G when using theUMACindex, but this same state is indeed detected as a local transition using the more refined description provided by theRMACindex. As a result, we can safely conclude that in the case of complex 1,all functionals considered provide a correct description of the excited states due to the limited range of charge transfer. Analogously, also in the case of 2, only very few states are detected to be wrongly predicted when using theUMACindex, two G at the PBE0 level (S10 and S11) and few spurious states at B3LYP, PBE0, and CAM-B3LYP levels—see Table SI.2. All these states, however, appear relatively high in energy, being all above 2.97 eV. Indeed, also in the case of complex 2, when using relaxed densities, no ghost or spurious states are detected in all excited states, show- ing either a local or real CT character, as shown in the right panel of Fig. 3. The difference between the relaxed and unrelaxed densi- ties can be negligible or very important depending on the basis set and system considered as clearly shown by the comparison of left and right panels in Fig. 3, suggesting that unrelaxed densities can be used for a first screening using theUMACindex and that relaxed den- sities could then be used to analyze in further details the real nature of the so-detected G and S states to get a definitive answer on their reliability. The benefits of the MACdiagnostic analysis are most appre- ciable when employed to investigate the electronic transitions of heteroleptic complexes in which substituents of various kinds give rise to transitions of different characters and CT extents. Complexes 3,4,and 5are exemplary in this regard as reported in Fig. 4. Among the MLCTs, some involve the transfer of one electron from the metal center to the ancillary ligands (bipyridine and terpyridine). Others, most represented, involve the main ligand (dppz in 3, tpphz in 4,and pbbzim/tpy-HImzPh3 in 5). Further sub-classification of the MLCT transitions can be made considering the localization of the trans- ferred charge on a specific part of the ligands—see, for instance, the list of structure and abbreviations reported in Figs. SI.1 and SI.2 of the supplementary material. These varieties of transitions also extend to ligand-to-ligand excitations (see, for instance, Figs. SI.5–SI.7 of the supplementary material, which show the ensemble of the NTOs associated with each vertical excitation computed with the different functionals). With the abundance of electronic transitions of different nature, the number of ghost and spurious states increases significantly. As previously discussed for complexes 1and 2, the use ofUMAC allows for a first screening of potentially non-physical states whose real nature may be refined using relaxed densities and, thus,RMAC. Analysis of the data collected in Tables SI.3–SI.5 and Figs. SI.8 and SI.10 shows that for these complexes, very few states differently classified using one or the other definition. In particular, among the270 computed transitions across complexes 3,4,and 5, only 18 are characterized as S or G at theUMAClevel but not atRMACone: only two states (ES16 and ES28) for complex 3are misleadingly com- puted at theUMAClevel as spurious (S) at B3LYP and one (ES14) at CAM-B3LYP and PBE0 levels and three states (ES18, ES21, and ES29) at the CAM-B3LYP level for complex 4. In the case of complex 5,the number of states differently predicted with the two approaches is slightly larger (one ghost at the B3LYP level, six spurious states and one ghost at the CAM-B3LYP level, and three spurious and one ghost at PBE0), but only three states appear at low energies, below 2.2 eV—see Tables SI.3–SI.5. Due to these small differences in the following, we will com- ment only on the descriptors computed using unrelaxed densities since theUDCTindex has the advantage of being computed on the fly without any extra cost. The classification of the lowest 30 excited states reported in Fig. 4 shows that for these systems, the spurious and ghost states all fall into a particular range of DCTvalues, that is, theUDCTwin- dow between 2.2 and 7.5 Å for complexes 3and 4and up to 12 Å for5, in accordance with the increased length of the molecule. Spu- rious and ghost states both appear in the same region of DCTvalues, which prevents the identification of a clear limit of distinction of one or the other category in terms of CT extent only. Most of the states labeled S or G in Fig. 4 appear when applying GH functionals with low percentage of HF exchange, such as B3LYP and PBE0. The use of CAM-B3LYP drastically reduces their presence, especially for com- plexes 3and 4, as shown in Fig. 4—right panel. Using GH, several states are identified as a ghost. We can appropriately comment on them inspecting their density change upon excitation. Consider, for instance, the first excited state (ES1) in complex 3, which, according to the MACanalysis, is assigned as a ghost state when calculated using B3LYP and PBE0 (Table SI.3). The difference in density between the ground and the last excited states—reported in Fig. SI.8—shows that this state is an MLCT in which the excited electron is highly delocalized over the entire dppz ligand, while at CAM-B3LYP level, the third excited state is computed to be more strongly localized on the tpy core of the dppz ligand. Thus, based on the MACanalysis, we deduce that such an over-delocalization observed with GH is an artifact of the level of theory used, implying also that this state erroneously appears too low in energy. Analogous consideration applies to the first excited state (ES1) calculated using GH function- als for complex 4(see Table SI.4 and Fig. SI.9 of the supplementary material). On the other hand, in complex 5, the first ghost state appears higher in energy at both the B3LYP and PBE0 levels. At the latter level of theory, the first ghost is computed indeed at 550 nm (2.25 eV; see Table SI.5). The corresponding different density plots reported in Fig. SI.10 show that at the GH level, this is an MLCT transition, which extends over both pbbzim and tpy-HImzPh3 ligands. The same is also true for the ghost states computed at higher energy corresponding to S17 (at 3.29 eV) and S21 (at 3.35 eV) at the PBE0 level. Overall, all these ghost states have the characteristic feature of being largely delocalized, with the limited hole/particle overlap. Nonetheless, ghost states do not contribute to alter the spec- tral shape. The case of the transitions classified as spurious is more controversial. These transitions correspond to excited states that are only predicted at—unphysically—lower energy but still present a non-zero oscillator strength, thus contributing to the final shape of the absorption profiles. Correcting the energy positioning of these J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 154, 204102-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp last shall thus influence the calculated spectra. Several excitations, mostly among the MLCT and ILCT/LLCT transitions of complexes 3,4,and 5,are classified as spurious according to the MACdiag- nostics. A simple way of correcting the energy of these states is to consider the computed MACvalues as a lower bound estimate for the energy of both ghost and spurious states. Using this assumption, it is possible to recompute the absorption profiles of all complexes. Figure 5 shows the “corrected” profiles—calculated at the PBE0 level—where ghost and spurious states are shifted at higher energy, as determined by their corresponding MACvalue. The same analy- sis carried out at B3LYP and CAM-B3LYP levels is reported in the supplementary material, Figs. SI.11 and SI.12. Independently of the functional, no spectral modification is expected for those complexes not displaying S states, i.e., in the case of1and 2, while larger effects appear for the absorption spectra of compounds 3,4,and 5calculated using GH, which have a larger number of spurious states.Let us consider, for instance, complex 3: at the PBE0 level, as previously mentioned, three transitions, corresponding to popula- tion of excited states 4, 9, and 10 and computed at 428, 401, and 393 nm, respectively, significantly contribute to the first absorption band, having oscillator strength values ranging between 0.15 and 0.23. Based on the MACanalysis (performed both using unrelaxed and relaxed densities; see Table S1.3), two of these excited states cor- respond to spurious states (ES4 and ES10). Both states correspond to MLCT transitions involving a large delocalization of the excited electron over the dppz ligand (see Fig. SI.5 of the supplementary material) and a sizable DCTvalue (larger than 3.1 and 2.4 Å, respec- tively). The MACindex analysis indicates that indeed they should appear at higher energies: in the 218–278 nm range, ES4, and in the 328–348 nm range, ES10, based on the unrelaxed and relaxed correc- tions, respectively. Clearly, both these M ACvalues are a lower bound to transition energies, and as such, they are intrinsically approximate and further errors are associated with the use of Eq. (2) to estimate FIG. 5. Absorption spectrum (solid lines) of 1,2,3,4, and 5computed at the PBE0 level of theory—see the supplementary material, along with the corresponding experimental spectra (black dotted line), retrieved from the literature.47,49–52 The dotted lines correspond to corrected spectra where the spurious CT excitations are translated in the spectra at the wavelength corresponding to their MACindex value. J. Chem. Phys. 154, 204102 (2021); doi: 10.1063/5.0050680 154, 204102-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp them. Correcting the spectra on their basis determines the absence of an asymmetric tail in the first band and a larger contribution to the second and third band in the spectra. The situation is exactly the same in the case of the spectrum predicted at the B3LYP level (see Table SI.3 and Fig. SI.5), two of the three electronic transitions contributing to the first band also involving spurious states (ES4 predicted at 455 nm and ES10 at 418 nm). Again, in this case, these are two MLCT transitions show- ing a large delocalization over the dppz ligand and high DCTvalues that, based on the M ACindices, should appear at higher energies (in the 206–257 nm range and in the 452-436 nm range, respectively). On the hand, at the CAM-B3LYP level, no spurious transitions are found among the ones contributing to the spectra in the UV–Vis range; thus, no correction is applied. Of note, by construction, cor- rections, if present, always determine a blueshift of the transition energies. The situation for complex 4is qualitatively similar: of the three transitions contributing to the first band computed at the PBE0 level (at 417, 401, and 353 nm), only the first one corresponds to a spuri- ous state (see Table SI.4) and should appear in the 241–300 nm range (see Table SI.4). Once again, the transition corresponds to a state (ES4) characterized by a high DCTand a significant delocalization on the tppz ligand (see Fig. SI.6). A similar analysis and conclusion can be drawn for B3LYP computed transitions, while using CAM- B3LYP, the most intense transition at 357 nm, which gives rise to the first absorption band, is not a spurious state and while also possess- ing an MLCT character (see Fig. SI.6), it is actually involving only the tetraazatriphenylene moiety (Tat see Scheme 2 of the supplementary material) of the tppz ligand and is thus showing a smaller CT character ( DCTof 1.5 Å; see Table SI.4). Interestingly, while one could expect that due to their lower energies, the lowest transitions of 5could be related to spurious states, this is actually not the case. Indeed, as also shown by the right column of Fig. 4, the number of spurious states of 5,at both the GH and RSH levels, is very limited compared to compounds 3and 4, despite the larger size of complex 5. Indeed, many of the com- puted transitions while possessing an MLCT character (see Fig. SI.7) are actually characterized by relatively low CT distances (see data in Table SI.5) and are therefore defined as local (L) in Fig. 4. As a consequence, the most intense transition contributing to the first absorption band at both B3LYP and PBE0 (that is the one associated with ES11) is not involving a spurious state as it corre- sponds to an MLCT transition with a very limited CT character and localized on the tpy ligand. The same holds for CAM-B3LYP. There- fore, for this system, the first band is not affected by the correction unlike the second peak in the spectrum (predicted around 400 nm), which results from several closely lying spurious states, presenting larger CT values and involving more delocalized orbitals. V. CONCLUSIONS We have shown how, in the case of simple TMC, the pres- ence of ghost and spurious states cannot be excluded using hybrid functionals and may affect the interpretation of both photophysical and photochemical processes. A diagnostic analysis tool, the MAC index, that can be used for their detection and correction has been proposed and proven effective to assess the quality of the excited states also in the case of TMCs. The analysis performed shows thatspurious and ghost states can appear in a wide spectral range and that it is difficult to detect them only on the basis of the CT length or the degree of delocalization of the state. Indeed, neither very delocal- ized nor charge transfer states are necessarily spurious or ghost, this criterion alone being neither a necessary nor a sufficient condition to identify a state as spurious. An index, as the MACindex, combining information on both the degree of locality and CT, can thus provide relevant insights to detect unphysical CT states. Of note, the evaluation of the MACindex is rather computation- ally inexpensive. We based our discussion on unrelaxed densities since the classification of the excited-state levels can be made on the fly and for all transitions at once in such a case. Relaxed densities provide, nonetheless, a refined description of excited states possess- ing a sizable CT character.68Using a suitably large basis set reduces the difference between the descriptions provided using relaxed and unrelaxed densities.68Nonetheless, since relaxation effects are also expected to be dependent on the system considered, a careful case- by-case examination based on relaxed density of all the spurious and ghost states predicted at the unrelaxed density level is strongly encouraged, especially when aiming at using the MACvalues to correct the computed spectra. SUPPLEMENTARY MATERIAL See the supplementary material for tables reporting all raw rata computed with the different strategies for all compounds, including excitation energies, oscillator strength fosc,UDCTandRDCT,UMAC andRMAC, and relative labels of the lowest 30 excited states; a selec- tion of natural transition orbitals and difference density plots; and absorption spectra andRDCTvs energy, as referenced in the main text. ACKNOWLEDGMENTS I.C., F.M., M.C., and J.S.-G. acknowledge the European Research Council (ERC) for funding (ERC Consolidator Grant STRIGES to I.C., Grant No. 648558). M. Casida is acknowledged for interesting discussions during the CECAM workshop, Theo- retical and Computational Inorganic Photochemistry: Methodolog- ical Developments, Applications and Interplay with Experiments, Toulouse (France). DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1C. J. Cramer and D. G. 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5.0054333.pdf

Surface plasmon enhanced photoluminescence of monolayer WS 2on ion beam modified functional substrate Cite as: Appl. Phys. Lett. 118, 263103 (2021); doi: 10.1063/5.0054333 Submitted: 16 April 2021 .Accepted: 9 June 2021 . Published Online: 30 June 2021 Lingrui Chu,1 Ziqi Li,1,2,a) Han Zhu,1Rang Li,1,3,a) Feng Ren,4 and Feng Chen1,a) AFFILIATIONS 1School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China 2Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore 3Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, Dresden 01328, Germany 4Department of Physics, Center for Ion Beam Application and Center for Electron Microscopy, Wuhan University, Wuhan 430072,China a)Authors to whom correspondence should be addressed: ziqi.li@ntu.edu.sg ;r.li@hzdr.de ; and drfchen@sdu.edu.cn ABSTRACT Developing efficient methods for boosting light–matter interactions is critical to improve the functionalities of two-dimensional (2D) transitio n metal dichalcogenides toward next-generation optoelectronic devices. Here, we demonstrate that the light–matter interactions in tungsten disul - fide (WS 2) monolayer can be significantly enhanced by introducing an air-stable functional substrate (fused silica with embedded plasmonic Ag nanoparticles). Distinctive from conventional strategies, the Ag nanoparticles are embedded under the surface of fused silica via ion implantatio n, forming a functional substrate for WS 2monolayer with remarkably environmental stability. A tenfold photoluminescence enhancement in WS 2 monolayer has been achieved due to the plasmonic effect of Ag nanoparticles. This work offers a strategy to fabricate the plasmon-2D hybridsystem at low cost and large scale and paves the way for their applications in optoelectronics and photonics. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0054333 The demand for the next-generation optoelectronic and photonic devices has triggered intensive research in two-dimensional (2D) tran-sition metal dichalcogenides (TMDs) due to their extraordinary optical response at visible and near-infrared (NIR) band. 1–3However, the rel- atively low light absorption and emission efficiency restricts their prac- tical applications in phototransistors, sensors, etc. A number of effective strategies have been proposed to enhance the interactions of light in layered materials, such as chemical treatment,4–6optical wave- guides,7,8and localized surface plasmons.9–21One of the most promis- ing strategies is to utilize localized surface plasmon resonance (LSPR) for enhancement of the light–matter interactions in 2D materials. Within the wavelength band of the resonance, the collective oscillation of the conduction electrons in metallic nanoparticles (NPs) or nano- structures significantly enhance the optical response at the nanoscale vicinity to achieve stronger light–matter interactions. Taking the advantages of LSPR effect, giant photoluminescence (PL) enhancement has been demonstrated with diverse patterned nanostructures produced by nanofabrication techniques, such aselectron-beam lithography (EBL).17,18Apart from nanostructures, dis- persion of NPs using chemical approaches is also an effective route to realize PL enhancement.22,23Nevertheless, the stability of PL enhance- ment may not be guaranteed with the nanostructures or NPs exposing in surrounding environment. In addition, to fabricate large-area, uni- form nanostructures at low cost is crucial for practical applications ofplasmon-2D TMDs hybrids. 24,25Therefore, exploring new strategies to address these challenges is a long-sought goal for 2D optoelectronic devices. As a powerful and unique tool to modify the properties ofmaterials, ion implantation has been widely applied in a variety offields, including production of large-scale integrated circuits, fabrica- tion of diverse waveguide structures, 26fabricating NPs in dielectrics,27 modifications of layered materials,28–31etc. Remarkably, large-scale (up to 4-in.-diameter) wafers of lithium-niobate-on-insulator (LNOI) have been successfully manufactured and commercially available,32,33 enabling construction of groundbreaking on-chip photonic devices toward applications in telecommunications, light manipulation, and quantum science and technology.34By using ion implantation, Appl. Phys. Lett. 118, 263103 (2021); doi: 10.1063/5.0054333 118, 263103-1 Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplplasmonic NPs are embedded under the surface of dielectric substrates up to wafer-scale, exhibiting excellent environmental stability, and also meanwhile change the optical properties of dielectrics significantly. The dielectrics with embedded plasmonic NPs have shown great potentials as functional substrates to achieve enhanced light–matter interactions for a broad variety of 2D materials. In this work, we propose a Ag NPs-tungsten disulfide (WS 2) hybrid system and demonstrate a tenfold PL intensity enhancement of WS 2monolayer on fused silica (SiO 2) substrate with embedded Ag NPs structures in a centimeter-scale area. The embedded Ag NPs in functional substrate supplies a promising material platform to achieve enhanced performance in TMDs-based optoelectronic devices at low cost and large-scale. Figure 1(a) schematically shows the fabrication of embedded Ag NPs in fused silica. The Agþions at an energy of 160 keV are implanted into SiO 2wafer with the surface area of 1 /C21c m2(by the ion-implanter LC22-1C0-01). The ion fluence is 1 /C21017ions/cm2 that satisfies the criteria of supersaturation of Agþions in SiO 2.A f t e r the processes of nucleation, growth, and Ostwald ripening, the discrete Agþions spontaneously aggregate to form NPs. To characterize the morphology of Ag NPs, cross-sectional high-resolution transmission electronic microscopy (HRTEM) and element mapping is imple- mented by Tecnai G2 F20 S-Twin (FEI) at an accelerating voltage of 200 kV. A quasi-dual layer structure is observed from HRTEM image and element mapping in Fig. 1(b) . The Ag NPs are mainly divided into two layers with narrow size distribution: shallow layer (layer 1) and deep layer (layer 2).35For the shallow layer, the depth of smallN P si sw i t h i n1 5n m ,w h i l ef o rt h ed e e pl a y e r ,t h el a r g eN P sa r ea tt h e depth of 80 nm. As displayed in Figs. 1(b) and1(c), small Ag NPs with diameter around 5 nm are mainly distributed in shallow layer, whilelarge Ag NPs are mainly distributed in a deep layer with diameter around 40 nm. The size distribution of embedded Ag NPs extracted from HRTEM image is presented in Fig. 1(d) . The selected area electron diffraction (SAED) analysis, which is depicted in the inset of Fig. 1(c) , further confirms the formation of Ag NPs. Two marks of dif- fraction pattern in SAED analysis can be assigned to (111) and (200) planes of Ag with lattice spacing of 0.234 and 0.204 nm, respectively.Fused silica is chemically inert, has high working and melting temper- atures, and exhibits a low coefficient of thermal expansion, which could well protect the embedded Ag NPs for long time preservation. 27 The environmental stability of functional substrate could be guaran-teed. After the fabrication of embedded Ag NPs, single-crystalline WS 2 monolayer flakes are grown on sapphire substrates by chemical vapor deposition (CVD)36using WO 3powder as a precursor with an evapo- ration temperature close to 1000/C14C. Then, by wet transfer method, WS 2monolayers are transferred on SiO 2substrate (WS 2:SiO 2)a n d SiO 2substrate embedded with Ag NPs (WS 2:AgNPs:SiO 2), respec- tively. Figure 1(e) presents the extinction spectra of three fabricated samples by a UV-Vis-NIR spectrophotometer (Agilent, Carry 5000). Two distinct extinction peaks at the wavelength of 420 and 620 nm are o b s e r v e di nA gN P se m b e d d e dS i O 2substrate (AgNPs:SiO 2). Due to the interactions between Ag NPs, a new resonance peak (620 nm)appears in Ag NPs with a high packing density. 35,37For sample WS 2:SiO 2, two peaks in the extinction spectral signature arise from FIG. 1. (a) The schematic of Agþion implantation. (b) HRTEM image of Ag NPs in SiO 2substrate (left side) and mapping of Ag element (right side). (c) The HRTEM image of Ag NPs. Inset is the SAED pattern of Ag NPs. (d) Size distribution of Ag NPs embedded in SiO 2substrate. (e) Experimental extinction spectra of WS 2:SiO 2, AgNPs:SiO 2, and WS 2:AgNPs:SiO 2.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 263103 (2021); doi: 10.1063/5.0054333 118, 263103-2 Published under an exclusive license by AIP Publishingspin–orbit coupling of the direct bandgap in WS 2monolayer, labeled as A, B excitons. Extinction peaks at /C24615 (A excitons) and /C24517 nm (B excitons) from WS 2monolayer in sample WS 2:SiO 2and WS 2:AgNPs:SiO 2are in agreement with literature values.38 The PL excitation in Ag NPs-WS 2hybrid structure is schemati- cally presented in Fig. 2(a) . To demonstrate the enhanced light–matter interactions in WS 2monolayer on modified SiO 2, PL spectroscopy and PL spatial mapping of fabricated samples are performed in theconfocal microscope configuration using a 532 nm laser with a powerof 130 lW. The excitation laser is focused on the WS 2layer, and the laser spot size on the sample surface is about 2 lm. All the compara- tive tests are carried out under the same conditions. As presented in Fig. 2(b) , due to the LSPR effect, the PL intensity of WS 2:AgNPs:SiO 2 is about one order of magnitude higher than that of WS 2:SiO 2.T h eP L peak at /C24620 nm (2.0 eV) is in accordance with the direct A exciton transitions in WS 2monolayer. The same peak position at 620 nm in PL spectra of two samples further certificates that two pieces of WS 2 are monolayers, considering the fact that distinct redshift of emission peak will occur in WS 2bilayers or multilayers.39The inset of Fig. 2(b) presents the measured PL intensity of WS 2:SiO 2and WS 2:AgNP:SiO 2 as a function of excitation power density. The highest applied excita- tion power intensity is far below the saturation of WS 2monolayer and hybrid nanostructure. Large-scale PL enhancement is achieved in WS 2 monolayer flakes on the ion beam modified SiO 2substrate with anarea of 1 /C21c m2.Figures 2(c) and2(d) present the PL mappings of two monolayer flakes on SiO 2and AgNPs:SiO 2substrates. The aver- aged PL intensity of WS 2:AgNPs:SiO 2is increased by about one order of magnitude compared with WS 2:SiO 2, indicating relatively uniform PL enhancement by ion beam modified substrate. The enhancementof PL intensity in WS 2:AgNPs:SiO 2remains unchanged after four- month preservation at room temperature, exhibiting great environ- mental stability. By varying the excitation wavelength at 473 nm, a 14-fold PL enhancement is observed in WS 2:AgNPs:SiO 2inFig. 3(a) ,w h i c hi s attributed to stronger local field enhancement at a different excitation wavelength. The Raman measurement is also carried out by the same configuration in PL tests. Figure 3(b) displays the Raman spectra of all hybrid structures. Two characteristics Raman modes of WS 2mono- layer are observed at 357 and 417.4 cm/C01, which suggests that WS 2 is monolayer in nature.40The Raman signal intensity of A 1gand E1 2gmodes in WS 2:AgNPs:SiO 2is enhanced by a factor of 9. The functional substrate may be used a surface-enhanced Raman spec-troscopy for 2D material characterization, 41where two characteris- tics modes show no blueshift or redshift with the contact ofsubstrate. By harnessing plasmon excitations, the Raman signal of the 2D materials is enhanced. The Raman intensity enhancement factor is proportional to jEj 4/jE0j4,w h e r e E0represents the incident electric field and Erefers to the electric field around the embedded FIG. 2. (a) The schematic of PL excitation in Ag NPs-WS 2hybrid structure. (b) Measured PL spectra of sample WS 2:SiO 2and WS 2:AgNPs:SiO 2. Inset is the PL intensity of two samples as a function of excitation power density. (c) and (d) are measured PL mappings of sample WS 2:SiO 2and WS 2:AgNPs:SiO 2, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 263103 (2021); doi: 10.1063/5.0054333 118, 263103-3 Published under an exclusive license by AIP PublishingNPs.42,43The averaged near field enhancement factor ( jEj/jE0j)a t the substrate surface could be estimated about 1.7 at 473 nm. Aneightfold enhancement of Raman signal in WS 2:AgNPs:SiO 2is also observed with 633-nm laser excitation, demonstrating the LSPRinduced near field enhancement at emission wavelength (Fig. S1),which is essential for PL enhancement. 44 The PL enhancement is determined by two main elements: LSPR enhanced excitation rate (i.e., absorption of pump laser) and emissionefficiency (i.e., Purcell factor). The overall PL enhancement factorcould be approximately given by 45 IEF/C25Iexc/C2Iemi; (1) where IEFis the overall PL enhancement factor; Iexcand Iemirepresent excitation rate enhancement and emission efficiency enhancement,respectively. Stronger local field contributes to both excitation rateenhancement and emission efficiency enhancement. For excitationrate enhancement, the absorption efficiency is proportional to jEj 2;Iexc could be expressed as45 Iexc/jEj2=jE0j2: (2) For emission efficiency enhancement, the intensified local field and increased density of states will lead to large changes in thespontaneous emission rate. 44From Eq. (1), the emission efficiency enhancement factor could be estimated as about 5 inWS 2:AgNPs:SiO 2. To obtain a detailed understanding about the enhanced PL in WS 2:AgNPs:SiO 2, we use COMSOL MultiphysicsVRprogram to simulate the LSPR effect in embedded Ag NPs. Two layers ofembedded NPs may contribute to the LSPR induced PL enhance-ment: Ag NPs in a shallow layer (small NPs, especially in depthwithin 5 nm under the SiO 2substrate surface), and Ag NPs in a deep layer (large NPs, embedded in depth at 80 nm). Near electricfield distributions in the y–zplane around small NPs and large NPs are obtained at the excitation wavelength of 473, 532, andemission wavelength of 620 nm, respectively (see more simulationdetails in supplementary material ). For the large Ag NPs, the nearfield around NPs will decay dramatically within 20 nm, which does not exhibit near field enhancement at substrate surface (Fig. S2).Experimentally, the embedded depth of large NPs could be dynam-ically tuned to 100 nm at ion energy of 200 keV, while Ag NPs in ashallow layer is barely influenced. 46With further PL spectroscopy measurements, the PL enhancement in WS 2monolayer keeps unvaried on the ion beam modified functional substrate with adeeper embedded depth of large Ag NPs, confirming that largeNPs in a deep layer do not play significant roles in PL enhance-ment (Fig. S3). In the shallow layer, the small NPs could be isolated or in proximity with each other, depending on the value of ion fluencein the fabrication process. From the HRTEM image [ Fig. 1(c) ], we observe that the neighboring Ag NPs in the shallow layer are inproximity due to the high packing density (i.e., in the case of ionfluence of 1 /C210 17ions/cm2). We may take a Ag NPs array as the model to simulate near field distributions. For Ag NPs in proxim-ity with each other, near field enhancement around NPs are simu-lated in the y–zplane at excitation wavelength (473, 532 nm) and emission wavelength (620 nm), as displayed in Figs. 4(a)–4(c) .T h e calculated field enhancement factors at the substrate surface areshown in Figs. 4(d)–4(f) . We could observe distinguished near field enhancement at 473, 532 , and 620 nm, which agrees with theobtained experimental results. For isolated NPs, the interparticledistance is set as 10 nm to avoid interactions between NPs. Thenear field distributions in the y–z plane are shown in Figs. 4(g)–4(i) . Near field enhancement factors are also calculated at the substrate surface [see Figs. 4(j)–4(l) ]. Different from Ag NPs in high packing density, there is distinctive near field enhancement at473 nm, while nearly no field enhancement at 532 and 620 nm. Itcould be further confirmed by the PL measurements with excita-tion wavelength on or off resonance with isolated plasmonic AgNPs in another two fabricated samples (Fig. S4). Therefore, smallAg NPs in the shallow layer and interactions between them areresponsible for the LSPR effect induced tenfold PL enhancement inWS 2:AgNPs:SiO 2. FIG. 3. (a) PL spectra of WS 2:SiO 2and WS 2:AgNPs:SiO 2at excitation wavelength of 473 nm. (b) Measured Raman spectra of sample WS 2:SiO 2and WS 2:AgNPs:SiO 2by a 473 nm laser excitation.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 263103 (2021); doi: 10.1063/5.0054333 118, 263103-4 Published under an exclusive license by AIP PublishingIn conclusion, we have demonstrated large-area, low-cost fabrica- tion of embedded Ag NPs for boosting the light–matter interactions inWS 2monolayer through the ion implantation technique. The PL intensity of WS 2monolayer is enhanced by one order of magnitude on the SiO 2substrate embedded with Ag NPs. The ion beam modifiedfunctional substrate exhibits excellent environmental stability and sim- plifies the fabrication process of plasmon-2D hybrid nanostructures.This work paves the way for practical applications of ion beam modi-fied functional dielectric substrates for enhanced light–matter interac-tions in 2D materials system. FIG. 4. (a)–(c) are simulated near field distributions around Ag NPs in the y–zplane; the interparticle distance is 0.5 nm. (d)–(f) are calculated near field enhancement factors at the substrate surface along ydirection. (g)–(i) are simulated near field distributions around isolated Ag NPs in the y–zplane, where the interparticle distance is set as 10 nm. (j)–(l) are calculated near field enhancement factors at the substrate surface along yposition. Solid line refers to the substrate surface where WS 2is layered, the radius of Ag NPs is 2.5 nm, and the center of Ag NPs is set at 3 nm under substrate surface.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 263103 (2021); doi: 10.1063/5.0054333 118, 263103-5 Published under an exclusive license by AIP PublishingSee the supplementary material for more detailed PL spectros- copy measurements, Raman spectroscopy measurements, and simula- tion results by COMSOL Multiphysics program. This work was supported by the National Natural Science Foundation of China (Grant No. 11535008) and the Taishan Scholars Program of Shandong Province (No. tspd20210303). 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5.0039542.pdf

Appl. Phys. Lett. 118, 093101 (2021); https://doi.org/10.1063/5.0039542 118, 093101 © 2021 Author(s).Strong interlayer excitons in PtSe2/ZrS2 van der Waals heterobilayer Cite as: Appl. Phys. Lett. 118, 093101 (2021); https://doi.org/10.1063/5.0039542 Submitted: 04 December 2020 . Accepted: 06 February 2021 . Published Online: 01 March 2021 Longjun Xiang , Qingyun Zhang , and Youqi Ke ARTICLES YOU MAY BE INTERESTED IN Engineering Schottky-to-Ohmic contact transition for 2D metal–semiconductor junctions Applied Physics Letters 118, 091601 (2021); https://doi.org/10.1063/5.0039111 Helicity-dependent all-optical switching based on the self-trapped triplet excitons Applied Physics Letters 118, 093301 (2021); https://doi.org/10.1063/5.0035217 Electron mobility in monolayer WS 2 encapsulated in hexagonal boron-nitride Applied Physics Letters 118, 102105 (2021); https://doi.org/10.1063/5.0039766Strong interlayer excitons in PtSe 2/ZrS 2 van der Waals heterobilayer Cite as: Appl. Phys. Lett. 118, 093101 (2021); doi: 10.1063/5.0039542 Submitted: 4 December 2020 .Accepted: 6 February 2021 . Published Online: 1 March 2021 Longjun Xiang,1,2,3Qingyun Zhang,1,2,3,a) and Youqi Ke1,2,3,a) AFFILIATIONS 1School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China 2Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China 3University of Chinese Academy of Sciences, Beijing 100049, China a)Authors to whom correspondence should be addressed: zhangqy2@shanghaitech.edu.cn andkeyq@shanghaitech.edu.cn ABSTRACT Capturing interlayer excitons with large binding energy plays a pivotal role in exploring the quantum Bose gas and developing excitonic devices at high temperature. In this work, we combine first-principles Kohn–Sham density functional theory and many-body perturbationtheory to investigate the electronic and excited-state properties of two-dimensional van der Waals heterobilayer PtSe 2/ZrS 2, with the consid- eration of spin–orbit coupling. We find that the PtSe 2/ZrS 2heterobilayer possesses a strong interlayer interaction and exhibits a type-II band alignment. We obtain the optical absorption spectrum by solving the Bethe–Salpeter equation with the inclusion of electron-hole interactionand observe emerged absorption peaks in the low-energy region compared to their constituent monolayers. According to the layer-resolvedband structure and the interband transition weights in reciprocal space, we further confirm that these excitons are spatially separated into dif- ferent constituent layers, featuring the landscape of interlayer excitons. Importantly, the binding energy for the lowest-energy interlayer exci - ton is estimated as large as 350 (meV), establishing PtSe 2/ZrS 2as a promising candidate toward the realization of room temperature coherent phenomena and for the development of signal processing devices based on excitons. Published under license by AIP Publishing. https://doi.org/10.1063/5.0039542 In the past few decades, based on III–V and II–VI semiconduct- ing coupled quantum wells (CQWs),1,2interlayer excitons (IEs) have been extensively investigated to study the quantum Bose gas and develop excitonic devices.3The observed IEs in CQWs present a long lifetime due to the large spatial separation, but the low binding ener- gies owing to the strong dielectric screening in bulk materials suppress the operation temperature. For example, the operation temperature ofAlAs/GaAs CQWs with an IE binding energy of about 10 meV is lim- ited below 100 K. 1Therefore, finding materials with appreciable IE binding energy is desirable to explore the quantum Bose gas in semi- conducting materials and develop the excitonic devices at high temperature. Two-dimensional (2D) semiconducting materials, such as phos- phorene4and transition metal dichalcogenides (TMDCs),5–15exhibit strong excitonic effects due to the reduced dielectric screening and the quantum confinement.16,17For example, the absorption spectrum for group VI TMDC monolayer MoS 2has been shown to be dominated by excitonic states with a binding energy near 1 eV.18Although the binding energy in the monolayer is favorably huge, the exciton lifetime is short because the electron and hole are confined in the same layer,displaying the landscape of the intralayer exciton, as shown in Fig. 1(III) . However, stacking different monolayers to constitute a van der Waals (vdW) heterostructure19offers new opportunities to realize the IEs with appreciable binding energy and long lifetime. The landscape of IEs in the vdW heterostructure is illustrated in Fig. 1(III) ,i nw h i c h the electron and hole are located in different layers, resulting in a larger spatial separation than the intralayer exciton. Currently, the pursuit ofIEs in the community of 2D vdW heterostructures is mainly on top of group VI TMDC monolayers, such as MoSe 2/WSe 220and MoS 2/ WS 2,21to investigate spin-valley physics and Moir /C19ep o t e n t i a l s .22The observation of interlayer excitons in the MoSe 2/WSe 2heterobilayer was first reported by Rivera et al.20using photoluminescence and pho- toluminescence excitation spectroscopy, in which the lifetime of the interlayer exciton is measured to be about 1.8 ns, an order of magni- tude longer than that of intralayer excitons in monolayers. Moreover, the experimental study conducted by Wilson et al.23has demonstrated that MoSe 2/WSe 2possessed an interlayer exciton binding energy more than 200 meV, an order of magnitude higher than that in CQWs.24 Recently, the group IV and X semiconducting TMDC mono- layers received much attention due to their unique electronic Appl. Phys. Lett. 118, 093101 (2021); doi: 10.1063/5.0039542 118, 093101-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplproperties, such as high carrier mobility25,26and tunable bandgap by the film thickness.27Unlike the group VI H-phase TMDC monolayer with a trigonal prismatic structure, the group IV and X TMDC mono- layers favorably crystallize in the T-phase, in which the metal atom isoctahedrally coordinated by six chalcogen atoms and, hence, presentsdifferent mechanical, electronic, and optical properties. For example,the group X TMDC homobilayer PtS 2in the T-phase presents a stron- ger interlayer interaction than the group VI TMDC homobilayerMoS 2in the H-phase, reported by Zhao et al.27In addition, the strongly bound excitons in group X TMDC monolayers PtS 2and PtSe 2have been predicted.28On the other hand, the electronic proper- ties and the optical excitations of the group IV TMDC ZrS 2and HfS 2 monolayers and their vdW heterostructure have been investigated,29 in which the optical absorption spectra for monolayers are dominatedby excitonic effects, while the type-I band alignment in their heterobi-layer hinders the formation of IEs. Therefore, creating vdW hetero-structures based on T-phase monolayers, featuring type-II bandalignment and strong interlayer interaction, may enable the realizationof the IEs with a long lifetime and appreciable binding energy at thesame time. In this work, we combine first-principles KS-DFT and many- body perturbation theory (MBPT) to investigate the electronic andexcited-state properties of the PtSe 2/ZrS 2vdW heterobilayer. In partic- ular, we consider the influence of spin–orbit coupling (SOC) at all the-oretical levels. First, we compare the interlayer charge accumulation of PtSe 2/ZrS 2with the T-phase homobilayer PtS 2and demonstrate the existence of strong interlayer interaction in PtSe 2/ZrS 2.S e c o n d ,w ec a l - culate their band structures at both DFT and G 0W0levels and clarify the band alignments and state characteristics according to the layer-resolved and orbital-resolved band dispersion. Furthermore, based onthe G 0W0quasiparticle (QP) energies and screened Coulomb interac- tion, the Bethe–Salpeter equation (BSE) is solved to obtain the absorp-tion spectra. Finally, the features of the emerged excitons in theabsorption spectrum are analyzed in detail by combining the layer-resolved band structure and the interband transition weights in recip- rocal space. All the first-principles calculations (KS-DFT, G 0W0,a n dB S E ) are performed using the Vienna Ab initio Simulation Package(VASP).30,31The generalized gradient approximation for the exchange-correlation functional, as parameterized by Perdew, Burke, and Ernzerhof (PBE), is employed.32The heterobilayer PtSe 2/ZrS 2is built by AA stacking the monolayer primitive cells PtSe 2 (a¼b¼3:72 ˚A)28and ZrS 2(a¼b¼3:69 ˚A)29due to their almost same lattice constants, as shown in Figs. 1(I) and1(III) . In particular, a vacuum space of 20 ˚A is adopted to avoid the interaction between two periodic images along the [001] direction. The atomic positions and cell parameters are relaxed with inclusion of semiempirical vdW cor- rections from the DFT-D3 scheme.33For the ground-state calculation, an energy cutoff of 400 eV was chosen and a 15 /C215/C21k - m e s hh a s been utilized. To obtain a well-converged G 0W0QP gap, the energy cutoff of the response function ( Ec) is set to 150 eV and the total num- ber of bands ( Nb) to 480. The convergence test for the G 0W0QP bandgap at CforEcand Nbis shown in section Iof the supporting materials. The G 0W0QP band structure along the assigned high sym- metry k-path in reciprocal space was obtained through Wannier inter- polating.34The optical properties with excitonic effects were calculated by solving the BSE with a 15 /C215/C21k - m e s hu n d e rT a m m – D a n c o f f approximation, using 8 unoccupied bands and 8 occupied bands to converge the investigated excitons. The indirect transitions are not considered in this work, and the imaginary part of the dielectric func- tion is plotted as the absorption spectrum. In particular, the average method over multiple grids has been utilized to verify the convergenceof the BSE spectra calculated on the 15 /C215/C21k - m e s h . 35,36 To access the role of interlayer interaction in the vdW heterobi- layer PtSe 2/ZrS 2, we compare the charge density accumulation of PtSe 2/ZrS 2with that of the homobilayer PtS 2,i nw h i c ht h eS - Sq u a s i - bonding of PtS 2brings sufficient electrons localized in the interlayer region and displays covalent characteristics,27as shown in Fig. 2(I) .A s expected, a significant charge accumulation is also observed in the interlayer region of PtSe 2/ZrS 2,a ss h o w ni n Fig. 2(II) . This significant charge accumulation manifests the existence of strong interlayer inter- action in PtSe 2/ZrS 2, resulting in a shorter interlayer distance than the vdW layered materials stacked by H-phase TMDC monolayers.37In particular, the interlayer distance between PtSe 2and ZrS 2is 2.88 ˚A, which is a bit larger than T-phase homobilayer PtS 2(2.54 ˚A) but smaller than H-phase homobilayer MoS 227(3.06 ˚A). This strong inter- layer interaction will be beneficial for the formation of IEs with appre- ciable binding energy in PtSe 2/ZrS 2. InFigs. 2(III) and2(IV) , the band structures of PtSe 2/ZrS 2at both PBE and G 0W0levels are shown. It is found that the heterobilayer PtSe 2/ZrS 2at both levels presents an indirect gap, in which the conduc- tion band minimum (CBM) is at M, whereas the valence band maxi- mum (VBM) is between CandM. With the inclusion of SOC, we find that the global bandgap will be decreased by 0.14 eV at PBE and 0.19 eVat G 0W0. In addition, the degeneracy at the valence band edge (VBE) along the C-Kdirection is lifted with SOC due to the broken inversion symmetry between constituent layers. For completeness, in Table I ,w e list the global gaps and the minimal direct gaps at PBE and G 0W0levels, without and with the inclusion of SOC, for heterobilayer PtSe 2/ZrS 2and its constituent monolayers. In particular, we note that the SOC will reduce the gap about 0.18 eV in monolayer PtSe 2and, hence, manifest the importance of including SOC to obtain reliable absorption onsets in PtSe 2/ZrS 2. In addition, we note that the G 0W0results in PtSe 2/ZrS 2, PtSe 2,a n dZ r S 2will promote the PBE gap about 1 eV, displaying the importance of electron–electron interaction in 2D materials. FIG. 1. (I) The top view of the PtSe 2/ZrS 2van der Waals heterobilayer. (II) The Brillouin zone for PtSe 2/ZrS 2and the high symmetry k-path. (III) The side view of PtSe 2/ZrS 2and the schematic illustration of intralayer and interlayer excitons.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 093101 (2021); doi: 10.1063/5.0039542 118, 093101-2 Published under license by AIP PublishingIt was well known that the band alignment and state characteris- tics around the minimal direct gap are important for the formation of IEs. To figure out the band alignment and the state characteristics inPtSe 2/ZrS 2, we further investigate the layer-resolved and orbital- resolved band structures, as shown in Figs. 2(V) and2(VI) .F r o mt h elayer-resolved band structure, we conclude that PtSe 2/ZrS 2exhibits a well-defined type-II band alignment, in which the conduction band edge (CBE) retains the characteristics of the ZrS 2layer, whereas the valence band edge (VBE) is dominated by the PtSe 2layer. In particu- lar, we find that the CBE is contributed by the d-orbitals of the Zr atom, while the VBE around Cis dominated by p-orbitals of the Se atom, as shown in Fig. 2(VI) . Owing to the enhanced Coulomb interaction in reduced dimen- sionality, the optical spectra of PtSe 2/ZrS 2and its constituent mono- layers are calculated by solving BSE with the explicit inclusion of electron–hole interaction. In particular, considering that the SOC will affect the bandgap and also lift the band degeneracy along C-K,w e include SOC in all the calculated BSE spectra to obtain reliable absorp- tion onset and features. In Fig. 3 , the optical spectra of the heterobilayer PtSe 2/ZrS 2and its constituent monolayers are shown. It is found that the total spectrum cannot be approximated simply by summing up the spectra of the constituent monolayers. In particular, four emerged absorption peaks, indicated with vertical red lines, are observed in Fig. 3(I) .A m o n gt h e m ,t h el o w e s t - e n e r g ye x c i t o n Adisplays the largest oscillator strength. For convenience, we mark these emerging peaks as A(1.53 eV), B(1.59 eV), C(1.66 eV), and D(1.73 eV) in energy order. Correspondingly, the exciton binding energies (defined as the differ-ence between the involved direct G 0W0gap and the BSE eigenvalue) are estimated to be 350 (meV), 300 (meV), 304 (meV), and 306 (meV). They are lower than the binding energy of the intralayer exciton in monolayers PtSe 2(/C24600 meV)28and ZrS 2(600–800 meV)29but larger than that of IEs in the vdW heterobilayers stacked with group VI H- phase TMDC monolayers. In addition, we note that the lowest-energy absorption peak and exciton binding energy (0.56 eV) of PtSe 2are in good agreement with those in Ref. 28(0.58 eV). For ZrS 2,t h ed o u b l e absorption peaks are in line with those of Ref. 29. To figure out the character and composition for these emerged excitons, we analyze in detail the electron–hole amplitude matrix AS vck for each exciton state Si(i¼A;B;C;D), in which v,c,a n d krepresent valence-band states, conduction-band states, and the k-point in the Brillouin zone, respectively. In particular, we plot the interband transi- tion weights in reciprocal space for the A,B,C,andDexcitons, as shown inFigs. 3(II)–3(V) . First of all, we note that all these peaks are fully con- tributed by the transitions between the highest occupied band (HOB)and the lowest unoccupied band (LUB). Second, we note that the domi- nant transitions are distributed along the C-Mdirection in reciprocal space and display a C 6rotation symmetry. Moreover, the transition pat- terns are different between the four emerged excitons, as shown in Figs. 3(II)–3(IV) . Recalling the layer-resolved band structure shown in Fig. 2(V) ,i nw h i c ht h eH O Ba l o n g C-Mis mainly contributed by PtSe 2, whereas the LUB along C-Mis dominated by ZrS 2,w ec o n c l u d et h a ta l l these excitons are located in different constituent monolayers in real space, featuring the landscape of the interlayer exciton. Moreover, this conclusion can be drawn visually from the G 0W0QP band structures with fat-band features, in which the excitations contributed by the tran- sitions between HOB and LUB are distributed along the high symmetryline, as shown in Figs. 3(VI)–3(IX) . Although we focus on the lowest- energy emerged interlayer excitons A–D,w en o t et h a tt h es p e c t r u mo f PtSe 2/ZrS 2also exhibits the landscape of intralayer excitons. For exam- ple, the first absorption peak above D[marked as Eby red vertical line inFig. 3(I) ], with strong oscillator strength, displays the feature of the intralayer exciton, which is confirmed in Sec. II of the supplementary FIG. 2. The differential charge density for the vdW homobilayer (I) PtS 2and hetero- bilayer (II) PtSe 2/ZrS 2. The isosurface value of 0.0001 electron/ ˚A3is adopted, and only the charge accumulation is shown. The (III) PBE and (IV) G 0W0band structure with (blue line) and without (red line) the consideration of spin–orbit coupling. (V) The layer-resolved and (VI) orbital-resolved band structure of PtSe 2/ZrS 2. TABLE I. The global gaps and the minimal direct gaps for the PtSe 2/ZrS 2vdW heter- obilayer and its constituent monolayers PtSe 2and ZrS 2at both PBE and G 0W0lev- els, without and with the consideration of SOC. Bandgaps (eV) PBE PBE þSOC G 0W0G0W0þSOC PtSe 2/ZrS 2Global 0.379 0.239 1.386 1.197 Direct 0.926 0.736 2.027 1.875 PtSe 2 Global 1.359 1.169 2.186 2.009 Direct 1.578 1.429 2.433 2.309 ZrS 2 Global 1.240 1.197 2.625 2.587 Direct 1.700 1.654 3.222 3.202Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 093101 (2021); doi: 10.1063/5.0039542 118, 093101-3 Published under license by AIP Publishingmaterial . However, the intralayer excitons in PtSe 2/ZrS 2are located in the higher energy region and, hence, hardly affect the properties of theemerged interlayer excitons A–D. It should be emphasized that the IEs presented in the PtSe 2/ZrS 2 vdW heterobilayer feature large binding energies Eb. In particular, the binding energy for the lowest-energy IE Awith a large oscillator strength is estimated as large as 350 (meV). It has been revealed thatthe IE excitons with large binding energy are important to explore the quantum Bose gas and develop the excitonic devices at high tempera- ture. On the one hand, the operation temperature for the IEs is belowE b=kB(where kBis the Boltzmann constant); on the other hand, the temperature of quantum degeneracy, achieved with increasing densitybefore exciton dissociation to electron–hole plasma, also scales propor-tionally to E b.12,41InFig. 4 , we compare the IE binding energies for dif- ferent materials. Compared with other experimental (red bars) andtheoretical studies (green bars), we find that the PtSe 2/ZrS 2vdW heter- obilayer presents an appreciable IE binding energy. Therefore, ourresults manifest that the PtSe 2/ZrS 2heterobilayer offers a promising candidate toward the realization of high-temperature coherent phe- nomena and excitonic devices. In conclusion, we have utilized first-principles KS-DFT and MBPT to investigate the electronic and excited-state properties ofvdW heterobilayer PtSe 2/ZrS 2. We demonstrate the existence of strong interlayer interaction from interlayer charge accumulation and identifythe type-II band alignment from the layer-resolved band structure.Also, we study the absorption spectra by solving the BSE and observefour emerged excitons in PtSe 2/ZrS 2. Furthermore, we analyze the composition and features and confirm that these emerged excitons exhibit the landscape of the interlayer exciton. Importantly, the bind-ing energy for the lowest-energy interlayer exciton with a large oscilla-tor strength is estimated as large as 350 (meV). Therefore, our resultsmanifest that the PtSe 2/ZrS 2vdW heterobilayer offers a promising platform toward the realization of high-temperature coherent phe-nomena and for the development of excitonic devices.FIG. 4. The binding energies for AlAs/GaAs CQW,1bulk MoTe 2,9MoSe 2/WSe 2,23 WSe 2/WS 2,38MoSe 2/WSe 2,11MoS 2/WSe 2,39MoS 2/MoSe 2,40and PtSe 2/ZrS 2. Here, the bars with red color indicate the experimental studies, the bars with green color the published theoretical results, and the bar with blue color our theoretical result. FIG. 3. (I) The imaginary parts of the transverse dielectric constant /C15? 2as a function of photon energy for vdW heterobilayer PtSe 2/ZrS 2. Here, the red vertical lines are labeled asA,B,C,D, and E, respectively. (II–V) The electron-hole amplitude plots for A,B,C, andDin PtSe 2/ZrS 2, in which all the dominant excitations are contributed by the transi- tions between HOB and LUB. (VI–IX) The G 0W0QP band structures with fat-band fea- tures for A,B,C,a n d D. Here, the size of the solid circle in (II–V) and empty circle in (VI–IX) is proportional to the magnitude of transition weight.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 093101 (2021); doi: 10.1063/5.0039542 118, 093101-4 Published under license by AIP PublishingSee the supplementary material for two sections: (I) convergence test of G 0W0calculations and (II) interlayer exciton E. Y.K. acknowledges the support from the NSFC with Grant No. 11874265 and the ShanghaiTech start-up, and Q.Z. acknowledges the financial support from the NSFC with Grant No. 11704121. The authors thank the HPC platform of ShanghaiTech University forproviding a computational facility. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Zrenner, P. Leeb, J. Sch €afer, G. B €ohm, G. Weimann, J. Worlock, L. Florez, and J. Harbison, “Indirect excitons in coupled quantum well structures,” Surf. Sci.263, 496–501 (1992). 2Z. V €or€os, D. Snoke, L. Pfeiffer, and K. 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