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PhysRevB.96.054436.pdf

PHYSICAL REVIEW B 96, 054436 (2017) Enhanced spin pumping near a magnetic ordering transition Behrouz Khodadadi,1Jamileh Beik Mohammadi,1Claudia Mewes,1Tim Mewes,1 M. Manno,2C. Leighton,2and Casey W. Miller3,4,* 1Department of Physics and Astronomy, MINT Center, The University of Alabama, Tuscaloosa, Alabama 35487, USA 2Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA 3School of Chemistry and Materials Science, Rochester Institute of Technology, Rochester, New York 14623, USA 4Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden (Received 17 May 2017; revised manuscript received 21 July 2017; published 25 August 2017) We study the temperature-dependent static and dynamic magnetic properties of polycrystalline bilayers of permalloy (Ni 80Fe20, or Py) and gadolinium (Gd) bilayers using DC magnetometry and broadband ferromagnetic resonance. Magnetometry measurements reveal that the 3-nm-thick Gd layers undergo a magnetic orderingtransition below 100 K, consistent with finite size suppression of their Curie temperature. Upon cooling below thisGd ordering temperature, ferromagnetic resonance spectroscopy reveals a sharp increase in both the gyromagneticratio (γ) and effective Gilbert damping parameter ( α eff) of the neighboring Py layers. The increase of γis attributed to the onset of strong antiferromagnetic coupling between the Gd and Py layers as the Gd orders magnetically.We argue that the increase of α eff, on the other hand, can be explained by spin pumping into the rare-earth layer when taking into account the increase of γ, the decrease of the Gd spin diffusion length as it orders, and, most significantly, the corresponding increase of the Py/Gd interfacial spin mixing conductance in the vicinity of themagnetic ordering transition. We propose that these observations constitute a qualitative confirmation of a recenttheoretical prediction of spin sinking enhancement in this situation. DOI: 10.1103/PhysRevB.96.054436 I. INTRODUCTION Magnetization dynamics of magnetic multilayers, in addition to being an interesting fundamental research subjectin its own right, also has many implications for the design and development of spintronic devices. While the Landau-Lifshitz- Gilbert equation of motion [ 1,2] has been very successful in describing magnetization dynamics in ferromagnet-containingsystems, the origin and underpinnings of the damping param-eterαcontinue to be a major focus of current research [ 3]. A number of different physical mechanisms have the functionalform of the damping term introduced by Gilbert, including, for example, spin-orbit relaxation [ 4,5], magnon-phonon relaxation [ 6], eddy current damping [ 7–9], and spin pumping [10–12]. In magnetic multilayers and heterostructures, spin pumping is often found to be one of the dominant contributionsto magnetic relaxation [ 13,14]. Because spin pumping enables the generation of pure spin currents in normal metals adjacentto ferromagnets [ 15,16], which in turn can be detected using the inverse spin Hall effect [ 17–20], this relaxation mechanism has been studied in great detail in ferromagnet/nonmagnetic metal systems. Recent theoretical work has suggested anew avenue of research in this general area by predictingthat the interfacial spin-mixing conductance, and thusmagnetic relaxation, can be significantly enhanced when spinpumping is performed not into a conventional nonmagneticmetal, but rather into a metallic system undergoing a paramagnet to ferromagnetic transition on cooling [ 21]. This essentially expands this research into the situation where spinpumping occurs into a metal with strong ferromagnetic spincorrelations, which is the subject of this paper. Broadband ferromagnetic resonance (FMR) is a technique that has been widely used to investigate the magnetic relaxation *Corresponding author: cwmsch@rit.edumechanisms in ferromagnetic thin films, as well as spinpumping into adjacent nonmagnetic layers [ 22–26]. In this work, we employ FMR to probe the above mentioned situationof spin pumping into a layer undergoing a ferromagneticordering transition by studying transition metal/rare earthbilayers. Such a multilayer system is one natural choice torealize this situation, using the transition metal as a relativelysimple, low damping, high Curie temperature ferromagnet,and the rare earth as a high atomic number, low Curietemperature adjacent layer. Specifically, we employ herebilayers of permalloy (Ni 80Fe20, or Py) and gadolinium (Gd) bilayers with Py thicknesses of 5 and 10 nm, and a Gdthickness of only 3 nm, thus suppressing the rare-earth orderingtemperature by finite size effects. By studying the temperaturedependence of the FMR of the Py in these bilayers, weare thus able to investigate the influence of the magneticordering transition in the Gd [ 27,28] on the magnetization dynamics and relaxation in the Py. We find that the Gdparamagnetic to ferromagnetic transition indeed has a stronginfluence on the gyromagnetic ratio and effective Gilbertdamping parameter of Py. Clear enhancement is seen in bothquantities across the transition, which we interpret in termsof strong antiferromagnetic interlayer coupling, as well asthe critical temperature dependence of the Py/Gd spin mixingconductance and Gd spin diffusion length. These results arethen compared to recent theoretical predictions, which webelieve we qualitatively confirm. II. EXPERIMENTAL DETAILS The Py/Gd bilayers were deposited on Si /a-SiO xsubstrates using a multi-source RF magnetron sputtering system witha base pressure better than 2 ×10 −8Torr. The layers were sequentially sputtered, with no break in vacuum, in 3 mTorr ofultra-high-purity Ar gas at ambient temperature. This was done 2469-9950/2017/96(5)/054436(8) 054436-1 ©2017 American Physical SocietyBEHROUZ KHODADADI et al. PHYSICAL REVIEW B 96, 054436 (2017) from a Ni 80Fe20target at a deposition rate of 0.044 nm /s, and from an elemental Gd target at a deposition rate of 0.057 nm /s. We note that the growth of high quality Gd in this sputtertool has been previously accomplished [ 29], with protocols published elsewhere [ 30]. The specific structures grown here were Py(5 or 10 nm)/Gd(3 nm)/Al(5 nm). The Al cappinglayer was chosen as an oxidation barrier as any remainingunoxidized Al metal should not significantly influence themagnetization dynamics under investigation, due to the lowspin-orbit coupling and relatively large spin diffusion lengthof Al. With similar concerns in mind, we also avoided oftenused seed layer metals such as Ta, as those could serveas unintentional spin sinks, and thus a source of additionalmagnetization damping. In-plane magnetometry characterization of these bilayers was performed between 5 and 400 K, in fields up to 90 kOe,using a Quantum Design Physical Property Measurement Sys-tem (PPMS) equipped with a vibrating sample magnetometer(VSM) option. Some out-of-plane measurements were alsoperformed using a Quantum Design Magnetic Property Mea-surement System (MPMS) XL7 SQUID magnetometer. In thatcase, samples were mounted using concentric self-tighteningpolyethylene straws to minimize background signals andartifacts [ 31]. For both orientations of the magnetic field, samples were cooled in a +90 kOe magnetic field, and high- field linear background subtraction was applied to removecontributions from the holder and substrate. Importantly, themagnetization values reported below are all based on thecombined nominal thickness of the Py and Gd, i.e., we plot the moment per total volume of Py and Gd layers. Temperature-dependent broadband FMR measurements were carried out using a custom designed setup capable ofmeasuring in the frequency range 1–40 GHz [ 32–34]. The raw FMR spectra were fit using a derivative of a Lorentzian lineshape [ 35] in order to determine the resonance field ( H res) and the peak-to-peak linewidth ( /Delta1H) as functions of frequency and temperature. As recently shown by Shaw et al. [36], broadband FMR is useful for reducing experimental error margins whendetermining gyromagnetic ratios and damping parameters. III. RESULTS AND ANALYSIS In the sections below, we provide first the results of magne- tometry characterization, followed by temperature-dependentFMR measurements. A. Magnetometry In-plane magnetization hysteresis loops were first measured at temperatures Tbetween 10 and 400 K, in magnetic fields up to 90 kOe. Hysteresis loops of the Py(10 nm)/Gd(3 nm)bilayer taken at 10, 30, 40, 50, 60, and 200 K are shown inFig. 1. The magnetic response is seen to be quite complex, due to the superposition of Gd finite size effects and strongPy-Gd interlayer coupling, which is known to generate highsaturation fields [ 27,37,38] antiferromagnetic interactions, and noncollinear spin states [ 39]. Specifically, Py(10 nm)/Gd(3 nm) bilayers exhibit soft ferromagnetic behavior above200 K with negligible coercivity and 95% of the saturationmagnetization M sbeing reached in only 1 kOe. Below 100 K,FIG. 1. Representative in-plane hysteresis loops of the Py(10 nm)/Gd(3 nm) bilayer at multiple temperatures. Magnetization was determined using the combined thickness of the Py and Gd layers, as also emphasized in the text. Magnetic field sweeps up to 90 kOe are show in (a), while expanded 10 kOe magnetic field sweeps are shown in (b). (Inset) Comparison of the film’s magnetic response toin-plane and out-of-plane magnetic fields at 10 K. however, the soft, low-field character remains, but new features appear. First, a high-field component emerges [Fig. 1(a)], with the apparent Msrising by more than 50% as Tis decreased to 10 K. We ascribe this to the onset of thermallystable ferromagnetism in the Gd, adding to the existing softferromagnetism of Py. While bulk Gd has a Curie temperaturenear ambient ( ∼292 K) [ 40,41], finite size effects are well- known in general, and specifically in Gd films, multilayers,and nanoparticles, and can produce ordering temperatures wellbelow 300 K [ 27,38,42]. The second noticeable feature is that the additional low Tmagnetization saturates quite slowly with increasing field. In the Py(10 nm)/Gd(3 nm) bilayer shown inFig.1(a), saturation is reached only after applying 80 kOe at 10 K. This is yet more pronounced in the Py(5 nm)/Gd(3 nm)case (data not shown), to the point that 90 kOe was not 054436-2ENHANCED SPIN PUMPING NEAR A MAGNETIC . . . PHYSICAL REVIEW B 96, 054436 (2017) sufficient to completely saturate the sample at low T.T h i r d l y , the rise in Mson cooling below 100 K is also accompanied by distinct changes in the low-field behavior (below 10 kOe).As best seen in Fig. 1(b), the low-field magnetization in fact decreases monotonically on cooling, the magnetization at / 1 kOe, for example, M 1 kOe, falls by 37% from 100 to 10 K. Closer examination of the 10-K data further reveals rapidapparent saturation at low fields, followed by a kink at ∼3k O e , then a gradually increasing magnetization until true saturationis achieved near 80 kOe. We note additionally at this pointthat while in-plane anisotropy clearly dominates, out-of-planemagnetization measurements at 10 K do reveal both finitecoercivity ( ∼120 Oe) and remanence [inset to Fig. 1(b)]. This indicates a weak out-of-plane component to the anisotropy. We interpret the behavior displayed in Fig. 1in terms of antiferromagnetic coupling between Py and Gd. This has beenpreviously observed in various transition metal /rare-earthsystems, including Fe/Gd [ 43], Fe/Gd-Fe [ 44], Co/Gd [ 45,46], and even Py/Gd [ 39], the system studied here. Using the 10-K field sweep in Fig. 1to illustrate this, as Happroaches 90 kOe, the magnetizations of the Gd and Py layers are apparentlyaligned near parallel to each other, and to the magneticfield. (We note that whether true saturation is achieved in90 kOe is not completely clear here, as noncollinear statesdue to competition between antiferromagnetic coupling andZeeman energies can result in very large saturation fieldsin Py/Gd, with spin angles varying over larger distancesthan our film thicknesses [ 39]). As the applied field is decreased, the antiferromagnetic coupling between the Py andGd causes the Gd spins to rotate progressively away fromthe Py magnetization, which remains predominantly alignedwith the field. This leads to a reduction of the low-fieldmagnetization upon cooling, i.e., as the Gd becomes orderedand the antiferromagnetic coupling dominates [Fig. 1(b)]. In this picture, the kink observed near 3 kOe at 10 K [Fig. 1(b)] is likely a spin-flop-type transition, as seen in Fe/Gd-Fe [ 44] and Co/Gd multilayers [ 46]. In order to further elucidate the antiferromagnetic coupling in these Py/Gd bilayers, Fig. 2plotsM sandM1 kOefor both the Py(10 nm)/Gd(3 nm) and Py(5 nm)/Gd(3 nm) samples. As wecould not clearly saturate the Py(5 nm)/Gd(3 nm) bilayer in90 kOe, we plot the maximum magnetization ( M max) for this sample, as an approximation to Ms. These plots clearly reflect the two main conclusions from the magnetometry discussedabove: ferromagnetic order in the Gd layers sets in around60–80 K, and antiferromagnetic coupling between the Py andGd simultaneously emerges. The former is reflected in theclear increase in M s(orMmax) on cooling, while the latter is indicated by the corresponding decrease in M1 kOe. A quantitative analysis reveals further interesting features, particularly with respect to the absolute magnetization values.The bulk M svalues of Py and Gd are ∼800 emu /cm3 atT=290 K [ 41], and ∼2100 emu /cm3when T<10 K [27,37,40], respectively, resulting in a total averaged expected Msof 1100 emu /cm3for the Py(10 nm)/Gd(3 nm) bilayer at 10 K. Reference to Fig. 2(a) immediately reveals that these bilayers have significantly reduced Msin comparison to these simple expectations. This is true even at high T, above the Gd ordering temperature, with the 400-K Msvalues for the Py(10 nm)/Gd(3 nm) and Py(5 nm)/Gd(3 nm) bilayers beingFIG. 2. Temperature dependence of the saturation magneti- zation ( Ms) and the magnetization at 1 kOe ( M1k O e) for the (a) Py(10 nm)/Gd(3 nm) and (b) Py(5 nm)/Gd(3 nm) bilayers. As discussed in the text, the Py(5 nm)/Gd(3 nm) bilayer did not completely reach saturation even in 90 kOe, leading us to report themaximum magnetization achieved ( M max)r a t h e rt h a n Ms. 44% and 56% lower, respectively, than expectations for Py alone. This reduction in magnetization is maintained at lowT, with the 3-nm Gd layers displaying magnetizations about 60%–70% lower than simple bulk estimates. Consideringpossible explanations for this, trivial possibilities such asmiscalibration of magnetometers or deposition rates wereruled out; these would need to involve ∼50% inaccuracies, far above our estimates. Another possibility is ineffectiveAl capping, leading to partial oxidation of the underlyingmagnetic bilayers. We stress, however, that we did not detectany signatures (including exchange bias) of the presence ofNi-O, Fe-O, and/or Gd-O phases. A third possibility, at leastpartially supported by prior work, is that some fraction ofthe Gd layer has a significantly proximity-enhanced Curietemperature, and is antiferromagnetically coupled to the Pylayer even at 400 K, thus reducing the apparent M s. Strong proximity effects enhancing the Curie temperature of Gdlayers in contact with transition metal layers have in factbeen clearly elucidated in systems such as Fe/Gd [ 47] and Ni 1−xFex/Gd [48]. A recent x-ray magnetic circular dichroism (XMCD) study of Ni 1−xFex/Gd multilayers by Ranchal et al. , for example, showed that this interfacial Gd layer is 1 – 2.5-nmthick, aligned antiparallel to the Ni 1−xFexmagnetization [ 48], 054436-3BEHROUZ KHODADADI et al. PHYSICAL REVIEW B 96, 054436 (2017) with an enhanced ordering temperature of up to 550 K. If we assume that our observed reduced Msat high Tis associated with such an ordered interfacial Gd layer, we estimate this layerto be∼1.6-nm and 0.9-nm thick for the Py(10 nm)/Gd(3 nm) and Py(5 nm)/Gd(3 nm) bilayers, respectively. This matchesvery well with Ranchal et al. ’s observations. This would result in Gd near the Al cap layer of the heterostructure (i.e., farfrom the Py interface) that is magnetically disordered at highT, with 1.4-nm thickness in the Py(10 nm)/Gd(3 nm) bilayer, and 2.1-nm thickness in the Py(5 nm)/Gd(3 nm) bilayer. Theincrease in M s(orMmax) on cooling below 100 K would then be due to the ferromagnetic ordering of this “cap-side” Gd(still antiferromagnetically coupled to the Py). Assuming thatthe rise in M s(orMmax) at 10 K is indeed due to the onset of ferromagnetism in this noninterfacial Gd layer, its thicknesscan be estimated to be ∼1.1 nm for the Py(10 nm)/Gd(3 nm) case, and 0.8 nm for the Py(5 nm)/Gd(3 nm) case, reasonablyconsistent with the above arguments. We thus consider thisa possibility in this case, although this would require verystrong antiferromagnetic coupling between the interfacial Gdlayer and the Py, and a Gd Curie point enhanced well above400 K. Future work with element-resolved techniques such asXAS and XMCD could potentially test this. B. Ferromagnetic resonance measurements Having characterized the static magnetic response of these Py/Gd bilayers across the Gd ordering temperature, dynamiccharacterization was performed via temperature-dependentFMR. These FMR results show significant temperature de-pendence to both the Py gyromagnetic ratio and the effectivedamping parameter, which we treat below in turn. Due tothe significant broadening of the resonance with decreasingtemperatures, we were limited in our ability to measureferromagnetic resonance spectra to temperatures above 65 and50 K for the 5-nm and 10-nm Py films, respectively. We alsonote that no discernible resonance features were observed atlower temperatures, down to 10 K. 1. Gyromagnetic ratio As shown in the inset to Fig. 3(a), the Py FMR spectra are strongly temperature dependent at a given frequency (inthis case, 10 GHz), both the resonance field and linewidthchanging substantially on cooling. Figure 3(a)shows that the relationship between resonance field and frequency ffollows the usual Kittel behavior at all temperatures in these Py/Gdbilayers. Least square fits using Kittel’s formula [ 49], f=γ /prime/radicalbig H(H+4πM eff), (1) where used to extract the gyromagnetic ratio γ/prime=gμB hand the effective magnetization Meffat each T. Here, gis thegfactor, μBthe Bohr magneton, and hthe Planck constant. As evident from the fit curves shown in Fig. 3(a),E q .( 1) provides an excellent description of the experimental data over the entire frange. Although the significant broadening of the resonance with decreasing T[see inset to Fig. 3(a)] limited our ability to measure spectra at the lowest T, as shown in Fig. 3(b), we find a considerable increase in the gyromagnetic ratio on cooling,particularly below about 50–70 K. In terms of gfactors, at 300 K, we find g=2.116 for the 10 nm and g=2.141 for the FIG. 3. (a) FMR frequency as a function of resonance field for the Py(5 nm)/Gd(3 nm) bilayer at various temperatures; solid lines are fits to the Kittel equation at 65 and 300 K as examples. (Inset) Example FMR spectra at 10 GHz and three representative temperatures.(b) Temperature dependence of the gyromagnetic ratio for 5-nm and 10-nm Py films capped with 3 nm of Gd. 5 nm Py, which are both somewhat larger than the bulk value ofgNiFe,bulk=2.109 determined by Shaw et al. using similar broadband FMR techniques [ 36]. We note that the increase is stronger for the thinner NiFe film, consistent with an interfacialorigin of the enhancement. The apparent divergence of the gyromagnetic ratio at low Tresembles the increase of this ratio seen in ferrimagnetic alloys near a compensation point [ 50–53]. The effective gyromagnetic ratio for ferrimagnetic alloys is the ratio of the net magnetization to the net angular momentum, S[51]. In the case of antiferromagnetic coupling between two sublattices(or in the present case between Py and Gd layers) thenet angular momentum is S=M Py/γ/prime Py−MGd/γ/prime Gd, where the subscripts indicate the contributions from the individuallayers. In this case, where M Pyis essentially independent of 054436-4ENHANCED SPIN PUMPING NEAR A MAGNETIC . . . PHYSICAL REVIEW B 96, 054436 (2017) temperature, an increasing MGdwith decreasing temperature reduces the net angular momentum. As Sapproaches zero at the angular momentum compensation temperature, theeffective gyromagnetic ratio thus diverges. No such diver-gence of the gyromagnetic ratio would be expected in asystem with ferromagnetic coupling between layers. As themagnetometry measurements in Sec. III A clearly revealed antiferromagnetic coupling between the Py/Gd bilayers, this therefore qualitatively explains the strong Tdependence of γ /prime. The temperatures for the divergence in γ/primeindeed roughly correspond to the increases in Msin Figs. 2(a)and2(b). One limitation to the above is that an underlying assumption used to derive Eq. ( 1) is that the ferromagnet is saturated. However, the magnetometry data indicate that the low- Tmag- netization in these bilayers still varies with applied magnetic field, even for the largest fields in the FMR experiments, indicating that the samples are not fully saturated at low T. While it is generally difficult to obtain closed-form expressionsfor the ferromagnetic resonance condition in unsaturatedsamples, one can obtain valuable insight by simply assumingthat the effective magnetization increases linearly with theapplied field H[see Fig. 1(b)], i.e., M eff=Meff(H)=Meff,0+qH, withq/greaterorequalslant0. (2) The parameter qis related to the field-dependent net mag- netization as the two independent layers’ magnetizations alignwith increasing field, a consequence of the antiferromagneticcoupling between the Py and Gd. Using the linearized modelcaptured by Eq. ( 2) one obtains a modified Kittel equation: f=˜γ /prime/radicalBig H(H+4π˜Meff), (3) where the modified effective gyromagnetic ratio ˜ γ/primeand effective magnetization ˜Meffare defined as ˜γ/prime=γ/prime/radicalbig 1+4πq, ˜γ/prime/greaterorequalslantγ/prime(4) ˜Meff=Meff,0 1+4πq,˜Meff/lessorequalslantMeff,0. (5) Equation ( 3) has the same functionality as the Kittel equation, which explains the excellent fit to the experimentaldata in Fig. 3(a), despite the lack of saturation. While more detailed information would be needed to develop a fullyquantitative model, the simplified model resulting in Eq. ( 4), taken together with the magnetometry data discussed earlier,shows that the gyromagnetic ratio ˜ γ /primeextracted using the Kittel equation can be expected to be systematically larger thanthe intrinsic value γ /primeif the effective magnetization probed by FMR increases with the applied field. While achievingan exact deconvolution of the factors that contribute to theobserved increase of the gyromagnetic ratio with decreasingTis beyond the scope of this work, we simply point out that antiferromagnetic coupling between the NiFe and Gd layers isthe essential element. This is true for both the picture based ona compensation point of the net angular momentum, and forthe field-dependent effective magnetization model. FIG. 4. (a) FMR linewidth as a function of frequency for Py(5 nm)/Gd(3 nm) at various temperatures. Lines are fits to the experimental data used to extract the effective Gilbert dampingparameter. (b) Temperature dependence of the effective Gilbert damping parameter for 5-nm and 10-nm Py films capped with 3 nm of Gd. The red triangle indicates the intrinsic damping parameter forPy thin films at room temperature [ 54]. 2. Damping parameter Figure 4(a)reveals that there also exist strong temperature and frequency dependencies of the FMR linewidth. In the caseof Gilbert-type relaxation of the magnetization, the isothermalfrequency dependence of the linewidth is expected to be givenby [2] /Delta1H=/Delta1H 0+2√ 3αeff γ/primef, (6) where the slope is proportional to the effective damping parameter αeff, and the zero-frequency offset /Delta1H 0is typically associated with inhomogeneities; /Delta1H 0was less than 2 Oe for these samples. Using the gyromagnetic ratio extracted from theKittel analysis at each temperature [Fig. 3(b)], fits to the data 054436-5BEHROUZ KHODADADI et al. PHYSICAL REVIEW B 96, 054436 (2017) in Fig. 4(a)using Eq. ( 6) yield the temperature-dependent αeff shown in Fig. 4(b). For both bilayers, the effective damping pa- rameter increases significantly on cooling, particularly around50–70 K, similar to what was observed for the gyromagneticratio. Note, however, that the change in α effis roughly an order of magnitude between 300 K and the lowest measurabletemperatures [Fig. 4(b)], which is much more significant than the 10% increase seen in the gyromagnetic ratio [Fig. 3(b)]. Overall the effective damping parameter is larger for thethinner Py layer, which is consistent with the Gd orderingaffecting the Py spin dynamics across the interface. As discussed in Sec. I, spin pumping is known to contribute to the magnetic relaxation of ferromagnetic films in proximityto normal metal films, or other ferromagnetic films [ 10,11,13]. Given this, and the fact that the Tdependence of the effective damping parameter in these Py/Gd bilayers [Fig. 4(b)] exhibits clear similarities to the Tdependence shown in Fig. 2,w en o w consider the possibility that spin pumping into the Gd is centralto our observed damping enhancements. We note immediatelythat while the T-dependent increase in the gyromagnetic ratio will translate into some increase in the effective damping parameter, this is simply too small an effect to explain theorder of magnitude increase seen in α eff. Spin pumping from the Py into the Gd should also be affected by the change withtemperature of the spin diffusion length in the Gd, however, aswell as the change of the spin mixing conductance across thePy/Gd interface. These are both expected to be large effects asthe Gd undergoes magnetic ordering on cooling, and they arediscussed below in turn. We note that the effective dampingat room temperature is enhanced compared to establishedvalues for Py [ 54]. However, because for Gd the total magnetic moment is due to spin, with no orbital magnetic moment [ 55], the enhancement is small, as can be expected due to the lackof spin-orbit interaction in this material [ 55]. Consider first the contribution to the effective damping parameter related to the Tdependence of the Gd spin diffusion length. Based on the theoretical model of spin pumping byTserkovnyak et al. [10,11], the effective Gilbert damping parameter of a ferromagnetic film adjacent to a spin sinkcan be expressed as the sum of the intrinsic Gilbert dampingparameter α 0and a spin pumping contribution αeff=α0+/bracketleftbigg 1+g↑↓τSFδSD/h tanh(L/λSD)/bracketrightbigg−1gg↑↓ 4πμ, (7) where g↑↓is the interfacial spin mixing conductance, μis the film’s total magnetic moment (the product of magnetizationper unit volume, interface area, and film thickness), τ SFis the mean time between spin-flip collisions in the spin sink, δSDis the effective energy-level spacing of the states that contribute tothe spin-flip scattering events, Lis the thickness of the adjacent metal layer, and λ SDis the spin diffusion length of that spin sink metal layer. The termτSFδSD/h tanh(L/λSD)is commonly referred to as the back-reflection factor. The maximum possible enhancement of the effective Gilbert damping parameter by ordering of anadjacent ferromagnet occurs when this back-reflection factoris zero, yielding α /prime max=gg↑↓ 4πμ. (8)Approximating the high-temperature spin mixing conduc- tance for our interface at 10 nm−2[56,57], the maximum damping would then be of the order 0.025 and 0.050 for the5-nm and 10-nm-thick Py layers, respectively. Given that theseα /prime maxestimates are lower by a factor of two than the increases we observed experimentally, this explanation clearly has somelimitation. However, it should be emphasized that the existenceof proximity-ordered Gd at the Py interface implies that thismechanism likely does contribute to the damping enhancementseen here, throughout the temperature range studied. Note thatone can use a more detailed form of Eq. ( 7), such as the one used in the work by Y . Tserkovnyak et al. [12] to express the back-reflection parameter in terms of the Sharvin interfaceresistance, and eventually the resistance of Gd. However,this does not alter the estimate for the maximum increasein damping expected. Looking in closer detail, the model associated with Eq. ( 7) suggests that damping will increase when the spin diffusionlength of the spin sink layer is reduced, e.g., below amagnetic ordering transition. Since the spin diffusion lengthλ SDof a ferromagnetic or normal metal can be described by [58] λSD=/radicalbigg/parenleftbig 1−β2 f/parenrightbigλsfλt 6, (9) where λtis the electron mean free path, λsfis the spin- flip length (the product of the Fermi velocity and τSF), andβfis an asymmetry coefficient related to the spin polarization (which scales as the magnetic order parameter),the temperature dependence of damping should then vary ina manner controlled by the magnetic order parameter [bycombining Eqs. ( 7) and ( 9)]. This is inconsistent with the data in Fig. 4(b) though, which resembles a divergence at the ordering temperature, as opposed to gradual growth asorder develops. The heterogeneity of magnetic ordering ofthe Gd with both depth and temperature clearly complicatesthe situation, however; unfortunately, we are aware of nomeasurements of Gd spin diffusion length in this or simplersituations. Even if we were measuring fully ordered Gd,existing measurements of the spin polarization of Gd suggestthat it is not particularly large, which would limit the impacton damping (via β f). For example, in the pioneering work of Tedrow and Meservey [ 59] the reported spin polarization of Gd at 0.4 K was only 4.3%, and in temperature dependent spin-polarized photoemission studies of Gd(0001) the measuredspin polarization did not exceed 30% even well below theCurie temperature [ 60]. While these different measurement techniques probe different spin polarizations, all reported val-ues for Gd are low. We therefore expect that this contributionhas a relatively minor effect on the temperature dependence ofdamping. More consistent with the major features of our data, a recent study by Ohnuma et al. [21] offers another mechanism, based on the spin mixing conductance, that could explain theenhanced damping as Tapproaches the Gd phase transition from above. That work extends an alternative theory of spinpumping first proposed by E. Simanek and B.Heinrich [ 61] and uses a linear-response theory to clarify the role of spinfluctuations in spin sinking, for a metallic ferromagnet near its 054436-6ENHANCED SPIN PUMPING NEAR A MAGNETIC . . . PHYSICAL REVIEW B 96, 054436 (2017) Curie temperature. In this theory, spin pumping is enhanced asTis lowered toward the Curie temperature of the spin sink layer, due to an increase of the spin mixing conductance acrossthe interface. This is caused by the proportionality of the spinmixing conductance to the momentum sum of the imaginarypart of the dynamical transverse spin susceptibility in the spinsink layer: g ↑↓∝/summationdisplay kImχR k(ωrf), (10) where χR kis the dynamical transverse spin susceptibility and ωrfis the microwave frequency. This magnetic susceptibility diverges as Tis lowered toward the Curie temperature at a standard second-order paramagnetic to ferromagnetic phasetransition, as expected for Gd (at least from dc measure-ments) [ 62,63]. Thus Eq. ( 10) suggests that the spin mixing conductance should increase dramatically on cooling as themagnetic ordering temperature is approached. Assuming asmall back-reflection term, one would expect the dampingparameter to increase with a susceptibilitylike Tdependence [from Eqs. ( 7) and ( 10)], qualitatively consistent with our data. While the complicated depth- and temperature-dependentordering of the Gd layer, and the lack of independent datafor the spin diffusion length, prevent us from quantitativelydetermining the two contributions to the Tdependence of the damping enhancement in our case, the qualitative featuresof our data do suggest that the spin mixing conductanceis the dominant factor. This is clear from the appearanceof (susceptibility-like) divergence of α effat the Gd ordering temperature, as opposed to (order-parameter-like) gradualgrowth in α effon cooling below the Curie temperature, thus providing a qualitative confirmation of the prediction ofOhnuma et al. A quantitative test of the exact temperature dependence in Fig. 4(b) would require careful T-dependent measurements of the Gd spin polarization, spin diffusionlength, and dynamical spin susceptibility, which is clearlydemanding. We note however that the linewidth data shownin Fig. 4(a)deviate from the linear dependence described by Eq. (6), which was used for all datasets to enable a quantitativedata analysis with a minimal set of free parameters. These deviations are expected, due to the field dependence of thesusceptibility [see Fig. 1(b)] and thus provide further evidence that this mechanism contributes significantly to the observedtemperature dependence of the effective damping. IV. SUMMARY In summary, we combined temperature-dependent dc mag- netometry and broadband ferromagnetic resonance to studyPy/Gd bilayers both above and below the temperature at whichthe Gd orders magnetically. We find finite-size-suppressedGd ordering temperatures, strong antiferromagnetic couplingbetween the transition metal and rare earth layers, andevidence of a proximity ordered interfacial component. FMRmeasurements then reveal clear increases in the gyromagneticratio and Gilbert damping parameter of the Py on coolingtoward the phase transition temperature of the Gd. The increasein the gyromagnetic ratio can be attributed to the antiferro-magnetic coupling between Py and Gd, conceptually similarto compensation points in rare-earth/transition metal alloys.The strong temperature dependence of the effective dampingparameter, on the other hand, is caused by enhancement ofspin pumping, which we argue to originate from the combinedeffects of an increasing gyromagnetic ratio, a decreasing Gdspin diffusion length, and most significantly, an increasing spinmixing conductance when approaching the paramagnetic-to-ferromagnetic phase transition of Gd. The latter provides aqualitative confirmation of a recent theoretical prediction. ACKNOWLEDGMENTS Work at RIT was supported by NSF-CAREER 1522927. Work at the University of Alabama was supported by NSF-CAREER Award No. 0952929 and by NSF-CAREER AwardNo. 1452670. Work at the University of Minnesota wassupported by NSF under Award No. DMR-1507048. Theauthors thank T. 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PhysRevApplied.15.064004.pdf

PHYSICAL REVIEW APPLIED 15,064004 (2021) Editors’ Suggestion Skyrmion-Based Programmable Logic Device with Complete Boolean Logic Functions Z.R. Yan,1,2Y.Z. Liu,1,2Y. Guang ,1,2K. Yue,3J.F. Feng,1,2R.K. Lake ,4G.Q. Yu ,1,2,5, *and X.F. Han1,2,5, † 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China 3Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, California 90089, USA 4Department of Electrical and Computer Engineering, University of California, Riverside, California 92521, USA 5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China (Received 4 February 2021; accepted 7 May 2021; published 2 June 2021) A skyrmionic programmable logic device (SPLD) with complete Boolean logic functions is proposed and analyzed by micromagnetic simulations. The SPLD is based on an antiferromagnet-ferromagnet bilayer structure, in which the antiferromagnetic layer supports the interfacial Dzyaloshinskii-Moriyainteraction and the out-of-plane exchange-bias field for stabilizing a zero-field skyrmion. By varying the local exchange-bias field, artificial pinning sites are introduced to trap skyrmions. Depending on the input currents and initial position of skyrmions at different pinning sites, different logic functions can be realized.Micromagnetic simulations show that the proposed SPLD has a robust performance, even under thermal fluctuations and inhomogeneity effects. Our work can provide insights for the design of programmable spin-logic devices. DOI: 10.1103/PhysRevApplied.15.064004 I. INTRODUCTION Magnetic order that can be harnessed via a magnetic field or electrical current is widely explored for memory, logic, and sensor applications [ 1–5]. The demand for scal- ing down the size of magnetic devices grows with devel- opments in information processing and storage technolo- gies. To this end, nanosized magnetic textures have been extensively studied with promising prospects on dense magnetic devices. A magnetic skyrmion is a newly dis- covered topological magnetic texture, which has received much attention due to its prominent properties, such as nanoscale size, nontrivial topology, and low driving cur- rent density [ 5–10]. These properties lead to proposals for a range of prototypes of skyrmionic devices, such as skyrmion racetrack memory [ 5,8,10–12] and skyrmion logic [ 13–17]. Boolean logic devices comprise the basic elements in modern electronic circuits. Conventional logic devices can be classified into two broad categories: a fixed logic device (FLD) and a programmable logic device (PLD) [ 18]. In *guoqiangyu@iphy.ac.cn †xfhan@iphy.ac.cnrecent years, many FLDs that are based on magnetic devices have been proposed. For example, a single mag- netic domain, domain wall, or skyrmion can be employed as information bits to achieve different logic functionalities in FLD [ 13–17,19–26]. However, the logic functional- ity of the FLD is fixed once it has been manufactured, which limits flexibility in applications. In contrast, a PLD can be reconfigured and, hence, possesses multiple func- tionalities. Nonetheless, there are only a few research works reporting the realization of a skyrmion-based PLD [16]. Similar to the racetrack device, in a skyrmionic programmable logic device (SPLD), the input and out- put binary data bits “1” and “0” are encoded by the state with and without a skyrmion, respectively. Based on the voltage-controlled magnetic anisotropy effect, some SPLDs have been proposed [ 16,17]. However, racetrack- based devices may suffer from the skyrmion Hall effect (SKRHE), thermal fluctuations, and edge roughness, espe- cially in constricted devices [ 7,12,27–31]. Here, we report a SPLD with complete Boolean logic functions (16 functions, including AND,NAND ,XOR,e t c . ) based on artificially induced skyrmion pinning sites and current-driven skyrmion motion. Depending on the ini- tial skyrmion position and applied currents, full logic 2331-7019/21/15(6)/064004(9) 064004-1 © 2021 American Physical SocietyZ. R. YAN et al. PHYS. REV. APPLIED 15,064004 (2021) functionalities can be achieved. The proposed scheme does not suffer from the SKRHE and skyrmion-edge repulsion, since the working principle does not involve long-range skyrmion diffusion and skyrmion-edge interactions. Fur- thermore, simulations with thermal fluctuations and inho- mogeneity effects also demonstrate the robustness of the proposed SPLD device. II. MODEL AND METHOD The proposed logic device is shown in Fig. 1(a). The antiferromagnet (AFM)-based heterostructure is patterned into a crossbar with a center region of 400×450 nm2. The antiferromagnetic layer provides the interfacial Dzyaloshinskii-Moriya interaction (DMI) and an exchange-bias field to the ferromagnetic (FM) layer, which endows zero-field stability to the skyrmion. Elec- trical currents are designed as input signals with identical amplitude, I1and I2, which are orthogonal to each other. Due to the spin Hall effect in the AFM layer [ 32–37], the applied current generates spin-orbit torque and drives skyrmion motion in the FM layer [ 8,38,39]. To achieve dif- ferent logic functions, the skyrmions must be initializedin the ferromagnetic layer by injecting a spin-polarized current through four different magnetic tunnel junctions (MTJs) marked as A,B,C,a n d D.M T J Dis also used to read the skyrmion as an output signal via the tunneling magnetoresistance (TMR) [ 9,10]. As shown in Figs. 1(b) and 1(c), the FM layer that hosts the skyrmions serves as the free layer of the MTJ, and the TMR is determined by the relative magnetization state between the FM layer and the fixed layer of the MTJ. The presence (absence) of a skyrmion can be defined as the high- (low-) resistance state. The local exchange-bias field can be manipulated by scanning-tip-based field cooling [ 40], x-ray exposure [ 41], or an electron beam to achieve effective pinning for the skyrmion [ 42]. Figure 1(d) shows the layout of pinning sites and MTJs. These pinning sites are designed for posi- tioning the skyrmions. The input currents determine the direction of skyrmion motion associated with the SKRHE [7,30,31]. In the proposed device, we set the skyrmion Hall angle ( θSKRHE ) to about 45° for clarity, and this parameter can be optimized for different materials. Figures 1(e)–1(h) show the direction of skyrmion motion under different input current schemes. The influence of different θSKRHE values will be discussed later. (b) (a) (c)(d) (e) (f) (g) (h) FIG. 1. Setup of device and motion of skyrmions under input current. (a) Illustration of the device configuration in FM-AFM bilayer. AFM layer (blue) is patterned into cross-shaped Hall bar, and FM layer (gray) cover the center region. Two orthogonal currents with identical amplitude, I1and I2, are injected into AFM layer as input signals. Four magnetic tunnel junctions (MTJs), marked as A,B,C, and D, are used for writing skyrmions. MTJ Dis also used for reading output voltage. (b),(c) Spin structures in MTJ of parallel (P) and antiparallel (AP) states, respectively. Spins in AFM layer are reoriented into local in-plane order, resulting in local exchange-bias field and pinning sites (PSs). (d) Layout of pinning sites and MTJs. (e)–(h) Skyrmion motion in FM layer under different input currents.Jrepresents overall current direction, and vrepresents skyrmion velocity. I 1=0a n d I2=0 indicate no input current (or equivalently, input 0), leading to no skyrmion motion. I1=0(I1=1) and I2=1(I2=0) indicate applied in-plane current flowing along y(x)a x i s and skyrmion moving along vector ( −1, 1) [(1, 1)]. I1=1a n d I2=1 indicate current flow along vector (1, 1) and skyrmion moving along yaxis. 064004-2SKYRMION-BASED PROGRAMMABLE . . . PHYS. REV. APPLIED 15,064004 (2021) To simulate skyrmion motion and design a skyrmionic programmable logic device, the micromagnetic simulation package Mumax3 is employed [ 43]. In the sim- ulation, the magnetic system is discretized into a mesh of dimensions 1 ×1×1n m3. The material-related parameters are as follows: saturation magnetization (M S), 1.0 ×106A/m; exchange stiffness constant ( A), 1.5×10−11J/m; Dzyaloshinskii-Moriya constant ( D), 2m J / m2; perpendicular magnetic anisotropy (PMA) con- stant ( Ku), 0.9×106J/m3; spin Hall angle ( θSHE), 0.4; and damping coefficient ( α), 0.3. The dipole-dipole interaction is also considered. The working principle of the proposed device is irrespective of the edge and, hence, we do not consider the edge effect. The periodic boundary condi- tion is adopted to improve the simulation efficiency (for more details of simulations with larger areas and open boundary conditions, see Appendix B). The pinning site induced by the local exchange-bias field (LEBF) serves as an effective potential well for the skyrmion. Such a pinning effect has been systematically investigated in our previous work [ 12]. It is shown that the pinning strength can be engineered by the size of the potential well ( DE) and the intensity of the exchange-bias field ( HE). For large DEand HE, the pinning effect is strong and the skyrmion is eas- ily trapped by the pinning site. In our model, DEis 60 nm and the nearest distance between different pinning sites is 240/√ 2 nm. The effective local exchange-bias field, HE, in the circle is 0.02 T (field direction along the yaxis because the Néel orders are reconfigured by local thermal excitations induced by x-ray exposure [ 41]o ra ne l e c t r o n beam [ 42] and finally point to the +ydirection), while HE outside the circular pinning region is 0.02 T (field direc- tion along the zaxis because the Néel orders in the AFM layer are initially uniformly pointing to the +zdirection). We consider a layered AFM and assume that the LEBF is not affected by magnetization dynamics in FM layer [41,42]. Thermal fluctuations ( ∼300 K) are included. The effect of inhomogeneity is introduced by a 5% variation of saturation magnetization, exchange stiffness constant, DMI constant, and anisotropy in different grains (grain size approximately10 nm). III. SKYRMION-BASED PROGRAMMABLE LOGIC GATES We first explain the working principle of the logic opera- tion for a XOR gate as an example. As shown in Fig. 2(a1), the clean operation is first executed by applying a large current pulse ( ∼1012A/m2) to annihilate skyrmions on all MTJ sites. This operation can wipe out previously configured functions. To initialize the XOR function, two skyrmions are created at positions Band Cby local injec- tion of a spin-polarized current [Fig. 2(a2)]. Then, elec- tric current pulses with duration tp=3 ns and density Jp=2×1011A/m2are injected as inputs. For the casewith input signal I1=0a n d I2=0 [Fig. 2(a3)], no electric current is injected and the skyrmions stay in their origi- nal position. Therefore, a low-resistance state in MTJ Dis read out, showing an output signal of “ R=0,” as shown in Fig. 2(a4). Figures 2(b1), 2(c1), and 2(d1) show the results for different combinations of input signals. If the input signal is I1=0(I1=1) and I2=1(I2=0), as shown in Fig. 1(b3) [Fig. 1(c3)], the applied in-plane current flows along the y(x) axis, and the skyrmions both move along the vector ( −1, 1) [(1, 1)]. After injection of the input current pulse, the skyrmion in the B(C) site moves to the Dsite. Finally, the high-resistance state in MTJ Dis read out, showing an output signal of “ R=1” [Figs. 2(b4) and2(c4)]. As for the case with I1=1a n d I2=1, which is similar to the case of I1=0a n d I2=0, no skyrmion moves into the Dsite, as a result R=0 [Fig. 2(d4)]. In Fig. 3, we show the simulation results for several other logic gates. Different initial skyrmion configurations correspond to different logic operations. For example, ini- tializing skyrmions at Band Csites leads to the function of the XOR logic gate, as discussed above. The initializa- tion of skyrmions at sites A,D,AD,BC,ABC ,a n d BCD leads to AND,NOR,NXOR ,XOR,OR,a n d NAND logic gates, respectively. It should be noted that the device also has a stable performance under thermal fluctuations (300 K) and inhomogeneity effects. The proposed device can realize 16 logic functions (see Table I), including material impli- cation [ Bfor inverse implication (NIMP), Cfor reverse- inverse implication (RNIMP) ,ACD for inverse implication (IMP), and ABD for reverse implication (RIMP)], inoper- ation ( ABfor I1and ACfor I2),NOT gate ( BDfor NOT I2and CDfor NOT I1), false (no skyrmion set), and true (ABCD ). IV . DISCUSSION A. Influences of θSKRHE andθSHE The examples discussed above show that the skyrmion- shift operation determines the output signal, indicating that the logic operation relies on stable skyrmion positioning and precise skyrmion-shift operation. Thus, θSKRHE is an important parameter in this device. For specific values of θSKRHE , the design of the layout of pinning sites and MTJs should be further optimized, as shown in Appendix A [Figs. 5(d),5(h),a n d 6]. On the other hand, θSHEalso influ- ences the amplitude and duration of the input current pulse and is closely related to the operation frequency. To further understand the role of θSHEand the magnitude of the input current pulse Jp, we employ Thiele’s approach [8,38,39]. Their relationship can be obtained from the Thiele equation: /vectorG×/vectorv−α↔D·/vectorv+4πB↔R·/vectorJ=0. (1) 064004-3Z. R. YAN et al. PHYS. REV. APPLIED 15,064004 (2021) (a)(a1) (a2) (a3) (a4) (b)(b1) (b2) (b3) (b4) (c)(c1) (c2) (c3) (c4) (d)(d1) (d2) (d3) (d4)XOR FIG. 2. Simulated logic device with XOR function. Complete process includes four sets of inputs, (a) I1=0a n d I2=0, (b) I1=0a n d I2=1, (c) I1=1a n d I2=0, (d) I1=1a n d I2=1, and four operations, clean, initialization (initial for short), input, and read. (a1)–(d1) “Clean” operation wipes out all skyrmions in FM layers by using a large current pulse. (a2)–(d2) “Initialization” operation creates skyrmions in specified pinning sites to initialize specific logic function. (a3)–(d3) “Input” operation injects input signals (currents) and drives skyrmion motion. Black arrows indicate input currents. Arrows in green, red, and blue show the trajectory of skyrmionsafter injecting current. (a4)–(d4) “Read” operation reads output voltage signal from MTJ D. Output signals, R, are indicated by dashed arrows. Here, /vectorG=(0, 0, g)is gyrocoupling vector with g=4π, /vectorv=(v x,vy)is the velocity of the skyrmion, and /vectorJ=(Jp,0)is the current density.↔D=/bracketleftbigg D00 0 D0/bracketrightbigg is the dissipative tensor, and↔R=/bracketleftbigg cosϕ0 sinϕ0 −sinϕ0cosϕ0/bracketrightbigg is an in-plane rotation matrix with ϕ0=0 for the Néel skyrmion. αis the Gilbert damping and coefficient B= [(γ0/planckover2pi1θSHE LscIρ)/(2eM sL)] is linked to the spin Hall effect, where γ0the gyromagnetic ratio, /planckover2pi1is the reduced Planck constant, θSHEis the spin Hall angle, eis the electron charge, M sis the saturation magnetization, Lis the thick- ness of the ferromagnetic layer, Lscis the scaling length equal to the strip width, and Iρis the shape factor of the skyrmion. The solution of Eq. (1)is given (without boundary conditions) as−gvy−αD0vx+4πBJ p=0, gvx−αD0vy=0.(2) Then, we can obtain the relationship between the velocity of the skyrmion and the input current pulse: v=/radicalBig v2x+v2y=4πBJ p/radicalBig α2D2 0+g2.( 3 ) Therefore, the shift of the skyrmion between nearest pin- ning sites can be described as S=vtp,( 4 ) where Sis the distance between the nearest pinning sites and tpis the duration of input current pulse. From Eqs. (3) and(4), one obtains 064004-4SKYRMION-BASED PROGRAMMABLE . . . PHYS. REV. APPLIED 15,064004 (2021) A (AND)D (NOR)AD (NXOR)BC (XOR)ABC (OR)BCD (NAND)I2I1FIG. 3. Simulated results of different logic functions. Func- tion is configured by settingskyrmions in specified pinning sites. Corresponding initial states and logic functions (written inbrackets) are shown in the first row. 1” and 0 indicate output signals from MTJ D.A r r o w s in green, red, and blue show trajectory of skyrmions after injecting different input currents. tpJpθSHE=C,( 5 ) where C=SeM sL/radicalBig α2D2 0+g2/2πγ0/planckover2pi1IρLscis a constant for a given device. In our system, for tp=3n s , Jp= 2×1011A/m2,a n dθSHE=0.4, C=1.2×1011ns A/m2isobtained. The relationship between tpand Jpis shown in Fig. 4. It is obvious that tpis inversely proportional to Jp. As mentioned above, large DEand HElead to a rela- tively strong pinning effect. Although the LEBF potential well can pin the skyrmion, it also causes a problem when driving the skyrmion. Thus, Jpshould be larger than the TABLE I. Sixteen logic functions programmed by different skyrmion configurations. Gray cells indicate input values ( I1and I2). Corresponding initial states for different logic functions (written in brackets) are shown in the first row. Orange cells represent common logic functions. Blue cells indicate material implication. Green cells indicate inoperations and a NOT gate. Pink cells represent false and true. A(AND) B(NIMP) C(RNIMP ) D(NOR) AB(I1) AC(I2) AD(NXOR) BC(XOR) ABC (OR) ABCD (True) --(False) ABD (RIMP) ACD (IMP) BCD (NAND) BD(NOT I2) CD(NOT I1)I2I1 I2I1 064004-5Z. R. YAN et al. PHYS. REV. APPLIED 15,064004 (2021) (Fe-Mn) FIG. 4. Duration of input current ( tp) as a function of current amplitude ( Jp) for various θSHE. Curves are plotted according to Eq. (6) with different values of θSHEmarked by blue 0.02 for Fe-Mn [ 49], green (0.3 for IrMn 3[35]), and black (0.08 for Pt-Mn [ 34]). critical driving current ( Jc) to overcome the trapping force and drive the skyrmion out of the LEBF potential well. The influence of DEand HEon Jcare similar to that of pinning strength. Therefore, DEand HEneed to be engi- neered to obtain optimal Jcand Jp[12]. The spin Hall angle also plays an important role in the device. Antiferro- magnets with large spin Hall angles, such as IrMn 3[35]o r Pt-Mn [ 34], are expected to increase the speed and energy efficiency of the device. To build a robust device with high performance and energy efficiency, an antiferromagnetic metal with a large spin-orbit coupling (providing large enough values of DMI andθSKRHE ) and exchange-bias effects at room temperature is necessary. This specific requirement limits the variety of alternative materials. IrMn 3and Pt-Mn are promising candidates for the antiferromagnetic layer, but other anti- ferromagnets can also be explored. On the other hand, the initialization of skyrmions leads to extra energy con- sumption and longer operating time, which need to be optimized. One solution is to increase the spin polariza- tion of the electric current to lower the critical current for writing of the skyrmions. The dipolar field originating from the fixed layer of MTJ is neglected in our simula- tion for simplicity, but for sub-100-nm MTJ the dipolar field may affect the pinning strength and needs to be fur- ther modified [ 12]. By using an optimized MTJ structure, the energy consumption of the initialization operation can be greatly reduced [ 44]. It is also possible to employ the electric-field-assisted method to further optimize the device [ 45].B. More inputs of instructions For a programmable logic device, additional inputs of instructions are needed for more functions. In general, the number of logic functions ( NF) is determined by the number of instructions ( Nins). For binary instruction, a device with 16 complete logic functions needs four addi- tional inputs of instructions. The four terminals in our proposal correspond to the four MTJs (at the A,B,C, and Dsites). For other types of magnetic logic device [13,46–48], it is found that more terminals of instructions lead to more functions, indicating the importance of Nins. Such multi-instruction schemes can also be implemented in logic devices based on other physical systems, providing a guideline for the design of a programmable logic device with multiple functions. V . CONCLUSION We propose a skyrmionic programmable logic device that includes 16 types of logic functions. The logic func- tions are configured by initializing the layout of skyrmions. The pinning sites can enhance the stability and certainty of skyrmion motion, and the device thus shows significant robustness under the effect of inhomogeneity and thermal fluctuations. Our work could stimulate the design of a PLD with complete functions and serve as a candidate for future skyrmionic logic applications. ACKNOWLEDGMENTS This work is funded by the National Key Research and Development Program of China (Grant No. 2017YFA020 6200), the Science Center of the National Science Foun- dation of China (Grant No. 52088101), the National Nat-ural Science Foundation of China (NSFC, Grants No. 11874409 and No. 11804380), the Beijing Natural Sci- ence Foundation (Grant No. Z190009), the NSFC-Science Foundation Ireland (SFI) Partnership Programme (Grant No. 51861135104), and the K. C. Wong Education Foun- dation (Grant No. GJTD-2019-14). Y.L. also acknowl- edges support from the Institute of Physics, Chinese Academy of Sciences, through an International Young Sci- entist Fellowship (Grant No. 2018001). R.K.L. acknowl- edges support from Spins and Heat in Nanoscale Electronic Systems (SHINES), an Energy Frontier Research Cen- ter funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE- SC0012670. APPENDIX A: LAYOUT OF PINNING SITES AND MTJS The skyrmion Hall angle determines the direction of motion of the skyrmion under an applied current. This machine thus influences the layout of pinning sites and 064004-6SKYRMION-BASED PROGRAMMABLE . . . PHYS. REV. APPLIED 15,064004 (2021) (a) (b) (c) (d) (e) (f) (g) (h)v vJ Jv vv vJ JJ Jy yy yy C B B ACD/RAD/R y–x –xx xx xqSKRHE qSKRHE qSKRHEqSKRHEqSKRHEqSKRHE FIG. 5. Sketch of the layout of pinning sites and MTJs in different skyrmion Hall angle. (a)–(c),(e)–(g) Skyrmion motion with small and large skyrmion Hall angles, respectively. (d),(h) Layouts of pinning sites and MTJs. Solid blue lines in (c),(g) are parallel to thosein (d),(h). MTJs in the proposed skyrmionic programmable logic device, as shown in Figs. 5(d) and5(h).F o rθSKRHE ≈45°, the layout is shown in Fig. 1(d).F o r θSKRHE >45° (θSKRHE <45°), the layout can be obtained by com- plete counterclockwise (clockwise) rotation of the one at θSKRHE=45°. For example, when the damping constant is XOR XOR XOR XOR FIG. 6. Simulation results for XOR gate with damping α=0.01.0.01, the skyrmion Hall angle is close to 90°. Thus, the layout of the device is rotated about 45° clockwise. The simulation results are shown in Fig. 6. The performance of the proposed device is still robust. This also implies that damping should be taken into account when optimizing the device. XOR XOR XOR XOR FIG. 7. Simulation results for XOR gate with sample size 800×800×1n m3under open boundary conditions. 064004-7Z. R. YAN et al. PHYS. REV. APPLIED 15,064004 (2021) APPENDIX B: MICROMAGNETIC SIMULATIONS WITH LARGE AREA AND OPEN BOUNDARY CONDITIONS Figure 7shows the simulation results of the XOR gate with sample size 800 ×800×1n m3and open boundary conditions. The results are still robust and similar to those shown in Fig. 2. [1] A. Wang and W. D. Woo, Static magnetic storage and delay line, J. Appl. Phys. 21, 49 (1950). [2] M. Kestigian, A. B. Smith, and W. R. 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PhysRevB.103.024442.pdf

PHYSICAL REVIEW B 103, 024442 (2021) Domain-wall dynamics in a nanostrip with perpendicular magnetic anisotropy induced by perpendicular current injection I. L. Kindiak,1,2P. N. Skirdkov ,2,3K. A. Tikhomirova ,4K. A. Zvezdin ,2,3 E. G. Ekomasov ,5,6,7and A. K. Zvezdin3 1Moscow Institute of Physics and Technology, Institutskiy pereulok 9, 141700 Dolgoprudny, Russia 2New Spintronic Technologies, Russian Quantum Center, Bolshoy Bulvar 30, building 1, 121205 Moscow, Russia 3Prokhorov General Physics Institute of the Russian Academy of Sciences, Vavilova 38, 119991 Moscow, Russia 4Skolkovo Institute of Science and Technology, Skoltech, Building 3, 143026 Moscow, Russia 5University of Tyumen, Institute of Physics and Technology, Volodarskogo 6, 625003 Tyumen, Russia 6Bashkir State University, Institute of Physics and Technology, Zaki Validi 32, 450076 Ufa, Russia 7South Ural State University (National Research University), Laboratory of Functional Materials of the Scientific and Educational Center “Nanotechnology”, Lenin Avenue 76, 454080 Chelyabinsk, Russia (Received 11 October 2019; revised 20 November 2020; accepted 11 January 2021; published 26 January 2021) A numerical and analytical study of the dynamics of domain walls (DWs) in a magnetic tunnel junction with perpendicular magnetic anisotropy in a free layer is presented. Equilibrium states of the domain wall areobtained for various widths of the structure. The corresponding symmetries of the components of spin transfertorques T ST Tand the polarizer directions favoring stable DW motion under perpendicular current injection are obtained. The DW steady motion with velocities up to 200 m /s at current densities below 106A/cm2is reported. The Walker breakdown is demonstrated, and the dynamics of the postthreshold DW motion is investigated forvarious configurations of torques and polarizer directions. To have analytical insight into the investigated regimesof DW dynamics a theoretical model is developed and verified by micromagnetic simulations. DOI: 10.1103/PhysRevB.103.024442 I. INTRODUCTION Recently, studies of domain-wall (DW) dynamics in ferro- magnetic nanowires have attracted much attention [ 1–5]. This is due to both purely fundamental interests and promising ap-plications. Controlled DW dynamics can be used in spintronicdevices such as racetrack memory [ 6], logic units [ 7–10], and spintronic memristors [ 11–13], which imitate neuronal synapses [ 14] and have great potential for use as the hard- ware basis of neuromorphic computational architectures. In this context, materials with perpendicular magnetic anisotropy(PMA) are of particular interest. Compared to materialswith in-plane magnetic anisotropy, magnetic tunnel junctions(MTJs) with PMA [ 15–17] have attractive advantages, such as a lower critical excitation current, higher thermal stabil-ity [18–20], and a smaller DW width. Thus, PMA-based MTJs have the potential to be next-generation, energy-efficient, high-density spintronic devices. Early DW-based spintronic device concepts required in- duced magnetic fields to control DW dynamics [ 1,21,22]. However, it was found that this approach is hardly suitablefor close-packed arrays of nanoscale devices due to significantcross-talk effects. Spin-orbit torques (see the review in [ 23]) are another possible way to excite DW. Possible types ofspin-orbit torque required for the excitation of steady DWmotion for various anisotropy and DW types were first ana-lyzed numerically in [ 24]. Such an approach of spin-orbit DW excitation was recently considered in detail for PMA materi-als [25–28]. In addition, spin-orbit coupling in ferromagnetscan lead to the anomalous Hall effect and anisotropic mag- netoresistance, which can also move the DW [ 29]. However, spin-orbit structures are more difficult to use to determinethe position of a DW compared to magnetic tunnel junc-tions, with which it is possible to determine the position of aDW through the tunneling magnetoresistance. Thus, the spin-orbital torques, despite the higher efficiency of DW excitation,may be less attractive for neuromorphic and other real-lifeapplications [ 14]. An alternative approach based on current-induced DW mo- tion has been the subject of many experimental [ 2,4,30–32] and theoretical [ 33–37] studies. In these works, nanostruc- tures were usually represented by a long and narrow magneticnanostrip containing a DW. For this geometry, there are twopossible current configurations: current in plane (CIP), whenspin-polarized current flows in the plane of the magnetic film,and current perpendicular to the plane (CPP), when it flowsperpendicular to the magnetic film surface. The CIP case wasanalyzed in detail for both planar and perpendicular magneticanisotropies [ 38–41]. For the case of CPP geometry, it was demonstrated numerically [ 42] and experimentally [ 43,44] that the DW velocities can be up to two orders of magnitudehigher than in the CIP configuration, provided that the cur-rent densities are equal. Thus, the CPP configuration requiresrelatively low current densities for efficient DW dynamicsexcitation [ 45,46]. The drawback of this configuration is the higher electric current required for efficient DW motion in alarge cross-section area, which, however, can be addressedby using local current injection [ 47]. A detailed analytical 2469-9950/2021/103(2)/024442(8) 024442-1 ©2021 American Physical SocietyI. L. KINDIAK et al. PHYSICAL REVIEW B 103, 024442 (2021) description of DW dynamics under the CPP injection with analysis of various polarizers was presented recently [ 48]. However, all these results correspond to the in-plane or evenzero magnetic anisotropy, while the case of the CPP geom-etry in combination with the PMA ferromagnetic nanostripremains unclear. Based on the above arguments, perpendicular current injec- tion seems to be the most efficient way of DW excitation, withthe exception of spin-orbit torques, which, however, are lim-ited by the difficulty of reading the DW position and are notas effective for neuromorphic applications [ 14]. Indeed, the maximum velocities and minimum required current densitieshave been reported for the CPP geometry of current injec-tion [ 42,44–46,48]. However, all these results were obtained for the case of in-plane anisotropy, while PMA may furtherimprove energy efficiency. At the same time, despite the largenumber of above-mentioned works devoted to domain walls innanostrips, there is no clear understanding of the DW motionin MTJs upon perpendicular current injection into a ferromag-net with PMA. To date, there has been only one experimentaldemonstration [ 14] of DW motion in an MTJ with PMA in the CPP geometry, which encourages detailed numericaland analytical study of the mechanisms and features of DWdynamics. In this paper, we investigate in detail the DW dynamics induced by the perpendicularly injected spin-polarized currentin an MTJ with PMA. We report the results of micromagneticmodeling on the stable DW states in a free layer of variouswidths. We also study in detail the influence of different po-larizer directions and torque types on the DW dynamics con-ditions and features. The DW steady motion with velocities upto 200 m /s at current densities below 10 6A/cm2is reported. We analyze DW transformation during motion and demon-strate the Walker breakdown for each DW steady-motionregime. Finally, we provide analytical insight that shows goodagreement with micromagnetic simulations and helps to ana-lyze the effect of different parameters on the results. II. SYSTEM AND METHODS Let us consider an MTJ nanostrip [Fig. 1(a)], which con- sists of a ferromagnetic polarizer layer, a spacing insulator,and a ferromagnetic free layer with a single domain wall.The following geometrical parameters of MTJ were cho-sen: free-layer thickness h=2.2 nm, length L=5000 nm, widthw=10–300 nm. The magnetic parameters were cho- sen according to experimental results [ 14]: uniaxial magnetic anisotropy along the zaxis, saturation magnetization M S= 1050 emu /cm3, anisotropy constant K=7×106erg/cm3, exchange constant A=2×10−6erg/cm, damping parameter α=0.005. The magnetization dynamics in a nanostrip can be described by the Landau-Lifshitz-Gilbert equation with anadditional term responsible for the spin transfer torque: ˙M=−γM×H eff+α MSM×˙M−TST T, (1) where Mis the magnetization vector, γis the gyromagnetic ratio,αis the Gilbert damping constant, MSis the saturation magnetization, and Heffis the effective field consisting of the magnetostatic, exchange, anisotropy, and demagnetizationfields. The spin transfer torque can be written [ 49–51]a s FIG. 1. (a) Schematic representation of the considered MTJ structure with a single domain wall. (b) The dependence of the angles ϕbetween the xaxis and the average DW magnetization and /Phi1 between the xaxis and the DW plane and their difference in the relaxed state on the MTJ width in the xdirection. Inset: Néel and Bloch domain walls. Red and blue correspond to the perpendicular component of M, white is zero, and arrows show the direction and magnitude of planar components. a sum of two orthogonal components, TST T=TST+TFLT, where the Slonczewski torque (ST) equals TST=−γaJM× [M×mref]/MSand the fieldlike torque (FLT) equals TFLT= −γbJ[M×mref]. Here, mrefis a unit vector along the magne- tization direction of the polarizer layer; aJ≈¯hjP/(2heM S), where his the thickness of the free layer, ¯ his the reduced Planck constant or Dirac constant, jis the current density, e>0 is the charge of the electron, and P=0.4i st h es p i n polarization of the current; bJ=ξCPPaJ, where ξCPPis taken to be about 0.4 [ 45]. Micromagnetic simulations consisting of Eq. ( 1) numerical integration on a 2 ×2 nm rectangular grid were performed using our SPINPM micromagnetic finite-difference code based on the fourth-order Runge-Kutta method with adaptive timestep control for time integration. To focus on the effect of thespin-polarized current, we ignored the Dzyaloshinskii-Moriyainteraction, Oersted fields, and thermal fluctuations. III. MODELING RESULTS AND DISCUSSION First of all, micromagnetic modeling of relaxation in a free layer with a single DW was carried out for different widths.One can relate DW evolution from Néel to Bloch type tothe magnitude of ϕ, which is the angle between the xaxis and the average DW magnetization in the relaxed state. TheNéel domain wall corresponds to ϕ=90 ◦, and the Bloch wall corresponds to ϕ=0◦. The range of ϕvalues from 90◦to 0◦ corresponds to the hybrid state of the DW. The dependence of the angle ϕon the free-layer width in the xdirection is illustrated in Fig. 1(b). For widths up to 110 nm, the mag- netization relaxes to the Néel DW, for a width of more than110 nm, the wall becomes a hybrid, and for a width above300 nm, the Bloch-type DW is an equilibrium state. Theseresults are consistent with an experimental work [ 14]. Indeed, 024442-2DOMAIN-WALL DYNAMICS IN A NANOSTRIP WITH … PHYSICAL REVIEW B 103, 024442 (2021) TABLE I. Summary of the DW motion (no, no spin current in- duced DW motion; slow motion, extremely slow DW motion, which can be neglected; fast motion, steady DW motion). mref=(1,0,0)mref=(0,1,0)mref=(0,0,1) Néel TFLT slow motion no fast motion TST fast motion no slow motion Bloch TFLT no slow motion fast motion TST no fast motion slow motion although DW magnetization at widths of about 300 nm is not parallel to the xaxis, the DW plane also has a certain tilt at an angle /Phi1from the xaxis; therefore, the type of DW should be defined by the difference ϕ−/Phi1, which is almost zero for the mentioned widths. The reason for the tilt of the DWplane is a rather strong magnetostatic field. Our simulationdemonstrates that at half the saturation magnetization, the DWplane is perfectly aligned with the xaxis, and the Bloch-type DW becomes stable at widths of about 150 nm. It is alsopossible to avoid this tilt by designing a low magnetostaticshape, for example, with the half-ring geometry, as in [ 45]. To identify the conditions for DW motion in the outlined MTJ structure, we studied the effect of the spin current whiletaking into account T STandTFLTseparately. We chose 50- and 300-nm-wide nanostrips (with Néel and Bloch initialDWs in a free layer, respectively) as the system for the study.For each geometry, we considered a polarizer magnetizedin the x,yand zdirections. Thus, six configurations (three polarizations and two torques) for each DW type (width) wereconsidered (see Table I). Let’s start with planar polarizers. In the case of m ref= (1,0,0), a Bloch-type DW cannot be excited by either TST orTFLT. A Néel-type DW starts a considerably fast steady motion with TSTaction. However, TFLT also leads to the Néel DW steady motion, but the velocity in this case is ex-tremely low ( <1m/s). The origin of such DW behavior will be explained later. In the case of m ref=(0,1,0) we have the opposite situation: a Néel-type DW cannot be excited byeither T STorTFLT, while a Bloch-type DW demonstrates a considerably fast steady motion under the action of TSTand an extremely slow steady motion under TFLTaction. It is worth noting that in both no-motion cases, the corresponding mref directions are parallel to the DW magnetization in the DW core, and the vector product of the torques is zero. In the case of a perpendicular polarizer [ mref=(0,0,1)], both initial DW types start very slow ( <1m/s) steady motion under the action of TST. At the same time, TFLT induces steady motion of significant velocity for both initial DW types. As seen from Table I, four steady-motion configurations were observed: TSTexcitation with mref=(1,0,0) for Néel andmref=(0,1,0) for Bloch and TFLT excitation with mref=(0,0,1) for both DW types. Now let’s focus on DW dynamics in these torque and polarizer configurations. In allthese cases, after the DW motion is initiated by the spincurrent, which reaches its amplitude instantaneously, ϕstarts to tilt gradually during the acceleration of the DW until thesteady-motion regime is reached and the DW is transformedinto a hybrid-type DW [see, for example, Fig. 2(a)]. There- after, the DW continues its steady motion with constant ϕ FIG. 2. Displacement of the Néel DW (70 nm wide) vs time in the case of (a) steady motion before Walker breakdown for j=0.04×106A/cm2(in the case of a perpendicular polarizer), (b) oscillatory DW motion after Walker breakdown for j=0.9× 106A/cm2(in the case of a perpendicular polarizer), and (c) DW shift and stop after Walker breakdown for j=1.2×106A/cm2(in the case of a planar polarizer). until it reaches the edge of the nanostrip. It is worth noting that planar and perpendicular polarizers induced motion inopposite directions, which will be explained by the analyticalmodel later. With an increase in the current density, the steady-motion velocity and ϕalso increase until the current density reaches the critical value j wand Walker breakdown occurs. However, this breakdown has different behavior for planar andperpendicular polarizers. In the case of a perpendicular polar-izer, once the critical value j wis reached, ϕbegins to change continuously, and the DW starts to oscillate with some averagemotion along the nanostrip [see, for example, Fig. 2(b)]. For the planar polarizer case [with m ref=(1,0,0) for Néel DW andmref=(0,1,0) for Bloch DW], with an increase in the current density from zero to jw, the angle ϕchanged from ϕ0(ϕ0=0 for Bloch DW and ϕ0=π/2 for Néel DW) to ϕ0±π/2, the DW type is transformed into the opposite one, and the DW stops after a transitional shift [see, for example,Fig.2(c)]. The difference between Walker breakdowns can be easily explained by comparing the above mentioned behavior 024442-3I. L. KINDIAK et al. PHYSICAL REVIEW B 103, 024442 (2021) FIG. 3. Average DW’s velocity for different widths vs current density for (a) perpendicular polarizer and (b) planar polarizer. Solid lines with solid dots represent steady motion, dashed lines and open dots show averaged motion during oscillations, and crossed dotsillustrate post-Walker steady motion caused by magnetostatic stabi- lization due to the finite size of the nanostrip. with Table I. Indeed, according to Table I, for a planar po- larizer, when DW changes its type to the opposite one, it canno longer be excited by any of the torques. At the same time,for the case of a perpendicular polarizer, motion is excitedfor both DW types, so the DW continues to move, and ϕ continues to change, which causes the DW oscillations andits continuous transformation. Once the symmetries of the components of spin trans- fer torques ( T ST T) and the polarizer directions that favor the stable DW motion have been obtained, we proceed withfurther quantitative analysis. To this end, series of micromag-netic simulations were carried out at various current densitiesand widths with favorable combinations of polarization andtorque. Modeling results demonstrate that DW velocity de-creases with increasing DW width. Because of this, here, wedisplay in detail the cases of 10, 30, 50, and 70 nm sincefor large widths the velocities become too small, although thedependencies remain the same. The dependences of the average DW velocity, obtained by micromagnetic modeling for the cases of different polar-izer directions and nanostrip widths, are shown in Figs. 3(a) and3(b). The velocity increases with the current density for both planar and perpendicular polarizers [see solid lines withsolid dots in Figs. 3(a) and3(b.) ]. At a low current density, it increases almost linearly, and at higher currents, the growthbecomes nonlinear, and the velocity tends to saturation. Then,upon reaching the critical current density j w, Walker break- down occurs. In the case of a perpendicular polarizer, above the critical current density jw, the DW starts to oscillate with some av- erage displacement along the nanostrip. The average motionduring these oscillations is shown by the dashed line andopen dots in Fig. 3(a). This is not a steady motion, but it is still possible to obtain a certain average velocity whichdoes not depend on the nanostrip length. At the same time,the nanostrip length plays an important role in the case ofcurrent densities slightly higher than j w. In the case of an infinite nanostrip, the DW starts to oscillate immediately afterbreakdown (see the dashed line), while in the case of a finitenanostrip ( L=5000 nm), the DW structure is stabilized due to the magnetostatic interaction near the edge of the nanostrip.Indeed, at current densities slightly higher than j w, DW tends to make one oscillation before reaching the nanostrip edge, soDW should stop and then change the velocity direction forthe first time somewhere near the edge. At the same time,micromagnetic modeling demonstrates that the magnetostaticinteraction prevents this stopping and subsequent velocitydirection change, as it tries to expel DW from the nanostripin the region near the edge. This leads to additional steadymotion above the Walker breakdown [see crossed dots inFig.3(a)] for finite-size samples. Our simulation demonstrates that the range of this post-Walker steady motion decreaseswith an increase of length L. In the case of a planar polarizer [Fig. 3(b)] we observe a different behavior. Here, when the DW reaches its maximumvelocity at the critical current density j w, it transforms into another type DW and then stops. However, this transitionalprocess of acceleration and deceleration requires a certaintime, during which the DW manages to shift significantly(up to several thousand nanometers in our case) or even toreach the edge of the nanostrip. A similar behavior of DWs inshort nanostrips above the Walker breakdown was observedin [46]. Here, we managed to achieve the DW steady-motion velocities up to 200 m /s at current densities below 10 6A/cm2. It is important to note that although these low levels of currentdensity may become relatively close to the noise level, theyare similar to those used experimentally in a similar DW-based system [ 14], which makes them feasible. However, in future experiments, one should be aware of the possibility toencounter noise-related problems. The symmetry of torque considered in this work, in prin- ciple, can be obtained not only in the case of CPP injection.In the first approximation, the cases of a planar polarizer are similar to spin-orbit torques [ 23]. In this regard, the direction of the polarizer can be associated with the direction of po-larization of the current in the spin-orbit layer, and ST andFTL can be associated with the spin Hall and direct Rashbaeffects, respectively. With this in mind, one can find a perfectcorrelation between Table Iand the DW motion conditions in the case of spin-orbit excitation [ 24]. The dependence of the velocity on the current density for the spin-orbit case andCPP injection case with a planar polarizer is also rather close.At the same time, the effect of the perpendicular polarizer andthe corresponding torque symmetry cannot be achieved by thespin-orbit effect due to in-plane polarization of the carriers.This makes the perpendicular polarizer a unique condition foronly CPP injection. In addition, it should be noted that the case of the perpendicular polarizer is the most important in terms ofapplications. Indeed, in this case, no one needs additionallayers for magnetoresistance-based DW detection since thefree layer is perpendicularly magnetized. However, in thecase of a planar polarizer and spin-orbit-based excitation, it isnecessary to add a perpendicularly magnetized analyzer layer.At the same time, the perpendicular polarizer demonstratesrather high DW velocities. Moreover, in this case, after Walkerbreakdown, DW oscillation with some mean displacementstarts, which makes it possible to use this configuration evenat currents exceeding the Walker limit. 024442-4DOMAIN-WALL DYNAMICS IN A NANOSTRIP WITH … PHYSICAL REVIEW B 103, 024442 (2021) TABLE II. Spin transfer torque components in spherical coordi- nates for x,y,zpolarizers. mref (1,0,0) (0,1,0) (0,0,1) Tθ −γaJsinϕγ aJcosϕγ bJsinθ −γbJcosθcosϕ −γbJcosθsinϕ Tϕ γaJcosϕsinθcosθγ aJsinϕsinθcosθ−γaJsin2θ −γbJsinθsinϕ +γbJsinθcosϕ IV . ANALYTICAL MODEL For analytical insight into DW dynamics in our case, let us consider Eq. ( 1) in spherical coordinates ( θandϕare the azimuth and polar angles, respectively) with the energywithin the framework of a one-dimensional model repre-sented by ε=ε exch+εmagn+εan+εsh.an., where exchange energy εexch=A[(∇θ)2+sin2θ(∇ϕ)2],Ais the exchange constant, magnetostatic energy εmagn=2πκM2 ssin2θsin2ϕ, anisotropy energy εan=Ksin2θ,Kis a constant of perpen- dicular magnetic anisotropy, and shape magnetic anisotropyenergy ε sh.an.=−(2π−ζ/2)M2 Ssin2θ. The parameter ζ/lessmuch 1 describes the difference between the demagnetizing fac-tor in the zdirection N zin the considered case and in the case of an infinite film ( Nz=4π−ζ). The parameter κde- termines the difference between magnetostatic fields in thecases of a nanostrip and an infinite film. It can be calcu-lated approximately, considering the demagnetization factorsand magnetostatic interaction of each domain. However, here,we obtain it more accurately by micromagnetic modeling asκ=| /angbracketleftH ms/angbracketright|/4πMS, where /angbracketleftHms/angbracketrightis the magnetostatic field distribution from the micromagnetic simulation, which wasaveraged along the xaxis (across the nanostrip). With this in mind, Eq. ( 1) takes the following form: sinθ˙ϕ−α˙θ=−ωQδ 2 0θyy+ωsinθcosθsin2ϕ +ωQδ2 0sinθcosθ(ϕy)2 +ωQsinθcosθ+Tθ, sinθ˙θ+αsin2θ˙ϕ=ωQδ2 0sin2θϕyy −ωsin2θsinϕcosϕ+Tϕ, (2) where ω=4πγκ MS,Q=K⊥/2πκM2 S,δ0=√A/K⊥,K⊥= K−(2π−ζ/2)M2 S. The spin transfer torque components Tθ andTϕdepend on the polarizer direction mrefand are summa- rized in Table II. Now, let us use a Walker-like assumption. We will look for a solution in the form ϕ=ϕ(t),θ=θ[(y−q(t))/δ(ϕ)], where δ(ϕ)=δ0//radicalbig 1+Q−1sin2ϕandq(t) is the position of DW. Under these assumptions Eq. ( 2) can be reduced to ∂2θ ∂y2−1 δ2(ϕ)sinθcosθ=−1+α2 ωQδ2 0sinθ∂ϕ ∂t −α Qδ2 0sinθsinϕcosϕ +1 ωQδ2 0/parenleftBig Tθ+αTϕ sinθ/parenrightBig .(3)The solution to the homogeneous equation corresponding to Eq. ( 3) in this case is well known and is a kink solution that describes the shape of the domain wall: θ0=2a r c t a n/bracketleftbigg exp/parenleftbigg ±y0+y−q(t) δ(ϕ)/parenrightbigg/bracketrightbigg . (4) Consider the solution to Eq. ( 3)t ob e θ=θ0+θ1, where θ1/lessmuch1. Neglecting small values and bearing in mind that ∂2θ0/∂y2−sinθ0cosθ0/δ2(ϕ)=0, we can rewrite Eq. ( 3)a s ˆLθ1=f(θ0),ˆL=∂2 ∂y2−cos 2θ0 δ2(ϕ), f(θ0)=−1+α2 ωQδ2 0sinθ0∂ϕ ∂t−α Qδ2 0sinθ0sinϕcosϕ +1 ωQδ2 0/parenleftBig Tθ+αTϕ sinθ0/parenrightBig . (5) According to the Fredholm alternative, this equation has a solution if and only if the right side f(θ0) of the equation is orthogonal to the eigenfunction of operator ˆLwith zero eigenvalue, which can be found from the equation ˆLθ(0) 1= 0. For the present problem the required eigenfunction takesthe form θ (0) 1=∂θ0/∂y. Then, considering that ∂θ0/∂y= ±sinθ0/δ(ϕ), the solvability condition is /angbracketleftsin2θ0/angbracketright/parenleftbigg (1+α2)∂ϕ ∂t+αωsinϕcosϕ/parenrightbigg =T, (6) where /angbracketleft ···/angbracketright means integration over yand T=/angbracketleftTθsinθ0+ αTϕ/angbracketrightvaries for different polarizers. After integration over y, one can obtain 2(1+α2)∂ϕ ∂t+2αωsinϕcosϕ=1 δ(ϕ)Ti, (7) where the impact of different polarizer directions Tihas the form Tx=2πγδ(ϕ)s i nϕ(−aJ−αbJ), Ty=2πγδ(ϕ) cosϕ(aJ+αbJ), Tz=2γδ(ϕ)(bJ−αaJ). (8) Let us now consider the stationary motion of the DW. In this case ∂ϕ/∂ t=0 and Eq. ( 7) can be solved directly for each polarizer. There are five stationary solutions up to a period: ϕx1=arccos/parenleftbigg−aJ−αbJ HWx/parenrightbigg at (aJ+αbJ)2<H2 Wx, ϕx2=0, ϕy1=arcsin/parenleftbiggaJ+αbJ HWy/parenrightbigg at (aJ+αbJ)2<H2 Wy, ϕy2=π/2, ϕz=1 2arcsin/parenleftbiggbJ−αaJ HWz/parenrightbigg at (bJ−αaJ)2<H2 Wz,(9) where HWx=HWy=4ακMSand HWz=2πακ MS. To find the velocity of the DW for each ϕicase, recall that ∂θ0/∂t= ∓Vsinθ0/δ(ϕ), which follows from Eq. ( 4). At the same time, Eq. ( 4) will be a solution of Eqs. ( 2) under the following con- dition: Tθ,Tϕ,α,∂φ/∂ t→0. In this case, the second equation 024442-5I. L. KINDIAK et al. PHYSICAL REVIEW B 103, 024442 (2021) in Eqs. ( 2)g i v e s ∂θ0/∂t=−ωsinθ0sinϕcosϕ, which leads to the dependence of the velocity on ϕ: V=ωδ0sinϕcosϕ/radicalbig 1+Q−1sin2ϕ. (10) Using it, the corresponding DW velocities for each solution ϕi are Vx1=πγδ 0 α−(aJ+αbJ)/radicalBig H2 Wx−(aJ+αbJ)2 HWx/radicalbigg 1+(QH Wx)−1/radicalBig H2 Wx−(aJ+αbJ)2, Vx2=0, Vy1=πγδ 0 α(aJ+αbJ)/radicaltp/radicalvertex/radicalvertex/radicalbtH2 Wy−(aJ+αbJ)2 H2 Wy+Q−1(aJ+αbJ)2, Vy2=0, Vz=γδ0 αbJ−αaJ/radicalbigg 1+(2Q)−1−(2QH Wz)−1/radicalBig H2 Wz−(bJ−αaJ)2. (11) It should be noted that the Vx1case is in good agreement with the analytical results for spin Hall effect induced DWmotion, analyzed in [ 27]. This correlates with the similarity in symmetry between the spin Hall effect and the Slonczewskitorque with the x-axis polarizer. Taking into account that αis small and only the term with the torque amplitude ( a JorbJ) without factor αcontributes toϕandVsignificantly [ 52], these solutions fully correspond to Table I. Indeed, in the case of an xpolarizer, there are two solutions: a Bloch DW with zero velocity ϕx2and a moving DW, which starts from an ideal Néel DW at zero current andmoves with a decrease in ϕ x1fromπ/2 with increasing current until Walker breakdown occurs at ϕ=0 and DW stops. After that, the DW becomes an ideal Bloch DW with no motion. Inthe case of a ypolarizer, there are also two solutions: a Néel DW with zero velocity ϕ y2and a moving DW, which starts from an ideal Bloch DW at zero current and moves with ϕy1 increasing from ϕ=0 with increasing current until Walker breakdown occurs at π/2 and the DW stops. After that, the DW becomes an ideal Néel DW with no motion. In both thesecases of planar polarizer, the DW moves under the action ofthe Slonczewski torque, and the contribution of the fieldliketorque is negligible ( a J/greatermuchαbJ); however, FLT can lead to ultraslow motion, which was also observed in micromagneticsimulations. For the zpolarizer, both Néel and Bloch DWs begin to move mostly under the action of the fieldlike torque (sinceb J/greatermuchαaJand the Slonczewski torque leads only to ultraslow motion, which also agrees with the modeling results) withϕ zchanging with increasing current until Walker breakdown occurs at ϕ=π/4+πn,n∈Z. However, in contrast to the case of planar polarizers, when approaching the Walker limit(π/2 or 0), the velocity vanishes; here, Walker breakdown oc- curs at ϕ=π/4, which corresponds to the maximum velocity according to Eq. ( 10). Due to this, the DW does not stop in this case but switches to the nonstationary-motion regime. It FIG. 4. Dependence of ϕoscand the corresponding velocity Vosc on time for HJ=1.2HWzand a nanostrip width of 50 nm, obtained from the equations. is also worth noting that the analytical results also demonstrate the opposite direction of DW motion for zandxpolarizers, the same as in micromagnetic simulations. This strictly followsfrom the opposite signs in V zandVx1. To describe the nonstationary-motion regime, consider Eq. ( 7). In the case of ( bJ−αaJ)2>H2 Wzand a zpolarizer, this equation can easily be integrated with the initial conditionϕ(t=0)=π/4 since it is exactly at this angle that the non- stationary regime starts. The resulting dependence of ϕabove the Walker breakdown, up to a period, is represented by ϕ osc=arctan⎧ ⎨ ⎩/radicalBig H2 J−H2 Wz HJtan⎡ ⎣arctan⎛ ⎝HJ+HWz/radicalBig H2 J−H2 Wz⎞ ⎠ −γt/radicalBig H2 J−H2 Wz⎤ ⎦−HWz HJ⎫ ⎬ ⎭, (12) where HJ=bJ−αaJ. Using this and Eq. ( 10), we can nu- merically calculate the velocity of the DW above the Walkerbreakdown. For example, the time dependence of ϕ oscand the corresponding velocity Voscfor the case HJ=1.2HWzand a nanostrip width of 50 nm is demonstrated in Fig. 4. As can be seen, the analytical model also demonstrates oscillatorymotion above the Walker breakdown for the zpolarizer, which was observed in micromagnetic modeling. To calculate theaverage velocity /angbracketleftV osc/angbracketrightabove the Walker breakdown, this de- pendence can be averaged. It is important to note that the one-dimensional model does not take into account some additional magnetization excita-tions (for example, spin waves, x-axis DW deformations, etc.) considered in micromagnetic modeling. It is, in some aspects,possible to add these excitations indirectly by increasing thedamping parameter αsince additional excitations lead to 024442-6DOMAIN-WALL DYNAMICS IN A NANOSTRIP WITH … PHYSICAL REVIEW B 103, 024442 (2021) FIG. 5. Analytical dependence of the DW velocity on the current for different polarizer directions and widths (10, 30, 50, and 70 nm). In (a), the solid line corresponds to Vz, and the dashed line corre- sponds to /angbracketleftVosc/angbracketright. In (b), the solid line corresponds to Vx1. additional dissipation. For our analytical results, we increase it toα=0.012 to achieve a better fit with simulations. A summary of the analytically obtained DW velocity de- pendence on the current for every polarizer direction anddifferent widths (10, 30, 50, and 70 nm) is presented inFigs. 5(a) and5(b).I nF i g . 5(a) the solid line corresponds toV z, and the dashed one corresponds to /angbracketleftVosc/angbracketright;i nF i g . 5(b) the solid line corresponds to Vx1. We choose Vx1, but not Vy1, because for the considered widths we have the Néel DW atzero current, so for the ypolarizer we had to take V y2, which is zero. These results show good agreement with micromagneticsimulations (see Fig. 3). This proves the applicability and reli- ability of the analytical model. However, despite the fact thatthe analytical solutions perfectly match all regimes (exceptthe post-Walker finite-size steady regime, which obviouslycannot be reproduced by the one-dimensional model) and thevalues of velocities from the micromagnetic modeling, thecritical currents differ slightly from the simulation. This ismainly due to the fact that the magnetostatic interaction in themodel is rather simplified. As we mentioned earlier, we use the parameter κ, which is obtained from the magnetostatic field distribution. This creates a realistic difference betweencases with different widths, and without it the velocities andcritical currents would be the same for every width. However,micromagnetic analysis shows that the magnetostatic field,which determines κ, changes during the DW motion with a change in ϕ. Hence, our estimates of κfor the static DW case give only the correct order for it but not the dependence on thecurrent, which changes the critical currents. V . CONCLUSIONS In this paper, we reported a detailed study, both analyti- cally and by micromagnetic modeling, of DW dynamics ina nanostrip with PMA induced by a perpendicularly injectedspin-polarized current. The stable DW states depending on thenanostrip width were presented. The influence of different po-larizer directions and torque configurations on the dynamicswas analyzed in detail. The polarizer and torque configura-tions at which the steady-state DW motion is feasible werepresented. The Walker breakdown was demonstrated, and thedynamics of the postthreshold DW motion was investigatedfor various configurations of torques and polarizer directions.An analytical model of the system under consideration wasproposed and verified by micromagnetic simulations. Ourresults show the possibility of efficient excitation of DW innanostripes with PMA with velocities up to 200 m /s at current densities less than 10 6A/cm2in the case of perpendicular current injection. Based on all these, one may expect thatthe thin-film PMA-based DW spintronic structures will formthe basic platform for the next generation of magnetic logicdevices, memristors, racetrack memories, etc., possessing avery high energy efficiency. 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PhysRevB.102.180402.pdf

PHYSICAL REVIEW B 102, 180402(R) (2020) Rapid Communications Combing the helical phase of chiral magnets with electric currents Jan Masell ,1Xiuzhen Yu ,1Naoya Kanazawa ,2Yoshinori Tokura ,1,2,3and Naoto Nagaosa1,2 1RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan 2Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan 3Tokyo College, University of Tokyo, Tokyo 113-8656, Japan (Received 29 July 2020; accepted 8 October 2020; published 2 November 2020) The competition between the ferromagnetic exchange interaction and antisymmetric Dzyaloshinskii-Moriya interaction can stabilize a helical phase or support the formation of skyrmions. In thin films of chiral magnets, thecurrent density can be large enough to unpin the helical phase and reveal its nontrivial dynamics. We theoreticallystudy the dynamics of the helical phase under spin-transfer torques that reveal distinct orientation processes,driven by topological defects in the bulk or induced by edges, limited by instabilities at higher currents. Ourexperiments confirm the possibility of on-demand switching the helical orientation by current pulses. This helicalorientation might serve as a novel order parameter in future spintronics applications. DOI: 10.1103/PhysRevB.102.180402 Introduction. In magnetic metals, the magnetization acts on the conduction electrons as a local magnetic field and inducesa spin polarization. When a current is induced, this couplinghas consequences for both the electrons and the magnetizationbeyond the anomalous Hall effect: On the one hand, the spinof the conduction electron locally adapts to the magnetization,which can lead to phenomena such as a topological Hall effectfrom picking up a real-space Berry phase [ 1] or an anisotropic magnetoresistance which reflects the anisotropic magnetic or-der. On the other hand, the magnetization can be spatiallyinhomogeneous and experiences a spin-transfer torque (STT)due to the local reorientation of the spin-polarized current[2,3]. This electrical control of magnetic states is impor- tant for both fundamental research and potential applications[4]. For example, it is exploited in commercially available STT-MRAM devices [ 5] and can be used to move magnetic domain walls [ 6,7], which might lead to shift register memory devices [ 8]. More recently, it was found that magnetic skyrmions can be stabilized in chiral magnets [ 9–11] and arouse great in- terest because of their nanometer size [ 12,13], nontrivial real-space topology [ 14,15], and high mobility [ 16–19], which is interesting for various applications [ 20,21]. In simple chi- ral ferromagnets like FeGe, spin-orbit coupling induces anantisymmetric Dzyaloshinskii-Moriya exchange interaction(DMI) [ 22,23] which can stabilize skyrmion lattices at certain magnetic fields and temperatures. However, the predominantmagnetic phase is not a skyrmion lattice but a topologicallytrivial (multidomain) helical phase [ 24–26]. Figure 1shows how the magnetization in the helical phase winds in the planeperpendicular to the qvector, which defines the orientation of the phase. When applying a magnetic field H,qand the orien- tation of the helical phase can be rotated as H/bardblqminimizes the energy [ 27]. When applying an electric current density j, in turn, the helical phase and its orientation usually stay pinned. The reason is that, in contrast to the easily manip-ulable skyrmion lattice, the helical phase features one extratranslation invariant direction perpendicular to its qvector. In this direction the helical phase is softer against deforma-tions [ 28], which leads to stronger pinning at defects [ 29,30] such that very high currents are required for depinning. How-ever, even when depinned, the dynamics of the helix are notdetermined by the state of lowest energy as the system ispumped out of equilibrium. Instead, its dynamics are governedby the Landau-Lifshitz-Gilbert-Slonczewski equation (LLGS)[2,3,31,32] dˆ m dt=−γˆm×Beff+αˆm×dˆm dt +PμB eMs(1+β2)[(j·∇)ˆm−βˆm×(j·∇)ˆm], (1) where ˆm=M/Msis the normalized magnetization, γis the gyromagnetic ratio, Beffis the effective magnetic field, αis the Gilbert damping, Pis the spin polarization, e>0i st h e electron charge, βis the nonadiabatic damping parameter, andjis the current density. So far, studies of the dynamics were limited to the time-reversal symmetry-breaking effectof the current, which induces a finite cone angle φ[33–35], FIG. 1. In the helical phase, the magnetization rotates in the plane perpendicular to the qvector. Spin-transfer torques drive the helical state out of equilibrium into a conical state where the magne- tization tilts towards the qaxis. 2469-9950/2020/102(18)/180402(6) 180402-1 ©2020 American Physical SocietyJAN MASELL et al. PHYSICAL REVIEW B 102, 180402(R) (2020) FIG. 2. Snapshots of the magnetization at times tas indicated. (a) With periodic boundary conditions, the dynamics are dominated by defects which order the helix with q/bardblj. (b) In a finite-size system, the old pattern is pushed out of the system and replaced by a helical phase withq⊥j. The color encodes the local orientation ˆqof the helix. Additionally, darker color encodes a larger mz>0 and lighter color encodes a larger cone angle. Results are obtained for j=1.6×1011A/m2in a system of size 4 .47×4.47μm2. schematically shown in Fig. 1, irrespective of whether the helix is pinned or mobile. In this Rapid Communication, we study the current- induced dynamics of the moving helical phase in chiralmagnets. Our analytical analysis and numerical simulationsshow a transition from a multidomain to single domain helicalphase with q/bardbljdeep in the bulk opposed to q⊥jat the edge of the system. Various instabilities add to the interestingdynamics. Our experiments confirm the current-induced reori-entation in a thin specimen of FeGe which could be exploitedin novel storage devices, e.g., MRAM cells [ 5] or memristors [36], which measure an orientation-dependent resistance. Results. The orientation of the helical phase is usually pinned by anisotropies, which leads to a multidomain statewhen cooling below the Curie temperature [ 25,27]. For our theoretical analysis, we consider large current densitiessuch that dynamical effects dominate over such orientationalanisotropies or pinning by defects. We also neglect the effectof the spin-orbit coupling induced torque in chiral magnets[37]. However, the multidomain character turns out to be crucial for the current-induced dynamics. We therefore modelthe magnetization far below the Curie temperature by a simpleisotropic two-dimensional nonlinear sigma model E[ˆm]=/integraldisplay d 2r/bracketleftbiggJ 2(∇ˆm)2+Dˆm·(∇׈m)/bracketrightbigg , (2) where J=17.5p J/m is the magnetic stiffness and D= 1.58 mJ/m2is the DMI in FeGe [ 38]. As a starting point, we use a multidomain helical phase (see first panels of Fig. 2), which is prepared via directly minimizing the energy of atessellation with random orientations of the helix. The evolution of the magnetization during our simula- tions [ 39] is shown in Fig. 2(a) and Supplemental Material Movie S1 [ 40] where we apply a current density of j=1.6× 10 11A/m2using periodic boundary conditions. On large timescales, the initially multidomain helical phase transformsinto a monodomain phase with q/bardblj. This ordering process is driven by the dynamics of defects in the helical texture whichcarry a nonquantized topological charge Q=/integraldisplay/integraldisplay /Omega1ˆm·/parenleftbiggdˆm dx×dˆm dy/parenrightbigg dr∈R (3) and naturally arise at the interfaces between differently ori- ented helical domains [ 41]. Here, /Omega1is an adequately chosen finite area around the defects which can comprise discli-nations [ 41], dislocations [ 42], and skyrmions [ 43]. The topological charge distribution for Fig. 2(a) (att=0.47μs) is shown in Fig. 3, including a magnified view on a positively and a negatively charged dislocation. At this relatively small FIG. 3. The driven helical phase (a) is combed by defects such as the dislocations in (b) (blue frame) and (c) (red frame), using the color code of Fig. 2. Panels (d)–(f) show the corresponding topological charge density, Eq. ( 3), with Q>0 (blue) to Q<0 (red). The white arrow in (e) and (f) indicates the Hall motion of the charged defects. 180402-2COMBING THE HELICAL PHASE OF CHIRAL MAGNETS … PHYSICAL REVIEW B 102, 180402(R) (2020) FIG. 4. Snapshots of the helical phase after simulating a sufficiently long time span with (a) periodic boundary condition or (b) open (Neumann) boundary conditions until a steady state is established. The current densities are indicated in each panel. Setup and color code are the same as in Fig. 2. In the last panel of (b), the magnetization in the white area is (almost) polarized with ˆm=− ˆj. current density, the motion of defects is confined to lanes defined by the helical background. This background movesuniformly at a velocity v∝−jparallel to the current, whereas a simple Thiele estimate [ 44] reveals that defects experience a transverse velocity component sgn(ˆz×v)=− sgn(α−β)s g n (Q)s g n ( j)( 4 ) similar to the skyrmion Hall effect [ 45,46]. This extra trans- verse velocity is indicated in Figs. 3(e) and3(f) and can be observed in Supplemental Material Movie S2 [ 40]. Due to their transverse motion, the defects comb their confining lanessuch that q/bardblj. Moreover, oppositely charged defects can annihilate and equally charged defects can form skyrmionsthat eventually decay under pressure [ 47], which decreases the number of defects as well as the total winding numberQ. In additional simulations with current densities down to j=10 10A/m2, the dynamics are slower but qualitatively not different. At the edges of a finite-size system we observe different dynamics: As shown in Fig. 2(b) and Supplemental Material Movie S3 [ 40], the collective sliding motion pushes the initial magnetic texture over one edge out of the system. On theopposite edge, the empty space is filled by a newly enteringphase with q⊥juntil the entire system is again in a mon- odomain state. This is also the case at lower current densities,where the initial texture is only partially expelled from thesystem. In Fig. 2(b), the current density is large enough to expel almost all of the initial texture until the entire systemis in a monodomain state. Exceptions are observed at thetransverse edges where defects might enter because of theircharge-induced dynamics. We confirmed the order q⊥jalso for other orientations of the current relative to the edge.For larger current densities, the helical phase becomes un- stable. One critical current is set by the analog of the Walker breakdown [48] of magnetic domain walls, which closes the cone angle φ→0 (see Fig. 1). The corresponding fixed point of the LLGS equation, Eq. ( 1), yields the critical current j Walker c=α(1+β2) |α−β|γe μB(2D−Jq), (5) which for FeGe evaluates to the orientation-dependent critical current jWalker c·ˆq≈2.5×1012A/m2if the wavelength is the equilibrium wavelength λ≈70 nm. A stretched wavelength above the equilibrium value increases the critical current upto a factor 2, whereas decreasing the wavelength reduces thecritical current. Detailed calculations [ 49] reveal that in the helix with q⊥jlongitudinal modes soften already below the Walker breakdown, which triggers the reduction of the wavenumber. An ideal helix would therefore undergo a series ofinstabilities to larger wavelengths until it finally saturates atthe Walker breakdown. However, in a more realistic setup withdefects, these instabilities can be locally activated. As a result,defects occasionally detach from their helical ties, leaving thesystem with only a short-range order [see Fig. 4(a), first panel, and Supplemental Movie S4 [ 40]]. At higher currents, more defects proliferate and the helical background becomes moretransparent, leading to a gradually shorter-ranged order [seeFig. 4(a)], which establishes instead of the in-plane polarized state. Another instability of the driven helical phase is owed even more directly to its low-dimensional texture, namely, thetranslational invariance and thus softer excitation spectrumperpendicular to q[28]. In fact, anyfinite current perpendicu- lar to qtriggers an instability [ 49] which spontaneously breaks 180402-3JAN MASELL et al. PHYSICAL REVIEW B 102, 180402(R) (2020) FIG. 5. Lorentz TEM images of a 150-nm-thick film of FeGe at 120 K before and after a current pulse. (a) The initial magnetization shows a multidomain helical phase (stripes) and a small skyrmioncluster (dots on left side). (b) After a current pulse with j=1.3× 10 9A/m2for 0.5 ms in the horizontal direction the magnetization is in an almost defect-free helical state, ordered with q⊥j. the continuous translational symmetry with k⊥∝j·(ˆz׈q), (6) but the timescale for building up the instability scales only as ( j/jc)−4, which is very long for small currents [ 49]. In Fig. 4(b) and the corresponding Supplemental Material Movies S5–S8 [ 40] we show the steady-state magnetization obtained from simulations with successively larger currentsfor the system known from Fig. 2(b). In the first panel, the inherent instability is not observed as its timescale is smallerthan the time needed to pass once through the system, similarto Fig. 2(b). In the second panel, this instability occurs faster and thus can be observed close to the edge, which results inthe proliferation of dislocations and skyrmions [ 50,51]. In the third panel, where j>j Walker c , the length scale of the inherent instability is too small to be observed. Instead, patches ofthe also unstable but more slowly decaying phase with q/bardblj eventually burst from the edge and decay into the fluctu-ating background via the Walker breakdown. For a current j>2j Walker c , fourth panel, we finally observe a large-scale in-plane polarized phase, here shown in white, which seedsat the edge but is unstable against both the seemingly laminarand turbulent phases that enter from the transverse edges. Thefinal state after turning off the current depends on the timedependence of the current strength. However, quickly turningoff the current can result in a strongly disordered phase. We also experimentally confirm the possibility of current- induced order in the helical phase, using the experimentalsetup from Ref. [ 52] where current pulses can be applied through a 20 ×20×0.15μm 3film of FeGe. Figure 5(a) shows an underfocused Lorentz TEM image of the initialstate after cooling to 120 K. The magnetization appears tobe in a multidomain helical phase, and also a skyrmion clus-ter can be spotted. After applying a single current pulse of 1.3×10 9A/m2for 0.5 ms, the helical phase is ordered with q⊥jand includes only very few dislocation defects [see Fig. 5(b)]. This observation is in agreement with our theo- retical prediction on the edge-induced order for the unpinnedhelical phase at small currents. However, we do not observethe defect-induced order q/bardbljpredicted in our simulations as the sample size is much too small. Conclusions. In this work, we have analyzed the spin- transfer torque induced dynamics of the helical phase of chiralmagnets, how it orders at small currents, and how it turnsdisordered above a critical current. For large systems, wetheoretically predict a reorientation transition from an initiallymultidomain helical phase to a monodomain phase with q/bardblj, driven by the dynamics of topological defect. At the one edgeof the system, however, we expect a new helical phase withq⊥jto enter. Our experimental observation in a thin plate of FeGe confirms this edge-induced ordering mechanism. Abovethe critical current, where defects are no longer bound to heli-cal lanes, the ordering mechanism in the bulk breaks down butedge-induced order can still be obtained. However, this edge-induced order is intrinsically unstable and shows a cascadeof possible instabilities at large currents as shown in Fig. 4(b). Albeit our study is focused on chiral magnets, the competitionbetween vertical skyrmion-charge-induced motion and paral-lel translation is expected to be rather ubiquitous. The current-induced helical orientation can be used to all- electrically imprint an anisotropic pattern onto the magnetiza-tion. Such a pattern shows an anisotropic magnetoresistancedependent on the helical orientation q. This effect might be exploited for novel MRAM-like cells which make use of thehelical orientation as an order parameter. More unconven-tional ideas can also exploit that the information encoded inthe helical orientation is not binary. In principle, a currentcan be applied in any direction to realize any orientation ofthe helix in a device with more than only two logic states.Moreover, in a larger cell we can use the readout currents tosimultaneously induce fractions of new helical order at theedge while probing the system. As a result, every readoutoperation lowers the resistance of the element, which is a keyelement for memristive computing [ 36]. Finally, large pulses above the instability can be used to reset the device. In conclusion, the helical phase of chiral magnets seemed featureless compared to magnetic skyrmions which appear inthe same class of materials. We disprove this prejudice, reveal-ing the nontrivial dynamics which will be analyzed further inthe future and unleash the helical orientation as a complexorder parameter for future applications. Acknowledgments. J.M. thanks V . Kravchuk and M. Garst for the helpful discussions. This work was financially sup-ported by Grants-in-Aid for Scientific Research (A) (GrantsNo. 18H03676 and No. 19H00660) from the Japanese Societyfor the Promotion of Science (JSPS) and the Japan Scienceand Technology Agency (JST) CREST program (Grant No.JPMJCR1874). J.M. acknowledges financial support by JSPS(Project No. 19F19815) and the Alexander von HumboldtFoundation. 180402-4COMBING THE HELICAL PHASE OF CHIRAL MAGNETS … PHYSICAL REVIEW B 102, 180402(R) (2020) [1] B. Binz and A. Vishwanath, Chirality induced anomalous-Hall effect in helical spin crystals, Phys. B (Amsterdam, Neth.) 403, 1336 (2008) . [2] J. C. Slonczewski, Current-driven excitation of magnetic multi- layers, J. Magn. Magn. Mater. 159, L1 (1996) . [3] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B 54, 9353 (1996) . [4] C. Chappert, A. Fert, and F. Van Dau, The emergence of spin electronics in data storage, Nat. 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Müller, Magnetic Skyrmions and Topological Domain Walls, Doctoral dissertation, University of Cologne, Cologne, 2018. 180402-6

PhysRevB.93.134413.pdf

PHYSICAL REVIEW B 93, 134413 (2016) Mutual synchronization of nanoconstriction-based spin Hall nano-oscillators through evanescent and propagating spin waves T. Kendziorczyk*and T. Kuhn Institut f ¨ur Festk ¨orpertheorie, Universit ¨at M ¨unster, Wilhelm-Klemm-Straße 10, 48149 M ¨unster, Germany (Received 29 July 2015; revised manuscript received 21 March 2016; published 11 April 2016) We perform a micromagnetic study of the synchronization dynamics in nanoconstriction-based spin Hall nano- oscillator (SHNO) arrays. The simulation reveals that efficient synchronization in this kind of system is possible,and indicates that the synchronization is mediated by a combination of linear coupling through the overlap oflocalized modes and parametric coupling through propagating spin waves which are excited by the secondharmonic oscillation in the SHNOs. Due to the anisotropic spin wave dispersion in the studied system, the synchro-nization properties decisively depend on the geometrical alignment of the SHNO array with respect to the externalfield. We find that, by utilizing the directional spin wave emission and correspondingly optimizing the alignmentof the SHNO array, the synchronization is enhanced with a significant increase of the phase-locking bandwidth. DOI: 10.1103/PhysRevB.93.134413 I. INTRODUCTION Since the first theoretical predictions of the spin-transfer torque (STT) [ 1,2], current-induced magnetization dynamics has been a very active research topic, leading to the devel-opment of nanoscale magnetic auto-oscillators—the so calledspin-torque nano-oscillators (STNOs) [ 3–5]. STNOs are of high fundamental interest, because they form a testbed fordifferent branches of physics, such as spin-dependent transporttheory, magnonics, and nonlinear dynamics. FurthermoreSTNOs are promising for application as future GHz oscilla-tors [ 6]. STNOs have some attractive advantages compared to conventional CMOS based oscillators, such as their nanoscaledimensions, a high frequency tunability, and ultrafast modula-tion [ 7]. However, they are also subject to several drawbacks, such as low output power and comparably large linewidths atroomtemperature. It is believed that these limitations can beovercome by implementing arrays of multiple synchronizedSTNOs [ 8–15]. Although much effort has been expended on this topic to date, a complete synchronization has beenachieved only for a maximum number of five STNOs [ 16], mainly due to technological limitations which prevent areliable fabrication of larger arrays [ 14]. Recently a new class of spintronic devices based on the spin Hall effect (SHE) [ 17,18] emerged. It has been demon- strated in several experiments that this effect can providea sufficiently large STT to efficiently induce magnetizationdynamics such as domain wall motion, magnetic switching,and the manipulation of magnetic damping [ 19–27]. Recently also autonomous oscillators similar to conventional STNOshave been found [ 28–31]. These oscillators are referred to as spin Hall nano-oscillators (SHNOs). SHNOs are basedon a bilayer system consisting of a ferromagnet and anadjacent metal layer with strong spin orbit coupling. Herea perpendicular pure spin current is induced by the SHE inthe metal by spin-dependent scattering of an in-plane electriccharge current. Therefore a fully planar design with the samelateral geometry of both layers becomes possible [ 31]. Due to the reduced fabrication complexity, this kind of system could *t.kend@uni-muenster.debring arrays of synchronized magnetic oscillators a step closerto the targeted application. In this paper we provide a numerical analysis of the synchronization in a SHNO array by means of micromagneticsimulations. We find that, although the excited modes inthe single SHNOs are localized, spin waves contribute tothe synchronization. These spin waves are excited at thesecond harmonic frequency of the SHNO and mediate the synchronization by means of parametric excitation [ 32]. This is a mechanism which is different from previously studied mu-tually synchronized STNOs and also highly interesting fromthe general viewpoint of nonlinear physics, since here we havea simultaneous synchronization of oscillators by two modeswith different frequencies and different localization behavior. II. SYSTEM AND THEORETICAL FRAMEWORK Figure 1(a) shows the studied SHNO array. The single SHNO geometry is chosen to match the experiment fromRef. [ 31]. The SHNOs consist of 150 nm wide nano- constrictions in a Pt(8)/Py(5) (thickness in nm) bilayer system.Each constriction is defined through two cutoff wedges witha curvature radius of 50 nm and an opening angle of 11 ◦.I n the general case we consider an array of two SHNOs withrelative lateral shifts /Delta1xand/Delta1y. The bilayer is traversed by an in-plane current with a total amplitude I SHNO . An external magnetic bias field with the amplitude H0=450 Oe is applied under an angle of α=30◦with respect to the xaxis. We perform the numerical study with our finite differ- ences time domain code [ 15], which solves the Landau- Lifshitz-Gilbert equation for the Py layer including the SHEinduced torque term τ SHE=β M2 0M×(M×[ez×jPt]) [19]. The prefactor β=gμBαH/(2etPy) contains the gfactor, the electron-charge e, the layer thickness tPy, and the spin Hall angle αH. The electric current distribution and the Oersted fields are obtained by numerical solution of the electrostaticPoisson equation, together with Ohm’s law for the two-layersystem and employing Amp `ere’s law [ 33]. As system parameters we use a saturation magnetization of M 0=600 kA /m, a Gilbert damping constant of αG=0.015, an exchange stiffness of A=1.3×10−11J/m, and a spin Hall 2469-9950/2016/93(13)/134413(6) 134413-1 ©2016 American Physical SocietyT. KENDZIORCZYK AND T. KUHN PHYSICAL REVIEW B 93, 134413 (2016) FIG. 1. (a) Schematic representation of the SHNO array. (b) Current density and (c) internal field in the Py layer for a single constriction and a total current of ISHNO=2m A . angle of αH=0.08 [19]. The electric conductivities are set to σPy=3.07×106S/m and σPt=8.93×106S/m[31]. III. SINGLE OSCILLATOR IN THE FREE-RUNNING REGIME In the first part of our micromagnetic study we consider only one constriction defining a single free-running SHNO.We calculate the current distribution, the Oersted field, andthe equilibrium orientation of the magnetization with the corresponding internal field. Figure 1(b) shows the current density distribution in the Pt layer for a single constriction. Anidentical distribution is obtained in the Py layer due to the samelateral geometry; however, more than 82% of the total currentflows through the Pt layer. For a total current I SHNO=2m A the maximal current density is 2 .4×1012A/m2. The maxima are located close to the borders of the nano-constriction. Figure 1(c) shows the internal field which is calculated within a micromagnetic simulation without the STT term,but including the Oersted fields. The internal field in theconstriction is strongly reduced in an elliptical-shaped regionwith its long axis oriented perpendicular to the external field. Ithas two strong minima near the boundaries of the constriction. We begin the analysis of the system dynamics with Fig. 2(a), which shows the power spectral density (PSD) calculatedfrom the time trace of the spatially averaged dynamicalmagnetization in the center of the SHNO, after instantaneouslyapplying the current I SHNO . We find an oscillation with the frequency f=4.95 GHz at the critical current IC1= 1.5 mA. The frequency shows a nonlinear redshift with df/dI =− 80 MHz /mA. In agreement with the experiment we observe a second critical current IC2=4.5 mA, where a second peak in the frequency spectrum appears and theoscillation becomes highly irregular with a strong linewidthbroadening. Furthermore in the regime I SHNO<IC2the PSD shows a significant spectral feature at the nonlinear secondharmonic corresponding to the frequency 2 f. This oscillation is particularly pronounced for currents well above the criticalcurrent. The maximum of the ratio between the second-and first-order harmonic is larger than 30% and is reachedatI SHNO=3.5 mA. We restrict our study in this paper to the single-mode regime IC1<I<I C2. In order to shine more light on the nature of the excited mode, we calculate FIG. 2. (a) Power spectral density (PSD) calculated from the spatially averaged dynamic magnetization in a single SHNO. (b)–(f) Spatial amplitude distribution of the mzcomponent for ISHNO=2 mA at the fundamental frequency fand second harmonic 2 f. (b) and (c) Mode amplitude profiles in the constriction region on a linear color scaling. (d) and (e) Power distribution on a logarithmic color scaling for the wholesurrounding film. The dashed line shows the direction perpendicular to the external field ( ⊥H 0). (f) Radial profiles in the direction with angles ϕ, which are defined with respect to the xaxis. (g) Angular profiles for a constant distance r=350 nm as a function of ϕ. 134413-2MUTUAL SYNCHRONIZATION OF NANOCONSTRICTION- . . . PHYSICAL REVIEW B 93, 134413 (2016) its spatial profile at the auto-oscillation frequency ffor ISHNO=2 mA. Figure 2(b) shows the amplitude distribution /tildewidemz(x,y,f ) resulting from the temporal Fourier transform of the zcomponent of the dynamical magnetization in a small region close to the constriction. Additionally we plotthe spatial power distribution /tildewidem 2 zfor the whole layer in Fig. 2(d). We can observe a localized mode with a size (FWHM with respect to the power) of 240 nm perpendicularto the external field (dashed line) and 110 nm parallel to theexternal field. Therefore the mode has a strong elongationin the direction perpendicular to the external field, consistentwith the experiment [ 31]. Furthermore the mode is much larger than the width of a typical self-localized bullet mode [ 34], whose size was determined in previous studies as about80 nm [ 35]. By comparing the spatial amplitude distribution [Fig. 2(b)] with the internal field [Fig. 1(c)], we clearly identify that the shape of the mode is very similar to the profileof the internal field. The two maxima of the amplitude areexactly located in the regions with low internal fields. Theseresults support the interpretation [ 31] that the studied SHNO is based on a localized mode in the effective static magneticpotential created by the inhomogeneous internal field ratherthan on a self-localized bullet mode, which can be observed inhomogeneous ferromagnetic films. In the following we analyze the oscillation at the second harmonic 2 f. Figure 2(c) shows the spatial amplitude of these dynamics in the region of the constriction. Similarly to thespatial profile of the first harmonic, the amplitude has twopronounced maxima close to the borders of the constriction.However, here we observe an additional maximum locatedin the center of the constriction. This can be explained byinterference of spin waves which can be emitted due to thefact that the frequency 2 fis well above the ferromagnetic resonance f FMR=5.31 GHz, in contrast to the frequency f of the localized mode. By providing further analysis of thespatial distribution in a larger region outside the constriction[see Figs. 2(e)–2(g)], we can show that an efficient emission of spin waves is possible. We perform a fit of the spin waveintensity [ 36]t o/tildewidem 2 z∝1 re−2r/ξ, where ris the distance from the center of the SHNO and ξdenotes the decay length of the spin wave amplitude. The fit is performed to a radial profilethrough the intensity emitted perpendicular to the external fieldwhich corresponds to an angle ϕ=120 ◦with respect to the xaxis [see Fig. 2(f)]. This yields ξ2f=930 nm, which is more than twice the decay length of the evanescent wave withξ f=404 nm. These values depend strongly on the emission direction, which can be seen in Fig. 2(g), where the angular profile of the spin wave intensity as a function of ϕwith a radius of r=350 nm is plotted. The emitted spin waves have a strong directionality with emission perpendicular to the external field, which is an expected result consideringthe anisotropic dispersion in the studied in-plane magnetizedsystem [ 36,37]. While the intensity of the evanescent spin waves has a single maximum for emission in the directionϕ=120 ◦, the main peak in the intensity for the spin waves 2 f is shifted to smaller angles and there are several maxima andminima present. This is because the two observed maxima inthe amplitude inside the constriction [see Fig. 2(c)] provide two separate coherent sources for spin waves which arewell separated. Therefore the observed spatial profile of the FIG. 3. (a)–(d) Spatial amplitude profile of mzfor two synchro- nized SHNOs with ISHNO=2.0 mA for (a),(b) /Delta1x=0n ma n df o r (c),(d) /Delta1x=dH⊥. (e) and (f) Sum of the PSDs calculated from signals of two SHNOs for (e) /Delta1x=0 nm and (f) /Delta1x=dH⊥. emitted spin waves can be explained in terms of spin wave interference. In order to obtain the wavelengths of the emitted spin waves, we furthermore analyze the spatial distribution of the phasesof the emitted spin waves. The analysis yields values of λ= 125 nm for the emission parallel to the external field andλ=180 nm for the emission perpendicular to the external field, which is in very good agreement with the spin wavedispersion [ 37] of an infinitely extended Py layer. This gives additional proof that the SHNO can indeed excite propagatingspin waves at the frequency 2 f. IV . SYNCHRONIZATION OF TWO OSCILLATORS In the remainder of the paper we analyze the synchro- nization dynamics of an array with two SHNOs. We proposean array geometry which is designed to increase the mutualexchange of spin waves by utilizing the strongly directionalemission of the spin waves perpendicular to the externalfield. Therefore, for two SHNOs with a distance of /Delta1yiny direction, we compare two different structures: a conventional,aligned one ( /Delta1x=0) and one which includes a shift of /Delta1x= d H⊥=/Delta1ytanα[see Fig. 1(a)]. In the latter case the direction perpendicular to the external field forms a direct connectionline through the points in the center of the nanoconstrictions.The spin wave amplitudes for I SHNO=2 mA and /Delta1y= 350 nm are plotted in Figs. 3(a) and 3(b) for/Delta1x=0n m and in Figs. 3(c) and 3(d) for/Delta1x=dH⊥=202 nm. As expected from the single SHNO propagation properties [seeFig. 2(f)], in the case /Delta1x=d H⊥we find a much larger amplitude of the evanescent spin wave and the interferencepattern produced by the propagating spin waves in betweenboth SHNOs. In order to investigate the influence of theachieved increased mutual spin wave emission, we study the synchronization dynamics as a function of d=/radicalbig /Delta1x2+/Delta1y2. The sum of the two PSDs calculated individually from theoscillation signal produced by both SHNOs is shown inFigs. 3(e) and 3(f). We have introduced a small mismatch 134413-3T. KENDZIORCZYK AND T. KUHN PHYSICAL REVIEW B 93, 134413 (2016) FIG. 4. (a) and (b) Sum of the PSDs calculated from the signals of two SHNOs under the influence of a nanowire with Iwirefor (a) /Delta1x=0n ma n d( b ) /Delta1x=150 nm, ISHNO=2.0 mA. (c) Size of the synchronization region /Delta1I wireas a function of the angle ϕfor different values of ISHNO .( d )/Delta1I wireas a function of the SHNO distance for a fixed value ϕ=120◦,ISHNO=2.0 mA. The insets show the spatial amplitude profile for the second harmonic /tildewidemz(2f). The lines are guides to the eye. in the free-running frequencies by increasing the width of one of the constrictions to 170 nm. This ensures that we can clearlydistinguish synchronized regions (single peak in the frequencyspectrum) from unsynchronized regions (multiple peaks in thefrequency spectrum). In the simple geometry [Fig. 3(e)]w e find synchronization for distances below d< 350 nm and in a very small region close to d=450 nm. In the case of the optimized geometry [Fig. 3(f)], synchronization is possible for much larger distances, up to d=750 nm. Interestingly the frequency spectra reveal a pronounced periodic behaviorwith a periodicity corresponding to the wavelength of the spinwaves emitted at the second harmonic, which will be analyzedin more detail below [see the discussion of Fig. 4(d)]. For larger distances the simulation results also show synchronizedand unsynchronized regions alternately as a function of thedistance. Both features are expected from the theory ofmutual phase locking in auto-oscillators [ 10,38] and previous micromagnetic simulations [ 11,15] if the synchronization is mediated by propagating spin waves. Therefore the simulationof the SHNO array indicates that the spin waves emitted at thesecond harmonic contribute to the mutual synchronization.This can be explained by parametric coupling of the emittedspin waves by each SHNO with the auto-oscillation modeof the other SHNO respectively. Experimental proof forparametric synchronization of nanocontact STNOs has beenprovided already in Ref. [ 32]. However, in contrast to the present paper, the synchronization was achieved by excitationof the auto-oscillator with an external microwave signal witha frequency of twice the auto-oscillator frequency. Here thissignal intrinsically appears as the nonlinear second harmonicof the oscillation from the SHNOs themselves and providesa channel for mutual synchronization via spin waves. Theobserved mutual parametric coupling is a distinct feature forthe studied SHNO array, which is caused by the fact that herethe fundamental mode is below the spin wave spectrum and hasa strong attenuation. The second harmonic has still quite lowrelaxation frequencies and and has a large amplitude comparedto the localized mode for reasonable distances. In contrast tothis, in typical nanocontact STNOs the fundamental mode isalready propagating—the so called Slonczewski mode [ 39,40]. These spin waves are exchange dominated and the secondharmonic frequency is so high that due to the strong attenuationno significant contribution to the synchronization is expected. The focus of the last part of the paper is to investigate the ori- gin of the synchronization and to show which synchronizationmechanisms are particularly important based on the choice ofthe system parameters. This can be achieved by analyzing thephase-locking bandwidth for different conditions, which favoreither linear synchronization or parametric synchronizationmediated by spin waves. The phase-locking bandwidth isdefined as the maximum frequency detuning for whichsynchronization is possible. In contrast to systems based onnanocontact STNOs with independent bias currents, here itis not directly possible to change the frequency detuning.We solve this problem in the simulation by introducing anadditional external magnetic field with a spatial gradient whichmainly influences one of the SHNOs. This could be realizedfor example by the Oersted field produced by a nanowire,which is sketched in Fig. 1(a). Figures 4(a) and 4(b) show that the detuning can be controlled by the current I wirethrough the nanowire. While the frequency of the SHNO close to thewire experiences a linear shift with about 42 MHz /mA, the frequency of the second SHNO is barely affected. Thereforewe can identify the phase-locking region /Delta1I wire, which is approximately proportional to the phase-locking bandwidth. In Fig. 4(c) we examine the value /Delta1Iwireas a function of ϕ, which is the angle between the center-to-center connection line between the SHNOs with the xaxis. We perform these calculations for two different parameter sets: (i) the squaresshow the simulation results for a distance of 500 nm with acurrent I SHNO=2.0 mA and (ii) the points show the data for 800 nm with ISHNO=3.0 mA. These parameters are chosen to prefer the linear synchronization by the overlap of the localizedmodes in case (i) and the spin wave mediated parametricsynchronization in case (ii). This is because the influence ofthe parametric coupling increases for larger distances due to 134413-4MUTUAL SYNCHRONIZATION OF NANOCONSTRICTION- . . . PHYSICAL REVIEW B 93, 134413 (2016) the larger decay length ξ2f>ξf[see Fig. 2(e)]. Furthermore the parametric coupling becomes more important for largercurrent densities because the ratio between the power of theevanescent waves and propagating waves increases in thiscase [see Fig. 2(a)]. For better visibility the data for the larger distance (ii) have been multiplied by a factor 1 .7t o compensate for the reduced synchronization bandwidth at themaximum, which is simply related to the spatial attenuationof both the localized and propagating modes. The data forparameter set (i) shows a pronounced maximum at an angle ofϕ=125 ◦, which is close to the expected value of ϕ=120◦, corresponding to /Delta1x=dH⊥(alignment of the center-to-center connection line between the SHNOs perpendicular to theexternal field). The value for the phase locking-bandwidth atthe conventional, aligned geometry with ϕ=90 ◦is reduced by a factor of 8 in comparison to the main maximum. Incontrast to these results, the phase-locking bandwidth as afunction of ϕfor the parameter set (ii) has multiple pronounced maxima at different values of ϕ. This can be explained by the fact that the propagating spin waves, which are generated bytwo sources at each constriction, can interfere. Therefore theresults are reminiscent of the angular behavior obtained for asingle SHNO [see Fig. 2(g)], which is characterized by one maximum for the localized mode and multiple maxima forthe propagating modes. However, in the case of two SHNOsadditional quantization effects can occur due to the finiteextension of the permalloy film between the wedges definingthe top and bottom constrictions. Therefore more complicatedinterference patterns are possible for the propagating spinwaves [see Figs. 3(b) and 3(d)]. These observations are consistent with the increased number of side maxima in thephase-locking bandwidth as a function of ϕ. The same method for obtaining the phase-locking band- width can be used to obtain the latter as a function of theSHNO distance. Figure 4(d) shows periodic features in the phase-locking bandwidth. To understand the reason for thisperiodicity in more detail, we show the spatial amplitudeprofile for the second harmonic /tildewidem z(2f) in the insets of Fig. 4(d). These spatial profiles are calculated for SHNO distances corresponding to the values where we obtain maximain the phase-locking bandwidth (see the arrows in the insets).For all distances we can see an interference pattern created bythe spin waves emitted at the second harmonic. Comparing thenumber of maxima in the interference pattern for the chosendistances we obtain three, five, and seven maxima betweenthe SHNOs. Therefore the difference in the distance betweentwo maxima in the phase-locking bandwidth corresponds toone additional wavelength, which is again evidence for theparametric synchronization mediated by the spin waves at the second harmonic. The obtained periodic behavior of the phase-locking band- width can explain the alternate appearance of synchronizedand unsynchronized regions as a function of the distancein Figs. 3(e) and 3(f), because it shows that the phase- locking bandwidth is a nonmonotonic function, which incombination with a constant detuning between the SHNOsleads to the observed features. Furthermore this emphasizesthe importance of taking into account the second harmonicspin waves in the design of an SHNO array, because for certaingeometries the bandwidth can be significantly reduced. V . CONCLUSION In conclusion, we have performed a micromagnetic study of the synchronization in SHNO arrays. We found that,in contrast to previously studied nanocontact STNOs, thesynchronization is composed of two mechanisms: linearsynchronization through the overlap of the localized modesand parametric synchronization mediated by propagating spinwaves which are excited by the nonlinear second harmonicof the SHNOs. The synchronization mechanism based onpropagating spin waves can be favored in two ways. One ofthese ways is based on the geometry of the array. By choosinglarger distances between the SHNOs, the influence of thelocalized modes is reduced due to their strong attenuationcompared to the propagating spin waves. The second way isto operate the SHNOs at larger current densities. In this casethe ratio between the amplitude of the second harmonic andthe fundamental mode can be increased. Furthermore our study revealed that it is important to take into account the directionality of the spin wave emission in thedesign of the SHNO array geometry. By proposing a geometrywhich maximizes the overlap of the spin wave modes, weshow that the synchronization bandwidth can be substantiallyimproved. This result applies for both synchronization medi-ated by the propagating mode and the localized one. Our simulations therefore provide multiple general predic- tions for the operation of synchronized SHNOs which shouldbe carefully considered in the design of future devices basedon SHNO arrays. ACKNOWLEDGMENT The authors are grateful to S. O. Demokritov and V . E. Demidov for fruitful discussions. [1] J. C. Slonczewski, J. Magn. Magn. Mater. 159,L1(1996 ). [2] L. Berger, Phys. Rev. B 54,9353 (1996 ). [3] M. Tsoi, A.-G.-M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V . Tsoi, and P. Wyder, Phys. Rev. Lett. 80,4281 (1998 ). [4] S.-I. Kiselev, J.-C. Sankey, I.-N. Krivorotov, N.-C. Emley, M. Rinkoski, C. Perez, R.-A. Buhrman, and D.-C. Ralph, Phys. Rev. Lett.93,036601 (2004 ). [5] W.-H. Rippard, M.-R. Pufall, S. Kaka, S.-E. Russek, and T.-J. Silva, P h y s .R e v .L e t t . 92,027201 (2004 ).[6] P. Villard, U. Ebels, D. Houssameddine, J. Katine, D. Mauri, B. Delaet, P. Vincent, M.-C. Cyrille, B. Viala, J.-P. Michel et al. ,IEEE J. Solid-State Circuits 45,214 (2010 ). [7] R. K. Dumas, S. R. 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PhysRevC.103.034601.pdf

PHYSICAL REVIEW C 103, 034601 (2021) Investigating high-energy proton-induced reactions on spherical nuclei: Implications for the preequilibrium exciton model Morgan B. Fox ,1,*Andrew S. Voyles ,1,2,†Jonathan T. Morrell ,1Lee A. Bernstein ,1,2Amanda M. Lewis ,1 Arjan J. Koning ,3Jon C. Batchelder,1Eva R. Birnbaum,4Cathy S. Cutler,5Dmitri G. Medvedev ,5 Francois M. Nortier,4Ellen M. O’Brien ,4and Christiaan Vermeulen4 1Department of Nuclear Engineering, University of California, Berkeley, Berkeley, California 94720, USA 2Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3International Atomic Energy Agency, P . O. Box 100, A-1400 Vienna, Austria 4Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA 5Brookhaven National Laboratory, Upton, New York 11973, USA (Received 14 October 2020; accepted 16 December 2020; published 1 March 2021) Background: A number of accelerator-based isotope production facilities utilize 100- to 200-MeV proton beams due to the high production rates enabled by high-intensity beam capabilities and the greater diversity of isotopeproduction brought on by the long range of high-energy protons. However, nuclear reaction modeling at theseenergies can be challenging because of the interplay between different reaction modes and a lack of existingguiding cross-section data.Purpose: A Tri-lab collaboration has been formed among the Lawrence Berkeley, Los Alamos, and Brookhaven National Laboratories to address these complexities by characterizing charged-particle nuclear reactions relevantto the production of established and novel radioisotopes.Method: In the inaugural collaboration experiments, stacked-targets of niobium foils were irradiated at the Brookhaven Linac Isotope Producer ( E p=200 MeV) and the Los Alamos Isotope Production Facility (Ep=100 MeV) to measure93Nb(p,x) cross sections between 50 and 200 MeV. First measurements of the 93Nb(p,4n)90Mo beam monitor reaction beyond 100 MeV are reported in this work, as part of the broadest energy-spanning dataset for the reaction to date.93Nb(p,x) production cross sections are additionally reported for 22 other measured residual products. The measured cross-section results were compared with literature dataas well as the default calculations of the nuclear model codes TALYS, CoH, EMPIRE, and ALICE.Results: The default code predictions largely failed to reproduce the measurements, with consistent underesti- mation of the preequilibrium emission. Therefore, we developed a standardized procedure that determines thereaction model parameters that best reproduce the most prominent reaction channels in a physically justifiablemanner. The primary focus of the procedure was to determine the best parametrization for the preequilibriumtwo-component exciton model via a comparison to the energy-dependent 93Nb(p,x) data, as well as previously published139La(p,x) cross sections. Conclusions: This modeling study revealed a trend toward a relative decrease for internal transition rates at intermediate proton energies ( Ep=20–60 MeV) in the current exciton model as compared to the default values. The results of this work are instrumental for the planning, execution, and analysis essential to isotope production. DOI: 10.1103/PhysRevC.103.034601 I. INTRODUCTION The continued rise of nuclear medicine to study physiolog- ical processes, diagnose, and treat diseases requires improvedproduction routes for existing radionuclides, as well as newproduction pathways for entirely novel radioisotopes [ 1]. The implementation of these new methodologies or products in nuclear medicine relies on accurate and precise nuclear re- action cross-section data in order to properly inform andoptimize large-scale creation for clinical use [ 2–7]. A primary *morganbfox@berkeley.edu †asvoyles@lbl.govcomponent in obtaining these data is a suitable reaction mon- itor, defined as a long-lived radionuclide with a well-knowncross section as a function of incident beam energy that canaccurately describe beam properties during a production irra-diation [ 2,5,8–10]. In the case of high-energy proton-induced reactions, which are important production routes at national accelerator fa-cilities on account of the high beam intensities and largeprojectile range in targets [ 4,5,7], the 93Nb(p,4n)90Mo re- action is emerging as a valuable new monitor candidate asevidenced by Voyles et al. [2]. In this work, proton-induced reaction cross sections for 93Nb were measured for energies 50–200 MeV using the stacked-target activation technique. The results include the 2469-9985/2021/103(3)/034601(34) 034601-1 ©2021 American Physical SocietyMORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) first cross-section measurements for93Nb(p,4n)90Mo be- yond 100 MeV within the most comprehensive dataset for thereaction to date, spanning over the broadest energy range. In addition to the ( p,4n) channel, production cross sections were extracted for 22 additional reaction products. This ex-tensive body of data forms a valuable tool to study nuclearreaction modeling codes and assess the predictive capabili-ties for proton reactions on spherical nuclei up to 200 MeV[6,11–15], which have been studied less than neutron-induced reactions [ 16]. It was demonstrated that default modeling predictions from TALYS, CoH, EMPIRE, and ALICE codesfailed to reproduce the measured niobium data and requiredmodifications to improve [ 17–20]. In this manuscript, we set forth a systematic algorithm to determine the set of re-action model input parameters, in a scientifically justifiablemanner, that best reproduces the most prominent reactionchannels. The algorithm is built in the TALYS modelingframework and sets a premier focus on determining the bestparametrization of the two-component exciton model in or-der to gain insight into high-energy preequilibrium reactiondynamics [ 11,17,21]. The algorithm was then further applied to existing high-energy 139La(p,x) data. Taken together, this work suggests that the default internal transition rates of theexciton model must be modified as a function of excitonnumber and total system energy when considering residualproduct data from high-energy proton-induced reactions. The fitting methodology proposed in this work aims to improve an accepted approach in cross-section measurementliterature where too few observables are used to guide model-ing parameter adjustments, thereby potentially subjecting themodeling to compensating errors. The results of this work should benefit the experimental and theoretical calculations central to isotope production plan-ning and execution, as well as help inform the physical basisof the exciton model. II. EXPERIMENTAL METHODS AND MATERIALS The charged-particle irradiations in this work were per- formed as part of a Tri-lab collaboration between LawrenceBerkeley National Laboratory (LBNL), Los Alamos Na-tional Laboratory (LANL), and Brookhaven National Lab-oratory (BNL). The associated experimental facilities werethe 88-Inch Cyclotron at LBNL for proton energies ofE p<55 MeV, the Isotope Production Facility (IPF) at LANL for 50 <Ep<100 MeV, and the Brookhaven Linac Isotope Producer (BLIP) at BNL for 100 <Ep<200 MeV. A. Stacked-target design The stacked-target activation technique was employed in this work, where three separate target stacks were constructedand irradiated, each at a different accelerator facility. Thestacked-target approach requires a layered ensemble of thinfoils such that induced activation on these foils by a well-characterized incident charged-particle beam allows for themeasurement of multiple energy-separated cross-section val-ues per reaction channel. Monitor foils are included amongthe thin foil targets in order to properly assess the beam inten-sity and energy reduction throughout the depth of the stack. Degraders are additionally interleaved throughout the stackto reduce and selectively control the primary beam energyincident on each target foil [ 2,6,9]. 1. LBNL stack and irradiation The initial primary motivation for these Tri-lab stacked- target experiments was to determine residual nuclide produc-tion cross sections for 75As(p,x) from threshold to 200 MeV, with a specific focus on the production of68Ge and72Se for PET imaging. However, the76Se compound system is non- spherical, which could necessitate the use of coupled-channelscalculations in the reaction modeling. Deformed systems mayalso require the use of a modified Hauser-Feshbach code thatextends angular momentum and level-density considerationsto include nuclei spin projections on the symmetry axis. Thismodification is presented in Grimes [ 22] and suggests an increased accuracy for deformed nuclei calculations versusthe assumption of spherical symmetry inherent to the standardHauser-Feshbach formalism. Yet these deformation aspectslie beyond the scope of this current paper and in turn, theresults from the 75As(p,x) measurements will be presented in a separate publication. Consequently, the LBNL stack in this campaign focused only on arsenic targets and did not contain niobium foils. Theexperimental setup and procedure at this site will therefore notbe discussed in this work. 2. LANL stack and irradiation The IPF stack utilized 25- μmnatCu foils (99.999%, LOT: U02F019, Part: 10950, Alfa Aesar, Tewksbury, MA01876, USA), 25- μm natAl foils (99.999%, LOT: Q26F026, Part: 44233, Alfa Aesar), 25- μmnatNb foils (99.8%, LOT: T23A035, Alfa Aesar), and thin metallic75As layers elec- troplated onto 25- μmnatTi foil backings (99.6%, TI000205/ TI000290, Goodfellow Metals).natNb is 100%93Nb isotopic abundance. Ten copper, niobium, and aluminum foils each were cut into 2.5c m×2.5 cm squares and their physical dimensions were characterized by taking four length and width mea-surements using a digital caliper (Mitutoyo America Corp.)and four thickness measurements taken at different locationsusing a digital micrometer (Mitutoyo America Corp.). Mul-tiple mass measurements at 0.1-mg precision were takenafter cleaning the foils with isopropyl alcohol. Ten tita-nium foils were cut to the same approximate sizes but thesame dimensioning and weighting techniques could not beused due to the chemical and mechanical constraints of thecollaboration-developed electroplating process. Instead, thenominal manufacturer thickness and density were acceptedfor the titanium, with confidence and uncertainties gatheredfrom separate physical measurements of extra titanium foilsnot used in the stack. The creation and characterizationof the accompanying 2.25-cm-diameter arsenic depositionsused in this stack will be described in detail in a futurepublication dedicated to the arsenic irradiation products.This characterization involved dimensional measurements, 034601-2INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 1. A top view of the assembled LANL target stack showing the 10 target “compartments” separated by aluminum degraders. The beam enters through a 0.411-mm aluminum entrance window on theright-hand side of the target box. electron transmission, and reactor-based neutron activation analysis. The electroplated arsenic targets, as well as the niobium foils, were sealed using LINQTAPE PIT0.5S-UT Series Kap-ton polyimide film tape composed of 12 μm of silicone adhesive on 13 μm of polyimide backing (total nominal 3.18 mg/cm 2). The copper and aluminum foils were not en- capsulated in any tape. The electroplated arsenic foils were attached to 10 acrylic frames (1.5 mm in thickness), which protected the foils dur-ing handling and centered them in the bombardment positionafter the stack was fully arranged. The 10 copper foils weretreated in an identical manner. The aluminum and niobiumfoils were paired up and mounted on the front and backof the same frames due to physical space limitations of themachined 6061-T6 aluminum IPF target box. Nine aluminum1100 series degraders were characterized in the same manneras the Cu, Nb, and Al foils and included in the stack to yield10 different beam energy “compartments” for cross-sectionmeasurements. In each compartment, one 93Nb+natAl target, one75As+natTi target, and onenatCu target were placed and bundled together using baling wire. The baling wire, attachedat the top of the frames and not obstructing any target material,was necessary to aid the removal of the foils from the targetbox following irradiation using the hot cell’s telemanipulators.The assembled stack in the IPF target box can be seen inFig.1, where it is also noted that the box has a 0.411 mm aluminum beam entrance window and is specially designedto be watertight since the IPF target station is located un-derwater. Additionally, stainless steel plates (approximately100 mg/cm 2) were placed in the front and back of the stack. Postirradiation dose mapping of the activated stainless platesusing radiochromic film (Gafchromic EBT3) was used to de-termine the spatial profile of the beam entering and exiting thestack [ 2,6].The upstream beamline components at IPF have a sig- nificant effect on beam energy that must be taken intoaccount [ 23]. Two materials exist upstream of the target box entrance window: the beam window separating beamline vac-uum from the target chamber and a single cooling waterchannel defined by the distance between the beam windowand the aluminum target box window during operation. Theinstalled beam window is 0.381-mm-thick Inconel alloy 718and it is precurved toward the vacuum side of the beamline by1.3 mm. However, under the hydrostatic and vacuum loadingpressures experienced during operation, the beam windowfurther elastically deforms toward the vacuum side. Duringoperation at low beam currents, typical of this work, the beamwindow elastically deforms toward the vacuum side by ap-proximately 0.12 mm. Given the geometry of the target box,this information implies that the proton beam travels througha cooling water channel 7.414 mm thick [ 23]. The combined upstream effects total an approximate effective degrader arealdensity of 1165 mg/cm 2. The full detailed target stack ordering and properties for the LANL irradiation are given in Table Vin Appendix A. The stack was irradiated for 7203 s with an H+beam of 100-nA nominal current. The beam current, measured usingan inductive pickup, remained stable under these conditionsfor the duration of the irradiation. The mean beam energyextracted was 100.16 MeV at a 0.1% uncertainty. 3. BNL stack and irradiation The target stack for the BNL irradiation was composed of 25-μmnatCu foils (99.95%, CU000420, Goodfellow Met- als, Coraopolis, PA 15108-9302, USA), 25- μmnatNb foils (99.8%, LOT: T23A035, Alfa Aesar), and thin metallic75As layers electroplated onto 25- μmnatTi foil backings (99.6%, TI000205/TI000290, Goodfellow Metals). The arsenic targetswere again produced by members of this collaboration andcharacterized similarly to the arsenic targets created for theLANL experiment. The copper, niobium, and titanium foilsfor BNL were prepared according to the process outlined forthe same foils in Sec. II A 2 . Seven targets of each material were prepared for this irra- diation and six copper degraders were in turn characterized tocreate seven energy compartments within the stack. The electroplated arsenic targets were sealed using the same LINQTAPE PIT0.5S-UT Series Kapton polyimide filmtape described in Sec. II A 2 . The copper and niobium foils were encapsulated with DuPont Kapton polyimide film tape of43.2μm of silicone adhesive on 50 .8μm of polyimide back- ing (total nominal 11 .89 mg/cm 2). The foils were mounted to plastic frames, with copper and niobium foils paired dueto space limitations of the BLIP target box. Similarly tothe LANL irradiation, baling wire was used to secure one natCu+93Nb target and one75As+natTi target together in each energy compartment of the stack between degraders. TheBNL target box, also specially designed to be watertight sincethe BLIP target station is located underwater, has a 0.381-mmaluminum beam entrance window. A single stainless steelplate could only be included at the beginning of the stack in 034601-3MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) this experiment to assess the physical beam profile postirradi- ation due to space constraints. BLIP facility upstream beamline components that influ- ence beam properties were also included into the stackconsiderations. Beryllium and AlBeMet windows exist tofacilitate the beamline vacuum connections; two stainlesssteel windows and two water cooling channels are also inplace [ 24]. Together, these components give an approximately 1820-mg/cm 2system that the proton beam must traverse be- fore reaching the target box’s aluminum window. Unlike IPF,possible deformation of the BLIP upstream windows underhydrostatic and vacuum loading conditions are not measuredand may introduce unknown uncertainties to the stack charac-terization. Though the effect of these uncertainties is expectedto be small due to the lower stopping power at a higher beamenergy, corrections for potential changes to these upstreamconditions are considered in the stack transport calculationsin Sec. II C. The BNL target stack (Table VIin Appendix A)w a s irradiated for 3609 s with an H +beam of 200-nA nominal cur- rent. The beam current during operation was recorded usingtoroidal beam transformers and remained stable under theseconditions for the duration of the irradiation. The mean beamenergy extracted was 200 MeV at a 0.2% uncertainty [ 25]. B. Gamma spectroscopy and measurement of foil activities The collaborative nature of this work prompted the use of different types of germanium detectors and data acquisitionsystems to measure the induced activities of target foils. 1. LANL The LANL counting took place at two locations. One ORTEC IDM-200-VTM High-Purity Germanium (HPGe)detector and one ORTEC GEM p-type coaxial HPGe de-tector (model GEM20P-PLUS) were used to capture short-and intermediate-lived activation species directly at theIPF site of target irradiation. The IDM is a mechani-cally cooled coaxial p-type HPGe with a single, large-area85 mm diameter ×30 mm length crystal and built-in spec- troscopy electronics. The energy and absolute photopeakefficiency of the detectors were calibrated using standard 152Eu,207Bi, and241Am sources as well as a mixed γsource containing57Co,60Co,109Cd, and137Cs. The efficiency model used in this work is taken from the physical model presentedby Gallagher and Cipolla [ 26]. The LANL countroom was fur- ther commissioned to perform longer counts over a multiweekperiod, which was not possible at IPF. The countroom usesp-type ORTEC GEM series HPGes with aluminum windows. Following the irradiation, the IPF target box was removed from the beamline and raised into the IPF hot cell. Telema-nipulators were used to disassemble the stack and extract thefoils. The radiochromic film showed that an ≈1-cm diameter proton beam was fully inscribed within the samples through-out the stack. All target frames were wrapped in one layerof Magic Cover clear vinyl self-adhesive to fix any surfacecontamination. Due to elevated dose rates, only the arsenic,titanium, and copper targets were made available for countingon the day of irradiation. Initial data were acquired from 10- to 20-min counts of the targets starting approximately 2 h afterthe end-of-bombardment (EoB) at distances of 15 and 17 cmfrom the GEM detector face and 55 cm and 60 cm from theIDM face. One day postirradiation, within 19 h of EoB, thealuminum and niobium targets were accessible and countedmultiple times along with the other targets throughout the dayat positions of 15, 17, 25, 55, and 60 cm from the detectorfaces. Once appropriate statistics had been acquired to eitherestablish necessary decay curves for induced products or char-acterize monitor reaction channels, all targets were packagedand shipped to the LANL countroom. In the dedicated counting laboratory, the 40 available tar- gets were first repeatedly cycled in front of detectors at10–15 cm capturing 1 h counts over the course of a week. Thecountroom curators varied the foil distance from the detectorface on a regular basis to optimize count rate and dead time.The calibration data for each detector used, at each countingposition, were collected each day and made available withthe foil data. Over the following 6 weeks, cycling of thetarget foils in front of the detectors continued and count timeswere increased to 6–8 h to capture the longest-lived activationproducts. 2. BNL The BNL γspectroscopy setup incorporated two EU- RISYS MESURES 2 Fold Segmented “Clover” detectors inaddition to two GEM25P4-70 ORTEC GEM coaxial p-typeHPGe detectors and an ORTEC GAMMA-X n-type coaxialHPGe detector (model GMX-13180). All detector efficiencieswere calculated using a combination of 54Mn,60Co,109Cd, 137Cs,133Ba,152Eu, and241Am calibrated point sources, with the Gallagher and Cipolla [ 26] physical model. One GEM detector was situated in the BLIP facility at the irradiation sitewhile the remaining detectors were in a counting laboratory ina neighboring building. Within 2 h of EoB at BLIP, the copper foils and electro- plated arsenic targets were removed from the hot cell andcounted for over 10 min each using the GEM detector in thefacility. The observed beam spot size on targets was ≈1c m in diameter. Once the niobium foils had been pulled from theBLIP hot cell, all targets were transported to the nearby count-ing laboratory. There, the copper and arsenic foils were cycledfirst through 10- to 30-min counts, followed by hour-longcounts, on the Clovers and GEM at 10–15 cm from the detec-tor faces. The niobium foils were assigned a similar countingscheme starting approximately 20 h after EoB. Cycling andcounting of the foils continued for an additional 24 h. Within two weeks of EoB, all targets were shipped back to LBNL. The subsequent γ-spectroscopy at the 88-Inch Cyclotron utilized an ORTEC GMX series (model GMX-50220-S) HPGe, which is a nitrogen-cooled coaxial n-typeHPGe with a 0.5-mm beryllium window and a 64.9-mm diam-eter×57.8-mm-long crystal. Multiday to week-long counts of the copper, arsenic, and niobium foils were performed withthe LBNL GMX over the course of 2 +months to ensure that all observable long-lived products could be quantified. 034601-4INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 2. Example γ-ray spectrum from the induced activation of a niobium target in the LANL stack at approximately Ep=91 MeV. The spectrum was taken approximately 20 h after EoB, and the smooth fits to the peaks of interest shown are produced by the NPAT package [ 27]. 3. Activation analysis While the specifications of counting equipment and pro- cedure varied between irradiations, the data analysis for themeasurement of induced target activities and cross sectionsfollowed a standardized approach. The procedure is well de-scribed in Voyles et al. [2] and Morrell et al. [6] but is included here for clarity and completeness. Theγemission peaks from decaying activation products were identified from the previously described γ-ray spectra. These photopeaks were fit using the NPAT code packagedeveloped at UC Berkeley [ 27]. Example fits are shown in Fig.2for a spectrum collected from the LANL Nb-SN1 target of the stack in Table V(see Appendix A). The activity Afor each activation product of interest at a delay time t dsince the end-of-bombardment to the start of counting was then determined from the net counts foundN cafter corrections for γintensity Iγ, detector efficiency /epsilon1, dead time, counting time, and self-attenuation within the foilsaccording to: A(t d)=Ncλ (1−e−λtreal)Iγ/epsilon1treal tliveFatt, (1) where λis the decay constant for the radionuclide of interest, trealandtlivedescribe the real and live time for detector ac- quisition, respectively, and Fattis the photon self-attenuation correction factor. Fattis calculated using photon attenuation cross sections retrieved from the XCOM database [ 28] and takes the convention that all activity is assumed to be made atthe midplane of the foils. The EoB activity A 0for a given radionuclide was subse- quently found from a fit to the relevant Bateman equation.Moreover, the benefit of repeated foil counts in this work andthe use of multiple γrays is evidenced here by providing mul- tiple radionuclide activities at numerous t d, which establish a consistent decay curve. Through a regression analysis ofdecay curves, it is possible to extract the A 0for each activation product in a more accurate manner than simply basing its cal-culation on a single time point and a single γ-ray observation. If an activation product of interest is populated without contribution from the decay of a parent radionuclide, then theEoB activity is found from a fit to the first-order Bateman equation: A(t d)=A0e−λtd. (2) Typically, if it is needed to calculate EoB activities within a feeding chain in this work, then the required calculation isonly second order. This is the case for isomeric to groundstate conversions as well as two-step β-decay chains. In these circumstances, the decay curve is given by: A 2(td)=A0,1Brλ2 λ2−λ1(e−λ1td−e−λ2td)+A0,2e−λ2td,(3) where A2(td) is still found from Eq. ( 1),Bris the decay branching ratio, and the 1 and 2 subscripts denote the parentand daughter nuclides, respectively, in the two-step decaychain. This two-step fit to calculate A 0,2uses the indepen- dently determined A0,1from Eq. ( 2) when possible, but otherwise both variables are fit together. The decay curveregressions in this work were additionally performed withthe NPAT code package [ 27]. A regression example for the 86Zr→86Ydecay chain is shown in Fig. 3. The total uncertainties in the determined EoB activities had contributions from uncertainties in fitted peak areas, evaluatedhalf-lives and γintensities, and detector efficiency calibra- tions. Each contribution to the total uncertainty was assumedto be independent and was added in quadrature. The impactof calculated A 0uncertainties on final cross-section results is detailed in Sec. II D. C. Stack current and energy properties The methods of current monitoring during beam operation discussed in Secs. II A 2 andII A 3 provide valuable informa- tion for the experimental conditions, but their output is notsufficient to precisely describe the beam energy and intensityevolution throughout a target stack [ 2,6,9,10]. Instead, more detailed calculations must be retrieved from monitor foil ac-tivation analysis, where known reaction cross sections can beused to measure beam current in the multiple energy positionsof a stack. 034601-5MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) FIG. 3. Example of initial activity fitting for two-step beta-decay chain of86Zr feeding86Yas residual products in the niobium irradiations. The relevant proton fluence monitor reactions used in the irradiations were LANL: (i)natCu(p,x)56Co,58Co,62Zn (ii)natTi(p,x)48V (iii)natAl(p,x)22Na BNL: (i)natCu(p,x)58Co where only reactions with IAEA-recommended data in the relevant proton energy ranges have been considered [ 29]. In the BNL irradiation, the lack of reliable data for high- proton-energy reactions precluded the use of most monitorchannels and as a result only the 58Co activation product was taken to extract the beam current. However,natCu(p,x)56Co has significant data in this high-energy region and was prelim-inarily used as a validation of the beam current derived fromthe 58Co calculations. TheA0for the monitor reaction products were calculated according to the formalism presented in Sec. II B 3 . Since the beam was constant throughout the irradiation period, theproton beam current I pwas calculated at each monitor foil position by the relation: Ip=A0 (ρN/Delta1r)(1−e−λtirr)¯σ, (4) where Ipis output in units of protons per second, (1 −e−λtirr) corrects for decay that occurred during the beam-on irradi-ation time t irr,ρN/Delta1ris the relevant measured areal number density calculated from Tables VandVI(see Appendix A), and ¯σis the flux-weighted production cross section. The ¯σformalism is needed to account for the energy width broadening resulting from energy straggle of the beam as itis propagated toward the back of the stack [ 2,6,9,10]. Using the IAEA-recommended cross-section data σ(E) for the rele- FIG. 4. Visualization of the calculated proton energy spectrum for each niobium foil in the LANL stack. vant monitor reactions [ 29], the flux-weighted cross section is calculated from: ¯σ=/integraltext σ(E)φ(E)dE/integraltext φ(E)dE, (5) where φ(E) is the proton flux energy spectrum. φ(E)w a s determined here using an Anderson and Ziegler-based MonteCarlo code, as implemented in NPAT [ 27,30]. The calculated energy spectrum resulting from the Anderson and Zieglercalculation in the LANL irradiation is shown in Fig. 4as an example. The implementation of this monitor foil deduced current, following Eqs. ( 4) and ( 5), is shown for each irradiation site in Fig. 5. Included in Fig. 5are weighted averages of all the available monitor foils for the fluence at each stack position.The weighted averages account for data and measurement cor-relations between the reaction channels in each compartment.An uncertainty-weighted linear fit is also included for eachsite as a global model to impose a smooth and gradual fleuncedepletion. Included in the results of Fig. 5is a reduction in system- atic uncertainty using the “variance minimization” techniquepresented in Graves et al. [9], Voyles et al. [2], and Morrell et al. [6]. This technique was applied, as partial disagree- ment between the initial proton fluence predictions from eachmonitor channel in each energy compartment of the stack ateach experiment site was observed. The disagreement wasmost noticeable near the rear of each stack where contribu-tions of poor stopping power characterization, straggling, andsystematic uncertainties from upstream components becamemost compounded. The independent measurements of protonfluence from the monitor reactions should all theoreticallybe consistent at each energy position given accurate monitorreaction cross sections and foil energy assignments. The vari-ance minimization technique is a corrective tool applied tothe stopping power in simulations to address this discrepancythrough the treatment of the effective density of the Al/Cu 034601-6INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 5. Plots of the proton beam current measured by monitor reactions in the LANL and BNL stacks following adjustments madeby the variance minimization technique. The natCu(p,x)56Co moni- tor reaction is plotted for BNL but its data were not used for any of the BNL fluence calculations or the variance minimization. degraders in each stack as a free parameter. This is reasonable because the majority of the stopping power for the beamoccurs in the thick degraders. The free parameter can thenbe optimized by a reduced χ 2minimization technique for the global linear fit of the monitor fluence data. For both stacks, the degraders’ effective densities were varied uniformly in the stopping power simulations by a factorof up to ±25% of nominal values. The resulting reduced χ 2 in each case is given in Fig. 6. Figure 6indicates that a change in degrader density, which is equivalent to a linear changein stopping power, of +4.35% and −1.84% compared to nominal measurements for the LANL and BNL stacks, respec-tively, minimizes the monitor foil disagreement in each case.Previous stacked-target work has always shown a modest pos-itive enhancement to the stopping power of +2−5%, which makes the BNL optimization interesting [ 2,6]. It is likely that the negative adjustment in the BNL case is mostly due tocompensation for the less well-known characterizations of theupstream cooling water channel and window deformation. Itis also possible that some of this effect may be attributedto the use of copper degraders at BNL versus the aluminumdegraders used at LANL and LBNL. FIG. 6. Result of χ2analysis used in the variance minimization technique to determine the required adjustment to stopping power within the proton energy spectrum calculations per stack. Monitor reactions that threshold in the energy region of the stack, such as56Co near the LANL stack rear, are ex- tremely valuable in this minimization approach as they are most sensitive to changes in stopping power and energy as- signment thereby providing physical limits for the problem.The relative shallowness of the BNL χ 2curve is most likely due to the limitation of minimizing using just one monitorreaction. Note that this degrader density variation procedureis a computation tool to correct for poorly characterized stop-ping power at these energies and does not mean that theactual degrader density was physically different than what wasmeasured [ 6]. The minimized reduced χ 2also provides optimized beam energy assignments for each foil in a stack from the cor-rected transport simulation. The energy assignments are theflux-averaged energies using φ(E) with uncertainties per foil taken as the full width at half maximum. These en-ergy assignments for the niobium targets are provided inTable I. In the BNL fluence results, the optimized global linear model provides an interpolation to each individual niobiumfoil with a better accuracy and uncertainty than just utilizingthe sole 58Co fluence prediction in each compartment. In the LANL fluence results, the linear fit was used for the varianceminimization but the correlation-weighted-average values ineach compartment were directly used for calculating produc-tion cross sections. This is possible without any need forinterpolation or worry of model selection influence because ofthe contributions from multiple available monitor reactions. D. Cross-section determination Given the activity, weighted-average beam current and en- ergy, timing, and areal density factors previously discussed,the flux-averaged cross sections for products of interest in this 034601-7MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) TABLE I. Summary of cross sections measured in this work. Subscripts ( i)a n d( c) indicate independent and cumulative cross sections, respectively. Uncertainties are listed in the least significant digit, that is, 119.8 (10) MeV means 119.8 ±1.0 MeV. 93Nb(p,x) Production cross sections (mb) Ep(MeV) 192.38 (73) 177.11 (77) 163.31 (81) 148.66 (86) 133.87 (92) 119.8 (10) 104.2 (11) 91.21 (52) 79.32 (58) 72Se(c) 0.066 (13) 0.0193 (26) −−−− − − − 73As(c) 1.15 (30) 0.77 (18) −−−− − − − 74As(i) 0.182 (12) 0.1071 (71) −−−− − − − 75Se(c) 1.443 (76) 0.963 (25) 0.603 (21) 0.200 (24) −− − − − 81Rb(c) −−−−−− − 2.99 (55) − 82mRb(i) 10.55 (36) 9.28 (27) 8.39 (22) 6.86 (24) 4.93 (18) 3.65 (20) 3.49 (18) 3.07 (13) 1.06 (15) 83Rb(c) 40.0 (22) 36.8 (17) 35.0 (19) 30.9 (19) 27.0 (19) 15.97 (71) 5.59 (41) 6.27 (47) 7.12 (53) 83Sr(c) 32.3 (20) 29.1 (17) 27.1 (15) 25.0 (16) 20.5 (13) 13.2 (11) 3.64 (42) 3.88 (61) 5.13 (75) 84Rb(i) 3.11 (17) 2.89 (16) 2.64 (14) 2.32 (13) 2.06 (11) 1.701 (94) 0.699 (40) 0.563 (37) 0.436 (31) 85mY(c) −−−−−− − 26.1 (28) 18.8 (24) 86Rb(i) − 0.256 (21) −−−− − − − 86Y(i) 45.2 (11) 43.88 (93) 44.77 (84) 44.21 (84) 42.64 (80) 38.67 (88) 29.31 (78) 33.4 (13) 42.7 (15) 86Zr(c) 20.3 (18) 21.5 (19) 22.3 (19) 22.5 (19) 23.0 (19) 18.4 (16) 9.91 (90) 16.4 (15) 23.5 (20) 87Y(c) 106.5 (27) 110.3 (26) 112.9 (24) 115.7 (24) 120.2 (26) 123.7 (30) 103.2 (30) 106.1 (48) 56.2 (25) 87mY(c) 86.5 (57) 89.4 (58) 92.5 (59) 94.6 (61) 98.4 (63) 99.2 (65) 82.4 (55) 87.9 (41) 47.1 (21) 88Y(i) 18.36 (52) 18.71 (46) 18.63 (40) 18.39 (38) 18.22 (39) 17.84 (41) 17.18 (47) 19.07 (62) 14.86 (48) 88Zr(c) 85.9 (48) 91.5 (50) 95.9 (51) 101.1 (54) 109.0 (58) 117.6 (64) 136.5 (77) 159 (12) 141.5 (95) 89Zr(c) 108.6 (36) 114.4 (35) 125.2 (43) 136.2 (52) 145.5 (50) 159.5 (59) 177.3 (63) 196 (15) 249 (16) 90Nb(i) 69.4 (22) 76.2 (21) 84.7 (21) 90.4 (24) 102.8 (25) 110.5 (31) 131.2 (39) 155.1 (46) 174.4 (49) 90Mo (i) 4.54 (33) 5.01 (34) 5.46 (32) 6.55 (59) 7.70 (70) 9.64 (88) 12.3 (11) 17.9 (11) 22.8 (14) 91mNb(c) 14.1 (22) 14.7 (23) 14.7 (23) 17.3 (27) 17.3 (27) 20.5 (32) 22.0 (34) 25.8 (40) 27.3 (42) 92mNb(i) 25.9 (12) 29.5 (13) 30.9 (13) 32.4 (14) 35.4 (15) 37.8 (16) 41.4 (19) 45.4 (24) 47.8 (26) 93mMo (i)−−−−−− − 1.069 (71) 0.75 (10) Ep(MeV) 72.52 (62) 67.14 (65) 63.06 (68) 60.08 (71) 57.47 (73) 55.58 (75) 53.62 (77) 51.61 (80) 83Rb(c) 5.32 (39) 2.31 (19) 0.71 (11) 0.19 (11) −− − − 83Sr(c) 4.31 (68) 1.40 (59) 1.04 (55) −−− − − 84Rb(i) 0.625 (43) 0.637 (44) 0.533 (39) 0.368 (31) 0.250 (25) 0.143 (21) 0.078 (14) − 85mY(c) 5.8 (13) −−−−− − − 86Y(i) 43.5 (15) 32.7 (12) 21.8 (10) 10.02 (61) 4.38 (46) −−− 86Zr(c) 28.0 (23) 22.1 (18) 12.3 (13) 5.9 (10) 2.50 (64) 1.58 (72) −− 87Y(c) 61.5 (23) 78.3 (26) 101.1 (32) 115.3 (43) 116.2 (56) 109.3 (41) 97.3 (31) 86.9 (36) 87mY(c) 50.6 (23) 64.7 (30) 83.6 (39) 93.8 (43) 96.5 (45) 90.6 (43) 80.3 (38) 69.7 (36) 88Y(i) 11.82 (41) 9.60 (35) 9.15 (34) 9.55 (36) 10.93 (60) 10.53 (40) 11.45 (42) 13.34 (47) 88Zr(c) 92.0 (75) 45.2 (56) 27.3 (41) 24.0 (41) 25.4 (70) 27.6 (42) 31.9 (42) 41.0 (47) 89Zr(c) 309 (21) 328 (17) 296 (21) 205 (15) 171 (23) 136 (14) 80.3 (86) 54.6 (77) 90Nb(i) 201.0 (58) 225.0 (62) 271.2 (79) 307.2 (85) 350.7 (97) 369 (10) 394 (11) 429 (12) 90Mo (i) 28.5 (17) 36.2 (22) 48.9 (36) 63.7 (37) 83.3 (46) 91.7 (51) 103.3 (57) 118.9 (63) 91mNb(c) 30.7 (47) 31.0 (48) 34.0 (53) 36.3 (56) 37.0 (62) − 36.9 (57) 40.6 (63) 92mNb(i) 51.3 (28) 51.2 (32) 54.7 (30) 58.3 (30) 58.2 (31) 56.6 (30) 57.7 (29) 61.7 (32) 93mMo (i) 1.19 (12) 1.11 (14) 1.33 (15) 1.59 (20) 1.45 (24) 1.25 (19) 1.86 (25) 1.76 (18) work were calculated using Eq. ( 6): σ=A0 Ip(ρN/Delta1r)(1−e−λtirr). (6) The93Nb(p,x) cross-section results are given in Table I, which reports measurements for93mMo,92mNb,91mNb,90Mo, 90Nb,89Zr,88Zr,88Y,87mY,87Y,86Zr,86Y,86Rb,85mY,84Rb, 83Sr,83Rb,82mRb,81Rb,75Se,74As,73As, and72Se. The 75As(p,x) data in addition to thenatCu(p,x) andnatTi(p,x) results will be detailed in a future publication.A distinction is made in this work between cumulative, ( c), and independent, ( i), cross-section values. Numerous reaction products in these irradiations were produced both directlyand from decay feeding. Where the decay of any precursorscould be measured and the in-growth contribution separated,or where no decay precursors exist, independent cross sec-tions for direct production of a nucleus are reported. Wherethe in-growth due to parent decay could not be deconvolved,due to timing or decay property limitations, cumulative crosssections are reported. The final uncertainty contributions to the cross-section measurements include uncertainties in evaluated half-lives 034601-8INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) TABLE II. Default models implemented in reaction codes. Reaction code Proton/neutron optical model Alpha optical model Level density Preequilibrium TALYS-1.9 Koning-Delaroche [ 47] Avrigeanu (2014) [ 48] Gilbert-Cameron constant temperature and Fermi gasmodel [ 17]Two-component exciton model [ 11] CoH-3.5.3 Koning-Delaroche Avrigeanu (1994) [ 49] Gilbert-Cameron constant temperature and Fermi gasmodelTwo-component exciton model EMPIRE-3.2.3 Koning-Delaroche Avrigeanu (2009) [ 50] Enhanced Generalized Superfluid Model [ 19]PCROSS one-component exciton model [ 19] ALICE-20 Becchetti-Greenlees [ 20,51] Igo (1959) [ 52] Shell-dependent Kataria-Ramamurthy model [ 20]Hybrid Monte-Carlo Simulation precompound decay [ 20] (0.1–0.8%), foil areal density measurements (0.05–0.4%), proton current determination calculated from monitor flu-ence measurements and variance minimization (2–4%), andA 0quantification that accounts for efficiency uncertainty in addition to other factors listed in Sec. II B 3 (2–10%). These contributions were added in quadrature to give uncertainty inthe final results at the 3–6% level on average (Table I). III. RESULTS AND DISCUSSION The experimentally extracted cross sections are compared with the predictions of nuclear reaction modeling codesTALYS-1.9 [ 17], CoH-3.5.3 [ 18], EMPIRE-3.2.3 [ 19], and ALICE-20 [ 20], each using default settings and parameters, to initially explore variations between the codes and theirsensitivity to preequilibrium reaction dynamics. Where mea-sured cumulative cross sections are plotted, the correspondingcode calculations shown also include the necessary parentproduction to estimate cumulative yields. Note, however, thatALICE-20 is not suited to calculate independent isomer orground state production due to a lack of detailed angularmomentum modeling. Furthermore, in the code comparisons, the TALYS and ALICE codes account for potential deuteron, 3He, and triton emissions at all incident proton energies. Default EMPIRElimits these emissions and CoH ignores these effects alto-gether. The TALYS output provides total production crosssections for these emission channels that can be used to es-timate their influence. In TALYS, the cumulative deuteron, 3He, and triton cross section is calculated as 3.1%, 3.5%, and 11.8% of the combined proton and neutron productionat 50, 100, and 200 MeV, respectively. At each energy, thedeuteron production dominates over 3He and triton emissions. Therefore, while the inclusion of these more complex emis-sion types accounts for mostly a small effect, it is a pointof difference between the code calculations. A summary ofthe key default models implemented in each code is given inTable II. Comparisons with the TENDL-2019 library [ 21] are also made. Additionally, the cross-section measurements in thiswork are compared to the existing body of literature data,retrieved from EXFOR [ 2,12,31–46].The cross sections and code comparisons for four residual products of interest are described in detail below. The re-maining cross-section figures are given in Appendix B(Figs. 44–62). A.93Nb(p,4n)90Mo cross section As presented in Voyles et al. [2], the93Nb(p,4n)90Mo reaction is compelling as a new, higher energy proton monitorreaction standard. The 93Nb(p,4n) reaction channel is inde- pendent of any ( n,x) contaminant production that could be due to secondary neutrons stemming from ( p,xn) reactions and requires no corrections for precursor decays.90Mo decays with seven intense γlines ranging from near 100 to 1300 keV that allow for easy delineation on most detectors [ 53]. Further, the90Mo 5.56 ±0.09 hr half-life is fairly flexible for a monitor reaction [ 53], as the isotope can still be readily quantified more than one day postirradiation, as was done inthe counting for these experiments. The cross-section results here, shown in Fig. 7, align very well with the Voyles et al. [2] measurements in FIG. 7. Experimental and theoretical cross sections for90Mo pro- duction, peaking near 120 mb around 50 MeV. 034601-9MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) FIG. 8. Experimental and theoretical cross sections for90Nb pro- duction, peaking near 425 mb around 50 MeV. predicting a peak cross section of approximately 120 mb near 50 MeV. The Ditrói et al. [33] data in Fig. 7predicts a compound peak of less than half the magnitude observed in this workand Voyles et al. [2]. This underprediction appears as a trend across numerous reaction products and can be seen in theremaining excitation function plots shown in Appendix B.T h e Titarenko et al. [32] dataset is also slightly inconsistent with this work, as it too implies a smaller peak, though not as smallas that put forth by Ditrói et al. [33]. Only CoH and ALICE reproduce the peak magnitude of the cross section, while TALYS, EMPIRE, and TENDL predicta smaller magnitude similar to Ditrói et al. [33]. Further, the TALYS and EMPIRE default calculations misplace thecompound peak centroid relative to the other calculations.Although CoH and ALICE perform best, neither properly ac-counts for the increased production on the peak’s high-energyfalling edge due to a preequilibrium “tail” contribution. This work gives the first measurements of 93Nb(p,4n)90Mo above 100 MeV and is the broadest energy-spanning dataset for the reaction to date. A recentproton irradiation with niobium targets was conducted ina separate experiment at LBNL for energies from 55 MeVto threshold in order to fully characterize the remaininglow-energy side of the compound peak. These results will bediscussed in a subsequent publication. B.93Nb(p,p3n)90Nb cross section 90Nb is the most strongly fed observed residual product stemming from proton reactions on niobium in this investi-gation, accounting for ≈30% of the total nonelastic reaction value at its peak. The 90Nb cross-section data in this work were measured independently through a two-step β-decay chain fit that accounted for contributions from its90Mo parent. The93Nb(p,p3n)90Nb results of this work (Fig. 8) agree very well with the prior literature data and provide a well-characterized, significant extension beyond 75 MeV. FIG. 9. Experimental and theoretical cross sections for cumu- lative89Zr production, showing peaks for both93Nb(p,αn)a n d 93Nb(p,2p3n) formation mechanisms. No code matches the large compound peak magnitude of the experimental data. CoH and EMPIRE come the closestbut suffer from their misplacement of the peak’s energy by ap-proximately 5 MeV. The shapes of default TALYS, TENDL,and CoH show some affinity for the very pronounced high-energy preequilibirum tail in 90Nb production whereas default ALICE and EMPIRE lack in this regard. The mispredictionfrom ALICE here is in stark contrast to its close prediction ofthe neighboring ( p,4n) reaction. It is particularly concerning for the global predictive power of 93Nb(p,x) modeling that no code adequately reproduces this dominant reaction channel. Moreover, the proton emittedin the ( p,p3n) channel is likely to result from preequilibrium emission at higher energies due to its suppression from theCoulomb barrier. The poor default predictions of this chan-nel thereby suggest a systematic issue in the preequilibriummodeling of these codes. C.93Nb(p,x)89Zr cross section The lifetimes of89Zr precursor feeding nuclei (89Mo, 89mNb,89Nb,89mZr) were too short to be able to quantify their production in these irradiations given the counting proceduresdescribed in Secs. II B 1 andII B 2 [54]. As a result, the mea- surement of 93Nb(p,x)89Zr, provided in Fig. 9, is cumulative and includes contributions from all of these precursors as wellas the ground state of 89Zr. 89gZr is a useful positron emitting isotope for radiolabelling monoclonal antibodies to provide an accurate picture ofdose distribution and targeting effectiveness in immunoPET[1,55,56]. Its 78.41 ±0.12 hr half-life meshes nicely with the typical 2–4 day pharmacokinetic properties of monoclonalantibodies in tumors [ 54,55]. Further, zirconium is especially attractive for this application because of existing commer-cially available chelating agents for labeling, which have beenproven to remain bound in vivo . Production of 89gZr via 93Nb(p,x) using 200 MeV protons may offer an attractive 034601-10INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) alternative to the established89Y(p,n)89Z rr o u t eu s e di nl o w - energy cyclotrons, potentially facilitating89Zr production in locations such as IPF and BLIP [ 55]. However, the coproduc- tion of88Zr (t1/2=83.4±0 . 3d[ 57]) in the93Nb(p,x) path may make the low-energy ( p,n) route more viable. This work gives the most complete description of the cu- mulative higher-energy production peak near 67 MeV andgreatly extends the cross-section information beyond 75 MeV,where only two prior data points existed. The larger higher-energy peak is indicative of independent 89Zr formation through the93Nb(p,2p3n) mechanism in contrast to the lower-energy compound peak around 25 MeV, denoting for-mation by 93Nb(p,αn). The measured values agree well with Steyn et al. [34] on the higher-energy peak rising edge, but predict a peak value of approximately 325 mb, which is largerthan both Steyn et al. [34] and Titarenko et al. [32]. The Ditrói et al. [33] magnitude discrepancy is noticeable in this measurement where the dataset underpredicts both the risingedge and peak relative to all the other literature. It is difficult to comment on the performance of the codes here due to the feeding from the three nuclei, and multiple iso-meric states, involved in the calculations. It can be noted thatthere is still the persistent difficulty in properly modeling thepreequilibrium effect throughout these nuclei though, whichmanifests in these codes as both a shift in the centroids for thehigher-energy peak and a missing high-energy tail. D.93Nb(p,x)86Ycross section The LANL and BNL irradiations in this investigation al- lowed for a measurement of86Yproduction from reaction threshold to near 200 MeV. As specifically referenced inFig.3, the cumulative 86Zr production could be directly de- termined, which then enabled an independent quantificationof 86Y. The 33% β+decay mode of86Yalong with its 14.74 ±0.02 h half-life make it a promising surrogate for imaging the biodistribution and studying the absorbed doseof 90Y(100% β−) for bone palliative treatments [ 58,59]. However, compared to the established86Yproduction routes using strontium targets, a niobium target based pathway intro-duces long-lived 88Y(t1/2=106.626±0.021 d [ 57]) isotopic impurities and suffers a lower yield, making it less advanta-geous [ 60]. The extracted excitation function (Fig. 10) is in excellent agreement with the measurements of Voyles et al. [2] and Titarenko et al. [32]. This wide-spanning dataset, similar to the Michel et al. [31] work, characterizes the full compound behavior as well as the high-energy preequilibrium compo-nent. However, where there is good agreement to the Michelet al. [31] work below 100 MeV, our dataset predicts lower values for the remainder of the preequilibrium tail by 10–15mb. 86Yis not a strongly fed residual product channel, which gives some explanation to the variation between differentcode calculations. The theoretical predictions are sensitive tocompensating effects from miscalculations in more dominantreaction channels. As a result, no code properly reproducesboth the experimentally determined magnitude and shape ofthe excitation function using default parameters. CoH pre- FIG. 10. Experimental and theoretical cross sections for86Ypro- duction, spanning from reaction threshold to near 200 MeV. dicts the compound peak with the closest magnitude, though the peak centroid, falling edge, and preequilibrium shape areincorrect. TALYS and TENDL perhaps best represent theoverall shape but are far lower in magnitude than the exper-imental data. Other notable cross-section results in this work include 82mRb,83Sr, and84Rb production, where data had been ex- tremely sparse but now have their excitation functions wellcharacterized beginning from threshold. These cross-sectionresults, along with the measurements of all other observednuclei, are detailed in Appendix B. IV . HIGH-ENERGY PROTON REACTION MODELING The large body of data measured here, in addition to the ex- isting93Nb(p,x) literature data, presents a good opportunity to study high-energy proton reaction modeling on sphericalnuclei. Our approach is to follow the procedure establishedfor modeling high-energy ( n,x) reactions by comprehensively fitting the most prominent residual product channels first, fol-lowed by the weaker channels. A critical focus in developinga consistent fitting procedure is to gain insight into preequilib-rium reaction dynamics in an attempt to isolate shortcomingsin the current theoretical understanding. As a note, the fitting work presented here is based in the TALYS reaction code. TALYS has widespread use in thenuclear community and is an accessible code-of-choice forreaction cross-section predictions. Further, TALYS incorpo-rates the widely employed two-component exciton model forpreequilibrium physics, which means that any outcomes de-rived in this work can be applied broadly by the nuclearreaction data evaluation community [ 17,61–63]. A. Preequilibrium in TALYS-1.9 The currently used two-component exciton model in TALYS-1.9 was constructed through an extensive global pree-quilibrium study by Koning and Duijvestijn [ 11]. Their work 034601-11MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) relied on virtually all existing angle-integrated experimen- tal continuum emission spectra for ( p,xp), (p,xn), (n,xn), and ( n,xp) reactions for A/greaterorequalslant24 spanning incident energies between 7 and 200 MeV. No double-differential or residualproduct cross sections were included in the semiclassicaltwo-component model development, but these results wereexpected to fall out naturally from globally fitting the emissionspectra. The decision to adopt the exciton model over otherpotential preequilibrium calculation methods is detailed byKoning and Duijvestijn [ 11]. The significant updates made by Koning and Duijvestijn [ 11] to previous two-component models include using a more recent optical model potential (OMP) forneutrons and protons, a new and improved determinationof collision probabilities for intranuclear scattering to moreor less complex particle-hole states, surface interactionsspecific to projectiles and targets, and greater detailapplied to multiple preequilibrium emission. The mostnoteworthy of these changes is the collision probabilities,which use a new parametrization of the phenomenologicalsquared matrix element for the effective exciton residualinteraction applicable across the entire 7- to 200-MeV energyrange [ 11,62,64]. Moreover, in the two-component exciton master equation used by Koning and Duijvestijn [ 11], which describes the temporal development of the composite system for projectile-target interaction in terms of exciton states characterized byproton and neutron particle and hole numbers, internal tran-sition rates are defined to model particle-hole creation ( λ +), conversion ( λ0), and annihilation ( λ−). These transition rates govern the evolution of the total exciton state and are crit-ical pieces for the overall preequilibrium energy-differentialcross-section calculation [ 62–64]. Formally, the model is ap- proximated to disregard pair annihilation where it has beenshown that decay rates to less complex exciton states aresmall compared to other processes in the preequilibrium partof the reaction and can be neglected [ 11,62]. Transition rates are calculated from collision probabilities, determined usingtime-dependent perturbation theory and Fermi’s golden ruleto give expressions such as Eq. ( 7)f o rap r o t o n( π)-proton (π) collision λ ππ, leading to an additional proton particle-hole pair (1 p)[17]: λ1p ππ=2π ¯hM2 ππω. (7) In the collision probability definition given in Eq. ( 7),ω is the particle-hole state density as a function of the exci-ton state configuration and excitation energy, as formulatedby Dobeš and B ˇeták [ 64]. An exciton state configuration is defined by ( p π,hπ,pν,hν) with the proton (neutron) particle number as pπ(pν) and the proton (neutron) hole number as hπ(hν).M2 ππ, and the other corresponding proton and neu- tron (ν) permutations ( M2 πνetc.), are average squared matrix elements of the residual interaction inside the nucleus thatdepend only on the total energy of the composite nucleusto describe two-body scattering to exciton states of differentcomplexity [ 17]. In TALYS-1.9, the matrix element variations for like and unlike nucleons can be cast in terms of a total FIG. 11. Illustration of the initial stages of reaction in the pree- quilibrium exciton model from Selman [ 65]. Solid horizontal lines are representative of single particle states in a potential well. Particlesare shown as solid circles while holes are empty dashed circles [ 63]. average M2by: M2 xy=RxyM2, (8) with xandydenoting some combination of πandν.Rxyis a free parameter with default values in TALYS-1.9 such asR πν=1.0[17]. Given the complete body of experimental emission spectra data, the following semiempirical expression for the total av-erage squared matrix element is implemented in TALYS-1.9for incident energies 7–200 MeV [ 17]: M 2=C1Ap A3⎡ ⎣7.48C2+4.62×105 /parenleftbigEtot nAp+10.7C3/parenrightbig3⎤ ⎦, (9) where C1,C2, and C3are adjustable parameters, Ais the target mass, Apis the mass number of the projectile, nis the total exciton number, and Etotis the total energy of the composite system. In particle-hole creation, the change in state excitonnumber is /Delta1n=+2, while in a conversion transition /Delta1n=0. For an incident proton projectile, a simplified visualization of the scattering with target nucleons defined by the exci-ton model is shown in Fig. 11. Additionally, a schematic of the two-component transitions from an initial exciton stateconfiguration of (1,0,0,0) to more complex states is given inFig.12[64]. Each state in Fig. 12has an associated mean lifetime τ(p π,hπ,pν,hν) defined as the inverse sum of the various internal transition rates and the total emission rate [ 17]. As a result, the parametrization of M2is an essential compo- nent of the state lifetime calculation. Moreover, it can benoted from the representation in Fig. 12that in order to calculate the overall energy differential preequilibrium crosssection, the exciton model calculation must keep track ofall emissions in addition to the part of the preequilibriumflux that has survived emission and now passes throughnew configurations. This survival population is generally de-noted by P(p π,hπ,pν,hν) and is also calculated on the basis of the M2parametrization. The total emission rate Wfor an ejectile kof emission energy Ekis not a function of M2but is instead calculated from the optical model and ω [17]. 034601-12INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 12. Scheme of the two-body interaction pathways in the two-component exciton model where individual exciton states arecharacterized by ( p π,hπ,pν,hν). The particle-hole annihilation pathways to less complex states are neglected here. The single arrows represent particle-hole creation transitions and the double arrowsrepresent conversion transitions. The hooked arrows represent the chance for particle emission to the continuum at the given exciton number n,w h e r e nis the sum of all present particles and holes in a configuration. Given these considerations, the energy differential preequi- librium cross section can be calculated by [ 17]: dσPE k dEk=σCFpmax π/summationdisplay pπ=p0πpmax ν/summationdisplay pν=p0νWk(pπ,hπ,pν,hν,Ek) ×τ(pπ,hπ,pν,hν)P(pπ,hπ,pν,hν),(10) where σCFis the compound nucleus formation cross section, also calculated from the optical model; pmax πand pmax νare particle numbers representing the equilibration limit for thescattering interactions at which point the Hauser-Feshbachmechanism handles the reaction calculations. In the case ofmultiple preequilibrium emissions, additional proton and neu-tron number dependencies are introduced into the excitonmodel, though M 2and the internal transition rates play similar critical roles [ 11]. Ultimately, given that the level-density and optical model parameters at high energies are well characterized comparedto the relative paucity of information surrounding preequi-librium dynamics, it can be argued that an exploration ofpreequilibrium emission resulting from the exciton model inTALYS is centrally an exploration of the effective squaredmatrix element parametrization. TALYS’s abundance of ad-justable keywords related to M 2make it an ideal tool to investigate this parametrization using measured residual prod-uct excitation function data. However, it will not be possible toentirely neglect the effects of level-density and optical modeladjustments on reaction observables and it is necessary tobe cognizant of these additional degrees of freedom in anyattempt to isolate M 2effects [ 11]. B. Residual product-based standardized fitting procedure The approach pursued in this work to accurately reproduce production probabilities for high-energy proton-induced re-actions on spherical nuclei using TALYS and its associatedadjustable parameters is outlined in the flow chart of Fig. 13. This fitting procedure prioritizes an examination of excitonmodel physics to help identify trends and biases within thecurrent calculation technique. A further motivation of this procedure is to avoid the compensating errors caused by current nonevaluation fittingmethods that utilize too-few experimental data and/or too-simplistic parameter changes, which may ultimately hindermodeling as a whole. Particularly, simplistic or arbitrary pa-rameter adjustments in TALYS, tuned to provide a better fit fora singular reaction channel of interest, are nonunique and maynot hold a global physical basis because neighboring reaction Select base level density model Sensitivity study of exciton model parameters to optimize pre-equilibriumIdentify strongest reaction channels with existing cross section data Optical model adjustments Local level density adjustments Improved cross section calculations Validation of ad justed parameters via unused channels Global weighted goodness-of-fit comparisonDefault TALYS code Default residual product cross sections FIG. 13. Proposed standardized reaction modeling code parameter adjustment procedure, reliant on residual product excitation function data, built to best fit multiple dominant reaction channels and gain justified insight into the preequilibrium mechanism. 034601-13MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) FIG. 14. Evidence for nonunique modeling solution when only considering one reaction channel. Ten sets of different parameter changes are shown to reproduce similar improvement over the defaultprediction, with the three dashed cases performing best as assessed by a statistical test. channels can suffer from the fit choice [ 12,14,16,42,43,56,66– 70]. Nevertheless, these adjustment methods are represen- tative of a norm in nonevaluation modeling work andcan have real-world implications such as incorrect pre-dicted yields during medical radioisotope production, high-level coproduction of an unwanted contaminant, or poorparticle transport calculations. Even with a foundationalunderstanding of the level-density, OMP, and exciton modelparameter adjustments, the interplay between the permuta-tions and combinations of changes in each component is notwell understood [ 11]. In turn, it is difficult to determine the most physically justifiable modeling parameters if the datafrom every open reaction channel is not known. For example, consider the numerous modeling possibilities for the large residual product channel 93Nb(p,p3n)90Nb, as shown in Fig. 14. The list of parameter adjustments in each modeling case is described in Appendix C(Table VII). It is qualitatively seen that 10 different models, with arbitrarychoices of which simplistic or complex parameters are ad-justed, can reproduce similar improvement over the defaultprediction. Still, it could be argued that one set of changes is quan- titatively the best to model this channel. A χ 2test using the experimental data demonstrates that models 1, 5, and 10give the largest improvements over default. These models areindicated with dashed/dotted lines in Fig. 14and the χ 2result of each parameter set is listed as well in Appendix C.G i v e n these best fits, it consequently seems logical to search formeaning in the altered parameters and attribute their need tolacking physics in this charged-particle problem. However,simply applying these best fit models to surrounding reactionchannels proves that these sets of parameter changes in fact donot improve the model’s predictive capabilities. For example,in the 93Nb(p,4n)90Mo channel, which also makes up a large FIG. 15. Extension of model adjustments, optimized to singu- larly reproduce the ( p, p3n ) channel, to a neighboring channel demonstrating poor fit behavior, especially for the three dashed casesthat previously performed best. share of the reaction cross section, models 1, 5, and 10 from Fig.14perform extremely poorly, as shown in Fig. 15. Instead, a more useful and realistic modeling approach should involve many prominent cross-section channels andsensitivity studies. The inclusion of more experimental dataand increased detail in the analysis process will yield a moreunique and global solution along with the capability to justifythe set of adjusted parameters while providing physics contextfor the predictions. As outlined in Fig. 13, this suggested improved fitting pro- cedure for spherical nuclei begins by identifying and havingaccurate experimental data for numerous prominent residualproduct channels. This approach is anchored in examining themost probable outcomes where it is possible to best isolatethe impact of model changes. Experimental data for weakerproduction channels are still involved and relevant but areweighted less heavily due to their high sensitivity to the be-havior of the dominant reactions. Once the largest reaction channels have been identified, the following step is to select a level-density model for all thenuclei involved in the interaction being studied such that thereis a concrete foundation, based on the well-established com-pound nucleus model, to build model adjustments on and puttheir effects in context. TALYS-1.9 provides six level-densitymodels, three that are microscopic calculations, which arepreferred in this procedure for their better care of the physicsinvolved and use in predictive scenarios versus the remainingthree phenomenological models [ 17]. At this point, the pro- posed fitting approach reaches the key step of an explorationof the exciton model parameter space. Notably, the preequilib-rium dynamics are adjusted the most in this suggested method.Both the OMP and exciton model parametrizations are basedon very large global studies. However, deviations from theoptical model default values represent a much greater changeto the physics of the situation than tuning for the exciton 034601-14INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) model [ 11,17,47,71]. The optical model fundamentally affects the nature of the particle-nucleus reaction while changing theexciton model parameters maintains the same preequilibriumphysics basis but shifts evolution and emission rates withinthe model, which are not known precisely at the outset. In thismanner, this fitting mechanism is specifically suited to isolateand gain insight into preequilibrium modeling for high-energyproton-induced reactions. The most significant of the available exciton model free parameters within TALYS are M2constant ,M2limit , and M2shift , which adjust C 1,C2, and C3, respectively, in Eq. ( 9). M2constant ,M2limit , and M2shift are set to 1.0 as de- fault in TALYS [ 17]. A decrease in M2constant reduces the transition rate to more complex exciton states, therebyincreasing preequilibrium emission in the initial interactionstages and creating an overall harder emission spectrum withan increased high-energy tail. The opposite effect appliesfor an increase in M2constant .T h e M2limit controls the asymptotic behavior of M 2and its increase leads to scattering to more complex states at high energies, thereby preventing anoverestimation of the high-energy tail, which pulls reactioncross section from the evaporation peak [ 11]. The M2shift affects the total system energy and can shift the excitonmodel strength along the projectile energy axis. Other param-eters that alter the preequilibrium effects to a lesser degreealso exist such as Rgamma, Cstrip, Rnupi, preeqspin, gpadjust , etc., which are all described in the TALYS-1.9 manual and should be considered as well [ 17]. Once the components of the exciton model are set accord- ing to the behavior of the largest reaction channels, there isan opportunity to perform some studies of OMP and level-density parameters. These aspects can help optimize the fitfounded on the exciton model changes for smaller residualproduction channels or localized outstanding discrepanciesbetween theory and experiment. The OMP and level-densityadjustments here are minor corrective factors to the broaderdeduced preequilibrium modeling. These adjustments mayrequire some iterations to reach convergence [ 16]. Last, a validation step is an important conclusion to this procedure. If the exciton, OMP, and level-density adjustmentsset by the breadth of reaction channels considered are uniqueand correct, their application to channels not included in theinitial sensitivity studies should yield appropriate fits. Cumu-lative excitation functions are good examples of unused data,where they may have large cross sections but the ambiguityfrom contributions of a chain of multiple nuclei and emissionchannels is not ideal for the initial sensitivity study. This isa test of the predictive capability of this procedure. Finally,a descriptive metric, such as a global χ 2test, can be applied to compare the adjusted fit in all utilized channels from thisprocedure to the default calculation [ 16,72,73]. Ideally, the metric is properly weighted to reflect the emphasis on the mostprominent reaction channels. Formulae for these weights arediscussed in Sec. IV C. C. Fitting procedure applied to93Nb(p,x) This work demonstrates the procedure outlined in Fig. 13for high-energy proton reactions on niobium. Atpresent, this sensitivity study work is performed man- ually to better gauge the physical effects of differentparameters and to mimic typical cross-section parame-ter adjustment work. Nine reaction channels are con-sidered: 93Nb(p,x)93m,90Mo,92m,90Nb,88,87,86Zr,88,86Y, with 90Nb,90Mo, and88Zr production as the most prominent. In the base level-density model choice step, the mi- croscopic models were indeed found to have greater pre-dictive power than the phenomenological models. The 93Nb(p,4n)90Mo reaction was found to be most sensitive to the level-density model. Only the microscopic calculationsfrom Goriely’s tables using the Skyrme effective interaction(ldmodel 4) could produce a fit magnitude in the vicinity of the experimental data while maintaining adequate predictivepower in the other considered channels [ 17]. The apparent sensitivity of 90Mo production to angular momentum distri- butions in nuclei closer to the target93Nb therefore made it the constraint for a level-density choice. Once the level-density model was chosen, the adjustment of preequilibrium could take place. The sensitivity study of theexciton model parameters showed that reducing M2constant from its default 1.0 value could best benefit high-energy tailbehavior across the prominent residual product cross sections.The tail-shape improvement came at the cost of unwantedreduced compound peak magnitudes, which could be compen-s a t e db ya ni n c r e a s ei n M2limit and a decrease in M2shift . Marginal variations of the three M2parameters relative to each other given these constraints demonstrated a best fit forthe largest available channels when M2constant =0.875, M2limit =4.5, and M2shift =0.6. Furthermore, this preequilibrium correction for the larger channels intro-duced a cascade effect that improved the compound peakbehavior of smaller cross-section channels, giving confidencethat these adjustments were globally beneficial. The numer-ous other additional scaling factors and modeling choices forpreequilibrium available in TALYS were also explored butwere shown to be insensitive relative to the M2parameters or physically inconsistent across the nine considered reactionshere. However, while compound peak improvement was seen in the weaker far-from-target channels, issues arose with theirhigher-energy cross-section predictions deviating from the ex-perimental data. This applies to nuclei such as 87,86Zr and86Y, which exist on the other side of the N=50 shell gap rela- tive to the target93Nb. The base level-density model choice, which served calculations for the niobium and molybdenumexcitation functions well, proved to be a root cause for theseunpredictable emission issues further from the target nucleus.The level densities of all nuclei involved in this charged-particle interaction are not perfectly modelled by the basechoice and may require specific variations, as outlined inFig.13. Adjusting the level-density model for niobium and molybdenum nuclei relevant to emissions for these far-from-target residual products from ldmodel 4 to the Hilaire com- binatorial calculation using the Skyrme force ( ldmodel 5) was tested. This change produced a sufficient compensat-ing effect to quell the incorrect high-energy behavior in themajority of the far-from-target channels [ 17,68]. Note that 93Mo and92Mo needed to remain modelled by ldmodel 4 034601-15MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) FIG. 16. TALYS default and adjusted calculation for90Nb. as these were key nuclei in the90Mo angular momentum constraint discovered earlier in the base level-density choicestudy. Minor deviations to the optical model could then be consid- ered to address outstanding discrepancies between predictionand experimental data. The key discrepancies remaining atthis point in the analysis included a slight underpredictionof the 90Nb production compound peak and falling edge versus a slight overprediction of the same aspects in90Mo, as well as an incorrect competition between86Zr and86Y production, where the former was overestimated and pulled reaction flux from the latter. The zirconium and yttrium chan-nels are inherently difficult to predict accurately as they areweaker reactions (with peak cross sections nearly an orderof magnitude lower than the dominant channels comprisingthe initial tuning set) susceptible to large variations fromcompounding effects in the modeling. The larger 90Nb and 90Mo reactions were therefore the primary constraints for FIG. 17. TALYS default and adjusted calculation for90Mo. FIG. 18. TALYS default and adjusted calculation for88Zr. OMP parameter adjustments. Exploring the real and imagi-nary volume components of the OMP is the most physicallysensible course for correcting the fit versus experimental datamagnitude discrepancies, as these parameters directly affectparticle flux loss and emission. The sensitivity study of theTALYS OMP volume terms revealed a significant relianceon only rvadjust p /n/a(multipliers to energy-independent radial factors of volume potentials) and w1adjust p (direct multiplier to proton imaginary volume potential well depth)in this charged-particle reaction setting [ 17,47]. The other volume potential parameters may be relevant in a differentcontext but are difficult to assess without double differen-tial scattering information. Marginal changes to rvadjust p/n/aandw1adjust p demonstrated that only w1adjust pwas needed to best improve the 90Nb peak magnitude and falling edge. w1adjust p affects the overall proton reactivity and emission. An increase to w1adjust p from its 1.0 default FIG. 19. TALYS default and adjusted calculation for93mMo. 034601-16INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 20. TALYS default and adjusted calculation for92mNb. to a value of 2.2 increased the cross section reasonably of all channels but most noticeably for90Nb production, especially relative to the93Nb(p,4n)90Mo reaction. A slight errant local competition between90Nb and90Mo still existed that could be improved by manually adjustinglevel densities using the ctable andptable TALYS com- mands. This level-density table adjustment can be appliedto an individual nuclide and when adjusted by reasonableamounts only has sensitivity for the selected nuclide andits neighbors, thereby maintaining the good global behaviorset by all the previous parameter changes. 90Mo required a ctable decrease to bring its production down while increas- ing the competing90Nb channel, allowing both predictions to align well with experimental data. The zirconium and yt-trium competition issues also required ctable decreases to be resolved and even prompted a slight 87Zr level-density decrease as well. Adjusting the level densities in this manner FIG. 21. TALYS default and adjusted calculation for88Y. FIG. 22. TALYS default and adjusted calculation for87Zr. for far-from-target nuclei holds a less clear physical meaning as the changes are potentially brought on by more complexreaction aspects, hidden from this sensitivity study work, thatare lumped into this compensating correction. This is a part ofthe procedure described in Fig. 13but it should be emphasized that the most clear application of this approach is for dominantreaction channels. All of the final derived parameter changes for 93Nb(p,x) are listed in Appendix D(Table VIII). The adjusted fits ac- companying this more detailed parameter study are showncompared to the default TALYS calculation for the nine con-sidered reaction channels in Figs. 16–24. The fits shown apply from 0 to 200 MeV. 1. Parameter adjustment validation A crucial aspect in this suggested approach is val- idation of the derived parameters to ensure that it is FIG. 23. TALYS default and adjusted calculation for86Zr. 034601-17MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) FIG. 24. TALYS default and adjusted calculation for86Y. justified to attribute physical meaning to their values. The 93Nb(p,x)89Zr,89Nb,87Y, and84Rb reaction channels, with all but84Rb being cumulative data, were used for this purpose. The adjusted fit shown in Figs. 25–28continues to show improved behavior over the default in these cases, especiallyin the compound peak regions. The total chi-squared, χ 2 tot, used to compare the default and adjusted TALYS fit across all utilized and validation channelsis given by: χ 2 tot=1 NcNc/summationdisplay c=1χ2 cwc, (11) where Ncis the number of reaction channels considered, χ2 cis the chi-squared value per channel, and wcis the weighting per FIG. 25. TALYS default and adjusted extended to89Zr. FIG. 26. TALYS default and adjusted extended to89Nb. channel [ 16,73]. Each χ2 cis defined by: χ2 c=1 NpNp/summationdisplay i=1/parenleftbiggσi T−σi E /Delta1σi E/parenrightbigg2 , (12) where Npis the number of data points from all experimental datasets in a given channel, σi Eare the experimental cross sections with /Delta1σi Euncertainty, and σi Tis the TALYS cross- section calculation [ 16,73]. No exclusions or preference was given to the quality of data beyond weighting by uncertainty,which is in opposition to techniques typically used in anevaluation [ 16,74]. Two weighting calculations were consid- ered in this application, both of which tried to emphasizethe importance of fits to the most prominent channels. Oneweighting methodology is to use the cumulative cross sectionof the TALYS calculation in a given channel relative to the FIG. 27. TALYS default and adjusted extended to87Y. 034601-18INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 28. TALYS default and adjusted extended to84Rb. sum of all channels’ cumulative cross sections: wc=/summationtextNp i=1σci T(E) /summationtextNc c=1/summationtextNp i=1σci T(E). (13) The above “cumulative σ” weighting potentially poses a risk of washing out the importance of large compound peaksthat were significant to parameter adjustment studies but falloff at high energies such as the case with 90Mo production. This issue could be resolved with an alternative “maximumσ” weighting that considers the maximum production cross section reached in each channel relative to the sum of allchannels’ maximums: w c=σc T,max/summationtextNc c=1σc T,max. (14) Theχ2 totresults based on both weighting methods are given in Table III. In this case both weighting techniques yield similar results, which clearly show that the adjusted param-eters fit performs much better for high-energy proton-inducedreactions on niobium than the default prediction. Ultimately,this more realistic analysis method, even as a manual search,has produced a fit with a better performance than the defaultcalculations with a justifiable limited set of parameter changesbuilt from measured experimental data. This analysis is there-fore an improved standard over the one-channel adjustment norm and can be a reasonable expectation for future parameter optimization data work. TABLE III. Global χ2metric describing goodness-of-fit for the default and adjusted TALYS calculations of93Nb(p,x). Weighting method Default χ2 tot Adjusted χ2 tot Cumulative σ 15.6 3.37 Maximum σ 16.0 3.28 FIG. 29. TALYS default and adjusted calculation for135Ce. D. Fitting procedure applied to139La(p,x) The same fitting approach detailed for niobium was also applied to high-energy proton-induced reactions on lan-thanum. Eight reaction channels were used in the study: 139La(p,x)137m,137g,135,134,133m,132Ce,135La,133mBa, with 135Ce,134Ce,137mCe, and135La production as the most promi- nent. The cross-section data for139La(p,x) are more limited than what was available in the niobium case. These eightchannels only contain the three datasets of Tárkányi et al. [75], Becker et al. [3], and Morrell et al. [6], with the latter two characterizations utilizing stacked-target activation at LANLand LBNL, respectively, consistent with the work performedhere. In addition to a sparser body of data, there is a limited diversity of reaction products, where only the 135La produc- tion gives insight into proton emission behavior and only the FIG. 30. TALYS default and adjusted calculation for134Ce. 034601-19MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) FIG. 31. TALYS default and adjusted calculation for135La. 133mBa production gives insight into alpha emission behavior. The measured cerium channels, comprising the bulk of theavailable data, are solely ( p,xn) reactions. That being said, the restricted dataset makes 139La(p,x) a valuable application of the suggested fitting procedure as it can show the amount ofpredictive power that can be gained even from reactions thatare being partially measured for the first time. Note that the default TALYS calculations for lanthanum were significantly better than for niobium, whose dominantchannels were predicted with extremely discrepant shapes,magnitudes, and positioning from the experimental data. Asa result, the amount of parameter adjustments, fine tuning,and iteration needed to properly model the niobium can beconsidered higher than typical. First, the application of microscopic level densities over phenomenological ones in the lanthanum calculations pro-vided immediate benefit, matching the observed rising edgesand shapes of the dominant 135Ce and134Ce compound peaks FIG. 32. TALYS default and adjusted calculation for133mBa. FIG. 33. TALYS default and adjusted calculation for133mCe. quite well. Similarly to the niobium, ldmodel 4 performed best and was chosen, though there was no apparent constrain-ing residual product in this case and ldmodel 5w a sac l o s e next best choice. The preequilibrium portion of the procedure revealed a need for adjustments of M2constant =0.85, M2limit =2.5, andM2shift =0.9 to the exciton model matrix parametriza- tion. It should be noted that these parameters are all shiftedin the same directions as in the niobium case, simply to alesser extent, which emphasizes the better initial default guesshere. A last additional preequilibrium change also includedCstrip a =2.0, where Cstrip a affects the transfer reac- tion contribution of ( p,α) to the overall preequilibrium cross section. This helps to increase 133mBa production without much noticeable effect to the other considered channels. For OMP fine tuning, the135La and133mBa channels nec- essarily played important roles due to their particle emission FIG. 34. TALYS default and adjusted calculation for137mCe. 034601-20INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 35. TALYS default and adjusted calculation for137gCe. diversity. The prevailing discrepancies in these two channels at this point included a slight overprediction of135La produc- tion and a minor underprediction of the133mBa compound peak falling edge. A testing of the available TALYS OMPparameters demonstrated that rvadjust p andrvadjust a held the most sensitivity. The most accurate behavior wasextracted solely using rvadjust p =0.96. Finally, there was a small local competition error between 135Ce and134Ce that could be corrected by a ctable increase to135Ce. There were far fewer confounding level-density changes for the lan-thanum relative to the niobium. The total derived parameter changes for 139La(p,x)a r e listed in Appendix D(Table IX). The adjusted TALYS fits from this procedure are given in Figs. 29–36compared to the default calculation and EXFOR data for the eight usedreaction channels [ 3,6,75]. Given that the experimental data do not extend beyond 100 MeV, the fits are shown only up tothis point. FIG. 36. TALYS default and adjusted calculation for132Ce. FIG. 37. TALYS default and adjusted extended to139Ce. 1. Parameter adjustment validation Validation of this adjusted fit is performed via compar- ison to the139La(p,x)139Ce,133La,133g,131Ba,132Cs chan- nels, which were not used in the fitting approach due totheir magnitudes or ambiguity/lack of data [ 76–78]. How- ever, even in these channels, the adjusted fit is shown inFigs. 37–41to have impressive predictive power versus the de- fault. Specifically, the predictive success for the single-particleout 139La(p,n)139Ce reaction, necessarily heavily influenced by preequilibrium, instills confidence in the adjusted parame-ters. Theχ 2 totresults comparing the adjusted and default fit glob- ally based on both weighting methods described in Sec. IV C 1 are given in Table IV. Again, both weighting methodolo- gies yield similar results, and it is evident that the adjustedfit outperforms the default prediction. In both the niobiumand lanthanum presented cases of this work, the suggested FIG. 38. TALYS default and adjusted extended to133La. 034601-21MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) FIG. 39. TALYS default and adjusted extended to133gBa. standardized fitting procedure has produced improved results over the TALYS default in a comprehensive and justifiablemanner. E. Interpretation of parameter adjustments The success of this fitting approach suggests that phys- ical meaning could be inferred from the adjustments madeto the exciton model parameters. Moreover, the consistentadjustments made to the M2exciton parameters in both the niobium and lanthanum cases appears to reveal a system-atic trend in how residual product excitation functions forhigh-energy proton-induced reactions on spherical nuclei aremiscalculated in the current exciton model scheme. Acrossthe prominent reaction channels explored in this work, therewas a consistent underprediction of both the high-energy pree-quilibrium tails and compound peak magnitudes. It was seenthat enforcing M2constant <1.0 could improve lacking tail FIG. 40. TALYS default and adjusted extended to131Ba. FIG. 41. TALYS default and adjusted extended to132Cs. behavior while M2limit >1.0 with M2shift <1.0 helped compensate for the increased tail by creating more productionin the compound peak. It is possible to further visualize andquantify this trend by plotting the magnitude of the squared ef-fective interaction matrix element within the ( E tot,n) reaction phase space. Specifically, defining /Delta1adj-def as the difference of normalized M2between the adjusted fit and the default calculation by: /Delta1adj-def=M2(Etot,n)adj M2(Etot,n)adj,max−M2(Etot,n)def M2(Etot,n)def,max,(15) the relative strength of M2for the adjusted case can be com- pared to the relative strength of M2in the default case across all of the reaction phase space. The /Delta1adj-def results for both the 93Nb(p,x) and139La(p,x) modeling are plotted in Fig. 42. It is seen that the adjustments for both targets exhibit thesame trend that better modeling fits were achieved when therewas a relative decrease for internal transition rates at inter-mediate proton energies ( E p=20–60 MeV) in the exciton model as compared to default values. The relative decrease re-duces the probability of formation of complex exciton states,and in turn the compound nucleus equilibration limit, infavour of preequilibrium emission. Furthermore, the locationof the relative decrease in reaction phase space indicates thatthere is difficulty transitioning between the Hauser-Feshbachand exciton models for nuclear reactions. These exciton TABLE IV. Global χ2metric describing goodness-of-fit for the default and adjusted TALYS calculations of139La(p,x). The very large improvement in χ2for the adjusted case may imply that the applied weights were too large, contributing to an inflated change versus the default. Weighting method Default χ2 tot Adjusted χ2 tot Cumulative σ 87.8 1.89 Maximum σ 96.4 3.34 034601-22INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 42. Visualization of impact from preequilibrium parameter adjustments across reaction phase space on the exciton model squared matrix element for the effective residual interaction. A consistent pattern is seen in the adjustments for the niobium and lanthanum cases, withmore pronounced behavior for the niobium. The color scale is a mapping of the zaxis in each case. adjustments appear to act as a surrogate for better damping into the compound nucleus system. The results of Fig. 42are additionally interesting because of the variation between the /Delta1adj-def magnitudes for93Nb(p,x) and139La(p,x). The /Delta1adj-def for139La(p,x)a r es m a l l e ra sa function of the better initial default residual product calcula-tions in TALYS compared to 93Nb(p,x). However, the rootcause of this more pronounced default model failure in the niobium case is unknown, especially given that both niobiumand lanthanum are structurally similar. In total, the modeling adjustments in this work suggest the need to incorporate residual product excitation function datain some capacity into future exciton model parametrizations.Further, this trend applies for proton-induced reactions and 034601-23MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) perhaps implies a need to release the strict generality of having the same exciton model formulas for both incident protons andneutrons [ 11]. F. Future considerations Residual product excitation functions were not used in the initial exciton model parametrization by Koning and Dui-jvestijn [ 11] because of the complexity and uncertainties brought in by the additional level-density and transmissioncoefficient models. This study has included this complexityand tried to isolate for these competing issues and uncertain-ties through the order of the fitting procedure and the focuson fitting many of the prominent channels, though difficultiesstill remain with their incorporation. Furthermore, the adjusted parameters lead not only to changes in specific product reaction channels, but to the totalnonelastic channel as well. Consider the difference in totalnonelastic cross section for protons incident on niobium be-tween the TALYS default, other evaluation databases, andthe TALYS adjusted case, as given in Fig. 43(a) [79–83]. The adjusted case argues for an increased high-energy crosssection. While below 50 MeV, the adjusted calculation seemsquite reasonable, above 50 MeV it is evident that there is alarge discrepancy between it and the other predictions. How-ever, it should be noted that the evaluations are all heavilyconstrained by a single high-energy data point, which maynot fully represent reality. Nonetheless, they suggest that thereshould be less confidence in extension of the adjusted TALYSfit to far-from-target residual products such as Kr, Se, andAs. It is possible that the poorer fit at high energies is alsoa reflection of the deterioration in the quality of level-densitypredictions in general at such high excitations. It is likely thatthe employed microscopic models used in the fitting are lessappropriate at such high energies than a more simple stochas-tic model such as a Fermi gas calculation, though this modeltoo may break down near 200-MeV excitation energy [ 84]. This is a difficult consideration to experimentally check butmight be a more realistic cause for error than the shell gapeffects discussed in Sec. IV C. A further neglected effect, which may be relevant to the code mispredictions seen at high energies for far-from-targetproducts, is the incorporation of isospin conservation inthe modeled reactions. The theoretical calculations ofGrimes et al. [85] and Robson et al. [86] using a modified Hauser-Feshbach formalism including isospin effects andthe experimental findings from works such as Lu et al. [87] and Kalbach-Cline et al. [88] explored this factor. They demonstrate that isospin conservation yields cross sectionsand particle emission spectra different from the Bohr inde-pendence hypothesis of compound nuclear decay includingonly angular momentum and from the typical exciton modelfor preequilibrium decay. Particularly, Grimes et al. [85] and Lu et al. [87] show that isospin selection rules for proton-induced reactions result in enhanced proton emission.These publications explored proton bombardment energies inthe 10- to 20-MeV range. Although the adjusted modeling fitsin this work were appropriate at those incident energies, it ispossible that the choice of level-density parameters were anunknowing compensating factor for neglected isospin effects, FIG. 43. Comparison of experimental, evaluated, and theoretical nonelastic cross sections. The filled error bands are associated with the TENDL data. which did not remain effectively compensating at higher energies. It is also possible that isospin effects are simplysmall for the target mass and energies under considerationhere. We believe it would be a worthwhile experimentfor the community to explore these isospin considerationsthrough a study of particle emission spectra resulting fromboth p+ 93Nb and α+90Zr irradiations. Specifically, these reactions populate the same94Mo compound system with different isospins and the proximity of94Mo to the N=50 shell gap may mean that pure isospin states exist that canbe well defined, making the compound system a suitablecandidate for this type of structure investigation. Unfortunately, it is not possible to derive any 93Nb(p,non) data points from summed residual product cross sections measured in this work for a more in-depth fit comparison. The presented cross-section results are not exhaustive enough 034601-24INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) for this calculation since stable and very short-lived iso- tope production was not measured. This potential nonelasticcross-section issue, or the possible high-energy theoreticalshortcomings, do not discredit the procedure shown here butinstead emphasize that the approach suggested in this work is not meant to be on par with complete reaction evaluations. In general, this approach is a holistic and realistic methodology,grounded on observables and experimental data, that exper-imenters can perform to benefit theory and support furtherpredictive work. Although, it is clear that the niobium fittingis an extreme case and looking at the total nonelastic crosssection for protons incident on lanthanum in Fig. 43(b) instills more confidence in this overall fitting process [ 79]. A worthwhile different way of continuing study on the departure of equal matrix elements for neutron-induced orproton-induced reactions may be to systematically study onereaction channel, instead of all reaction channels simultane-ously as in this work. Hence, one could investigate whether(p,n) reactions for different nuclides would show the same exciton adjustment trends discovered here. In the future, this fitting procedure could expand to in- clude emission spectra and double-differential data to try andimprove the elastic versus nonelastic competition and poten-tially determine other corrective parameter adjustments thatare simply not sensitive in the purely residual product dataanalysis [ 47]. Including the extra datasets can help clarify effects between level-density models, the optical model, and preequilibrium parametrizations. Such a procedure could be an inspiration and act as a stepping stone to the developmentof a charged-particle evaluated data database [ 89]. Although the sensitivity work performed in this paper was a manual search, it would be useful to incorporate automation,such as search techniques within a Bayesian framework, withthe acquired exciton adjustment knowledge. This would helpto more accurately determine a global minimum for parameter optimization and to better express the resolving power of dif- ferent parameters and channels in a more quantitative fashion. V . CONCLUSIONS This work reports 23 sets of measured93Nb(p,x) residual product cross sections between 50 and 200 MeV as part ofa Tri-lab collaboration between LBNL, LANL, and BNL.The reported cross sections greatly extend the datasets for numerous products and are of higher precision than a major- ity of previous measurements. The 93Nb(p,4n)90Mo monitor reaction of particular interest for intermediate proton energystacked-target activation experiments was characterized be-yond 100 MeV for the first time. Given the measured data, an in-depth investigation of reaction modeling and preequilibrium mechanisms was con- ducted. A standardized parameter adjustment fitting proce- dure to improve default code predictions in a physicallyjustifiable manner was proposed and applied to 93Nb(p,x) and 139La(p,x) cross-section data as tests. The fitting approach focused on the current parametrization of the squared matrixelement in the preequilibrium two-component exciton model.A systematic trend for the exciton parameter adjustments to correct high-energy tails and compound peak magnitudes was seen that implied the current parametrization is not whollycorrect. This result suggests the need to incorporate resid- ual product excitation function data in some capacity intofuture exciton model parametrizations and potentially createdifferent parametrizations altogether for incident protons andneutrons. The focus of this work was on presenting and interpreting the results from ( p,x) reactions on spherical target nuclei (Nb and La). Subsequent papers will discuss additional data resultsfrom the Tri-lab collaboration for 75As(p,x) reactions as well as the production and characterization of thin arsenic targets. Theγ-ray spectra and all other raw data created during this research are openly available [ 90]. On publication, the experimentally determined cross sections will be uploaded tothe EXFOR database. ACKNOWLEDGMENTS This research was supported by the Isotope Program within the U.S. Department of Energy’s Office of Science, carried outunder Lawrence Berkeley National Laboratory (Contract No.DE-AC02-05CH11231), Los Alamos National Laboratory(Contract No. 89233218CNA000001), and BrookhavenNational Laboratory (Contract No. DEAC02-98CH10886).The authors acknowledge the assistance and support of BrienNinemire, Scott Small, Nick Brickner, Devin Thatcher, andall the rest of the operations, research, and facilities staff ofthe LBNL 88-Inch Cyclotron. We also thank David Reass andMike Connors at LANSCE-IPF, the LANL C-NR Countroomoperators, and the LANSCE Accelerator Operations staff.The authors acknowledge Deepak Raparia, head of thePre-Injector Systems group at CAD-BNL, for LINAC beamtuning for the experiment and all members of the BNLMedical Isotope Research and Production group for theirassistance. We are grateful to Patrick Sullivan and John Aloiof the BNL Radiological Control Division for the HealthPhysics support. Sumanta Nayak is acknowledged for theengineering and Frank Naase for the IT support. APPENDIX A: TARGET STACK DESIGNS Details of the stacked-targets irradiated in this work are given in Tables VandVI. TABLE V. Target stack design for irradiation at IPF. The proton beam initially hits the stainless steel plate (SS-SN1) after passingthrough the upstream Inconel beam entrance window, a water cool- ing channel, and the target box aluminum window. The thickness and areal density measurements are prior to any application of thevariance minimization techniques described in this work. Areal Areal Thickness density density Target layer ( μm) (mg/cm2) uncertainty (%) SS-SN1 Profile Monitor 130.0 100.12 0.07 Al-SN1 27.33 7.51 0.21 Nb-SN1 25.75 23.08 0.12As-SN1 4.27 2.45 8.2 Ti-SN1 25.00 11.265 1.0 Cu-SN1 24.33 19.04 0.13 034601-25MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) TABLE V. ( Continued .) Areal Areal Thickness density density Target layer ( μm) (mg/cm2) uncertainty (%) Al Degrader 01 6307.0 1702.89 0.001 Al-SN2 26.67 7.58 0.32 Nb-SN2 24.75 22.67 0.08As-SN2 4.30 2.46 8.3 Ti-SN2 25.00 11.265 1.0 Cu-SN2 24.00 18.90 0.36Al Degrader 02 3185.5 860.09 0.02 Al-SN3 26.67 7.38 0.22 Nb-SN3 24.50 22.83 0.03As-SN3 3.62 2.07 9.0 Ti-SN3 25.00 11.265 1.0 Cu-SN3 23.33 19.38 0.11Al Degrader 03 2304.5 622.22 0.06 Al-SN4 28.00 7.34 0.18 Nb-SN4 25.50 22.57 0.16As-SN4 3.54 2.03 9.2 Ti-SN4 25.00 11.265 1.0 Cu-SN4 24.67 19.24 0.11Al Degrader 04 1581.3 426.94 0.04 Al-SN5 27.00 7.48 0.44 Nb-SN5 24.75 22.78 0.12As-SN5 3.90 2.23 8.7 Ti-SN5 25.00 11.265 1.0 Cu-SN5 25.00 19.09 0.17 Al Degrader 05 1033.8 279.11 0.06 Al-SN6 28.67 7.44 0.25Nb-SN6 25.25 22.80 0.08 As-SN6 3.11 1.78 10 Ti-SN6 25.00 11.265 1.0Cu-SN6 24.33 19.50 0.16 Al Degrader 06 834.8 225.38 0.22 Al-SN7 28.33 7.56 0.15Nb-SN7 25.50 22.62 0.06 As-SN7 2.79 1.59 9.2 Ti-SN7 25.00 11.265 1.0Cu-SN7 23.67 18.79 0.04 Al Degrader 07 513.5 138.65 0.10 Al-SN8 27.67 7.56 0.10Nb-SN8 25.50 22.95 0.45 As-SN8 2.20 1.26 9.0 Ti-SN8 25.00 11.265 1.0Cu-SN8 24.00 19.06 0.23 Al Degrader 08 517.3 139.66 0.43 Al-SN9 27.00 7.47 0.36Nb-SN9 25.00 22.53 0.24 As-SN9 2.57 1.47 9.9 Ti-SN9 25.00 11.265 1.0Cu-SN9 26.33 19.19 0.12 Al Degrader 09 517.8 139.79 0.09 Al-SN10 28.00 7.41 0.17Nb-SN10 24.75 22.82 0.02 As-SN10 1.94 1.11 10 Ti-SN10 25.00 11.265 1.0 Cu-SN10 25.67 18.87 0.18 SS-SN10 Profile Monitor 130.0 100.12 0.07TABLE VI. Target stack design for irradiation at BLIP. The pro- ton beam initially hits the stainless steel plate after passing through the upstream beam windows, water cooling channels, and target boxaluminum window. The thickness and areal density measurements are prior to any application of the variance minimization techniques described in this work. Areal Areal Thickness density density Target layer ( μm) (mg/cm2) uncertainty (%) SS Profile Monitor 120.2 95.16 0.58 Cu-SN1 26.00 22.34 0.10 Nb-SN1 25.75 22.75 0.25 As-SN1 1.89 1.08 9.9 Ti-SN1 25.00 11.265 1.0 Cu Degrader 01 5261.1 4708.07 0.02Cu-SN2 26.75 22.41 0.11 Nb-SN2 24.75 22.91 0.19 As-SN2 2.94 1.68 9.0Ti-SN2 25.00 11.265 1.0 Cu Degrader 02 4490.7 4018.99 0.04 Cu-SN3 26.50 22.26 0.05 Nb-SN3 24.00 22.67 0.31 As-SN3 3.06 1.75 10Ti-SN3 25.00 11.265 1.0 Cu Degrader 03 4501.8 4028.84 0.03 Cu-SN4 26.00 22.29 0.15Nb-SN4 24.75 22.70 0.23 As-SN4 4.85 2.78 9.9 Ti-SN4 25.00 11.265 1.0Cu Degrader 04 4243.9 3797.96 0.03 Cu-SN5 25.50 22.35 0.04 Nb-SN5 25.00 22.54 0.12As-SN5 7.26 4.15 12 Ti-SN5 25.00 11.265 1.0 Cu Degrader 05 3733.8 3341.56 0.03Cu-SN6 26.25 22.34 0.08 Nb-SN6 25.00 22.36 0.24 As-SN6 4.93 2.82 9.0 Ti-SN6 25.00 11.265 1.0 Cu Degrader 06 3783.0 3385.41 0.04Cu-SN7 25.75 22.26 0.09 Nb-SN7 25.75 22.62 0.10 As-SN7 12.62 7.22 9.3Ti-SN7 25.00 11.265 1.0 APPENDIX B: MEASURED EXCITATION FUNCTIONS Plots of extracted cross sections in this work are given with reference to existing literature data, TENDL-2019, andreaction modeling codes TALYS-1.9, EMPIRE-3.2.3, CoH-3.5.3, and ALICE-20 using default parameters [ 2,12,31–46]. Subscripts ( i) and ( c) in figure titles indicate independent and cumulative cross sections, respectively. 034601-26INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 44. Experimental and theoretical cross sections for72Se production. FIG. 45. Experimental and theoretical cross sections for73As production. FIG. 46. Experimental and theoretical cross sections for74As production. FIG. 47. Experimental and theoretical cross sections for75Se production. FIG. 48. Experimental and theoretical cross sections for81Rb production. FIG. 49. Experimental and theoretical cross sections for82mRb production. 034601-27MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) FIG. 50. Experimental and theoretical cross sections for83Rb production. FIG. 51. Experimental and theoretical cross sections for83Sr production. FIG. 52. Experimental and theoretical cross sections for84Rb production. FIG. 53. Experimental and theoretical cross sections for85mY production. FIG. 54. Experimental and theoretical cross sections for86Rb production. FIG. 55. Experimental and theoretical cross sections for86Zr production. 034601-28INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) FIG. 56. Experimental and theoretical cross sections for87Y production. FIG. 57. Experimental and theoretical cross sections for87mY production. FIG. 58. Experimental and theoretical cross sections for88Y production. FIG. 59. Experimental and theoretical cross sections for88Zr production. FIG. 60. Experimental and theoretical cross sections for91mNb production. FIG. 61. Experimental and theoretical cross sections for92mNb production. 034601-29MORGAN B. FOX et al. PHYSICAL REVIEW C 103, 034601 (2021) FIG. 62. Experimental and theoretical cross sections for93mMo production. APPENDIX C: NONUNIQUE TALYS PARAMETER ADJUSTMENTS Table VIIoutlines the ambiguity surrounding TALYS pa- rameter adjustments when modeling is based on a singleexcitation function. TABLE VII. Details of modeling cases used to reproduce similar behavior for93Nb(p,p3n)90Nb reaction, shown in Figs. 14and15. Model number Parameter adjustments χ2 ν Default – 57.9 1 ldmodel 5 24.0 strength 4 preeqmode 3 2 ldmodel 2 50.3 strength 1 M2constant 1.8avadjust p 0.85 rvadjust p 1.35 3 ldmodel 1 118.9 strength 2 M2constant 3.0 M2shift 2.2M2limit 2.0 4 ldmodel 3 298.4 strength 2 M2constant 7.0 M2shift 0.1 M2limit 5.0preeqmode 1 w1adjust p 1.5 v1adjust p 1.1 rvadjust p 1.33TABLE VII. ( Continued .) Model number Parameter adjustments χ2 ν 5 ldmodel 6 34.5 strength 8M2constant 0.95M2shift 0.95M2limit 3.0w1adjust p 1.4ctable 41 90 0.15 6 ldmodel 4 57.8 strength 5M2constant 2.3M2shift 0.6M2limit 0.8w1adjust p 1.3rvadjust n 1.3rvadjust a 0.85 7 ldmodel 1 46.9 strength 2M2constant 1.7w1adjust p 1.2v1adjust p 1.05rvadjust p 1.25 8 jlmomp y 67.3 preeqmode 3lwadjust 1.08 9 ldmodel 1 45.1 strength 2M2constant 0.85localomp nrvadjust n 0.85v1adjust n 1.25ctable 42 90 −1.0 10 ldmodel 5 23.5 strength 4M2constant 3.3ctable 42 88 −1.2 ctable 42 87 −1.2 ctable 41 90 1.6ctable 41 86 −1.0 ctable 40 86 −1.8 APPENDIX D: TALYS PARAMETER ADJUSTMENTS FROM FITTING PROCEDURE The derived parameter adjustments from the fitting proce- dure applied to the93Nb(p,x) and139La(p,x) data are listed in Tables VIII andIX. 034601-30INVESTIGATING HIGH-ENERGY PROTON-INDUCED … PHYSICAL REVIEW C 103, 034601 (2021) TABLE VIII.93Nb(p,x) best-fit parameter adjustments derived from proposed procedure. The equidistant keyword adjusts the width of excitation energy binning and will be a default in updated TALYS versions. The strength keyword selects the γ-ray strength model and has little impact in this charged-particle investigation, so it is chosen as one of the available microscopic options. Parameter Value ldmodel 4 594−86Nb 594Mo,91−86Mo strength 5 equidistant y M2constant 0.875 M2limit 4.5 M2shift 0.6w1adjust p 2.2 ctable 39 86 −0.6 40 86 −0.35 40 87 −0.85 42 90 −0.5 ptable 39 86 2.0TABLE IX.139La(p,x) best-fit parameter adjustments derived from proposed procedure. Parameter Value ldmodel 4 strength 5equidistant y M2constant 0.85 M2limit 2.5 M2shift 0.9 cstrip a 2.0rvadjust p 0.96 ctable 58 135 0.6 [1] National Research Council and Institute of Medicine of the National Academies, Advancing Nuclear Medicine Through Innovation (The National Academies Press, Washington, DC, 2007). [2] A. S. Voyles, L. A. Bernstein, E. R. Birnbaum, J. W. Engle, S. A. Graves, T. Kawano, A. M. 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PhysRevB.91.104405.pdf

PHYSICAL REVIEW B 91, 104405 (2015) Mode-coupling mechanisms in nanocontact spin-torque oscillators Ezio Iacocca,1,2Philipp D ¨urrenfeld,1Olle Heinonen,3,4Johan ˚Akerman,1,2,5and Randy K. Dumas1,2 1Physics Department, University of Gothenburg, 412 96, Gothenburg, Sweden 2NanOsc AB, Electrum 205, 164 40, Kista, Sweden 3Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA 4Northwestern-Argonne Institute for Science and Engineering, Evanston, Illinois 60208, USA 5Material Physics, School of ICT, Royal Institute of Technology, Electrum 229, 164 40, Kista, Sweden (Received 18 December 2014; revised manuscript received 19 February 2015; published 11 March 2015) Spin-torque oscillators (STOs) are devices that allow for the excitation of a variety of magnetodynamical modes at the nanoscale. Depending on both external conditions and intrinsic magnetic properties, STOs canexhibit regimes of mode hopping and even mode coexistence. Whereas mode hopping has been extensivelystudied in STOs patterned as nanopillars, coexistence has been only recently observed for localized modes innanocontact STOs (NC-STOs), where the current is confined to flow through a NC fabricated on an extendedpseudo spin valve. By means of electrical characterization and a multimode STO theory, we investigate thephysical origin of the mode-coupling mechanisms favoring coexistence. Two coupling mechanisms are identified:(i) magnon-mediated scattering and (ii) intermode interactions. These mechanisms can be physically disentangledby fabricating devices where the NCs have an elliptical cross section . The generation power and linewidth from such devices are found to be in good qualitative agreement with the theoretical predictions, as well as provideevidence of the dominant mode-coupling mechanisms. DOI: 10.1103/PhysRevB.91.104405 PACS number(s): 75 .30.Ds,75.78.−n,75.78.Cd I. INTRODUCTION Spin-torque oscillators (STOs) [ 1,2] are devices in which magnetization dynamics in a thin-film ferromagnet can beexcited by means of the spin transfer torque (STT) effect [ 3,4]. Depending on the geometry and magnetic properties of the freelayer, propagating spin waves (SWs) [ 5–7], solitonic modes [8–11], vortex gyration [ 12,13], and magnetic dissipative droplets [ 14–17] have been observed. STOs can also be used to generate SWs in physically extended thin films, which isof particular interest for future magnonic applications [ 18,19]. To date the generation of SWs using STT into extended filmshas been achieved in two types of structures: the spin Hallnano-oscillator, where the current is confined by patterningAu needles [ 20,21] and the spin polarization provided by the spin Hall effect [ 22], and in nanocontact spin-torque oscillators (NC-STOs) [ 1], where the current is confined by patterning a metallic NC on top of an extended spin valve. NC-STOs with permalloy free layers exhibit a particularly rich and reproducible dynamical phase space, featuring atransition between a propagating mode and two localizedmodes depending only on the out-of-plane external magneticfield angle θ. The effect of the current-induced Oersted field H Oehas been recently shown to be critical in determining the SW propagation direction and mode volume [ 11,23]. Moreover, the energy landscape induced by the Oerstedfield was shown to determine the position of the localizedmodes [ 11,24]. Such localization can be observed from micromagnetic simulations by calculating the mode energydistribution or the spatial representation of the power spectraldensity, shown in Figs. 1(a) and1(b), where black indicates zero power and light yellow indicates the (scaled) maximumpower. In particular, a low-frequency mode, identified asa spin-wave bullet [ 8,10], is primarily located at the local field minimum (B), while a high-frequency mode is primarilylocated at the local field maximum (HF). This situation can alsobe schematically represented by the frequency landscape along X=0 [Fig. 1(c)]. Additionally, such a spatial localization was shown to promote mode coexistence in these devices [ 11]. In a recent multimode theory developed for STOs [ 25–28], a coexistence regime was found to be favored by modecoupling. However, both the experimental and theoreticalstudies to date emphasize the mode-hopping regime, i.e.,when only one mode exists at a particular instance of time.Consequently, the physical origin and coupling mechanismsbehind mode coexistence are not well understood. To elucidatethe characteristics of, and conditions for, mode coexistencewe study NC-STO devices specifically tailored to test thepredictions of the multimode theory for STOs. In this paper, we derive an expression for the genera- tion linewidth of a multimode STO, in particular, for thecase of coexisting modes of generally different frequencies. We identify two mode-coupling mechanisms: (i) magnon- mediated scattering, originating from the interaction betweenthe dominant modes and a bath of thermally excited magnons;and (ii) intermode exchange, originating from the nonlocaldipole or exchange coupling between the dominant, finite-volume modes. The resulting linewidth is found to have a verydifferent dependence on these mode-coupling mechanisms. Totest our theoretical predictions, NC-STOs were fabricated with elliptical NCs that break the symmetry and thus disentangle the mode-coupling mechanisms. The electrical characterizationof these devices provides evidence for the underlying mode-coupling mechanisms and their effects on the magnetodynam-ics. Furthermore, these NC-STOs represent a model system tounderstand the effects of multimode generation and coupling,and allow one to propose novel methods to further tune the magnetodynamics at the nanoscale. In particular, the coupling between well-defined modes can play a fundamental role inthe development of novel computation methods based onthe interaction between magnetic solitons and propagatingSWs [ 19]. 1098-0121/2015/91(10)/104405(7) 104405-1 ©2015 American Physical SocietyEZIO IACOCCA et al. PHYSICAL REVIEW B 91, 104405 (2015) FIG. 1. (Color online) Mode energy distribution for the bullet (a) and high-frequency (b) modes for a NC-STO with a circular NC. The relative orientation of the current-induced Oersted field (yellow)and in-plane applied field component (white) are indicated in (b). (c) Schematic representation of the local frequency landscape along theYdirection in a NC-STO modified by the current-induced Oersted field (black line). The red dashed lines indicate the edges of the NC. The bullet mode, B (high-frequency mode, HF) sits at the local frequency minimum (maximum). Black arrows represent the possibleflow of magnons due to the finite mode volume of both B and HF. (d) Generation linewidth due to magnon-mediated scattering and intermode exchange mechanisms, by virtue of Eq. ( 2). Inset: Magnon-scattering event between the dominant modes ( ω 1andω2) and thermal magnons. The paper is organized as follows: The multimode STO the- ory extended for coexistent modes with different frequenciesis discussed in Sec. IIas well as the approximate Lorentzian linewidth expected from the generated dynamics. In Sec. III, the physical origin of the mode-coupling mechanisms inNC-STOs and their influence on the linewidth are discussed.The experiments and simulations performed to investigate thetheoretical predictions are described in Sec. IV. In Sec. V,t h e experimental results are discussed and interpreted by meansof the multimode STO theory predictions. Finally, we provideour conclusions in Sec. VI. II. MULTIMODE THEORY FOR COEXISTING MODES: POWER AND LINEWIDTH The magnetodynamics of NC-STOs can be described by means of the Landau-Lifshitz-Gilbert-Slonczewski (LLGS)equation dˆm dt=−γˆm×/vectorHeff+αˆm×dˆm dt −γμoσ(I)/epsilon1ˆm×(ˆm׈M), (1) where γ/2π=28 GHz /T is the gyromagnetic ratio; ˆmand ˆMare the normalized free and fixed-layer magnetizationvectors, respectively; αis the Gilbert damping; and σ(I)= /planckover2pi1IPλ/μ oMSeVis the dimensionless spin-torque coefficient, where /planckover2pi1is the reduced Planck constant, Iis the spin-polarized current, Pis the polarization, /epsilon1=λ/[1+λ+(λ−1)ˆm·ˆM] is the spin asymmetry factor as a function of the spin-torqueasymmetry λ,μ ois the vacuum permeability, MSis the saturation magnetization, eis the electron charge, and Vis the free-layer volume. The effective field Heffincludes the exchange, demagnetizing, anisotropy, and external fields aswell as the current-induced Oersted field. Throughout thispaper, we will use the convention in which a negative currentpolarity opposes the damping and corresponds to electronsflowing from the free to the fixed layer. Due to the nonlinearity of Eq. ( 1), analytical studies are usually performed by recasting the LLGS equation in a reducedbasis [ 29,30]. Such approaches assume a single-mode genera- tion, which is translated into a “winner-takes-all” strategy forthe available energy resources [ 31]. The multimode theory of STOs [ 25–28] differs in that it considers the existence of at least two excited modes. As a consequence, regimes of modehopping [ 25,32], periodic mode transitions [ 33], and coexis- tence [ 11] are obtained in addition to the single-mode regime. The mode-coexistence regime is of particular relevance due to its recent experimental observation [ 11] and the develop- ment of novel STO geometries [ 20,21] that can potentially sustain multimode generation. In order to understand the features of this regime, it is important to relate the theory to experimental observables. The generation linewidth is oneof the parameters that can be easily and accurately measuredexperimentally by straightforward electrical characterization.Consequently, we derive the generation linewidth predictedfrom the multimode STO theory for a coexistence regime formodes oscillating at generally different frequencies, ω 1andω2. A simple approach is to assume that the phase difference between the modes is a linear function of time, equivalent toa constant “velocity.” Hence, it is possible to invoke Galileaninvariance to generalize the equations of motion, as detailed inAppendix A. We obtain a bounded phase space as a function of the energy and the phase difference ψ(in the “moving” frame) between the excited modes. In the theory, there is alinear coupling of amplitude Kbetween the modes [ 25]. In the case of strong mode coupling K> >/Gamma1 p, where /Gamma1pis the total restoration rate in STOs, the resulting phase spacehas stable fixed point solutions for ψ=2nπ, where nis an integer starting at 0, and equal energy share [ 28]. This indicates that the coexistence regime is characterized by an equal splitof the energetic resources in the system, in contrast to thesingle-mode scenario. The generation linewidth can be obtained for each mode by computing the autocorrelation as a function of the timelagτ. Under Galilean invariance, the resulting autocorrelation function (Appendix B) predicts identical linewidths for both modes. To first order in τ, the Lorentzian linewidth is given by /Delta1f L≈/Delta1ωo/parenleftBigg p2 oN2/parenleftbig ω−1 1−ω−1 2/parenrightbig2 2(2 ˜αK+4K2)+1/parenrightBigg , (2) where pois the mode free-running power, /Delta1ωo∝1/pois the linear generation linewidth, Nis the nonlinear frequency shift, ˜α=(1+a2)/a, anda=√ω1/ω2. 104405-2MODE-COUPLING MECHANISMS IN NANOCONTACT SPIN- . . . PHYSICAL REVIEW B 91, 104405 (2015) Equation ( 2) predicts that in this mode-coexistence regime the linewidth asymptotically decreases as a function of thecoupling strength K. This result is in strong contrast to the mode-hopping regime [ 28], where the linewidth was predicted to increase as a function of K. It is noteworthy that these opposing trends suggest a smooth transitionbetween the mode-coexistence and mode-hopping regimesat an intermediate value of K. Consequently, Eq. ( 2)i s only valid for mode coexistence and, as Kapproaches zero, mode-hopping [ 28] or single-mode (e.g., Ref. [ 30]) frameworks should be used instead. III. MODE-COUPLING MECHANISMS IN NC-STOS In the previous section, we obtained the theoretical prediction of the linewidth behavior as a function of thecoupling strength K[27]. The origin of this term has two important ingredients. The first one is the existence of a bathof thermally excited magnons that allows for on-the-shellscattering events that can couple two modes. Such a scatteringevent is schematically depicted in the inset of Fig. 1(d), where the two dominant modes with frequencies ω 1andω2can couple through a conservative four-magnon scattering eventinvolving two thermal magnons with frequencies ω th,1and ωth,2. The second is a nonlocal coupling between modes 1 and 2 mediated by magnetostatic interactions (we can safely assume that the speed of light cis infinite) or by exchange and magnetostatic coupling [ 27]. By carefully controlling the experimental geometry and the applied current, these twointeractions can be manipulated in different ways, giving riseto different experimental signatures. First, through the STTand its pumping action, thermal magnons become easier toexcite as a function of current, opening a growing numberof scattering channels and therefore increasing the couplingbetween the two stable modes. By virtue of Eq. ( 2), this then leads to a linewidth decrease as a function of the bias current,as schematically shown by the solid black line in Fig. 1(d).W e stress that the local Joule heating originating from electronictransport can also excite thermal magnon [ 34]. However, for the investigated current range, such a temperature contributionis not expected to be a dominant effect. Second, the mode coupling through the nonlocal interac- tions between the two dominant modes can be manipulatedthrough the finite volume of the modes and the separationbetween them. In fact, the profile of the bullet mode wasderived to be limited by the size of the NC [ 8]. In a coexistence regime, the NC must be sufficiently large to accommodate boththe bullet and the high-frequency mode, which is possible dueto the preferential spatial location of each mode induced bythe Oersted field [Figs. 1(a)–1(c)]. Their mode volumes can overlap, leading to a coupling mediated by exchange. We willrefer to this mechanism as intermode coupling. Clearly, theOersted field plays a fundamental role in separating the modesand thus tuning the exchange and magnetostatic intermodecoupling. This suggests the possibility to control the couplingby means of the bias current. Specifically, larger currentsinduce a stronger Oersted field, which in turn will furtherspatially separate the modes [ 11] and reduce the intermode coupling strength. Invoking Eq. ( 2), the linewidth generated solely by this mechanism is expected to increase as a functionof the bias current, as schematically shown by the solid red line in Fig. 1(d). IV . DISENTANGLING MODE-COUPLING MECHANISMS The coupling mechanisms described above occur simul- taneously in a NC-STO and have opposite effects on thegeneration linewidth, making it difficult to experimentallyidentify the expected features. However, considering thespatial dependence of the intermode coupling, it is possibleto disentangle these mechanisms by breaking the symmetryat the nanoscale. This is achieved by fabricating NCs with anelliptical cross section. The fabricated NC-STOs are based on a pseudo spin valve [ 11,35] with the layer structure Co(8 nm) /Cu (8 nm) /Py(4.5 nm), where the Co (fixed layer) acts as a relatively static current spin polarizer, whereas the Py(Ni 80Fe20, free layer) is subject to STT-induced magnetody- namics. Devices with both circular NCs and elliptical NCswith an aspect ratio r=1.5 are defined via electron-beam lithography. A scanning electron microscopy (SEM) image ofan elliptical NC with minor axis of 100 nm is shown in theinset of Fig. 2(a). Magnetization dynamics are excited by a dc current and electrically measured in a probe station featuringan external magnetic field set by a rotatable Halbach array ofpermanent magnets, creating a fixed and highly uniform fieldofμ oHa=0.9 T. The generated dynamics are separated from FIG. 2. (Color online) Field angle dependent spectra obtained (a) experimentally and (b) numerically using an elliptical NC with tiltφNC=0◦a n db i a s e da t −18 mA. The ferromagnetic resonance (FMR) frequency is shown by a yellow solid line. In both cases, thecritical angle θ c≈60◦defines the onset of mode-hopping regime, which coincides with the onset of the bullet mode. A second critical angle θL≈42.5◦is also observed when both the bullet (B) and high-frequency (HF) modes are localized, corresponding to a coexistence regime. The inset in (a) shows an SEM image of a fabricated elliptical contact with a minor axis of 100 nm. The insetin (b) schematically shows the convention of φ NCwith respect to the in-plane component of the applied field, H/bardbl. 104405-3EZIO IACOCCA et al. PHYSICAL REVIEW B 91, 104405 (2015) the dc current using a 0 .1−40 GHz bias T and converted to the frequency domain by an R&S FSV40 spectrum analyzer afterusing a 32 dB gain low-noise amplifier with a bandwidth of18−40 GHz. In order to fine tune and control the energy landscape, the elliptical NCs were patterned at differentin-plane angles φ NC. In the measurement framework, such angles are measured between the ellipses major axes and thein-plane field component H /bardbl=Hasin(θ), as schematically shown in the inset of Fig. 2(b). The fabricated devices have three different in-plane angles, namely, φNC=0◦,45◦, and 90◦. Representative spectra as a function of the external out- of-plane field angle θare shown in Fig. 2(a) for an elliptical NC of minor axis 70 nm, rotated φNC=0◦and biased at Idc=−18 mA. Each spectrum was averaged ten times with a video bandwidth of 10 kHz in order to minimize the noise. Inagreement with our circular NC-STOs (not shown) and previ-ous results [ 8,10,11,36], the spectra show a transition between the propagating spin-wave mode predicted by Slonczewski [ 5] and a mode-hopping regime. This transition occurs at awell-defined critical angle θ c, corresponding to the bullet-mode onset [ 10,36]. Additionally, we observe a second critical angle θL, where both the bullet mode and a high-frequency mode are observed, maintaining an approximately constant frequencydifference [ 11]. The experimental spectra are well reproduced by micro- magnetic simulations performed on the graphic-processing- unit-based code MUMAX 2[37]. For the simulations, we assume standard parameters for the Py layer and the spin polarizationof the Co layer, namely, saturation magnetization μ oMs= 0.88 T, exchange stiffness A=11 pJ/m3, Gilbert damping α=0.01, spin polarization P=0.35, and no out-of-plane torque, as customary for pseudo spin valves with metallicspacers. The simulations shown here are performed using asymmetric torque λ=1, although similar results are obtained with a more general antisymmetric torque, λ> 1. The tilt of the Co fixed layer is calculated for each external field angleθby solving the magnetostatic boundary condition in a thin film approximation, considering a Co saturation magnetization FIG. 3. (Color online) Mode energy distribution for the bullet and high-frequency modes for NC-STOs with elliptical NCs tilted φNC=0◦,4 5◦, and 90◦. The relative orientations of the current- induced Oersted (yellow) and in-plane field component (white) areindicated in the lower left panel.ofμoMs,p=1.5 T. The simulated spectra, estimated from 10 ns long time traces [Fig. 2(b)], show excellent agreement with the experimental results. We stress that the frequencyresolution is insufficient to accurately determine the powerand linewidth from the simulated spectra. However, this isoutside the scope of the present work as we determine suchquantities experimentally. The excellent spectral characteristics obtained from the micromagnetic simulations justify a closer inspection of thespatial energy distribution of the generated dynamics. Inparticular, we focus on the modes observed below θ L≈42.5◦. In Fig. 3, the spatial extent of the bullet (high-frequency) mode for elliptical NCs tilted 0◦,45◦,and 90◦is observed to closely follow the edge of the NC (white), where thein-plane field component is minimal (maximal). Consequently,the elliptical NC tilt allows us to control the strength ofthe intermode coupling mechanism, as indicated in Fig. 3. By virtue of the analytical predictions of Sec. II, markedly different dependencies of the linewidth are expected forelliptical NCs tilted φ NC=0◦and 90◦from the in-plane applied field. V . EXPERIMENTAL RESULTS To interpret the experimental results by means of the multimode STO theory, we must first determine the mode- coexistence regime. In the frequency domain, indirect evidence of coexistence can be provided by a low-frequency ( f< 2 GHz) intermodulation feature [ 11]. However, in our exper- iments, the bandwidth of the amplifier attenuates such lowfrequencies. Another approach is to confirm that the energy, orpower, is approximately equally divided between the modes,as predicted by the fixed point solutions of the multimodeSTO theory [ 27,28]. In our electrical measurements, this corresponds to an integrated power share between the modes.The integrated power p Iis directly extracted from the spectra by fitting Lorentzian lineshapes for each mode. Figure 4(a) shows the integrated power ratio between the high-frequencyand bullet mode, p I,HF/pI,B, as a function of field angle. The average values are obtained from elliptical NCs tilted 0◦ (blue circles), 45◦(red diamonds), and 90◦(black squares) biased at currents ranging from −16 to−26 mA. This average is possible to perform due to the similar current and field-dependent characteristics of the fabricated devices. Clearly,the energy is best shared for the elliptical NC tilted φ NC=0◦, suggesting a stronger coupling. For elliptical NCs tilted awayfrom 0 ◦, the ratio is larger although small enough to be still con- sidered a coexistence regime. Note that the ratio diverges forθ> 40 ◦for all cases, corresponding to the transition between a coexistence regime and a mode-hopping dominated regime. The corresponding generation linewidth of a particular set of devices as a function of applied field angle is shown inFig. 4(b), also obtained from the Lorentzian fits. Here, we observe a markedly different behavior between the linewidth ofthe bullet mode and that of the high-frequency mode. In partic-ular, the bullet mode exhibits a much lower linewidth than thehigh-frequency mode, whose minimum is nearly independentofφ NC. Such a behavior can be understood from the schematic energy landscape shown in Fig. 1(c). The bullet mode is pinned at a local energy minimum, and is thus robust against 104405-4MODE-COUPLING MECHANISMS IN NANOCONTACT SPIN- . . . PHYSICAL REVIEW B 91, 104405 (2015) FIG. 4. (Color online) (a) Integrated power ratio between the high-frequency and bullet mode, pI,HF/pI,B, for devices at φNC=0◦ (blue circles), 45◦(red diamonds), and 90◦(black squares), biased at currents between −16 and −26 mA. The modes have an integrated power ratio close to 1, denoting mode coexistence. Above θ=40◦a divergence is observed, corresponding to the onset of a mode-hopping regime. (b) Linewidth of the high-frequency (HF) and bullet (B) mode for a particular device biased at −18 mA. The HF mode exhibits variations between the devices with different φNC, while the B mode remains mostly stable. fluctuations. On the other hand, the high-frequency mode is perched upon a local energy maximum, an archetypal unstablecondition, where it is more prone to fluctuations in frequencyand thus to linewidth variations due to mode coupling. Conse-quently, the high-frequency mode can be used as a probe forthe aforementioned coupling mechanisms. For both the bulletand high-frequency modes, the linewidth when θ> 40 ◦corre- sponds to the transition to a mode-hopping dominated regime. We now investigate the mode-coupling mechanisms and their influence on the generation linewidth of the high-frequency mode. In order to directly relate the schematicrepresentation of Fig. 1(d)to experiments, we perform current- dependent measurements at a fixed out-of-plane field angle ofθ=30 ◦. Figures 5(a)and5(b) show the high-frequency mode FIG. 5. (Color online) Integrated power (a) and linewidth (b) at θ=30◦for devices with elliptical NCs satisfying φNC=0 (blue circles) and 90◦(black squares). The data are obtained from averaging the results obtained from five nominally identical devices, exhibiting a similar threshold current of ≈−15 mA. The integrated power in both cases decreases linearly, in agreement with the Oersted fieldinduced mode separation. However, the linewidth exhibits different behaviors in correspondence with the mode-coupling mechanisms in these devices. In particular, for φ=90 ◦, the magnon-mediated scattering mechanism dominates, as observed from the average linewidth behavior.integrated power and linewidth averaged from five devices with elliptical NCs at φNC=0 (blue circles) and 90◦(black squares) and similar threshold current for auto-oscillations≈−15 mA. In Fig. 5(a), a linear decrease in integrated power is observed as a function of current magnitude. This is consistentwith the Oersted field induced intermode separation. In fact, asthe modes move towards the boundaries of the NC, the current-carrying region cannot probe the entire magnetodynamics,leading to a reduced power [ 11,24]. Within the theoretical framework described above, this observation provides directevidence that the coexisting modes become increasingly moreseparated from one another as a function of current. Further-more, the absolute value of the power is consistently smallerfor the elliptical contact tilted at 90 ◦, suggesting an increased physical distance between the modes, as expected from themicromagnetic simulations shown in Fig. 3. This indicates that the intermode coupling mechanism when φ=90 ◦is negligibly small compared to the magnon-mediated coupling. The corresponding average linewidths [Fig. 5(b)]s h o wt w o markedly different features. In the case of the elliptical NCstilted at 0 ◦(blue circles), the linewidth is bounded between 200 and 400 MHz, exhibiting a complex dependency that reflectsthe sample-to-sample variations and the competition betweenthe two coupling mechanisms described above. On the otherhand, the elliptical NCs tilted at 90 ◦(black squares) exhibit a decrease in linewidth corresponding to the dependency expected from a purely magnon-mediated coupling. These observations provide direct proof of the mode-coexistenceregime and its relation to the dominant coupling mechanismsin these devices. VI. CONCLUSIONS NC-STOs were fabricated with elliptical NCs with the goal of breaking the symmetry of the system and finetune the energy landscape at the nanoscale. By electricallycharacterizing these devices, it was possible to find evidenceof mode coexistence and the underlying coupling mechanismsby means of a multimode STO theory. The obtained results areconsistent with two distinct coupling mechanisms, namely,one dominated by magnons and the other dominated byintermode overlap. This study shows that the multimode STOtheory is capable of predicting qualitatively the behaviorof complex magnetodynamics, such as mode hopping andmode coexistence. We expect this theory to be valuablein interpreting experimental results and to be extended toquantitatively describe the growing observations of multimodemagnetodynamics. Furthermore, the ability to control theenergy landscape and coupling mechanisms at the nanoscale isrelevant for the development of magnonic applications basedon the interaction between solitonic modes and propagatingspin waves. ACKNOWLEDGMENTS This work was supported by the European Commission FP7-ICT-2011, Contract No. 317950 “MOSAIC.” The workby O.H. was funded by the Department of Energy Office ofScience, Materials Sciences and Engineering Division. Sup-port from the Swedish Research Council (V .R.), the Swedish 104405-5EZIO IACOCCA et al. PHYSICAL REVIEW B 91, 104405 (2015) Foundation for Strategic Research (S.S.F.), and the Knut and Alice Wallenberg Foundation is gratefully acknowledged. APPENDIX A: AUTO-OSCILLATOR COMPLEX AMPLITUDE UNDER GALILEAN INVARIANCE The dynamics of a nonlinear auto-oscillator can be repre- sented by its complex amplitude [ 30] c=√peiφ, (A1) where pandφare the (generally time-dependent) power and phase. Furthermore, it is possible to cast the phase as φ= ωt+φ0, for a well-defined oscillation frequency ω. In the case of two coupled oscillators, we can write the phase difference and addition as ϕ=φ2−φ1=/Delta1ωt+ϕ0 and/Psi1=φ2+φ1=/Omega1t+/Psi10, where /Delta1ω=ω2−ω1and/Omega1= ω2+ω1. Without loss of generality, we can set ϕ0=/Psi10= 0. For the coupling regimes discussed in the main text andRef. [ 28], the phases of the oscillators are perturbed, leading toϕ=/Delta1ωt+δϕand/Psi1=/Omega1t+δϕ. Here, we will assume that/Omega1t/greatermuchδϕand that the reference frame of the coupled system is established by the well-defined frequency difference/Delta1ω. In other words, we establish that the phase-difference fluctuations δϕare solved under Galilean invariance. Finally, the complex amplitudes for the coupled oscillators are c 1=√p1ei(/Omega1t−/Delta1ωt−δϕ)/2, (A2a) c2=√p2ei(/Omega1t+/Delta1ωt+δϕ)/2. (A2b) APPENDIX B: AUTOCORRELATION FOR COEXISTENT MODES The autocorrelation function of the coupled auto- oscillators, K=/angbracketleft[c1(t)+c2(t)],[c∗ 1(t/prime)+c∗ 2(t/prime)]/angbracketright, (B1)provides the required information to determine the generation linewidth. Solving Eq. ( B1) by using the definitions of Eq. ( A2), we obtain K=p1eiω1te−/angbracketleftδϕ(t),δϕ(t/prime)/angbracketright/2 =p2eiω2te−/angbracketleftδϕ(t),δϕ(t/prime)/angbracketright/2. (B2) The most interesting feature of Eq. ( B2) is the fact that the phase autocorrelation is identical for both modes, indicatingan identical linewidth. To obtain the linewidth, it is possible to follow the same method introduced in Ref. [ 28], where the coupled power and phase equations are linearized and solved by standard matrixalgebra. Further assuming that in the case of coexistence thepower is equally split and /angbracketleftϕ−/Delta1ωt/angbracketright=0, we are left with /angbracketleftδϕ(t),δϕ(t /prime)/angbracketright=/Delta1ωo Kβ/parenleftbiggp2 oC2 (P+Kα)2−(Kβ)2+2/parenrightbigg eKβτ, (B3) where /Delta1ωois the linear linewidth for STOs, Kis the coupling strength, C=−N/2(ω−1 2−ω−1 1),Nis the nonlinear frequency shift, Pis a term proportional to the STO bias, α= (1+a2)/a,β=(1−a2)/a,a=√ω1/ω2, andτ=|t−t/prime|is the time lag. Expanding the exponential of Eq. ( B3) to first order, we obtain the Lorentzian contribution to the linewidth,cast in Eq. ( 2). As for the case of mode hopping, a Gaussian contribution is also present by expanding the exponential tosecond order. However, we are interested here in the qualitativebehavior of the linewidth with coupling strength K, so that Lorentzian fits are sufficient to study the behavior of theexperimentally measured spectra. [1] T. Silva and W. 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PhysRevApplied.10.034063.pdf

PHYSICAL REVIEW APPLIED 10,034063 (2018) Macromagnetic Simulation for Reservoir Computing Utilizing Spin Dynamics in Magnetic Tunnel Junctions Taishi Furuta,1Keisuke Fujii,2,3,*Kohei Nakajima,3,4Sumito Tsunegi,5Hitoshi Kubota,5 Yoshishige Suzuki,1,6and Shinji Miwa1,6,7, † 1Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan 2Graduate School of Faculty of Science, Kyoto University, Sakyo, Kyoto 606-8502, Japan 3PRESTO, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan 4Graduate School of Information Science and Technology, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan 5National Institute of Advanced Industrial Science and Technology (AIST), Spintronics Research Center, Tsukuba, Ibaraki 305-8568, Japan 6Center for Spintronics Research Network, Osaka University, Toyonaka, Osaka 560-8531, Japan 7The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan (Received 25 May 2018; revised manuscript received 21 August 2018; published 27 September 2018) The figures-of-merit for reservoir computing (RC), using spintronics devices called magnetic tunnel junctions (MTJs), are evaluated. RC is a type of recurrent neural network (RNN). The input information isstored in certain parts of the reservoir and computation can be performed by optimizing a linear transform matrix for the output. While all the network characteristics should be controlled in a general RNN, such optimization is not necessary for RC. The reservoir only has to possess a nonlinear response with memoryeffect. In this paper, macromagnetic simulation is conducted for the spin dynamics in MTJs for RC. It is determined that the MTJ system possesses the memory effect and nonlinearity required for RC. With RC using 5–7 MTJs, high performance can be obtained, similar to an echo-state network with 20–30 nodes,even if there are no magnetic and/or electrical interactions between the magnetizations. DOI: 10.1103/PhysRevApplied.10.034063 I. INTRODUCTION The magnetization direction of ferromagnetic metallic film is determined by the magnetic anisotropy energy, which causes nonvolatility. This property can be used for magnetic random access memory devices [ 1]. In magnetic tunnel junction (MTJ) devices consisting of ferromagnetic and dielectric thin films, the magnetization direction in the ferromagnet can be detected by the change in device resistance originating from the tunneling magnetoresis- tance (TMR) effect [ 2–5]. Moreover, the magnetization direction can be electrically controlled by the spin torque [6–9]. Therefore, MTJ devices are suitable for construct- ing nonvolatile high-density memory devices. In addition to a long-term memory effect, the magnetization preces- sional dynamics appear to possess a short-term memory effect with nonlinear behavior. Such additional magnetiza- tion dynamics properties may be suitable for computation using MTJ devices. The recurrent neural network (RNN) [ 10,11]i sa machine learning method. It is a mathematical model, *fujii@qi.t.u-tokyo.ac.jp †miwa@issp.u-tokyo.ac.jpwhich emulates the nerve system in the human brain. TheRNN concept is depicted in Fig. 1(a). The model con- sists of three layers, input, middle (node), and output. In the RNN, the information of the middle layer recursively propagates in itself. The middle-layer state is determined by the present input and past middle-layer state, i.e., the middle layer in the RNN possesses the memory effect. All the weight matrices for the input ( W in), middle ( W), and output ( Wout) should be precisely trained to obtain the desired output. However, when the middle layer has sufficient memory effect and nonlinearity, it is feasible to perform computation by optimizing only the output matrix (Wout). This type of simple RNN is called reservoir com- puting (RC) [ 12–14]. In RC, as system training is simple, it is easy to construct large-scale systems. The RC con- cept is depicted in Fig. 1(b). It has been reported that RC can be implemented in real physical systems, such as atomic switches [ 15–18], optoelectronic architecture [19–21], and the mechanical bodies of soft and compliant robots [ 22–24]. While it is possible to perform RC with such classical systems, RC using quantum dynamics can show higher figures-of-merit [ 25]. Recently, voice recogni- tion by RC using an MTJ [ 26] was reported; however, the figures-of-merit for RC using MTJs are not quantitatively 2331-7019/18/10(3)/034063(9) 034063-1 © 2018 American Physical SocietyTAISHI FURUTA et al. PHYS. REV. APPLIED 10,034063 (2018) (a) (b)S SInput Output Input Output FIG. 1. Concept of (a) recurrent neural network (RNN) and (b) reservoir computing (RC). understood. In this paper, we report the quantitative anal- ysis of the figures-of-merit for RC [ 27,28] using MTJ devices. We employ macromagnetic simulation for the study. II. METHODS A. Reservoir computing The RC method is as follows. A Boolean-type input sin(T) is employed. Tis an integer variable that represents time. sin(T) randomly assumes “0” or “1” in every time step: sin(T)=0o r1 . ( 1 ) The middle-layer state is defined as a node vector x(T) consisting of Nelements: x(T)=⎛ ⎜⎝x1(T) ... xN(T)⎞ ⎟⎠.( 2 ) The time evolution of the node is determined from the input at the present time and the past state of the node: x(T+1)=f[x(T),sin(T+1)]. (3) Here, fis a function for time evolution. Then the output yout(T) is defined as the inner product of thetime-independent weight vector Wout,and the node vector x(T): yout(T)=N/summationdisplay i=1Wixi(T)=Woutx(T) =(W1··· WN)⎛ ⎜⎝x1(T) ... xN(T)⎞ ⎟⎠,( 4 ) Wout=(W1··· WN).( 5 ) The training data ytrain(T) is prepared to optimize the sys- tem. Woutis determined for yout(T), reproducing ytrain(T), and is selected to minimize the mean squared error ( MSE ) between yout(T)a n d ytrain(T). The MSE is expressed as follows: MSE=1 LL/summationdisplay T=1[ytrain(T)−yout(T)]2 =1 LL/summationdisplay T=1[ytrain(T)−Woutx(T)]2.( 6 ) Woutis optimized using Ltime steps. In this paper, Lis 2000. A pseudoinverse matrix x−1is used for optimization. Wt out=x−1ytrain.( 7 ) The optimization of the output weight vector Woutis called learning . In this paper, a time-independent constant is added to xN+1(T) as a bias term, in addition to x1(T)t o xN(T). B. Figures-of-merit for reservoir computing In this paper, two types of tasks are employed for learn- ing. One is a short-term memory (STM) task [ 28]f o r characterizing the memory effect in the system. The train- ing data for the short-term memory task is expressed as follows: ytrain, STM (T,Tdelay)=sin(T−Tdelay).( 8 ) Here, sin(T) are random pulses, which are described later. It is feasible to obtain a finite memory effect, even if the sys- tem is completely linear. Therefore, we need another task to characterize the computing capability. In this paper, we additionally employ the parity check (PC) task [ 28]. The training data for the PC task requests the parity of the input sum. PC is used for characterizing the type of nonlinearity in the system and is expressed as follows: ytrain, PC (T,Tdelay)=sin(T−Tdelay)+sin(T−Tdelay+1) +···+ sin(T)(mod2).( 9 ) 034063-2MACROMAGNETIC SIMULATION FOR RESERVOIR . . . PHYS. REV. APPLIED 10,034063 (2018) After learning with the training data, the correlation between the output and training data is evaluated using the following equation: Cor(Tdelay)2=Cov[ ytrain(T,Tdelay),yout(T)]2 Var[ ytrain(T,Tdelay)]Var[ yout(T)]. (10) Here, Cor, Cov, and Var are the correlation, covariance, and variance, respectively. In this paper, Cor2is evaluated during 500 time steps, after learning with 2000 time steps. Cor2assumes values from 0 to 1, with a larger Cor2indi- cating better learning. Moreover, the capacity, C, is defined as the integration of Cor2, which can be used for evaluating the figures-of-merit for RC. Capacity : C≡Tdelay,max/summationdisplay Tdelay=1Cor(Tdelay)2. (11) Tdelay, max should be sufficiently large. In our calculation, Cor is always less than 0.01, when Tdelayis more than 10. Therefore, we set Tdelay, max =30. In this paper, we define CSTMas the capacity for short-term memory and CPCas the capacity for PC. C. Magnetic tunnel junction system Figure 2shows the schematics of the RC simulation using MTJs. An MTJ contains an insulating tunneling barrier layer with two ferromagnetic layers. For ferromag- netic layer 1 called the reference layer, the magnetization direction is designed to be fixed. This can be done withthe exchange bias effect using antiferromagnetic materi- als, such as PtMn and IrMn [ 29], or magnetic anisotropy energy. In this study, the magnetization direction of layer 1 is fixed perpendicular to the film plane. For ferromagnetic layer 2 called the free layer, the magnetization direction is not fixed and can be controlled by the current [ 6,7]- or voltage [ 8]-driven spin torque. The MTJ device resistance reflects the magnetization direction of s 2. The spin dynamics in ferromagnetic layer 2 follows the Landau-Lifshitz-Gilbert (LLG) equation with spin-transfertorque [ 30], where a thermal fluctuation in ferromagnetic layers [ 31] is not included: ds2 dt=−γ0s2×Heff−αs2×ds2 dt +P 1+P2s1·s2I −e/planckover2pi1 2s2×(s1×s2). (12) Here, s1and s2represent the unit spin vectors for ferro- magnetic layers-1 and -2, respectively. γ0(<0) is the gyro magnetic ratio. αis the Gilbert damping constant. Pis the spin polarization in the vicinity of the Fermi level in the ferromagnetic layers. I(=Vin/R) is the electric current, where Vinand Rare the input voltage and device resistance of the MTJ, respectively. Heffis the effective magnetic field ins2: Heff=−1 γ0∇U. (13) Here, Uis the magnetization energy for ferromagnetic layer 2, which includes the external magnetic field Hextand the magnetic anisotropy tensor ˆHani: U=μ0MSA/parenleftbigg Hext·s2+1 2st 2·ˆHani·s2/parenrightbigg =1 2μ0MSAst 2·ˆHani·s2(∵Hext=0), (14) ˆHani=⎛ ⎝00 0 00 0 00 Hazz⎞ ⎠. (15) Here, μ0,MS,a n d Aare the magnetic permeability in a vacuum, saturation magnetization, and volume of the fer- romagnetic layer 2, respectively. In the simulation, the external magnetic field is not applied. We assume uni- axial anisotropy perpendicular to the film plane. Here, Hazz>0(Hazz<0) shows in-plane (perpendicular) mag- netic anisotropy. The device resistance of the MTJ ( R) varies as a function of the relative angle between the spins (a) (b) s2s2s2 s1s1s1s2 s1FIG. 2. (a) Schematic of a RC system using the spin dynamics in a magnetic tunnel junction (MTJ) and (b) system with multiple MTJs. In the MTJs, the spindirection of the ferromagnetic layer 2 ( s 2) can be con- trolled by the input bias voltage Vin, whereas that of the ferromagnetic layer 1 ( s1) is fixed. 034063-3TAISHI FURUTA et al. PHYS. REV. APPLIED 10,034063 (2018) (a) (b) dcFIG. 3. (a) Input sin(T), input bias voltage to the MTJ device Vin, and MTJ device resistance as a function of time. Typical characteristics during the learn- ing and evaluation processes. Wedefine the virtual nodes [ x 1(T) to xN(T)] as shown in the inset and the (b) MTJ device resistanceas a function of the static input dc bias voltage. The black and red plots indicate the resistanceswhen the values of the uniax- ial anisotropy fields are 500 and 1000 Oe, respectively. V 0and V1 are voltages that render the device resistance constant. in the free and pinned layers: R=RAPRP (RAP+RP)+(RAP−RP)(s1·s2). (16) RAPand RPare the resistances when s1and s2are paral- lel and antiparallel, respectively. The time evolution of the MTJ resistance is characterized by sequential calculation using the fourth Runge-Kutta method. For evaluating the STM and PC capacities, an input pulse voltage, Vin,c o r - responding to the computational input, sin(T), is applied to the MTJs, as depicted in Fig. 3(a). Figures 2(a) and2(b) display the schematics of circuits with single and multiple MTJs, respectively. In this paper, the physical parame- ters listed in Table Iare employed. It almost follows our previous experimental research [ 32]. D. Reference calculation with echo-state network Additionally, an echo-state network [ 13] is introduced for comparison with the system using MTJs, where the following function is employed for Eq. (3): x(T)=tanh[ Wx(T−1)+Winsin(T)]. (17) The tanh function is used for component-wise projec- tion. Wand Winare matrices whose components are time- independent random values from ( −1) to 1. We normalize by dividing each component of Wby the spectral radius, r, obtained by singular value decomposition [ 14]. III. RESULTS AND DISCUSSION Figure 3(a)depicts an example of the MTJ device resis- tance under an input voltage, Vin. We employ a pulse voltage with binary values of V0(=−44 mV) and V1 (=+44 mV) as Vin. These binary values of Vincorrespond to 0 and 1 in sin(T), respectively, in the RC learning andevaluation processes [Eq. (1)]. The pulse width [20 ns in Fig. 3(a), for instance] corresponds to the discrete unit time step T. Because the device resistance is scalar, the node dimension is only one. However, the number of nodes can be increased by employing virtual nodes [ 15,33]. As shown in the inset of Fig. 3(a), the virtual nodes x1toxNare defined; these virtual nodes are further defined as a node vector x(T/prime). Figure 3(b)depicts the dc bias voltage dependence of the static MTJ device resistance. Under a dc bias voltage, the MTJ device resistance is collected after the spin dynamics are damped. Under a positive bias voltage, the spin- polarized current flows from the free layer s2, to the pinned layer s1. Then the spin-transfer effect induces auto- oscillation [ 34,35]i n s2. The relative magnetization angle between s1and s2increases, and an antiparallel-like mag- netization configuration is realized. Therefore, the device resistance increases when a positive bias voltage is applied. Under a negative bias voltage, a parallel-like magnetiza- tion configuration is induced, and the device resistance TABLE I. Physical parameters’ set of ferromagnetic layer 2 for RC with a single MTJ [Fig. 2(a)]. Parameter Value Gilbert damping constant (layer 2): α0.009 Uniaxial anisotropy (layer 2): Hazz1000 Oe Saturation magnetization (layer 2): MS1375 emu/cc Volume (layer 2): A 23 500 nm3 (φ122 nm ×2n m ) Resistance in parallel: RP 210/Omega1 Resistance in antiparallel: RAP390/Omega1 034063-4MACROMAGNETIC SIMULATION FOR RESERVOIR . . . PHYS. REV. APPLIED 10,034063 (2018) (a) (b) (c) (d)S S Cor2Cor2 FIG. 4. (a) Input sin, output for training ytrain, STM [Eq. (8)with Tdelay=1], and trained output youtfor evaluating the short-term memory task, (b) Input sin, output for training ytrain, PC ,[ E q . (9)with Tdelay=1], and trained output youtfor evaluating the parity check task, (c) Correlation using Eq. (10); the integrated values are defined as the short-term memory capacity ( CSTM), (d) Correlation using Eq.(10); the integrated values are defined as the parity check capacity ( CPC). The input-voltage pulse width =20 ns and the number of virtual nodes N=50. decreases. For the input pulse voltage in Fig. 3(a), the binary values V0and V1are defined as voltages that render the device resistance constant. As shown in Fig. 3(b), V0and V1vary as a function of the uniaxial magnetic anisotropy Hazz. A. Figures-of-merit for RC using a single MTJ In this section, we present the figures-of-merit for RC using a single MTJ device. The uniaxial magnetic anisotropy of the free layer s2is fixed as Hazz=1000 Oe. Here, the positive value of Hazzshows the magnetic cell in MTJ is in-plane magnetized. Figure 4shows the simulated data for evaluating the short-term memory and parity check capacities for a single MTJ. In Fig. 4, the input-voltage pulse width is 20 ns and the number ofvirtual nodes, N, is 50. Figure 4(a) shows the typical sim- ulation results for an input sin(T), training data for the short-term memory task ytrain,STM(T), and trained output yout(T) as a function of the time step. Similarly, Fig. 4(b) shows the input sin(T), training data for PC task ytrain, PC(T), and trained output yout(T). Here, training data for the short-term memory task ytrain,STM(T) and PC task ytrain,PC(T) are defined using Eqs. (8)and(9)atTdelay=1, respectively. The output is calculated using the simu- lated MTJ resistance (see Fig. 3)a n d Woutusing Eq. (4). Woutis trivially calculated using the definitions given by Eqs. (5)–(7). Figures 4(c) and4(d) depict the correlations [Eq. (10)] between youtand the training data as a function of Tdelay. We use ytrain, STM as the training data for short-term mem- ory and ytrain, PC for the parity check. CSTM and CPCare (a) (b)(c) (d)FIG. 5. Results of RC using single MTJ. (a) Short- term memory capacity ( CSTM) as a function of the input-voltage pulse width, (b) Parity check capacity(C PC) as a function of the input-voltage pulse width; Nis number of virtual nodes in the MTJ, (c) CSTM, and (d) CPCas functions of the virtual-node number, where the input-voltage pulse width is fixed to 20 ns. 034063-5TAISHI FURUTA et al. PHYS. REV. APPLIED 10,034063 (2018) (a) (b) (c)s2 s1s2 s1s2 s1s2 s1 FIG. 6. Results of RC using multiple MTJs. (a) Example of a parameter set for multiple MTJs, when the ratio of the uniax-ial magnetic anisotropy in each MTJ ( H azz, k/Hazz, k+1)=2a n d the number of MTJs ( M)=4, (b) Short-term memory capac- ity ( CSTM), and (c) parity check capacity ( CPC) as functions of Hazz, k/Hazz, k+1. The input-voltage pulse width =20 ns and the virtual nodes for each MTJ ( N)=50. defined as the numerical integration of the correlation and the capacity using training data for the short-term memory and parity check, respectively. Figure 5shows the CSTMand CPC,respectively, as func- tions of the input-voltage pulse width [Figs. 5(a)and5(b)] and the number of virtual nodes N[Figs. 5(c) and5(d)]. From Figs. 5(a) and5(b), both CSTMand CPCincrease as the pulse width increases. When the pulse width is greater t h a n2 0n s , CSTM and CPCare nearly constant because when the pulse width is less than 20 ns, the change in the magnetization direction is very small and the spin dynam- ics cannot work as a reservoir. In Figs. 5(c) and5(d), the dependence of CSTMand CPC, respectively, on the number of virtual nodes N, are displayed when the pulse width is fixed at 20 ns. From Figs. 5(c) and5(d), we find that it is better to set the number of virtual nodes greater than 20.TABLE II. Variation for uniaxial magnetic anisotropy for RC with multiple MTJs [Fig. 2(b)]. Hazz, k/ Hazz, k+1 Hazz, 1 Hazz, 2 Hazz, 3 ... Hazz, 7 1.0 1000 Oe 1000 Oe 1000 Oe . . . 1000 Oe 1.1 1000 Oe 909.1 Oe 826.4 Oe . . . 564.5 Oe1.2 1000 Oe 833.3 Oe 694.4 Oe . . . 334.9 Oe ... ... ... ... ... ... 2.9 1000 Oe 344.8 Oe 118.9 Oe . . . 1.7 Oe3.0 1000 Oe 333.3 Oe 111.1 Oe . . . 1.4 Oe B. Figures-of-merit for RC using multiple MTJs When multiple MTJs are employed for RC, higher figures-of-merit can be obtained. A schematic of a mul- tiple MTJ circuit for RC is depicted in Fig. 2(b). Multiple MTJs are placed in parallel, and an identical pulse voltage is applied to all the MTJs. To construct nodes for RC, spa- tial multiplexing [ 36] is employed. The node vector x(T) is defined as a vector with M×Nelements, where Mis the number of MTJs and Nis the number of virtual nodes in a MTJ: x1(T)=⎛ ⎜⎝x11(T) ... x1N(T)⎞ ⎟⎠,x2(T)=⎛ ⎜⎝x21(T) ... x2N(T)⎞ ⎟⎠,...,xM(T) =⎛ ⎜⎝xM1(T) ... xMN(T)⎞ ⎟⎠ → x(T)≡[x11(T)··· x1N(T)x21(T) ··· x2N(T)··· xM1(T)··· xMN(T)]t. (18) Figure 6shows the CSTM and CPCwith multiple MTJs. The uniaxial anisotropy Hazz,of ferromagnetic layer 2 in each MTJ is listed in Table II. For instance, when four MTJs and Hazz, k/Hazz, k+1=2 are employed, the uni- axial anisotropies of the MTJs are 1000, 500, 250, and 125 Oe, respectively, as shown in Fig. 6(a). Such variations in the anisotropies can be obtained by voltage-controlled magnetic anisotropy in the MTJs [ 37]. In this study, ther- mal fluctuation in ferromagnetic layer 2 is not included. For instance, thermal fluctuation energy at room temper- ature (26 meV) is negligibly small when compared to the magnetization energy from Eq. (14)(approximately 10 eV) when the magnetic anisotropy energy is Hazz=1000 Oe. Therefore, thermal fluctuation can be comparable or less than the magnetization energy of ferromagnetic layer 2 atH azz<3 Oe. In such a region, simulations assuming the ground state are not very correct, and a random magnetic field to reproduce the thermal fluctuation [ 31] should be included in the simulation. 034063-6MACROMAGNETIC SIMULATION FOR RESERVOIR . . . PHYS. REV. APPLIED 10,034063 (2018) (a) (b)(c) (d)FIG. 7. Results of RC using multiple MTJs. (a) Short-term memory capacity ( CSTM), and (b) Parity check capacity ( CPC) with seven MTJs (M=7) as functions of the input-voltage pulse width, (c) CSTM, and (d) CPCas functions of the virtual node number of each MTJ ( N) under an input-voltage pulse width of 20 ns. The uniaxial magnetic anisotropy ratio ( Hazz, k/Hazz, k+1)=1.6. Similar to a single MTJ, the binary values V0and V1 for the input voltage are determined as shown in Fig. 3(b). Note that V0and V1vary as a function of the uniaxial anisotropy field, and the smallest absolute values of the saturation voltages are employed as V0and V1for RC with multiple MTJs; i.e., V0and V1are determined for the MTJ with the smallest uniaxial magnetic anisotropy field. Fig- ures 6(b) and6(c) display the CSTMand CPC, respectively, as functions of the anisotropy ratio Hazz, k/Hazz, k+1. In the simulation, the input-voltage pulse width is 20 ns and the number of virtual nodes for each MTJ is 50 for all MTJs. The maximum value of CSTMincreases as the number of MTJs ( M) increases. Because each MTJ has a different uniaxial magnetic anisotropy field Hazz, it has a differ- ent response speed to an external voltage/current. This variation in the response speed increases the CSTMof the system. On the other hand, the increase in CPCis insignif- icant compared to that of the CSTM. Note that as there is no electric and/or magnetic interaction between the free layers of the MTJs, the CPCis insignificant. In Fig. 6(c), when Hazz, k/Hazz, k+1is large, the CPCusing multiple MTJs is less than that using a single MTJ. This is because the input-voltage pulse width of 20 ns is the best condition only for the parameters of a single MTJ ( Hazz=1000 Oe, V1=44 mV, V0=−44 mV).Figure 7shows the CSTMand CPCunder various con- ditions. In Fig. 7,Hazz, k/Hazz, k+1is fixed to 1.6. This is the best condition for the CSTMwith M=7. From Fig. 7(a), the CSTMis maximum, around a pulse width of 20 ns. When the pulse width is less than 20 ns, the change in the magne- tization direction by the spin-transfer torque is too small to perform as a reservoir. When the pulse width is greater than 20 ns, the spin dynamics are almost damped during a unit time step, and such a condition is not preferable for RC. From Fig. 7(b), the best conditions for the CSTMand CPC are not identical. This is because a relatively long pulse is required to induce nonlinearity in the spin dynamics in multiple MTJs. Figures 7(c) and7(d) depict the Ndepen- dences of the CSTM and CPC, respectively, when M=7, Hazz, k/Hazz, k+1=1.6, and the pulse width =20 ns. When Nis greater than four, both the CSTMand CPCare nearly constant. C. Comparison with the echo-state network The CSTMand CPC, using a multiple MTJ system, are summarized in Fig. 8(a); the pulse width =20 ns and the virtual node number N=50 for each MTJ. The data points from top to bottom are the data when Hazz, k/Hazz, k+1 (a) (b) Number NumberFIG. 8. Plots showing CSTMvs CPCin the (a) MTJ system and (b) echo-state network with Eq. (17).I n the MTJ system, the pulse width and virtual nodes of each MTJ ( N) are fixed to 20 ns and 50, respectively. 034063-7TAISHI FURUTA et al. PHYS. REV. APPLIED 10,034063 (2018) changes from 1.1 to 3.0. The result of the echo-state net- work, using the tanh function, is shown in Fig. 8(b).T h e data points from top to bottom are the data when the spec- trum radius, r,o f W[see Eq. (17)] varies from 0.05 to 2.0. From Fig. 8, it can be observed that a high perfor- mance can be obtained for RC using 5–7 MTJs, similar to an echo-state network with 20–30 nodes. In terms of the total number of virtual nodes in the system ( M×N), 35 nodes (7 ×5) of an MTJ system are comparable to 20–30 nodes of an echo-state network [see Figs. 7(c) and 7(d)]. Although the CPCincreases slightly as Mincreases, we can obtain a large CPCif there are magnetic and/or electrical interactions between the free layers in each MTJ. IV . CONCLUSION In this research, we demonstrate RC using the spin dynamics in MTJs. With RC using 5–7 MTJs, we can obtain a high performance similar to that of an echo-state network using tanh functions with 20–30 nodes. If there are magnetic and/or electrical interactions between the free layers in each MTJ, higher performance can be obtained. ACKNOWLEDGMENTS We thank E. Tamura, K. Shimose, S. Hasebe, and M. Goto of Osaka University and T. Taniguchi of AIST for the discussions. Part of this work was supported by JSPS KAKENHI (Grants No. JP18H03880, No. JP26103002, No. JP16H02211, No. JP16KT0019, No. JP15K16076, and No. JP26880010) and the Ministry of Internal Affairsand Communications. This work was also supported by JST-ERATO (Grant No. JPMJER1601), JST-CREST (Grant No. JPMJCR1673), and JST-PRESTO (Grants No. JPMJPR1668 and No. JPMJPR15E7), Japan. [1] H. Yoda, T. Kishi, T. Nagase, M. Yoshikawa, K. Nishiyama, E. Kitagawa, T. Daibou, M. Amano, N. Shimomura, S. Takahashi, T. Kai, M. Nakayama, H. Aikawa, S. Ikegawa, M. Nagamine, J. Ozeki, S. Mizukami, M. Oogane, Y. Ando,S. Yuasa, K. Yakushiji, H. Kubota, Y. Suzuki, Y. Nakatani, T. Miyazaki, and K. 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PhysRevB.91.214416.pdf

PHYSICAL REVIEW B 91, 214416 (2015) Comparison of spin-orbit torques and spin pumping across NiFe/Pt and NiFe/Cu/Pt interfaces Tianxiang Nan,1,*Satoru Emori,1,*,†Carl T. Boone,2Xinjun Wang,1Trevor M. Oxholm,1John G. Jones,3Brandon M. Howe,3 Gail J. Brown,3and Nian X. Sun1,‡ 1Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA 2Department of Physics, Boston University, Boston, Massachusetts 02215, USA 3Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, Ohio 45433, USA (Received 15 March 2015; revised manuscript received 21 May 2015; published 11 June 2015) We experimentally investigate spin-orbit torques and spin pumping in NiFe/Pt bilayers with direct and interrupted interfaces. The dampinglike and fieldlike torques are simultaneously measured with spin-torqueferromagnetic resonance tuned by a dc-bias current, whereas spin pumping is measured electrically through theinverse spin-Hall effect using a microwave cavity. Insertion of an atomically thin Cu dusting layer at the interfacereduces the dampinglike torque, fieldlike torque, and spin pumping by nearly the same factor of ≈1.4. This finding confirms that the observed spin-orbit torques predominantly arise from diffusive transport of spin currentgenerated by the spin-Hall effect. We also find that spin-current scattering at the NiFe/Pt interface contributes toadditional enhancement in magnetization damping that is distinct from spin pumping. DOI: 10.1103/PhysRevB.91.214416 PACS number(s): 75 .76.+j,75.78.−n I. INTRODUCTION Current-induced torques due to spin-orbit effects [ 1–3] potentially allow for more efficient control of magnetizationthan the conventional spin-transfer torques [ 4,5]. The spin-Hall effect [ 6] is reported to be the dominant source of spin-orbit torques in thin-film bilayers consisting of a ferromagnet (FM)interfaced with a normal metal (NM) with strong spin-orbitcoupling. Of particular technological interest is the spin-Hall“dampinglike” torque that induces magnetization switching[7–10], domain-wall motion [ 11–14], and high-frequency magnetization dynamics [ 15–20]. Although this spin-Hall torque originates from spin-current generation within the bulkof the NM layer, the magnitude of the torque depends onthe transmission of spin current across the FM/NM interface[3]. Some FM/NM bilayers with ∼1-nm-thick FM exhibit another spin-orbit torque that is phenomenologically identicalto a torque from an external magnetic field [ 21–28]. This “fieldlike” torque is also interface dependent because it mayemerge from the Rashba effect at the FM/NM interface [ 2] or the nonadiabaticity [ 4] of spin-Hall-generated spin current transmitted across the interface [ 3,23–25]. To understand the influence of the FM/NM interface on magnetization dynamics, many studies have experimentallyinvestigated resonance-driven spin pumping from FM toNM [ 29,30], detected with enhanced damping [ 31–35]o r dc voltage due to the inverse spin-Hall effect [ 36–45]. The parameter governing spin-current transmission acrossthe FM/NM interface is the spin-mixing conductance G ↑↓ (Ref. [ 46]). Simultaneously investigating spin pumping and spin-orbit torques, which are theoretically reciprocal effects[5], should reveal the interface dependence of the observed torques in FM/NM. Here we investigate spin-orbit torques and magnetic res- onance in in-plane magnetized NiFe/Pt bilayers with directand interrupted interfaces. To modify the NiFe/Pt interface, *These authors contributed equally to this work. †s.emori@neu.edu ‡n.sun@neu.eduwe insert an atomically thin dusting layer of Cu that does notexhibit strong spin-orbit effects by itself. We use spin-torqueferromagnetic resonance (ST-FMR) [ 47,48] combined with a dc bias current to extract the dampinglike and fieldlike torquessimultaneously. We also independently measure the dc voltagegenerated by spin pumping across the FM/NM interface. Theinterfacial dusting reduces the dampinglike torque, fieldliketorque, and spin pumping by the same factor. This finding isconsistent with the diffusive spin-Hall mechanism [ 3,32]o f spin-orbit torques where spin transfer between NM and FMdepends on the interfacial spin-mixing conductance. II. EXPERIMENTAL DETAILS A. Samples The two film stacks compared in this study are sub/ Ta(3)/Ni80Fe20(2.5)/Pt(4) (NiFe/Pt) and sub/ Ta(3)/Ni80 Fe20(2.5)/Cu(0.5)/Pt(4) (NiFe/Cu/Pt) where the numbers in parentheses are nominal layer thicknesses in nanometers and sub is a Si(001) substrate with a 50-nm-thick SiO 2 overlayer. All layers were sputter deposited at an Ar pressure of 3×10−3Torr with a background pressure of /lessorsimilar1×10−7Torr. The atomically thin dusting layer of Cu modifies the NiFe/Ptinterface with minimal current shunting. The Ta seed layerfacilitates the growth of thin NiFe with a narrow resonancelinewidth and near-bulk saturation magnetization [ 31,33]. We measured the saturation magnetization M s=(5.8± 0.4)×105A/m for both NiFe/Pt and NiFe/Cu/Pt with vibrat- ing sample magnetometry. From four-point measurements onvarious film stacks and assuming that individual constituentlayers are parallel resistors, we estimate the resistivities ofTa(3), NiFe(2.5), Cu(0.5), and Pt(4) to be 240, 90, 60, and40μ/Omega1cm, respectively. Approximately 70% of the charge current thus flows in the Pt layer. In the subsequent analysis,we also include the small dampinglike torque and the Oerstedfield from the highly resistive Ta layer (see Appendix A). B. Spin-torque ferromagnetic resonance We fabricated 5- μm-wide, 25- μm-long microstrips of NiFe/Pt and NiFe/Cu/Pt with Cr/Au ground-signal-ground 1098-0121/2015/91(21)/214416(9) 214416-1 ©2015 American Physical SocietyTIANXIANG NAN et al. PHYSICAL REVIEW B 91, 214416 (2015) -80 -60 -40 -20 0 20 40 60 80-150-100-50050100150 Vxim(µV) µ0H(mT) 4.5 GHz 5.5 GHz 6.5 GHz 30 40 50 60 70-80-60-40-2002040 0 mA-2 mA 2 mA Vxim)Vµ( µ0H(mT)(b) (c)(a) Ref. InputLIA IRF IDCVmix DLTDamping torqueFLTH m Field torque FIG. 1. (Color online) (a) Schematic of the dc-tuned ST-FMR setup and the symmetry of torques acting on the magnetization m. Through spin-orbit effects, the charge current in the normal metal generates two torques in the ferromagnet: dampinglike torque (DLT) and fieldlike torque (FLT). (b) and (c) ST-FMR spectra of NiFe/Pt at (b) different frequencies and (c) dc-bias currents. electrodes using photolithography and liftoff. We probed magnetization dynamics in the microstrips using ST-FMR(Refs. [ 47,48]) as illustrated in Fig. 1(a): An rf current drives resonant precession of magnetization in the bilayer, and therectified anisotropic magnetoresistance voltage generates anFMR spectrum. The rf current power output was +8 dBm and modulated with a frequency of 437 Hz to detect the rectifiedvoltage using a lock-in amplifier. The ST-FMR spectrum [e.g.,Fig. 1(b)] was acquired at a fixed rf driving frequency by sweeping an in-plane magnetic field |μ 0H|<80 mT applied at an angle of |φ|=45◦from the current axis. The rectified voltage Vmixconstituting the ST-FMR spectrum is fit to a Lorentzian curve of the form Vmix=SW2 (μ0H−μ0HFMR )2+W2 +AW(μ0H−μ0HFMR ) (μ0H−μ0HFMR )2+W2, (1) where Wis the half-width-at-half-maximum resonance linewidth, HFMR is the resonance field, Sis the symmetric Lorentzian coefficient, and Ais the antisymmetric Lorentzian coefficient. Representative fits are shown in Fig. 1(c). The line shape of the ST-FMR spectrum, parametrized by the ratio of StoAin Eq. ( 1), has been used to evaluate the ratio of the dampinglike torque to the net effective field from theOersted field and fieldlike torque [ 26,48–52]. To decouple the dampinglike torque from the fieldlike torque, the magnitude ofthe rf current in the bilayer would need to be known [ 48,51]. Other contributions to V mix(Refs. [ 53–55] )m a ya l s oa f f e c tt h e analysis based on the ST-FMR line shape.We use a modified approach where an additional dc-bias current Idcin the bilayer, illustrated in Fig. 1(a), transforms the ST-FMR spectrum as shown in Fig. 1(c). A high-impedance current source outputs Idc, and we restrict |Idc|/lessorequalslant2m A (equivalent to the current density in Pt |Jc,Pt|<1011A/m2) to minimize Joule heating and nonlinear dynamics. Thedependence of the resonance linewidth WonI dcallows for quantification of the dampinglike torque [ 48,54–60], whereas the change in the resonance field HFMR yields a direct measure of the fieldlike torque [ 52]. Thus, dc-tuned ST-FMR quantifies both spin-orbit torque contributions. C. Electrical detection of spin pumping The inverse spin-Hall voltage VISH due to spin pumping was measured in 100- μm-wide, 1500- μm-long strips of NiFe/Pt and NiFe/Cu/Pt with Cr/Au electrodes attached onboth ends, similar to the submillimeter-wide strips used inRef. [ 60]. These NiFe/(Cu/)Pt strips were fabricated on the same substrate as the ST-FMR device sets described inSec. II B. The sample was placed in the center of a rectangular TE 102microwave cavity operated at a fixed rf excitation frequency of 9.55 GHz and rf power of 100 mW. A bias field H was applied within the film plane and transverse to the long axisof the strip. The dc voltage V dcacross the sample was measured using a nanovoltmeter while sweeping the field as illustrated inFig. 2(a). The acquired V dcspectrum is fit to Eq. ( 1)a ss h o w n by a representative result in Fig. 2(b). The inverse spin-Hall voltage is defined as the amplitude of the symmetric Lorentziancoefficient Sin Eq. ( 1)( R e f s .[ 38–41,44]). We note that the antisymmetric Lorentzian coefficient is substantially smaller, 214416-2COMPARISON OF SPIN-ORBIT TORQUES AND SPIN . . . PHYSICAL REVIEW B 91, 214416 (2015) (a) (a) JS H mVdc hrf 80 100 120 140 160 µ0H (mT)012 V cd(µV)NiFe/Pt H +90°Vdc VISH VAHE(b) ║ FIG. 2. (Color online) (a) Schematic of the dc spin-pumping (inverse spin-Hall effect) voltage measurement. (b) Representative dc voltage spectrum. The inverse spin-Hall signal VISHdominates the anomalous Hall effect signal VAHE. indicating that the voltage signal from the inverse spin-Hall effect dominates over that from the anomalous Hall effect. III. RESULTS AND ANALYSIS A. Magnetic resonance properties Figure 3(a) shows the plot of the ST-FMR linewidth Was a function of frequency ffor NiFe/Pt and NiFe/Cu/Pt at Idc=0 and±2 mA. The Gilbert damping parameter αis calculated for each sample in Fig. 3(a) from W=W0+2πα |γ|f, (2) where W0is the inhomogeneous linewidth broadening, fis the frequency, and γis the gyromagnetic ratio. With the Land ´e g-factor gL=2.10 for NiFe (Refs. [ 31,33,42,61]),|γ|/2π= (28.0 GHz /T)(gL/2)=29.4 GHz /T. From the slope in Fig. 3(a) atIdc=0,α=0.043±0.001 for NiFe/Pt and α= 0.027±0.001 for NiFe/Cu/Pt. The reduction in damping with interfacial Cu dusting is consistent with prior studies on FM/Ptwith nanometer-thick Cu insertion layers [ 31,33,35,42,44]. Afi to f H FMR versus frequency at Idc=0 to the Kittel equation, μ0HFMR=1 2[−μ0Meff+/radicalbig (μ0Meff)2+4(f/γ)2] −μ0Hk+μ0/Delta1HFMR (Idc), (3)shown in Figs. 3(b) and 3(c), gives the effective magnetiza- tionMeff=5.6×105A/m for NiFe/Pt and 5 .9×105A/m for NiFe/Cu/Pt with the in-plane anisotropy field |μ0Hk|< 1m T.M effandMsare indistinguishable within experimen- tal uncertainty, implying negligible perpendicular magneticanisotropy in NiFe/(Cu/)Pt. When I dc/negationslash=0, the linewidth Wis reduced for one current polarity and enhanced for the opposite polarity as shownin Fig. 3(a). The empirical damping parameter defined by Eq. ( 2) changes with I dc(see Appendix B), which indicates the presence of a current-induced dampinglike torque. Similarly,I dc/negationslash=0 generates an Oersted field and a spin-orbit fieldlike torque that together shift the resonance field HFMR as shown in Figs. 3(b) and 3(c). We discuss the quantification of the dampinglike torque in Sec. III B and the fieldlike torque in Sec. III E . B. Dampinglike torque Figure 4(a) shows the linear change in Was a function of Idcat a fixed rf frequency of 5 GHz. Reversing the external field (from φ=45◦to−135◦) magnetizes the sample in the opposite direction and reverses the polarity of the dampingliketorque. Wis related to the current-dependent effective damping parameter α effat fixed f, α eff=|γ|/(2πf)(W−W0). The magnitude of the dampinglike torque is parametrized bythe effective spin-Hall angle θ DL, proportional to the ratio NiFe/Pt 0 mA2 mA -2 mA 4.0 4.5 5.0 5.5 6.0 6.5 f (GHz)4.0 4.5 5.0 5.5 6.0 6.5 f (GHz)NiFe/Cu/Pt 0 mA2 mA -2 mA(b) (c) 203040506070 µ0HRMF)Tm( 5.5 6.05.5 6.04550 4550 01234560.02.04.06.08.010.012.0NiFe/Pt 0 mA2 mA -2 mANiFe/Cu/Pt 0 mA2 mA -2 mA f (GHz)W(mT)(a) H +45°║ FIG. 3. (Color online) (a) Resonance linewidth Wversus frequency fat different dc-bias currents. (b) and (c) Resonance field HFMR versus frequency fat different dc-bias currents for (b) NiFe/Pt and (c) NiFe/Cu/Pt. 214416-3TIANXIANG NAN et al. PHYSICAL REVIEW B 91, 214416 (2015) αffe(10-2) W(mT) 4.0 4.5 5.0 5.5 6.0 6.50.000.020.040.060.080.100.120.14 NiFe/Pt NiFe/Cu/PtθLD f (GHz)NiFe/Pt H 45° H -135°NiFe/Cu/Pt f=5GHz -2 -1 0 1 24.44.85.25.67.68.08.4 2.552.702.853.904.054.204.35 Idc(mA)(a)(b) ║ ║ H 45° H -135°║ ║ H 45° H -135°║ ║ H 45° H -135°║ ║ FIG. 4. (Color online) (a) Resonance linewidth Wversus dc-bias current Idcatf=5 GHz. (b) Effective spin-Hall angle θDLcalculated at several frequencies. of the spin-current density Jscrossing the FM/NM interface to the charge-current density Jcin Pt.θDLat each frequency, plotted in Fig. 4(b), is calculated from the Idcdependence of αeff(Refs. [ 48,62]), |θDL|=2|e| /planckover2pi1/parenleftbig HFMR+Meff 2/parenrightbig μ0MstF |sinφ|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/Delta1α eff /Delta1Jc/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (4) where t Fis the FM thickness. Assuming that the effective spin- Hall angle is independent of frequency, we find θDL=0.087± 0.007 for NiFe/Pt and θDL=0.062±0.005 for NiFe/Cu/Pt. These values are similar to recently reported θDLin NiFe/Pt bilayers [ 39,42,48,51,54–56,59]. θDLof NiFe/(Cu/)Pt is related to the intrinsic spin-Hall angle θSHof Pt through the spin diffusion theory used in Refs. [ 3,32]. For a Pt layer much thicker than its spin diffusion length λPt,θDLis proportional to the real part of the effective spin-mixing conductance Geff ↑↓, θDL=2R e [Geff ↑↓] σPt/λPtθSH, (5) where σPtis the conductivity of the Pt layer and Geff ↑↓= G↑↓(σPt/λPt)/(2G↑↓+σPt/λPt) includes the spin-current backflow factor [ 30,32]. Assuming that λPt,σPt, andθSHin Eq. ( 5) are independent of the interfacial Cu dusting layer, Geff ↑↓ is a factor of 1 .4±0.2 greater for NiFe/Pt than NiFe/Cu/Pt based on the values of θDLfound above. C. Reciprocity of dampinglike torque and spin pumping Figure 5shows representative results of the dc inverse spin-Hall voltage induced by spin pumping, each fitted tothe Lorentzian curve defined by Eq. ( 1). Reversing the bias field reverses the moment orientation of the pumped spincurrent and thus inverts the polarity of V ISH, consistent with the mechanism of the inverse spin-Hall effect. By averaging measurements at opposite bias field polarities for different samples, we find |VISH|=1.5±0.2μV for NiFe/Pt and |VISH|=2.6±0.2μV for NiFe/Cu/Pt. The inverse spin-Hall voltage VISHis given by [ 38] |VISH|=h |e|Geff ↑↓|θSH|λPttanh/parenleftbiggtPt 2λPt/parenrightbigg fRsLP/parenleftbiggγhrf 2αω/parenrightbigg2 , (6)where Rsis the sheet resistance of the sample, Lis the length of the sample, Pis the ellipticity parameter of magnetization precession, and hrfis the amplitude of the microwave excitation field. The factor γhrf/2αωis equal to the precession cone angle at resonance in the linear (small-angle) regime.By collecting all the factors in Eq. ( 6) that are identical for NiFe/Pt and NiFe/Cu/Pt into a single coefficient C ISH,E q .( 6) is rewritten as |VISH|=CISHRsGeff ↑↓ α2. (7) We note that the small difference in Mefffor NiFe/Pt and NiFe/Cu/Pt yields a difference in P[Eq. ( 6)] of∼1%, which we neglect here. From Eq. ( 7), we estimate that Geff ↑↓of the NiFe/Pt interface is greater than that of the NiFe/Cu/Pt interface by a factor of1.4±0.2. The dc-tuned ST-FMR and dc spin-pumping voltage measurements therefore yield quantitatively consistent results,confirming the reciprocity between the dampinglike torque(driven by the direct spin-Hall effect) and the spin pumping(detected with the inverse spin-Hall effect). The fact that thediffusive model captures the observations supports the spin-Hall mechanism leading to the dampinglike torque. D. Interfacial damping and spin-current transmission Provided that the enhanced damping αin NiFe/(Cu/)Pt [Fig. 3(a)] is entirely due to spin pumping into the Pt layer, the real part of the interfacial spin-mixing conductance can be 60 80 100 120 140 160 180 µ0H (mT)60 80 100 120 140 160 180 µ0H (mT)-3-2-10123 V cd(µV)NiFe/Pt NiFe/Cu/Pt(b) (a) H +90°║ H -90°║ H +90°║ H -90°║ FIG. 5. (Color online) (a) and (b) dc voltage Vdcspectra, dom- inated by the inverse spin-Hall voltage VSH, measured around resonance in (a) NiFe/Pt and (b) NiFe/Cu/Pt. 214416-4COMPARISON OF SPIN-ORBIT TORQUES AND SPIN . . . PHYSICAL REVIEW B 91, 214416 (2015) calculated by Re[Geff ↑↓]=2e2MstF /planckover2pi12|γ|(α−α0). (8) Using α0=0.011 measured for a reference film stack sub/ Ta(3)/NiFe(2 .5)/Cu(2.5)/TaOx(1.5) with negligible spin pumping into the top NM layer of Cu, we obtain Re[ Geff ↑↓]= (11.6±0.9)×1014/Omega1−1m−2for NiFe/Pt and (5 .8±0.5)× 1014/Omega1−1m−2for NiFe/Cu/Pt. This factor of 2 difference for the two interfaces is significantly greater than the factor of≈1.4 determined from dc-tuned ST-FMR (Sec. III B ) and elec- trically detected spin pumping (Sec. III C ). This discrepancy implies that the magnitude of Re[ G eff ↑↓] of NiFe/Pt, calculated from enhanced damping, is higher than that calculated for spininjection. In addition to spin pumping, interfacial scattering effects [44,63–65], e.g., due to proximity-induced magnetization in Pt [13,35,66] or spin-orbit phenomena at the NiFe/Pt interface [67], may contribute to both stronger damping and lower spin injection in NiFe/Pt. Assuming that this interfacial scattering issuppressed by the Cu dusting layer, ≈0.010 of αin NiFe/Pt is not accounted for by spin pumping. The corrected Re[ G eff ↑↓]f o r NiFe/Pt is (8 .1±1.2)×1014/Omega1−1m−2, which is in excellent agreement with Re[ Geff ↑↓] calculated from first principles [65]. Using Geff ↑↓quantified above and assuming λPt≈1n m [26,32,33,43,49–51,54,55], the intrinsic spin-Hall angle θSH of Pt and the spin-current transmissivity T=θDL/θSHacross the FM/NM interface can be estimated. We obtain θSH≈0.15 andT≈0.6 for NiFe/Pt and T≈0.4 for NiFe/Cu/Pt. These results, in line with a recent report [ 26], indicate that the dampinglike torque (proportional to θDL) may be increased by engineering the FM/NM interface, i.e., by increasing Geff ↑↓. For practical applications, the threshold charge-current densityrequired for switching or self-oscillation of the magnetizationis proportional to the ratio α/θ DL. Because of the reciprocity of the dampinglike torque and spin pumping, increasing Geff ↑↓ would also increase αsuch that it would cancel the benefit of enhancing θDL. Nevertheless, although spin pumping inevitably increases damping, optimal interfacial engineeringmight minimize damping from interfacial spin-current scat-tering while maintaining efficient spin-current transmissionacross the FM/NM interface. E. Fieldlike torque We now quantify the fieldlike torque from the dc-induced shift in the resonance field HFMR , derived from the fit to Eq. ( 3) as shown in Figs. 3(b) and3(c).Meffis fixed at its zero- current value so that /Delta1HFMR is the only free parameter [ 68]. Figure 6shows the net current-induced effective field, which is equivalent to√ 2/Delta1HFMR in our experimental geometry with the external field applied 45◦from the current axis. The solid lines show the expected Oersted field μ0HOe≈0.08 mT /mA for both NiFe/Pt and NiFe/Cu/Pt based on the estimatedcharge-current densities in the NM layers H Oe=1 2(Jc,PttPt+ Jc,CutCu−Jc,TatTa) where the contribution from the Pt layer dominates by a factor of >6.-2 -1 0 1 2 Oersted field Δµ0HRMFnis/φ)Tm( NiFe/Pt NiFe/Cu/Pt -0.8-0.40.00.40.8 Idc(mA) H 45°║ H -135°║ H 45°║ H -135°║ FIG. 6. (Color online) Net current-induced effective field, de- rived from resonance field shift /Delta1HFMR normalized by the field direction angle |sinφ|=1/√ 2. The solid lines denote the estimated Oersted field. Although the polarity of the shift in HFMR is consistent with the direction of HOe, the magnitude of√ 2/Delta1HFMR exceeds HOefor both samples as shown in Fig. 6. This indicates the presence of an additional current-induced effective fielddue to a fieldlike torque μ 0HFL=0.20±0.02 mT /mA for NiFe/Pt and μ0HFL=0.10±0.02 mT /mA for NiFe/Cu/Pt. Analogous to θDLfor the dampinglike torque, the fieldlike torque can also be parametrized by an effective spin-Hall angle[26], |θ FL|=2|e|μ0MstF /planckover2pi1/vextendsingle/vextendsingle/vextendsingle/vextendsingleHFL Jc,Pt/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (9) Equation ( 9) yields θFL=0.024±0.003 for NiFe/Pt and 0.013±0.003 for NiFe/Cu/Pt, comparable to recently re- ported results in Ref. [ 23]. The ultrathin Cu layer at the NiFe/Pt interface reduces the fieldlike torque by a factor of 1 .8±0.5, which is in agree- ment within experimental uncertainty to the reduction in thedampinglike torque (Sec. III B ). This suggests that both torques predominantly originate from the spin-Hall effect in Pt. Recentstudies on FM/NM bilayers using low-frequency measurementtechniques [ 23–25] also suggest that the spin-Hall effect is the dominant source of the fieldlike torque. Since the fieldliketorque scales as the imaginary component of G eff ↑↓(Refs. [ 3–5]), the Cu dusting layer must modify Re[ Geff ↑↓] and Im[ Geff ↑↓] identically. We estimate Im[ Geff ↑↓]=(θFL/θDL)Re[Geff ↑↓]t ob e (2.2±0.5)×1014/Omega1−1m−2for NiFe/Pt and (1 .2±0.3)× 1014/Omega1−1m−2for NiFe/Cu/Pt. Because of the relatively large error bar for the ratio of the fieldlike torque in NiFe/Pt and NiFe/Cu/Pt, our experimentalresults do not rule out the existence of another mechanismat the FM/NM interface, distinct from the spin-Hall effect.For example, the Cu dusting layer may modify the interfacialRashba effect that can be an additional contribution tothe fieldlike torque [ 2,3,24]. Also, the upper bound of the 214416-5TIANXIANG NAN et al. PHYSICAL REVIEW B 91, 214416 (2015) (a) (b) 0.000.020.040.060.080.100.120.14 θDL 4.0 4.5 5.0 5.5 6.0 6.5 f (GHz)4.0 4.5 5.0 5.5 6.0 6.5 f (GHz)NiFe/Pt NiFe/Cu/Pt w/o FLT NiFe/Pt NiFe/Cu/Pt w FLT H 45°║ H -135°║ H 45°║ H -135°║ H 45°║ H -135°║ H 45°║ H -135°║ FIG. 7. (Color online) (a) and (b) Effective spin-Hall angle θeff SH,rf extracted from ST-FMR line-shape analysis, disregarding the (a) fieldlike torque and taking into account the (b) fieldlike torque. fieldlike torque ratio is close to the factor of ≈2 reduction in damping with Cu insertion, possibly suggesting a correlationbetween the spin-orbit fieldlike torque and the enhancement indamping at the FM-NM interface. Elucidating the exact rolesof interfacial spin-orbit effects in FM/NM requires furthertheoretical and experimental studies. F. Comparison of the dc-tuned and line-shape methods of ST-FMR Accounting for the fieldlike torque, we determine the effective spin-Hall angle θrf DLin NiFe/Pt and NiFe/Cu/Pt from the line shape [Eq. ( 1)] of the ST-FMR spectra at Idc=0( R e f s .[ 26,48–52]). The coefficients in Eq. ( 1)a r e S=Vo/planckover2pi1Js,rf/2|e|μ0MstFandA=VoHrf√1+Meff/HFMR , where Vois the ST-FMR voltage prefactor [ 48] and Hrf≈ βJc,rfis the net effective rf magnetic field generated by the rf driving current density Jc,rfin the Pt layer. θrf DL=Js,rf/Jc,rf is calculated from the line-shape coefficients SandA, /vextendsingle/vextendsingleθrf DL/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingleS A/vextendsingle/vextendsingle/vextendsingle/vextendsingle2|e|μ 0MstF /planckover2pi1β/radicalBigg 1+Meff HFMR. (10) Figure 7(a) shows |θrf DL|obtained by ignoring the field- like torque contribution, i.e., β=tPt/2. This underestimates |θrf DL|, implying identical dampinglike torques in NiFe/Pt and NiFe/Cu/Pt. Using β=tPt/2+HFL/Jc,Ptextracted fromTABLE I. Parameters related to spin-orbit torques. NiFe/Pt NiFe/Cu/Pt θDL 0.087±0.007 0 .062±0.005 θFL 0.024±0.003 0 .013±0.003 Re[Geff ↑↓]( 1 014/Omega1−1m−2)8 .1±1.25 .8±0.5 Im[Geff ↑↓]( 1 014/Omega1−1m−2)2 .2±0.51 .2±0.3 CISHRe[Geff ↑↓]( a.u.)1 .4±0.21 α−α0 0.032±0.001 0 .016±0.001 Fig. 6,θrf DL=0.091±0.007 for NiFe/Pt and 0 .069±0.005 for NiFe/Cu/Pt plotted in Fig. 7(b) are in agreement with θDLdetermined from the dc-tuned ST-FMR method. The presence of a non-negligible fieldlike torque in thin FM may account for the underestimation of θrf DLbased on the line-shape analysis compared to θDLfrom dc-tuned ST-FMR as reported in Refs. [ 54,55]. IV . CONCLUSIONS We have experimentally demonstrated that the spin-orbit dampinglike and fieldlike torques scale with interfacial spin-current transmission. Insertion of an ultrathin Cu layer atthe NiFe/Pt interface equally reduces the spin-Hall-mediatedspin-orbit torques and spin pumping, consistent with diffusivetransport of spin current across the FM/NM interface. Param-eters relevant to spin-orbit torques in NiFe/Pt and NiFe/Cu/Ptquantified in this paper are summarized in Table I.W eh a v e also found an additional contribution to damping at theNiFe/Pt interface distinct from spin pumping. The dc-tunedST-FMR technique used here permits precise quantification ofspin-orbit torques directly applicable to engineering efficientspin-current-driven devices. ACKNOWLEDGMENTS This work was supported by the Air Force Research Laboratory through Contract No. FA8650-14-C-5706 and, inpart, by Contract No. FA8650-14-C-5705, the W.M. KeckFoundation, and the National Natural Science Foundationof China (NSFC) Grant No. 51328203. Lithography was 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.00.000.020.040.060.080.100.120.14 f (GHz)-2 -1 0 1 2-0.3-0.2-0.10.00.10.20.3 f = 6.5 GHz Idc(mA)-0.6 -0.3 0.0 0.3 0.6 -1.0-0.50.00.51.0 |θLD| ∆W(mT) α∆ ffe(10-3)Jc, Ta (1011A/m2) (a)(b) H 45°║ H -135°║ H 45°║ H -135°║ FIG. 8. (Color online) (a) Change in resonance linewidth Wversus dc-bias current Idcin Ta/NiFe at f=6.5 GHz. (b) Effective spin-Hall angleθDLcalculated at several frequencies. 214416-6COMPARISON OF SPIN-ORBIT TORQUES AND SPIN . . . PHYSICAL REVIEW B 91, 214416 (2015) performed in the George J. Kostas Nanoscale Technology and Manufacturing Research Center. S.E. thanks X. Fan and C.-F.Pai for helpful discussions. T.N. and S.E. thank J. Zhou andB. Chen for assistance in setting up the ST-FMR system andV . Sun for assistance in graphic design. APPENDIX A: DAMPINGLIKE TORQUE CONTRIBUTION FROM TANTALUM With the same dc-tuned ST-FMR technique described in Sec. II B, we evaluate the effective spin-Hall angle θDLof Ta interfaced with NiFe. Because of the high resistivity of Ta, thesignal-to-noise ratio of the ST-FMR spectrum is significantlylower than in the case of NiFe/Pt, thus making precisedetermination of θ DLmore challenging. Nevertheless, we are able to obtain an estimate of θDLfrom a 2- μm-wide, 10- μm- long strip of subs/ Ta(6)/Ni80Fe20(4)/Al2O3(1.5) (Ta/NiFe). The estimated resistivity of Ta(6) is 200 μ/Omega1cm and that of NiFe(4) is 70 μ/Omega1cm. Figure 8(a) shows the change in linewidth /Delta1W (or/Delta1α eff) due to dc-bias current Idc. The polarity of /Delta1W against Idcis the same as in NiFe capped with Pt [Fig. 4(a)]. Because the Ta layer is beneath the NiFe layer, this observedpolarity is consistent with the opposite signs of the spin-Hallangles for Pt and Ta. Here we define the sign of θ DLfor Ta/NiFe to be negative. Using Eq. ( 4) with Ms=Meff= 7.0×105A/m and averaging the values plotted in Fig. 8(b), we arrive at θDL=− 0.034±0.008. This magnitude of θDLis substantially smaller than θDL≈− 0.1 in Ta/CoFe(B) [ 8,12] and Ta/FeGaB [ 60] but similar to reported values of θDLin Ta/NiFe bilayers [ 41,42]. For the analysis of the dampinglike torque in Sec. III B , we take into account the θDLobtained above and the small charge-current density in Ta. In theTa/NiFe/(Cu/)Pt stacks, owing to the much higher conductivityof Pt, the spin-Hall dampinglike torque from the top Pt(4)layer is an order of magnitude greater than the torque from thebottom Ta(3) seed layer. APPENDIX B: DC DEPENDENCE OF THE EMPIRICAL DAMPING PARAMETER Magnetization dynamics in the presence of an effective field Heffand a dampinglike spin torque is given by the Landau- Lifshitz-Gilbert-Slonczewski equation, ∂m ∂t=− |γ|m×Heff+αm×∂m ∂t+τDLm×(σ×m), (B1)-2 -1 0 1 22.22.42.62.83.84.04.24.44.6 Idc(mA)αf/ W(10-2) NiFe/Pt NiFe/Cu/Pt H 45°║ H -135°║ H 45°║ H -135°║ FIG. 9. (Color online) Empirical damping parameter αW/ f as a function of dc-bias current Idc. where τDLis a coefficient for the dampinglike torque (pro- portional to θDL) and σis the orientation of the spin moment entering the FM. Within this theoretical framework, it is notpossible to come up with a single Gilbert damping parameteras a function of bias dc current I dcthat holds at all frequencies. However, at Idc=0 we empirically extract the damping parameter αfrom the linear relationship of linewidth Wversus frequency f[Eq. ( 2)]. 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PhysRevB.92.094438.pdf

PHYSICAL REVIEW B 92, 094438 (2015) Mixed Brownian alignment and N ´eel rotations in superparamagnetic iron oxide nanoparticle suspensions driven by an ac field Saqlain A. Shah,1,2Daniel B. Reeves,3R. Matthew Ferguson,4,1John B. Weaver,3,5and Kannan M. Krishnan1,* 1Materials Science and Engineering, University of Washington, Seattle, Washington 98195, USA 2Department of Physics, Forman Christian College (University), Lahore, Pakistan 3Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA 4LodeSpin Labs, P .O. Box 95632, Seattle, Washington 98145, USA 5Department of Radiology, Geisel School of Medicine, Hanover, New Hampshire 03755, USA (Received 11 May 2015; revised manuscript received 13 August 2015; published 24 September 2015) Superparamagnetic iron oxide nanoparticles with highly nonlinear magnetic behavior are attractive for biomedical applications like magnetic particle imaging and magnetic fluid hyperthermia. Such particles displayinteresting magnetic properties in alternating magnetic fields and here we document experiments that showdifferences between the magnetization dynamics of certain particles in frozen and melted states. This effect goesbeyond the small temperature difference ( /Delta1T∼20 ◦C) and we show the dynamics to be a mixture of Brownian alignment of the particles and N ´eel rotation of their moments occurring in liquid particle suspensions. These phenomena can be modeled in a stochastic differential equation approach by postulating log-normal distributionsand partial Brownian alignment of an effective anisotropy axis. We emphasize that precise particle-specificcharacterization through experiments and nonlinear simulations is necessary to predict dynamics in solution andoptimize their behavior for emerging biomedical applications including magnetic particle imaging. DOI: 10.1103/PhysRevB.92.094438 PACS number(s): 75 .30.Gw,75.75.−c,75.78.−n,75.60.Jk I. INTRODUCTION Superparamagnetic iron oxide nanoparticles (SPIOs) made of magnetite can possess magnetic moments that saturatein biologically relevant magnetic fields of the order of tensof milliteslas. This strong magnetization response allowsnoninvasive control and readout during biomedical appli-cations. Because SPIOs are biocompatible, they have beenextensively used to realize drug delivery, cell separation,magnetic resonance imaging (MRI), localized hyperthermiatherapy [ 1], and, most recently, magnetic particle imaging (MPI) [ 2], which exploits the nonlinear response of magnetic nanoparticles to oscillating magnetic fields as a signal. In MPIand most biomedical applications (separation being a notableexception), the particles are activated with an alternatingmagnetic field, and thus magnetization reversal dynamicsplays a critical role [ 3–8]. There are two possible rotation mechanisms: N ´eel rotation [ 9] governs the restructuring of electronic spin states to allow the magnetic moment to reorientirrespective of the orientation of the whole particle, andBrownian rotation [ 10] occurs when the particle itself rotates in the solution, carrying with it the magnetic moment fixedin a direction relative to the particle’s crystal lattice. As anillustrative instance of why both mechanisms are important,hyperthermia therapy usually relies on N ´eel rotations that locally heat when the response of the moment lags behindthe driving field, yet several studies now show the influence ofparticle alignment or orientations on the heating capabilities,indicating Brownian rotations may be useful, if not inherentlyused, as a mechanism of heating [ 11–13]. In this paper, combining experiments and modeling, we have uncovered interesting solution-phase-dependent mag-netic dynamics through rigorous testing of magnetization *kannanmk@uw.eduresponses in various frozen and melted configurations. For ex-ample, we observed a change in magnetic response of a dilute suspension of particles to an alternating field upon freezing, which reversed upon melting. We attribute differences betweenthe liquid and frozen responses to the additional (Brownian)rotational freedom of the particles. To be clear, we assume thatin the liquid suspension the particles can reorient their easyaxes to align with the applied field, and this Brownian rotationis not possible in the frozen state. When a static magnetic fieldwas applied concurrently with the freezing process, possiblyimparting a net alignment of the easy axes, further variationin magnetic behavior was observed. The basic idea of the titlephrase “Brownian alignment and N ´eel rotation” is shown in Fig. 1: (1) the entire crystal rotates slightly to align one of its easy axes and (2) the subsequent magnetization rotation withthe N ´eel mechanism is different than the unaligned case. We compare these results to nonlinear stochastic simu- lations and find the behaviors can be replicated when thedirection of the effective easy axis is no longer random butis partially aligned in the field. From these observations, weconclude that the relative rotational freedom of nanoparticles,and the orientation of easy axes with respect to an applied field,can have a significant impact on the response to an oscillatingfield even when the N ´eel reversal process dominates. This can impact biomedical applications such as magnetic particleimaging or magnetic fluid hyperthermia. II. EXPERIMENTAL MATERIALS AND METHODS A. Materials characterization Magnetite nanoparticles (sample LS-002-2) were obtained from LodeSpin Labs, LLC. The nanoparticles contained crys-talline magnetite cores that were coated with a poly(ethyleneglycol) (PEG)-based amphiphilic polymer. Sample LS-002-2 was coated with poly(maleic anhydride-alt-1-octadecene) 1098-0121/2015/92(9)/094438(11) 094438-1 ©2015 American Physical SocietySHAH, REEVES, FERGUSON, WEA VER, AND KRISHNAN PHYSICAL REVIEW B 92, 094438 (2015) FIG. 1. (Color online) The magnetocrystalline energy surface for cubic magnetite (negative K1), which has easy directions in the [111] family. After (a) applying an arbitrary magnetic field vector Happ in a fixed direction, particles in solution undergo Brownian rotation through a small angle until (b) the nearest easy axis is aligned with the applied field. (PMAO, Mn 40 000 Da) PEG (Mn 20 000 Da). The PMAO was loaded with PEG such that 25% of the available carboxylatesin the PMAO were bonded to a PEG molecule. For thisstudy, sample LS-002-2 was dispersed in dimethyl sulfoxide(DMSO), which freezes at 19 ◦C. The hydrodynamic diameter (Zaverage, i.e., the intensity-based harmonic mean) of nanoparticles in liquid DMSO was 68 ±25 nm, measured by dynamic light scattering (DLS). The diameter of the samesample dispersed in water was 61 ±20 nm. Magnetic performance was the same in DMSO and water, which have similar viscosities (for reference, the viscosityof DMSO is 1.996 cP, while water is 0.894 cP). Transmis-sion electron microscopy (TEM) images in Fig. 2showed the nanoparticle samples to be monodisperse, with mediandiameter of 26 ±1.5 nm. Multiple images (6000 particles) were analyzed to determine size distribution using IMAGEJ , an open-source image-processing software developed by the FIG. 2. (Color online) SPIOs were characterized by complemen- tary methods to obtain magnetic, size, and structural information: (a) VSM curve, (b) TEM image of nanoparticles showing monodis- perse size distribution, (c) DLS plot, and (d) x-ray diffractionplot.National Institutes of Health. Shape anisotropy of the particles was estimated from TEM images, also using IMAGEJ . Each particle measured for size determination was fit with an ellipseand the ratio of long axis to short axis determined. The resultinghistogram was fit with a log-normal distribution to determinethe median aspect ratio (1 .04±0.03). This equates to a typical elongation of approximately 1 nm. Figure 2also shows the vibrating sample magnetometer (VSM) curve and the log-normal size distribution of nanoparticles obtained by fitting themagnetization curve to the Langevin function using Chantrell’smethod [ 14]. We calculate that the median magnetic core di- ameter is 29 .1±1.5 nm, with σ nw=0.05 (number weighted), where exp( σnw) is the geometric standard deviation of the log- normal distribution. The measured saturation magnetization of263 kA /m was used for fitting to determine the magnetic size from VSM measurements. Powder x-ray diffraction patternsmatched a magnetite reference [Fig. 1(d)], although we note the saturation magnetization was about 57% of bulk magnetite,which may be due to some phase impurity. Magnetite isferrimagnetic, having an inverse spinel crystalline structure,and displays a cubic magnetocrystalline anisotropy, with easyaxes along the [111] directions, i.e., the body diagonals of acube. The angles between the eight equivalent easy axes in aperfect cube are 70 .53 ◦. B. Immobilization of SPIOs Sample SPIOs were suspended in DMSO and immobilized by freezing in the absence or presence of a static magnetic field.To prepare samples, a solution containing 200 μL of magnetic nanoparticles (MNPs) with concentration 1.22 g Fe/L wastransferred to a 0.6-mL microcentrifuge tube and sonicatedfor 5 min at room temperature (23 ◦C). After sonication, the DMSO solution was visibly in the liquid phase. The samplewas immobilized by freezing at −20 ◦C inside a commercial freezer. During freezing, a magnetic field was applied to thesample to orient the nanoparticles by aligning their magneticeasy axes. The magnetic field was arranged with field linesparallel, transverse, or diagonal to the reference axis, defined tobe along the axis of the sample tube (see Fig. 3). The magnetic field was generated by a NdFeB permanent magnet (K&Jmagnetics, grade N42; dimensions: 1 .5×1×0.125 in. 3). The liquid sample was frozen in three different magnetic fields (5,10, and 15 mT) with a zero-field control sample, and for thethree respective orientations. For each freezing condition, thesample was cooled to −20 ◦C under the applied field and kept at that temperature for 30 min. After freezing, the sample tubewas transferred to an ice bath (0 ◦C) to maintain the frozen state of the ferrofluid during magnetic particle spectroscopy (MPS) measurements. Repeated MPS measurements showed that the freezing process was fully reversible and the sample behaviorupon melting was the same as before freezing. In anotherstudy, a dilution series (0.05, 0.1, 0.5, 1, and 1.22 g Fe/L) wasprepared and analyzed with the same measurement procedureafter freezing in different fields (0, 10, and 60 mT) to de-termine whether the freezing process introduced interparticleinteractions into the magnetic behavior of the system. The magnetic field profile of the NdFeB magnet was measured using a handheld gauss meter (Lakeshore model410). Distance was measured from the center of the magnet 094438-2MIXED BROWNIAN ALIGNMENT AND N ´EEL ROTATIONS . . . PHYSICAL REVIEW B 92, 094438 (2015) FIG. 3. (Color online) (a) The simulated magnetic field of the permanent magnet (lines of equal field strength are shown in the figure). The test tube containing the SPIOs was frozen in this orientation at various distances to achieve the varying magnetic field amplitudes. In the example illustrated in (a), the aligning field was applied perpendicular to the direction of the oscillating (drive) field, which was always applied parallel to the long axis of the sample tube. The surface field for these magnets is 0.12 T, so we see that at 5 cm the field is in the single milliteslarange as shown in experiment. Notably at this distance we can see the variations in the field are minimal within the test tube. (b) Measured variation of field strength with distance from the magnet. Error bars represent the variation in field within the sample tube for the stated field orientation, and they differ because the tube geometry is anisotropic. to the center of the sample volume. Uncertainties in the magnetic field profile, represented by error bars in Fig. 3(b), were estimated based on the range of field values measuredover the sample volume along the magnetic field vector(maximum 1.4 cm, for the parallel orientation). The largestfield corresponded to the volume of sample closest to themagnet, and the smallest field to the volume farthest fromthe magnet. The parallel orientation admitted the largestuncertainty, as the sample volume was longest in this direction. C. Magnetic measurements A custom-built magnetic particle spectrometer was used to determine the magnetization response of MNPs [ 3,15]. The system applies an oscillating magnetic field to the sample ofMNPs and measures the voltage signal induced in a receivercoil from the response of the magnetic particles. The signalis thus the time derivative of the magnetization and must beintegrated numerically to obtain the magnetization response. D. Landau-Lifshitz-Gilbert simulations with effective anisotropy Simulations of magnetization reversal were incorporated to support experiments. There have been many modelsintroduced in the literature to model the rotational dynamicsof magnetic particles [ 16–19]. Here, state-of-the-art simula- tions integrate the stochastic Landau-Lifshitz-Gilbert (LLG)equation, Eq. ( 1). The equation describes a classical magnetic dipole with a phenomenological damping term and can be usedto discuss N ´eel rotations of magnetic particles [ 16,20]: dm dt=−γ 1+α2{m×[H+αm×H]}. (1)In the LLG dynamics, the magnetization mevolves over time depending on the gyromagnetic ratio γand the unitless LLG damping parameter α. To model magnetic spectroscopy of SPIOs with core volume Vcand magnetic moment μ, we considered the total field Hto contain contributions from an applied oscillating excitation field of magnitude H0 and frequency f=ω/2π, an effective uniaxial anisotropy direction with magnitude K, and easy axis in direction n. Particles in liquid suspension are not fixed spatially andundergo Brownian motion; in all samples the concentrationof nanoparticles was on the order of 1 g/L, making the volumefraction of particles 0.025%. For both of these reasons wemodeled the particles as noninteracting dipoles. We could thusignore translational movement in the model. However, it wasessential to introduce thermal fluctuations of the direction ofthe magnetization using a random field h(t) so that the total field of the model was H=H 0cos(ωt)ˆz+2KVc μ(m·n)n+h(t), (2) with a zero-mean, δ-time correlated random field /angbracketlefth(t)/angbracketright=0/angbracketlefthi(t)hj(t/prime)/angbracketright=2kBT μγ(1+α2) αδijδ(t−t/prime)( 3 ) so that the field is Markovian in time and i,j∈x,y,z describes the lack of correlation between the random fields in differentCartesian directions. The thermal energy k BTis written in terms of Boltzmann’s constant and temperature T. The model assumes the particles are effectively single do- main, which is a reasonable assumption given that the criticaldiameter for multiple domains in magnetite is approximately85 nm [ 21]. A distribution for the magnetic moments was determined from the measured saturation magnetization andthe measured core size distribution. The anisotropy is of 094438-3SHAH, REEVES, FERGUSON, WEA VER, AND KRISHNAN PHYSICAL REVIEW B 92, 094438 (2015) particular interest and because the underlying mechanism is not fully understood, we used an effective anisotropydistribution that could incorporate crystalline anisotropy andshape anisotropies potentially caused by minor chaining (e.g.,dimerization). This approach was taken by Jamet et al. to discuss multiple anisotropic contributions in clusterednanoparticles and by Tamion et al. to numerically account for the distribution of anisotropy constants [ 22,23]. We expect that in solutions, the particles were rotating with the Brownianmechanism to align one of their cubic axes along the appliedfield. The required angle of rotation is small on averagebecause there are several equivalent axes in a cubic magnetiteparticle, and any one could align along the applied field.To simulate this behavior, we made nonrandom choices forthe direction of an effective easy axis of each particle. Westress that the strength of the anisotropy was also considereda log-normal distribution and that the alignment with the fieldwas modeled as a distribution of the angle of each particle’seasy axis dependent on the extent of the alignment. Forexample, in the freezing experiments, we imagine a singleparticle immersed in a static field potentially allowing oneof its easy axes—geometrically, the closest—to rotate andalign with the field. The ensemble of particles thus developsa distribution of alignments that are locked in place when thesolution is subsequently frozen. The alignment then persistedduring spectroscopy. Coupled differential equations describingthe Brownian and N ´eel rotation can be found in the works of Shliomis and Stepanov [ 24] as well as Coffey, Cregg, and Kalmykov [ 25], though because of the foggy nature of the anisotropy in this work and the excellent fit of the experimentaldata with the current model, these approaches were not used. E. Numerical implementation We use the Heun scheme to integrate the stochastic differen- tial equation. Excellent descriptions of numerical methods forstochastic problems can be found in Gardiner’s handbook [ 26]. Heun’s method converges in the sense of Stratonovich [ 27] and can be written in general for the change in magnetization as dm=f(m,t)dt+g(m,t)dW. (4) Here the numerical integration scheme has predictor ¯m(t+/Delta1t)=m(t)+f(m,t)/Delta1t+g(m,t)/Delta1W (5) and then the true value is m(t+/Delta1t)=m(t)+f(m,t)+f(¯m,t+/Delta1t) 2/Delta1t +g(m,t)+g(¯m,t+/Delta1t) 2/Delta1W, (6) where an increment of the Wiener process is defined by /Delta1W=√ /Delta1tN(0,1) where N(0,1) is a vector of normal or Gaussian- distributed random numbers each with mean zero and unitvariance. The predicted size distributions for particle diameters are log-normal (see Fig. 2)[14]. This means the probability ofhaving a particle with a certain radius rcan be modeled by p(r)=1 √ 2π1 rσrexp⎡ ⎢⎣−/braceleftBig ln/parenleftBig r mr/radicalBig 1+s2r m2r/parenrightBig/bracerightBig2 2σ2r⎤ ⎥⎦ (7) with /integraldisplay∞ 0p(r)dr=1. (8) We have used that the mean and standard deviation of the distribution are mrandsr, respectively, and the scale parameter σr=/radicalBig ln(1+s2r m2r). An equivalent log-normal distribution for the anisotropy constants allows for a mixture of effects, in- cluding crystalline anisotropy and chaining. Simulations werealways performed at the correct temperature [implementedin the thermal field, Eq. ( 3)] corresponding to the desired experimental procedure. The LLG damping parameter washeld constant at α=1. Changing this value did not change the dynamics but required more time steps. It is typical to choosethe value between 0.01 and 1 [ 18,28,29], though new studies are shedding light on the variation of this parameter during fer-romagnetic resonance [ 30]. The integration required 2 12time steps and 105nanoparticles, or equivalently 105simultaneous integrations of the stochastic equation. We included a signal atthe background frequency to mimic any feedthrough from theapparatus (at most 10% of the magnitude of the nanoparticlesignals and usually below 5%). The magnetite density was4.9 g/cm 3and the saturation magnetization was found from experiment to be 50 emu/g or equivalently 263 kA /m. The measurement techniques are discussed in Sec. II C.W eu s e d the bulk value for the gyromagnetic ratio of 1.3 GHz/T asopposed to the single electron value [ 31] .T h eb e s tfi tt o experimental data was achieved using a mean anisotropyconstant of 3.4 kJ /m 3w i t ha3k J /m3standard deviation, which is consistent with recent measurements of similar samples [ 32]. In a suspension of particles, the easy axis, n, for each particle is typically assumed to be in a random direction. However, theexperiments showed that, by freezing the particles in a staticfield, their subsequent magnetic dynamics were different. Twopossible physical mechanisms were considered: the Brownianalignment of the closest magnetocrystalline easy axes, orformation of short chains in the direction of the applied field.Either way, this would replace the cubic anisotropy of theparticles by an effective uniaxial anisotropy. To simulate thisbehavior, we allowed that an effective single easy axis foreach particle was partially aligned with the direction of thestatic field. This was implemented by choosing a randomorientation for each particle, adding a vector in the direction ofalignment, and then normalizing again. Three realizations ofthis procedure are shown in Fig. 4: a fully randomized sample, a sample that has been aligned 30% to the transverse directionof the oscillating field, and a sample that has been aligned 50%parallel to the direction of the oscillating field. III. RESULTS Freezing the sample altered its dynamic magnetization, as shown in Fig. 5, which compares the response of a single sample in liquid and frozen states. Measurements were taken 094438-4MIXED BROWNIAN ALIGNMENT AND N ´EEL ROTATIONS . . . PHYSICAL REVIEW B 92, 094438 (2015) FIG. 4. (Color online) Distribution of the effective easy axes of 500 particles. This figure should aid the visualization of the asymmetry that was included in the simulations in order to mimic partial alignment of an effective easy axis with the static field. for a drive field of f=26 kHz and H0=50 mT/ μ0.B y convention, only half of the full period (forward scan) is shownin all plots of the differential susceptibility χ diff=dm/dH vs field with respect to the field amplitude. The fluid samplehad greater maximum differential susceptibility χ max diff, andχdiff decayed to its minimum at lower field magnitude. When the sample was frozen in the presence of an applied field, both χdiffand its integrated value, the magnetization m(H), varied with the angle of the applied field with respect to the reference axis (Fig. 6). Among the field-freezing data, the parallel orientation had the highest value of χmax diff[i.e., the maximum value of dm/dH (m3), which is proportional to the maximum MPS signal intensity (V)] and highestcoercivity, H c. Freezing in the transverse orientation showed the minimum values of χmax diff andHcand the diagonal orientation showed intermediate values. The magnetic response also varied with the intensity of the freezing field for a given angular orientation. Figures 6(c) and 6(d) show the magnetization response of MNPs frozenin different magnetic fields along parallel orientation. Among the field-freezing data, the smallest freezing field led to thesmallest values of χ max diffandHc. All these values increased with increasing freezing fields (Fig. 6), although the variation was relatively small, most probably due to the small range offreezing fields (15 mT/ μ 0maximum) that were achievable with our experimental apparatus. χmax diffwas the highest at 15 kHz and it decreased slightly as the frequency was increased. CoercivityH cincreased for increasing frequencies. A summary of immobilized sample behavior is provided in Fig. 7, which plots variation of the peak value of the differential susceptibility χmax diff=dm/dH and coercive field, Hc, for various combinations of freezing fields, orientations, and drive frequencies. Parallel freezing gave the highest valuesforχ max diffas well as Hc, whereas transverse freezing showed the smallest values. Diagonal freezing gave intermediateresponse. For field-frozen samples, χ max diffandHcincreased with greater freezing field for parallel and diagonal orientations,but decreased for transverse orientation. However, increasing FIG. 5. (Color online) MPS results of the same sample measured in fluid state (at 20◦C) and frozen (in zero magnetic field, 0 mT/ μ0). The excitation field was 50 mT/ μ0amplitude at 26 kHz: (a) χdiffand (b) m(H) curves calculated by integrating data in (a). 094438-5SHAH, REEVES, FERGUSON, WEA VER, AND KRISHNAN PHYSICAL REVIEW B 92, 094438 (2015) FIG. 6. (Color online) MPS (26 kHz, 50 mT/ μ0) signal parameters of MNPs, field frozen (10 mT/ μ0) along different directions with respect to the MPS applied field. (a) χdiffand (b) integrated m(H) curves. MPS (26 kHz, 50 mT/ μ0) signal parameters of MNPs, frozen in different magnetic fields along parallel orientation: (c) χdiffand (d) m(H) curves. MPS (50 mT/ μ0, different frequencies) signal parameters of MNPs, frozen in 5 mT/ μ0along parallel orientation: (e) χdiffand (f) m(H)c u r v e s . the freezing field from 10 to 15 mT/ μ0did not produce a monotonic change in the response. The field-frozen sampleresponse varied with the excitation field frequency [Figs. 6(e) and6(f)], in agreement with previous investigations [ 3]. Here drive field frequencies were varied but amplitude was fixedat 50 mT/ μ 0.χmax diffvaried slightly with frequency but with no clear trend, while coercivity increased monotonically withfrequency. We note some variation in χ max diffcould have resulted from undersampling due to the finite sample acquisition rateof our system (2M samples/s), which is exacerbated at higherfrequencies.The MPS response of the liquid sample (dispersed in water) was measured at room temperature and 0 ◦C. We note the slight variation in magnetic response due to temperature,particularly in the differential susceptibility [Fig. 8(a)]. While the variation is close to the experimental error, the slightlyhigher value of χ max diffand reduced coercive field observed in the warmer sample may be expected in terms of the relaxationdynamics of the particles. When colder, magnetic reversalrequires slightly greater energy, and therefore greater appliedfield, since thermal energy is reduced. These phenomena canbe seen in Fig. 8, particularly in the differential susceptibility 094438-6MIXED BROWNIAN ALIGNMENT AND N ´EEL ROTATIONS . . . PHYSICAL REVIEW B 92, 094438 (2015) FIG. 7. (Color online) Comparison plot of MNPs response for various combinations of freezing fields, orientations, and drive frequencies: (a)χmax diffand (b) coercive field. curve. The MPS responses of water-dispersed and liquid DMSO-dispersed samples were identical (data not shown). A dilution series was prepared and frozen to observe how the freezing process affected sample magnetization dynamics.MPS analysis showed linear variation of signal intensity withthe concentration for all freezing fields, as shown in Fig. 9,b u t with greater intensity per unit concentration at higher freezingfields. We found good agreement between simulations andexperimental results and in particular show that the frozenparticles act as if they are partially aligned by the field-freezingprocedure. The linear change with concentration indicates thata combination of Brownian alignment and N ´eel rotations is possible in these particles, and, to first order, the formation ofchains can be ruled out. In Fig. 10, we show that the hysteresis loops change dramatically when frozen in the directionsparallel to and perpendicular to the oscillating field that isapplied. The simulations are identical except for an averagereorientation of the easy axes. In each case, we simulate a50% alignment in the direction of the static field in which theparticles were frozen. In Fig. 11we show that the simulated hysteresis loops are quite different based on the phase of the suspending liquid, aneffect much beyond simply changing the temperature, that hasbeen previously studied in cobalt nanoparticle solutions [ 33]. In particular, the melted sample shows an increase in the amount of saturation. This indicates potentially that the easyaxes become slightly polarized in the direction of the oscil- lating field due to Brownian rotation. In this simulation, weaccount for partial (30%) alignment to the field. In the frozensample, however, the Brownian rotations should be restricted,and the fit is the most accurate without any alignment. To show that increasing the strength of the applied static field affects the simulations in a similar way to the experiment,we simulated the hysteresis loops and the susceptibility ofthe magnetization at various applied field strengths, leadingto various alignment percentages (Fig. 12). As the Brownian alignment increased, we saw the expected increase in the slopeof the hysteresis as well as the delay in the peak of the slopecaused by the anisotropy field parallel to the applied field. Thesusceptibility data were smoothed using a Gaussian window toavoid the artifacts that arise from taking a numerical derivative. IV . DISCUSSION The most significant result of this work was the ob- served difference in response between immobilized and liquidsamples, indicating that both N ´eel relaxation and Brownian alignment mechanisms are possible in the liquid sample. Infact, for these particles, if the relaxation times are computedusing the standard zero-field expressions for N ´eel and Brow- nian relaxation, they are of the same order of magnitude.diff [m3/gFe] 051015106 H [mT-1 0]20 10 0 10 20 (a) m [Am2/gFe] 510505105 H [mT-1 0]20 10 0 10 20 (b) FIG. 8. (Color online) MPS (26 kHz, 20 mT/ μ0) response of SPIOs in water at room temperature and 0◦C (liquid state). Error bars in (a) represent the standard deviation of three measurements. 094438-7SHAH, REEVES, FERGUSON, WEA VER, AND KRISHNAN PHYSICAL REVIEW B 92, 094438 (2015) FIG. 9. (Color online) χmax diffdata (at 26 kHz and 20 mT/ μ0) of different concentrations: (a) at different freezing fields and (b) at 10 mT/ μ0 parallel freezing. Admittedly, these expressions are inappropriate to describe the dynamic response of particles forced to oscillate in amagnetic field, but if the relaxation times were many ordersof magnitude different an adiabatic approximation would bereasonably made to ignore the slower mechanism [ 34]. Though both mechanisms seem to occur, the N ´eel mechanism appears to dominate, because a significant response can be measuredfrom immobilized samples in which Brownian rotation shouldbe quenched. We hypothesized that variation between liquid and im- mobilized samples could result from significant magneticanisotropy in the nanoparticles. To investigate our hypothesis,we immobilized the SPIOs in DMSO, since the DMSO-nanoparticle suspension could be reproducibly frozen andmelted to orient and reorient the nanoparticles. Furthermore,DMSO stabilized the nanoparticles in solution and solidphases, with the nanoparticles displaying behaviors consistentwith noninteracting particles (see Fig. 9for the linear increase in magnetization with concentration). d (nm)20 30 40# 010203040 K (kJ/m3)02 0 4 0# 020406080 -40 -20 0 20 40<m> -1-0.8-0.6-0.4-0.200.20.40.60.81Parallel -40-20 0 20 40Transverse Data Simulation H0 (mT/ 0) FIG. 10. (Color online) Comparison between simulation and ex- periment for nanoparticles that were frozen while exposed to a 10 mT/ μ0static field aligned parallel or perpendicular to the oscillating field. Then spectroscopy was performed using a 40-kHz, 20 mT/ μ0field. The log-normal distributions of particle diameter and anisotropy constant are shown on the right by number in arepresentative 1000-particle subsample.To complement the experimental results, we simulated the dynamic particle magnetizations using a model basedon the stochastic Landau-Lifshitz-Gilbert equation [ 20]. From the modeling, we saw almost identical magnetizationswhen the easy axes were partially aligned and a log-normalsize distribution for the particle sizes, and effective uniaxialanisotropy constant was included. We interpret the agreementwith the effective uniaxial simulations to mean that an initialBrownian alignment causes one of the easy cubic axes to alignwith an applied static field, at which point the particles arefrozen and fixed spatially. Subsequent N ´eel rotations depend on the direction of the alignment. The distribution of theanisotropy constant allows for several possible contributions tothe anisotropy, and the form of the anisotropy energy dependson the constant and simply how much the magnetization isaligned with the effective easy axis. The parallel field-freezing alignment leads to higher slopes in the hysteresis curves (higher differential susceptibility χ diff) and wider hysteresis loops (larger coercive fields Hc) than the diagonally or perpendicularly aligned particles. This behavior H0 (mT/ 70)-20 0 20<m> -1-0.8-0.6-0.4-0.200.20.40.60.81Frozen -20 0 20Melted Data Simulationd (nm)02 0 4 0# 010203040 K (kJ/m3)02 0 4 0# 020406080100 FIG. 11. (Color online) Comparison between simulation and ex- periment for nanoparticles either in either a frozen or melted state.Spectroscopy was performed with a 26-kHz, 31 mT/ μ 0field. The log-normal distributions of particle diameter and anisotropy constant are shown on the right by number in a representative 1000-particlesubsample. 094438-8MIXED BROWNIAN ALIGNMENT AND N ´EEL ROTATIONS . . . PHYSICAL REVIEW B 92, 094438 (2015) -20 -10 0 10 20<m> -1-0.8-0.6-0.4-0.200.20.40.60.81 A -20 -10 0 10 20-0.500.511.522.533.5 0% 71% 98% B(a) (b) H0 (mT/ 0) H0 (mT/ 0) diff (a.u.)<m> FIG. 12. (Color online) Magnetizations were simulated using a 40-kHz, 20 mT/ μ0field at room temperature to show how the (a) differential magnetic susceptibility in arbitrary units χdiff(arb. units) and (b) hysteresis loops are affected by increasing the percentage of particles aligned in the direction of the applied field. The size and anisotropy distribution are identical to that in other simulations. can be conceptualized by imagining a two-state system where the energy barrier to magnetization reversal is the highestwhen the magnetic field is aligned with the easy axis as in theparallel case. The result that the fluid samples displayed steeper hysteresis curves is potentially more surprising and can be interpretedwith simultaneous combinations of the N ´eel and Brown dynamics. It is possible to imagine that because the rotational timescales are comparable, the oscillating field might inducealignment of one of the easy crystalline axes. This effect isnot extreme—the simulations only needed 30% alignment insome cases—but it is absolutely noticeable in the data [see,for example, Figs. 6(a) and7]. The origin of the anisotropy is not fully clear. Magne- tocrystalline anisotropy is expected for these particles andmay be sufficient to explain the observed behavior, but onlytrue simultaneous in situ experiments such as high-resolution electron microscopy of liquid samples under applied fieldscould rule out the possibility of chaining. Moreover, it ishard to quantify the shape anisotropy of the particles fromTEM imaging, though in the data of Fig. 2it is clear they are quite monodisperse and any elongation is minor.Notwithstanding the fact that a TEM image renders a two-dimensional projection of a three-dimensional nanoparticle,a rough estimate of shape anisotropy of the particles wasmeasured from TEM images using IMAGEJ . Each particle measured for size determination was fit with an ellipse andthe ratio of the long axis to the short axis determined. Theresulting histogram was fit with a log-normal distributionto determine the median aspect ratio (1 .04±0.03). This equates to a typical elongation of approximately 1 nm for the26-nm-diam particles. We can estimate the anisotropy energydue to this minor elongation by considering the particles tobe prolate ellipsoids of revolution with axes a,b,c , where a>b =c. In this case, the critical field for switching due to shape anisotropy is H k=Ms(Nb−Na)[35], where Ms is the saturation magnetization (263 kA /m), and NaandNbare the demagnetization factors along the aandbdirections, respectively. For the measured aspect ratio of 1 .04±0.03, Hkis about 5.2 mT/ μ0. For magnetocrystalline energy Hk would be expected of the order K/M s,o r1 3m T / μ0, assuming K=3.4k J / m3; specifically, Hk=4K/3Msfor cubic systems, Hk=2K/M sfor uniaxial systems when the field is aligned antiparallel to the moment, and Hk=K/M sfor uniaxial systems when the field is aligned perpendicular to the moment,with the moment always assumed initially to be directed alongan easy axis [ 36]. Since the magnetocrystalline anisotropy energy is greater, it should dominate in this sample, but shapeanisotropy may contribute a slight uniaxial character. Themagnetocrystalline and shape anisotropies would be similarwith approximately 1.1 average particle aspect ratio. In addition to alignment of individual particles, chain formation can be thought of as contributing to the effectiveanisotropy. Saville et al. showed that magnetite particles coated with 20-kDa PEG (with M s=264 kA /m, core diameter of 24 nm, and hydrodynamic diameter of 170 nm) formed chainsaveraging 3 μm long were observed after applying a steady field of 270 mT/ μ 0for 30 s [ 37]. Chain formation may have occurred during the field-freezing experiments reported here,although it is less likely to have affected the MPS of liquidsamples, since only weak oscillating fields were applied andour MPS has a 50% duty cycle (the field is switched off for 0.5 s every 1 s). The magnetically optimized, monodisperse SPIOs used in this study showed highly nonlinear behavior, producing asharp magnetization response useful for sensing and imaging applications that use magnetic induction, such as MPI. With uniformly sized magnetic cores and well-defined hydrody-namic size, the samples were also ideal for studies of how particle behavior responds to variations in the drive field, since variations in response can be assumed to come fromuniform changes in the response of each particle, rather thanfrom multiple fractions of varying size, as in clustered-core particles like Resovist. 094438-9SHAH, REEVES, FERGUSON, WEA VER, AND KRISHNAN PHYSICAL REVIEW B 92, 094438 (2015) V . CONCLUSIONS This work arose from the observation that immobilized (frozen) SPIOs displayed distinctly different rotational dynam-ics than the same particles in a liquid state. This phenomenonwas well beyond the respective change in the relaxation time purely due to the temperature. It was previously expected that the particles would reorient their magnetic moment internally,with the N ´eel mechanism, and thus the local environment (e.g., the phase of the solution) should have had no impact. The largechange in the dynamics is equivalent to an effective uniaxialanisotropy that could be due to small chain formations orlocal reorientation of the particles themselves. For sphericalnanoparticles with cubic symmetry, only a slight physicalrotation, referred to here as Brownian alignment, is required toalign one of the magnetocrystalline easy axes along the appliedfield direction. The dynamics of the particles in a randomlyfrozen state were still different from the liquid-state dynamics.This indicates that even while N ´eel oscillations occur, a general alignment through Brownian rotation is possible in the liquidsample. All of these dynamical behaviors could be replicatedusing Langevin equation simulations. The partial alignment ofthe easy axes could be incorporated into the model through theeffective field of the Landau-Lifshitz-Gilbert equation. We also found that while the magnetic particles are predicted to havecubic anisotropy, a single effective axis with a large distributionover the anisotropy constant is adequate to account for theobservable dynamical effects. In the future, it appears wise to specifically characterize any magnetic particles used in dynamical applications, astypical equilibrium calculations of relaxation times alone areinadequate predictors of rotational mechanisms. We havedemonstrated that an initial Brownian alignment, leadingto an effective uniaxial anisotropy, plays an important rolefor magnetic nanoparticles conventionally thought to onlyrotate with the N ´eel mechanism. This result has far-reaching implications for emerging biomedical technologies such asMPI. ACKNOWLEDGMENTS This work was supported by NIH Grant No. 2R42EB013520-02A1, a UW/CGF commercialization grant,and a Commercialization Fellowship (R.M.F.). 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PhysRevB.103.054435.pdf

PHYSICAL REVIEW B 103, 054435 (2021) Domain-wall dynamics in multisegmented Ni /Co nanowires V . O. Dolocan* Aix-Marseille Université, CNRS, IM2NP UMR7334, F-13397 Marseille Cedex 20, France (Received 2 December 2020; revised 10 February 2021; accepted 15 February 2021; published 23 February 2021) The current-induced motion of transverse magnetic domain walls (DWs) in a multisegmented Co /Ni nanowire is investigated numerically. We find that the phase diagram current pulse length magnitude presents a rarediversity of behaviors depending on the segment’s length and material parameters. We show that by changingonly the pulse shape, in a range of parameters we obtain the controlled motion of the DW with or withoutpolarity change. The polarity change arises in the simplest case from the birth and propagation of an antivortexalong the width of the nanowire. The antivortex can be displaced over long distances depending on the pulsecharacteristics and boundary conditions. The systematic motion of the DW with polarity flip is found to be stableat room temperature. Moreover, by modifying the material parameters through alloying, the phase diagram canbe engineered, decreasing the depinning current and paving the way for storage or logic applications. DOI: 10.1103/PhysRevB.103.054435 I. INTRODUCTION An increasing number of magnetic nanostructures are stud- ied nowadays for their storage or logic properties. Functionaldevices are designed that use the magnetic domains as digitalbits which are displaced in a controlled manner inside thenanostructure. These devices take advantage of the domainwall (DW) motion induced by a magnetic field or a spin polar-ized electric current [ 1,2] in a flat or cylindrical nanowire. The central ingredient is the proper manipulation of the DW’s mo-tion between well-defined positions (pinning centers) in thenanowire in a reproducible manner [ 3,4]. Several approaches were used to tailor the DW energy landscape like artificialconstrictions or protuberances [ 5–9], bending [ 10], or local modifications of the magnetic properties like anisotropy or thesaturation magnetization [ 11–13]. Recently, multisegmented nanowires (sometimes referred to as multilayered nanowires), which are formed by alter-nating regions of different composition and structure alongthe longitudinal axis, were investigated for spintronic applica-tions. Ferromagnetic (FM) /FM [ 14,15]o rF M /nonmagnetic (NM) [ 16] multisegmented nanowires were studied for shift registers [ 17], magnetic nano-oscillators [ 18], and flexible spin caloritronic devices [ 19]. In most cases, cylindrical nanowires were fabricated by electrochemical deposition inalumina templates [ 20], which readily allows for modifica- tions of geometrical aspect or material properties. MetallicFM/FM multisegmented cylindrical nanowires composed of Co and Ni segments were shown experimentally to effi-ciently pin the vortex DWs at the Co /Ni interface [ 15]. It was shown that individual Co nanowires can be prepared assingle crystals with hcp structure and controlled easy-axis orientation [ 21]. A FM nanowire with segments of different materials and different orientations of the magnetization easyaxis can therefore be easily engineered. However, the complex *voicu.dolocan@im2np.frmagnetic states (multivortex) observed in these wires and theDW dynamics are not completely understood. In this paper, we address this issue by studying the system- atic motion of a magnetic transverse DW in a Co /Ni multiseg- mented nanostrip (flat nanowire) when submitted to ultrashortspin-polarized current pulses at low and room temperature.When the magnetocristalline anisotropy easy axis is orientedout of plane for the Co segments, a complex periodic pinningpotential is created in which an individual transverse DWmoves between pinning sites (the Ni regions) for a large rangeof geometrical characteristics (width /length /number) of seg- ments. The stray field generated at the interfaces between Coand Ni regions strongly deform the DW, which behaves likea string in a tilted washboard potential [ 22]. When a current pulse is applied, the DW is tilted out of plane (tilting angle ψ), which is usually suppressed by the effective wall anisotropyK ⊥. A small variation in K ⊥can cause an exponentially large change in ψat the segment edges [ 23] and, for certain pulse characteristics and material parameters, an antivortex appears[24] that can traverse the nanostrip width, switching the trans- verse DW polarity. More complex structures can appear as avortex-antivortex pair depending on the exchange energy vari-ation and boundary conditions at the interfaces. We show thatby varying only the current pulse shape, in a certain parameterrange, the transverse DW can be reliably displaced with orwithout polarity change with potential applications for storagedevices. Our results show that the systematic DW displace-ment between pinning regions with a polarity switch is stableat room temperature in a Co /Ni multisegmented nanowire. We also determine the impact of the material parameter vari-ation, for example, by alloying, on the depinning current andphase diagrams. A 1D model is developed that agrees with thenumerical one only in the parameter range where no vortex-antivortex structures appear. Our study brings physical insightinto DW motion in multisegmented nanowires and could leadto future theoretical and applied studies. This paper is organized as follows. In Sec. II, we present the numerical simulations and stochastic 1D model used to 2469-9950/2021/103(5)/054435(11) 054435-1 ©2021 American Physical SocietyV . O. DOLOCAN PHYSICAL REVIEW B 103, 054435 (2021) FIG. 1. (a) Multisegmented Co /Ni nanowire with an in-plane transverse DW. The Ni regions (40 nm length) are shaded and the Co regions (80 nm length) have a magnetocrystalline anisotropy along thezaxis. The arrows indicate the direction of the magnetization. The enlarged section shows the central regions with the transverse domain wall. (b) Expanded part of the multisegmented nanowire when the Co regions have a magnetocrystalline anisotropy along theyaxis. The length of the regions have been inverted: Co regions of 40 nm and Ni regions of 80 nm. The nanowire is 1300 nm long, 60 nm wide, and 5 nm thick. calculate the DW displacement. In Sec. III, we compute and investigate the phase diagram of the DW dynamics in aninfinite nanostrip at T=0 K and at room temperature. Dis- cussion and concluding remarks are presented in Sec. IV. II. MODEL The dynamics of the DWs is governed by the Gilbert form of the Landau-Lifschitz equation (LLG) with adiabatic andnonadiabatic spin-transfer torques (STTs) [ 25], ˙m=−γ 0m×Heff+α(m×˙m) −(u·∇)m+βm×(u·∇)m, (1) where mis the normalized magnetization, γ0is the gyro- magnetic ratio, u=jePμB/eMsis the spin drift velocity, P the spin polarization of conduction electrons (taken as 0.7in numerical simulations), μ Bthe Bohr magneton, and jethe applied current density. The temperature is added to the LLGequation in the form of a thermal field with zero average anduncorrelated in time and space. We study the dynamics of a transverse DW in a multiseg- mented nanostrip of Co and Ni regions. The type of DW inconfined nanostructures depends on the nanostrip dimensions[26]. The length of the regions /segments (along the longi- tudinal xaxis) is varied between 20 and 40 nm for Ni and between 50 nm and 90 nm for Co. The width (along yaxis) and the thickness of the nanostrip are kept constant at 60 nmand, respectively, at 5 nm, while the total strip length is variedaround the value of 1300 nm to ensure that the end regionsare always made of Co [Fig. 1(a)]. The following parameters are used for Ni: saturation magnetization M s=477 kA /m, exchange stiffness parameter A =8.6 pJ /m, and damping parameter α=0.05, and for Co: saturation magnetizationMs=1400 kA /m, exchange stiffness parameter A =28.1 pJ/m, damping parameter α=0.005, and uniaxial magne- tocrystalline anisotropy (MCA) K u=4.5×105J/m3[27]i f not specified otherwise. When the Co MCA easy axis is di-rected along the perpendicular zaxis, the transverse DW is pinned in the Ni regions for the geometrical dimensions ofthe regions chosen [see Fig. 1(a)], with the magnetization in plane. When the Co MCA is directed along the transversey-axis a single DW is no longer a stable state and the DWs are pinned in the Co regions as shown in Fig. 1(b).A st h e reliable control of the DW position is a prerequisite for mem-ory applications, only the case with the Co regions havingperpendicular z-axis MCA is studied below. A series of periodic spin-polarized current pulses are ap- plied along the stripe longitudinal axis ( xdirection) to displace the DW, which is initiated in the central Ni region. The currentpulse has a trapezoidal geometry with t r,ts, and t fthe rise, stable, and fall times [ 3]. The periodic pulses are separated by tz, the zero-current time which ensures that the DW reaches its equilibrium position before another pulse is applied. Thenonadiabatic parameter is set to β=2αif not specified other- wise. The DW dynamics was computed mainly using 3D micro- magnetic simulations with the MUMAX3 package [ 28]; some simulations were also conducted with the OOMMF package[29]. The strip was discretized into a mesh with a cell size of 2×3×2.5n m 3, inferior to the exchange length. A one- dimensional model [ 30] was also developed to understand the DW displacement in the multisegmented wire. The analyticalequations of motion used are based on the 1D model of theDW (collective coordinates: average DW center position X and azimuthal angle ψ)[31,32], (1+α 2)˙X=−αγ/Delta1 2μ0MsS∂E ∂X+γ/Delta1 μ0MsK⊥sin 2ψ +qpγ 2μ0MsS∂E ∂ψ+(1+αβ)u+ηψ−αηX, (1+α2)˙ψ=− qpγ 2μ0MsS∂E ∂X−γα μ0MsK⊥sin 2ψ −αγ 2/Delta1μ 0MsS∂E ∂ψ+qpβ−α /Delta1u+ηX+αηψ, (2) with/Delta1the DW width, K ⊥the transverse anisotropy, ηXand ηψrepresent stochastic Gaussian noise with zero mean value and correlations /angbracketleftηi(t)ηj(t/prime)/angbracketright=(2αkBT)/(μ0Ms/Delta1S)δijδ(t− t/prime).Eis the potential energy of the DW that includes the internal energy and the pinning energy. The azimuthal angleof the DW ψrepresents the conjugate momentum in the Lagrangian formulation. q= 1 π/integraltext dx∂xψ=± 1 represents the chirality or the topological charge of the DW and is related tothe direction of rotation of the in-plane magnetization whentraversing the DW and p=± 1 represents the direction of the magnetization at the DW center along the yaxis (width). The product Q=q·pis always equal to +1 for a head-to-head DW and to −1 for a tail-to-tail DW. In the present situation, the DW moves in an effective pinning potential that arises due to the magnetic parametersvariation between the segments. The 1D model still gives 054435-2DOMAIN-WALL DYNAMICS IN MULTISEGMENTED Ni /Co … PHYSICAL REVIEW B 103, 054435 (2021) FIG. 2. Micromagnetic phase diagrams (pulse stable time—pulse amplitude) of DW motion under a periodic current pulse in a multiseg- mented Ni /Co nanowire at T=0 K. The periodic pulse is formed of two identical current pulses of opposite amplitude (forward-backward motion) separated by a waiting time of 5 ns. The length of the Ni and Co regions is varied as—from the top row to the bottom row—the Ni region’s length increases from 20 nm (top row) to 40 nm (bottom row) in steps of 10 nm, while the Co region’s length increases from left column to right column from 50 nm (left column) to 90 nm (right column) in steps of 10 nm. The thickness of the nanowire is 5 nm and its totallength is around 1300 nm (varies slightly to always have Co regions at the ends). The observed DW motion is classified in bands numbered as follows: positive bands correspond to the DW moving in the direction of the electron flow, negative numbers to the DW moving contrary to the electron flow, zero state correspond to the DW staying pinned at initial position, and u to the unintended states in which the DW does notcome back to the initial state after the periodic pulse. good quantitative results (see Ref. [ 33]), if the azimuthal angleψstays well below π/4. The pinning potential can be determined from micromagnetic simulations together with thenumerical integration of the DW energy, allowing for contin-uous x-axis variation of the material parameters (K, A, M s, α,β)[12,13] contrary to the micromagnetic model where an abrupt transition between segments is considered exceptingthe exchange energy (the details of the 1D calculation arepresented elsewhere). One parameter that varies locally andcan introduce numeric artifacts near interfaces is the exchangeenergy calculation. In its usual implementations, the exchangeenergy is approximated between nearest or second-nearestneighbors or interpolated between nearest grid points withthe Neumann or Dirichlet boundary conditions [ 34]. The in- terfaces can lead to singularities in the magnetization due toa jump in the exchange stiffness Aand its effect was stud- ied for some time in connection with grain boundary effects[35,36]. The effective exchange between adjacent pairs of atoms is usually overestimated, particularly in soft magnetswhen considering the same effective exchange strength overthe interface (see Ref. [ 35] and references therein). In mi- cromagnetic packages, the effective exchange interaction isusually assumed identical on both sides of the interface andcalculated by default as a harmonic mean of the values oneach side of the interface. A reduced interface exchange maybe closer to reality as a real interface may have impuritiesor mixing phases or be amorphous and was considered be-low. The details of the exchange stiffness variation in theinterface region are described in the Supplemental Material[33].III. RESULTS In this section, we study the impact on the DW dynamics of the segment’s length and material parameter variation at zeroand room temperature. The phase diagram (stable time t s— current amplitude j e) obtained when periodic current pulses are applied shows a rich variety of stable states, includingsystematic displacement with or without DW polarity change. A. Influence of the segment’s length on the phase diagrams at T=0K To describe the DW dynamics in a Co /Ni multisegmented nanowire [Fig. 1(a)], we use the analysis previously used for the DW dynamics in a notched nanowire [ 3,4]. A transverse DW pinned in a Ni region in the center of the nanowire issubmitted to a series of periodic trapezoidal spin-polarizedcurrent pulses. The Ni regions represent the pinning sites forDW displacement. For a reliable control of DW propagation,the DW should move forward and backward to the sameposition after a periodic current pulse. The current pulses arevaried in length, amplitude, or shape and the DW motion isextracted after several periodic current pulses. The relativeDW position, after the series of periodic pulses, is representedin a phase diagram (stable time t s—current amplitude j e). The regular DW displacements were shown to form bands depend-ing on the pulse characteristics in notched nanowires [ 3,4]. In Fig. 2, the influence of the segments length variation on the phase diagram is shown at T=0 K. The Ni segments were varied between 20 and 40 nm (top row to bottom row), 054435-3V . O. DOLOCAN PHYSICAL REVIEW B 103, 054435 (2021) FIG. 3. DW displacement in a 40-nm Ni /80-nm Co segmented nanowire under a current pulse when t s=400 ps, j e=5.5A/μm2(a) or ts=1050 ps, j e=6.4A/μm2(b) was applied. Both cases correspond to a state +1. The Ni regions are indicated by darker colors. In panel (a), an antivortex is observed crossing the nanowire width (first row, blue circle) while the TDW changes polarity when moving to the nearest Ni region. In panel (b), the antivortex does not cross the nanowire width and is expelled on the same edge it entered. The TDW does not change its polarity when moving to the nearest Ni region. The second row in panel (a) shows the mxcomponent of magnetization, illustrating the DW deformation in the process. while the Co segments were varied between 50 and 90 nm (left column to right column). The bands were indexed asfollows: band 0 if the DW stays pinned in the initial Nisegment, band +1 if the DW was displaced periodically to the nearest-neighbor Ni segment in the direction of the STTand band −1 if the DW was displaced to the nearest-neighbor Ni segment in the opposite direction of the STT. The superiorbands ( ±-2 etc.) correspond to the DW displacement between the initial pinning site and the second neighboring site andso on. The u state represents an unintended state where theDW does not move periodically back and forth to the initialpinning site or depins completely. The pinning potential varies strongly with the segment’s length [ 33]. For a small length, the pinning barrier is small and the DW can move easily between the pinning sites as shownin Fig. 2(a). In this case, the DW depins easily and the bands are thin and a reliable control cannot be achieved. Increasingthe Ni or Co segment length by 10 nm strongly increases thestability and the band width (Figs. 2(b) and2(f)). The pinning potential of the Ni segments becomes important as soon asthe segment width increases above the DW width (25 nm inNi). Increasing the Ni width even more diminishes the slopeof the pinning potential without a clear modification of thepinning barrier height (details in Ref. [ 33]). Increasing the Co length increases the barrier height but not the pinning potentialslope. The depinning current has a complex dependence onthe Ni and Co segment length. It increases with increasingCo or Ni length as long as the DW motion forms regularbands in the phase diagram. Starting from (20-nm Ni) /(90- nm Co) (Fig. 2(e)), (30 nm Ni) /(70 nm Co) (Fig. 2(h)) and (40 nm Ni) /(60 nm Co) (Fig. 2(l)), the depinning does not start with regular DW bands but with nonregular (chaoticlike)pockets /branches of the bands that appear in the phase dia- grams. In these cases, the depinning current decreases withincreasing segment length.In the areas of the phase diagrams where the nonregular branching of the bands appear, the DW starts to depin due tothe apparition of an antivortex at the boundary between the Niand Co regions, similar to the depinning from a notch [ 24,37]. This depinning mechanism is detailed in Fig. 3for (40-nm Ni)/(80-nm Co) multisegmented nanowire [phase diagram in Fig.2(m) ]. The DW behaves like a string in a tilted washboard potential [ 22] and cannot be treated as 1D any longer. The position and azimuthal angle depend on the ycoordinate. It was shown in Ref. [ 23] that the azimuthal angle ψof a 2D DW segment varies as a solution of a Schrodinger-type equation,with the DW anisotropy K ⊥playing the part of the difference between the potential barrier energy and the particle energy.The exponential-like solutions of the Schrodinger equationapply to the azimuthal angle, and a small change in the DWanisotropy at the lower and upper edges induces an expo-nential increase of ψat the boundaries. The DW anisotropy changes drastically at the boundary between the Ni and Coregions, therefore a tilting of the azimuthal angle out of theplane is favored. The antivortex appears at the lower edgewhere the DW center first touches the segment’s boundary.The antivortex may be followed by other complex structures(vortex-antivortex pair, half antivortices) depending on theeffective exchange strength over the interface [ 33]. The nucleation and dynamics of the antivortex are more complex here than in the case of a transverse DW movementabove the Walker breakdown in an homogeneous strip. Asdetailed in Figs. 3and4(see movie in Ref. [ 33]), the DW first pinned in the Ni region starts to move after the applicationof a current pulse without deformation. After just 20 ps, theDW begins to deform, the lower edge moving faster than theupper edge (contrary to the motion in a homogeneous strip),as shown in the position of the DW center on the top andbottom edges in the first row of Fig. 4. The lower part of the DW center first reaches the Ni /Co boundary, passing slightly 054435-4DOMAIN-WALL DYNAMICS IN MULTISEGMENTED Ni /Co … PHYSICAL REVIEW B 103, 054435 (2021) FIG. 4. Time variation of the DW upper (dotted line) and lower (full line) center position (first row) and azimuthal angle (second row) in a 40-nm Ni /80-nm Co multisegmented nanowire. The first column corresponds to a forward followed by a backward pulse with characteristics ts=400 ps, j e=5.5A/μm2s e p a r a t e db yt z=5 ns, the second column to forward-backward pulses with characteristics t s=1050 ps, j e= 6.4A/μm2,tz=5 ns, the third column to forward-backward pulses with characteristics t s=250 ps, j e=8.5A/μm2,tz=5 ns, and the last column to to forward-backward pulses with characteristics t s=800 ps, j e=6.4A/μm2,tz=5 ns. The last row shows the temporal antivortex trajectory (time variation from dark to lighter colors) with the arrows indicating the current pulse end. The horizontal dotted lines in the first row and the vertical dotted lines in the last row illustrate the Ni regions edges. The vertical dotted lines in the first row pinpoint the current pulse end. in the Co region and being pulled back (also the upper edge moves back in the same time). The magnetization angle startsturning out of plane in the −zdirection at the lower and upper DW edges and along all the DW center line (nanostrip width)as seen in Fig. 3(a). An antivortex is formed at the lower edge with its core polarity along the −zdirection (not to be confounded with the transverse DW polarity which is alongthe±ydirection), while the magnetization of the upper edge turns gradually along the in-plane direction (after reachingan angle of −24 ◦for t s=400 ps, j e=5.5A/μm2or−32◦ for t s=1050 ps, j e=6.4A/μm2). For currents above 7.9 A/μm2, an antivortex can also form at the upper edge, but these cases are fewer and form smaller pocket bands than theones found below the current values that are discussed here.In the case of a total pulse duration of 0.41 ns for a currentamplitude of 5.5 A /μm 2[Figs. 3(a) and4, first column], the antivortex moves slowly to the right and up across the stripwidth with the lower and upper parts of the DW still pinnedon the Ni /Co boundary. At the pulse end, the upper edge of the DW moves continuously to the right while the lower edgeis pushed in the other direction, strongly deforming the DW.The DW center line has a wavelike dynamical deformation.The antivortex at the center of the wall stays pinned and movesfor a short time only in the +ydirection. The lower DW edge becomes much larger (still being in the Ni region) and at thesame time the upper edge shrinks (being in the Co region). Theantivortex begins to move to the right and up (see the trajec-tory in the bottom row of the Fig. 4), following the upper edgeand being attracted to it, and the lower DW edge moves to the right displacing the DW now with opposite polarity to the nextNi region. Bursts of spin waves are ejected along the edges asthe antivortex comes out of the nanowire. If the current pulseis applied in the opposite direction, the antivortex will format the upper edge (image not shown) with a polarity along the+zdirection and the DW will travel to the next Ni region to the left, changing its polarity. As discussed below (Sec. III C ), the DW displacement with polarity switching is stable at roomtemperature. A particularity of the DW motion in the Ni /Co multiseg- mented nanowire is the possibility of systematic displacementwith or without polarity switching by changing the pulse char- acteristics (magnitude and length) for all types of interfacesconsidered. As observed from the phase diagrams in Fig. 2, when the length of the segments is increased, the homoge-neous bands are displaced to the right until they disappearand on the left the chaoticlike pockets of bands appear. In thedifferent pockets corresponding to the +1 state, the antivortex always appears, but it only traverses the nanowire width forlow pulse duration. This constitutes a unique case, where the motion of a transverse DW between pinning sites takesplaces with or without polarity switching depending on thepulse length. Figure 3(b) and the second column of Fig. 4 (movie in Ref. [ 33]) display the case when the pulse length is long enough (t s=1050 ps, j e=6.4A/μm2but same behavior for j e=5.5A/μm2) for the antivortex to traverse the next-nearest Ni region completely and come close to the 054435-5V . O. DOLOCAN PHYSICAL REVIEW B 103, 054435 (2021) second to the right Ni region before the pulse ends. The an- tivortex comes out at the lower edge where it entered and theDW is displaced to the next pinning region without polarityswitching, contrary to the shorter pulse case. The antivortexstrongly deforms when passing the Co /Ni and Ni /Co bound- aries with spin-wave emissions (or half antivortices) along theedges. The antivortex moves almost linearly in the Ni andin the middle of the Co regions and shows an inertial dis-placement after the pulse end continuing in the ±y direction depending on its previous displacement. Therefore, timing thepulse length just right when the antivortex goes up or downin the Co regions (close to the Ni boundaries) will makethe antivortex traverse the nanowire width or not and variatethe DW polarity. The last column of Fig. 4shows another possibility of DW polarity switching due to the antivortextraversing the nanowire, where the DW travels to the secondpinning Ni region to the right ( +2 state). As the pulse ends when the antivortex moves upward, it continues to move untilgoing out at the upper edge, switching the DW polarity. As shown elsewhere [ 3,4,38–40], the DW shows auto- motion (inertial transient displacement) due to the currentpulse shape. Here, the automotion appears in some phasediagrams (negative bands) with an example being shown inthe third column of Fig. 4. For a pulse width t s=250 ps and je=8.5A/μm2, the DW is first pushed to the right (in the direction of the STT) and afterward is pulled back by thepotential well. An antivortex is created at the upper edge anda second one at the lower edge, joining across the nanowirewidth and both moving to the left at the pulse end. Theantivortices separate and the upper one is ejected while thelower one enters the nanowire, oscillates in size, emitting spinwaves, and comes out of the nanowire. Both antivortices comeout on the same edge they entered, while the DW moves to thenext Ni region to the left, keeping its polarity. The motion of a vortex or antivortex in a nanowire can be described by the Thiele equation [ 30,41], −G×dX dt−ˆDdX dt+∂E ∂X=0, (3) where X=(X,Y) is the antivortex core position, E(X)i st h e potential energy of the antivortex, Gis the gyrovector, and ˆDis the damping tensor. The first term of Eq. ( 3) suggests a Magnus-type force that acts in a perpendicular directionto the antivortex velocity (gyration) pushing it to the edges.The gyrovector is along the zaxis and is expressed as G= − 2πμ 0Msqp tˆz γ, with qthe topological charge ( =− 1 for an an- tivortex), pthe polarization of the antivortex core ( ±1 along thezaxis), and tthe sample thickness. Knowing the potential energy of the antivortex, its motion can be calculated analyt-ically [ 42,43] and shows a gyrotropic motion under applied magnetic field or current. In the multisegmented Ni /Co nanowire, the gyrovector is oriented in the −zdirection for the initial antivortex. To understand better the motion of the antivortex, a longer currentpulse with t s=5 ns is applied (Fig. 5). The Oersted field created by the current pulse is not taken into account althoughit may become important for longer pulses, influencing theantivortex motion [ 42]. The displacement of the antivortex along the nanowire follows a clockwise gyrotropic motion ineach Co region crossed. The gyration appears to stop in the NiFIG. 5. Temporal antivortex trajectory (time variation from dark to lighter colors) in a 40-nm Ni /80-nm Co multisegmented nanowire for a 5-ns current pulse with magnitude j eof 5.5 A /μm2in (a) and 6.4 A/μm2in (d). The temporal evolution of the demagnetizing (full line) and exchange (dashed line) energies and the antivortex yposi- tion corresponding to panel (a) are shown in (b) and (c), respectively.The vertical dotted lines in (a) and (d) illustrate the Ni regions edges. The nanowire is 1300 nm long. regions and to restart again when the antivortex reenters the Co region. The average velocity of the antivortex (not shown) 054435-6DOMAIN-WALL DYNAMICS IN MULTISEGMENTED Ni /Co … PHYSICAL REVIEW B 103, 054435 (2021) increases to around 400 m /s when approaching a Ni region (on the downward ymotion) and decreases to around 250 m/s when the antivotex leaves the Ni region (on the upper ymotion). As seen in Fig. 5(b), the demagnetizing energy and the exchange energy have a similar oscillatory behavior eachtime the antivortex moves across the Co /Ni region correlating with its motion [compare Figs. 5(b) and5(c)]. The antivortex pursues the potential energy shape, implying that, due to thedemagnetizing and exchange energy variation, the gyrotropicmotion is lessened whenever the antivortex comes close toaC o/Ni boundary (the demagnetizing field changes signs at each Ni /Co boundary). If the magnitude of the current pulse is too high [see Fig. 5(d)], the gyroforce (which depends on the M sthrough the gyrovector and on velocity) is strong enough to push the antivortex out of the wire each second Co regionbefore the antivortex could reach the Ni region. The antivortexis recreated again (comes back into the wire) at the next Ni /Co boundary and this periodic motion continues until it reachesthe end of the wire. It should be pointed out that the antivortexdoes not change its core polarization during its motion in themultisegmented nanowire (for the pulse characteristics used). B. Influence of material parameters To control and understand the depinning current in the Ni/Co multisegmented nanowire, we altered the potential en- ergy landscape in which the DW moves by modifying themagnetization, the exchange stiffness, or the anisotropy ofthe Co regions (along with the damping constant αand the nonadiabatic parameter β) and the effective exchange at the Ni/Co interfaces. While the magnetization of a thin film is measured by magnetometry, the exchange constant is esti-mated indirectly assuming particular models [ 27,44], leading to a higher incertainty. Therefore, we varied the exchangeconstant Aon a larger range from 15 to 50 pJ /m (higher values suggested recently [ 45,46]) and the magnetization M s between 1300 and 1500 kA /m for the Co regions (details in Ref. [ 33]). We observed that the depinning current varies oppositely to the depinning field when varying Aor M s,t h e depinning current decreasing with increasing both Aor M sup to a threshold value. Lowering the magnetization has a higherimpact on the potential barrier height for the DW, while thevariation of the exchange constant influences more the formof the potential well. In Fig. 6, we present the consequences of the exchange stiffness constant variation (exchange lengthand DW width scale with√ A) on the phase diagrams for a multisegmented 40-nm Ni /80-nm Co nanowire. For the low- est exchange constant value [Fig. 6(a)], the regular bands are still visible to the right of the phase diagram but large pocketsof disordered bands appear to the left with a large −1 pocket band. Increasing further on the exchange stiffness constant,the regular bands shift to the right of the phase diagramand disappear, leaving only the branching pocket bands thatcorrespond to the antivortex presence. For the largest Avalue, the pocket bands seem to become more regular for lowercurrent values as the +1 band is observed continuously up to a pulse length t s=850 ps, followed by superior bands (still a+1 band branches to higher currents). The dynamics of the DW and the antivortex also changes compared with the casepresented above [shown in Fig. 6(c), identical with Fig. 2(n)]. FIG. 6. Micromagnetic phase diagrams of DW motion under a periodic current pulse in a multisegmented 40-nm Ni /80-nm Co nanowire at T=0 K. Only exchange stiffness constant Ais varied for the Co regions from 15 pJ /m (a) to 20 pJ /m (b), 28.1 pJ /m( c ) ,3 5 pJ/m( d ) ,4 0p J /m (e), and 50 pJ /m (f). The observed DW motion is classified in bands numbered as follows: positive bands correspond to the DW moving in the direction of the electron flow, negativenumbers to the DW moving contrary to the electron flow, zero state correspond to the DW staying pinned at initial position, and u to the unintended states in which the DW does not come back to the initialstate after the periodic pulse. In this case, the branch of the +1 band appearing at lower currents (to the left of the phase diagram) does not correspondanymore to the DW movement with polarity change. Insidethis branch, the DW moves to the next ( +1) Ni region with or without polarity change, as the two different branches seenin Fig. 6(c) (for A=28.1p J/m) coalesce. For a current of j e=4.0A/μm2, the DW moves with polarity change up to a pulse length of t s=750 ps from which the motion takes place without DW polarity change [ 33]. The antivortex with core polarity in the −zdirection that is created at the lower edge traverses the nanowire width faster and disappears at theupper edge in the +1 Ni region, while the DW moves to the +1 Ni region and remains pinned. Above t s=700 ps, another antivortex with core polarity in the +zdirection is formed at the upper edge in the +1 Ni region and, for pulses between 750 ps and 850 ps, traverses the nanowire width almost verti-cally in the Ni region (with the DW staying pinned) changingthe DW polarity to the initial one ( +ydirection). For longer pulses, the second antivortex moves toward the +2N ir e g i o n before going out of the sample, displacing the DW to the +2 Ni region. The DW motion with polarity change is still stableat room temperature (see next section) even for the increasedstiffness constant. When the effective exchange energy is lowered, the DW motion with polarity change still arises, with the antivortexmoving along the Ni /Co interface and reversing the po- larity of the DW. The bandlike states become larger andmore ordered and the systematic DW motion is more stable 054435-7V . O. DOLOCAN PHYSICAL REVIEW B 103, 054435 (2021) [33]. When the effective inter-region exchange is increased (considering the same value on both sides of the interface),the antivortex traverses the nanowire width faster and othercomplex structures appear as vortex-antivortex pairs or half-antivortices along the edges. The state bands are intermixedand less regular, but the DW motion with polarity changingstill occurs depending on the pulse characteristics. When varying the magnitude of the magnetocristalline anisotropy constant K ufor the Co segments, the phase di- agram is modified greatly [ 33]. For a 40-nm Ni /80-nm Co wire, when the anisotropy constant is lowered to a value of5×10 4J/m3, from 4 .5×105J/m3, the regular bands are still visible with small band pockets to the left, almost identicalto the case of shorter Co segments [of 60 nm, shown inFig. 2(l)]. The states /bands are displaced to the left (lower currents) in the phase diagram. Even when the the anisotropyconstant of the Co regions is turned off (equal to zero as inthe Ni regions), the phase diagram is similar to the one ofthe reduced anisotropy constant with band pockets between3.1 and 8.2 A /μm 2and regular bands above. However, in the band pockets, the DW motion with polarity switchingdoes not happen for all the points in the same band pocketand pairs of vortex-antivortex can enter the nanowire even forlow currents, making the DW motion with or without polarityswitching unstable. When the anisotropy constant is increasedto a value of 1 ×10 6J/m3, the regular bands and the pocket bands almost disappear, as the DW is highly distorted, beingpinned at the boundary of the Ni /Co regions and transforming in several perpendicular DWs when moving across the Co re-gions (uniaxial anisotropy higher than demagnetizing energy).The systematic DW displacement is no longer stable in thiscase. Altering the pulse shape, varying the rise time or the fall time has the same impact on the phase diagrams as for anotched nanowire [ 3], displacing the states at higher currents and does not improve the dynamics discussed here. To be able to compare to other real materials, whose param- eters were already measured, we replace the Co regions withaC o 50Ni50alloy with the following parameters: M s=1070 kA/m,A=22.7p J/m, and K u=7.5×104J/m3[44]. The damping constant αis varied between the values of Ni and Co [47,48]. Alloying the Co regions with Ni decreases the DW potential barrier between the regions and thus it should lowerthe depinning current. The results are presented in Fig. 7. For a 30-nm Ni /50-nm Co 50Ni50multisegmented nanowire [Fig. 7(a)], the depinning current decreases to 1.9 A /μm2 from 2.8 A /μm2for the case of pure Co regions [Fig. 2(f)], while the bands become larger and more regular. The samesituation is observed for the 20-nm Ni /60-nm Co 50Ni50mul- tisegmented nanowire, with the depinning current being 2.2A/μm 2(as compared to 3.4 A /μm2for pure Co). Varying the segment length, we observe that the pocketlike bands dueto the presence of the antivortex appear for longer segmentlengths. The regular bands are still visible for the 40-nmNi/80-nm Co 50Ni50mulstisegmented nanowire (Fig. 7(c)), even when a lower damping parameter (equal to the one ofCo) is used for Co 50Ni50segments (Fig. 7(d)). In this case, for both damping parameters, the same behavior is observedin the pocket bands as for the Ni /Co multisegmented wire: For the lower t spockets of the +1 band, the DW changes FIG. 7. Micromagnetic phase diagrams of DW motion under a periodic current pulse in a multisegmented Ni /Co50Ni50nanowire at T=0 K. The length of the regions varies as (a), 30-nm Ni /50-nm Co50Ni50; (b), 20-nm Ni /60-nm Co 50Ni50; (c), (d) 40-nm Ni /80-nm Co50Ni50.F r o m( a ) – ( c ) , α=0.05 and β=2αfor both region types, while in (d) αCo50Ni50=0.005. The observed DW motion is classified in bands numbered as follows: positive bands correspond to the DW moving in the direction of the electron flow, negative numbers to theDW moving contrary to the electron flow, zero state correspond to the DW staying pinned at initial position, and u to the unintended states in which the DW does not come back to the initial state afterthe periodic pulse. polarity when moving to the +1 pinning site while, for larger t spockets of the +1 band, the DW does not change its polarity when moving to the next pinning site (the an-tivortex enters and comes out of the nanowire on the loweredge). C. Temperature dependence The temperature is taken into account by introducing a stochastic thermal field in the effective field term of the LLGequation in both the precession and damping terms [ 49]. The effect of temperature was computed micromagneticallymainly for the +1 band on 100 realizations per point. A symmetric current pulse (t r=tf=5 ps) was applied after an initial relaxation time of 5 ns, followed by another relaxationtime of 5 ns. These results are valid at low temperatures com-pared to the Curie temperature of materials, as the saturationmagnetization is considered constant. At larger temperatures,the alternatives are the Landau-Lifschitz-Bloch equation or anatomistic model [ 50,51]. In Fig. 8, the probability distributions at T=293 K for positioning the DW to the nearest notch ( +1 band) for dif- ferent Co /Ni or Co 50Ni50/Ni multisegmented nanowires are shown. The probability distribution for a regular +1 band (no antivortex present) is shown in the first two columns,while in the last column the probability distribution when theantivortex appears and traverses the nanowire. In Fig. 8(a),f o r a 30-nm Ni /50-nm Co nanowire, the maximum of the proba- bility distribution is of 87% obtained for a pulse with t s=150 ps and j e=6.1A/μm2. The maximum of the probability distribution increases to 96% for the multisegmented 20-nm 054435-8DOMAIN-WALL DYNAMICS IN MULTISEGMENTED Ni /Co … PHYSICAL REVIEW B 103, 054435 (2021) FIG. 8. Probability of DW motion in +1 bands at T=293 K for a multisegmented nanowire of Ni and Co or Co 50Ni50regions: (a) 30-nm Ni/50-nm Co, (b) 20-nm Ni /60-nm Co, (c) 40-nm Ni /80-nm Co, (d) 30-nm Ni /50-nm Co 50Ni50, (e) 20-nm Ni /60-nm Co 50Ni50, and (f) 40-nm Ni/80-nm Co 50Ni50. Ni/60-nm Co nanowire [Fig. 8(b)], obtained for the pulse characteristics j e=9.1A/μm2and t s=100 ps. The maxi- mum probability increases to 100% for the multisegmented40-nm Ni /50-nm Co nanowire (image not shown) at the same pulse characteristics. In the case of a Ni /Co 50Ni50wire, the maximum of the distribution probability is of 96% for j e=8.2 A/μm2and t s=100 ps for 30-nm Ni /50-nm Co 50Ni50seg- mented wire [Fig. 8(d)], and of 97% for j e=6.1A/μm2and ts=150 ps for 20-nm Ni /60-nm Co 50Ni50segmented wire [Fig. 8(e). In the case of DW displacement with polarity change, the maximum of the probability distribution is of 100% for40-nm Ni /80-nm Co multisegmented nanowires [Fig. 8(d)] for several points centered on j e=5.5–6.1A/μm2and t s= 350–400 ps (also obtained for other values of effective ex-change). This indicates that the systematic DW motion withpolarity change is stable at room temperature in a reason-able parameter range and can lead to practical applications.For the regular DW displacement without polarity change,the maximum of the probability distribution is reached forj e=6.4A/μm2and t s=1050 ps and is of 82% in the same multisegmented nanowire. The motion with polarity changeis more stable at room temperature than the motion withoutfor the chosen parameters. The maximum probability dimin-ishes to 75% in the second band and to 28% for the −1 band (not shown). The 100% maximum probability for DWmotion with probability change is also realized for the samemultisegmented wire with an increased stiffness constant A of 50 pJ /m for the Co regions for slightly lower pulse values (j e=5.2–5.5A/μm2and t s=350–400 ps). A high probabil- ity of DW displacement (96%) is also obtained for a 40-nmNi/70-nm Co wire at j e=5.5A/μm2and t s=400 ps for the+1 band, corresponding to polarity changing DW motion, similarly as above. On the contrary, for the 40-nm Ni /80-nm Co50Ni50segmented wire [Fig. 8(f)], the maximum rate of success of 100% is realized for the DW motion without polar-ity change around j e=6.1A/μm2and t sbetween 1100–1150 ps, while the success rate decreases to 68% for the polaritychanging motion (j e=6.4A/μm2and t s=500 ps). This behavior can be understood by the larger pocket of the +1band (at lower t s), corresponding to polarity changing motion, for the Co /Ni wire than for the Ni /Co50Ni50wire. IV . DISCUSSION AND CONCLUSIONS A multisegmented Ni /Co nanowire constitutes a particular case in which a transverse magnetic DW can be displacedsystematically between well-defined pinning sites (Ni seg-ments), with or without polarity switching between initial andfinal positions. This system brings an additional degree offreedom (the DW polarity) to the DW motion compared to thehomogeneous nanowire with artificial constrictions studiedextensively [ 3]. Depending on the particular geometric dimen- sions of the segments and material parameters as discussedbelow, the DW can be displaced with polarity flip controlledby the current pulse characteristics. The DW polarity flip isprimarily due to the birth and propagation of an antivortexalong the DW width similar to the DW motion above theWalker breakdown in homogeneous nanostrips. The antivor-tex is created at the boundary between the Ni /Co regions and has in part the expected gyrotropic motion governed bythe complex potential along the wire. The polarity switchis determined by the current pulse length (and magnitude),the antivortex keeping its previous motion direction afterthe pulse end. The antivortex can be displaced long dis-tances along the multisegmented nanowire having a sinusoidalmotion depending on the boundary conditions and effectiveexchange. Several parameters play an important role in the emergence of complex textures in the Ni /Co multisegmented nanowire. The total energy of the system comprises the exchange energy,the anisotropy energy which for Co segments is given by theeffective anisotropy constant K eff=Ku−Nzμ0M2 s/2, the ef- fective DW transverse anisotropy K ⊥sin2ψ[52] where K ⊥= Ky−Kxfor the Co regions with perpendicular anisotropy and K⊥=Kz−Kyfor the Ni regions with in-plane anisotropy (Ki—effective anisotropy constants) and the pinning energy. It was shown previously that a step in the anisotropy (K eff) [12] pins the DW and that the depinning field depends on the anisotropy step. Under an applied polarized current, the 054435-9V . O. DOLOCAN PHYSICAL REVIEW B 103, 054435 (2021) stable Neel wall in the Ni region will start to move and tilt (increasing ψ). The rotation anisotropy barrier to the tran- sition to a Bloch wall depends on the rotation anisotropyconstant K ⊥, the DW width /Delta1, and the azimuthal angle ψ [13]. This rotation anisotropy barrier suppresses the tilting of the DW when K ⊥is large [ 22,23]. In the studied case, at the Ni /Co boundary there is a large fluctuation in K ⊥due to the large variation in the saturation magnetization and thesuppression of the tilting no longer applies. The DW trans-verse anisotropy constant K ⊥fluctuation is a main ingredient that leads to the birth of an antivortex and to the DW mo-tion with polarity switching. We also observed that, for thesegment lengths studied, the antivortex starts to emerge whenthe transverse anisotropy K ⊥changes sign for the Co regions. In the opposite case, when the rotation anisotropy barrier istoo low (increasing K uin Co) the DW starts to rotate and transforms from Neel to Bloch wall when moving, leading toa nonreliable systematic movement between pinning regions.Even though the antivortex emerges without a perpendicularmagnetocrystalline anisotropy (PMA) present (K u=0i nC o ) , the presence of a moderate PMA favors the emergence andstability of a singular antivortex at lower currents and supportsthe large +1 band pockets with or without a homogeneous DW polarity switching. Another principal ingredient is theexchange energy variation between segments, as detailed inpart in Fig. 6: A moderate exchange constant Apreserves the different +1 band pockets while an increase barrier will collapse the different pockets and reduce the possibility ofDW motion with or without polarity switching. A low effec-tive exchange stiffness at the Ni /Co interfaces stabilizes the DW motion and diminishes the apparition of other complextextures, leading to larger state bands [ 33]. In the case of only an adiabatic STT (nonadiabatic parame- terβ=0 in all segments), an antivortex still forms (see figure in Ref. [ 33]), but for 40-nm Ni /80-nm Co segments, the lower and upper +1 band pockets collapse and form only one band with very few states without DW polarity motion reducing theapplications. The DW motion with polarity switching between neigh- boring pinning segments ( +1 band) is room-temperaturecompatible as the success rate is 100% in a reasonable current range depending on the segment’s length. For 40-nm-longNi segments, the success rate is only 76% for DW motionwith polarity flip and 71% without polarity flip for 60-nmCo segments. The maximum probability increases systemat-ically with the Co segment length, being of 96% and 88%with and without polarity switch for 70-nm Co segments andup to 100% and 92% for 90-nm Co segments. The room-temperature stability paves the way for possible memoryapplications. Our results predict that the best possible implementation for memory devices (like the racetrack memory) is obtainedfor Ni segments with widths comparable with the DW widthhaving neighboring segments with materials parameters notvery different from those of Ni having a lower variation at theinterfaces. In this respect, a Co-Ni alloy with a magnetization,exchange stiffness, and anisotropy closer to the Ni values (anda lowered effective exchange at the interface) reduce the ap-parition of complex textures (like antivortices or vortices) andstabilize the DW motion, leading to lower depinning currentsand larger bands, increasing the range of reliable systematicDW displacement. This could be the case in real structures,where the grain boundaries are far from perfect. In summary, transverse DW motion with or without polar- ity switching was shown to occur in multisegmented Ni /Co nanowires depending only on the current pulse shape. Thepolarity switching is mainly due to the emergence of complextextures that traverse the nanowire width. The motion is room-temperature compatible and can be engineered by alloying.The additional degree of freedom in the DW motion can openthe way for logical or memory devices based on the informa-tion encoding into the magnetization direction of a DW. ACKNOWLEDGMENTS This work was granted access to the HPC resources of Aix- Marseille Université financed by the project Equip@Meso(ANR-10-EQPX-29-01) of the program Investissementsd’Avenir supervised by the Agence Nationale pour laRecherche. [1] D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, a n dR .P .C o w b u r n , Science 309, 1688 (2005) . [2] S. S. P. Parkin, M. 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PhysRevB.88.064402.pdf

PHYSICAL REVIEW B 88, 064402 (2013) Double resonance response in nonlinear magnetic vortex dynamics J.-S. Kim and M. Kl ¨aui* Institut f ¨ur Physik, Johannes Gutenberg-Universit ¨at Mainz, 55128 Mainz, Germany M. V . Fistul Theoretische Physik III, Ruhr-Universt ¨at Bochum, 44801 Bochum, Germany J. Yoon and C.-Y . You Department of Physics, Inha University, Incheon 402-751, Republic of Korea R. Mattheis Institut f ¨ur Photonische Technologien e.V ., 07745 Jena, Germany C. Ulysse and G. Faini Phynano Team, Laboratoire de Photonique et de Nanostructures, CNRS, 91460 Marcoussis, France (Received 7 November 2012; published 5 August 2013) We present experimental evidences for the dynamical bifurcation behavior of ac-driven magnetic vortex core gyration in a ferromagnetic disk. The dynamical bifurcation, i.e., appearance and disappearance of two stabledynamical states in the vortex gyration, occurring as the amplitude of the driving Oersted field increases toB Oe>Bcr Oe, manifests itself in a double resonance response in the dependence of homodyne the dc-voltage signal on the frequency ωof the applied microwave current. We find that the frequency range δωbetween the two resonance features strongly increases with the excitation power. Our analysis based on the model of a lowdissipative nonlinear oscillator subject to a resonant alternating force is in good agreement with the experimentalresults. This allows us to determine quantitatively key parameters of magnetic vortex dynamics, i.e., the criticalvalue of the driving Oersted field B cr Oefor nonlinear dynamics to occur, the resonant frequency, and the quality factor as well as damping of the magnetic vortex gyration. DOI: 10.1103/PhysRevB.88.064402 PACS number(s): 75 .78.−n, 05.45.−a, 74.25.Ha, 74 .40.De Nonlinear effects are ubiquitous in nature. The model of a nonlinear oscillator has been used to explain fascinatingdiverse phenomena in various fields of physics. 1The dynamics of a nonlinear oscillator becomes especially interesting inanonautonomous case when a resonant alternating force is applied. Well-known examples of resonant nonlinear phe-nomena are the “foldover effect”, 2multiphoton resonances and antiresonances,3,4and resonantly induced suppression of the potential barrier,5just to name a few. Recently the dynamical bifurcation between two oscillating states hasbeen experimentally observed in resonantly driven Josephsonjunctions. 6This effect has been used for amplifying a signal in a so-called Josephson bifurcation amplifier.7 In patterned thin film ferromagnets, diverse nonlinear dynamical effects have been identified in the motion of a singledomain wall. Interesting examples of this are thermally as-sisted and quantum depinning of a magnetic domain wall. 8,9In the presence of applied alternating current, such phenomena asstochastic resonance, 10the microwave field induced resonant dip of the magnetic depinning field,11and resonant amplifica- tion of magnetic domain wall motion12,13have been observed. A particularly exciting and well-controlled nonuniform magnetization configuration is the magnetic vortex state , which has curling in-plane magnetization (chirality) around anout-of-plane vortex core (VC). 14The defects distributed in the film generate a localized vortex pinning, and the disk geometrygenerates a circular confining potential. The confining poten-tial in combination with the application of in-plane magneticfield allows one to shift the equilibrium position of a VC, and the various oscillation (gyrotropic) states of a VC can be easilyexcited by, e.g., the application of an alternating spin-polarizedcurrent. 15–17Moreover, employing a microwave current recti- fying effect, i.e., the homodyne detection scheme,11,18allows one to quantitatively study the gyrotropic motion of a VC.Indeed, in the presence of an applied alternating currentI=ηcos(ωt), the oscillating anisotropic magnetoresistance (AMR) reads as R AMR=A(ω) cos[ωt+φ(ω)], where A(ω) andφ(ω) are the amplitude of the oscillatory VC gyration and the phase shift between the injected microwave current andAMR, accordingly. The homodyne voltage signal can bewritten as V DC∝A(ω) cos(ϕ). In the linear regime, as the amplitude of microwave current ηis small, the homodyne dc- voltage signal displays the usual resonant amplification, wherethe resonance frequency ω 0is determined by the confining and pinning potential.18The amplitude of the VC oscillations grows with η, and, in general, due to the anharmonicity of the potential, the resonance frequency ω0also varies with η.18In a strongly nonlinear regime as η>η crthe gyrotropic motion of the VC has to display the dynamical bifurcation , i.e., appear- ance and disappearance of two dynamical oscillating states.2,19 In this paper, we present experimental evidence of such nonlinear behavior. We show that the dynamical bifurcationmanifests itself in a peculiar double resonance response in the dependence of homodyne dc-voltage signal on the frequencyω. Due to an inhomogeneous distribution of the applied alternating current along the film, the excitation power ∝η 2 064402-1 1098-0121/2013/88(6)/064402(7) ©2013 American Physical SocietyJ.-S. KIM et al. PHYSICAL REVIEW B 88, 064402 (2013) -100 -50 0 50 100-80-60-40-200204060ΔΔR (mΩ) Applied field (mT) 1 μm V IAC VDC Bias-T x y θ(a) (b) FIG. 1. (Color online) (a) SEM image of the magnetic disk and electric contacts, and schematic configuration of homodyne detection. The distance between two electric contacts is about 500 nm. (b) Results of AMR measurements as a function of the external field.The blue (red) line is the result for applied in-plane magnetic field parallel to the current ( B ext//I) [applied field perpendicular to current (Bext⊥I)]. can be tuned by varying the VC equilibrium position, i.e., by the application of an in-plane magnetic field Bext. We quantify our experimental results by determining the dependence ofthe frequency range between two resonant features, δω,o n the amplitude of the microwave current η. We analyze our result using the model of a low dissipative nonlinear oscillatorsubject to a resonant alternating force. Our analysis is in goodagreement with experimental observations, and this modelalso allows us to quantitatively determine the key parametersof the nonlinear dynamics of VC such as the critical valueof excitation power for the dynamical bifurcation η 2 cr,t h e resonant frequency ω0, and the quality factor, i.e., the ratio 2ω0/α⋆, where α⋆is an effective dissipation constant of the nonlinear oscillator. 1-μm-diameter Permalloy (Py) disks [Py (37 nm) /Ru (2 nm)] were fabricated by electron beam lithographyfollowed by sputter deposition and lift-off [see scanningelectron microscopy (SEM) picture Fig. 1(a)]. Electric contacts Ti (10 nm) /Au (110 nm) with a coplanar waveguide [not visible in Fig. 1(a)] are defined for microwave current injection. Figure 1(a) also shows the schematic configuration of the measurements using the homodyne detectionscheme. 11,18For the measurement of VDC, microwave currents are applied via a bias-tee, and a lock-in amplifier is usedto acquire V DC. All measurements are carried out at room temperature and for a high injected microwave power 0 dBm,corresponding to a current density J=1.0×10 11A/m2as a root-mean-square value (see Appendix). In order to move theVC, a static magnetic field B extis applied, and the VC moves perpendicularly to the direction of the external field (directiondepending on the chirality of the vortex state). Figure 1(b) shows AMR measurements with angles θ=0 ◦and 90◦ showing the nucleation and annihilation fields of the vortex state. To keep the VC inside the disk, the external field waslimited to B ext±15 mT for the measurements.16When we apply a microwave current with Bext, the temperature of the sample can be changed since the magnetoresistance changesas a function of the VC position. However, /Delta1Ris about 0.1% and it can be neglected for the current distribution. When a spin-polarized microwave current is injected into the magnetic disk via lateral electrical contacts, an in-planealternating Oersted (Oe) field B Oealong the xdirection [see Fig. 1(a)] is generated. It occurs due to the inhomogeneouscurrent distribution resulting from the conductivity and thick- ness mismatch between the magnetic disk and the electriccontacts 15,20(see Appendix). We have shown previously that the VC gyration in such devices is primarily excited by this Oefield and not the spin transfer torque. 18In this system, the Oe field contribution as a main driving force makes up about 75%of the force exerted on the VC as described in the Appendix.This opens up the possibility to tune the excitation power ∝B 2 Oe continuously by simply moving the VC closer or further away from the electric contacts, while in our experimental setup,we cannot vary the power continuously due to instrumentallimitations. Furthermore by keeping the excitation powerconstant, we also keep the sample temperature constant, whichotherwise leads to varying magnetic properties that impede agood analysis of the effect. The exact strength of the Oerstedfield depends on the geometry of the system and can be easilycalculated. Indeed, as an in-plane magnetic field B extis applied in the direction parallel to the ac current ( θ=90◦), the VC moves along the xdirection and the excitation power of the Oe field is rather constant. On the other hand, for the case ofB extperpendicular to the ac current ( θ=0◦), the VC moves closer to electrical contacts, and thus the excitation power ofthe Oe field strongly increases. Figure 2(a)shows the size of the homodyne signal V DC(≡ Vmax DC−Vmin DC) as a function of the in-plane applied magnetic field (5 mT <B ext<15 mT) for both angles θ=0◦and θ=90◦. As expected, the homodyne signal Vmax DC−Vmin DCfor the case of θ=0◦(>59μVa tBext=±15 mT) is significantly larger than Vmax DC−Vmin DCfor the case of θ=90◦(<28μV atBext=±15 mT). But the difference in the size of the VC gyration between the two angles is not purely a resultof the Oe field effect generated by the inhomogeneous currentdistribution. A small increase of the homodyne signal with B ext observed in the case of θ=90◦can be attributed to an in-plane inhomogeneous current distribution occurring due to the cir-cular shape of the disk that increases the size of the homodynesignal as the vortex core is pushed to an off-center position. However, in contrast to the usual resonant homodyne signal obtained previously for lower excitation power, 18we observe a double resonance response in the dependence of the homodynesignal on the frequency ωof applied microwave current. This behavior is shown in Fig. 2(b) for a particular value of B ext= 15 mT applied perpendicular to the ac-current direction. As onecan see, there is a rather wide frequency region δω, where the homodyne signal is small. The absolute value of the homodynesignal abruptly decreases on the boundaries of this region andsmoothly decreases outside of this region. This observed double resonant response indicates the dy- namical bifurcation behavior, i.e., appearance (disappearance)of two dynamical states in VC gyrations. 12In the frequency rangeδf, there are two dynamical states with small and large amplitudes of the VC gyration. The gyration of the VC is deter-mined by numerous switchings between these two dynamicaloscillating states. Since these two states exhibit a homodynesignal of a different sign, the observed tiny homodyne signalgives direct evidence for a dynamical bifurcation. 6,7,12 To quantify our measurements, we first plot the δω/2πas a function of in-plane dc field Bext[see Fig. 3(a)]. To find the equilibrium VC position as a function of Bext,m i c r o - magnetic simulations using the object-oriented micromagnetic 064402-2DOUBLE RESONANCE RESPONSE IN NONLINEAR ... PHYSICAL REVIEW B 88, 064402 (2013) -5 -10 -1501020304050607080 51 0 1 5θ=0o θ=90oVmax DC-VminDC (μV) Applied field (mT)θ=0o θ=90o 200 300 400 500-30-15015304560VDC(μV) Frequency (MHz)δω/2π= 76 MHzBext = +15 mT with θ = 0o)b( )a( FIG. 2. (Color online) (a) The dependence of homodyne signal ( Vmax DC−Vmin DC) and the frequency region δfon an in-plane applied external field (5 mT <H ext<15 mT) for two angles θ=0◦(blue squares) and 90◦(red circles). (b) The dependence of homodyne signal on the frequency of applied microwave current. The in-plane applied magnetic field Hext=+15 mT parallel to the direction of the ac current ( θ=0◦) was used. framework (oommf)21are performed with material parameters for Py yielding reliable results as previously confirmed.18,22 Pinning sites for the vortex do exist in our magnetic disk due to surface roughness and the defects resulting from the 4 6 8 1 01 21 41 620304050607080δω/2π (MHz) Applied field (mT)θ=0o θ=180o 020406080 θ=0o θ=180o δδω/2π (MHz) Oe field (mT)(a) (b) -200 0 2000.450.500.550.60Oersted field (mT) y position (nm) 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 FIG. 3. (Color online) (a) The δfas a function of in-plane applied field (blue squares for θ=0◦and red circles for θ=180◦). Inset shows the calculated oscillating Oe fields as a function of the positionin the ydirection.(b) The δfas a function of the calculated power of in-plane oscillating Oe field (blue squares for θ=0 ◦and red circles for θ=180◦). Blue and red solid lines are fitted curves from the time-dependent classical nonlinear oscillator in nonequilibrium states for θ=0◦andθ=180◦, respectively. The fitting curves are calculated by Eq. (2).polycrystalline nature of the material. This means that the positions of the VCs in our experiments include a degreeof stochasticity, but this averages out when we study thesystematic dependencies of the position on field. Furthermorewe see that for the cases of B ext=±5∼±15 mT, the AMR curves increase and decrease gradually showing that pinningsites are not dominating in our samples. This indicates that theVC gradually moves as a function of B ext. So although these defects can lead to small additional experimental errors inFigs. 2(a) and 3(a), it is clear that the conclusions are not altered by the small pinning. After that we also calculatethe dependence of Oe field on the coordinate y(yaxis is parallel to the direction of ac current) by means of a threedimensional simulation. 23This dependence is shown in the inset of Fig. 3(a). The minimum Oe field at y=0 is about 0.43 mT, and the maximum Oe fields at y=±250 nm are about 0.61 mT. By combining the experimental data ( δω/2π) and the calculated values of the magnitude of the oscillating Oe fieldB Oewe obtain the dependence δω/2πonBOe[see Fig. 3(b), experimental data is shown by squares and circles] which isused to quantitatively compare measurements and analysis. Although the magnetic vortex core dynamics is described by Thiele’s equation 18,24we show that all important experi- mental features can be reproduced in a generic model of a low-dissipative nonlinear oscillator subject to a resonant alternatingforceF=ηcos(ωt), where ηandωare the amplitude and frequency of F, accordingly. The classical dynamics of a nonlinear oscillator is characterized by time-dependent gen-eralized coordinate x, which satisfies the equation of motion 19 ¨x+α⋆˙x+ω2 0x+γx3=ηcos(ωt), (1) where α⋆is the effective dissipation constant, ω0is the fre- quency of self-oscillations of a magnetic vortex in the pinningpotential, and the parameter γdescribes the deviation of pin- ning potential from a harmonic one, i.e., we choose the pinningpotential in the following form: U(x)=ω 2 0x2/2+γx4/4. The solution of Eq. (1)is written as x(t)=Acos(ωt+φ) and the response Rof a nonlinear oscillator to the alternating force Fis obtained as R=Acos(φ). Note here that the homodyne signal VDCobserved in experiments is just proportional to R. A main result of our analysis is that the response Ras a function of frequency displays a double resonant feature. It is 064402-3J.-S. KIM et al. PHYSICAL REVIEW B 88, 064402 (2013) well known for a low-dissipative nonlinear oscillator,19i.e., as α⋆/lessmuchω0, that there is a particular interval of frequencies δω where Eq. (1)has three different solutions [two solutions with small and large amplitudes A(ω) are stable, and one solution is unstable]. On the boundaries of this frequency region, a secondstable dynamic state appears (disappears), and this type ofbifurcation is characterized by a sharp jump (positive or nega-tive) in the dependence of the response Ron the frequency ω. Moreover, the response Ris positive for the low frequency res- onance, and it changes sign for the high frequency resonance. Introducing the parameter β= 3η2γ 32(α⋆ω0)3that is proportional to the power of the alternating force, we obtain the power depen- dence of the width of the frequency region δωbetween the two resonant features as (details are explained in the Appendix) δω=ω+−ω−=α⋆(4β−3β1/3). (2) This equation was used to fit the experimental data for δω/(2π). The fitting curves [blue and red solid lines in Fig. 3(b)] are in good agreement with the experimental data. As one can see, there is a critical value of theamplitude of the resonant alternating force for which thisdouble resonant structure vanishes. From Eq. (2)we obtain η cr2=4√ 3(α⋆ω0)3/γin line with expectations. Using a direct mapping between resonances in the dynamics of a nonlinear oscillator and gyrations of a magnetic vortex,we obtain the effective damping α ⋆=2αω0D/G 0, where α, D, andG0are the Gilbert damping, the damping tensor, and the magnitude of the gyrovector, accordingly.18Using the fitting procedure for δω/(2π) we extract the key parameters characterizing the VC gyration, i.e., the value of the frequencyof self-oscillations ω 0/similarequal2π×250 MHz ( ω0/similarequalω−), the dis- sipation parameter α⋆=2π×52 MHz, and therefore, the quality factor of magnetic vortex gyration as 2 ω0/α⋆=10. The parameter D/G 0/similarequal3.3 for our samples was estimated in Ref. 18. Taking into account this value, we estimate the Gilbert damping as α=(α⋆/(2ω0))(G0/D)=0.03 in good agreement with the values expected for vortices in permalloyand an increase of the damping for vortices higher than thebulk value for permalloy as theoretically predicted. 25 Similarly, the amplitude of the resonant alternating force is written as η=ωγGBOer/2, where γGis the gyromagnetic factor, and ris the radius of the magnetic disk. The parameter β reads as β=(3/4)3/2(BOe/Bcr Oe)2, where Bcr Oe=2ηcr/(ωrγG) is the critical amplitude of alternating Oe field for which thebifurcation occurs. In our experiments, we find B cr Oe/similarequal0.4 mT. Notice here that the double resonant feature was not seen in ourprevious measurements 11,18because the exciting Oe field was below this critical value of 0.4 mT. We note that the appliedmagnetic field B Oeis inhomogeneous on the scale of the disk diameter, but as we only investigate small vortex gyrationamplitudes, the relevant field is very homogeneous across thearea of the gyration. In particular the amplitude of VC gyrationcan be estimated for the largest excitations as x∼=η/α ∗ω0= γGBOe/2α∗∼=0.15r/lessmuchr. Thus, the vortex gyration occurs on the scale which is much smaller than the disk diameter.From the calculations of the Oe field we see that one canconsider the ac-driving force applied to the vortex corehomogeneous across the relevant gyration area.Inside in this specific frequency region ω −<ω<ω +the two dynamic states coexist, and the dynamics of the VC is char-acterized by numerous switchings between driven oscillatingstates. 26These switchings are induced by thermal fluctuations, and thus, the averaged response /angbracketleftR/angbracketrightis small and it smoothly decreases with frequency as seen for the signal in the δωregion [Fig. 2(b)]. The effective potential barriers separating the two dynamical states for a nonlinear oscillator can be estimated asUeff/similarequal8mω3 3γ(δω)(m=G0/ωis the effective mass of the magnetic vortex),26or in the units characterizing the magnetic vortex gyration, Ueff/similarequalG0(δω)ω (α∗)3(Bcr OeRγG)2. Thus, we obtain the typical values of potential barriers Ueff/similarequal10−3eV , and since the switching time τis determined by expression τ/similarequal exp[Ueff/(kBT)] there are numerous switchings between two dynamical states at room temperature. It is also plausible to as-sume that individual transitions between dynamical states canbe observed by dynamic measurements at low temperatures ask BT/lessmuchUeff. Also using a microwave pulse technique instead of continuous microwave current might allow one to observeindividual transitions between the states, 7but this involves time resolved measurements of magnetic vortex gyration. In conclusion, using a microwave current rectifying effect, we study experimentally the nonlinear dynamics of a singlemagnetic vortex subject to a resonant microwave magneticfield. We could tune excitation power of alternating Oe fieldby varying the VC position. We observe that the amplitudeof alternating Oe field is larger than the critical value B cr Oe; the homodyne signal response displays a double resonant feature . The frequency range δω=ω+−ω−between these resonances strongly increases with the excitation power ofalternating Oe field. In the frequency region ω −<ω<ω +the homodyne signal is rather small, and this behavior indicatesthe bifurcation between two dynamical states (with small andlarge amplitudes) in nonlinear VC gyrations. Our quantitativeanalysis of this effect based on a generic model of nonlinearoscillator is in good agreement with experimental features, andimportant parameters characterizing the nonlinear dynamics ofmagnetic vortex such as the quality factor and the frequencyof self-oscillations were obtained. For the future, the observedeffect could be used in order to resonantly amplify magneticvortex gyrations. This work is supported by the EU STREP project MAG- WIRE (FP7-ICT-2009-5 257707), the DFG (SFB 767, GrantNo. KL 1811 and SPP 1459 “Graphene”), the EuropeanResearch Council via a Starting Independent Researcher Grant(Grant No. ERC-2007-Stg 208162), and NRF funds (GrantNos. 2010-0023798 and 2010-0022040) of the Ministry ofEducation, Science, and Technology of Korea. APPENDIX: METHOD TO DETERMINE A CURRENT DENSITY INTO THE MAGNETIC DISK AND TO CALCULATE CURRENT DISTRIBUTION The current density was determined from the resistance in- crease induced by the Joule heating when the microwave powerincreases. We corrected for the microwave frequency depen-dent power reflection and losses and then the current density inthe magnetic disk is constant for all microwave frequencies. 18 064402-4DOUBLE RESONANCE RESPONSE IN NONLINEAR ... PHYSICAL REVIEW B 88, 064402 (2013) xy FIG. 4. (Color online) The three dimensional surface current distribution ( xcomponent) is calculated by means of the Comsol multiphysics package. Two Au contacts (1 μm×1μm and thickness =110 nm) are located on both sides and 1 μm Py disk (thickness = 37 nm) is located at the center of the geometry. The inhomogeneous current distribution can be calculated by numerical methods. The three dimensional current distributionis calculated by means of the Comsol multiphysics package 23 depicted in Fig. 4. As expected, an inhomogeneous current distribution exists on the surface of the disk and thus thedifference in the size of the VC gyration between the twoangles in the main text is not purely a result of the Oe fieldeffect generated by the inhomogeneous current distribution. 1. Numerical calculations for the Oersted field contribution vs spin-transfer torque effects When a spin-polarized current is applied into the magnetic disk, the Oersted (Oe) field can be generated by theinhomogeneous current distribution. The Oe field contributionas a main VC driving force is discussed. First the generated Oefield profile with the current density J=1.0×10 11A/mw a s calculated by implementing numerical simulation. By usingthe object-oriented micromagnetic framework (OOMMF), 21 the trajectories of the VCs have been calculated with only STTeffect, only Oe field contribution, and including both effects(Oe and STT). Figure 5shows the VC displacements of the VCs along the xdirection as a function of the time up to 50 ns. There are two major aspects of the comparison between theOe field contribution and the STT effect. First, the diameterof the VC gyration due to the Oe field is 2.5 times larger thanthe diameter due to the STT effect. In general, there is a phasedifference between the Oe field and the STT effect driven VCgyration. As shown in Fig. 5, the phase difference between only Oe field contribution and including both (Oe +STT) effects is about 15 ◦, and thus we deduced that the Oe field contribution of the VC gyration in this system is about 75%. 2. Analytical expression of the nonlinear dynamics: double resonance response In order to quantitatively analyze the dynamics of a magnetic vortex in the presence of both a pinning potential andan externally applied alternating magnetic field of microwavefrequency, and a double resonance structure of the microwave0 1 02 03 04 05 0-120-60060120X displacement (nm) Time (ns) STT only Oe only STT + Oe FIG. 5. (Color online) The displacements of the VC along the x direction as a function of the time ( t=0∼50 ns). Red, black, and blue lines indicate the ydisplacements only Oe field contribution, only STT effect, and Oe field contribution +STT effect, respectively. The diameters of the VC trajectories are 64 nm (only STT effect),160 nm (only Oe field contribution), and 180 nm (STT +Oe field). The phase difference between only the Oe and the combined STT and Oe field is about 15 ◦. induced the VC response, we consider a generic model of a nonlinear oscillator subject to resonant time-dependentperturbation force F=ηcos(ωt), where η,ωare the amplitude and frequency of F, accordingly. The classical dynamics of a nonlinear oscillator is characterized by time-dependentcoordinate x, which satisfies the equation of motion: 19 ¨x+α˙x+ω2 0x+γx3=ηcos(ωt), (A1) where αis the dissipation constant, ω0is the frequency of self- oscillations of a magnetic vortex in the pinning potential, andthe parameter γdescribes the deviation of pinning potential from a harmonic one, i.e., we choose the pinning potential inthe following form: U(x)=ω 2 0x2/2+γx4/4. Now, we search for a solution of Eq. (A1) in the following form: x(t)=bexp (iωt)+cexp (−iωt). (A2) Note here that this approximated solution is a proper one as we consider a resonant case, i.e., ω/similarequalω0. The amplitudes of other nonresonant harmonics are small in respect to |b|,|c|and can be neglected. Substituting (A2) in(A1) we obtain that b=c∗, and the amplitude |b|ofx(t) can be found from a transcendent equation (this method to obtain amplitudes bandcwas used in Refs. 19,27and28): |b|2/bracketleftbig/parenleftbig −ω2+ω2 0+3γ|b|2/parenrightbig2+α2ω2/bracketrightbig =η2 4. (A3) Moreover, the response of a magnetic vortex to an externally applied microwave radiation is obtained as R=2Re(b)=2η−ω2+ω2 0+3γ|b|2 /parenleftbig −ω2+ω2 0+3γ|b|2/parenrightbig2+α2ω2.(A4) There is a particular interval of frequencies δωwhere Eq. (A3) has three different solutions (two solutions are stable, andone solution is unstable). Outside of this frequency region, theEq.(A3) has one solution only. Therefore, on the boundaries of the frequency region, the response (A4) displays two resonant 064402-5J.-S. KIM et al. PHYSICAL REVIEW B 88, 064402 (2013) features. In the resonant limit as |ω0−ω|/lessmuchω0, we simplify Eq.(A3) as z[(−ω+ω0+z)2+α2/4]=3η2γ 32ω3 0, (A5) where z=3γb2/(2ω0). In this case the upper boundary of resonant frequency region ω+can be obtained when the condition ( −ω+ω0+z) is closed to zero, and therefore, ω+=ω0+α3η2γ 8α3ω3 0. (A6) On the other hand, the lower boundary of the resonant fre- quency region is determined from a condition that ( −ω+ω0+ z)/greatermuchαand therefore, ω−=ω0+α(9η2γ 32α3ω3 0)1/3. Introducing the parameter β=3η2γ 32α3ω3 0that is proportional to the power of ac magnetic field, we obtain the power dependence of the width of the frequency region δωbetween two resonant features as δω=ω+−ω−=α(4β−3β1/3). (A7) Thus, as we can see in good accord with experimental data, the upper resonance substantially shifts to higher frequenciesas the microwave power increases, but the position of lowresonance does not shift very much with the microwavepower. Also, for the power region available in the experiments,Eq. (A7) fits the experimental data very well. Moreover, by fitting the experimental data with Eq. (A7) , we obtain the resonant frequency ω 0/similarequal2π×250 MHz, the dissipation parameter α=2π×52 MHz and the quality factor of mag- netic vortex motion as 2 ω0/α=10. We also can determine the critical alternating magnetic field H2 cr=0.6432α3ω3 0 3γ[or β=0.64(H/H cr)2] for which this double resonant structure vanishes. In experiments, this corresponds to Hcr/similarequal4O e . Thus, we obtain that two dynamic states (with small and large amplitudes b) can coexist in a specific frequency region ω−<ω<ω +. On the boundaries of this region, a second stable dynamic state occurs, and an appearance of such a stateis characterized by a sharp jump (positive or negative) in thedependence of the response Ron the frequency ω.I nt h e frequency region δωthe dynamics of a nonlinear oscillator is characterized by numerous switchings between these states.These switchings are induced by thermal fluctuations, and thus, the averaged response /angbracketleftR/angbracketrightsmoothly decreases with frequency. Quantitatively, it can be analyzed by making use ofthe equations /angbracketleftR/angbracketright=R(b 1)W1+R(b2)W2, (A8) where b1andb2are the amplitudes of two stable dynamic states, and W1andW2are the probability of these states. In the thermal fluctuation regime, the probability W1andW2has been obtained in Refs. 26and27, and it reads as W1/W 2=exp[(Q1−Q2)/(kBT)],W1+W2=1( A 9 ) and Q1=4ω3 0/Delta1ω 3γ/bracketleftbigg 1−2(α δω)3/4β1/4/bracketrightbigg , (A10) Q2=16ω3 0/Delta1ω 3γ/parenleftbiggα δω/parenrightbigg3/2/radicalbig β, where /Delta1ω=ω−ω0. Thus, one can see that in the vicinity of ω−,t h eQ2>Q 1andW1<W 2, and the response is mostly determined by the dynamic state with a large amplitude ofoscillations. In this case the response /angbracketleftR/angbracketrightdecreases, but it is still a positive one. However, as the frequency increasesand it approaches the ω +,t h eQ2<Q 1andW2<W 1, and the response is mostly determined by the dynamic state withsmall amplitude of oscillations. In this case the response /angbracketleftR/angbracketright becomes a negative one. Finally, we discuss the correspondence of the nonlinear oscillator model subject to a resonant time-dependent force toan ac-driven single magnetic vortex gyration. The magneticvortex gyration is described properly by Thiele’s equations,where indeed the second derivative term is absent. However,Thiele ´s equations are two dimensional ones, for XandY that are coordinates of a magnetic vortex core. 13In our study, we carried out a mapping of these equations to a single one-dimensional nonlinear equation. With this mapping we expressthe model parameters of a nonlinear oscillator ( ω 0,γ,α⋆,η) through real parameters characterizing the magnetic vortexgyration ( α,D,G). In this case the generalized coordinate x has a physical meaning of the “distance from the equilibriumposition.” *klaeui@uni-mainz.de 1S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, MA, 1994). 2J. Guckenheimer and P. 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Faini, and L. Vila, Phys. Rev. Lett. 99, 146601 (2007). 12K. S. Buchanan, M. Grimsditch, F. Y . Fradin, S. D. Bader, and V. N ovo s a d , Phys. Rev. Lett. 99, 267201 (2007). 13L. Thomas, M. Hayashi, X. Jiang, R. Moriya, Ch. Rettner, and S. Parkin, Science 315, 1553 (2007). 064402-6DOUBLE RESONANCE RESPONSE IN NONLINEAR ... PHYSICAL REVIEW B 88, 064402 (2013) 14T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289, 930 (2000). 15M. Bolte, G. Meier, B. Kr ¨uger, A. Drews, R. Eiselt, L. Bocklage, S. Bohlens, T. Tyliszczak, A. Vansteenkiste, B. Van Waeyenberge,K. W. Chou, A. Puzic, and Hermann Stoll, P h y s .R e v .L e t t . 100, 176601 (2008). 16S. Kasai, Y . Nakatani, K. Kobayashi, H. Kohno, and T. Ono, Phys. Rev. Lett. 97, 107204 (2006). 17A. Bisig, J. Rhensius, M. Kammerer, M. Curcic, H. Stoll, G. Sch ¨utz, B. Van Waeyenberge, K. W. Chou, T. Tyliszczak, L. J. Heyderman,S. Krzyk, A. von Bieren, and M. Kl ¨aui,Appl. Phys. Lett. 96, 152506 (2010). 18J.-S. Kim, O. Boulle, S. Verstoep, L. Heyne, J. Rhensius, M. Kl ¨aui, L. J. Heyderman, F. Kronast, R. Mattheis, C. Ulysse, and G. Faini,P h y s .R e v .B 82, 104427 (2010). 19L. D. Landau and E. M. Lifshitz, Mekhanika (Mechanics) (Nauka, Moscow, 1973) [English translation (Pergamon Press, Oxford,1976).20Y . Nakatani and T. Ono, Appl. Phys. Lett. 99, 122509 (2011). 21http://math.nist.org/oommf/ . 22A saturation magnetization MS=8×105A/m, an exchange constant A=1.3×10−11J/m, and a Gilbert damping constant α=0.01 are used. 23http://www.comsol.com/ . 24A. A. Thiele, P h y s .R e v .L e t t . 30, 230 (1973). 25S. Zhang and Steven S.-L. Zhang, P h y s .R e v .L e t t . 102, 086601 (2009). 26M. I. Dykman and M. A. Krivoglaz, Sov. Phys. JETP 50,3 0 (1979). 27A. P. Dmitriev, M. I. D’yakonov, and A. F. Ioffe, Sov. Phys. JETP63, 838 (1986). 28E. Ilichev, M. V . Fistul, B. A. Malomed, H. E. Hoenig, and H.-G. Meyer, Europhys. Lett. 54, 515 (2001); M. V . Fistul, A. Wallraff, and A. V . Ustinov, Phys. Rev. B 68, 060504(R) (2003). 064402-7

PhysRevB.89.094428.pdf

PHYSICAL REVIEW B 89, 094428 (2014) Photoinduced spin angular momentum transfer into an antiferromagnetic insulator Y . Fan,1X. Ma,1F. Fang,1J. Zhu,2Q. Li,2T. P. Ma,2Y. Z . Wu ,2Z. H. Chen,2H. B. Zhao,3,*and G. L ¨upke1,† 1Department of Applied Science, College of William & Mary, Williamsburg, VA 23187 2Department of Physics, State Key Laboratory of Surface Physics and Advanced Materials Laboratory, Fudan University, Shanghai 200433, China 3Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Department of Optical Science and Engineering, Fudan University, Shanghai, 200433, China (Received 20 January 2014; revised manuscript received 11 March 2014; published 31 March 2014) Spin angular momentum transfer into an antiferromagnetic (AFM) insulator is observed in a single-crystalline Fe/CoO/MgO(001) heterostructure by time-resolved magneto-optical Kerr effect. The transfer process is mediated by the Heisenberg exchange coupling between Fe and CoO spins. Spin angular momentum transfer toordered AFM spins is independent of the external magnetic field and enhances the spin precession damping inFe, which remains nearly invariant with temperature. DOI: 10.1103/PhysRevB.89.094428 PACS number(s): 75 .78.−n The spin angular momentum transfer effect is crucial for next generation spintronic devices, including spin random ac-cess memory (RAM) [ 1]. To write information into spin RAM, a spin-polarized current is usually applied to transfer spinangular momentum into a ferromagnetic (FM) metallic layer.This spin-transfer-torque (STT) effect has been demonstratedto drive ultrafast spin precession, which causes the magneti-zation to switch when the current density exceeds a thresholdvalue of 5 ×10 6A/cm2in tunnel junctions [ 2–4]. However, the critical current density still needs to be reduced for wideapplications of spin RAM with high integration density. Alternatively, antiferromagnetic (AFM) spins can also be modulated by the STT effect [ 5–7]. Spin angular momentum transfer into AFM metals has been observed via modificationof the exchange bias in FM /AFM systems [ 8–10]. The spin- polarized current creates a nonequilibrium spin density in theAFM layer, which alters the exchange field and thus generatesa spin torque. Furthermore, the torque exerted on each AFMspin is weak, causing a long decay length of the STT effect,which results in the transfer of spin angular momentum deepinto the AFM layer [ 6]. This is in contrast to FM metals where the STT effect is a near interface effect; hence, a lower currentdensity may be required for AFM spin reversal [ 7]. Spin angular momentum transfer might be achieved in FM-metal /AFM-insulator heterostructure, which would lead to a pure spin current with the advantage of avoiding a heateffect [ 11–13]. One possible approach to generate such a transfer is to excite FM spin precession, which exerts a torqueon AFM spins due to FM-AFM exchange coupling. As aresult of angular momentum transfer, the FM spin precessionwould have larger damping, which can be observed in the timeor frequency domain. However, to observe such an effect, asingle-crystalline heterostructure is preferred since it wouldsignificantly reduce interface roughness to prevent magnon-magnon scattering (MMS) [ 14,15] which will also contribute to the damping. Recently, single-crystal Fe /CoO(001) het- erostructures have been grown by molecular beam epitaxy(MBE) with the crystallographic axis Fe [100]//CoO[110] *hbzhao@fudan.edu.cn †luepke@wm.edu[16–18]. The large crystalline magnetic anisotropy energy of 3m e V /Co2+in CoO [ 19,20] may suppress the nucleation of multi-AFM domains and local exchange fluctuations. Thestatic magnetization measurements reveal that AFM spinsfavor a collinear coupling with Fe magnetic moments [ 17], which is a favorable configuration for studying the spin angularmomentum transfer effect. In this paper, we report on time-resolved magneto-optical Kerr effect (TRMOKE) measurements to investigate opticallyexcited coherent spin precession in single-crystalline Fe /CoO heterostructure. The damping behavior in the Fe film changesabruptly at the N ´eel temperature ( T N) of the CoO layer and becomes independent of the applied magnetic field at lowtemperature. This observation is distinct from MMS or thedephasing effect and indicates that the spin angular momentumis transferred to the CoO spins via the FM-AFM Heisenbergexchange coupling. The Fe /CoO bilayer is deposited on MgO(001) substrate by MBE at room temperature (RT). CoO layers with thicknessesof 1 nm, 2.5 nm, and 4 nm are grown by the reactive depositionof Co with oxygen [ 16,17] at a pressure of 2 ×10 −6Torr. [ 18]. A 5-nm-thick epitaxial Fe film is then deposited on top of theCoO layer [Fig.1(a)]. The single-crystal structure of the CoO and Fe films is verified by reflection high-energy electrondiffraction [ 18]. For comparison, a 5-nm-thick Fe film is directly grown on MgO (Fe /MgO). All of the samples are covered by a 3-nm-thick MgO protection layer. Coherent spin precessions in the Fe films are investigated by TRMOKE [ 21] in a canted magnetization configuration where the magnetic field ( H) is applied along Fe [110],a s depicted in Fig. 1(a). An intense pump laser excites the mag- netization ( M) precession in the Fe film via fast modulation of the anisotropy fields [ 22–25]. The precession dynamics are recorded by the polarization change of a time-delayed probebeam[Fig.1(b)]. The TRMOKE measurements are carried out using a Ti:sapphire amplifier laser system delivering 150 fspulses at 800-nm wavelength with a repetition rate of 1 kHz.The pump beam is focused on the sample with a diameterof/223c0.2 mm and a pulse energy density of /223c0.28 mJ /cm 2. The probe beam has a smaller diameter of /223c0.1 mm and a pulse energy density of /223c0.05 mJ /cm2. All measurements are performed after field cooling the sample from RT to /223c80 K. 1098-0121/2014/89(9)/094428(6) 094428-1 ©2014 American Physical SocietyY. FA N et al. PHYSICAL REVIEW B 89, 094428 (2014) 90 180 270 3600200400600 T(K)Hc&He(Oe)Hc He(b)(c) (d)-1.0 -0.5 0.0 0.5 1.082 K H//[010]Kerr Signal (arb. units.) Field(KOe)82 K H//[100] 330 K H//[100]Fe MgOMgO CoOMH Fe [110][001](a) 0 150 30020 30 40TRMOKE (a.u.) Time Delay (ps)f( G H z )TRMOKE (arb. units.) FIG. 1. (Color online) (a) Schematic presentation of magnetiza- tion ( M) precession in Fe /CoO heterostructure with the magnetic fieldHapplied along the Fe [110]. (b) TRMOKE data at 78 K measured with a magnetic field of H=600 Oe. The solid line is a fit by the damped sine function, and the inset shows the Fourier power spectrum. (c) Longitudinal hysteresis loops for magnetic field along uniaxial easy axis [100]and hard axis [010]at 82 K and 330 K. (d) Temperature dependence of coercivity ( Hc) and exchange biasing field (He) obtained from easy axis loops. All data are measured from Fe film grown on 2.5-nm-thick CoO (001) layer. Longitudinal MOKE measurements indicate a negligible exchange bias ( He)i nF e /CoO from /223c80 K to above RT, as shown in Figs. 1(c) and1(d). The absence of exchange biasis likely due to a very small number of uncompensated AFM spins at the smooth interface. However, the field cooling leadsto the alignment of AFM spins along CoO /angbracketleft110/angbracketrightdirections, which is collinear to Fe /angbracketleft100/angbracketright. Due to the exchange coupling of the Fe magnetization with the AFM spins, a uniaxial anisotropyappears below T N(discussed below), and the coercivity ( Hc) increases with decreasing temperature down to /223c130 K, below which Hcdecreases because less AFM spins are dragged to switch with FM spins [ 26]. As shown in Fig. 1(c), the hysteresis loop is almost square for the field along the easy axis andis much sharper at 80 K than 330 K; however, it is hard toreach saturation magnetization at the largest field, restrictedby our electromagnet, along the hard axis perpendicular to thecooling field. Such results indicate a well-defined easy axisand homogeneous anisotropy of the sample. Figure 1(b) presents TRMOKE data from a Fe film grown on a 2.5-nm-thick CoO [Fe/CoO(2.5 nm) ]layer measured at 78 K with a magnetic field of H=600 Oe. The data is fitted to a damped sine function, Aexp(−t/τ)cos(2 πft), with precession amplitude A, time delay t, decay rate 1 /τ, and precession frequency f. The inset shows the Fourier power spectrum, ver- ifying the uniform precession mode of the Fe magnetization. The spin precession dynamics in Fe /CoO(2.5 nm) is investigated as a function of the magnetic field at 78 K, 240 K,and 300 K (Fig. 2). The results are compared with TRMOKE data from Fe /MgO. The effective Gilbert damping parameter αis determined from the decay rate 1 /τ,u s i n g[ 27,28] α=2/[τγ(2Hcos(δ−φ)+H a+Hb)], Ha=4πMs+2K⊥/Ms−2Kusin2φ/M s (1) +K1(2−sin2(2φ))/Ms, Hb=2K1cos(4φ)/Ms+2Kucos(2φ)/Ms, 051015300 K Fe/CoO Fe/MgO 24240 K 2478 K α(x 10-2) 51015 102030 H( k O e )0 1 2 0 1 2 0 1 2102030f( G H z )(a) (b) (c) (d) (e) (f) FIG. 2. (Color online) Magnetic field Hdependence of (a)–(c) effective Gilbert damping αobtained from Eq. ( 1) and (d)–(f) precession frequency fwithHapplied along Fe [110]at 78 K, 240 K, and 300 K, respectively. The curves in (d)–(f) are fits of Eq. ( 2). The damping and frequency data obtained by TRMOKE from Fe film grown on 2.5-nm-thick CoO (001) layer and Fe /MgO are presented by /squaresolidand/triangledownsld, respectively. 094428-2PHOTOINDUCED SPIN ANGULAR MOMENTUM TRANSFER . . . PHYSICAL REVIEW B 89, 094428 (2014) derived by solving the Landau-Lifshitz-Gilbert equation. γ= γeg/2 is the gyromagnetic ratio (for Fe, g=2.09 and γe=1.76×107Hz/Oe).δandφare the angles of Hand in-plane equilibrium Mwith respect to the Fe [100]axis,Ms is the Fe-saturated magnetization, and Ku,K1, andK⊥are the in-plane uniaxial, crystalline cubic, and out-of-plane magneticanisotropies, respectively. K uandK1have the easy axis along Fe[100], andK⊥has the easy plane of Fe(001). The values ofKu,K1, andK⊥are determined by fitting the precession frequency fas a function of HFigs. 2(d)–2(f)], according to [27,28] 2πf=γ{[Hcos(δ−φ)+Ha][Hcos(δ−φ)+Hb]}1/2. (2) Figure 2compares the field dependence of the effective Gilbert damping αin Fe/CoO(2.5 nm) ( /squaresolid) and Fe /MgO (/triangledownsld). In Fe/CoO(2.5 nm), αis nearly independent of the applied field at 78 K, and it has a weak and broad peak around600 Oe at 240 K. We simulate the effective damping dueto the dephasing caused by nonuniformity of the exchangecoupling-induced anisotropy field. The results (see Appendix)indicate that the damping enhances with increasing field in thefield range of 0–1920 Oe for both temperatures. In particular,the damping increases by a factor of four at 240 K. Therefore,our experimental observations rule out the dephasing processesas the dominant extrinsic damping mechanism below T Nin Fe/CoO(2.5 nm). Furthermore, we can also exclude MMS as the major damping source in our sample partially becauseit is induced by the local FM-AFM exchange fluctuations,as observed in polycrystalline or amorphous FM /AFM het- erostructures [ 14,15,29–32]. Moreover, the in-plane MMS- induced damping increases with precession frequency becauseof the increased spin wave degeneracy [ 14,33,34]. However, this disagrees with the fact that the damping is invariant with H and precession frequency in single crystalline Fe /CoO(2.5 nm) at 78 K [ 35]. The reduced sample roughness may suppress MMS in the single-crystalline heterostructure. In contrast, the field dependence of αin Fe/MgO ( /triangledownsld)a t 78 K reveals a dephasing effect accompanied by the field-independent intrinsic damping. Inhomogeneities in the Fe filmcause variations in the local magnetic anisotropy fields, whichlead to the dephasing effect. A clear indication of this effectis that the maximum in αnearly coincides with the minimum inf, which occurs at a minimum in the effective field H eff when the external field reaches a strength equal to that of the cubic anisotropy field of /223c600 Oe. The weak Heffleads to a large variation in the Fe spin orientation, resulting in a largedephasing effect. The dephasing effect in Fe /MgO is almost identical at 80 K, 240 K, and RT [Figs. 2(a)–2(c)].T h i si s in agreement with the similar anisotropy fields at differenttemperatures, as can be seen from the comparable precessionfrequencies shown in Figs. 2(d)–2(f). The dephasing effect is nearly absent in Fe /CoO(2.5 nm) at 78 K due to the strong uniaxial magnetic anisotropy(UMA) field 2 K u/Ms, as revealed by the enhanced pre- cession frequency ( /squaresolid) shown in Fig. 2(d) [36]. This effect increases with temperature and appears at 240 K, causinga weak dependence of α(/squaresolid)o nH, as shown in Fig. 2(b). Since the temperature is slightly below the N ´eel temperature (T N=255 K) of the CoO thin film [ 37], the thermal energybecomes comparable to the AFM exchange energy. This causes some fluctuations in AFM spin orientation and further inducesvariations in FM-AFM exchange field causing the dephasingeffect. At RT, the dephasing effect is dominant, and thedamping exhibits a strong dependence on H,a ss h o w ni n Fig.2(c). The exchange interaction near the Fe /CoO interface forces a small fraction of CoO spins to form disordered AFMspin clusters [ 38], which introduce a random exchange field enhancing the dephasing effect. From the previous discussions, we can exclude MMS and dephasing effects as the dominant damping processes inFe/CoO(2.5 nm) below the N ´eel temperature. Furthermore, the CoO insulating layer eliminates the spin pumping ef-fect [ 39], which can occur with a normal metal adjacent to a FM layer. The fact that αis independent of H[Fig.2(a)]and is enhanced with respect to the intrinsic damping in Fe /MgO, as revealed at high external fields, indicates that the dampingprocess in Fe /CoO(2.5 nm) involves both the intrinsic spin relaxation and the transfer of Fe spin angular momentum toCoO spins via FM-AFM exchange coupling and then intothe lattice by spin-orbit coupling. Such a transfer processwould also be independent of the external field, similar tothe intrinsic damping, because the FM-AFM spin exchangestiffness, the AFM order, and the AFM spin-orbital couplingare all independent of H. To gain further insight into the spin relaxation mechanism in Fe/CoO, we performed temperature-dependent TRMOKE measurements. Figure 3(a) reveals a sudden jump of α(/trianglesolid) atT NwithH=1920 Oe. Above the N ´eel temperature, the population of AFM spins would be very small, which limitsthe spin angular momentum transfer effect. Thus, the damping 024 125 2501020302Ku/Ms(kOe)2Ku/Ms 2468 α(x 10-2)α@ 1920 Oe α@5 0 0O e TNf( G H z ) T( K )TN(a) (b) FIG. 3. (Color online) Temperature Tdependence of (a) effec- tive Gilbert damping αand UMA field 2 Ku/Msand (b) precession frequency of Fe film grown on 2.5-nm-thick CoO (001) layer. /trianglesolid and♦present the damping data in (a) and frequency data in (b), measured using H=1920 Oe and 500 Oe applied along Fe [110], respectively. o represents the UMA field. The dashed line indicates the N ´eel temperature TN. The solid line is the calculated αusing Eq. ( 4) with η=1. 094428-3Y. FA N et al. PHYSICAL REVIEW B 89, 094428 (2014) is close to the intrinsic Gilbert damping of Fe. At the N ´eel transition, the number of AFM spins significantly increases.The precessional Fe spins can induce precession of AFM spinsvia the FM-AFM coupling. This process transfers spin angularmomentum into the CoO layer and thus enhances the dampingof Fe spin precession. AtH=500 Oe, αexhibits a large drop at T N[Fig. 3(a)]. Above TN, the dephasing effect caused by random exchange fields dominates as the weak Hcannot align the disordered AFM spin clusters. While at TN, the ordered AFM state nearly eliminates the dephasing effect, which causes the dropofα. As temperature decreases, the strong UMA field (o) further aligns AFM spin orientation, thus αapproaches the value measured at high field. Even though the UMA field2K u/Ms, calculated from the measured precession frequencies [Fig. 3(b)]using Eq. ( 2), strongly increases with decreasing temperature, the extracted cubic anisotropy field 2 K1/Msstays nearly unchanged. Next, we calculate the effective Gilbert damping αof Fe magnetization precession by including the FM-AFM exchangecoupling in the Landau-Lifshitz-Gilbert equation, dM/dt=−γ(M×H eff)+(αFe/M)(M×dM/dt) −(γ/M )(M×HFM-AFM )×M. (3) Here,αFe≈0.003 is the Fe intrinsic Gilbert damping [ 40,41]. The last term in Eq. ( 3) is analogue to the STT effect [ 5], de- scribing the transfer of the Fe spin angular momentum to AFMspins via FM-AFM exchange coupling, where H FM-AFM = ηj m Coe[100]is the FM-AFM exchange field; ηis the percentage of CoO AFM spins to which spin angular momentum istransferred; j=J/(2a 2tFeMsμB) is the coupling coefficient between Fe and Co magnetic moments with Fe-CoO Heisen-berg exchange coefficient J=2.87×10 −17erg [ 42], CoO lattice constant a=4.27 ˚A, Fe film thickness tFe=5n m , and Bohr magneton μB;mCo=3.8μB[19] is the Co magnetic moment in CoO; and e[100]is the unit vector along Fe [100].T h e solution of Eq. ( 3) yields the expression of effective Gilbert damping as α≈αFe+ηj γ m Cocosφ/2πf, (4) where φ=15°–20°is the angle between Fe magnetization and AFM spins determined by TRMOKE measurements.Equation ( 4) captures important features of α(/trianglesolid)a tH= 1920 Oe, as shown by the calculated curve (solid line) inFig. 3(a). Above T N,ηis small, and the second term is negligible in Eq. ( 4), hence α≈αFe.A tt h eN ´eel transition, the AFM order is established, and the value of ηcannot be neglected, which causes the jump of αatTN. We calculate αusing Eq. ( 4) with the above values and η=1 below TN. The result is shown in Fig. 3(a). We note a slight decrease of the calculated αdue to the increase of precession frequency with lowering temperature. However, the measured dampingis almost invariant with temperature below T N.T h i ss m a l l difference may be due to a varied number of rotatable andfrozen AFM spins [ 16,43] as well as the increase of effective m Cowith decreasing temperature. The decrease of damping measured at H=500 Oe is caused by the reduction of the dephasing effect. Here, we need to point out that thedamping induced by the slow relaxer mechanism, observed in Exchange CouplingSpin-orbital Coupling Spin Angular Momentum Transfer(a) (b)24 051015 0.0 0.5 1.0 1.5 2.0102030 0.0 0.5 1.0 1.5 2.051015 α(x 10-2)78 K 1n m 4n m300 K CoO thicknessf( G H z ) H(KOe) FIG. 4. (Color online) (a) Precession frequency and damping in Fe films grown on 1-nm- and 4-nm-thick CoO layer at 78 K and300 K. (b) Schematic channel of spin angular momentum transfer from FM to AFM spins and to lattice. The exchange coupling between FM and AFM spins and the spin-orbital coupling between AFM spinsand lattice are represented by springs. polycrystalline FM /AFM heterostructures, has strong thermal dependence due to large variation of the relaxation time ofthe AFM grains [ 44,45]. Thus, it may not account for the enhanced damping below T Nin our single-crystalline Fe /CoO sample. We found similar damping and precession frequency for a Fe film grown on 4-nm-thick CoO, as shown in Fig. 4(a). However, for a Fe film on 1-nm-thick CoO layer, which istoo thin to establish exchange torque because of the lack ofAFM order, its damping and precession frequency are similarto Fe/MgO. Therefore, we can also exclude that the enhanced damping just below T Nof CoO may be caused by the presence of submonolayer FeO at the top and buried interfaces [ 17], although it may slightly affect the overall damping of the Feprecession. These results confirm that the uniform exchangetorque exerted on the Fe magnetization by the ordered AFMspins in CoO forms the prerequisite for the observed spinangular momentum transfer. A simple model of the spin angular momentum transfer channel in Fe /CoO is depicted in Fig. 4(b).A F Ms p i n s experience a torque from the precessional FM magnetizationthrough FM-AFM exchange coupling (first spring). Theexchange coupling has an interaction distance of /223c5n m[ 46], which directly transfers FM spin angular momentum to AFMbulk spins. Considering that the frozen spins in CoO mayprecess at frequency in the THz region because they experiencevery large anisotropy field, they may not be in resonance of 094428-4PHOTOINDUCED SPIN ANGULAR MOMENTUM TRANSFER . . . PHYSICAL REVIEW B 89, 094428 (2014) the FM spin precession. However, the rotatable AFM spins have much lower anisotropy energy and may be dragged by theprecessing magnetization, thus offering the channel for angularmomentum transfer. The transferred angular momentum ofrotatable AFM spins quickly relaxes via spin-orbital coupling(second spring) to the lattice. The CoO AFM spins havea precession lifetime estimated to be /223c16 ps [ 47], which is very short compared to the FM spin precession lifetime(/223c150 ps in Fe films as measured by TRMOKE), and serve as an efficient “sink” to drain spin angular momentum fromthe FM layer. In addition, some AFM domains with spinorientation perpendicular to the direction of the coolingfield may probably be formed in CoO during the fieldcooling process, thus producing AFM domain walls that haveresonance frequency in the GHz region. Such domain wallmotion can also be a possible source to drain the angularmomentum. In conclusion, we have observed a sharp increase of the Fe spin precession damping in single-crystal Fe /CoO heterostructure just below the N ´eel transition temperature. The enhanced damping is field-independent and depends onthe thickness of the CoO layer, which is consistent with spinangular momentum transfer into AFM insulator driven by theFM-AFM exchange coupling. The TR-MOKE experiments, data analysis, simulations, and discussions performed at the College of William andMary were sponsored by the DOE through Grant No. DE-FG02-04ER46127. The work at the Department of Physics,Fudan University, was supported by NSFC with Grants No.10925416, No. 11274074, No. 91121007, and No. 11225417and 973 projects of China (Project No. 2011CB925600). Thework at the Department of Optical Science and Engineering,Fudan University, was supported by the NSFC with GrantsNo. 61222407 and No. 11074044 and NCET (11-0119). APPENDIX: SIMULATION OF DEPHASING-INDUCED DAMPING BELOW TN In the case of magnetic disorder/dispersion ( /Delta1Ku), the frequency broadening is given as /Delta1ωd=(|∂ω/∂K u|+|∂ω/∂φ |·|∂φ/∂K u|)/Delta1Ku,(A1) where ∂ω/∂K u=ω(−sin2φ/Ha+cos2φ/Hb)/Ms, ∂ω/∂φ =ω[(−K1sin4φ/M s−Kusin2φ/M s+Hsin(δ− φ)/2)/Ha−(4K1sin4φ/M s+2Kusin 2φ/M s+Hsin(δ− φ)/2)/Hb], calculated from Eqs. ( 1) and ( 2), and ∂φ/∂K u= −sin 2φ/(2K1cos4φ+2Kucos 2φ+HM scos(φ−δ)), ob- tained from the relationship between the equilibrium mag-netization orientation φand the UMA K u. The broadening of precession frequency, /Delta1ωd, results in a dimensionless(a) (b)0.0 0.5 1.0 1.5 2.0123456 78 K 240 Kαe (x10-2) H( K O e ) 100 150 200 250123456 1900 Oe 2000 Oeαe (x10-2) T (K) FIG. 5. (Color online) Simulation of dephasing-induced damp- ing as a function of (a) field and (b) temperature below TN. dephasing-induced damping term, αd=/Delta1ωd/[γ(2Hcos(δ− φ)+Ha+Hb)], by solving the Landau-Lifshitz-Gilbert equation. Therefore, the effective Gilbert damping is αe=α0+/Delta1ωd/[γ(2Hcos(δ−φ)+Ha+Hb)],(A2) where α0is the intrinsic Gilbert damping constant. Assuming a magnetic disorder below TN, i.e.,/Delta1Ku, which is linearly proportional to Ku, we can simulate /Delta1Ku-induced effective damping (dephasing). From the precession frequency,we obtain the temperature dependence of anisotropy constantK uandK1. 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PhysRevB.94.220404.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 94, 220404(R) (2016) Interaction between a domain wall and spin supercurrent in easy-cone magnets Se Kwon Kim and Yaroslav Tserkovnyak Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA (Received 6 October 2016; published 21 December 2016) A domain wall and spin supercurrent can coexist in magnets with easy-cone anisotropy owing to simultaneous spontaneous breaking of Z2and U(1) symmetries. Their interaction is theoretically investigated in quasi-one- dimensional ferromagnets within the Landau-Lifshitz-Gilbert phenomenology. Specifically, the spin supercurrentcan exert the torque on a domain wall and thereby drive it. We also show, as a reciprocal phenomenon, afield-induced motion of a domain wall can generate spin supercurrent. DOI: 10.1103/PhysRevB.94.220404 Introduction. Spins in magnets see the crystal lat- tice through overlap of electron orbitals, which engendersanisotropy energy. In particular, crystal lattices with a singleaxis of high symmetry, e.g., hexagonal crystals with the axisof sixfold rotational symmetry, endow magnets with uniaxial anisotropy [ 1]. Uniaxial anisotropy energy is invariant under two operations on spins: the time reversal and the rotations ofspins around the axis, which can be characterized by discreteZ 2and continuous U(1) symmetries, respectively. When the symmetry axis is easy axis, there are two ground states, in which all the spins are either parallel or antiparallelto the axis. The ground states break the Z 2symmetry, but respect the U(1) symmetry. Spontaneous breaking of the discrete symmetry in a continuous field theory entails adomain wall, which is a topological soliton that smoothlyinterpolates two distinct ground states [ 2]. Such domain walls in easy-axis magnets have been extensively investigated [ 3] due to a fundamental interest as well as practical motivationsexemplified by the racetrack memory [ 4]. One of the main results of these studies is a collection of various means to drive a domain wall, which includes a magnetic field [ 5] and a spin-polarized electric current [ 6]. When the symmetry axis is hard direction for spins, there are continuously degenerate ground states: uniform spin statesin the easy plane perpendicular to the symmetry axis. Theground states break the U(1) symmetry while maintaining theZ 2symmetry. In a classical field theory, a continuous symmetry of the system implies the existence of a conservedquantity according to Noether’s theorem [ 2]. For easy-plane magnets, the conserved quantity is the spin angular momentumprojected onto the symmetry axis. In particular, when thebroken symmetry is U(1), the conserved quantity can betransported in the form of superfluid. Easy-plane magnets thus can support superfluid spin transport, which is realized by spiraling spin texture within the easy plane [ 7]. Spin superfluidity has been gaining attention in spintronics as anefficient spin-transport channel owing to its slower decayingthan spin transport by quasiparticles such as magnons [ 8–10]. Some magnetic systems have uniaxial anisotropy that is neither easy-axis nor easy-plane anisotropy. Exam-ples of such systems include bilayer Co/Pt [ 11], multi- layer Ta/Co 60Fe20B20/MgO [ 12], and hexagonal compound HoMn 6Sn4Ge2[13] under their favorable conditions, some of which have been theoretically and experimentally investigatedfor a memory unit owing to the ease of switching [ 12,14]. The (a) (b) (c) FIG. 1. (a) Two arrows represent two ground states of easy-axis magnets. (b) The unit circle in the xyplane represents continuously degenerate ground states of easy-plane magnets; one exemplary spin direction is shown as an arrow. (c) Two cones represent the ground- state manifold of easy-cone magnets; two example spin directions aredepicted as arrows. ground states of those magnets are uniform spin states that tilt away from the symmetry axis. The ground states thereby formtwo disconnected cones on the unit sphere, which are referredto as easy cones (see Fig. 1for schematic illustrations of the ground states in uniaxial magnets for comparison of easy-axis,easy-plane, and easy-cone anisotropy. The ground states ineasy-cone magnets break both the Z 2and U(1) symmetries, whereby a domain wall and spin superfluidity can coexist (seeFig. 2for schematic illustrations of a domain wall and spiraling spin texture carrying finite spin supercurrent). In this RapidCommunication, we theoretically study the interaction of adomain wall and spin superfluidity in these systems withinthe Landau-Lifshitz-Gilbert treatment. Specifically, we showthat a domain wall can be driven by spin supercurrent byidentifying a spin-transfer torque from the spin supercurrent tothe domain wall. We also study its reciprocal phenomenon that xyz xyz(a) (b) FIG. 2. Schematic illustrations of (a) a domain wall and (b) spiraling spin texture within the upper cone, which carries finitespin supercurrent. 2469-9950/2016/94(22)/220404(5) 220404-1 ©2016 American Physical SocietyRAPID COMMUNICATIONS SE KWON KIM AND Y AROSLA V TSERKOVNY AK PHYSICAL REVIEW B 94, 220404(R) (2016) FIG. 3. A schematic geometrical setup for a domain-wall motion driven by spin supercurrent. Blue and red colors represent positive and negative spin components projected onto the zaxis, respectively. The black arrow is the spin direction at the center of the domainwall. The charge currents through two adjacent heavy metals inject spin current into the magnet via spin Hall effect. The polarization directions of injected spin are depicted as circles in the metals. Thespin is transported to the domain wall by spin supercurrent in the form of spiraling spin texture. The domain wall moves by absorbing the transported spin. spin supercurrent can be generated by the field-induced motion of a domain wall. We conclude the Rapid Communication bydiscussing other possible consequences of the coexistence ofa domain wall and spin superfluidity. Let us present here one of our main findings, a domain- wall motion driven by spin supercurrent (see Fig. 3for the schematic geometrical setup). The source and drain of spinare realized at the left and right boundaries by sandwichingthe magnet with heavy metals such as Pt or Ta. Via spin Halleffect [ 15], the two-dimensional (2D) charge current density j lin the left metal injects the spin current (polarized along the zaxis)js l=ϑjlsin2θcinto the ferromagnet [ 16], where the coefficient ϑparametrizes the efficiency of conversion from the charge current to the spin current [ 17] andθcis the angle that the easy cones make with the symmetry axis. Likewise,the charge current density j rin the right metal injects the spin current js r=−ϑjrsin2θcinto the magnet. We shall show below that the domain wall absorbs the injected spin currenttransported by spin supercurrent, and thereby moves at thevelocity v=g g2+α2ηvηωsin2θcϑ(jl−jr). (1) Here, g≡2scosθcis the gyrotropic coupling constant be- tween the translational motion of the domain wall and theglobal spin precession about the zaxis [ 18], where sis the scalar spin density per unit volume; αis the Gilbert damping constant; η vandηωare the coefficients that characterize energy dissipation associated with the linear dynamics ofthe domain wall and the global precessional dynamics ofspins, respectively, whose explicit definitions will be givenlater. In the absence of damping, all of the spin current istransported by spin superfluid to the domain wall, which inturn moves at the velocity v=tanθ csinθcϑ(jl−jr)/2sas a consequence of the conservation of spin angular momentum.Finite damping causes partial loss of spin due to the spinprecession ( ∝η ω) and the domain-wall motion ( ∝ηv), which decreases the domain-wall speed.Upadhyaya et al. [19], including us, recently showed that spin supercurrent flowing through an easy-plane magnet candrive a domain wall in an easy-axis magnet, when two magnetsare exchange coupled. In the proposal, the domain-wall speedincreases as the spin current increases in the linear regime.There is, however, a critical spin current that is proportionalto the exchange-coupling strength, above which the domain-wall speed decreases significantly by entering the nonlinearregime. Differing from that, in easy-cone magnets, the domain-wall speed keeps increasing linearly as spin current increaseswithout any breakdown as long as superfluid spin transport isstable [ 20]. Easy-cone magnets. Our model system is a quasi-one- dimensional ferromagnet with easy-cone anisotropy. When theambient temperature is much below the magnetic orderingtemperature, the state of the system is described by theunit vector ˆnalong the local spin density s≡sˆn.I ti s convenient to parametrize ˆnin spherical coordinates θandφ for our discussions: ˆn≡(sinθcosφ,sinθsinφ,cosθ). The potential energy of the system is given by U=/integraldisplay dV[A{(∇θ) 2+sin2θ(∇φ)2}/2+Kf(θ)], (2) where AandKparametrize the spin-direction stiffness and the easy-cone anisotropy, respectively, and the high symmetry axisis defined as the zaxis. Here, a dimensionless function f(θ)i s arbitrary except for the following conditions: it is invariantunder n z/mapsto→−nz, i.e., f(π−θ)=f(θ), and it attains the local minimum only at two points 0 <θc<π / 2 and π−θc. Without loss of generality, it is assumed that the anisotropyenergy vanishes at the minimum points, e.g., f(θ c)=0. A class of the functions given by f(θ)=(sin2θ−sin2θc)2will be used when providing a concrete example. The characteristiclength and energy-density scales of the problem are√ A/K and√ AK, respectively, in which we shall work henceforth. When the ferromagnet is narrow compared to the characteristic lengthscale, variations of the order parameter across the ferromagnetcan be neglected: θ(r,t)=θ(x,t) andφ(r,t)=φ(x,t). The system has two symmetries: the spin-reflection sym- metry through the xyplane, θ(x,t)/mapsto→π−θ(x,t), and the spin-rotational symmetry about the zaxis,φ(x,t)/mapsto→φ(x,t)+ δφ, which we shall refer to as Z 2and U(1) symmetries, respectively. The ground-state manifolds are two cones in theunit sphere that make the angle θ cwith the zaxis [see Fig. 1(c)]. A ground state lies in one of the two cones, whereby it breakstheZ 2symmetry; it takes an arbitrary azimuthal angle φ, whereby it breaks the U(1) symmetry. Our system has two disconnected ground-state manifolds, and thus can harbor a domain wall interpolating betweenthem [ 2]. It is an extremum of the potential energy, which satisfies δU/δθ =−θ /prime/prime+sinθcosθφ/prime2+∂θf=0, (3a) δU/δφ =− (sin2θφ/prime)/prime=0, (3b) with the boundary condition θ(x=− ∞ )=θcand θ(x=∞ )=π−θc. The equilibrium domain-wall solution is 220404-2RAPID COMMUNICATIONS INTERACTION BETWEEN A DOMAIN W ALL AND SPIN . . . PHYSICAL REVIEW B 94, 220404(R) (2016) implicitly given by x−x0=/integraldisplayθ0(x) π/2dθ√2f(θ),φ (x)≡φ0, (4) where x0is the center of the domain wall and φ0is an arbitrary reference angle. The solution θ0(x) can be explicitly obtained for certain cases. For example, when f(θ)=(sin2θ−1/2)2, we have θ0(x)=π−arctan[coth( x/2)]. The explicit solution forθ0(x) is not necessary for our main discussion on the interaction between a domain wall and spin supercurrent,which shall be shown later, and thus we content ourselveswith the implicit solution here. Next, our system has the ground-state manifold with U(1) spin-rotational symmetry, and thus can support spinsupercurrent. To discuss the dynamic steady state that carriesfinite spin supercurrent, let us employ the Landau-Lifhistz-Gilbert (LLG) equations: −ssinθ˙φ−αs˙θ=−θ /prime/prime+sinθcosθφ/prime2+∂θf,(5a) ssinθ˙θ−αssin2θ˙φ=− (sin2θφ/prime)/prime. (5b) The latter equation in the absence of damping α=0 can be interpreted as the continuity equation of the spin angularmomentum projected onto the zaxis: the time evolution of the spin density, s˙n z=−ssinθ˙θ, and the divergence of the spin current density, js≡− sin2θφ/prime, add up to zero. We shall set s=1 hereafter by using s/K as the unit time. We are interested in the nonequilibrium steady state close to the ground state withthe constant polar and azimuthal angles θ(x)≡θ candφ/prime≡0, and thus we expand the LLG equations to the linear order inδ θ≡θ(x,t)−θc,φ/prime, and ˙φ, which results in −sinθc˙φ−α˙δθ=κδθ, (6a) sinθc˙δθ−αsin2θc˙φ=− sin2θcφ/prime/prime, (6b) where κ≡∂2 θf(θc) parametrizes the curvature of the anisotropy at the local minimum point θ=θc. The spin current can be injected by sandwiching the magnet with heavy metals(see Fig. 2for the schematic geometrical setup), the effects of which can be captured by the following boundary conditionsfor the spin current density (projected onto the zaxis): j s(0)=sin2θ(0)[ϑjl−γ˙φ(0)], (7a) js(L)=sin2θ(L)[ϑjr+γ˙φ(L)], (7b) within the linear response [ 9]. Here, ϑis the coefficient parametrizing the dampinglike torque on the magnet due tothe charge current at the interfaces, which is related to theeffective interfacial spin Hall angle /Theta1viaϑ=/planckover2pi1tan/Theta1/2et withtthe thickness of the metals and −ethe charge of electrons; γ≡/planckover2pi1g ↑↓/4πparametrizes the spin pumping at the interfaces with g↑↓the effective interfacial spin-mixing conductance [ 17]. The steady-state solution to the linearized LLG equations ( 6) with the above boundary conditions isgiven by ˙φ(x,t)≡ω=ϑ(jl−jr) 2γ+αL, (8a) js(x,t)=sin2θc[ϑjl−(γ+αx)ω], (8b) with the uniform polar angle δθ(x,t)≡− sinθcω/κ. Note that, by taking the limit θc→π/2, we can recover the result for the global spin-precession frequency in the case of easy-planeferromagnets [ 9]. Domain-wall motion. With the understanding of the physical manifestation of the broken Z 2symmetry—a do- main wall—and that of the broken U(1) symmetry—spinsuperfluidity—now let us turn to our main interest: theinteraction between a domain wall and spin supercurrent.First, we study the motion of a domain wall driven by spinsupercurrent (see Fig. 3for an illustration). Specifically, we look for a steady-state solution to the LLG equations ( 5) that contains a domain wall moving at the velocity vwithin the linear response regime. To that end, we go to the frame movingat the velocity v, which can be implemented by replacing ∂ t by∂t−v∂xin the laboratory-frame LLG equations ( 5). To the linear order in v,˙φ(x,t)≡ω, andφ/prime, the resultant LLG equations are −sinθω+αsinθθ/primev=−θ/prime/prime+∂θf, (9a) −sinθθ/primev−αsin2θω=− (sin2θφ/prime)/prime. (9b) To obtain the equations for vandω, we multiply the former equation by θ/primeand integrate both equations over the spatial dimension, which results in −gω+αηvv=0, (10a) gv+αηωω=[sin2θcφ/prime]x=L x=0, (10b) to the linear order in δθ(0)≡θ(0)−θcandδθ(L)≡θ(L)− (π−θc) (which turned out to not appear in the result). Here, g≡2 cosθcis the gyrotropic coupling constant between v andω[18], and ηv≡/integraldisplayπ−θc θcdθ/radicalbig 2f(θ), (11a) ηω≡sin2θcL+/integraldisplayπ−θc θcdθsin2θ−sin2θc√2f(θ)(11b) parametrize energy dissipation associated with vandω, respectively. In deriving these results, we used θ/prime 0(x)=√2f[θ0(x)] to change the integration over the spatial variable xto the one over the angle variable θ. The latter equation ( 10b) represents the conservation of the spin angular momentum.The right-hand side is the net injection of the spin angularmomentum into the magnet, j s(0)−js(L). The corresponding addition of the spin angular momentum translates into themotion of the domain wall, gv. The Gilbert damping causes a partial loss of spin, αη ωω, which is proportional to the global precession frequency. The former equation representsthe absence of a force on the domain wall. By solving 220404-3RAPID COMMUNICATIONS SE KWON KIM AND Y AROSLA V TSERKOVNY AK PHYSICAL REVIEW B 94, 220404(R) (2016) Eqs. ( 10) subjected to the boundary conditions ( 7), which is invariant under the transformation x/mapsto→x−vtwithin the linear response, we obtain the self-consistent solution for v andω: v=g g2+α2ηvηωsin2θcϑ(jl−jr), (12a) ω=αηv g2+α2ηvηωsin2θcϑ(jl−jr). (12b) This is our first main result (see Fig. 3for the schematic plot for the spatial profile of the spin current js, whose rapid drop in the domain wall represents the spin-transfer torque from thespin current j sto the domain wall). Spin-current generation. Next, as a reciprocal phenomenon, we study spin-current generation by the field-induced domain-wall motion. An external magnetic field in the zdirection en- genders a Zeeman term in the potential energy, −h/integraltext dxcosθ, which creates an extra term, hsinθ, in the right-hand side of the LLG equation ( 5a). The modified equations for vandω are given by −gω+αη vv=2hcosθc, (13a) gv+αηωω=0, (13b) in the absence of the charge current in the attached metals. Here, 2 hcosθcis the force on the domain wall. Solving the above equations with the boundary conditions ( 7), we obtain v=αηω g2+α2ηvηω2hcosθc, (14a) ω=−g g2+α2ηvηω2hcosθc. (14b) The dynamics of spins at the interface injects spin current into the adjacent metals via spin pumping [ 16]. The amount of spin injected into the left and right metals is equal andgiven by −j s(0)=js(L)=γsin2θcω, (15) which can be inferred by measuring induced charge current in the metals via inverse spin Hall effect [ 15]. This is our second main result. Discussion. Let us discuss how easy-cone anisotropy can arise with an example of bilayer Co/Pt [ 11]. The anisotropy energy of the system can be effectively written as κ1sin2θ+ κ2sin4θ. The coefficient of the first term is positive, κ1>0, when the cobalt film is so thin (e.g., 0.5 nm thick) that the termis dominated by the interfacial easy-axis anisotropy. It becomesnegative, κ 1<0, due to the easy-plane shape anisotropy, when the cobalt film is thick enough to have negligible interfaceeffect. The second term comes from the bulk crystallineanisotropy and its coefficient remains positive, κ 2>0, in- dependently of the cobalt thickness. When the thickness istuned to satisfy −2κ 2<κ 1<0, the cobalt has easy-cone anisotropy with the canting angle θc=arcsin√|κ1|/2κ2.F o r example, when the Co and Pt thicknesses are 0.7 nm and1.5 nm, respectively, the coefficients are κ 1=− 30 kJ/m3 xτ(a) (b) xτ Q=−2 3Q=+1 3 FIG. 4. Schematic illustrations of two types of vortices. Spins are in the upper cone away from the vortex centers. Qis the skyrmion charge of a vortex. See the main text for discussions. andκ2=120 kJ /m3, which yields the equilibrium cone angle θc=20◦[11]. To make a simple quantitative estimate for the domain- wall speed induced by spin supercurrent, let us take thefollowing parameters: the saturation magnetization densityM s∼106A/m and the equilibrium cone angle θc∼20◦ measured in bilayer Co 0.7nm/Pt1.5nm [11,21], and the spin Hall angle/Theta1∼0.1 measured in NiFe|Pt interfaces [ 22]. Then, the 2D charge current densities jl=105A/m and jr=0 through the 5-nm-thick platinums will yield the domain-wall speed ofv∼7m/s, when neglecting the Gilbert damping. We have studied the interaction between a domain wall and spin supercurrent in quasi-one-dimensional easy-cone ferro-magnets. The coexistence of spin superfluidity and a domainwall can lead to other possibly interesting phenomena. Forexample, two-dimensional magnets with easy-cone anisotropysupport vortices, and topological defects associated with U(1)symmetry, which can interact with a domain wall. Sincevortices cause phase slips disturbing spin supercurrent [ 7], their interaction may have an interesting effect on phase-slip-induced resistances of spin supercurrent [ 23]. In addition, easy-cone magnets support two types of magnetic vortices,which have different skyrmion-charge magnitudes due to thebroken Z 2symmetry (see Fig. 4for illustrations). These skyrmion charges have important effects on the dynamics ofvortices by determining the gyrotropic coupling between twospatial coordinates [ 18]. V ortices in easy-cone magnets will thus show the two distinct gyrotropic dynamics, which cannotbe observed in easy-plane magnets that can only supportvortices with the skyrmion charges of the same magnitude,Q=± 1/2. Some frustrated magnets also have the ground states characterized by Z 2×U(1), where Z2is often associated with the spontaneous selection of the chirality and U(1) representsthe spin-rotational symmetry about a certain axis [ 24]. It would be a natural question to ask if our theory for the motion ofdomain walls induced by spin-current injection is applicableto those magnets. Acknowledgments. We thank Daniel Hill, Satoru Nakatsuji, and Pramey Upadhyaya for useful discussions. This work wassupported by the Army Research Office under Contract No.911NF-14-1-0016. 220404-4RAPID COMMUNICATIONS INTERACTION BETWEEN A DOMAIN W ALL AND SPIN . . . PHYSICAL REVIEW B 94, 220404(R) (2016) [1] C. Kittel, Introduction to Solid State Physics , 8th ed. (Wiley, New York, 2005), and references therein. [2] M. Kardar, Statistical Physics of Fields (Cambridge University Press, Cambridge, 2007), and references therein. [3] A. Kosevich, B. Ivanov, and A. Kovalev, Phys. 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Kim, and Y . Tserkovnyak, arXiv:1608.00683 [Phys. Rev. Lett. (to be published)]. [20] The velocity of spin in superfluid spin transport, which is the ratio of spin supercurrent to spin density, cannot be larger thanthe speed of spin waves according to the Landau criterion [ 7]. [21] E. Barati, M. Cinal, D. M. Edwards, and A. Umerski, Phys. Rev. B90,014420 (2014 ). [22] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106,036601 (2011 ). [23] S. K. Kim, S. Takei, and Y . Tserkovnyak, Phys. Rev. B 93, 020402(R) (2016 ); S. K. Kim and Y . Tserkovnyak, Phys. Rev. Lett. 116,127201 (2016 ). [24] S. E. Korshunov, Phys. Usp. 49,225 (2006 ), and references therein; O. A. Starykh, Rep. Prog. Phys. 78,052502 (2015 ), and references therein. 220404-5

PhysRevB.101.184404.pdf

PHYSICAL REVIEW B 101, 184404 (2020) Second-order topological solitonic insulator in a breathing square lattice of magnetic vortices Z.-X. Li,1Yunshan Cao,1X. R. Wang,2,3and Peng Yan1,* 1School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China 2Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 3HKUST Shenzhen Research Institute, Shenzhen 518057, China (Received 15 January 2020; accepted 16 April 2020; published 4 May 2020) We study the topological phase in a dipolar-coupled two-dimensional breathing square lattice of magnetic vortices. By evaluating the quantized Chern number and Z4Berry phase, we obtain the phase diagram and identify that the second-order topological corner states appear only when the ratio of alternating bond lengthsgoes beyond a critical value. Interestingly, we uncover three corner states at different frequencies ranging fromsubgigahertz to tens of gigahertz by solving the generalized Thiele’s equation, which has no counterpart in naturalmaterials. We show that the emerging corner states are topologically protected by a generalized chiral symmetryof the quadripartite lattice, leading to particular robustness against disorder and defects. Full micromagneticsimulations confirm theoretical predictions with great agreement. A vortex-based display device is designed asa demonstration of the real-world application of the second-order magnetic topological insulator. Our findingsprovide a route for realizing symmetry-protected multiband corner states that are promising to achieve spintronichigher-order topological devices. DOI: 10.1103/PhysRevB.101.184404 I. INTRODUCTION Over the past few years, the concept of higher-order topological insulators (HOTIs) [ 1–7] has attracted significant attention from the community for the peculiar symmetry-protected states emerging in device corners and hinges. HOTIshave been studied in the broad fields of acoustics [ 8–12], pho- tonics [ 13–17], mechanics [ 18–20], electric circuits [ 21–24], and recently, spintronics [ 25–27]. As a new member of topo- logical insulators (TIs) [ 28,29], HOTIs go beyond the conven- tional bulk-boundary correspondence and are characterized bya few new topological invariants [ 3–5,30,31]. Very recently, it has been suggested that the Z NBerry phase quantized to 2π/Nis a useful tool to characterize the higher-order topolog- ical phase [ 20,26,32–35]. Since topologically protected corner states are robust against disorder and defects, they can actas localized oscillators. Previous work, however, focused oncorner states only with a single frequency [ 8,9,13,14,21,23]. It is intriguing to pursue topologically stable multimode cornerphases for practical applications, such as imaging [ 36]. The chiral edge state of spin waves has been well stud- ied in magnetic systems [ 37–39]. Similar to other (quasi- )particles, the collective motion of magnetic textures, suchas vortex [ 40,41], bubble [ 42,43], and skyrmion [ 44,45], can also exhibit the behavior of waves [ 46–53]. Notably, single- band topological chiral edge states have been demonstratedin a two-dimensional honeycomb lattice of magnetic vortices(or bubbles) by solving the massless Thiele’s equation thatdescribes the coupled dynamics of magnetic solitons [ 54]. We recently generalized the approach by including both a *Corresponding author: yan@uestc.edu.cnsecond-order inertial term and a third-order non-Newtoniangyroscopic term to interpret the emerging multiband nature ofchiral edge states observed in a honeycomb lattice of magneticskyrmions [ 55]. More recently, we have predicted the second- order TIs in a breathing lattice of magnetic vortices [ 25,26] and have shown that the generalized chiral symmetry protectsthe topological corner states. However, the high-frequency-band corner modes have not been well addressed. A prominentdemonstration of the practical application of corner states isstill lacking. In addition, the second-order topological phase ina quadripartite lattice is yet to be reported, although the issuehas received thorough investigations in both tripartite [ 25] and sexpartite [ 26] vortex lattices. In this paper, we present both analytical and numerical studies of the collective dynamics of dipolar-coupled mag-netic vortices in a two-dimensional breathing square lattice.By solving the generalized Thiele’s equation, we obtain theband structure of the collective vortex gyrations. We derivethe full phase diagram of the system by evaluating the topo-logical invariants Chern number and Z 4Berry phase. Two different phases are identified: the trivial insulating phaseand the second-order topological phase with the phase tran-sition crossing the border d 2/d1=1. Here d1andd2are the alternating intercellular and intracellular bond lengths,respectively, as shown in Fig. 1(a). We discover three corner states possessing very different frequencies in a finite latticewhen the system is in the HOTI phase. Full micromagneticsimulations are performed to verify theoretical predictionswith excellent agreement. Finally, we design a vortex-baseddisplay device, as a demonstration of the realistic applicationof the HOTI state. The paper is organized as follows: The model and method are presented in Sec. II. The band structure and associated 2469-9950/2020/101(18)/184404(9) 184404-1 ©2020 American Physical SocietyLI, CAO, WANG, AND YAN PHYSICAL REVIEW B 101, 184404 (2020) FIG. 1. (a) Illustration of the breathing square lattice of magnetic vortices, with d1andd2denoting the alternating lengths of intercellular and intracellular bonds, respectively. The dashed black rectangle is the unit cell for calculating the band structure and topological invariants,witha 1anda2denoting the basis vectors. The inset shows the micromagnetic structure of the vortex, with thickness w=10 nm and radius r= 50 nm. (b) The first Brillouin zone, with the high-symmetry points /Gamma1,K,a n d Mlocating at ( kx,ky)=(0,0), (π a,π a), and (π a,0), respectively. The (lowest-four) band structures along the path /Gamma1-K-M-/Gamma1for different geometric parameters: d1=3.6r,d2=2.4r(c),d1=d2=3.6r(d), andd1=2.08r,d2=3.6r(e). (f) Dependence of Chern number and Z4Berry phase on the ratio d2/d1with d1b e i n gfi x e dt o3 r. topological invariants (Chern number and Z4Berry phase) are evaluated. Section IIIgives the main results, including a theoretical computation of the corner states (Sec. III A ), micromagnetic simulations (Sec. III B ), and HOTI display device design (Sec. III C ). The discussion and conclusion are presented in Sec. IV. The model parameters and theory of generalized chiral symmetry are given in the Appendices. II. MODEL AND METHOD We consider a breathing square lattice of magnetic nan- odisks with vortex states [see Fig. 1(a)]. The generalized Thiele’s equation [ 25,26,55] is adopted to describe the col- lective dynamics of the vortex lattice: G3ˆz×d3Uj dt3−Md2Uj dt2+Gˆz×dUj dt+Fj=0,(1) where Uj=Rj−R0 jis the displacement of the jth vortex core from the equilibrium position R0 j;G=−4πQwMs/γis the gyroscopic constant, with Q=1 4π/integraltext/integraltext dxdy m·(∂m ∂x×∂m ∂y) being the topological charge [ Q=+1/2 for our vortex configuration shown in Fig. 1(a)];mis the unit vector along the local magnetization direction; wis the thickness of the nanodisk; Msis the saturation magnetization; γis the gyromagnetic ratio; Mis the effective mass of the magnetic vortex [ 42,56,57]; andG3is the third-order non- Newtonian gyroscopic coefficient [ 58–60]. The conservativeforce can be expressed as Fj=−∂W/∂Uj.H e r eWis the total potential energy due to both the confinement from a single disk and the interaction between nearest-neighbor disks: W=/summationtext jKU2 j/2+/summationtext j/negationslash=kUjk/2 with Ujk= I/bardblU/bardbl jU/bardbl k−I⊥U⊥ jU⊥ k[54,61,62]. Here,Kis the spring con- stant, and I/bardblandI⊥are the longitudinal and transverse coupling constants, respectively. By imposing Uj=(uj,vj) and defining ψj=uj+ivj,w eh a v e ˆDψj=ωKψj+/summationdisplay k∈/angbracketleftj/angbracketright,l(ζlψk+ξlei2θjkψ∗ k), (2) where the differential operator ˆD=iω3d3 dt3−ωMd2 dt2−id dt, ω3=G3/|G|,ωM=M/|G|,ωK=K/|G|,ζl=(I/bardbl,l− I⊥,l)/2|G|, and ξl=(I/bardbl,l+I⊥,l)/2|G|, in which l=1 (l=2) represents the intercellular (intracellular) connection. θjkis the angle of the direction ˆ ejkfrom the xaxis, where ˆejk=(R0 k−R0 j)/|R0 k−R0 j|and/angbracketleftj/angbracketrightis the set of nearest intercellular and intracellular neighbors of j. Parameters G3, M, andKcan be determined from micromagnetic simulations (see Appendix Afor details). The analytical expression of I/bardbl andI⊥on the distance dbetween vortices has been obtained in a simplified two-nanodisk system [ 25,26]. We then recast the complex variable ψjas ψj=χj(t)e x p (−iω0t)+ηj(t)e x p ( iω0t), (3) withω0the eigenfrequency of a single vortex gyration. For vortex gyrations with Q=+1/2, one can justify |χj|/lessmuch|ηj|. 184404-2SECOND-ORDER TOPOLOGICAL SOLITONIC INSULATOR … PHYSICAL REVIEW B 101, 184404 (2020) By substituting ( 3)i n t o( 2), we have ˆDψj=/parenleftbigg ωK−ξ2 1+ξ2 2 ¯ωK/parenrightbigg ψj+ζ1/summationdisplay k∈/angbracketleftj1/angbracketrightψk+ζ2/summationdisplay k∈/angbracketleftj2/angbracketrightψk−ξ1ξ2 2¯ωK/summationdisplay s∈/angbracketleft/angbracketleftj1/angbracketright/angbracketrightei2¯θjsψs−ξ2 2 2¯ωK/summationdisplay s∈/angbracketleft/angbracketleftj2/angbracketright/angbracketrightei2¯θjsψs−ξ2 1 2¯ωK/summationdisplay s∈/angbracketleft/angbracketleftj3/angbracketright/angbracketrightei2¯θjsψs, (4) where ¯ ωK=ωK−ω2 0ωM,¯θjs=θjk−θksis the relative angle from the bond k→sto the bond j→kwith kbetween jand s, and/angbracketleftj1/angbracketrightand/angbracketleftj2/angbracketright(/angbracketleft/angbracketleftj1/angbracketright/angbracketright,/angbracketleft/angbracketleftj2/angbracketright/angbracketright, and/angbracketleft/angbracketleftj3/angbracketright/angbracketright) are the set of nearest (next-nearest) intercellular and intracellular neighbors of j, respectively. For an infinite lattice, the dashed black rectangle indicates the unit cell, as shown in Fig. 1(a).a1=aˆxanda2=aˆyare two basis vectors, with a=d1+d2. We then obtain the matrix form of the Hamiltonian in momentum space by considering a plane-wave expansion of ψj=φjexp(iωt)e x p [ i(nk·a1+mk·a2)], where kis the wave vector, and nandmare two integers: H=⎛ ⎜⎜⎜⎝Q 0 ζ2+ζ1exp(−ik·a1)ζ2+ζ1exp(ik·a2) Q1 ζ2+ζ1exp(ik·a1) Q0 Q2 ζ2+ζ1exp(ik·a2) ζ2+ζ1exp(−ik·a2) Q∗ 2 Q0 ζ2+ζ1exp(−ik·a1) Q∗ 1 ζ2+ζ1exp(−ik·a2)ζ2+ζ1exp(ik·a1) Q0⎞ ⎟⎟⎟⎠, (5) with elements explicitly expressed as Q 0=ωK−ξ2 1+ξ2 2 ¯ωK−ξ1ξ2 ¯ωK[cos(k·a1)+cos(k·a2)], Q1=ξ1ξ2 ¯ωK[exp( ik·a2)+exp(−ik·a1)]+ξ2 1 ¯ωKexp[ik·(a2−a1)]+ξ2 2 ¯ωK, Q2=ξ1ξ2 ¯ωK[exp( ik·a2)+exp(ik·a1)]+ξ2 1 ¯ωKexp[ik·(a2+a1)]+ξ2 2 ¯ωK. (6) A topological invariant Chern number can be used to judge whether the system is in the first-order TI phase [ 63,64]: C=i 2π/integraldisplay/integraldisplay BZdkxdkyTr/bracketleftbigg P/parenleftbigg∂P ∂kx∂P ∂ky−∂P ∂ky∂P ∂kx/parenrightbigg/bracketrightbigg ,(7) where Pis the projection matrix P(k)=φ(k)φ(k)†, with φ(k) being the normalized eigenstate (column vector) of ( 5), and the integral is over the first Brillouin zone (BZ). However,to determine whether the system allows the HOTI phase, adifferent topological invariant should be considered. In the presence of fourfold rotational ( C 4) symmetry, topological invariant Z4Berry phase is a powerful tool to characterize the HOTI state [ 26]: θ=/integraldisplay L1Tr[A(k)]·dk(mod 2 π), (8) where A(k) is the Berry connection: A(k)=i/Psi1†(k)∂ ∂k/Psi1(k). (9) Here,/Psi1(k)=[φ1(k),φ2(k),φ3(k)] is the 4 ×3 matrix com- posed of the eigenvectors of Eq. ( 5) for the lowest three bands [the corner states exist between the third and fourth bands;see Figs. 2(a) and3(a)].L 1is an integral path in momentum space K/prime→/Gamma1→K/prime/prime[see the green line segment in Fig. 1(b)]. Here, we use the Wilson-loop approach to evaluate the Berryphase θso that the difficulty of the gauge choice can be avoided [ 1,2]. In addition, because of the C 4symmetry, the four high-symmetry points K,K/prime,K/prime/prime, and K/prime/prime/primein the first BZ are equivalent [see Fig. 1(b)]. Therefore, there are three other equivalent integral paths ( L2:K/prime/prime→/Gamma1→K/prime/prime/prime,L3:K/prime/prime/prime→ /Gamma1→K,L4:K→/Gamma1→K/prime) giving rise to identical θ.I ti salso obvious that the integral along the path L1+L2+L3+ L4is zero. Thus, the Z4Berry phase must be quantized as θ=2nπ 4(n=0,1,2,3). Therefore, by simultaneously quan- tifying the Chern number and the Z4Berry phase, we can accurately characterize the emerging topological phases andtheir transition. Of particular interest are the corner states that are related to the symmetry of the system Hamiltonian. One can show thatthe emergence of topological corner states is protected by thegeneralized chiral symmetry of the quadripartite lattice. Thedetailed proof is presented in Appendix B. III. SECOND-ORDER MAGNETIC TOPOLOGICAL INSULATOR A. Theoretical results To investigate different phases allowed in the system, we calculate the bulk band structures with various geometricparameters ( d 1andd2), as shown in Figs. 1(c)–1(e) [we only plot the lowest four bands there]. For d1=d2=3.6r[see Fig. 1(d)], we find that all bands merge together, leading to a gapless band structure. However, when d1/negationslash=d2, two gaps open and locate between the first and second bands, and thethird and fourth bands, respectively [see Figs. 1(c) and1(e)]. Interestingly, the second and third bands are always mergedno matter what values d 1andd2take. To further distinguish whether these insulating phases are topologically protected,we examine simultaneously the topological invariants Chernnumber and Z 4Berry phase. Figure 1(f)plots the dependence of the Chern number C and the Z4Berry phase θon the ratio d2/d1with d1fixed to 3r. In the calculations, the material parameters of Py [ 65,66] 184404-3LI, CAO, WANG, AND YAN PHYSICAL REVIEW B 101, 184404 (2020) FIG. 2. (a) Eigenfrequencies of collective vortex gyrations under different ratios d2/d1in a finite lattice of the size (3 d1+4d2)× (3d1+4d2). We set d1to 2.08rin the calculation. The schematic plot of the vortex lattice in (a) for two limit cases, d2→∞ (b) and d1→∞ (c). FIG. 3. (a) Eigenfrequencies of the square-shaped vortex lattice with d1=2.08randd2=3.6r. The spatial distribution of vortex gyrations for the corner (b), edge (c), and bulk (d) states with the frequency 0.939, 0.986, and 1.047 GHz, respectively.are adopted. We can clearly see that the Z4Berry phase is quantized to 0 when d2/d1<1 and to πotherwise, showing thatd2/d1=1 is the phase transition point separating the trivial and topological phases. In our model, by diagonalizingEq. ( 5), one can obtain the “eigenfrequencies” of the system. However, the left side of Eq. ( 4) is a cubic equation of frequency. Therefore, the true eigenfrequencies of the systemcan be divided into three branches, around 0.939, 11.945, and14.192 GHz, respectively [see Fig. 2(a)], but they share the same eigen wave function. Since the calculations of the Z 4 Berry phase for different band branches are based on the same wave function, all corner states thus have the same topologicalproperties and the Z 4invariant associated to different band branches is identical. Furthermore, it is worth mentioningthat the (total) Chern number in Fig. 1(f)is the sum of the lowest three bands. Since it vanishes, there is no first-ordertopological insulating phase in the bulk gap between the thirdand fourth bands. We indeed have calculated the total Chernnumber and found it vanishes below the bulk gap. The higher-frequency edge states are therefore topologically trivial, too.Therefore, we conclude that the system is in the HOTI phasewhen d 2/d1>1, and in the trivial phase when d2/d1<1. It is worth noting that this conclusion holds independent of thed 1value we choose. To further confirm the existence of corner states in our system, we calculate the eigenfrequencies of collective vortexgyration as a function of d 2/d1in a finite square-shaped lattice [see Figs. 2(b) and 2(c)]. Numerical results are presented in Fig. 2(a). By analyzing the spatial distribution of the eigenfunctions, we identify three different phases: bulk state,edge state, and corner state, marked by black, blue, and redarrows in Fig. 2(a), respectively. The frequencies of three corner states are equal to those of a single vortex gyration(see Appendix A). To provide an intuitive understanding why these corner states only appear in the special parameter region(d 2/d1>1), we plot the configuration of the lattice in the zero-correlation length limit, as shown in Fig. 2(b) [Fig. 2(c)] ford2→∞ (d1→∞ ). On the one hand, from the phase diagram [Fig. 1(f)], we can infer that the configuration shown in Fig. 2(b) is in the HOTI phase. In such a case, we clearly identify four isolated vortices at the corners of the lattice. Onecan thus observe localized corner states. On the other hand,in the limit d 1→∞ [see Fig. 2(c)], there are no uncoupled vortices, thus no corner states. The system is therefore in thetrivial phase. To visualize the second-order corner states, we consider the square-shaped vortex lattice with d 1=2.08randd2=3.6r, as indicated by the vertical dotted black line in Fig. 2(a).F i g - ure3shows the computed eigenfrequencies and eigenmodes. It is found that there exist three corner states with differentfrequencies (0.939, 11.945, and 14.192 GHz), representedby red balls in Fig. 3(a). The spatial distribution of the corner state shows that its oscillation is highly localized atfour corners [see Fig. 3(b)]. We also identify the edge state, denoted by blue balls [see Fig. 3(a)]. The spatial distribution of the edge state are confined along the lattice boundary, asshown in Fig. 3(c). However, these edge states are Tamm- Shockley type [ 67,68] and are not topologically protected because of the vanishing Chern number [see Fig. 1(f)]. The spatial distribution of the bulk state [black balls in Fig. 3(a)] 184404-4SECOND-ORDER TOPOLOGICAL SOLITONIC INSULATOR … PHYSICAL REVIEW B 101, 184404 (2020) FIG. 4. (a) Eigenfrequencies of the square-shaped vortex lattice in the absence of deformations (black balls) and in the presence ofdeformations (red balls). The blue circle indicates the topologically stable corner state with the inset showing the corresponding spatial distribution of vortex gyrations in pristine lattice. (b) Eigenfrequen-cies of the square-shaped vortex lattice under different disorder strengths. is also plotted in Fig. 3(d), where vortices at corners and edges do not participate in the oscillation. The topologically protected corner states are predicted to be robust against disorders in the bulk which do not close theband gap, but sensitive to perturbations on the corner sites. Toexamine whether the corner states emerging in Fig. 3have a topological nature, we calculate the eigenfrequencies of thesquare-shaped vortex lattice under deformations and disorder,with numerical results presented in Figs. 4(a) and4(b),r e - spectively. Here, the deformations are introduced by assuminga shift to the coupling parameters ζandξ, i.e., ζ→1.2ζ andξ→0.8ξ. The disorder is introduced by assuming the resonant frequency ω Kundergoing a random variation, i.e., ωK→ωK(1+δZ), where δis the strength of disorder, and Z is a uniformly distributed random number between −1 and 1, which apply to all vortices except for those in corners.Numerical calculations have been averaged by 100 realiza-tions. From Fig. 4(a), we can clearly see that the frequency of the lowest corner state is perfectly pinned to 0.939 GHz,while the frequencies of edge and bulk states are significantlymodified. Meanwhile, with the increasing of disorder strength,the corner state is very robust [see Fig. 4(b)]. We therefore conclude that the corner states emerging in our system aretopologically stable. It is worth mentioning that the other twohigh-frequency corner states around 11.945 and 14.192 GHzshare similar properties (not shown here). B. Micromagnetic simulations To verify the theoretical predictions, we perform full mi- cromagnetic simulations. All material parameters used inthe simulations are the same as those for the theoreticalcalculations in Fig. 3(a): the saturation magnetization M s= 0.86×106Am−1, the exchange stiffness A=1.3×10−11 Jm−1, and the Gilbert damping constant α=10−4.T h e micromagnetic package MUMAX 3[69] is adopted to simulate the collective dynamics of the vortex lattice. The cell size isset to 2 ×2×10 nm 3. To obtain the full spectrum, we apply a sinc-function magnetic field H(t)=H0sin[2πf(t−t0)]/ FIG. 5. (a) The temporal Fourier spectra of the vortex oscilla- tions at different positions. (b1)–(d3) The spatial distribution of os-cillation amplitude under the exciting field with different frequencies indicated in (a). Since the oscillation amplitudes of the vortex core are too small, we have magnified them by different times labeled ineach figure. [2πf(t−t0)] along the xdirection with H0=10 mT, f= 30 GHz, and t0=1 ns, over the whole system for 1 μs. The position of vortex cores Rj=(Rj,x,Rj,y) in all nan- odisks are recorded every 20 ps. Here, Rj,x=/integraltext/integraltext x|mz|2dxdy/integraltext/integraltext |mz|2dxdy andRj,y=/integraltext/integraltext y|mz|2dxdy/integraltext/integraltext |mz|2dxdy, with the integral region confined in thejth nanodisk. For the model of magnetic vortex, only a very small part of the magnetic moments has the componentsperpendicular to the plane. The closer to the center, the greaterthe component perpendicular to the plane. Therefore, |m z| can be the weight function to define the position of magneticvortices, computationally cheaper than the topological chargedensity. We analyze the temporal Fourier spectra of the vortex oscillations at different positions [labeled by arabic numbers1, 2, and 3; see Fig. 2(b)]. Figure 5(a) shows the spectra, with black, blue, and red curves representing the position ofbulk, edge, and corner bands, respectively. We have magnifiedthe fast Fourier transform power by 20 times at the high- 184404-5LI, CAO, WANG, AND YAN PHYSICAL REVIEW B 101, 184404 (2020) frequency part ( >11 GHz, as marked in the figure) because the vortex-oscillation amplitudes are weak in this region.From Fig. 5(a), we can clearly see that near the eigenfrequen- cies of a single vortex gyration (0.939 and 11.941 GHz), thespectrum for the corner has two very strong peaks, which donot exist for edge and bulk bands. We thus infer that theyare two corner states. Similarly, we identify the frequencyrange supporting the bulk and edge states, around 0.756 GHz(11.102 GHz) and 0.843 GHz (11.433 GHz). Interestingly, forthe 14.189 GHz peak, although the spectrum in the cornerhas a strong peak, the oscillation amplitude at the edge issizable as well, which indicates a strong coupling betweenedge and corner oscillations. Similar mode hybridizationoccurs at 14.025 and 14.213 GHz, too. To visualize thespatial distribution of vortex oscillations for different modesmentioned above, we stimulate the dynamics of the vortexlattice by applying a sinusoidal field h(t)=h 0sin(2πft)ˆx with h0=0.1 mT for 100 ns. Then we plot the spatial distribution of oscillation amplitude for different frequenciesin Figs. 5(b1) –5(d3) . One can distinguish the bulk states [Figs. 5(b1) and5(c1) ], edge states [Figs. 5(b2) and5(c2) ], and corner states [Figs. 5(b3) and 5(c3) ]. The hybridized modes are observed as well: bulk & edge state [Fig. 5(d1) ] and edge & corner state [Figs. 5(d2) and5(d3) ]. It is worth noting that the mode hybridization results from the fact thatthe frequencies for these different states are so close [seeFig.5(a)]. To verify the robustness of the corner states, we perform further micromagnetic simulations. We consider the case inwhich there are two nanodisks that do not host magneticvortices (the nanodisks are uniformly magnetized), which canrepresent a kind of defect. From the micromagnetic simula-tions (not shown), we find that the corner states are still veryrobust against the defect of “null” vortex. C. HOTI display device From the application aspect, it is still a challenging issue to realize the stable display function in natural and artificialmaterials. The main obstacle lies in the difficulty for preciselycontrolling both the position and the frequency of the localoscillations. Topological insulators provide a new route forthat purpose. The chiral topological edge states can perfectlylocalize the energy at the boundary of the system [ 28,29,70– 74]. If the boundary of the system is set to a specific shape, the display function can be realized under the external excitationof chiral edge modes. However, the proposal suffers somedisadvantages. On the one hand, to form a clear picture, a verylarge system needs to be conceived. On the other hand, thetarget picture must be continuous and it is difficult to displaydiscrete pictures. The emerging HOTI can well solve theseproblems. To demonstrate a practical application of topologically stable corner states, we design a display device based on thevortex lattice. The desired display “H” in the HOTI phase issurrounded by another vortex lattice in the trivial phase, asshown in Fig. 6(a). Concretely, we set the distance between the two nearest nanodisks of “tetramer and dimer” to be 2 .08r, while the shortest distance between the display disk and thesurrounding ones is set to be 3 .02r(3.02r/2.08r=1.45>1), FIG. 6. (a) The schematic plot of the vortex lattice for “H” display. (b) Micromagnetic simulation of vortex gyrations with fre- quency f=0.939 GHz. The oscillation amplitudes of the vortex core have been magnified by three times. such that the desired display “H” is in the HOTI phase, surrounded by vortex lattices in the trivial phase. The displaypoints are marked by arabic numbers 1–7. Then we stimulatethe collective dynamics of the whole system by applyinga sinusoidal magnetic field with frequency f=0.939 GHz and amplitude h 0=0.1 mT along the xdirection for 80 ns. Figure 6(b) plots the spatial distribution of the oscillation amplitudes, from which we can clearly see that only thevortices at the desired display points have sizable oscillations,while the other vortices do not participate in the display. Wetherefore demonstrate an “H” display device. Furthermore,we point out that, even without topological corner states, thevortex disks corresponding to “H” are still expected to supportsome localized modes. However, without the topology, the lo-calized mode will be very fragile against external frustrationssuch as disorder and defects. We expect that other displayshapes can be realized by a similar method, too. It is worthnoting that the second corner state (around 11.941 GHz) canbe used for display application as well (not shown). However,the highest corner state (around 14.19 GHz) is not ideal forthis purpose, because of the hybridization between differentmodes [see Figs. 5(d1) –5(d3) ]. IV . DISCUSSION AND CONCLUSION To conclude, we have studied the collective dynamics of a breathing square lattice of magnetic vortices. The full phasediagram was obtained theoretically by computing the Chernnumber and Z 4Berry phase. Two different phases (the trivial and the second-order topological phases) were identified withthe phase transition point at d 2/d1=1. By including higher- order modifications in Thiele’s equation, we predicted theexistence of three corner states with different frequenciesvarying from subgigahertz to tens of gigahertz. The emergingcorner states are shown to be robust against moderate defectsand disorder because of the topological protection from thegeneralized chiral symmetry in quadripartite lattices. Full mi-cromagnetic simulations matched theoretical predictions withan excellent agreement. We finally designed a HOTI displaydevice, as a demonstration of practical applications. From anexperimental point of view, the artificial vortex lattices canbe easily fabricated by electron-beam lithography [ 48,49,75], and the nanometer-scale vortex orbits can be tracked by using 184404-6SECOND-ORDER TOPOLOGICAL SOLITONIC INSULATOR … PHYSICAL REVIEW B 101, 184404 (2020) the ultrafast Lorentz microscopy technique in a time-resolved manner [ 76]. A very recent experiment has created the artifi- cial triangular and square magnetic-soliton lattices patternedby x-ray illumination [ 77]. Up to now, all HOTIs reported are single-frequency states. Our findings of multifrequencycorner states thus shift this well-established paradigm, whichcertainly has theoretical interest. Further, multifrequency cor-ner states can provide a broadband response for topologicallyrobust oscillation of magnetic solitons, which are promisingto design flexible and diverse spintronic higher-order topo-logical devices. The topologically protected multifrequencycorner states also have great potential applications in nano-oscillators. On the one hand, the novel nano-oscillators aretopologically stable against disorder and defects. On the otherhand, the multifrequency nano-oscillators provide a widefrequency range for modulation. We envision the existence ofthird-order topological states in a three-dimensional breathingsquare lattice of magnetic vortices, which is an interestingissue for future study. In addition, the topological property oftwisted bilayer of magnetic soliton lattice is also an appealingresearch topic. ACKNOWLEDGMENTS We thank Zhenyu Wang for helpful discussions. This work was supported by the National Natural Science Foundationof China (NSFC) (Grants No. 11604041 and No. 11704060),and the National Key Research Development Program underContract No. 2016YFA0300801. X.R.W. was supported byHong Kong RGC (Grants No. 16300117, No. 16301518, andNo. 16301619). Z.-X.L. acknowledges the financial supportof the China Postdoctoral Science Foundation (Grant No.2019M663461) and the NSFC (Grant No. 11904048). APPENDIX A: MODEL PARAMETERS The non-Newtonian gyroscopic coefficient G3, effective mass M, and spring constant Kare important parameters for evaluating the band structures. Here we determine theseparameters by considering the dynamics of a single vortexconfined in a nanodisk. We start with the generalized Thiele’sequation ( 1). In this case, the potential energy reads W= W 0+KU2 j/2. Assuming ψj(t)=ψjeiω0t,E q .( 1) can be simplified to G3ω3 0+Mω2 0−Gω0−K=0. (A1) In our system, we have G=−3.0725×10−13Jsr a d−1m−2. Moreover, the three eigenfrequencies for a single vortexoscillation in a nanodisk are [ 25,26]ω 0,1/2π=+0.939 GHz,ω0,2/2π=−11.945 GHz, and ω0,3/2π=+14.192 GHz. Here, the sign +(−) represents the vortex gyration di- rection to be anticlockwise (clockwise). Solving Eq. ( A1) with the three eigenfrequencies, we obtain G3=−4.6488× 10−35Js3rad−3m−2,M=9.3061×10−25kg, and K= 1.8356×10−3Jm−2. APPENDIX B: GENERALIZED CHIRAL SYMMETRY IN QUADRIPARTITE LATTICES Here, we prove that the emerging topological corner states in our system are protected by the generalized chiralsymmetry for quadripartite lattice. First of all, because ξ2 1+ξ2 2 ¯ωK+ξ1ξ2 ¯ωK[cos(k·a1)+cos(k·a2)]/lessmuchωK, the diagonal element of Hcan be regarded as a constant Q0=ωK, which is the “zero energy” of the Hamiltonian. Then we generalizethe chiral symmetry for a unit cell containing four sites bydefining [ 9,78] /Gamma1 −1 4H1/Gamma14=H2, /Gamma1−1 4H2/Gamma14=H3, /Gamma1−1 4H3/Gamma14=H4, H1+H2+H3+H4=0, (B1) with the chiral operator /Gamma14a diagonal matrix to be determined, andH1=H−Q0I. By combining the last equation with the previous three in Eqs. ( B1), we have /Gamma1−1 4H4/Gamma14=H1, indicating that [ H1,/Gamma14 4]=0; so that /Gamma14 4=I, via the reasoning completely analogous to the Su-Schrieffer-Heegermodel [ 79]. In addition, Hamiltonians H 1,2,3,4each have the same eigenvalues λ1,2,3,4. The eigenvalues of /Gamma14are given by 1,exp(2πi/4),exp(πi), and exp(6 πi/4). We thus have /Gamma14=⎛ ⎜⎜⎜⎝10 0 0 0e2πi/400 00 eπi0 00 0 e6πi/4⎞ ⎟⎟⎟⎠, (B2) in suitable bases. 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PhysRevB.97.224410.pdf

PHYSICAL REVIEW B 97, 224410 (2018) Magnetoelastic excitation of single nanomagnets for optical measurement of intrinsic Gilbert damping W. G. Yang, M. Jaris, D. L. Hibbard-Lubow, C. Berk, and H. Schmidt* School of Engineering, University of California Santa Cruz, 1156 High Street, Santa Cruz, California 95064, USA (Received 17 March 2018; published 12 June 2018) We report an alternative all-optical technique to drive and probe the spin dynamics of single nanomagnets. Optically generated surface acoustic waves (SAWs) drive the magnetization precession in nanomagnets viamagnetoelastic (MEL) coupling. We investigate the field-swept dynamics of isolated Ni nanomagnets at variousSAW frequencies and show that this method can be used to accurately determine the intrinsic Gilbert damping ofnanostructured magnetic materials. This technique opens a new avenue for studying the spin dynamics of nanoscaledevices using nonthermal (“cold”) excitation, enabling direct observation of the MEL driven dynamics. DOI: 10.1103/PhysRevB.97.224410 Nanostructured magnetic devices have been identified as leading candidates for next-generation data storage, quantumcomputation, low-power magnetic logic [ 1], and neuromorphic computing [ 2]. The operation characteristics of these technolo- gies are closely related to their dynamic magnetic properties,such as the Gilbert damping parameter ( α), which directly determines the switching-power threshold and operating speedof spin-transfer torque memory [ 3]. Thus, the direct measure- ment of a single nanomagnet is crucial to the development ofspintronic devices, but many conventional techniques do notpossess the sensitivity to resolve the small-angle magnetizationprecession of isolated nanostructures. One approach to over-come this problem is to measure an array of nanomagnetic de-vices. This, however, yields an ensemble signal that is heavilyinfluenced by the magnetostatic interactions of neighboringelements and dynamic dephasing between elements, both ofwhich mask the intrinsic spin dynamics [ 4–6]. Instead, a variety of measurement techniques have been developed to probe thedynamic behavior of magnetic nanostructures [ 7–15], but only a few of these approaches, such as cavity-enhanced magneto-optic Kerr effect (CE-MOKE) [ 16], spin-torque ferromagnetic resonance (ST-FMR) [ 17], and heterodyne magneto-optical microwave microscopy [ 18] (H-MOMM), have been used to accurately determine the damping parameter. Here, weintroduce an alternative technique to excite and probe the mag-netization dynamics of a single nanomagnet. Surface acousticwaves (SAWs) are optically generated on a one-dimensionalnanowire array and used to excite narrow-band magnetizationprecessions in a single, remote nanomagnet via magnetoelastic(MEL) coupling [Fig. 1(a)]. These nonthermal (“cold”) excita- tions can be measured with time-resolved MOKE spectroscopyand are qualitatively different from all-optical TR-MOKEdynamics. Specifically, we use this approach to measure theintrinsic Gilbert damping in single Ni nanomagnets and showthatαcan be extracted directly from the range of applied fields over which the magnetic precession is excited by the SAW. This *hschmidt@soe.ucsc.edudemonstrates that the larger effective damping αeffobserved in optically excited single nanomagnets is due to thermal effects. Previous studies have used SAWs to excite and detect the magnetization dynamics of films [ 19–21] and nanomagnet arrays [ 22–24] and found that a magnetoelastic resonance occurs, which pins magnetic response to the SAW frequency(f SAW) over a range of applied fields. Thus, magnetoelastic coupling has been established as an alternative means to studythe dynamic properties of magnetic. To investigate the SAW-driven magnetization dynamics of a single nanomagnet, a nickel (Ni) nanocylinder (200 nmdiameter, 30 nm thick) was defined between two sets ofidentical, nonmagnetic aluminum (Al) bars [Fig. 1(b)]o na (100) Si substrate capped by a 110-nm-thick hafnium ox-ide antireflection (AR) coating [ 25–27] utilizing multilevel electron beam lithography (EBL), electron beam evaporation,and lift-off processes. Four sets of Al bars (width =100, thickness =30 nm) were fabricated, each with a differ- ent pitch ( p=250,300,350,and 400 nm), corresponding to a distinct SAW frequency ( f SAW=11.45,9.75,8.65, and 7.75 GHz) that is determined by the relation: fSAW=vs/p [22], where vsis the speed of sound along the sample surface. To excite SAWs, the Al bars are illuminated by an ultrafastpump pulse (modulated at 1 kHz), which causes impulsivethermal expansion of the elements that produces a periodicelastic strain along the surface, launching SAWs that propagateat a velocity v s∼3k m/s to the Ni nanomagnet. This method of SAW excitation has distinct advantages over other approaches,such as phononic Bragg mirrors [ 20] and interdigitated trans- ducers (IDTs) [ 21,28]. The former requires complex deposition techniques (e.g., molecular beam epitaxy) to grow super-latticestructures, and has yet to be demonstrated in conjunction withnanopatterned structures. However, IDTs require the use ofexpensive piezoelectric materials that simultaneously generateacoustic and electromagnetic waves when an RF voltage isapplied, which complicates the ensuing magnetization dynam-ics [21]. In addition, these approaches may prohibit the use of an antireflection surface, which is known to enhance themagneto-optic sensitivity [ 27]. Here, two counterpropagating strain waves are generated by identical pump pulses and 2469-9950/2018/97(22)/224410(6) 224410-1 ©2018 American Physical SocietyYANG, JARIS, HIBBARD-LUBOW, BERK, AND SCHMIDT PHYSICAL REVIEW B 97, 224410 (2018) FIG. 1. (a) Schematic plot of all-optical “cold” magnetization ex- citation in single Ni nanomagnet (b) SEM image of a Ni nanomagnet embedded between an Al nanowire array (pitch =300 nm). consequently form a standing wave at the nanomagnet. We use two sets of bars to maximize the SAW strain amplitude[29], and thus the magnetoelastic field ( H MEL), while avoiding ablation of the aluminum bars. The pump is generated bysecond harmonic generation of an ultrafast Ti:Sapphire laserand is subsequently split into two pulses using a beam splitter.Both beams are then focused onto different positions of thesample surface using a microscope objective (M =100X, N.A.=0.9,λ=400 nm, pulse width =150 fs, FWHM ∼ 3.5μm). The center of each pulse is at least 3 μm away from the nanomagnet to ensure there is negligible photoexcitation ofthe spin system. The magnetization dynamics were studied us-ing the time-resolved magneto-optic Kerr effect (TR-MOKE)technique. A mechanically delayed probe pulse [ 22–24,30,31] (λ=800 nm, pulsewidth =150 fs, FWHM ∼0.58μm) is fo- cused onto the nanomagnet and experiences a gyrotropicpolarization rotation upon reflection. Lock-in detection at thepump modulation frequency is used to record the Kerr rotation(magnetic) as well as the elastic motion (nonmagnetic) usingthe difference and sum signal of a balanced photodetector setup[32]. We first discuss the optically excited magnetization dynam- ics of an unpatterned Ni film and an isolated nanomagnet(no Al bars) measured using a standard TR-MOKE setupwith overlapped pump and probe pulses. The pump quasi-instantaneously demagnetizes the sample, and within picosec-onds the magnetization is restored and subsequently follows ahelical trajectory back to the equilibrium orientation [Figs. 2(a) and2(b)][33–35]. The signal is transformed into the frequency domain by applying a Hamming window function and takingthe discrete Fourier transform (DFT) of the time-evolution.The response is the well-known field-dependent Kittel mode[Figs. 2(c)and2(d)]. Now, we turn to our approach of triggering spin dynamics in the single nanomagnet using a magnetoelastic driving force(“cold” excitation), which yields dramatically different results.We plot the MEL driven dynamics in Fig. 2(e)(f mag=fSAW) and observe an increase in the precession amplitude between0</Delta1 t< 1.5 ns. To better understand this behavior, we simul- taneously monitor the transient reflectivity signal [Fig. 2(f)], which reveals that the amplitude of the nonmagnetic oscillationis largest around a pump-probe delay of /Delta1t∼1n s ,a f t e r which the oscillation decays with a characteristic lifetime of1.7 ns. The delayed response matches the time it takes thestrain wave to travel from the center of the pump ( ∼3μm) at the speed of sound ( v s∼3k m/s) to the nanomagnet. It is worth noting that the MEL driven oscillation slightly lagsthe strain, which is consistent with the observations presented in Ref. [ 19]. Furthermore, the magnetic precession persists almost four times longer than the dynamics instigated by ultra-fast demagnetization, indicating highly efficient coupling andqualitatively different magnetization dynamics. In Figs. 2(g) and2(h) we plot the field-dependent, SAW-driven spin and elastic (nonmagnetic) Fourier spectra of the Ni nanomagnetplaced between the Al bars. The oscillatory strain in thenanomagnet generates a magnetoelastic field at the acousticfrequency, resulting in a peak Fourier amplitude when thetwo systems are on resonance. Any field dependence of themagnetic precession frequency has been completely removed,as can be seen in Fig. 2(g). To estimate the MEL field in the nanomagnet, we followed the multistep simulation procedureoutlined in Ref. [ 36] and estimate a peak H MELfield on the order of 100 Oe. Cold excitation does not only provide a different method to drive the magnetization precession, it also allows for adifferent analysis of magnetic material properties. To illustratethis, we consider the damping behavior of the optically excitedspin dynamics. The magnetic precession at each field is fitusing a damped harmonic function: sin(2 πf mag/Delta1t+φ)e−t/τeff, where fmagis the Kittel frequency and τeffis the effective lifetime of the magnetic oscillation. The frequency and lifetimedirectly determine the effective damping via the relation:α eff=(2πfmagτeff)−1. For the film and nanomagnet measured using the conventional TR-MOKE setup, τeffis determined by as follows: 1 /τeff=1/τGilb+1/τext, where τGilbis the lifetime determined by the intrinsic Gilbert damping ( α) in the LLG equation, and τextrepresents extrinsic damping mechanisms that inevitably reduce τeffand typically depend on Happ. In the case of an obliquely magnetized isotropic thin film,such as the unpatterned Ni, scattering between the uniformprecession and degenerate spin waves, known as two-magnonscattering (TMS), is a dominant source of extrinsic damping[37]. Therefore, to accurately estimate αfrom measurements of thermally excited spin dynamics, TMS contribution to thedamping must be extracted from the field dependence of α eff [38]. Using the model presented in Ref. [ 39], the effective damping can be expressed as [ 39–42] αeff=1 2πfmag/bracketleftbigg1 τGilb+1 τTMS/bracketrightbigg ≈1 2πfmag/bracketleftbiggαγ(H1+H2) 2 +N0/summationdisplay kC(k) 2πfmagIm{(4π2(fmag,k2−fmag2 +iδf mag,kfmag)−1}/bracketrightbigg , (1) where H1,2are effective fields that include contributions from the external field and demagnetizing field, N0,C(k),fmag,k, and δfmag,kare the scattering intensity, correlation function, spin wave frequency, and wave-vector-dependent inverse lifetime.The scattering intensity and correlation function are deter-mined via the following: N 0=γ4 3(H1+H2)/angbracketlefth/prime2/angbracketright, (1a) C(k)=2πξ2 [1+(kξ)2]3/2, (1b) 224410-2MAGNETOELASTIC EXCITATION OF SINGLE … PHYSICAL REVIEW B 97, 224410 (2018) FIG. 2. TR-MOKE time traces of the optically (OPT) excited (a) Ni film and (b) isolated nanomagnet (NM), and (c), (d) the corresponding field-dependent Fourier spectra of the film and NM, respectively, measured with a fixed field angle θH=30o. (e) TR-MOKE trace of the MEL driven nanomagnet ( Happ=3.7 kOe), and (f) nonmagnetic signal of the acoustically modulated nanostructure. The illustrations in panels (a), (b) and (e), (f) indicate the pump-probe configurations used and the applied field geometry, respectively. The field-dependent Fourier spectra of the acoustically driven NM shown in (g), which reveals strong MEL enhancement when the magnetic and nonmagnetic modes are degenerate.SAWs are identified by monitoring the transient reflectivity, shown in (h), and therefore do not depend on the applied field. where h/primeis the magnitude of the random inhomogeneous field arising from sample defects, and ξis the correlation length ofh/prime.I nF i g . 3we show the result of fitting Eq. ( 1)t ot h e data allowing h/prime,ξ,andαto vary as fitting parameters and find excellent agreement between theory and experiment using thefollowing values: h /prime=46 Oe,ξ=60 nm ,and most impor- tantlyα=0.03. We refer the reader to Ref. [ 39] for a complete description of the TMS calculation. The measurements on the FIG. 3. Field dependence of the effective damping of the Ni film and single nanomagnet (NM) measured using conventionalTR-MOKE (closed black circles and open pink squares, respectively). For comparison, the damping measured using the MEL approach is included (blue triangles). The effective damping of the film was fitusing Eq. ( 1) to estimate the intrinsic damping (0.03) (red line).isolated single nanomagnet (no Al bars) [Fig. 2(f)] exhibited similar precession frequencies over an extended field range,but consistently larger effective damping values for all appliedfields (Fig. 3). It is worthwhile to discuss the discrepancy between the film and nanomagnet damping behavior, in light of previ-ous studies that have found the intrinsic damping parame-ter of magnetic materials is not affected by nanopatterning[17,18,43]. First, consider the heat generated by the pump pulse in the nanomagnet which can only dissipate into the HfO 2 film and the substrate beneath, whereas the excess thermalenergy in the film can spread laterally because the thermalconductivity ( W) of Ni is nearly 2 orders of magnitude larger (W Ni∼90,W HfO 2∼1W/m×K). Thus, the temperature of the nanomagnet is significantly higher than the film during themeasurement, which could explain the increase in α eff. Indeed, the enhanced damping in the nanostructure is consistent withrecent reports that have shown the intrinsic damping of NiFefilms increases monotonically with sample temperature [ 44]. In addition, we have observed evidence of thermally assistedoxide formation on the nanomagnet surface when illuminatedby the intense pump pulse, which could also contribute to theenhanced damping observed here [ 45,46]. Cold excitation at the SAW resonance frequency, however, produces a completely different result. Fig. 2(g) shows that the spin dynamics are strongly excited over a range of fieldscentered on the resonance field ( H res). We again take the DFT of the SAW driven dynamics at each applied field, butnow consider the complex Fourier amplitude at the excitationfrequency f SAW. The field dependence of the normalized real 224410-3YANG, JARIS, HIBBARD-LUBOW, BERK, AND SCHMIDT PHYSICAL REVIEW B 97, 224410 (2018) and imaginary parts of the DFT can be fitted using the following Lorentzian functions [ 24]: Im{F(mz(t))}=1 2π/Delta1Hp (Happ−Hres)2+/Delta1H2p/slashbig 2,(2) Re{F(mz(t))}=1 π16/Delta1Hp(Happ−Hres) 4(Happ−Hres)2+/Delta1H2p, (3) to determine Hresof the magnetoelastically driven magnetiza- tion dynamics as well as the pinning width ( /Delta1Hp). The latter is directly related to the damping and SAW frequency via therelation [ 24]: /Delta1H p=αeff(Happ)4π γfSAW. (4) We reiterate that for conventional TR-MOKE, the extracted αeffis field-dependent and only converges to the Gilbert damping at high fields [ 24,38]. For the case of cold excitation, however, the slope of the pinning width as a function offrequency can be used to directly evaluate α. We measured the field-dependent dynamics of four nominally identical Ninanomagnets, each surrounded by Al bars with differentpitch to produce a distinct f SAW. The field-dependence of the complex Fourier spectra taken for each sample were fitusing Eqs. ( 2) and ( 3) to extract /Delta1H pat each SAW frequency [Figs. 4(a)–4(d)]. Now, the pinning widths display a linear dependence on the resonance frequency as seen in Fig. 4(i). Per Eq. ( 4), this implies a single damping value for all applied fields. Indeed, we extract a damping value of 0.034, which isnearly identical to the intrinsic damping value determined fromthe fit to the unpatterned film data using Eq. ( 1). The damping extracted from the pinning width of the Ni nanomagnet doesnot depend on the applied field, unlike the conventional TR-MOKE measurements as shown in Fig. 3. This suggests that this technique can be used to directly determine the intrinsicGilbert damping of nanostructures from a single resonance,unobscured by extrinsic contributions which we will brieflydiscuss. First, we note the absence of the y-intercept parameter com- monly included in Eq. ( 4) that is ascribed to inhomogeneous broadening of the resonance. This omission is justified by thefact that the spin wave is necessarily homogenous for a singleeigen-mode, as previous studies on similar nanomagnets havereported [ 17,43,47,48]. Furthermore, we observe no evidence of TMS based on the pinning width analysis of the MELdriven nanomagnet. This can be understood by consideringthe set of discrete wave vectors in the nanomagnet determinedvia the relation k=nπ/L , where Lis the relevant lateral dimension and nis an odd integer as discussed in refs [ 49,50]. It is well known [ 51,52] that the spin wave frequencies increase monotonically for all kthat satisfy 2Aexk2 Ms/greatermuch1, where Aex∼6×107erg/cm is the exchange constant of Ni. For the 200 nm nanomagnet here, we calculate2Aexk2 Ms≈100n2, which suggests TMS is not operative because there are nodegenerate spin waves to facilitate scattering [ 53,54]. Last, we consider the extrinsic contribution to the damping arising from FIG. 4. Field dependence of the normalized complex Fourier spectra (imaginary Fourier component–circles; real Fouriercomponent–squares) of the MEL driven dynamics at (a)–(d) four distinct SAW frequencies, and (e)–(h) four distinct applied field geometries using a single sample (pitch =400 nm). (i) Pinning width determined by fitting both the real and imaginary Fourier spectra from panels (a)–(d) plotted against f SAW, including the fit to Eq. ( 3) (red dashed line) used to estimate the damping, and (j) summary of/Delta1H pfrom panels (a), (e)–(h) plotted against θH. The data exhibits no significant variation of the pinning width as a function of the applied field angle, which supports the interpretation that the relationshipbetween αand/Delta1H pfor the nanomagnet is not complicated by extrinsic mechanisms. intralayer spin-pumping between the center and edge modes in the nanomagnet [ 18]. Based on the single mode behavior shown in Fig. 2(f), we conclude that this mechanism is likely inconsequential here [ 37]. To further support the conjecture that this method provides a direct measurement of α, we now discuss the MEL driven dynamics measured at various θHfor the sample with Al bars on a pitch of 400 nm. The results are summarized in Fig. 4(j) and show nearly identical /Delta1Hpandαvalues for all θH, which is again consistent with the notion that there are no significantextrinsic contributions to the damping. These findings are instark contrast with Ref. [ 24], which found that α effdecreased monotonically with increasing SAW frequency (or increasingH app). However, in that study the initial conditions were vastly different because the nanomagnets being probed were also thesource of the SAWs, thus the magnetic and acoustic oscillationswere simultaneously excited by rapid thermalization. Hence,the advantages of our nonlocal, magnetoelastic approach areclear. This technique provides the single nanomagnet sensi-tivity of established TR-MOKE detection and well-defined,resonant excitation of the spin dynamics without heating themagnetic system, thereby yielding a direct measurement of theGilbert damping parameter. In conclusion, we have demonstrated an alternative tech- nique that utilizes nonlocally generated SAWs to drive nar-rowband “cold” excitation of the magnetization precession in 224410-4MAGNETOELASTIC EXCITATION OF SINGLE … PHYSICAL REVIEW B 97, 224410 (2018) a remote single nanomagnet. Using this method, we report the first time-resolved measurements of MEL driven magnetiza-tion dynamics of a single Ni nanomagnet. We showed thatthe intrinsic Gilbert damping can be directly extracted fromthe applied field range over which magnetic precessions areexcited. This finding is in stark contrast from optically excitedprecessions of single nanomagnets which show larger effectivedamping that is likely due to the thermal character. In additionto providing an alternative method to probe transient magneticbehavior in nanostructured materials, elastically driven spindynamics may be used to improve the switching behavior of magnetic elements for low-energy data storage and memorydevices. We acknowledge T. Yuzvinsky and the W. M. Keck Center for Nanoscale Optofluidics at UC Santa Cruz. This work wassupported by the National Science Foundation under GrantsNo. ECCS-1509020 and No. DMR-1506104 and the SGMIprogram by Samsung Inc. W.G.Y . and M.J. contributed equally to this work. [1] B. 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PhysRevB.102.184432.pdf

PHYSICAL REVIEW B 102, 184432 (2020) Nutation resonance in ferromagnets Mikhail Cherkasskii ,1,*Michael Farle ,2,3and Anna Semisalova2 1Department of General Physics 1, St. Petersburg State University, St. Petersburg 199034, Russia 2Faculty of Physics and Center of Nanointegration (CENIDE), University of Duisburg-Essen, Duisburg 47057, Germany 3Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk 660036, Russia (Received 31 August 2020; revised 6 October 2020; accepted 4 November 2020; published 30 November 2020) The inertial dynamics of magnetization in a ferromagnet is investigated theoretically. The analytically derived dynamic response upon microwave excitation shows two peaks: ferromagnetic and nutation resonances. Theexact analytical expressions of frequency and linewidth of the magnetic nutation resonance are deduced fromthe frequency-dependent susceptibility determined by the inertial Landau-Lifshitz-Gilbert equation. The studyshows that the dependence of nutation linewidth on the Gilbert precession damping has a minimum, whichbecomes more expressive with increase of the applied magnetic field. DOI: 10.1103/PhysRevB.102.184432 I. INTRODUCTION Recently, the effects of inertia in the spin dynamics of fer- romagnets were reported to cause nutation resonance [ 1–12] at frequencies higher than the conventional ferromagneticresonance. It was shown that inertia is responsible for thenutation, and that this type of motion should be considered to-gether with magnetization precession in the applied magneticfield. Nutation in ferromagnets was confirmed experimentallyonly recently [ 2], since nutation and precession operate at sub- stantially different timescales, and conventional microwaveferromagnetic resonance (FMR) spectroscopy techniques donot easily reach the high-frequency (sub-Terahertz) regimerequired to observe the inertia effect which in addition yieldsa much weaker signal. Similar to any other oscillatory system, the magnetization in a ferromagnet has resonant frequencies usually studiedby ferromagnetic resonance [ 13,14]. The resonant eigenfre- quency is determined by the magnetic parameters of thematerial and applied magnetic field. Including inertia of themagnetization in the model description shows that nutationand precession are complementary to each other and severalresonances can be generated. In this paper, we concentrate onthe investigation of the resonance characteristics of nutation. The investigation of nutation is connected to the progress made in studies of the spin dynamics at ultrashort timescales[15,16]. These successes led to the rapid development of a new scientific field, the so-called ultrafast magnetism [ 17–25]. The experimental as well as theoretical investigation of theinertial spin dynamics is at the very beginning, although it *Corresponding author: macherkasskii@hotmail.com Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.might be of significance for future high speed spintronics applications including ultrafast magnetic switching. Besides nutation driven by magnetization inertia, several other origins of nutation have been reported. Transient nu-tations (Rabi oscillations) have been widely investigated innuclear magnetic resonance [ 26] and electron spin resonance [27–29], they were recently addressed in ferromagnets [ 30]. A complex dynamics and Josephson nutation of a local spins=1/2 as well as large spin cluster embedded in the tunnel junction between ferromagnetic leads was shown to occur dueto a coupling to Josephson current [ 31–33]. Low-frequency nutation was observed in nanomagnets exhibiting a nonlinearFMR with the large-angle precession of magnetization wherethe onset of spin wave instabilities can be delayed due togeometric confinement [ 34]. Nutation dynamics due to inertia of magnetization in ferromagnetic thin films was observed forthe first time by Neeraj et al. [2]. The microscopic derivation of the magnetization inertia was performed in Refs. [ 3–7]. A relation between the Gilbert damping constant and the inertial regime characteristic timewas elaborated in Ref. [ 3]. The exchange interaction, damp- ing, and moment of inertia can be calculated from firstprinciples as shown in Ref. [ 7]. The study of inertia spin dynamics with a quantum approach in metallic ferromag-nets was performed in Ref. [ 8]. In addition, nutation was theoretically analyzed as a part of magnetization dynamicsin ferromagnetic nanostructure [ 9,10] and nanoparticles [ 11]. Despite these advances, exact analytical expressions for thehigh-frequency susceptibility including inertia had not beenderived yet. In Ref. [ 35], the inertial regime was introduced in the framework of the mesoscopic nonequilibrium thermodynam-ics theory, and it was shown to be responsible for the nutationsuperimposed on the precession of magnetization. Wegroweand Ciornei [ 1] discussed the equivalence between the in- ertia in the dynamics of uniform precession and a spinningtop within the framework of the Landau–Lifshitz–Gilbertequation generalized to the inertial regime. This equation 2469-9950/2020/102(18)/184432(5) 184432-1 Published by the American Physical SocietyCHERKASSKII, FARLE, AND SEMISALOV A PHYSICAL REVIEW B 102, 184432 (2020) was studied analytically and numerically [ 12,36]. Although these reports provide numerical tools for obtaining resonancecharacteristics, the complexity of the numerical solution ofdifferential equations did not allow to estimate the nutationfrequency and linewidth accurately. Also in a recent remark-able paper [ 37] a novel collective excitation–the nutation wave–was reported, and the dispersion characteristics werederived without discussion of the nutation resonance line-shapes and intensities. Thus, at present, there is a necessity to study the reso- nance properties of nutation in ferromagnets, and this paperis devoted to this study. We performed the investigation basedon the Landau-Lifshitz-Gilbert equation with the additionalinertia term and provide an analytical solution. It is well known that the Landau-Lifshitz-Gilbert equation allows finding the susceptibility as the ratio between the time-varying magnetization and the time-varying driving magneticfield (see, for example, Refs. [ 38,39] and references therein). This susceptibility describes well the magnetic response of aferromagnet in the linear regime, that is a small cone angle ofthe precession. In this description, the ferromagnet usually isplaced in a magnetic field big enough to align all atomic mag-netic moments along the field, i.e., the ferromagnet is in thesaturated state and the magnetization precesses. The applieddriving magnetic field allows one to observe FMR as soon asthe driving field frequency coincides with eigenfrequency ofprecession. Using the expression for susceptibility, one canelaborate such resonance characteristics as eigenfrequencyand linewidth. We will present similar expressions for thedynamic susceptibility, taking nutation into account. II. SUSCEPTIBILITY The ferromagnet is subjected to a uniform bias magnetic fieldH0acting along the z-axis and being strong enough to initiate the magnetic saturation state. The small time-varyingmagnetic field his superimposed on the bias field. The coupling between impact and response, taking into accountprecession, damping, and nutation, is given by the InertialLandau-Lifshitz-Gilbert (ILLG) equation dM dt=−|γ|M×/bracketleftbigg Heff−α |γ|M0/parenleftbiggdM dt+τd2M dt2/parenrightbigg/bracketrightbigg ,(1) where γis the gyromagnetic ratio, Mthe magnetization vector, M0the magnetization at saturation, Heffthe vector sum of all magnetic fields, external and internal, acting uponthe magnetization, αthe Gilbert damping, and τthe inertial relaxation time. For simplicity, we assume that the ferromag-net is infinite, i.e., there is no demagnetization correction,with negligible magnetocrystalline anisotropy, and only theexternally applied fields contribute to the total field. Thus,the bias magnetic field H 0and signal field hare included in Heff. We assume that the signal is small |h|/lessmuch|H0|,hence the magnetization is directed along H0. Our interest is to study the correlated dynamics of nutation and precession simultaneously; therefore we write the magne-tization and magnetic field in the generalized form using the Fourier transformation M(t)=M0ˆz+1√ 2π/integraldisplay∞ −∞dω/primem(ω/prime)eiω/primet, (2) Heff(t)=H0ˆz+1√ 2π/integraldisplay∞ −∞dω/primeh(ω/prime)eiω/primet, (3) where ˆ zis the unit vector along the zaxis. If we substitute these expressions in the ILLG equation and neglect the smallterms, then it leads to 1 √ 2π/integraldisplay∞ −∞dω/primeiω/primem(ω/prime)eiω/primet =1√ 2π/integraldisplay∞ −∞dω/primeeiω/primet[−|γ|M0ˆz×h(ω/prime) +|γ|H0ˆz×m(ω/prime)+iαω/primeˆz×m(ω/prime) −ατω/prime2ˆz×m(ω/prime)]. (4) By performing the Fourier transform and changing the order of integration, Eq. ( 4) becomes 1 2π/integraldisplay∞ −∞dω/prime/integraldisplay∞ −∞dtiω/primem(ω/prime)ei(ω/prime−ω)t =1 2π/integraldisplay∞ −∞dω/prime/integraldisplay∞ −∞dtei(ω/prime−ω)t ×[−|γ|M0ˆz×h(ω/prime)+|γ|H0ˆz×m(ω/prime) +iαω/primeˆz×m(ω/prime)−ατ(ω/prime)2ˆz×m(ω/prime)], (5) where the integral representation of the Dirac δfunction can be found. With the δfunction, Eq. ( 5) simplifies to iωm(ω)=−|γ|M0ˆz×h(ω)+|γ|H0ˆz×m(ω) +iαωˆz×m(ω)−ατω2ˆz×m(ω). (6) By projecting to Cartesian coordinates and introducing the circular variables for positive and negative circular polariza-tionm ±=mx±imy,h±=hx±ihy,one obtains −ατm+ω2+(−m++iαm+)ω+(ωHm+−ωMh+)=0, −ατm−ω2+(m−+iαm−)ω+(ωHm−−ωMh−)=0, (7) where ωH=|γ|H0is the precession frequency and ωM= |γ|M0.The small-signal susceptibility follows from these equations: m±=χ±h±, χ+=ωM ωH−ω−ατω2+iαω, χ−=ωM ωH+ω−ατω2+iαω. (8) It is seen that the susceptibility Eq. ( 8) is identical with the susceptibility for LLG equation, if one drops the inertial term,that is τ=0. 184432-2NUTATION RESONANCE IN FERROMAGNETS PHYSICAL REVIEW B 102, 184432 (2020) FIG. 1. (a) The FMR peak with nutation. (b) The nutation reso- nance. The calculation was performed for |γ|/(2π)=28 GHz T−1, μ0M0=1T,μ 0H0=100 mT ,α=0.0065, and τ=10−11s. Let us separate dispersive and dissipative parts of the sus- ceptibility χ±=χ/prime ±−iχ/prime/prime ±, χ/prime +=−ωM(ω−ωH+ατω2) D+,χ/prime/prime +=αωω M D+, χ/prime −=ωM(ω+ωH−ατω2) D−,χ/prime/prime −=αωω M D−, (9) D+=α2τ2ω4+2ατω3+(α2−2ατω H+1)ω2 −2ωHω+ω2 H, (10) D−=α2τ2ω4−2ατω3+(α2−2ατω H+1)ω2 +2ωHω+ω2 H. (11) The frequency dependence of the dissipative parts of sus- ceptibilities χ/prime/prime +andχ/prime/prime −is shown in Fig. 1. The plus and minus subscripts correspond to right-hand and left-hand directionof rotation. Since the denominators D +andD−are quartic polynomials, four critical points for either χ/prime/prime +orχ/prime/prime −can be expected. Two of them that are extrema with a clear physicalmeaning are plotted. In Fig. 1(a) the extremum, corresponding to FMR at ω /prime H≈|γ|H0is shown. Due to the contribution of nutation, the frequency and linewidth of this resonance areslightly different from the ones of usual FMR. The resonanceoccurs for right-hand precession, i.e., positive polarization. In Fig. 1(b) the nutation resonance possessing negative polarization is presented. Note that the polarizations of fer-romagnetic and nutation resonances are reversed. III. APPROXIMATION FOR NUTATION FREQUENCY Let us turn to the description of an approximation of the nutation resonance frequency. If we equate the denominatorD −to zero, solve the resulting equation, we obtain the ap- proximation from the real part of the roots. This is reasonable,since the numerator of χ /prime/prime −is the linear function of ω, and we are interested in ω/greatermuch1.Indeed, the equation α2τ2ω4−2ατω3+(α2−2ατω H+1)ω2 +2ωHω+ω2 H=0 (12) has four roots that are complex conjugate in pairs w−FMR 1,2=1±iα−/radicalbig 1−α2+4ατω H±2iα 2ατ, (13) wN1,2=1±iα+/radicalbig 1−α2+4ατω H±2iα 2ατ. (14)One should choose the same sign from the ±symbol in each formula, simultaneously. The real part of Eq. ( 13)g i v e s the approximate frequency for FMR, but in negative numbers,so the sign should be inversed. The approximate frequencyof FMR in positive numbers can be derived from equationD +=0.The approximate nutation frequency is obtained by the real part of Eq. ( 14). One takes half the sum of two conjugate roots wN1,2,neglects the high τterms, and obtains the nutation resonance frequency wN=1+√1+2ατ|γ|H 2ατ. (15) Note that the expression of wNis close to the approxima- tion given in Ref. [ 36]a tτ/lessmuch1/α|γ|H,namely, ωweak nu=√1+ατ|γ|H ατ. (16) The similarity of both approximations becomes clear, if we perform a Taylor series expansion and return to the notationω H, wN=1+√1+2ατω H 2ατ=1 ατ+ωH 2−ατω2 H 4 +1 4α2τ2ω3 H+O(α3τ3), ωweak nu=√1+ατω H ατ=1 ατ+ωH 2−ατω2 H 8 +1 16α2τ2ω3 H+O(α3τ3). IV. PRECISE EXPRESSIONS FOR FREQUENCY AND LINEWIDTH OF NUTATION The analytical approach proposed in this Letter yields pre- cise values of the frequency of nutation resonance and the fullwidth at half maximum (FWHM) of the peak. The frequencyis found by extremum, when the derivative of the dissipativepart of susceptibilities Eq. ( 9) is zero, ∂χ /prime/prime − ∂ω=0. (17) It is enough to determine zeros of the numerator of the derivative, that are given by 3α2τ2ω4−4ατω3+(α2−2ατω H+1)ω2−ω2 H=0. (18) Let us use Ferrari’s solution for this quartic equation and introduce the following notation: Ar=3α2τ2,Br=− 4ατ, Cr=α2−2ατω H+1, Er=−ω2 H,ar=Cr Ar−3B2 r 8A2r, br=−BrCr 2A2r+B3 r 8A3r,cr=B2 rCr 16A3r−3B4 r 256A4r+Er Ar.(19) In Ferrari’s method, one should determine a root of the nested depressed cubic equation. In the investigated case, theroot is written y r=−5ar 6+Ur+Vr, (20) 184432-3CHERKASSKII, FARLE, AND SEMISALOV A PHYSICAL REVIEW B 102, 184432 (2020) where Ur=3/radicalBigg −/radicalbigg P3r 27+Q2r 4−Qr 2,Vr=−Pr 3Ur, Pr=−a2 r 12−cr,Qr=1 3arcr−a3 r 108−b2 r 8. (21) Thus, the precise value of the nutation frequency is given by /Omega1N=−Br 4Ar+√ar+2yr 2+1 2/radicalBigg −3ar−2yr−2br√ar+2yr. (22) The performed analysis shows that approximate value of nutation resonance frequency is close to precise value. The linewidth of the nutation resonance is found at a half peak height. If one denotes the maximum by X/prime/prime −= χ/prime/prime −(ω=/Omega1N),the equation which determines frequencies at half magnitude is 1 2X/prime/prime −/bracketleftbig α2τ2ω4−2ατω3+/parenleftbig α2−2ατω H+1/parenrightbig ω2 +2ωωH+ω2 H/bracketrightbig −αωω M=0. (23) We repeat the procedure for finding solutions with Ferrari’s method introducing the following new notations: Alw=1 2α2τ2X/prime/prime −,Blw=−ατX/prime/prime −, Clw=1 2X/prime/prime −/parenleftbig α2−2ατω H+1/parenrightbig ,Dlw=ωHX/prime/prime −−αωM Elw=1 2ω2 HX/prime/prime −,alw=Clw Alw−3B2 lw 8A2 lw, blw=−BlwClw 2A2 lw+B3 lw 8A3 lw+Dlw Alw, clw=B2 lwClw 16A3 lw−3B4 lw 256A4 lw−BlwDlw 4A2 lw+Elw Alw. (24) A root of the nested depressed cubic equation ylwmust be found in the same way as provided in ( 20) with the corre- sponding replacement of variables, i.e. subscript ris replaced bylw. The difference between two adjacent roots gives the nutation linewidth /Delta1/Omega1 N=/radicalBigg −3alw−2ylw−2blw√alw+2ylw. (25) The explicit expression for the linewidth can be written using the Eqs. ( 19)–(25). The effect of the inertial relaxation time on the nutation linewidth is shown in Fig. 2. One can see that increasing iner- tial relaxation time leads to narrowing of the linewidth. Thisbehavior is expected and is consistent with the traditional viewthat decreasing of losses results in narrowing of linewidth. Since the investigated oscillatory system implements si- multaneous two types of motions, it is of interest to study theinfluence of the Gilbert precession damping parameter αon thenutation resonance linewidth. The result is presented in FIG. 2. The dependence of the nutation linewidth on the inertial relaxation time for μ0H0=100 mT ,μ 0M0=1T,andα=0.0065. Fig. 3and is valid for ferromagnets with vanishing anisotropy. One sees that the dependence of /Delta1/Omega1 Nonαshows a minimum that becomes more expressive with increasing bias magneticfield. In other words, the linewidth is parametrized by themagnitude of field. This nontrivial behavior of linewidthrelates with the nature of this oscillatory system, which per-forms two coupled motions. To elucidate the nontrivial behavior, one can consider the susceptibility ( 9) in the same way as it is usually performed for the forced harmonic oscillator with damping [ 40]. For this oscillator, the linewidth can be directly calculated from thedenominator of the response expression once the driving fre-quency is equal to eigenfrequency. In the investigated case ofmagnetization with inertia, the response expression is Eq. ( 9) with denominator Eqs. ( 10) and ( 11) written as D ±=α2τ2ω4±2ατω3+(α2−2ατω H+1)ω2 ∓2ωHω+ω2 H. (26) Since the applied magnetic field is included in this expres- sion as ωH=|γ|H0,the linewidth depends on the field. The obtained result can be generalized to a finite sample with magnetocrystalline anisotropy with method of effectivedemagnetizing factors [ 41,42]. In this case the bias magnetic FIG. 3. The dependence of nutation resonance linewidth on pre- cession Gilbert damping parameter at different magnetic fields H0for μ0M0=1Ta n d τ=10−11s. 184432-4NUTATION RESONANCE IN FERROMAGNETS PHYSICAL REVIEW B 102, 184432 (2020) fieldH0denotes an external field and in the final expressions this field should be replaced by Hi0=H0−(ˆNa+ˆNd)M0, where ˆNais the anisotropy demagnetizing tensor and ˆNdis the shape demagnetizing tensor. V. CONCLUSION In summary, we derived a general analytical expression for the linewidth and frequency of nutation resonance in ferro-magnets, depending on magnetization, the Gilbert damping,the inertial relaxation time and applied magnetic field. Weshow the nutation linewidth can be tuned by the appliedmagnetic field, and this tunability breaks the direct relation be-tween losses and the linewidth. This, for example, leads to the appearance of a minimum in the nutation resonance linewidthfor the damping parameter around α=0.15.The obtained results are valid for ferromagnets with vanishing anisotropy. ACKNOWLEDGMENTS We thank Benjamin Zingsem for helpful discussions. In part funded by Research Grant No. 075-15-2019-1886 fromthe Government of the Russian Federation, the DeutscheForschungsgemeinschaft (DFG, German Research Founda-tion), Projects No. 405553726 (CRC/TRR 270) and No.392402498 (SE 2853/1-1). [1] J.-E. Wegrowe and M.-C. Ciornei, Am. J. Phys. 80, 607 (2012) . [2] K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Zhou Hagström, S. S. P. K. Arekapudi, A. Semisalova, K. Lenz, B. Green,J. C. Deinert, I. Ilyakov, M. Chen, M. Bawatna, V . Scalera, M.d’Aquino, C. Serpico, O. Hellwig, J. E. Wegrowe, M. Gensch,and S. Bonetti, Nat. Phys. (2020), doi: 10.1038/s41567-020- 01040-y . [3] M. Fähnle, D. Steiauf, and C. Illg, Phys. Rev. B 84, 172403 (2011) . [4] D. Thonig, O. 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V Pisarev, A. M. Balbashov, and T. Rasing, Nature 435, 655 (2005) . [16] C. D. Stanciu, F. Hansteen, A. V Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, 047601 (2007) . [17] B. Koopmans, M. van Kampen, J. T. Kohlhepp, and W. J. M. de Jonge, P h y s .R e v .L e t t . 85, 844 (2000) . [18] B. Koopmans, J. J. M. Ruigrok, F. D. Longa, and W. J. M. de Jonge, P h y s .R e v .L e t t . 95 , 267207 (2005) . [19] A. Kirilyuk, A. V Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010) . [20] M. Battiato, K. Carva, and P. M. Oppeneer, Phys. Rev. Lett. 105, 027203 (2010) . [21] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. Dürr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, A.Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, and A. V . Kimel,Nature 472, 205 (2011) .[22] C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast, K. Holldack, S. Khan, C. Lupulescu, E. F. Aziz, and M. Wietstruk,Nat. Mater. 6, 740 (2007) . [23] B. Koopmans, G. 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PhysRevB.101.054438.pdf

PHYSICAL REVIEW B 101, 054438 (2020) Crystallization of optically thick films of Co xFe80−xB20: Evolution of optical, magneto-optical, and structural properties Apoorva Sharma ,1Maria A. Hoffmann ,2Patrick Matthes ,3Olav Hellwig ,1,4Cornelia Kowol,2Stefan E. Schulz,2,3 Dietrich R. T. Zahn ,1and Georgeta Salvan1 1Institute of Physics, Chemnitz University of Technology, 09126 Chemnitz, Germany 2Center for Microtechnologies, Chemnitz University of Technology, 09126 Chemnitz, Germany 3Fraunhofer Institute for Electronic Nanosystems, 09126 Chemnitz, Germany 4Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany (Received 10 October 2019; accepted 24 December 2019; published 26 February 2020) CoFeB alloys are highly relevant materials for spintronic applications. In this work, the crystallization of CoFeB alloys triggered by thermal annealing was investigated by x-ray diffraction techniques and scanningelectron microscopy, as well as spectroscopic ellipsometry and magneto-optical Kerr effect spectroscopy forannealing temperatures ranging from 300 to 600 ◦C. The transformation of ∼100-nm-thick Co xFe(80−x)B20films from amorphous to polycrystalline was revealed by the sharpening of spectral features observed in optical andmagneto-optical dielectric functions spectra. The influence of B on the dielectric function was assessed bothexperimentally and by optical modeling. By analyzing the Drude component of the optical dielectric function,a consistent trend between the charge-carrier scattering time/resistivity and the annealing temperature wasobserved, in agreement with the electrical investigations by means of the four-point-probe method. DOI: 10.1103/PhysRevB.101.054438 I. INTRODUCTION In the last few decades, 3 dtransition-metal borides have gained considerable interest due to their highly customiz-able mechanical, electrical, thermal, and magnetic propertiescompared to generic 3 dtransition metals and alloys [ 1–3]. One of such 3 dtransition-metal borides is CoFeB, which has received special attention not only from the fundamentalresearch point of view, but also in industrial applications[4–6]. The increasing interest in CoFeB alloys relates to their atypical properties, such as structurally smooth growth [ 7], soft magnetic properties [ 8], high spin polarization [ 9], and very low Gilbert damping [ 10], which makes them especially suitable for magnetic tunnel junction devices [ 8–11]. By ex- ploiting the benefits mentioned above, Ikeda et al. in the year 2005 presented a milestone improvement in the tunnel magne-toresistance ratio (TMR) in Co 40Fe40B20/MgO/Co 40Fe40B20 magnetic tunnel junction of 355% at room temperature (RT)[12]. The improved TMR ratio was ascribed to the improve- ment in the texture of the MgO barrier due to the CoFeBelectrodes. In the same year, Djayaprawira et al. reported that a 20% inclusion of B in the CoFe maintains it amorphousduring the deposition, thereby preventing any lattice mismatch issues at the interface with MgO [ 7]. This allows using thin MgO grown with a well-defined (001) texture as a template forthe CoFeB crystallization induced by a postdeposition thermaltreatment. In 2008, a TMR ratio of ∼600% at RT was reported in Co 20Fe60B20/MgO/Co20Fe60B20annealed at 525◦C[13]. On the other hand, it has also been observed that annealingat higher temperature could induce interlayer diffusion [ 14], resulting in a degradation of the TMR ratio. Therefore, itis important to understand the influence of temperature andcomposition on the crystallization of Co xFe(80−x)B20alloys in detail. Previous studies mainly focused on assessing thecrystallization of CoFeB alloys using x-ray diffractometry(XRD) [ 15], transmission electron microscopy (TEM) [ 16], resistivity [ 3,17], magnetoresistance [ 18], and magnetometry measurements [ 19]. However, these techniques inherit some limitations, regarding the sample volume required for ob-taining a reliable signal (e.g .,XRD), are invasive (TEM), require complex microfabrication processes for realizing de-vices (magnetoresistance measurements), or provide only anindirect indication of the crystallization (electrical and mag- netic measurements). In this work, we propose a nondestructive, high-precision, swift, and highly sensitive approach to probe the crystalliza-tion of Co xFe(80−x)B20based on spectroscopic ellipsometry (SE) and magneto-optical Kerr effect (MOKE) spectroscopy.Spectroscopic ellipsometry has proven to be a very sensitivemethod for investigating thin films [ 20], also providing the possibility of probing changes in the crystalline structure, asinvestigated for Si [ 21], diblock polymers [ 22], or organic photovoltaic devices [ 23]. Similar to SE, MOKE spectroscopy has also demonstrated its efficacy in probing not only thestructural changes, but also as being a highly sensitive toolto investigate the local environment effects. For instance,Bräuer et al. showed that MOKE spectroscopy can be used as a suitable method to determine the orientation of metal-free phthalocyanine molecules on various substrates [ 24]. Furthermore, the systematic study by Tikuišis et al. on a permalloy film shows that the MOKE spectroscopy and themagneto-optical dielectric function are strongly influenced bysurface oxidation [ 25], thus revealing a superior sensitivity of the MOKE spectroscopy to the changes in the surrounding of 2469-9950/2020/101(5)/054438(10) 054438-1 ©2020 American Physical SocietyAPOORV A SHARMA et al. PHYSICAL REVIEW B 101, 054438 (2020) the ferromagnetic material. So far, only few studies addressed the optical and magneto-optical properties of CoFe alloys [26–28] or CoFeB [ 29,30]. In a recent study, we reported on the sensitivity of SE and MOKE spectroscopy with re-spect to the crystallization of thin films of Co 60Fe20B20[31]. Here, we exploit this sensitivity to extract information relatedto the influence of the B content in the amorphous alloyson the crystallization process. In addition, we demonstratethat the Drude contribution to the dielectric functions and thecorresponding parameters (resistivity and scattering time) issusceptive to the crystallization onset. II. METHODS A. Structural and electrical investigations Magnetron sputtering was used to deposit 100-nm-thick CoxFe(80−x)B20films with x=40 and 60, as well as Co 50Fe50, from composite targets on silicon wafers with native siliconoxide. The layers were passivated with 5 nm of Pt to preventoxidation of the CoFeB. The deposition was performed atRT with a base pressure below 2 ×10 −4Pa and Ar working pressure of 0.35 Pa. The wafers were diced in 1 cm × 1 cm pieces, and each individual piece was then annealed for 30 min at temperatures in the range of 300 to 600◦C in steps of 50 K in an high vacuum (HV) (10−5Pa) oven. To improve thermal conductivity between the heater and sample a thinlayer of silver epoxy was used. The samples were investigatedex situ after each annealing step. The Co 50Fe50sample served as a standard, and no further annealing was performed. XRD measurements were conducted using a SmartLab diffractometer from Rigaku, equipped with a rotating Cuanode operated at 9 kW. XRD in θ-2θgeometry, grazing- incidence XRD (GIXRD), and x-ray reflectometry (XRR)measurements were performed to probe the crystallization,crystallite size, and thickness of the films while always usinga parallel beam. The crystallite size ( L) was calculated using the Scherrer formula L≈Kλ Cu /Delta1(2θ)cosθ, (1) with a shape factor of crystallites K≈0.9 considering cu- bic crystallites, the wavelength of x-ray radiation λCu≈ 0.154 nm, and /Delta1(2θ) as the full width at half maximum (FWHM) of the reflex at θ, given in radians [ 32]. The information about the thickness of the layers was extracted from simulating the reflectance scans using GENX [33]. The thickness values obtained from XRR of Co 50Fe50, Co40Fe40B20,C o 60Fe20B20, and Pt of as-deposited samples are shown in Table I. The cross-sectional morphologies of TABLE I. XRR-determined thickness of the Co xFe(80−x)B20, Co50Fe50, and Pt layers for the as-deposited samples. Thickness Sample ID tCoFeB (nm) tPt(nm) Co50Fe50 61.5±13 .64±1 Co40Fe40B20 97.4±54 .78±1 Co60Fe20B20 104.6±54 .81±1the Co xFe(80−x)B20films were inspected using an Auriga 60 scanning electron microscope (SEM) from Zeiss equippedwith a focused ion beam (FIB). The surface of the samples wasstudied by atomic force microscopy (AFM) in AC mode withan Agilent 5500 Scanning Probe Microscope, using reflectiveSi AFM probes. The sheet resistance of all samples annealed at different temperatures was measured with the four-point probe tech-nique using a home-built test bench, consisting of four gold-coated copper probes arranged in an equally spaced ( d∼ 1 mm) collinear manner. B. (Magneto-)optical methods and modeling Spectroscopic ellipsometry measurements were performed using an M-2000 ellipsometer from J. A. Woollam over thespectral range of 0.7 to 5 eV , with varying angles of incidence,in the range of 45 ◦–75◦in steps of 5◦. In order to determine the dielectric function ( εxx=ε1xx+iε2xx) of CoFeB from the measured /Psi1and/Delta1spectra, an optical model analogous to the physical layer structure was devised in the modelingand simulation tool Complete EASE ®. Thus, a “Si/SiO 2(1.8 nm)/CoFeB( tCoFeB )/Pt(tPt)/ surface roughness” layered optical model was built using the reported dielectric function of Si[34], SiO 2[34], and Pt [ 35] layers. Additionally, the layer thicknesses determined by XRR and the surface roughnessdetermined by AFM were used in the optical model andwere kept unchanged throughout the analysis. The unknowndielectric function of CoFeB was expressed in terms of theLorentz-Drude model [Eq. ( 2)] [36]. This model is composed of a Drude function to account for the free-charge-carriercontribution and two Lorentzian oscillators to describe thedispersion arising from interband transitions. This model wasfurther adjusted in terms of the Drude and Lorentzian param-eters to respond to the structural changes resulting from theannealing. ε(E)=ε Drude+εLorentz, (2) where ε(E) is the complex dielectric function, Eis the pho- ton energy, εDrude andεLorentz are the Drude and Lorentzian contribution to the dielectric spectrum. The version of the Lorentzian oscillator used here is a mathematical equation based on the Newton equation of mo-tion that defines the influence of the electric field on the boundelectrons. ε Lorentz (E)=AγEo E2o−E2−iEγ, (3) where A,γ, and Eoare the amplitude, the FWHM, and the center energy position of the oscillator, respectively. The classical Drude equation defines the free-charge- carrier concentration contribution to the dielectric function,which in its mathematical form is equivalent to a Lorentzianoscillator positioned at 0 eV: ε(E)=−¯h ε0ρ(τsE2+i¯hE), (4) where ε0is the vacuum dielectric constant, ¯ his the reduced Planck constant, τsis the mean scattering time of the free carriers between successive collisions and ρis the resistivity. 054438-2CRYSTALLIZATION OF OPTICALLY THICK FILMS OF … PHYSICAL REVIEW B 101, 054438 (2020) FIG. 1. X-ray diffraction patterns recorded for substrate/Co xFe(80−x)B20/Pt before and after annealing at the indicated temperatures for (a)x=40% and (b) x=60%. Additionally, the scan of the as-deposited Co 50Fe50/Pt sample is presented in black in (a) for reference. The respective reflexes of the constituent materials are marked by dashed lines along with the respective Miller indices. A home-built MOKE spectrometer in polar geometry (pMOKE) was utilized to measure the photon energy-dependent Kerr rotation ( θ K) and ellipticity ( ηK)[37]. Both θK(E) andηK(E) were recorded with an out-of-plane applied magnetic field of H∼1.8 T. By magnetizing the layers nor- mal to the sample surface and assuming optical isotropy of thestudied CoFeB alloys, the dielectric tensor can be formulatedas ε=⎛ ⎜⎝ε xxεxy 0 −εxyεxx 0 00 εxx⎞ ⎟⎠. (5) The diagonal components of the dielectric tensor ( εxx)a r e obtained from the SE measurements, while the off-diagonalcomponent ( ε xy=ε1xy+iε2xy), reflecting the magneto- optical response of the films, are calculated from the recordedθ KandηK, using a point-by-point fitting method described elsewhere [ 38], considering the same optical layer model as used in SE. III. RESULTS AND DISCUSSION A. X-ray diffractometry Figure 1presents the XRD θ-2θscans of the CoFeB samples annealed in vacuum at different temperatures. Thepronounced CoFe(110) reflex observed at 500 ◦C and above indicates crystallization of the films. In accordance with pre-vious studies, the creation of a crystalline alloy from theinitial CoFeB compound occurs while boron diffuses outof the lattice resulting in pure CoFe crystals surrounded byamorphous boron [ 39,40]. Furthermore, a closer look at the diffractograms indicates that Co 40Fe40B20crystallizes in a polycrystalline fashion, as the present (110) and (211) peakscorrespond to different crystallographic orientations of body-centered cubic (bcc) CoFe. Co 60Fe20B20, on the other hand, reveals a strong (110) texture, with more intense (110) and(220) peaks occurring. The strong (110) texture was alsoconfirmed by additional rocking-scan analysis of the (110) out-of-plane crystallite orientation distribution (see the fol-lowing discussion). At temperatures above 550 ◦C, a shift and broadening of the Pt(111) peak are found for both stoichiome-tries, most probably suggesting a degradation of the Pt layer,possibly due to alloying or intermixing at the interface withCoFeB. Here, it is worth mentioning that three stray reflexesat 61 ◦,4 3◦, and 97◦are from silicon (400) due to Cu-K β radiation, and Ag(200) as well as Ag(400) from silver epoxy,respectively. The vertical coherence lengths, corresponding to the crys- tallite size ( L) in the normal direction to the sample surfaces, were calculated from the FWHM of the Co 50Fe50(110) peak. For the investigated films, a maximum crystallite size ofaround (25 ±2) nm is obtained for annealing temperatures of 600 ◦C, as shown in Fig. 2(a), which is consistent with previ- ously reported studies on 100 nm thick CoFeB films [ 16]. As detected by cross-section scanning electron microscopy stud-ies [see Fig. 2(b)], the CoFe alloy does not fully crystallize within the 30 min applied annealing steps. The crystallizationstarts from the top interface with Pt and expands for 25–30nm, in agreement with the vertical coherence length of thecrystallites determined from XRD. The observation of a distinct out-of-plane (110) texture for- mation for the Co 60Fe20B20film in the Bragg scans of Fig. 1is further confirmed and supported by additional rocking scans(Omega scans) that directly reveal the actual crystallite orien-tation distribution. Figure 3shows the rocking scan profiles for the Co 60Fe20B20film, which exhibit a clear preferred out-of-plane (110) texture formation, once annealed to 500◦C. The transition from the polycrystalline structure to the (110)textured structure occurs very suddenly, as confirmed by thedramatic shape change of the rocking-scan profile from 450 to500 ◦C. After an initial FWHM of about 12◦after annealing to 500◦C the (110) crystallite out-of-plane alignment improves further to a FWHM of below 9° after annealing to 600◦C. In contrast to this strong texture formation of the Co 60Fe20B20 film above 450◦C, the Co 40Fe40B20film reveals the same 054438-3APOORV A SHARMA et al. PHYSICAL REVIEW B 101, 054438 (2020) FIG. 2. (a) Co 50Fe50crystallite sizes normal to the film plane calculated using the Scherrer expression for Co 40Fe40B20and Co 60Fe20B20 and the XRD patterns shown in Fig. 1. (b) SEM micrograph collage of substrate/Co 40Fe40B20/Pt before and after annealing recorded in the FIB trench at 36° stage tilt. polycrystalline rocking-scan characteristics for all annealing temperatures (not shown here). Only an overall strong in-crease in the rocking-scan intensity above 450 ◦C also con- firms for the Co 40Fe40B20film an increased coherence length, i.e., increased size of the randomly oriented crystallites. To investigate near-surface changes in the layer and to avoid the intense peak from the silicon substrate, GIXRDwas performed. The differences in the crystallization of bothCoFeB compositions become even more pronounced after theanalysis of GIXRD scans at fixed /Omega1=1 ◦, shown in Fig. 4.A l l the aforementioned CoFe peaks are present after annealing at500, 550, and 600 ◦Cf o rC o 40Fe40B20,a sw e l la sf o rC o 50Fe50 in the as-deposited state [see Fig. 4(a)], confirming the poly- crystalline nature of the CoFe alloy in this composition. On FIG. 3. Omega scans (rocking curve) measured at the Co50Fe50(110) reflex for the Co 60Fe20B20. The inset shows the FWHM determined by using a Gaussian fit to the measured rocking-scan profiles.the contrary, for Co 60Fe20B20shown in Fig. 4(b), none of the CoFe peaks are detected as the film is well (110) textured andthus the Bragg condition for CoFe crystallites is not fulfilledfor any detector angle due to the fixed incident angle of/Omega1=1 ◦andχ=ϕ=0◦. The Pt passivation layer deposited on the top of Co 40Fe40B20layer exhibits polycrystallinity even at the highest annealing temperature, whereas the Pt layer ontop of Co 60Fe20B20was noticed to transform to (200) texture upon annealing. B. Spectroscopic ellipsometry Using the optical model discussed in the experimental section, the complex dielectric functions ( ε1xxandε2xx)w e r e determined for the two investigated CoFeB stoichiometriesand Co 50Fe50. For the ease of discussion, the spectra can be divided into two main regions: (i) the near-infrared (NIR)region below 1.0 eV , accounting for intraband transitions,and (ii) the visible and ultraviolet (UV) region above 1.0 eV ,related mainly to interband contributions. The NIR regionof the spectrum is described by a Drude-type contribution,related to the free-electron absorption in Co 50Fe50, and will be discussed in more detail in the following. In the case of Co 50Fe50, the Drude contribution is followed by a broad structure centered at around ∼1.5e V i n t h e ε1xx spectra (corresponding feature at ∼2e V i n ε2xxspectra (see Fig.5). This feature was previously ascribed to the hybridiza- tion of panddorbitals, resulting in direct interband transitions from occupied dto unoccupied pstates in CoFe alloys with a bcc crystalline phase [ 26,28]. In order to understand the influence of B inclusions on the optical properties of the Co 50Fe50, the complex dielectric function of (Co 50Fe50)+B was simulated. For this purpose, the Bruggemann effective medium approximation approachwas used to calculate the optical constants of the mixedmaterial with the host matrix of Co 50Fe50with B inclusion. In this approach, 15% of the (Co 50Fe50)+B film volume is assumed to be a spherical inclusion of B in the metallicCo 50Fe50. However, it should be noted that this is only a 054438-4CRYSTALLIZATION OF OPTICALLY THICK FILMS OF … PHYSICAL REVIEW B 101, 054438 (2020) FIG. 4. GIXRD scans at /Omega1=1◦recorded before and after annealing at various temperatures for (a) Co 40Fe40B20and (b) Co 60Fe20B20.T h e expected positions of the XRD peaks are marked by the dotted lines along with the respective Miller indices. coarse approximation of the actual situation; previous studies suggested that B migrates to CoFe grain boundaries or tothe neighboring layers [ 39,40]. The most obvious change induced to the dielectric function spectra of Co 50Fe50by the B inclusion is visible in the ε2xxspectrum, namely a decrease of the absolute values. Since B is a nonmetallic material,its addition to the metallic Co 50Fe50increases the dielectric losses. Consistently, the values of ε1xxincrease, indicating an increase in the relative permittivity of CoFeB. A slightbroadening of the spectral features is also observed, but rathernegligible when compared with the changes in the features ofthe dielectric functions of the CoFeB alloys extracted from theexperimental ellipsometry spectra before and after annealing(see Fig. 6). The good correspondence between the simulated complex dielectric function of B incorporated in CoFe and thedielectric function determined for the Co 40Fe40B20indicates that during the crystallization process CoFe crystallites are FIG. 5. The complex dielectric function ( ε1xxandε2xx) spectra of the Co 50Fe50(red), Co 40Fe40B20(blue) annealed at 600◦Ca n dB (gray) [ 42], together with the simulated ε1xxandε2xxof (Co 50Fe50)+ B with 15% B content (yellow).formed and B migrates outside the crystallites, i.e., to the grain boundaries. This scenario is in line with the results of previousstudies of the local structure of CoFeB [ 39] and for crystalline CoCrPt-B alloys used for recording media in hard-disk drives[41]. The dielectric functions of the as-deposited CoFeB alloys present only weak and very broad spectral features, whichgradually become more pronounced with an increase in an-nealing temperature, as shown in Fig. 6. The characteristic spectral feature of Co 50Fe50at∼1.5 eV occurs in the ε1xx spectra for the samples annealed at 450◦C. This suggests that 450◦C is the onset temperature for crystallization. As the optical spectroscopy has an information depth limited toa few 10 nm, the changes visible in the spectra at 450 ◦C indicate that the crystallization takes place near the sur-face (Co xFe(80−x)B20/Pt interface), as supported by scanning electron microscopy images [cf. Fig. 2(b)] demonstrating a nucleation at the Pt interface. Noticeably, the pronouncedCoFe reflex was observed in XRD scans starting at 500 ◦C, indicating that the optical spectroscopy allows probing theincipient phase of crystallization with very small crystallites.In fact, a remarkable resemblance of the dielectric function ofthe Co 40Fe40B20after annealing at 600◦C and the as-deposited Co50Fe50is found, which is consistent with the similarities in the crystalline structure observed with XRD. This suggeststhat at 600 ◦C B diffuses completely out of the CoFe crystal- lites. The systematic decrease in ε2xxwith annealing tempera- ture is furthermore consistent with a greater ordering withinthe films due to crystallization. The characteristic spectralfeature in ε 1xxspectra of Co 60Fe20B20is redshifted relative to Co40Fe40B20, probably due to the difference in the stoichio- metric composition. Additionally, comparing the amplitudesof the ε 2xxspectra (mostly <1 eV) of the two stoichiometries, it is evident that the Co 40Fe40B20has lower dielectric losses due to the free electrons in comparison to Co 60Fe20B20.T h i s , in turn, implies that increasing Co concentration increasesthe charge-carrier concentration, which is consistent with theempirical finding that the resistivity of Co is almost half ofthat of Fe [ 43,44]. 054438-5APOORV A SHARMA et al. PHYSICAL REVIEW B 101, 054438 (2020) FIG. 6. Annealing temperature dependence of (a) ε1xxand (b) ε2xxspectra for Co 40Fe40B20(solid lines) and Co 60Fe20B20(dashed lines), and Co 50Fe50(black). The analysis of the Drude contribution to the dielectric function allows deriving the resistivity ( ρ) and scattering time (τs) of the investigated films. The resistivity and the scattering time ultimately relate to the ordering state of the films, ac-cording to the Fuchs size-effect theory [ 45]. These parameters are shown in Fig. 7(a) for both CoFeB stoichiometries. The resistivity remains barely unchanged until 400 ◦C, followed by a maximum at 450◦C and a subsequent decrease with increasing annealing temperature. It should be noted that thisevolution cannot be explained by the B diffusion since inRef. [ 14] we showed that the migration of B starts already at 200 ◦C. The decrease in resistivity can be ascribed to an increase in ordering and decrease in the number of defects,which, in fact, is consistent with the increase in the crystallitesize derived from the XRD measurements. The presence ofa maximum at 450 ◦C relates very likely to a temperature of nucleation of the crystallites, where the electrical resistiv-ity increases due to the formation of grain boundaries anddefects, originating from the low level of ordering of thecrystal. The poly-textured phase in Co 40Fe40B20(in contrast to the well-oriented phase in Co 60Fe20B20) could arguably also explain the difference in the resistivity between the twoalloys since more mismatched grain boundaries and defectswould lead to higher resistivity due to the shorter mean-freepath. Sheet resistance ( R /square) measurements were conducted on all the samples in order to investigate the influence of an-nealing on the electrical properties of the layers. The changein sheet resistance of the CoFeB samples with annealingtemperature is shown in Fig. 7(b). Up to 400 ◦C no significant FIG. 7. (a) Drude parameters resistivity ρand scattering time τsas a function of annealing temperature for Co 40Fe40B20(solid symbol) and Co60Fe20B20(empty symbol). The lines in the figure are guides to the eye. (b) Sheet resistance of the Co 40Fe40B20(filled circles in red) and Co60Fe20B20(unfilled circles in blue) layers passivated with a Pt thin film as a function of the annealing temperature. 054438-6CRYSTALLIZATION OF OPTICALLY THICK FILMS OF … PHYSICAL REVIEW B 101, 054438 (2020) FIG. 8. Polar Kerr effect measured (a) polarization rotation ( θK) and ellipticity ( ηK) spectra of as-deposited Co 50Fe50in comparison to literature values [ 46,47] and (b) calculated ( hν)2εxyas a function of photon energy, indicating present p-dhybridization. change in the sheet resistance is found. Above this tempera- ture, a monotonous decrease with increasing temperature isobserved, consistent with changes observed for the opticalresistivity parameter calculated from the Drude model [cf.Fig.7(a)]. Given the increase in crystallite size and ordering within the films with the annealing temperature revealed byXRD, a decrease in the scattering due to defects and grainboundaries is expected, which results in a decrease of thesheet resistance. The trend in R /squarefor both stoichiometries is noticeably similar to the change in ρobtained from the SE measurements. C. Magneto-optical spectroscopy Figure 8(a)shows the measured θKandηKMOKE spectra of Co 50Fe50, with a comparison of θKreported by Weller et al. for Co 48Fe52[46]. Even though the amplitude of θKisslightly lower than previously reported [ 46], the line shapes of both experiments resemble each other closely. In fact, thepresent data are closer to the first-principle calculations per-formed by Maurer et al. for this CoFe composition [ 47]. The off-diagonal dielectric function of Co 50Fe50was calculated and is shown in Fig. 8(b) as (hν)2εxy, in order to highlight the spectral features [ 48]. It is well established by theoret- ical studies that the spin-polarized density of states of 3 d transition metals and their alloys are fairly similar, resultingin similar electronic transitions in magneto-optical spectra[28]. These spectral features noticed in the optical region of the spectrum can be explained based on the theoreticalpredictions by K. J. Kim et al. for Fe 3Co and Co 3Fe using the tight-binding linear-muffin-tin orbitals method with thelocal spin-density approximation [ 26,47]. They assign the transition at 2 eV as originating mainly from transitions fromthe occupied minority-spin dtriplet states at lower energy into FIG. 9. Polar Kerr effect measured polarization rotation ( θK) spectra for the Pt capped 100-nm-thick CoFeB film before and after annealing at various temperatures for the two stoichiometries, (a) Co 40Fe40B20and (b) Co 60Fe20B20. 054438-7APOORV A SHARMA et al. PHYSICAL REVIEW B 101, 054438 (2020) FIG. 10. Calculated ( hν)2εxyas the function of photon energy for the two stoichiometry Co 40Fe40B20and Co 60Fe20B20for the samples annealed at 450 and 600◦C. the unoccupied minority-spin pstates. These d→ptransi- tions in the minority-spin bands become possible through p-d hybridization. Figure 9shows the evolution in θKspectra for the CoFeB samples annealed at different temperatures for the inves-tigated two stoichiometries. Similar to the SE spectra, nosignificant changes in θ KandηKspectra were observed up to 400◦C. Upon annealing at 450◦C, the characteristic line shape of the θKspectrum starts resembling that of Co 50Fe50. Annealing at higher temperatures results in the enhancementof spectral features at ∼2 and ∼4.7 eV. This is consistent with an increasing crystalline ordering of CoFe. It can alsobe observed that these features are slightly redshifted forCo 60Fe20B20compared to Co 40Fe40B20, contrary to previous theoretical calculations, where no significant differences werefound on the MOKE spectra of different CoFe content [ 28]. In fact, this shift may be as well related to the differencesfound by XRD in the crystalline structure of both compounds.In this context, we note that besides composition, also thecrystalline environment influences the magneto-optical prop-erties of the material significantly [ 49]. The larger amplitude of the spectral features of Co 40Fe40B20is, furthermore, an indication of higher magnetization for the lower Co con-tent, which is consistent with the calculated Slater-Paulingcurve [ 50]. The calculated ( hν) 2εxyas a function of the photon en- ergy for the two CoFeB compositions annealed at 450 and600 ◦C is shown in Fig. 10. Similar to the off-diagonal di- electric function of Co 50Fe50, the real part of εxyshows two main features in the measured spectral range, at ∼2 and ∼4.5e V . As discussed earlier, the features in the spectra reflect the density of states of the occupied part of the 3 dband and are ascribed to the transition to empty hybridized pzstates near the Fermi energy. The relative shifts in the positionsof the spectral features of Co 60Fe20B20to lower energies with respect to the Co 40Fe40B20case indicate that the density of states strongly depends on the stoichiometry. This shiftwas further explained by Liu and Singh with the theoreticalcalculation of the electronic structure of CoFe alloys [ 51]. Though the density of states near the Fermi energy is similarfor both Co and Fe, due to the higher electronegativity and asmaller exchange splitting of Co, the minority-spin orbitals ofCo are situated at lower energy as compared to the Fe orbitals.Hence a higher percentage of Co in the alloy will lead to ashift of the states to lower energies. The change in amplitudeof the spectral features relates, as mentioned previously, to themagnetization, increasing in the case of 600 ◦C annealing with increasing Fe content [ 50]. IV . CONCLUSION In this work, the optical- and magneto-optical properties of optically thick (100 nm) films of Co xFe(80−x)B20(x=40 and 60%) passivated with a 5 nm Pt cap layer were investigatedin the as-deposited amorphous state and upon subsequentannealing steps between 300 and 600 ◦C. The structural and electrical properties of the films were assessed by XRDand electrical four-point probe measurements, respectively.The comparison of the Co xFe(80−x)B20dielectric function ex- tracted from spectroscopic ellipsometry with that of Co 50Fe50 allowed us to identify CoFe specific spectral features and toanalyze the impact of B on the optical properties of the CoFeBalloys. The (magneto-)optical spectroscopic techniques are proven to be extremely sensitive to structural changes. The analysis of the Drude component of the dielectric function of CoFeB allowed extracting information regardingthe resistivity and charge-carrier scattering time, which isclosely related to the crystalline order in the films. It was thuspossible to identify 450 ◦C as the temperature at which nu- cleation of CoFe crystallites occurs. Corroborating the resultsof spectroscopic ellipsometry, SEM and XRD demonstratethat the nucleation of the crystallization starts at the interfacebetween the Co xFe(80−x)B20and the crystalline Pt capping layer. The magneto-optical off-diagonal component of the dielec- tric function of Co xFe(80−x)B20extracted from the MOKE 054438-8CRYSTALLIZATION OF OPTICALLY THICK FILMS OF … PHYSICAL REVIEW B 101, 054438 (2020) spectra shows significant changes with the composition of the alloy as well as with the amorphous to crystalline structuralevolution. This study underlines the utility of spectroscopic ellipsom- etry and MOKE spectroscopy for material optimization in thefield of metallic alloys for spintronic applications.ACKNOWLEDGMENT This work was supported by the Deutsche Forschungsge- meinschaft (DFG) under the project “Interfacial perpendicularmagnetic anisotropy for next-generation monolithic 3D TMRsensors” (Project No. 282193534) and by the FraunhoferInternal Programmes under Grant No. MEF 836303. [1] P. Mohn, J. Phys. C: Solid State Phys. 21,2841 (1988 ). [2] Y . Bourourou, L. Beldi, B. Bentria, A. Gueddouh, and B. Bouhafs, J. Magn. Magn. Mater. 365,23(2014 ). [3] W. Kettler, R. Wernhardt, and M. Rosenberg, J. Appl. Phys. 53, 8248 (1982 ). [4] T. Kawahara, K. Ito, R. Takemura, and H. Ohno, Microelectron. Reliab. 52,613(2012 ). [5] P. P. Freitas, R. Ferreira, and S. Cardoso, Proc. IEEE 104,1894 (2016 ). [6] C. Zheng, K. Zhu, S. Cardoso de Freitas, J.-Y . Chang, J. E. Davies, P. Eames, P. P. Freitas, O. Kazakova, C. 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PhysRevB.97.224514.pdf

PHYSICAL REVIEW B 97, 224514 (2018) Devil’s staircases in the IV characteristics of superconductor/ferromagnet/superconductor Josephson junctions M. Nashaat,1,2A. E. Botha,3and Yu. M. Shukrinov2,3,4,* 1Department of Physics, Cairo University, Cairo, 12613, Egypt 2BLTP, JINR, Dubna, Moscow Region, 141980, Russia 3Department of Physics, University of South Africa, Science Campus, Private Bag X6, Florida Park 1710, South Africa 4Dubna State University, 141982 Dubna, Russian Federation (Received 6 March 2018; revised manuscript received 10 May 2018; published 14 June 2018) We study the effect of coupling between the superconducting current and magnetization in the supercon- ductor/ferromagnet/superconductor Josephson junction under an applied circularly polarized magnetic field.Manifestation of ferromagnetic resonance in the frequency dependence of the amplitude of the magnetizationand the average critical current density is demonstrated. The IV characteristics show subharmonic steps that formdevil’s staircases, following a continued fraction algorithm. The origin of the found steps is related to the effectof the magnetization dynamics on the phase difference in the Josephson junction. The dynamics of our system isdescribed by a generalized RCSJ model coupled to the Landau-Lifshitz-Gilbert equation. We justify analyticallythe appearance of the fractional steps in IV characteristics of the superconductor/ferromagnet/superconductorJosephson junction. DOI: 10.1103/PhysRevB.97.224514 I. INTRODUCTION An important challenge, in superconducting spintronics dealing with the Josephson junctions coupled to magneticsystems, is the achievement of electric control over the mag-netic properties by the Josephson current and its counterpart,i.e., the achievement of magnetic control over the Josephsoncurrent [ 1–4]. In some systems, spin-orbit coupling plays a major role in the attainment of such control [ 5]. For example, a recent study showed a full magnetization reversal in a super-conductor/ferromagnet/superconductor (S/F/S) structure, withspin-orbit coupling, by adding an electric current pulse [ 6]. Such a reversal may be important for certain applications[6]. Another approach was followed in Refs. [ 7,8], where the authors demonstrated the interaction of a nanomagnetwith a weak superconducting link and the reversal of singledomain magnetic particle magnetization by an ac field. Thesuperconducting current of a Josephson junction (JJ) coupledto an external nanomagnet driven by a time-dependent mag-netic field both without and in the presence of an externalac drive were studied in Ref. [ 9]. The authors showed the existence of Shapiro-type steps in the IV characteristics of theJJ subjected to a voltage bias for a constant or periodicallyvarying magnetic field and explored the effect of rotation ofthe magnetic field and the presence of an external ac driveon these steps. Furthermore, a uniform precession mode (spinwave) could be excited by a microwave magnetic field, atferromagnetic resonance (FMR), when all the elementary spinsprecess perfectly in phase [ 10]. Finally, coupling between the Josephson phase and a spin wave was studied in the series ofpapers [ 4,11–16]. *shukrinv@theor.jinr.ruIn Josephson junctions driven by external microwave radia- tion the Shapiro steps [ 17] that appear in the IV characteristics can form the so-called devil’s staircase (DS) structure as aconsequence of the interplay between Josephson plasma andapplied frequencies [ 18–21]. The DS structure is a universal phenomenon and appears in a wide variety of different systems,including infinite spin chains with long-range interactions[22], frustrated quasi-two-dimensional spin-dimer systems in magnetic fields [ 23], and even in the fractional quantum Hall effect [ 24]. In Ref. [ 25] the authors considered symmetric dual-sided adsorption, in which identical species adsorb toopposite surfaces of a thin suspended membrane, such asgraphene. Their calculations predicted a devil’s staircase ofcoverage fractions for this widely studied system [ 25]. In Ref. [ 26] a series of fractional integer size steps was observed experimentally in the Kondo lattice CeSbSe. In this systemthe application of a magnetic field resulted in a cascade ofmagnetically ordered states—a possible devil’s staircase. Adevil’s staircase was also observed in soft-x-ray scatteringmeasurements made on single crystal SrCo 6O11, which con- stitutes a novel spin-valve system [ 27]. An extension of the investigation of this problem on the S/F/S Josephson junctionmight open new horizons in this field. The problem of coupling between the superconducting current and magnetization in the S/F/S Josephson junctionattracts much attention today (see Ref. [ 2] and the references therein). An intriguing opportunity is related to the connectionbetween the staircase structure and current-phase relation [ 28]. Particularly, the manifestation of the staircase structure inthe IV characteristics of S/F/S junctions might provide thecorresponding information on current-phase relation and, inthis case, serve as a novel method for its determination. Theappearance of the DS structure and its connection to thecurrent-phase relation in experimental situations has not yet 2469-9950/2018/97(22)/224514(6) 224514-1 ©2018 American Physical SocietyM. NASHAAT, A. E. BOTHA, AND YU. M. SHUKRINOV PHYSICAL REVIEW B 97, 224514 (2018) been investigated in detail. It stresses a need for a theoretical model which would fully describe the dynamics of the S/F/SJosephson junction under external fields, features of Shapiro-like steps and their DS staircase structures. In Ref. [ 13]t h e Josephson energy in the expression for the effective field wasnot considered. Consequently, the IV characteristics of theS/F/S junction at FMR only showed current steps at voltagescorresponding to even multiples of the applied frequency. Theauthors related these steps to the interaction of Cooper pairswith an even number of magnons [ 13]. In this paper we investigate the effect of coupling between the superconducting current and magnetization in the su-perconductor/ferromagnet/superconductor Josephson junctionunder an applied circularly polarized magnetic field. Takinginto account the Josephson energy in the effective field, wedemonstrate an appearance of odd and fractional Shapiro stepsin IV characteristics, in addition to the even steps that werereported in Ref. [ 13]. We demonstrate the appearance of devil’s staircase structures and show that voltages corresponding to thesubharmonic steps under applied circularly polarized magneticfield follow the continued fraction algorithm [ 19–21]. An analytical consideration of the linearized model, based on ageneralized RCSJ model and Landau-Lifshitz-Gilbert (LLG)equation, including the Josephson energy in the effectivefield, justifies the appearance of the fractional steps in IVcharacteristics, in agreement with our numerical results. Wealso show the manifestation of ferromagnetic resonance in the frequency dependence of the amplitude of the magnetization and the average critical current density. An estimation of themodel parameters shows that there is a possibility for theexperimental observation of this phenomenon. The plan of the rest of the paper is as follows. In Sec. II, we describe the model and present an explicit form ofthe equations. Ferromagnetic resonance is demonstrated inSec. III, where the effect of Gilbert damping is shown and a comparison with the linearized case is presented. This isfollowed by a discussion of the IV characteristics and observedstaircase structures in Sec. IV. In Sec. Vwe discuss the additional effect of an oscillating electric field on the Shapirosteps. Demonstration of different possibilities of the frequencylocking and discussion of the experimental realization of thefound effects is presented in Sec. VI. Finally, we conclude in Sec. VIIand specify our calculations for the linearized case [29]. II. MODEL AND METHODS The geometry of the S/F/S Josephson junction under an applied circularly polarized magnetic field is shown in Fig. 1. There is a uniform magnetic field of magnitude H0applied in the zdirection. Additionally, a circularly polarized mag- netic field, of amplitude Hacand frequency ω, is applied in the xyplane. The total applied field is thus H(t)= (Haccos(ωt),Hacsin(ωt),H0). A bias current Iflows in the xdirection. The microwave sustains the precessional motion of the magnetization in the presence of Gilbert damping. The mag-netic fluxes in the zandydirections are given by /Phi1 z(t)= 4πdL yMz(t)//Phi1 0,/Phi1y(t)=4πdL zMy(t)//Phi1 0, where Mzand Myare components of magnetization and dis the thickness ofd LyLzS SF (Hac cos ωt, H ac sin ωt, 0)Ho xyz IHac FIG. 1. Geometry of the S/F/S Josephson junction with cross- sectional area LyLzin uniform magnetic field H0and circularly polarized magnetic field Hac. ferromagnet. Using the equation ∇θ(y,z,t )=−2πd /Phi10B(t)×n, where nis a unit vector in xdirection, and the fact that two superconductors are thicker than London’s penetrationdepth, we obtain an expression for the gauge-invariant phase difference, θ(y,z,t )=θ(t)− 8π2dMz(t) /Phi10y+8π2dMy(t) /Phi10z, where /Phi10=h/(2e) is the magnetic flux quantum. Hence, within the framework of the modified RCSJ model, which takes intoaccount the gauge invariance including the magnetization ofthe ferromagnet [ 13], the electric current reads I/I 0 c=sin/parenleftbigπ/Phi1z(τ) /Phi10/parenrightbig sin/parenleftbigπ/Phi1y(τ) /Phi10/parenrightbig (π/Phi1z(τ)//Phi1 0)(π/Phi1y(τ)//Phi1 0)sinθ(τ) +dθ(τ) dτ+βcd2θ(τ) dτ2, (1) where τ=tωcis the normalized time, ωc=2πI0 cR//Phi1 0is the characteristic frequency, Ris the junction resistance, βc= RCω cis the McCumber parameter [ 30], and Cis the junction capacitance. In the present paper we will only consider theoverdamped case for which β c=0. The applied circularly polarized magnetic field in the xyplane causes precession of the magnetization Min the ferromagnetic (FM) layer. The dynamics of the magnetizationis described by the LLG equation[ 10] (1+α 2)dM dt=−γM×He−γα |M|M×(M×He),(2) where αis the Gilbert damping, γis the gyromagnetic ratio, andHeis an effective field. Taking into account that the phase difference depends on the magnetization components, we writethe total energy of our system as E=E s+EM+Eac, where Es=−/Phi10 2π/parenleftbigg θ(t)−8π2d /Phi10(Mz(t)y−My(t)z)/parenrightbigg I +EJ/bracketleftbigg 1−cos/parenleftbigg θ(t)−8π2d /Phi10(Mz(t)y−My(t)z)/parenrightbigg/bracketrightbigg , EM=−vH 0Mz(t), Eac=−vMx(t)Haccos(ωt)−vMy(t)Hacsin(ωt). (3) Here H0=ω0/γ,ω 0is the ferromagnetic resonance fre- quency, and vis the volume. When we switch on H0andHac, the phase difference starts to depend on M, and so does the Josephson energy. The addition of Esleads to the dependence of the effective field on the ratio EJ/EMand generalizes the considerations made in Ref. [ 13]. The effective field is now 224514-2DEVIL’s STAIRCASES IN THE IV CHARACTERISTICS … PHYSICAL REVIEW B 97, 224514 (2018) given by He=−1 v∇ME. (4) In dimensionless form, we write m=M/M 0,M0=|M|,he= He/H0,hac=Hac/H0,/Omega1=ω/ω c, and /Omega10=ω0/ωc,A f t e r integrating the total effective field over the junction area, ithas the following form h e=(haccos/Omega1τ)ˆex+(hacsin/Omega1τ+Γyz/epsilon1Jcosθ)ˆey +(1+Γzy/epsilon1Jcosθ)ˆez, (5) where /epsilon1J=EJ/(vM 0H0) and Γyz=sin(φsymz) my(φsymz)/bracketleftbigg cos(φszmy)−sin(φszmy) (φszmy)/bracketrightbigg , Γzy=sin(φszmy) mz(φszmy)/bracketleftbigg cos(φsymz)−sin(φsymz) (φsymz)/bracketrightbigg , (6) withφsy=4π2LydM 0//Phi1 0, andφsz=4π2LzdM 0//Phi1 0.I fw e setΓyz=Γzy=0, our system reduces to that of Ref. [ 13]. We note that the first term for Esin Eq. ( 3) does not contribute to the effective field after integration over the junction area andtaking the derivative with respect to the magnetization. TheLLG equation in the dimensionless form reads dm dτ=−/Omega10 (1+α2)(m×he+α[m×(m×he)]).(7) The magnetization and phase dynamics of the considered S/F/S Josephson junction is determined by Eqs. ( 1) and ( 7). To solve this system and calculate the IV characteristics,we assume a constant bias current and calculate the voltagefrom the Josephson relation V(τ)=dθ/dτ.W ee m p l o ya fourth-order Runge-Kutta integration scheme which conservesthe magnetization magnitude in time. The dc bias current Iis normalized to the critical current I 0 c, and the voltage V(t)t o ¯hωc/(2e). As a result, we find the temporal dependence of the voltage in the JJ at a fixed value of bias current I. Then, the current value is increased or decreased by a small amount, δI (the bias current step), to calculate the voltage at the next pointof the IV characteristics. We use the final phase and voltageachieved at the previous point of the IV characteristics as theinitial condition for the next current point. The average of the voltage V(τ) is given by V= 1 Tf−Ti/integraltextTf TiV(τ)dτ, where Tiand Tfdetermine the interval for the temporal averaging. Further details of the simulation procedure are described in Ref. [ 31]. The initial conditions for the magnetization components are assumed to be mx=0,my=0.01, and mz=/radicalbigg 1−m2 x−m2 y, while for the voltage and phase we take zeros. The numerical parameters (if not mentioned) are α=0.1,hac=1,φsy= φsz=4,/epsilon1J=0.2, and /Omega1=/Omega10=0.5. III. FERROMAGNETIC RESONANCE First we show that the system displays ferromagnetic res- onance. Its manifestation, in the frequency dependence of theamplitude of the magnetization component m yand the average critical current density, is presented in Fig. 2, where we see that the maximum in both cases occurs at the resonance frequency/Omega1=/Omega1 0=0.5. Furthermore, the oscillation amplitude is notΩLinearized φs= 0.4 Non-Linearized φsy= 4, φsz= 2hac= 0.2, α=0.1 (a) (b) (c) FIG. 2. (a) Manifestation of the FMR in the frequency depen- dence of the maximum of magnetization component myand the average critical current density at bias current I=1.16. Lines added to guide the eye. (b) Frequency dependence of the maximum of magnetization component myat different damping αand amplitude of circularly polarized magnetic field hac. Other parameters are the same as in (a). (c) Comparison with a linearized case at /epsilon1J=0.02. symmetric relative to /Omega10, which reflects the influence of Hs in the effective field. The behavior of the amplitude of the magnetization component mxis qualitatively the same. 224514-3M. NASHAAT, A. E. BOTHA, AND YU. M. SHUKRINOV PHYSICAL REVIEW B 97, 224514 (2018) In Fig. 2(b) we show the frequency dependence of the max- imum of magnetization component myat different damping α and amplitude of circularly polarized magnetic field hac.W e see that the resonance line width changes with changing hac (curves with label 1 and 2) and α(curves with label 2 and 3). For comparison, we also demonstrate the manifestation of theferromagnetic resonance in the linearized case [ 29,32]. In the linearized case, the RSJ equation reduced to I/I c=sin(φsmy) (φsmy)sinθ(t)+dθ(t) dt, (8) where Ic=I0 csin(φsy)/(φsy),φsy=4π2LydMz//Phi1 0and the expression for the ycomponent of magnetization has a form my=−2α/Omega12 /Omega12 0cos(/Omega1t)+/parenleftbig 1−η1/Omega12 /Omega12 0/parenrightbig sin(/Omega1t) /parenleftbig 1−η2/Omega12 /Omega12 0/parenrightbig2+/Delta1J/parenleftbig 1−η1/Omega12 /Omega12 0/parenrightbig +4α2/Omega12 /Omega12 0, (9) where /Delta1J=/epsilon1Jφ2 szcosθ(t)/3,η1=1−α2, andη2=1+α2. Results of calculations based of these formulas are pre- sented in Fig. 2(c). We see a qualitative agreement of the ferromagnetic resonance features in both cases. IV . DS STRUCTURE IN THE IV CHARACTERISTICS Let us now discuss the S/F/S junction at FMR, when the coupling between Josephson and magnetic system is strongest.In Fig. 3(a) the IV characteristic demonstrates current steps at V=m/Omega1 0, with minteger, and also some fractional steps. In the case of conventional JJs the widths of the first Shapiro step is larger than the second. In the present case, we see thatthe width of the first step is much narrower than that of thesecond. So, the width of the harmonics are different for evenand odd m: Large steps are at even mand small steps at odd m. In Ref. [ 13], which did not consider the Josephson energy in the expression for the effective field, only the steps with even m were observed. In our case, taking into account the Josephsonenergy in the effective field, we have obtained additional stepswith odd and fractional values of m, as we see in Fig. 3(a). The structure of those fractional steps can be clarified by analysis of their positions on the voltage scale, using analgorithm based on the generalized continued fraction formula[19–21]: V=⎛ ⎝N±1 n±1 m±1 p±..⎞ ⎠/Omega1, (10) where N,n,m,p, ... are positive integers. The locking of the Josephson frequency to the frequency of magnetic precessionoccurs due to the additional terms ( /Gamma1 yz/epsilon1Jcosθ,/Gamma1zy/epsilon1Jcosθ)i n the effective field, as given by Eq. ( 5). Figures 3(b) and3(c) demonstrate the enlarged parts of the IV characteristic shown inFig.3(a). There are the fractional current steps between V=0 andV=0.5 which can be described by the continued fractions of second level [ 19](N−1)+1/nandN−1/nwithN=1 in both cases [see Fig. 3(b)]. In addition, there is a manifestation of two third-level continued fractions ( N−1)+1/(n−1/m) withN=1,n=2 (shown in the inset) and n=3. The steps between V=0.5 andV=1 follow the continued fractions of second level ( N−1)+1/nandN−1/nwithN=2 in bothIV 0 0.5 1 1.5 2 2.5 300.511.522.5Ω=ΩO = 0.5, φsz=φsy=4 α=0.1, hac = 1, εJ=0.2 See in (Fig.3-b)See in (Fig.3-c)(a) with εJ without εJ IV 0.2 0.3 0.4 0.5 0.6 0.7 0.80.20.40.60.81 2/5(N-1)+1/n N=27/4 4/5(b) 2/33/4N-1/n N=2 6/54/3 (N-1)+1/(n-1/m) N=1,n=33/2 3/25/3 see insetN-1/n N=1 1/21IV 0.35 0.40.30.4(N-1)+1/(n-1/m) N=1,n=2 2/3 3/54/7 IV 1.3 1.4 1.5 1.6 1.7 1.8 1.9 211.21.41.61.82 (N-1)+1/n N=4 N-1/n N=316/5 5/2(c) 14/57/2 8/311/410/3 13/4 FIG. 3. (a) IV characteristic of S/F/S junction at ferromagnetic resonance. The case in Ref. [ 13] is shown by the dashed line for comparison, shifted by 0.8 to the right for clarity; (b) and (c) enlarge the parts of IV characteristic marked by rectangles in (a). cases. In Fig. 3(c) we see clearly the manifestation of second level continued fractions N−1/nwithN=3 and ( N−1)+ 1/nwithN=4 between voltage steps V=1 andV=2. 224514-4DEVIL’s STAIRCASES IN THE IV CHARACTERISTICS … PHYSICAL REVIEW B 97, 224514 (2018) FIG. 4. IV characteristics of the S/F/S junction at ferromagnetic resonance without oscillating electric field ( A=0) and two charac- teristics at amplitudes A=0.3a n d A=1. Here /Omega1r=ωr/ωc,o t h e r parameters are the same as in Fig. 3. For clarity, the curves at A=0.3 andA=1h a v eb e e ns h i f t e dt ot h er i g h t ,b y /Delta1I=0.6a n d/Delta1I=1.2, respectively, relative to the IV characteristic at A=0. V . EFFECT OF OSCILLATING ELECTRIC FIELD The ac field can affect the Josephson junction directly and not only via the oscillating magnetization. The effect of anoscillating electric field from microwave radiation is usuallytaken into account by adding the term Asin/Omega1 rtin Eq. ( 1), where Ais the amplitude and /Omega1r=ωr/ωc—the frequency of the external electromagnetic radiation. Figure 4shows the IV characteristics without the effect of the oscillating electricfield (i.e., for A=0) and two curves at amplitudes A=0.3 andA=1. In comparison to A=0, where the width of the first step at V=0.5 is smaller relatively to the step at V=1 (a signature of the S/F/S IV characteristics), we see that atA=0.3 the first step has widened in comparison to the second step at A=1. But, even in this case, the IV characteristics show the unusual behavior of Shapiro step widths for a conventionalJosephson junction, specifying width of odd and even steps. VI. DISCUSSION We have also found that one can control the structure of the devil’s staircase by tuning the frequency of the ac-magneticfield out of resonance. Of course, the width of the subharmonicsteps is largest at the FMR. The step structure depends on thejunction parameters (Gilbert damping, cross-section, etc). Themain parameter determining the appearance of the DS structureis the ratio of the Josephson to magnetic energy. If this ratio isclose to zero, we observe only even steps. In the present workthe appearance of the fractional steps and the formation of thedevil’s staircases in the IV curve are consequences of includingthe Josephson energy in the effective field, i.e., the term /epsilon1 J in (5). We justify this claim by solving the linearized LLG equation analytically using well known mathematical methodsSyncronization PosibilitiesEven integer step Even integer + Odd integer + Fractional stepsIf He=Hac+H0±2(m-n) Ω ±2(m+n+1) Ω ±2(m-n) Ω±2(m+n+1) Ω±[(m-n)/(k-r)]Ω ±[(m-n)/(k-r+1)]Ω ±[(m+n+1)/(k-r)]Ω±[(m+n+1)/(k-r+1)]Ω±[2(m-n)/(1-2(k-r))]Ω±[2(m-n)/(2(k-r)+1)]Ω ±[2(m+n+1)/(2k-r+1)]Ω ±[2(m+n+1)/(1-2(k-r))]ΩIf H e=Hac+H0+Hs FIG. 5. Different possibilities of the frequency locking excluding (red) and including (blue) the Josephson energy in the effective magnetic field. [29,33,34]. As demonstrated in Fig. 5, our proposed model shows different possibilities of the frequency locking leadingto even, odd, and fractional current steps in IV characteristics ofS/F/S junction under an external circularly polarized magneticfield. This fact is in an agreement with the results presented inFig. 3. Let us now discuss the possibility of experimentally ob- serving the effects found in this paper. The main param-eter which controls the appearance of the current steps is/epsilon1 J=EJ/(vM 0H0). Using typical junction parameters d= 5n m,Ly=Lz=75 nm, critical current I0 c≈160μA, satu- ration magnetization M0≈4×105A/m,H 0≈26 mT, and gyromagnetic ration γ=3πMHz/T, we find the value of φsy(z)=4π2Ly(z)dM 0//Phi1 0=3.6 and /epsilon1J=0.18, which are very close to the values we used in our simulations andjustify the choice of parameters: φ sy,sz=4,/epsilon1J=0.2. With the same junction parameters one can control the appearanceof the subharmonic steps by tuning the strength of the constantmagnetic field H 0. Estimations show that, for H0=90 mT, the fractional subharmonic steps disappear at /epsilon1J=0.05. For junctions with Ly=Lz=50 nm ,H 0=10 mT, we find φsy(z)=2.4 and/epsilon1J=1.05, which are rather good for the step manifestation. Of course, in general, the subharmonic stepsare sensitive to junction parameters, Gilbert damping, and thefrequency of the magnetic field. VII. CONCLUSION The S/F/S Josephson junction is of considerable importance for the development of certain spintronic applications/devices.Motivated by physical considerations, our paper has presenteda major advance in modeling the S/F/S Josephson junction,by including a previously neglected physical effect, i.e., ofthe Josephson energy on the effective magnetic field. Ourcalculations predict that the addition of the Josephson energyshould manifest itself (measurably) through the appearanceof devil’s staircase structures in the IV characteristics, thusproviding insight into the precise nature of the current-phaserelation and opportunities for potential applications. In our paper we have developed a model which fully describes the dynamics of the S/F/S Josephson junction underan applied circularly polarized magnetic field. Manifestationof ferromagnetic resonance in the frequency dependence of theamplitude of the magnetization and the average critical current 224514-5M. NASHAAT, A. E. BOTHA, AND YU. M. SHUKRINOV PHYSICAL REVIEW B 97, 224514 (2018) density was demonstrated. The IV characteristics showed subharmonic steps which formed devil’s staircase structures,following the continued fraction algorithm [ 19]. The origin of the found steps was related to the effect of the magnetizationdynamics on the phase difference in the Josephson junction.Analytical considerations of the steps were in agreement withthe numerical results. An interesting question appears about whether the man- ifestation of the staircase structure in the IV characteristicscan provide information on the current-phase relation of theS/F/S Josephson junction and, in some cases, serve as a novelmethod for its determination. The results on the developedmodel might serve for better understanding of the couplingbetween the superconducting current and magnetization inthe S/F/S Josephson junction. The appearance of the staircasestructure in experimental situations and its connection with the current-phase relation may open horizons in this field. Theobserved features might also find application in some fields ofsuperconducting spintronics. ACKNOWLEDGMENTS The authors thank A. Buzdin, S. Maekawa, S. Takahashi, S. Hikino, I. Rahmonov, K. Kulikov, I. Bobkova, and A. Bobkovfor useful discussions, and D. Kamanin, V . V . V oronov, andH. El Samman for supporting this work. The reported studywas partially funded by RFBR according to the research project18-02-00318, and the SA-JINR and Egypt-JINR collabora-tions. Y .M.S. and A.E.B. thank the visiting researcher programat the University of South Africa for financial support. [1] I.Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76,323 (2004 ). [2] J. Linder and J. W. A. Robinson, Nat. Phys. 11,307(2015 ). [3] A. A. Golubov, M. Yu. Kupriyanov, and E. Il’ichev, Rev. Mod. Phys. 76,411(2004 ). [4] S. Mai, E. Kandelaki, A. F. V olkov, and K. B. Efetov, Phys. Rev. B84,144519 (2011 ). [5] A. Buzdin, Phys. Rev. Lett. 101,107005 (2008 ). [6] Yu. M. Shukrinov, I. R. Rahmonov, K. Sengupta, and A. Buzdin, Appl. Phys. Lett. 110,182407 (2017 ). [7] L. Cai and E. M. Chudnovsky, P h y s .R e v .B 82,104429 (2010 ). [8] L. Cai, D. A. Garanin, and E. M. Chudnovsky, Phys. Rev. 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PhysRevApplied.13.044080.pdf

PHYSICAL REVIEW APPLIED 13,044080 (2020) Voltage-Controlled Anisotropy and Current-Induced Magnetization Dynamics in Antiferromagnetic-Piezoelectric Layered Heterostructures P.A. Popov ,1,2,*A.R. Safin ,1,3,†A. Kirilyuk,1,4,5S.A. Nikitov,1,2,6I. Lisenkov,7V. Tyberkevich,8 and A. Slavin8 1Kotel’nikov Institute of Radio-Engineering and Electronics of RAS, Moscow 125009, Russia 2Moscow Institute of Physics and Technology, Dolgoprudny 141701, Moscow Region, Russia 3Moscow Power Engineering Institute, Moscow 111250, Russia 4Institute for Molecules and Materials, Radboud University, 6525 AJ Nijmegen, Netherlands 5FELIX Laboratory, Radboud University, 6525 AJ Nijmegen, Netherlands 6Laboratory “Metamaterials”, Saratov State University, Saratov 410012, Russia 7Winchester Technologies LLC, Burlington, Massachusetts 01803, USA 8Department of Physics, Oakland University, Rochester, Michigan 48309-4479, USA (Received 21 January 2020; revised manuscript received 19 February 2020; accepted 31 March 2020; published 30 April 2020) It is shown theoretically that in a layered heterostructure comprising piezoelectric, dielectric antiferro- magnetic crystal, and heavy metal (PZ/AFM/HM), it is possible to control the anisotropy of the AFM layer by applying a dc voltage across the PZ layer. In particular, we show that by varying the dc voltage across the heterostructure and/or the dc current in the HM, it is possible to vary the frequency of the antiferro-magnetic resonance of the AFM in a passive (subcritical) regime and, also, to reduce the threshold of the current-induced terahertz-frequency generation. Our analysis also shows that, unfortunately, the voltage- induced reduction of the generation threshold leads to the proportional reduction of the amplitude of theterahertz-frequency signal generated in the active (supercritical) regime. The general results are illustrated by a calculation of the characteristics of experimentally realizable PZT-5H/NiO/Pt. DOI: 10.1103/PhysRevApplied.13.044080 I. INTRODUCTION In the recent years, antiferromagnetic (AFM) spintron- ics has attracted considerable attention from researchers [1,2]. Antiferromagnets are interesting for spintronic appli- cations because the magnetic order in these substances does not require an external bias magnetic field, as it is supported by a very large internal magnetic field of an exchange nature, which, along with the moderate magnetic field of the AFM anisotropy, determines the frequency of the antiferromagnetic resonance (AFMR) lying in the sub- terahertz to terahertz frequency range [ 3–6]. Thus, AFMs can be considered as promising materials for the devel- opment of signal generators operating in the terahertz frequency range. It has been shown previously [ 7–9] that spin-torque nano-oscillators (STNOs) driven by a spin-transfer torque and based on ferromagnets (FMs) can generate signals in the frequency range 1–30 GHz, which is mainly lim- ited from above by the practically possible values of the applied bias magnetic field necessary to saturate the FM *paavali.popov@gmail.com †arsafin@gmail.comfree layer of the STNO. The introduction of AFMs into current-driven signal generators could raise the generated frequencies to those of the typical range of AFMR frequen- cies, i.e., to the terahertz range [ 1,10–12]. In particular, in Ref. [ 13] it has been theoretically shown that if an AFM material is used as a magnetic layer of a spin Hall auto- oscillator, it would be possible to generate signals with fre- quencies of 0.1–2.0 THz and amplitudes of approximately 1 V/cm, using driving electric currents that have densities in the range 108–109A/cm2. Note that driving currents of such densities, mainly determined by the magnitude of the easy-plane anisotropy of the AFM, have previously been attained in experiments with STNO devices based on FMs (see, e.g., Ref. [ 9]). In Ref. [ 14], it has been shown that the emission power of AFM-based STNOs may reach 1 μW, which is comparable to that of FM-based STNOs. For the AFM-based STNO from Ref. [ 13], if we assume that the Pt layer has a height of 10 nm, a length and width of 10μm, a resistivity of 10.6 ×10−8/Omega1m, and a current density of around 108A/cm2, such an emission power cor- responds to an efficiency of η∼10−5. This efficiency can be compared to that of FM-based STNOs η∼10−3[15]. The efficiency of AFM-based STNOs can be increased by using a high- Qsubterahertz frequency resonator, such as in 2331-7019/20/13(4)/044080(11) 044080-1 © 2020 American Physical SocietyP.A. POPOV et al. PHYS. REV. APPLIED 13,044080 (2020) Refs. [ 16]a n d[ 17] and as discussed in Ref. [ 14], and then a subterahertz waveguide [ 18]. Another way to increase the efficiency is to synchronize an array of STNOs [ 19]. Another important property of AFMs is the absence of net magnetization in AFM substances, which makes them insensitive to stray dipolar magnetic fields and, there- fore, makes the placement of AFM elements in close proximity to each other possible without significant cross- interference in all the possible signal-processing applica- tions. At the same time, this insensitivity of AFMs to external magnetic fields makes it rather difficult to control the magnetization dynamics in AFMs externally. Of course, there exist means other than external mag- netic fields to control the dynamics of magnetization in magnetically ordered substances. It has been shown theo- retically that a spin-transfer torque created by a dc current flowing in a layer of a heavy metal (e.g., Pt) adjacent to the AFM layer could reduce the magnetic anisotropy and, therefore, the AFMR frequency in the AFM layer [20], and, at a sufficiently large current magnitude, could excite the terahertz-frequency magnetization dynamics in AFMs [ 13]. These analytical predictions have recently been confirmed in micromagnetic modeling [ 21,22]. It has been also confirmed experimentally that an external spin- transfer torque could switch the direction of the AFM order parameter (the Néel vector) in dielectric AFMs [ 23,24]. Also, it has been shown previously that the magneti- zation dynamics in ferromagnetic (FM) heterostructures containing piezoelectric and/or multiferroic layers could be controlled by the application of a static electric field across the structure thickness (magnetoelectric and mag- netoelastic effects) [ 25,26]. The application of a transverse voltage (or electric field) creates a stress in the PZ layer, which, due to the magnetoelastic effect in a FM layer, changes the FM-layer magnetization and/or anisotropy . The effect of the voltage-induced magnetic anisotropy has been seen in ultrathin FM layers and heterostructures sub- jected to the action of a perpendicular voltage [ 27–29]. This effect has been studied in a recent paper [ 30], where it has been suggested that voltage control can be used to influence the terahertz-frequency magnetization dynamics in AFMs. In this paper, we apply similar methods to monocrys- talline AFM, study the influence of the simultaneous appli- cation of a transverse dc electric field and a longitudinal dc current to the PZ/AFM/HM layered heterostructure, and evaluate theoretically how such an application can influ- ence the AFM anisotropy and affect the magnetization dynamics in the structure. The studied heterostructure is, in principle, similar to the AFM-based STNOs considered in Refs. [ 13]a n d[ 14], where they are shown as possi- ble terahertz-frequency signal generators. The PZ/AFM stack in our structure corresponds to the AFM layer from Refs. [ 13]a n d[ 14], where it serves as a source of terahertz- frequency dynamics.We expect that the voltage applied to the PZ layer of the heterostructure will create a stress at the PZ/AFM bound- ary, which, due to the magnetoelastic effect in AFMs, willcreate a variation in the AFM anisotropy. This anisotropy variation, in the first approximation, will be linearly pro- portional to the applied electric field and will depend on the field direction, so that the mechanical stress at the PZ/AFM boundary could either increase or decrease the AFM anisotropy. Since the heterostructure is mainly dielectric, this electric control of the AFM anisotropy could be performed without significant transverse currents that increase dissipation and, since both the AFMR fre- quency [ 3–6] and the threshold of the current-induced generation of periodic signals in AFM-based STNOs [13,14] are proportional to the AFM anisotropy, the appli- cation of a transverse voltage to the PZ/AFM/HM lay- ered structure could allow us the possibility of tuning the frequency of the AFMR in the subcritical regime. The heterostructure could then be used as a resonant ele- ment in a passive receiver of terahertz-frequency signals and to reduce the magnitude of the generation threshold, if the heterostructure could be used in the supercritical regime to generate terahertz-frequency periodic signals. The electric control (reduction) of the generation thresh- old could be of substantial practical importance, as it may help to realize terahertz-frequency generation in AFMs experimentally. The paper has the following structure. In Sec. I,a n introduction to voltage- (or electric-field-) controlled AFM spintronics is presented. In Sec. II, we consider a PZ/AFM bilayer with a transverse electric field applied across the bilayer. In such a situation, we derive a general linear ten- sorial relation between the vector of the applied electric field and the field-induced change in the AFM anisotropy tensor. In Sec. III, we consider the PZ/AFM/HM lay- ered heterostructure under the simultaneous action of a transverse dc electric field applied to the PZ layer and a longitudinal dc current flowing in the HM layer. Using the approximation of a “sigma model” [ 6], we derive a general equation describing the magnetization dynam- ics in the AFM layer, in which the AFM anisotropy is affected by both the external dc current and the external dc electric field. In Sec. IV, we apply the general theory developed above to a particular case of a monocrystalline bianisotropic dielectric AFM—nickel oxide (NiO)—and derive explicit expressions for the AFMR frequency in the subcritical regime, the generation threshold in the critical regime, and the amplitude and frequency of the generated periodic signal in the supercritical regime as functions of the external dc current and the external dc electric field in a particular case in which monocrystalline NiO is used asthe AFM material, a lead zirconate titanate-based piezoce- ramics PZT-5H is used as a PZ material, and Pt is used as a HM. Discussion of the obtained results is presented in Sec. V, while the conclusions are given in Sec. VI. 044080-2VOLTAGE-CONTROLLED ANISOTROPY... PHYS. REV. APPLIED 13,044080 (2020) II. VOLTAGE-CONTROLLED ANISOTROPY IN THIN AFM LAYERS In this section, we consider the PZ/AFM bilayer shown in Fig. 1, and derive the equations describing the influ- ence of a dc electric field Eapplied to the PZ layer on the magnetic parameters of the AFM layer. The AFM layer is assumed to be much thinner than the PZ layer, so that it will not affect any of the mechanical properties of the PZ layer. Also, we assume an ideal acoustic contact between the layers, i.e., we assume that the displacement is continu- ous across the interlayer interface [ 32]. Here and below, we denote σPZ ij,ePZ ijandσij,eijas the elastic stress and strain tensors of the PZ and AFM layers, respectively. We also use the Einstein summation convention. An application of the dc electric field Ekto the PZ layer produces the following strain in the layer: ePZ ij=dijkEk,( 1 ) where dijkis the strain-piezoelectric tensor of the PZ layer [see Eq. (2.11) in Ref. [ 33] ]. Here, we can use the bulk strain-piezoelectric constants of the PZ layer, because the AFM layer is much thinner than the PZ layer. Thus, the mechanical feedback from the AFM layer to the PZ layer is negligible. The Hooke’s law relating the volume stress ekland strain σijinduced in the AFM layer can be written as σij=Cijklekl,( 2 ) where Cijklis the stiffness tensor of the AFM layer. FIG. 1. A schematic view of the considered PZ/AFM bilayer under the influence of a bias electric voltage V:al a y e ro ft h e AFM material (e.g., NiO) and a layer of the PZ material (e.g., PZT-5H) in a conductive coating. The interface between the AFM and the PZ layers is considered as an ideal acoustic con-tact. The AFM layer is chosen to be much thinner than the PZ layer, so that the mechanical properties of the PZ layer are not affected by the AFM layer. The conductive coating allows oneto apply the transverse electric field Eto the PZ layer (such a design has been used in the experimental paper by Sadovnikov et al. [31]).An ideal acoustic contact between the PZ and AFM lay- ers implies that the in-plane strain is continuous across the AFM/PZ interface: PikePZ klPlj=PikeklPlj=eIF ij,( 3 ) where Pij=Iij−ninjis the projector on the AFM/PZ interface, Iijis the identity matrix, and niis the vector nor- mal to the AFM/PZ interface. It should be noted that the net mechanical result of the application of the bias electric voltage (or electric field) to the PZ layer is the creation of the mechanical strain eIF ijat the PZ/AFM interface. The opposite interface of the AFM layer is considered to be free, so the normal component of the stress σijat this interface of the PZ layer should vanish: σijni=0. (4) To find the components of the strain induced in the AFM layer by the electric field applied to the PZ layer, we need to find the simultaneous solution of the system of Eqs. (2)–(4)comprising the Hooke’s law and the two boundary conditions at the boundaries of the AFM layer. In general, the solution of the system of Eqs. (2)–(4)for an arbitrary symmetry of the tensor Cijklis rather cumber- some. Nonetheless, it can be found in the following general form: eij=GijkleIF kl.( 5 ) In the case in which the stiffness tensor of the AFM layer Cijklhas a cubic symmetry (as in NiO), the tensor Gijklin Eq.(5)can be simplified to the following form (see the Appendix ): Gijkl=(δikδjl−ηninjδkl),( 6 ) where η=C12/C11,C12,a n d C11are the components of the stiffness tensor Cijklin the Voigt notation. In fact, the existence of strain components dependent on the parameter ηis a result of the Poisson effect. The strain applied to the AFM layer produces the mag- netoelastic (ME) energy in this layer, which can be written as [see Eq. (5.66) in Ref. [ 34]] WME=bijklmα imαjekl,( 7 ) where bijklis the magnetoelastic tensor and mα iis the mag- netization unit vector of the AFM sublattice, denoted by the index “ α.” The magnetoelastic energy term is quadratic inmα iin exactly the same way as the terms describing the first-order magnetocrystalline anisotropy: Wanis=Kijmα imαj,( 8 ) where Kijis the tensor of the magnetocrystalline anisotropy. 044080-3P.A. POPOV et al. PHYS. REV. APPLIED 13,044080 (2020) Therefore, we can formally take into account the mag- netoelastic interaction in the AFM layer by introducing the effective magnetic anisotropy tensor containing a magne- toelastic part KME ij(Ef)that is dependent on the applied electric field: Keff ij=Kij+KME ij(Ef).( 9 ) This magnetoelastic part of the second-rank AFM anisotropy tensor can be written as KME ij(Ef)=bijklGklpq(PprdrsfPsqEf)=ZijfEf. (10) Here, Zijfis a third-rank tensor, which describes the voltage-induced magnetic anisotropy effect in the AFM layer by linearly coupling the vector of the applied elec- tric field Efto the second-rank tensor KME ijof the magnetic anisotropy tensor induced by an applied electric field in an arbitrary direction. Equations (9)and(10) represent a main general result of this paper, describing how a vectorial external dc elec- tric field Ecan be used to control the magnetic anisotropy in AFM (and/or FM) materials. The tensor Zijfdescribing this effect is defined by the piezoelectric properties of the PZ layer and the magnetoelastic and elastic properties of the AFM layer, as well as by the relative crystallographic orientation of the layers. It is worth mentioning that in addition to the strain cre- ated by the applied electric field, the AFM layer is also affected by the “ground-state” strain caused by the intrinsic interaction between the magnetic and elastic subsystems of the AFM material in the free state (see pp. 173 and 182 in Ref. [ 35]). The magnetic anisotropy caused by this strain is usually included in the magnetocrystalline anisotropy con- stants found in the experiments. Thus, we do not consider this effect in any detail here. III. CURRENT-INDUCED MAGNETIZATION DYNAMICS IN AN AFM HETEROSTRUCTURE BIASED BY A dc ELECTRIC FIELD The magnetization dynamics in the current-driven AFM/HM bilayer in both the active (generation) [ 13,14] and the passive (reception) [ 20] regimes have been dis- cussed previously. In this section, we study the current- driven AFM magnetization dynamics in the PZ/AFM/HM layered heterostructure in the case in which the PZ layer is biased by the dc voltage V(see Fig. 2). In such a struc- ture, the electric current jflowing along the HM layer becomes spin polarized due to the spin Hall effect in the HM, creating a perpendicular spin current. This spin cur-rent, polarized along p, creates a spin-transfer torque (STT) acting on the magnetization sublattices in the AFM layer [13]. The magnetization dynamics in the AFM layer are described by the “sigma-model” equation for the AFMFIG. 2. A schematic view of the three-layer PZ/AFM/HM het- erostructure, where the piezoelectric (PZ) layer is biased by the dc voltage Vwhile the layer consisting of a heavy metal (HM) is driven by the longitudinal current of the density j. When the current density in the HM layer exceeds a certain threshold deter- mined by the easy-plane anisotropy in the AFM, the spin-transfer torque created by the current in the HM due to the spin Hall effect generates terahertz-frequency rotation of the magnetization sub-lattices in the AFM, which can be detected in the HM layer due to the inverse spin Hall effect (for details, see Ref. [ 13]). The voltage bias applied to the PZ layer can change the anisotropy inthe AFM layer and, therefore, the AFMR frequency in the AFM layer in the subcritical (passive) regime, the generation threshold current, and the amplitude of the signal generated in the AFMlayer in the supercritical (active) regime. order parameter—the AFM Néel vector l[see Eq. (14) in Ref. [ 13] ] which in our case has the following form: l×/parenleftbigg1 ωex¨l+α˙l+ˆ/Omega1·l+τp×l/parenrightbigg =0, (11) where ωex=γHex,γis the gyromagnetic ratio, Hexis the internal exchange field in the AFM, αis the damp- ing parameter, τ=σjis the STT strength [ 13],σis the electric current to STT proportionality coefficient [ 13], ˆ/Omega1=(2γ/Ms)ˆKeffis the anisotropy matrix in frequency units, and Msis the saturation magnetization of the AFM sublattices. Below, we study the small-amplitude dynamics of the AFM magnetization; therefore, we can represent the Néel vector as l=λ+se−iωt, where λis a static part describing the ground state and sis the vector describing the excita- tion,λ·s=0. Using this representation in Eq. (11),w e obtain two equations: one describing the static state of the AFM magnetization and the other one describing its small-amplitude dynamics. The “static” equation has the following form: λ×/parenleftBig ˆ/Omega1·λ+τp×λ/parenrightBig =0, (12) which can be rewritten as ˆ/Omega1·λ+τp×λ=hλ, (13) 044080-4VOLTAGE-CONTROLLED ANISOTROPY... PHYS. REV. APPLIED 13,044080 (2020) where his the effective static staggered field. The solution to Eq. (12) defines a magnetic equilibrium ground state of Néel vector λ. The “dynamic” equation can be written in the following form: λ×/bracketleftbigg −ω2 ωexs+/parenleftBig ˆ/Omega1−hˆI/parenrightBig ·s+τp×s/bracketrightbigg =0. (14) The solution to this “dynamic” equation determines the frequencies of small oscillations around the static equilib- rium. Simplifying the above “dynamic” equation, we can rewrite it in the form of a standard eigenvalue problem for the excitation vector s: ω2 ωexs= ˆ/Omega10·s+τ(λ·p)(p×s), (15) where ˆ/Omega10=ˆPλ/parenleftBig ˆ/Omega1−hˆI/parenrightBig ˆPλ, (16) ˆPλ=ˆI−λ⊗λ, (17) and the symbol ⊗denotes a tensor product. The eigenfrequencies ωcan be formally found as solu- tions to the eigenvalue problem (15). It is important to note that the obtained eigenvalue problem (15) in the case of the AFM dynamics has a form similar to that of a lin- earized vector Landau-Lifshitz (LL) equation derived in Refs. [ 36]a n d[ 37]. This similarity is natural, because both the sigma-model equation (15) and the linearized LL equa- tions in Refs. [ 36]a n d[ 37] describe the behavior of a unit vector on a sphere. Equations (12)–(15) represent a general formulation of the problem of the small-amplitude magnetization dynam- ics in AFMs obtained in the framework of a sigma model and can be used for a wide variety of different AFM crystals and piezoelectrics. IV. EXAMPLE: VOLTAGE-CONTROLLED DYNAMICS OF A CURRENT-DRIVEN BIANISOTROPIC DIELECTRIC ANTIFERROMAGNET—NiO To illustrate the general formalism for the AFM mag- netization dynamics developed above [Eqs. (12)–(15)], we apply this formalism to the case of a PZT-5H/NiO/Pt lay- ered heterostructure, choosing an AFM crystal (NiO) with cubic symmetry and piezoelectric ceramics PZT-5H polar-ized in the direction perpendicular to the PZ layer surface (n=x). The STT in this structure, created by the driving dc current in the Pt layer, is assumed to be polarized along the hard-anisotropy axis of the NiO layer ( p=z).In such a geometry, the tensor ˆKMEof the AFM mag- netic anisotropy induced by the applied transverse dc volt-age V(or the transverse dc electric field E) in the cubic crystal NiO has the form (see the Appendix ) ˆKME=⎛ ⎝−2η00 01 0 00 1⎞ ⎠2b1d2E, (18) where d2and b1are the components of the strain- piezoelectric and the magnetoelastic tensors. The effective magnetic anisotropy tensor ˆ/Omega1for NiO, which contains the magnetocrystalline and induced anisotropies and enters the eigenvalue equation (15), has the form ˆ/Omega1=ωhz⊗z−ωey⊗y+κE(−2ηx⊗x+y⊗y+z⊗z), (19) where κ=4γb1d2/Ms. Since the ground-state vector λlies in the x-yplane, it is convenient to represent it in the form λ=xcosφ+ ysinφ. Solving the “static” Eq. (12), we find the angle φ determining the equilibrium orientation of the static Néel vector λ, φ=1 2arcsin2τ ωe−κE(1+2η), (20) and the value of the staggered field h, h=(κE−ωe)cos2φ−2ηEκsin2φ. (21) The solution of the “dynamic” eigenvalue problem (15) yields two AFM eigenmode frequencies: ω1=/radicalbig ωex[ωe−κE(1+2η)]cos 2φ, (22) ω2=/radicalBig ωex/bracketleftbig ωh+κE+(ωe−κE)cos2φ+2ηκEsin2φ/bracketrightbig , (23) which are the frequencies of the lower and higher AFMR modes of the voltage- and current-driven PZT-5H/NiO/Pt layered heterostructure. Since ωh/greatermuchωe, the influence of the voltage-induced strain and the current-induced STT on the high-frequency mode of the AFMR is negligible. Therefore, in the following we will only discuss in detailthe properties of the low-frequency AFMR mode. Using Eqs. (20) and(22), it is possible to find an explicit expression for the frequency of the low-frequency AFMR mode as a function of the density of the driving current j 044080-5P.A. POPOV et al. PHYS. REV. APPLIED 13,044080 (2020) and the magnitude of the bias electric field E: ω1(j,E)=/radicaltp/radicalvertex/radicalvertex/radicalbt ωexωeff(E)/radicalBigg 1−4(σj)2 ωeff(E)2, (24) where ωeff(E)=ωe(1−βE) (25) and β=κ(1+2η)/ω e. (26) The line width /Delta1ω of the found AFMR mode in the small-amplitude approximation does not depend on the magnitudes of jand Eand for the intrinsic Gilbert damping parameter is [ 38] /Delta1ω=αωex≈2π×18 GHz. (27) In the case of E=0a n d j=0, it corresponds to the quality factor Q=ω//Delta1ω ≈12. The obtained expression (24) for the frequency of the low-frequency AFMR mode is valid for the values of the driving-current density below the thresholds of the genera- tion regime j<jth1,2[see Eqs. (4) and (5) in Ref. [ 13]] .I n the absence of the bias electric field ( E=0), an equation similar to Eq. (24)has been derived in Eq. (10) in Ref. [ 20]. In the presence of the electric field, the expressions for the generation thresholds jth1,2 are modified using the replacement ωe→ωeff(E)=ωe(1−βE): jth1=ωe(1−βE) 2σ=jth0(1−βE), (28) jth2=2α πσ/radicalbig ωexωe(1−βE). (29) This means that for the proper sign and magnitude of the bias electric field, the “easy-plane” anisotropy of the AFM can be substantially reduced and, consequently, the threshold of the terahertz-frequency signal generation in a current-driven AFM, which is proportional to the ωe→ ωeff(E), can also be reduced. Unfortunately, this reduction of the generation threshold also leads to a reduction in the amplitude of the gener- ated signal that is proportional to the time derivative of the azimuthal angle of the Néel vector lin the AFM. In the supercritical (generation) regime [for j>jth1(E)], the expression for the derivative of the azimuthal angle ˙φhas been derived in Eq. (6) in Ref. [ 13] and for the case of biasing by an electric field Ewhenωe−→ωeff(E), we obtain ˙φ=ω(j) 2+ωexωe(1−βE) 4/radicalbig (αω ex)2+ω2(j)cosω(j)t, (30) where ω(j)=2σj/α.In our particular case of the PZT-5H/NiO/Pt layered heterostructure, it is possible to calculate all the material coefficients explicitly and we obtain the following value of the coefficient βdetermining the dependence of the NiO easy-plane anisotropy on the dc electric field applied to the PZT piezoelectric layer: β=4γb1d2(1+2η)/ω eMs= 1.64×10−2(kV/cm )−1. In the same structure, the thresh- old current density of the terahertz-frequency signal gen- eration in the absence of the bias electric field is evaluated asjth0=ωe/2σ=2.02×108A/cm2, which is the current density that has been reached in many previous experi- ments performed in FM-based spin-torque and spin Hall auto-oscillators [ 7–9]. It should be noted that the magnitude of the coefficient βthat quantifies the dependence of the AFM anisotropy on the applied electric field is rather small [approximately 10−2(kV/cm )−1] and that large electric fields, close to those characteristic of dielectric breakdown, are needed to substantially influence the AFM anisotropy and, there- fore, all the other properties of an AFM crystal. Thus, for practical applications of the developed formalism, it is very important to find PZ/AFM pairs that have the largest possible coefficient β. V. DISCUSSION The results of our study of the influence of the bias dc electric field Eapplied to the piezoelectric PZ layer and the density jof the driving electric current supplied into the HM layer of the PZ/AFM/HM heterostructure on the magnetic properties of the AFM layer of the same structure are summarized in Figs. 3and4. Figure 3demonstrates that the application of the dc electric field can, in principle, substantially reduce the AFM anisotropy and, therefore, the threshold current den- sity necessary for the generation of periodic terahertz- frequency signals in current-driven AFM crystals. This result opens a way to the practical development of terahertz-frequency signal generators based on AFM sub- stances. Another notable theoretical result is illustrated in the inset to Fig. 3and shows that for a sufficiently large bias electric field, the “extinction” threshold current, determined by the Gilbert damping in the AFM crystal [see Eq. (5) in Ref. [ 13] ] could become larger than the “ignition” threshold, determined by the “easy-plane” AFM anisotropy [see Eq. (4) in Ref. [ 13] ] and, therefore, the threshold of generation in AFMs will be determined by the AFM Gilbert damping. Figure 4shows the frequency of oscillations in the AFM crystal as a function of the driving current in both the subcritical (passive) and the supercritical (active, or gener-ation) regimes for two values of the electric field. It is clear that in the subcritical regime the frequency of the low- frequency mode of the AFMR is reduced with an increase of the driving current and is also reduced with an increase 044080-6VOLTAGE-CONTROLLED ANISOTROPY... PHYS. REV. APPLIED 13,044080 (2020) FIG. 3. The dependence of the effective easy-plane anisotropy ωeff(left-hand axis) and the threshold current jth1(right-hand axis) in NiO on the electric field Eapplied to the PZT layer. Note that both the AFM anisotropy and the generation threshold could be tuned and reduced by the application of an electric fieldto the PZT layer. The inset demonstrates that for a sufficiently large value of E, the “elimination” threshold j th2can exceed the “ignition” threshold jth1[for details, see Eqs. (4) and (5) in Ref. [ 13]]. of the applied bias electric field. On the other hand, in the supercritical regime, the frequency of the generated signal is simply proportional to the supplied current density jand is practically independent of the bias electric field E. Figure 5again demonstrates that the frequency of the low-frequency AFMR mode in the subcritical(passive) regime is decreased with an increase of the bias electric FIG. 4. The dependence of the eigenfrequency of the low- frequency AFMR mode (22) in the passive subcritical ( j<jth1) regime and the frequency of generated oscillations ω(j)in the active supercritical ( j>jth1) regime on the density jof the driv- ing electric current, calculated for two values of the electricfield Eapplied to the PZT layer. Note that the AFMR fre- quency in the subcritical regime is reduced with an increase of the driving-current density j[see also Eq. (10) in Ref. [ 20]] .FIG. 5. The dependence of the eigenfrequency of the low- frequency AFMR mode (22) in the passive subcritical regime on the electric field Eapplied to the PZT layer, calculated for two magnitudes of the driving electric current density j[j1,2< jth1(E)]. field, so that in this regime an AFM crystal biased by a dc electric field can be used as an electrically tunable resonator working at subterahertz frequencies. Figure 6shows that the amplitude of the periodic high- frequency signal generated in an AFM in the supercritical regime is reduced both with an increase of the driving- current density and with an increase of the bias electric field. This means that by applying the bias electric field,it is possible to reduce the generation threshold, but the price one will has pay for this advantage is the reduced amplitude of the generated high-frequency signal. Finally, Fig. 7demonstrates the dynamic behavior of FIG. 6. The dependence of the amplitude of oscillations of the frequency ω(j)generated in the active supercritical regime on the density jof the driving current for three different values of electric field Eapplied to the PZT layer. Vertical dashed lines show the position of the generation threshold j=jth1(E).N o t e that amplitude of the generated oscillations is reduced with the increase of both the driving current jand the bias electric field E. 044080-7P.A. POPOV et al. PHYS. REV. APPLIED 13,044080 (2020) the PZ/AFM/HM layered heterostructure in Fig. 2under the influence of the dc bias current jand the bias elec- tric field E. Using the dynamic equation (11), we calculate numerically the time dependence of the angular velocity of the AFM Néel vector lin the (a) subcritical and (b) supercritical regimes. In the subcritical regime [Fig. 7(a)], we assume that the bias current is absent ( j=0) and that the magnetization dynamics start when the Néel vec- torlis displaced from its equilibrium state ( l=λ) with the initial condition l(t=0)=λ+s0,∂l/∂t|t=0=0. In this case, the magnetization dynamics are described by damped oscillations with a frequency close to the AFMR frequency and this frequency is reduced with an increase of the bias electric field E[see Eq. (24)]. In the supercrit- ical regime [Fig. 7(b)]j>jth1(E)(j=2×108A/cm2), the dynamics are described by self-sustained oscillations, the frequency of which is determined by the bias current j (for details, see Ref. [ 13]), while the electric field Edeter- mines only the threshold current jth1and the amplitude of the angular-velocity oscillations [see Eq. (30)]. (a) (b) FIG. 7. The numerically calculated dynamical behavior of the PZ/AFM/HM layered heterostructure in Fig. 2under the influ- ence of a dc bias current and a dc electric field in the (a)subcritical and (b) supercritical regimes. The angular velocity of the rotating AFM Néel vector lis presented as a function of time.VI. CONCLUSIONS The results presented above demonstrate that the dynamic magnetic properties of AFM crystals in PZ/AFM/HM layered heterostructures can be effectively controlled by both the driving current flowing along the HM layer and the transverse dc electric field applied to the PZ layer of the heterostructure. One of the main results of our current work is the deriva- tion of the general tensorial equations (9)and(10) relating the tensor of anisotropy in an AFM crystal to the vec- tor of the dc electric field applied to the PZ layer. Using these tensorial equations in a particular case of a trans- verse electric field applied to a NiO AFM crystal (having cubic symmetry), we consider the current-induced dynam- ics of magnetization in NiO and obtain an expression (25) describing the dependence of the “easy-plane” anisotropy in NiO on the applied dc electric field, and the explicit expression (24) for the frequency of the AFMR acoustic mode as a function of the driving-current density jand the bias electric field Ein the subcritical regime j<jth1,2. We also obtain the expression (30) for the derivative of the azimuthal angle ˙φof the Néel vector lin the super- critical (generation) regime for j>jth1(E)in the presence of the bias electric field, which allows us to evaluate the amplitude of the high-frequency signal generated in the AFM layer as a function of jand E(see6). We believe that the obtained explicit equations could be used for theoretical analysis in a variety of practical spintronic problems that require AFM-anisotropy manipu- lation and switching, such as control of magnetic random- access memories (MRAMs) [ 39,40], the development of terahertz-frequency signal generators [ 1,12,13]a n dr e s o - nance monochromatic receivers [ 1,20,22] based on AFM crystals, the control and excitation of spin waves in AFM and FM magnonic devices [ 41], and many others. The obtained equations can also be useful in the study of the influence of elastic strain on nonmagnetic systems (e.g., Ref. [ 42]). ACKNOWLEDGMENTS This work was supported in part by the U.S. National Science Foundation (Grants No. EFMA-1641989 and No. ECCS-1708982), by the U.S. Air Force Office of Scien- tific Research under the Grant No. FA9550-19-1-0307, and by the Oakland University Foundation. Studies of the voltage-induced magnetic anisotropy effect were carried out with the support of the Russian Science Foundation (Grant No. 19-19-00607) and a grant from the Gov- ernment of the Russian Federation for the state support of scientific research conducted under the guidance ofleading scientists in Russian higher-education institutions, research institutions, and state research centers of the Rus- sian Federation (Project No. 2019-220-07-9114). Studies of the magnetic dynamics in antiferromagnetic STNOs 044080-8VOLTAGE-CONTROLLED ANISOTROPY... PHYS. REV. APPLIED 13,044080 (2020) with voltage-induced magnetic anisotropy were supported by the Russian Foundation for Basic Research (RFBR) under Grants No. 18-37-20048, No. 18-29-27018, No. 18- 57-76001, No. 18-07-00509 A, and No. 18-29-27020, and by the grants of the President of the Russian Federation (Grants No. MK-283.2019.8 and No. MK-3607.2019.9). S.A.N. acknowledges support from the Government of the Russian Federation [Grant No. 074-02-2018-286 for the “Terahertz Spintronics” laboratory of the Moscow Institute of Physics and Technology (MIPT)]. A.R.S. acknowledges the support from the RFBR (Grant No. 19-29-03015). APPENDIX: MATERIAL CONSTANTS In this appendix, we present the magnitudes of all the material constants used in the calculations presented above and show explicitly the tensorial formalism leading to the derivation of the general equation (10). Here, in addition to the quantities σPZ ij,ePZ ij,σij,a n d eij defined above, which denote the stress tensors and the elas- tic strain tensors in the PZ and AFM layers, respectively, we also introduce the quantities σPZ,ePZ,σ,a n d eas the six-dimensional vector representations of these tensors in the Voigt notation (see p. 66 in Ref. [ 33]). We introduce the six-dimensional vectors in order to represent the tensors dkijand Cijklin more explicit and transparent forms. In our calculations, we use the following magnetic parameters of the NiO crystal: He=6.2×102G[38],ωe=γHe=2π×1.75 GHz, Hh=1.6×104G[38],ωh=γHh=2π×43.9 GHz, Hex=9.8×106G[38],ωex=γHex=2π×27.5 THz, Ms=3.5×102Oe [43]. The following values of the NiO material constants are taken from Ref. [ 13]: α=3.5×10−3; σ=2π×4.32×10−4Hz/(A/m2). The following parameters of the ME interaction and components of the NiO stiffness tensor are taken from Ref. [ 44]: WME=bijklmimjekl, bijkl=b0δijδkl+b1(δikδjl+δilδjk), b0=5.93×105J/m3, b1=−17.7×105J/m3; Wel=1 2Cijkleijekl, Cijkl=C0δijδkl+C1(δikδjl+δilδjk), C0=7.85×1010J/m3,C1=5.39×1010J/m3.In our calculations, we assume that the piezoelectric ceramic PZT-5H is polarized along the xaxis (see Fig. 2): ˆd=/parenleftBig ˆCPZ/parenrightBig−1 ˆe=⎛ ⎜⎜⎜⎜⎜⎝d 100 d200 d200 000 00 d4 0 d40⎞ ⎟⎟⎟⎟⎟⎠, (A1) ⎛ ⎝d 1 d2 d4⎞ ⎠=⎛ ⎝5.94 −2.75 7.39⎞ ⎠×10−10(V/m)−1, (A2) where ˆCPZandˆeare the stiffness and stress-piezoelectric tensors of PZT-5H, presented on p. 278 in Ref. [ 33]. Assuming that the dc voltage is applied perpendicular to the interface, so that the dc electric field is directed along thexaxis E=Ex, the PZ strain ePZcan be calculated as ePZ=ˆdE=(d1,d2,d2,0 ,0 ,0 )T×E. (A3) Then, the strain at the PZ/AFM interface eIFentering Eq.(3)can be represented as eIF=PikePZ klPlj=(0,d2,d2,0 ,0 ,0 )T×E. (A4) Let us now explicitly derive the tensor Gijklfor the case of a NiO crystal. The tensor Cijkl, written in the Voigt notation, has the following form: ˆC=⎛ ⎜⎜⎜⎜⎜⎝C 11 C12 C12 000 C12 C11 C12 000 C12 C12 C11 000 000 C44 00 0000 C44 0 00000 C44⎞ ⎟⎟⎟⎟⎟⎠, (A5) ⎛ ⎝C 11 C12 C44⎞ ⎠=⎛ ⎝C0+2C1 C0 C1⎞ ⎠. (A6) Then, the boundary condition (4)σijni=0 takes the fol- lowing explicit form, ⎧ ⎪⎨ ⎪⎩C12eyy+C12ezz+C11exx=0, 2C44exz=0, 2C44exy=0,(A7) while the boundary condition (3)PikeklPlj=eIF ijcan be written as ⎧ ⎪⎨ ⎪⎩eyy=ePZ yy, ezz=ePZ zz, 2eyz=2ePZ yz.(A8) 044080-9P.A. POPOV et al. PHYS. REV. APPLIED 13,044080 (2020) Taking Eq. (A4) into account, we can rewrite the equation eij=GijkldmklEmfor NiO in the following form: eij=⎛ ⎝−2η00 01 0 00 1⎞ ⎠d2E, (A9) where η=C12/C11=C0/C0+2C1. We can then obtain an explicit expression for the tensor KME ijof induced magnetic anisotropy in the form KME ij=bijklekl=/parenleftbig b0δijδkl+b1(δikδjl+δilδjk)/parenrightbig ekl =⎛ ⎝−2η00 01 0 00 1⎞ ⎠2b1d2E. 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PhysRevB.82.214403.pdf

Detection and quantification of inverse spin Hall effect from spin pumping in permalloy/normal metal bilayers O. Mosendz,1,*V . Vlaminck,1J. E. Pearson,1F. Y . Fradin,1G. E. W. Bauer,2S. D. Bader,1,3and A. Hoffmann1,† 1Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 2Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands 3Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, USA /H20849Received 27 August 2010; published 1 December 2010 /H20850 Spin pumping is a mechanism that generates spin currents from ferromagnetic resonance over macroscopic interfacial areas, thereby enabling sensitive detection of the inverse spin Hall effect that transforms spin intocharge currents in nonmagnetic conductors. Here we study the spin-pumping-induced voltages due to theinverse spin Hall effect in permalloy/normal metal bilayers integrated into coplanar waveguides for differentnormal metals and as a function of angle of the applied magnetic field direction, as well as microwavefrequency and power. We find good agreement between experimental data and a theoretical model that includescontributions from anisotropic magnetoresistance and inverse spin Hall effect. The analysis provides consistentresults over a wide range of experimental conditions as long as the precise magnetization trajectory is takeninto account. The spin Hall angles for Pt, Pd, Au, and Mo were determined with high precision to be0.013/H110060.002, 0.0064 /H110060.001, 0.0035 /H110060.0003, and −0.0005 /H110060.0001, respectively. DOI: 10.1103/PhysRevB.82.214403 PACS number /H20849s/H20850: 72.25.Rb, 75.47. /H11002m, 76.50. /H11001g I. INTRODUCTION Information in semiconductor electronic devices and data storage technologies is mainly transported and manipulatedby charge currents. With advancing miniaturization, heat dis-sipation and power consumption become significant ob-stacles to further technological advances. Alternative tech-nologies that solve or at least circumvent these problems areneeded. One promising candidate to replace existing charge-based technologies is based on using spin currents; an effortreferred to as spintronics. 1Magnetoelectronic devices em- ploying spin-polarized charge currents are already actively inuse in hard drive read-heads and nonvolatile magnetic ran-dom access memories. Pure spin currents that are not accompanied by a net charge current, may offer additional advantages inapplications, 2,3such as reduced power dissipation, absence of stray Oersted fields, and decoupling of spin and chargenoise. Furthermore, undisturbed by charge transport, purespin currents can provide more direct insights into the basicphysics of spin-dependent effects. Pure spin currents can becreated by, for instance: /H20849i/H20850nonlocal electrical injection from ferromagnetic contacts in multiterminal structures; /H20849ii/H20850opti- cal injection using circularly polarized light; /H20849iii/H20850spin pump- ing from a precessing ferromagnet; and /H20849iv/H20850spin Hall effect. The last possibility is particularly interesting since ferromag-nets are not involved. 4–6The spin Hall effect is caused by the spin-orbit interactions of defect scattering potentials or thehost electronic structure. The efficiency of this spin-chargeconversion can be quantified by a single material-specificparameter, viz., the spin Hall angle /H9253, which is defined as the ratio of the spin Hall and charge conductivities7and can be measured by magnetotransport measurements.8–12Previous experimental studies report quite different /H9253values for nomi- nally identical materials. For example, for Au a giant /H9253 =0.113 was reported11while subsequent experiments found values that are one or even two orders of magnitudesmaller.13,14Similarly, for Pt different experiments10,14,15re- sulted in /H9253values that vary between 0.0037 and 0.08. Recently we demonstrated a robust technique14to mea- sure spin Hall angles with high accuracy in arbitrary conduc-tors. Our approach is based on the combination of spinpumping, which generates pure spin currents, and measure-ments of electric voltages due to the inverse spin Hall effect/H20849ISHE /H20850. 16Here we present a detailed discussion of the mea- surements in Ref. 14and examine the validity of the theo- retical model used to describe the voltages induced in theNi 80Fe20/H20849Py/H20850/normal metal /H20849N/H20850bilayers. In particular, we measure the ISHE voltage as a function of angle of the ap-plied magnetic field, and microwave frequency and power.We find excellent agreement between model calculations andexperimental results. Accounting for the proper magnetiza-tion trajectory is important for a quantitative interpretation ofthe results. Good agreement between the theoretical modeland experiments for a wide range of controlled experimentalparameters implies that our approach is robust and can beused to determine the magnitude and sign of spin Hall effectsin more conductors than included in the present study. II. COUPLING BETWEEN SPIN AND CHARGE CURRENTS A. Spin pumping in Py/N bilayers Spin pumping generates pure spin currents in N, when they are in contact with a ferromagnet with time-dependentmagnetization induced, e.g., by ferromagnetic resonance /H20849FMR /H20850. 17–21The instantaneous spin-pumping current js0at the Py/N interface is given by22,23 js0s/H6023=/H6036 8/H9266Re/H208492g↑↓/H20850/H20875m/H6023/H11003/H11509m/H6023 /H11509t/H20876, /H208491/H20850 where m/H6023is the unit vector of the magnetization, s/H6023is the unit vector of the spin current polarization in N, and Re /H20849g↑↓/H20850isPHYSICAL REVIEW B 82, 214403 /H208492010 /H20850 1098-0121/2010/82 /H2084921/H20850/214403 /H2084910/H20850 ©2010 The American Physical Society 214403-1the real part of the spin-mixing conductance. The spin cur- rent generated by spin pumping is polarized perpendicular tothe instantaneous magnetization direction m /H6023and its time de- rivative /H11509m/H6023//H11509t/H20849see Fig. 1/H20850. Note that this spin current always has a polarization component along H/H6023dcand propagates into N normal to the interface. The spin current generated at the Py/N interface accumu- lates a spin density /H9262/H6023Ninside the N layer. In the ballistic limit /H20849i.e., no spin relaxation in N /H20850the spin current reaching the N/vacuum interface is fully reflected and reabsorbedupon returning to the Py/N interface, without influencing themagnetization dynamics of the bilayer system. In real systems, pure spin currents are not conserved, since spins relax over length scales given by the spin diffu-sion length /H9261 sdin N, and the accumulated spin density moves across the N layer via spin diffusion limited by momentumscattering /H20849leading to electrical resistance /H20850and spin-flip scat- tering /H20849leading to loss of spin angular momentum /H20850by spin- orbit coupling or magnetic impurities. The spin-diffusionequation describes the dissipative propagation of the spinaccumulation /H20849difference in local electrochemical potentials of up and down spins /H20850 /H9262/H6023Nin the N layer, i/H9275/H9262/H6023N=D/H115092/H9262/H6023N /H11509z2−1 /H9270sf/H9262/H6023N, /H208492/H20850 where /H9275is the angular frequency, /H9270sfis the spin-flip time, zis the coordinate normal to the interface, and D=vF2/H9270el/3 is the electron diffusion constant, with /H9270elthe electron momentum relaxation time.24The solutions of Eq. /H208492/H20850depend on the boundary conditions. For a single magnetic layer structurePy/N the boundary condition at the Py/N interface is givenby 23 js0s/H6023/H20849z=0 /H20850=−D/H20879/H11509/H9262/H6023N /H11509z/H20879 z=0/H208493/H20850 while for the outer interface we use the free magnetic mo- ment condition /H20849full spin current reflection /H20850/H20879/H11509/H9262/H6023N /H11509z/H20879 z=L=0 . /H208494/H20850 Equations /H208492/H20850–/H208494/H20850can be solved analytically to yield the de- cay of the spin accumulation as a function of the distancefrom the Py/N interface. This decay results in spin accumu-lation profile in the N layer, which decays as a function ofthe distance from the interface, thus driving a spin currentwith a dc contribution, j s/H20849z/H20850=js0sinh /H20851/H20849z−tN/H20850//H9261sd/H20852 sinh /H20849tN//H9261sd/H20850, /H208495/H20850 where tNis the thickness of the N layer. The spin accumula- tion in N gives rise to spin backflow into the ferromagnet,which effectively reduces the spin pumping current, whichcan be accounted for by replacing g ↑↓in Eq. /H208491/H20850with an effective spin mixing conductance geff↑↓.24 In FMR experiments the absorption of the microwave field that excites the magnetization is monitored. The mag-netization dynamics in ferromagnetic films can be describedby the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation of motion, 1 /H9253g/H11509m/H6023 /H11509t=− /H20851m/H6023/H11003H/H6023eff/H20852+/H9251G /H9253g/H20875m/H6023/H11003/H11509m/H6023 /H11509t/H20876, /H208496/H20850 where /H9253g=ge /2mcis the absolute value of the gyromagnetic ratio and /H9251Gis the dimensionless Gilbert damping parameter. The first term on the right-hand side represents the preces- sional torque due to the effective internal field H/H6023eff, which for the case of permalloy with small anisotropy is approxi-mately equal to the externally applied magnetic field H dc. The second term in Eq. /H208496/H20850represents the Gilbert damping torque.25,26The spin pumping can be accounted for in the LLG equation of motion by adding a spin pumping contri-bution /H9251spto/H9251G, i.e., the effective damping becomes /H9251eff =/H9251G+/H9251sp. The damping can be quantified by measuring the FMR linewidth /H9004H, half width at half maximum /H20849HWHM /H20850, of the imaginary part of the rf susceptibility /H9273/H11033, which is commonly measured at a constant microwave frequency bysweeping the dc magnetic field H dc. In case of Gilbert damp- ing,/H9004Hdepends linearly on the microwave angular fre- quency /H9275f, i.e.,/H9004H=/H9251eff/H9275f//H9253g. The difference in the damp- ing parameter, determined by the FMR linewidth, forsamples without capping layer and samples in which the cap-ping N layer is sufficiently thick to fully dissipate thepumped magnetic moment, is attributed to the loss of spinmomentum in Py due to relaxation of the spin accumulationin N. This permits the determination of the additional inter-face damping due to spin pumping, 27which in turn fixes the interfacial spin-mixing conductance to geff↑↓=4/H9266/H9253gMstPy g/H9262B/H9275f/H20849/H9004HPy/H20841N−/H9004HPy/H20850, /H208497/H20850 where tPyis the Py layer thickness, Msis the Py saturation magnetization, and /H9262Bis the Bohr magneton. Note that Eq. /H208497/H20850is only applicable when the damping is governed by the Gilbert phenomenology or /H9004H/H11008/H9275f, i.e., when inhomoge- neous linewidth broadening is negligible. Otherwise geff↑↓can still be determined from the additional Gilbert-type damping Py N ssjj/c114/c114 SSdcH/c114 m/c114||m/c114 dtmd/c114 FIG. 1. /H20849Color online /H20850Schematic model of spin pumping in Py/N bilayer. s/H6023shows that the polarization of the spin current is oscillating in time with the frequency of the magnetization preces-sion. The polarization of the spin current is perpendicular to theinstantaneous magnetization direction m /H6023and the rate of magnetiza- tion change /H11509m/H6023//H11509t.MOSENDZ et al. PHYSICAL REVIEW B 82, 214403 /H208492010 /H20850 214403-2contribution /H9251sp=/H9251Py/H20841N−/H9251Py, where the latter two are ob- tained from the linewidth difference that scales linear withfrequency, i.e., /H9004H=/H9004H ih+/H9251eff/H9275f//H9253g, where /H9004Hihis the sample-dependent inhomogeneous linewidth, measured asthe zero-frequency intercept. The dc component of the spin current pumped into N is polarized parallel to the equilibrium magnetization and haspreviously been detected via a dc voltage normal to the Py/N interface. 28Under a simple circular precession of the Py magnetization the time averaging of the spin current fromEq. /H208491/H20850for small precession cone angles /H9258reads js,dc0,circ=/H6036/H9275f 4/H9266Regeff↑↓sin2/H9258. /H208498/H20850 In thin magnetic films the trajectory of the magnetization precession is not circular but elliptic due to the strong de-magnetizing fields, which force the magnetization into thefilm plane. The time-dependent cone angle /H9258modifies the dc component of the pumped spin current by an ellipticity cor-rection factor Pas derived and measured by Ando et al. 29 For an in-plane equilibrium magnetization js,dceff=P/H11569js,dccircwith P=2/H9275f/H20851/H9253g4/H9266Ms+/H20881/H20849/H9253g4/H9266Ms/H208502+/H208492/H9275f/H208502/H20852 /H20849/H9253g4/H9266Ms/H208502+/H208492/H9275f/H208502. /H208499/H20850 Equation /H208499/H20850is a nonmonotonic function of /H9275f, and Pcan become slightly larger than 1, but tends toward 1 for highfrequencies, i.e., large applied fields. B. Inverse spin Hall effect Spin-orbit coupling or magnetic impurities give rise to different scattering directions for electrons with oppositespin. In their presence, a spin current in N induces a trans-verse Hall voltage. This ISHE transforms spin currents intoelectrical voltage differences over the sample edges. Spinpumping generates dc and ac components to the spin current: j s,dceffand an rf component transverse to the equilibrium mag- netization direction. In this paper we address only the ISHE effect generated by the dc component js,dceff. The dc ISHE transverse charge current reads j/H6023cISH/H20849z/H20850=/H9253/H208492e//H6036/H20850js,dceff/H20851n/H6023/H11003/H20855s/H6023/H20856/H20852, /H2084910/H20850 where /H9253is the spin Hall angle, n/H6023is the unit vector normal to the interface, and /H20855s/H6023/H20856is the polarization vector of the dc spin current. For js,dceffthe spin polarization /H20855s/H6023/H20856is along the equi- librium magnetization direction in Py. The dc electric fieldlies in N in the plane of the films and perpendicular to theequilibrium magnetization of Py. 14,16,30 III. EXPERIMENTAL RESULTS Here we elucidate our method to obtain voltage signals due to the ISHE in various Py/N combinations under FMRconditions, thereby determining the spin Hall angle /H9253with high accuracy. The measured voltage signals scale with thesample length and, therefore, can be increased readily bymaking the samples longer. We identify two contributions tothe dc voltage: one stems from the anisotropic magnetoresis-tance /H20849AMR /H20850and the second from the ISHE, which can be distinguished by their symmetries with respect to the fieldoffset from the resonance field. Furthermore, we present atheoretical model for the spin Hall angle contribution andtest its functional dependence of several parameters that canbe controlled experimentally. A. Experimental setup The Py/N bilayers were integrated into coplanar waveguides with additional leads in order to measure the dcvoltage along the sample. This is shown in Fig. 2for a Py/Pt bilayer, with lateral dimensions of 2.92 mm /H1100320 /H9262m and 15-nm-thick individual layers. The bilayer was prepared byoptical lithography, sputter deposition, and lift-off on a GaAssubstrate. Subsequently, we prepared Ag contacts for thevoltage measurements, covered the whole structure with 100nm of MgO /H20849for dc insulation between bilayer and wave- guide /H20850, and defined a 30- /H9262m-wide and 200-nm-thick Au co- planar waveguide on top of the bilayer. Similar samples wereprepared with Pd, Au, and Mo layers replacing Pt. The high bandwidth of the coplanar waveguide setup en- abled us to carry out measurements with microwave excita-tions in the frequency interval of 4–11 GHz. The power ofthe rf excitation was varied from 15 to 150 mW. For a givenfrequency, experiments were carried out as a function of ex- ternal magnetic field H /H6023dc, with an in-plane orientation that could be rotated to arbitrary angles /H9251with respect to the central axis of the coplanar wave guide. While the FMRsignal was determined from the impedance of thewaveguide, 31the dc voltage was measured simultaneously with a lock-in modulation technique as a function of H/H6023dc. B. FMR measurements The FMR frequency vs peak position for the Py/Pt sample is shown in Fig. 3/H20849a/H20850. Fitting the data to the Kittel formula, /H20849/H9275f//H9253g/H208502=Hdc/H20849Hdc+4/H9266Ms/H20850/H20849 11/H20850 results in the saturation magnetization for Py of Ms =852 G. Figure 3/H20849b/H20850shows the FMR line width as a func- tion of frequency. The linear behavior of the FMR linewidthindicates that damping in Py is governed by the intrinsicGilbert phenomenology and any extrinsic effects are small. Py/Pt waveguid e(a) (b)Contacts V200um20um FIG. 2. /H20849Color online /H20850Experimental setup: /H20849a/H20850Optical image of the Py/Pt bilayer integrated into the coplanar waveguide. /H20849b/H20850Con- tacts are added at the end of the bilayer to measure the voltagealong the waveguide direction.DETECTION AND QUANTIFICATION OF INVERSE SPIN … PHYSICAL REVIEW B 82, 214403 /H208492010 /H20850 214403-3Figure 4shows FMR spectra for a Py/Pt bilayer and a Py single layer at 4 GHz excitation frequency. The FMR peakpositions for the two samples are similar. The main differ-ence between the spectra is their FMR linewidth. The FMRlinewidths /H20849HWHM /H20850extracted from fits to Lorentzian ab- sorption functions are /H9004H Pt/Py=16.9 Oe for Py/Pt and /H9004HPy=12.9 Oe for Py. The difference in /H9004Hcan be attrib- uted to the loss of pumped spin momentum in the Pt layer.The thickness of the Pt layer is 15 nm, which is larger than /H9261 sdPt=10/H110062 nm.32Thus, all pumped spin momentum is dis- sipated in the Pt layer and we can extract the value of the spin mixing conductance geff↑↓from the increased linewidth. Using Eq. /H208497/H20850we calculate a spin mixing conductance geff↑↓ =2.1/H110031019m−2at the Py/Pt interface. This experimental value is somewhat smaller than the previously reported2.58/H1100310 19m−2.24,33Caoet al.34showed that for high power rf excitation, the spin mixing conductance can be reduceddue to the loss of coherent spin precession in the ferromag-net. This could be the case here, since the cone angle for theFMR at 4 GHz is relatively large, and a slightly larger mix-ing conductance for the smaller precession angles at 11 GHzwould lead to more consistent frequency-dependent values ofthe spin Hall angles as discussed below. C. dc voltage due to ISHE and AMR effects Figure 5shows the dc voltage measured along the samples with an external field applied at 45° from the copla-nar waveguide axis. For the Py/Pt sample we observe a reso-nant increase in the dc voltage along the sample at the FMRposition. However, the line shape is complex: below theresonance field the voltage is negative, it changes sign justbelow the FMR resonance field, and has a positive tail in thehigh-field region. In contrast, the single-layer Py sample,which is not affected by spin pumping, shows a voltage sig-nal that is purely antisymmetric with respect to the FMRposition and thus mirrors the derivative FMR signal shownin Fig. 4/H20849b/H20850. The voltage generated by the ISHE depends only on the cone angle /H9258of the magnetization precession /H20851see Eq. /H208498/H20850/H20852and thus must be symmetric with respect to the FMR resonance position. This means that the voltage mea-sured in the Py/Pt sample has two contributions: /H20849i/H20850a sym- metric signal due to the ISHE and /H20849ii/H20850an antisymmetric sig- nal of the same origin as that in the Py control sample. The antisymmetric voltages observed in both Py and Py/Pt depend on the cone angle /H9258of the magnetization dy-(a) (b) FIG. 3. /H20849Color online /H20850Experimental data /H20849symbols /H20850for FMR peak positions and FMR linewidths of a Py/Pt bilayer are shown asa function of rf frequency in /H20849a/H20850and /H20849b/H20850, respectively. The solid line in/H20849a/H20850represents the fit to Eq. /H2084911/H20850and results in M s=852 G. The solid line in /H20849b/H20850represents a linear fit to the linewidth vs frequency dependence. FIG. 4. /H20849Color online /H20850Derivative FMR spectra at 4 GHz for /H20849a/H20850 Py and /H20849b/H20850Py/Pt. Solid lines are Lorentzian line-shape fits.-300-1500150300 Voltage (/c109V)Py 100 150 200 250 30 0-2000200400600 Voltage (/c109V) Hdc(Oe)Py/Pt(a) (b) FIG. 5. /H20849Color online /H20850V oltage measured along the samples vs field Hdcfor Py and Py/Pt at 4 GHz is shown with symbols in /H20849a/H20850 and /H20849b/H20850, respectively. Only the AMR contribution is present in the Py sample. Solid line in /H20849a/H20850shows a fit to Eq. /H2084912/H20850. Both AMR and ISHE effects are observed in the Py/Pt. Dotted and dashed-dottedlines show the AMR and ISHE contributions, respectively, whichare extracted from fitting the data to Eqs. /H2084912/H20850and /H2084918/H20850; the solid line in /H20849b/H20850shows the combined fit for the Py/Pt sample.MOSENDZ et al. PHYSICAL REVIEW B 82, 214403 /H208492010 /H20850 214403-4namics since they vary rapidly around the FMR resonance position. This suggests that the antisymmetric signal origi-nates from the AMR. 35–37Although the MgO provides dc insulation between the sample and the waveguide, see Fig.6/H20849a/H20850, the strong capacitive coupling allows some leakage of the rf driving current I rf=Irfmsin/H9275ftinto the sample, Irf,S, which flows along the waveguide direction. Its magnitudecan be estimated from the ratio between the waveguide re-sistance R wgand the sample resistance RS:Irf,S=IrfRwg/RS since the capacitive coupling impedance is negligible for /H9275f/H114074 GHz. Furthermore, due to the strong capacitive cou- pling /H2084950 pF /H20850between sample and waveguide, both rf cur- rents in the sample and waveguide are for all practical pur-poses in phase, i.e., the relative phase shift is expected to beat most 10 −3/H9266. Indeed, experiments with single-layer Py samples14and with a MgO layer inserted between the Py and Pt layers39are consistent with a pure AMR signal, as de- scribed below, without any appreciable phase shift. The precessing magnetization in the Py /H20851see Fig. 6/H20849b/H20850/H20852 results in a time-dependent RS/H20851/H9274/H20849t/H20850/H20852=R0−/H9004RAMRsin2/H9274/H20849t/H20850due to the AMR given by /H9004RAMR, where R0is the sample resistance with the magnetization along the waveguide axisand /H9274is the angle between the instantaneous magnetization m/H6023and the waveguide axis /H20851see Fig. 6/H20849b/H20850/H20852.35/H9004RAMR can be experimentally determined by static magnetoresistance mea-surements under rotation of an in-plane field sufficientlylarge to saturate the magnetization. Since the AMR contribu-tion to the resistance oscillates at the same frequency as therf current, but phase shifted, a homodyne dc voltage devel-ops and is given by 14 VAMR=IrfmRwg RS/H9004RAMRsin/H208492/H9258/H20850 2sin/H208492/H9251/H20850 2cos/H92720, /H2084912/H20850 where /H92720is the phase angle between magnetization preces- sion and driving rf field, and the relation between /H9258,/H9251, and/H9274 is illustrated in Fig. 6/H20849b/H20850. Well below the FMR resonance the phase angle /H92720is zero, it becomes /H9266/2 at the peak, and is /H9266 far above the resonance.38Thus, cos /H92720changes sign upon going through the resonance, which gives rise to an antisym-metric V AMR, as is observed in both the Py and Py/Pt samples. Following Guan et al.38we calculate the cone angle /H9258and sin /H92720as a function of the applied field Hdc, FMR resonance field Hr, FMR linewidth /H9004Hand rf driving field hrf, /H9258=hrfcos/H9251 /H9004H/H208811+/H20875/H20849Hdc−Hr/H20850/H20849Hdc+Hr+4/H9266Ms/H20850 /H9004H4/H9266Ms /H208762/H2084913/H20850 and sin/H92720=1 /H208811+/H20875/H20849Hdc−Hr/H20850/H20849Hdc+Hr+4/H9266Ms/H20850 /H9004H4/H9266Ms /H208762./H2084914/H20850 The anisotropic magnetoresistance was determined by dc magnetoresistance measurements with fields applied alongthe hard axis as /H9004R AMR=0.95%. This allows us to fit the Py data /H20851see Fig. 5/H20849a/H20850/H20852with only one adjustable parameter hrf =4.5 Oe using Eqs. /H2084912/H20850–/H2084914/H20850. In order to understand the symmetric contribution to the Py/Pt voltage data we have to include an additional voltagedue to the ISHE. In principle, an inductive coupling /H20849if any /H20850 could result in a symmetric voltage contribution to the sig-nal. However, this type of coupling is unlikely in oursamples due to the fact that sample and transmission line areprepared as a stack with a thin insulator in the middle. Fur-thermore, our recent work 39showed that if spin pumping is suppressed by insertin ga3n mM g O layer at the Py/Pt in- terface, then the symmetric part of the voltage vanishes. Thisunambiguously shows that the symmetric part of the mea-sured voltage is related to spin accumulation in N, whichappears due to the ISHE. The absence of a symmetric con-tribution for Py alone also suggests that inductive effects arenegligible. In an open circuit, an electric field E /H6023is generated leading to a total current density xyz /c113 /c121dcH/c114 ||m/c114m/c114(a) /c97 V/c97(b) Nt LNISH cj/c114s/c114 sj(c)SrfI, rfh NWaveguidedcH/c114 MgO SrfI,wgrfI, rfI Py FIG. 6. /H20849Color online /H20850Splitting of rf current due to capacitive coupling is schematically shown in /H20849a/H20850together with the directions of the applied dc magnetic field H/H6023dcand the rf driving field h/H6023rfwith respect to the bilayer and waveguide. /H20849b/H20850Schematic of m/H6023precessing in Py. m/H6023precesses around its equilibrium direction given by H/H6023dcat the driving frequency /H9275fand with a phase delay /H92720with respect to hrf./H9251is the angle between H/H6023dcand the waveguide axis /H20849along y/H20850,/H9258 is the cone angle described by m/H6023and/H9274is the angle between m/H6023and the waveguide axis. Due to the strong capacitive coupling part of Irf flows through the Py given by Irf,S./H20849c/H20850Geometry of the dc compo- nent of the pumped spin current with polarization direction /H20855s/H6023/H20856 along the equilibrium magnetization direction m/H6023/H20648. The charge cur- rent due to ISHE j/H6023cISHis orthogonal to the spin current direction /H20849normal to the interface /H20850and /H20855s/H6023/H20856. The voltage due to the ISHE is measured along y/H20849waveguide axis /H20850. Solid arrows indicate the spin accumulation inside N, which decays with the spin-diffusion length/H9261 sd.DETECTION AND QUANTIFICATION OF INVERSE SPIN … PHYSICAL REVIEW B 82, 214403 /H208492010 /H20850 214403-5jc/H6023/H20849z/H20850=jcISH/H20849z/H20850/H20849xˆcos/H9251+yˆsin/H9251/H20850+/H9268E/H6023, /H2084915/H20850 where /H9268is the charge conductivity and xˆ,yˆare defined in Fig. 6. Since there is no current flowing in the open circuit, /H20885 −tPytN jc/H6023/H20849z/H20850dz=0 . /H2084916/H20850 When the wire is much longer than thick, the electric field in the wire is constant. On the other hand, voltage generationoccurs only in the Pt layer /H20849more precisely in a skin depth of the spin-flip length in which the ISHE emf is generated /H20850, while the Py layer acts as a short, which decreases the volt-age difference at the sample terminals. Solving the system ofEqs. /H2084915/H20850and /H2084916/H20850, we obtain the component of the electric field along the measurement direction yas E y=−Pgeff↑↓sin/H9251sin2/H9258/H9253e/H9275f/H9261sd 2/H9266/H20849/H9268NtN+/H9268PytPy/H20850tanh/H20873tN 2/H9261sd/H20874, /H2084917/H20850 where /H9268Nand/H9268Pyare the charge conductivities in the N layer /H20849e.g., Pt /H20850and Py, and tNandtPyare the thicknesses of the N and Py layers. Using Eq. /H2084917/H20850we calculate the voltage due to the ISHE generated along the sample with length L, VISH=−/H9253eLP/H9275f/H9261sdgeff↑↓sin/H9251sin2/H9258 2/H9266/H20849/H9268NtN+/H9268PytPy/H20850tanh/H20873tN 2/H9261sd/H20874./H2084918/H20850 Note that this voltage is proportional to Land, thus, suffi- ciently large voltage signals can be measured even for small /H9253values by increasing the sample length. Furthermore, note that for the case of the normal layer thickness tNbeing com- parable to the spin-diffusion length /H9261sdthe measured voltage depends only very weakly on either value, sincet N//H9261sdtanh /H20849tN/2/H9261sd/H20850is approximately constant. One of the input parameters in VISHis the ellipticity cor- rection factor P. At 4 GHz excitation, FMR occurs at Hdc /H11015200 Oe. Therefore, the magnetization precession trajec- tory is highly elliptical and a correction to the dc voltagecomponent due to the ellipticity is significant. Figure 7 shows Pas a function of microwave frequency as calculated using Eq. /H208499/H20850. In the range from 4 to 13 GHz, Pchangesalmost by a factor of 3 and, therefore, has to be taken into account. At frequencies above 10 GHz Pbecomes larger than 1 and reaches a maximum value of 1.3 at /H1101528 GHz before it slowly decreases toward 1 for higher frequencies.This means that the most effective pumping of dc componentof spin current is achieved not for circular precession butrather for some elliptical trajectory of magnetization preces-sion. We used Eqs. /H2084918/H20850and /H2084912/H20850to fit the voltage measured for the Py/Pt sample /H20851see the solid line in Fig. 5/H20849b/H20850/H20852. The dashed and dotted lines in Fig. 5/H20849b/H20850are the AMR and ISHE contri- butions, respectively. By using a literature value for Pt of/H9261 sd=10/H110062 nm,32the only remaining adjustable parameters are the rf driving field hrfand the spin Hall angle /H9253 /H110150.011/H110060.002. Note that through the cone angle /H9258,hrfaf- fects both the AMR and ISHE contributions. In fact, as seenfrom the fit to the control Py sample, h rfis already deter- mined by the negative and positive tails of the AMR part. We carried out additional measurements of the spin Hall angle as a function of the microwave frequency. Since thespin pumping is proportional to the time derivative of themagnetization /H20851as manifested by the factor /H9275fin Eq. /H2084918/H20850/H20852 and the ellipticity correction factor P, which increases with frequency /H20849see Fig. 7/H20850, the voltages due to the ISHE are expected to increase at higher microwave frequencies. How-ever, the cone angle of magnetization precession for a con-stant power of rf excitation decreases due to higher reso-nance fields. Since spin pumping is proportional to sin 2/H9258an overall decrease in the voltage due to the ISHE is observed.However, the relative strength of the antisymmetric /H20849AMR /H20850 and symmetric /H20849ISHE /H20850parts of the signal change since the AMR decreases faster than the ISHE contribution as a func-tion of the frequency. This effect is illustrated in Figs. 8/H20849a/H20850 and8/H20849b/H20850. Figure 8/H20849c/H20850shows the values of the spin Hall angle /H9253extracted from the fits, which are essentially constant for all frequencies, except for a slight decrease in /H9253at lower frequencies. Equation /H208499/H20850is strictly valid only for small pre- cession cones and constant power of rf excitation. From thefitting of the data we can extract the cone angles of the mag-netization precession. At 4 GHz the fitted value /H9258/H1101510° at the resonance while at 11 GHz /H9258/H110152.5°. For 4 GHz excita- tion nonlinear effects may start to play a role, possiblyslightly changing the estimated value of /H9253. Our model was further tested by varying the angle /H9251of the applied field with respect to the microwave transmissionline. Note that both Eqs. /H2084912/H20850and /H2084918/H20850have besides the ex- plicit dependence on /H9251an additional dependence through the implicit /H9251dependence of /H9258given by Eq. /H2084913/H20850. For small cone angles /H9258this results in both VAMR andVISHbeing propor- tional to sin /H9251cos2/H9251. The dependence on the dc magnetic field direction is shown in Fig. 9. The measured voltage pro- file is consistent with the theoretical model and results in aconsistently constant fitted value of the spin Hall angle in Pt.Due to the specific geometry of the sample we were not ableto measure at angles close to /H9251=90°, at which the magneti- zation dynamics cannot be excited, because the componentofh rfperpendicular to the magnetization vanishes and FMR cannot be excited. But in the range of angles from −5° to45°, excellent agreement between experiment and theory wasachieved.048 1 20.00.30.60.91.2Correction factor P Frequenc y(GHz ) FIG. 7. Elliptical precession trajectory results in a time- dependent cone angle of magnetization precession that modifies thedc component of pumped spin current. The ellipticity correctionfactor Pfor the dc component of spin current is calculated as a function of microwave frequency according to Eq. /H208499/H20850.MOSENDZ et al. PHYSICAL REVIEW B 82, 214403 /H208492010 /H20850 214403-6The other adjustable parameter in our measurements is the power of the microwave excitation. The rf microwave fieldamplitude increases as a square root of the power. Accordingto Eq. /H2084913/H20850the cone angle /H9258of the magnetization precession increases linearly with driving field. The voltage due to theISHE is quadratic in /H9258and, thus, is expected to be propor- tional to the power. After fitting the data, we extracted thesymmetric part due to the ISHE, which is shown in Fig. 10.The maximum values of the measured voltage depend lin- early on power, as expected by theory, except for the highestpower of about 150 mW, at which the system is driven intothe nonlinear regime. We observe a deviation of the FMRpeak position as a well as a deviation of the FMR spectrafrom a Lorentzian shape. It is known that at high rf powerother modes beside the uniform FMR mode are excited. In120 160 200 240 280-50005001000Voltage ( /c109V) Field (Oe)4G H z 1100 1200 1300 1400 1500-1000100200Voltage ( /c109V) Field (Oe)11 GHz 369 1 20.0000.0040.0080.0120.016Spin Hall angle Frequenc y(GHz )Pt/Py(a) (b) (c) FIG. 8. /H20849Color online /H20850/H20849a/H20850and /H20849b/H20850show voltages measured at 4 and 11 GHz. V oltages measured at 11 GHz are smaller due to adecreased precession cone angle. Note that the ratio between theISHE and AMR contributions change due to a faster decrease in theAMR voltage at high frequencies. /H20849c/H20850Spin Hall angle in Pt as a function of frequency. The slightly decreased values at lower fre-quencies may be due to nonlinear effects and a concomitant de-crease in spin pumping at large angles of magnetization precession.(a) (b)100 150 200 250 30 0-1000100200300400500600 45 35 25 15 5 -5Voltage ( /c109V) Field (Oe)Py/PtA n g l eo fHdc(deg) -20 0 20 400.0000.0050.0100.015Spin Hall angle Angle(degree)Py/Pt FIG. 9. /H20849Color online /H20850/H20849a/H20850V oltage measured at 4 GHz as a func- tion of angle /H9251of the external magnetic field with respect to the coplanar waveguide axis. Experimental data and fits are shown withsymbols and solid lines, respectively. /H20849b/H20850Spin Hall angle extracted from the fits. The theoretical model correctly takes into account theangular dependence for the ISHE and AMR contributions. 100 150 200 250 300 35 00200400 0 50 100 1500100200300Voltage ( /c109V) Power (mW)151 100 81 63 49 35 25 16VISHE(/c109V) Field (Oe)Py/Pt ISHEPower (mW) FIG. 10. /H20849Color online /H20850Power dependence of the symmetric ISHE voltage contribution measured at 4 GHz. The inset shows thatthe maximum of the ISHE signal is linear vs rf excitation power.The highest power /H20849150 mW /H20850deviates from the linear behavior due to excitation of nonuniform modes. This deviation is also observedin the corresponding FMR spectra.DETECTION AND QUANTIFICATION OF INVERSE SPIN … PHYSICAL REVIEW B 82, 214403 /H208492010 /H20850 214403-7this case one expects a substantial line broadening and even saturation of the FMR absorption, as observed. D. Spin Hall angle in Pd, Au, and Mo Since the sample structure in our experiments is just a bilayer of Py with the nonmagnetic material of interest, thistechnique can be readily applied to determine /H9253in any con- ducting material. In Fig. 11we show voltages measured for Py/Pd, Py/Au, and Py/Mo measured at 11 GHz excitation.The spin Hall contributions in Au and Mo are smaller than inPt, and note that for Mo the spin Hall contribution has theopposite sign. Fitting of the data enabled us to extract thevalues of /H9253for Pd, Au, and Mo /H20849see Table I/H20850. The effective mixing conductance at the intermetallic Py/N interface isgoverned by N and we adopt the value obtained by the ex-perimentally measured increased damping in Py/N. Note thatthe determination of /H9253furthermore requires /H9268Nand/H9261sdas input parameters. /H9268Nwas obtained using four-probe mea- surements for all samples. Reported values for /H9261sdvary con- siderably. We choose literature values for Pt and Pd fromRef. 32and Au from Ref. 19, and for Mo we assumed that /H9261 sdis comparable to that for Au. Even though this latter assumption may not necessarily hold, the sign change is con-sistent with earlier measurements. 12We furthermore note that the/H9253values in Table Idiffer from the previously reported ones in Ref. 14, where we assumed circular precession and therefore underestimated /H9253by a factor of roughly 2. In ad- dition, Table Iis based on 11 GHz data, which due to the smaller cone angles is less susceptible to deviations stem-ming from nonlinear effects, and therefore should be morereliable. Our values for /H9253are in good agreement with values re- ported by Otani et al.12,40from measurements in lateral spin valves but conflict with values reported by other groups.11,15 We note that in lateral spin valves it is important to also understand the charge current distribution in order to rule outor correct for additional nonlocal voltage contributions. 13A distinct advantage of our approach is that the measured volt-age signal scales with the sample dimension and no addi-tional charge current is directly applied to the sample thatcould result in unwanted spurious voltage signals. We also gain insights into spin-orbit coupling in nonmag- netic metals, which ultimately give rise to spin Hall effects.Even for nonmagnetic materials, that are next to each otherin the periodic table /H20849Pt and Au /H20850the spin Hall angle differs almost by a factor of 4. On the other hand, Mo has a signifi-cantly smaller spin Hall angle with opposite sign. This signchange can be rationalized by a simple s-dhybridization model and is supported by first-principles calculations, 41in- dicating that the spin Hall angle should be negative for lessthan half-filled dbands, and positive for more than half-filled ones, consistent with our experimental results. Pd in spite ofbeing a lighter element than Au has a spin Hall angle whichis almost 2 times larger. First-principles calculations areagain consistent with the experimental observation of /H9253be- ing larger for Pt and Pd compared to Au.42 IV . CONCLUSIONS We presented a spin pumping technique that enables mea- suring spin Hall angles in various materials, which has clearadvantages over standard dc electrical spin injection in Hallbar microstructures. Our results for Pt, Pd, Au, and Mo show1000 1200 1400 160 0-40-2002040Voltage ( /c109V) Field (Oe)Py/Au 1000 1200 1400 1600-60-40-2002040Voltage ( /c109V) Field (Oe)Py/Mo1000 1200 1400 1600-2000200400Voltage ( /c109V) Field (Oe)Py/Pd (a) (b) (c) FIG. 11. /H20849Color online /H20850V oltages measured at 11 GHz for /H20849a/H20850 Py/Pd, /H20849b/H20850Py/Au, and /H20849c/H20850Py/Mo. Shown are data /H20849symbols /H20850, com- bined fits /H20849black lines /H20850, and individual AMR and ISHE contribu- tions, with dotted green and dashed-dotted blue lines, respectively.Note the opposite sign of the ISHE contribution for Mo comparedto those for Au and Pd.TABLE I. Spin Hall angle /H9253determined using /H9261sdand/H9268Nfrom data measured at 11 GHz. Normal metal/H9261sd /H20849nm /H20850/H9268N 1//H20849/H9024m/H20850 /H9253 Pt 10 /H110062 /H208492.4/H110060.2 /H20850/H110031060.013/H110060.002 Pd 15 /H110064 /H208494.0/H110060.2 /H20850/H110031060.0064 /H110060.001 Au 35 /H110063 /H208492.52/H110060.13 /H20850/H110031070.0035 /H110060.0003 Mo 35 /H110063 /H208494.66/H110060.23 /H20850/H11003106−0.0005 /H110060.0001MOSENDZ et al. PHYSICAL REVIEW B 82, 214403 /H208492010 /H20850 214403-8that spin Hall angles are rather small, with the largest value found to be 0.013 for Pt. Our approach provides a uniformspin current across a macroscopic sample. The voltage signalfrom the inverse spin Hall effect can readily be increased viathe use of longer samples since V ISH/H11008L. We verified this relationship by fabricating Py/Pd samples with lengths of0.5, 1, and 3 mm and obtained spin Hall angles that werewithin the error bar identical to each other. Furthermore, by using an integrated coplanar waveguide architecture we can control parameters, such as the rf drivingfield distribution, microwave frequency, and power of rf ex-citation. This enabled a quantitative analysis of the data and a test of the theoretical model under various experimentalconditions. Our model accounts for both the anisotropicmagnetoresistance and the inverse spin Hall effect contribu-tions and agrees with experiments for a wide range of con-trollable parameters. We demonstrated the existence of sym-metric /H20849ISHE /H20850and antisymmetric /H20849AMR /H20850voltages and could model the frequency, magnetic field direction, and excitationpower dependence well. The AMR voltage in our experi-ments originates from capacitive coupling between the wave- guide and the sample and is consistent with the parameterscharacteristic for the ferromagnetic resonance. Our methodwill enable additional studies of spin Hall effects in othermaterials, and, therefore, will be useful to further understandthe spin-orbit coupling mechanism in metals. This is neces-sary in order to develop and optimize the spin Hall effect asa method to generate and detect spin currents in various cir-cumstances, such as in the spin Seebeck effect. 43 ACKNOWLEDGMENTS We thank R. Winkler and G. Mihajlovi ćfor valuable dis- cussions, and C.-M. Hu specifically for pointing out that theangular dependence of the AMR and ISHE contributions areidentical. This work was supported by the U.S. Departmentof Energy—Basic Energy Sciences under Contract No. DE-AC02-06CH11357 and EU-IST through project MACALO/H20849Grant No. 257159 /H20850. *Present address: San Jose Research Center, Hitachi Global Storage Technologies, San Jose, California 95135, USA. †hoffmann@anl.gov 1S. D. Bader and S. S. P. Parkin, Annu. Rev. Cond. Matter Phys. 1,7 1 /H208492010 /H20850. 2C. Chappert and J.-V . 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PhysRevApplied.15.034046.pdf

PHYSICAL REVIEW APPLIED 15,034046 (2021) Symmetry of the Magnetoelastic Interaction of Rayleigh and Shear Horizontal Magnetoacoustic Waves in Nickel Thin Films on LiTaO 3 M. Küß ,1,*M. Heigl ,2L. Flacke ,3,4A. Hefele ,1A. Hörner,1M. Weiler,3,4,5M. Albrecht,2and A. Wixforth1 1Experimental Physics I, Institut of Physics University of Augsburg, Augsburg 86135, Germany 2Experimental Physics IV, Institut of Physics, University of Augsburg, Augsburg 86135, Germany 3Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching 85748, Germany 4Physics-Department, Technical University Munich, Garching 85748, Germany 5Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, Kaiserslautern 67663, Germany (Received 1 December 2020; revised 30 January 2021; accepted 10 February 2021; published 16 March 2021) We study the interaction of Rayleigh and shear horizontal surface acoustic waves (SAWs) with spin waves in thin Ni films on a piezoelectric LiTaO 3substrate, which supports both SAW modes simulta- neously. Because Rayleigh and shear horizontal modes induce different strain components in the Ni thinfilms, the symmetries of the magnetoelastic driving fields, of the magnetoelastic response, and of the transmission nonreciprocity differ for both SAW modes. Our experimental findings are well explained by a theoretical model based on a modified Landau-Lifshitz-Gilbert approach. We show that the symmetriesof the magnetoelastic response driven by Rayleigh and shear horizontal SAWs complement each other, which makes it possible to excite spin waves for any relative orientation of magnetization and SAW prop- agation direction and, moreover, can be utilized to characterize surface strain components of unknownacoustic wave modes. DOI: 10.1103/PhysRevApplied.15.034046 I. INTRODUCTION Owing to the wealth of useful properties of surface acoustic waves (SAW) combined with the ease of launch- ing and detecting SAWs on a piezoelectric crystal and low cost fabrication processes, SAW technology is employed in manifold ways in our daily life as rf filters [ 1], sen- sors [ 2], and lab-on-a-chip applications [ 3]. However, basic research also benefited very profoundly from the use of SAWs, ranging from quantum phenomena in low- dimensional electron systems [ 4] to acoustically operated nanophotonic devices [ 5]. In recent years increasing attention has been paid to the coupling of SAWs with thin magnetic films. On the one hand, it was demonstrated that this coupling makes a type of magnetic field sensor with an excellent signal-to-noise ratio possible [ 6]. On the other hand, SAWs can excite spin waves (SWs) in magnetic films, which turns out to be a fruitful playground for studying the SAW-SW coupling mechanism itself [ 7–12], characterizing the SW disper- sion relations [ 13,14] or even developing alternate kinds of “acoustic isolators” [ 14–19] based on nonreciprocity. *matthias.kuess@physik.uni-augsburg.deAlthough SAW propagation is in general reciprocal, i.e., invariant under inversion of the propagation direction, the coupling mechanism with the SW, and the SW propagation itself can be nonreciprocal. First, a pronounced nonrecipro- cal SW dispersion relation is obtained, among other things, due to the interfacial Dzyaloshinskii-Moriya interaction in a ferromagnetic-heavy metal bilayer [ 14,18]. Secondly, the nonreciprocity of the SAW-SW coupling mechanism arises because of a helicity mismatch between the magnetic driv- ing fields, induced by the SAW and the fixed, right-handed rotational sense of the magnetic moments [ 8,14,15]. Nevertheless, both the observation and possible techno- logical application of these interesting effects are limited to certain experimental geometries, defined by the orientation of the static magnetization Mwith respect to the SW wave vector kSW, which is assumed to be determined by the wave vector of the SAW kSAW=kSW=k[14]. This ori- entation dependence is caused by the SAW mode-specific symmetry of the magnetoelastic driving fields [ 8]. So far, mainly Rayleigh-type ( R) SAWs on piezoelectric LiNbO 3 substrates have been studied [ 7,8,10,13–15,17,20], which show vanishing magnetoelastic SW excitation efficiency for the often discussed backward volume magnetostatic SW mode ( M/bardblk) and magnetostatic surface SW mode (M⊥k). Previous experiments, using shear horizontal 2331-7019/21/15(3)/034046(8) 034046-1 © 2021 American Physical SocietyM. KÜSS et al. PHYS. REV. APPLIED 15,034046 (2021) (SH) SAW modes were not focused on resonant coupling of the SH waves with SWs [ 6,21,22]. In this study, we demonstrate in detail how the SAW mode shape determines the symmetry of the magnetoe- lastic interaction and its nonreciprocal behavior, caused by the SAW-SW helicity mismatch effect. Since we use a well-established 36◦-rotated Y-cut X-propagation LiTaO 3 substrate [ 2,23], which simultaneously supports both R- and SH-wave excitations, we can directly compare the symmetry of the magnetoelastic response of both SAW modes. Because the symmetry of the magnetoelastic driv- ing fields of Rand SH waves complement each other, efficient SW excitation is possible for any in-plane field geometry, which in fact could be a technologically rel- evant aspect. In particular, the SH wave allows efficient magnetoelastic coupling for M/bardblkand M⊥k. II. THEORY AND FEM SIMULATION First, we discuss the nonreciprocal transmission char- acteristics of the magnetoacoustic sample displayed in Fig. 1. A SAW is excited on the piezoelectric substrate, once an alternating voltage with the resonance frequency f=cSAW/λof the interdigital transducer (IDT) is applied. Here, cSAWis the propagation velocity of the SAW and λis kS12 kS21 IDTH y ϕ0ϕH x kS21M ϕ0 z Ni(10 nm)/Al(5 nm) LiTaO3 SAW Rayleigh mode ( R) Shear wave mode (SH) εxxεxy εxxεxy zxy yxz FIG. 1. Schematic illustration of the experimental setup. R- wave and SH-wave modes can both be excited on the LiTaO 3 substrate. The FEM eigenfrequency simulation of the SAW modes shows the magnitude of the strain /epsilon1xx,/epsilon1xyin false colors (green represents low strain, blue and red represent large negative and positive strain) and the exaggerated lattice deformation forthe LiTaO 3/Ni(10 nm)/Al (5n m)layer stack and a wavelength of 1.17μm. The 1000 μm long Ni film is placed centered between the 1600 μm distant IDTs. The nonreciprocal behavior of both SAW modes is characterized by different transmission ampli- tudes/Delta1S21and/Delta1S12for oppositely propagating SAWs kS21and kS12.the wavelength, given by the periodicity of the IDT. In this study, we use a 36◦-rotated Y-cut X-propagation LiTaO 3 substrate, which has been extensively exploited for build- ing high-frequency bandpass filters [ 23] and for biosensing applications [ 2]. This substrate supports predominantly a SH mode ( cSAW=4112 m/s), but additionally a Rmode (cSAW=3232 m/s) [ 24], resulting in different IDT reso- nance frequencies. The lattice displacement of both modes is depicted in Fig. 1for the LiTaO 3/Ni(10 nm)/Al (5n m) layer stack. Depending on the wave mode, either SAW will induce specific strain components in the thin ferro- magnetic Ni film and dynamically modulate the magnetic free energy due to inverse magnetostriction. Because of the high magnetoelastic coupling efficiency of Ni, we neglect nonmagnetoelastic interaction, like magneto-rotation cou- pling [ 9,10,14], spin-rotation coupling [ 11,25,26], or gyro- magnetic coupling [ 12]. The SAW-SW interaction can be described by dynamic magnetoelastic driving fields, which exert a torque on the static magnetization M[7]. The resulting attenuated pre- cession of Mis then given by the Landau-Lifshitz-Gilbert equation. As shown in Fig. 1, an external magnetic field H with the direction φHis applied to align the static magne- tization Mto the angle φ0in the film plane. According to Ref. [ 14], the magnetoelastic driving fields h(x,t)with the normalized out-of-plane component ˜h1and in-plane com- ponent ˜h2, both being perpendicular to M, are a function of the SAW power PSAW: h(x,t)=/parenleftbigg˜h1 ˜h2/parenrightbigg/radicalbigg k2 RωW/radicalbig PSAW(x)ei(kx−ωt).( 1 ) Here, kandωare the wave vector and angular frequency of the SAW, respectively, Wis the aperture of the IDT, andRis a constant factor, depending on the type of the SAW mode. Following Dreher et al. [8], the symmetry of the normalized magnetoelastic driving fields ˜h1and˜h2for vanishing strain /epsilon1yyare /parenleftbigg˜h1 ˜h2/parenrightbigg =/parenleftBigg˜hRe 1+i˜hIm 1 ˜hRe 2+i˜hIm 2/parenrightBigg =2 μ0/parenleftbigg bxzcosφ0+byzsinφ0 bxxsinφ0cosφ0−bxycos(2φ0)/parenrightbigg .( 2 ) The magnetoelastic parameters are bkl=b1˜akl(kl∈ {xx,xy,xz,yz}) with an isotropic magnetoelastic coupling constant b1=b2for polycrystalline films. The complex amplitudes of the normalized strain ˜akl= /epsilon1kl,0/(|k||uz,0|), where uz,0is the amplitude of the lattice displacement in the zdirection, can be determined by a finite element method (FEM) simulation [ 14]. Results and parameters of the FEM simulation are given in Table I.I n Figs. 2(a) and2(b) we show the calculated strain compo- nents of the Rand SH waves in the center plane of the 034046-2SYMMETRY OF THE MAGNETOELASTIC INTERACTION. . . PHYS. REV. APPLIED 15,034046 (2021) Ni film, as simulated for the LiTaO 3/Ni(10 nm)/Al (5n m) structure and for the resonance frequencies of the IDTs. Because the longitudinal strain /epsilon1xxis dominating in the R mode, the main symmetry of the driving field in Eq. (2) is proportional to sin φ0cosφ0[7,8,13]. In contrast, /epsilon1xyis dominating for the SH mode and the expected symme- try of the main driving field component is proportional to cos(2φ0). We thus expect qualitatively different dependen- cies of SAW absorption on the magnetization direction φ0 for magnetoacoustic resonance driven by Rand SH SAWs. Smaller strain components potentially cause nonrecip- rocal SAW transmission due to the SAW-SW helicity mismatch effect [ 8,14,15]. For the Rwave (SH wave), the strain components /epsilon1xy,xz(/epsilon1xx,yz) are phase shifted by +90◦(−90◦)with respect to the main strain component /epsilon1xx(/epsilon1xy). Therefore, the corresponding amplitudes of /epsilon1kl,0, ˜akl,a n d ˜h1,2are complex and we can separate the real and imaginary parts of ˜h1,2with˜hRe 1,2≡Re(˜h1,2)and˜hIm 1,2≡Im(˜h1,2). By reversing the propagation direction of the SAW ( kS21→kS12), the phase difference between the com- plex and the main strain components becomes inverted. Thus, the helicity of the driving fields changes. This is expressed by the inversion of the sign of the complex ˜akl in Table I. In combination with the fixed, right-handed rotational sense of the magnetization precession, the SAW- SW helicity mismatch effect arises, inducing nonreciprocal efficiency of SW excitation and SAW absorption. We now expand the theory, as outlined in Ref. [ 14], in terms of generalized driving field components of Eq. (2). This model is based on the “Landau-Lifshitz-Gilbert approach” of Ref. [ 8] and thus considers implicitly the backaction of the precessing magnetization on the SAW for low excitation amplitudes [ 8]. For zero Dzyaloshinskii- Moriya interaction, the absorbed power of the SAW, which is used to drive magnetization precession, is expressed by Pabs=P0/bracketleftbigg 1−exp/parenleftbigg −CM sαHω [H2ω(1+α2)−H11H22]2+[αHω(H11+H22)]2{[H2 ω(1+α2)+H2 11][(˜hRe 2)2+(˜hIm 2)2] +[Hω(H11+H22)][2(˜hRe 1˜hIm 2−˜hIm 1˜hRe 2)]+[H2 ω(1+α2)+H2 22][(˜hRe 1)2+(˜hIm 1)2]}/parenrightbigg/bracketrightbigg .( 3 ) With respect to the initial power P0, the power of the traveling SAW is exponentially decaying while propagat- ing through the magnetic thin film. The decay rate depends on the effective SW damping constant αand C,Hω,H11, H22, which are defined in Ref. [ 14]. Equation (3)is derived by taking into account (i) the Zeeman energy, (ii) a uniaxial in-plane magnetic anisotropy field Hani, under an angle φani with the xaxis, (iii) an out-of-plane magnetic anisotropy field Hk, counteracting the magnetic shape anisotropy, (iv) the dipolar fields of the SW [ 27], (v) the magnetic exchange interaction, and (vi) the magnetoelastic driving fields of Eq. (1). Finally, to directly fit the exponent of Eq. (3)to the experimentally determined relative change of the SAW transmission /Delta1Sijon the logarithmic scale, the fit equation TABLE I. Results of the FEM simulation. The normalized complex amplitudes of the strain tensor are ˜akl=/epsilon1kl,0/(|uz,0||k|) with kl∈{xx,xy,xz,yz}. The errors are assumed to be of the order of ±10% of ˜axx(˜axy)f o rt h e Rwave (SH wave). f (GHz)cSAW (m/s) ˜axx ˜axy ˜axz ˜ayz R 4.47 3105 0.613 ±i0.024 ±i0.037 0 SH 3.47 4075 ∓i0.53 4.85 −0.18 ∓i0.21is given by /Delta1Sij=10 lg/parenleftbiggP0−Pabs P0/parenrightbigg .( 4 ) The symmetry of /Delta1Sijis determined by the symmetry of the driving fields, as discussed before. We obtain, for the main symmetry of Rwaves (SH waves), /Delta1Sij∝ (sinφ0cosφ0)2(/Delta1Sij∝sin22φ0). Employing the FEM study, the real and imaginary terms in Eq. (2)can be iden- tified. For Rwaves and SH waves, the expected leading term that causes nonreciprocity ( /Delta1S21−/Delta1S12/negationslash=0) is pro- portional to ˜hIm 1˜hRe 2. This nonreciprocity for Rwaves (SH waves) is mediated by the strain component /epsilon1xz(/epsilon1yz) with the symmetry of the nonreciprocity being proportional to sinφ0cos2φ0(sinφ0cos 2φ0). III. EXPERIMENTAL METHODS To prove the theoretical predictions for Rand SH waves, we fabricate a magnetic thin film sample, as depicted in Fig. 1. IDTs with a periodicity of 3.4 μm and a 200 μm aperture are e-beam lithographically defined on a 36◦- rotated Y-cut X-propagation LiTaO 3substrate, evaporating 5 nm Ti and 70 nm of Al. The rectangular-shaped Ni(10 nm)/Al(5 nm) film is deposited by dc magnetron sputter- deposition (base pressure less than 10−8mbar) at room 034046-3M. KÜSS et al. PHYS. REV. APPLIED 15,034046 (2021) (a) (b)Strain Strain FIG. 2. FEM eigenfrequency simulation of the LiTaO 3/Ni(10 nm)/Al (5n m )structure, carried out with COMSOL [28] to determine the phase and normalized magnitude of the relevant strain components for magnetoelastic SAW-SW interaction. The lengths of the simulation geometries that correspond to one wavelength λof the (a) Rwave and (b) SH wave are adjusted to match the resonance frequencies of the experiment. The thin-film parameters that are used for theFEM simulation for Ni (Al) are: density ρ=8900 kg /m 3[29] (2700 kg /m3[30]), Young’s modulus E=218 GPa [ 31]( 7 0 GPa [ 30]), and Poisson’s ratio ν=0.3 [ 31] (0.33 [ 30]). The parameters for the anisotropic single-crystal LiTaO 3are taken from Ref. [ 28]. temperature and positioned in the middle between the two 1600-μm-spaced IDTs. The Ar sputter pressure is kept constant at 3.5 ×10−3mbar and the sample holder is rotated during sputtering. We carried out superconducting quantum interference device-vibrating sample magnetometry (SQUID-VSM) measurements to determine the saturation magnetiza- tion ( Ms=408 kA/m). Additionally, broadband ferro- magnetic resonance (FMR) measurements are performed to obtain values for the g-factor, the out-of-plane mag- netic anisotropy HFMR k, and the effective damping con- stantαFMR eff=μ0/Delta1Hγ/(2ω)+αFMR[14], which includes Gilbert damping αFMRand inhomogeneous line broaden- ing/Delta1H. To characterize the delayline sample, and to measure the magnitude of the complex transmission signal Sijwith ij∈{21, 12}, we employ standard network analyzer mea- surements [ 32,33]. Nonreciprocal effects are studied by comparing the Sijobtained for oppositely propagating SAWs with kS21and kS12. IV . DISCUSSION The acoustic wave transmission magnitude S21in the time domain as a function of frequency is characterized in Fig. 3(a) at a quite high magnetic field of −200 mT, and, therefore, far off the SW resonance. The obtained spectro-gram contains electromagnetic crosstalk at t≈0, acoustic bulk waves, SAWs, and also some higher harmonic reso- nances, as described in more detail in the caption of Fig. 3. By comparing the SAW propagation velocities c SAW=(a) (b) FIG. 3. (a) Various acoustic wave modes are visible in the S21(t,f,μ0H=−200 mT )spectrogram. Signal components that we identified are (i) electromagnetic crosstalk at about 0 ns, (ii) SH* mode at 349 ns (discussed later), (iii) two harmonic res- onances of the SH mode at 390 ns, (iv) bulk waves that are multiple times reflected on the upper and lower sides of the LiTaO 3substrate at 475 ns, and (v) three harmonic resonances of the Rmode and additionally bulk waves at 515 ns. (b) Line cuts of (a) at 3.5 and 4.5 GHz, which correspond to the third and fifth harmonic resonance frequencies of the SH and Rmodes. The adjusted time gates for the SAW modes are depicted in gray. The peak at 515 ns and 3.5 GHz does not show a magnetoelastic response and is thus attributed to bulk waves. 1600μm/twith the results from the FEM simulation given in Table I, we identify the Rmode at 515 ns (3107 m/s) and the SH mode at 390 ns (4103 m/s). Now, we turn to the detailed study of the symmetry of the magnetoacoustic response and its nonreciprocity for both different SAW modes. To do so, we use adjusted time gates for both modes, as depicted in Fig. 3(b).T h e nw e apply inverse Fourier transformation to solely measure the peak transmission of each individual SAW mode in the fre- quency domain at 4.5 GHz for the Rmode and at 3.5 GHz for the SH mode. The relative change of the background- corrected SAW transmission magnitude, which is caused by the SAW-SW interaction, is defined as /Delta1Sij(μ0H)= Sij(μ0H)−Sij(−200 mT ). In Figs. 4(a) and4(b) we show /Delta1Sijfor the Rmode as a function of the external magnetic field magnitude H and direction φH. Since the resonance fields Hresare much higher ( >30 mT) than the uniaxial in-plane anisotropy (1.4 mT, fit results in Table II),Mand Hare approxi- mately parallel ( φ0≈φH)f o r Hresand the symmetry of the main driving field shows up in Figs. 4(a) and4(b).A s expected from theory, we observe the fourfold symmetry /Delta1Sij∝(sinφ0cosφ0)2for the Rwave [ 7,8,13]. Equations (1)–(4)are used to fit the experimental results of Figs. 4(a) and4(b), following the curve fitting proce- dure described in Ref. [ 14]. Because we do not know the parameter Rin Eq. (1), the fitting parameters are α,Hk, bkl/√ Rwith kl=xx,xy,xz,yz, and the uniaxial in-plane magnetic anisotropy. The fits in Figs. 4(d) and4(e) show 034046-4SYMMETRY OF THE MAGNETOELASTIC INTERACTION. . . PHYS. REV. APPLIED 15,034046 (2021) (a) (b) (c) (d) (e) (f) FIG. 4. Transmission nonreciprocity of the Rwave at 4.47 GHz. The experimental data /Delta1S21(a) and /Delta1S12(b) demonstrate the expected fourfold symmetry, caused by dominant longitudi- nal strain /epsilon1xx. Additionally, the nonreciprocal behavior /Delta1S21− /Delta1S12in (c) is fourfold and induced by the strain component /epsilon1xz. Experiment and fit (d)–(f) show excellent agreement. excellent agreement with the experiment. Furthermore, the fit results, as summarized in Table II, are in accordance with the FMR data for HFMR kandαFMR eff. Note that the FMR experiments were performed on reference samples (same sputter deposition run as SAW samples) 20 months after the SAW measurements had been carried out, explaining slight deviations due to possible degeneration of the Ni thin film. The experimentally determined symmetry of the non- reciprocity /Delta1S21−/Delta1S12of the Rmode is depicted in Fig. 4(c). As expected from theory, the nonreciprocity is caused by the vertical shear strain /epsilon1xzand is proportional to cos2(φ0)sin(φ0). Excellent agreement between Figs. 4(c) and4(f), which is obtained by subtracting the fit curves of Figs. 4(d) and4(e), further validates the theoretical model. The results for the SH wave are shown in Fig. 5. Since the main strain component of the SH wave /epsilon1xyinduces driving fields with a symmetry proportional to cos (2φ0), the experimental response /Delta1Sijin Figs. 5(a) and5(b) dif- fers, but complements the symmetry of the Rwave. The(a) (b) (c) (d) (e) (f) FIG. 5. Transmission nonreciprocity of the SH wave at 3.47 GHz. The experimental data /Delta1S21(a) and /Delta1S12(b) show the expected symmetry for SH waves with the dominant strain /epsilon1xy. A complicated nonreciprocal behavior is observed in (c), which is attributed to the nonzero strain component /epsilon1yz. Experiment and fit (d)–(f) show excellent agreement. fit results in Figs. 5(d) and 5(e) reproduce the experi- ment again very well. Moreover, the fit parameters Hkand the effective damping α, which depends on the SAW fre- quency, are in good agreement with the fit parameters of the Rwave and with the FMR measurements given in Table II. The experimentally determined nonreciprocity of the SH wave in Fig. 5(c)has a different symmetry with a lower magnitude than the Rwave. As expected from theory, the strain /epsilon1yzcauses the SAW-SW helicity mismatch effect with the symmetry being proportional to sin φ0cos(2φ0), as observed in the experiment. Again, the difference of the fits/Delta1S21−/Delta1S12in Fig. 5(f)agrees well with the nonre- ciprocity of the experiment, also confirming the theoretical model for SH waves. An approximate estimation of the magnitude of the dominant strain component gives /epsilon1xx,0≈12×10−6for the Rmode and /epsilon1xy,0≈23×10−6for the SH mode [ 35,36]. With these values we estimate the in-plane magnetization precession amplitude to be of the order of φM,IP≈0.6◦for theRmode and φM,IP≈3◦for the SH mode [ 37], which is TABLE II. Summary of the results obtained by fitting the SAW transmission /Delta1S21of Figs. 4(a) and5(a), and of the FMR mea- surements. Additional parameters that are used for the fit are Ms=408 kA/m (obtained by SQUID-VSM), γ=193.5×109rad/(s T) (obtained by FMR), and the exchange constant A=7.7 pJ/m [ 34]. Further fit results are the direction φani=(83.6±3.6)◦and magnitude μ0Hani=(1.4±1)mT of the in-plane uniaxial anisotropy easy axis. SAW FMR f(GHz) Hk/parenleftbigkA m/parenrightbig α(10−3)bxx√ R/parenleftbigg μT√ J/m3/parenrightbigg bxy√ R/parenleftbigg μT√ J/m3/parenrightbigg bxz√ R/parenleftbigg μT√ J/m3/parenrightbigg byz√ R/parenleftbigg μT√ J/m3/parenrightbigg HFMR k/parenleftbigkA m/parenrightbig αFMR eff/parenleftbig 10−3/parenrightbig R 4.47 158.2 ±0.1 69 ±1 20.80 ±0.02 +i(1.68±0.04)+i(1.03±0.02)0.03±0.11 127.4 ±0.2 75 ±4 SH 3.47 161.7 ±0.1 76 ±2−i(5.64±0.13)15.23±0.01 −(0.06±0.23)−i(0.55±0.03)127.4±0.2 87 ±4 034046-5M. KÜSS et al. PHYS. REV. APPLIED 15,034046 (2021) (a) (b) (c) FIG. 6. Experimental results for the transmission nonreciproc- ity of the SH* wave at 3.52 GHz, revealing a similar symmetryas the SH-wave response shown in Fig. 5. one order of magnitude lower compared to Ref. [ 38]. For both SAW modes, the transmission /Delta1S21does not change with the output power of the vector network analyzer PVNA in the studied range of −15 to+15 dBm. Thus, SAW prop- agation and SW excitation of the presented experiments (PVNA=15 dBm) are in the linear regime. So far, we have presented the results for the fifth and third harmonic resonance frequencies of the Rand SH modes. We perform similar measurements for all transmis- sion peaks visible in Fig. 3(a). Since only surface modes are expected to induce considerable magnetoelastic driv- ing fields, we make use of this assumption and identify the surface modes by looking at the magnitude of the absorbed SAW power, caused by the SAW-SW interac- tion/Delta1Sij(μ0H,φH). None of the other transmission peaks in Fig. 3(a) shows a magnetoelastic response, except the weak but still detectable SAW mode at t=349 ns. This mode cannot be identified from either a literature search [24] or our FEM eigenfrequency simulation based on its propagation velocity. We still name this mode the SH* mode. Given the example of this SH*-wave mode, we demon- strate that an unknown SAW mode can also be char- acterized in terms of its strain components by employ- ing SAW-driven SW spectroscopy. The magnetoelastic response /Delta1Sijof this SH* mode is depicted in Fig. 6. Because the symmetry of /Delta1Sijis an unambiguous indi- cation of the strain component /epsilon1xyin Eq. (2), we con- clude and experimentally confirm that this mode must be a shear-horizontal-type wave. The smaller strain compo- nents, which are phase shifted with respect to the main strain component, show up in the nonreciprocal response in Fig. 6(c). Despite a low signal-to-noise ratio, the nonre- ciprocity of the SH* wave agrees with the nonreciprocity of the SH wave, caused by the phase-shifted strain /epsilon1yz.W e infer that the SH* mode is also a surface acoustic wavewith low transmission and strain tensor elements, which indicate it to be similar to the SH mode. To discover further details about the SH* mode, we carry out additional time-dependent FEM simulations,(a) (b) Expt. Expt. FIG. 7. Comparison of the experiment results and FEM sim- ulation of the normalized strain components for the (a) Rwave and (b) SH wave. using the exact geometry of the LiTaO 3sample. These simulations include, in contrast to the eigenfrequency FEM simulation, acoustic wave reflections, and secondary induced acoustic waves due to electromagnetic crosstalk. The time-dependent simulation shows that electromagnetic crosstalk causes low-amplitude secondary acoustic wave excitation at the edge of the magnetic film, close to the exciting IDT. Because of the reduced propagation path, the propagation time of this secondary mode is lowered by about 300 μm/(4075 m /s)=74 ns in comparison to the SH mode. This, however, does not agree with the experi- mental findings for the SH* mode (time delay of 42 ns in Fig.3). Since the propagation time of none of the acoustic wave modes of the simulation matches with the propaga- tion time of the SH* mode of 349 ns, we have to conclude that we cannot reproduce the SH* mode in the FEM simu- lations. Furthermore, because we observe the SH* mode in the off-resonant condition in Fig. 3, this mode cannot be a secondary elastic wave generated by the backaction of the magnetization precession, as described in Ref. [ 39]. Finally, we discuss the magnitude of the magnetoe- lastic driving fields of the fits bkl/√ R(Table II)b y comparing these values with ˜akl,0of the FEM simula- tion (Table I). Thus, the normalized strain components of the fit (/epsilon1kl,0//epsilon1xx,0)=(bkl,0/bxx,0)and of the simulation (/epsilon1kl,0//epsilon1xx,0)=(˜akl,0/˜axx,0)are shown for the Rwave and SH wave (normalized to /epsilon1xy,0) in Figs. 7(a) and 7(b), respectively. Note that the sign of ˜akl,0of the simulation is in accordance with the experimental result for bkl,0and we plot absolute values. The excellent agreement, except for /epsilon1xx,0(SH), between the FEM simulation and experimental results again con- firms the theory. Because of additional, nonmagnetoelas- tic coupling mechanisms like magneto-rotation coupling [9,10,14] or spin-rotation coupling [ 11,25,26], corrections for the driving fields are potentially the reason for the devi-ation of /epsilon1 xx,0of the SH wave. Furthermore, we assume in the theory section that the SAW mode shape is fixed [Eq. (1)], but only the amplitudes decay exponentially with increasing SAW propagation in the magnetic film 034046-6SYMMETRY OF THE MAGNETOELASTIC INTERACTION. . . PHYS. REV. APPLIED 15,034046 (2021) [Eq. (3)]. Because the shape of the acoustic wave trans- forms from a SH wave with an extremely large penetration depth in the zdirection at the bare LiTaO 3surface to a SH wave with a much shorter penetration depth at the shorted LiTaO 3/Ni(10 nm)/Al (5n m)surface [ 24], this assumption holds only for long propagation distances through the mag- netic film and also explains deviations between the FEM eigenfrequency study and the experiment. V . CONCLUSION In conclusion, we have extended the theoretical model of SAW-driven SW spectroscopy [ 14] in terms of arbitrary magnetoelastic driving fields. This model now describes the experimental results for both R- and SH-wave trans- missions on a LiTaO 3/Ni(10 nm)/Al (5n m)sample in an excellent way. Additionally, the fit values were cross checked with FMR measurements and FEM simulations, reasonably confirming the theoretical model. The SAW- SW helicity mismatch effect causes nonreciprocal SAWtransmission for the SH and Rmodes, with symmetries proportional to sin φ 0cos(2φ0)and sin φ0cos2(φ0), respec- tively. Since the symmetry of the induced magnetoelastic driving fields of all relevant strain components is ambigu- ous, the SAW type can in general be identified by studying the symmetry behavior of the magnetoelastic response. Furthermore, the symmetry of the magnetoelastic driv- ing fields of Rand SH waves complement each other. Thus, a large SAW-SW interaction is observed for any in-plane field geometry on the LiTaO 3sample. This is especially interesting because processing and transport of data with SWs in magnonics are usually carried out in either the M⊥kor the M/bardblkgeometry [ 40]. 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[35] To estimate the approximate amplitude of the induced strain /epsilon1 xx,0for the Rwave ( /epsilon1xy,0for the SH wave), we assume that both IDTs have the same efficiency in SAW excitation and detection. Since the delayline has a total insertion loss of about 82 dB for the Rmode (58 dB for the SH mode), the loss to excite the SAW in the middle of the delayline is approximately −41 dB ( −29 dB). Because the output power of the vector network analyzer is 15 dBm, the power of the Rmode (SH mode) in the middle of the delayline isPSAW≈−26 dBm ( PSAW≈−14 dBm). Together with Eq. (1) of Ref. [ 36] and the normalized displacement pro- file u(x,z)from the FEM simulation we estimate the strain /epsilon1xx,0(/epsilon1xy,0) to be of the order of 12 ×10−6(23×10−6). [36] B. Croset, I. S. Camara, J.-Y. Duquesne, L. Largeau, L. Thevenard, and P. Rovillain, Vector network analyzer mea-surement of the amplitude of an electrically excited surface acoustic wave and validation by x-ray diffraction, J. Appl. Phys. 121, 044503 (2017). [37] To estimate the magnetization precession amplitude φ M,IP, we expand the theory of Ref. [ 8] by the dipolar fields of the SW [ 14,27]. The parameters used for the estima- tion are (i) the strain components discussed before, (ii) isotropic magnetoelastic coupling constant b1=b2=23 T[8], and (iii) the magnetic film properties given in Table II. [38] B. Casals, N. Statuto, M. Foerster, A. Hernández-Mínguez, R. Cichelero, P. Manshausen, A. Mandziak, L. Aballe, J. M. Hernàndez, and F. Macià, Generation and Imaging of Magnetoacoustic Waves Over Millimeter Distances, Phys. Rev. Lett. 124, 137202 (2020). [39] A. V. Azovtsev and N. A. Pertsev, Dynamical spin phe- nomena generated by longitudinal elastic waves traversingCoFe 2O4films and heterostructures, P h y s .R e v .B 100, 224405 (2019). [40] A. A. Serga, A. V. Chumak, and B. Hillebrands, YIG magnonics, J. Phys. D: Appl. Phys. 43, 264002 (2010). 034046-8

PhysRevApplied.14.034063.pdf

PHYSICAL REVIEW APPLIED 14,034063 (2020) Spin-Wave Diode and Circulator Based on Unidirectional Coupling Krzysztof Szulc ,1,*Piotr Graczyk ,2Michał Mruczkiewicz ,3,4Gianluca Gubbiotti ,5and Maciej Krawczyk1,† 1Faculty of Physics, Adam Mickiewicz University, Pozna´ n, Uniwersytetu Pozna´ nskiego 2, Pozna´ n 61-614, Poland 2Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, Pozna´ n 60-179, Poland 3Institute of Electrical Engineering, SAS, Bratislava 841 04, Slovakia 4Centre for Advanced Materials Application CEMEA, Slovak Academy of Sciences, Dúbravská cesta 5807/9, Bratislava 845 11, Slovakia 5Istituto Officina dei Materiali del CNR (CNR-IOM), Sede Secondaria di Perugia, c/o Dipartimento di Fisica e Geologia, Università di Perugia, Perugia I-06123, Italy (Received 20 November 2019; revised 20 June 2020; accepted 18 August 2020; published 25 September 2020) In magnonics, a fast-growing branch of wave physics characterized by low energy consumption, it is highly desirable to create circuit elements useful for wave computing. However, it is crucial to reach the nanoscale so as to be competitive with the electronics, which vastly dominates in computing devices.Here, based on numerical simulations, we demonstrate the functionality of the spin-wave diode and the circulator to steer and manipulate spin waves over a wide range of frequency in the GHz regime. They take advantage of the unidirectional magnetostatic coupling induced by the interfacial Dzyaloshinskii-Moriya interaction, allowing the transfer of the spin wave between thin ferromagnetic layers in only one direction of propagation. Using the multilayered structure consisting of Py and Co in direct contact with heavy metal, we obtain submicrometer-size nonreciprocal devices of high efficiency. Thus, our work con-tributes to the emerging branch of energy-efficient magnonic logic devices, giving rise to the possibility of application as a signal-processing unit in the digital and analog nanoscaled spin-wave circuits. DOI: 10.1103/PhysRevApplied.14.034063 I. INTRODUCTION A diode and a circulator are electronic and microwave components, which have found wide applications in many devices for signal processing. A diode allows the flow of signal in only one direction, and for microwaves, it is also known as an isolator. It already has equivalents in optics [1], heat transfer [ 2,3], acoustics [ 4,5], and spin Seebeck effect [ 6]. Diodes for spin waves (SWs) relying on the dipolar [ 7–9] or interfacial Dzyaloshinskii-Moriya interac- tion (IDMI) [ 10] were recently proposed. In circulators, the signal going from one port is always directed only to the nearest port, according to the same sense of rotation. It usu- ally consists of three or four ports. Apart from microwaves and photonics, where the circulators have found applica- tions [ 11–14], they have been recently demonstrated also for acoustic waves [ 15], while a demonstration for SWs is still missing. Circulators used in industry are mostly macroscopic devices. Their miniaturization with the pos- sibility of implementation to real-life systems is a crucial point of the present studies. *krzysztof.szulc@amu.edu.pl †krawczyk@amu.edu.plAntisymmetric exchange interaction was described by Dzyaloshinsky [ 16] and Moriya [ 17] about 60 years ago. Recently, it has found interest due to induced chirality of the magnetization configuration [ 18,19] and nonreciproc- ity in the SW propagation [ 20–25]. The DMI can exist in bulk noncentrosymmetric crystals [ 26] or at the interface between ferromagnetic and heavy-metal layers (IDMI). The IDMI is of high interest due to a larger DMI constant value [ 27,28], flexibility in shaping its strength, and the possibility of working at the nanoscale. In this paper, we propose a layered sequence of ultrathin ferromagnetic films where the presence of IDMI over one layer leads to asymmetric or even unidirectional coupling of SWs between the layers. Interestingly, the multilayer composition can work as a SW diode or a three- or four- port SW circulator, in dependence on the particular struc- turization. The proposed SW diode, based on Py (Ni 80Fe20) and Co ultrathin films, offers isolation of SW signal in the reverse direction reaching 22 dB with respect to the transmission in the forward direction. From the application point of view, the functionality of the device is preserved for a broad GHz-frequency range. We investigate the cou- pling between SWs in a heterogeneous ultrathin bilayer by numerical frequency-domain and time-dependent sim- ulations. Then we discuss the coupling strength and the 2331-7019/20/14(3)/034063(12) 034063-1 © 2020 American Physical SocietyKRZYSZTOF SZULC et al. PHYS. REV. APPLIED 14,034063 (2020) spin-wave propagation(a) (b) HM FIG. 1. (a) Schematic representation of the multilayer stack and the geometry considered. The layer sequence consists of twoferromagnetic films, FM1 and FM2, separated by the nonmag- netic layer (NM). In FM2, the IDMI is induced by the proximity with the heavy metal (HM). Generally, this structure under-lies the unidirectional coupling in a wide range of frequency. (b) Definition of the transmission lengths x trandxtr2on the basis of the SW in the coupled FM bilayer system. SW transmission between the layers in the framework of the coupled-mode theory. Finally, we present possi- ble realizations of the SW devices—the SW diode and the four-port circulator, with in-depth analysis of their efficiency. The considered multilayer stack consists of two ferro- magnetic (FM) layers separated by a nonmagnetic spacer, and heavy-metal layer in contact with one of the FM layers [Fig. 1(a)]. We consider SW propagation in the Damon-Eshbach geometry, where the magnetization M and the external magnetic field H0are aligned in plane of the films and perpendicular to SW propagation defined by the wavevector k. II. THEORETICAL MODEL Magnetization dynamics in the systems under investiga- tion are described by the Landau-Lifshitz-Gilbert equation: ∂M ∂t=−γμ 0M×Heff+α MSM×∂M ∂t,( 1 ) where M=(mx,my,mz)is the magnetization vector, γis the gyromagnetic ratio, μ0is the magnetic permeability of vacuum, and Heffis the effective magnetic field, which is given as follows: Heff=H0ˆz+2Aex μ0M2 S∇2M+2D μ0M2 S/parenleftbigg ˆz×∂M ∂x/parenrightbigg −∇ϕ, (2) where Aexis the exchange stiffness constant, Dis the IDMI constant, and ϕis the magnetic scalar potential fulfillingMaxwell equations in a magnetostatic approximation: ∇2ϕ=∇· M.( 3 ) Equations (1)and (3)are solved numerically in the linear approximation, i.e., assuming mx,my/lessmuchmz≈MS, where MSis saturation magnetization, using the finite- element method in COMSOL Multiphysics environment [29]. Frequency-domain simulations are carried out to cal- culate the SW dispersion relation in the system of coupled FM layers. Time-domain simulations are performed to demonstrate the functionality of the designed devices. A dynamic magnetic field is used to excite the system sinu- soidally at the desired frequency. We use triangular mesh with a maximum element size of 1 nm inside the FM lay- ers and a growth rate of 1.15 outside of the FM layers. We assume that the NM spacer is made from a dielectric material. The metallic layer, such as Cu or Au, can screen the dipolar microwave field [ 30], causing the reduction of the dipolar interaction between the layers and changing the dispersion relation. However, the effect is negligible for the thin spacer. In the first step of calculations, we consider a multilayer of the Py (3n m)/NM(5n m)/Co(2n m)/Pt composition. For the Co layer we assume MS=956 kA /m, exchange stiffness constant Aex=21 pJ/m[31], Gilbert damping constant α=0.05, IDMI constant D=−0.7 mJ/m2[32], and for Py layer MS=800 kA/m, Aex=13 pJ/m, α= 0.005, D=0. External static magnetic field H0is fixed to 50 mT. A. Coupled-mode theory with damping The SWs propagating in the system composed of two FM layers separated by a NM layer are magnetostatically coupled. We can describe this phenomenon using general coupled-mode theory [ 33,34] based only on the wave prop- erties. To describe the interaction between propagating modes, we use coupling-in-space formalism. The differen- tial equation describing the scalar wave ψlpropagating in a single layer lis dψl dx=−iβlψl,( 4 ) with βl=k/prime l−iαlk/prime/prime l (5) denoting the complex wavevector, where the real part cor- responds to the propagation, and the imaginary part to the attenuation of the wave. For the waves propagating in two coupled layers, we get the mutually dependent differential equations: dψ1 dx=−iβ1ψ1+κ12ψ2,( 6 ) 034063-2SPIN-WAVE DIODE AND CIRCULATOR... PHYS. REV. APPLIED 14,034063 (2020) dψ2 dx=−iβ2ψ2+κ21ψ1,( 7 ) where for the codirectional coupling, i.e., coupling of the waves propagating in the same direction κ12=−κ21=1 2(|kP−kCP|−|k1−k2|) (8) are the coupling coefficients. kPandkCPare wavevectors of the in-phase and in-counterphase modes of the coupled bilayered system, respectively. Generally, the waves can be described by the complex numbers with the coupling magnitude described with the right side of Eq. (8). In our case, we are only interested in the magnitude of coupling, and not the phase of the wave, which derives from the argument of κ12. The system of differential equations [Eqs. (6)and(7)] can be reduced to the homogeneous linear equations. Assuming that the solutions are in the form of e−iβx, the solvability condition requires that β2−(β1−β2)β+(β1β2+κ12κ21)=0. (9) The solutions of this equation are β±=¯β±B, (10) where ¯β=β1+β2 2,B=/radicalBig /Delta1β2+|κ12|2,a n d /Delta1β=β1−β2 2. Substituting the solutions of Eq. (10) to Eqs. (6)and(7) and assuming the initial conditions as ψ1(0)=Aand ψ2(0)=0, we end with the general solutions for the coupled wave functions ψ1(x)=A/parenleftbigg cosBx−i/Delta1β BsinBx/parenrightbigg e−i¯βx, (11)ψ2(x)=Aκ21 BsinBx e−i¯βx. (12) In the synchronous state k/prime 1=k/prime 2=k/prime, so we can determine transmission length xtrof the wave from layer 1 to layer 2 [see Fig. 1(b)] from zeroing of the term in the brackets in Eq.(11): xtr=1 B/parenleftbiggπ 2−arctani/Delta1β B/parenrightbigg . (13) In the synchronous state, /Delta1β=−i(α1k/prime/prime 1−α2k/prime/prime 2), so the term in the arctangent is real. In the case when the wave is transferred from the layer with lower damping to the layerwith higher damping, the transmission length becomes larger, while in the opposite case, it becomes smaller. If −/Delta1β 2>|κ12|2, then the parameter Bbecomes imaginary, and if α1k/prime/prime 1<α 2k/prime/prime 2then xtr<0 and complete transmission cannot be achieved (the structure behaves like an over- damped harmonic oscillator), while if α1k/prime/prime 1>α 2k/prime/prime 2then xtr>0 and complete transmission can be achieved but only once. We can also extract “there and back transmission” length xtr2considering the length at which the wave transfers from layer 1 to layer 2 and then transfers back from layer 2 to layer 1 [see Fig. 1(b)]. The solution comes from zeroing of the sine term in Eq. (12). The lowest positive solution is xtr2=π B. (14) At this point, we have to introduce the SW parameters to the coupled-mode theory. Knowing that ω/prime=vphk/primeand ω/prime/prime=vgrk/prime/prime[35], where vphis the phase velocity and vgr– the group velocity of the SW, Eq. (5)is transformed into βl=1 vph,lω/prime l−iαl vgr,lω/prime/prime l, (15) where the real ω/primeand imaginary ω/prime/primeparts of the frequency of a single layer in the Damon-Eshbach geometry are defined as [ 21,35] ω/prime=γμ 0⎛ ⎝/radicalBigg/parenleftbigg H0+MS 4+2Aex μ0MSk2/parenrightbigg/parenleftbigg H0+3MS 4+2Aex μ0MSk2/parenrightbigg −e−4|k|dM2 S 16/parenleftbig 1+2e2|k|d/parenrightbig +2D μ0MSk⎞ ⎠, (16) ω/prime/prime=γμ 0/parenleftbigg H0+MS 2+2Aex μ0MSk2+2D μ0MSk/parenrightbigg . (17) 034063-3KRZYSZTOF SZULC et al. PHYS. REV. APPLIED 14,034063 (2020) The value of the κ12is determined from the dispersion rela- tion of the coupled bilayer system obtained in the numer- ical simulations. The parameters in Eq. (8)are calculated for the given frequency ω/prime. B. Coupling parameters To describe the coupling between the SWs propagating in a bilayered structure, we define the two coupling param- eters between the FM layers. The first is the power-transfer factor FP, which relies upon the dispersion relation. From the coupled-mode theory, we get that the power-transfer factor is [ 34] FP=f2 coup f2coup+/Delta1f2, (18) where /Delta1f=|f1−f2|, and (19) fcoup=fP−fCP−/Delta1f, (20) f1=ω1/2π,f2=ω2/2πare frequencies of the SWs in the single layers, fPandfCPare frequencies of SWs, related to the in-phase and in-counterphase oscillations of the ampli- tude in the coupled layers, respectively, discussed later in more details. The second parameter is the energy-distribution factor FE. We assume that the mode energy of fully coupled SWs is shared equally between both FM layers. The mode energy for uncoupled SWs is accumulated only in one of the layers. The total energy density in the ith layer is Ei=Ei,dip+Ei,ex, (21) where the dipolar-energy density Edipis defined as Ei,dip=1 Li1 2μ0/integraldisplay/integraldisplay Sim·∇ϕdy dx , (22) and the exchange-energy density Eex Ei,ex=1 LiAex M2 S/integraldisplay/integraldisplay Si(∇m)2dy dx , (23) where Liis the length of the FM layer in the simulations, Si=diLi, where diis the thickness of FM layer, and m= (mx,my)is a dynamical component of the magnetization. The energy-distribution factor is defined as follows: FE=1−1 2/vextendsingle/vextendsingle/vextendsingle/vextendsingleE P 1−EP 2 EP 1+EP 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1 2/vextendsingle/vextendsingle/vextendsingle/vextendsingleE CP 1−ECP 2 ECP 1+ECP 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (24) Values of F PandFEare in the range [0,1], where we inter- pret 0 as no coupling and 1 as a full coupling between SWs propagating in the FM layers.III. RESULTS A. Unidirectional coupling in the wide range of frequency The first step of the investigation of the SW dynamics is the calculation of a dispersion relation. In Fig. 2,w e plot the dispersion relations of the Py(3)/NM(5)/Co(2)/Ptmultilayer (solid lines) and uncoupled Co(2)/Pt (dashed lines) and Py(3) (dotted lines) layers for two different values of IDMI constant. For D=0 [Fig. 2(a)] all disper- sion relations are almost symmetric with respect to k=0 with only small asymmetry related to dipolar interaction. A small change of the dispersion relation for the multi- layer, in comparison to the uncoupled layers, is the effect of weak coupling between the FM layers. Taking the nonzero IDMI constant, we introduce strong nonreciproc- ity to the SW dispersion of the mode related to the Co layer. Interestingly, for D=−0.7 mJ/m 2[see Fig. 2(b)] the dispersion relation for the Co layer almost overlaps with the dispersion relation for Py in the broad range of positive wavevector. Since both modes have almost the same frequency (resonance) and wavevector (phase match- ing), one can expect strong interaction between them in the multilayer system [ 34]. Two interacting modes are hybridized forming collective excitations, with in-phase (at frequency fP) [see inset 3 in Fig. 2(b)] and in-counterphase (at frequency fCP) [see inset 4 in Fig. 2(b)] SW modes at a higher and lower frequency, respectively [ 29,36]. Indeed, we can see the repulsion of the dispersion branches related to the in-phase and in-counterphase modes for pos- itive kin the multilayer [see the red and blue curves in Fig. 2(b)], being the effect of strong dipolar coupling between modes in Py and Co. For the negative wavevec- tors, the dispersions for the uncoupled FM layers are well separated, and in the multilayer, they follow the same lines pointing at the weak coupling between FM layers [see insets 1 and 2 in Fig. 2(b)]. Comparing both dis- persions in Figs. 2(a) and2(b), we conclude that adding IDMI to the Co layer can lead to strong SW coupling between FM layers for the waves propagating in one ( +k) direction, while in the structure without IDMI, the cou- pling is weak and symmetrical. The general procedure for achieving unidirectional coupling is described in the Appendix. At this point, we can look at the SW propagation in the Py(3)/NM(5)/Co(2)/Pt multilayer. The model of the inves- tigated structure is shown in Fig. 2(c). The antenna located in the Py layer excites the SW at 15.2-GHz frequency. Over the antenna, we made the indent in the Co/Pt layer to avoid the excitation coming from the dipolar field. It comes from the dispersion relation in Fig. 2(b) that a SW propa- gating in +xdirection should be influenced by the strong coupling between Co and Py layer while propagating in −xdirection should go through the Py layer only weakly interacting with the Co layer. Indeed, on the right side 034063-4SPIN-WAVE DIODE AND CIRCULATOR... PHYS. REV. APPLIED 14,034063 (2020) (a) (c)(b) FIG. 2. (a),(b) Dispersion relation of SWs as a function of wavevector kin the Py(3)/NM(5)/Co(2)/Pt multilayer for the IDMI constant in the Co layer (a) D=0a n d( b ) D=−0.7 mJ /m2. For reference, we show the dispersion relation of SWs in the uncoupled Pt/Co and Py layers with dashed and dotted lines, respectively. In the insets in (b), we show the mxamplitude of the SWs propagating in both directions at 15.2 GHz. For D=0, the dispersion relation is almost symmetrical with respect to k=0. For D=−0.7 mJ /m2, the IDMI breaks the symmetry leading to strongly coupled modes in the +krange (insets 3 and 4) and single-layer excitation in the−krange (insets 1 and 2). The highest coupling occurs in the region of overlapping of the dispersion relation of the uncoupled layers. (c) Propagation of the SW at 15.2-GHz frequency in the Py(3)/NM(5)/Co(2)/Pt multilayer. The antenna is located in the Pylayer, below the indent in the Co/Pt layer. SW propagating in the +xdirection transfers back and forth between Py and Co layer. The transmission to the Co layer in the −xdirection is weak, and most of the SW intensity remains in the Py layer. of Fig. 2(c) a SW appears alternately in Co and Py layer being the effect of the interference between in-phase and in-counterphase modes. On the left side, most of the SW intensity remains in the Py layer, with only weak transfer to the Co layer being the effect of weak dynamic coupling. We term this effect as a unidirectional coupling. For further investigations, the determination of the SW coupling in a broad spectrum is the crucial point. For this purpose, we use the coupling parameters defined in Eqs. (18) and(24). In Fig. 3, we plot FP(vertical axis) and FE(color of the points) in the Pt/Co(2)/NM/Py(3) multi- layer with D=−0.7 mJ/m2for different thicknesses of NM layer in dependence on the wavevector of the SW. On the positive kside, both coupling parameters are very close to the maximum value in the range between two dispersion crossing points (1.6 ×107and 6.3 ×107m−1). That means the SWs are nearly fully coupled in a wide range of wavevector and frequency. On the negative kside, the coupling is significant only in the long-wavelength range, reaching its maximum for k≈−2×107m−1.T h e increase of the thickness of the NM spacer leads to a decrease of coupling parameters, except the range of strong coupling between the dispersion crossing points. It isascribed to the weaker dipolar interaction between the layers. Another important parameters associated with the cou- pling between the two layers are the transmission lengths defined in Eqs. (13) and (14). Many parameters affect these physical quantities. We focus on two of them, which are important in our study—the damping constant and the NM-layer thickness. In Fig. 4(a), we show the trans- mission length in the Py(3)/NM(5)/Co(2)/Pt multilayer depending on the damping constant in the Co layer. In the simulations, the SW source emitting the SW at fre- quency ω/prime/2π=15.2 GHz is located in the Py layer. From Eq.(17),w eg e t ω/prime/prime/2π=20.6 GHz for the Co layer and 19.2 GHz for the Py layer. Results from the numer- ical simulations are compared with Eqs. (13) and (14) derived from the coupled-mode theory. We get a satis- fying agreement between these approaches. Both xtrand xtr2are increasing with the increase of the damping con- stant. However, xtris growing faster than xtr2leading to the conclusion that the transmission length from the layer with higher damping (Co layer) to the layer with lower damping (Py layer) is decreasing with the increase of the damping constant. In Fig. 4(b), we show the transmission 034063-5KRZYSZTOF SZULC et al. PHYS. REV. APPLIED 14,034063 (2020) NM-layer thickness Energy-distribution factorPower-transfer factor FIG. 3. Power-transfer factor FP(the vertical axis) and energy-distribution factor FE(color scale) as a function of the wavevector in the Py(3)/NM( t)/Co(2)/Pt multilayer with the IDMI constant in the Co layer D=−0.7 mJ /m2for three differ- ent thicknesses of the NM layer. We reach coupling parameters close to a maximum value of 1 in the range between 1.6 ×107 and 6.3 ×107m−1(marked with dashed lines). In the negative k range, the coupling is weak and reduces with the increase of the NM-layer thickness. length in the Py(3)/NM( t)/Co(2)/Pt multilayer depending on the NM-layer thickness t. We assume αCo=αPy=0. The transmission length is increasing with the exponen- tial character of growth. When /Delta1β=0, Eq. (13) reduces to the form xtr=π/(2|κ12|). Thus, the coupling coeffi- cient is decreasing exponentially with the increase of the separation between the layers [ 37]. B. Spin-wave diode Taking into account the unidirectional coupling dis- cussed above, we can design the SW diode. The proposed structure is shown in Fig. 5. It consists of continuous Py film, which is the medium where the SWs propagate from the input to the output and Co/Pt stripe, which is a functional element of a diode where IDMI introduces non- reciprocal interaction. They are separated by a 5-nm-thick NM spacer, which is sufficient to neglect Ruderman-Kittel- Kasuya-Yosida interaction. We chose the frequency of the SW from the crossing point of the dispersion relation of uncoupled layers shown in Fig. 2(b) to get the full cou- pling between FM layers. The width of the Co/Pt stripe is matched to the transmission length xtr, which is related to the coupling strength and the damping in the layers. To determine the efficiency of the device, we calculate the power loss dP=10 log (Ein/Eout), where Einis the energy measured in the steady state in front of the device and Eout behind the device, calculated according to Eq. (21). The operation of the diode is depicted in Figs. 5(a) and5(b), which shows results from the time-domain sim- ulations of SW continuously excited at the 15.2-GHz fre- quency in Py at the antenna (A). We fix the width of the(a) (b) NM-layer thickness FIG. 4. Transmission-length dependence on (a) the damping constant in the Co layer in the Py(3)/NM(5)/Co(2)/Pt multilayer with the damping in the Py αPy=0.005 and (b) the thickness of NM layer in the Py(3)/NM( t)/Co(2)/Pt multilayer. The source of the SW of 15.2 GHz is located in the Py layer. In (a), the simulations (Sim) results are compared with Eqs. (13) and(14) from the coupled-mode theory (CMT). In (b), we present only the simulation results for xtr. Co/Pt stripe to 320 nm. The signal for efficiency analy- sis is collected from the areas marked as input and output ports, which are located at a distance of 20 nm from the Co/Pt stripe edges. Due to weak coupling between the SWs propagating in the −xdirection [Fig. 5(a), see also the animation, Movie S1, within the Supplemental Material [38] ], the transmission to the Co stripe is small, and the SW passes the diode retaining its intensity. The total power loss in this direction reaches 3.3 dB, and it is mainly due to the Gilbert damping in Py (2.2 dB). On the other hand, the SW propagating in the +xdirection [Fig. 5(b), see also the animation, Movie S2, within the Supplemental Material [38] ] transfers almost entirely to the Co stripe where it is strongly attenuated due to the high damping. Some residual intensity at the output is the effect of incomplete transfer toCo and return transfer from Co after reflections from the boundaries of the stripe. In fact, along the reverse direc- tion, the total power loss increases to 25 dB. To sum up, the difference in the SW energy in the forward and reverse 034063-6SPIN-WAVE DIODE AND CIRCULATOR... PHYS. REV. APPLIED 14,034063 (2020) (a) (b) (c) (d) Amplitude (arb. units) FIG. 5. (a),(b) Propagation of the SW ( mxcomponent) in a diode at 15.2-GHz frequency in Py(3)/NM(5)/Co(2)/Pt in (a) for- ward and (b) reverse direction for the width of the Co/Pt stripe320 nm. In (a), the SW propagation direction is opposite to the coupling direction, so that the SW transfers weakly to the Co stripe, and we get a signal of high intensity in the output. In (b), the SW propagation direction is the same as the coupling direc- tion, so that the SW transfers to the Co stripe, where is stronglydamped, leading to the low intensity of the signal at the output. (c) The power loss in the forward and reverse direction in depen- dence on the frequency. In a broad frequency range of 7 GHz,the SW diode preserves its strong isolating properties. (d) The power loss in the forward and reverse direction in dependence on Co stripe width.direction equals 21.7 dB. The change of the position of the ports impacts the power loss only by the damping in the Py layer, thus it is not changing the difference between the forward and reverse direction. We investigate the efficiency of the Py(3)/NM(5)/Co(2)/ Pt SW diode in the wide range of frequency. Obtained results of the power loss in both directions of propaga- tion are collected in Fig. 5(c). Although SW transmission length varies in dependence on the frequency, the struc- ture preserves strongly asymmetric transmission in a broad range of frequency. In the forward direction, the diode works as well as in nominal frequency. The power loss is decreasing with the frequency due to the decrease of the coupling for negative wavevectors as shown in Fig. 3.T h e power loss in the reverse direction is reduced but remains significantly higher than in the forward direction. Esti- mated relative frequency range in which the device works is/Delta1ω/prime/ω/prime 0≈0.5 (at ω/prime 0/2π=15.2 GHz). Additional simulations are made to check the efficiency of the diode for different widths of the Co stripe at 15.2 GHz. The results are presented in Fig. 5(d). The power loss in the forward direction is mainly associated with the damping in the Py layer. A small negative slope reflects the increasing distance between the input and output. In the reverse direction, the most substantial effect on the results come from the SW transmission between the Py and Co layers. The curve reaches the first minimum for 300 nm, being close to xtr=320 nm, then increases up toxtr2=515 nm, and decreases again, reaching the sec- ond minimum near xtr2+xtr=835 nm. The power loss in the reverse direction at xtr2is significantly higher than in the forward direction because the SW intensity strongly decreases during the propagation through the Co stripe.Moreover, we perform detailed investigations in the vicin- ity of x trin Fig. 5(d) to check if the resonance effect in Co stripe plays any role in the coupling. Indeed, the local max- ima and minima in both forward and reverse directions are present with an approximate period of λ/2=50 nm, con- firming the influence of the resonance on the power-loss value. However, its impact is small in comparison with the interlayer SW transmission. In conclusion, we show that the SW diode is efficient in a wide range of the Co stripe width. Moreover, we investigate the SW diode working with SWs of longer (390 nm) wavelength, which should sim- plify the detection of the effect experimentally. We select another crossing point from Fig. 2(b), located at 8.2-GHz frequency. The width of the Co stripe is set to 190 nm. We obtain a power loss of 6.7 dB in the forward direc- tion and 14.6 dB in the reverse direction. In this case, we distinguish three mechanisms responsible for the smaller efficiency of the diode. First, SWs of longer wavelength are coupled stronger than SWs of shorter wavelength. This effect is shown in Fig. 3. The SW at 8.2 GHz corresponds 034063-7KRZYSZTOF SZULC et al. PHYS. REV. APPLIED 14,034063 (2020) tok=1.6×107m−1. The coupling for negative kreaches its maximum in the vicinity of this point. This effect leads to an additional decrease in the signal in the forward direc- tion. Second, the width of the Co stripe is too small to attenuate the SWs in reverse direction effectively. Third, the SW tends to reflect partially inside the Py layer in the points where the Co layer has its boundaries, which leads to additional losses. Besides these limitations, which can be further optimized, the structure is still efficient enough to be considered as a diode. C. Four-port spin-wave circulator Next, we exploit the unidirectional coupling further to design a SW circulator. The schematic structure of the four-port circulator is shown in Fig. 6. As compared with the structure of the diode, an additional FM layer is present on the opposite side of the stripe, playing the role of two additional ports. To get the functionality of the circula- tor, we need the stripe, which is unidirectionally coupled with both top and bottom layers but in opposite directions of SW propagation. We achieve this condition by taking identical outer layers having opposite IDMI constant and the inner stripe lacking IDMI. In our case, we propose to use Py as an IDMI-free coupling stripe and Co/Pt as guiding layers with swapped order in the bottom and top layers. The separation between the stripe and the layer is increased to 15 nm to reduce the dipolar coupling between the Co layers. We keep the width of the Py stripe suffi- cient to transfer the SW fully from one layer to another, thus for 15.2 GHz, we assume 440 nm. The SW is excited by antenna A, and the SW energy is measured by the ports located 20 nm from the device. Moreover, we perform sim- ulations with assuming no damping to check the efficiency in the ideal case, while the effect of the damping constant is presented further. The structure has a center of symmetry, therefore, the ports on the same diagonal, i.e., P1 and P3 as well as P2 and P4, work identically, and it is sufficient to investigate only two cases—propagation in the coupling and the noncoupling direction. In the noncoupling case, antenna A is located in the upper-right corner and emits the SWs at 15.2 GHz prop- agating to the left, as shown in Fig. 6(a) [see also the animation, Movie S3, within the Supplemental Material [38] ]. We observe very weak transfer of energy to the Py stripe, so the SW propagates mainly in the top Co layer. The SWs of low intensity in the bottom Co layer result from direct magnetostatic coupling between Co lay- ers. In the lossless structure, the power loss in port P2 reaches 0.2 dB, port P3—12.8 dB, and port P4—19.4 dB. The coupling direction is shown in Fig. 6(b) [see also the animation, Movie S4, within the Supplemental Material [38] ]. Here, antenna A is located in the upper-left corner. The SW is transferred to the Py stripe, and it reflects from the right edge of the stripe. After the reflection, the SWis coupled with the bottom Co layer, and, as a result, is transferred to it. In the lossless structure, the power loss in port P3 reaches 0.1 dB, port P4—24.9 dB, and port P1—16.1 dB. The SW circulator can also be used as a SW diode. How- ever, it benefits the mechanism of the redirection rather than the attenuation of a SW. Considering port P1 and P2 as the input-output ports, the transmission from port P1 to port P2 works as a forward direction and the transmission from port P2 to port P1 as a reverse direction. In that case, the difference in the SW energy in the forward and reverse direction equals 15.9 dB. Figures 6(c) and6(d) show the power loss in the circu- lator as the function of NM-layer thickness for the input port P1 and P2, respectively. We assume αCo=αPy=0t o focus on the principle transmission properties of the sys- tem. The width of the Py stripe is set to the transmission length, which is plotted in Fig. 4(b). In the noncoupling case [Fig. 6(c)], the power loss in the target port P2 is decreasing, reaching almost no loss for about 10 nm, while in port P3 and P4, we see the oscillations. This is the result of the resonance in the Py stripe. This behavior is even more relevant in Fig. 6(d) representing the coupling case. The power loss in the target port P3 is oscillating in coun- terphase with respect to port P1. The points with large power loss in port P3 correspond to the width wof the Py stripe fulfilling the resonance condition w=Nλ/2, where λ=100 nm. In the resonance, the SW is reflecting from the left side of the Py stripe, and it is coupled with the top Co layer. As a result, we observe the increase of the inten- sity of the SW in port P1 and, simultaneously, decrease of the intensity in port P3. Interestingly, the effect of negative power loss occurs in Fig. 6(d). It comes from the unwanted effect of the weak direct coupling between Co layers. For the thin NM layer, the coupling is significant enough to reach weak SW transmission from the top to bottom Co layer. In that case, we measure the SW energy in the range where we get the maximum value of the transmission. Moreover, the method of calculating the SW energy does not distinguish between the SW propagating in the left and right direction, which can fix this misleading effect. More- over, because of the weak direct transmission between Co layers, the power loss can vary depending on the posi- tion of the antenna, as well as the position of the ports. However, the effect is relatively small for assumed separa- tion between the outer layers, and the circulator preserves its properties even for ports located far away from the Py stripe edges. The effect of the damping in the Co layers on the SW circulator functionality is shown in Figs. 6(e)and6(f).T h e damping constant in the Py layer is set to 0.005. In thenoncoupling direction [Fig. 6(e)], the power loss in port P4 is almost constant, ultimately reaching the value of power loss in target port P2 for α Co≈0.035. The mechanism is identical to the one described in Fig. 6(d). The Co layers 034063-8SPIN-WAVE DIODE AND CIRCULATOR... PHYS. REV. APPLIED 14,034063 (2020) (a) (b) (c) (d) (e) (f)NM-layer thickness NM-layer thickness FIG. 6. (a),(b) Model of the four-port SW circulator based on the multilayered structure with unidirectional coupling. Propaga- tion of the SW ( mxcomponent) of 15.2-GHz frequency in the lossless Pt/Co(2)/NM(15)/Py(3)/NM(15)/Co(2)/Pt circulator in the (a) noncoupling direction and (b) coupling direction. In (a), the antenna A is located in the upper-right corner. The SW transfers weakly to the Py layer, so it goes mainly to port P2. In (b), the antenna A is located in the upper-left corner. The SW transfers from theupper Co layer to Py, and after the reflection from the right side of Py, it transfers to the lower Co layer, reaching port P3 in the end. A small-amplitude signal visible at isolated ports is a result of weak direct magnetostatic coupling between Co layers. (c),(d) The power loss measured in the output ports in regard to the input port in dependence on the NM-layer thickness for the input located in (c) portP1 and (d) port P2. In (c), the power loss in the target port P2 is increasing and in the rest of the ports is decreasing with the increase of NM-layer thickness. In (d), the power loss in the target port P3 and port P1 is fluctuating due to the resonance in the Py stripe. The minima of the power loss are corresponding to the Py stripe width fulfilling the resonance condition. (e),(f) The power loss measuredin dependence on the damping constant in the Co layer for the input located in (e) port P1 and (f) port P2. In (e), the power loss in port P4 is almost constant, ultimately reaching the value in the target port P2. In (f), the power loss in the target port P2 is significantly larger than in the other ports. The SW circulator is not working properly for α Co>0.025. are coupled, which leads to the additional SW energy in port P4 coming from the SW going into the circulator. The power loss in port P4 exceeds 35 dB when the SW going into the circulator is excluded from the calculationsof the power loss using the reference simulations without the Py stripe. In the coupling direction [Fig. 6(f)], the power loss in port P1 is decreasing with increasing αCoand approaches the value of the power loss in target port P3. 034063-9KRZYSZTOF SZULC et al. PHYS. REV. APPLIED 14,034063 (2020) Py PtPt Co CoNM coupling direc/g415oncoupling direc/g415onP3 P1 P2 Py PtPt Co CoNM coupling direc/g415oncoupling direc/g415onP3 P1 P2P3 P1 P2P3 P1 P2P3 P1 P2 P3 P1 P2P3 P1 P2P3 P1 P2spin-wave propaga/g415on(a) (b) (c) (d) (e) (f) (g) (h) FIG. 7. Three-port SW circulator in (a)–(d) easy-input and (e)–(h) easy-output port P3 configurations. We present the models (a),(e) and the way of acting when the input of SWs is localized in (b),(f) port P1, (c),(g) port P2, and (d),(h) port P3. The coupling direction determines the direction in which the SW will be transmitted from one layer to another. The orange arrows directing from one layer to another denote the SW transmission. This effect comes from the fact that the SW has to be trans- mitted from Co to Py as well as from Py to Co. The sign of/Delta1βin Eq. (13) is opposite for these two cases, and with increasing the difference between the transmission lengths in both directions, the circulator becomes less efficient. In comparison with the SW diode, the damping is, in gen- eral, an adverse effect in the SW circulator, so the materials with strong IDMI and low damping are highly desirable. A recent paper indicates Co/Ru as a possible low-damping alternative of Co/Pt bilayers [ 28,32]. The SW circulator remains efficient for the damping constant αCo=0.017 reported in Ref. [ 32]. D. Three-port spin-wave circulator Along with the four-port circulator, we can propose the three-port circulator with a slightly modified structure, as shown in Fig. 7. As compared to the structure of the four- port circulator, here the upper Co/Pt layer length is reduced and consists now from one port. Furthermore, this layer has to fully cover the Py stripe to preserve the possibility of complete transfer of the SW between the layers. The three- port device does not have any symmetries, therefore, we have to take three cases into account independently. More- over, the number of cases is doubled due to the reversal of the coupling directions. We can distinguish them by con- sidering the efficiency of port P3 localized in the upper Co layer. If the coupling direction between the upper Co layer and Py stripe is directed onto the port P3, we can consider it as an easy-output port because the SW coming from the Py stripe will be directed straight into port P3 [Fig. 7(f)]. On the other hand, with the opposite coupling direction,the SW coming from port P3 will be transmitted directly to the Py stripe [Fig. 7(d)], and port P3 becomes an easy-input port. Figure 7shows that the wave propagating from port P2 or P3 in the easy-input or easy-output configuration,respectively, have to reflect two times before reaching the output port [Figs. 7(c)and7(h)]. On the other hand, in the four-port circulator, only one reflection is needed. It means that the phenomenon in the three-port circulator requires very efficient reflections of SWs from the edges of the stripe. Further investigations are required to optimize its functionality. IV . CONCLUSIONS To sum up, we show the effect of unidirectional magne- tostatic coupling between the SW modes, which arises due to the IDMI-induced nonreciprocity in the ultrathin multi- layer system. The modes related to each layer are strongly coupled in only one direction of the SW propagation in a broad GHz range of frequency. In the opposite direction, within the same range of frequency, the SW modes propa- gate in only one layer. We propose to exploit this effect for the realization of the magnonic devices in the submicrome- ter scale. In the Py/NM/Co/Pt structure, limiting the Co/Pt stripe width to the length required to transfer the SW from the Py layer to the stripe, the possibility to get the diode effect is arising. In the forward direction, the SW propa- gates through the stripe area with small losses associated mainly with the Gilbert damping in Py, while in the reverse direction, the SW transmits to the Co stripe, in which the strong damping significantly reduces the SW intensity in the output. The device works efficiently in a broad range of microwave frequencies, as well as a broad range of Co stripe width. The SW diode can be further improved by opening the possibility to control the magnetization direc- tion in the Co layer and thus becoming the SW transistor[39–45]. The main advantage of the transistor based on the unidirectional coupling is that it will work immediately, so that the time of operation is limited only by the SW velocity and the time needed to reach the steady state. 034063-10SPIN-WAVE DIODE AND CIRCULATOR... PHYS. REV. APPLIED 14,034063 (2020) Another proposed type of the magnonic device, which bases on the same effect, is the SW circulator. It uses the two extended Co layers with Pt inducing IDMI as the waveguides with the input and output ports and the Py stripe in between as a coupler. In each possible case, we get efficient SW transfer to the target port with the strongly suppressed signal at the other ports. The high damping in the Co/Pt films suppresses transmission to the required port of the circulator. To preserve the functionality, the damp- ing constant in the layer shall be smaller than 0.025 and as close as possible to the damping of the Py stripe. In the circulator, the isolation effect in the selected output ports is achieved without involving losses. Moreover, the SW circulator can also work as a diode. A diode and a cir- culator take a place among the signal processing devices, thus demonstrating unidirectional coupling and proposing magnonic devices open the possibility for further devel- opment of energy-efficient, miniaturized beyond-CMOS, magnonic logic components [ 46–49]. ACKNOWLEDGMENTS The study has received financial support from the National Science Center of Poland, Projects Nos. UMO- 2018/30/Q/ST3/00416 and UMO-2018/28/C/ST3/00052. M.M. acknowledges funding from the Slovak Grant Agency APVV, No. APVV-16-0068 (NanoSky) and APVV-19-0311 (RSWFA). G.G. acknowledges the finan- cial support by the European Metrology Programme for Innovation and Research (EMPIR), under the Grant Agree- ment 17FUN08 TOPS. APPENDIX: PROCEDURE FOR ACHIEVING THE UNIDIRECTIONAL COUPLING IN A WIDE FREQUENCY RANGE The effect of unidirectional coupling of SWs in a wide frequency range can be obtained according to the following procedure. We limit our approach to the Landau-Lifshitzequation consisting of the Zeeman, exchange, magneto- static, and Dzyaloshinskii-Moriya terms [Eq. (1)]. At first, we assume that the external magnetic field is uniform. Next, one should fulfill a condition, that ifM S,FM1>( < ) MS,FM2, then Aex,FM1/MS,FM1>( < ) Aex,FM2/MS,FM2. It yields the noncrossing of the dispersion relation between noninteracting bilayers. If this condition is not fulfilled, we always will get crossing of the dispersion relations, and the coupling can be only asymmetric rather than unidi- rectional. Moreover, it is difficult to obtain the effect ofcoupling in a wide frequency range without fulfilling this condition. 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PhysRevB.96.144406.pdf

PHYSICAL REVIEW B 96, 144406 (2017) Current-driven second-harmonic domain wall resonance in ferromagnetic metal/nonmagnetic metal bilayers: A field-free method for spin Hall angle measurements M. R. Hajiali,1M. Hamdi,2S. E. Roozmeh,1and S. M. Mohseni2,* 1Department of Physics, University of Kashan, 87317 Kashan, Iran 2Faculty of Physics, Shahid Beheshti University, Evin, 19839 Tehran, Iran (Received 4 May 2017; revised manuscript received 24 July 2017; published 5 October 2017) We study the ac current-driven domain wall motion in bilayer ferromagnetic metal (FM)/nonmagnetic metal (NM) nanowires. The solution of the modified Landau-Lifshitz-Gilbert equation including all the spin transfertorques is used to describe motion of the domain wall in the presence of the spin Hall effect. We show that thedomain wall center has a second-harmonic frequency response in addition to the known first-harmonic excitation.In contrast to the experimentally observed second-harmonic response in harmonic Hall measurements of spin-orbittorque in magnetic thin films, this second-harmonic response directly originates from spin-orbit torque drivendomain wall dynamics. Based on the spin current generated by domain wall dynamics, the longitudinal spinmotive force generated voltage across the length of the nanowire is determined. The second-harmonic responseintroduces additionally a practical field-free and all-electrical method to probe the effective spin Hall angle forFM/NM bilayer structures that could be applied in experiments. Our results also demonstrate the capability ofutilizing FM/NM bilayer structures in domain wall based spin-torque signal generators and resonators. DOI: 10.1103/PhysRevB.96.144406 I. INTRODUCTION Current-induced domain wall (DW) dynamics in magnetic nanostructures raised great research interest in the field ofspintronics because of its prospective applications in noveldevices including DW-based shift register [ 1], logic [ 2], racetrack memory [ 3], and DW collision spin wave emitters [4]. Current-induced DW motion (CIDWM) in magnetic nanowires can occur thanks to the spin transfer torque (STT)effect due to exchange coupling between local magnetizationof DWs and spin-polarized currents [ 5–9]. There are two types of STTs acting on a noncollinear magnetization texture,e.g., a DW, when a spin-polarized current flows through it,namely, adiabatic and nonadiabatic terms. It has been showntheoretically [ 5,10] that the initial DW velocity is mostly controlled by the adiabatic term, while the nonadiabatic termis responsible for its terminal velocity. In ferromagnetic metal/nonmagnetic metal (FM/NM) bi- layers, a spin current generated by the spin Hall effect (SHE)through the adjacent NM layer can be injected into the FMlayer [ 11] that produces another type of STT, named spin-orbit torque (SOT), which can result in magnetization dynamics andCIDWM [ 12–16]. For instance, SOT can significantly reduce the critical dc current density to depinning a DW from a pin-ning potential, therefore causing lowering of the energy con-sumption for operation of DW-based electronic devices [ 17]. Investigations of the dynamical response of DW to ac currents have resulted in many achievements including de-termination of the mass of DW [ 18,19], micrometer range DW displacement [ 18,19], and resonant control of DW movement [20–22]. Furthermore, depinning of a DW benefits more from the ac current than from the dc current because ofcurrent-induced DW resonance [ 23]. Therefore, to benefit from *Corresponding author: m-mohseni@sbu.ac.ir; majidmohseni@gmail.com.such a dynamically rich feature, uncovering the dynamics of DWs under ac current and SOT is important and remainsunclear. Since the discovery of SHE in semiconductors [ 24,25] and metals [ 26,27], various techniques have been developed to determine the conversion rate of charge currents to spincurrents, named the spin Hall angle, θ SH. All these methods including cavity ferromagnetic resonance (C-FMR) spec-trometry [ 28,29], spin pumping (SP) [ 30], spin-torque FMR (ST-FMR) [ 31,32], hybrid phase-resolved optical-electrical FMR (OE-FMR) [ 33], and determining the DW velocity using magneto-optical microscopy [ 34,35] require almost strong bias magnetic fields to saturate the FM layer or in somecases a complicated measurement setup. Hence, a field-freemeasurement technique of θ SHis a challenge. In this paper, we investigate the ac CIDWM in FM/NM bilayers (Fig. 1) based on a collective coordinate approach. The FM layer has perpendicular magnetic anisotropy andthe NM layer has a strong spin-orbit interaction responsiblefor the SHE. All current-induced STTs and the SOT areincluded in our study. By solving equations of DW motion, weobtained second-harmonic motion in addition to the knownfirst-harmonic motion. This second-harmonic response hasa different nature in comparison with that observed in thesecond-harmonic Hall measurements of SOT [ 36–41] in single domain FM/NM structures. The former, which we study here,originates from SOT exerted on a DW due to the ac SHE occur-ring in the NM layer, while the latter originates from anoma-lous Hall and spin Hall magnetoresistance effects [ 36–41]. Finally, we calculated the spin motive force (SMF) voltage induced by DW motion. Our results propose a field-freemethod to measure the effective spin Hall angle for suchbilayer structures. In addition, second-harmonic DW motionin the presence of SHE and ac current can further be usedin recently reported DW and skyrmion based high-frequency signal generators and resonators [ 20,21,42] to obtain higher frequencies. 2469-9950/2017/96(14)/144406(6) 144406-1 ©2017 American Physical SocietyHAJIALI, HAMDI, ROOZMEH, AND MOHSENI PHYSICAL REVIEW B 96, 144406 (2017) FIG. 1. Schematic illustration of a FM/NM bilayer system. An in-plane ac current density Jacgenerates a perpendicular spin current, which exerts SOT on FM. II. MODEL A. ac current-induced domain wall motion in presence of spin Hall effect We considered a bilayer strip wire with dimensions 2 L× w×(tF+tN) along x,y, andzdirections, where 2 Lis the length and wis the width of the bilayer wire, tFandtNrepresent the thickness of the FM and NM layers, respectively, andL/greatermuchw, t F,tN. An in-plane ac current applied along the x direction generates conventional adiabatic and nonadiabaticSTTs and SOT acting on a DW within the FM. The geometryof the structure is shown in Fig. 1. The modified Landau- Lifshitz-Gilbert (LLG) equation including all the STTs isgiven by ∂m ∂t=−γm×Heff+αm×∂m ∂t−aJm×/parenleftbigg m×∂m ∂t/parenrightbigg −nJm×∂m ∂x−θSHcJm×(m׈y), (1) where mis the unit vector along the magnetization direction, γis the gyromagnetic ratio, Heffis effective field including the exchange, anisotropy, and external fields, αis the Gilbert damping constant, aJ=(¯hγP/ 2eMs)JFis the magnitude of adiabatic STT in velocity dimension, nJ=βaJis the magnitude of nonadiabatic STT and βis the nonadiabaticity coefficient, θSHcJ=(θSH¯hγJN/2eMstF) is the magnitude of SOT, where θSHis an effective spin Hall angle for the bilayer system, ¯ his the reduced Planck constant, Pis the spin polarization in FM, eis electron charge, Msis the saturation magnetization of FM, and JF(JN) is applied current density in FM (NM). JFandJNare determined by Ohm’s law for two parallel resistors as JF=J(σF σF+σN) andJN=J(σN σF+σN), where Jis the average current density in a bilayer nanowire andσF(σN) is the conductivity of FM (NM). Introducing the magnetization direction as m= (sinθcosϕ,sinθsinϕ,cosθ), the micromagnetic energy den- sityuin polar coordinates θandϕis given by u=Aex/bracketleftbigg/parenleftbigg∂θ ∂x/parenrightbigg2 +/parenleftbigg sinθ∂ϕ ∂x/parenrightbigg2/bracketrightbigg +Kusin2θ +Kdsin2θsin2ϕ, (2)where Aexis the exchange stiffness coefficient and Kuand Kdare perpendicular and in-plane anisotropy constants, re- spectively. A dynamic DW structure is considered as θ(x,t)= 2arctan[exp(x−q(t) /Delta1(t))] and ϕ(x, t)=φ(t), where q(t) and/Delta1(t) represent the DW center position and width, respectively [ 43]. The chirality of the DW is determined by angle φ(t) and distortion of the wall width is small during wall motion,hence we suppose /Delta1(t)∼=√ A/K u. The restoring energy density arising from geometric notch [ 20,44], impurity or demagnetization field [ 23] induced pinning is assumed to be of the form ur=kq2/2. Therefore, equations of motion for two collective coordinates qandφin rigid DW limit, by neglecting the small coefficients of nonlinear terms, are given by m∂2q ∂t2+b∂q ∂t+kq=−2MS γ/parenleftbiggnJ /Delta1+1 γ/Delta1H d ×/parenleftbigg α∂nJ ∂t+∂aJ ∂t/parenrightbigg +αaJnJBSH γ/Delta12Hd/parenrightbigg , (3) m∂2φ ∂t2+b∂φ ∂t+kφ=kaJ γ/Delta1H d+2MSα /Delta12γ2Hd∂aJ ∂t −2MS /Delta12γ2Hd∂nJ ∂t, (4) where Hd=(2Kd/Ms)=4πMsis the demagnetization field, BSH=θSHcJ(π/Delta1eM s/PJ F¯hγ)=(πθSH/Delta1JN/2tFPJF), m=(1+α2)2Ms γ2/Delta1Hd, andb=α(2Ms γ/Delta1+k γHd). Considering an ac applied current with frequency ω,E q .( 3) describes forced damped oscillations. Therefore, the steady-state solution to qis of the form q(t)=q 1ω(t)+q2ω(t), q1ω(t)=Aωcos[ωt−(δ−ρ)], (5) q2ω(t)=A2ωcos[2ωt−ξ], where q1ω(t),q2ω(t) are the first- and second-harmonic components of DW motion. AωandA2ωare the first- and second-harmonic DW motion amplitudes, respectively, andare given by A ω=−2MS γ⎛ ⎜⎝/radicalBig n2 /Delta12+(αn+a)2ω2 (4πγ/Delta1M S)2/parenleftBig σF σF+σN/parenrightBig J /radicalBig (k−mω2)2+ω2b2⎞ ⎟⎠, (6) A2ω=−2MS γ⎛ ⎜⎝αnaB SH 4πγ/Delta12MS/parenleftBig σF σF+σN/parenrightBig2 J2 /radicalBig (k−4mω2)2+4ω2b2⎞ ⎟⎠. (7) φ(t) has no second-harmonic dynamics and is obtained as φ(t)=φωcos[ωt−(δ+η)], (8) where the chirality amplitude is φω=2MS γ⎛ ⎜⎜⎝/radicalbigg/parenleftBig ak 8π/Delta1M2 S/parenrightBig2 +(n−αa)2ω2 (4πγ/Delta12MS)2/parenleftBig σF σF+σN/parenrightBig J /radicalBig (k−mω2)2+ω2b2⎞ ⎟⎟⎠.(9) 144406-2CURRENT-DRIVEN SECOND-HARMONIC DOMAIN W ALL . . . PHYSICAL REVIEW B 96, 144406 (2017) The phases δ,ρ,ξ, andηare in the form of δ=arctan [ bω/(k−mω2)], ρ=arctan [( αn+a)ω/(4πMSγn)], (10) ξ=arctan [2 bω/(k−4mω2)], η=arctan [2 MSω(n−αa)//Delta1γak ]. According to Eqs. ( 6), (7), and ( 9) the first-harmonic DW motion amplitude Aωand chirality amplitude φωare proportional to the applied current density J, while the second- harmonic amplitude is proportional to the J2due to ac SOT. B. Spin currents and voltages A given dynamical noncollinear magnetization texture, m(r,t), generates a spin current which is given by [ 45–48] Js i=μB¯h 2e2/summationdisplay kσc ik/bracketleftbigg/parenleftbigg∂m ∂t×∂m ∂rk/parenrightbigg ·m +β∂m ∂t·∂m ∂rk/bracketrightbigg , (11) where Js iis the ith component of spin current, Js= (Js x,Js y,Js z), which flows through the idirection ( i,j,k= x,y,z ) and its polarization is determined by the instantaneous local magnetization direction, m(r,t),σc ikis theik-element of the electrical conductivity tensor of FM, rk=x, y, z, and μBis the Bohr magneton. All other parameters are previously determined. Such spin current generates electrical currents orvoltages in FM/NM bilayers through two dominant mecha-nisms: (i) spin motive force (SMF) within the FM layer and(ii) inverse spin Hall effect (ISHE) within the NM layer. SMFis related to the s-dexchange coupling between conduction electrons and localized moments within FM; whereas, ISHEis the reciprocal effect of SHE, i.e., an injected spin currentinto NM is converted to a transverse electrical current orvoltage. In the case of our study and considering the dynamicDW structure introduced earlier, the only nonvanishing spatialderivative of magnetization is ∂m(r, t)/∂x. Furthermore, most of the common FMs have a cubic or hexagonal crystal structureand hence their conductivity tensor is diagonal. Therefore,according to Eq. ( 9), all components of spin current vanish except J s xandJs=(Js x,0,0). As the z-direction flowing component of the spin current, Js z=0,no spin current enters the NM layer, hence no charge current or voltage is generatedthrough ISHE. The local current density generated by the SMFmechanism is given by J c i,SMF=eP μBJs i. (12) Here,Jc i,SMFis the charge current density flowing through theidirection generated by SMF. Using Eqs. ( 9) and ( 10) and a dynamic DW structure with ∂θ/∂t =− (∂θ/∂x )(∂q/∂t ) and ∂θ/∂x =(sinθ//Delta1 ), one can obtain the charge current density induced by SMF in polar coordinates as Jc x,SMF=P¯hσF 2e/bracketleftbigg −∂θ ∂x∂ϕ ∂tsinθ+β∂θ ∂x∂θ ∂t/bracketrightbigg =−P¯hσF 2e/bracketleftbigg1 /Delta1∂φ ∂t+β /Delta12∂q ∂t/bracketrightbigg sin2θ. (13)Averaging SMF charge current density over the wire length, /angbracketleftJc x,SMF/angbracketright=1 2L/integraltextL −LJc x,SMFdxand considering the first- and second-harmonic components of DW motion in Eq. ( 5), SMF voltage can be decomposed to first harmonic, Vω SMF, and second harmonic, V2ω SMF, components as Vω SMF(t)=Vωsin[ωt−ψ], (14) V2ω SMF(t)=V2ωsin[2ωt−ξ], where the voltage amplitudes VωandV2ωand the first- harmonic voltage phase ψare given by Vω=¯hPω e/Delta1/radicalBig β2A2ω+/Delta12φ2ω+2β/Delta1A ωφωcos[ρ+η], tanψ=βAωsin[δ−ρ]−/Delta1φωsin[δ+η] βAωcos[δ−ρ]+/Delta1φωcos[δ+η], (15) V2ω=2¯hPωβ e/Delta1A2ω. (16) III. RESULTS AND DISCUSSIONS A. DW dynamics Implementing the realistic geometric and material param- eters introduced in Table Iand the current density value of J=20×1013A/m2[13,15,23,49] into Eqs. ( 5)–(7), we plot the DW motion amplitudes and components. Figure 2(a) shows Aω(blue line) and A2ω(red line) as functions of the applied current frequency, ω. There is a resonance peak in both the first- and second-harmonic amplitudes of 390 and 30 nm,respectively. The inset shows the second-harmonic amplitudeseparately for clarity. The resonance frequencies for first- andsecond-harmonic motion are current independent and for usedmaterial parameters they are obtained as ω r,1=59.9 GHz and ωr,2=29.4 GHz, respectively. It is obvious that the second- harmonic component of DW motion and its related effectsare significant at second-harmonic resonance frequency, ω r,2. Therefore, we used ω=ωr,2to perform all other calculations hereafter. Figure 2(b) shows the time dependence of the TABLE I. Realistic material parameters and geometry dimen- sions for numerical calculations [ 13,15,23,49]. Symbol Value Unit Saturation magnetization of FM Ms 0.5 T Exchange stiffness coefficient Aex 3.37×10−11J/m Perpendicular anisotropy constants Ku 1.5×105J/m3 Gilbert damping constant α 0.02 Nonadiabaticity of STT β 0.05 Restoring force constant k 1.25×106T2/m Conductivity of FM (NM) σF 1.5×107s/m (σN)( 1 .89×107) Spin polarization in the FM P 0.5 Spin Hall angle θSH 0.15 Domain wall width /Delta1 15 nm Thickness of FM (NM) tF(tN) 0.6 (2) nm Applied current density J (5−20)×1013A/m2 Width of bilayer wire w 5n m Half-length of bilayer wire L 10×10−6m 144406-3HAJIALI, HAMDI, ROOZMEH, AND MOHSENI PHYSICAL REVIEW B 96, 144406 (2017) 0 50 100 150 200 2500100200300400 0 50 100 150 2000102030 (a)r,1=59.9 r,2=29.4A (nm) (GHz)A A2A2(nm) (GHz) 0.0 0.1 0.2 0.3 0.4-200-1000100200(b)q(nm) t(ns)q1(t) q2(t) q(t) FIG. 2. (a) Frequency dependence of first-harmonic ( Aω) and second-harmonic ( A2ω) amplitudes, respectively. The inset shows the second-harmonic motion amplitude separately. (b) Time dependence of first harmonic ( q1ω), second harmonic ( q2ω), and total ( q) DW motion at second-harmonic resonance frequency, respectively. Applied current density is 20 ×1013A/m2. first- (blue line) and second- (red line) harmonic components and the total (black line) DW motion, respectively. B. SMF voltages Using Eqs. ( 15) and ( 16) and the material parameters in Table Iwe obtain the current dependence of the first- and second-harmonic SMF voltage amplitudes at ωr,2which are shown in Figs. 3(a) and 3(b), respectively. As mentioned earlier, Aωandφωlinearly depend on the applied current, J, hence the first-harmonic voltage amplitude, Vω, also depends linearly on J; whereas, the second-harmonic voltage ampli- tude,V2ω, quadratically grows with Jdue to the quadratic dependence of A2ωonJ. C. Field-free spin Hall angle measurement based on electrical detection of second-harmonic DW resonance Based on our results, we propose a field-free practical experimental method to measure the effective spin Hall anglefor a bilayer FM/NM structure based on electrical detectionof the current-driven second-harmonic DW resonance. In thistechnique, one can apply an ac current with constant amplitudeto a bilayer strip of FM/NM structure and sweep the frequencyand simultaneously probe the SMF second-harmonic voltage along the wire to find second-harmonic resonance frequency,ω r,2. Then, by fixing the frequency to second-harmonic reso- nance frequency and sweeping the current amplitude, one thusmeasures the current-dependent second-harmonic voltage. Fit-ting the result with Eq. ( 16), and using the appropriate material parameters, the effective spin Hall angle θ SHcan be deter- mined. This could be a powerful field-free and all-electricalmethod to measure spin Hall angle for FM/NM structures incomparison with any existing field required techniques such asC-FMR spectrometry [ 28,29], SP [ 30], ST-FMR [ 31,32], and hybrid phase-resolved OE-FMR [ 33]. All of these FMR-based techniques require large magnetic fields of the order of afew Teslas to saturate the FM layer and provide the FMRconditions. Along with this, the DW velocity measurementbased techniques require a complicated measurement setupsuch as magneto-optical Kerr microscopy [ 34,35]. Therefore, our proposed technique represents the advantages of bothfield-free and simple all-electrical measurements. In addition,second-harmonic generation in FM/NM structures, by a DW ora noncollinear magnetization texture in general, could be use-ful in designing DW-based spin-torque signal generators andresonators. 4 6 8 1 01 21 41 61 82 023456789 (a)V(v) J (1013A/m2)4 6 8 1 01 21 41 61 82 00.00.51.01.52.0V2(v) J (1013A/m2)(b) FIG. 3. Applied current density dependence of (a) first-harmonic ( Vω) and (b) second-harmonic ( V2ω) SMF voltage amplitudes at second- harmonic resonance frequency. 144406-4CURRENT-DRIVEN SECOND-HARMONIC DOMAIN W ALL . . . PHYSICAL REVIEW B 96, 144406 (2017) IV . CONCLUSIONS Using the real material parameters and reasonable current densities, we predict a second-harmonic DW resonance inFM/NM bilayer nanowires originating from ac SOT exerted ona FM layer. The longitudinal SMF-induced second-harmonicvoltage is calculated in the range of 0 .15−2μV, which is measurable in laboratories. 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PhysRevB.76.104404.pdf

Strong spin-orbit-induced Gilbert damping and g-shift in iron-platinum nanoparticles Jürgen Kötzler, Detlef Görlitz, and Frank Wiekhorst Institut für Angewandte Physik und Zentrum für Mikrostrukturforschung, Universität Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany /H20849Received 23 June 2007; revised manuscript received 1 August 2007; published 6 September 2007 /H20850 The shape of ferromagnetic resonance spectra of highly dispersed, chemically disordered Fe 0.2Pt0.8nano- spheres is perfectly described by the solution of the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation excluding effects by crystalline anisotropy and superparamagnetic fluctuations. Upon decreasing temperature, the LLG damping /H9251/H20849T/H20850and a negative g-shift, g/H20849T/H20850-g0, increase proportional to the particle magnetic moments determined from the Langevin analysis of the magnetization isotherms. These features are explained by the scattering of theq→0 magnon from an electron-hole /H20849e/h/H20850pair mediated by the spin-orbit coupling, while the sdexchange can be ruled out. The large saturation values, /H9251/H208490/H20850=0.76 and g/H208490/H20850/g0−1=− 0.37, indicate the dominance of an overdamped 1 meV e/hpair which seems to originate from the discrete levels of the itinerant electrons in the dp=3 nm nanoparticles. DOI: 10.1103/PhysRevB.76.104404 PACS number /H20849s/H20850: 76.50. /H11001g, 78.67.Bf I. INTRODUCTION The ongoing downscaling of magneto electronic devices maintains the yet intense research of spin dynamics in ferro-magnetic structures with restricted dimensions. The effect ofsurfaces, interfaces, and disorder in ultrathin films, 1multilay- ers, and nanowires2has been examined and discussed in great detail. On structures confined to the nm scale in allthree dimensions, like ferromagnetic nanoparticles, the im-pact of anisotropy 3and particle-particle interactions4on the Neél-Brown-type dynamics, which controls the switching ofthe longitudinal magnetization, is now also well understood.On the other hand, the dynamics of the transverse magneti-zation, which, e.g., determines the externally induced, ul-trafast magnetic switching in ferromagnetic nanoparticles, isstill a topical issue. Such fast switching requires a large, i.e.,a critical value of the Landau-Lifshitz-Gilbert /H20849LLG /H20850damp- ing parameter /H9251.5This damping has been studied by conventional6,7and, more recently, by advanced8ferromag- netic resonance /H20849FMR /H20850techniques, revealing enhanced val- ues of /H9251up to the order of one. By now, the LLG damping of bulk ferromagnets is almost quantitatively explained by the scattering of the q=0 magnon by conduction electron-hole /H20849e/h/H20850pairs due to the spin-orbit coupling /H9024so.9According to recent ab initio band-structure calculations,10the rather small values for /H9251/H11015/H9024so2/H9270result from the small /H20849Drude /H20850relaxation time /H9270of the electrons. For nanoparticles, the Drude scattering and also the wave-vectorconservation are ill-defined, and ab initio many-body ap- proaches to the spin dynamics should be more appropriate.Numerical work by Cehovin et al. 11considers the modifica- tion of the FMR spectrum by the discrete level structure ofthe itinerant electrons in the particle. However, the effect of the resulting electron-hole excitation, /H9280p/H11011vp−1, where vpis the nanoparticle volume on the intrinsic damping has not yetbeen considered. Here we present FMR spectra recorded at /H9275/2/H9266 =9.1 GHz on Fe 0.2Pt0.8nanospheres, the structural and mag- netic properties of which are summarized in Sec. II. In Sec.III the measured FMR shapes will be examined by solutionsof the LLG equation of motion for the particle moments. Several effects, in particular those predicted for crystallineanisotropy 12and superparamagnetic /H20849SPM /H20850fluctuations of the particle moments13will be considered. In Sec. IV , the central results of this study, i.e., the LLG-damping /H9251/H20849T/H20850 reaching values of almost 1 and a large g-shift, g/H20849T/H20850-g0, are presented. Since both /H9251/H20849T/H20850andg/H20849T/H20850increase proportional to the particle magnetization, they can be related to spin-orbit damping torques, which, due to the large values of /H9251and/H9004g are rather strongly correlated. It will be discussed which fea-tures of the e/hexcitations are responsible for these correla- tions in a nanoparticle. A summary and the conclusions aregiven in the final section. II. NANOPARTICLE CHARACTERIZATION The nanoparticle assembly has been prepared14following the wet-chemical route by Sun et al. .15In order to minimize the effect of particle-particle interactions, the nanoparticleswere highly dispersed. 14Transmission electron microscopy revealed nearly spherical shapes with mean diameter dp =3.1 nm and a rather small width of the log-normal size distribution, /H9268d=0.17 /H208493/H20850. Wide angle x-ray diffraction pro- vided the chemically disordered fcc structure with a lattice constant a0=0.3861 nm.14 The mean magnetic moments of the nanospheres /H9262p/H20849T/H20850 have been extracted from fits of the magnetization isotherms M/H20849H,T/H20850, measured by a SQUID magnetometer /H20849Quantum Design, MPMS2 /H20850in units emu /g=1.1 /H110031020/H9262B/g, to ML/H20849H,T/H20850=Np/H9262p/H20849T/H20850L/H20873/H9262pH kBT/H20874. /H208491/H20850 Here, L/H20849y/H20850=coth /H20849y/H20850−y−1with y=/H9262p/H20849T/H20850H/kBTis the Lange- vin function and Npis the number of nanoparticles per gram. The fits shown in Fig. 1/H20849a/H20850demanded for a small paramag- netic background, M−ML=/H9273b/H20849T/H20850H, with a strong Curie-type temperature variation of the susceptibility /H9273b, signalizing the presence of paramagnetic impurities. According to the insetof Fig. 1/H20849b/H20850this 1/ Tlaw turns out to agree with the tempera-PHYSICAL REVIEW B 76, 104404 /H208492007 /H20850 1098-0121/2007/76 /H2084910/H20850/104404 /H208498/H20850 ©2007 The American Physical Society 104404-1ture dependence of the intensity of a weak, narrow magnetic resonance with gi=4.3 depicted in Figs. 2and3. Such nar- row line with the same gfactor has been observed by Berger et al.16on partially crystallized iron-containing borate glass and could be traced to isolated Fe3+ions. The results for /H9262p/H20849T/H20850depicted in Fig. 1/H20849b/H20850show the mo- ments to saturate at /H9262p/H20849T→0/H20850=/H20849910±30 /H20850/H9262B. This yields a mean moment per atom in the fcc unit cell of /H9262¯/H208490/H20850 =/H9262pa03/4vp=0.7/H9262Bcorresponding to a spontaneous magneti- zation Ms/H208490/H20850=5.5 kOe. According to previous work by Men- shikov et al.17on chemically disordered Fe xPt1−xthis corre- sponds to an iron concentration of x=0.20. Upon rising temperature the moments decrease rapidly, which above40 K can be rather well parametrized by the empirical powerlaw, /H9262p/H20849T/H1135040 K /H20850/H11011/H208491−T/TC/H20850/H9252revealing /H9252=2 and for the Curie temperature TC=/H20849320±20 /H20850K. This is consistent with TC=/H20849310±10 /H20850K for Fe 0.2Pt0.8emerging from a slight ex- trapolation of results for TC/H20849x/H113500.25 /H20850of Fe xPt1−x.18No quan- titative argument is at hand for the exponent /H9252=2, which is much larger than the mean-field value /H9252MF=1/2. W e believe that/H9252=2 may arise from a reduced thermal stability of the magnetization due to strong fluctuations of the ferromagneticexchange between Fe and Pt in the disordered structure andalso to additional effects of the antiferromagnetic Fe-Fe andPt-Pt exchange interactions. In this context, it may be inter-esting to note that for low Fe concentrations, x/H113490.3, bulk Fe xPt1−xexhibits ferromagnetism only in the disordered structure,17while structural ordering leads to paraferromag-netism or antiferromagnetism. Recent first-principle calcula- tions of the electronic structure produced clear evidence forthe stabilizing effect of disorder on the ferromagnetism inFePt. 19 From the Langevin fits in Fig. 1/H20849a/H20850we obtain for the particle density Np=3.5/H110031017g−1. Based on the well-known mass densities of Fe 0.2Pt0.8and the organic matrix, we find by a little calculation20for the volume concentration of the particles cp=0.013 and, hence, for the mean interparticle dis- tance, dpp=dp/cp1/3=13.5 nm. This implies for the maximum /H20849i.e., T=0/H20850dipolar interaction between nearest particles, /H9262p2/H208490/H20850/4/H9266/H92620dpp3=0.20 K, so that at the present temperatures, T/H1135020 K, the sample should act as an ensemble of indepen- FIG. 1. /H20849a/H20850Magnetization isotherms of the nanospheres fitted to the Langevin model plus a small paramagnetic background /H9273bH;/H20849b/H20850 temperature dependences of the magnetic moments of the nanopar-ticles /H9262pand of the background susceptibility /H9273b/H20849inset units are emu/g kOe /H20850fitted to the indicated relations with TC=/H20849320±20 /H20850K. The inset shows also data from the intensity of the electron para-magnetic resonance /H20849EPR /H20850at 1.45 kOe, see Figs. 2and3. FIG. 2. /H20849a/H20850Derivative of the microwave /H20849f=9.095 GHz /H20850absorp- tion spectrum recorded at T=52 K, i.e., close to magnetic saturation of the nanospheres. The solid and dashed curves are based on fits toEqs. /H208494/H20850and /H208498/H20850, respectively, which both ignore SPM fluctuations and assume either a g-shift and zero anisotropy field H A/H20849/H9004g-FM /H20850or /H9004g=0 and a randomly distributed HA=0.5 kOe /H20849a-FM /H20850, Eq. /H208499/H20850. Also shown are fits to predictions by Eq. /H2084911/H20850, which account for SPM fluctuations, with HA=0.5 kOe and /H9004g=0/H20849a-SPM /H20850and, using Eq. /H2084911/H20850, for/H9004g/HS110050 and HA=0/H20849/H9004g-SPM /H20850. The weak, narrow reso- nance at 1.45 kOe is attributed to the paramagnetic backgroundwith g=4.3±0.1 indicating Fe 3+/H20849Ref. 16/H20850impurities. FIG. 3. Derivative spectra at some representative temperatures and fits to Eq. /H208494/H20850. The LLG-damping, g-shift, and intensities are depicted in Fig. 4.KÖTZLER, GÖRLITZ, AND WIEKHORST PHYSICAL REVIEW B 76, 104404 /H208492007 /H20850 104404-2dent ferromagnetic nanospheres. Since also the blocking temperature, Tb=9 K, as determined from the maximum of the ac susceptibility at 0.1 Hz in zero magnetic field,20turned out to be low, the Langevin analysis in Fig. 1/H20849a/H20850is valid. III. RESONANCE SHAPE Magnetic resonances at a fixed X-band frequency of 9.095 GHz have been recorded by a homemade microwavereflectometer equipped with field modulation to enhance thesensitivity. A double-walled quartz tube containing thesample powder has been inserted to a multipurpose, gold-plated V ARIAN cavity /H20849model V-4531 /H20850. Keeping the cavity at room temperature, the sample could be either cooled bymeans of a continuous flow cryostat /H20849Oxford Instruments, model ESR 900 /H20850down to 15 K or heated up to 500 K by an external Pt-resistance wire. 20At all temperatures, the inci- dent microwave power was varied in order to assure the lin-ear response. Some examples of the spectra recorded below the Curie temperature are shown in Figs. 2and3. The spectra have been measured from −9.5 kOe to +9.5 kOe and proved to beindependent of the sign of H and free of any hysteresis. Thiscan be expected due to the completely reversible behavior ofthe magnetization isotherms above 20 K and the low block-ing temperature of the particles. Lowering the temperature,we observe a downward shift of the main resonance accom-panied by a strong broadening. On the other hand, the posi-tion and width of the weak narrow line at /H208491.50±0.05 /H20850kOe corresponding to g i/H110614.3 remain independent of temperature. This can be attributed to the previously detected paramag-netic Fe 3+impurities16and is supported by the integrated intensity of this impurity resonance Ii/H20849T/H20850evaluated from the amplitude difference of the derivative peaks. Since the inten- sity of a paramagnetic resonance is given by the paramag- netic susceptibility, Ii/H20849T/H20850/H11011/H20848 dH/H9273xx/H11033/H20849H,T/H20850/H11011/H9273i/H20849T/H20850can be com- pared directly to the background susceptibility /H9273b/H20849T/H20850, see inset to Fig. 1/H20849b/H20850. The good agreement between both tem- perature dependencies suggests to attribute /H9273bto these Fe3+ impurities. An analysis of the fitted Curie constant, Ci =5 emu K/g kOe, yields Ni=164/H110031017g−1for the Fe3+ concentration, which corresponds to 50 Fe3+per 1150 atoms of a nanosphere. With regard to the main intensity, we want to extract the maximum-possible information, in particular, on the intrinsicmagnetic damping in nanoparticles. Unlike the conventionalanalysis of resonance fields and linewidths, as applied, e.g.,to Ni 6and Co7nanoparticles, our objective is a complete shape analysis in order to disentangle effects by the crystal-line anisotropy, 12by SPM fluctuations,13by an electronic g-shift,21and by different forms of the damping torque R/H6023.22 Additional difficulties may enter the analysis due to non- spherical particle shapes, size distributions and particle inter-action, all of which, however, can be safely excluded for thepresent nanoparticle assembly. The starting point of most FMR analyses is the phenom- enological equation of motion for a particle moment /H20849see, e.g., Ref. 13/H20850d dt/H9262/H6023p=/H9253H/H6023eff/H11003/H9262/H6023p−R/H6023, /H208492/H20850 using either the original Landau-Lifshitz /H20849LL/H20850damping with damping frequency /H9261L, R/H6023L=/H9261L Ms/H20849H/H6023eff/H11003/H9262/H6023p/H20850/H11003s/H6023p, /H208493/H20850 or the Gilbert damping with the Gilbert damping parameter /H9251G, R/H6023G=/H9251Gd/H9262/H6023p dt/H11003s/H6023p, /H208494/H20850 where s/H6023p=/H9262/H6023p//H9262pdenotes the direction of the particle mo- ment. In Eq. /H208492/H20850, the gyromagnetic ratio, /H9253=g0/H9262B//H6036,i sd e - termined by the regular g-factor g0of the precessing mo- ments. It should perhaps be noted that the validity of themicromagnetic approximation underlying Eq. /H208492/H20850has been questioned 23for volumes smaller than /H208512/H9261sw/H20849T/H20850/H208523, where /H9261sw=2a0TC/Tis the smallest wavelength of thermally ex- cited spin waves. For the present particles, this estimate leadsto a fairly large temperature of /H110110.7T Cup to which micro- magnetics should hold. At first, we ignore the anisotropy being small in cubic FexPt1−x,24,25so that for the present nanospheres the effective field is identical to the applied field, H/H6023eff=H/H6023. Then, the solu- tions of Eq. /H208492/H20850for the susceptibility of the two normal, i.e., circularly polarized modes, of Npindependent nanoparticles per gram take the simple forms /H9273±L/H20849H/H20850=Np/H9262p/H92531/H11007i/H9251 /H9253H/H208491/H11007i/H9251/H20850/H11007/H9275/H208495/H20850 forR/H6023=R/H6023Lwith/H9251=/H9261L//H9253Msand for the Gilbert torque R/H6023G, /H9273±G/H20849H/H20850=Np/H9262p/H92531 /H9253H/H11007/H9275/H208491+i/H9251G/H20850. /H208496/H20850 For the LL damping, the experimental, transverse suscepti- bility, /H9273xx=1 2/H20849/H9273++/H9273−/H20850, takes the form /H9273xxL/H20849H/H20850=Np/H9262p/H9253/H9253H/H208491+/H92512/H20850−i/H9251/H9275 /H20849/H9253H/H208502/H208491+/H92512/H20850−/H92752−2i/H9251/H9275/H9253H. /H208497/H20850 As the same shape is obtained for the Gilbert torque with /H9251=/H9251G, the damping is frequently denoted as the LLG param- eter. However, the gyromagnetic ratio in Eq. /H208497/H20850must be replaced by /H9253//H208491+/H92512/H20850, which only for /H9251/lessmuch1 implies also the same resonance field Hr. Upon increasing the damping up to /H9251/H110150.7/H20849the regime of interest here /H20850, the resonance field Hr of/H9273xxG/H20849H/H20850, determined by d/H9273/H11033/dH=0, remains constant, HrG /H11015/H9275//H9253, while HrLdecreases rapidly. After renormalization /H9253//H208491+/H92512/H20850the resonance fields and also the shapes of /H9273L/H20849H/H20850 and/H9273G/H20849H/H20850become identical. This effect should be observed when determining the gfactor from the resonance fields of broad lines. It becomes even more important if the down-ward shift of H ris attributed to anisotropy, as done recently for the rather broad FMR absorption of Fe xPt1−xnanopar- ticles with larger Fe content, x/H113500.3.26STRONG SPIN-ORBIT-INDUCED GILBERT DAMPING … PHYSICAL REVIEW B 76, 104404 /H208492007 /H20850 104404-3In order to check here for both damping torques, we se- lected the shape measured at a low temperature, T=52 K, where the linewidth proved to be large /H20849see Fig. 2/H20850and the magnetic moment /H9262p/H20849T/H20850was close to saturation /H20851Fig. 1/H20849b/H20850/H20852. Neither of the damping terms could explain both, the ob- served resonance field Hrand the linewidth /H9004H=/H9251/H9275//H9253, and, hence, the line shape. By using R/H6023G, the shift of Hrfrom /H9275//H9253=3.00 kOe was not reproduced by HrG=/H9275//H9253, while for R/H6023Lthe resonance field HrL, demanded by the linewidth, be- came significantly smaller than Hr. This result suggested to consider as next the effect of a crystalline anisotropy field H/H6023Aon the transverse susceptibil- ity, which has been calculated by Netzelmann from the freeenergy of a ferromagnetic grain. 12Specializing his generalansatz to a uniaxial H/H6023Aoriented at angles /H20849/H9258,/H9278/H20850with respect to the dc field H/H6023/H20648e/H6023zand the microwave field, one obtains by minimizing F/H20849/H9258,/H9278,/H9277,/H9272/H20850=−/H9262p/H20877Hcos/H9277+1 2HA/H20851sin/H9277sin/H9258− cos /H20849/H9272−/H9278/H20850 + cos/H9258cos/H9277/H208522/H20878 /H208498/H20850 the equilibrium orientation /H20849/H92770,/H92720/H20850of the moment /H9262/H6023pof a spherical grain. After performing the trivial average over /H9278, one finds for the transverse susceptibility of a particle withorientation /H9258, /H9273xxL/H20849/H9258,H/H20850=/H9253/H9262p 2/H20849F/H92770/H92770+F/H92720/H92720/tan2/H92770/H20850/H208491+/H92512/H20850−i/H9251/H9262 p/H9275/H208491 + cos2/H92770/H20850 /H208491+/H92512/H20850/H20849/H9253Heff/H208502−/H92752−i/H9251/H9275/H9253/H9004H. /H208499/H20850 Here Heff2=/H20851F/H92770/H92770F/H92720/H92720−F/H92770/H927202//H20849/H9262psin/H92770/H208502/H20852and /H9004H =/H20849F/H92770/H92770+F/H92720/H92720/sin2/H92770/H20850//H9262pare given by the second deriva- tives of Fat the equilibrium orientation of /H9262/H6023p. For the ran- domly distributed H/H6023AofNpindependent particles per gram one has /H9273xxL/H20849H/H20850=/H20885 0/H9266/2 d/H20849cos/H9258/H20850/H9273xxL/H20849/H9258,H/H20850. /H2084910/H20850 In a strict sense, this result should be valid at fields larger than the so-called thermal fluctuation field HT=kBT//H9262p/H20849T/H20850, see, e.g., Ref. 13, which for the present case amounts to HT=1.0 kOe. Hence, in Fig. 2we fitted the data starting at high fields, reaching there an almost perfect agreement withthe curve a-FM. The fit yields a rather small H A=0.5 kOe which implies a small anisotropy energy per atom, EA =1 2/H9262p/H208490/H20850HA=1.0/H9262eV. This number is smaller than the cal- culated value for bulk fcc FePt, EA=4.0/H9262eV,25most prob- ably due to the lower Fe concentration /H20849x=0.20 /H20850and the strong structural disorder in our nanospheres. We emphasize, that the main defect of this a-FM fit curve arises from the finite value of d/H9273xx/H11033/dHatH=0. By means of Eq. /H208499/H20850one finds/H9273xx/H11033/H20849H→0,/H9258/H20850/H11011HAH//H92752,which remains finite even af- ter averaging over all orientations according to /H9258/H20851Eq. /H2084910/H20850/H20852. The finite value of the derivative of /H9273xx/H11033/H20849H→0/H20850should disappear if superparamagnetic /H20849SPM /H20850fluctuations of the particles are taken into account. Classical work27predicted the anisotropy field to be reduced by SPM, HA/H20849y/H20850 =HA/H208511/L/H20849y/H20850−3/y/H20852, which for y=H/HT/lessmuch1 implies HA/H20849y/H20850 =HAy/5 and, therefore, /H9273xx/H11033/H20849H→0/H20850/H11011H2. A statistical theory for/H9273xxL/H20849H,T/H20850which considers the effect of SPM fluctuations exists only to first order in HA/H.21The result of this linear model /H20849LM/H20850inHA/Hwhich generalizes Eq. /H208494/H20850, can be cast in the form/H9273±LM/H20849/H9258,H/H20850=Np/H9262p/H9253L/H20849y/H20850/H208491+A/H11007i/H9251A/H20850 /H9253/H208491+B/H11007i/H9251B/H20850H/H11007/H9275./H2084911/H20850 The additional parameters are given in Ref. 21and contain, depending on the symmetry of HA, higher-order Langevin functions Lj/H20849y/H20850and their derivatives. Observing the validity of the LM for H/greatermuchHA=0.5 kOe, we fitted the data in Fig. 2 to Eq. /H2084911/H20850with/H9273xxLM/H20849/H9258,H/H20850=1 2/H20849/H9273+LM+/H9273−LM/H20850at larger fields. There, one has also H/greatermuchHT=1.0 kOe and the fit, denoted as a-SPM, agrees with the ferromagnetic result /H20849a-FM /H20850. How- ever, increasing deviations appear below fields of 4 kOe. Byvarying H Aand/H9251, we tried to improve the fit near the reso- nance Hr=2.3 kOe and obtained unsatisfying results. For low anisotropy, HA/H113493 kOe, the resonance field could be re- produced only by significantly lower values of /H9251, which are inconsistent with the measured width and shape. For HA /H110223 kOe, a small shift of Hroccurs, but at the same time the line shape became distorted, tending to a two-peak structurealso found in previous simulations. 13Even at the lowest tem- perature, T=22 K, where the thermal field drops to HT =0.4 kOe, no signatures of such inhomogeneous broadening appear /H20849see Fig. 3/H20850. Finally, it should be mentioned that all of the above attempts to incorporate the anisotropy in the dis-cussion of the line shape were based on the simplest non-trivial, i.e., uniaxial symmetry, which for FePt was also con-sidered by the theory. 25For cubic anisotropy, the same qualitative discrepancies were found in our simulations.20 This insensitivity with respect to the symmetry of HAorigi- nates from the orientational averaging in the range of the HA values of relevance here. As a finite anisotropy failed to reproduce Hr,/H9004H, and also the shape, we tried the ansatz for the magnetic resonance ofnanoparticles by introducing a complex LLG parameter,KÖTZLER, GÖRLITZ, AND WIEKHORST PHYSICAL REVIEW B 76, 104404 /H208492007 /H20850 104404-4/H9251ˆ/H20849T/H20850=/H9251/H20849T/H20850−i/H9252/H20849T/H20850. /H2084912/H20850 According to Eq. /H208494/H20850this is equivalent to a negative g-shift, g/H20849T/H20850−g0=−/H9252/H20849T/H20850g0, which is intended to compensate for the too large downward shift of HrLdemanded by /H9273xxL/H20849H/H20850due to the large linewidth. In fact, inserting this ansatz in Eq. /H208495/H20850, the fit, denoted as /H9004g-FM in Fig. 2, provides a convincing description of the line shape down to zero magnetic field. Itmay be interesting to note that the resulting parameters, /H9251 =0.56 and /H9252=0.27, revealed the same shape as obtained by using the Gilbert susceptibilities, Eq. /H208496/H20850. In spite of the agreement of the /H9004g-FM model with the data, we also tried to include here SPM fluctuations by using /H9251ˆ/H20849T,H/H20850=/H9251ˆ/H20849T/H20850/H208511/L/H20849y/H20850−1/y/H20852/H20849Ref. 21/H20850forHA=0. The result, designated as /H9004g-SPM in Fig. 2agrees with the /H9004g-FM curve for H/greatermuchHTwhere /H9251ˆ/H20849T,H/H20850=/H9251ˆ/H20849T/H20850, but again significant deviations occur at lower fields. They indicate that SPM fluc- tuations do not play any role here, and this conclusion is alsoconfirmed by the results at higher temperatures. There, thethermal fluctuation field, H T=kBT//H9262p/H20849T/H20850, increases to values larger than the maximum measuring field, H=10 kOe, so that SPM fluctuations should cause a strong thermal, homo- geneous broadening of the resonance due to /H9251ˆ/H20849H/greatermuchHT/H20850 =/H9251ˆ2HT/H. However, upon increasing temperature, the fitted linewidths, /H20849Fig. 3/H20850and damping parameters /H20849Fig. 4/H20850display the reverse behavior. IV. COMPLEX DAMPING In order to shed more light on the magnetization dynam- ics of the nanospheres we examined the temperature varia-tion of the FMR spectra. Figure 3shows some examples recorded below the Curie temperature, T C=320 K, togetherwith fits to the a-FM model outlined in the preceding section. Above TC, the resonance fields and the linewidths are tem- perature independent revealing a mean gfactor, g0 =2.16±0.02, and a damping parameter /H9004H/Hr=/H92510 =0.18±0.01. Since g0is consistent with a recent report on g values of Fe xPt1−xforx/H113500.43,21we suspect that this reso- nance arises from small Fe xPt1−xclusters in the inhomoge- neous Fe 0.2Pt0.8structure. Fluctuations of g0and of local fields may be responsible for the rather large width. Thisinterpretation is supported by the observation that above T C the line shape is closer to a Gaussian than to the Lorentzian following from Eq. /H208497/H20850for small /H9251. The temperature variation for both components of the complex damping, obtained from the fits below TCto Eq. /H208497/H20850, are shown in Fig. 4. Clearly, they obey the same power law as the moments, /H9262p/H20849T/H20850, displayed in Fig. 1/H20849b/H20850, which implies /H9251ˆ/H20849T/H20850=/H20849/H9251−i/H9252/H20850ms/H20849T/H20850+/H92510. /H2084913/H20850 Here ms/H20849T/H20850=/H9262p/H20849T/H20850//H9262p/H208490/H20850,/H9251=0.58, and /H9252=−/H9004g/H208490/H20850/g0 =0.39 denote the reduced spontaneous magnetization and the saturation values for the complex damping, respectively. Itshould be emphasized that the fitted intensity I/H20849T/H20850of the spectra, shown by the inset to Fig. 4/H20849b/H20850, exhibits the same temperature variation I/H20849T/H20850/H11011 /H9262p/H20849T/H20850. This behavior is pre- dicted by the ferromagnetic model, Eq. /H208497/H20850, and is a further indication for the absence of SPM effects on the magneticresonance. If the resonance were dominated by SPM fluctua-tions, the intensity should decrease like the SPM Curie sus- ceptibility, I SPM/H20849T/H20850/H11011/H9262p2/H20849T/H20850/T, following from Eq. /H2084911/H20850, be- ing much stronger than the observed I/H20849T/H20850. At the beginning of a physical discussion of /H9251ˆ/H20849T/H20850,w e should point out that the almost perfect fits of the line shape to Eq. /H208497/H20850indicate that the complex damping is related to an intrinsic mechanism and that eventual inhomogeneous ef-fects by distributions of particle sizes and shapes in the as-sembly, as well as by structural disorder are rather unlikely.Since a general theory of the magnetization dynamics innanoparticles is not yet available, we start with the currentknowledge on the LLG damping in bulk and thin film ferro-magnets, as recently reviewed by Heinrich. 5Based on ex- perimental work on the archetypal metallic ferromagnets andon recent ab initio band-structure calculations 10there is now rather firm evidence that the damping of the q=0 magnon is associated with the torques T/H6023so=m/H6023s/H11003/H20858j/H20849/H9264jL/H6023j/H11003S/H6023/H20850on the spin S/H6023due to the spin-orbit interaction /H9264jat the lattice sites j. The action of the torque is limited by the finite lifetime /H9270of an e/hexcitation, the finite energy /H9280of which may cause a phase, i.e., a g-shift. As a result of this magnon- e/h-pair scattering, the temperature-dependent part of the LLG damp-ing parameter becomes /H9251ˆ/H20849T/H20850−/H92510=/H9261L/H20849T/H20850 /H9253Ms/H20849T/H20850=/H20851/H9024soms/H20849T/H20850/H208522 /H9270−1+i/H9280//H60361 /H9253Ms/H20849T/H20850. /H2084914/H20850 For intraband scattering, /H9280/lessmuch/H6036//H9270, the aforementioned nu- merical work10revealed /H9024so=0.8/H110031011s−1and 0.3 /H110031011s−1as effective spin-orbit coupling in fcc Ni and bcc Fe, respectively. Hence, the narrow unshifted /H20849/H9004g=0/H20850bulk FMR lines in pure crystals, where /H9251/H1134910−2, are related toFIG. 4. Temperature variation /H20849a/H20850of the LLG damping /H9251and /H20849b/H20850 of the relative g-shifts with g0=2.16 /H20849following from the resonance fields at T/H11022TC/H20850. Within the error margins, /H9251/H20849T/H20850and/H9004g/H20849T/H20850and also the fitted intensity of the LLG shape /H20849see inset /H20850display the same temperature dependence as the particle moments in Fig. 1/H20849b/H20850.STRONG SPIN-ORBIT-INDUCED GILBERT DAMPING … PHYSICAL REVIEW B 76, 104404 /H208492007 /H20850 104404-5intraband scattering with /H9280/lessmuch/H6036//H9270and to electronic /H20849momen- tum/H20850relaxation times /H9270smaller than 10−13s. Following Eq. /H2084914/H20850, we discuss at first the temperature variation, which implies a linear dependence, /H9251ˆ/H20849T/H20850−/H92510 /H11011ms/H20849T/H20850. Obviously, both, the real and imaginary part of /H9251ˆ/H20849T/H20850−/H92510, agree perfectly with the fits to the data in Fig. 4,i f the relaxation time /H9270remains constant. It may be interesting to note here that the observed temperature variation of thecomplex damping /H9261 L/H20849T/H20850is not predicted by the classical model28incorporating the sd-exchange coupling Jsd. Accord- ing to this model, which has been advanced recently to fer-romagnets with small spin-orbit interaction 29and ferromag- netic multilayers,30Jsdtransfers spin from the localized 3 d moments to the delocalized s-electron spins within their spin-flip time /H9270sf. From the mean-field treatment of their equations of motion by Turov,31we find a form analogous to Eq. /H2084914/H20850, /H9251sd/H20849T/H20850=/H9024sd2/H9273s /H9270sf−1+i/H9024˜sd1 /H9253Ms/H20849T/H20850, /H2084915/H20850 where /H9024sd=Jsd//H6036is the exchange frequency, /H9273sthe Pauli susceptibility of the selectrons, and /H9024˜sd//H9024sd=/H208491 +/H9024sd/H9273s//H9253Md/H20850. The same form follows from more detailed considerations of the involved scattering process /H20849see, e.g., Ref. 5/H20850. As a matter of fact, the LLG damping /H9251sd =/H9261sd//H9253Mdcannot account for the observed temperature de- pendence, because /H9024sdand/H9273sare constants. The variation of the spin torques with the spontaneous magnetization ms/H20849T/H20850 drops out in this model, since the sdscattering involves tran- sitions between the 3 dspin-up and spin-down bands due to the splitting by the exchange field Jsdms/H20849T/H20850. By passing from the bulk to the nanoparticle ferromagnet, we use Eq. /H2084914/H20850to discuss our results for the complex /H9251ˆ/H20849T/H20850, Eq. /H2084913/H20850. Recently, for Co nanoparticles with diameters 1–4 nm, the existence of a discrete level structure near /H9280F has been evidenced,32which suggests to associate the e/h energy /H9280with the level difference /H9280pat the Fermi energy. From Eqs. /H2084913/H20850and /H2084914/H20850we obtain relations between /H9280and the lifetime of the e/hpair and the experimental parameters /H9251and/H9252, /H9270−1=/H9251 /H9252/H9280 /H6036, /H2084916a/H20850 /H9280 /H6036=/H9252 /H92512+/H92522/H9024so2 /H9253Ms/H208490/H20850. /H2084916b/H20850 Due to /H9251//H9252=1.5, Eq. /H2084916a/H20850reveals a strongly overdamped excitation, which is a rather well-founded conclusion. Theevaluation of /H9280, on the other hand, depends on an estimate for the effective spin-orbit coupling, /H9024so=/H9257L/H9273e1/2/H9264so//H6036where /H9257Lrepresents the matrix element of the orbital angular mo- mentum between the e/hstates.5The spin-orbit coupling of the minority Fe spins in FePt has been calculated bySakuma, 24/H9264so=45 meV, while the density of states D/H20849/H9280F/H20850 /H110151//H20849eV atom /H20850/H20849Refs. 24and33/H20850yields a rather high suscep- tibility of the electrons, /H9257L/H9273e=/H9262B2D/H20849/H9280F/H20850=4.5/H1100310−5. Assum- ing/H9257L/H110051, both results lead to /H9024so/H110153.5/H110031011s−1, which isby one order of magnitude larger than the values for Fe and Ni mentioned above. One reason for this enhancement andfor a large matrix element, /H9257L/H110051, may be the strong hybrid- ization between 3 dand 4 dPt orbitals24in Fe xPt1−x.B yi n - serting this result into Eq. /H2084916b/H20850we find /H9280=0.8 meV. In fact, this value is comparable to an estimate for the level differ-ence at /H9280F,32/H9280p=/H20851D/H20849/H9280F/H20850Np/H20852−1which for our particles with Np=/H208492/H9266/3/H20850/H20849dp/a0/H208503=1060 atoms yields /H9280p=0.9 meV. Re- garding the several involved approximations, we believe that this good agreement between the two results on the energy ofthee/hexcitation, /H9280/H11015/H9280p, may be accidental. However, we think, that this analysis provides a fairly strong evidence forthe magnon scattering by this excitation, i.e., for the gap inthe electronic states due to confinement of the itinerant elec-trons to the nanoparticle. V. SUMMARY AND CONCLUSIONS The analysis of magnetization isotherms explored the mean magnetic moments of Fe 0.2Pt0.8nanospheres /H20849dp =3.1 nm /H20850suspended in an organic matrix, their temperature variation up to the Curie temperature TC, the large mean particle-particle distance Dpp/greatermuchdp, and the presence of Fe3+ impurities. Above TC, the resonance field Hrof the 9.1 GHz microwave absorption yielded a temperature independentmean gfactor, g 0=2.16, consistent with a previous report21 for paramagnetic Fe xPt1−xclusters. There, the line shape proved to be closer to a Gaussian with rather large linewidth,/H9004H/H r=0.18, which may be associated with fluctuations of g0and local fields both due to the chemically disordered fcc structure of the nanospheres. Below the Curie temperature, a detailed discussion of the shape of the magnetic resonance spectra revealed a numberof unexpected features. /H20849i/H20850Starting at zero magnetic field, the shapes could be described almost perfectly up to highest field of 10 kOe bythe solution of the LLG equation of motion for independentferromagnetic spheres with negligible anisotropy. Signaturesof SPM fluctuations on the damping, which have been pre-dicted to occur below the thermal field H T=kBT//H9262p/H20849T/H20850, could not be realized. /H20849ii/H20850Upon decreasing temperature, the LLG damping in- creases proportional to /H9262p/H20849T/H20850, i.e., to the spontaneous mag- netization of the particles, reaching a rather large value /H9251 =0.7 for T/lessmuchTC. We suspect that this high intrinsic damping may be responsible for the absence of the predicted SPMeffects on the FMR, since the underlying statistical theory 13 has been developed for /H9251/lessmuch1. This conjecture may further be based on the fact that the large intrinsic damping field/H9004H= /H9251/H9275//H9253=2.1 kOe causes a rapid relaxation of the trans- verse magnetization /H20849q=0 magnon /H20850as compared to the effect of statistical fluctuations of HTadded to Heffin the equation of motion, Eq. /H208492/H20850.13 /H20849iii/H20850Along with the strong damping, the line-shape analy- sis revealed a significant reduction of the gfactor, which also proved to be proportional to /H9262p/H20849T/H20850. Any attempts to account for this shift by introducing uniaxial or cubic anisotropy fields failed, since low values of H/H6023Ahad no effect on theKÖTZLER, GÖRLITZ, AND WIEKHORST PHYSICAL REVIEW B 76, 104404 /H208492007 /H20850 104404-6resonance field due to the orientational averaging. On the other hand, larger H/H6023A’s, by which some small shifts of Hr could be obtained, produced severe distortions of the calcu- lated line shape. The central results of this work are the temperature varia- tion and the large magnitudes of both /H9251/H20849T/H20850and/H9004g/H20849T/H20850. They were discussed by using the model of the spin-orbit induced scattering of the q=0 magnon by an e/hexcitation /H9280, well established for bulk ferromagnets, where strong intrabandscattering with /H9280/lessmuch/H6036//H9270proved to dominate.5In nanopar- ticles, the continuous /H9280/H20849k/H6023/H20850spectrum of a bulk ferromagnet is expected to be split into discrete levels due to the finite num- ber of lattice sites creating an e/hexcitation /H9280p. According to the measured ratio between damping and g-shift, this e/h pair proved to be overdamped, /H6036//H9270p=1.5/H9280p. Based on the free-electron approximation for /H9280p/H20849Ref. 32/H20850and the density of states D/H20849/H9280F/H20850from band-structure calculations for FexPt1−x,24,33one obtains a rough estimate /H9280p/H110150.9 meV for the present nanoparticles. Using a reasonable estimate of theeffective spin-orbit coupling to the minority Fe spins, this value could be well reproduced by the measured LLG damp-ing, /H9251=0.59. Therefore, we conclude that the unexpected results of the dynamics of the transverse magnetization re-ported here are due to the presence of a broad e/hexcitation with energy /H9280p/H110151 meV. Deeper quantitative conclusions, however, must await more detailed information on the realelectronic structure of nanoparticles near /H9280F, which are also required to explain the overdamping of the e/hpairs, as it is inferred from our data. ACKNOWLEDGMENTS The authors are indebted to E. Shevchenko and H. Weller /H20849Hamburg /H20850for the synthesis and the structural characteriza- tion of the nanoparticles. One of the authors /H20849J.K./H20850thanks B. Heinrich /H20849Burnaby /H20850and M. Fähnle /H20849Stuttgart /H20850for illuminat- ing discussions. 1G. Woltersdorf, M. Buess, B. Heinrich, and C. H. Back, Phys. Rev. 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PhysRevB.78.064423.pdf

Electrical detection of spin pumping: dc voltage generated by ferromagnetic resonance at ferromagnet/nonmagnet contact M. V. Costache,1,2,*S. M. Watts,1C. H. van der Wal,1and B. J. van Wees1 1Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands 2Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA /H20849Received 21 January 2008; revised manuscript received 23 July 2008; published 26 August 2008 /H20850 We describe electrical detection of spin pumping in metallic nanostructures. In the spin pumping effect, a precessing ferromagnet attached to a normal metal acts as a pump of spin-polarized current, giving rise to aspin accumulation. The resulting spin accumulation induces a backflow of spin current into the ferromagnetand generates a dc voltage due to the spin dependent conductivities of the ferromagnet. The magnitude of suchvoltage is proportional to the spin-relaxation properties of the normal metal. By using platinum as a contactmaterial we observe, in agreement with theory, that the voltage is significantly reduced as compared to the casewhen aluminum was used. Furthermore, the effects of rectification between the circulating rf currents and themagnetization precession of the ferromagnet are examined. Most significantly, we show that using an improvedlayout device geometry, these effects can be minimized. DOI: 10.1103/PhysRevB.78.064423 PACS number /H20849s/H20850: 72.25.Ba, 72.25.Hg, 73.23. /H11002b, 85.75. /H11002d I. INTRODUCTION During the last several years there has been a continuing interest in high-frequency phenomena in spintronic devices,as they are expected both to provide applications for micro-wave signal processing, and to become a powerful new toolfor fundamental studies of spin dynamics in magneticnanostructures. 1–5 It was predicted by Slonczewski1and Berger2that angular momentum is transferred from spin-polarized currents to themagnetization of the ferromagnets when charge currents aresent trough spin valves with noncollinear magnetizations/H20849i.e., spin torque effect /H20850. This can excite and even switch the magnetization direction of the softer ferromagnet. Experi-ments with pillar-type structures 6–8confirmed these predic- tions. It is natural to expect that if a spin current can induce magnetization motion the reciprocal process may also bepossible: a moving magnetization in a ferromagnet can emita spin current into an adjacent conductor. This effect is theso-called spin pumping, proposed by Tserkovnyak andco-workers. 9,10Spin pumping is a mechanism where a pure spin current, which does not involve net charge currents, isemitted at the interface between a ferromagnet with a pre-cessing magnetization and a normal-metal region. It is animportant mechanism to generate spin currents, since otherelectronic methods based on driving an electrical currentthrough a ferromagnet/semiconductor interface are stronglylimited by the so-called conductance mismatch. 11Berger12 proposed a similar mechanism to generate a dc voltage byferromagnetic resonance /H20849FMR /H20850, based on spin flip scattering in the ferromagnet as induced by spin waves. Recently, spin pumping has been demonstrated in ferro- magnetic resonance experiments with thin multilayers, whereit appears as an enhancement of the Gilbert damping con-stant of magnetization dynamics, 13–17and using time re- solved magneto-optic Kerr effect.18Although these experi- ments are very important in providing evidence for the spinpumping mechanism, the detection technique can be viewed as an indirect method to measure the spin pumping effect.Several experimental methods have been proposed to electri-cally detect the spin pumping mechanism. 19,20The general problem of these methods is the rectification effects at theferromagnet/normal-metal contact, which can suppress ormimic the spin pumping signal. 21–23Thus, the identification of these spurious effects is crucial and represents one of themain themes of this paper. In a recent paper, 24we have demonstrated spin pumping with a single permalloy strip in an electronic device, inwhich it is directly detected as a dc voltage signal. In thispaper, we describe additional experiments on spin pumpingeffect, designed explicitly to eliminate the rectification ef-fects. We explain in more detail the theoretical prediction forthe voltage, and we identify and quantify different contribu-tions of the rectification effect. Importantly, we show that byusing appropriate device geometry, these effects can be mini-mized. II. SPIN PUMPING EFFECT As discussed above, the emission of a spin current into a conductor by a moving magnetization of an adjacent ferro-magnet is essentially the reciprocal of the spin torque mecha-nism in spin valves, where the magnetization is excited by aspin current. A simplified picture of the process is schematically shown in Fig. 1. We consider a F/Njunction at equilibrium, where inFexist a larger population of spins in the direction of magnetization, than antiparallel. When the magnetization di-rection is suddenly switched, the bands instantaneously shiftin energy. However, in order to go back to the equilibriumsituation there has to be spin transfer from one-spin popula-tion to another /H20849i.e., spin relaxation /H20850.I fFis in contact with N, this transfer of spins can go via N. Thus the spin- relaxation process for Fis modified when it is in contact with an adjacent N, and depends on the spin-relaxation propertiesPHYSICAL REVIEW B 78, 064423 /H208492008 /H20850 1098-0121/2008/78 /H208496/H20850/064423 /H208499/H20850 ©2008 The American Physical Society 064423-1onN. As a result, an ac spin current is emitted into Nwhen the magnetization is switched back and forth under an oscil-lating magnetic field. Tserkovnyak et al. 9analyzed the case of circular precession of the magnetization and found that inaddition to the ac current, a dc spin current is also emitted. Away to periodically change the magnetization direction is toputFinto FMR, where circular precession of the magneti- zation can be resonantly excited by a small applied rf mag-netic field. 21,26 The transfer of spin angular momentum by the precessing magnetization of Fin contact with N/H20849spin pumping /H20850was first described9using the formalism of parametric charge pumping27developed in the context of mesoscopic scattering problems. The main points of this description are discussedbelow. Spin currents at the interface: As illustrated in Fig. 2,a spin current I spumpis pumped by the /H20849resonant /H20850precession of a ferromagnet magnetization into an adjacent normal-metalregion. Assuming the Fat FMR state Tserkovnyak et al. 9 have calculated the spin pumped current using a scattering matrix approach and based on the microscopic details of theinterface, I spump=/H6036 4/H9266g↑↓m/H11003dm dt, /H208491/H20850 where mrepresents the magnetization direction. g↑↓is the real part of the mixing conductance,28,29a material parameterwhich describes the transport of spins that are noncollinear to the magnetization direction at the interface and is propor-tional to the torque acting on the ferromagnet in the presenceof a noncollinear spin accumulation in the normal metal. 30,31 This equation shows that the spin current, which goes into N, is perpendicular both to the magnetization direction mand to the change of min time. This current has ac and dc compo- nents, but in the limit /H9275/H9270N/H112711/H20849see later discussion /H20850, the time-averaged pumping current reads10/H20841/H20855Ispump/H20856t/H20841=Idc =/H6036/H9275g↑↓sin2/H9258/4/H9266. Depending on the spin related properties of the N, the spin-current emission has two limiting regimes. When the N is a good “spin sink” /H20849in which spins relax fast /H20850, the injected spin current is quickly dissipated and this corresponds to aloss of angular momentum and an increase in the effectiveGilbert damping of the magnetization precession. This hasbeen observed experimentally in nanopillar structures. 13–17 The total spin current is given by Ispump. The opposite regime is when the spin-flip relaxation rate is smaller than the spin injection rate. In this case, a spinaccumulation /H9262sbuilds up in the normal metal /H20849Fig.2/H20850. The spin accumulation can diffuse away from the interface, butcan also diffuse back into the F. This back flow current is given by I sback=g↑↓ 2/H9266N/H20851/H9262s−m/H20849m·/H9262s/H20850/H20852, /H208492/H20850 where Nis the one-spin density of states. The total spin current in this case is IsF=Ispump+Isback. Spin battery: A spin battery operated by FMR has been proposed by Brataas et al.10in the limit of weak spin-flip scattering in the F. The spin accumulation in Ncan be cal- culated by solving the spin-diffusion equation /H11509/H9262s /H11509t=DN/H115092/H9262s /H11509x2−/H9262s /H9270N/H208493/H20850 where /H9270Nis the spin-flip time and DNis the diffusion coeffi- cient in N. We assume that the spin-diffusion length in Nis much larger than the spin precession length l/H9275/H11013/H20881DN//H9275/H20849/H9275is precessional frequency /H20850, i.e., /H9261N=/H20881DN/H9270N/H11271l/H9275, or equivalent by/H9275/H9270N/H112711. This means that if the length of Nis larger than l/H9275, the x,ycomponents of spin accumulation are fully aver- aged /H20849due to dephasing /H20850and the remaining zcomponent is constant and along the static magnetic-field direction.10The time-averaged spin accumulation /H20855/H9262s/H20856t=/H9262zzin the Nclose to the interface reads10 /H92620,z=/H6036/H9275sin2/H9258 sin2/H9258+/H9257, /H208494/H20850 where /H9258is the precession cone angle and /H9257is a reduction factor determined by the ratio between injection time andspin-flip relaxation time. Intuitively, the spin accumulation can be measured elec- trically using a second ferromagnet as a spin-dependent con-tact, placed at a shorter distance compared to the spin-fliplength. 10,32–34 Voltage at F/Ninterface: Importantly,35have predicted a more direct way to detect the spin pump effect in which theFIG. 1. Simplified picture of the spin pumping process. /H20849a/H20850 Population of spin-up and spin-down bands in equilibrium. /H20849b/H20850Situ- ation after sudden reversal of the magnetization direction. The ar-rows denote spin flow from one-spin population to another one. /H20849c/H20850 Equilibrium situation again but with magnetization in opposite di-rection /H20849Adapted from Ref. 25/H20850. F N H Ispump Isback xyz /c113mmdmxdt l/c119/c109s FIG. 2. The F/Nstructure in which the resonant precession of the magnetization direction mpumps a spin current Ispumpinto N. The spin pumping builds up a spin accumulation /H9262sNinNthat drives a spin current Isbackback into the F. The component of the Isbackparallel to mcan enter into F. Since the interface and the bulk conductances of Fare spin dependent, this can result in a dc voltage across the interface.COSTACHE et al. PHYSICAL REVIEW B 78, 064423 /H208492008 /H20850 064423-2precessing ferromagnet acts also as the detector. We have to take into account that the spin accumulation /H9262sin a diffusive metal drives the spin current Isbackback into the F. The com- ponent parallel to mcan enter F. Moreover, since the inter- face and the bulk conductances of Fare spin dependent, this can result in charge accumulation, close to the interface, andthereby a dc voltage across the interface. The chemical-potential difference across the interface has been calculatedby Wang et al. 35following the lines of the Brataas et al.10 model, but including the spin diffusion back into Fand spin relaxation in F. As mentioned above, the relevant length scale for the averaging of the transverse /H20849x,y/H20850components of the spin current is l/H9275. Therefore for a device with dimensions larger than l/H9275the spin-up /H20849down /H20850effective conductances g/H9275↑/H20849↓/H20850 of the interface are composed of the interface conductances g↑/H20849↓/H20850in series with a conductance of the bulk Nover a length scale of l/H9275. These relations are given by g/H9275↑/H20849↓/H20850=g↑/H20849↓/H20850//H208491+g↑/H20849↓/H20850/g/H9275/H20850and the mixing conductance g/H9275↑↓ =g↑↓//H208491+g↑↓/g/H9275/H20850, where g/H9275=/H20849/H9268NA/H20850/l/H9275/H20849Ais the area of the interface /H20850. Polarization p/H9275=/H20849g/H9275↑−g/H9275↓/H20850//H20849g/H9275↑+g/H9275↓/H20850is also intro- duced. In the limit of large spin-flip in Fand the size of N/H11271/H9261 N and for small-angle precession /H20849/H9258→0/H20850, the chemical- potential difference is given by35 /H9004/H92620=p/H9275g/H9275↑↓ 2/H208491+gN gF/H20850/H208491−p/H92752/H20850/H20849g/H9275↑+g/H9275↓/H20850+2gN/H92582/H6036/H9275, /H208495/H20850 where gN/H20849gF/H20850is the conductance of the bulk N/H20849F/H20850over a length scale of /H9261N/H20849/H9261F/H20850. For a thorough review of the above discussion see Ref. 35. Interface currents matching: In this section, we describe a simple way to find the voltage /H20851similar to Eq. /H208495/H20850/H20852using spin-current matching at the interface. By writing all the cur-rents involved in the process and matching them at the inter-face, all components of the spin accumulation at the interfacecan be determined. It is convenient to transform the equa-tions into a rotating frame of reference in which the uniformmagnetization motion can be formally eliminated, and the unit magnetization vector is mˆ/H20849sin /H9258,0,cos /H9258/H20850. Basically, forthis problem we have to consider three currents with their components. First, the spin pumping current /H20851Eq. /H208491/H20850/H20852is given by Is,/H11036pump=g↑↓sin/H9258/H6036/H9275. /H208496/H20850 Second, the back flow current consists of components paral- lel and perpendicular to mˆ, and can be written in terms of spin accumulation /H92620at the interface, Is,/H20648back=gF/H92620,/H20648; Is,/H11036back=g↑↓/H92620,/H11036. /H208497/H20850 The sum of Eqs. /H208496/H20850and /H208497/H20850gives the total spin current on theFside of the interface. Third, the spin current on the N side of the interface is found by solving the Bloch equationsfor the spin accumulation in N, and from this the current at the interface is given by 36 IsN=g/H9275/H20898/H92620,x−/H92620,y /H92620,x+/H92620,y gN g/H9275/H92620,z/H20899, /H208498/H20850 in terms of /H92620at the interface. This current has three com- ponents. The zcomponent is determined only by the usual spin-relaxation process. For the xand ycomponents, two effects are important: precession, which results in mixing ofthe two components, depending on the time spent in N; and averaging, which reduces the total amplitude of the compo-nents. The spin accumulation /H92620is determined by matching the currents at the interface IsN=IsF=Isback+Ispump. The dc volt- age at the interface is proportional to the projection of /H92620 onto mˆ, and for the limit g↑↓/H11350g/H9275is given by V=−p/H92620·mˆ/H11229−pg/H9275 gF/H208731−gN g/H9275/H20874cos/H9258sin2/H9258/H6036/H9275. /H208499/H20850 The simple form of Eq. /H208499/H20850results from the relative indepen- dence of the dc voltage on g↑↓/H20849the physical argument of this result remains to be clarified /H20850. For our devices /H20849N=Al and F=Py/H20850, using /H9268F=6.6·106/H9024−1m−1,/H9268N=3.1·107/H9024−1m−1, /H9261F=5 nm, /H9261N=500 nm and l/H9275=300 nm, we estimate the conductances at room temperature: gF/A=/H9268F//H9261F/H112291·1 015/H9024−1m−2 g/H9275/A=/H9268N/l/H9275/H112291·1 014/H9024−1m−2 gN/A=/H9268N//H9261N/H112298·1 013/H9024−1m−2. /H2084910/H20850 Moreover, according to Xia et al.37g↑↓/A /H112295·1014/H9024−1m−2. For a quantitative assessment of the re- lations /H208495/H20850and /H208499/H20850we assume /H9258/H110155°/H20849sin2/H9258=0.01 /H20850and/H9275 =1011s−1/H20849/H6036/H9275=65/H9262eV/H20850. First Eq. /H208499/H20850, by using p=0.4 we find dc voltage /H1101520 nV. Second Eq. /H208495/H20850, by using p/H9275 =0.06, g↑/A=0.31 /H110031015/H9024−1m−2and g↓/A=0.19 /H110031015/H9024−1m−2/H20849from Ref. 38/H20850, the dc voltage is of the same order of magnitude /H1101520 nV. FIG. 3. /H20849a/H20850Schematic of the device. On the lower side, through the shorted end of a coplanar strip a current Irfgenerates an rf magnetic field, denote by the arrows. The Py strip in the centerproduces a dc voltage /H9004V=V +−V−. H denotes the static magnetic field applied along the strip. /H20849b/H20850Scanning electron microscope pic- tures of the central part of the devices.ELECTRICAL DETECTION OF SPIN PUMPING: DC … PHYSICAL REVIEW B 78, 064423 /H208492008 /H20850 064423-3III. EXPERIMENTAL PROCEDURES Our detection technique is based on the asymmetry in the spin pumping effect between two contacts in a device geom-etry where a ferromagnet is contacted with two normal-metalelectrodes. The largest such asymmetry is obtained when oneof the metal electrodes is a spin sink such as Pt, for which weexpect a negligible contribution, while the other has a smallspin-flip relaxation rate, such as Al. Therefore, we anticipatea net dc voltage across a Py strip contacted by Pt and Alelectrodes when the ferromagnet is in resonance. Additional, we studied control devices where the Py strip is contacted by the same material Pt and Al. For these de-vices we expected no signal because: /H20849i/H20850The voltages for identical interfaces are the same and their contribution to /H9004V cancels. /H20849ii/H20850Pt has a very short spin-diffusion length, result- ing in a small spin accumulation, a small backflow and thusa lower signal. Figure 3/H20849a/H20850shows a schematic illustration of the lateral devices used in the present study. The central part of thedevice is a ferromagnetic strip of permalloy /H20849Ni 80Fe20,o rP y /H20850 connected at both ends to normal metals, Al and/or Pt /H20849V−and V+contacts /H20850. The devices are fabricated on a Si /SiO 2 substrate using e-beam lithography, material deposition, and liftoff. A 25 nm thick Py strip with 0.3 /H110033/H9262m2lateral size was e-beam deposited in a base pressure of 1 /H1100310−7mBar. Prior to deposition of the 30 nm thick Al or/and Pt contactlayers, the Py surface was cleaned by Ar ion milling, using an acceleration voltage of 500 V with a current of 10 mA for30 s, removing the oxide and a few nanometers of Py mate-rial to ensure transparent contacts. We measured in total 17devices /H20849this includes 4 devices with a modified contact ge- ometry, described later in the paper /H20850. The different contact material configurations are shown in Fig. 3/H20849b/H20850. Figure 4/H20849a/H20850illustrates the experimental setup for the mea- surements. We measured the dc voltage generated betweentheV +,V−electrodes as a function of a slowly sweeping magnetic field /H20849H/H20850applied along the Py strip, while applying an rf magnetic field /H20849hrf/H20850perpendicular to the strip. We have recently shown that a submicron Py strip can be driven into the uniform precession ferromagnetic resonancemode 21,26by using a small perpendicular rf magnetic field created with an on-chip coplanar strip waveguide39/H20849CSW /H20850 positioned close to Py strip /H20849similar geometry as shown in Fig. 3/H20850. For the applied rf power 9 dBm, an rf current of /H1101512 mA rms passes through the shorted end of the coplanar strip waveguide and creates an rf magnetic field with an am-plitude of h rf/H110151.6 mT at the location of the Py strip.40We confirmed with anisotropic magnetoresistance /H20849AMR /H20850 measurements21that on resonance the precession cone angle is/H110155°. In order to reduce the background /H20849amplifier /H20850dc offset and noise we adopted a lock-in microwave frequency modu-lation technique. During a measurement where the staticmagnetic field is swept from −400 to +400 mT, the rf field isperiodically switched between two different frequencies and FIG. 4. Schematic of the experimental setup and of the micro- wave frequency modulation method. /H20849a/H20850A TTL signal at a reference frequency fref/H2084917 Hz /H20850, generated by a lock-in amplifier /H20849master device /H20850, is first fed into a frequency doubler. Then, the TTL at 2/H11003frefis fed into a CW microwave generator. At each TTL input, the CW generator provides frequency hopping of the rf currentswitching between f highandflowatfref. The dc voltages produced by the device are amplified and detected by the lock-in amplifier as adifference /H9004V=V/H20849f high/H20850−V/H20849flow/H20850./H20849b/H20850At the bottom, the resonant frequency dependence on the static magnetic field is shown. Next,the diagrams of the dc voltage vs static magnetic field correspond-ing to the resonance at high and low frequencies. On top, the mea-sured difference in dc voltage between the two frequencies,/H9004V=V/H20849f high/H20850−V/H20849flow/H20850is plotted.FIG. 5. The dc voltage /H9004Vgenerated by a Pt/Py/Al device in response to the rf magnetic field plotted as a function of the staticmagnetic field. The frequencies of the rf field are as shown. Thepeaks /H20849dips /H20850correspond to resonance at f high/H20849flow/H20850. The data are offset vertically for clarity /H20849Ref. 24/H20850.COSTACHE et al. PHYSICAL REVIEW B 78, 064423 /H208492008 /H20850 064423-4we measured the difference in dc voltage between the two frequencies /H9004V=V/H20849fhigh/H20850−V/H20849flow/H20850using a lock-in amplifier. For all the measurements the lock-in frequency is 17 Hz andthe difference between the two microwave frequencies is 5GHz. A diagram of the measurement method is shown inFig.4/H20849b/H20850. IV . RESULTS AND DISCUSSION A. Detection of Spin pumping Here, we describe precise, room-temperature measure- ments of the dc voltage across a Py strip contacted by Pt andAl electrodes when the ferromagnet is in resonance. Figure 5 shows the electric potential difference /H9004Vfrom a Pt/Py/Al device. Sweeping the static magnetic field in a range −400 to+400 mT, a peak and a diplike signals are observed at bothpositive and negative values of the static field. Since wemeasured the difference between two frequencies, the peakcorresponds to the high resonant frequency /H20849f high/H20850and the dip to the low resonant frequency /H20849flow/H20850, see Fig. 4/H20849b/H20850. For the opposite sweep direction, the traces are mirror image. Wemeasured 8 devices with contact material Pt/Py/Al. The mea-sured resonances are all in the range +100 to +250 nV. No-tably, the dc voltages are all of the same sign /H20849always a peak forf high/H20850, meaning that for Pt/Py/Al devices, the Al contact atresonance is always more negative than the Pt contact. First, we look at the peak/dip position dependence of the rf frequency. In Fig. 6/H20849a/H20850, the dc voltage in gray scale is plotted versus static field for different high /H20849low /H20850frequencies of the rf field. Figure 6/H20849b/H20850shows the fitting of the peak/dip position dependence of the rf field frequency /H20849dotted curve /H20850 using Kittel’s equation for a small-angle precession of a thin-strip ferromagnet: 42 f=/H9253/H92620 2/H9266/H20881/H20849H+N/H20648MS/H20850/H20849H+N/H11036MS/H20850/H20849 11/H20850 where /H9253is the gyromagnetic ratio, N/H20648,N/H11036are in-plane /H20849along the width of the strip /H20850and out-of-plane demagnetization fac- tors and MSis the saturation magnetization. The fit to this equation /H20851see Fig. 6/H20849b/H20850/H20852gives/H9253=176 GHz /T, and N/H20648/H92620MS =60 mT, N/H11036/H92620MS=930 mT, consistent with earlier reports.26,43The fit confirms that the dc voltage appears at the uniform ferromagnetic resonance mode of the Py strip. Themeasured amplitude of the dc voltage as a function of thesquare of the applied rf current, at 13 GHz and 139 mT isshown in Fig. 6/H20849c/H20850. Here, we observe a linear dependence on the square of the rf current, consistent with the prediction ofthe spin pumping theory, see Eqs. /H208495/H20850and /H208499/H20850. FIG. 6. /H20849a/H20850Gray scale plot of the dc voltage /H9004V, measured function of static field for different high /H20849low /H20850frequencies of the rf field from the Pt/Py/Al device /H20849Ref. 41/H20850. The dark /H20849light /H20850curves denote resonance at flow/H20849fhigh/H20850./H20849b/H20850The static magnetic-field dependence of the resonance frequency of the Py strip /H20849dots /H20850. The curve is a fit to Eq. /H2084911/H20850./H20849c/H20850The amplitude of the dc voltage from a Al/Py/Al device as a function of the square of the rf current, at 13 GHz and 139 mT /H20849dots /H20850. The line shows a linear fit /H20849Ref. 24/H20850.ELECTRICAL DETECTION OF SPIN PUMPING: DC … PHYSICAL REVIEW B 78, 064423 /H208492008 /H20850 064423-5Further, we studied several control devices where both electrodes are of the same nonmagnetic material, Al /H208495 de- vices /H20850or Pt /H208494 devices /H20850. Here we expected no signal because of the reasons mentioned above. The results from Al/Py/Aldevices show smaller signals than Pt/Py/Al devices, with alarge scatter in amplitude and both with positive and nega-tive sign for the resonance at f high. Values for the 5 devices are −100 /H20851shown in Fig. 7/H20849a/H20850/H20852, +25, +30, +75, and +110 nV. In contrast, all 4 Pt/Py/Pt devices exhibit only weak signalsless than 20 nV, with resonance signals barely visible, as inFig.7/H20849b/H20850. The overall values of the dc voltages as a function of different contact materials are shown in Fig. 8. We summa- rize the results as follow: First, the Pt/Py/Al devices have signals that are always positive, on average 150 nV, and with a scatter comparable inamplitude to that of Al/Py/Al devices around zero. This scat- ter in the signal amplitude can be due to: /H20849i/H20850samples varia- tion, due to different interface quality, not identical contacts/H20851i.e., different overlap between the Nelectrodes and the Py strip, see Fig. 3/H20849b/H20850/H20852and a small variation in distance between the Py strip and the CSW; /H20849ii/H20850different rf power at the end of CSW, due to different positions and contact resistance of themicrowave probe on the CSW. These characteristics are dif-ficult to estimate for each device. Second, we attribute the signals from Al/Py/Al devices to the asymmetry of the two contacts, possibly caused by smallvariation of the interfaces and in the contact geometry. De-pending on the asymmetry, the signals therefore have a scat-ter around zero. Third, in the Pt/Py/Pt devices, independent of possible asymmetry, we expected and found very small signals.Therefore, we conclude that the signals measured with thePt/Py/Al devices arise mainly from the Al/Py interface. B. Spin pumping vs rectification effects We now discuss the rectification effects. As we have shown recently,21due to capacitive and inductive coupling between the CSW and the Py strip, rf currents /H20851I/H20849t/H20850 =I0cos/H9275t/H20852can be induced in the detection circuit. The rf currents in combination with a time-dependent AMR /H20851R/H20849t/H20850 /H11229/H9004Rcos/H20849/H9275t+/H9272/H20850/H20852can give a dc effect due to rectification effect /H20849Vdc/H11229/H20855I·R/H20856t/H20850. However, for rectification to occur, the resistance R/H20849t/H20850must have first harmonic components, which is not true assuming circular or even elliptical precession ofthe magnetization. There are two ways to have first harmonic components: /H20849i/H20850an offset angle between the applied field and the long axis of the Py strip, namely bulk rectification effect; /H20849ii/H20850an offset angle between the circulating rf currents and the magnetiza-tion. When the rf circulating currents enter and leave thestrip, they can pass through a large angle relative to the mag-netization. Asymmetry in the entry and exit paths, due todifferent conductivities of the two contacts, in combinationwith the time-dependent AMR, can lead to a rectificationeffect at the contacts, which we call the contact rectificationeffect. 44 Even if we can accurately control the offset angle between the applied field and the Py strip, we cannot rule out thecontacts effect that may also contribute to the data presentedin the previous section, see Fig. 5. A small contribution from rectification effects on top of spin pumping signal can alsoexplain the asymmetric peak/dip shape which does not havea Lorentzian shape as expected from Eq. /H208495/H20850. In order to study these effects we prepared a new set of 4 devices verysimilar to the one shown in Fig. 3/H20849b/H20850, but now with contacts at the ends of the Py strip, extending along the long axis ofthe strip, see Fig. 9/H20849d/H20850for a SEM image. In this geometry, the induced rf current flows through the contacts predominantlyparallel to the magnetization direction. This suppresses thepossible contribution to the measured dc voltages from arectification effect at the contacts. 45 We first align the devices with Py strip parallel to the applied field and measure the dc voltage function of the field,FIG. 7. The dc voltage /H9004Vgenerated across the /H20849a/H20850Al/Py/Al and /H20849b/H20850Pt/Py/Pt devices as a function of the static magnetic field. The frequencies of the rf field are as shown /H20849Ref. 24/H20850. FIG. 8. Overall distribution of the amplitude of the dc voltages as a function of different contact materials. Different symbols rep-resent different batches of samples. This includes four devices withlongitudinal electrode device geometry, indicated by symbol /H20849/H17034/H20850 and discussed in Sec. IV B.COSTACHE et al. PHYSICAL REVIEW B 78, 064423 /H208492008 /H20850 064423-6as explained above. The measurements are shown in Figs. 9/H20849a/H20850and9/H20849b/H20850for Pt/Py/Al and Pt/Py/Pt configurations. These results are consistent with the above discussion, as the Pt/Py/Al devices show signals equal to the average value mea-sured in the previous device geometry, while the Pt/Py/Ptdevices show no signal as expected. Figure 9/H20849c/H20850shows a comparison between the voltages of two Al/Py/Pt deviceswith longitudinal /H20849bold line /H20850and transverse /H20849normal line /H20850 contacts geometry, at f high=18.5 GHz, flow=13.5 GHz. Par- ticularly, devices with the longitudinal contacts exhibit, inaddition to the main peak, a series of peaks at higher fields.An exact explanation of these observations is not yet clear.We assume these are related to end-mode resonances, sincein this contacts geometry we are sensitive also to the mag-netic structure of the ends of the Py strip. Moreover, wefound no significant difference in the measured dc voltagesbetween these two contacts geometries, Figs. 3/H20849b/H20850and9/H20849d/H20850. In order to confirm the above assumptions and to quantify the bulk rectification effect, we misaligned the direction ofthe static field with respect to the Py strip long axis by 5°/H20849and 10° /H20850and measured the voltage at f high=18 GHz, flow=13 GHz. The results for Pt/Py/Al and Pt/Py/Pt devices are shown in Fig. 10. Note that we see significant contributions from the bulk rectification effect only at offset angles largerthan 5°. This rules out that small offset angles which may bepresent in the other geometry caused significant effects in theresults, at most 10–20 nV. In the following, the above results are analyzed taking into account that the voltages measured in Pt/Py/Al devicesare due to two effects: /H20849i/H20850spin pumping, and /H20849ii/H20850bulk recti- fication effect for a nonzero offset angle. Of these two effectsonly the bulk rectification depends on the sign of the offsetangle. This means that if we take the sum of the voltagesmeasured at + /−10°, V/H2084910°/H20850+V/H20849−10° /H20850, we obtain two times the contribution from the spin pumping effect with a Lorent-zian peak shape. And in contrast we expect no signal if wedo the same operation for Pt/Py/Pt devices. These results areshown in Fig. 11/H20849a/H20850. On the other hand, if we subtract, V/H2084910°/H20850−V/H20849−10° /H20850,w e obtain two times the contribution from the bulk rectificationeffect. Figure 11/H20849b/H20850shows the resulting data, which are prac- tically the same for both devices, Pt/Py/Al and Pt/Py/Pt. FIG. 9. The dc voltage generated by /H20849a/H20850Al/Py/Pt and /H20849b/H20850Pt/Py/Pt devices for a device geometry shown in part /H20849d/H20850./H20849c/H20850The comparison of the signals for the two Pt/Py/Al devices, one with longitudinal electrode device geometry /H20849bold line /H20850and other with transverse electrode device geometry. /H20849d/H20850SEM picture of an longitudinal electrode geometry, Pt/Py/Al device.ELECTRICAL DETECTION OF SPIN PUMPING: DC … PHYSICAL REVIEW B 78, 064423 /H208492008 /H20850 064423-7Such a result is expected because the bulk rectification effect does not depend on the contact material. We now consider a quantitative assessment of possible contribution to the measured signal from rectification effects,namely bulk and contact rectification effect. Note, the con-tact rectification effect in principle should cancel for equiva-lent contacts. Both rectification effects depend primarily onthe rf circulating current, which varies from device to device, depending on position of the picoprobe and rf current fre-quency. In a similar device geometry 21we have estimated the rf currents to be up to 30 /H9262A. With this value we obtain the following. /H20849i/H20850Bulk: A rough estimate of an upper bound contribution, assuming an offset angle of 2°, gives 15 nV. The data shownin Fig. 10/H20849b/H20850/H20849for zero degree /H20850is less than this value. /H20849ii/H20850Contact: This contribution, which is present only in devices with the transverse electrode geometry, is estimatedat 30 nV. 44 The sum of these contributions can have any value be- tween −45 and 45 nV, and thus can add or subtract to theaverage spin pumping signal /H20849150 nV /H20850, given the rise to extra scatter in the data, see Fig. 8. In addition to signal magnitude analysis, it is also impor- tant to discuss the difference in signal shape due to theseeffects. It should be noted that each of the rectification ef-fects discussed above can have a signal shape which can beany combination of absorptive and dispersive peak shape. Incontrast, the spin pumping signal is only absorptive with aLorentzian shape. V . CONCLUSION We have presented dc voltage caused by the spin pumping effect, across the interface between Al and Py at ferromag-netic resonance. We found that the devices where the Alcontact has been replaced by Pt show a voltage close to zero,in good agreement with theory. We observed a quadratic de-pendence of dc voltage function of precession cone angle, inagreement with the discussed theory. Theoretical predictedspin pumping voltage /H2084920 nV /H20850is less than the values ob- served experimentally /H20849in average 150 nV /H20850. This underesti- mation might arise from the fact that the model does notconsider device geometry or disorder at the interface, and assumes a homogeneous magnetization in the ferromagnet. Furthermore, to rule out a possible contribution from rec- tification effects to the measured signal, we have studieddevices with different electrode geometries. We observedthat for a nonzero offset angle, between the static field andthe Py strip, the measured voltages are due to two differenteffects, namely spin pumping and rectification effects. Byusing appropriate device geometry, these effects can be quan-tified and for the zero offset angle the rectification effects areminimized. This work demonstrates a means of directly converting magnetization dynamics of a single nanomagnet into an elec-trical signal which can open new opportunities for techno-logical applications. ACKNOWLEDGMENTS We thank M. Sladkov, J. Grollier, A. Slachter, G. Visa- nescu, and J. Jungmann for discussion and assistance in thisproject and B. Wolfs and S. Bakker for technical support.This work was supported by the Dutch Foundation for Fun-damental Research on Matter /H20849FOM /H20850, the Netherlands Orga- nization for Scientific Research /H20849NWO /H20850, the Nanoned, and the EU project DYNAMAX.FIG. 10. The dc voltage at fhigh=18 GHz, flow=13 GHz for /H20849a/H20850 Pt/Py/Al and /H20849b/H20850Pt/Py/Pt devices for different angles between the static field and the long axis of the strip. FIG. 11. To isolate the contribution from different effects we performed: /H20849a/H20850The sum between the voltage at offset angle of 10° and −10°, V/H2084910°/H20850+V/H20849−10° /H20850, vs static field for Pt/Py/Al and Pt/Py/Pt devices. The result represents two times contribution from spinpumping effect. /H20849b/H20850The difference between the voltages, V/H2084910°/H20850 −V/H20849−10° /H20850, which represents two times contribution from bulk rec- tification effect, as explained in the text.COSTACHE et al. PHYSICAL REVIEW B 78, 064423 /H208492008 /H20850 064423-8*costache@mit.edu 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850 425, 380 /H208492003 /H20850. 4A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Mae- hara, K. Tsunekawa, D. D. Djayaprawira, N. 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PhysRevB.93.134420.pdf

PHYSICAL REVIEW B 93, 134420 (2016) Reconfigurable wave band structure of an artificial square ice Ezio Iacocca,1,2,3,*Sebastian Gliga,4,5Robert L. Stamps,6and Olle Heinonen7,8 1Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA 2Department of Physics, Division for Theoretical Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden 3Physics Department, University of Gothenburg, 412 96 Gothenburg, Sweden 4Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland 5Laboratory for Micro- and Nanotechnology, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland 6School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom 7Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA 8Northwestern-Argonne Institute for Science and Engineering, Evanston, Illinois 60208, USA (Received 14 October 2015; revised manuscript received 6 March 2016; published 18 April 2016) Artificial square ices are structures composed of magnetic nanoelements arranged on the sites of a two- dimensional square lattice, such that there are four interacting magnetic elements at each vertex, leading togeometrical frustration. Using a semianalytical approach, we show that square ices exhibit a rich spin-waveband structure that is tunable both by external magnetic fields and the magnetization configuration of individualelements. Internal degrees of freedom can give rise to equilibrium states with bent magnetization at the elementedges leading to characteristic excitations; in the presence of magnetostatic interactions these form separatebands analogous to impurity bands in semiconductors. Full-scale micromagnetic simulations corroborate oursemianalytical approach. Our results show that artificial square ices can be viewed as reconfigurable and tunablemagnonic crystals that can be used as metamaterials for spin-wave-based applications at the nanoscale. DOI: 10.1103/PhysRevB.93.134420 I. INTRODUCTION Spin waves, or magnons, are fundamental excitations in magnetic thin films and nanostructures. Because of theirpotential applications in information technology [ 1–3] and computation [ 4], means to control magnon dispersion and band gap have been studied intensively over the past fewdecades. The term magnonics has been coined to describe this field of study [ 5,6]. One pathway to control magnon dispersions is to construct magnonic crystals [ 7,8] that are metamaterials with a spatial modulation of the magneticproperties on length scales comparable to relevant magnonicwavelengths [ 9–11]. Patterned thin magnetic films [ 12,13] or topographically modulated thin films have been used tomanipulate the magnon spectra [ 14]. This approach is similar to superlattices in photonics and, fundamentally, to the crystalstructure of semiconductors. A paradigm that is the focus ofrecent investigation consists of actively modifying the bandstructure of magnonic crystals [ 15]. This has been achieved to date by use of Meander-type structures [ 16] and, more recently, via heating [ 17] in one-dimensional ferromagnets. Artificial spin ices [ 18–20] are another class of structures based on an organized array of nanosized magnetic elementsthat have been shown to support a wealth of static, dynamic,and emergent magnetic phenomena [ 20–22]. Artificial spin ices are geometrically frustrated : the geometry of the elements and the lattice is such that all interaction energies cannot besimultaneously minimized. Examples of artificial spin ices arethe square ice [ 18] and the kagome ice [ 23]. The square ice is composed of magnetic stadium-shaped nanoislands positionedon the sites of a two-dimensional square lattice with latticeconstant d,F i g . 1(a), and obeys the “ice rules” in which *ezio.iacocca@colorado.edulow-energy states are characterized by the magnetization in two nanoislands pointing into a vertex and out of the vertex inthe two other nanoislands. Dynamically, correlated excitationsare supported in spin ices because of the magnetostatic inter-actions between nanoislands [ 24]. Because of their intrinsic periodicity and wealth of static states, artificial spin ices offerinteresting opportunities as programmable magnonic crystalsto control the magnon dispersion and band gap [ 20]. The resonant mode spectrum of square ices has been studied numerically by means of micromagnetic simulations,demonstrating the observable effects of magnetic defects [ 24]. More recently, a detailed numerical study has shown that edgemodes arising from the internal degrees of freedom of themagnetization equally have observable consequences in theresonant spectrum in sufficiently thick nanoislands [ 25]. In fact, edge modes efficiently couple neighboring nanoislands,influencing the collective oscillations [ 20]. This is reminiscent of impurity states in semiconductors that locally modify theenergy landscape and give rise to shallow electronic bands[26]. Recent experimental results have explored the excitation spectrum of artificial spin ices [ 27–29], but the existence and dependencies of the band diagram in square ices has notbeen explored to date. To close the gap between the fieldsmagnonics and artificial spin ices, we examine square icesfrom the perspective of magnonics, including bands arisingfrom the edge modes as well as the bulk modes. In this paper, we study long-range dipolar-mediated two- dimensional magnon dispersion in square ices in the spirit ofa tight-binding model. In contrast to similar procedures onsimpler structures [ 30,31], we account for the internal degrees of freedom resulting from edge modes in the nanoislands.Consequently, we are able to calculate the magnon dispersionas a function of local equilibrium states as well as its fieldtunability, including edge mode bands. Our semianalyticalapproach provides enough degrees of freedom to qualitatively 2469-9950/2016/93(13)/134420(8) 134420-1 ©2016 American Physical SocietyIACOCCA, GLIGA, STAMPS, AND HEINONEN PHYSICAL REVIEW B 93, 134420 (2016) FIG. 1. (a) Square ice lattice with lattice constant dand with magnetic stadia of width w, length l, and thickness t.T h es t a b l e magnetization directions (black arrows) of the magnetic elements in a unit cell are shown for the (b) ground (vortex) and (c) remanent states (the gray-colored stadia are not part of the unit cell and are shown here for clarity). Edge states have two stable configurations as(d)Cand (e) Sstates. estimate the band structure of an extended square ice lattice while being computationally tractable. We focus on the small-amplitude excitations in two experi- mentally accessible configurations of a square ice, namely thevortex and remanent states, Figs. 1(b) and 1(c), respectively. The vortex state is the ground state of the system, achieved bythermal relaxation [ 32], and the remanent state can be obtained by saturating the system in an external field along the ( ˆx,ˆy) direction, and then slowly removing the external field, lettingthe system relax. In each configuration, the magnetization canbend close to the nanoisland edges [ 25], providing a local “impurity.” In square ices, two stable edge configurationssatisfy the minimization of dipolar fields in the ground state,resulting in C and S states [ 33–35], Figs. 1(d) and1(e). II. ANALYTICAL FORMALISM The small-amplitude dynamics in square ices can be approached semianalytically using a Hamiltonian formalism[36]. The same approach has been used and shown to be accurate in many dynamical regimes to date [ 37–43]. In this formalism, the Landau-Lifshitz equation of motion describingconservative magnetization dynamics is cast as a function ofa complex amplitude a, using a Holstein-Primakoff transfor- mation. By expanding the resulting equation in Taylor series,the linear dynamics for an ensemble of complex amplitudes a can be generally expressed (see Appendix A)a s da dt=−id da∗A†HA=−id da∗A†/parenleftbigg H(1,1)H(1,2) H(2,1)H(2,2)/parenrightbigg A, (1) where the dagger denotes the complex transpose, A is an array of 2 ncomplex amplitudes AT=[aT,a†]= [a1,..., a n,a∗ 1,..., a∗ n] and His the 2 n×2nHamiltonian. The right-hand side of Eq. ( 1) includes terms up to second order in a, corresponding to linear excitations. Beacuse of the lattice perodicity, propagating waves are Bloch waves with atime dependence gvien by a→ae iωt. This allows us to reduceEq. ( 1) to an eigenvalue problem by means of Colpa’s grand dynamical matrix [ 44] ωψ=/parenleftbigg H(1,2)H(2,2) H(1,1)H(2,1)/parenrightbigg ψ, (2) from which we obtain the eigenvalues ω, and the eigenvectors ψ. Due to the complex conjugate definition of A, we observe thatH(1,1)=H(2,2)andH(1,2)=H(2,1), leading to conjugate eigenvalues in Eq. ( 2). The Hamiltonian is related to the magnetic field /vectorHviaH= −γδW/ (2MS), where δW=−/integraltext/vectorH(/vectorM)·d/vectorMis the energy functional, γis the gyromagnetic ratio, /vectorMis the magnetization vector, and MS=| |/vectorM||is the saturation magnetization. We consider field contributions from shape anisotropy, dipolarinteractions, and intra-element exchange as well as an externalapplied field. Each field contribution can be reduced to aHamiltonian matrix as detailed in Appendix B. Of particular importance are the dipolar interactions, which are the onlysource of inter-element coupling in our framework and theconcomitant magnon dispersion. The dipolar energy betweena nanoisland jin cell τand all the other nanoislands kin cells τ /primecan be expressed as Hd=−V 4π/summationdisplay k,τ/prime/bracketleftbigg3(/vectorRjk,ττ/prime·/vectorMj,τ/prime)(/vectorRjk,ττ/prime·/vectorMj,τ) (/vectorRjk,ττ/prime)5 −/vectorMj,τ/prime·/vectorMj,τ (/vectorRjk,ττ/prime)3/bracketrightbigg , (3) where Vis the volume of the magnetic element and /vectorRjk,ττ/prime is the translation vector between the nanoisland jin cell τ and the nanoisland kin cell τ/prime. Considering the Bloch wave /vectorMj,τ=/vectorMj,τ/primeei/vectorq/vectorRjk, where /vectorqis the wave vector, it is possible to recast Eq. ( 3) for the unit cell in terms of the lattice ( Sβ where β=ˆx,ˆy,ˆz) and cross-direction ( Sc) summations. As an example, the resulting Hamiltonian matrices for the groundstate in the absence of exchange interactions are H(1,1) d=⎛ ⎜⎜⎜⎝S11 xS12 cS13 xS14 c S21 cS22 yS23 cS24 y S31 xS23 cS33 xS34 c S41 cS24 yS34 cS44 y⎞ ⎟⎟⎟⎠−ˆSz, (4a) H(1,2) d=D+⎛ ⎜⎜⎜⎝0S12 cS13 xS14 c S21 c 0S23 cS24 y S31 xS32 c 0S34 c S41 cS42 yS43 c 0⎞ ⎟⎟⎟⎠+ˆS z, (4b) where Dis a diagonal matrix containing inter-island interac- tions (the expressions for Dand the lattice summations are shown in the Appendix B). The reduction of the dipolar field to Hamiltonian matrices is a key result of this work. The magnon dispersion can be numerically calculated by solving the eigenvalue problem of Eq. ( 2). We consider a square ice composed of Permalloy stadia with dimensions280 nm ×120 nm ×20 nm, saturation magnetization M S= 770 kA /m, and center-to-center separation of d=395 nm. Exchange is implemented as an additional degree of freedomin a nanoisland divided by three equidistant spins coupled 134420-2RECONFIGURABLE W A VE BAND STRUCTURE OF AN . . . PHYSICAL REVIEW B 93, 134420 (2016) FIG. 2. Band structure of the vortex state. The insets show the magnetization vector configuration of the unit cell for each band, showcasing their excitation symmetry. by the constant J=0.016, which parametrizes the exchange in Permalloy J=cA/2, where c=0.33 nm is the lattice constant and A=10 pJ/m is the exchange stiffness (see Appendix C). This approximation for the exchange interaction is applicable for the low-energy sector of the magnon bands, asdemonstrated below by the good quantitative agreement withfull-scale micromagnetic simulations. III. BAND STRUCTURE IN SQUARE ICES It is instructive to consider first the band structure neglecting internal degrees of freedom, or “macrospin” approximation.A typical band structure for the macrospin vortex state isshown in Fig. 2. There are four bands consistent with the available degrees of freedom in the system, one for each island.From the corresponding eigenvectors, it is possible to identifythe location and symmetry of each mode. A snapshot of themagnetic configurations at the /Gamma1point for each band (labeled fromM1t oM4) are shown above Fig. 2. We notice that M1 has pair of islands in phase and a phase difference of ±π between each pair, whereas M4 represents a mode with all islands excited in phase. Furthermore, M1(M4) has positive (negative) group velocity. M2 and M3 are close in energy and consist of modes with a pairwise phase difference of±π/2. Note that the pairwise difference make these bands non-degenerate, resulting in anticrossings close to the /Gamma1and Mpoints. These latter two modes form narrow bands that separate away form the /Gamma1point, and establish a band gap reaching ≈195 MHz between the /Gamma1andXpoints of M1 and M2, respectively. Bands effectively touch at the /Gamma1andM points. However, we did not observe band inversion in any calculation. We now include exchange interactions in our framework. By dividing each magnetic island into three equidistant spins,we now have access to 12 bands. In the ground state, threeconfigurations are stable: homogeneous or onion [ 25],C, andSstates. The corresponding band diagrams are shown in Fig. 3. The additional degrees of freedom give rise to lower frequency bands identified as edge modes (black dashedlines), also showing anticrossing behavior. We observe thatthe bulk modes (blue lines) maintain their qualitative features. FIG. 3. Band diagram for the vortex state in (a) onion, (b) C, and (c) Sstates. The bulk (edge) modes are depicted in blue (dashed black) lines. The schematic of each static configuration is also shownfor each case. However, the band gaps are enhanced due to the additional energy incorporated into the system. Furthermore, the par-ticular magnetic configuration quantitatively modifies theband diagram, indicating that edge bending can be comparedto impurity states in semiconductor materials. Because atransition between CandSstates can be induced by, e.g., temperature [ 25], this can be used as another avenue to program the magnonic response of the square ice. In the remanent state,the unit cell is composed of two magnetic islands, Fig. 1(c). The band diagrams for a macrospin and stable onion and S configurations are shown in Fig. 4, exhibiting similar features as discussed above. FIG. 4. Band diagram for the remanent state in (a) macrospin, (b) onion, and (c) Sstates. The bulk (edge) modes are depicted in blue (dashed black) lines. The inset shows the magnetization vector configuration of the unit cell for each band. The schematic staticconfiguration is also shown for each case. 134420-3IACOCCA, GLIGA, STAMPS, AND HEINONEN PHYSICAL REVIEW B 93, 134420 (2016) IV . INFLUENCE OF AN IN-PLANE EXTERNAL FIELD We now explore the effect of an applied field /vectorHeon the square ice. We consider a feasible experimental scenarioof an in-plane field along the ˆxdirection and detection of coherent excitations (at the /Gamma1point) by means of resonance measurements (the effect of the external field angle is shown in Appendix D). Note that in our framework, the stable magnetization direction of the magnetic nanoislands is set andassumed ap r i o r i , i.e., only small amplitude variations are accessible. In fact, large fields induce imaginary eigenvalues,denoting decaying modes and thus the breakdown of ourmodel. We study the effect of field magnitudes between 0 < |/vectorH e|<100 Oe which maintains real eigenvalues. The results obtained for both vortex and remanent states under macrospinapproximation are shown in Figs. 5(a) and 5(b). In the case of the vortex state, we observe that the coherent modes, M1 andM4, have positive and negative tunabilities, respectively, whereas M2 and M3 exhibit only slight tunability. In the case of the remanent state we observe either a positive ornegligible tunability. The strongly tunable modes can be tracedto those magnetic elements parallel to the applied field. This isalso consistent with the Landau-Lifshitz equation predictinga blueshift (redshift) of frequencies when the internal field increases (decreases). The modes with negligible tunabilitycorrespond to magnetic elements perpendicular to the field. Byconsidering edge bending, a richer behavior for the tunabilityof both the vortex and remanent states is obtained, Figs. 5(c) and 5(d). For both the vortex in an onion state and the remanent Sstate, we observe similar tunabilities for the bulk and low-frequency edge modes. In all cases, the slope of eachband is generally different, leading to band crossings, andimplying that the bandgaps in square ices can be manipulatedby an applied magnetic field. FIG. 5. Magnon frequencies at the /Gamma1point as a function of an external field applied along the ˆxaxis for the (a) macrospin vortex, (b) macrospin remanent, (c) onion vortex, and (d) remanent Sstates. The bulk (edge) modes are depicted in blue (dashed black) lines. Thered dots are obtained from micromagnetic simulations.Full-scale micromagnetic simulations were performed for comparison with the semianalytical model. We used a compu-tational system containing eight islands and imposing periodicboundary conditions consistent with the geometry describedabove. The system was discretized into a mesh of size1.25 nm ×1.25 nm ×5 nm and the magnetostatic interactions calculated using fast Fourier transform with periodic boundaryconditions applied in the plane. The system was first set in anapproximate local equilibrium state with each island homo-geneously magnetized along their easy axis, approximatingthe vortex or remanent states. These initial configurationswere then relaxed by integrating the Landau-Lifshitz-Gilbertequation for the micromagnetic spins using a dimensionlessdamping of α=0.25. The magnetization of the islands in a vortex (remanent) states then relaxed into an onion (S) state. The relaxed configuration was then subjected to a uniform external field pulse of magnitude approximately 10 Oe in the(−ˆx,−ˆy) direction for 50 ps and the full Landau-Lifshitz- Gilbert equation integrated in time steps of 0.25 ps for 10 nsusing a damping of α=0.02. Magnetization configurations were sampled every 25 ps; the average magnetization at eachtime slice was Fourier transformed to yield 1D spectra of themagnetization components as functions of frequency, and thesequence of 2D time slices was Fourier transformed to yieldfull 2D amplitude and phase maps for each frequency. Thesecalculations were performed for constant external fields of 0,50, and 100 Oe along the (1,0) direction for the vortex andremanent states. The results are shown as red circles in Fig. 5(note that the micromagnetic modeling only returns modes that areeven in the unit cell because the exciting field is uniform,while the semianalytical model captures all modes irrespectiveof symmetry). For the vortex state, a good agreement forthe bulk modes is obtained from the macrospin model.Further comparison with the extended semianalytical modelalso shows excellent agreement with the low-frequency edgemodes. For the remanent state, the macrospin model yieldsa good qualitative agreement with the micromagnetic results.A three-spin S-state model also yields good agreement with the micromagnetic low-frequency modes, especially in viewof the simplistic treatment in the three-spin model of thesmooth static equilibrium magnetization in the micromagneticmodel. We remark that in both real and micromagnetically modeled nanoislands, there are many higher-order modes, beyond whatcan be described by the three-spin model considered here,because of the large number of internal degrees of freedom.Such higher-order modes are characterized by multiple internalnodal lines of the magnetization eigenmodes. Therefore, the magnetostatic fields emanating from such modes decay rather quickly in space. This results in a weak coupling betweendifferent islands, so the magnonic bands arising from suchmodes are nondispersive with no phase or group velocity, andare not of interest here. It is also noteworthy that a strongvariation of the islands’ aspect ratio can significantly affectthe excited frequencies, i.e., in the nanowire and circular-dotlimits. Moreover, we expect the thickness to play an importantrole in the ultrathin film regime, where the anisotropy becomesperpendicular or in thicker films, where the vortex state insideeach element is favored. 134420-4RECONFIGURABLE W A VE BAND STRUCTURE OF AN . . . PHYSICAL REVIEW B 93, 134420 (2016) V . CONCLUSIONS In summary, we have calculated the magnon band structure and the mode tunability at the /Gamma1point for a square ice in two equilibrium states, the vortex (or ground) state, andthe remanent state, using a model that includes internal degrees of freedom of the magnetization in the nanoislands as well as edge bending. The good quantitative agreementwith micromagnetic simulations confirms the accuracy ofthe small-amplitude semianalytical model while avoiding thecomputational limitations intrinsic to fully three-dimensionalmicromagnetic simulations. These results show that themagnon spectra, and therefore group and phase velocities aswell as band gap, can be manipulated by external fields. Inparticular, the edge modes give rise to separate magnon bandsallowing for a larger parameter space in terms of magnoncontrol. This suggests that square ices can be consideredmetamaterials for spin waves. In addition, the square ice is inprinciple reconfigurable in that the magnetization in individualislands can be changed by the application of external fields(e.g., from vortex to remanent state) or temperature, or byusing more sophisticated techniques such as using spin torqueby patterning nanocontacts on the elements or making theelements part of magnetic tunnel junctions. This opens up thepossibility of two-dimensional reprogrammable magnoniccrystals consisting of an artificial square spin ice. ACKNOWLEDGMENTS E.I. acknowledges support from the Swedish Research Council, Reg. No. 637-2014-6863. The work by O.H. wasfunded by the Department of Energy Office of Science,Materials Sciences and Engineering Division. E.I. was partlysupported by the U.S. Department of Energy, Office of Science, Materials Sciences and Engineering Division through the Materials Theory Institute. We gratefully acknowledge thecomputing resources provided on Blues, a high-performancecomputing cluster operated by the Laboratory ComputingResource Center at Argonne National Laboratory. The workby R.L.S. was funded by EPSRC EP/L002922/1. APPENDIX A: HAMILTONIAN FORMALISM The magnetization dynamics can be described by means of the Landau-Lifshitz equation d/vectorM dt=−γ/vectorM×/vectorHeff, (A1) where /vectorMis the magnetization vector, γis the gyromagnetic ratio, and Heffis an effective field. In the Hamiltonian formalism proposed by Slavin and Tiberkevich [ 36], Eq. ( A1) is recast as a function of the complex amplitude adefined through a Holstein-Primakoff transformation a=m1+im2√2MS(MS+m3), (A2) where m3is the magnetization component parallel to /vectorHeff,m1 andm2are perpendicular to /vectorHeff, andMS=| |(m1,m2,m3)||is the saturation magnetization.By expanding the resulting equation in Taylor series, the linear dynamics can be written as da dt=−id da∗H(a,a∗), (A3) where His the Hamiltonian of the system. In the main text, we generalize Eq. ( A3) to an array of complex amplitudes a, so that Hbecomes a matrix. APPENDIX B: HAMILTONIAN MATRICES Here, we outline the expressions for the Hamiltonian matrices for the field contributions specified in the main text. 1. Anisotropy field We assume that the anisotropy field in the magnetic elements is dominated by shape; this is certainly the case for thePermalloy islands that are commonly used. The demagnetizingfactors in thin films are defined as N,M, andLin the 1, 2, and 3 directions, respectively [see Eq. ( A2)]. Computing the energy functional for every island leads to the diagonal Hamiltonianmatrices H (1,1) an=γMS(N−M) 2I, (B1) H(1,2) an=γMS[4L−2(M+N)] 4I, (B2) where Iis the 4 ×4 identity matrix. 2. External field The external field is considered to be homogeneous throughout the spin ice structure, with magnitude |/vectorH|=Ho and an arbitrary direction in space. To second order in a,t h e Hamiltonian takes a diagonal form with terms proportionalto the stable magnetization direction of each island. In otherwords, only fields parallel to each magnetic element’s easyaxis will affect linear spin waves. Since we consider thinfilms, only H x=/vectorHˆxandHy=/vectorHˆysurvive, and we are left with the matrices H(1,1) ext=O, (B3) H(1,2) ext=− 2⎛ ⎜⎜⎜⎝Hy 00 0 0Hx 00 00 −Hy 0 00 0 −Hx⎞ ⎟⎟⎟⎠, (B4) where Ois the 4 ×4 zero matrix. 3. Dipolar field The derivation for the dipolar field is outlined in the main text. The Hamiltonian matrices obtain for the ground state are H(1,1) d=⎛ ⎜⎜⎜⎝S11 xS12 cS13 xS14 c S21 cS22 yS23 cS24 y S31 xS23 cS33 xS34 c S41 cS24 yS34 cS44 y⎞ ⎟⎟⎟⎠−ˆSz (B5) 134420-5IACOCCA, GLIGA, STAMPS, AND HEINONEN PHYSICAL REVIEW B 93, 134420 (2016) H(1,2) d=D+⎛ ⎜⎜⎜⎝0S12 cS13 xS14 c S21 c 0S23 cS24 y S31 xS32 c 0S34 c S41 cS42 yS43 c 0⎞ ⎟⎟⎟⎠+ˆSz.(B6) The elements of the above Hamiltonian matrices contain contributions between a particular island and every other islandin the structure but itself. Consequently, we can divide themin two terms: S β,ττ/primecontaining them sum between island j at cell τand every island in cell τ/prime, and the elements Sjk β,jk containing the component-wise products between the islands in cell τ. Using the notation where /vectorRjk,ττ/primeis the distance between island jin cell τand island kin cell τ/primeand/vectorRjkis the distance between islands jandkin island τ, these terms along the Cartesian direction are defined as Sˆx,ττ/prime=/summationdisplay k,τ/prime/negationslash=τ/bracketleftBigg 3(/vectorRjk,ττ/prime·ˆx)2 /vectorR5 jk,ττ/prime−1 /vectorR3 jk,ττ/prime/bracketrightBigg e−i/vectorq/vectorRjk,(B7a) Sˆy,ττ/prime=/summationdisplay k,τ/prime/negationslash=τ/bracketleftBigg 3(/vectorRjk,ττ/prime·ˆy)2 /vectorR5 jk,ττ/prime−1 /vectorR3 jk,ττ/prime/bracketrightBigg e−i/vectorq/vectorRjk,(B7b) Sˆz,ττ/prime=/summationdisplay k,τ/prime/negationslash=τ/bracketleftBigg −1 /vectorR3 jk,ττ/prime/bracketrightBigg e−i/vectorq/vectorRjk, (B7c) Sjk ˆx,jk=/bracketleftBigg 3(/vectorRjk·ˆx)2 /vectorR5 jk−1 /vectorR3 jk/bracketrightBigg , (B7d) Sjk ˆy,jk=/bracketleftBigg 3(/vectorRjk·ˆy)2 /vectorR5 jk−1 /vectorR3 jk/bracketrightBigg , (B7e) Sjk ˆz,jk=/bracketleftBigg −1 /vectorR3 jk/bracketrightBigg . (B7f) The summation on the cross direction ˆx,ˆycan be similarly divided into two contributions, defined as Sc,ττ/prime=/summationdisplay k,τ/prime/negationslash=τ/bracketleftBigg 3(/vectorRjk,ττ/prime·ˆx)(/vectorRjk,ττ/prime·ˆy) /vectorR5 jk,ττ/prime/bracketrightBigg e−i/vectorq/vectorRjk,(B8) Sjk c,jk=/bracketleftBigg 3(/vectorRjk·ˆx)(/vectorRjk·ˆy) /vectorR5 jk/bracketrightBigg . (B9) Finally, the diagonal matrix Din Eq. ( B6) can be labeled from 1 to 4, taking the values D1=Sˆx,ττ/prime+2(Sˆy,13+Sc,14−Sc,12), (B10a) D2=Sˆy,ττ/prime+2(Sˆx,24+Sc,23−Sc,21), (B10b) D3=Sˆx,ττ/prime+2(Sˆy,31+Sc,32−Sc,34), (B10c) D4=Sˆy,ττ/prime+2(Sˆx,42+Sc,43−Sc,41). (B10d)In the case of the remanent state, we note that the components of the Hamiltonian are 4 ×4 matrices with a similar form as the matrices in the vortex state Hamiltonian.In fact, taking the first two rows and columns of the abovematrices and considering the structure’s translation vector for a remanent state, /vectorR r=(dˆx,dˆy), leads to the correct Hamiltonian matrices. APPENDIX C: EXCHANGE INTERACTION Intra-island exchange interactions between noncollinear spins are important to correctly describe the dynamics ofspin ices, especially the modes that arise because of internaldegrees of freedom. For our analytical model, we considereach island to be composed of N exmacrospins interacting with their nearest neighbors by using an effective, discreteHeisenberg Hamiltonian model. The exchange Hamiltonianmatrix blocks defined above will now have a dimension(8N ex)×(8Nex) to take into account the internal spins in each island. Consequently, the elements of the Hamiltonianmatrices above must be also expanded by replacing each ofthem by an N ex×Nexblock. Furthermore, we can introduce an arbitrary direction for each spin, so that the complex amplitude of a particular spin is a=[(1−2|a|2) cosθ+/radicalbig 1−|a|2(a+a∗)|sinθ|]ˆx +[(1−2|a|2)s i nθ+/radicalbig 1−|a|2(a+a∗)|cosθ|]ˆy −i/radicalbig 1−|a|2(a−a∗)ˆz, (C1) where θis the angle with respect to the ˆxaxis, and the absolute values represent the isotropic nature of deviations from themagnetic elements’ easy axis. We consider N ex=3, i.e., one bulk and two edge spins. For the particular example of the vortex square ice, the exchangeHamiltonian takes the form H (1,1) ex=O|12×12, (C2) H(1,2) ex=I⊗C, (C3) where O|12×12indicates a 12 ×12 zero matrix, and Cis expressed as a function of the exchange constant Jand the array of spin angles θ. For example, for the case where all spins are collinear in a single magnetic element, the matrix C takes the form C=− 2JM2 S⎛ ⎝1−10 −12 −1 0−11⎞ ⎠. (C4) The dominant Hamiltonian matrices described above can be extended simply by completing the diagonal terms in Hanand Hextand calculating the summations between the new spins of different islands in Hd. Noncollinear spins can be also easily included in the model by computing the products originatingfrom the definition of Eq. ( C1). 134420-6RECONFIGURABLE W A VE BAND STRUCTURE OF AN . . . PHYSICAL REVIEW B 93, 134420 (2016) APPENDIX D: ANGLE DEPENDENCE In the main text, we explored the effect of an applied field along the ˆxdirection. Varying the angle of such a field provides the means to explore the symmetry of thesquare ices. As discussed in the main text, CandSstates are energetically stable in the nanoislands. A vortex state with C-state magnetic elements has a fourfold symmetry, while both vortex and remanent states with S-state magnetic elements have a twofold symmetry. This can be readily shown bycalculating the angle dependence of the spin waves at the/Gamma1point with an applied field of 50 Oe. Figure 6(a)-6(b) clearly displays these symmetries for each case, focusing on the M2 andM3 bulk modes for clarity. On the other hand, the remanent state in an onion state or in a macrospin approximationhas a twofold symmetry with elements magnetized at 90degrees. The resulting angle dependence shown in Fig. 6(c) follows these symmetries as well. Such an angle dependencerepresents a valuable tool to experimentally manipulate themagnon spectra, and to infer the magnetic configuration ofsquare ices. Coupled with the field tunability, it is possible tounambiguously determine the dominant static state throughoutthe structure. FIG. 6. 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PhysRevB.76.092402.pdf

Enhancement of the Gilbert damping constant due to spin pumping in noncollinear ferromagnet/ nonmagnet/ferromagnet trilayer systems Tomohiro Taniguchi1,2and Hiroshi Imamura2 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan /H20849Received 24 April 2007; revised manuscript received 23 May 2007; published 5 September 2007 /H20850 We analyzed the enhancement of the Gilbert damping constant due to spin pumping in noncollinear ferromagnet/nonmagnet/ferromagnet trilayer systems. We show that the Gilbert damping constant depends bothon the precession angle of the magnetization of the free layer and on the direction of the magnetization of thefixed layer. We find the condition to be satisfied to realize strong enhancement of the Gilbert damping constant. DOI: 10.1103/PhysRevB.76.092402 PACS number /H20849s/H20850: 75.70.Cn, 72.25.Mk, 76.50. /H11001g, 76.60.Es There is currently great interest in the dynamics of mag- netic multilayers because of their potential applications innonvolatile magnetic random access memory /H20849MRAM /H20850and microwave devices. In the field of MRAM, much effort hasbeen devoted to decreasing power consumption through theuse of current-induced magnetization reversal /H20849CIMR /H20850. 1–7 Experimentally, CIMR is observed as the current perpendicu- lar to plane-type giant magnetoresistivity of a nanopillar, inwhich the spin-polarized current injected from the fixed layerexerts a torque on the magnetization of the free layer. Thetorque induced by the spin current is utilized to generatemicrowaves. The dynamics of the magnetization Min a ferromagnet under an effective magnetic field B effis described by the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation dM dt=−/H9253M/H11003Beff+/H92510M /H20841M/H20841/H11003dM dt, /H208491/H20850 where /H9253and/H92510are the gyromagnetic ratio and the Gilbert damping constant intrinsic to the ferromagnet, respectively.The Gilbert damping constant is an important parameter forspin electronics since the critical current density of CIMR isproportional to the Gilbert damping constant 8,9and fast- switching time magnetization reversal is achieved for a largeGilbert damping constant. 10Several mechanisms intrinsic to ferromagnetic materials, such as phonon drag11and spin- orbit coupling,12have been proposed to account for the ori- gin of the Gilbert damping constant. In addition to theseintrinsic mechanisms, Mizukami et al. 13,14and Tserkovnyak et al.15,16showed that the Gilbert damping constant in a non- magnet /H20849N/H20850/ferromagnet /H20849F/H20850/nonmagnet /H20849N/H20850trilayer system is enhanced due to spin pumping. Tserkovnyak et al.17also studied spin pumping in a collinear F/N/F trilayer system andshowed that enhancement of the Gilbert damping constantdepends on the precession angle of the magnetization of thefree layer. On the other hand, several groups who studied CIMR in a noncollinear F/N/F trilayer system in which the magnetiza-tion of the free layer is aligned to be perpendicular to that ofthe fixed layer have reported the reduction of the criticalcurrent density. 5–7Therefore, it is intriguing to ask how theGilbert damping constant is affected by spin pumping in noncollinear F/N/F trilayer systems. In this Brief Report, we analyze the enhancement of the Gilbert damping constant due to spin pumping in noncol-linear F/N/F trilayer systems such as that shown in Fig. 1. Following Refs. 15–18, we calculate the spin current induced by the precession of the magnetization of the free layer andthe enhancement of the Gilbert damping constant. We showthat the Gilbert damping constant depends not only on theprecession angle /H9258of the magnetization of a free layer but also on the angle /H9267between the magnetizations of the fixed layer and the precession axis. The Gilbert damping constantis strongly enhanced if angles /H9258and/H9267satisfy the condition /H9258=/H9267or/H9258=/H9266−/H9267. The system we consider is schematically shown in Fig. 1. A nonmagnetic layer is sandwiched between two ferromag-netic layers, F 1and F 2. We introduce the unit vector mito represent the direction of the magnetization of the ith ferro- magnetic layer. The equilibrium direction of the magnetiza-tionm 1of the left free ferromagnetic layer F 1is taken to exist along the zaxis. When an oscillating magnetic field is applied, the magnetization of the F 1layer precesses around thezaxis with angle /H9258. The precession of the vector m1is expressed as m1=/H20849sin/H9258cos/H9275t,sin/H9258sin/H9275t,cos/H9258/H20850, where /H9275 FIG. 1. /H20849Color online /H20850The F/N/F trilayer system is schemati- cally shown. The magnetization of the F 1layer /H20849m1/H20850precesses around the zaxis with angle /H9258and angular velocity /H9275. The magne- tization of the F 2layer /H20849m2/H20850is fixed with tilted angle /H9267. The pre- cession of the magnetization in the F 1layer pumps spin current Ispumpinto the N and F 2layers and creates the spin accumulation /H9262N in the N layer. The spin accumulation induces the backflow spin current Isback /H20849i/H20850/H20849i=1,2 /H20850.PHYSICAL REVIEW B 76, 092402 /H208492007 /H20850 1098-0121/2007/76 /H208499/H20850/092402 /H208494/H20850 ©2007 The American Physical Society 092402-1is the angular velocity of the magnetization. The direction of the magnetization of the F 2layer, m2, is assumed to be fixed and the angle between m2and the zaxis is represented by /H9267. The collinear alignment discussed in Ref. 17corresponds to the case of /H9267=0,/H9266. Before studying spin pumping in noncollinear systems, we shall give a brief review of the theory of spin pumping ina collinear F/N/F trilayer system. 17Spin pumping is the in- verse process of CIMR where the spin current induces theprecession of the magnetization. Contrary to CIMR, spinpumping is the generation of the spin current induced by theprecession of the magnetization. The spin current due to theprecession of the magnetization in the F 1layer is given by Ispump=/H6036 4/H9266g↑↓m1/H11003dm1 dt, /H208492/H20850 where g↑↓is a mixing conductance18,19and/H6036is the Dirac constant. Spins are pumped from the F 1layer into the N layer and the spin accumulation /H9262Nis created in the N layer. Spins also accumulate in the F 1and F 2layers. In the ferromagnetic layers, the transverse component of the spin accumulation isassumed to be absorbed within the spin coherence length defined as /H9261 tra=/H9266//H20841kFi↑−kFi↓/H20841, where kFi↑,↓is the spin-dependent Fermi wave number of the ith ferromagnet. For ferromag- netic metals such as Fe, Co, and Ni, the spin coherencelength is a few angstroms. 20Hence, the spin accumulation in theith ferromagnetic layer is aligned to be parallel to the magnetization, i.e., /H9262Fi=/H9262Fimi. The longitudinal component of the spin accumulation decays on the scale of spin diffu- sion length /H9261sdFi, which is of the order of 10 nm for typical ferromagnetic metals.21 The difference in the spin accumulation of ferromagnetic and nonmagnetic layers, /H9004/H9262i=/H9262N−/H9262Fimi/H20849i=1,2 /H20850, induces a backflow spin current, Isback /H20849i/H20850, flowing into both the F 1and F 2 layers. The backflow spin current Isback /H20849i/H20850is obtained using circuit theory18as Isback /H20849i/H20850=1 4/H9266/H208752g↑↑g↓↓ g↑↑+g↓↓/H20849mi·/H9004/H9262i/H20850mi+g↑↓mi/H11003/H20849/H9004/H9262i/H11003mi/H20850/H20876, /H208493/H20850 where g↑↑and g↓↓are the spin-up and spin-down conduc- tances, respectively. The total spin current flowing out of the F1layer is given by Isexch=Ispump−Isback /H208491/H20850.17The spin accumu- lation /H9262Fiin the F ilayer is obtained by solving the diffusion equation. We assume that spin-flip scattering in the N layer is so weak that we can neglect the spatial variation of the spin current within the N layer, Isexch=Isback /H208492/H20850. The torque /H92701 acting on the magnetization of the F 1layer is given by /H92701 =Isexch−/H20849m1·Isexch/H20850m1=m1/H11003/H20849Isexch/H11003m1/H20850. For the collinear system, we have /H92701=g↑↓ 8/H9266/H208731−/H9263sin2/H9258 1−/H92632cos2/H9258/H20874m1/H11003dm1 dt, /H208494/H20850 where /H9263=/H20849g↑↓−g*/H20850//H20849g↑↓+g*/H20850is the dimensionless parameter introduced in Ref. 17. The Gilbert damping constant in the LLG equation is enhanced due to the torque /H92701as /H92510→/H92510+/H9251/H11032, with/H9251/H11032=gL/H9262Bg↑↓ 8/H9266M1dF1S/H208731−/H9263sin2/H9258 1−/H92632cos2/H9258/H20874, /H208495/H20850 where gLis the Landé gfactor, /H9262Bis the Bohr magneton, dF1is the thickness of the F 1layer, and Sis the cross section of the F 1layer. Next, we move on to the noncollinear F/N/F trilayer sys- tem with /H9267=/H9266/2, in which the magnetization of the F 2layer is aligned to be perpendicular to the zaxis. Following a similar procedure, the LLG equation for the magnetizationM 1in the F 1layer is expressed as dM1 dt=−/H9253effM1/H11003Beff+/H9253eff /H9253/H20849/H92510+/H9251/H11032/H20850M1 /H20841M1/H20841/H11003dM1 dt,/H208496/H20850 where /H9253effand/H9251/H11032are the effective gyromagnetic ratio and the enhancement of the Gilbert damping constant, respec-tively. The effective gyromagnetic ratio is given by /H9253eff=/H9253/H208731−gL/H9262Bg↑↓/H9263cot/H9258cos/H9274sin/H9275t 8/H9266Md F1S/H9280 /H20874−1 , /H208497/H20850 where cos /H9274=sin/H9258cos/H9275t=m1·m2and /H9280=1−/H92632cos2/H9274−/H9263/H20849cot2/H9258cos2/H9274− sin2/H9274+ sin2/H9275t/H20850. /H208498/H20850 The enhancement of the Gilbert damping constant is ex- pressed as /H9251/H11032=gL/H9262Bg↑↓ 8/H9266Md F1S/H208731−/H9263cot2/H9258cos2/H9274 /H9280 /H20874. /H208499/H20850 It should be noted that, for noncollinear systems, both the gyromagnetic ratio and the Gilbert damping constant aremodified by spin pumping, contrary to what occurs in collin-ear systems. The modification of the gyromagnetic ratio andthe Gilbert damping constant due to spin pumping can beexplained by considering the pumping spin current and thebackflow spin current /H20851see Figs. 2/H20849a/H20850and2/H20849b/H20850/H20852. The direction of the magnetic moment carried by the pumping spin current I spumpis parallel to the torque of the Gilbert damping for both collinear and noncollinear systems. The Gilbert damping constant is enhanced by the pumping spin current Ispump.O n the other hand, the direction of the magnetic moment carried by the backflow spin current Isback /H208491/H20850depends on the direction of the magnetization of the F 2layer. As shown in Eq. /H208493/H20850, the backflow spin current in the F 2layerIsback /H208492/H20850has a projection onm2. Since we assume that the spin current is constant within the N layer, the backflow spin current in the F 1layer Isback /H208491/H20850also has a projection on m2. For the collinear system, bothIspumpandIsback /H208491/H20850are perpendicular to the precession torque because m2is parallel to the precession axis. How- ever, for the noncollinear system, the vector Isback /H208491/H20850has a projection on the precession torque, as shown in Fig. 2/H20849b/H20850. Therefore, the angular momentum injected by Isback /H208491/H20850modi- fies the gyromagnetic ratio as well as the Gilbert damping inthe noncollinear system. Let us estimate the effective gyromagnetic ratio using re- alistic parameters. According to Ref. 17, the conductancesBRIEF REPORTS PHYSICAL REVIEW B 76, 092402 /H208492007 /H20850 092402-2g↑↓and g*for a Py/Cu interface are given by g↑↓/S =15 nm−2and/H9263/H112290.33, respectively. The Landé gfactor is taken to be gL=2.1, magnetization is 4 /H9266M=8000 Oe, and thickness dF1=5 nm. Substituting these parameters into Eqs. /H208497/H20850and /H208498/H20850, one can see that /H20841/H9253eff//H9253−1/H20841/H112290.001. Therefore, the LLG equation can be rewritten as dM1 dt/H11229−/H9253M1/H11003Beff+/H20849/H92510+/H9251/H11032/H20850M1 /H20841M1/H20841/H11003dM1 dt. /H2084910/H20850 The estimated value of /H9251/H11032is of the order of 0.001. However, we cannot neglect /H9251/H11032since it is of the same order as the intrinsic Gilbert damping constant /H92510.22,23 Experimentally, the Gilbert damping constant is measured as the width of the ferromagnetic resonance /H20849FMR /H20850absorp- tion spectrum. Let us assume that the F 1layer has no aniso- tropy and that an external field Bext=B0zˆis applied along the zaxis. We also assume that the small-angle precession of the magnetization around the zaxis is excited by the oscillating magnetic field B1applied in the xyplane. The FMR absorp- tion spectrum is obtained as follows:24 P=1 T/H20885 0T dt/H9251/H9253M/H90242B12 /H20849/H9253B0−/H9024/H208502+/H20849/H9251/H9253B0/H208502, /H2084911/H20850 where /H9024is the angular velocity of the oscillating magnetic field, T=2/H9266//H9024, and /H9251=/H92510+/H9251/H11032. Since /H9251is very small, the absorption spectrum can be approximately expressed as P/H11008/H92510+/H20855/H9251/H11032/H20856and the highest point of the peak proportional to /H208551//H20849/H92510+/H9251/H11032/H20850/H20856, where /H20855/H9251/H11032/H20856represents the time-averaged valueof the enhancement of the Gilbert damping constant. In Fig. 3/H20849a/H20850, the time-averaged value /H20855/H9251/H11032/H20856for a noncollinear system in which /H9267=/H9266/2 is plotted by the solid line as a function of the precession angle /H9258. The dotted line represents the en- hancement of the Gilbert damping constant /H9251/H11032for the collin- ear system given by Eq. /H208495/H20850. The time-averaged value of the enhancement of the Gilbert damping constant /H20855/H9251/H11032/H20856takes its maximum value at /H9258=0,/H9266for the collinear system /H20849/H9267=0,/H9266/H20850. Contrary to the collinear system, /H20855/H9251/H11032/H20856of the non- collinear system in which /H9267=/H9266/2 takes its maximum value at/H9258=/H9266/2. As shown in Fig. 2/H20849b/H20850, the backflow spin current gives a negative contribution to the enhancement of the Gilbertdamping constant. This contribution is given by the projec- tion of the vector I sback /H208491/H20850onto the direction of the torque of the Gilbert damping, which is represented by the vectorm 1/H11003m˙1. Therefore, the condition to realize the maximum value of the enhancement of the Gilbert damping is satisfied if the projection of Isback /H208491/H20850ontom1/H11003m˙1takes the minimum value; i.e., /H9258=/H9267or/H9258=/H9266−/H9267. We can extend the above analysis to the noncollinear sys- tem with an arbitrary value of /H9267. After performing the appro- priate algebra, one can easily show that the LLG equation forthe magnetization of the F 1layer is given by Eq. /H208496/H20850with /H9253eff=/H9253/H208751−gL/H9262Bg↑↓/H9263sin/H9267sin/H9275t/H20849cot/H9258cos/H9274˜− csc/H9258cos/H9267/H20850 8/H9266MdS/H9280˜ /H20876−1 , /H2084912/H20850 M2Ispump F2 F1 Gilbert dampingprecessionIsback(1) backflow M1(a) (b) FIG. 2. /H20849Color online /H20850/H20849a/H20850Top view of Fig. 1. The dotted circle in F 1represents the precession of magnetization M1and the arrow pointing to the center of this circle represents the torque of the Gilbert damping. The arrows in IspumpandIsback /H208491/H20850represent the mag- netic moment of spin currents. /H20849b/H20850The backflow Isback /H208491/H20850has com- ponents aligned with the direction of the precession and the Gilbertdamping.0 0.00270.00320.0037 (a) (b)0.0037 0.0032 0.0027π/2 π θ<α'> 0 π/2 π ρ0π/2π θ<α'> FIG. 3. /H20849Color online /H20850/H20849a/H20850The time-averaged value of the en- hancement of the Gilbert damping constant /H9251/H11032is plotted as a func- tion of the precession angle /H9258. The solid line corresponds to the collinear system derived from Eq. /H208499/H20850. The dashed line corresponds to the noncollinear system derived from Eq. /H208495/H20850./H20849b/H20850The time- averaged value of the enhancement of the Gilbert damping constant /H9251/H11032of the noncollinear system is plotted as a function of the preces- sion angle /H9258and the angle /H9267between the magnetizations of the fixed layer and the precession axis.BRIEF REPORTS PHYSICAL REVIEW B 76, 092402 /H208492007 /H20850 092402-3/H9251/H11032=gL/H9262Bg↑↓ 8/H9266MdS/H208751−/H9263/H20849cot/H9258cos/H9274˜− csc/H9258cos/H9267/H208502 /H9280˜ /H20876, /H2084913/H20850 where cos /H9274˜=sin/H9258sin/H9267cos/H9275t+cos/H9258cos/H9267=m1·m2and /H9280˜=1−/H92632cos2/H9274˜−/H9263/H20851/H20849cot/H9258cos/H9274˜− csc/H9258cos/H9267/H208502− sin2/H9274˜ + sin2/H9267sin2/H9275t/H20852. /H2084914/H20850 Substituting the realistic parameters into Eqs. /H2084912/H20850and /H2084914/H20850, we can show that the effective gyromagnetic ratio /H9253effcan be replaced by /H9253in Eq. /H208496/H20850and that the LLG equation reduces to Eq. /H2084910/H20850. Figure 3/H20849b/H20850shows the time-averaged value of the enhancement of the Gilbert damping constant /H20855/H9251/H11032/H20856of Eq. /H2084913/H20850. Again, the Gilbert damping constant is strongly en- hanced if angles /H9258and/H9267satisfy the condition that /H9258=/H9267or /H9258=/H9266−/H9267.In summary, we have examined the effect of spin pump- ing on the dynamics of the magnetization of magnetic mul-tilayers and calculated the enhancement of the Gilbert damp-ing constant of noncollinear F/N/F trilayer systems due tospin pumping. The enhancement of the Gilbert damping con- stant depends not only on the precession angle /H9258of the mag- netization of a free layer but also on the angle /H9267between the magnetizations of the fixed layer and the precession axis, asshown in Fig. 3/H20849b/H20850. We have shown that the /H9258and/H9267depen- dences of the enhancement of the Gilbert damping constantcan be explained by analyzing the backflow spin current. Thecondition to be satisfied to realize strong enhancement of theGilbert damping constant is /H9258=/H9267or/H9258=/H9266−/H9267. The authors would like to acknowledge the valuable dis- cussions they had with Y . Tserkovnyak, S. Yakata, Y . Ando,S. Maekawa, S. Takahashi, and J. Ieda. This work was sup-ported by CREST and by a NEDO Grant. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850 425, 380 /H208492003 /H20850. 4A. Deac, K. J. Lee, Y . Liu, O. Redon, M. Li, P. Wang, J. P. Noziéres, and B. Dieny, J. Magn. Magn. Mater. 290-291 ,4 2 /H208492005 /H20850. 5A. D. Kent, B. Ozyilmaz, and E. del Barco, Appl. Phys. Lett. 84, 3897 /H208492004 /H20850. 6K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86, 022505 /H208492005 /H20850. 7T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl. Phys. Lett. 89, 172504 /H208492005 /H20850. 8J. Z. Sun, Phys. Rev. B 62, 570 /H208492000 /H20850. 9J. Grollier, V . Cros, H. Jaffres, A. Hamzic, J. M. George, G. Faini, J. B. Youssef, H. L. LeGall, and A. Fert, Phys. Rev. B 67, 174402 /H208492003 /H20850. 10R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett. 92, 088302 /H208492004 /H20850. 11H. Suhl, IEEE Trans. Magn. 34, 1834 /H208491998 /H20850. 12V . Kamberský, Can. J. Phys. 48, 2906 /H208491970 /H20850. 13S. Mizukami, Y . Ando, and T. Miyazaki, J. Magn. Magn. Mater.239,4 2 /H208492002a /H20850. 14S. Mizukami, Y . Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 /H208492002 /H20850. 15Y . Tserkovnyak and A. Brataas and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 /H208492002 /H20850. 16Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 /H208492002 /H20850. 17Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 67, 140404 /H20849R/H20850/H208492003 /H20850. 18A. Brataas, Y . V . Nazarov, and G. E. W. Bauer, Eur. Phys. J. B 22,9 9 /H208492001 /H20850. 19A. Brataas, Y . V . Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. 84, 2481 /H208492000 /H20850. 20M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 /H208492002 /H20850. 21J. Bass and W. P. Pratt Jr., J. Phys.: Condens. Matter 19, 183201 /H208492007 /H20850. 22J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 23F. Schreiber, J. Pflaum, Th. Mühge, and J. Pelzl, Solid State Com- mun. 93, 965 /H208491995 /H20850. 24Ferromagnetic resonance , edited by S. V . V onsovskii /H20849Israel Pro- gram for Scientific Translations Ltd., Jersalem, 1964 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 76, 092402 /H208492007 /H20850 092402-4

PhysRevB.98.014521.pdf

PHYSICAL REVIEW B 98, 014521 (2018) Spin torques and magnetic texture dynamics driven by the supercurrent in superconductor/ferromagnet structures I. V . Bobkova,1,2A. M. Bobkov,1and M. A. Silaev3 1Institute of Solid State Physics, Chernogolovka, Moscow reg., 142432 Russia 2Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia 3Department of Physics and Nanoscience Center, University of Jyväskylä, P .O. Box 35 (YFL), FI-40014 University of Jyväskylä, Finland (Received 4 April 2018; revised manuscript received 17 May 2018; published 26 July 2018) We introduce the general formalism to describe spin torques induced by the supercurrents injected from the adjacent superconducting electrodes into the spin-textured ferromagnets. By considering the adiabaticlimit for the equal-spin superconducting correlations in the ferromagnet, we show that the supercurrent cangenerate both the fieldlike spin-transfer torque and the spin-orbital torque. These dissipationless spin torquesare expressed through the current-induced corrections to the effective field derived from the system energy.The general formalism is applied to show that the supercurrent can either shift or move the magnetic domainwalls depending on their structure and the type of spin-orbital interaction in the system. These results canbe used for the prediction and interpretation of the experiments studying magnetic texture dynamics insuperconductor/ferromagnet/superconductor Josephson junctions and other hybrid structures. DOI: 10.1103/PhysRevB.98.014521 I. INTRODUCTION It has been commonly recognized that reducing Joule heating effects and power consumption are among the mainpriorities for the development of electrically controlled mag-netic memory devices [ 1–4]. Since the first spin-transfer torque (STT) experiments [ 5,6], much effort has been invested to optimize the switching currents, thermal stability, and tunnelmagnetoresistance of the magnetic tunnel junctions [ 3,7,8]. Thermal effects are also of the crucial importance for the op-eration of the other type of STT memory—magnetic racetrackmemory [ 9–11] based on the electrical control over the domain wall (DW) motion. The progress in improving these spin mem-ory devises depends crucially on the competition between thethermal stability of DWs and large current densities requiredto overcome the pinning forces [ 4,12–18]. As an alternative route to the low-power manipulation of magnetic textures, thecurrent-driven magnetic skyrmion dynamics has attracted largeinterest [ 19–21]. In applications that require very large currents, for example, in powerful magnets, using superconducting materials havebeen proven to be an effective solution to eliminate Jouleheating effects. In view of the energy-saving spintronics, itis quite appealing to employ the spin torques generated bythe dissipationless spin-polarized superconducting currents(supercurrents). The existence of spin-polarized supercurrentsis ubiquitous to the spin-textured superconductor/ferromagnet(SC/FM) hybrid structures resulting from long-range spin-triplet proximity [ 22–25]. Recently, there have been many works studying spin- polarized supercurrents in various SC/FM systems (for thereview see Refs. [ 26,27]). However, the supercurrent-induced spin torques have been characterized theoretically only inseveral model systems: in Josephson junctions through single-domain magnets [ 28–32], two [ 33–35] and three [ 36]F Mlayers, and in ferromagnetic spin-singlet [ 37] and spin- triplet superconductors [ 38]. The general understanding of the supercurrent-spin texture interaction has been lacking sincethere is no direct connection between the above examples andpractically interesting systems—bulk nonhomogeneous FMs.That is, the possibility of moving DWs and skyrmions byinjecting the supercurrent in real ferromagnets has been anopen question for a long time despite of the large attention tothe subject. This challenging question is addressed in the present paper. We employ the adiabatic approximation, which is widely usedfor the description of kinetic processes in metallic ferromag-nets with spin textures including the calculation of conductivity[19] and spin-transfer torques [ 39] in the inhomogeneous FMs. We bring this approach to the realm of superconducting sys-tems to describe their transport properties governed by equal-spin superconducting correlations. For that, we go beyondthe commonly used quasiclassical theory of hybrids [ 25,40], which has been designed to treat only weak ferromagnetswith an exchange splitting much less than the Fermi energy.Instead of that, we employ the recently developed approachof generalized quasiclassical theory [ 41], which allows for the description of proximity effect in strong ferromagnets with anexchange splitting much larger than other energy scales andcomparable to the Fermi energy. We show that the spin-polarized superconducting current can induce magnetization dynamics, described in general bythe Landau-Lifshitz-Gilbert (LLG) equation ˙M=−γM×H eff+α MM×˙M, (1) where γ=2μBis the electron gyromagnetic ratio. The second term in the r.h.s. is the Gilbert damping. The superconductingspin current Jcan induce two types of spin torques which can be written as the correction to effective field −γMטH eff= 2469-9950/2018/98(1)/014521(9) 014521-1 ©2018 American Physical SocietyI. V . BOBKOV A, A. M. BOBKOV , AND M. A. SILAEV PHYSICAL REVIEW B 98, 014521 (2018) Nst+Nso. The first term here is the adiabatic spin-transfer torque [ 15,42,43], while the second term Nsois the spin-orbital (SO) torque [ 44,45]. The nonadiabatic (antidamping) STT [ 14] is not produced by the supercurrent since it breaks the time-reversal symmetry of LLG equation and can be considered asa correction to the dissipative Gilbert damping [ 46]. That is, the antidamping STT should be connected with the quasipar-ticle contribution, which is beyond the scope of our presentstudy. The paper is organized as follows. In Sec. II, the general equations for spin dynamics and spin torques generated bysupercurrent are considered. In Sec. III, we derive expressions for the spin-transfer and spin-orbital torques using the general-ized quasiclassical theory. In Sec. IV, we derive the Josephson energy in SFS junctions and use it to provide an alternativederivation of supercurrent spin torques. Section Vis devoted to the DW dynamics in SFS Josephson junctions induced bythe supercurrent spin torques. Our conclusions are given inSec. VI. II. SPIN TORQUE GENERATED BY THE SUPERCURRENT We use an s-dmodel with a localized magnetization M and that of the itinerant electrons Ms=−μBs, where sis the electron spin and μBis the Bohr magneton. The dynamics of localized spins is determined by the usual LLG equation witha contribution to the effective field resulting from exchangeinteraction with conductivity electrons [ 47]: ˙M=−γM×H eff+α MM×˙M−JsdM×Ms. (2) The last term here is the source of spin torque and should be found from the kinetic equation for conductivity electrons. The kinetic theory for the conduction electrons in metals can be formulated in terms of the matrix Green’s function ˇG= ˇG(r1,r2,t1,t2),which has the following explicit structure in the Keldysh space ˇG=(ˇGR ˇGK 0 ˇGA),where ˇGR/A/Kare the retarder/advanced/Keldysh components. The general quantum kinetic equation reads i{∂t,ˇG}t−[ˆH,ˇG]t,r=ˇI, (3) ˆH(t,r)=−ˆ/Pi12 r 2mF+(ˆσh(r,t))ˆτ3−i(ˆσˆBˆ/Pi1r), (4) ˇI=(ˇ/Sigma1◦ˇG−ˇG◦ˇ/Sigma1)(r1,r2,t1,t2). (5) Here, we define the ◦-product as ( ˆA◦ˆB)(t1,t2)=/integraltext∞ −∞dtˆA(t1,t)ˆB(t,t2). The commutator is defined as [ˆH,ˇG]t=ˆH(t1,r1)ˇG−ˇGˆH(t2,r2), ˆ/Pi1r=∇− ieˆτ3A(r), ˆσiand ˆτiare Pauli matrices in spin and Nambu spaces, respectively. The exchange field is determined by localizedmoments h=−J sdM/2μB. The last term in Eq.( 4)i st h e general form of a linear in momentum spin-orbit coupling(SOC) determined by the constant tensor coefficient ˆB.T h e collision integral in the right-hand side (r.h.s.) of Eq. ( 3)i s given by Eq. ( 5). The self-energy term ˇ/Sigma1includes the effects related to disorder scattering as well as the off-diagonalsuperconducting self-energies.The conduction electron spin polarization s, charge jand spinJ icurrents are given by s(r,t)=−i 8Tr4[ˆσˆτ3ˆGK]|r1,2=r,t1,2=t, (6) j(r,t)=Tr4/bracketleftBigg/parenleftbigˆ/Pi1r1−ˆ/Pi1r2/parenrightbig 8mFˆτ3ˇGK/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle r1,2=r,t1,2=t, (7) Jk(r,t)=Tr4/bracketleftBigg/parenleftbigˆ/Pi1r1−ˆ/Pi1r2/parenrightbig 16mFˆσkˇGK/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle r1,2=r,t1,2=t.(8) The strategy of studying magnetization dynamics consists of solving the coupled LLG ( 2) and kinetic equations ( 3)– (5) together with the expression for the magnetic moment (6). However, the general problem is too complicated for the analysis. In the next section, (Sec. III), we discuss the simplification of the kinetic equation using the so-calledgeneralized quasiclassical approximation [ 41] adopted to treat the nonstationary problems. Besides that, a significant simplification can be obtained in the linear response limit when the dynamics of magnetizationis slow so that the characteristic frequency is small as comparedto the energy gap in the quasiparticle spectrum. In this case, wecan make use of the quasistationary equation for the electronmagnetization, which is obtained from Eq. ( 3). Multiplying it byˆσfrom the left and taking the trace we obtain −∂ ts+∇jJj=Jsd μB(M×Ms) +2mF(Bj×Jj)+Tr4[ˆσˇI]K 8.(9) Here, we introduce the vector Bj=(Bxj,Byj,Bzj), which is determined by the jth coordinate component of the tensor ˆB, andJj=(Jx j,Jy j,Jz j)i st h e j-coordinate component of the spin current ( 8). Next, the driving term in the LLG equation (2) can be found neglecting the term with time derivative in Eq. ( 9): Jsd μB(M×Ms)=∇ jJj−2mF(Bj×Jj)−Tr4[ˆσˇI]K 8. (10) In this work, we are interested in the quasiequilibrium spin torques generated solely by the supercurrent without thecontribution of nonequilibrium quasiparticles. That means thenormal component of the current and the electric field areassumed to be absent. Generally, the last term with the collisionintegral in Eq. ( 10) has contributions both from the off-diagonal order parameter and the spin-orbital scattering self-energy. Thepresent work is based on the following simplifying assump-tions allowing to put Tr 4[ˆσˇI]K=0. First, we are interested in the spin torques occurring in the normal metal ferromagneticinterlayer where the order parameter is absent. Second, theexchange splitting between spin subbands is assumed to belarge enough to suppress spin-flip transitions between them.Below, we demonstrate that in this regime the first term in ther.h.s. of Eq. ( 10) produces the adiabatic STT [ 39,42,48], while the second term yields the spin-orbit torque [ 44,45,49]. In order 014521-2SPIN TORQUES AND MAGNETIC TEXTURE DYNAMICS … PHYSICAL REVIEW B 98, 014521 (2018) to find these contributions, we calculate the spin supercurrent through the spatially-inhomogeneous ferromagnet. In the nextsection, it is shown that in the adiabatic limit valid for thedescription of strong ferromagnet, this calculation can bedone analytically in the most general way using the techniquedeveloped in Ref. [ 41]. III. GENERALIZED QUASICLASSICAL THEORY A. Eilenberger equation for equal-spin correlations To find the results in the adiabatic approximation, it is convenient to work in the local reference frame, where thespin quantization axis is aligned with the local direction of theexchange field in the ferromagnet. Then we use the transfor-mation ˇG loc=ˆU†ˇGˆU, where ˆU=ˆU(r,t) is in general the time- and space-dependent unitary 2 ×2 matrix that rotates the spin quantization axis zto the local frame determined by the exchange field, so that h/bardblz. To implement the adiabatic approximation, we introduce the equal-spin (ES) pairing components of the GF ˆGσ ES=1 4/summationdisplay iˆτiTr[ ˇγσiˇGloc]. (11) Here, the projection operators to spin-up and spin-down states defined by the index σ=± 1a r eg i v e nb y ˇ γσ0=ˆτ0ˆσ0+ σˆτ3ˆσ3,ˇγσ1=ˆτ1ˆσ1−σˆτ2ˆσ2,ˇγσ2=ˆτ2ˆσ1+σˆτ1ˆσ2,ˇγσ3= ˆτ3ˆσ0+σˆτ0ˆσ3. The generalized quasiclassical theory is formu- lated in terms of the spinless propagators ˆgσ(np,r)=−/contintegraldisplaydξpσ πiˆGσ ES(p,r), (12) where ˆGσ ES=ˆGσ ES(p,r) is the GF in the mixed representation, ξpσ=p2/2mF+σh−μ, and the notation/contintegraltext means that the integration takes into account the poles of GF near the corre-sponding Fermi surface. Then, in the adiabatic approximation,which neglects the coupling between equal-spin and mixed-spin correlations [ 41], we obtain the generalized Keldysh- Eilenberger equation: i{ˆτ 3∂t,ˇg}t+ivσˆ∂rˆgσ−[ˆ/Sigma1σ,ˆgσ]t=0, (13) ˆ∂r=∇− ie[Aˆτ3,.]t+iσ[Zˆτ3,.]t. (14) Here, the spin-dependent Fermi velocities v±=√2(μ±h)/mFare determined on each of the spin-split Fermi surfaces. The spin-dependent gauge field is givenby the superposition of two terms Z=Z m+Zso, where Zm i=−iTr( ˆσzˆU†∂iˆU)/2 is the texture-induced part and the termZso i=mF(mBi) (where m=M/M), which appears due to the SOC. One can see that the Eilenberger-type equations for the spin- up/down correlations contain an additional U(1) gauge field Z which is added to the usual electromagnetic vector potential Awith the opposite effective charges for spin-up and spin- down Cooper pairs. On a qualitative level, it is equivalent tothe adiabatic approximation in the single-particle problems thatallows to describe the quantum system evolution in terms ofthe Berry gauge fields [ 50].B. Charge and spin currents The Eilenberger equations ( 13) are supplemented by the expressions for the charge current jand the spin current Jk, where kdenoted the spin index. The former is given by j(t)=−πe 4/summationdisplay σ=±νσ/angbracketleftbig vσTr/bracketleftbig ˆτ3ˆgK σ(t,t)/bracketrightbig/angbracketrightbig , (15) where νσare the spin-resolved DOS and /angbracketleft ···/angbracketright denotes the averaging over the spin-split Fermi surface. The spin current in rotated frame is given by ˜Jz(t)=−π 8/summationdisplay σ=±σνσ/angbracketleftbig vσTr/bracketleftbig ˆτ3ˆgK σ(t,t)/bracketrightbig/angbracketrightbig . (16) C. Diffusive limit Let us consider the system with large nonmagnetic impurity scattering rate as compared to the superconducting energiesdetermined by the bulk energy gap /Delta1. In this experimentally relevant diffusive limit, it is possible to derive the generalizedUsadel theory with the help of the normalization condition (ˆg σ◦ˆgσ)(t1,t2)=ˆδ(t1−t2),which holds due to the commu- tator structure of the quasiclassical equations ( 13). The impurity self-energy in the Born approximation is given byˆ/Sigma1σ=/angbracketleftˆgσ/angbracketright/2iτσ. In the dirty limit, we have 2τσ(vσˆ∂r)ˆgσ=− [/angbracketleftˆgσ/angbracketright,ˆgσ]t. (17) The solution of Eq. ( 17) can be found as ˆgσ=/angbracketleftˆgσ/angbracketright+ ˆga σpσ/pσ, where the anisotropic part of the solution ˆga σis small with respect to /angbracketleftˆgσ/angbracketright. Making use of the relation {/angbracketleftˆgσ/angbracketright,ˆga σ}t= 0, which follows from the normalization condition, one obtains ˆga σ=−τσvσ/angbracketleftˆgσ/angbracketright◦ˆ∂r/angbracketleftˆgσ/angbracketright. (18) Substituting to Eq. ( 13) and omitting the angle brackets, we get the diffusion equation {ˆτ3∂t,ˆgσ}t−Dσˆ∂r(ˆgσ◦ˆ∂rˆgσ)=0, (19) where Dσare the spin-dependent diffusion coefficients, in the isotropic case given by Dσ=τσv2 σ/3. This equation is a spin- scalar equation, but cannot describe conventional spin-singletsuperconducting correlations unlike the standard spin-scalarform of the nonstationary Usadel equation [ 51]. It is only applicable for strong ferromagnets and describes equal-spintriplet correlations residing at one and the same Fermi surface.Therefore this equation is a nonstationary generalization ofthe corresponding equations for homogeneous strong ferro-magnets [ 52] and inhomogeneous strong ferromagnets [ 41]. The current and spin current are obtained by substituting expansion ( 18)t oE q s .( 15) and ( 16): j=πe 4/summationdisplay σ=±νσDσTr[ ˆτ3ˆgσ◦ˆ∂rˆgσ], (20) ˜Jz=π 8/summationdisplay σ=±σνσDσTr[ ˆτ3ˆgσ◦ˆ∂rˆgσ]. (21) Further simplification can be obtained as follows. First, due to the normalization condition, we introduce the parametriza-tion of Keldysh component in terms of the distributionfunction ˆg K σ=ˆgR σ◦ˆfσ−ˆfσ◦ˆgA σ. Then, switching to the mixed representation in time-energy domain ˆgσ(t1,t2)= 014521-3I. V . BOBKOV A, A. M. BOBKOV , AND M. A. SILAEV PHYSICAL REVIEW B 98, 014521 (2018) /integraltext∞ −∞ˆgσ(ε,t)e−iε(t1−t2)dε/2π, where t=(t1+t2)/2, we keep only the lowest order terms in the time derivatives. As an example of the above procedure, one can obtain from (20) the charge current in the normal state j=e2(ν+D+− ν−D−)Eedriven by the emergent electric field [ 19,53]Ee= −∂tZ. We, however, will neglect these effects and take into account only the quasiequilibrium contributions to the currentsgiven by j=/summationdisplay σ=±eνσDσ 8/integraldisplay∞ −∞dεf 0Tr/parenleftbig ˆτ3ˆJRA σ/parenrightbig , (22) ˜Jz=/summationdisplay σ=±σνσDσ 16/integraldisplay∞ −∞dεf 0Tr/parenleftbig ˆτ3ˆJRA σ/parenrightbig , (23) where ˆJRA σ=ˆgR σˆ∂rˆgR σ−ˆgA σˆ∂rˆgA σis the spectral current and f0(ε)=tanh(ε/2T) is equilibrium distribution function. D. Supercurrent-induced torque In the quasiequilibrium regime when the time derivative of the GF in the mixed representation can be neglected, Eqs. ( 13) or (19) yield the conservation of spin current in rotating frame ∇·˜Jz=0. The spin current in the laboratory frame is given byJk=Rkz˜Jz, which can be written in the form Jk(r)=mk(r)˜Jz. (24) It is not conserved due to the spatially-dependent magnetiza- tion of delectrons m=m(r). Substituting Eq. ( 24) into Eq. ( 10), we obtain the torque, induced by the supercurrent in the quasiequilibriumregime: J sdMs×M=Nst+Nso, (25) Nst=2μB(˜Jz∇)m, (26) Nso=4μBmF(m×Bj)˜Jz j. (27) Here, Nstis the supercurrent spin-transfer torque, which takes only the form of the adiabatic torque in the consideredapproximation, and N sois the spin-orbit torque. Its particular structure strongly depends on the type of the spin-orbit cou-pling, realized in the system. Below we show that due to thecoherent nature of the spin-polarized superconducting currentthe same result can be obtained from the energy functional ofthe system yielding the correction to the effective field. IV . SUPERCURRENT SPIN TORQUES AS CORRECTIONS TO THE EFFECTIVE FIELD Above, we have derived general expressions ( 26) and ( 27) for the superconducting spin torques starting from the kineticequation treated in the adiabatic limit. For any particularsystem, one can find the spin torques solving generalized Eilen-berger/Usadel equations for the quasiclassical propagators andcalculating the spin current according to Eq. ( 21). An alternative approach to obtain superconducting spin torques is based on the description of magnetization dynamicsin terms of the phenomenological expression for the effec-tive field H eff=−δF/δ M, where F=F(M) is the systemenergy as a functional of the magnetization distribution. The LLG equation without dissipation terms is given by ˙M=−γM×Heff. (28) This approach cannot be applied to derive spin-transfer torques in the normal state where the conduction electron magne-tization is not coherent. In contrast to the normal system,superconducting electrons are in the macroscopically coherentstate. Therefore the total energy of the system written in termsof the macroscopic variables describes the interaction betweenthe condensate spin and the ferromagnetic order parameter. Based on the above discussion one can conclude that the superconducting spin-transfer torques ( 26) and ( 27) can be obtained from the energy arguments. To demonstrate this, weconsider a generic example of the Josephson system consistingof superconducting leads coupled through the ferromagnetwith nonhomogeneous magnetization texture. In general, thistask is rather complicated and requires extensive numericalcalculations for each particular system considered. However,in strong ferromagnets, the general expressions for Josephsonspin and charge currents for different magnetic textures ofthe interlayer can be obtained using the machinery of thegeneralized quasiclassical theory [ 41]. We consider the 1D magnetic texture M=M(x)i nt h e interlayer of the thickness dbetween two superconducting interfaces, located at x=±d/2. The superconducting order parameter phase difference between them is χ. The current- phase relation for this setup has been found [ 41]a st h e superposition of partial currents carried by the spin-up andspin-down Cooper pairs: j(χ)=/summationdisplay σ=±jσsin/parenleftbigg χ+2σ/integraldisplayd/2 −d/2Zxdx/parenrightbigg . (29) The amplitudes jσare determined by the boundary condi- tions at FM/SC interfaces and the overlap factor of the equal-spin correlations injected from the opposite SC electrodesj σ∝e−d/ξNσ, where ξNσ=√Dσ/Tis the spin-dependent normal metal correlation length [ 41]. The other characteristic scale of the problem is the characteristic length of the magneticinhomogeneity. In the case of the domain wall, it is the wallsized w. If we are interested in the domain wall motion and consider the situation when the DW is located inside theinterlayer ( d w<d) and not in the vicinity of S/F interfaces, the amplitudes jσdo not depend on dwat all. However, this scale enters the Josephson current via the effective gauge field Z= −mx(my∇mz−mz∇my)/2m2 ⊥, where m⊥=/radicalBig m2y+m2z.Z is a crucial factor giving rise to the DW dynamics, as it is explained below. In more general case, when the DW is wided w>d or the DW is located in the vicinity of a S/F interface, the amplitudes jσalso depend on dwvia boundary conditions [41], but consideration of the DW dynamics presented below is not applicable in this case. The rotated-frame spin current ˜Jz≡˜Jz xis given by the difference of the partial spin-up/down currents ˜Jz(χ)=1 2e/summationdisplay σ=±σjσsin/parenleftbigg χ+2σ/integraldisplayd/2 −d/2Zxdx/parenrightbigg . (30) 014521-4SPIN TORQUES AND MAGNETIC TEXTURE DYNAMICS … PHYSICAL REVIEW B 98, 014521 (2018) The Josephson energy can be obtained according to the usual relation FJ=const−1 2e/summationdisplay σ=±jσcos/parenleftbigg χ+2σ/integraldisplayd/2 −d/2Zxdx/parenrightbigg .(31) The current-phase relation ( 29)i sg i v e nb y j(χ)= 2e(dFJ/dχ). Therefore, calculating the correction to effective field ˜Heff=−δFJ/δM(see details in Appendix), we obtain γMטHeff=2μB˜Jz(2mFBx×m−∇xm). (32) Substituting the result ( 32) to the LLG equation, we get the spin-transfer torques identical to Eqs. ( 25)–(27). The above energy consideration demonstrates that a direct coupling between the magnetization and superconducting cur-rent exist even in the limit when the spontaneous charge currentis absent. Indeed, the spontaneous phase shift of the Josephsoncurrent-phase relation ( 29) is given by tan χ 0=tanϕ(j+− j−)/(j++j−), where ϕ=2/integraltextd/2 −d/2Zxdx. Therefore χ0=0i n the limit when the spin subbands are formally degenerate,j +=j−. At the same time, the Josephson spin current ( 30) and correspondingly the spin torque in Eq. ( 32) are nonzero. This result generalizes the previously suggested mechanism of the supercurrent-induced spin-orbital torque stemming fromtheχ 0=χ0(m) dependence [ 31,32]. In our case, it is not only the phase shift, but in addition the overall critical cur-rent in Eq. ( 29), which depends on the magnetization j c=√ j2 ++j2 −+2j+j−cos(2ϕ) through ϕ=ϕ(m). This provides the nonzero effective field even in the case of degenerate bands. V . SUPERCURRENT DRIVEN MAGNETIC TEXTURE DYNAMICS A. General case of the texture dynamics driven by the adiabatic STT The first striking consequence of the dissipationless super- current spin torques is the possibility to realize the quasiequi-librium magnetic texture dynamics driven solely by the adi-abatic STT generated by the superconducting current. In theabsence of dissipation, the LLG equation ( 1) has a solution in the form of a traveling wave m=m(x−ut) with a constant velocity determined by the spin current u=2μ B˜Jz/M.F o ra periodic magnetic structure, e.g., magnetic helix, this yieldsa locally rotating magnetization with a frequency definedbyω∼u/L, where Lis the period. However, these time- dependent quasiequilibrium solutions do not correspond to theground state. It can be reached only in the presence of theGilbert damping, which transforms the magnetic texture insuch a way to compensate the effective field generated by thespin-polarized supercurrent. Therefore, eventually the systemswill stop at the stationary state when H eff=0. In the absence of dissipation, the same quantity udetermines the characteristic velocity of the domain wall motion by the adiabatic STT inthe system. In principle, current-driven motion of DWs inJosephson junctions with strong ferromagnets can be realizedin different systems with high enough critical current densities.High critical currents through strong ferromagnets are typicallycarried by equal-spin triplet correlations, which decay on thelength scale ξ Nσinside the ferromagnet [ 25]. The Josephson current carried through strong ferromagnets by equal-spintriplet pairs was experimentally reported in different systems [54–58]( s e ea l s oR e f .[ 26] for review), which are the promising elements for the dissipationless superconducting spintronics.Here, we can estimate ufor the parameters of half-metallic CrO 2nanostructures [ 56]. The maximal Josephson current density through the CrO 2nanowire is jc∼109A/m2, which determines the spin current ˜Jz=jc/(2e). Taking into account the saturation magnetization M=4.75×105A/m, we get the speed of the order of u=1 m/s. As we show below, in case if the initial state contains DW, the ground state modified bythe supercurrent can correspond either to the distorted DW orto the homogeneous state when the DW is eliminated fromthe sample. The dynamics of the initial state containing a DWunder the applied supercurrent in the presence of the Gilbertdamping in the LLG equation is also considered below. B. Domain wall motion Now we consider the magnetic texture of the ferromagnet in the form of the DW. We are interested in its dynamics inducedby the supercurrent spin torques, discussed above. The twoparticular types of DW are considered: head-to-head DW andNeel DW. The particular shape of the DW is dictated by the combi- nation of the anisotropy energy and the exchange energy. Westart with the head-to-head DW. In this case, the correspondingenergy term can be written as follows: F=1 2/integraldisplay d3r/bracketleftbig K⊥m2 y−Km2 x+Aex(∇xm)2/bracketrightbig , (33) where K> 0 andK⊥>0 are the anisotropy constants for the easy and hard axes, respectively. Aexis the constant describing the inhomogeneous part of the exchange energy. The effectivemagnetic field H eff=(1/M)(Kmxx−K⊥myy+Aex∇2 xm). It is convenient to parametrize the magnetization as follows: m=(cosθ,sinθcosδ,sinθsinδ), (34) where in general the both angles depend on ( x,t). At zero applied supercurrent, the equilibrium shape of the DW is givenbyδ=π/2 and cosθ=± tanh[(x−x 0)/dw], (35) where dw=√Aex/K is the DW width. The above ansatz corresponds to the head-to-head DW, lying in the xzplane. The tail-to-tail DW can be obtained by θ→θ+π. Let us consider the behavior of the head-to head DW under the applied supercurrent and the presence of SOC given bythe superposition of the Rashba-type term 2 m FμBBx/M= (0,−βR,0) and the Dresselhaus-type term 2 mFμBBx/M= (βD,0,0). First, we follow the Walker’s procedure [ 59] by assuming thatδ=δ(t) and the DW is moving according to the time- dependent shift x0(t)=/integraltextt 0v(t/prime)dt/primein Eq. ( 35). Substituting this ansatz to the LLG equation, we obtain that this type of thesolution exists only in the absence of Rashba SOC, β R=0. We assume that the distortion of the wall is small duringthe wall motion, that is, δ=π/2+δ 1, where |δ1|/lessmuch 1. In this case, taking into account that dw∇xθ=sinθfor the DW, 014521-5I. V . BOBKOV A, A. M. BOBKOV , AND M. A. SILAEV PHYSICAL REVIEW B 98, 014521 (2018) we obtain ∂tδ1=−αv dw−2˜JzβD, (36) (1+α2)v−u=γdwK⊥δ1 M−2dw˜JzαβD. (37) In this case, Eqs. ( 36) and ( 37) yield the following equation forv(t): ∂tv+γαK ⊥ M(1+α2)v=−2dwγK⊥uβD M(1+α2). (38) Taking into account the initial condition determined by Eq. ( 37), (1+α2)v(t=0)=u−2dwαβD˜Jz, which follows fromδ1(t=0)=0, we determine the solution of Eq. ( 38)i n the form v(t)=/bracketleftbigg u+2dw˜JzβD α/bracketrightbigge−t/td (1+α2)−2dw˜JzβD α, (39) δ(t)=π 2+td(1−e−t/td) 1+α2/bracketleftbigguα dw−2˜JzβD/bracketrightbigg . (40) where td=(1+α2)M/(αγK ⊥) is the characteristic time scale. The solution for the moving DW expressed by Eqs. ( 39) and ( 40) exactly coincides with the solution found for the DW motion in normal ferromagnets under the influence ofthe adiabatic and nonadiabatic torques [ 60]. But, nevertheless, there is an important physical difference between the spin-orbittorque, considered here, and the nonadiabatic spin torque. Asit can be seen from Eq. ( 27), the SO torque is equivalent to the torque, generated by an external applied field γH= −4μ BmF˜Jz jBj/M. Consequently, it moves DWs of opposite types ( +/−and−/+) to opposite directions as opposed to the action of the nonadiabatic torque, which moves all the DWs inone and the same direction. At the same time, it is seen fromEqs. ( 36) and ( 37) that the Rashba SO torque is equivalent to the field perpendicular to the wall plane, therefore it does notmove the DW and only distorts it. The solution ( 39) and ( 40)i s only valid for small enough electric and, correspondingly, spincurrents, applied to the system. If the current is large enough,the condition |δ 1|/lessmuch 1 is violated and Eqs. ( 36) and ( 37)a r e not valid. It was shown [ 15] that in this regime for ˜Jz>˜Jcrit the DW can be moved even by the adiabatic torque only. We consider the regime of arbitrary values of the applied current numerically by solving Eq. ( 2) together with the expressions for the torque, Eqs. ( 25)–(27), and the effective fieldHeff, found from Eq. ( 33). The results for the case of small enough applied currents, when our analytical solutionsare valid, are represented in Fig. 1. The figure demonstrates the displacement of the DW center as a function of time.The black curve corresponds to the case of no spin-orbittorque. The blue and pink curves are for the Rashba case,β R/negationslash=0,βD=0. They demonstrate that the Rashba spin-orbit torque does not move the DW in this case, as it was mentionedabove. The green and red curves demonstrate the influence ofthe Dresselhaus SO torque on the DW motion. In agreementwith our analytical calculations, the numerics gives that att/greatermucht dthe DW moves with a constant velocity. The direction of the motion is determined by the sign of βDor, in other words, by the sign of the effective magnetic field. In this case,0 2 4 6 8 10−0.2−0.10.00.10.20.30.4x/dw tγK/M FIG. 1. The displacement of the DW as a function of time in the regime below the threshold current ˜Jcrit.βR=βD=0 (black), βR= −0.05,βD=0 (blue), βR=0.05,βD=0 (pink), βR=0,βD= −0.05 (green), βR=0,βD=0.05 (red). The other parameters are K⊥/K=3.0,α=0.2a n d ˜Jz=− 0.1Kdwfor all the curves. it is possible that the DW reverses the direction of its motion if the adiabatic spin torque tends to displace it in the directionopposite to the one dictated by the effective field. This case isillustrated by the red curve in Fig. 1. The regime of large applied currents ˜J z>˜Jcrit, when the DW can be moved by the adiabatic torque only, is shownin Fig. 2. We have obtained that the value of ˜J critis rather close to Kdw. Therefore the critical electric current density is of the order eKd w/¯h∼1010A/m2, which is an order of magnitude larger than the Josephson critical current obtainedin experiment [ 56]. Again, the black curve in Fig. 2shows the displacement of the DW in the absence of the SO torques. Theinitial dynamics of the DW at small tcoincides with Fig. 1, but at larger values of tthe situation changes, so that in this regime the DW moves, but its velocity is not constant. TheRashba SO torque does not cause any essential influence onthe DW dynamics, as in the case of the small applied currents.However, the effect of Dresselhaus SOC is significant and at thefirst glance unexpected. Indeed, as shown in Fig. 1, the torque generated by this type of SOC, e.g., for β D<0, moves the DW to the direction x> 0. However, in the above-threshold regime, it can also reduce the averaged DW velocity (green 5 10 15 205101520x/dw tγK/M FIG. 2. The displacement of the DW as a function of time in the regime above the threshold current ˜Jz>˜Jcrit.βR=βD= 0 (black), βR=− 0.05,βD=0 (blue), βR=0.05,βD=0 (pink), βR=0,βD=− 0.05 (green), βR=0,βD=0.05 (red). The other parameters are K⊥/K=3.0,α=0.2a n d ˜Jz=− 1.5Kdwfor all the curves. 014521-6SPIN TORQUES AND MAGNETIC TEXTURE DYNAMICS … PHYSICAL REVIEW B 98, 014521 (2018) 5 10 15 205101520(a)x/dw tγK/M 5 10 15 20010203040 (b)x/dw tγK/M FIG. 3. The displacement of the DW as a function of time in the regime above the threshold current ˜Jcritfor the case of βR=0a n d different values of the Dresselhaus SO coupling. (a) βD=0 (black), 0.025 (pink), 0.05 (blue), 0.075 (red); (b) βD=0 (black), −0.025 (dashed black), −0.05 (green), −0.075 (dashed green), −0.1 (tan), −0.15 (dashed tan). The other parameters are as in Fig. 2. curve in Fig. 2), that is, the combined action of the adiabatic ST torque and SO torque cannot be viewed just as a simple sumof independent motions due to the both reasons. Vice versa, theSO torque generated at β D>0, which by itself tends to move the DW to the direction x< 0, can slightly enhance the average DW velocity, as it is demonstrated by the red curve. The influence of the Dresselhaus SOC on the DW average velocity is represented in Fig. 3in more detail. Figure 3(a) demonstrates the displacement of the DW as a function oftimetfor several values of β D>0. It is seen that there is a weak increase of the average velocity at βD>0, but the more important and pronounced effect is that increasing βDleads to the decrease of the velocity oscillation period. The case βD<0 is shown in Fig. 3(b), where one can see that the dependence of the average DW velocity on βDis nonmonotonous. While at smaller values of |βD|the average velocity is indeed reduced with respect to the case βD=0, at larger values of |βD|, the velocity starts to increase and exceed its value at βD=0 considerably. This behavior can be understood in the framework of the analogy between the SO torque and the magnetic-fieldinduced torque. For the situation when the DW moves underthe combined action of the current-induced torque and field-induced torque, it is known that the steady motion of the DWwith ˙δ=0 is only possible for a range of fields and currents [61]. The lines in the ( ˜J z,H) plane, separating the regions of steady motion and precession motion ˙δ/negationslash=0, are called by the Walker-like stability lines [ 61]. This limit condition for thesteady motion is strictly equivalent to the Walker breakdown [59] condition in the case where only an external magnetic field is applied. For the problem under consideration, theincrease of β Dabsolute value at fixed current is equivalent to the increase of the applied field (at fixed current). When atzeroβ Dthe system is in the precession regime, as in Fig. 2,t h e increase of |βD|atβD<0 moves the system towards the steady motion region, where the wall velocity is higher. Therefore thetransition from the precession regime to the steady regime inFigs. 1and2is analogous to crossing the Walker-like stability lines for problem of DWs motion under the combined actionof the current-induced torque and field-induced torque. Let us now consider the Neel DW. In this case, the combi- nation of the anisotropy energy and the exchange energy takesthe form F=1 2/integraldisplay d3r/bracketleftbig K⊥m2 z−Km2 y+Aex(∇xm)2/bracketrightbig . (41) It is convenient to parametrize the magnetization as m=(sinθsinδ,cosθ,sinθcosδ). (42) At zero applied supercurrent, the equilibrium magnetization profile is described by Eqs. ( 35) andδ=π/2. It can be shown that the problem of the Neel DW motion in the presenceof the Rashba SO coupling is mathematically equivalent tothe considered above motion of the head-to-head DW in thepresence of the Dresselhaus SO coupling with the substitutionβ D→−βR. Therefore, in this case, the Rashba SO torque plays the part of the field-induced torque moving DWs. The above analysis demonstrates that the dynamics of a DW under an applied supercurrent depends strongly (i) onthe particular type of the DW and (ii) on the particular typeof the SO coupling, which induces the spin-orbit torque. Thestationary motion of the DWs induced by small supercurrentsis possible even in the absence of the nonadiabatic torque ifthe spin-orbit torque is present in the system. Due to the presence of the Gilbert damping, the motion of a DW by a supercurrent is not a disspationless process.Interestingly, the DW motion generates voltage across thejunction in the regime when the charge current is fixed butits magnitude is smaller than the Josephson critical current ofthe system. In this situation, the voltage can manifest itselfas an additional step at the current-voltage characteristics ofthe junction at j<j c, where jcis the critical current of the junction. The voltage amplitude Vcan be roughly estimated from the balance of the energy dissipation rate in the magneticsubsystem due to the Gilbert damping and the power put in bythe current source. The characteristic energy dissipation ratecan be estimated as ˙F∼/Delta1F/t d, where /Delta1Fis the difference between the free energies of the equilibrium state of the DWat zero current and the nonequilibrium state of the distortedwall in the presence of the current. Our quasiequilibriumconsideration of the DW dynamics is strictly valid only if eV is small with respect to the characteristic inverse timescale ofthe problem γK/M . For small distortions of the DW, /Delta1Fcan be obtained as follows: /Delta1F=1 2/integraldisplay d3r[K⊥sin2θ+Aex(∇xθ)2cos2θ]δ2 1.(43) 014521-7I. V . BOBKOV A, A. M. BOBKOV , AND M. A. SILAEV PHYSICAL REVIEW B 98, 014521 (2018) Substituting the equilibrium profile of the DW θ(x) given by Eq. ( 35) into Eq. ( 43), we obtain /Delta1F=Spdw(K⊥+K/3)δ2 1, (44) where Spis the cross-section area of the ferromagnet. The voltage, generated at the Josephson junction can be estimatedasV∼˙F/S pjc, which yields V∼γδ2 1αdwK⊥(K⊥+K/3) jcM¯h, (45) where we have assumed that α/lessmuch1. For estimations, we use the material parameters of the CrO 2nanostructures [ 62], which are the promising systems for the dissipationless spintronics [ 56]. Taking the maxi- mal Josephson current density through the CrO 2nanowire to be jc∼109A/m2, the saturation magnetization M= 4.75×105A/m ,dw=10−6cm,K=1.43×105erg/cm3, andK⊥=3 K, we obtain V∼0.1δ2 1mV, where we took into account the typical values of the Gilbert damping α∼ 0.01. The amplitude of DW distortion angle can be varied in wide limits, e.g., δ2 1∼10−4–10−3for the red curve in Fig. 1, δ2 1∼10−3for the green curve in Fig. 1andδ2 1∼10−1for the dashed green curve in Fig. 3(b). The estimated values of the induced voltage Vare small with respect to the characteristic superconducting scales ∼0.1 mV for Al superconductors, therefore our assumption of quasiequilibrium quasiparticledistribution works rather well. From the other hand, the strictcalculation of the voltage induced at the Josephson junctionrequires accounting for dynamics of the superconducting phaseinduced by the DW motion in the current-phase relation. Thisis beyond the scope of the present paper and will be doneelsewhere. VI. CONCLUSION To conclude, we have calculated the spin-transfer torques acting on the magnetic textures from the spin-polarized super-conducting current flowing through the ferromagnetic mate-rial. For this, we take the advantage of the widely used adiabaticapproximation, bringing it from the realm of single-electrondynamics into the field of superconductivity governed by thepropagation of the spin-triplet Cooper pairs generated at theSC/FM interface. This approximation enables us to find theanalytical expression for the spin torques in the most generalcase of the spin texture and develop the efficient formalism ofthe generalized quasiclassical theory for calculating the chargeand spin supercurrents through the inhomogeneous magneticsystems. We show that the supercurrent-driven dynamics ofDWs crucially depends on the type and magnitude of thespin-orbital coupling. The obtained results demonstrate thatthe DW motion by the supercurrent is a phenomenon realistic for the recently developed Josephson junctions through CrO2nanowires. ACKNOWLEDGMENTS This work was supported by the Academy of Finland Research Fellow (Project No. 297439) and RFBR Grant No.18-02-00318. We thank Jan Aarts and Tero Heikkila for interestingdiscussions that initiated this project. APPENDIX: CALCULATION OF THE EFFECTIVE FIELD ( 32) From Eqs. ( 30) and ( 31), we obtain ∂FJ ∂M=2˜Jz x Mδ/integraltextd/2 −d/2Zxdx δm. (A1) Let us consider the following form of the unitary matrix ˆU=exp[−iσx(δ 2+π 4)] exp[ −iσy(θ 2+π 4)], which yields the texture part of the gauge field Zm=− cosθ∇δ/2, where mx=cosθand tan δ=mz/myso that ∇δ=(my∇mz− mz∇my)/m2 ⊥, where m⊥=√ m2 y+m2 z. Then we get δ δmx/integraldisplayd/2 −d/2Zmdx=−x 2m2 ⊥(my∇mz−mz∇my), (A2) δ δmy/integraldisplayd/2 −d/2Zmdx=−y 2m2 ⊥mz∇mx, (A3) δ δmz/integraldisplayd/2 −d/2Zmdx=z 2m2 ⊥my∇mx. (A4) Hence 2/parenleftBigg M×δ/integraltextd/2 −d/2Zxdx Mδm/parenrightBigg y =1 m2 ⊥[mz(mz∇xmy−my∇xmz)−mymx∇xmx]=∇ xmy. (A5) Treating analogously other components and the spin-orbital part of the gauge field, we get 2/parenleftBigg M×δ/integraltextd/2 −d/2Zm xdx Mδm/parenrightBigg =∇ xm, (A6) 2/parenleftBigg M×δ/integraltextd/2 −d/2Zso xdx Mδm/parenrightBigg =− 2mFBx×M. (A7) Combining that into the total effective field yields Eq. ( 32). [1] A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,372 (2012 ). [2] N. Locatelli, V . Cros, and J. Grollier, Nat. 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PhysRevApplied.10.054046.pdf

PHYSICAL REVIEW APPLIED 10,054046 (2018) Near- TcFerromagnetic Resonance and Damping in FePt-Based Heat-Assisted Magnetic Recording Media Daniel Richardson,1Sidney Katz,1J. Wang,2Y. K. Takahashi,2Kumar Srinivasan,3Alan Kalitsov,3 K. Hono,2Antony Ajan,3and Mingzhong Wu1,* 1Department of Physics, Colorado State University, Fort Collins, Colorado 80523, USA 2National Institute for Materials Science, Sengen 1-2-1, Tsukuba 305-0047, Japan 3Western Digital, San Jose, California 95131, USA (Received 2 August 2018; revised manuscript received 29 September 2018; published 20 November 2018) High-temperature ferromagnetic resonance (FMR) in FePt-based media materials is studied for the first time. The FMR linewidth ( /Delta1H) as a function of temperature ( T), field angle ( θH), and the volume fraction ( x) of carbon in the material is determined, and the effective Gilbert damping constant and the Bloch−Bloembergen relaxation time are estimated. The data suggest that at temperatures 10–45 K below the Curie temperature, two-magnon scattering and spin-flip magnon-electron scattering make compara-ble contributions to /Delta1H. With a decrease in T,/Delta1Hincreases due to enhancement of the two-magnon scattering. /Delta1Hcan be tuned via varying xand shows a maximum at θ H≈45° when varying θH. DOI: 10.1103/PhysRevApplied.10.054046 I. INTRODUCTION Heat-assisted magnetic recording (HAMR), the most promising technology for next-generation hard disk drives, makes use of a laser to heat the recording media to an elevated temperature, near the Curie temperature ( Tc), to significantly reduce the coercivity of the media material and thereby ease the switching of the magnetization [ 1–4]. HAMR drives are expected to be released to the market in the near future, but understanding of the damping at high temperatures near Tcin HAMR media has not been real- ized yet, although it is of great fundamental and practical interest. Fundamentally, the physical mechanisms underlying the near- Tcdamping in HAMR media are unclear, although there have been interesting experimental stud- ies on damping properties at room temperature (RT) in perpendicular recording media materials including HAMR media [ 5–9]. Further, it is also unknown which macroscopic model is more suitable to describe the near- Tcdamping in HAMR media, although several dif- ferent models have been previously used to analyze damping at high temperature ( T), including the Gilbert equation [ 10,11], the Bloch-Bloembergen (BB) equation [12], the Landau-Lifshitz-Bloch equation [ 13–16], the Xu-Zhang equation [ 17,18], and the Tzoufras-Grobis equation [ 19]. Practically, the nature and strength of the damping in the HAMR media is directly related to the switching time and the signal-to-noise ratio of *mwu@colostate.eduthe reading, which significantly impact the hard drive performance. Previous experimental efforts on investigating damping in perpendicular recording media include three studies on damping in L10ordered FePt thin films [ 7–9], which have been widely accepted as the media material for next- generation HAMR drives. Those studies used the same approach, the optical pump-probe technique, to measure the effective Gilbert damping constant ( αeff), but the αeff values obtained are inconsistent, possibly due to the dif- ferences in the sample properties, such as the degree of L10order and the strength of the perpendicular anisotropy, or the experimental details. Specifically, Mizukami et al. found αeff=0.055 and also emphasized that the intrin- sic damping should be smaller than this value [ 7], while Becker et al . found a much larger value, αeff=0.1, and claimed that this value was mostly intrinsic and contained little extrinsic contribution if any [ 8]. Lee et al. obtained an even larger value which was 0.21 [ 9]. Although these studies represent the first attempts on exploring damp- ing in FePt-based HAMR media, the measurements were all carried out at RT, rather than at the elevated tem- peratures at which the writing operation occurs. Possible relaxation routes in FePt media include spin-flip magnon- electron scattering (SF MES, interband scattering) [20–23], magnon-electron scattering associated with Fermi surface breathing (intraband scattering) [ 20–23], two- magnon scattering (TMS) [ 5,6,24–26], and magnon- phonon scattering [ 27]. As these relaxation processes all exhibit strong Tdependence, the damping value near Tcin FePt HAMR media may differ significantly from the RT values cited above. 2331-7019/18/10(5)/054046(9) 054046-1 © 2018 American Physical SocietyDANIEL RICHARDSON et al. PHYS. REV. APPLIED 10,054046 (2018) This paper reports an experimental study of near- Tc damping in FePt-C granular films that have structure and properties very similar to practical L10FePt-based HAMR media. Specifically, ferromagnetic resonance (FMR) experiments are performed along three distinct dimensions of important relevance to HAMR applications, (1) the volume fraction of carbon ( x) in the media, (2) the media temperature ( T), and (3) the angle of the external magnetic field ( θH) relative to the film normal direction, at temperatures just below Tc. The FMR linewidth ( /Delta1H) data as a function of T,x,a n dθHare determined, and the effec- tive damping constant αeffin the Gilbert model [ 10,11]a n d the transversal relaxation time T2in the BB model [ 12,27] are estimated. The data indicate that at temperatures about 10–45 K below Tc, relevant to Tin a HAMR writing oper- ation, the TMS and SF MES processes coexist and make comparable contributions to /Delta1H. With a decrease in T, however, /Delta1Hincreases, mostly due to the enhancement of the TMS process. Via varying x,/Delta1H,αeff,a n d T2can be tuned as large as a factor of four near Tc. When varying θH,/Delta1Handαeffshow a maximum at about 45°, which is an angle relevant to the actual HAMR writing operation. Theαeffvalues obtained are smaller than those measured at RT in previous works [ 7–9]. II. STRUCTURE AND STATIC MAGNETIC PROPERTIES OF SAMPLES The samples are grown on single-crystal (001) MgO substrates by dc magnetron sputtering and consist of a 10-nm-thick FePt-C granular layer with the carbon vol- ume fraction x=0%, 10%, 20%, or 30% and a 3-nm-thick carbon capping layer. The base pressure of the sputtering chamber is 5.0 ×10−7Pa or lower. Prior to the sputtering growth, the MgO substrate surface is thermally cleaned at 600 °C for 1 h. After that, the MgO substrate is maintained at the same temperature and a FePt-C granular film is deposited by cosputtering a FePt alloy target and a carbon target under an Ar pressure of 0.48 Pa at a deposition rate of about 0. 2 Å/s. The Fe:Pt atomic ratio in the FePt films is nearly 1:1. The alternating-layer deposition technique is used to suppress the growth of the secondary mis- oriented FePt grains and obtain single-columnar, highly- (001)-textured FePt-C nanogranular films [ 28]. Following the FePt growth, a 3-nm-thick carbon over-coating layer is deposited at RT, which works as the protection layer for the FePt film. The microstructure properties of the samples are measured by transmission electron microscopy (TEM) using an “FEI Tecnai T20” TEM system at an electron accelerating voltage of 200 kV. The magnetization curves and hysteresis loops are measured by a superconductingquantum interference device vibrating sample magnetome- ter using magnetic fields of up to ±70 kOe. More details about the sample growth and characterization are provided in Refs. [ 28–31].Figure 1presents the main data about the microstructure and static magnetic properties of the samples. Figure 1(a) shows the TEM images of the four samples, with the car- bon volume fractions xindicated at the top left corners. Figures 1(b)–1(d) give the average grain size ( d), coerciv- ity ( H c) measured with perpendicular fields, and the ratio of the in-plane remnant magnetization ( MrIP) to the out- of-plane remnant magnetization ( MrOP), respectively, as a function of x. The data are all measured at RT. The data in Figs. 1(a)and1(b) show that the addition of carbon to FePt can effectively break big FePt grains with d≈75 nm into much smaller grains with d≈10 nm; the higher the carbon volume fraction is, the smaller the grains are. The data in Fig. 1(c) indicate that the introduction of 10% carbon can result in a significant increase in Hc, which is mostly due to the formation of physically sepa- rated, vertically oriented, small-size columnar FePt grains and a corresponding transition from domain wall motion- type magnetization reversal to rotation reversal. However, an increase in xto 20% and then 30% results in a notable decrease in Hc, which is mainly due to the size effect of the grains, namely, that the smaller the grains are, the stronger the role the thermal energy plays in magnetization rever- sal. In contrast, the MrIP/MrOPratio exhibits a completely different trend – it increases very little when xincreases from 0% to 10% and then 20%, but increases substantially when xis raised to 30%, as shown in Fig. 1(d). This result suggests that an increase in xfrom 0% to 10% and then 20% results in big changes in both dand Hc, but not in the (001) orientation of the FePt grains, while an increase to x=30% leads to the presence of some misoriented grains in the FePt layer. This degrading of the (001) orientation also explains in part the relatively low Hcvalue measured for x=30%. These results together clearly indicate that one can use the carbon volume fraction as a very effective tool to widely tune the microstructural and magnetic prop- erties of the FePt media, as well as the FMR and damping properties as described below. III. HIGH-TEMPERATURE FMR EXPERIMENTS Figure 2shows the high- TFMR approach which is used to study the near- Tcdamping in the above-described samples. Figure 2(a) shows a schematic diagram of the experimental system. The main components include a rectangular microwave cavity (purple), a diamond rod (yellow) with a diameter of 2 mm that loads the sam- ple (red) into the cavity, and a ceramic heater (gray) that heats the sample through the diamond rod. These components are housed in a high-vacuum chamber, and the measurements are performed at a pressure of about6.7×10 −3Pa (or about 5 ×10−5Torr) to prevent changes in sample properties due to oxygen during high- Tmea- surements. For the FMR data presented in this paper, the microwave frequency ( f) is kept constant at 13.7 GHz, 054046-2NEAR- TcFERROMAGNETIC RESONANCE . . . PHYS. REV. APPLIED 10,054046 (2018) (a) (b) (c) (d) FIG. 1. Microstructural and static magnetic properties of four FePt media samples. (a) TEM images. (b) Average grain size as a function of the carbon-volume fraction ( x). (c) Coercivity as a function of x. (d) The ratio of in-plane remnant magnetization ( MrIP)t o out-of-plane remnant magnetization ( MrOP) as a function of x. which is also the resonant frequency of the microwave cavity, while the external magnetic field is swept. Field modulation and lock-in detection are used, so all the FMR profiles presented in this paper are the derivatives of the FMR power absorption. The sample temperature ( T)i s calibrated through separate measurements using a thermo- couple. Prior to placing the sample in the FMR system, the sample is saturated by an out-of-plane field of 80 kOe at RT. After placing the sample in the FMR system and heating it, prior to each FMR measurement, the sample is saturated by a field of 15 kOe. Figure 2(b) presents the FMR data (blue dots) measured atT=648 K on the “ x=20%” sample and a numerical fit (red curve) to the derivative of a Lorentzian trial func- tion. The Lorentzian fitting peak-to-peak FMR linewidth /Delta1Hand field HFMR are indicated in the figure. A fit (green curve) to the derivative of a Gaussian trial func- tion is also included in the figure. One can see that the Lorentzian fit is better than the Gaussian fit, indicat- ing that the inhomogeneity line-broadening contribution to/Delta1H,if any, is small. In the case where a film sam- ple has strong spatial inhomogeneity and the associated line broadening is large, the Gaussian function would fit the data better than the Lorentzian function [ 32]. It should be noted that the inhomogeneity line-broadeningcontribution may be significant at RT due to the presence of very strong anisotropy. Note also that one can carry out frequency-dependent FMR measurements to determine the inhomogeneity line-broadening contribution, as reported in Refs. [ 33]a n d[ 34]. (a) (b) FIG. 2. High- TFMR. (a) Schematic of experimental setup. (b) Representative FMR power absorption data (blue dots), and Lorentzian fit (red curve) and Gaussian fit (green curve) of the data. The data are measured on the FePt media sample withx=20% at 648 K. The Lorentzian fitting-yielded FMR field H FMRand peak-to-peak FMR linewidth /Delta1Hare indicated in (b). 054046-3DANIEL RICHARDSON et al. PHYS. REV. APPLIED 10,054046 (2018) IV . TEMPERATURE DEPENDENCE OF FMR PROPERTIES Turn now to the high- TFMR data measured using the approach described above. Figure 3presents the data measured on the “ x=20%” sample at six different T. Figure 3(a) gives the FMR data (blue dots) measured at four different T, as indicated, and the corresponding Lorentzian fits (red curves). Figures 3(b) and3(c) plot the HFMRand/Delta1Hobtained by Lorentzian fitting, respectively, as a function of T. Figure 3(d) shows the saturated mag- netic moment ( ms) and coercivity ( Hc) as functions of T. The big blue arrows in Figs. 3(b)and3(c)indicate the over- all trends, while the blue rectangle in Fig. 3(d) indicates the Trange of the FMR measurements. All the measure- ments are taken with a perpendicular magnetic field, that is,θH=0. Prior to discussing the data in Fig. 3, it should be noted that the FMR measurements are carried out over a T range of 634–673 K, as indicated by the blue rectangle in Fig.3(d). No FMR measurements are taken at T<634 K. This is because with a decrease in T, the FMR profile shifts to negative fields, which can be inferred from the trend shown in Fig. 3(b); and the signal-to-noise ratio of the FMR data also become small at low Tdue to linewidth enhancement, which is shown in Fig. 3(c).A t T>673 K, the FMR signal becomes nondetectable, mainly due to asignificant drop in mswhich is evident from the red curve in Fig. 3(d). Note that although the highest measurement temperature of 673 K is still about 22 K below Tc, which is about 695 K as shown in Fig. 3(d), it is already near or close enough in terms of HAMR applications in which the writing operation occurs at temperatures about 10–25 K below Tc. T h ed a t ai nF i g . 3(b) show an overall increase of HFMR with T, and this result suggests that with an increase in Tover 634–673 K, the effective perpendicular anisotropy field Hudrops by a larger amount than the saturation mag- netization 4 πMsdoes. The Kittel equation for the FMR concerned here can be written as 2πf=|γ|(HFMR+Hu−4πMs),( 1 ) where |γ|is the absolute gyromagnetic ratio. One can see that for a given f, an increase in HFMR will mean a decrease in Hu-4πMs. Since the Tdependences of Huand 4πMsdiffer in different samples due to differences in the microstructures, one would expect different HFMR vsT trends in the four samples. This expectation is discussed below. The data in Fig. 3(c)suggest an overall decrease of /Delta1H with increasing T. This result may indicate that TMS is a dominant relaxation mechanism in the Trange consid- ered here. Generally speaking, the damping mechanisms (a) (b) (c) (d) FIG. 3. High- TFMR data measured on the “ x=20%” sample with the field angle θH=0. (a) Representative FMR profiles (blue dots) and corresponding Lorentzian fits (red curves). (b) FMR field HFMRas a function of T. (c) FMR linewidth /Delta1Has a function of T. (d) Saturated magnetic moment (red, left axis) and coercivity (blue, right axis) as a function of T. The blue rectangle in (d) indicates theTrange of the FMR measurements. 054046-4NEAR- TcFERROMAGNETIC RESONANCE . . . PHYS. REV. APPLIED 10,054046 (2018) in FePt medium samples should include SF MES [ 20–23], magnon-electron scattering associated with Fermi surface breathing [ 20–23], TMS [ 24–26], and magnon-phonon scattering [ 27] as listed in the introduction section. Practi- cally, the contributions to /Delta1Hfrom both the Fermi surface breathing-associated magnon-electron scattering and the magnon-phonon scattering should be notably smaller than those from the SF MES and TMS processes. The damping due to the Fermi surface breathing-associated scattering usually decreases with an increase in T, so it is large at low T,but can be ignored near Tc[20–22]. The magnon-phonon scattering generally plays important roles in relaxation in magnetic insulators, such as Y 3Fe5O12and BaFe 12O19 [35,36], but in metallic systems, it usually makes much smaller contributions to the damping than the magnon- electron scattering processes [ 27]. Thus, for the FMR data in this work, one can approximately write /Delta1H=/Delta1HSFMES+/Delta1HTMS,( 2 ) where /Delta1HSFMES and/Delta1HTMSdenote the contributions of the SF MES and TMS processes, respectively. It is known that both /Delta1HSFMES and/Delta1HTMS exhibit strong Tdependences. In general, /Delta1HSFMES increases with T. This is because the SF MES process requires both momentum and energy conservations, which can be sat- isfied more easily at high T[20–22]. In contrast, /Delta1HTMS usually decreases with an increase in Tin magnetic thin films with perpendicular anisotropy. This is because the damping due to the TMS generally scales with the square ofHu[25], while the latter drops as Tapproaches Tc.F o r this reason, the data in Fig. 3(c) seem to indicate that /Delta1HTMSmay be dominant over /Delta1HSFMES over the Trange of 634–673 K. This result is further discussed below. V . EFFECTS OF CARBON-VOLUME FRACTION ON FMR PROPERTIES To confirm the above conclusions and also evaluate the effects of the carbon-volume fraction x, the same FMR measurements and analyses are performed on the other three samples. Figure 4summarizes the main results of all four samples. Note that the measurement tempera- ture range is different for different samples due to the temperature limitations mentioned above. The data in Fig. 4(a) show that the four samples share the same trend, namely, that HFMRincreases with T. This result indicates that in all the samples, Hudrops by a larger amount than 4 πMswhen Tincreases, as discussed above. The data also show that the “ x=10%” and “ x=30%” sam- ples exhibit much stronger Tdependences than the other two samples. This suggests that with an increase in T, Hu-4πMsdrops faster in the “ x=10%” and “ x=30%” samples. In other words, Hudecreases by a larger amount than 4πMsdoes in the “ x=10%” and “ x=30%” samples, (a) (b) FIG. 4. Comparison of high- TFMR data of four samples with different carbon-volume fractions as indicated, measured at the field angle θH=0. (a) FMR field HFMRa saf u n c t i o no f T.( b ) FMR linewidth /Delta1Ha saf u n c t i o no f T. The vertical arrows indi- cate the Tcvalues of the four samples, with the colors matching those of the data sets. but not as large an amount as in the other two samples. This result supports the above-drawn conclusion that one can effectively manipulate the magnetic properties of the HAMR media via tuning the carbon-volume fraction in the media. T h ed a t ai nF i g . 4(b) show that /Delta1Hdecreases with an increase in Tfor all four samples, indicating that the TMS process is a dominant relaxation mechanism in all the samples. The data also indicate that the “ x=10%” and “x=30%” samples show much stronger Tdependences than the other two samples. This result is consistent with the above-described result on the Tdependences of HFMR. This consistency supports the above conclusion that the TMS is a dominant damping mechanism. In general, HFMR increases with a decrease in Huas shown in Eq. (1)while /Delta1HTMSscales with H2 uas discussed in Ref. [ 25]. For this reason, a larger drop in Huwill give rise to a larger increase inHFMRand a larger decrease in /Delta1H. VI. FIELD-ANGLE DEPENDENCE OF FMR PROPERTIES The FMR data presented in Figs. 2–4are all mea- sured with a perpendicular magnetic field ( θH=0), but in actual HAMR applications, the writing operation occurs at θH=35°–45°. For this reason, high- TFMR measurements are also performed at different field angles, with the major results presented in Fig. 5. Note that the largest angle used in the experiments is 65°, and the physical constraints of the FMR system do not allow for measurements at larger angles ( θH>65°). The data in Fig. 5(a) indicate that the HFMR vsT responses show different trends for different θH. This is because the roles of Huand 4πMsin the FMR strongly depend on the equilibrium direction of the magnetization 054046-5DANIEL RICHARDSON et al. PHYS. REV. APPLIED 10,054046 (2018) (a) (b) (c) FIG. 5. Field angle-dependent high- TFMR data measured on the “ x=20%” sample. (a) and (b) show the FMR field HFMR and linewidth /Delta1H, respectively, as a function of Tmeasured at six different θH. (c) shows /Delta1Has a function of θHfor two different T,a s indicated. The vertical arrows in (a) and (b) indicate the Tcof the sample. vector in the materials, as described by [ 37] /parenleftbigg2πf |γ|/parenrightbigg2 =[HFMRcos(θH−θM) +(Hu−4πMs)cos(2θM)] ·[HFMRcos(θH−θM) +(Hu−4πMs)cos2(θM)], (3) where θMis the angle of the equilibrium magnetization rel- ative to the film normal direction. It is expected that as Tincreases toward Tc,θH−θM,Hu,a n d4 πMsapproach zero and HFMRbecomes closer to (2πf)/|γ|=4.89 kOe. This trend for HFMRis somewhat shown in Fig. 5(a). The data in Figs. 5(b) and5(c) together indicate that /Delta1Hshows a clear θHdependence, and this dependence is very strong at lower T,but less pronounced at higher T. These results agree with the expectations of the TMS pro- cess. Specifically, the strength of the TMS strongly relies on the spin-wave manifold, while the latter varies with the magnetic field direction. This gives rise to a strong θH dependence of /Delta1HTMS[5,6]. With an increase in T, how- ever,/Delta1HTMS decreases due to its proportionality to H2 u [25], and the weight of /Delta1HTMSin/Delta1Hdecreases accord- ingly, leading to a weaker θHdependence. It should be noted that the /Delta1HvsθHdata in Fig. 5(c) seem to show more than one peak. This multipeak behavior differs from the TMS-associated single-peak responses discussed in Refs. [ 5]a n d[ 6]. Future studies that could confirm the existence of the second peak at θH>65° and explore its physical origin are of great interest. Further, the data in Figs. 5(b) and 5(c) also suggest that near Tc, the SF MES process makes notable contribu- tions to /Delta1Hand/Delta1HSFMES is comparable to /Delta1HTMS. This result is supported by two observations. First, the extrapo- lation of the data shown in Fig. 5(b) toTcseems to give a nonzero /Delta1Hvalue, as indicated by the dashed gray lines in the figure. This nonzero contribution is mostly from theSF MES process, because /Delta1HTMSdecreases to zero when Tapproaches Tc[25], while /Delta1HSFMES usually increases with Tand makes a notable contribution near Tc[20–22]. Second, the data in Fig. 5(c) indicate a nontrivial compo- nent of /Delta1Hthat does not vary with θH. This component is most likely /Delta1HSFMES as it is known to exhibit a very weakθHdependence [ 5,6]. Thus, one can see that the SF MES and TMS processes coexist and make comparable contributions to /Delta1HatT=675 K. One can draw four main conclusions from the above- discussed results on the T,x,a n dθHdependences of the FMR data. (1) At temperatures about 10–45 K below Tc, which are the temperatures relevant to the HAMR writing operation, the TMS and SF MES processes coexist in the FePt-based HAMR media and make comparable contribu- tions to /Delta1H. (2) With a decrease in T,/Delta1Hincreases due to the enhancement of the TMS process. (3) When θHis var- ied,/Delta1Hshows a maximum at about 45°, which is an angle relevant to the HAMR writing operation. (4) The strength of the Tdependence of /Delta1Hcorrelates with that of the T dependence of Hu-4πMs, while the latter can be effectively tuned by the carbon-volume fraction. VII. ESTIMATION OF DAMPING CONSTANTS One can use the /Delta1Hdata to estimate the effective damping constant αeffin the Gilbert model [ 10,11]a s αeff=√ 3|γ|/Delta1H 2(2πf)(4) as well as the transversal relaxation time T2in the BB model [ 12]a s T2=1√ 3|γ|/Delta1H(5) where |γ|/(2π)=2.8 MHz/Oe. The estimated values together with the /Delta1Hdata are listed in Tables IandII. 054046-6NEAR- TcFERROMAGNETIC RESONANCE . . . PHYS. REV. APPLIED 10,054046 (2018) TABLE I. Comparison of near- TcFMR linewidth ( /Delta1H), effec- tive Gilbert damping constant ( αeff), and the BB transversal relaxation time ( T2) for four samples with different carbon- volume fractions ( x). The data are measured at a field angle of θH=0. xT c(K) T(K) /Delta1H(Oe) αeff T2(ns/rad) 0% 680 670 48 0.0084 0.684 10% 710 675 195 0.0345 0.168 20% 695 673 88 0.0155 0.373 30% 705 660 86 0.0152 0.382 It should be mentioned that the reason the αeffvalues are estimated and listed here is that the Gilbert model has been widely considered in previous studies on damping in perpendicular media [ 5–9,11], not that the Gilbert model is the most appropriate model to describe near- Tcmag- netization dynamics. It is known that the Gilbert model assumes a conserved magnetization vector length during the relaxation, but most likely this is not the case at high T. In comparison, the BB model appears to be a better model because it involves two separate relaxation pro- cesses – the longitudinal relaxation or the T1process, and the transversal relaxation or the T2process, and thereby does not assume a conserved magnetization vector [ 12,27]. Note that the TMS relaxation involves a decrease in the magnetization vector length; it could be described by the T2process, but not the Gilbert model [ 27]. There are five important points that should be made about the data listed in Tables IandII. (1) The data in Table Iindicate that by varying the carbon-volume frac- tion xone can tune the /Delta1H,αeff,a n d T2parameters of the FePt-based HAMR media by as much as a factor of four. (2) The data in Table IIshow that the damping at field angles relevant to the HAMR writing operation is relatively larger. For example, the damping at θH=45° is about 1.8 times that at θH=0°. (3) The αeffvalues in Tables Iand IIrepresent the upper limit of the Gilbert damping constant, as /Delta1Hmay include a small contri- bution due to inhomogeneity line broadening [ 32]. This contribution was ignored during the estimation for the reason mentioned in the discussion about the numerical fits in Fig. 2(b).( 4 )T h e αeffvalues listed are all smaller than the values (0.055–0.21) measured on FePt materi- als at RT in previous studies [ 7–9]. Possible reasons for this inconsistency include that the TMS process [ 25]a n d the Fermi surface breathing-associated damping [ 20–22] make stronger contributions to the overall damping when Tis decreased. (5) Strictly speaking, one cannot use Eq. (4)to obtain the αeffvalues listed in Table II. In addi- tion to the fact that /Delta1Hincludes a contribution from the TMS which cannot be described by the Gilbert model as explained above, the calculations also assume θH=θM.TABLE II. Comparison of near- TcFMR linewidth ( /Delta1H), effec- tive Gilbert damping constant ( αeff), and the BB transversal relaxation time ( T2) for six different field angles ( θH). The data are measured on the “ x=20%” sample. θH(°) T(K) /Delta1H(Oe) αeff T2(ns/rad) 0 673 88 0.0155 0.373 30 675 102 0.0180 0.32245 675 157 0.0277 0.209 50 675 90 0.0159 0.365 55 675 98 0.0173 0.33565 675 182 0.0321 0.180 The difference between these two angles can be small near Tc, but it is definitely nonzero. VIII. CONCLUSIONS AND OUTLOOK In summary, the near- TcFMR of FePt-C HAMR media is studied in this work. The FMR linewidth ( /Delta1H)d a t aa sa function of the sample temperature ( T), the carbon-volume fraction ( x) in the media, and the magnetic field angle ( θH) are determined, and the effective damping constant in the Gilbert model and the transversal relaxation time in the BB model are estimated. The data indicate that at temperatures about 10–45 K below Tc, the TMS and SF MES processes coexist and make comparable contributions to /Delta1H. With a decrease in T,/Delta1Hincreases due to the enhancement of the TMS process. The strength of the Tdependence of /Delta1H correlates with that of the Tdependence of Hu-4πMs, while the latter can be effectively tuned by x. As a result, via vary- ingx,one can tune the relaxation parameters by a factor of four. The FMR linewidth and damping parameters exhibit a strong θHdependence, showing a maximum at θH=45°. It should be noted that although the contributions of the TMS and SF MES processes to the damping are found to be comparable near Tc, they are not quantified in this work. Future work that takes FMR measurements over a wider angle range (0°–90°) as well as frequency-dependent FMR measurements and then numerically fits the angle- and frequency-dependent linewidth data to separate and quan- tize those two contributions as in previous studies [ 5,6,38] will be of great interest. Possible approaches for enabling frequency-dependent FMR measurements at high temper- atures include (1) the use of multiple microwave cavities that have different dimensions, and therefore, have differ- ent resonant frequencies and (2) the replacement of the microwave cavity with a hot-resistant, coplanar waveguide structure that, with the help of a vector network analyzer, can allow for broadband FMR measurements [ 39–41]. The development of such broadband high- TFMR spectrome- ters is of practical interest to the magnetics community in general and the HAMR community in particular. Finally, it should be mentioned that it is currently still unclear 054046-7DANIEL RICHARDSON et al. PHYS. REV. APPLIED 10,054046 (2018) whether the Gilbert and BB equations represent appro- priate models for near- Tcmagnetization dynamics or not, although the corresponding damping parameters have been estimated in this work. Future studies that compare the suitability of various models in terms of describing near- Tc damping are of great interest. ACKNOWLEDGMENTS This work was supported by the Advanced Storage Technology Consortium (ASTC). In addition, the work at CSU was also supported by the U.S. National Science Foundation under Grants No. EFMA-1641989 and No. DMR-1407962 and the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Grant No. DE- SC0018994. 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PhysRevB.79.212410.pdf

Theory of spin accumulation and spin-transfer torque in a magnetic domain wall Tomohiro Taniguchi,1,2Jun Sato,1and Hiroshi Imamura1 1Nanotechnology Research Institute, AIST, Tsukuba, Ibaraki 305-8568, Japan 2Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan /H20849Received 23 February 2009; revised manuscript received 7 May 2009; published 24 June 2009 /H20850 We studied the spin accumulation and spin-transfer torque in a magnetic domain wall by solving the Boltzmann equation for spin accumulation with the diffusion approximation. We obtained analytical expres-sions of spin accumulation and spin-transfer torque. Both the adiabatic and the nonadiabatic components of thespin-transfer torque oscillate with the thickness of the domain wall. We showed that the oscillating componentplays a dominant role in the nonadiabatic torque when the domain wall is thinner than the spin-flip length. Wealso showed that the magnitude of the nonadiabatic torque is inversely proportional to the thickness of thedomain wall. DOI: 10.1103/PhysRevB.79.212410 PACS number /H20849s/H20850: 72.25.Ba, 75.45. /H11001j, 75.60.Ch Spin-dependent electron transport in magnetic nanostruc- tures results in many interesting phenomena such as the giantmagnetoresistance effect 1and current-induced magnetization dynamics.2,3Recently, such spin-dependent phenomena in magnetic domain walls have been investigated due to greatinterest in their potential application to spin-electronics de-vices such as spin-motive-force memory 4,5and racetrack memory.6In these devices, data are stored by moving the domain wall using spin-transfer torque. In 2004, Zhang and Li showed that spin-transfer torque in a domain wall can be decomposed into two parts, the adia-batic and nonadiabatic torque. 7The adiabatic torque lies along the spatial gradient of the local magnetization whilethe nonadiabatic torque is perpendicular to this direction. As-suming that the spin accumulation obeys the phenomenologi-cal diffusion equation and is spatially independent of thedomain wall, Zhang and Li 7showed that the ratio of the magnitudes of the adiabatic and nonadiabatic torques is de-termined by the precession frequency of the spin accumula-tion due to the exchange coupling and the spin-flip scatteringtime, and that the nonadiabatic torque is about two orders ofmagnitude smaller than the adiabatic torque. The thickness of a domain wall is determined by the com- petition of the exchange coupling between the localizedmagnetizations and the magnetic anisotropy, and is usuallyon the order of 100 nm for conventional ferromagnetic met-als such as Fe, Co, Ni, and their alloys. However, recentdevelopments in the processing technology for nanostruc-tures have allowed the production of a domain wall whosethickness is on the order of 1–10 nm, by reinforcing theshape anisotropy of a magnetic nanowire 8or trapping a do- main wall in a current-confined-path geometry.9These devel- opments motivated us to study the transport phenomena in athin domain wall. 10–12For such a thin domain wall, we can- not assume the spatial independence of the spin accumula-tion. Thus, it is important to estimate spin-transfer torque bytaking into account the spatial variation in the spin accumu-lation, which would be different from the estimation byZhang and Li. 7Recently, Vanhaverbeke and Viret13calcu- lated spin-transfer torque in a thin domain wall by numeri-cally solving the time-dependent Larmor equation of the spinaccumulation in a moving frame and showed that the nona-diabatic torque is one order of magnitude larger than thatestimated by Zhang and Li 7when the thickness of the do- main wall is comparable to the Lamor precession length. In this Brief Report, we study spin accumulation and spin- transfer torque in a domain wall by solving the Boltzmannequation with a diffusion approximation. We obtained theanalytical expressions of spin accumulation and spin-transfertorque. Both the adiabatic and the nonadiabatic componentsof the spin-transfer torque oscillate with the thickness of thedomain wall. We show that the oscillation plays a dominantrole in the nonadiabatic torque when the domain-wall thick-ness is less than the spin-flip length, which is defined by theproduct of the Fermi velocity and the spin-flip scatteringtime. For a domain wall that is much thinner than the spin-flip length, the nonadiabatic torque is about one order ofmagnitude smaller than the adiabatic torque, which is oneorder of magnitude larger than that estimated by Zhang andLi 7and qualitatively consistent with the results of Vanhaver- beke and Viret.13We also showed that the magnitude of the nonadiabatic torque is inversely proportional to the thicknessof the domain wall. We considered electron transport in a one-dimensional magnetic nanowire with a 180° domain wall which lies over−d/2/H11349x/H11349d/2, where dis the thickness of the domain wall. We assume that the interaction between the conducting/H20849s-like /H20850electrons and the localized /H20849d-like /H20850electrons is de- scribed by an sdexchange interaction, Hˆ sd=−/H20849J/2/H20850/H9268ˆ·Sˆ, where /H9268ˆis the vector of the Pauli matrices, Jis the sd ex- change coupling constant, and Sˆ/H20849x/H20850=/H208490,−sin /H9258,cos/H9258/H20850is the unit vector pointing along the direction of the localized spinangular momentum. The angle /H9258/H20849x/H20850is given by 0 for x/H11021−d/2, /H20849/H9266/d/H20850/H20849x+d/2/H20850for − d/2/H11021x/H11021d/2, and /H9266for x/H11022d/2, respectively. Following Simanek and Rebei,14,15we employ the rotat- ing frame where basic unit vectors are defined as ex=/H9251−1Sˆ/H11003/H20849/H11509Sˆ//H11509x/H20850,ey=−/H9251−1/H11509Sˆ//H11509x, and ez=Sˆ, respectively. We assume that the direction of the localized spin variesslowly compared to the Fermi wavelength /H9261 F, i.e., /H9251=d/H9258/dx/H112702/H9266//H9261F; thus, we could neglect the higher-order terms of /H9251in the following calculation. The spin accumula- tion and spin-transfer torque in a domain wall are obtainedby solving the Boltzmann equation for the Wigner function defined as fˆ/H20849x,p x/H20850=/H20851f/H20849x,px/H208501ˆ+g/H20849x,px/H20850·/H9268ˆ/H20852/2, where f/H20849x,px/H20850PHYSICAL REVIEW B 79, 212410 /H208492009 /H20850 1098-0121/2009/79 /H2084921/H20850/212410 /H208494/H20850 ©2009 The American Physical Society 212410-1andg/H20849x,px/H20850represent the charge and spin-distribution func- tions, respectively. The spin accumulation sand the spin cur- rent density jare defined as s=/H20885gd3p /H208492/H9266/H6036/H208503, /H208491/H20850 j=/H20885vxgd3p /H208492/H9266/H6036/H208503, /H208492/H20850 respectively. It should be noted that the dimensions of sand jare density and density times velocity, respectively. The diffusion approximation, /H20848vx2gd3p//H208492/H9266/H6036/H208503/H11229/H20849vF2/3/H20850s,i sa p - plied to the Boltzmann equation.14Up to the first order of /H9251, the transverse components of the spin accumulation, sxand sy, and spin current, jxandjy, obey the following equations: /H11509sx /H11509x=−1 2Djx+/H9275JT˜ Djy, /H208493/H20850 /H11509sy /H11509x=−/H9275JT˜ Djx−1 2Djy, /H208494/H20850 /H11509jx /H11509x−/H9275Jsy+2 /H9270sfsx=0 , /H208495/H20850 /H11509jy /H11509x+/H9275Jsx+2 /H9270sfsy=/H9251jz, /H208496/H20850 where /H9275J=J//H6036is the Larmor precession frequency, T˜is the momentum relaxation time, /H9270sfis the spin-flip scattering time andD=vF2T˜/3 is the diffusion constant.14,15The longitudinal spin current jzin Eq. /H208496/H20850is given by jz=/H9252je//H20849−e/H20850, where /H9252 andjeare the spin-polarization factor and the electric current density, respectively. In our definition, the positive electriccurrent corresponds to the electron flow along the − xdirec- tion. The physics behind Eqs. /H208493/H20850–/H208496/H20850are as follows. Traveling through the domain wall, the conducting electrons vary thedirection of their spin along the localized spin angular mo- mentum Sˆ. Then, spin accumulation and spin current polar- ized along the ydirection /H20849/H11008 /H11509Sˆ//H11509x/H20850are induced /H20851see Eq. /H208496/H20850/H20852. The accumulated spins precess around Sˆdue to the sdex- change coupling with the precession frequency /H9275J. Then, the xcomponents of the spin accumulation and spin current are induced /H20851see Eq. /H208495/H20850/H20852. Equations /H208493/H20850and /H208494/H20850relate the spin accumulation and spin current by the diffusion constant. Before estimating the spin accumulation and spin-transfer torque in a domain wall, we should emphasize the validity ofour calculations. Since Eqs. /H208493/H20850–/H208496/H20850are obtained by applying the diffusion approximation to the Boltzmann equation, they are valid for d/H11350l mfp, where lmfp=vFT˜is the mean-free path of the conducting electrons. For a domain wall the thicknessof which is much smaller than the mean-free path, i.e.,d/H11270l mfp, the Boltzmann equation should be solved without the diffusion approximation. Moreover, in such a very thindomain wall, we cannot neglect the higher-order terms of /H9251.We assume that the transverse spin accumulation and spin current are such that they vanish at the limit of /H20841x/H20841→/H11009and are continuous at x=/H11006d/2. Then, solving Eqs. /H208493/H20850–/H208496/H20850, the transverse spin accumulations in the domain wall are ob-tained as s x=Re /H20851s+/H20852andsy=Im /H20851s+/H20852, where s+=/H9266/H208491+i/H9256/H20850jz /H9275Jd/H208491+/H92562/H20850/H208751 − exp/H20873−d 2/H5129/H20874cosh/H20873x /H5129/H20874/H20876. /H208497/H20850 Here/H9256=2 //H20849/H9275J/H9270sf/H20850and/H5129is given by 1 /H5129=/H208811 2D/H208491+2 i/H9275JT˜/H20850/H20873i/H9275J+2 /H9270sf/H20874, /H208498/H20850 where kr=Re /H208511//H5129/H20852andki=Im /H208511//H5129/H20852characterize the oscilla- tion and damping of sxandsydue to the sdexchange cou- pling and the spin-dependent scattering, respectively. Asshown in Eq. /H208497/H20850, the transverse spin accumulations can be decomposed into spatially independent /H20849first /H20850and dependent /H20849second /H20850parts. Figure 1shows the spatial dependence of the transverse spin accumulations, s xandsy, for thick /H20849d=100 nm /H20850and thin /H20849d=10 nm /H20850domain walls, respectively. For convenience, sx andsyare divided by jz//H9275J/H20851see Eq. /H208497/H20850/H20852. The parameters are taken to be J=1.0 eV, /H9270sf=10−4ns, and lmfp=3.0 nm, re- spectively. The Fermi velocity is given by vF=/H208812/H9255F/m, where the Fermi energy is taken to be /H9255F=5.0 eV. These are typical values for the conventional transition ferromagneticmetals. 16As shown in Figs. 1/H20849a/H20850and1/H20849b/H20850, for a thick domain wall, the spin accumulation in the domain wall is nearlyspatially independent except at the boundaries of the domainwall x=/H11006d/2. On the other hand, as shown in Figs. 1/H20849c/H20850and 1/H20849d/H20850, for a thin domain wall, the spin accumulations vary in the domain wall and we cannot assume the spatial indepen-dence of the spin accumulations. Let us estimate spin-transfer torque in the domain wall, which is defined asFIG. 1. The spatial variation in the transverse spin accumula- tions for a thick /H20849d=100 nm /H20850and thin /H20849d=10 nm /H20850domain wall; /H20849a/H20850 sxford=100 nm, /H20849b/H20850syford=100 nm, /H20849c/H20850sxford=10 nm, and /H20849d/H20850syford=10 nm, respectively. The magnitudes of sxandsyare divided by jz//H9275J/H20851see Eq. /H208497/H20850/H20852.BRIEF REPORTS PHYSICAL REVIEW B 79, 212410 /H208492009 /H20850 212410-2/H9270=/H20885 −d/2d/2 /H9275Js/H11003Sˆdx. /H208499/H20850 The Landau-Lifshitz-Gilbert equation for the localized mag- netization Mˆ=−Sˆwith the torque /H9270is given by /H11509Mˆ /H11509t=−/H9253Mˆ/H11003B+/H92510Mˆ/H11003/H11509Mˆ /H11509t +/H9253/H6036 2/H9266M/H9270y/H11509Mˆ /H11509x+/H9253/H6036 2/H9266M/H9270xMˆ/H11003/H11509Mˆ /H11509x, /H2084910/H20850 where /H9253is the gyromagnetic ratio, Bis the effective mag- netic field, Mis the magnitude of the magnetization and /H92510is the Gilbert-damping constant. /H9270y=ey·/H9270and/H9270x=ex·/H9270corre- spond to the adiabatic and nonadiabatic torque, respectively.By using Eq. /H208497/H20850, we find that /H9270y=/H9266/H9252je e/H208491+/H92562/H20850−/H9266/H9252je/H20851kr−e−krd/H20849krcoskid−kisinkid/H20850/H20852 ed/H208491+/H92562/H20850/H20849kr2+ki2/H20850 −/H9266/H9256/H9252je/H20851ki−e−krd/H20849kicoskid+krsinkid/H20850/H20852 ed/H208491+/H92562/H20850/H20849kr2+ki2/H20850, /H2084911/H20850 /H9270x=−/H9266/H9256/H9252je e/H208491+/H92562/H20850−/H9266/H9252je/H20851ki−e−krd/H20849kicoskid+krsinkid/H20850/H20852 ed/H208491+/H92562/H20850/H20849kr2+ki2/H20850 +/H9266/H9256/H9252je/H20851kr−e−krd/H20849krcoskid−kisinkid/H20850/H20852 ed/H208491+/H92562/H20850/H20849kr2+ki2/H20850. /H2084912/H20850 The first terms of Eqs. /H2084911/H20850and /H2084912/H20850are identical to the adiabatic and nonadiabatic torque estimated by Zhang andLi, 7respectively. These first terms arise from the spatially independent part of the spin accumulation, i.e., the first termof Eq. /H208497/H20850. It should be noted that these terms are indepen- dent of the thickness of the domain wall d. For a thick do- main wall, these first terms are dominant for spin-transfertorque and the ratio of the magnitude of the adiabatic andnonadiabatic torque, /H20841 /H9270x//H9270y/H20841, is given by /H9256/H1122910−2.7On the other hand, the second and third terms of Eqs. /H2084911/H20850and /H2084912/H20850 arise from the spatial variation in the spin accumulation, i.e.,the second term of Eq. /H208497/H20850. As shown in Figs. 1/H20849c/H20850and1/H20849d/H20850, for a thin domain wall, we cannot neglect the spatial varia-tion in the transverse spin accumulation and these secondand third terms dominate the spin-transfer torque. It shouldbe noted that these terms are inversely proportional to thethickness d. Thus, for a thin domain wall, the strength of the spin-transfer torque is considerably different from that esti-mated by Zhang and Li. 7 Figure 2shows the strength of the adiabatic torque /H9270yand the nonadiabatic torque /H9270xrenormalized by /H9266/H9252je/eagainst the thickness of the domain wall d. We denote the torque for d/H11349lmfpby the dotted line because our calculations are re- stricted for d/H11350lmfp; thus, the torques for d/H11349lmfpare not valid. As shown in Fig. 2, for d/H1135030 nm, spin-transfer torque is nearly independent of the thickness d. On the other hand, for d/H1127030 nm, the strength of the nonadiabatic torque increases as the thickness ddecreases. For d/H1134910 nm, /H20841/H9270x//H9270y/H20841/H1122910−1, which is 1 order of magnitude larger than thatestimated by Zhang and Li.7Moreover, for d/H1134910 nm, the spin-transfer torque oscillates against the thickness dwith the period of the oscillation given by 2 /H9266/ki. Let us reveal the parameters which characterize the above behavior of the spin-transfer torque. Assuming that kr/H11229/H208813//H208494lmfp/H20850/H11270ki/H11229/H208813/H9275J/vF/H20849Ref. 15/H20850and d/H11271lmfp,w e find that /H9270y/H11229/H9266/H9252je/eand/H9270x/H11229−/H9266/H9256/H9252je/e−/H9266/H9252je//H20849edk i/H20850, re- spectively. Thus, for d/H11271lmfp, the adiabatic torque is nearly independent of the thickness. On the other hand, for d/H112701//H20849/H9256ki/H20850/H11229lsf//H208492/H208813/H20850/H1122940 nm, where lsf=vF/H9270sfis the spin- flip length, the torque due to the spatial variation in the spinaccumulation is dominant for the nonadiabatic torque. For athin domain wall, the ratio of the adiabatic and nonadiabatic torque is characterized by vF//H20849/H208813/H9275Jd/H20850, which is the ratio of the precession frequency of the electrons’ spin around thelocalized spin angular momentum due to the sdexchange coupling and the angular velocity of the rotation of the ex-change field in the domain wall. For d=10 nm, vF//H20849/H208813/H9275Jd/H20850/H1122910−1. The oscillation period is given by 2/H9266/ki/H112292/H9266vF//H20849/H208813/H9275J/H20850/H112292.5 nm. These estimations can be confirmed by the plots shown in Fig. 2. When the precession frequency of the electrons’ spin around the exchange field, /H9275J, is comparable to the angular velocity of the rotation of the exchange field in space, /H9266vF/d, the direction of the electrons’ spin cannot vary their direction adiabatically, and the nonadiabaticity, which issometimes called the mistracking effect, plays an importantrole in the spin-dependent transport phenomena. For ex-ample, the terminal velocity of the domain-wall motion isproportional to the ratio of the adiabatic and nonadiabatictorque. 17The nonadiabaticity is characterized by a dimen- sionless parameter /H9264=/H9266vF//H208492d/H9275J/H20850.18As shown above, the(a) (b) 01 0 2 03 0 4 0 thickness of the domain wall d (nm)1.01.2 0.8 0.6 0.4 0.2 -0.6-0.4-0.20 0τy/(πβje/e) τx/(πβje/e)01 0 2 03 0 4 0 thickness of the domain wall d (nm) FIG. 2. /H20849a/H20850The strength of the adiabatic torque /H9270yrenormalized by/H9266/H9252je/eagainst the thickness of the domain wall is shown. /H20849b/H20850 The strength of the nonadiabatic torque /H9270xrenormalized by /H9266/H9252je/e against the thickness of the domain wall is shown. Ford/H11349l mfp=3 nm, the diffusion approximation cannot be applied to the Boltzmann equation, and thus, torque below d/H11349lmfpdenoted by the dotted line is not valid.BRIEF REPORTS PHYSICAL REVIEW B 79, 212410 /H208492009 /H20850 212410-3ratio of the adiabatic and nonadiabatic torques for a thin domain wall, vF//H20849/H20881d/H9275Jd/H20850, is the first order of /H9264while the magnetoresistance due to the mistracking effect or spin ac-cumulation is on the second order of /H9264.10,14,15For conven- tional ferromagnetic metals with d/H11350lmfp,/H9264is less than unity. Thus, the nonadiabaticity plays an important role in thedynamics of the localized magnetization compared to themagnetoresistance. It should be noted that for a thickdomain wall the nonadiabatic torque is characterized by /H9256=2 //H20849/H9275J/H9270sf/H20850, not/H9264, as shown by Zhang and Li.7 We compare our results with those of Vanhaverbeke and Viret.13In Ref. 13, a time-dependent phenomenological Lar- mor equation for the magnetic moment in a moving frame issolved numerically and showed that the nonadiabatic torqueis shown to be one order of magnitude larger than that esti-mated by Zhang and Li 7when the thickness of the domain wall is comparable to the Larmor precession length/H9261 L=vF//H208492/H9266/H9275 J/H20850, which is on the order of a few nanometers. On the other hand, we consider the spin diffusion in thedomain wall in a steady state by solving the Boltzmannequation in the rotated frame and analytical expressions ofthe spin accumulation and spin-transfer torque are obtained.We show that the strength of the nonadiabatic torque in-creases as the thickness of the domain wall decreases for d/H11349l sf//H208492/H208813/H20850. Note that the condition is determined by the spin-flip length lsfinstead of the Larmor precession length /H9261L. We also find that the strength of the nonadiabatic torque is characterized by the first order of the nonadiabatic param-eter/H9264/H110081/d. The nonadiabatic torque does not change its sign, as shown in Fig. 5 in Ref. 13. Since the diffusion approximation is applied to the Bolt- zmann equation, the present theory is not applicable to theballistic region d/H11349l mfp. The spin-transfer torque in the bal- listic region is obtained by Waintal and Vilet.19One can eas- ily confirm that our Eq. /H208497/H20850reduces to Eq. /H2084912/H20850of Ref. 19in the limit of T˜,/H9270sf→/H11009, where they assume that the local spin-transfer torque is proportional to the spin accumulation.One might expect a simple connection formula between bal-listic and diffusive spin-transfer torquelike Wexler’s formulafor conductance. 20However, this is beyond the scope of the present Brief Report. In conclusion, we studied spin-transfer torque in a domain wall by solving the Boltzmann equation for spin accumula-tion and found their analytical expressions. For a thin do-main wall whose thickness is much thinner than the spin-fliplength, the ratio of the magnitude of the adiabatic and nona-diabatic torque is about 10 −1, which is one order of magni- tude larger than that estimated in Ref. 7and consistent with that in Ref. 13. We also found that the strength of the nona- diabatic torque is inversely proportional to the thickness ofthe domain wall. The authors would like to acknowledge the valuable dis- cussions they had with P. M. Levy, J. Ieda, H. Sugishita, K.Matsushita, and N. Yokoshi. This work was supported byJSPS and NEDO. 1M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas,Phys. Rev. Lett. 61, 2472 /H208491988 /H20850. 2J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 3L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 4Concepts in Spin Electronics , edited by S. Maekawa /H20849Oxford Science, 2006 /H20850, Chap. 7. 5S. E. Barnes, J. Ieda, and S. Maekawa, Appl. Phys. Lett. 89, 122507 /H208492006 /H20850. 6S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 /H208492008 /H20850. 7S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004 /H20850. 8U. Ebels, A. Radulescu, Y. Henry, L. Piraux, and K. Ounadjela, Phys. Rev. Lett. 84, 983 /H208492000 /H20850. 9H. F. Fuke, S. Hashimoto, M. Takagishi, H. Iwasaki, S. Ka- wasaki, K. Miyake, and M. Sahashi, IEEE Trans. Magn. 43,2848 /H208492007 /H20850. 10P. M. Levy and S. Zhang, Phys. Rev. Lett. 79,5 1 1 0 /H208491997 /H20850. 11J. Sato, K. Matsushita, and H. Imamura, IEEE Trans. Magn. 44, 2608 /H208492008 /H20850. 12K. Matsushita, J. Sato, and H. Imamura, IEEE Trans. Magn. 44, 2616 /H208492008 /H20850. 13A. Vanhaverbeke and M. Viret, Phys. Rev. B 75, 024411 /H208492007 /H20850. 14E. Simanek, Phys. Rev. B 63, 224412 /H208492001 /H20850. 15E. Simanek and A. Rebei, Phys. Rev. B 71, 172405 /H208492005 /H20850. 16B. A. Gurney, V. S. Speriosu, J.-P. Nozieres, H. Lefakis, D. R. Wilhoit, and O. U. Need, Phys. Rev. Lett. 71, 4023 /H208491993 /H20850. 17A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 /H208492005 /H20850. 18C. H. Marrows, Adv. Phys. 54, 585 /H208492005 /H20850. 19X. Waintal and M. Viret, Europhys. Lett. 65, 427 /H208492004 /H20850. 20G. Wexler, Proc. Phys. Soc. London 89, 927 /H208491966 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 79, 212410 /H208492009 /H20850 212410-4

PhysRevB.72.174416.pdf

Effect of dipolar interactions on the magnetization of a cubic array of nanomagnets Marisol Alcántara Ortigoza, *Richard A. Klemm,†and Talat S. Rahman‡ Department of Physics, Kansas State University, Manhattan, Kansas 66506, USA /H20849Received 30 January 2005; revised manuscript received 9 August 2005; published 11 November 2005 /H20850 We investigated the effect of intermolecular dipolar interactions on an ensemble of 100 three-dimensional systems of 5 /H110035/H110034 nanomagnets, each with spin S=5, arranged in a cubic lattice. We employed the Landau- Lifshitz-Gilbert equation to solve for the magnetization curves for several values of the damping constant, theinduction sweep rate, the lattice constant, the temperature, and the magnetic anisotropy. We find that thesmaller the damping constant, the stronger the maximum induction required to produce hysteresis. The shapeof the hysteresis loops also depends on the damping constant. We find further that the system magnetizes anddemagnetizes at decreasing magnetic field strengths with decreasing sweep rates, resulting in smaller hysteresisloops. Variations of the lattice constant within realistic values /H208491.5–2.5 nm /H20850show that the dipolar interaction plays an important role in the magnetic hysteresis by controlling the relaxation process. The temperaturedependencies of the damping constant and of the magnetization are presented and discussed with regard torecent experimental data on nanomagnets. Magnetic anisotropy enhances the size of the hysteresis loops forexternal fields parallel to the anisotropy axis, but decreases it for perpendicular external fields. Finally, wereproduce and test a previously reported magnetization curve for a two-dimensional system /H20851M. Kayali and W. Saslow, Phys. Rev. B 70, 174404 /H208492004 /H20850/H20852. We show that its hysteretic behavior is only weakly dependent on the shape anisotropy field and the sweep rate, but depends sensitively upon the dipolar interactions. Althoughin three-dimensional systems, dipole-dipole interactions generally diminish the hysteresis, in two-dimensionalsystems, they strongly enhance it. For both square two-dimensional and rectangular three-dimensional latticeswith B /H20648/H20849xˆ+yˆ/H20850, dipole-dipole interactions can cause large jumps in the magnetization. DOI: 10.1103/PhysRevB.72.174416 PACS number /H20849s/H20850: 75.40.Mg, 75.60.Ej, 75.75. /H11001a, 75.50.Xx I. INTRODUCTION The need of smaller memory storage devices,1–14the in- terest in developing quantum computing,15and the hope for understanding the relationship between the macroscopic andmicroscopic magnetic behaviors has led intense research intothe properties of nanoscale magnets. 1–32Many issues still remain unclear and serious problems must be overcome in order for them to be technologically useful. Prominentamong these is the loss of memory during magnetic relax-ation. Ferromagnetic nanodots are complex systems consisting of up to hundreds of magnetic atoms within a singledot. 5,11,12In this case, interparticle interactions along with anisotropy effects dominate the dynamics of the systems, andcontrol the magnetization processes. 8Moreover, since inter- dot exchange interactions are negligibly small, the dynamicsof the ferromagnetic nanodot arrangements are strongly in-fluenced by dipolar interdot interactions. 13,14 Single molecule magnets /H20849SMM’s /H20850consist of clusters of only a few magnetic ions, and are thus among the smallestand simplest nanomagnets, but are also well-characterizedsystems exhibiting magnetic hysteresis. 27In SMM’s, the one-body tunnel picture of the magnetization mostly explainsthis phenomenon in the sense that the sequence of discretesteps in those curves provides evidence for resonant coherentquantum tunneling. 28–30Nevertheless, this one-body tunnel model neglects intermolecular interactions, and is not alwayssufficient to explain the measured tunnel transitions. 31,32A close examination of the magnetization curves reveals finestructures which cannot be explained by that model. Werns-dorfer et al. suggest that these additional steps are due tocollective quantum processes, called spin-spin cross relax- ation /H20849SSCR /H20850, involving pairs of SMM’s which are coupled by dipolar and/or exchange interactions. 31,32If dipolar and/or exchange interactions cooperate in the relaxation process,then one might hope to be able to better control such loss ofmagnetic memory. Analyzing the relaxation of the magnetization is difficult for both SMM’s and ferromagnetic nanodots. Besides dipolarinteractions, many other factors may be involved in suchprocesses. Geometric features, such as the shape and volumeof the magnets, as well as the type of lattice on which theyare placed, can directly influence the anisotropy barriers andthe easy axis directions. In the case of SMM’s, a quantumtreatment must be considered to show that resonant tunnelingof the magnetization results in the discrete steps appearing inthe low temperature Tmagnetization curves. Although in many SMM’s the intercluster exchange interactions are neg-ligible, as for ferromagnetic nanodots, in other SMM’s, suchinteractions are comparable in strength to the dipolarinteractions. 32Besides the quadratic Heisenberg and qua- dratic anisotropic intramolecular exchange interactions,some SMM’s are thought to contain intramolecular interac- tions of the Dzyaloshinskii-Moriya type. 33Additional higher order, anisotropic spin exchange interactions further compli-cate the problem. Therefore, by studying models that dealwith each one of these factors separately, one hopes to sim-plify the problem, to build up gradually a more realistic sys-tem, and at the same time, to elucidate how each of thesefactors contributes to the magnetization process. With regard to SMM’s, there have been recent approaches to the quantum dynamics of the low- Trelaxation. 17,34–39 Prokof’ev and Stamp assumed a single relaxation mode,34inPHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 1098-0121/2005/72 /H2084917/H20850/174416 /H2084913/H20850/$23.00 ©2005 The American Physical Society 174416-1which the dipolar and hyperfine fields are frozen unless an SMM flips its spin. Then by assuming the effective fieldaround each SMM is that of randomly placed dipoles, theyobtained an expression for the low- Tdecrease proportional tot pof the magnetization of each SMM from its fully mag- netized state,34,40,41where p/H110150.5–0.7, but pmight be as large as 0.7.34–37This procedure was restricted to very small deviations of the magnetization from its saturated value, so itis not useful for studying the central portion of the hysteresiscurves, for which the magnetization can be small. Moreover,as first argued for ferromagnets by Anderson, 42the spin-spin and spin-lattice relaxation times can be very different, so thatsuch simple behavior is not expected. In fact, experiments onSMM’s have shown that an exponential relaxation of themagnetization is consistent with the data, 38,39so that as a minimum, one requires two distinct relaxation times forSMM’s, which could be very different from one another. 42 The most commonly studied model of spin dynamics con- taining two distinct relaxation parameters is the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation. 43,44Using the LLG equa- tion, Kayali and Saslow /H20849KS/H20850investigated the hysteresis curves at T=0 for two-dimensional /H208492D/H20850square arrays of 4 to 169 ferromagnetic nanodots subject to dipole-dipole inter-actions and a magnetic field applied in various directionswithin the array’s xyplane. 45They included anisotropy ef- fects via an effective field proportional to the zcomponent of each dot’s dipole moment. Earlier studies of square planarlattices of 9 to 36 ferromagnetic dots were made by Stampsand Camley. 46In addition, Zhang and Fredkin /H20849ZF/H20850studied the LLG model to obtain the zero-field time decay of theeasy-axis magnetization of a three-dimensional /H208493D/H20850cubic lattice of 12 /H1100312/H1100312 Stoner-Wohlfarth particles interacting with each other via dipole-dipole interactions. 14Since the size /H20849or radius /H20850of the Stoner-Wohlfarth particles was taken to be much less than the lattice constant, they could betreated as pointlike magnetic moments, the classical analogof SMM’s. Here we study only the effects of the intermolecular dipole-dipole interactions upon the magnetization curves foran ensemble of N c=100 3D cubic crystals each containing N=5/H110035/H110034 nanomagnets, all with the same magnetic mo- ment. As in the ZF model of Stoner-Wohlfarth particles, wetake the lattice parameter to be much greater than the nano-magnet size or radius. Except when a strong anisotropy fieldis present, we assume that there is no long-range order in theTregime of interest, so that in the absence of an external magnetic field, the magnetization of each nanomagnet crystalis essentially zero. We note that long-range ordering wasclaimed to exist in such a system with Ising spinanisotropy. 47,48In our studies with a strong anisotropy field HA, hysteresis curves exhibiting a substantial zero-field mag- netization were obtained for the applied magnetic inductionB /H20648HAafter the system had been fully magnetized by B. The strength of the dipole interactions is primarily determined bythe lattice constant, a, which we vary from 1.25 nm to 2.5 nm. The dynamics of each nanomagnet are assumed to begiven by the LLG equation, which includes precession anddamping relaxation processes, the damping coefficient /H9251of which can also depend upon T.49,50Then, the magnetic mo- ment Micof the ith nanomagnet within the cth crystal of our ensemble obeysdMic dt/H11013/H9253Mic/H11003Bic,eff−/H9251 MsMic/H11003/H20849Mic/H11003Bic,eff/H20850, /H208491/H20850 Bic,eff=B+/H20849Bic/H20850dip, /H208492/H20850 where /H9253=g/H9262Bis the gyromagnetic ratio, Ms=g/H9262BSis the magnetic moment of an individual nanomagnet, and /H20849Bic/H20850dip is the contribution to the effective magnetic induction Bic,eff at the ith nanomagnet within the cth crystal arising from dipole-dipole interactions between it and the other nanomag-nets within the same crystal, /H20849B ic/H20850dip=/H92620 4/H9266/H20858 j/H110323/H20849Mjc·rij/H20850rij−rij2Mjc rij5, /H208493/H20850 where the prime indicates that the j=iterm is omitted. The second term of Eq. /H208491/H20850, the damping term, was first intro- duced by Landau and Lifshitz43and later by Gilbert to give a phenomenological description of the relaxation of the mag-netization. They did not derive it from first principles due tothe enormous complexity of summarizing all of the relax-ation processes into a single term. As noted above, whenferromagnetic interactions are present, /H9251//H9253is generally ex- pected to be /H112701.42By extending to electronic spin systems the Wangsness-Bloch model of nuclear spin magnetic relax-ation by magnetic dipole coupling to a heat bath, 49Fredkin and Ron showed that the damping term could be derived forlarge spin values and /H9260=q/H9253H/kBT/H112701, where qandkBare Planck’s constant divided by 2 /H9266and Boltzmanns’ constant, respectively, and in our case H=Bic,eff.50To the extent that electric quadrupole interactions could be neglected, /H9251varies inversely with Tfor/H9260/H112701, but depends upon /H9260otherwise.50 More recently, a different derivation of the Gilbert damp- ing term was derived from a spin Hamiltonian containing theinteraction between the spin and the radiation field, which isinduced by the precessing magnetization itself. 51,52In that case, no explicit Tdependence of /H9251was given. We remark that rather complex explicit expressions for /H9251for the differ- ent system of local spin moments arising from p-dkinetic- exchange coupling of the itinerant-spin subsystem in ferro-magnetic semiconductor alloys have been given recently. 53 In any event, the damping coefficient /H9251at some Tvalue must be determined experimentally for each system. In order to study the magnetization of ferromagnetic dots, KS used an extremely large value for the damping coeffi-cient, /H9251//H9253=0.6, a huge sweep rate, /H9004B//H9004t/H110113000 T/s, and a small maximum external induction Bmax=2/H92620Ms.I no u r studies of nanomagnet arrays, we used values of /H9251//H9253that varied from these values to values 12 orders of magnitudesmaller. Depending upon the /H9251values, we also varied the sweep rate /H9004B//H9004tfrom those values to the much smaller /H1101110−3T/s, and varied Bmaxfrom much larger values /H208492T/H20850, comparable to those reported in SMM experiments,23,26to those used by KS. Similarly, the lattice constants reported inthe present work are in accordance with the near neighborseparation in the most extensively studied SMM crystals.Quantum processes within the individual SMM’s will betreated in a separate presentation. 54ALCÁNTARA ORTIGOZA, KLEMM, AND RAHMAN PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-2The present paper is organized as follows. In Sec. II, we present our model system and a brief description of the over-all calculation procedure that we followed. In Sec. III, wesolve the LLG for each nanomagnet subject to both the ex-ternal and the combined dipolar inductions. In Sec. IV , wepresent and discuss our main results for the magnetizationcurves, which are evaluated at various values of the sweeprate, T,a, the anisotropy field, and /H9251//H9253. When spin aniso- tropy is present, we study the cases B/H20648HAandB/H11036HA.I n Sec. V , we reproduce one of the KS magnetization curves forsquare 2D lattices, and vary some of their parameters toshow that the results are almost independent of the sweeprate over the range /H11011300–6000 T/s. Analogously, we show that the anisotropy field does not affect significantly the mag-netization curves until its magnitude is comparable toB max//H92620. By varying the lattice constant, we also show that the results of KS are very sensitive to the strength of thedipolar fields, which mainly govern the behavior of the mag-netization of such systems. Finally, in Sec. VI, we summa-rize our main conclusions. II. MODEL SYSTEM In the present work we consider an ensemble of Nc=100 cubic crystals /H20849or configurations /H20850, each configuration consist- ing of N=5/H110035/H110034=100 nanomagnets, each with ground state spin S=5, which interact with one another only via dipolar interactions. Each of the Ncsystem configurations c =1,…,Ncis constructed to have a starting total magnetic moment Mc/H110150a t B=0. The hysteresis curves are obtained for each configuration, and these are then averaged over the Ncconfigurations. One then obtains the magnetization M/H6023/H20849B/H20850 curves, where M/H6023=/H20855Mc/H20856c/Vis the configuration averaged magnetization, Vis the crystal volume, and B=/H20841B/H20841. A. Ensemble of random spin configurations In order to proceed, we first find a large number Ncof random spin configurations cofN=100 spins, such that for each configuration, Mc/Ms/H110150a tB=0 and as T→/H11009, where the total magnetic moment Mc/H20849t,B/H20850=/H20858 i=1N=100 Mic/H20849t,B/H20850. /H208494/H20850 At the start of the iteration, we take t=0,B=0, and T→/H11009in the absence of the dipole-dipole /H20849or any other inter-spin /H20850in- teractions for configuration c. Then we select those configu- rations for which /H20841Mc/H20841/Ms/H333550.1, which we deem sufficiently close to Mc/H110150. Our resulting magnetization curves are based upon the average over Nc=100 configurations, each one containing N=100 similarly chosen nanomagnets. We reiterate that Nis the number of nanomagnets in each configuration, and Ncis the number of configurations stud- ied. Although we have chosen both of these numbers to be100 in order to obtain reliable statistics, NandN chave com- pletely different meanings. Finding many /H20849Nc/H20850configura- tions, each of which has an almost vanishing initial magne- tization consumes a significant amount of computer time,especially if the number Nof nanomagnets per configuration is not very large. However, choosing a rather small numberNof nanomagnets reduces the time required to calculate the dipolar field at each nanomagnet due to all of its neighbors,which must be calculated at each integration time step of theLLG equation, offsetting the large amount of computer timerequired to set up N cinitially nearly nonmagnetic configura- tions. B. Evolution of the magnetization versus field curves In this model one increases the external magnetic induc- tion Bin discrete steps /H9004B, until B=Bmax, where Bmax =/H20841Bmax/H20841must be large enough to align every nanomagnet in its direction. How large Bmaxmust be generally depends upon T, the field sweep rate /H9004B//H9004t/H11013/H20841/H9004B/H20841//H9004t, the lattice parameter a, and the crystal structure.27In addition, the steps /H20841/H9004B/H20841must be small enough to give rise to numerically smooth M/H20849B/H20850curves. We therefore set /H20841/H9004B/H20841=Bmax/NB, where the number of steps NB/H112711 should be on the order of 103. After each magnetic step, one allows each of the nano- magnets to relax for a fixed amount of time /H9004t, which is chosen to be sufficiently small that the nanomagnets do notreach equilibrium. Otherwise, in the absence of a sufficientlystrong anisotropy field, no hysteresis would result. First, we choose one of our configurations c/H20849e.g., c=1/H20850 and set the moments of the nanomagnets equal to their val- ues in this initially nonmagnetic configuration, /H20853M ic=1/H20849t =0,B=0/H20850/H20854i=1,…,N. That is, just after we turn on the magnetic induction in the xdirection by the amount B=/H9004B, the nano- magnets have not yet precessed from their initial configura-tion. Then, we calculate the effective magnetic induction B ic=1,effat each of the i=1,…,Nnanomagnets for c=1. T o do so, we must calculate the dipolar induction in Eq. /H208493/H20850due to the presence of all the other nanomagnets. Then, we let each of the nanomagnets evolve in the pres- ence of its effective magnetic induction for a chosen fixedtime interval /H9004t. To do this accurately, we break /H9004tup into N tintervals dt=/H9004t/Nt. Obviously, this is extremely time con- suming, because it is necessary to recalculate the effectiveinduction at each nanomagnet after each time-integration step of width dt. Once the whole set of moments /H20853M i1/H20849t =/H9004t,B=/H9004B/H20850/H20854i=1,…,Nis obtained, we proceed to calculate the configuration magnetic moment, Mc=1/H20849/H9004t,/H9004B/H20850, for this choice of fixed sweep rate, /H9004B//H9004t, from Mc/H20849t,B/H20850=/H20858 i=1N Mic/H20849t,B/H20850exp/H20851−/H9252Hic/H20849t,B/H20850/H20852 /H20858 i=1N exp/H20851−/H9252Hic/H20849t,B/H20850/H20852, /H208495/H20850 Hic/H20849t,B/H20850=−Bic,eff/H20849t,B/H20850·Mic/H20849t,B/H20850, /H208496/H20850 by setting c=1, t=/H9004t, and B=/H9004B, where Bic,eff/H20849t,B/H20850is given from Eqs. /H208492/H20850and /H208493/H20850,/H9252=1/kBT, and kBis Boltzmann’s con- stant. Since /H20849Bic/H20850dipas given by Eq. /H208493/H20850inBic,eff/H20849t,B/H20850contains a self-fieldless single sum, there is no site overcounting in Eq. /H208496/H20850.EFFECT OF DIPOLAR INTERACTIONS ON THE … PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-3We are interested in Mc/H20849B,/H9004B//H9004t/H20850. In this nonequilib- rium situation, the Mic,Bic,effand hence Hicfor each nano- magnet change after each time step dtat which they are evaluated, but the statistical weighting in Eqs. /H208495/H20850and /H208496/H20850is only evaluated at the end of each fixed interval /H9004t, which has a one-to-one correspondence with /H9004B. Thus, this single- configuration average can be directly compared to those indifferent configurations after the same number of intervals.Moreover, since /H9004t=/H9004B/H20849/H9004B//H9004t/H20850 −1,Mc/H20849t,B/H20850for our purpose can be written as Mc/H20851/H20849B//H9004B//H9004t/H20850,B/H20852, which is effectively a function of Band/H9004B//H9004t. Next, we increase the external magnetic induction by an- other equal step /H9004B, and let the nanomagnets precess during another equal time interval, /H9004t, under the action of the new effective induction. We continued increasing Bin this equal step fashion a total of NBtimes, until B=Bmax. At this point, the incremental induction direction is reversed, setting B =Bmax−/H9004Bfor the same time interval /H9004t, repeating the pro- cedure 2 NBtimes, until B=−Bmax. After that, we reverse the incremental induction direction once again, setting B =−Bmax+/H9004Bfor the same time interval /H9004t, etc., and continue NBtimes until B=0 is reached, or until the configuration magnetization hysteretic loop /H20849if it exists /H20850is closed. Then, one repeats the entire procedure above described for each ofthe other N c−1 configurations c=2,…,Nc, keeping the time intervals /H9004tand the subintervals dtconstant for each step in each configuration. Once all of the calculations for each oftheN cconfiguration are finished, we average the results over theNcconfigurations, obtaining, /H20883Mc/H20873B /H9004B//H9004t,B/H20874/H20884c=1 Nc/H20858 c=1Nc Mc/H20873B /H9004B//H9004t,B/H20874. /H208497/H20850 Then, the magnetization M/H6023is easily calculated. Having tabulated M/H6023for every external magnetic induction step with fixed/H9004B//H9004t,T,a,Ms,/H9251, and Bmax, we generate the mag- netization curve M/H20849B/H20850for this set of parameters. C. Variation of experimental parameters Unlike the parameters such as Bmaxand dt, which are details of the theoretical calculation, other parameters can inprinciple be varied in experiments in a variety of materials.Using the same initial dipole configurations we repeat thewhole procedure with different values of /H9251,/H9004B//H9004t,T, and a. The only parameters that can be experimentally varied instudies on a particular sample are /H9004B//H9004tandT, since the other parameters are fixed. Nevertheless, the possibility ofsetting the nanomagnets further apart by varying the compo-sition of the nonmagnetic ligand groups in SMM’s, for ex-ample, justifies the study of the variation of a. Also, given that the damping term appearing in the LLG equation is phe-nomenological, and that in most cases /H9251should be deter- mined experimentally, we have also examined its variation.We note that /H9251is expected to depend inversely upon T, un- less Tis sufficiently low that thermal processes no longer dominate the relaxation.14We keep Msfixed.III. INTEGRATION OF THE LLG EQUATION FOR ONE NANOMAGNET The magnetic moment of each nanomagnet is obtained by numerically integrating the LLG equation. The time evolu-tion of one nanomagnet must be determined synchronouslywith all its neighbors in order to calculate the dipolar induc-tion acting on each of them at a given time. To solve theLLG equation for the ith nanomagnet in the cth crystal, we first rotate its coordinates at each time integration step such that B ic,eff/H20849t/H20850/H20648zˆ/H20849t/H20850. We then solve the resulting differential equations for either the coordinate spherical angles /H9258/H20849t/H20850,/H9278/H20849t/H20850, or the components of Mic/H20849t/H20850, as shown in the Appendix. The quantity relevant to each spherical angle or component of Mic/H20849t/H20850is/H20848t0td/H9270/H20841Bic,eff/H20849/H9270/H20850/H20841, which explicitly involves the past history of /H20841Bic,eff/H20849t/H20850/H20841. In order to decrease the computation time, we approximate this integral for small time integration steps dt=t−t0/H11270t0, /H20885 t0t d/H9270/H20841Bic,eff/H20849/H9270/H20850 /H20841/H11015/H20841 Bic,eff/H20849t0/H20850/H20841dt. /H208498/H20850 In order to assure numerical accuracy of our results for the greatly different experimental parameters studied, we hadto make appropriate choices for the numerical parametersused in the calculations, as discussed in the Appendix. Gen-erally, calculations with slow sweep rates require corre-spondingly small /H9251//H9253values. For the calculations leading to the results presented in Figs. 1, 2, and 6–9, we take thenumerical parameters dt=1/H1100310 −4s,Bmax=2.0 T, Nt=1000, andNB=500, 1000, and 4000, respectively, for the different sweep rates studied. For the calculations presented in Figs.3–5, we take dt=6/H1100310 −12s,Bmax=22.5 mT, Nt=1000, and NB=1250. IV . RESULTS AND DISCUSSION A. Effects of damping and maximum induction values on the hysteresis We first neglect any spin anisotropy effects. In Fig. 1, we plotted the average over Nc=100 configurations of the nor- malized magnetization at the lattice constant a=1.5 nm, sweep rate /H9004B//H9004t=0.005 T/s, and temperature T=0.7 K for the four weak damping rates /H9251//H9253=3/H1100310−n, where n=10, 11, 12, and 13. These results appear, respectively, from left toright /H20849right to left /H20850in the upper /H20849lower /H20850part of Fig. 1. The magnetization curves show hysteresis for all four of these /H9251 values. For the smallest /H9251value we studied, /H9251//H9253=3.0 /H1100310−13, the hysteresis only occurs for external induction magnitudes exceeding 3.0 T, observed by setting Bmaxabove that value, which is well beyond the domain pictured in Fig.1. We also note that in Fig. 1, the central regions for/H20841/H20855M/H20856//H20849NM s/H20850/H20841/H110210.8 are nonhysteretic. For each of these four parameter value choices, the initial curve describing the first increase of the average magnetization from essentially 0 toits saturation value is indistinguishable from subsequentsimilar curves obtained after completing the full hysteresispaths. Hence, in this case, the main consequence of thechoice of N c=100 configurations is the improvement in theALCÁNTARA ORTIGOZA, KLEMM, AND RAHMAN PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-4statistics, reducing the noise that remains most evident in the curves corresponding to the smallest /H9251values. From the inset to Fig. 1, we see that although the height /H20851in/H20855M/H20856//H20849NM s/H20850/H20852of the hysteretic region decreases with de- creasing /H9251, the width /H20849inB/H20850of the hysteretic region increases faster with decreasing /H9251, so that the overall area of the hys- teretic region increases with decreasing /H9251. From a computa- tional standpoint, for the parameter values studied in Fig. 1,the smaller the value of /H9251, the larger the required value of Bmaxto produce hysteresis. We also noticed that in these magnetization curves, the hysteresis sets in at the point of anabrupt change in slope in the initial curve, which describesthe first increase of the average magnetization from 0 to itssaturation value. Moreover, we conclude that B maxmust be chosen to guarantee that the system reaches saturation at B /H33355Bmax, because of the different nature of the hysteresis present in each curve. For example, in Fig. 1 the hysteresiscan occur only after saturation, but with smaller avalues, if the system has not saturated by B=B max, then the magneti- zation will keep increasing for a number of subsequent /H9004B steps, even though the direction of /H9004B/H20849but not of B/H20850has been reversed. B. Effect of temperature on the hysteresis 1. Temperature independent /H9251 We first investigate the role of temperature that arises only from the statistics, Eq. /H208495/H20850, and present our results for a T independent /H9251in Fig. 2. In this figure, we have replotted the inset of Fig. 1, excluding the curve for /H9251//H9253=3/H1100310−13, for which the hysteresis occurred for Btoo large to display on the same plot. Otherwise, the parameters are the same as inFig. 1, except that we have compared our results /H20849gray curves /H20850forT=0.7 K shown in Fig. 1 with those /H20849black curves and circles /H20850forT=0.1 K. Since the evolution of the magnetization with Bin this model is independent of T,w e note from Fig. 2 that the departures of the magnetizationcurves from the points of saturation are the same at both Tvalues, so that the widths /H20849inB/H20850of the hysteretic regions are nearly the same. However, the height in /H20855M/H20856//H20849NM s/H20850of each hysteretic region decreases strongly with decreasing T,s o that the overall area of each hysteretic region decreases withdecreasing T. This particular result is in strong contrast to the existing experimental results on SMM’s. Nevertheless, ourresults are reasonable from the point of view of the LLGequation and the way Tenters our calculation. We reiterate that we have so far neglected quantum and spin anisotropyeffects, the latter of which will be discussed in the following. We remark that in Fig. 2, Tonly enters into the equations of motion when the average magnetic moment is evaluatedfrom Eq. /H208495/H20850. As for the Brillouin function that describes the magnetization of a paramagnet in the absence of the dipoleinteractions, the initial slope of the magnetization at low B increases as Tis lowered. This increases the alignment of the moment of each nanomagnet at decreasing T, so that the dipole-dipole interactions tend to be maximized, enhancingthe effect. 2. Temperature-dependent /H9251 We now consider the effect of the temperature depen- dence of the damping constant /H9251upon the magnetic hyster- esis, focussing upon the case of correspondingly fixed veryhigh sweep and damping rates. We assume that our choice ofspin value, S=5 for each nanomagnet, satisfies S/H112711. In this limit, Fredkin and Ron showed that the damping of thenuclear spin precession by magnetic dipole coupling to aheat bath, as derived under the assumption of spin-orbit fac-torization by Wangsness and Bloch, could be readily ex-tended to the spins in magnetic systems. 49,50ForS/H112711, they found /H9251/H20849T/H20850//H9253/H11015T0/T, /H208499/H20850 where T0=2/H6036/H9021111/H208491−e−/H9260/H20850S2/kB/H9260,50and/H9021111is a rate constant /H20849with units of s−1/H20850, the expression for which is a complicated orbital integral arising from the interaction of the local spinwith its surrounding molecular electronic orbitals in second- FIG. 1. Magnetization curves for Nc=100, a=1.5 nm, /H9004B//H9004t =0.005. From left to right for M/H110220,/H9251//H9253=3/H1100310−10/H20849dashed /H20850,3 /H1100310−11/H20849thin dark gray /H20850,3/H1100310−12/H20849light gray /H20850,3/H1100310−13/H20849thick black /H20850. The inset highlights the hysteretic region of the first three of these curves. FIG. 2. Shown is the upper hysteretic region of the normalized magnetization curves at T=0.7 K /H20849gray /H20850and T=0.1 K /H20849black, circles /H20850. The T-independent damping constants /H9251//H9253are 3/H1100310−12 /H20849a/H20850,3/H1100310−11/H20849b/H20850, and 3 /H1100310−10/H20849c/H20850. The other parameters are the same as in Fig. 1.EFFECT OF DIPOLAR INTERACTIONS ON THE … PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-5order perturbation theory,49and/H9260=/H6036/H9253Beff//H20849kBT/H20850. For /H9260 /H112701,T0→2/H6036/H9021111S2/kB, which can be taken to be independent ofTandBeff, so that /H9251/H11008T−1, but for /H9260/H112711,/H9251/H110081/Beff, which would completely change its effect. Here we only considerthe case /H9260/H112701, for which Eq. /H208499/H20850holds for constant T0.W e note that, as in Figs. 1 and 2, Talso affects the results for the magnetization from the statistics, Eq. /H208495/H20850. In Fig. 3, we have shown our results, averaged over Nc =100 configurations, of the normalized magnetization as a function of Bin mT, for a=1.5 nm, /H9004B//H9004t=3000 T/s, /H9251/H20849T/H20850//H9253=T0/T,T0=0.3 K, and T=5 K. For the calculations presented in this figure, we used the numerical parameters dt=6/H1100310−12s,Bmax=22.5 mT, Nt=1000, and NB=1250. Note that although ahas the same value as in Figs. 1 and 2, the sweep and damping /H20851/H9251/H20849T/H20850//H9253=0.06 /H20852rates are six and at least eight orders of magnitude larger than in those figures. For these parameters, there are three regions of hysteresis inthe pictured magnetization curve. The left inset is an enlarge-ment of the upper hysteretic region, the mirror image ofwhich occurs in the lower region of the pictured magnetiza-tion curve. In contrast to the behavior shown in Figs. 1 and 2,at the top of the upper hysteretic region, the magnetizationdoes not rise abruptly to its saturation value, but first goesthrough an extended nonhysteretic region. In addition, thereis a central hysteretic region, an enlargement of which isshown in the right inset, along with an enlargement of thesame central hysteretic region obtained at T=0.25 K with the same set of parameters. We note that at T=5 K, the onset magnetization averaged over N c=100 configurations, pic- tured by the thin curve in the lower portion of the right inset,does not coincide with the thick curve corresponding to thecentral hysteresis loop region obtained subsequently to theattainment of the saturation value by the magnetization. Inaddition, we note that the thick central hysteresis loop exhib-its reproducible oscillations with B-independent frequency fatT=5 K, which oscillations have disappeared at the lower T=0.25 K value, for which /H9251/H20849T/H20850//H9253=1.2, pictured in the up- per portion of the right-hand inset. In order to investigate further the differences between the starting magnetization curves and the curves obtained subse-quent to saturation, in Fig. 4, we have shown the correspond-ing central hysteresis loop portion of the magnetization ob-tained for two individual configurations, using the sameexperimental and numerical parameters as in Fig. 3 exceptthatT=10 K for which /H9251/H20849T/H20850//H9253=0.03. As in Fig. 3 Tenters both through the statistical averaging and through the damp- ing/H9251/H20849T/H20850. In Fig. 4 the solid and open circles correspond to the starting magnetizations of the two configurations, and the coincident thick black and thin light gray curves correspondto the central hysteresis loop region of their respective mag-netization curves obtained after saturation. Note that after theinitial noisy regions, the starting magnetizations for thesetwo configurations exhibit comparably large amplitude oscil-lations at the frequency f/2, the phases of which are very different. However, after the attainment of the saturationmagnetization, these large amplitude oscillations are absent,and replaced by smaller amplitude oscillations at the fre-quency f, which are similar to the oscillations present in our results obtained at T=5 K shown in the lower curves in the right-hand inset of Fig. 3. We note that in the first oscillationpresent on both sides of the central post-saturation hysteresisloops obtained with these parameters at T=5 and 10 K show additional small amplitude, higher frequency oscillations,which may be higher harmonics of f. In addition, the ampli- tudes of the fifth and sixth oscillations are larger at 10 K inFig. 4 than a t5Ki nt h e lower right inset of Fig. 3. We remark that the large amplitude oscillations present in the starting magnetizations shown in Fig. 4 are absent in Fig.3. This occurs due to the randomness of the oscillationphases, which is averaged out in the N c=100 configurations studied in Fig. 2. FIG. 3. The magnetization curves for Nc=100 at T=5 K, a =1.5 nm, and /H9004B//H9004t=3000 T/s with Bmax=22.5 mT and /H9251/H20849T/H20850//H9253 =T0/Tfor/H6036g/H9262BBeff//H20849kBT/H20850/H112701 and T0=0.3 K are shown /H20849Ref. 50 /H20850. Left-hand inset, details of the upper portion of the curve. Right-hand inset, details of the central hysteretic portion of the curveshown, along with the central portion of the corresponding curve atT=0.25 K. The thin curves beginning near to the origin represent the magnetization onsets. These curves are offset for clarity, withthe scales on the right-hand /H20849left-hand /H20850sides corresponding to the lower /H20849upper /H20850curves, respectively. FIG. 4. The central loop and starting magnetization curves for two separate configurations, each with Nc=1 /H20849open and filled circles /H20850atT=10 K, a=1.5 nm, and /H9004B//H9004t=3000 T/s with /H9251/H20849T/H20850//H9253=T0/Tfor/H6036g/H9262BBeff/kBT/H112701 and T0=0.3 K are shown /H20849Ref. 50/H20850. The thin gray and thick black curves represent the identical behaviors of the central hysteretic loop portion of the magnetizationfor the same two configurations obtained after saturation. The ar-rows indicate the direction of the magnetization hysteresis. HereB max=22.5 mT. See text.ALCÁNTARA ORTIGOZA, KLEMM, AND RAHMAN PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-6From Fig. 4, we therefore conclude that our starting con- figurations that were chosen to have /H20841M/H20841/Ms/H333550.1, appropri- ate for SMM’s, lead to starting magnetization curves that arevery different from those that start at the saturation magne-tization, but are subsequently identical. That is, after the at-tainment of saturation, all configurations are identical. 3. External field directed towards the crystal corners with /H9251„T… We now consider the 3D case of the external magnetic induction directed from the crystal center to one of its cor- ners, B=B/H20849xˆ+yˆ/H20850//H208812, the /H20849110 /H20850direction. In Fig. 5, we show the resulting central hysteresis region obtained from our cal- culations for Nc=50, N=5/H110035/H110034,T=10 K, a=1.5 nm, and/H9004B//H9004t=3000 T/s with /H9251/H20849T/H20850//H9253=0.03. In this case, it is sufficient to set Bmax=22.5 mT, which leads to full satura- tion. We note that for this field direction, a small /H20849−6 mT /H11021B/H110216m T /H20850hysteresis region appears on either side of the origin, which is rather central to the full magnetization curve, but vanishes over a small region close to the origin. Thereare also tiny hysteresis regions near to saturation that appearas dots in the inset depicting the full magnetization curve. The nearly central hysteretic regions shown in Fig. 5 ex- hibit reproducible jumps at specific Bvalues, similar to those observed at low Tin SMM’s. However, we note that in this figure, we have taken T=10 K, and have used a classical spin model. We also note that we have used a rather smallsample /H20849N=100 /H20850with a fast sweep rate and a large damping coefficient in our calculations, and caution that such behavior might not be present in large single crystals, especially withmuch slower sweep rates. Nevertheless, this figure demon-strates that steps in the magnetization do not necessarily havea quantum origin, and that the sample shape can lead tounusual hysteresis effects. C. Effect of sweep rate on the hysteresis From the curves obtained using the same numerical pa- rameters as in Fig. 1 for different induction sweep rates at afixed, small damping rate in Fig. 6, it is clear that stronger hysteresis is found for higher sweep rates, in agreement withexperiments on a variety of nanomagnets, including SMM’s.This shows that the reversibility of the process depends onhow close to equilibrium the sweep rate allows the nanomag-net spins to reach. That is, although for different sweep ratesthe external induction is increased by the same amount /H9004B, at higher sweep rates, the time /H9004tallowed for the nanomag- nets to evolve towards equilibrium is less. This makes theprocess less reversible and the hysteresis loops larger. We also note that at the much higher sweep and damping rates studied in Figs. 3 and 4 the magnetization also exhibitsa central hysteretic region, which exhibits oscillations at T values not too low and/or damping constants not too large. D. Effect of lattice constant on the hysteresis In Fig. 7, we show hysteresis curves for two different values of the lattice constant a, obtained using the same nu- merical parameters as in Fig. 1. For weaker dipole-dipoleinteractions /H20849larger a/H20850, the rise in the magnetization is steeper with increasing B, and the rapid decrease in the magnetiza- FIG. 5. The central loops /H20849solid curves /H20850of the magnetization curve for Nc=50, N=5/H110035/H110034=100, with Balong the /H20851110 /H20852direc- tion /H20851B/H20648/H20849xˆ+yˆ/H20850//H208812/H20852atT=10 K, with a=1.5 nm, Bmax=22.5 mT, and/H9004B//H9004t=3000 T/s with /H9251/H20849T/H20850//H9253=T0/T, and T0=0.3 K. The dashed curve is the starting magnetization curve. The arrows indi-cate the direction of the hysteresis. Inset, the full magnetizationcurve. See text. FIG. 6. Hysteretic region of M/H20849B/H20850at 0.7 K, /H9251//H9253=3/H1100310−12, anda=1.5 nm, for the sweep rates /H9004B//H9004t=0.04 T/s /H20849thin black /H20850, 0.02 T/s /H20849dark gray /H20850, and 0.005 T/s /H20849thick light gray /H20850. The inset shows the entire curves. FIG. 7. Magnetization curves for lattice constants a=1.5 nm /H20849gray /H20850and a=2.5 nm /H20849black /H20850./H9004B//H9004t=0.04 T/s,/H9251=3/H1100310−12/H9253, T=0.7 K.EFFECT OF DIPOLAR INTERACTIONS ON THE … PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-7tion from its saturation value upon decreasing Boccurs at a smaller value of /H20841B/H20841. Furthermore, we shall see that dipolar interactions do not promote hysteresis in these systems, butsuppress it. Actually, the same conclusion was found recentlyfor magnetic nanoparticles in the framework of the general-ized mean-field approximation. 55 This peculiar hysteresis is easily understood by analyzing the LLG equation. If the nanomagnet magnetization Micis parallel to its local magnetic induction Bic,eff,dMic/dt= 0 ,a si t will remain thereafter, so that Michas reached equilibrium. The only chance for the system to decrease its magnetizationfrom its saturation value is through the combined weak di-polar induction, which strengthens with decreasing latticeparameter a. The dipolar induction can oppose the system from remaining completely magnetized, since it has small,but nonvanishing components. Therefore, even when B reaches its maximum /H20849finite /H20850amplitude B maxand the mis- alignments of each Micwith Bare negligible, dynamic equi- librium will not generally have been attained due to the lim-ited time allowed for relaxation before the next change in B. There will remain a slight deviation between the directions of the B ic,effand the Micdue to the presence of the Bic,dip, which is especially important when Bdecreases from Bmax. Of course, it is harder to decrease Mat the very begin- ning of the induction cycle. This is precisely the cause of the hysteretic behavior, given that changes in Micare propor- tional to /H20841Mic/H11003Bic,eff/H20841, which nearly vanishes when the direc- tion of the incremental induction has just been reversed. Weconclude then that the smaller the lattice parameter /H20849the stronger the dipolar induction /H20850, the greater the deviation of M icfrom the direction of Bieff. Hence, the easier it is to de- crease M, making the magnetization curve less hysteretic. This is shown in Fig. 7, in which the magnetization curvesresemble those obtained for Mn 4SMM’s.31Those data show an abrupt decrease in Mat nearly zero external induction that is not evident in the magnetization curves of otherSMM’s. 28 It is important to note that the curves in Figs. 1–7 do not show the strong hysteresis observed experimentally in mostSMM’s, which is especially large in the central region of theM/H20849B/H20850curves. We remind the reader of our intent to focus upon the effects of the dipole-dipole interactions, whereas the most important features of SMM’s involved in their low-Trelaxation of the magnetization are generally thought to be their quantum structure and magnetic anisotropy. Neverthe-less, for this entirely classical and magnetically isotropic sys-tem, we are indeed finding hysteretic curves. In addition, thesweep rates in Figs. 1, 2, 6, and 7 are comparable to thoseused in experimental SMM studies. At much larger sweeprates, such as were studied in Figs. 3–5, an hysteretic centralregion was found. However, the sizes and Tdependencies of these hysteretic regions were still, respectively, much smallerand qualitatively different than observed in SMM’s. E. Effect of spin anisotropy upon the hysteresis It is straightforward to generalize our model to include some of the effects of magnetic spin anisotropy. Here weassume the nanomagnets contain sufficiently many spins thattheir quantum nature can be neglected. We note that SMM’s at low Tvalues behave as quantum entities, because of the small number of spins in each nanomagnet. In those systems,most workers have assumed that in addition to the isotropicHeisenberg and Zeeman interactions, the magnetic aniso-tropy terms could also be written in terms of components ofthe global spin operator S, with the overall dominant terms often written as − DS z2−E/H20849Sx2−Sy2/H20850.56However, portions suffi- ciently large for model comparison of the low- Tmagnetiza- tion curves of two Fe 2SMM dimers have been studied experimentally.57,58In neither antiferromagnetic dimer case was any evidence for either of those types of spin anisotropypresent. 59In contrast, in one of those cases, strong evidence for a substantial amount of local, single-ion spin anisotropy,in which the individual spins within a dimer align relative tothe dimer axis, is present in the data. 57,59In addition, the global anisotropy in the ferromagnetic SMM Mn 6is ex- tremely weak.60Since the precise quantum nature of more complicated SMM’s appears therefore to be poorly under-stood, we shall investigate the quantum features of the mag-netic hysteresis curves in SMM’s, including some effects oflocal spin anisotropy, in a subsequent presentation. 54 We therefore restrict our investigations of the role of mag- netic anisotropy upon the magnetization curves of arrays ofnanomagnets to the simplest classical model of spin aniso-tropy, B ic,eff=B+Bi,dipc+/H92620HA, /H2084910/H20850 where we take B=Bxˆand studied the cases HA=HAxˆand HA=HAzˆ. This is the 3D analog of the model studied by KS.45In this model, the magnetic anisotropy of each of the nanomagnets points in the same direction, and in our finitesized crystal consisting of 5 /H110035/H110034 nanomagnets on a cubic lattice, our chosen direction is one of the most general ones.We first performed two studies of the magnetic hysteresis inthis model, for which the anisotropy field H Ais directed, respectively, along /H20849100 /H20850,/H20648B, and /H20849001 /H20850,/H11036B, and our results are shown in Figs. 8 and 9, respectively. For both anisotropyfield directions, we take N c=100, N=5/H110035/H110034=100, /H9251//H9253 FIG. 8. Parallel 3D magnetization curves including different an- isotropy field HA=HAxˆstrengths, with the external induction B/H20648HA./H92620HA=0 /H20849thin black /H20850, 0.2 T /H20849dark gray /H20850, and 1.0 T /H20849thick dashed /H20850, respectively. For each curve, Nc=100, N=5/H110035/H110034=100, /H9251//H9253=3/H1100310−12,a=1.5 nm, /H9004B//H9004t=0.04 T/s, T=0.7 K, and Bmax =2.0 T.ALCÁNTARA ORTIGOZA, KLEMM, AND RAHMAN PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-8=3/H1100310−12,a=1.5 nm, /H9004B//H9004t=0.04 T/s, T=0.7 K, and Bmax=2.0 T. The sweep rates used in Figs. 8 and 9 are slightly faster than those used in SMM experiments butmuch slower than those used in the calculations of KS. Sincea=1.5 nm in these curves, these curves also represent the strongest realistic dipolar interaction we studied. In Fig. 8, we show the portions of the parallel magnetiza- tion curves with B /H20648HA/H20648xˆ, that exhibit the resulting regions of magnetic hysteresis for three HAvalues. For the /H92620HA =0, 0.2, and 1.0 T values shown, all three curves are hyster- etic, but the two lower HAvalues do not lead to a central hysteresis region. Nevertheless, the largest anisotropy value,H A=1.0 T, leads to a strong central hysteresis. We remark that the trends shown in Fig. 7 are rather different from thoseobtained for a single magnetic particle with magneticanisotropy. 5 In Fig. 9, we show the portions of the 3D perpendicular magnetization curves exhibiting the resulting regions ofmagnetic hysteresis for the five anisotropy fields /H92620HA =0,1 mT, 12 mT , 0.5 T , and 1.0 T , with the magnetic induc- tion B/H20648xˆ/H11036HA/H20648zˆ. In each case, hysteresis occurs near to magnetic saturation, but is absent in the central region forsmall magnetic induction. At /H92620HA=1.0 T, this is distinctly different from the large central hysteretic region observed forparallel anisotropy. Note that the slope dM/dBat small Bis nonmonotonic with increasing H A, as it has a minimum at curve /H20849c/H20850, corresponding to /H92620HA=12 mT. Thus, we conclude that it is possible to obtain a central hysteresis region using this classical model of dipolar inter-actions with constant spin anisotropy. However, our resultssuggest that such central hysteresis regions only arise for themagnetic induction parallel to the spin anisotropy direction, and for sufficiently strong anisotropy fields, H A/H33356HAmin, where 1.0 T /H11022/H92620HAmin/H110220.2 T. V . DIPOLAR INTERACTION, INDUCTION SWEEP RATE, AND ANISOTROPY DEPENDENCIES FOR A 2D SYSTEM To estimate the importance of the dipolar induction /H20849espe- cially when it becomes comparable to the external induc-tion /H20850, the anisotropy and the sweep rate, we have reproduced one of the 2D calculations of KS.45The KS calculation we chose to reproduce was pictured in their Fig. 2 /H20849i/H20850, and is shown here as the left panel of Fig. 10. Then, we changedsome experimental parameters to see how the results dependon the anisotropy strength, sweep rate, and lattice parameter. Our calculations for a cubic lattice consisting of four 25- particle layers differ from those of KS in many ways. 45They used a 2D square lattices of cylindrical nanodots /H20849here, we take their 5 /H110035 lattice with external induction aligned along an array’s diagonal /H20850, included a shape-dependent anisotropy field perpendicular to the lattice, performed their calculationsatT=0, used a much larger damping constant than we gen- erally did for 3D systems, and did not average their resultsover an ensemble of 2D samples, because such systems donot show variations in the resulting hysteresis loops for dif-ferent initial states. Nevertheless, we both integrated theLLG equation using the Runge-Kutta algorithm, and surpris-ingly, KS’s system turned out to be very sensitive to thedipolar field strength. The effective induction they consid-ered can be written as B ic,eff=B+Bi,dipc+/H92620HAzˆ. /H2084911/H20850 For lattice constant a=1.5 nm, spin S=5, and V/a3=0.5, where Vis the volume of the nanomagnet, the saturation magnetization is Ms/H1101555 Oe. Then, they took the dimension- less dt=5/H1100310−3, which implies a real time interval dt =5.17/H1100310−12s. If the system evolves during 700 time steps dtfor some fixed value of B, then Bis changed every /H9004t /H110153.62/H1100310−9s. On the other hand, KS chose a maximum external induction Bmax=2/H92620Ms/H110151.1/H1100310−2T. In addition, they took fixed induction steps of magnitude /H9004B=2 /H1100310−3/H92620Ms/H110151.1/H1100310−5T. Therefore, we estimate their re- sulting sweep rate to be /H9004B//H9004t/H110153/H11003103T / s ,a si no u r3 D results shown in Figs. 3 and 4. In the absence of any specific information, we then had to induce the value of the anisotropy field that KS used to ob-tain their figure. Fortunately, as discussed in the following,the results are rather insensitive to it, unless H Abecomes comparable to Bmax//H92620. In the right panel of Fig. 10, our 2D FIG. 9. Upper region of the 3D perpendicular magnetization curves with the external induction B=Bxˆ/H11036HA=HAzˆ, for different values of HA. Curves /H20849a/H20850–/H20849e/H20850correspond to /H92620HA=0, 1 /H1100310−3, 1.2/H1100310−2, 0.5, 1.0 T, respectively. The other parameters are the same as in Fig. 7. The arrows indicate the directions of thefield sweeps. FIG. 10. /H20849Left /H20850Hysteresis loop M/H20849B/H20850in units of Ms, for a weakly coupled array of 5 /H110035 ferromagnetic nanodots in a square lattice on the xyplane, from Fig. 2 /H20849i/H20850of KS. The external induc- tion is applied along the array diagonal /H2084945° from the xaxis /H20850/H20849Ref. 45/H20850./H20849Right /H20850Our results calculated for Nc=1 with 5 /H110035 nanomag- nets on a square lattice, /H9251//H9253=0.6, T=0 K, /H9004B//H9004t=3000 T/s, /H92620HA=7.5/H1100310−4T,B=B/H20849xˆ+yˆ/H20850//H208812.EFFECT OF DIPOLAR INTERACTIONS ON THE … PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-9calculations with /H92620HA=0.75 mT are shown, and by compar- ing that figure with Fig. 2 /H20849i/H20850of KS pictured in the left panel of Fig. 10, we see that the agreement is remarkably good. Very recently, Takagaki and Ploog /H20849TP/H20850used a fourth- order Runge-Kutta procedure to integrate the LLG equationswith N/H11003N2D nanomagnet lattices with magnetic anisotropy and dipole-dipole interactions. 61They used a fixed time in- terval dt=0.1q//H20849/H9253Ms/H20850, 20 times as fast as that used by KS,45 and continued interating until no further changes in the na- nomagnet spin configurations were obtained. They obtainedresults for N=5 which they described as considerably differ- ent from those of KS, with a somewhat different magnetiza-tion loop and a larger area of the hysteretic regions. 61In addition, in the absence of anisotropy fields, the spontaneousmagnetization and the area of their hysteretic region did notdecrease substantially as the Nincreased to 57, the maximum value they studied, 61in apparent contradiction to the predic- tion by Prakash and Henley that the infinite system withdipolar interactions alone is infinitely degenerate, and hencedoes not exhibit a spontaneous magnetization. 62Although TP claimed that their fourth-order procedure was intrinsicallymore accurate than the second-order one used by KS and byus, the apparent contradiction in the infinite system limit andthe specific fact that we obtained the excellent agreementpictured in Fig. 10 and with H A=0 in Fig. 11 with one of the N=5 results of KS both suggest that the procedure used by TP might have been less accurate than they claimed.61 A. Anisotropy field dependence of the hysteresis We first investigated the effects of changing the strength of the anisotropy fields, and presented our results in Fig. 11.The most important issue about the results shown in Fig. 11is the fact that the curve obtained by KS /H20849the left panel of Fig. 10 /H20850is basically independent of the anisotropy field H A for sufficiently small HA. That is, there are no essential dif-ferences between that curve reproduced in the right panel of Fig. 10 with /H92620HA=7.5 mT, and the one with HA=0. Strong deviations from these essentially identical curves appear for /H92620HA/H333564 mT, however. Since identical behavior is obtained without any anisotropy, this implies that all hysteretic fea-tures /H20849including the stepped magnetization and demagnetiza- tion /H20850are due to the dipolar interaction. H Abecomes impor- tant only when it is comparable to Bmax//H92620and tends to close the hysteresis loops, starting from the lower and upperloops. We note that by comparing Fig. 11 with Fig. 9, the details of the hysteresis obtained with H A=0 for Balong the /H20849110 /H20850 direction are different in 3D and 2D samples. The hysteresisis much larger in the 2D case pictured in Fig. 11, and has alarge loop in the central region that does not vanish at theorigin, plus large loops that extend up to saturation. In the3D case constructed from four 2D planes each equivalent tothat used in the calculation shown in Fig. 11, the magnitudeof the hysteresis is reduced and its details have been greatlyaltered. B. Induction sweep rate dependence of the hysteresis In Fig. 12, we show our results for a single configuration of a square 2D lattice with N=5/H110035 for different sweep rates, keeping the other parameters fixed at /H92620HA=0.75 mT, /H9251//H9253 =0.6, a=1.5 nm, S=5, T=0, and B=B/H20849xˆ+yˆ/H20850//H208812. From Fig. 12, we note that the hysteresis is nearly independent of induction sweep rate over the range 300 to 6000 T/s, dis-tinctly different from the strong dependence found in 3Dsystems shown in Fig. 5. C. Lattice parameter dependence of the hysteresis In Fig. 13, we have illustrated the effect of the lattice constant aupon the hysteresis. In this figure, we kept the other parameters fixed at S=5, T=0,/H9004B//H9004t=3000 T/s, /H92620HA=0.75 mT, /H9251//H9253=0.6, and B=B/H20849xˆ+yˆ/H20850//H208812. As ais var- ied from 2.0 to 1.25 nm, the upper portions of the hysteresis FIG. 11. Hysteresis loops for different strengths of HAfor 5 /H110035 nanomagnets on a square lattice with Nc=1.S=5, T=0 K, a =1.5 nm, /H9004B//H9004t=3000 T/s,/H9251//H9253=0.6, B=B/H20849xˆ+yˆ/H20850//H208812. The thin gray and thick black curves with /H92620HA=0,0.75 mT, respectively, are nearly indistinguishable. The small gray circles and dashedcurves correspond to /H92620HA=4.0,5.5 mT, respectively. The inset shows the entire curves, which are symmetric with respect to theorigin. FIG. 12. Hysteresis loops for different induction sweep rates with 5 /H110035 nanomagnets on a square lattice with Nc=1, S=5, T =0 K, a=1.5 nm, /H92620HA=0.75 mT, /H9251//H9253=0.6. The dashed gray, thick black, and light solid gray curves correspond to /H9004B//H9004t =300,1500,6000 T/s, respectively. The inset shows the entirecurves. B=B/H20849xˆ+yˆ/H20850//H208812.ALCÁNTARA ORTIGOZA, KLEMM, AND RAHMAN PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-10curves appear from left to right, respectively. From Fig. 13, it is readily seen that the magnetization curves are very sensi-tive to aand hence to the strength of the dipolar interaction, which is proportional to a −3. Our results for a=2.5 nm ex- hibit a smaller hysteresis shifted further to the left, and allindications of steps have disappeared. Although not shown inFig. 13, as ais increased further to 3.0 nm, the hysteresis almost disappears entirely. We deduce that stronger dipolarinteractions /H20849smaller a/H20850result in larger hysteresis loops con- taining increased widths of additional steps. We then infer that contrary to the conclusion found for the 3D systems /H20849based upon much smaller damping coefficients and much slower sweep rates /H20850, the dipolar interactions pro- mote a hysteretic behavior in this 2D system. VI. SUMMARY AND CONCLUSIONS We first found Nc=100 sample configurations with an overall magnetization close to 0. We then solved the Landau-Lifshitz-Gilbert equation for a 3D cubic lattice of N=5/H110035 /H110034 nanomagnets, subject to dipole-dipole interactions and spin anisotropy. These results should be relevant for an arrayof Stoner-Wolfarth nanomagnets, and to some extent, singlemolecule magnets, although the quantum nature of the latterhas so far been neglected. In the absence of spin anisotropy,we varied the magnetic induction sweep rate /H9004B//H9004t, the damping constant /H9251, the lattice constant a, and the tempera- ture T. We also considered the effects of a T-dependent damping constant of the form /H9251/H20849T/H20850//H9253=T0/Tsuggested by Fredkin and Ron. For slow sweep rates and small /H9251relevant for experimental studies on single molecule magnets, mag-netic hysteresis appears in the regions of the magnetizationcurves near to saturation, the area and onset magnetic induc-tion strength of which increases with decreasing /H9251and in- creasing sweep rate. With decreasing T, the onset magneti- zation magnitude of the hysteretic regions near to saturationdecreases. With decreasing acorresponding to increased dipole-dipole interaction strengths, the onset of the hysteresisregions near to saturation appears at increasing magnetic in-duction magnitude.At much larger sweep rates and damping constants, the magnetization curves attain saturation at much smaller ap-plied magnetic induction strengths. The hysteretic regionsjust below saturation have moved somewhat below satura-tion, and a new central hysteretic region appears. As onefollows the magnetization curve for a single configuration,the starting curve exhibits oscillations at a rather constant/H20849magnetic induction independent /H20850frequency f/2, but the phase of the magnetization oscillations is a random functionof the configuration. After the attainment of magnetic satu-ration, this central hysteretic region exhibits oscillations at f, twice that frequency, possibly with weak higher harmonics,forTnot too low, which are independent of the configura- tion. When the applied magnetic induction is in the /H20849110 /H20850di- rection /H20849from the sample center to one of its corners /H20850, mag- netic hysteresis exhibiting steps and jumps appears withinthe central region, but vanishes at and very near to the origin. Although these steplike features are suggestive of the behav-ior seen in single molecule magnets, they are present atrather high Tvalues, as they arise from the classical sample shape effects. In the presence of the magnetic anisotropy field H A,a n applied magnetic induction parallel to the anisotropy axisleads to a large central hysteresis region, provided that themagnitude of the spin anisotropy is sufficiently large. For theapplied magnetic induction perpendicular to the magnetic an-isotropy, no central hysteresis region is present, although asmall amount of hysteresis near to saturation persists for suf-ficiently small spin anisotropy strength, and the slope of themagnetization curve at the origin is nonmonotonic, exhibit-ing a maximum at a particular small value of the spin aniso-tropy strength. These effects for the spin anisotropy arequalitatively in agreement with those in many types of arraysof nanomagnets, including single molecule magnets. As a test of our numerical procedure, we studied the sim- plified 5 /H110035 2D square lattice with a perpendicular spin an- isotropy field H Ausing the same procedure, and for a par- ticular set of parameters, obtained quantitative agreementwith a hysteresis curve obtained for that system by Kayaliand Saslow. 45We showed that their hysteresis curve is basi- cally independent of HAuntil/H92620HAis on the order of the external induction. We also demonstrated that the magnetichysteresis does not depend significantly upon the magneticinduction sweep rate, as opposed to the dependence wefound in our 3D system. In addition, we found that the mag-netization of the 2D system is very sensitive to variations inthe lattice parameter a. Finally, we noticed that although di- polar interactions also oppose the magnetization process in2D systems, increasing the onset magnetic induction strengthfor the attainment of saturation as in 3D systems, they in-crease the area of the hysteresis, a behavior opposite to thatfound for the 3D system with a much smaller damping co-efficient and much slower sweep rate. We expect our results to be relevant to the magnetization processes in a variety of nanomagnet arrays, especially thoseapproximating arrays of Stoner-Wolfarth particles. In addi-tion, some of the features we obtained should be relevant tosingle molecule magnets, although the temperature depen-dence of the effects is not in agreement with experiments on FIG. 13. Hysteresis loops for lattice parameters a=2.5 nm /H20849solid black /H20850,a=2.0 nm /H20849dashed black /H20850,a=1.5 nm /H20849solid gray /H20850, and a =1.25 nm /H20849dotted-dashed black /H20850, for 5/H110035 nanomagnets on a square lattice with Nc=1, S=5, T=0 K, /H9004B//H9004t=3000 T/s,/H92620HA=7.5 /H1100310−4T,/H9251//H9253=0.6. The inset shows the entire curves. B=B/H20849xˆ +yˆ/H20850//H208812.EFFECT OF DIPOLAR INTERACTIONS ON THE … PHYSICAL REVIEW B 72, 174416 /H208492005 /H20850 174416-11those materials. Further studies of the magnetic hysteresis using quantum models of the nanomagnets and their variousanisotropy types is warranted, and will be addressedsubsequently. 54 ACKNOWLEDGMENT This work was supported in part by the NSF under Grant No. NER-0304665. APPENDIX We rotate our reference frame at every integration time step in such a way that Bic,effis along the zaxis. In this case, we can easily solve the LLG equation, Eq. /H208491/H20850. For simplicity of notation, we drop the subscripts iand superscripts c, and remember that we are describing the precession of the ith nanomagnet in the cth crystal. We define the axes to describe the magnetization direction of this particular nanomagnet, Mˆ,/H9258ˆ, and/H9278ˆ, where /H9278ˆ=Mˆ/H11003/H9258ˆ, and then write Beff=Bzzˆ=Bz/H20849Mˆcos/H9258−/H9258ˆsin/H9258/H20850=MˆBM+/H9258ˆB/H9258./H20849A1/H20850 Since the magnitude of the dipole moment Msis conserved, in spherical coordinates Eq. /H208491/H20850leads to dMˆ dt=/H9258ˆd/H9258 dt+/H9278ˆsin/H9258d/H9278 dt=/H9258ˆ/H9251B/H9258+/H9278ˆ/H9253B/H9258. /H20849A2/H20850 Finally, from d/H9258 dt== −/H9251/H20841Beff/H20841sin/H9258, /H20849A3/H20850 d/H9278 dt=−/H9253/H20841Beff/H20841, /H20849A4/H20850 we obtain for a very small time interval dt, /H9278/H20849t0+dt/H20850/H11015/H9278/H20849t0/H20850−/H9253/H20841Beff/H20849t0/H20850/H20841dt, /H20849A5/H20850 /H9258/H20849t0+dt/H20850/H11015/H9258/H20849t0/H20850−/H9251/H20841Beff/H20849t0/H20850/H20841sin/H20851/H9258/H20849t0/H20850/H20852dt. /H20849A6/H20850 These equations were used in our numerical calculations. In order to relate the angles to measurable quantities, however,we note that it is possible to integrate Eqs. /H20849A3/H20850and /H20849A4/H20850 exactly, obtaining/H9258/H20849t/H20850= cos−1/H20875tanh/H20873tanh−1/H20853cos/H20851/H9258/H20849t0/H20850/H20852/H20854+/H9251/H20885 t0t d/H9270/H20841Beff/H20849/H9270/H20850/H20841/H20874/H20876, /H20849A7/H20850 /H9278/H20849t/H20850=/H9278/H20849t0/H20850−/H9253/H20885 t0t d/H9270/H20841Beff/H20849/H9270/H20850/H20841, /H20849A8/H20850 which is equivalent to that obtained using a somewhat dif- ferent technique.3We note that by expanding Eqs. /H20849A7/H20850and /H20849A8/H20850to leading order in dt, we recover Eqs. /H20849A6/H20850and /H20849A5/H20850, respectively. However, these more general forms for /H9258/H20849t/H20850and/H9278/H20849t/H20850lead to a more physical interpretation of our method. Since the dimensionless magnetization components along and perpen-dicular to B effare Mz=cos/H9258,Mx=sin/H9258cos/H9278, and My =sin/H9258sin/H9278, we have Mz/H20849t/H20850= tanh/H20873tanh−1/H20851Mz/H20849t0/H20850/H20852+/H9251/H20885 t0t d/H9270/H20841Beff/H20849/H9270/H20850/H20841/H20874,/H20849A9/H20850 Mx/H20849t/H20850=/H208811− /H20851Mz/H20849t/H20850/H208522cos/H20851/H9278/H20849t/H20850/H20852, /H20849A10 /H20850 My/H20849t/H20850=/H208811− /H20851Mz/H20849t/H20850/H208522sin/H20851/H9278/H20849t/H20850/H20852. /H20849A11 /H20850 Independent of the coordinates, we must assure that /H20849for theith nanomagnet in the cth configuration /H20850Mchanges its direction smoothly, in order to obtain a reliable calculation for the overall M/H6023. Since each component of Mcannot change dramatically, we must therefore require /H9258/H112702/H9266and /H9278/H112702/H9266. These restrictions then require us to set the time integration step width dtsufficiently small. If, for example, /H9253//H9251were on the order of 10+11and /H20841Beff/H20841were in the range 10−3–10−2T, we would require dt/H1102110−11s. For sweep rate /H9004B//H9004t/H1101510−2T/s, where /H9004t=Ntdt/H1101510−4s,Ntmust be on the order of 107. Since we would need to recalculate the direction of the magnetization of each nanomagnet Nttimes in each /H9004Bstep, this would be a significant challenge with present day computers. One thing we can do to make our calculations feasible for the sweep rates used in SMM studies is to set /H9251extremely small, say /H9251//H9253/H1135110−10, although such small /H9251values have not been reported in experiments. Otherwise, to study muchlarger but perhaps more reasonable /H9251values, we would have to use much faster sweep rates, as in KS.45 *Electronic address: alcantar@phys.ksu.edu †Electronic address: klemm@phys.ksu.edu ‡Electronic address: rahman@phys.ksu.edu 1D. Newns, W. Donath, G. Martyna, M. Schabes, and B. Lengs- field, J. Appl. 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PhysRevB.92.180412.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 92, 180412(R) (2015) Electron-magnon scattering in magnetic heterostructures far out of equilibrium Erlend G. Tveten*and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Yaroslav Tserkovnyak Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA (Received 1 April 2015; published 23 November 2015) We present a theory of out-of-equilibrium ultrafast spin dynamics in magnetic heterostructures based on the s-d model of ferromagnetism. Both in the bulk and across interfaces, the exchange processes between the itinerantsand the localized delectrons are described by kinetic rate equations for electron-magnon spin-flop scattering. In our treatment, the magnon distribution function remains nonthermalized on the relevant time scales of thedemagnetization process, and the relaxation of the out-of-equilibrium spin accumulation among itinerant electronsprovides the principal channel for dissipation of spin angular momentum from the combined electronic system. DOI: 10.1103/PhysRevB.92.180412 PACS number(s): 72 .25.Mk,72.10.Di,72.20.Pa,75.40.Gb Controlling spin flow in magnetic heterostructures at ultrafast time scales using femtosecond laser pulses opensintriguing possibilities for spintronics [ 1]. These laser-induced perturbations [ 2,3] stir up the most extreme regime of spin dynamics, which is governed by the highest energy scaleassociated with magnetic order: the microscopic spin exchangethat controls the ordering temperature T C. In contrast, at microwave frequencies the ferromagnetic dynamics in thebulk are well described by the Landau-Lifshitz-Gilbert (LLG)phenomenology [ 4], which has been successfully applied to the problem of the ferromagnetic resonance (FMR) [ 5]. At finite temperatures below T C, the spin Seebeck and Peltier effects [ 6,7] describe the coupled spin and heat currents across interfaces in magnetic heterostructures. Despite theirdifferent appearances, the microwave, thermal, and ultrafastspin dynamics are all rooted in the exchange interactionsbetween electrons. It is thus natural to try to advance amicroscopic understanding of the ultrafast dynamics basedon the established phenomena at lower energies. Although some attempts have been made [ 8,9] to extend the LLG phenomenology to describe ultrafast demagnetizationin bulk ferromagnets, no firm connection exists between theultrafast spin generation at interfaces and the microwavespin-transfer and spin-pumping effects [ 10] or the thermal spin Seebeck and Peltier effects. In this Rapid Communication, weunify the energy regimes of microwave, thermal, and ultrafastspin dynamics in magnetic heterostructures from a commonmicroscopic point of view, so that the parameters that controlthe high and low energy limits of spin relaxation originatefrom the same electron-magnon interactions. In addition to theunified framework, this Rapid Communication’s unique con-tributions are the history-dependent, nonthermalized magnondistribution function and the crucial role of the out-of-equilibrium spin accumulation among itinerant electrons asthe bottleneck that limits the dissipation of spin angularmomentum from the combined electronic system. The first reports on ultrafast demagnetization in Ni [ 11] challenged the conventional view of low-frequency magneti-zation dynamics at temperatures well below T C. A multitude of *Corresponding author: erlend.tveten@ntnu.nomechanisms and scenarios have been suggested to explain the observed quenching of the magnetic moment. Some advocatedirect coherent spin transfer induced by the irradiating laserlight as the source of demagnetization [ 12]. Alternative theo- ries argue that ultrafast spin dynamics arise indirectly throughincoherent heat transfer to the electron system [ 13,14]. Recent experiments have demonstrated that nonlocal laser irradiationalso induces ultrafast demagnetization [ 15], and atomistic modeling [ 16] supports the view that heating of magnetic materials is sufficient to induce ultrafast spin dynamics. Terahertz (THz) magnon excitations in metallic ferromag- nets have recently been proposed as an important elementof ultrafast demagnetization by several authors [ 17,18]. The elementary interaction that describes these excitations is theelectron-magnon scattering. Our theory is based on kineticequations for the low-frequency spin and charge transportassociated with the microwave magnetization dynamics inheterostructures [ 19] and with the linear spin-caloritronic response [ 7,20]. We extend these theories to treat far-from- equilibrium spin dynamics, in which transport is dominatedby magnons and hot electrons. Electron-magnon scatteringplays a critical role in this regime. We base our understandingof this interaction on the transverse spin diffusion [21]i nt h e bulk and the spin-mixing physics , e.g., spin transfer and spin pumping [ 19,22], at the interfaces. In our approach, we assume that the localized spins that result in the experimentally detectable macroscopic magne-tization [ 23] are distinct from the itinerant electrons at the energy scales of interest. According to the accepted descriptionof relaxation in ferromagnetic metals, the loss of energy andangular momentum from localized delectrons is mediated by the exchange interaction to the itinerant selectrons. The spin transfer from dtosstates is accompanied by the relaxation of the selectron spins to the lattice through an incoherent spin-flip process caused by the spin-orbit coupling. Mitchellformulated such a model several decades ago to describe thelongitudinal relaxation of ferromagnetic metals [ 24]. A similar description was later employed to describe Gilbert damping inferromagnets at low frequencies [ 25,26]. In the following, we start by outlining the basic quantum- kinetic formalism for ultrafast spin dynamics in bulk fer-romagnetic metals. Later, we show that the ferromagnet 1098-0121/2015/92(18)/180412(5) 180412-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS TVETEN, BRATAAS, AND TSERKOVNY AK PHYSICAL REVIEW B 92, 180412(R) (2015) (F)|normal-metal (N) interfacial spin transport due to electron-magnon interactions follows a similar essential struc-ture, unifying the bulk and interfacial spin dynamics inmagnetic heterostructures. The Hamiltonian that describes FisˆH=ˆH 0+ˆHsd, where ˆH0consists of decoupled s- andd- electron energies, including the kinetic energy of the itinerantelectron bath, the d-dexchange energy, dipolar interactions, and the crystalline and Zeeman fields. The s-dinteraction is ˆH sd=Jsd/summationdisplay jSd j·s(rj), (1) where Jsdis the exchange energy and Sd j[s(rj)] is the d- electron ( s-electron) spin vector (spin density) at lattice point j. We express the s-dinteraction in terms of bosonic and fermionic creation and annihilation operators: ˆHsd=/summationdisplay qkk/primeVqkk/primeaqc† k↑ck/prime↓+H.c., (2) where a† q(aq) is the Holstein-Primakoff creation (annihilation) operator for magnons with wave number qandc† kσ(ckσ) is the creation (annihilation) operator for selectrons with momentum kand spin σ.ˆHsddescribes how an electron flips its spin while creating or annihilating a magnon with momentumqand spin /planckover2pi1. The scattering strength is determined by the matrix element V qkk/prime. In Eq. ( 2), we have disregarded terms of the form ∼a† qaq/primec† kσck/primeσ, which describe multiple-magnon scattering and do not contribute to a net change in magnetization along thespin-quantization axis. We have also disregarded higher-orderterms associated with the Holstein-Primakoff expansion. Fullyaddressing magnonic correlation effects in the ultrafast regimewould require a rigorous approach, e.g., using nonequilibriumKeldysh formalism [ 27]. However, when the s-dcoupling ( 1)i s not the dominant contribution to ˆH, we follow a mean-field ap- proach and use Fermi’s golden rule to compute the spin transferbetween the sanddsubsystems. We assume that all relevant energy scales are much smaller than the Fermi energy /epsilon1 F≡ kBTFof the itinerant selectrons. In this limit, the electronic continuum remains largely degenerate, with electron-holepairs present predominantly in the vicinity of the Fermi level. We orient the coordinate system such that the localized spin density points in the negative zdirection at equilibrium, with saturation value S(in units of /planckover2pi1). In the presence of a magnon density n d, the longitudinal spin density becomes Sz=nd− S. The magnons are assumed to follow a quadratic dispersion relation /epsilon1q=/planckover2pi1ωq=/epsilon10+Aq2, where /epsilon10is the magnon gap andAparametrizes the stiffness of the ferromagnet. /angbracketlefta† qaq/prime/angbracketright= n(/epsilon1q)δqq/primedefines the magnon distribution function n(/epsilon1q), which is related to the total magnon density through nd=/integraltext/epsilon1b /epsilon10d/epsilon1qD(/epsilon1q)n(/epsilon1q), where D(/epsilon1q)=√/epsilon1q−/epsilon10/(4π2A3/2)i s the magnon density of states. The integral over D(/epsilon1q) is cut off at an energy corresponding to the bandwidth, /epsilon1b∼kBTC, which is the magnon energy at the edge of the Brillouin zone. Because of the s-dinteraction ( 1), the itinerant selectrons have a finite spin density at equilibrium (see Fig. 1). One of the key driving forces of the out-of-equilibrium spin dynamics isthe spin accumulation μ s≡δμ↑−δμ↓. The bands for spin-up and spin-down electrons are split by /Delta1xc∼JsdSa3, where ais FIG. 1. (Color online) (a) Sketch of the density of selectron states in a ferromagnetic metal with saturation spin density S.A t equilibrium, the exchange splitting /Delta1xcshifts the bands for spin-up and spin-down electrons. (b) A laser pulse heats the selectron bath. The out-of-equilibrium spin accumulation μs=δμ↑−δμ↓results from two different mechanisms: (1) electron-magnon scattering induces a spin density among the selectrons, and (2) the mean- field exchange splitting is shifted by δ/Delta1xcby the induced magnon density nd. the lattice constant of F. By introducing a dynamic exchange splitting, we can write μs=δns/D+δ/Delta1xc[28], where δns is the out-of-equilibrium spin density of the selectrons, D=2D↑D↓/(D↑+D↓), and D↑(↓)is the density of states for spin-up (spin-down) electrons at the Fermi level. Becausethe mean-field band splitting due to the s-dexchange vanishes when the dorbitals are fully depolarized, δ/Delta1 xc//Delta1xc=±nd/S, where the sign determines whether the sanddorbitals couple ferromagnetically ( −) or antiferromagnetically ( +). The rate of spin transfer (per unit volume) between the s anddsubsystems due to electron-magnon spin-flop processes is determined from Eq. ( 2) by Fermi’s golden rule [ 22]: Isd=/integraldisplay/epsilon1b /epsilon10d/epsilon1q/Gamma1(/epsilon1q)(/epsilon1q−μs)D(/epsilon1q)[nBE(/epsilon1q−μs)−n(/epsilon1q)], (3) where /Gamma1(/epsilon1q) parametrizes the scattering rate at energy /epsilon1q. In the derivation of Eq. ( 3)w eh a v ea s s u m e dt h a tt h e kinetic energy of the itinerant electrons and the empty states(holes) thermalize rapidly due to Coulombic scatteringand that they are distributed according to Fermi-Diracstatistics. Correspondingly, after standard manipulations [ 29], it can be shown that the electron-hole pairs follow theBose-Einstein (BE) distribution function, n BE(/epsilon1q−μs)= {exp[βs(/epsilon1q−μs)]−1}−1, at the electron temperature Ts=1/(kBβs). The number of available scattering states is influenced by the spin accumulation μs, as expected. In contrast to the low-energy treatment in Ref. [ 22], the derivation of Eq. ( 3) does not constrict the form of the magnonic distribution n(/epsilon1q) to the thermalized BE distribution 180412-2RAPID COMMUNICATIONS ELECTRON-MAGNON SCATTERING IN MAGNETIC . . . PHYSICAL REVIEW B 92, 180412(R) (2015) function. When the time scale of the s-dscattering is faster than the typical rates associated with magnon-magnon interactions,magnons are notinternally equilibrated shortly after rapid heat- ing of the electron bath, as also predicted by atomistic mod-eling [ 30]. Consequently, the occupation of the magnon states can deviate significantly from the BE distribution on the timescale of the demagnetization process. Our treatment of thiscentral aspect differs from that of Ref. [ 31], in which the ex- cited magnons are assumed to be instantly thermalized with aneffective spin temperature and zero chemical potential and thethermally activated electron bath is assumed to be unpolarized. Thes-dscattering rate can be phenomenologically ex- panded as /Gamma1(/epsilon1 q)=/Gamma10+χ(/epsilon1q−/epsilon10), where /Gamma10(which van- ishes in the simplest Stoner limit [ 21]) parametrizes the scattering rate of the long-wavelength magnons and χ(/epsilon1q− /epsilon10)∝q2describes the enhanced scattering of higher-energy magnons due to transverse spin diffusion [ 21]. In general, one might expect other terms of higher order in qto be present in this expansion as well. We will, however, limit ourselves toextrapolating /Gamma1(/epsilon1 q) up to the bandwidth /epsilon1b, which should be sufficient for qualitative purposes. Neglecting any direct relaxation of magnons to the static lattice or its vibrations (i.e., phonons), ∂tnd=Isd//planckover2pi1.T h e equations of motion for the s-electron spin accumulation and thed-electron magnon distribution function are ∂tμs=−μs τs+ρ /planckover2pi1Isd, (4) ∂tn(/epsilon1q)=/Gamma1(/epsilon1q) /planckover2pi1(/epsilon1q−μs)[nBE(/epsilon1q−μs)−n(/epsilon1q)],(5) where ρdetermines the feedback of the demagnetization onμsandτsis the spin-orbit relaxation time for the s- electron spin density relaxing to the lattice. τsis typically on the order of picoseconds [ 32] and represents the main channel for the dissipation of angular momentum out ofthe combined electronic system. In general, τ salso depends on the kinetic energy of the hot electrons after laser-pulseexcitation. However, this discussion is beyond the scope of thisRapid Communication, and we assume that τ sis independent of energy. ρ=ρD+ρ/Delta1=− 1/D±/Delta1xc/Sincludes effects arising from both the out-of-equilibrium spin density andthe dynamic exchange splitting. For ferromagnetic ( −)s-d coupling, these effects add up, whereas for antiferromagnetic(+) coupling, they compete. At low temperatures, low-frequency excitations result in purely transverse spin dynamics. In the classical pictureof rigid magnetic precession, the transverse relaxation timeτ 2is determined by the longitudinal relaxation time τ1as follows: 1 /τ2=1/(2τ1)=αω, where αis the Gilbert damping parameter and ωis the precession frequency. Indeed, in the limit (q,Ts)→0, Eq. ( 3) yields ∂tnd→−/Gamma10 /planckover2pi1/epsilon10nd, (6) which is identical to the LLG phenomenology, indicating that/epsilon10=/planckover2pi1ωand thus /Gamma10=2α. This result establishes the important link between the scattering rate /Gamma10in this treatment and the Gilbert damping parameter that is accessible throughFMR experiments. FIG. 2. (Color online) Numerical solutions of Eqs. ( 4)a n d( 5) afterTsis increased from 102to 103K(TC) within 50 fs with a decay time of 2 ps. /epsilon10=5m e V ,A=0.6m e Vn m2,ρ=6m e Vn m3,τs= 2p s ,a n d α∗=10α=0.1. (a) The itinerant electron-hole pair distribution nBE(/epsilon1−μs) is rapidly depleted by the spin accumulation μsthat is built up via electron-magnon scattering. (b) In the magnon distribution n(/epsilon1q) the high-energy magnon states are rapidly populated, whereas the low-energy states remain unaffected on short time scales. (c) Time evolution of the spin accumulation μs(t)a n d (d) the longitudinal spin density −Sz(t) with different decay times of Ts: 0.15, 0.5, and 2 ps. In the opposite high-frequency limit, pertinent to ultrafast demagnetization experiments, we consider F to be in a low-temperature equilibrium state before being excited by a THzlaser pulse at t=0, upon which the effective temperature of the itinerant electron bath instantly increases such thatT s/greaterorsimilarTC. This regime is clearly beyond the validity of the LLG phenomenology, which is designed to address the low-energy extremum of magnetization dynamics. Dissipation inthe LLG equation, including relaxation terms based on thestochastic Landau-Lifshitz-Bloch treatment [ 14,33], is subject to a simple Markovian environment without any feedback orinternal dynamics. This perspective must be refined for highfrequencies when no subsystem can be viewed as a featurelessreservoir for energy and angular momentum. To appreciate the nonthermalized nature of the excited magnons, we consider the limit in which μ sis small compared with/epsilon10and no magnons are excited [ n(/epsilon1q)=0] for t<0. After rapid heating of the itinerant electrons at t=0, the time evolution of the magnonic distribution follows n(/epsilon1q,t)≈nBE(/epsilon1q,t)[1−e−/Gamma1(/epsilon1q)/epsilon1qt//planckover2pi1]. (7) This result implies that, initially, the high-energy states are populated much faster than low-energy states. When μsbe- comes sizable, the coupled equations ( 4) and ( 5) must be solved subject to a suitable Ts(t). Figures 2(a) and 2(b) present nu- merical solutions of ( 4) and ( 5) when Tsis increased from 102 to 103K within 50 fs with a decay time of 2 ps. By comparison, internal magnon-magnon interactions equilibrate the distribu-tion function on the time scale τ −1 eq∼/planckover2pi1−1/epsilon1m[/epsilon1m/(kBTC)]3[22], where /epsilon1mis a characteristic energy of the thermal magnon cloud. For short times, Isd[Eq. ( 3)] dominates the magnon dy- namics, and we expect the magnon population to significantlydiffer from the thermalized BE distribution. 180412-3RAPID COMMUNICATIONS TVETEN, BRATAAS, AND TSERKOVNY AK PHYSICAL REVIEW B 92, 180412(R) (2015) When Ts>TC, the thermally excited electron-hole pairs are populated in accordance with the classical Rayleigh-Jeansdistribution, n BE(/epsilon1q−μs)→kBTs/(/epsilon1q−μs). Assuming, for simplicity, that the expansion for /Gamma1(/epsilon1q) is valid throughout the Brillouin zone, Eq. ( 3) yields ∂tnd|t→0=Isd(0)//planckover2pi1=[/Gamma10+ 3χ(/epsilon1b−/epsilon10)/5]kBTsS//planckover2pi1. Thus, the demagnetization rate is initially proportional to the temperature of the electron bath butis reduced by the lack of available scattering states for high-energy magnons within the time scale of the demagnetizationprocess. This finding conflicts with the results obtainedfrom a Langevin treatment of the LLG equation [ 34], in which the magnetization relaxation rate is proportional tothe temperature difference at all times . Figures 2(c) and2(d) illustrate the time evolution of the out-of-equilibrium spinaccumulation μ s(t) and the longitudinal spin density −Sz(t) for different decay times of Ts. In the ultrafast regime, the electron-magnon spin-flop scattering is governed by the effective Gilbert damping parameter α∗≡χ(/epsilon1b−/epsilon10). Experimental investigations of the magnon relaxation rates on Co and Fe surfaces confirm thathigh-qmagnons have significantly shorter lifetimes than low- q magnons [ 18]. It is reasonable to assume that the same effects are also present in the bulk. The initial relaxation time scale inthe ultrafast regime is τ i∼(α∗/planckover2pi1−1kBTs)−1. This generalizes the result of Koopmans et al. [8] for the ultrafast relaxation of the longitudinal magnetization to arbitrary α∗based on the transverse spin diffusion [ 21]. The notion of magnons becomes questionable when the intrinsic linewidth approachesthe magnon energy, which corresponds to α ∗∼1. Staying well below this limit and consistent with Refs. [ 18,21], we use α∗=0.1. ForTC=103K the initial relaxation time scale τi∼ 102(TC/Ts) fs, which is generally consistent with the demagne- tization rates observed for ultrafast demagnetization in Fe [ 35]. We now show that the interfacial scattering follows a struc- ture similar to that of the bulk scattering in a unified descriptionbased on the electron-magnon interaction. Figure 3presents a schematic illustration of an F |N interface. In magnetic het- erostructures and for stand-alone ferromagnets on a conductingsubstrate, the demagnetization dynamics of F are also affectedby the spin accumulation in N μ N(x), which can impact how nonlocal laser irradiation (e.g., the heating of N alone) inducesultrafast demagnetization of F [ 15]. By adding terms of the form∼/summationtext qkk/primeUqkk/primeaq˜c† k↑˜ck/prime↓toˆHsd, where ˜c† k↑(˜ck/prime↓) describes the creation (annihilation) of an electron with spin up (down)at the F |N interface, the interfacial spin transfer (per unit area) due to electron-magnon spin-flop scattering is [ 22] I i=/integraldisplay/epsilon1b /epsilon10d/epsilon1q/Gamma1i(/epsilon1q)/parenleftbig /epsilon1q−μ0 N/parenrightbig D(/epsilon1q)/bracketleftbig nBE/parenleftbig /epsilon1q−μ0 N/parenrightbig −n(/epsilon1q)/bracketrightbig , (8) where μ0 N≡μN(0) is the spin accumulation at the interface and/Gamma1i(/epsilon1q) parametrizes the interfacial scattering rate. The scattering of coherent long-wavelength magnons at the F|N interface can be described in the language of spin pumping/spin Seebeck effects [ 22], parametrized by the spin- mixing conductance g↑↓(per unit area) [ 19]. Motivated by /Gamma1(/epsilon1q) in the bulk, we write for the interfacial scattering rate /Gamma1i(/epsilon1q)=g∗ ↑↓(/epsilon1q)/(πS), where g∗ ↑↓reduces to g↑↓for low- energy scattering, /epsilon1q→/epsilon10. The interface scattering [Eq. ( 8)] FIG. 3. (Color online) Sketch of a metallic ferromagnet (F) cou- pled to a normal metal (N). In the ultrafast regime, both the rapid heating of selectrons in F by /Delta1Ts[labeled (1)] and the heating of Nb y /Delta1T N[labeled (2)] can demagnetize F. Isd[Eq. ( 3)] induces the spin accumulation μsin F, whereas Ii[Eq. ( 8)] induces the spin accumulation μ0 Nat the F |N interface. Subsequently, μN(x) diffuses into N until it vanishes due to spin-flip dissipation to the lattice. The additional interfacial spin current IsN, due to the thermodynamic biases δμ=μs−μ0 NandδT=Ts−TN, can be described by conventional thermoelectric parameters for longitudinal spin-dependent transport [ 36]. dominates the microwave spin relaxation in thin ferromagnetic layers of thickness dF/lessorsimilar10 nm [ 19,37]. This trend should continue for higher frequencies and is relevant for ultrafastspin dynamics in thin magnetic layers in heterostructures [ 1]. We expect the energy dependence of the effective spin-mixingconductance to be relatively weak compared to that of thebulk scattering /Gamma1(/epsilon1 q), which can be severely constrained at low energies due to momentum conservation [ 21]. For a finite temperature bias δTacross the interface and for magnons thermalized at the temperature T< T C, the connection to the thermal spin Seebeck and Peltier effects is made by identifyingS=∂ TIiand/Pi1=TS//planckover2pi1[20] as the Seebeck and Peltier coefficients, respectively. In conclusion, we have extended the concepts of trans- verse spin diffusion in bulk ferromagnets and the interfacialspin-mixing physics to address the ultrafast spin dynamicsobserved in rapidly heated magnetic heterostructures. Inthe ultrafast regime, the relative importance of the bulkscattering, parametrized by α ∗, and the interfacial scattering, parametrized by g∗ ↑↓, can be extracted from measurements of demagnetization strength and spin currents in magneticheterostructures. For metallic ferromagnets in the bulk, ouranalysis shows that treating the magnonic subsystems asquasiequilibrated and parametrized by an effective tempera-ture is insufficient to describe the far-from-equilibrium spindynamics induced by pulsed laser heating. 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PhysRevB.88.104426.pdf

PHYSICAL REVIEW B 88, 104426 (2013) Spin-torque switching efficiency in CoFeB-MgO based tunnel junctions J. Z. Sun,1,*S. L. Brown,1W. Chen,2E. A. Delenia,3M. C. Gaidis,1J. Harms,2G. Hu,1Xin Jiang,3R. Kilaru,1W. Kula,2 G. Lauer,1L. Q. Liu,1S. Murthy,2J. Nowak,1E. J. O’Sullivan,1S. S. P. Parkin,3R. P. Robertazzi,1P. M. Rice,3 G. Sandhu,2T. Topuria,3and D. C. Worledge1 1IBM-Micron MRAM Alliance, IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA 2IBM-Micron MRAM Alliance, Micron, Boise, Idaho 83707, USA 3IBM-Micron MRAM Alliance, IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120, USA (Received 7 July 2013; published 26 September 2013) It is convenient to define the spin-torque switching efficiency in nanostructured magnetic tunnel junctions as the ratio between the free-layers thermal activation barrier height Eband the threshold switching current Ic0. Recent device exploration has led to occasional observations of spin-torque induced magnetic switching efficiency in magnetic tunnel junctions that exceeds the macrospin limit by a factor of 2–10. In this paper weexamine the possible origins for such enhancement, and materials properties that may allow the full realizationof such enhancements. DOI: 10.1103/PhysRevB.88.104426 PACS number(s): 75 .78.Jp, 85.75.Dd, 75 .76.+j, 73.40.−c I. INTRODUCTION Spin-transfer torque switched magnetic tunnel junction is a basic building block for a new class of solid-state technologiesas represented by the so-called spin-transfer-torque magneticrandom access memory, or STT-MRAM. 1For the approach to be competitive commercially, it is important to scale the devicestructures below 30 nm while maintaining a strong magneticanisotropy energy for data retention. At the same time, oneneeds to keep the switching current minimized for reducingthe load on the bit-selection transistor and for power savings.The magnetic anisotropy energy E bneeds to stay above 60 or so kBTwhere Tis the ambient temperature. The spin- torque switching involves a threshold current Ic0(or voltage Vc0for a tunnel junction) whose value is, in an ideal macrospin limit, proportional to Eb. A macrospin limit is equivalent to setting the exchange energy Aex→+ ∞ . For an ideal tunnel junction in a pure uniaxial anisotropy potential, the thresholdvoltage is symmetric in magnitude for parallel-to-antiparallel (P-AP) and antiparallel-to-parallel (AP-P) switching. 2,3At low tunnel junction bias when the conductances are nearly biasindependent, this voltage is approximately 2,3 Vc0≈/parenleftbigg2e ¯hη/parenrightbigg/parenleftbiggα GP/parenrightbigg mHk, (1) where mis the total magnetic moment of the macrospin, Hkis the uniaxial anisotropy field of the macrospin, whose axis is here assumed to be in collinear alignment with thespin current’s polarization, and η=√ mr(mr+2)/2(mr+1) is the spin-polarization factor related to the tunnel magne-toresistance (TMR) value m r=(Rap−Rp)/Rp.4Obviously, Ic0≈GpVc0, and the threshold current density Jc0≈/parenleftbigg4e ¯hη/parenrightbigg α/parenleftbigg1 2MsdHk/parenrightbigg ≈/parenleftbigg4e ¯hη/parenrightbigg α/Sigma1 eff, (2) where Msis the free-layer film’s saturation magnetization, and dis its thickness. Note the quantity1 2MsdHk→/Sigma1effcan be viewed sometimes as an areal density of anisotropy energy,especially in the ultrathin film limit discussed below.The spin-torque switching efficiency κis empirically defined as the ratio κ=E b/Ic0. In the macrospin limit, κ≡Eb Ic0=/parenleftbigg¯h 4e/parenrightbigg/parenleftbiggη α/parenrightbigg ≈/parenleftbigg¯h 4e/parenrightbigg√mr(mr+2) 2α(mr+1)(3) for the P-AP switching.3 The validity of the macrospin assumption depends on a comparison of length scales. A finite exchange energy Aexof the ferromagnet gives rise to several length scales, originatingfrom the competition between an exchange energy penalty dueto spatially inhomogeneous magnetization orientation and thechange of various other energy terms. A basic expression ofthe total energy density at a given point rinside a ferromagnet with order parameter M=M snm(r)is Kv(nm,r)=Kani−Msnm·Heff+Aex|∇nm|2, (4) where Msis the ferromagnet’s saturation magnetization and is assumed to be uniform throughout the material, nm(r) is the local magnetization direction, assumed to be position (r) dependent. Kaniis the total materials-related anisotropy which gives rise to a local nmdependent energy, and Heff is the total effective magnetic field acting on the volume element at location r, including any applied field Happland dipolar field Hdipole from the ferromagnetic moment elsewhere. Note that Kaniand Heffcan both be position rdependent by themselves in addition to the position dependence in nm. The spatial derivative in the exchange energy density term is5 |∇nm|2=(∇nmx)2+(∇nmy)2+(∇nmz)2in a Cartisian coor- dinates system {ex,ey,ez}where ∇=∂xex+∂yey+∂zez, and nm=nmxex+nmyey+nmzez. Within the effective magnetic field Heffin the context of a perpendicularly magnetized thin film nanomagnet, an important contributor is the effectivedemagnetization field from n m, which in the thin-film limit where film thickness dis small compared to the rest of the structures is often reduced to a simple −4πMsin the direction normal to the film thickness when nmis perpendicular, giving rise to a so-called demagnetization energy 2 πM2 sto the−Msnm·Heffterm in Eq. (4). The competition between exchange energy and this demagnetization gives rise to a 104426-1 1098-0121/2013/88(10)/104426(9) ©2013 American Physical SocietyJ. Z. SUN et al. PHYSICAL REVIEW B 88, 104426 (2013) dipolar exchange length for such thin films that is λd=/radicalBig Aex/2πM2s. (5) In thin films with perpendicular magnetic anisotropy (PMA), Kaniin Eq. (4)contains a driving anisotropy term that is perpendicular to film surface, and is to the lowest orderuniaxial in symmetry, meaning the total energy varies as sin 2θ, where θis the angle between nmand the film normal. This may originate from either the bulk or the interfaces of the thin film.One can in either case write an effective uniaxial volume PMAenergy density as K ⊥. Then it is convenient to write the total perpendicular anisotropy for such a thin film as Kpmain the form of/braceleftBigg Kpma=K⊥−2πM2 s=1 2MsHk, Q=K⊥/slashbig/parenleftbig 2πM2 s/parenrightbig =Hk 4πMs+1,(6) where one has defined the thin film’s net measurable hard- axis anisotropy field as Hk, and the “quality factor” Qthat describes the strength of PMA in relation to demagnetization.Obviously, Q=1 describes the strength of K ⊥=2πM2 s which is necessary to bring the moment out of the plane. The larger Qis, the stronger the total PMA density. The competition of the exchange energy with this total PMAstrength leads to a PMA exchange length λ pma=/radicalBigg Aex Kpma=/radicalBigg 2Aex MsHk. (7) It follows that λpma=λd/√Q−1. An ultrathin-film limit is defined as the film thickness d/lessmuchmin(λpma,λd). For a typical CoFeB-based thin-film MTJ considered here, one hasd≈2n m , M s≈900 emu /cm3,Hk≈4.5 kOe, with Aex within a range 2–6 ×10−6erg/cm,6thus one has Q∼1.4, λd≈6–10 nm, and λPMA≈10–17 nm. So d/lessmuchλdis satisfied, and one is within the ultrathin-film limit where the magneticmoment can normally be considered position independentalong the direction of the film norm. In more detailed analysis, the locational nature of K ⊥ could also be significant, especially when K⊥is dominated by interface -originated anisotropy energies. In such a situation, an interface anisotropy energy density /Sigma1swould introduce another nontrivial length scale: λspma=Aex /Sigma1s(8) whose significance will become more apparent in later discussions. When device sizes are comparable or larger than these length scales, the free-layer typically exhibits nonmacrospinbehavior, which also causes a coherent thermal fluctuationlength that is shorter than the device size, resulting in a reducedthermal activation energy. For such relatively large devices, thethermal activation energy is limited by shorter length-scalefluctuations mostly related to λ pma,6–9that could nucleate magnetic reversal events, and is not simply proportional tothe lateral area of the device. As discussed above, λ pmafor our CoFeB-based free layer is of the order of 10–17 nm—asmall lateral size only became lithographically accessiblefairly recently.For the smaller devices, the more fundamental macrospin- dictated switching efficiency Eq. (3)is being approached. 10,11 Since Eq. (3)contains only two relatively well-known materi- als and device parameters ( α∼0.005–0 .01 for most free layer materials in use today, and mr≈0.5–1.0 for a high-MR, low- RA MTJ), the macrospin-limit efficiency typically evaluates toκ≈1k BT/μA where Tis the ambient temperature of 300 K. For devices with sizes below 30 nm (conductances below about 0.1 1 /k/Omega1), a range of efficiencies between 1 and 10kBT/μA has been seen. The lower end of this range, about 1kBT/μA, agrees well with the macrospin-based estimate Eq.(3). The high end of the efficiencies are about three times higher than the macrospin model would comfortably expect.Similarly high efficiency values have also been reportedin recent literature on individual junction devices by otherresearchers. 12,13 This paper will attempt to analyze the dependence of efficiency κon MTJ size and other properties, and propose a few plausible mechanisms that may account for the largerthan macrospin values for κ. II. EXPERIMENT Tunnel junctions with MgO barrier and CoFeB as free layer with perpendicular magnetic anisotropy were producedin ways similar to what has been reported earlier. 11These include the sputter deposition, at ambient temperature, aCoFeB-based MTJ materials stack. The wafers are thenpostdeposition annealed in vacuum at 300 ◦C for 1 h prior to being lithographically patterned down to sizes ranging fromabout 15 nm to >100 nm in diameter for circular-shaped devices in this series of samples. A reactive ion etch is usedfor the main junction etching step, followed by a low-energy(<200 eV), grazing incidence Ar ion-beam etch for trimming the junction sides to the desired dimensions. The finishedstructures are characterized for their spin-torque switchingproperties using methods described in Ref. 8. These junction devices are used to deduce a set of junction- size dependent quantities including thermal activation energyE b, spin-torque switching threshold voltage Vc0, spin-torque switching threshold current Ic0, as evaluated from Vc0/Rp, and the switching efficiency Eb/Ic0. A detailed description of these measurements in our experiments can be found in Refs. 7and8. Figure 1shows a set of switching efficiency vs parallel-state junction conductance Gpand vs junction TMR plots. Data include devices from multiple wafers of the general materialsparameters as described above, with the dominant variationofG poriginating from junction size differences, although a junction RA variation of about 5–20 /Omega1μm2is also part of the variables convoluted in this summary. For larger devices whereE bis relatively constant against size variations (consistent with earlier findings7,8), their efficiency κis inversely proportional to the junction area, reflecting a relatively size-independent Eb, and a junction Ic0scaling with junction area. In the small Gp region below 0.1 (k/Omega1)−1, corresponding roughly to junctions of about 30 nm or so in size and below, the data in Fig. 1(a) suggest a change of slope, caused by a decrease of Ebfor smaller junction size. The exact mapping of junction size at thesmall-size end from G phowever may be unreliable, as there are variations of device-level junction resistance-area product 104426-2SPIN-TORQUE SWITCHING EFFICIENCY IN CoFeB-MgO ... PHYSICAL REVIEW B 88, 104426 (2013) FIG. 1. An overview plot of the measured efficiencies of spin-torque switchable MTJs of different sizes. (a) The dependence of observed efficiency κvs junction size. Here the parallel-state conductance Gpis used to represent the junction area, with an averaged junction resistance-area product ranging between 5 and 20 /Omega1μm2. Junctions are all of circular shape. (b) The same set of devices showing the efficiency vs junction’s MR. The two dashed lines are comparisons to Eq. (3)withα=0.005 (upper) and 0 .015 (lower). that are process and junction size dependent. Regardless, the switching efficiency is seen to increase with decreasing devicesize. This is expected as the device size is decreased towardsthe magnetic exchange lengths. Data in Fig. 1suggest that for the majority of devices the macrospin-level efficiency ofκ> 1k BT/μA is not reached. Most of that is due to the device size being larger than the exchange length. However,for those very small devices at the lowest conductance end inFig. 1(a), and on the top boundary of Fig. 1(b),t h e yh a v e κ values exceeding 3 k BT/μA, some even with apparent values exceeding 5 kBT/μA. For such small devices it is important to unambiguously establish the physical size of the samejunction the transport measurements were performed on, soas to ascertain the relationship between E b,κ, and device size. Figure 2gives a subset of devices shown in Fig. 1. Here the transport-measured properties are directly plottedagainst their individual physical device sizes as determinedfrom cross-sectional transmission-electron microscopy (TEM)images obtained post-transport measurement. Figures 2(a)and 2(b) indicate the thermal barrier E bdecreases with decreasing device size when devices are smaller than about 40–50 nm indiameter. For such small devices, Fig. 2(b)shows that E bscales almost linearly with device diameter, rather than with devicearea—as a comparison with Fig. 2(a) would demonstrate. On the other hand, for switching current threshold I c0,F i g . 2(c) shows a scaling over this entire junction size range that isessentially linear with device area, with a near-zero intercept.This particular set of experimental observations, namely withE b∝aandIc0∝a2(where ais the device diameter), leads to an efficiency κ∝1/a, and is the apparent reason the observed efficiency keeps increasing with decreasing junction size a,a s shown in Fig. 2(d). At the small device sizes around 17 nm, one observes an efficiency around 3.5 kBT/μA which stretches the best estimate one could expect from a macrospin-based modelaccording to Eq. (3). In sections following, one examines these two size- dependent scaling behaviors in more detail—that of E bvsa, andIc0vsa2. This examination leads one to conclude that the Eb∝ascaling results mostly from an edge-demagnetization field correction, whereas the Ic0∝a2scaling may be indicativeof a spin-wave instability threshold that is governed by the combined interface and bulk anisotropy area density. III. NEARLY LINEAR DEPENDENCE OFEbON DIAMETER In the thin-film limit it is often assumed that the total PMA energy density described by Eq. (6)is sufficient to estimate the total thermal activation barrier height in the macrospinlimit. This would give rise to an E b=Kpmad(π 4)a2for a circularly shaped film disk where dis the film thickness and a is the film diameter. Such Ebshould scale with disk area. The experimentally observed Ebhowever shows predominantly a linear dependence on afor the size ranges between around 15 and 50 nm with nearly zero intercept, as shown in Fig. 2(b). There could be several plausible sources for an Ebthat would depart from simple area dependence on disk size. Belowone evaluates a few of the most likely causes quantitatively toidentify the leading cause. A. Reversal through a domain-wall sweep across the disk Here one is concerned mostly with disk-shaped ferromag- netic thin films with diameter aapproaching that of the exchange length λd. In this size range one candidate for magnetic reversal is a domain-wall-like object sweeping acrossthe disk. 14This would produce a total thermal activation energy roughly proportional to the length of such a wall-like structure, and could potentially explain the linear sizedependence of the observed E b. A closer examination of this mechanism however shows that it is unlikely to be thedominant reason for the linear size dependence of E bin this set of experiments. The maximum energy point of such a wall sweep process is the symmetric situation illustrated in the inset of Fig. 3, where the center of the wall-like structure crosses the disk,lying along its diameter in, say, the e ydirection. In fact, if one ignores long-range dipolar interaction, but includesthe related thin-film demagnetization into the local PMA 104426-3J. Z. SUN et al. PHYSICAL REVIEW B 88, 104426 (2013) FIG. 2. Size dependence of junction properties. Here for most devices [except the relatively large ones marked in gray pentagons in (a)–(c)], the size of the actual junction is confirmed post-transport measurement with cross-section transmission electron microscopy (TEM) images. (a) The dependence of thermal activation energy on junction area, showing the initial rise followed by saturation at larger junction area. Notethat the initial rise does not follow a simple linear area dependence with zero intercept but rather there is a significant intercept at zero device area. (b) The dependence of E bon the diameter of junction, showing a large region of nearly linear dependence with zero intercept before reaching saturation Eb(Refs. 6–9) as indicated by line (3). (c) The dependence of switching current on junction area showing the expected linear dependence with zero intercept. (d) The dependence of spin-torque efficiency κ=Eb/Ic0on junction area. The gray scale represents the actual measured resistance-area product RA of the junction. It shows that the effect of RA on κis minor compared to κ’s very strong size dependence. energy, the one-dimensional domain-wall solution of θ(x,y)= 2t a n−1[e(x/λ)]16would be a good approximation for a thin-film disk with uniform total perpendicular anisotropy density Kpma. Hereθis the polar angle of the local magnetization with respect to the easy-axis (film-perpendicular) direction of ez=ex×ey; −a/2<x<a / 2 with xbeing the horizontal coordinate, and λ=λpma=/radicalbigAex/Kpma. Integrating throughout the volume of the disk in the thin-film limit with thickness d/lessmuchλ, and one gets a wall-sweeping activation energy (measured in reducedunit of K pmadλ2) as a function of the disk diameter aas the straight line 0-A in Fig. 3. The cross point at 4 .63λ is the diameter above which domain-wall sweep yields alower barrier height energy than macrospin rotation, and thusbecomes the preferred mode of reversal. This reveals a significant problem with the estimate. The crossover point thus estimated (point A in Fig. 3) lies well above the subvolume saturation value of E b. This is illustrated in Fig. 3. The subvolume saturation Eb, roughly estimated to be of the order of 4 πAexd(which was already an overestimate when compared with experiment), is seen to be crossedby the macro-spin branch of the E b(a) at point B, with a∼4λ< 4.63λ, the latter being what is for crossing over into domain-wall sweeplike reversal. If this simple picture werecorrect, one would not see the presence of a domain-wall sweep-mediated reversal mode, and the macrospin modewould directly cross into subvolume reversal with saturatedE b∼4πAexd. Thus, if the domain-wall sweep mode is what gave rise to the linear Ebvsascaling, to be consistent with experimental observations, the Eb(a) relationship has to have a shallower slope than predicted by this simple model describedabove, and intersect the macrospin reversal E bat a much shorter length than 4 .63λas illustrated in Fig. 3. Although a more careful inclusion of long-range dipolar interactions into such a picture could moderate the DW-likestructure’s maximum energy, it is still unlikely that it couldbe the leading-order effect to explain our observation. This isespecially the case when an alternative factor appears to bemore significant in controlling the leading order E bvs size behavior, as will be discussed below. B. Edge-enhancement of Kpma As described by Eq. (6), the net PMA energy volume den- sityKpmais a balance between the perpendicular anisotropy energy K⊥(bulk or interface, here represented in volume density; if interface, then K⊥would contain a 1 /ddependence 104426-4SPIN-TORQUE SWITCHING EFFICIENCY IN CoFeB-MgO ... PHYSICAL REVIEW B 88, 104426 (2013) aEb , ~4bs a t u r a t e d e xEA d π (),~4bD W e x p m aE AK a d2 ,4bM S p m aEd a Kπ=~4 . 6 3a λ ~aπλ ~4a λA 00B C experimentθ= 0 θ= πxy FIG. 3. A sketch of the various model predictions of thermal activation energy vs disk diameter acompared with experimen- tal observations. Here dis the disk thickness. Eb,MS is the macrospin energy barrier, Eb,DW is the energy barrier for a domain- wall-like object sweeping across the disk, Eb,saturated ≈4πAexdis the subvolume related saturation activation energy (Refs. 6–9). The gray curve illustrates our experimentally observed behavior. The inset shows the energy maximum state of a domain-wall-like objectsweeping across the disk. The gray scale represents the polar angle ofn mwith disk-film normal ez=ex×ey. The choice of Kpmahere, and the related relationship between the macrospin branch and theexperimental curve is only for illustrating the difference in trend. Scaling relations with “ ∼” symbols are only crude estimates, while those with “ =” are exact predictions. A more quantitative comparison is given by the model calculation presented in Fig. 2. inside) and the shape-dictated dipolar demagnetization energy density −2πM2 s. Such definitions are based on extended films over a length scale far greater than the film thickness. In anypatterned structures with finite size, the edge demagnetizationfield is reduced. Based simply on symmetry, a half-infiniteplane of film would have the edge demagnetization field onlyof 1/2 the value of that deep inside the film. Thus an edge demagnetization energy density at most would be only about−πM 2 s. Therefore, even if the intrinsic PMA energy density K⊥is uniform and position independent, one would have a total PMA energy density Kpmashowing an enhancement near sample edges due to the reduction of demagnetization. One may estimate the amount of PMA gain from this mech- anism crudely by integrating the edge-fringe field related lossto dipolar demagnetization energy, assuming the disk diameteris far greater than the film thickness. The demagnetizationfield distortion near the film edge extends at least the order ofa film thickness dinto the lateral direction of the film. Thus the total energy gain from this mechanism is of the order ofη gπ2(Msd)2awhere ηgis a geometry dependent integration constant of the order ∼1. This would give Eb≈/parenleftbiggπ 4/parenrightbigg a2dKPMA+ηga(πMsd)2(9) giving a leading order linear dependence in the small alimit of δEb δa=ηgπ2(Msd)2. (10)The geometry constant ηgcan be roughly estimated by using a long stripe thin film with width L→+ ∞ , thickness d, with perpendicularly magnetized moment density Ms. In such a case, with −L<x< 0 and x=0 defined as one edge, the near edge-dipole related field profile can be calculated to be,for the edge near x=0, H(x)≈/bracketleftbigg 2π+4 cos −1/parenleftbiggd√ d2+4x2/parenrightbigg/bracketrightbigg Ms (11) and thus the per-unit edge-length dipole demagnetization energy to be, at distance sinto the plate in the −xdirection, Eed=−/parenleftbiggdM2 s 2/parenrightbigg/integraldisplay0 −sH(x)dx =/parenleftbig 2πM2 s/parenrightbig (sd)−(Msd)2/bracketleftbigg 1+ln/parenleftbigg2s d/parenrightbigg/bracketrightbigg ,(12) where the first term is the uniform demagnetization field 4πMs’s contribution, and the second term is the edge-dipole related energy reduction per unit length. If one crudely assumesthe integration depth sto be around the radius ( a/2) of our disk, this gives an estimate to the η gvalue in Eq. (9)as ηg≈1 π/bracketleftbigg 2+ln/parenleftbigga d/parenrightbigg/bracketrightbigg . (13) For MTJs in this study, Ms≈900 emu /cm3,d≈2 nm. Thus δEb δa≈1.3kBT/nm. While this is an extremely crude estimate considering the logarithmic nature of the edge dipole energy’sdistance dependence, it is of the same order of magnitudeas the experimental value of around 2 k BT/nm, as shown in Fig. 2(b). A slightly more refined way of estimating the edge-dipole field reduction of a disk geometry is to directly integrate outthe total demagnetization energy for the disk shape. Thiscan be done numerically, again assuming a uniform localmagnetization orientation leading to an effectively uniformsurface magnetic charge density M s, giving a magnetic field inside the disk in the ezdirection along the disk axial direction asHz(r2)=Ms/integraltext 2(r12·ez/|r12|3)d2r1, where r12=r2−r1 is the relative position vector. r1resides on one of the disk surfaces while r2is inside the disk. With a rigid local magnetization assumed to be perpendicularly aligned, oneonly concerns the e zcomponent of the demagnetization field. Integrating once more over the disk’s entire volume for r2, one has the disk’s total contribution to energy barrier height due todemagnetization as U demag (d,a)=−/parenleftbig 2πM2 s/parenrightbig (da2π/4)/bracketleftbigg3 2Nz/parenleftbigg2d a/parenrightbigg −1 2/bracketrightbigg , with Nz(ξ)=1 2π2/integraldisplay2(r12·ez) |r12|3d2r1d3r2, (14) where d2r1gets integrated through a surface of the disk, and d3r2gets integrated through the volume of the disk. The integral runs through a disk of normalized radius 1 and thenormalized thickness ξ=2d/a, and can be numerically eval- uated. lim ξ→0N(ξ)=1i st h e ezdirection demagnetization factor of the disk in the limit of zero thickness. For the firstl i n ei nE q . (14) one also employed an approximate sum rule 17 of 2Nx+Nz=1 for the demagnetization factors as a way to offset the effect of in-plane demagnetization of the disk at 104426-5J. Z. SUN et al. PHYSICAL REVIEW B 88, 104426 (2013) finite thickness when the moments are rigidly aligned in the plane of the disk—a position that corresponds to the energymaximum during reversal of a macrospin state. This estimate would give, in place of Eq. (9): E b=(πa2d/4)K⊥+Udemag (d,a). (15) For our experimental data, assuming Ms≈900 emu /cm3, d=2 nm, and K⊥=5.5×106erg/cm2, one has the cal- culated Ebfrom Eq. (15) shown in Figs. 2(a) and2(b) as the short-dashed line labeled (2) in Fig. 2(b). This produces a size dependence far from quadratic in diameter a, and agrees with the experimental data. The gray curve (1) inFig. 2(b) is the E bof the same parameter set, but assuming a simple uniform demagnetization energy of −2πM2 s.O ft h i s model calculation the only adjustable parameter without priorindependent measurement is the value of K ⊥. Based on these estimates, one concludes that the edge dipo- lar field reduction-related PMA enhancement is significant,and it could largely explain the observed size dependenceofE bin our device size range. Further refinement of the energy calculation would need to include both exchange andthe edge enhancement that would produce more realistic sizedependencies over wider size ranges. These however aretoo complex for analytical estimates, and would have to beevaluated with micromagnetic computation. IV . SCALING OF SWITCHING THRESHOLD Ic0ON JUNCTION DIAMETER The previous sections have established, both in terms of ex- perimental observations and in investigations of mechanisms,a thermal activation energy’s scaling relation with device sizethat is not areal but more linear. It then begs the question ofhow would the spin-torque switching threshold’s current scalewith device size. In the experimental data shown in Fig. 2(c), the junction’s switching current I c0is seen to scale robustly as a2. This implies a constant current density scaling, without being much moderated by the reduction of the measuredE bin small devices. This is clearly an unexpected behavior from a macrospin point of view, as in macrospin limit, theswitching current is expected to scale as the barrier height E b. The observed nonareal scaling of Ebapparently does not get translated into a nonareal scaling of Ic0. IfIc0is not scaling with the barrier height as macrospin would dictate, what determines it? The slope of Ic0vs device area of the data shown in Fig. 2(d) gives a threshold current density of Jc0≈5.4×106A/cm2. The question then becomes, what determines this experimentally observedthreshold current density, if not E b? The rest of this section is going to address this question from a spin-wave excitationthreshold point of view. Before going into such detailed model considerations, one simple observation can be made here with the experimentaldata. If one uses the J c0expression derived based on macrospin type of model as in Eq. (2), one sees that Jc0can be related to an areal anisotropy energy density /Sigma1eff. In macrospin limit, that is the areal density of the net uniaxial anisotropy Kpmadwhere dis the film thickness. In our case here, using the slope from Fig. 2(c) andmr∼0.5,α∼0.005 as assumed before, and using Eq. (2), it appears the experimentally obtained effectiveareal energy density is about /Sigma1eff∼0.66 erg /cm2. Note that our extended-film measured Hk∼4.5 kOe would give a net PMA energy density of /Sigma1pma=MsdHk/2∼0.41 erg /cm2, and the demagnetization energy density corresponding toM s∼900 emu /cm3is/Sigma1demag≈1.02 erg/cm2. In an extended film limit, /Sigma1eff=/Sigma1⊥−/Sigma1demag , which leads to an intrinsic PMA energy density of the order /Sigma1⊥/definesK⊥d≈1.43 erg/cm2. These areal energy densities are within a factor of 2 or so ofwhat is deduced from J c0, possibly implying a correlation. A. Spin-torque excitation with interface and bulk perpendicular anisotropy As well known in the past,19,20magnetic anisotropies concentrated at interfaces of thin-film ferromagnets can altertheir spin-wave excitation behaviors, and under favorableconditions encourage the formation of interface spin waves.More recently, interface-localized perpendicular anisotropyhas been shown to induce a particular type of softening inspin-wave modes that could reduce the spin-torque excitationthreshold for inducing spin-wave instabilities in thin slabsof YIG crystals. 21Below the same methodology is explored to relate our observations above to such interface-anisotropyrelated possible softening of spin-wave modes. To this end in a classical-dynamics limit one can start with the continuous-medium Landau-Lifshitz-Gilbert equation fora description of the magnetodynamics of the free layer ofthe MTJ: −1 γ/bracketleftbigg/parenleftbigg∂nm ∂t/parenrightbigg −αnm×/parenleftbigg∂nm ∂t/parenrightbigg/bracketrightbigg =nm×H+/parenleftbigg2Aex Ms/parenrightbigg nm×∇2nm, (16) where nm=nm(x,y,z )is the unit vector of the local magnetic moment. At an interface of the ferromagnet with a nonmagnet,one writes the boundary conditions for n mto be19–21 nm×/bracketleftbigg/parenleftbigg2γAex Ms/parenrightbigg/parenleftbigg∂nm ∂n/parenrightbigg +/parenleftbigg2γ/Sigma1s Ms/parenrightbigg (nm·n)n −hs(ns×nm)]=0, (17) where nis the interface normal unit vector pointing into the ferromagnet. /Sigma1sis the interface anisotropy energy. /Sigma1s>0i n this sign convention of ndefines a perpendicular interface anisotropy. hsis the spin current in units of (γ/M s)Js where Jsis the conventionally defined magnetic moment flow density, in magnetic moment per unit area per unit time. Oneassumes here that the spin-torque term applies only to thebottom interface—a simplifying assumption that is justified bythe very short spin-decoherence length in ferromagnets withstrong s-dexchange coupling. γ=gμ B/¯h≈2μB/¯h> 0i s the gyromagnetic ratio, here defined as a positive number. Tokeep the problem to a manageable size, one ignores dipolarinteractions between the magnetic free layer and the referencelayers inside an MTJ, and focuses instead on the basic dynamicproblem of the free layer itself. Further for simplicity, here one only examines the result in the thin-film limit, with film thickness d/lessmuchmin{λ d,λspma}, and attempts to find the characteristics of spin-torque exci-tation threshold at such an interface as it depends on the 104426-6SPIN-TORQUE SWITCHING EFFICIENCY IN CoFeB-MgO ... PHYSICAL REVIEW B 88, 104426 (2013) materials parameters. To this end, assume that nm∼n/bardblez, with ez=ndefined as the film-normal direction. Assume further that only the bottom interface has a perpendicularanisotropy through /Sigma1 s. The top interface is assumed to be free of either spin-current or interface anisotropy, for simplicity. Equations (16) and(17) are linearized in the small am- plitude limit of nmxy in the x-yplane, with |nmxy|/lessmuch1, and nm≈ez+nmxy. Fourier transform them into a plane- wave state and write nmxy=nm0exp[i(ωt+k·r)],t o - gether with its accompanied oscillating magnetic field22 hxy=−4πMs(k·nmxy)(k/k2), satisfying the conditions of ∇·(H+4πMsnm)=0 and ∇× H=0 as required by the Maxwell equations, with H=(Heff−4πMs)ez+hxythe total magnetic field in Eq. (16), and Heff=Hathevolume uniaxial anisotropy field of the film, also assumed to be alonge z. The eigenvalue problem for this set of equations gives ω(k)dispersion relations whose imaginary parts’ zeros yield the onset of magnetic instability. When expanded to the firstorder in damping αand spin current h sand second order in the in-plane spin-wave wave vector kx, the resulting instability threshold for spin-current density Jsc=(Ms/γ)hsis Jsc=2α/parenleftbig 2πM2 sd/parenrightbig/bracketleftbigg Q+Q1−1+k2 xQ/parenleftbiggAexd /Sigma1s/parenrightbigg ×/parenleftbigg 1+2Q+Q1−1 2Q2/parenrightbigg/bracketrightbigg =2αd/bracketleftbig Kpma+(2νAex)k2 x/bracketrightbig (18) withQ=/Sigma1s/(2πM2 sd), and Q1=Ha/(4πMs) describing the interface and bulk perpendicular anisotropy quality factorfor the thin film, respectively. The second line in Eq. (18)writes the total PMA anisotropy K pma=/Sigma1s/d+Ku−2πM2 sas dis- cussed before, and ν=1/2+[Q+(Q1−1)/2]/Q2∼1a s a dimensionless parameter depending on details of QandQ1. B. Spin-wave modes laterally confined by an edge enhancement of Kpma Atkx=0, Eq. (18) leads to Jsc=2αd[/Sigma1s+Ms(Ha− 4πMs)/2], which is the macrospin threshold spin-current density. For finite kxexcitations the threshold current density increases according to the dispersion relation of kxas expected. Assuming a circular disk with diameter a, the net charge- threshold current from Eq. (18) for the lowest spin-wave branch of wavelength corresponding to 2 ais Ic≈/parenleftbigg4eα ¯hη/parenrightbigg/bracketleftbigg Kpma/parenleftbiggπ 4a2d/parenrightbigg +π3ν 2Aexd/bracketrightbigg =Ic0+Ic1, (19) where Ic0=(4eα ¯hη)Kpma(π 4a2d), and Ic1=(4eα ¯hη)π3ν 2Aexd, which is the first available confined spin-wave mode, and its threshold current is an additional Ic1above that of a macrospin. Thus in normal conditions, the macrospin mode with kx=0i s the lowest-lying mode in terms of its spin-torque threshold.23 The situation could be different if Kpmais strongly position dependent. As discussed above, in the case where edge-demagnetization reduction plays a significant role in deter-mining the total PMA energy, K pmacan become significantly enhanced near the edges of a circular disk. Consider an extremelimit, where most of the PMA energy for Ebis concentrated near the edge of the disk, and for the interior of the disk Kpma is essentially zero. In such a limit, the threshold for exciting the first spin-wave mode would be of the order Ic1, while the macrospin-threshold current for Ebwould be Ic0. Assuming an excitation of the spin-wave mode allows sufficient spin trans-port into the precessing moments that could lead to an eventualreversal of the total magnetization, then, even though the mo-ments are edge-pinned to a total potential of E b, the threshold current for negative-damping related thermal activation wouldbecome I c1, whose ratio to the macrospin threshold is Ic0 Ic1≈Eb π3ν 2Aexd. (20) This could contribute to a possible efficiency enhancement over a certain size range—due to the excitation of the firstfinite wavelength mode spin wave, which under the specificconditions of a K pmabeing weak near disk center could result inIc1<Ic0. This mechanism however has the fundamental difficulty of Ic1being a size-independent quantity, and could thus not be consistent with the range of data observed in our experimentsdescribed above, where the threshold involves a relativelyconstant current density , while E bscales roughly linearly with sample diameter. This dilemma could not be easily resolvedby invoking the low-lying spin-wave modes of nonzero k x, confined by an edge-enhanced PMA potential. C. Interface PMA-related mode softening Another materials-related possibility is a significant reduc- tion of the exchange constant Aexnear the interfaces, which under certain conditions could create a softened interfacespin-wave mode. 21Assume an interface layer of thickness d1 separating the bottom interface which Eq. (17) addresses and the main slab of the ferromagnet with a thickness of d/greatermuchd1. Further assume the bottom d1interface layer has an apparent Aexthat is reduced from the interior of the slab by a factor ofβwith 0 <β/lessmuch1. Solving the linearized LLG equations with the properly defined boundaries gives a threshold currentdensity J scfor the simpler case of kx=0: Jsc≈2α/parenleftbig 2πM2 sd/parenrightbig/bracketleftbigg Q+Q1−1+/parenleftbiggβc β/parenrightbigg/parenleftbiggQ 2+Q1−1/parenrightbigg/bracketrightbigg , (21) where Q=/Sigma1s/(2πM2 sd),Q1=Ku/(2πM2 s) with Kubeing the volume PMA of the interior of the ferromagnetic slab notincluding demagnetization energy, and β c≈2Qdd 1/λ2 d/lessmuch1 is the critical softening value of βfor the interface exchange energy βAex. If the anisotropy strengths are such that Q1<1−Q/2< 1, as β→βc, a large amount of reduction in Jscdue to this spin-wave mode softening could be expected, hence anincrease of the efficiency factor κ. Quantitatively, if one writes J scmsas the macrospin threshold current, and Jscswthat of the spin-wave mode, then Jscms Jscsw≈Q+Q1−1 Q+Q1−1+/parenleftbigβc β/parenrightbig/parenleftbigQ 2+Q1−1/parenrightbig (22) could become greater than 1 as β→βc. 104426-7J. Z. SUN et al. PHYSICAL REVIEW B 88, 104426 (2013) The origin of this mode softening is the coupling strength limit of the weakened bottom layer exchange βAexin maintaining perpendicular moment of the full slab. Theeigenfrequency of the full slab at zero spin current can bewritten as ω≈γ(4πM s)/parenleftbigg Q+Q1−1−βc βQ/parenrightbigg . (23) It shows the effect of β< 1 and approaching βcas a reduction of the resonance frequency. A more quantitative analysis would however need to proceed cautiously, bearing in mind the uncertainties of thematerials model and parameters as well as the myriads ofassumptions made for such a simplified model. For one thing,to safely stay within the ultrathin-film limit one would needto confine the discussion to a length scale where the interfaceanisotropy and exchange energy related length scale is beyondthat of the materials thicknesses. That is to require Aexβ /Sigma1s/greatermuchd1, orβ/greatermuchd1/λspma. Since one could write βc=2d1/λspma,i t follows that staying within the thin-film limit requires β/greatermuchβc, thus the region of β→βccannot be approached too closely without violating the ultrathin-film assumption used in this toymodel. In reality of course the situation is even more complex. The softening of the interface spin-wave mode would have someconsequences on the total thermal activation energy barrier aswell which is not investigated here. However this extremelysimplified analysis points to the possible importance of aweakened exchange energy at the interface region where botha majority portion of the PMA is induced as well as where thespin torque is absorbed—an optimal match of this exchangeenergy with the rest of the film would likely provide a thresholdcurrent less than that of the simple macrospin threshold. Such lowering of the threshold spin current for negative damping instability does not necessarily translate into a low-ering of threshold current for the reversal of the magnetic mo- ment in the layer. What this simple analysis showed is only a re-duction of the threshold for inducing an instability in the linearregime. The large-angle dynamics are too complex to be ana-lytically predicted. Numerical simulations would be necessaryto quantify the parameter space within which such reductionof threshold instability could correspond to complete reversal. In addition to relating the linear instability threshold to reversal, for finite-time magnetic switching, there remains thefundamental requirement of angular-momentum conservation,and the slope of switching speed’s dependence on spin current.The simple and robust relationship between switching time τ and driving spin current I s(Ref. 24) remains. In the limit of Is/greatermuchIsc,τIs∼mwhere mis the total magnetic moment of the free layer, although the detailed form of the finite-temperatureswitching probability vs the spin-current intensity might besignificantly altered by such interface spin-wave excitations.Nevertheless, this scenario provides a plausible explanation to the fact of a threshold current that is area scaling, asthe spin wave dictates a threshold current density regardless of the actual thermal activation barrier height’s size-scalingbehavior—even if the activation energy is saturated at largedevice sizes by subvolume excitations. 6–9This could account for the separate scaling properties of Eband that of Ic0 observed experimentally as discussed earlier. Furthermore, this conceptual model points out some basic trends on how efficiency enhancement may depend on ma-terials parameters. This type of spin-wave-softening inducedefficiency increase only occurs when interface PMA dominatesand the bulk PMA is insufficient by itself to overcomedemagnetization (i.e., Q 1<1, but Q+Q1>1). Within this constraint, an increase of bulk PMA strength (increasing Q1) would reduce the amount of possible reduction in Jscsw, and thus lessen the enhancement of the efficiency κ. These qualitative trends may be of value for materials and deviceoptimization work. V . CONCLUSIONS An in-depth examination of spin-torque switched MTJs’ threshold current in relation to same-device thermal activa-tion energy suggests that the experimentally evaluated spin-torque switching efficiency borders on the maximum valuea macrospin model would yield. In all likelihood additionalmechanisms are at play. A set of size-dependence scalingbehaviors were experimentally observed. For the thermalactivation energy E b, it is primarily linearly dependent on junction diameter. For threshold current, it scales with thearea of the junction, over the size range 15 to over 100 nm.The linear scaling of E bcan be semiquantitatively accounted for by including an edge-fringe-field related reduction ofdemagnetization energy. The area scaling of threshold currentis consistent with a spin-wave dictated threshold that mayinvolve additional softening due to the interplay between aninterface perpendicular magnetic anisotropy term and a netin-plane magnetic anisotropy from the interior of the free layer.An interface-dominant PMA with a net easy-plane anisotropyfor the interior could provide a mechanism for a softenedspin-wave mode that reduces its threshold current density tovalues below what is required by a macrospin. ACKNOWLEDGMENTS Work done with the MRAM group at IBM T. J. Watson Research Center was supported in part by partnerships withTDK/Headway Technologies and with Micron Technologies.Also acknowledged are fruitful scientific discussions withProf. A. D. Kent’s group at New York University, and with Dr.G. Chavez-Flynn, now at New Jersey Institute of Technology. *jonsun@us.ibm.com 1N. D. Rizzo, D. Houssameddine, J. Janesky, R. Whig, F. B. Mancoff, M. DeHerrera, J. J. Sun, K. Nagel, S. Deshpande, M. L. Schneider,H.-J. Chia, S. M. Alam, T. Andre, S. Aggarwal, and J. M. Slaughter, IEEE Trans. Magn. 49, 4441 (2013). 2J. C. Slonczewski, Phys. Rev. B 71, 024411 (2005). 104426-8SPIN-TORQUE SWITCHING EFFICIENCY IN CoFeB-MgO ... PHYSICAL REVIEW B 88, 104426 (2013) 3J. C. Slonczewski and J. Z. Sun, J. Magn. Magn. Mater. 310, 169 (2007). 4Strictly speaking, Eqs. (1)and (2)are approximations taken in the limit of large mr/greatermuch1. See discussion in, for example, Ref. 3. 5C. Kittel, Phys. Rev. 110, 836 (1958). 6M. Yamanouchi, A. Jander, P. Dhagat, S. Ikeda, F. Matsukura, and H. Ohno, IEEE Magn. Lett. 2, 3000304 (2011). 7J. Z. Sun, R. P. Robertazzi, J. Nowak, P. L. Trouilloud, G. Hu, D. W. Abraham, M. C. Gaidis, S. L. Brown, E. J. O’Sullivan, W. J.Gallagher, and D. C. Worledge, P h y s .R e v .B 84, 064413 (2011). 8J. Z. Sun, P. L. Trouilloud, M. J. Gajek, J. Nowak, R. P. Robertazzi, G. Hu, D. W. Abraham, M. C. Gaidis, S. L. Brown, E. J. O’Sullivan,W. J. Gallagher, and D. C. Worledge, J. Appl. Phys. 111, 07C711 (2012). 9H. Sato, M. Yamanouchi, K. Miura, S. Ikeda, R. Koizumi,F. Matsukura, and H. Ohno, IEEE Magn. Lett. 3, 3000204 (2012). 10S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan,M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno,Nat. Mater. 9, 721 (2010). 11M. Gajek, J. J. Nowak, J. Z. Sun, P. L. Trouilloud, E. J. O’Sullivan, D. W. Abraham, M. C. Gaidis, G. Hu, S. Brown, Y . Zhu, R. P.Robertazzi, W. J. Gallagher, and D. C. Worledge, Appl. Phys. Lett. 100, 132408 (2012). 12S. Yuasa (unpublished). 13G. Jan, Y .-J. Wang, T. Moriyama, Y .-J. Lee, M. Lin, T. Zhong, R.-Y . Tong, T. Torng, and P.-K. Wang, Appl. Phys. Exp. 5, 093008 (2012).14For devices of larger sizes compared to the exchange length, a curling mode equivalent to the formation of a vortex is often theenergetically favored mode of reversal (Ref. 15). When the disk diameter is close to or shorter than the exchange length however,the curling mode becomes energetically less favorable as it wouldinvolve inhomogeneities at twice the spatial period as that of asingle domain wall. 15R. Skomski, H.-P. Oepen, and J. Kirschner, Phys. Rev. B 58, 3223 (1998). 16D. Craik, Magnetism: Principles and Applications (John Wiley & Sons, Chichester, England, 1995), p. 96. 17This sum rule is strictly true for three-dimensional ellipsoids asderived by, e.g., Stoner (Ref. 18). 18E. C. Stoner, Philos. Mag. Ser. 7 36, 803 (1945). 19G. T. Rado and R. J. Hicken, J. Appl. Phys. 63, 3885 (1988). 20B. Hillebrands, P h y s .R e v .B 41, 530 (1990). 21J. Xiao and G. E. W. Bauer, Phys. Rev. Lett. 108, 217204 (2012). 22C. Herring and C. Kittel, Phys. Rev. 81, 869 (1951). 23There is a narrow parameter space of QandQ1,w h e n −1/2< Q< 0a n d1 <Q 1+Q< 1−Q−2Q2when the coefficient for kxin Eq. (18) is negative, implying a reduction of finite wavelength spin-wave excitation by spin current, due to the competition ofan in-plane interface anisotropy and that of a bulk perpendicularmoment arrangement. In the ultrathin-film limit, which is the caseone considers here, this parameter space is very narrow and is oflittle practical interest since the combined perpendicular anisotropyQ+Q 1<1.125 would severely limit the total PMA available. 24J. Z. Sun, Phys. Rev. B 62, 570 (2000). 104426-9

RevModPhys.86.855.pdf

Spin-dependent phenomena and device concepts explored in (Ga,Mn)As T. Jungwirth Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnick á10, 162 53 Praha 6, Czech Republicand School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD,United Kingdom J. Wunderlich Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnick á10, 162 53 Praha 6, Czech Republicand Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Kingdom V. Nov ák and K. Olejn ík Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnick á10, 162 53 Praha 6, Czech Republic B. L. Gallagher, R. P. Campion, K. W. Edmonds, and A. W. Rushforth School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom A. J. Ferguson Microelectronics Group, Cavendish Laboratory, University of Cambridge, CB3 0HE,United Kingdom P. N ěmec Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3, 121 16 Prague 2, Czech Republic (published 11 July 2014) Over the past two decades, the research of (Ga,Mn)As has led to a deeper understanding of relativistic spin-dependent phenomena in magnetic systems. It has also led to discoveries of new effects anddemonstrations of unprecedented functionalities of experimental spintronic devices with generalapplicability to a wide range of materials. This is a review of the basic material properties that make(Ga,Mn)As a favorable test-bed system for spintronics research and a discussion of contributions of(Ga,Mn)As studies in the general context of the spin-dependent phenomena and device concepts.Special focus is on the spin-orbit coupling induced effects and the reviewed topics include theinteraction of spin with electrical current, light, and heat. DOI: 10.1103/RevModPhys.86.855 PACS numbers: 75.50.Pp, 75.70.Tj, 75.76.+j, 75.78. −n CONTENTS I. Introduction 856 II. Test-bed Material for Spintronics Research 858 A. Electronic structure and magnetism in (Ga,Mn)As 858 1. Curie point singularities 8602. Localization effects in transport 861 B. Doping trends in basic magnetic and transport properties of (Ga,Mn)As 863 1. Low Mn-doped bulk materials 8632. Synthesis of high Mn-doped epilayers 8643. Curie temperature and conductivity 8664. Micromagnetic parameters 867 III. Phenomena and Device Concepts for Spintronics 868 A. Nonrelativistic versus relativistic based spintronics concepts 868 B. Interaction of spin with electrical current 8701. Anomalous and spin Hall effects 870 2. Anisotropic magnetoresistance 870 3. Tunneling anisotropic magnetoresistance 872 4. Transistor and chemical potential anisotropy devices 874 5. Spin torques and spin pumping 8776. Current-induced spin-transfer torque 8797. Current-induced spin-orbit torque 880 C. Interaction of spin with light 883 1. Magneto-optical effects 8832. Optical spin-transfer torque 8853. Optical spin-orbit torque 885 D. Interaction of spin with heat 887 1. Anomalous Nernst effect 8872. Anisotropic magnetothermopower 8883. Tunneling anisotropic magnetothermopower 8884. Spin-Seebeck effect 890REVIEWS OF MODERN PHYSICS, VOLUME 86, JULY –SEPTEMBER 2014 0034-6861 =2014=86(3)=855(42) 855 © 2014 American Physical SocietyIV. Summary 890 List of Symbols and Abbreviations 891Acknowledgments 891References 891 I. INTRODUCTION Under equilibrium growth conditions the incorporation of magnetic Mn ions into III-As semiconductor crystals islimited to approximately 0.1%. To circumvent the solubility problem a nonequilibrium, low-temperature molecular- beam-epitaxy (LT-MBE) technique was employed which led to the first successful growths of (In,Mn)As and (Ga,Mn)As ternary alloys with more than 1% Mn and to the discovery offerromagnetism in these materials ( Ohno et al. , 1992 ,1996 ; Munekata et al. , 1993 ;Hayashi et al. , 1997 ,2001 ;Shen et al. , 1997 ;Van Esch et al., 1997 ;Ohno, 1998 ;Shimizu et al., 1999 ). The compounds qualify as ferromagnetic semiconductors to the extent that their magnetic properties can be altered bythe usual semiconductor electronics engineering variables, such as doping, electric fields, or light. The achievement of ferromagnetism in an ordinary III-V semiconductor with Mnconcentrations exceeding 1% demonstrates on its own the sensitivity of magnetic properties to doping. Several experi- ments have verified that changes in the carrier density and distribution in thin (III,Mn)As films due to an applied gate voltage can induce reversible changes of the Curie temper-ature T cand other magnetic and magnetotransport properties (Ohno et al. , 2000 ;Chiba et al. , 2003 ,2008 ,2013 ;Chiba, Matsukura, and Ohno, 2006 ;Wunderlich, Jungwirth, Irvine et al. , 2007 ;Olejn íket al. , 2008 ;Stolichnov et al. , 2008 ; Owen et al. , 2009 ;Riester et al. , 2009 ;Sawicki et al. , 2010 ; Mikheev et al. , 2012 ;Niazi et al. , 2013 ). Experiments in which ferromagnetism in a (III,Mn)As system is turned on and off optically or in which recombination of spin-polarized carriers injected from the ferromagnetic semiconductor yields emission of circularly polarized light clearly demonstrated theinteraction of spin and light in these materials ( Koshihara et al. , 1997 ;Munekata et al. , 1997 ;Ohno et al. , 1999 ). (Ga,Mn)As has become a test-bed material for the research of phenomena in which charge carriers respond to spin and vice versa. By exploiting the large spin polarization of carriersin (Ga,Mn)As and building on the well-established hetero- structure growth and microfabrication techniques in semi- conductors, high-quality magnetic tunnel junctions have beendemonstrated showing large tunneling magnetoresistances (TMRs) ( Tanaka and Higo, 2001 ;Chiba, Matsukura, and Ohno, 2004 ;Mattana et al. , 2005 ;Saito, Yuasa, and Ando, 2005 ). In the studies of the inverse magnetotransport effects, namely, spin-transfer torques (STTs) in tunnel junctions(Chiba et al. , 2004 ) and domain walls (DWs) ( Yamanouchi et al. , 2004 ,2006 ;Wunderlich et al. , 2007 ;Adam et al. , 2009 ; Wang et al. , 2010 ;Curiale et al. , 2012 ;De Ranieri et al. , 2013 ), the dilute-moment p-type (Ga,Mn)As is unique for its low saturation magnetization and strongly spin-orbitcoupled valence band ( Sinova, Jungwirth et al. , 2004 ;Garate, Gilmore et al. , 2009 ;Hals, Nguyen, and Brataas, 2009 ). Compared to common transition-metal ferromagnets this implies a more significant role of the fieldlike (nonadiabatic) STT complementing the anti-damping-like (adiabatic) STTand lower currents required to excite magnetization dynamics. Moreover, the leading role of magnetocrystalline anisotropies over the dipolar shape anisotropy fields allows for the control of the direct and inverse magnetotransport phenomena bytuning the lattice strains ex situ by microfabrication ( Wenisch et al. , 2007 ;Wunderlich et al. , 2007 )o rin situ by piezoelectric transducers ( Goennenwein et al. , 2008 ;Overby et al. , 2008 ; Rushforth et al. , 2008 ;De Ranieri et al. , 2013). In general, TMR ( Julliere, 1975 ;Miyazaki and Tezuka, 1995 ;Moodera et al. , 1995 ) and STT ( Berger, 1996 ; Slonczewski, 1996 ;Zhang and Li, 2004 ) are examples of spin-dependent phenomena which can be understood withinthe basically nonrelativistic two-channel model of conduction in ferromagnets ( Mott, 1964 ), and in which spins are trans- ported between at least two noncollinear parts of a nonuniform magnetic structure with the magnetization in one part serving as a reference to the other one. Besides these more commonlyconsidered spintronic effects, (Ga,Mn)As studies have exten- sively focused on relativistic phenomena which in principle can be observed in uniform magnetic structures and where the spin dependence of the transport stems from the internal spin-orbit coupling in carrier bands. An archetypical exampleamong these effects is the anisotropic magnetoresistance (AMR) discovered by Kelvin more than 150 years ago in wires of Ni and Fe ( Thomson, 1857 ). Research in (Ga,Mn)As led to the observation of a tunneling anisotropic magneto- resistance (TAMR) ( Brey, Tejedor, and Fern ández-Rossier, 2004 ;Gould et al. , 2004 ). Unlike the TMR which corresponds to the different resistances of the parallel and antiparallel magnetizations in two magnetic electrodes separated by thetunnel barrier, the TAMR relies on the rotation of the magnetization in a single magnetic electrode while the other electrode can be nonmagnetic. Large and electrically tunable relativistic anisotropic magnetotransport phenomena were observed in the Coulomb blockade (CB) devices in which(Ga,Mn)As formed the island or the gate electrode of a single- electron transistor (SET) ( Wunderlich et al. , 2006 ;Schlapps et al. , 2009 ;Ciccarelli et al. , 2012 ). The TAMR and CB-AMR were subsequently reported in other systems including common transition-metal ferromagnets and antiferromagnets(Gao et al. , 2007 ;Moser et al. , 2007 ;Park et al. , 2008 ,2011 ; Bernand-Mantel et al. , 2009 ). For the inverse magnetotransport effects, the relativistic counterpart of the STT is the current-induced spin-orbit torque (SOT) ( Bernevig and Vafek, 2005 ;Manchon and Zhang, 2008 ). Similar to the TAMR and CB-AMR, the SOT can be observed in uniform magnets, the seminal experiment was performed in (Ga,Mn)As ( Chernyshov et al. , 2009 ), and subsequently the phenomenon was reported in other systems including transition-metal ferromagnets ( Miron et al. , 2010 ). For the SOT, the above-mentioned favorable characteristics of (Ga,Mn)As, namely, the strong spin-orbit coupling in the carrier bands and exchange coupling of carrier spins withthe dilute local moments, combine with the broken space- inversion symmetry in the host zinc-blende lattice. The broken space-inversion symmetry is a necessary condition for observ- ing the relativistic SOT ( Bernevig and Vafek, 2005 ;Manchon and Zhang, 2008 ). Theoretical studies of the intrinsic nature of the anomalous Hall effect (AHE) ( Luttinger, 1958 ;Jungwirth, Niu, and856 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014MacDonald, 2002 ;Onoda and Nagaosa, 2002 ) and experi- ments in (Ga,Mn)As interpreted by this theory ( Jungwirth, Niu, and MacDonald, 2002 ;Nagaosa et al. , 2010 )h a v e inspired a renewed interest in the AHE in a broad class offerromagnets ( Nagaosa et al. , 2010 ). Simultaneously they led to predictions of a directly related intrinsic spin Hall effect (SHE) ( Murakami, Nagaosa, and Zhang, 2003 ;Sinova, Culcer et al. , 2004 ) in which the spin-dependent transverse deflection of electrons originating from the relativistic band structureoccurs in a nonmagnetic conductor. The intrinsic SHE proposal triggered an intense theoretical debate and prompted the experimental discovery of the phenomenon ( Kato et al. , 2004 ;Wunderlich et al. , 2005 ). The SHE has become a common tool to electrically detect or generate spin currents (Jungwirth, Wunderlich, and Olejnik, 2012 ) and the intrinsic SHE combined with the STT can allow for an in-plane current-induced switching of the free magnetic electrode ina TMR magnetic tunnel junction ( Liuet al. , 2012 ). An intense discussion ensued on the alternative, SHE-STT based or SOT based interpretations of these in-plane current-induced spin reorientation effects ( Miron et al. , 2011 ;Liuet al. , 2012 ; Garello et al. , 2013 ). Research in (Ga,Mn)As continues to contribute to this area in a distinct way; experimental and theoretical studies in (Ga,Mn)As have uncovered the fact that the intrinsic SHE and SOT can be linked by a common microscopic origin ( Kurebayashi et al. , 2014 ), the same one that was originally proposed for interpreting the AHE data in(Ga,Mn)As ( Jungwirth, Niu, and MacDonald, 2002 ). The SHE, STT, and SOT phenomena are at the forefront of the research field of electrically controlled spin manipulationand play an important role in the development of a new generation of magnetic random access memories (MRAMs), tunable oscillators, and other spintronic devices ( Chappert, Fert, and Dau, 2007 ;Ralph and Stiles, 2008 ). Optical excitations of magnetic systems by laser pulses have tradi-tionally represented a complementary research field whose aim is to explore magnetization dynamics at short time scales and enable ultrafast spintronic devices ( Kirilyuk, Kimel, and Rasing, 2010 ). The optical counterparts of the STT and SOT, in which current carriers are replaced by photocarriers andwhich have been identified in laser induced spin-dynamics studies in (Ga,Mn)As ( Fern ández-Rossier et al. , 2003 ;Núñez et al. , 2004 ;Nemec et al. , 2012 ;Tesarova et al. , 2013 ), build a bridge between these two important fields of spintronics research. The direct-gap GaAs host allowing for the gener-ation of a high density of photocarriers, optical selection rules linking light and carrier-spin polarizations, and the carrier spins interacting with magnetic moments on Mn via exchangecoupling make (Ga,Mn)As a unique ferromagnetic system for exploring the interplay of photonics and spintronics. Thermopower, also known as the Seebeck effect, is the ability of conductors to generate electric voltages from thermal gradients. A subfield of spintronics, termed spincaloritronics, explores the possibility of controlling charge and spin by heat and vice versa ( Bauer, Saitoh, and van Wees, 2012 ). In (Ga,Mn)As, experiments on the anomalous Nernst effect (ANE) ( Puet al. , 2008 ), which is the spin-caloritronics counterpart to the AHE, confirmed the validity of the Mottrelation between the off-diagonal electrical and thermal trans- port coefficients in a ferromagnet ( Wang et al. , 2001 ). Theexperiments also firmly established the intrinsic nature of both the AHE and ANE in metallic (Ga,Mn)As. The anisotropic magnetothermopower (AMT) ( Ky, 1966 ) is a phenomenon in which the Seebeck coefficient of a uniform magnetic con-ductor depends on the angle between the applied temperature gradient and magnetization. Measurements of this counterpart to the AMR electrical-transport effect in (Ga,Mn)As ( Puet al. , 2006 ) initiated renewed interest in the phenomenon in a broad class of magnetic materials ( Wisniewski, 2007 ;Tang et al. , 2011 ;Anwar, Lacoste, and Aarts, 2012 ;Mitdank et al. , 2012 ). The spin-caloritronic counterpart of the TMR effect in magnetic tunnel junctions is observed when the voltagegradient across the junction is replaced with a temperature gradient. The resulting tunneling magnetothermopower (TMT) represents the difference between the Seebeck coef- ficients for the parallel and antiparallel magnetizations of the tunnel junction electrodes ( Liebing et al. , 2011 ;Walter et al. , 2011 ). The relativistic analog in a tunnel junction with only one magnetic electrode is the tunneling anisotropic magneto- thermopower (TAMT) whose observation was reported in (Ga,Mn)As ( Naydenova et al. , 2011 ), reminiscent of the discovery of the TAMR ( Gould et al. , 2004 ). Another spin- caloritronics effect which is distinct from the magnetothermo- power (magneto-Seebeck) phenomena is the spin-Seebeck effect ( Uchida et al. , 2008 ,2010 ;Jaworski et al. , 2010 ; Sinova, 2010 ). Here the thermal gradient in a ferromagnet induces a spin current which is then converted into electricalvoltage via, e.g., the SHE in an attached nonmagnetic electrode (Uchida et al. , 2008 ,2010 ;Jaworski et al. , 2010 ;Sinova, 2010 ). Experiments in (Ga,Mn)As ( Jaworski et al. , 2010) provided a direct evidence that, unlike the Seebeck effect in normal conductors, the spin-Seebeck effect does not originate from charge flow. The intriguing origin of the spin-Seebeck effect has been extensively debated ( Bauer, Saitoh, and van Wees, 2012 ;Tikhonov, Sinova, and Finke ľstein, 2013) since these seminal experiments. In Sec. IIwe provide an overview of the material properties of (Ga,Mn)As with the emphasis on characteristics that make (Ga,Mn)As a favorable model system for spintronics research. For more detailed discussions of the materials aspects of theresearch of (Ga,Mn)As in the context of the family of (III,Mn) V and other magnetic materials we refer to other compre- hensive review articles ( Matsukura, Ohno, and Dietl, 2002 ; Dietl, 2003 ;Jungwirth, Sinova et al. , 2006 ;Sato et al. , 2010 ; Dietl and Ohno, 2014 ). The focus of this review is the spin- dependent phenomena and devices concepts explored in (Ga,Mn)As, and their relevance within the broad spintronics research field. These are discussed in Sec. III. Our aim is to find conceptual links between the seemingly diverse areas of spintronic studies in (Ga,Mn)As. Simultaneously, we attempt to provide intuitive physical pictures of the spin-dependent phenomena and functionalities for not only describing the specific observations in the ferromagnetic semiconductor(Ga,Mn)As but also for highlighting their applicability to other materials including the common transition-metal ferro- magnets, and other types of magnetic order such as anti- ferromagnets. While (Ga,Mn)As and the related ferromagnetic semiconductors have so far failed to allow for practicalspintronic functionalities at room temperature, transition- metal ferromagnets are commonly used in commercialT. Jungwirth et al. : Spin-dependent phenomena and device concepts … 857 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014spintronic devices ( Chappert, Fert, and Dau, 2007 ) and antiferromagnets can readily combine room-temperature oper-ation with not only metal but also semiconductor electronicstructure ( Jungwirth et al. , 2011 ). In Sec. IVwe provide a brief summary of the spintronics research directions inspired by (Ga,Mn)As. II. TEST-BED MATERIAL FOR SPINTRONICS RESEARCH A. Electronic structure and magnetism in (Ga,Mn)As The elements in the (Ga,Mn)As compound have nominal atomic structures ½Ar/C1383d104s2p1for Ga, ½Ar/C1383d54s2for Mn, and ½Ar/C1383d104s2p3for As. This circumstance correctly sug- gests that the most stable position of Mn in the GaAs hostlattice, at least up to a certain level of Mn doping, is on the Ga site where its two 4selectrons can participate in crystal bonding in much the same way as the two Ga 4selectrons. Because of the missing valence 4pelectron, the substitutional Mn Gaimpurity acts as an acceptor. In the electrically neutral state, the isolated Mn Gahas the character of a local moment with zero angular momentum and spin S¼5=2(Landé gfactor g¼2) due to the five 3delectrons and a moderately bound hole. GaAs is an intermediate band-gap III-V semi-conductor, with E g¼1.5eV at low temperatures. The exper- imental acceptor binding energy of an isolated Mn impuritysubstituting for Ga is of an intermediate strength E 0a≈0.1eV (Chapman and Hutchinson, 1967 ;Blakemore et al. , 1973 ; Bhattacharjee and à la Guillaume, 2000 ;Madelung, Rössler, and Schulz, 2003 ;Yakunin et al. , 2004 ). The perturbation of the crystal potential of GaAs due to a single Mn impurity has three main components as shown in Fig. 1(Mašeket al. , 2010 ). (i) The first is the long-range hydrogeniclike potential of a single acceptor in GaAs whichalone produces a bound state at about 30 meV above thevalence band ( Marder, 2000 ). (ii) The second contribution is a short-range central-cell potential. It is specific to a given impurity and reflects the difference in the electronegativity of the impurity and the host atom ( Harrison, 1980 ). For a conventional nonmagnetic acceptor Zn Ga, which is the firstnearest neighbor of Ga in the periodic table, the atomic plevels are shifted by ∼0.25eV which increases the binding energy by ∼5meV. For Mn, the sixth nearest neighbor of Ga, thep-level shift, is ∼1.5eV which when compared to Zn Ga implies the central-cell contribution to the acceptor level of Mn Ga∼30meV ( Bhattacharjee and à la Guillaume, 2000 ). (iii) The remaining part of the Mn Gabinding energy is due to the spin-dependent hybridization of Mn dstates with neigh- boring As pstates. Its contribution, which has been directly inferred from spectroscopic measurements of uncoupled Mn Gaimpurities ( Schneider et al. , 1987 ;Linnarsson et al. , 1997 ;Bhattacharjee and à la Guillaume, 2000 ), is again comparable to the binding energy of the hydrogenic single- acceptor potential. Combining (i) –(iii) accounts for the exper- imental binding energy of the Mn Gaacceptor of 0.1 eV. An important caveat to these elementary considerations is that the short-range potentials alone of strengths inferred in (ii) and(iii) would not produce a bound state above the top of the valence band but only a broad region of scattering states inside the valence band. The low-energy degrees of freedom in (Ga,Mn)As materials are the orientations of Mn local moments and the occupationnumbers of acceptor levels near the top of the valence band. The number of local moments and the number of holes may differ from the number of Mn Gaimpurities in the GaAs host due to the presence of charge and moment compensating defects. Hybridization between Mn dorbitals and valence As=Gasporbitals, mainly the As porbitals on the neighbor- ing sites, leads to an antiferromagnetic exchange inter- action between the spins that they carry ( Schneider et al. , 1987 ;Linnarsson et al. , 1997 ;Okabayashi et al. , 1998 ; Bhattacharjee and à la Guillaume, 2000 ). At concentrations ≪1%of substitutional Mn, the average distance between Mn impurities (or between holes bound to Mn ions) is much larger than the size of the bound holecharacterized approximately by the impurity effective Bohr radius. These very dilute (Ga,Mn)As systems are insulating, with the holes occupying a narrow impurity band, and paramagnetic. Experimentally, ferromagnetism in (Ga,Mn) As is observed when Mn doping reaches approximately 1%and the system is still below but near the insulator-to-metal transition ( Ohno, 1999 ;Potashnik et al. , 2002 ;Campion et al. , 2003 ;Jungwirth et al. , 2007 ). (x¼1%Mn doping corre- sponds to Mn density c¼4x=a 3¼2.2×1020cm−3, where a is the lattice constant in Ga 1−xMnxAs.) At these Mn concentrations, the localization length of the holes is extended to a degree that allows them to mediate, via thep-dhybridization, ferromagnetic exchange interaction between Mn local moments, even though the moments are dilute. Beyond a critical Mn doping, which in experiments is about 1.5%, Mn-doped GaAs exhibits a transition to a state in which the Mn-impurity levels overlap sufficiently strongly thatthe ground state is metallic, i.e., that states at the Fermi level are not bound to a single or a group of Mn atoms but are delocalized across the system ( Matsukura, Ohno, and Dietl, 2002 ;Jungwirth, Sinova et al. , 2006 ;Jungwirth et al. , 2007 ). In the metallic regime Mn can, like a shallow acceptor (e.g., C, Be, Mg, Zn), provide delocalized holes with a low-temperature density comparable to Mn density FIG. 1 (color online). Schematic illustration of the long-range Coulomb and the two short-range potentials each contributing∼30meV to the binding energy of the Mn Gaacceptor. From Mašeket al. , 2010 .858 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014(Ruzmetov et al. , 2004 ;Jungwirth et al. , 2005 ;MacDonald, Schiffer, and Samarth, 2005 ). The transition to the metallic state occurs at Mn density which is about 2 orders of magnitude larger than in GaAs doped with shallow acceptors(Ferreira da Silva et al. , 2004 ). This is because of the central cell and p-dhybridization contributions to the binding energy which make Mn acceptors more localized than the shallow acceptors. A crude estimate of the critical metal-insulator transition density can be obtained with a short-range potentialmodel, using the experimental binding energy and assuming an effective mass of valence band holes m /C3¼0.5me. This model implies an isolated acceptor level with effective Bohr radiusa 0¼ðℏ2=2m/C3E0aÞ1=2¼10Å. The radius a0then equals the Mn-impurity spacing scale c−1=3atc≈1021cm−3. This explains qualitatively the higher metal-insulator-transition critical density in Mn-doped GaAs compared to the case of systems doped with shallow, more hydrogeniclike acceptorswhich have binding energies E 0a≈30meV ( Madelung, Rössler, and Schulz, 2003 ;Ferreira da Silva et al. , 2004 ). Unlike the metal-insulator phase transition, which is sharply defined in terms of the temperature T¼0limit of the conductivity, the crossover in the character of states nearthe Fermi level in semiconductors with increased doping is gradual ( Shklovskii and Efros, 1984 ;Lee and Ramakrishnan, 1985 ;Paalanen and Bhatt, 1991 ;Jungwirth, Sinova et al. , 2006 ;Dietl, 2007 ,2008 ). At very weak doping, the Fermi level resides inside a narrow impurity band (assuming somecompensation) separated from the valence band by an energy gap of a magnitude close to the impurity binding energy. In this regime strong electronic correlations are an essentialelement of the physics and a single-particle picture has limited utility. Well into the metallic state, on the other hand, the impurities are sufficiently close together, and the long-range Coulomb potentials which contribute to the binding energy of an isolated impurity are sufficiently screened, in which thesystem can be viewed as an imperfect crystal with disorder- broadened and shifted host bands. In this regime, electronic correlations are usually less strong and a single-particle picture often suffices. The short-range components of the Mn binding energy in GaAs, which are not screened by thecarriers, move the crossover to higher dopings and contribute significantly to carrier scattering in the metallic state. The picture of disorder-broadened and shifted Bloch bands has to be applied, therefore, with care even in the most metallic (Ga,Mn)As materials. While for some properties it mayprovide even a semiquantitatively reliable description, for other properties it may fail, as we discuss in more detail below. Although neither picture is very helpful for describing the physics in the crossover regime which spans some finite range of dopings, the notion of the impurity band on the lower doping side from the crossover and of the disordered exchange-split host band on the higher doping side from the crossover still have a clear qualitative meaning. The formerimplies that there is a deep minimum in the density of states between separate impurity and host band states. In the latter case the impurity band and the host band merge into one inseparable band whose tail may still contain localized states depending on the carrier concentration and disorder. Inmetallic ferromagnetic (Ga,Mn)As materials, hard x-ray angle-resolved photoemission ( Gray et al. , 2012 ) and thedifferential off- and on-resonance photoemission ( Di Marco et al. , 2013 ) data do not show a separation or intensity drop near the Fermi energy that would indicate the presence of a gap between the valence band and a Mn-impurity band. Thehost and impurity bands are merged in ferromagnetic (Ga,Mn)As according to these spectroscopic measurements. Note that terms overlapping and merging impurity and valence bands describe the same basic physics in (Ga,Mn)As. This is becausethe Mn-acceptor states span several unit cells even in the verydilute limit and many unit cells as the impurity band broadens with increasing doping. The localized and delocalized states then have a similarly mixed As-Ga-Mn spd character. This applies to systems on either side of the metal-insulatortransition. By recognizing that the bands are merged, that is, overlapped and mixed, in ferromagnetic (Ga,Mn)As materials, the distinction between valence and impurity statesbecomes mere semantics which can lead to seemingly con-troversial statements on the material ’s electronic structure but has no fundamental physics relevance. A microscopic theory directly linked to the above quali- tative considerations is based on the spd tight-binding approximation (TBA) Hamiltonian of (Ga,Mn)As in which electronic correlations on the localized Mn dorbitals are treated using the Anderson model of the magnetic impurity(Mašeket al. , 2010 ). In Fig. 2we plot examples of the total and orbital resolved densities of states (DOSs) for 10% of Mn Gaimpurities. The Mn- dspectral weight is peaked at several eV below the top of the valence band, in agreementwith photoemission data ( Okabayashi et al. , 1998 ;Gray et al. , 2012 ;Di Marco et al. , 2013 ), and is significantly smaller near the Fermi energy E F. The Fermi level states at the top of the valence band have a dominant As(Ga) p-orbital character. The p-dcoupling strength N0β≡N0Jex¼Δ=Sx (N0¼1=Ωu:c., where Ωu:c.is the unit cell volume) ( Jungwirth, Sinova et al. , 2006 ), determined from the calculated valence band exchange splitting Δ(and taking S¼5=2), is close to the upper bound of the reported experimental range of N0β∼1–3eV (Matsukura et al. , 1998 ;Okabayashi et al. , 1998 ;Szczytko et al. , 1999 ;Bhattacharjee and à la Guillaume, 2000 ;Omiya 6 4 2 0 -2 -4 -64 0 -4 -8Density of states (arb.units) Energy (eV)total d(Mn) 2 0 -2 4 0 -4 -8 Energy (eV)s(Ga,Mn)p(Ga,Mn)p(As) FIG. 2 (color online). TBA-Anderson density of states of Ga0.9Mn 0.1As and its orbital composition. The position of the Fermi energy is indicated by a vertical line. FromMašeket al. , 2010 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 859 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014et al. , 2000 ). This is regarded as a moderately weak p-d coupling because the corresponding Fermi level states of the(Ga,Mn)As have a similar orbital character to the states in thehost GaAs valence band. These spectral features are amongthe key characteristics of the hole mediated ferromagnetism in(Ga,Mn)As. The effective Hamiltonian theory of (Ga,Mn)As, based on the kinetic-exchange (Zener) model ( Dietl, Haury, and d’Aubigne, 1997 ;Jungwirth et al. , 1999 ;Dietl et al. , 2000 ; Jungwirth, Sinova et al. , 2006 ), assumes also a value of N 0β within the above experimental range, namely, N0β¼1.2eV (Jex¼55meV nm3) which is closer to the lower experimen- tal bound ( Jungwirth, Sinova et al. , 2006 ). It is this moderate p-dhybridization that allows it to be treated perturbatively and to perform the Schrieffer-Wolff transformation fromthe microscopic TBA-Anderson Hamiltonian to the effectivemodel in which valence band states experience a spin-dependent kinetic-exchange field ( Jungwirth, Sinova et al. , 2006 ). Hence, the effective kinetic-exchange model and the microscopic TBA-Anderson theory provide a consistentphysical picture of ferromagnetic (Ga,Mn)As. These twomodels of the electronic structure of (Ga,Mn)As have repre-sented the most extensively used basis for analyzing thespin-dependent phenomena and device functionalities in(Ga,Mn)As. In Fig. 3we show DOSs over the entire Mn Gadoping range obtained from the generalized gradient approximation(GGA þU) density-functional calculations ( Mašeket al. , 2010 ;Sato et al. , 2010 ). The GGA þU, the TBA- Anderson, and the kinetic-exchange Zener theories all providea consistent picture of the band structure of ferromagnetic(Ga,Mn)As. Simultaneously, it is important to keep in mindthat the moderate acceptor binding energy of Mn Gashifts the insulator-to-metal transition to orders of magnitude higherdoping densities than in the case of common shallow non-magnetic acceptors, as mentioned above ( Jungwirth et al. , 2007 ;Mašeket al. , 2010 ). Disorder and correlation effects, therefore, play a comparatively more significant role in(Ga,Mn)As than in degenerate semiconductors with commonshallow dopants and any simplified one-particle band pictureof ferromagnetic (Ga,Mn)As can represent a proxy only to the electronic structure of the material. As seen in Fig. 3, the bands evolve continuously from the intrinsic nonmagnetic semiconductor GaAs via the degenerateferromagnetic semiconductor (Ga,Mn)As to the ferromagneticmetal MnAs. From this it can be expected that T cof MnAs, with the value close to room temperature [350 K for cubicMnAs inclusions in (Ga,Mn)As ( Yokoyama et al. , 2005 ; Kovacs et al. , 2011 )], sets the upper theoretical bound of achievable T c’s in (Ga,Mn)As across the entire doping range. In experiment, as discussed in Sec. II.B, the Mn Gadoping is limited to approximately 10% with corresponding Tcreaching 190 K in uniform thin-film crystals prepared by optimizedLT-MBE synthesis and postgrowth annealing. In these sam-ples the hole density is in the ∼10 20–1021cm−3range, i.e., several orders of magnitude higher than densities in com-monly used nonmagnetic semiconductors but also 1 –2 orders of magnitude lower than is typical for metals. 1. Curie point singularities Ferromagnetic (Ga,Mn)As with Mn doping ranging from ∼1%to∼10% is a very heavily doped compound semi- conductor or can be also regarded at these high Mn concen-trations as a random alloy. Quantities such as the residualresistivity are then inevitably affected by strong disordereffects. Even in the most metallic (Ga,Mn)As materials thehole mean free path is comparable to the separation of the Mnimpurities so the diffusivity is low. Typically, the product ofthe Fermi wave vector and the mean free path is k FΛ¼ ℏμk2 F=e∼1–10, estimated from the experimental mobilities μand hole densities ( Jungwirth et al. , 2007 ). For thermody- namic properties, as well as for the spintronics effectsdiscussed in Sec. III, the disordered nature of (Ga,Mn)As can, however, play a less significant role. This makes the spin-dependent phenomena and device functionalities discoveredand explored in (Ga,Mn)As applicable to a broad class ofmaterials beyond the dilute-moment ferromagnetic semicon-ductor compounds. An example of the seemingly surprising similarity between the basic magnetic characteristics of (Ga,Mn)As and thecommon transition-metal ferromagnets such as Ni is shownin Fig. 4. Here we illustrate that (Ga,Mn)As can have Curie point singularities ( Nov áket al. , 2008 ;Yuldashev et al. , 2010 ) which are typical of uniform itinerant ferromagnets (Shacklette, 1974 ;Joynt, 1984 ). Figure 4(a)shows remanent magnetization MðTÞwhich vanishes sharply at T→T −c. For the same 11% Mn-doped sample, Fig. 4(a) also shows the resistivity ρðTÞand its temperature derivative dρ=dT. While ρðTÞhas a broad shoulder near Tc,dρ=dT has a singularity atTcwhich precisely coincides with Tcinferred from the remanence measurement in the same (Ga,Mn)As material(Nov áket al. , 2008 ;Jungwirth et al. , 2010 ;Nemec et al. , 2013 ). We explain below that the Curie point singularity in dρ=dT is related to the singularity in the specific heat which was also detected in (Ga,Mn)As ( Yuldashev et al. , 2010 ) and is shown in Fig. 4(b). The specific heat measurements were performed in lower Mn-doped samples (Ga,Mn)As [2.6% Mndoping in Fig. 4(b)] and therefore the singularity occurs in these samples at a correspondingly lower T c.20 0 -20 20 0 -20 20 0 -20 4 0 -4 -8Density of states (arb.units) Energy (eV)10% Mn 20% Mn 50% Mn80 40 0 -40 -80 4 0 -4 -8 Energy (eV)100% Mn LDA+U FIG. 3 (color online). Density of states for (Ga,Mn)As mixed crystals with various content of Mn obtained in the GGA þU theory. The lines represent the total DOS while the shaded areasshow the partial density of Mn dstates. From Mašeket al. , 2010 .860 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014Since seminal works of de Gennes and Friedel (1958) and Fisher and Langer (1968) , critical behavior of resistivity has been one of the central problems in the physics of itinerant ferromagnets. Theories of coherent scattering from longwavelength spin fluctuations, based on the original paper by de Gennes and Friedel, have been used to explain the large peak in the resistivity ρðTÞatT cobserved in Eu-chalcogenide dense-moment magnetic semiconductors ( Haas, 1970 ). The emphasis on the long wavelength limit of the spin-spin correlation function, reflecting critical behavior of the mag- netic susceptibility, is justified in these systems by the smalldensity of carriers relative to the density of magnetic momentsand corresponding small Fermi wave vectors of carriers. As pointed out by Fisher and Langer (1968) , the resistivity anomaly in high carrier density transition-metal ferromagnetsis qualitatively different and associated with the critical behavior of correlations between nearby moments. Whenapproaching T cfrom above, thermal fluctuations between nearby moments are partially suppressed by short-rangemagnetic order. Their singular behavior is like that of theinternal energy and unlike that of the magnetic susceptibility.The singularity at T coccurs in dρ=dTand is closely related to the critical behavior of the specific heat. While Fisher andLanger expected this behavior for T→T þcand a dominant role of uncorrelated spin fluctuations at T→T−c, later studies of elemental transition metals found a proportionality betweendρ=dT and specific heat on both sides of the Curie point, as shown in the upper inset of Fig. 4(b) (Shacklette, 1974 ; Joynt, 1984 ). The character of the transport anomaly in (Ga,Mn)As is distinct from the critical contribution to transport in the dense-moment magnetic semiconductors ( Haas, 1970 ) and is reminiscent of the dρ=dT singularity in transition-metal ferro- magnets ( Shacklette, 1974 ;Joynt, 1984 ). Ferromagnetism in (Ga,Mn)As originates from spin-spin coupling between localMn moments and valence band holes JP iδðr−RiÞσ·Si (Dietl, Haury, and d ’Aubigne, 1997 ;Jungwirth et al. , 1999 ; Dietl et al. , 2000 ;Jungwirth, Sinova et al. , 2006 ). Here Si represents the local spin and σis the hole spin operator. This local-itinerant exchange interaction plays a central role intheories of the critical transport anomaly. When treated in theBorn approximation, the interaction yields a carrier scatteringrate from magnetic fluctuations, and the corresponding con-tribution to ρðTÞ, which is proportional to the static spin-spin correlation function ΓðR i;TÞ∼J2½hSi·S0i−hSii·hS0i/C138(de Gennes and Friedel, 1958 ). Typical temperature dependences of the uncorrelated part ΓuncorðRi;TÞ∼δi;0J2½SðSþ1Þ−hSii2/C138 and of the Fourier components of the correlation functionΓðk;TÞ¼P i≠0ΓðRi;TÞexpðk·RiÞare illustrated in the lower inset of Fig. 4(b) (Fisher and Langer, 1968 ). At small wave vectors, Γðk;TÞand correspondingly ρðTÞhave a peak at Tc.A tksimilar to the inverse separation of the local moments (kd↑−↑∼1) the peak broadens into a shoulder while the singular behavior at Tcis in the temperature derivative of the spin-spin correlator and, therefore, in dρ=dT. M2expansion providing a good fit to the magnetic con- tribution to the resistivity at T<T c(Nov áket al. , 2008 ) corresponds to the dominant contribution from Γuncor on the ferromagnetic side of the transition. The shoulder in ρðTÞon the paramagnetic side and the presence of the singularity indρ=dT suggest that large wave-vector components of Γðk;TÞ dominate the temperature dependence of the scattering intheT→T þccritical region ( Nov áket al. , 2008 ). The large k-vector limit is consistent with the ratio between hole and Mn local-moment densities approaching unity in high-quality(Ga,Mn)As materials with low charge compensation byunintentional impurities ( Nemec et al. , 2013 ). 2. Localization effects in transport While the sharp transport Curie point singularities highlight the fact that ferromagnetic (Ga,Mn)As epilayers can have ahigh degree of uniformity ( Kodzuka et al. , 2009 ) and can behave similarly to common, weakly disordered itinerantferromagnets, the magnitude of the resistivity at zero andfinite frequencies and over the broad temperature range issignificantly affected by the vicinity of the metal-insulator (a) (b) FIG. 4 (color online). (a) Temperature dependent remanent magnetization, resistivity, and temperature derivative of theresistivity of a nominally 11% Mn-doped (Ga,Mn)As. FromJungwirth et al. , 2010 . (b) Magnetic contribution to the specific heat of a 2.6% Mn-doped (Ga,Mn)As. Adapted from Yuldashev et al. , 2010 . Upper inset: Temperature derivative of resistance and a multiple of the specific heat plotted against temperature for Ni.From Joynt, 1984 . Lower inset: Schematic diagram of the spin- spin correlation function in low and large k-vector limits. From Fisher and Langer, 1968 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 861 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014transition in (Ga,Mn)As. The valence band calculations treating disorder in the first-order Born approximation over- estimate the experimental conductivities of metallic (Ga,Mn)As by up to a factor of 10 ( Jungwirth et al. , 2002 ;Sinova et al. , 2002 ). This discrepancy is removed by accounting for strong disorder and localization effects using, e.g., exact-diagonalization calculations ( Yang et al. , 2003 ;Jungwirth et al. , 2007 ). Even the most metallic (Ga,Mn)As materials with delocalized carriers at the Fermi level may containlocalized states in the valence band tail which modify the finite-frequency absorption spectra ( Burch et al. , 2006 ; Jungwirth et al. , 2007 ,2010 ;Chapler et al. , 2011 ). The low diffusivity of carriers implies that quantum interference and electron-electron interactions can produce sizable effects in (Ga,Mn)As. Weak localization (WL) quan-tum corrections are due to constructive interference between partial waves undergoing multiple scattering from a state with wave vector kto a state −kand partial waves traversing the time-reversed trajectory. The effect is also referred to as coherent backscattering and it leads to a reduction of the conductivity. A distinct, electron-electron interaction quantumcorrection to the conductivity ( Lee and Ramakrishnan, 1985 ) can arise in disordered conductors which often has a similar magnitude to the WL correction. This arises because electron-electron interactions cannot be treated independently of the disorder scattering for strong disorder. Explicit expressions for the WL corrections can be obtained forL Φ≫Λ≫λF, where LΦ,Λ, and λFare the carrier phase coherence length, mean free path, and Fermi wavelength. Thesecond condition can be rewritten as k FΛ≫1, where kFis the Fermi wave vector. The corrections are of the order of ðkFΛÞ−1 and so become important for small kFΛ. It has been argued that higher order corrections are small and that the condition kFΛ≫1can be relaxed to kFΛ>1. Application of a magnetic field can suppress the resistance enhancement dueto WL as it removes time-reversal invariance leading to negative magnetoresistance. The magnetic field begins to have a significant effect when l B∼LΦ, where lB¼ ðℏ=eBÞ1=2is the magnetic length, and the magnetic field completely suppresses WL when lB∼Λ. Since WL quantum corrections are suppressed by sufficiently large magneticfields one expects a similar suppression by the internal magnetization. For dense-moment ferromagnets like Fe, Ni, etc., μ 0M∼2T and the mean free path is usually quite large so WL is strongly suppressed. However, WL is observed, for example, in highly disordered Ni films ( Aprili et al. , 1997 ). For the dilute-moment ferromagnet (Ga,Mn)As, μ0M∼ 50mT while the field needed to suppresses WL, i.e., when lB∼Λ∼1nm, is∼1000 T. So one expects WL effects to be present, and since typically kFΛ∼1–10, they may be large. The identification of WL contributions to the temperature dependence of resistance is difficult as they generally coexistwith other temperature dependent contributions and because the expected functional form can be very different for the different possible phase breaking mechanisms. In disorderedferromagnets like (Ga,Mn)As, spin-disorder scattering can, e.g., produce large magnetoresistance, particularly close to the localization boundary ( Kramer and MacKinnon, 1993 ; Nagaev, 1998 ;Omiya et al. , 2000 ).For external magnetic fields less than the coercive fields the magnetoresistance response is usually dominated by AMR(see Secs. III.A andIII.B). At larger fields a negative isotropic magnetoresistance is observed which can be very large for lowconductivity material ( Matsukura et al. , 2004 ). This could be due to the suppression of spin disorder ( Lee, Stone, and Fukuyama, 1987 ). However, as shown in Fig. 5(Matsukura et al. , 2004 ), the negative magnetoresistance does not seem to saturate, even in extremely strong magnetic fields. It has beenargued ( Matsukura et al. , 2004 ) that the negative magneto- resistance arises from WL and gives a correction consistentwith the predicted form proportional to −B 1=2(Kawabata, 1980 ), which assumes a complete suppression of spin- disorder and spin-orbit scattering (see Fig. 5). The role of spin-orbit coupling in WL phenomena in (Ga,Mn)As has been extensively discussed ( Neumaier et al. , 2007 ;Rokhinson et al. , 2007 ;Garate, Sinova et al. , 2009 ). In the context of the spintronic phenomena and functionalities in (Ga,Mn)As and their applicability to other materials, dis-cussed in Sec. III, an important conclusion is drawn from numerical studies of WL in (Ga,Mn)As ( Garate, Sinova et al. , 2009 ). They showed that while WL corrections can signifi- cantly contribute to the absolute residual resistivity, the0.60.81.0 -9 -6 -3 0 3 6 90.81.21.62.0I // [100] H plane T = 2, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 125, 150, 200, 300 KGa0.95Mn0.05As (200 nm), compressive strainRsheet (k) H planeI // <110>Ga0.957Mn0.043As (200 nm), tensile strain µ0H (T) FIG. 5 (color online). Field and temperature dependences of re- sistance in Ga 0.95Mn 0.05As=GaAs (compressive strain, upper panel) and in tensile strained Ga 0.957Mn 0.043As=ðIn;GaÞAs (lower panel) for magnetic field perpendicular to the film plane. Start-ing from the top, subsequent curves at B¼0correspond to temperatures in K: 70, 60, 80, 50, 90, 40, 100, 30, 125, 20,2, 5, 10, 150, 200, 300 (upper panel) and to 50, 60, 40, 70,30, 80, 90, 20, 100, 2, 10, 5, 125, 150, 200, 300 (lowerpanel). From Matsukura et al. , 2004 .862 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014relative changes in resistivity associated with magnetization reorientations, namely, the AMR ratios, are nearly indepen- dent on whether the WL corrections are included or not (Garate, Sinova et al. , 2009 ). These results, which agree qualitatively with analytical considerations on simpler models (Bhatt, Wolfle, and Ramakrishnan, 1985 ), illustrate that the intrinsically strong disorder in (Ga,Mn)As can qualitatively play a minor role in not only the thermodynamic properties but also in the spintronic phenomena reflecting the inter-actions of carrier spins with electrical current, light, or heat. What determines these phenomena is primarily the magnetic exchange and spin-orbit fields acting on the carrier states.Disorder can mix the carrier states but as long as this mixing does not significantly alter the effects of the exchange field and spin-orbit coupling on the carriers the spintronic phenom- ena remain robust against disorder. This explains the quali- tative and often semiquantitative success, and justifies theapplicability, of microscopic theories of spintronic phenomena in (Ga,Mn)As starting from a Bloch-band description of the material ’s electronic structure. Simultaneously it should be noted that due to strong disorder and the vicinity of the metal- insulator transition a full quantitative description is unlikely tobe achievable within any of the existing theoretical models of (Ga,Mn)As. We conclude this section by discussing the universal conductance fluctuations (UCFs) in (Ga,Mn)As. These result from the interference between partial waves from scatteringcenters within a conductor. In the usual semiclassical theory of electron conduction this is neglected since it is assumed that such effects will be averaged away. However, for conductors ofsize comparable with L Φthe interference effects are intrinsi- cally non-self-averaging. This leads to corrections to the conductivity of order e2=h. Application of a magnetic field modifies the interference effects, giving reproducible but aperiodic UCFs ( Lee, Stone, and Fukuyama, 1987 ) of ampli- tude∼e2=h. One can think of a conductor with dimensions >LΦas made up of a number of independent phase coherent subunits leading to averaging. UCFs are then diminished for dimensions ≫LΦand only WL due to the coherent back- scattering may still contribute in macroscopic samples. At temperatures which are a significant fraction of the Curie temperature one expects spin-disorder and spin-orbit scatter- ing to lead to the phase coherence length LΦ∼Λ, strongly suppressing quantum corrections. However, in high-quality metallic (Ga,Mn)As it has been argued ( Matsukura et al. , 2004 ) that LΦneed not be very small at low temperatures because virtually all spins contribute to the ferromagnetic ordering and the large splitting of the valence band makes bothspin-disorder and spin-orbit scattering relatively inefficient. The strong magnetocrystalline anisotropies also tend to suppress magnon scattering at low temperatures. Recent observations ( Wagner et al. , 2006 ;Vila et al. , 2007 ) of large UCFs in (Ga,Mn)As microdevices, and the evidencefor the closely related Aharonov-Bohm effect (ABE) in (Ga,Mn)As microrings, confirm that L Φcan be large at low temperatures. Figure 6shows UCFs measured ( Wagner et al. , 2006 ) in (Ga,Mn)As wires of approximate width 20 nm and thickness 50 nm. Figure 6(a) shows that the UCF amplitude is∼e2=hin a 100 nm long wire at 20 mK. This directly demonstrates that LΦ∼100nm. Similar measurements inhigher conductivity (Ga,Mn)As give LΦ∼100nm at 100 mK. These are large values corresponding to a phaserelaxation time that is orders of magnitude larger than the elastic scattering time. Figure 7shows measurements ( Wagner et al. , 2006 ) of the magnetic field dependence of the conductivity of a litho-graphically defined 100 nm diameter (Ga,Mn)As ring com-pared to that of a 200 nm long (Ga,Mn)As wire. Additional small period oscillations are observed for the ring which the Fourier transform shows to be consistent with the expectedABE period. This confirms the long L Φindicated by the large amplitude UCFs and confirms that almost all spins areparticipating in the magnetic order with strong suppressionof spin scattering. B. Doping trends in basic magnetic and transport properties of (Ga,Mn)As 1. Low Mn-doped bulk materials Narrow impurity bands have been clearly observed in Mn-doped GaAs samples with carrier densities much lowerthan the metal-insulator transition density, for example, in equilibrium grown bulk materials with Mn density c¼ 10 17–1019cm−3(Brown and Blakemore, 1972 ;BlakemoreL( n m ) 100 200 300 1000 500 200 100 50 20(a) (b)0.5 0 00.20.40.60.82 FIG. 6 (color online). (a) Conductance fluctuations for three wires of different length L. For the shortest wire the amplitude of the conductance fluctuations is about e2=h, expected for con- ductors with all spatial dimensions smaller or comparable to LΦ. The inset shows an electron micrograph of a 20 nm wide wirewith a potential probe separation of ∼100nm. (b) Conductance vs magnetic field of the 200 nm wire for different temperaturesbetween 20 mK and 1 K. From Wagner et al. , 2006 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 863 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014et al. , 1973 ;Woodbury and Blakemore, 1973 ). The energy gap between the impurity band and the valence band Eacan be measured by studying the temperature dependence oflongitudinal and Hall conductivities, which show activatedbehavior because of thermal excitation of holes from the impurity band to the much more conductive valence band (Blakemore et al. , 1973 ;Woodbury and Blakemore, 1973 ; Marder, 2000 ). The activation energy decreases with increasing Mn density (Blakemore et al. , 1973 ). The lowering of impurity binding energies at larger c, which is expected to scale with the mean impurity separation, is apparent already in the equilibrium grown bulk materials with c¼10 17–1019cm−3. The degen- erate semiconductor regime was, however, not reached in thebulk materials. 2. Synthesis of high Mn-doped epilayers A comprehensive experimental assessment of basic doping trends including the regimes near and above the insulator-to-metal transition became possible since the late 1990s with the development of LT-MBE (Ga,Mn)As films ( Ohno, 1998 ). The epilayers can be doped well beyond the equilibrium Mnsolubility limit while avoiding phase segregation and main-taining a high degree of uniformity ( Kodzuka et al. , 2009 ). Because of the highly nonequilibrium nature of the heavily doped ferromagnetic (Ga,Mn)As, the growth and postgrowthannealing procedures have to be individually optimized foreach Mn-doping level in order to obtain films which are as close as possible to idealized uniform (Ga,Mn)As mixedcrystals with the minimal density of compensating and otherunintentional defects. This is illustrated in Fig. 8showing, side by side, basic electrical and magnetic characteristics of twomedium, 7% Mn-doped epilayers ( Nemec et al. , 2013 ). The left column shows data measured on a material which wasprepared under optimized conditions for the given nominal Mn doping. The sample has sharp Curie point singularities in magnetization and dρ=dT [Fig. 8(a)]. The magnetization precession damping factor and spin-wave resonances (SWRs)obtained from magneto-optical measurements [Figs. 8(b), and8(c)] confirm the high magnetic quality of the material. The initial decrease of the damping factor with frequencyfollowed by a frequency independent part [Fig. 1(b)] is typical of uniform ferromagnets ( Walowski et al. , 2008 ). It allows one to accurately separate the intrinsic Gilbert damping constant α, corresponding to the frequency independent part, from effects that lead to inhomogeneous broadening of ferromagneticresonance (FMR) linewidths. Similarly, the observed KittelSWR modes of a uniform ferromagnet [Fig. 1(c)] allows one to accurately measure the magnetic anisotropy and spin-stiffness parameters of (Ga,Mn)As. 100 nm(a) (c) (b) T = 20 mK Ring, d = 100 nm Wire, L = 200 nm B( T )G)h/ e(2h/e 0.7 T FIG. 7. (a) Electron micrograph of a (Ga,Mn)As ring sample with a diameter of ∼100nm. (b) Comparison of the magneto- conductance trace of the ring sample with the conductance of awire of comparable length and 20 nm width. (c) The correspond-ing Fourier transform taken from the conductance of the ring andwire. The region where ABE oscillations are expected is high-lighted. From Wagner et al. , 2006 .0 1 02 03 04 0051015f (GHz) 0Hext (mT)0 5 10 15 20012damping (10-2) f (GHz) 0 1 02 03 04 002468f (GHz) 0Hext (mT)0 5 10 15 200246damping (10-2) f (GHz) (c)0.00.51.0 02550M (emu/cm3) (d/dT)* 0.00.51.0 0.012.525.0 M (emu/cm3) (d/dT)* 0 50 100 150 2001.82.12.4 T (K) (10-3 cm) 0 25 50 75 1006789 T (K) (10-3 cm) (b)(a) (d) (e) (f) FIG. 8 (color online). (a) Magnetization M, temperature deriva- tive of the resistivity normalized to the peak value ðdρ=dTÞ/C3, and resistivity ρðTÞof an optimized 20 nm thick epilayer with 7% nominal Mn doping. (b), (c) Frequency dependence of thedamping factor and field dependence of the SWR frequenciesof the same sample. (d) –(f) Same as (a) –(c) for a material differing by having only one of the synthesis parameters notoptimized (epilayer thickness of 500 nm is too large). FromNemec et al. , 2013 .864 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014The right column data [Figs. 8(d)–8(f)] were measured on a 7% Mn-doped epilayer differing from the sample of the left column in only one of the synthesis parameters not being optimized. The stoichiometry, substrate growth temperature,postgrowth annealing temperature and time, and epilayerthickness are among the key synthesis parameters. All these parameters were equally optimized in the two samples except for the epilayer thickness. In the medium and high Mn-dopedsamples, full material optimization is possible only for film thicknesses ≲50nm. The epilayer whose measurements are shown in the right panels of Fig. 8is 500 nm thick. Its magnetization and transport Curie point singularities arelargely smeared out, the damping factor is strongly frequency dependent, and an alternating number of SWRs is observed with increasing applied field whose spacings are inconsistentwith Kittel modes. The material is nonuniform, the magneti-zation and transport data indicate strong moment and charge compensation by extrinsic impurities, and for this material it is impossible to reliably extract any of the intrinsic micro-magnetic parameters of (Ga,Mn)As. In Fig. 9we illustrate the fact that even in films thinner than 50 nm apparently small changes in the remaining keysynthesis parameters can significantly affect the material quality ( Nemec et al. , 2013 ). Staying near the 1:1 stoichio- metric As ∶ðGaþMnÞratio is favorable for the LT-MBE growth of (Ga,Mn)As ( Myers et al. , 2006 ;Wang et al. , 2008 ). Figure 9(a)shows the optimal growth temperature T Gfor the stoichiometric growth as a function of the nominal Mn doping x. The optimal TGremains near (from the lower temperature side) the 2D/3D growth-mode boundary which implies its strong dependence on x. Figure 9(b)shows Tcas a function of the annealing time for the optimal TG¼190°C for the 13% Mn-doped sample and for two annealing temperatures. One is the optimal annealing temperature TA¼160°C and the other one is 20° lower. The maximum Tc¼188K sample is obtained by simultaneously optimizing the annealing time andTA. Figures 9(c)and9(d)illustrate how the increasing Tc is accompanied by the improving material quality (reduction of extrinsic compensation and sample inhomogeneity) overthe annealing time for optimal T GandTA. The importance of the optimal TGduring the growth is highlighted in Figs. 9(e) and9(f)showing the same annealing sequence measurements as in Figs. 9(c)and9(d) on a 13% doped sample grown at a temperature of only 10° below the optimal TG. In contrast to the material grown at the optimal TG, the sample is insulating and paramagnetic in the as-grown state. Ferromagnetism and metallic conduction can be recovered by annealing; however, the compensation and inhomogeneity cannot be removed and the ferromagnetic transition temperature remains tensof degrees below the T cof the sample grown at the optimal TG. Similarly lower quality samples are obtained by growing at higher than optimal TG. Figures 8and9illustrate the following general conclusions drawn from extensive material optimization studies ( Nemec et al. , 2013 ). Inferring doping trends in basic material properties of (Ga,Mn)As from sample series mixing as-grown and annealed materials is unsuitable as the quality of thesamples may strongly vary in such a series. Choosing one a priori fixed T G,TA, and annealing time for a range of Mn dopings is unlikely to produce a high-quality, uniform and uncompensated (Ga,Mn)As material even for one of the considered dopings and is bound to produce low-qualitysamples for most of the studied Mn dopings. Finally, optimized (Ga,Mn)As samples require exceedingly long annealing times for film thicknesses ≳50nm and are impossible to achieve in ∼100nm and thicker films by the known (Ga,Mn)As synthesis approaches. When limited attention is paid to the details of the synthesis of the highly nonequilibrium (Ga,Mn)As alloy, seemingly contradictory experimental results can be found in these materi-als (Burch et al. ,2 0 0 6 ;Tang and Flatté, 2008 ;Dobrowolska, Liuet al. ,2 0 1 2 ;Dobrowolska, Tivakornsasithorn et al. ,2 0 1 2 ) as compared to measurements on samples prepared under the above optimized growth conditions ( Jungwirth et al. ,2 0 1 0 ; Wang et al. ,2 0 1 3 ). As an example we show in Fig. 10 measurements of T cversus hole density p(Dobrowolska, Tivakornsasithorn et al. ,2 0 1 2 ;Wang et al. ,2 0 1 3 ). The data are normalized to xeff(Neff¼4xeff=a3) representing the con- centration of Mn magnetic moments which contribute to the magnetic order. The results obtained by Dobrowolska, Tivakornsasithorn et al. (2012) indicated a strong suppression ofTcin (Ga,Mn)As layers with close to one hole per50 100 150 200 250-101(d/dT)* T (K)50 100 150 200 250246 as grown 0.5h 8h (10-3cm) T (K)50 100 150 200 250246 x 0.4x 0.04 (10-3cm) T (K)02468 1 0 1 2 1 4200250300TG x (%) 50 100 150 200 250-0.50.00.51.0(d/dT)* T (K)11 0140150160170180190TC (K) Annealing time (h)TA(b) (d)(e) (f)(a) (c) FIG. 9 (color online). (a) Optimal growth temperature TGas a function of the nominal Mn doping x. (b) Dependence of the Curie temperature Tcon the annealing time for two different annealing temperatures TAin a 15 nm thick (Ga,Mn)As epilayer with 13% nominal Mn doping grown at optimalT G. (c), (d) ρðTÞand ðdρ=dTÞ/C3in the x¼13% epilayer grown at optimal TGin the as-grown state, for optimal TAand annealing time 0.5 h, and for optimal TAand optimal annealing time of 8 h. (e), (f) Same as (c), (d) for a x¼ 13% epilayer grown at 10° below the optimal TG;ðdρ=dTÞ/C3is not plotted for the as-grown insulating and paramagneticsample. From Nemec et al. , 2013 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 865 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014substitutional Mn. It was thus suggested that Tcin ferromagnetic (Ga,Mn)As is determined by the location of the Fermi levelwithin a narrow impurity band, separated from the valence band.On the other hand, experiments on epilayers prepared under theoptimized growth conditions found that T ctakes its largest values in weakly compensated samples when pis comparable to the concentration of substitutional Mn acceptors. This is incon-sistent with models in which the Fermi level is located within anarrow isolated impurity band and corroborates predictions forT cof the above discussed microscopic theories (see Fig. 10)i n which valence and impurity bands are merged in ferromagnetic(Ga,Mn)As. Reliable measurements of systematic doping trends in intrinsic semiconducting and magnetic properties of materialswhich represent as close as possible idealized uniform(Ga,Mn)As mixed crystals with the minimal density ofcompensating and other unintentional defects require thecareful optimization of the synthesis. Many studies of thespintronics phenomena in (Ga,Mn)As, discussed in Sec. III, have also benefited from the high-quality optimized epi-layers. This applies, in particular, to experiments sensitive tosmall tilts of carrier spins from the equilibrium directionwhich is the case, e.g., of the magneto-optical phenomenaobserved in the pump-and-probe experiments discussed inSec. III.C. While for the detailed analysis the optimally synthesized and thoroughly characterized (Ga,Mn)Asepilayers are always favorable, many of the spintronicseffects and functionalities have been demonstrated inmaterials with extrinsic disorder not fully removed fromthe film. As shown in Figs. 8and9these materials can stillbe ferromagnetic and conductive and as discussed in Secs. II.A.1 and II.A.2 the spintronics phenomena can be, at least on a qualitative level, relatively robust againststrong disorder, whether intrinsic or extrinsic. 3. Curie temperature and conductivity Uniform (Ga,Mn)As materials with minimized extrinsic disorder can be divided into the following groups: at nominaldopings below ∼0.1%the (Ga,Mn)As materials are para- magnetic, strongly insulating, showing signatures of the activated transport corresponding to valence band –impurity band transitions at intermediate temperatures, and valenceband –conduction band transitions at high temperatures [see Fig.11(a) ](Jungwirth et al. , 2007 ;Nemec et al. , 2013 ). For higher nominal dopings, 0.5≲x≲1.5%, no clear signatures of activation from the valence band to the impurity band are seen in the dc transport, indicating that the bands start tooverlap and mix, yet the materials remain insulating. Atx≈1.5%, the low-temperature conductivity of the film increases abruptly by several orders of magnitude [seeFig. 11(b) ], and the system turns into a degenerate semi- conductor. The onset of ferromagnetism occurs already on the insulating side of the transition at x≈1%. All ferromagnetic samples over a broad nominal Mn-doping range can havesharp Curie point singularities when synthesized underFIG. 10 (color online). Curie temperature Tcvs hole density p normalized to xeff(Neff¼4xeff=a3) representing the concentra- tion of Mn magnetic moments which contribute to the magneticorder. The squares correspond to samples from Wang et al. (2013) prepared under optimized growth conditions where the holedensity pis obtained from high-field Hall measurements. The circles correspond to samples from Dobrowolska, Tivakornsasithorn et al. (2012) , where pis obtained from ion channeling measurements. The stars correspond to samples fromRushforth, Farley et al. (2008) prepared under optimized growth conditions, where pis obtained from ion channeling measure- ments. The line is the prediction of the microscopic calculation ofJungwirth et al. (2005) . 2468 1 0 1 210-310-1101103 (-1cm-1) 1000/T (K-1)x=0.05%1%2%7% 0123p (1021cm-3) 02468 1 0 1 2 1 4050100150200TC (K) x (%)(a) (d) 12 0255075100 M (emu/cm3)NMn (1021cm-3) 02468 1 0 1 2 1 410-310-1101103 (-1cm-1) x (%) 0 50 100 150 2000.00.51.0x=1.5% 13%(d/dT)* T (K)(b) (c) (e) (f) FIG. 11 (color online). (a) Temperature dependence of the conductivity σðTÞof optimized (Ga,Mn)As epilayers with depicted nominal Mn doping. Dashed lines indicate the activatedparts of σðTÞof the insulating paramagnetic (Ga,Mn)As with 0.05% Mn doping, corresponding to the Mn acceptor level andthe band gap, respectively. (b) Conductivity at 4 K as a functionof the nominal Mn doping. The open symbol corresponds to aparamagnetic sample. (c) Sharp Curie point singularities in thetemperature derivative of the resistivity in the series of optimizedferromagnetic (Ga,Mn)As epilayers with metallic conduction.(d)–(f) Hole density p, magnetization Mand corresponding Mn-moment density N Mn, and Curie temperature Tcas a function of the nominal Mn doping in the series of optimized (Ga,Mn)Asepilayers. From Nemec et al. , 2013 .866 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014individually optimized growth and postgrowth annealing conditions [see Fig. 11(c) ]. The hole concentration pcan be measured by the slope of the Hall curve at high fields with an error bar due to the multibandnature estimated to ∼20% (Jungwirth et al. , 2005 ). Within this uncertainty, the overall trend shows increasing pwith increas- ing doping in the optimized materials, as shown in Fig. 11(d) . Similarly, the saturation moment and T csteadily increase with increasing nominal doping up to x≈13%, as shown in Figs. 11(e) and11(f) . Assuming 4.5μBper Mn atom ( Jungwirth et al. , 2006 ) the density c≡NMnof uncompensated Mn Ga moments can be inferred from the magnetization data [see the leftyaxis in Fig. 11(e) ]. Since there is no apparent deficit of p compared to NMn, and since the interstitial Mn impurity (Edmonds et al. , 2002 ;Máca and Ma šek, 2002 ;Yuet al. , 2002 ) compensates one local moment but two holes it can be concluded that interstitial Mn, which is the key contributor toextrinsic disorder, is removed in the optimally grown andannealed epilayers. Hence, a broad series of optimized (Ga,Mn)As materials can be prepared with reproducible characteristics,showing an overall trend of increasing saturation moment withincreasing x, increasing T c(reaching 188 K), and increasing hole density. The materials have no measurable charge ormoment compensation of the substitutional Mn Gaimpurities and have a large degree of uniformity. Figure 12demonstrates that the intrinsic micromagnetic parameters of (Ga,Mn)As measured on the optimized materi-als show also a smooth monotonic doping dependence(Nemec et al. , 2013 ). As detailed below, their values are characteristic of common band ferromagnets and all the semiconducting and magnetic properties summarized in Figs. 11and 12are consistent with the microscopically established electronic structure of (Ga,Mn)As. The controland reproducibility of material properties of (Ga,Mn)As havebeen confirmed in the optimized films by multiple materialsynthesis and characterization experiments in different MBEchambers ( Nemec et al. , 2013 ;Wang et al. , 2013 ). 4. Micromagnetic parameters Micromagnetic parameters of (Ga,Mn)As and related (III,Mn)V ferromagnetic semiconductors were studied bymagnetization, magnetotransport, magneto-optical, or ferro- magnetic or spin-wave resonance (FMR or SWR) measure- ments ( Munekata et al. , 1993 ;Ohno, 1998 ;Abolfath et al. , 2001 ;Dietl, Ohno, and Matsukura, 2001 ;Potashnik et al. , 2002 ;Rappoport et al. , 2004 ;Sinova, Jungwirth et al. , 2004 ; Sawicki et al. , 2005 ;Gourdon et al. , 2007 ;Hümpfner et al. , 2007 ;Liu, Zhou, and Furdyna, 2007 ;Pappert et al. , 2007 ; Wang et al. , 2007a ;Wenisch et al. , 2007 ;Wunderlich et al. , 2007 ;Zhou et al. , 2007 ;Chiba et al. , 2008 ;Goennenwein et al. , 2008 ;Gould et al. , 2008 ;Khazen et al. , 2008 ;Overby et al. , 2008; Rushforth et al. , 2008; Rushforth, Wang et al. , 2008; Stolichnov et al. , 2008; Bihler et al. , 2009; Owen et al. , 2009; Zemen et al. , 2009; Cubukcu, von Bardeleben, Cantin, and Lemaitre, 2010 ;Cubukcu, von Bardeleben, Khazen et al. , 2010; Haghgoo et al. , 2010; Werpachowska and Dietl, 2010; De Ranieri et al. , 2013; Nemec et al. , 2013). A large experimental scatter of the measured micromagnetic param-eters can be found in the literature which partly reflects the issues related to the control of extrinsic disorder in the synthesis of (Ga,Mn)As. The experimental scatter also reflects, however, the favorable intrinsic tunability of (Ga,Mn)As properties by varying the temperature, hole andMn-moment densities, III-V substrate on which the (Ga,Mn) As film is deposited, or by alloying the magnetic film with other III or V elements, by device microfabrication, by applying electrostatic or piezoelectric fields on the film, etc. When measuring the micromagnetic parameters on the optimally and consistently synthesized series of bare (Ga,Mn)As epilayers on a GaAs substrate, fully reproducible and systematic trends can be inferred when simultaneouslydetermining the magnetic anisotropy K i, Gilbert damping α, and spin-stiffness Dconstants from one set of measurements. This has been demonstrated, e.g., on a series of ðGa;MnÞAs= GaAs epilayers over a broad range of Mn dopings by employ- ing the magneto-optical pump-and-probe technique, as shownin Fig. 12(Nemec et al. , 2013 ). The magnetic anisotropy fields are dominated by three components. The out-of-plane component K outis a sum of the thin-film shape anisotropy and the magnetocrystalline anisotropy due to the substrate lattice-matching growth strain.In (Ga,Mn)As grown on GaAs the strain in the (Ga,Mn)As epilayer is compressive and K outfavors for most Mn dopings in-plane magnetization [see Fig. 12(a) ]. However, when using an InGaAs substrate or adding phosphorus into the magnetic film, the growth strain can change from compressive to tensile,K outflips sign, and the film turns into an out-of-plane ferromagnet ( Abolfath et al. , 2001 ;Dietl, Ohno, and Matsukura, 2001 ;Yamanouchi et al. , 2004 ;Rushforth, Wang et al. , 2008 ;Cubukcu, von Bardeleben, Khazen et al. , 2010 ). This transition from an in-plane to an out-of-plane magnet has been exploited, e.g., in studies of the current- induced domain wall motion and spin-orbit torque discussed in Secs. III.B.6 andIII.B.7 (Yamanouchi et al. , 2004 ;Wang et al. , 2010 ;Fang et al. , 2011 ;Curiale et al. , 2012 ;De Ranieri et al. , 2013 ). The cubic magnetocrystalline anisotropy Kcreflects the zinc-blende crystal structure of the host semiconductor. The origin of the additional uniaxial anisotropy component alongthe in-plane diagonal K uis associated with a more subtle symmetry breaking mechanism introduced during the epilayer(a) (b) FIG. 12 (color online). (a) Dependence of magnetic anisotropy constants on nominal Mn doping. (b) Dependence of the Gilbertdamping constant αand the spin-stiffness constant Don nominal Mn doping. Measurements were performed at 15 K. From Nemec et al. , 2013 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 867 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014growth ( Kopecky et al. , 2011 ;Mankovsky et al. , 2011 ; Birowska et al. , 2012 ). The sizable magnitudes of Kcand Kuand the different doping trends of these two in-plane magnetic anisotropy constants [see Fig. 12(a) ] are crucial for the micromagnetics of the in-plane magnetized (Ga,Mn)As materials. The cubic anisotropy Kcdominates at very low dopings and the easy axis (EA) aligns with the main crystal axis [100] or [010]. At intermediate dopings, the uniaxial anisotropy Kuis still weaker but comparable in magnitude to Kc. In these samples the two equilibrium easy axes are tilted toward the ½1¯10/C138direction and their angle is sensitive to changes of temperature [the ratio of Ku=Kctends to increase with temperature ( Wang, Sawicki et al. , 2005 )] or externally applied electrostatic or piezovoltages which have been exploited in numerous studies of spintronics effects and device functionalities in (Ga,Mn)As ( Ohno et al. , 2000 ; Chiba et al. , 2003 ,2008 ;Goennenwein et al. , 2008 ; Olejn íket al. , 2008 ;Overby et al. , 2008 ;Rushforth et al. , 2008 ;Stolichnov et al. , 2008 ;Owen et al. , 2009 ;De Ranieri et al. , 2013 ). The origin of the magnetocrystalline anisotropies is in the spin-orbit coupling of the valence band holes mediating the ferromagnetic Mn-Mn coupling, as describedon a qualitative or semiquantitative level by the model, kinetic-exchange Hamiltonian theory ( Abolfath et al. , 2001 ; Dietl, Ohno, and Matsukura, 2001 ;Zemen et al. , 2009 ). A systematic doping trend of the Gilbert damping constant is also found across the series of optimized materials [seeFig.12(b) ]. The magnitudes of α∼0.1−0.01and the doping dependence are consistent with Gilbert damping constants in conventional transition-metal ferromagnets. In metals, α typically increases with increasing resistivity and is enhanced in alloys with enhanced spin-orbit coupling ( Ingvarsson et al. , 2002 ;Rantschler et al. , 2007 ;Gilmore, Idzerda, and Stiles, 2008 ). Similarly in (Ga,Mn)As the increase of αcorrelates with an increase of the resistivity in the lower Mn-dopedsamples. Moreover, the spin-orbit coupling effects tend to be stronger in the lower doped samples with lower filling of the hole bands and with the carriers closer to the metal-insulator transition. Theory ascribing magnetization relaxation to the kinetic-exchange coupling of Mn moments with the spin-orbit coupled holes yields a comparable range of values of αas observed in experiment [see Fig. 12(b) ](Sinova, Jungwirth et al. , 2004 ;Nemec et al. , 2013 ). The direct measurement of the spin stiffness requires a rather delicate balance between thin enough epilayers whosematerial quality can be optimized and thick enough films allowing one to observe the higher-index Kittel spin-wave modes ( Kittel, 1958 ) of a uniform thin-film ferromagnet. The magneto-optical pump-and-probe technique ( Nemec et al. , 2013 ) has an advantage that, unlike FMR, it is not limited to odd index spin-wave modes ( Kittel, 1958 ). The ability to excite and detect the n¼0, 1, and 2 resonances is essential for the observation of the Kittel modes in the optimized (Ga,Mn)As epilayers whose thickness Lis limited to ∼50nm. The modes in the optimized films show the expected quadratic scaling with nand with 1=L, and could be fitted by one set of magnetic anisotropy constants and spin-stiffness constant D (Nemec et al. , 2013 ). In the optimized series of (Ga,Mn)As epilayers a consistent, weakly increasing trend in Dwith increasing doping is observed [see Fig. 12(b) ] with values ofDbetween ∼2and 3meV nm 2. Similar to the Gilbert damping constant, the measured spin-stiffness constant inthe optimized (Ga,Mn)As epilayers is comparable to thespin stiffness in conventional transition-metal ferromagnets(Collins et al. , 1969 ). The values of the spin stiffness of the order meV nm 2are consistent with calculations based on the model kinetic-exchange and tight-binding Hamiltonians, ortheab initio electronic structure of (Ga,Mn)As ( König, Jungwirth, and MacDonald, 2001 ;Brey and Gómez-Santos, 2003 ;Bouzerar, 2007 ;Werpachowska and Dietl, 2010 ). To conclude Sec. II, the micromagnetic parameters of optimized (Ga,Mn)As epilayers are characteristic of commonband ferromagnets and the semiconducting and magneticproperties summarized in Figs. 11and12are consistent with the model Hamiltonian or ab initio theories of the electronic structure of (Ga,Mn)As. The materials research reviewed inSec. IIestablishes the overall view of (Ga,Mn)As as a well- behaved and understood degenerate semiconductor and band ferromagnet. Combined with the tunability of its electronic and magnetic properties, strong exchange and spin-orbitinteractions in the carrier bands, special symmetries of thehost zinc-blende lattice, and the compatibility with establishedIII-V semiconductor microfabrication techniques, this makes(Ga,Mn)As an ideal model system for spintronics research. III. PHENOMENA AND DEVICE CONCEPTS FOR SPINTRONICS A. Nonrelativistic versus relativistic based spintronics concepts Most of the spintronic devices discussed in Sec. IIIcan be associated with one of two basic physical principles. The firstone stems from Mott ’s two-spin-channel picture of transport in ferromagnets with exchange-split bands ( Mott, 1936 ) and we will label it a Mott spintronics principle. Phenomena whichfollow from the Mott picture can be typically understoodusing the nonrelativistic band structure with momentum-independent spin quantization axis. The second paradigm isdue to the quantum-relativistic spin-orbit coupling ( Strange, 1998 ) and we will label it a Dirac principle. Spintronics effects based on the Dirac principle stem from a relativistic bandstructure comprising states with momentum dependent spinexpectation values. Mott devices require that spins are trans-ported between at least two noncollinear parts of a nonuniformmagnetic structure with the magnetization in one part serving as a reference to the other one. Dirac devices, on the other hand, can rely on a single uniform magnetic component andthe reference for detecting or manipulating spins by chargecarriers is provided internally by the spin-orbit coupling. The archetype Ohmic Mott device, schematically illustrated in Fig. 13, is based on the giant magnetoresistance (GMR) of a ferromagnet –normal-metal –ferromagnet multilayer in which magnetizations in the ferromagnets are switched betweenparallel and antiparallel configurations ( Baibich et al. , 1988 ; Binasch et al. , 1989 ). The archetype Ohmic Dirac device (see Fig. 13), which is discussed in Sec. III.B.2 , is based on the relativistic AMR of a uniform magnetic conductor in whichmagnetization is rotated with respect to the current direction orcrystal axes ( Thomson, 1857 ;McGuire and Potter, 1975 ). In the early 1990s the AMR and subsequently the GMR sensors868 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014were introduced in hard disk drive read-heads launching the field of applied spintronics ( Chappert, Fert, and Dau, 2007 ). In these Ohmic devices, the exchange-split and, in the case ofthe AMR, spin-orbit coupled bands enter the physics of spin transport in a complex way via electron scattering which is often difficult to control and accurately model. A more direct connection between spin-dependent transport and band structure is realized in tunneling devices. Here theTMR stack with two ferromagnetic electrodes ( Julliere, 1975 ; Miyazaki and Tezuka, 1995 ;Moodera et al. , 1995 ) operates on the Mott principle and the TAMR stack with one magneticelectrode ( Brey, Fern ández-Rossier, and Tejedor, 2004 ;Gould et al. , 2004 ;Giraud et al. , 2005 ;Ciorga et al. , 2007 ;Gao et al. , 2007 ;Moser et al. , 2007 ;Sankowski et al. , 2007 ;Park et al. , 2008 ,2011 ), discussed in Sec. III.B.3 , is the corresponding Dirac spintronics device (see Fig. 13). The more direct connection between transport and electronic structure intunneling devices implies that tunneling spintronics effectscan be significantly larger than their Ohmic counterparts. Thelarge TMR signals are used, e.g., to represent logical 0 and 1in MRAMs ( Chappert, Fert, and Dau, 2007 ). CB-AMR devices discussed in Sec. III.B.4 represent an ultimate simplification in the relation between themagnetotransport and the relativistic exchange-split band structure. Transport is governed here by a single electronic structure parameter which is the magnetization-direction dependent chemical potential, resulting in a large magneto-resistance response of the device ( Wunderlich et al. , 2006 ). A CB-AMR device with the spin-orbit coupled magnet forming a gate electrode of the SET ( Ciccarelli et al. , 2012 ) illustrates the fact that the Dirac spintronics principle not only works without a spin current connecting two separate magneticelectrodes but also with the spin-orbit-coupled magnetic component completely removed from the transport channel (see Fig. 13). Such a spintronic device operating without spin current cannot be realized within the more commonly con- sidered Mott spintronics principle which may explain why it falls beyond the Wikipedia ’s definition of spintronics as “a portmanteau meaning spin transport electronics ”(http://en .wikipedia.org/wiki/Spintronics ). The Mott GMR and TMR effects have their spin- caloritronic counterparts in the giant magnetothermopower (GMT) ( Sakurai et al. , 1991 ) and TMT ( Liebing et al. , 2011 ; Walter et al. , 2011 ). A similar correspondence is between the Dirac electrical transport AMR and TAMR effects and thespin-caloritronic AMT ( Puet al. , 2006 ;Wisniewski, 2007 ; Tang et al. , 2011 ;Anwar, Lacoste, and Aarts, 2012 ;Mitdank et al. , 2012 ) and TAMT ( Naydenova et al. , 2011 ), discussed in Sec. III.D.3 . The distinction between Mott and Dirac spintronics can be analogously applied to the inverse magnetotransport effects (spin torques), discussed in Secs. III.B.6 andIII.B.7 . The STT (Berger, 1996 ;Slonczewski, 1996 ;Zhang and Li, 2004 ;Ralph and Stiles, 2008 ) applied to switch the magnetization of a free layer by a vertical current driven through the TMR stack is a Mott spin-torque effect. The in-plane current-induced SOT in a uniform magnet with a broken space-inversion symmetry (Bernevig and Vafek, 2005 ;Manchon and Zhang, 2008 ; Chernyshov et al. , 2009 ;Miron et al. , 2010 ) is the Dirac spin-torque counterpart. Similarly the optical STT and SOT (Fern ández-Rossier et al. , 2003 ;Núñez et al. , 2004 ;Nemec et al. , 2012 ;Tesarova et al. , 2013 ) reviewed in Sec. III.C can be viewed as Mott and Dirac phenomena arising from theinteraction of spin with light. Observations of the Ohmic AMR in an antiferromagnetic metal FeRh ( Marti et al. , 2014 ) and antiferromagnetic semi- conductor Sr 2IrO 4(Marti et al. , 2013 ), and of the TAMR in tunnel junctions with a magnetic electrode made of a metalantiferromagnet IrMn ( Park et al. , 2011 ;Wang et al. , 2012 ), illustrate the fact that the Dirac approach to spintronics can be equally applicable to spin-orbit coupled ferromagnets andantiferromagnets. The anisotropic magnetoresistance pheno- mena make in principle no difference between the parallel- aligned moments in ferromagnets and antiparallel-aligned moments in antiferromagnets because they are an even function of the microscopic magnetic moments. In nonmagnetic con-ductors the SHE is an example of a spintronic phenomenon converting a normal electrical current into a spin current or vice versa ( Kato et al. , 2004 ;Wunderlich et al. , 2005 ; Valenzuela and Tinkham, 2006 ;Jungwirth, Wunderlich, and Olejnik, 2012 ). It has a similar microscopic physics origin to the AHE ( Hall, 1881 ;Nagaosa et al. , 2010 ) in uniform spin- orbit coupled ferromagnets and the SHE can be therefore AMR GMR TMR TAMR Chemical potential AMR MOTT DIRAC FIG. 13 (color online). Schematic comparison of Ohmic Mott (GMR) and Dirac (AMR) devices and tunneling Mott (TMR)and Dirac (TAMR) devices. At the bottom a Dirac device isshown based on the chemical potential anisotropy which hasno immediate counterpart in Mott spintronics. The short thickarrows show the magnetization direction, and the long thin(thick) lines with arrows depict low (high) electrical current.The broken current lines illustrate stronger scattering. Non-relativistic (relativistic) densities of states are schematicallyillustrated in the TMR (TAMR, chemical potential AMR)panels.T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 869 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014regarded as an example of the Dirac spintronic phenomenon in nonmagnetic systems. The relevance of the research in(Ga,Mn)As to these Dirac spintronic phenomena observed inantiferromagnetic and nonmagnetic conductors will also bediscussed in the following sections. B. Interaction of spin with electrical current 1. Anomalous and spin Hall effects Advanced computational techniques and experiments in new unconventional ferromagnets have recently led to sig-nificant progress in coping with the subtle nature of themagnetoresistance effects based on relativistic spin-orbit coupling. There are two distinct relativistic MR coefficients in uniformly magnetized Ohmic devices, the AHE ( Hall, 1881 ) and the AMR ( Thomson, 1857 ). The AHE is the antisymmetric transverse MR coefficient obeying ρ xyðMÞ¼ −ρxyð−MÞ, where the magnetization vector Mis pointing perpendicular to the plane of the Hall bar sample. The AMR, discussed in Sec. III.B.2 , is the symmetric MR coefficient with the longitudinal and transverse resistivities obeyingρ xxðMÞ¼ρxxð−MÞandρxyðMÞ¼ρxyð−MÞ, where Mhas an arbitrary orientation. Note that in this review we use theterm transverse AMR rather than the alternative term planarHall effect ( Tang et al. , 2003 ) to clearly distinguish this symmetric off-diagonal magnetoresistance coefficient which is even in Mfrom the above antisymmetric off-diagonal Hall coefficient which is odd in M. (Ga,Mn)As has become one of the favorable test-bed systems for the investigation of the AHE. Here the unique position of (Ga,Mn)As ferromagnets stems from their tuna-bility and the relatively simple, yet strongly spin-orbit coupledand exchange-split carrier bands. The principles of the micro-scopic description of the AHE in the metallic (Ga,Mn)Asmaterials, based on the scattering-independent intrinsic mechanism ( Luttinger, 1958 ;Jungwirth, Niu, and MacDonald, 2002 ;Onoda and Nagaosa, 2002 ), have been successfully applied to explain the effect in other itinerant ferromagnets(Fang et al. , 2003 ;Haldane, 2004 ;Leeet al. , 2004 ;Yaoet al. , 2004 ;Dugaev et al., 2005 ;Kötzler and Gil, 2005 ;Sinitsyn et al., 2005 ), including conventional transition metals such as iron and cobalt, a pattern that has since then been repeated for other relativistic magnetotransport effects. The advances in the under-standing of the AHE are discussed in several reviews ( Chien and Westgate, 1980 ;Dietl et al.,2 0 0 3 ;Sinova, Jungwirth, and Černe, 2004 ;Jungwirth, Sinova et al.,2 0 0 6 ;Nagaosa et al., 2010 ). Here we recall the link between the AHE and SHE. Since the 1881 discovery of the AHE by Hall in Ni and Co, the phenomenon has been extensively employed in polarim-etry measurements of electron spins in ferromagnets. One lineof physical descriptions, illustrated in Fig. 14, associates the AHE with the same physical mechanism as the electron spin- dependent scattering from heavy nuclei which is used inpolarimetry of high-energy electron beams in accelerators.This relativistic spin-dependent skew-scattering mechanism isreferred to as Mott scattering ( Mott, 1929 ). [To avoid con- fusion we point out that Mott scattering ( Mott, 1929 )i s unrelated to the other work of Mott on the nonrelativistictwo-channel description of transport in ferromagnets ( Mott,1936 ) mentioned earlier; the AHE and SHE physics discussed here is relativistic in nature and falls within the family of Dirac spintronics phenomena, in the terminology used in theprevious section.] The applicability of the Mott skew- scattering mechanism to electrons scattering from heavy nuclei in the vacuum environment of accelerators as wellas to electrons scattering off impurities in the solid-stateenvironment of ferromagnets implies the presence of the same mechanism in nonmagnetic conductors. This was recognized by Dyakonov and Perel (1971) in their theoretical prediction of the skew-scattering SHE. A complementary line of research, also illustrated in Fig. 14and prompted by AHE experiments in the highly doped metallic (Ga,Mn)As epilayers ( Jungwirth, Niu, and MacDonald, 2002 ;Jungwirth et al. , 2003 ;Chun et al. , 2007 ; Glunk et al. , 2009 ), ascribes the AHE to a scattering- independent based mechanism in which the anomalous trans- verse component of the spin-dependent velocity stems directly from the spin-orbit coupled band structure in a clean crystal.In analogy with the skew-scattering AHE and SHE, a linkwas proposed between the scattering-independent mechanism of the AHE and a corresponding intrinsic SHE ( Murakami, Nagaosa, and Zhang, 2003 ;Sinova, Culcer et al. , 2004 ), followed by experimental discoveries of the SHE ( Kato et al. , 2004 ;Wunderlich et al. , 2005 ). We return to the physical description of these phenomena in Sec. III.B.7 where the link is extended from the AHE and SHE to the SOT. 2. Anisotropic magnetoresistance Phenomenologically, the AMR has “noncrystalline ”and “crystalline ”components ( Döring, 1938 ;McGuire and Potter, 1975 ). The former corresponds to the dependence of the resistance of the ferromagnet on the angle between magneti- zation and the direction of the electrical current while the latter depends on the angle between magnetization and crystal axes. d u d u + Skew (Mott) scattering Anomalous velocity AHE SHE FIG. 14 (color online). Schematic illustrations of the skew (Mott) scattering AHE and SHE (top panels) and the intrinsic AHE andSHE due to the anomalous transverse component of the spin-dependent velocity originating from the spin-orbit coupled bandstructure in a clean crystal (bottom panels). In the AHE, anelectrical current driven through a ferromagnetic conductor j seis spin polarized and the spin-dependent transverse deflection ofelectrons produces a transverse voltage. In the SHE, an unpo-larized electrical current j eis driven through a normal conductor and the spin-dependent transverse deflection of electrons pro-duces a transverse spin current. Opposite spins accumulate atopposite edges but unlike the AHE the transverse voltageremains zero.870 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014The noncrystalline AMR is the only component contributing to the AMR in polycrystalline samples in which the crystal axes directions average out. It is the component identified in Kelvin ’s seminal AMR measurements in Ni and Fe ( Thomson, 1857 ). The crystalline AMR components can be isolated in single-crystal materials patterned into a Corbino-disk micro- device geometry for which the averaging over the radial current lines eliminates all effects originating from a specific direction of the current. This was demonstrated in experimentsin (Ga,Mn)As ( Rushforth et al. , 2007 ). The measurements took advantage of the near perfect single-crystal epilayers of (Ga,Mn)As and, simultaneously, of the low carrier density andmobility (compared with single-crystal metals) resulting in large source-drain resistances compared with the contact resistances even in the short current-line Corbino geometry. Moreover, the strong spin-orbit coupling in the (Ga,Mn)As electronic structure yields sizable and tunable crystallineAMR components which in the lower conductive (Ga,Mn) As materials can even dominate over the noncrystalline AMR component ( Rushforth et al. , 2007 ). In contrast, crystalline AMR components in common transition-metal ferromagnets have been extracted indirectly from fitting the total AMRangular dependences ( van Gorkom et al. , 2001 ). Apart from the distinct phenomenologies there is also a qualitative difference between the microscopic origins of the noncrystalline and crystalline AMR components. Since the former component depends only on the angle betweenmagnetization and current, the effects of the rotating mag- netization on the equilibrium electronic structure of the ferromagnet do not contribute to the noncrystalline AMR.Instead, in the leading order, the noncrystalline AMR reflects the difference between transport scattering matrix elements of electrons with momentum parallel to the current for the current parallel or perpendicular to M. Unlike the noncrystalline AMR, the crystalline AMR originates from the changes in the equilibrium relativistic electronic structure induced by the rotating magnetization with respect to crystal axes. The picture applies not only to the Ohmic crystalline AMR but also to the TAMR and CB-AMR discovered in (Ga,Mn)As ( Gould et al. , 2004 ;Wunderlich et al. , 2006 ). In the CB-AMR case, the anisotropy of the electronic structure with respect to the magnetization angle, or more specifically the anisotropy of the DOS and the corre- sponding position of the chemical potential, provides a direct quantitative description of the measured transport effect(Wunderlich et al. , 2006 ;Ciccarelli et al. , 2012 ). In the case of the TAMR or the crystalline Ohmic AMR, the quantitative relativistic transport theory requires one to combine thecalculated DOS anisotropy with the tunneling or scattering matrix elements, respectively ( Jungwirth et al. , 2003 ;Brey, Tejedor, and Fern ández-Rossier, 2004 ;Giddings et al. , 2005 ; Elsen et al. , 2007 ). Because of the anisotropy of the electronic structure with respect to the magnetization angle the matrixelements may also change when magnetization is rotated. A physically appealing picture has been used to explain the positive sign of the noncrystalline AMR (defined as the relative difference between resistances for current parallel and perpendicular to M) observed in most transition-metal ferromagnets ( Smit, 1951 ;McGuire and Potter, 1975 ). The interpretation is based on the model of the spin-up andspin-down two-channel conductance corrected for perturba- tive spin-orbit coupling effects. In the model most of the current is carried by the light-mass selectrons which expe- rience no spin-orbit coupling and a negligible exchangesplitting but can scatter to the heavy-mass dstates. AMR is then explained by considering the spin-orbit potential which mixes the exchange-split spin-up and spin-down dstates in a way which leads to an anisotropic scattering rate of the currentcarrying sstates ( Smit, 1951 ;McGuire and Potter, 1975 ). Controversial interpretations, however, have appeared in the literature based on this model ( Smit, 1951 ;Potter, 1974 ) and no clear connection has been established between the intuitivepicture of the AMR the model provides and the numerical ab initio transport theories ( Banhart and Ebert, 1995 ;Ebert, Vernes, and Banhart, 1999 ;Khmelevskyi et al. , 2003 ). Among the remarkable AMR features of (Ga,Mn)As are the opposite sign of the noncrystalline component, as compared to most metal ferromagnets, and the sizable crystalline termsreflecting the rich magnetocrystalline anisotropies of (Ga,Mn)As (Baxter et al., 2002 ;Jungwirth et al., 2003 ;Tang et al., 2003 ; Matsukura et al. , 2004 ;Goennenwein et al. , 2005 ;Wang, Edmonds et al. , 2005 ;Limmer et al. , 2006 ;Rushforth et al. , 2007 ). In Fig. 15we show an example of AMR data from a systematic experimental and phenomenological study of the 0 100 200 300 400 Angle of rotation (degree)6.66.76.86.97.0 -101234 0.7 Tlong trans6.76.86.97.07.17.2long(10-3cm) -101234 0.25 TH in (001) plane6.86.97.07.17.2 -101234 0.1 T[110]__ [110]_ [110] [110]_ [110]__ FIG. 15 (color online). Measured longitudinal and transverse in- plane AMR curves at external fields smaller than the saturationfield (0.1 and 0.25 T) and larger than the saturation field (0.7 T).The solid lines represent fits to the experimental data. FromLimmer et al. , 2006 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 871 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014AMR coefficients in (Ga,Mn)As films grown on (001)- and ð113ÞA-oriented GaAs substrates at nonsaturating and saturat- ing in-plane and out-of-plane magnetic fields ( Limmer et al. , 2006 ). In the following we describe the AMR phenomenology in (Ga,Mn)As in more detail and explain the basic microscopicphysics origin of the noncrystalline AMR in (Ga,Mn)As. For simplicity we focus on the AMR in saturating magnetic fields, forMoriented in the plane of the device, and for (Ga,Mn)As films grown on the (001)-GaAs substrate. The phenomenological decomposition of the AMR of (Ga,Mn)As into various terms allowed by symmetry is obtainedby extending the standard phenomenology ( Döring, 1938 )t o systems with the cubic and in-plane uniaxial anisotropy. The corresponding AMR is then phenomenologically described as(Rushforth et al. ,2 0 0 7 ;De Ranieri et al. , 2008 ) Δρxx ρav¼CIcos2ϕþCUcos2ψþCCcos4ψ þCI;Ccosð4ψ−2ϕÞ; (1) where Δρxx¼ρxx−ρav,ρavis the ρxxaveraged over 360° in the plane of the film, ϕis the angle between the magnetization unit vector ˆMand the current I, and ψis the angle between ˆMand the [110] crystal direction. The four contributions are thenoncrystalline term, the lowest order uniaxial and cubic crystalline terms, and a crossed noncrystalline or crystalline term. The purely crystalline terms are excluded by symmetry forthe transverse AMR and one obtains ( Rushforth et al., 2007 ;De Ranieri et al. ,2 0 0 8 ) Δρxy ρav¼CIsin2ϕ−CI;Csinð4ψ−2ϕÞ: (2) Microscopic numerical simulations ( Jungwirth et al. , 2002 , 2003 ;Rushforth et al. , 2007 ;Vyborny et al. , 2009 ) consis- tently describe the sign and magnitudes of the noncrystallineAMR in (Ga,Mn)As materials with metallic conductivities and capture the presence of the more subtle crystalline terms (Jungwirth et al. , 2002 ;Matsukura et al. , 2004 ). Based on the numerical simulations the origin and sign of the noncrystallineAMR in (Ga,Mn)As was qualitatively explained using a simplified model in which carriers, represented by the heavy-hole Fermi surface in the spherical spin-textureapproximation (see Fig. 16), scatter off random Mn impurity potential approximated by ∝ðr1þˆM·sÞ. Here s¼j=3is the carrier spin operator in the spherical approximation with j representing the total angular momentum operator of heavyholes ( j¼3=2), and reffectively models the ratio of non- magnetic (Coulomb and central cell) and magnetic ( p-d kinetic exchange) parts of the Mn-impurity potential (Rushforth et al. , 2007 ;Trushin et al. , 2009 ;Vyborny et al. , 2009 ). The qualitative AMR considerations focus on scattering matrix elements of state with momentum along the current I and, in particular, on the strongest contribution to the transportlifetime which comes from backscattering (see Fig. 16) (Rushforth et al. , 2007 ;Trushin et al. , 2009 ;Vyborny et al. , 2009 ). When neglecting the nonmagnetic part of the impurity potential ( r¼0), nonzero backscattering matrix elementsoccur only for M∥Iand in the notation of Fig. 16they correspond to the elements h→jjxj→iand h←jjxj←i.F o r M⊥I, all backscattering elements h→jjyj→i¼0,h←jjyj→i¼ 0, etc., i.e., the backscattering is completely suppressed. The picture changes when the nonmagnetic part of the Mn- impurity potential is included, as illustrated in Fig. 16for r¼1=2.F o rM∥I, the coherent scattering of the nonmagnetic and magnetic parts interferes constructively or destructively leaving only one of the backscattering elements nonzero (see Fig.16). ForM⊥I, the nonmagnetic and magnetic parts do not interfere and now the nonmagnetic part of the scatteringpotential results in two nonzero backscattering elements (see Fig.16). As a result the resistivity ρ ∥ xxforM∥Iis larger than ρ⊥xx forM⊥Iwhen r¼0andρ∥ xxis smaller than ρ⊥xxwhen r¼1=2. The presence of the nonmagnetic part of the impurity potential can, therefore, flip the sign of the AMR from the positive which is seen in common transition-metal ferromagnets to the negative which is typical of (Ga,Mn)As. The negative sign isobtained in the above simplified model for r>1=ffiffiffiffiffi 20p which is safely satisfied in (Ga,Mn)As ( Rushforth et al. , 2007 ;Trushin et al. , 2009 ;Vyborny et al. , 2009 ). 3. Tunneling anisotropic magnetoresistance The electrical response to changes in the magnetic state is strongly enhanced in layered structures consisting of alter- nating ferromagnetic and nonmagnetic materials. The GMR and TMR effects which are widely exploited in metal spintronics technologies reflect the large difference between resistivities in configurations with parallel and antiparallel polarizations of ferromagnetic layers in magnetic multilayers,or trilayers like spin valves and magnetic tunnel junctions (Gregg et al. , 2002 ;Chappert, Fert, and Dau, 2007 ). The effect relies on transporting spin information between the layers. In M||x ky kx M||y I||x FIG. 16 (color online). Left panel: Cross section (parallel to the kx;kyplane) of the 3D radial spin texture belonging to the two heavy-hole bands of (Ga,Mn)As in a spherical approximation.Right top panel: Nonzero backscattering elements when neglect-ing the nonmagnetic part of the Mn-impurity potential. Thecorresponding AMR has a positive sign. The purely magnetic Mnimpurity is illustrated by a dot with an arrow. Right bottom panel:Nonzero backscattering elements for the same strengths of thenonmagnetic and magnetic parts of the Mn-impurity potential.The corresponding AMR has a negative sign. The combinedionized acceptor and magnetic nature of the Mn impurity is illustrated by a dot with a negative sign and an arrow. (Electrical current I∥x.) From Trushin et al. , 2009 .872 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014(Ga,Mn)As, functional magnetic tunnel junction devices can be built, as demonstrated by the measured large TMR effects (Tanaka and Higo, 2001 ;Brey, Tejedor, and Fern ández- Rossier, 2004 ;Chiba, Matsukura, and Ohno, 2004 ;Chiba et al. , 2004 ;Mattana et al. , 2005 ;Saito, Yuasa, and Ando, 2005 ;Saffarzadeh and Shokri, 2006 ;Elsen et al. , 2007 ;Ohya et al. , 2007 ;Sankowski et al. , 2007 ). Here we focus on the physics of the TAMR which was discovered in (Ga,Mn)As based tunnel devices ( Brey, Tejedor, and Fern ández-Rossier, 2004 ;Gould et al. , 2004 ;Giraud et al. , 2005 ;Rüster, Gould, Jungwirth, Sinova et al. , 2005 ; Saito, Yuasa, and Ando, 2005 ;Ciorga et al. , 2007 ;Elsen et al. , 2007 ;Sankowski et al. , 2007 ). TAMR, like AMR, arises from spin-orbit coupling and reflects the dependence of the tunnel- ing density of states of the ferromagnetic layer on the orientation of the magnetization. The effect does not rely on spin coherence in the tunneling process and requires only one ferromagnetic contact. In Fig. 17we show the TAMR signal which was measured in a ðGa;MnÞAs=AlO x=Au vertical tunnel junction ( Gould et al. , 2004 ;Rüster, Gould, Jungwirth, Girgis et al. , 2005 ). For the in-plane magnetic field applied at an angle 50° off the [100] axis the magnetoresistance is reminiscent of the conven-tional spin-valve signal with hysteretic high-resistance states at low fields and low-resistance states at saturation. Unlike the TMR or GMR, however, the sign changes when the field is applied along the [100] axis. Complementary superconducting quantum interference device (SQUID) magnetization mea-surements confirmed that for the sample measured in Fig. 17 the high-resistance state corresponds to magnetization in the (Ga,Mn)As contact aligned along the [100] direction and thelow-resistance state along the [010] direction, and that this TAMR effect reflects the underlying magnetocrystallineanisotropy between the M∥½100/C138andM∥½010/C138magnetic states of the specific (Ga,Mn)As material used in the study. Since the field is rotated in the plane perpendicular to the current, the Lorentz force effects on the tunnel transport can beruled out. Microscopic calculations consistently showed that the spin-orbit coupling induced density-of-states anisotropies with respect to the magnetization orientation can produce TAMR effects in (Ga,Mn)As of the order ∼1%to∼10% (Gould et al. , 2004 ;Rüster, Gould, Jungwirth, Girgis et al. , 2005 ). All-semiconductor TAMR devices with a single ferromag- netic electrode were realized in p-ðGa;MnÞAs=n-GaAs Zener- Esaki diodes ( Giraud et al. , 2005 ;Ciorga et al. , 2007 ). For magnetization rotations in the (Ga,Mn)As plane ( Ciorga et al. , 2007 ) comparable TAMR ratios were detected as in the ðGa;MnÞAs=AlO x=Au tunnel junction. About an order of magnitude larger TAMR (40%) was observed when magneti-zation was rotated out of the (Ga,Mn)As plane toward the current direction ( Giraud et al. , 2005 ). Several detailed numerical studies have been performed based on microscopic tight-binding or kinetic-exchange mod- els of the (Ga,Mn)As electronic structure and the Landauer-Büttiker quantum transport theory ( Brey, Tejedor, and Fern ández-Rossier, 2004 ;Giddings et al. , 2005 ;Elsen et al. , 2007 ;Sankowski et al. , 2007 ). Besides the Zener-Esaki diode geometry ( Sankowski et al. , 2007 ) the simulations consider magnetic tunnel junctions with two ferromagnetic (Ga,Mn)Ascontacts and focus on comparison between the TMR and TAMR signals in structures with different barrier materials and (Ga,Mn)As parameters ( Brey, Tejedor, and Fern ández- Rossier, 2004 ;Elsen et al. , 2007 ;Sankowski et al. , 2007 ). Figure 18shows the theoretical dependence of the TMR ratio for parallel and antiparallel configurations of the two (Ga,Mn)As contacts and Malong the [100] direction and the TAMR ratio for parallel magnetizations in the (Ga,Mn)Asfilms and Malong the [100] direction and the [001] direction (current direction) in a tunneling device with an InGaAs barrier ( Elsen et al. , 2007 ). The corresponding experimental measurements are shown in Fig. 19. There is an overall agreement between the theory and experiment, seen also intunnel junctions with other barrier materials, showing that the TMR is typically ten times larger than the TAMR. Both the theory and experiment also find that the TMR signal is always positive, i.e., the magnetoresistance increases as the field is swept from saturation to the switching field. The TAMR canhave both signs depending on the field angle but also depending on the parameters of the (Ga,Mn)As film such as the hole concentration and polarization, on the barriercharacteristics, or on the temperature ( Gould et al. , 2004 ; Elsen et al. , 2007 ). At very low temperatures and bias voltages large TAMR signals were observed ( Rüster, Gould, Jungwirth, Girgis et al. , 2005 )i na ðGa;MnÞAs=GaAs =ðGa ;MnÞAs tunnel junction which are not described by the one-body theories of anisotropic tunneling transmission coefficients. The obser- vation was interpreted as a consequence of electron-electron correlation effects near the metal-insulator transition ( Pappert et al. , 2006 ). Large anisotropic magnetoresistance effects were also measured in lateral nanoconstriction devices fabricated in ultrathin (Ga,Mn)As materials ( Rüster et al. , -30 -20 -10 0 102.882.943.003.063.123.183.24 Φ=30° T=20KT=15KT=5KT=1.6K Magnetic Field (mT)(a) (c)k( ecnatsiseR Ω) -30 -20 -10 0 10 20 3 02.922.963.003.04Φ=50° Magnetic Field (mT)2.922.963.003.04 Φ=0° 2.922.963.003.04 Φ=55° Hc2Hc1(b) k( ecnatsiseR Ω) FIG. 17 (color online). (a) Device schematic showing the contact geometry and the crystallographic directions. (b) Hystereticmagnetoresistance curves acquired at 4.2 K with 1 mV biasby sweeping the magnetic field along the 0°, 50°, and 55°directions. Spin-valve-like features of varying widths and signsare clearly visible, delimited by two switching events labeled H c1 andHc2. The magnetoresistance is independent of the bias direction or amplitudes up to 1 meV. (c) TAMR along 30° fortemperatures from 1.6 to 20 K, showing a change of sign of thesignal. The curves are vertically offset for clarity. From Gould et al. , 2004 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 873 Rev. Mod. Phys., Vol. 86, No. 3, July –September 20142003 ;Giddings et al. , 2005 ;Schlapps et al. , 2006 ). The comparison of the anisotropic magnetoresistance signals in the unstructured part of the device and in the nanoconstric-tion showed a significant enhancement of the signal in theconstriction ( Giddings et al. , 2005 ). Subsequent studies of these nanoconstrictions with an additional side gate patterned along the constriction, discussed in detail in Sec. III.B.4 (Wunderlich et al. , 2006 ;Wunderlich, Jungwirth, Irvine et al. , 2007 ;Wunderlich, Jungwirth, Nov áket al. , 2007 ; Schlapps et al. , 2009 ), indicated that single-electron charging effects were responsible for the observed large anisotropic magnetoresistance signals. Before moving on to the (Ga,Mn)As-based field-effect transistors we conclude this section with a remark on the impact of the TAMR discovery in (Ga,Mn)As on spintronicsresearch in other magnetic materials. Ab initio relativistic calculations of the anisotropies in the density of states predicted sizable TAMR effects in transition-metal ferromag- nets ( Shick et al. , 2006 ). Landauer-Büttiker transport theory calculations for a Fe =vacuum =Cu structure pointed out that apart from the density-of-states anisotropies in the ferromagnetic metal itself, the TAMR in the tunnel devices can arise from spin-orbit coupling induced anisotropies ofresonant surface or interface states ( Chantis et al. , 2007 ). Experimentally, several reports of metal TAMR devices havealready appeared in the literature including Fe, Ni, and Co lateral break junctions ( Bolotin, Kuemmeth, and Ralph, 2006 ; Viret et al. , 2006 ) which showed comparable ( ∼10%)l o w - temperature TMR and TAMR signals, Fe =GaAs =Au and Fe=n-GaAs vertical tunnel junctions ( Moser et al. , 2007 ; Uemura et al. , 2009 ) with a ∼1%TAMR at low temperatures reflecting the spin-orbit fields and symmetries at the metal/ semiconductor interface, a Co =Al 2O3=NiFe magnetic tunnel junction with a 15% TAMR at room temperature ( Grigorenko, Novoselov, and Mapps, 2006 ), reports of strongly bias dependent TAMRs in devices with CoFe ( Gao et al. , 2007 ) and CoPt electrodes ( Park et al. , 2008 ), and larger than 100% TAMRs in tunneling devices with an antiferromagnetic IrMn electrode ( Park et al. , 2011 ;Wang et al. , 2012 ). 4. Transistor and chemical potential anisotropy devices As mentioned in the Introduction, (In,Mn)As, (Ga,Mn)As, and (Ga,Mn)(As,P) based field-effect transistors were fabri- cated to demonstrate the electric-field control of ferromag- netism. It was shown that changes in the carrier density anddistribution in thin ferromagnetic semiconductor films due toan applied gate voltage can change the Curie temperature, asillustrated in Fig. 20, and thus reversibly induce the ferro- magnetic-paramagnetic transition ( Ohno et al. , 2000 ;Chiba, Matsukura, and Ohno, 2006 ;Stolichnov et al. , 2008 ;Riester et al. , 2009 ;Sawicki et al. , 2010 ). Another remarkable effect observed in these transistors is the electric-field control of themagnetization orientation ( Chiba et al. , 2003 ,2008 ,2013 ; Chiba, Matsukura, and Ohno, 2006 ;Wunderlich, Jungwirth, Irvine et al. , 2007 ;Olejn íket al. , 2008 ;Stolichnov et al. , 2008 ;Owen et al. , 2009 ;Niazi et al. , 2013 ). This functionality is based on the dependence of the magnetic anisotropies onthe gate voltage, again through the modified charge densityprofile in the ferromagnetic semiconductor thin film.-1000 -500 0 500 10000306090120150As-grown AnnealedTMR (%) Magnetic field (Oe) 0 30 60 90 120 150 180051015 As-grown AnnealedTAMR (%) )°( elgnA1E-3 0,01 0,1020406080100120140 As-grown AnnealedTMR (%) R.A ( ΩΩΩΩ.cm²)(a) (d)M⊥⊥⊥⊥I M//I(b) (c) 0 1 02 03 04 05 00,00,20,40,60,81,0 As-grown AnnealedTMR / TMR(4 K) Temperature (K) FIG. 19 (color online). (a) TMR measurements as a function of the magnetic field at 1 mV and 3 K for a 128μm2junction. (b) TMR measurements as a function of resistance area (RA)product at 3 K for four (un)annealed junctions. (c) TMR at 1 mVas a function of the temperature before and after annealing.(d) TAMR measurements as a function of the magnetic field at1 mV and 3 K. From Elsen et al. , 2007 . FIG. 18 (color online). Calculated (a) TMR values and (b) TAMR values represented as a function of the Fermi and spin splittingenergy for a 6 nm (In,Ga)As barrier with a band offset of450 meV. White lines represent the four bands at the center of theBrillouin zone. Gray lines indicate the Fermi energy for differenthole concentrations. From Elsen et al. , 2007 .874 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014For a spintronic transistor, the magnetoresistance is another key characteristic which should be controllable by the gate electric field. Large and voltage-dependent AMR effectswere reported in Ohmic (Ga,Mn)(As,P) channels with anintegrated polymer ferroelectric gate ( Mikheev et al. , 2012 ) and CB-AMR effects were demonstrated in (Ga,Mn)As SETs ( Wunderlich et al. , 2006 ;Wunderlich, Jungwirth, Irvine et al. , 2007 ;Wunderlich, Jungwirth, Nov áket al. , 2007 ;Schlapps et al. , 2009 ;Ciccarelli et al. , 2012 ), as illustrated in Fig. 21. In the conventional SET, the transfer of an electron from a source lead to a drain lead via a small, weakly coupled island is blocked due to the charging energy of e 2=2CΣ, where CΣis the total capacitance of the island ( Likharev, 1999 ). Applying a voltage VGbetween the source lead and a gate electrode changes the electrostatic energy function of the charge Qon the island to Q2=2CΣþQCGVG=CΣwhich has a minimum at Q0¼−CGVG. By tuning the continuous external variable Q0 toðnþ1=2Þe, the energy associated with increasing the charge Qon the island from netoðnþ1Þevanishes and electrical current can flow between the leads. Changing thegate voltage then leads to CB oscillations in the source-draincurrent where each period corresponds to increasing or decreasing the charge state of the island by one electron. The energy can be written as a sum of the internal, electrostaticcharging energy term and the term associated with, in general,different chemical potentials of the lead and of the island: U¼Z Q 0dQ0ΔVDðQ0ÞþQΔμ=e; (3)where ΔVDðQÞ¼ðQþCGVGÞ=CΣ. The Gibbs energy Uis minimized at Q0¼−CGðVGþVMÞ. The ferromagnetic SETs with (Ga,Mn)As in the transport channel of the transistor ( Wunderlich et al. , 2006 ;Schlapps et al. , 2009 ) were fabricated by trench isolating a side-gated narrow (tens of nm) channel in a thin-film (Ga,Mn)As epilayer. The narrow channel technique is a simple approach to realize a SET and was used previously to produce nonmagnetic thin- film Si and GaAs-based SETs in which disorder potential fluctuations create small islands in the channel without the need for a lithographically defined island ( Kastner, 1992 ; Tsukagoshi, Alphenaar, and Nakazato, 1998 ). The nonuniform carrier concentration produces differences between chemical potentials Δμof the lead and of the island in the constriction. There are two mechanisms through which Δμdepends on the magnetic field. One is caused by the direct Zeeman couplingof the external magnetic field and leads to a CB magneto- resistance previously observed in ferromagnetic metal SETs (Ono, Shimada, and Ootuka, 1997 ). The CB-AMR effect, discovered in the (Ga,Mn)As SETs, is attributed to the spin-orbit coupling induced anisotropy of the carrier chemical potential, i.e., to magnetization orientation dependent differences between chemical potentials of the lead and of the island in the constriction ( Wunderlich et al. , 2006 ). For the CB-AMR effect, the magnetization orientation dependent shift of the CB oscillations is given by V M¼ ½CΣ=CGΔμðMÞ/C138=e. Since jCGVMjhas to be of the order of jej to cause a marked shift in the oscillation pattern, the corresponding jΔμðMÞjhas to be similar to e2=CΣ, i.e., of the order of the island single-electron charging energy. The fact that CB-AMR occurs when the anisotropy in a band structure derived parameter is comparable to an independent scale (single-electron charging energy) makes the effect distinct and potentially much larger in magnitude as compared to the AMR and TAMR. Indeed, resistance variations by more than 3 orders of magnitude were observed in the (Ga,Mn) As SETs. The sensitivity of the magnetoresistance to the orientation of the applied magnetic field is an indication of the anisotropic magnetoresistance origin of the effect. This is confirmed by the observation of comparably large and gate-controlled magnetoresistance in a field-sweep experiment and whenthe saturation magnetization is rotated with respect to the crystallographic axes. The field-sweep and rotation FIG. 20 (color online). Top panel: Schematics of a capacitor with an ultrathin (3.5 nm) (Ga,Mn)As layer. Bottom panel: Exper-imental temperature dependence of the spontaneous moment forselected values of gate voltage. Temperatures at which themoment disappears define the Curie temperature T c, as marked by arrows. From Sawicki et al. , 2010 . FIG. 21 (color online). (a) Electron micrograph of the central part of a (Ga,Mn)As SET device. (b) Polar plot of thesource-drain resistance R sdat 1.6 K showing the strong anisotropy as a function of the magnetization direction. FromSchlapps et al. , 2009 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 875 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014measurements are shown in Figs. 22(c) and 22(d) and compared with analogous measurements of the Ohmic AMR in the unstructured part of the (Ga,Mn)As bar, plotted in Figs. 22(a) and22(b) (Wunderlich et al. , 2006 ). In the unstructured bar, higher or lower resistance states correspond to magnetization along or perpendicular to the currentdirection. Similar behavior is seen in the SET part of the device at, for example, V G¼−0.4V, but the anisotropic magnetoresistance is now largely increased and strongly depends on the gate voltage. The large magnetoresistance signals can also be hysteretic which shows that CB-AMR SETs can act as a nonvolatile memory-transistor element. In nonmagnetic SETs, the CB “on”(low-resistance) and “off”(high-resistance) states can represent logical “1”and“0”and the switching between the two states can be realized by applying a gate voltage, in analogy with a standard field-effect transistor. The CB-AMR SET can be addressed also magnetically with comparable on to off resistance ratios in the electric and magnetic modes. The functionality is illustrated in Fig. 23(Wunderlich, Jungwirth, Irvine et al. , 2007 ). The inset of Fig. 23(a) shows two CB oscillation curves corresponding to two different magnetiza- tion states M0andM1. As illustrated in Fig. 23(b) ,M0can be achieved by performing a small loop in the magnetic field B→B0→0, where B0is larger than the first switching field Bc1and smaller than the second switching field Bc2, andM1is achieved by performing the large field loop B→B1→0, where B1<−Bc2. The main plot of Fig. 23(a) shows that the high-resistance 0 state can be set by either the combinations ðM1;VG0ÞorðM0;VG1Þand the low-resistance 1 state by ðM1;VG1ÞorðM0;VG0Þ. One can therefore switch between states 0 and 1 either by changing VGin a given magnetic state (the electric mode) or by changing the magnetic state at fixed VG(the magnetic mode). Because of the hysteresis, the magnetic mode represents a nonvolatile memory effect. Figure 23(c) illustrates one of the new functionality concepts the device suggests in which low-power electrical manipula- tion and permanent storage of information are realized in one physical nanoscale element. Figure 23(d) highlights the possibility to invert the transistor characteristic; for example, the system is in the low-resistance “1”state at VG1and in the high-resistance “0”state at VG0(reminiscent of an n-type field-effect transistor) for the magnetization M1while the characteristic is inverted (reminiscent of a p-type field-effect transistor) by changing magnetization to M0. Chemical potential shifts in the relativistic band structure of solids have rarely been discussed in the scientific literature. This reflects the conceptual difficulty in describing the chemical potential shifts by quantitative theories, the lack of direct measurements of the effect, and the lack of proposals in which the phenomenon could open unconventional paths in microelectronic device designs. Wunderlich et al. (2006) , Shick et al. (2010) , and Ciccarelli et al. (2012) are among the few who attempted to quantify chemical potential anisotropies with respect to the spin orientation in semiconductor and metal magnets using relativistic model Hamiltonian or full-potential density-functional band structure calculations. The theories could account for chemical potential shifts due to the distortion in the dispersion of the spin-orbit coupled bands but for principle reasons omit possible shifts of the vacuum level 48495051RS[kΩ] B90⊥⊥⊥⊥IB0|| I B [T]-0.50 -0.25 0.00 0.25 0.50(a) (c) (d) B=5 T -90 0 90 180 27044454647 θ[deg]B=5 TRS[kΩ](b) VCVS VG I FIG. 22 (color online). (a) Resistance RS¼VS=Iof the unstructured bar (see schematic diagram) vs up and downsweeps of in-plane magnetic field parallel and perpendicular tothe current direction. (b) R Svs the angle between the current direction and an applied in-plane magnetic field of 5 T, atwhich M∥B. (c) Channel resistance R Cvs gate voltage and down sweep of the magnetic field parallel to current.(d)R Cvs gate voltage and the angle between the current direction and an applied in-plane magnetic field of 5 T. FromWunderlich et al. , 2006 . -0.08 0.00 0.08020RC[MΩ] -BC1-BC2BC2BC1 B90[T]POWER “OFF”Electrical operation mode “READ” : measure RC at VG=VG1“1”(M1) “0”(M0)“WRITE” permanently POWER “ON”POWER “OFF”Electrical operation modeElectrical operation mode “READ” : measure RC at VG=VG1“1”(M1) “0”(M0)“WRITE” permanently POWER “ON” M1 -M1±±±±M0 ±±±±M0VG=VG1=1 . 0 4 V1.00 1.01 1.02 1.03 1.0468101214161820 VG0VG1RC[MΩ] VG[V]electric modeelectric modemagneticmagnetic nonnon--volatilevolatile modemode0.6 0.8 1.002550 RC[MΩ] VG[V]““00”” ““11””MM00(a) (b)(c) MM11 Magnetic non-volatile mode VG=VG1:M0(“0”) ⇔M1(“1”) [[InverseInverse ::VG=VG0:M0(“1”) ⇔M1(“0”) ] M0:B B00BC1<B0<BC2 M1:B B10B1<- BC2Electric mode M=M1:VG0(“0”) ⇔VG1(“1”) [[InverseInverse ::M=M0:VG0(“1”) ⇔VG1(“0”)](d) Magnetic non-volatile mode VG=VG1:M0(“0”) ⇔M1(“1”) [[InverseInverse ::VG=VG0:M0(“1”) ⇔M1(“0”) ] M0:B B00BC1<B0<BC2 M1:B B10B1<- BC2Electric mode M=M1:VG0(“0”) ⇔VG1(“1”) [[InverseInverse ::M=M0:VG0(“1”) ⇔VG1(“0”)](d) c c c c e e e e e e e e FIG. 23 (color online). (a) Two opposite transistor characteristics in a gate-voltage range V(VG0) to 1.04 V ( VG1) for two different magnetization orientations M0andM1; corresponding Coulomb blockade oscillations in a larger range of VG¼0.6to 1.15 V are shown in the inset. Switching between low-resistance (1) andhigh-resistance (0) states can be performed electrically ormagnetically. (b) Hysteretic magnetoresistance at constant gatevoltage V G1illustrating the nonvolatile memory effect in the magnetic mode. (c) Illustration of integrated transistor (electricmode) and permanent storage (magnetic mode) functions in asingle nanoscale element. (d) The transistor characteristic forM¼M 1is reminiscent of an n-type field-effect transistor and is inverted (reminiscent of a p-type field-effect transistor) for M¼M0; the inversion can also be realized in the nonvolatile magnetic mode. From Wunderlich, Jungwirth, Irvine et al. , 2007 .876 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014with respect to band edges, in other words, possible shifts in band lineups in realistic heterostructure systems. In experiments described above and in other related measurements, the magnetic materials have been integrated in a conventional design of a magnetoelectronic device, i.e., embedded in the transport channel, and the chemicalpotential shifts could have been inferred only indirectly fromthe measured data ( Ono, Shimada, and Ootuka, 1997 ; Deshmukh and Ralph, 2002 ;Wunderlich et al. , 2006 ;van der Molen, Tombros, and van Wees, 2006 ;Bernand-Mantel et al. , 2009 ;Schlapps et al. , 2009 ;Tran et al. , 2009 ). One exception is the work discussed in more detail below, which has demonstrated direct measurements of chemical potential shifts in a spin-orbit coupled ferromagnet ( Ciccarelli et al. , 2012 ). The corresponding spintronic device operates without spin currents, i.e., it demonstrates a functionality which goesbeyond the common concepts of spintronics. The devicerepresents an unconventional spin transistor where the chargestate of the transport channel is sensitive to the spin state of itsmagnetic gate. The SET from Ciccarelli et al. (2012) has a micron-scale Al island separated by aluminum oxide tunnel junctions from Alsource and drain leads [Fig. 24(a) ]. It is fabricated on top of an epitaxially grown (Ga,Mn)As layer, which is electricallyinsulated from the SET by an alumina dielectric, and acts asa spin back gate to the SET. By sweeping the externally applied potential to the SET gate ( V g), one obtains the conductance oscillations that characterize the CB as shown in Fig. 24(b) . Because of the magnetic gate a shift is observed in theseoscillations by an applied saturating magnetic field whichrotates the magnetization in the (Ga,Mn)As gate. Figure 24(b) shows measurements for the in-plane ( Φ¼90°) and for theperpendicular-to-plane ( Φ¼0°) directions of magnetization. Alternatively, Fig. 24(c) shows the channel conductance as a function of the magnetization angle Φfor a fixed external potential V gapplied to the gate. The oscillations in Φseen in Fig.24(c) are of comparable amplitude as the oscillations in Vg in Fig. 24(b) . Since the (Ga,Mn)As back gate is attached to a charge reservoir, any change in the internal chemical potential of thegate induced by the rotating magnetization vector causes aninward, or outward, flow of charge in the gate, as illustrated inFig. 24(e) . This change in back-gate charge offsets the Coulomb oscillations [Fig. 24(b) ] and changes the conduct- ance of the transistor channel for a fixed external potentialapplied to the gate [Fig. 24(c) ]. In the case of the SET with the magnetic gate no capacitance scaling factors are required and the chemicalpotential shift may be directly read off as a shift in gatevoltage. This removes a source of systematic error, present inexperiments on the magneto-Coulomb effect ( Ono, Shimada, and Ootuka, 1997 ;Deshmukh and Ralph, 2002 ;van der Molen, Tombros, and van Wees, 2006 ) or chemical potential anisotropy in SETs with the ferromagnet forming part of thetransport channel (lead or island) ( Wunderlich et al. , 2006 ; Bernand-Mantel et al. , 2009 ;Schlapps et al. , 2009 ;Tran et al. , 2009 ), where the gate-voltage shift must be scaled due to the presence of a capacitive divider. In agreement with experiment, the theoretical chemical potential anisotropies in the studied (Ga,Mn)As epilayers withMn doping of several percent are of the order of 10–100μeV (Ciccarelli et al. , 2012 ). So far, the spin-gating technique was employed to accurately measure the anisotropic [and alsoisotropic Zeeman ( Ciccarelli et al. , 2012 )] chemical potential shifts in (Ga,Mn)As. However, the technique can be applied tocatalog these effects in other magnetic materials by the simplestep of exchanging the magnetic gate electrode. 5. Spin torques and spin pumping When spin-polarized carriers are injected into a magnetic region whose moments are misaligned with the injected spinpolarization of the carriers, STTs can act on the magneticmoments ( Ohno and Dietl, 2008 ;Ralph and Stiles, 2008 ). The phenomena belong to an important area of spintronicsresearch focusing on the means for manipulating magnetiza-tion by electrical currents and are the basis of the emergingtechnologies for scalable MRAMs ( Chappert, Fert, and Dau, 2007 ). Apart from STTs in nonuniform magnetic structures, whose research in (Ga,Mn)As is reviewed later in Sec. III.B.6 , experiments in (Ga,Mn)As devices established the presenceof current-induced spin torques in uniform magneticstructures originating from the internal spin-orbit coupling.These current-induced SOT phenomena are reviewed inSec. III.B.7 , and in Secs. III.C.2 andIII.C.3 we discuss the optical counterparts of the STT and SOT which were alsodiscovered in (Ga,Mn)As. A theoretical framework outlined inthis section can be used to highlight the key common anddistinct characteristics of all these spin-torque phenomena(Fern ández-Rossier et al. , 2003 ;Zhang and Li, 2004 ; Vanhaverbeke and Viret, 2007 ;Ralph and Stiles, 2008 ; Nemec et al. , 2012 ;De Ranieri et al. , 2013 ;Tesarova et al. ,-280 -140 0 140 2801.01.52.0 Vg(V)=90 deg =0 degG(S) 0 90 180 270 3601.01.52.0 V g=0 V V g=140 VG(S) (deg) (d)(c) (b) [1-10]M[001] θΦ [110]qVg µ1()MRes. BG AlOx - -+ +Res. BG AlOx - -+ +-+ µ2()M -+(e)Vsd Isd Vg AlOx GaAs(Ga,Mn)AsAl(a) SET SET FIG. 24 (color online). (a) Schematic showing the SET channel separated by AlO xdielectric from the ferromagnetic (Ga,Mn)As back gate (BG). The SET comprises Al leads and island, andAlO xtunnel barriers. (b) Coulomb oscillations for the SET on Ga0.97Mn 0.03As for two different polar angles Φof the magneti- zation. (c) Magneto-Coulomb oscillations shown by the sameSET by varying the angle of magnetization for two different gatevoltages. (d) Magnetization vector with respect to (Ga,Mn)Ascrystal axes. (e) Schematic explaining the spin-gating phenome-non: reorientation of the magnetization from M 1toM2causes a change in the chemical potential of the (Ga,Mn)As BG.This causes charge to flow onto the back gate from the reservoir(Res.). The net effect is to alter the charge on the back gate andtherefore the SET conductance. The externally applied electro-chemical potential on the gate μ ec¼qVgis held constant. From Ciccarelli et al. , 2012 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 877 Rev. Mod. Phys., Vol. 86, No. 3, July –September 20142013 ). At the end of this section we also introduce the Onsager related reciprocal effects to the STT (spin pumping) and to the SOT ( Tserkovnyak et al. , 2005 ;Hals, Brataas, and Tserkovnyak, 2010 ). The framework for describing spin-torque phenomena treats the nonequilibrium spin density of carriers sand magnetization of the ferromagnet as separate degrees offreedom and explores their coupled dynamics. The dilute- moment ferromagnetic semiconductor (Ga,Mn)As is a model system in which the separation is well justified microscopi- cally; magnetization is primarily due to Mn d-orbital local moments while the carrier states near the top of the valenceband (or bottom of the conduction band) are dominated by As porbitals (or Ga sorbitals). The carrier Hamiltonian can be written as H¼H 0þHexþHso; (4) where H0is the spin-independent part of the Hamiltonian, the kinetic-exchange term Hex¼JM·σ; (5) where Jis the exchange-coupling constant (in units of energy × volume), M¼cS ˆM(S¼5=2) is the spin density of Mn local moments, ˆMis the magnetization unit vector, σis the carrier spin operator, and Hsois the spin-orbit coupling Hamiltonian. The current-induced and optical STT phenom- ena are determined by the following dynamics equations for the nonequilibrium carrier-spin density sand for the magnetic moment density M, ds dt¼J ℏs×MþPn−s τs; (6) dM dt¼J ℏM×s: (7) The first term on the right-hand side of Eq. (6)is obtained from the Hamiltonian dynamics, dhσi dt¼1 iℏh½σ;H/C138i; (8) where h/C1 /C1 /C1i represents quantum-mechanical averaging over the nonequilibrium carrier states, hσi¼s, and Hsowas neglected in Hfor the STT effects which are basically nonrelativistic. The second term in Eq. (6)is the rate Pof carriers with spin polarization along a unit vector ninjected from an external polarizer. In the current-induced STT, the external polarizer may be, e.g., an adjacent magnetic layer in a multilayer structure. In the optical STT, Pandnof non- equilibrium photocarrier spins are governed again by the properties of an external polarizer which are the intensity,propagation axis, and helicity of the circularly polarized pump laser pulse. The last term in Eq. (6)reflects a finite spin lifetime of the nonequilibrium carriers in the ferromagnet. Two components of the STT can be distinguished when considering two limiting cases of Eq. (6)(Fern ández-Rossier et al. , 2003 ;Zhang and Li, 2004 ;Vanhaverbeke and Viret, 2007 ;Ralph and Stiles, 2008 ;Nemec et al. , 2012 ). One limitis when the carrier spin lifetime τ s≫τex, where the carrier precession time τex¼ℏ=JcS . In this limit the last term on the right-hand side of Eq. (6)can be neglected and introducing the steady-state solution of Eq. (6)(ds=dt¼0), s¼Pτexðn× ˆMÞ; (9) into Eq. (7)yields the antidamping adiabatic STT ( Berger, 1996 ;Slonczewski, 1996 ), dM dt¼PˆM×ðn× ˆMÞ: (10) (Recall that the form of this torque is the same as the damping term in the Landau-Lifshitz-Gilbert equation.) In this adiabaticSTT the entire spin-angular momentum of the injected carriersis transferred to the magnetization, independent of τ s,τex, and other parameters of the system. The adiabatic STT has been considered since the seminal theory works ( Berger, 1996 ; Slonczewski, 1996 ) on carrier induced magnetization dynam- ics which opened a large field ranging from metal magnetic tunnel junctions switched by the current to tunable oscillators (Ralph and Stiles, 2008 ) and ultrafast photomagnetic laser excitations of ferromagnetic semiconductors ( Fern ández- Rossier et al. , 2003 ;Nemec et al. , 2012 ). In the opposite limit of τs≪τex, the first term on the right- hand side of Eq. (6)can be neglected resulting in the fieldlike nonadiabatic STT ( Zhang and Li, 2004 ), dM dt¼τs τexPðˆM×nÞ. (11) The nonadiabatic STT is perpendicular to the adiabatic STT and only a fraction τs=τexof the injected spin-angular momentum is transferred to the magnetization. For intermedi- ate ratios τex=τs, both the nonadiabatic and adiabatic torques are present and the ratio of their magnitudes (nonadiabatic toadiabatic) is given by β¼τ ex=τs(Fern ández-Rossier et al. , 2003 ;Zhang and Li, 2004 ;Vanhaverbeke and Viret, 2007 ). The nonadiabatic STT plays a crucial role in current-induced DW motion ( Zhang and Li, 2004 ;Metaxas et al. , 2007 ; Mougin et al. , 2007 ;Vanhaverbeke and Viret, 2007 ) and, as we discuss, (Ga,Mn)As is a favorable material for exploring the effects of the nonadiabatic and adiabatic STTs. The SOT is distinct from the STT as it is a relativistic phenomenon in which magnetization dynamics is induced in auniform spin-orbit coupled ferromagnet in the absence of the external polarizer ( Linnarsson et al. , 1997 ;Bernevig and Vafek, 2005 ;Manchon and Zhang, 2008 ,2009 ;Chernyshov et al. , 2009 ;Endo, Matsukura, and Ohno, 2010 ;Garate and MacDonald, 2009 ;Fang et al. , 2011 ;Gambardella and Miron, 2011 ;Liu et al. , 2012 ;Kurebayashi et al. , 2014 ). The Hamiltonian spin dynamics described by Eq. (8)with the H soterm included in the carrier Hamiltonian implies that Eq.(6)is replaced with ds dt¼J ℏs×Mþ1 iℏh½σ;Hso/C138i: (12) The SOT is obtained by introducing the steady-state solution of Eq. (12) into Eq. (7),878 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014dM dt¼J ℏM×s¼1 iℏh½σ;Hso/C138i: (13) In the current-induced SOT the absence of an external polarizer implies that the effect can be observed when electrical current is driven through a uniform magnetic structure ( Linnarsson et al. , 1997 ;Bernevig and Vafek, 2005 ;Manchon and Zhang, 2008 ,2009 ;Chernyshov et al. , 2009 ;Endo, Matsukura, and Ohno, 2010 ;Garate and MacDonald, 2009 ;Fang et al. , 2011 ;Gambardella and Miron, 2011 ;Kurebayashi et al. , 2014 ). The optical SOT analogy of the absence of an external polarizer is that the nonequilibrium photocarriers are excited by helicity indepen- dent pump laser pulses which do not impart angular momen- tum ( Tesarova et al. , 2013 ). The electrical and optical SOTs may differ in the specific contributions to Hsowhich dominate the effect. This can be illustrated considering the Boltzmann linear-response transport theory of the current-induced SOT. Here h/C1 /C1 /C1i represents quantum-mechanical averaging constructed from the equilib- rium eigenstates of Hand with the nonequilibrium steady state entering through an asymmetric redistribution of the occupation numbers of these eigenstates on the Fermi surface due to the applied electrical drift and relaxation. Because of this specific form of the asymmetric nonequilibrium charge redistribution with a conserved total number of carriers, the current-induced SOT requires broken inversion symmetry terms in Hso(Manchon and Zhang, 2008 ,2009 ;Chernyshov et al. , 2009 ;Garate and MacDonald, 2009 ;Miron et al. , 2010 ; Fang et al. , 2011 ). The optical SOT is caused by optical generation and relaxation of photocarriers without an applied drift (without a defined direction of the carrier flow) and without conserving the equilibrium number of carriers in the dark. Therefore, the broken inversion symmetry in the crystal is not required, and inversion symmetric Hsoplus the time-reversal symmetry breaking the exchange-coupling term in the carrier Hamiltonian are sufficient for observing the optical SOT. In the STT, spin-angular momentum is transferred from the carriers to the magnet, applying a torque to the magnetization. Via the STT, the injected spin current is able to excite magnetization dynamics. A reciprocal effect to the STT is the spin-pumping phenomenon in which pure spin current is generated from magnetization precession ( Mizukami, Ando, and Miyazaki, 2001 ;Tserkovnyak et al. , 2005 ). The spin pumping has been measured, e.g., in ferromagnet –normal- metal –ferromagnet GMR structures ( Heinrich et al. , 2003 ; Woltersdorf et al. , 2007 ) or in ferromagnet –normal-metal bilayers ( Saitoh et al. , 2006 ;Czeschka et al. , 2011 ). In the latter structure, the inverse SHE in the spin-orbit coupled paramagnet adjacent to the ferromagnet serves as a spin- charge converter and provides a direct means for detecting the spin-pumping phenomenon electrically. Spin pumping can, therefore, be used not only for probing magnetization dynam- ics in ferromagnets but also spin physics in paramagnets, e.g., for measuring the SHE angles. Magnetization dynamics of ferromagnetic resonance also produces electrical signals in the ferromagnetic layer through galvanomagnetic effects. Experiments in a ðGa;MnÞAs=p-GaAs model system, where sizable galvanomagnetic effects are present, have demon- strated that neglecting the galvanomagnetic effects in theferromagnet can lead to a large overestimate of the SHE angle in the paramagnet. The study has also shown a method to separate voltages of these different origins in the spin- pumping experiments in the ferromagnet-paramagnet bilayers (Chen, Matsukura, and Ohno, 2013 ). The Onsager reciprocity relations imply that, as for the STT and spin pumping, there exists a reciprocal phenomenon of theSOT in which the electrical signal is generated from mag- netization precession in a uniform, spin-orbit coupled mag- netic system with broken spatial inversion symmetry ( Hals, Brataas, and Tserkovnyak, 2010 ;Tatara, Nakabayashi, and Lee, 2013 ). In this reciprocal SOT effect no secondary spin- charge conversion element is required and, as for the SOT, (Ga,Mn)As with broken inversion symmetry in its bulk crystalstructure and strongly spin-orbit coupled holes represents afavorable model system to explore this phenomenon. 6. Current-induced spin-transfer torque In this section we focus on the current-induced STT studies in (Ga,Mn)As. The dilute-moment ferromagnet (Ga,Mn)Ashas a low saturation magnetization, as compared to conven- tional dense-moment metal ferromagnets. Together with the high degree of spin polarization of carriers it implies thatelectrical currents required to excite magnetization by STT in(Ga,Mn)As are also comparatively low. In magnetic tunnel junctions with (Ga,Mn)As electrodes, STT induced switching was observed at current densities of the order 10 4–105Ac m−2 (Chiba et al. , 2004 ), consistent with theory expectations (Sinova, Jungwirth et al. , 2004 ). These are 1 –2 orders of magnitude lower current densities than in the STT experiments in common dense-moment metal ferromagnets. Current-induced DW motion in the creep regime at ∼105Ac m−2current densities was reported and thoroughly explored in perpendicularly magnetized (Ga,Mn)As thin-film devices, shown in Fig. 25(Yamanouchi et al. , 2004 ,2006 , (a) (b)(c) 60 µm2 0 µm 30 nm[110][-110] nominal temperature 100 K20 nm 05 1 0 1 5 2 001020304050 100 K 4.3 x 105 A/cm2 deff (µm) wp (µs)5 µm 3 µs 6 µs9 µs 12 µs 15 µs 18 µsj= 4.3 x 105A/cm2 j= 1.1 x 106A/cm2(I) (II) FIG. 25 (color online). (a) Layout of the device showing the 5μm mesa and step for DW pinning in perpendicular magnetic anisotropy (Ga,Mn)As film. (b) 7μm wide magneto-optical images with a 5μm mesa in the center show that DW moves in the opposite direction to the current independent of the initialmagnetization orientation, and that DW displacement is propor-tional to pulse duration (c). The lowest panel in (b) showsdestruction of the ferromagnetic phase by Joule heating. FromYamanouchi et al. , 2006 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 879 Rev. Mod. Phys., Vol. 86, No. 3, July –September 20142007 ;Chiba et al. , 2006 ). The perpendicular magnetization geometry was achieved by growing the films under a tensile strain on a (In,Ga)As substrate and allowed for a direct magneto-optical Kerr-effect imaging of the magnetic domains,as illustrated in Fig. 25. Alternatively, tensile-strained perpendicularly magnetized films for DW studies were grown on a GaAs substrate with P added into the magnetic film ( Wang et al. , 2010 ;Curiale et al. , 2012 ;De Ranieri et al. , 2013 ). In high crystal quality ðGa;MnÞðAs;PÞ=GaAs epilayers the viscous flow regime was achieved over a wide current range allowing one toobserve ( De Ranieri et al. , 2013 ) the lower-current steady DW motion regime separated from a higher-current precessional regime by the Walker breakdown (WB) ( Thiaville et al. , 2005 ; Metaxas et al. , 2007 ;Mougin et al. , 2007 ). This in turn enabled one to assess the ratio of adiabatic and nonadiabaticSTTs in the current driven DW motion. When the non-adiabatic STT is strong enough that β=α>1, where αis the DW Gilbert damping parameter, the mobility of a DW (velocity divided by the DW driving current) is larger belowthe WB. For β=α<1, on the other hand, the DW mobility is larger above the WB critical current. From the experiments in(Ga,Mn)(As,P) samples, shown in Fig. 26, it was concludedthat 1>β=α≳0.5(De Ranieri et al. , 2013 ), i.e., that the nonadiabatic STT plays a significantly more important role than in conventional transition metals where typically β=α≪1(Zhang and Li, 2004 ). Relatively large values of β¼τex=τs, compared to common dense-moment ferromag- nets, are both due to larger τexin the dilute-moment ferro- magnetic semiconductors and due to smaller τsof the strongly spin-orbit coupled holes in the ferromagnetic semiconductorvalence band ( Adam et al. , 2009 ;Garate, Gilmore et al. , 2009 ; Hals, Nguyen, and Brataas, 2009 ;Curiale et al. , 2012 ; De Ranieri et al. , 2013 ). The combination of low saturation moment and strong spin-orbit coupling has yet another key advantage which is the dominant role of magnetocrystalline anisotropy fields over theshape anisotropy fields. It allows one to control the internalDW structure and stability ex situ by strain relaxation in (Ga,Mn)As microstructures ( Wunderlich et al. , 2007 )o rin situ by a piezoelectric stressor attached to the ferromagneticsemiconductor epilayer ( De Ranieri et al. , 2013 ). As a result, the WB critical current can be tuned ( Roy and Wunderlich, 2011 ) resulting in the observed 500% variations of the DW mobility induced by the applied piezovoltage ( De Ranieri et al. , 2013 ). 7. Current-induced spin-orbit torque Following the theoretical prediction for III-V zinc-blende crystals with broken inversion symmetry ( Bernevig and Vafek, 2005 ), the experimental discovery of the SOT was reported in a (Ga,Mn)As device whose image is shown Fig. 27(a) (Chernyshov et al. , 2009 ). The sample was patterned into a circular device with eight nonmagnetic Ohmic contacts [Fig. 27(a) ]. In the presence of a saturating external magnetic (a) (b) (c) (d) FIG. 26. (color online). (a) Illustration of the steady-state non- equilibrium carrier spin polarization sand corresponding adia- batic STT (STT AD) acting on magnetization min the τs≫τex limit (left) and nonadiabatic STT (STT NA) in the τs≪τexlimit (right). (b) Schematic diagram of the predicted DW velocity as afunction of the driving current in the presence of adiabatic andnonadiabatic STTs and β=α<1orβ=α>1, and of the predicted shift of the WB threshold current j WBfor two values of the in- plane magnetocrystalline constant Ku;1<K u;2, controlled in situ by a piezostressor. (c) Measured DW velocity vs driving currentdensity at piezovoltages −200 orþ200V, strengthening or weakening the [ 1¯10] in-plane easy axis, respectively. Open symbols correspond to the [ 1¯10]-oriented microbar with less internally stable Néel DW and filled symbols to the [110]-oriented microbar with more internally stable Bloch DW. Thecharacter of the measured data, including the shift of the WBthreshold current, implies STTs with β=α<1. (d) Δv DW¼ vDWðþ200VÞ−vDWð−200VÞvs current density illustrates the piezoelectric control of the DW mobility achieved startingfrom lower currents in the [ 1¯10]-oriented microbar with less internally stable DW. From De Ranieri et al. , 2013 . (a) (c) (d)(b) FIG. 27 (color online). (a) Atomic force micrograph of the studied sample with eight nonmagnetic metal contacts. (b) Dia-gram of device orientation with respect to crystallographic axes,with easy and hard magnetization axes marked with dashed anddot-dashed lines, respectively. Measured directions of the H eff field are shown for different current directions. Orientation of the effective SOT field with respect to the current direction for(c) Dresselhaus and (d) Rashba spin-orbit interactions. FromChernyshov et al. , 2009 .880 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014field H, the magnetization of the (Ga,Mn)As sample is aligned with the field. For weak fields, however, the direction of magnetization is primarily determined by magnetic anisotropy. As a small field ( 5<H< 20mT) is rotated in the plane of the sample, the magnetization is realigned alongthe easy axis closest to the field direction. Such rotation of magnetization by an external field is demonstrated in Figs. 28(a) and28(b) . For the current I∥½1¯10/C138, the measured transverse AMR ( R xy) is positive for M∥½100/C138and negative for M∥½010/C138. The switching angles where Rxychanges sign are denoted as φðiÞ Hon the plot. The data can be qualitatively understood if one considers an extra current-induced effectivemagnetic field H eff, as shown in Fig. 27(b) . The symmetry of the measured Heffwith respect to the direction of current is sketched in Fig. 27(c) and this current-induced SOT field has been shown to allow for reversibly switching magnetizationbetween the [010] and [ ¯100] directions at a fixed magnetic field when applying positive and negative current pulses with the current I∥½1¯10/C138, as shown in Fig. 28(c) . It was also demonstrated that the SOT in (Ga,Mn)As can generate a 180°magnetization reversal in the absence of an external magnetic field ( Endo, Matsukura, and Ohno, 2010 ). Apart from the current-induced magnetization switching of a uniform ferro-magnet, the SOT was shown to provide the means fordeveloping an all-electrical broadband FMR technique appli- cable to individual nanomagnets ( Fang et al. , 2011 ). The SOT-FMR was used for determining micromagnetic param-eters of (Ga,Mn)As nanobars which were not accessible byconventional FMR techniques and simultaneously allowed toperform 3D vector magnetometry on the driving SOT fields (Fang et al. , 2011 ;Kurebayashi et al. , 2014 ). The SOT fields of the Dresselhaus and Rashba symmetries shown in Figs. 27(c) , and 27(d) , respectively, can arise in (Ga,Mn)As due to the following broken inversion symmetryterms in the spin-orbit-coupling Hamiltonian: H D;R so¼−3C4½σxkxðϵyy−ϵzzÞ−σykyðϵxx−ϵzzÞ/C138 −3C5½ðσxky−σykxÞϵxy/C138: (14) The first Dresselhaus term is due to the broken inversion symmetry of the host zinc-blende lattice combined with the growth-induced strain in the (Ga,Mn)As epilayer (ϵxx¼ϵyy≠ϵzz) while the second Rashba term combines the zinc-blende inversion asymmetry with a shear strain in theepilayer ( ϵ xy≠0)(Silver et al. , 1992 ;Chernyshov et al. , 2009 ;Stefanowicz et al. , 2010 ;Fang et al. , 2011 ;Kurebayashi et al. , 2014 ).Chernyshov et al. (2009) identified a Dresselhaus SOT field corresponding to a compressively strained (Ga,Mn)As epilayer grown on a GaAs substrate. Fang et al. (2011) observed a sign change of the Dresselhaus SOT field between ðGa;MnÞAs=GaAs and ðGa;MnÞðAs;PÞ= GaAs samples consistent with the change in the growth-induced strain in the epilayer from compressive in the formersample to tensile in the latter sample. A weaker Rashba SOTfield was also observed in these experiments ( Fang et al. , 2011 ). The shear-strain component which yields the Rashba SOT field is not physically present in the crystal structure of(Ga,Mn)As epilayers. It has been introduced, however, inmagnetization and SOT studies to effectively model the in-plane uniaxial anisotropy present in (Ga,Mn)As epilayers(Sawicki et al. , 2005 ;Zemen et al. , 2009 ;Fang et al. , 2011 ). The correspondence between the in-plane Dresselhaus and Rashba spin-orbit Hamiltonian terms in Eq. (14) and the in- plane SOT fields shown in Figs. 27(c) and 27(d) can be understood from Eq. (13) within the Boltzmann transport theory description of the nonequilibrium state. In this semi- classical transport theory, the linear response of the carriersystem to the applied electric field is described by thenonequilibrium distribution function of carrier eigenstateswhich are considered to be unperturbed by the electric field.The form of the nonequilibrium distribution function isobtained by accounting for the combined effects of the carrier acceleration in the field and of scattering. In particular, the nonequilibrium distribution function is used here to evaluatethe current-induced SOT. Equation (13) explicitly shows that the SOT is nonzero only when both the exchange and spin-orbit fields act on the carrier states. However, when evaluating the SOT fromðJ=ℏÞM×hσi , where part of the effect of the exchange field is explicitly factored out in the expression, an approximateform of the SOT can be obtained by considering in hσi eigenstates of the Hamiltonian HwithH exneglected. Since the resulting s¼hσi¼1 VX n;kσn;kgn;k (15) is independent of Mthis approximate form describes a pure fieldlike SOT whose origin is illustrated in Fig. 29for the(a) (b) (c) FIG. 28 (color online). (a), (b) Transverse anisotropic magneto- resistance Rxyas a function of external field direction φHfor H¼10mT and current I¼/C6 0.7mA. The angles φðiÞ Hmark magnetization switchings. (c) Magnetization switches betweenthe [010] and [ ¯100] directions when alternating I¼/C6 1mA current pulses are applied with the current I∥½1¯10/C138. The pulses have 100 ms duration and are shown schematically above thedata curve. R xyis measured with I¼10μA. Adapted from Chernyshov et al. , 2009 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 881 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014Rashba spin-orbit coupling (analogous images apply for the Dresselhaus or another broken inversion symmetryH so). The nonequilibrium spin density in the Hex¼0 approximation is a direct consequence of an electric field and scattering induced redistribution of carriers gn;kon the Fermi surface whose texture of spin expectation values σn;khas a broken inversion symmetry. For the Rashba spin-orbitcoupling, the in-plane nonequilibrium spin polarization is perpendicular to the applied electric field for all crystal directions of the electric field. For the Dresselhaus spin-orbitcoupling the relative angle between the in-plane nonequili-brium spin polarization and the applied electric field dependson the crystal direction of the electric field [see Fig. 27(c) ]. This current-induced spin-polarization phenomenon was discussed in nonmagnetic semiconductors ( Aronov and Lyanda-Geller, 1989 ;Edelstein, 1990 ;Ganichev et al. , 2002 ) prior to the SOT experiments in (Ga,Mn)As. An analogous fieldlike SOT mechanism was subsequently considered in nonmagnetic and ferromagnetic transition-metal bilayers with brokenstructural inversion symmetry at the interface ( Manchon et al. , 2008 ;Manchon and Zhang, 2009 ;Miron et al. , 2010 ). Studies of the SOT in (Ga,Mn)As have identified an addi- tional, antidamping SOT contribution which has a commonmicroscopic origin with the intrinsic SHE ( Kurebayashi et al. , 2014 ). Unlike the above scattering-related fieldlike SOT, described within the semiclassical Boltzmann theory, thepresence of an antidamping SOTwith a scattering-independentorigin is captured by the time-dependent quantum-mechanicalperturbation theory. Here the linear-response theory considersthe equilibrium distribution function and the applied electricfield perturbs the carrier wave functions. This can be visualizedby solving the Bloch equations of the carrier spin dynamics during the acceleration of the carriers in the applied electric field, i.e., between the scattering events, as shown in Fig. 30 (Kurebayashi et al., 2014 ). In the limit of large H excompared to Hsothe spins are approximately aligned with the exchange field in equilibrium. During the acceleration, the field acting on thecarriers acquires a time-dependent component due to H so,a s illustrated in Fig. 30(b) for the Rashba spin-orbit coupling. This yields a nonequilibrium spin reorientation. In the linearresponse, i.e., for small tilts of the spins from equilibrium, thecarriers acquire a time and momentum-independent out-of-plane component, resulting in a net out-of-plane spin densityproportional to the strength of the spin-orbit field and inverseproportional to the strength of the exchange field ( Kurebayashi et al. , 2014 ). As illustrated in Figs. 30(b) and30(c) , the nonequilibrium out-of-plane spin density s zdepends on the direction of the magnetization Mwith respect to the applied electric field. For the Rashba spin-orbit coupling it has a maximum for M(anti) parallel to Eand vanishes for Mperpendicular to E. For a general angle θM−Ebetween MandE,sz∼cosθM−E. The nonequilibrium spin polarization produces an out-of-plane field which exerts a torque on the in-plane magnetization given by Eq. (13). This intrinsic SOT is antidamping like, dM dt¼J ℏðM×szˆzÞ∼M×ð½E׈z/C138×MÞ: (16) For the Rashba spin-orbit coupling, Eq. (16) applies to all directions of the applied electric field with respect to crystalaxes. In the case of the Dresselhaus spin-orbit coupling, thesymmetry of the antidamping SOT depends on the direction of Ewith respect to crystal axes, as seen in Fig. 30(a) . To highlight the analogy between the intrinsic antidamping SOT and the intrinsic SHE ( Murakami, Nagaosa, and Zhang, 2003 ;Sinova, Culcer et al. , 2004 ) the solution of the Bloch equations in the absence of the exchange Hamiltonian term isillustrated in Fig. 30(d) (Sinova, Culcer et al. , 2004 ). In theFIG. 29 (color online). Left panel: Rashba spin texture in equilibrium with zero net spin density. Right panel: Nonequili-brium redistribution of eigenstates in applied electric fieldresulting in a nonzero spin density due to broken inversionsymmetry of the spin texture. (a) (b) (d) (c) FIG. 30 (color online). (a) Rashba and Dresselhaus spin textures. (b) For the case of a Rashba-like symmetry, the out-of- plane nonequilibrium carrier spin density that generates theintrinsic antidamping SOT has a maximum for E(anti)parallel toM. In this configuration the equilibrium effective field Beq eff and the additional field ΔBeff⊥Mdue to the acceleration are perpendicular to each other causing all spins to tilt in the sameout-of-plane direction. (c) For the case of a Rashba-like sym-metry, the out-of-plane nonequilibrium carrier spin density is zeroforE⊥Msince B eq effandΔBeffare parallel to each other. (d) The analogous physical phenomena for zero magnetization induces atilt of the spin out of the plane that has opposite sign for momentapointing to the left or the right of the electric field, inducing in thisway the intrinsic SHE. From Sinova, Culcer et al. , 2004 , and Kurebayashi et al. , 2014 .882 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014SHE case, the sense of the out-of-plane spin rotation depends on the carrier momentum resulting in a nonzero transversespin current but no net nonequilibrium spin density. The antidamping like SOT with the theoretically predicted symmetries was identified in measurements in (Ga,Mn)As, asshown in Fig. 31(Kurebayashi et al. , 2014 ). The all-electrical broadband SOT-FMR technique ( Fang et al., 2011 ) was applied which allowed one to perform 3D vector magnetometry on the driving SOT fields. Since the magnitudes of the measured out-of-plane and in-plane SOT fields are comparable, theantidamping SOT plays an important role in driving themagnetization dynamics in (Ga,Mn)As. The observation of the intrinsic antidamping like SOT in (Ga,Mn)As has direct consequences also for the physics of in-plane current-induced torques in the transition-metal bilayers(Miron et al. , 2011 ;Liuet al. , 2012 ). Here the antidamping like SOT considered at the broken inversion symmetry inter- face can compete with another, conceptually distinct mecha-nism in which the intrinsic SHE in the paramagnet generates aspin current which upon entering the ferromagnet exerts anantidamping STT on the magnetization ( Liuet al. , 2012 ). It has been mentioned above that the nonequilibrium spin density in the intrinsic antidamping SOT scales with thestrength of the spin-orbit field and with the inverse of thestrength of the exchange field. Similarly, the SHE spin current,which takes the role of the spin-injection rate Pin Eq. (9)for the nonequilibrium spin density sin the adiabatic STT, scales with the strength of the spin-orbit coupling in the para-magnetic metal ( Tanaka et al. , 2008 ) and sin the adiabatic STT is inverse proportional to the exchange field [Eq. (9)]. C. Interaction of spin with light 1. Magneto-optical effects Similar to the dc conductivity, the unpolarized finite- frequency absorption spectra ( Burch et al. , 2006 ;Jungwirthet al. , 2007 ,2010 ;Chapler et al. , 2011 ) show signatures of the vicinity of the metal-insulator transition and of strong disorder effects even in the most metallic (Ga,Mn)As materi- als, as illustrated in Fig. 32. Compared to a shallow-acceptor counterpart such as, e.g., C-doped GaAs [see the inset ofFig.32(c) ], the spectral weight in (Ga,Mn)As is shifted from the low-frequency Drude peak to higher frequencies. The ac conductivity scales with the dc conductivity over a broadrange of Mn dopings and does not strongly reflect the spin- dependent interactions in the system. Magneto-optical spectroscopies, on the other hand, provide a detailed probe into the exchange split and spin-orbit coupled electronic structure of (Ga,Mn)As ( Ando et al. , 1998 ,2008 ; Kuroiwa et al. , 1998 ;Beschoten et al. , 1999 ;Szczytko et al. , 1999 ;Komori et al. , 2003 ;Moore et al. , 2003 ;Kimel et al. , 2005 ;Lang et al. , 2005 ;Chakarvorty et al. , 2007 ;Acbas et al. , 2009 ;Tesarova, Nemec et al. , 2012 ;Tesarova, SubrtFIG. 31 (color online). Measured in-plane and out-of-plane SOT fields in (Ga.Mn)As. In-plane spin-orbit field and coefficients ofthe cos θ M−Eand sin θM−Efits to the angle dependence of the out-of-plane SOT field for our sample set. For the in-plane fields,a single sample in each microbar direction is shown (correspond-ing to the same samples that yield the out-of-plane data points).In the out-of-plane data, two samples are shown in eachmicrobar direction. The symmetries expected for the antidampingSOT, on the basis of the theoretical model for the Dresselhausterm in the spin-orbit interaction, are shown by shading. Alldata are normalized to a current density of 10 5Ac m−2. From Kurebayashi et al. , 2014 . FIG. 32 (color online). (a) Infrared absorption of a series of optimized ðGa;MnÞAs=GaAs epilayers with nominal Mn doping x¼0.1%–14% plotted from the measured optical transmissions of the samples ( T) and of the reference bare GaAs substrate ( T0). (b) Real part of the ac conductivity (lines) obtained from themeasured complex conductivity in the terahertz range (points)and from fitting the complex conductivity in the infrared rangeto the measured transmissions. (c) Comparison of the infraredabsorption in as-grown and annealed 4.5% doped sample. Inset:Comparison to GaAs:C samples with carbon doping densities2×10 19and 2×1020cm−3. (d) Height of the (Ga,Mn)As midinfrared absorption peak as a function of Mn doping.(e) Position of the peak inferred from the transmission measure-ments and from the fitted ac conductivities. (f) Zero frequencyconductivities obtained from dc transport measurements and fromextrapolated optical ac conductivities measured in the terahertzrange. From Jungwirth et al. , 2010 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 883 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014et al. , 2012 ;Tesarova et al. , 2014 ). It implies that they can be used as sensitive optical spin-detection tools, as illustrated in Fig. 33(Kimel et al. , 2005 ). For the light propagating in the perpendicular direction to the sample surface the magneto-optical effects can be clas- sified in the following way ( Tesarova et al. , 2014 ): The magnetic circular birefringence (MCB) is given by the realpart of the difference between refractive indices of two circularly polarized modes with opposite helicities and the magnetic circular dichroism (MCD) is given by its imaginarypart. These magneto-optical coefficients are sensitive to theout-of-plane component of the magnetization, are an odd function of M, and represent the finite-frequency counterparts of the AHE. The magnetic linear birefringence (MLB) is givenby the real part of the difference between refractive indices oftwo modes linearly polarized perpendicular and parallel to the magnetization and the magnetic linear dichroism (MLD) is given by its imaginary part. These magneto-optical coeffi-cients are sensitive to the in-plane components of themagnetization, are an even function of M, and represent the finite-frequency counterparts of the AMR. Both the circular and linear magneto-optical effects can cause a rotation (and ellipticity) of the polarization of a transmitted or reflected linearly polarized light. For therotation originating from the MCB and MCD the effects are referred to as the Faraday effect in transmission and the Kerr effect in reflection. For the rotation originating from the MLB and MLD the terminology is not unified across theliterature ( Tesarova et al. , 2014 ); however, it is clearly distinguishable from the Kerr (Faraday) rotation. While the Kerr (Faraday) rotation is independent of the polarizationangle of the incident light, the rotation originating from theMLB and MLD depends on the angle between the light polarization and the in-plane magnetization. There is a direct analogy between this magneto-optical effect and the trans-verse voltage in the noncrystalline off-diagonal AMRdescribed by Eq. (2). The transverse voltage in the latter case and the polarization rotation in the former case both have the ∼sinϕform, where ϕis the angle between the in-plane magnetization and the applied voltage in the transverseAMR case, and between the in-plane magnetization and the incident-light polarization in the case of the MLB and MLD induced rotation. Measurements in Fig. 33(b) used the dependence on the polarization angle to optically detect magnetization switch-ings between [100] and [010] crystal axes in a 2% Mn-doped(Ga,Mn)As sample with a dominant in-plane cubic anisotropy (Kimel et al. , 2005 ). Consistent with the phenomenology of the MLB and MLD induced rotation, the largest signal isobserved when the incident-light polarization is aligned withthe in-plane diagonal crystal axis. Figures 33(c) and33(d) highlight the fact that both the Kerr effect and the MLB and MLD induced rotation can be strong in (Ga,Mn)As for asuitably chosen frequency of the probe laser light. This allowsfor a sensitive optical detection of the in-plane and out-of- plane components of the magnetization. The decomposition of the magneto-optical signal into the MCB and MCD induced rotation due to the out-of-plane magnetization and the MLB and MLD induced rotation due toin-plane magnetization was also employed to quantitativelydetermine the three-dimensional magnetization vector trajec- tory in the time-resolved pump-and-probe magneto-optical measurements in (Ga,Mn)As, as shown in Fig. 34(Tesarova, Nemec et al. , 2012 ). The technique helped to experimentally identify different mechanisms by which photocarriers can induce magnetization dynamics in the pump-and-probe experiments in (Ga,Mn)As. The recombining photocarrierscan heat the lattice and the transient increase of temperaturecan trigger magnetization dynamics or, on much shorter time scales, the photocarriers can directly induce spin torques acting on the magnetization ( Oiwa, Takechi, and Munekata, 2005 ;Qiet al. , 2007 , 2009 ;Takechi et al. , 2007 ;Wang et al. , 2007b ;Hashimoto, Kobayashi, and Munekata, 2008 ; Hashimoto and Munekata, 2008 ;Rozkotov á, Nemec, Horodyska et al. , 2008 ,Rozkotov á,Němec, Tesa řováet al. , 2008 ;Kobayashi et al. , 2010 ;Nemec et al. , 2012 ;Tesarova, Nemec et al. , 2012 ;Tesarova et al. , 2013 ). These effects are reviewed in more detail in Secs. III.C.2 andIII.C.3 . We note that earlier magneto-optical pump-and-probe studies of photo-carriers exchange coupled to local magnetic moments havebeen performed in nonferromagnetic (II,Mn)VI diluted mag- netic semiconductors ( Baumberg et al. , 1994 ;Crooker et al. , 1996 ;Camilleri et al. , 2001 ).-10-50510 -1.0-0.50.00.51.0 1.4 1.6 1.8 2.0 2.2 2.4[010] [100]θ H (2)(4) (1)(3) (c) (d)(a) Eg+EFPolar Kerr effect Magnetic Linear Dichroism Energy (eV)AbsorptionPolar Kerr effect (mrad)MLD (mrad) -200 -100 0 100 200-1.0-0.8-0.6-0.4-0.20.00.20.40.60.8 αmax∆H θ=137°θ=107°θ=33°(2) (3) (1)(4)(b) Magnetic field (mT)Polarization rotation (mrad)H23 H12 T=5 KT=5 K FIG. 33 (color online). (a) Ga 0.98Mn 0.02As sample orientation with respect to the applied magnetic field and the four-stepmagnetization reversal process as consecutive 90° jumps (shownby dotted arrows) between the four easy directions (1) –(4). (b) Field dependences of the magnetic linear dichroism fordifferent angles θbetween the incident polarization and the [100] crystallographic direction, measured at a wavelength ofλ¼815nm. (1) –(4) corresponds to the magnetization directions indicated in (a). H 12andH23are the magnetic field values required for making jumps ð1Þ→ð2Þand ð2Þ→ð3Þ, respec- tively. (c) Spectra of the polar magneto-optical Kerr effect andmagnetic linear dichroism (at θ¼135 ∘); (d) absorption spectrum at 5 K. (c), (d) Fabry-Pérot oscillations in the signal due to thefinite buffer thickness have been removed numerically using abandpass filter. From Kimel et al. , 2005 .884 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 20142. Optical spin-transfer torque A direct observation of a nonthermal photocarrier induced spin torque was reported in a pump-and-probe optical experi-ment in which a coherent spin precession in a (Ga,Mn)As ferromagnetic semiconductor was excited by circularly polar- ized laser pulses at normal incidence ( Nemec et al. , 2012 ). During the pump pulse, the spin-angular momentum of photocarriers generated by the absorbed circularly polarizedlight is transferred to the collective magnetization of the ferromagnet, as described by Eqs. (4)–(11) and predicted in Fern ández-Rossier et al. (2003) andNúñez et al. (2004) . The time scale of the photoelectron precession due to the exchange field produced by the ferromagnetic Mn moments isτ ex∼100fs in (Ga,Mn)As ( Fern ández-Rossier et al. , 2003 ; Nemec et al. , 2012 ). The major source of spin decoherence of the photoelectrons in (Ga,Mn)As is the exchange interaction with fluctuating Mn moments. Microscopic calculations of thecorresponding relaxation time give a typical scale of tens of picoseconds ( Fern ández-Rossier et al. , 2003 ). The other factor that limits τsintroduced in Eq. (7)is the photoelectron decay time which is also approximately tens of picoseconds, asinferred from reflectivity measurements of the (Ga,Mn)Assamples ( Nemec et al. , 2012 ). Within the spin lifetime, the photoelectron spins therefore precess many times around theexchange field of ferromagnetic moments. In the correspond-ing regime of τ s≫τex, the steady-state photoelectron spin polarization is given by Eq. (9), i.e., is perpendicular to both the polarization unit vector of the optically injected carrierspins and magnetization, and the optical STT has the form ofthe adiabatic STT given by Eq. (10), as illustrated in the top inset of Fig. 35. The precession time of holes in (Ga,Mn)As is approximately tens of femtoseconds and the spin lifetime ofholes, dominated by the strong spin-orbit coupling, is esti-mated to be ∼1–10fs (Fern ández-Rossier et al. , 2003 ). Since τ s≲τexfor holes, their contribution in the experiment with circularly polarized pump pulse is better approximated by theweaker torque which has the form of the nonadiabatic STTgiven by Eq. (11) and can be neglected. The experimental observation of the magnetization pre- cession in (Ga,Mn)As excited by the optical STT, with thecharacteristic opposite phases of the oscillations excited bypump pulses of opposite helicities, is shown in the top panelof Fig. 35(Nemec et al. , 2012 ). Since the period of the magnetization precession (0.4 ns) is much larger than thepump-pulse duration, the action of the optical STT is reflectedonly in the initial phase and amplitude of the free precession ofthe magnetization. The decomposition of the magneto-opticalsignal in Fig. 35into MCB and MCD induced rotation due to the out-of-plane magnetization and the MLB and MLDinduced rotation due to in-plane magnetization shows ( Nemec et al. , 2012 ) that the initial tilt of the magnetization is in the out-of-plane direction, as expected from Eq. (10) for the adiabatic STT. The precisely opposite phase of the measuredmagneto-optical signals triggered by pump pulses withopposite helicities, shown in the top panel of Fig. 35, implies that the optical STT is not accompanied by any polarization-independent excitation mechanism. These were intentionallysuppressed in the experiment shown in the top panel of Fig. 35 by negatively biasing an attached piezostressor to the (Ga,Mn)As sample which modified the magnetic anisotropy of theferromagnetic film. At positive piezovoltage, on the other hand,the polarization-independent mechanisms ( Oiwa, Takechi, and Munekata, 2005 ;Qiet al. , 2007 ,2009 ;Takechi et al. , 2007 ; Wang et al. , 2007b ;Hashimoto, Kobayashi, and Munekata, 2008 ;Hashimoto and Munekata, 2008 ;Rozkotov á, Nemec, Horodyska et al. , 2008 ;Rozkotov á,Němec, Tesa řováet al. , 2008 ;Kobayashi et al. , 2010 ) start to act along with the optical STT, as illustrated in the bottom panel of Fig. 35(Nemec et al. , 2012 ). The polarization-independent optical excitation mech- anisms are discussed in the following section. 3. Optical spin-orbit torque In the optical STT reviewed previously, the external source for injecting spin-polarized photocarriers is provided by thecircularly polarized light at normal incidence which yields ahigh degree of out-of-plane spin polarization of injected FIG. 34 (color online). (a) Schematic diagram of the experi- mental setup for a detection of the magnetization precessioninduced in (Ga,Mn)As by an impact of the femtosecond laserpump pulse. Rotation of the polarization plane of reflectedlinearly polarized probe pulses is measured as a function ofthe time delay Δtbetween pump-and-probe pulses. The orienta- tion of magnetization in the sample is described by the in-planeangle φand the out-of-plane angle θ. The external magnetic field H extis applied in the sample plane at an angle φH. (b) Dynamics of the magneto-optical signal induced by an impact of the pumppulse on the sample that was measured by probe pulses withdifferent polarization orientations β. (c) Time evolution of the in- plane magnetization angle δφðtÞ, the out-of-plane angle δθðtÞ, and the magnitude δM sðtÞ=M 0; the dotted line depicts the in- plane evolution of the easy-axis position around which themagnetization precesses. (d) Orientation of magnetization at different times after the impact of the pump pulse; the sample plane is represented by the vertical line and the equilibriumposition of the easy axis is depicted by the gray spot. FromTesarova, Nemec et al. , 2012 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 885 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014photocarriers due to the optical selection rules in GaAs. Since large optical STT requires a large spin lifetime of injectedcarriers, i.e., spin-orbit coupling is detrimental for opticalSTT, the weakly spin-orbit coupled photoelectrons play a keyrole in this case. The optical SOT, on the other hand, originates from spin-orbit coupling of nonequilibrium photocarriers excited by polarization-independent pump laser pulses whichdo not impart angular momentum. Since the effect relies onthe strong spin-orbit coupling, the nonequilibrium photoholesgenerated in the (Ga,Mn)As valence band are essential for the optical SOT. The physical picture of the optical SOT in (Ga, Mn)As is based on the SOT formalism of Eqs. (12) and(13), and on the following representation of the nonequilibriumsteady-state spin polarization of the photoholes ( Tesarova et al. , 2013 ): The optically injected photoholes relax toward the hole Fermi energy of the p-type (Ga,Mn)As on a short (∼100fs) time scale ( Yildirim et al. , 2012 ) and the excitation and relaxation processes create a nonequilibrium excess holedensity in the spin-orbit coupled, exchange-split valence band. The increased number of nonequilibrium occupied hole states, as compared to the equilibrium state in dark, can generate anonequilibrium spin polarization of holes which is misaligned with the equilibrium orientation of Mn moments. This non- equilibrium photohole polarization persists over the time scale of the hole recombination ( ∼ps) during which it exerts a torque on the Mn local moments. Approximately, the non-equilibrium photoholes can be represented by a steady state which differs from the equilibrium state in the dark in that the distribution function has a shifted Fermi level corresponding to the extra density of the photoholes. In this approximation, the nonequilibrium spin polarization of holes which is mis-aligned with the equilibrium orientation of Mn moments, and the corresponding optical SOT, is determined by the hole density dependent magnetocrystalline anisotropy field (Tesarova et al. , 2013 ). The experimental identification of the optical SOT (Tesarova et al. , 2013 ) required to separate this nonthermal photomagnetic effect from the competing thermal excitation mechanism of magnetization dynamics ( Wang et al. , 2006 ; Kirilyuk, Kimel, and Rasing, 2010 ). The absorption of the pump laser pulse leads to photoinjection of electron-hole pairs. The nonradiative recombination of photoelectrons produces a transient increase of the lattice temperature which builds up on the time scale of ∼10ps and persists over ∼1000 ps. This results in a quasiequilibrium EA orientation which is tilted from the equilibrium EA. Consequently, Mn moments in (Ga,Mn)As will precess around the quasiequili- brium EA, as schematically illustrated in Fig. 36(a) , with a typical precession time of ∼100ps given by the magnetic anisotropy fields in (Ga,Mn)As. The EA stays in plane and the sense of rotation within the plane of the (Ga,Mn)As film with increasing temperature is uniquely defined by the different temperature dependences of the in-plane cubic and uniaxial anisotropy fields ( Zemen et al. , 2009 ;Tesarova et al. , 2013 ). In the notation shown in Fig. 36(c) , the change of the in-plane angle δφof the magnetization during the thermally excited precession can be only positive. The optical SOT, illustrated schematically in Fig. 36(b) , acts during the laser pulse (with a duration of 200 fs) and fades away within the hole recombination time ( ∼ps), followed by a free magnetization precession. It causes an impulse tilt of the magnetization which is a signature that allowed us to clearlydistinguish the optical SOT from the considerably slower thermal excitation mechanism. Moreover, the initial optical SOT induced tilt of magnetization can yield precession angles that are opposite to the initial tilt of the magnetization dynamics induced by the slower thermal mechanism. Examples of the direct observation of the thermally governed excitation of magnetization at a lower pump-pulse FIG. 35 (color online). Schematic illustration (top inset) of the optical spin-transfer torque induced by the rate Pof the photo- carrier spin injection along the light propagation axis ˆn(normal to the sample plane). The steady-state component of the non-equilibrium spin density sis oriented in the plane of the sample and perpendicular to the in-plane equilibrium magnetizationvector. The (Ga,Mn)As sample is placed on a piezoelectricstressor (lower inset) which allows one to control the magneticanisotropy in situ . Top panel: Precession of the magnetization induced in (Ga,Mn)As by σ þandσ−circularly polarized pump pulses. The points are the measured rotations of the polarization plane of the reflected linearly polarized probe pulse as a functionof the time delay between pump-and-probe pulses. The experi-ment was performed on the (Ga,Mn)As sample attached to apiezostressor at applied bias U¼−150V for which the σ þand σ−circularly polarized pump pulses produce signals with opposite sign corresponding to the opposite sign of the opticalSTT and no polarization-independent ( σ þþσ−) signal for this piezovoltage. Bottom panel: Same as the top panel for apiezovoltage U¼þ 150V. Here magnetization dynamics is excited by both the optical STT and a polarization-independentmechanism. Adapted from Nemec et al. , 2012 .886 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014intensity 6I0, where I0¼7μJc m−2, and of the excitation at a higher intensity 12I0with a strong contribution from the optical SOT are shown in Fig. 36(d) for a 3% doped (Ga,Mn) As sample ( Tesarova et al. , 2013 ). The distinct features of the optical SOT observed at pump intensity 12I0, namely, the impulse tilt and precession angles inaccessible by thermalexcitations seen at the lower intensity 6I 0, are clearly visible when comparing the two measured magnetization trajectories in Fig. 36(d) . We recall that both dynamical magneto-optical signals shown in Fig. 36(d) are independent of the polarization of pump pulses which distinguishes both the slower thermalmechanism and the fast optical SOT mechanism from theoptical STT. A complete suppression of the thermal mecha- nism and magnetization precession induced solely by the optical SOT was achieved by tuning the micromagnetics of the(Ga,Mn)As film ex situ by doping or in situ by applied magnetic fields ( Tesarova et al. , 2013 ). Magneto-optical pump-and-probe studies in (Ga,Mn)As demonstrated the possibility of studying STT and SOT onthe short time scales achievable by the optical techniques.The relativistic optical SOT should be observable in othersystems including, e.g., antiferromagnetic semiconductors,which unlike their ferromagnetic counterparts can have magnetic transition temperatures well above room temper- ature ( Jungwirth et al. , 2011 ). It is well established thatmagnetocrystalline anisotropies are equally present in spin- orbit coupled antiferromagnets as in ferromagnets and in Sec. III.B.3 we pointed out that the spin-orbit coupling induced anisotropic magnetotransport effects can also be strong in antiferromagnets. The optical SOT belongs to this family of relativistic effects and its exploration in antiferro- magnets may open a new direction of optical spin-torque studies beyond the ferromagnetic semiconductor (Ga,Mn)As. D. Interaction of spin with heat In Sec. III.A we outlined the distinction between the basically nonrelativistic Mott spintronic phenomena, such as the GMR or TMR, which depend on relative magnetization orientations in nonuniform magnetic structures, and the rela- tivistic Dirac effects, such as the AHE, AMR, or TAMR, in uniform spin-orbit coupled magnets. In this section we recall that the research of the relativistic spintronics effects in (Ga,Mn)As has led to seminal results not only in magneto- transport and magneto-optical studies but also in the research of magnetothermopower phenomena. 1. Anomalous Nernst effect In analogy to the AHE, we consider an experimental geometry for detecting the ANE in which the thermal gradient ∇T∥ˆx, magnetization M∥ˆz, and the Nernst signal is the M- antisymmetric electric field E∥ˆy. In nonmagnetic systems in zero magnetic field, the charge current density is given by jx¼σxxEx−αxx∂xT; (17) which for the open circuit geometry ( jx¼0) yields Ex¼αxx σxx∂xT¼Sxx∂xT; (18) where αxxis the diagonal Peltier coefficient and Sxxis the diagonal Seebeck (thermopower) coefficient. In the presence of the ˆz-axis magnetization, an off-diagonal Peltier current is generated resulting in the ANE, jy¼−αyx∂xTþσyxExþσxxEy; (19) and for jy¼0, Ey¼1 σxxðαyx−σyxSxxÞ∂xT¼Syx∂xT; (20) where αxyandSxyare the antisymmetric off-diagonal Peltier and Seebeck coefficients, respectively. Thermoelectric measurements on Hall bars fabricated in ðGa;MnÞAs=ðGa;InÞAs epilayers with perpendicular-to- plane easy axis were performed ( Puet al. , 2008 ) in order to test in a ferromagnet the validity of the Mott relation for the off-diagonal transport coefficients ( Wang et al. , 2001 ), αyx¼π2k2 BT 3e/C18∂σyx ∂E/C19 μ; (21) and to experimentally assess the microscopic mechanism of the AHE and ANE in (Ga,Mn)As. In the same devices, the(a) (c) (d)(b) FIG. 36 (color online). (a) Schematic illustration of the thermally excited precession of magnetization MðtÞaround the transient quasiequilibrium easy axis (EA). M0is the magnetization vector aligned with in-plane equilibrium EA before the pump pulse.(b) Schematic illustration of optical SOT induced by the in-planetransverse component s φof the nonequilibrium hole spin polari- zation. On the time scale of the magnetization precession, optical SOT causes an instantaneous tilt of the magnetization Mðt1Þ which allows one to clearly distinguish optical SOT from theconsiderably slower thermal excitation mechanism. The initialoptical SOT induced tilt of magnetization can yield precessionangles that are inaccessible in the thermally induced magnetiza-tion dynamics. (c) Definition of the coordinate system. (d) Timeevolution of the magnetization vector measured in a (Ga,Mn)Asmaterial with nominal Mn doping x¼3%. The direction of the time increase is depicted by arrows. Magnetization tilt angles δφ and δθare measured with respect to equilibrium EA. From Tesarova et al. , 2013 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 887 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014four thermoelectric coefficients ρxx,ρxy,Sxx, and Sxywere measured which allowed one to directly fit the experimentaldata by S yx¼ρxy ρxx/C18 Tπ2k2 B 3eλ0 λþð1−nÞSxx/C19 : (22) Equation (22)is obtained by introducing the Mott relation (21) into the expression for Syxfrom Eq. (20)and by considering a general power-law dependence of the AHE resistivity on the diagonal resistivity, ρxy¼σyx=ðσ2xxþσ2xyÞ≈σyx=σ2xx¼λMzρnxx: (23) Here the proportionality of the AHE to Mzis factored out explicitly in the power-law dependence, λis the remaining scaling factor [ λ0¼ð∂λ=∂EÞμ], and ρxx¼σxx=ðσ2xxþσ2xyÞ≈1=σxx: (24) The intrinsic AHE is characterized by the off-diagonal conductivity σyxwhich is independent of the scattering lifetime τ, i.e., independent of σxx. This corresponds to the above power-law scaling with n¼2. On the other hand, for the extrinsic skew-scattering AHE, σyx∼τ∼σxx, which corresponds to n¼1. The detection of both the AHE and ANE signals in (Ga,Mn)As Hall-bar samples is illustrated inthe top panels of Fig. 37. The measured ρ xx,ρxy,Sxx, and Sxy could be accurately fitted to Eq. (22) which confirmed the Mott relation between the AHE and ANE in a ferromagnet. Moreover, the inferred values of nfrom the fitting were close to 2 in all measured samples (see bottom panels of Fig. 37). This confirmed the intrinsic origin of the AHE and ANE in(Ga,Mn)As. Using Eq. (20) we can rewrite Eq. (22) as α yx¼σyx/C18 Tπ2k2 B 3eλ0 λþð2−nÞSxx/C19 ; (25) from which we directly obtain that for n¼2the intrinsic, scattering-independent AHE coefficient is accompanied by ascattering-independent ANE coefficient, σ yx¼λMz;αyx¼λ0MzTπ2k2 B 3e: (26) 2. Anisotropic magnetothermopower Besides ANE, the thermoelectric measurements in (Ga,Mn) As also revealed strong AMT signals, in particular, the spin-caloritronic analog of the noncrystalline AMR ( Puet al. , 2006 ). A noncrystalline AMT as high as 6% was measured in the longitudinal direction obeying the cos 2ϕdependence as for the noncrystalline longitudinal AMR, where ϕis the angle between magnetization and the applied electrical (thermal)voltage. Simultaneously, the transverse AMT was alsoobserved, as illustrated in Fig. 38, following the sin 2ϕ dependence of the corresponding transverse AMR coefficient. Experiments in (Ga,Mn)As marked a renewed interest in the AMT phenomenon ( Ky, 1966 ) which was subsequently identified in a broad class of magnetic materials, rangingfrom the strongly spin-orbit coupled uranium pnictides(Wisniewski, 2007 ) to transition-metal based oxides ( Tang et al. , 2011 ;Anwar, Lacoste, and Aarts, 2012 ), and nanowires and thin films of elemental transition-metal ferromagnets (Anwar, Lacoste, and Aarts, 2012 ;Mitdank et al. , 2012 ). 3. Tunneling anisotropic magnetothermopower Similar to uniform magnetic films, in the Ohmic GMR multilayers electrical and heat transport measurements can be performed in macroscopic samples in the current-parallel-to- plane geometry. This allowed one to observe the GMT effect(Sakurai et al. , 1991 ) shortly after the discovery of the GMR (Baibich et al. , 1988 ;Binasch et al. , 1989 ) in the same type of transition-metal-multilayer samples and to show that switch-ing from parallel to antiparallel magnetization configurations can lead to comparatively large changes in the thermopower (Sakurai et al. , 1991 ). Magnetothermopower measurements are significantly more challenging in the perpendicular-to-plane geometry of the-40 0 40-808 -30 0 30-1.50.01.5-30 0 30-1.50.01.5 -20 0 20-10010-40 0 40-606 -40 0 40-202 -30 0 30-505 -200 0 200-101 B (mT) B (mT)Syx (V / K)0 -66Syx (V / K)xy(m ) x=0.04* -2002002 x=0.05-20 AHEANEx=0.05* 2 -2x=0.07*-66 020K 5 0 38K -5 60K 1 0 -1-505 AHEANE 78K 0.000.0205 0 1 0 0 05 0 1 0 00.000.080 100 200 0.000.05 01 0 0 2 0 00.00.2x=0.05n=1.95 x=0.04* n=1.96 T (K)yx (A / m*K)x=0.05*n=1.82 x=0.07* n=2.0410 0.04 FIG. 37 (color online). Top eight panels: AHE and ANE loops at T¼10K for different samples (left column) and at different temperatures for the 4% annealed sample (right column). In theleft column, ANE data of 0.04*, 0.05*, and 0.07* samples weremultiplied by −1(* means that the sample was annealed). Bottom four panels: zero-field ANE coefficient. The solid lines are thebest fits using Eq. (22)[or equivalently Eq. (25)], and the dashed curves are the best fits with n¼1. Adapted from Puet al. , 2008 .888 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014magnetic tunnel junctions and the TMT effect was observed in transition-metal tunnel devices ( Liebing et al. , 2011 ;Walter et al. , 2011 ) more than 15 years after the discovery of the TMR ( Miyazaki and Tezuka, 1995 ;Moodera et al. , 1995 ). Similar to the electrical transport, the magnetothermopowerin the tunneling regime is much more closely related to theexchange-split electronic structure of the ferromagnets than inthe Ohmic regime of the GMR multilayers and correspond-ingly can be in principle much stronger in the tunnelingdevices ( Czerner, Bachmann, and Heiliger, 2011 ;Liebing et al. , 2011 ). The origin of the TMT effect is schematically illustrated in Fig.39(Walter et al. , 2011 ). Unlike electrical conductance of the tunneling device, G¼ e2 hZ TðEÞ½−∂EfðE;μ;TÞ/C138dE; (27) which in the linear response is governed by the transmission function TðEÞmultiplied by the derivative of the electron occupation function ∂EfðE;μ;TÞat temperature Tand electrochemical potential μ, the Seebeck coefficient, S¼−R TðEÞðE−μÞ½−∂EfðE;μ;TÞ/C138dE eTRTðEÞ½−∂EfðE;μ;TÞ/C138dE; (28) reflects the asymmetry in the energy dependence of the transmission around the chemical potential. As shown inFig. 39, the Seebeck coefficient is the geometric center of TðEÞ½−∂ EfðE;μ;TÞ/C138. When this changes from the parallel tothe antiparallel magnetization configurations the correspond- ing Seebeck coefficients are different in the two configurationsresulting in the TMT. The relativistic counterpart of the TMT in a tunnel junction with only one magnetic electrode is the TAMT. Observationsof the TMT ( Liebing et al. , 2011 ;Walter et al. , 2011 ) and TAMT ( Naydenova et al. , 2011 ) effects were reported independently and simultaneously and, reminiscent of the discovery of the TAMR ( Gould et al. , 2004 ), the TAMT was first identified in a (Ga,Mn)As-based tunnel junction(Naydenova et al. , 2011 ). The experiment was performed while rotating the magnetization in the plane of the (Ga,Mn) As layer, i.e., always perpendicular to the applied temperaturegradient across the tunnel junction. As shown in Fig. 40, four-4 -2 0 2 4 H (X102 Oe)IIV IIIII (c)ΦH(deg) Sxy (µV / K) -50-30-553050 20 40 60 Temperature (K) M Sxy / Sxx 01 (d) a.u.RH/ R(b)M-Txy IIII IV II [110] H -1 0 1-20020 H (X102 Oe)-15015 IIIIV II IRH (Ω)Sxy (µV/K) (a) FIG. 38 (color online). (a) Transverse AMT Sx;yand transverse AMR RHin a 3.9% Mn-doped (Ga,Mn)As. (b) The relative orientation of −∇T,Mand magnetic field H. The four directions marked as I, II, III, and IV are easy directions of M. (c) Angular dependence of the transverse AMT. (d) Comparison of Sxy=Sxx andRH=R, and sample magnetization Mmeasured by SQUID. Note that we use the terms transverse AMT and transverse AMRinstead of the alternative planar Nernst effect and planar Halleffect ( Puet al. , 2006 ) to clearly distinguish the fact that the effects shown here are the symmetric off-diagonal coefficientseven in M. From Puet al. , 2006 .FIG. 39 (color online). In magnetic tunnel junctions, thermal differences in the electron distributions and strong asymmetryin the spin-dependent tunneling channels are depicted. TðEÞ is the transmission of the full tunnel junction, for whicheither the ferromagnetic electrodes can be a highly spin-polarized half-metal or the combination of the barrier and theferromagnet exhibits half-metallic characteristics. The functionTðEÞ½−∂ EfðE;μ;TÞ/C138is given in a darker color. The thick line marks the resulting value of the geometric center determining theSeebeck coefficient in the parallel magnetization S Pand anti- parallel magnetization SAPof the electrodes. Note that we use the term TMT instead of the alternative magneto-Seebeck effect todistinguish it clearly from the spin-Seebeck effect discussed inSec. III.D.4 . Adapted from Walter et al. , 2011 . FIG. 40 (color online). Thermovoltage in a ðGa;MnÞAs=i− GaAs =GaAs∶Si tunnel junction as a function of the magnetiza- tion angle. 0 is along the [010] crystal axis. From Naydenova et al. , 2011 .T. Jungwirth et al. : Spin-dependent phenomena and device concepts … 889 Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014equivalent minima close to the [100] and [010] crystal axes and two sets of local maxima were observed. The symmetry of the observed TAMT reflects the competition of in-plane cubic and uniaxial magnetocrystalline anisotropies in the (Ga,Mn)As epilayer. The TAMT phenomenon originates from thechanges in the energy dependence of the tunneling density of states when changing the angle of the magnetization with respect to crystal axes, i.e., has the same spin-orbit-coupledband structure origin as magnetocrystalline anisotropies and the TAMR. 4. Spin-Seebeck effect Among the most intriguing spin-caloritronics effects is the spin-Seebeck effect ( Uchida et al. , 2008 ,2010 ;Jaworski et al. ,2010 ;Sinova, 2010 ;Bauer, Saitoh, and van Wees, 2012 ). Instead of directly generating electrical voltages from thermalgradients, as was the case of the above discussed magneto-thermopower effects, in the spin-Seebeck effect it is primarilythe difference between spin-up and spin-down chemicalpotentials μ ↑−μ↓, which is induced by the applied thermal voltage in a ferromagnet. An appealing picture was proposedfollowing the first experimental observation of the spin-Seebeck effect in NiFe in which the ferromagnet functionslike a thermocouple, but in the spin sector ( Uchida et al. , 2008 ). In this picture, instead of two different charge Seebeck coefficients in two metals forming the thermocouple, it is thedifferent carrier scattering and density and the correspondingSeebeck coefficient in the two spin channels which producethe nonzero difference μ ↑−μ↓. In this seminal work and in the subsequent experiments, the SHE in attached nonmagnetic electrodes was employed toconvert the difference in spin-dependent chemical potentialsinto electrical voltages ( Uchida et al. , 2008 ,2010 ;Jaworski et al. , 2010 ). Specifically, jμ ↑−μ↓jdecreases in the non- magnetic electrode from the interface with the ferromagnetalong the vertical direction. This results in a vertical spincurrent in the nonmagnetic electrode which is converted intoan in-plane electrical voltage via the SHE. Experiments in which the transition-metal ferromagnet was replaced with the layer of a metallic (Ga,Mn)As ( Jaworski et al. , 2010 ) ruled out the original picture of longitudinal diffusion of electrons in the two spin channels over macro-scopic distances in the ferromagnet. As shown in Fig. 41, the same electrical signals were detected on the SHE electrodesafter scratching out the conductive (Ga,Mn)As film in themiddle of the sample. The nonlocal character of the observedspin-Seebeck effect, i.e., the dependence of the measured SHEvoltage on the position of the electrode along the sample, hasbeen extensively discussed since the experiments in (Ga,Mn)As and the parallel observation of the spin-Seebeck effect in aferromagnetic insulator ( Uchida et al. , 2010 ). It has been argued that phonons or magnons in the ferromagnet-substratestructure may be responsible for the nonlocality of the spin-Seebeck effect ( Bauer, Saitoh, and van Wees, 2012 ;Tikhonov, Sinova, and Finke ľstein, 2013 ). IV. SUMMARY We have reviewed several areas of the rich physics of spintronics phenomena and device concepts explored in theferromagnetic semiconductor (Ga,Mn)As. The most exten-sively studied transport characteristics of (Ga,Mn)As arethe spin-orbit coupling related magnetoresistance effects.Experiments and calculations in (Ga,Mn)As have providedan unprecedented physical insight into the anomalous Halleffect which prompted a renewed interest and experimentaldiscovery of the spin Hall effect. Anisotropic magnetoresist- ance phenomena have been identified in (Ga,Mn)As-based tunneling devices and in devices sensing the anisotropy of thechemical potential. Apart from these direct magnetoresistancephenomena, (Ga,Mn)As has become a fruitful model systemfor exploring the inverse magnetotransport phenomena, i.e.,the current-induced spin torques. The studies have providednew insight into spin-transfer torques in domain walls and led(a) (c)(b) FIG. 41 (color online). Top panel: Measurement geometry of the spin-Seebeck effect. (a) Transverse voltage Vyas a function of the applied field Bfrom the strip contact 0.3 mm above the scratch (star) with an applied ΔTxof 0.63 K. (b) Spatial dependence of the spin-Seebeck coefficient Sxybefore and after the scratch. The scratched region is indicated by the shaded region. (c) Temper-ature dependence of S xyafter the scratch at various positions along the sample. Adapted from Jaworski et al. , 2010 .890 T. Jungwirth et al. : Spin-dependent phenomena and device concepts … Rev. Mod. Phys., Vol. 86, No. 3, July –September 2014to the discovery of the current-induced spin-orbit torques in uniform magnets. Moreover, optical counterparts of both the nonrelativistic spin-transfer and the relativistic spin-orbittorques have been identified in (Ga,Mn)As, allowing one to study these phenomena on time scales attainable in the optical pump-and-probe experiments. (Ga,Mn)As-based research hasalso made seminal contributions to the field of spin calori- tronics by discovering the Ohmic and tunneling anisotropic thermopower effects and helping to elucidate the origin of the spin-Seebeck effect. It is likely that (Ga,Mn)As and related ferromagnetic semi- conductors will continue to inspire new avenues of magnetic materials and spintronics research in the future. Many studies,in particular, of the relativistic phenomena in (Ga,Mn)As may become directly relevant to room-temperature magnetic systems with strong spin-orbit coupling and may thereforelead to new technological applications, independent of the existing limits of the Curie temperature in the ferromagnetic semiconductors. This knowledge transfer applies to room-temperature magnetic systems which include not only the conventional transition-metal ferromagnets but also, e.g., a class of metal and semiconductor antiferromagnets with high Néel temperatures. LIST OF SYMBOLS AND ABBREVIATIONS ABE Aharonov-Bohm effect AHE Anomalous Hall effect AMR Anisotropic magnetoresistance AMT Anisotropic magnetothermopower ANE Anomalous Nernst effect CB Coulomb blockade DOS Density of states DW Domain wall FMR Ferromagnetic resonance GGA Generalized gradient approximations GMR Giant magnetoresistance GMT Giant magnetothermopower LT-MBE Low-temperature molecular beam epitaxy MCB Magnetic circular birefringence MCD Magnetic circular dichroism MLB Magnetic linear birefringence MLD Magnetic linear dichroism MRAM Magnetic random access memory SET Single-electron transistor SHE Spin Hall effect SOT Spin-orbit torque SQUID Superconducting quantum interference device STT Spin-transfer torque SWR Spin-wave resonance TAMR Tunneling anisotropic magnetoresistance TAMT Tunneling anisotropic magnetothermopower TBA Tight-binding approximationTMR Tunneling magnetoresistance TMT Tunneling magnetothermopower UCF Universal conductance fluctuations WB Walker breakdown WL Weak localization ACKNOWLEDGMENTS We acknowledge support from the ERC Advanced Grant No. 268066, from the Praemium Academiae of the Academyof Sciences of the Czech Republic, from the Ministry ofEducation of the Czech Republic Grant No. LM2011026, andfrom the Czech Science Foundation Grant No. 14-37427G. 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PhysRevB.90.094401.pdf

PHYSICAL REVIEW B 90, 094401 (2014) Surface-acoustic-wave-driven ferromagnetic resonance in (Ga,Mn)(As,P) epilayers L. Thevenard,1,2,*C. Gourdon,1,2J. Y . Prieur,1,2H. J. von Bardeleben,1,2S. Vincent,1,2L. Becerra,1,2 L. Largeau,3and J.-Y . Duquesne1,2 1CNRS, UMR7588, Institut des Nanosciences de Paris, 4 place Jussieu, 75252 Paris, France 2Sorbonne Universit ´es, UPMC Universit ´e Paris 06, UMR7588 4 place Jussieu, 75252 Paris, France 3Laboratoire de Photonique et Nanostructures, CNRS, UPR 20, Route de Nozay, 91460 Marcoussis, France (Received 9 May 2014; revised manuscript received 7 July 2014; published 2 September 2014) Surface acoustic waves (SAW) were generated on a thin layer of the ferromagnetic semiconductor (Ga,Mn)(As,P). The out-of-plane uniaxial magnetic anisotropy of this dilute magnetic semiconductor is verysensitive to the strain of the layer, making it an ideal test material for the dynamic control of magnetizationvia magnetostriction. The amplitude and phase of the transmitted SAW during magnetic field sweeps showeda clear resonant behavior at a field close to the one calculated to give a precession frequency equal to theSAW frequency. A resonance was observed from 5 to 85 K, just below the Curie temperature of the layer.A full analytical treatment of the coupled magnetization/acoustic dynamics showed that the magnetostrictivecoupling modifies the elastic constants of the material and accordingly the wave-vector solution to the elasticwave equation. The shape and position of the resonance were well reproduced by the calculations, in particularthe fact that velocity (phase) variations resonated at lower fields than the acoustic attenuation variations. Wesuggest one reinterpret SAW-driven ferromagnetic resonance as a form of resonant, dynamic, delta- Eeffect, a concept usually reserved for static magnetoelastic phenomena. DOI: 10.1103/PhysRevB.90.094401 PACS number(s): 72 .55.+s,75.78.−n,75.50.Pp,62.65.+k I. INTRODUCTION Magnetostriction is the interaction between strain and magnetization, which leads to a change in a magnetic sample’sshape when its magnetization is modified [ 1]. The opposite effect, inverse magnetostriction, whereby magnetization canbe changed upon application of a strain, is particularlyrelevant to magnetic data storage technologies as a possibleroute towards induction-free data manipulation when useddynamically. It has been proposed for magnetization switchingthrough resonant [ 2,3] or nonresonant processes [ 4,5], the latter possibly at play in early results of surface acoustic wave(SAW)-induced lowering of coercivity in Galfenol films [ 6]. In the case of precessional (resonant) switching, two featuresare necessary: sizable magnetoacoustic coupling (to triggerprecession), and a highly nonlinear system (to force wide,noncircular precession needed for magnetization reversal).The first point has already been addressed in ferromagneticmetals, since the 1960s by ac electrical excitation of strainwaves [ 7–12], and more recently by optical excitation of acoustic pulses [ 13,14]. The electrical approach consists in the radio-frequency (rf) excitation of a piezoelectric emitter. Usingthis technique, the triggering of magnetization precession bysurface or bulk acoustic waves (BAWs) has been extensivelystudied in Ni-based films. Elegant data has also been obtainedmore recently on yttrium iron garnet, where BAWs were usedto build a magnetic field tunable acoustic resonator [ 12]. In this work, we evidence magnetoacoustic coupling in a different type of material, a dilute ferromagnetic semicon-ductor (DFS). The low Curie temperature (100–180 K) ofthese compounds makes applications somewhat remote fornow, but their magnetization precession frequencies are closeto accessible SAW frequencies (GHz) and their small and *thevenard@insp.jussieu.frtunable magnetic anisotropy make them good candidates to testSAW-assisted magnetization switching [ 2]. Moreover thanks to the semiconducting nature of host lattice [ 15,16], their magnetic properties can easily be band engineered and theirmagnetostrictive coefficients vary strongly with temperature,making them a good test-bench material to develop andvalidate theoretical models. In this paper, we show experimental evidence of SAW- driven ferromagnetic resonance in a thin film of DFS, in ourcase (Ga,Mn)(As,P). Both acoustic attenuation and velocityvariations are monitored in the time domain. Our experimentalapproach differs from previous work on metals in that wemainly use the temperature dependence of the effect to proveits resonant nature, as opposed to using different geometricalconfigurations (angle between magnetic field and SAW wavevector [ 9,11]). We solve the coupled magnetization and elastic dynamics equations and determine with a good match to theexperimental data (Sec. IV) the expected resonance fields versus temperature and acoustic resonance shape. Here ourmain addition to the existing theoretical literature on the topicis to derive the form of a depth-decaying SAW traveling on thesurface of a cubic medium, and to conclude on a very explicit dependence of the elastic constants on magnetization orienta-tion and magnetoelastic coefficients. This ultimately leads usto reinterpret SAW-driven ferromagnetic resonance (FMR) asa form of resonant, dynamic, delta- Eeffect, a concept usually reserved for static magnetoelastic phenomena [ 17]. II. EXPERIMENTAL METHODS Ad=50-nm-thick layer of (Ga 1−x,Mnx)(As 1−y,Py)w a s grown by molecular beam epitaxy. After a 1 h/250◦C anneal, its Curie temperature reached Tc=105 K and its active Mn concentration xeff≈3.5%. Since GaAs is only weakly piezoelectric, a 70 /250 nm bilayer of SiO 2/ZnO was sputtered onto the magnetic layer. Care was taken to keep the substrate 1098-0121/2014/90(9)/094401(8) 094401-1 ©2014 American Physical SocietyL. THEVENARD et al. PHYSICAL REVIEW B 90, 094401 (2014) FIG. 1. (Color online) Structure of the sample (not to scale). 50 nm ferromagnetic epilayer, 70 nm SiO 2buffer, and 250 nm piezoelectric ZnO. The IDTs have an aperture of 1 mm and are separated by 2 mm, but the effective length of the delay line is taken center to center of the IDTs, i.e., l=2.3 mm. Upper left: Definition of the ( x,y,z )a n d( 1 ,2,3) reference frames. holder at relatively low temperature (150◦C) during the ZnO deposition so as to not further anneal the magnetic layer. Thephosphorus ( y≈4%) was necessary to induce tensile strain in the layer, in order to obtain a dominantly uniaxial magneticanisotropy [ 16,18], spontaneously aligning the magnetization perpendicular to plane. The resulting lattice mismatch of thelayer to the substrate was of −0.161%. Cr/Au interdigitated transducers (60 pairs of fingers, thick- ness 10 /80 nm) were then evaporated and a window opened in the ZnO layer between the two IDTs (Fig. 1). The emitter (IDT 1) was excited by 550 ns bursts of rf voltage modulated at 1 kHz. After propagation along the [110] direction, the SAWwas detected by the receiver IDT 2and the signal was acquired with a digital oscilloscope over typically 4000 averages. Thistime-domain technique allowed us to (i) verify that the SAWswere indeed generated/detected in the sample, and (ii) clearlyseparate the antennalike radiation of IDT 1(traveling at the speed of light), from the acoustic echo (traveling at theRayleigh velocity), as shown in Fig. 2(a). The transit time lies around τ=693 ns, which immediately gives an experimental estimation of the Rayleigh velocity V r≈2886 m s−1.T h e transfer function of the device exhibited the typical band-passbehavior centered at the 549 MHz resonance frequency. The FIG. 2. (Color online) (a) Receiving IDT signal: The electro- magnetic field radiated by the emitter is shortly followed bythe transmitted surface acoustic wave. T=120 K, 549 MHz. (b) Acoustic attenuation changes and relative velocity variations at T=80 K. The opposite of /Delta1/Gamma1has been plotted in order to highlight the different resonant field from the velocity variations.FIG. 3. (Color online) (a) Variation of acoustic attenuation and (b) relative velocity change of the SAW between 5 and 90 K forthe field applied in plane, along the SAW wave vector. Insets: Field sweeps with the field applied perpendicular to plane at T=10 K. power applied to the IDT 1was of +20 dBm (100 mW) o na5 0 /Omega1load (30 dB conversion factor), resulting in an approximate strain [ 19]o fεxx≈− 2×10−5andεzz≈ 6×10−6. The excitation frequency was ω/2π=549 MHz, with the corresponding wavelength /Lambda1SAW=5μm. Unless specified, the field was applied in the plane of the sample, along the SAW wave vector, i.e., along a hardmagnetic axis. A phase detection scheme then yielded theamplitude Aand the phase φ=ωτof the transmitted SAW. The phase variations /Delta1φwere converted into relative velocity variations using /Delta1V/V 0=/Delta1φ ωτ0. The attenuation changes were computed using /Delta1/Gamma1=−20 llogA A0.A0is an arbitrary reference amplitude. l=2.3 mm and τ0=797 ns are the IDTs’ center- to-center distance and the corresponding transit time. III. EXPERIMENTAL RESULTS A typical sweep at T=80 K is shown in Fig. 2(b). Acoustic attenuation and velocity variations were both identical at lowand high fields, but showed a clear feature at a particular field,hereafter called the resonance field. The resonance disappearedabove 90 K. Measurements down to 5 K showed that theamplitude of the effect steadily increased with decreasingtemperature (Fig. 3). The resonance field was not, however, monotonous with temperature, lying within 35–94 mT witha maximum at 30–40 K. The resonance width followed thesame trend, within the bounds 9–17 mT. All curves shared thefollowing features: a fairly symmetrical, nonhysteretic reso-nance, with the velocity variations systematically resonating 094401-2SURFACE-ACOUSTIC-W A VE-DRIVEN FERROMAGNETIC . . . PHYSICAL REVIEW B 90, 094401 (2014) at a lower field than the amplitude variations. The maximum variation of acoustic attenuation, /Delta1/Gamma1=8.5d Bc m−1was observed at T=5 K. It remains weak compared to the value of 200 dB cm−1measured at 2.24 GHz on a similar device on nickel [ 10]. This is due to both the higher SAW frequency used by these authors, as the amplitude variationsare directly proportional to ω(see Ref. [ 11], for instance), and the much lower magnetostrictive constants found in DFS.These are defined as the fractional change in sample length asthe magnetization increases from zero to its saturation valueand their maximum values lie around |λ 100|≈9×10−6for (Ga,Mn)As [ 20] andλ100≈50×10−6for nickel [ 1]. IV . MODEL We have shown above that at a particular applied field, the transmitted SAW was slightly absorbed (by a 19% decreasein amplitude at 5 K), and delayed (by about...90 ps) throughits interaction with the magnetization of the (Ga,Mn)(As,P)layer. To confirm that this is indeed SAW-driven ferromagneticresonance, we calculate the expected resonance fields andshapes. Microscopically, the resonance may be seen as thecrossing of magnon and phonon dispersion curves at the wavevector imposed by the IDTs, k SAW. Macroscopically, the total energy of the system may then be written [ 21]a st h es u mo fa purely elastic contribution W, a purely magnetic energy (mag- netocrystalline, demagnetizing, and Zeeman contributions, inunits of field) F mc, and the magnetostrictive contribution Fms: Etot=W+MsFmc+MsFms (1) with W=1 2cijklεijεkl=W0+WSAW(t), (2) Fms=Fms,0+Fms,SAW (3) =/parenleftbigg εzz−εxx+εyy 2/parenrightbigg/bracketleftbig (A2ε+A4ε)m2 z +A4ε 2m4 z+A4ε/parenleftbig m4 x+m4 y/parenrightbig/bracketrightbig , (4) Fmc=−μ0/vectorH./vectorm+/bracketleftbiggμ0Ms 2−3Bc/bracketrightbigg m2 z+5 2Bcm4 z −Bc/parenleftbig m4 x+m4 y/parenrightbig +B2|| 4/parenleftbig m2 x−m2 y/parenrightbig . (5) The components of the unit magnetization vector are defined as mi=Mi/Ms(i=x,y,z ) where Msis the mag- netization at saturation and x/bardbl[110]. His the applied field, Bcthe cubic anisotropy constant, and B2/bardblthe uniaxial one, distinguishing in-plane [110] and [1 −1 0 ]a x e s[ 22].Fmsis the magnetoelastic contribution (in units of field) where themagnetostrictive coefficients A 2ε,A4εdepend on both the static strain felt by the layer ( εxx,0,εyy,0,εzz,0), and the dynamic SAW-induced strain [ εxx,SAW(t),εzz,SAW(t)]. The εxz,SAW(t) component of the SAW does not have any magnetostrictiveaction on the layer, and ε yy,SAW(t) is not excited by our setup. The total strain components are thus given by εii= εii,0+εii,SAW(t). AtT=5 K, the micromagnetic parameters areA2ε=35 T,A4ε=− 5T ,Bc=− 5m T , B2/bardbl=− 20 mT, andMs=36 kA/m. At low temperature, A2εis much biggerFIG. 4. (Color online) (a) Calculated precession frequency ver- sus field applied along [110], no sample tilt. The horizontal lineindicates the SAW frequency. (b) Measured (symbols) and simulated resonance fields (continuous line, sample tilt 1 .2 ◦, taking into account bothA2εandA4ε) versus temperature for the attenuation (black) and velocity (red) variations. thanA4ε, so magnetostrictive terms in A4εwill first be neglected to ease the reading. The full derivation includingA 4εis given in the Appendix, Sec. 5, and was used for the simulations shown in Figs. 4(b) and5. FIG. 5. (Color online) The variation of the acoustic attenuation (black) and relative variation of velocity (red) calculated with the T=40 K micromagnetic parameters, α=0.1a n d F=0.105. The simulations were done taking into account both the A2εandA4ε contributions, with (or without) a sample tilt in the ( x,z)p l a n e — symbols (full lines). The changes of attenuation and velocity withoutsample tilt have been divided by 20 for better visibility. 094401-3L. THEVENARD et al. PHYSICAL REVIEW B 90, 094401 (2014) The magnetization and acoustics dynamics are then obtained by solving the Landau-Lifshitz Gilbert equation[Eq. ( 6)] and the elastic wave equation [EWE, Eq. ( 7)]. A similar procedure has been used by other authors [ 11,23,24] but our main interests here are to derive explicitly (i) themodification of the elastic constants under the influence ofthe magnetoelastic coupling and (ii) the form of the solutionof a surface acoustic wave traveling on a cubic medium (which happens to be magnetostrictive). ∂/vectorm ∂t=γ Ms/vectorm×/vector∇/vectormEtot+α/vectorm×∂/vectorm ∂t, (6) ρ∂2Rtot,i ∂t2=∂σik ∂xk=∂ ∂xk∂E tot ∂εik. (7) αis an effective damping constant and γthe gyromagnetic factor. /vectorm=/vectorm0+/vectorm(t) is the sum of the equilibrium mag- netization unit vector and the rf magnetization and likewise for the displacement /vectorRtot=/vectorR0+/vectoru(t). The displacements are related to the strain by εij=∂Rtot,i ∂xj,ρis the material density, andcijklis the elastic constant tensor defined in the x,y,z frame (see Appendix, Sec. 2). Note that, as assumed by otherauthors [ 11,23] the exchange contribution was neglected in Eq. ( 6), as the typical SAW wave vector ( ≈1//Lambda1 SAW) is much smaller than the first spin-wave wave vector ( ≈1/d), leading to an essentially flat magnon dispersion curve for the frequenciesconsidered here. For this reason, although we should in all rigorbe talking about “spin-wave FMR,” we will use the shorter term“ferromagnetic resonance.” We first briefly recall the derivation of the Polder suscepti- bility and of the precession frequency [ 25]. Following Dreher et al. [11], we define a second reference frame (1,2,3) where /vectorm 3is aligned with the static magnetization [polar coordinates (θ0,φ0); see Appendix, Sec. 1]. We are then left with two sets of unknowns: ( m1,m2)(t) (magnetization dynamics) and (ux,uz)(t) (acoustic dynamics), as the transverse displacement uycannot be excited by our device. Solving Eq. ( 6)i nt h e linear approximation with mi(x,t)=m0,iei(/Omega1t−kx)leads to the following system: /parenleftbiggm1 m2/parenrightbigg =[χ]/parenleftbiggμ0h1 μ0h2/parenrightbigg . (8) The dynamic fields are defined by μ0hi=−∂Fms,SAW ∂mi|/vectorm=/vectorm3. Neglecting the A4εterms, the dynamic magnetoelastic energy then simply reads Fms,SAW=A2ε/Delta1ε(t)m2 z, so that μ0h1= A2ε/Delta1ε(t)(cos2θ0−sin2θ0m1) andμ0h2=0. The susceptibil- ity tensor [ χ] (given in the Appendix, Sec. 1) depends on the static magnetic anisotropy constants, the damping and theSAW excitation frequency ω. Canceling the determinant of [χ] −1yields the precession frequency (real part of /Omega1)(ωprec γ)2= (F11−F33)(F22−F33)−F2 12where the terms Fijstand for ∂2(Fmc+Fms,0) ∂mi∂mj. Figure 4(a) shows the field dependence of this precession frequency at various temperatures, calculated from the FMR anisotropy coefficients. fprec(μ0H) first decreases, crossing the SAW frequency of 549 MHz [full line in Fig. 4(a)] at a particular field. When the magnetization is aligned with thefield (saturated), f precreaches a minimum. After saturation, the resonance frequency increases with field, and crosses fSAW asecond time. We will show below that this second crossing does not give rise to any magnetoacoustic resonance. The crossoverfields of f prec(μ0H) with fSAW [Fig. 4(a)] can already give a good approximation of the expected resonance fields. It isnot, however, sufficient to explain why the resonance fieldsare different for relative variations of the SAW attenuation andvelocity. For this it is necessary to calculate how the SAW wavevector is modified by its interaction with the (Ga,Mn)(As,P)layer. We place ourselves in the semi-infinite medium approxima- tion and assume the SiO 2layer to be a small perturbation to the system since its thickness is much smaller than /Lambda1SAW(see the Appendix, Sec. 3 for details on this point) so that the generalform of displacement reads u η(x,t)=Uηe−βzexp[i(ωt−kx)] (η=x,y,z ). This point differs from the infinite medium approach of Dreher et al. [11], which considers plane acoustic waves but does not take into account the zdecay of the SAW, or the role of boundary conditions. Using the equilibriumconditions on the strain, the EWE [Eq. (7)] may be simplifiedinto /bracketleftbigg ρω 2+/parenleftbiggAχ 4−c11/parenrightbigg k2+c44β2/bracketrightbigg ux +/parenleftbigg c44+c13+Aχ 2/parenrightbigg βiku z=0, (9) /parenleftbigg c44+c13+Aχ 2/parenrightbigg βiku x +/bracketleftbig ρω2+(c33−Aχ)β2−k2c44/bracketrightbig uz=0.(10) Note that here, we treat the case of a cubic lattice. This differs from most related work [ 11,23], where the isotropic approxi- mation is kept, either because the material is polycrystalline,or to simplify calculations. In the above, we have introducedthe complex constant: A χ=MsA2 2εsin2(2θ0)χ11. (11) Two features come out. First, this system is the formal equivalent of the solution to the EWE in a cubic, nonmagne-tostrictive material, with three of the elastic constants modifiedas follows: c 13/mapsto→c/prime 13=c13+Aχ/2, c11/mapsto→c/prime 11=c11−Aχ/4, (12) c33/mapsto→c/prime 33=c33−Aχ. The elastic constants are modified through Aχwhich depends on the applied field, the anisotropy constants, andthe SAW frequency (through χ 11). The real part of Aχ represents at most ≈10% of the GaAs elastic constants. This parameter embodies the physics of the coupled magnon-phonon system as it modifies the elastic constants of thematerial. Equation ( 12), for instance, shows that the velocity of longitudinal phonons (proportional to/radicalbig c/prime 11) is modified through the magnetoacoustic interaction. TheAχparameter cancels out when the material ceases to be magnetostrictive ( A2ε=0) and/or when the magnetization is collinear or normal to the SAW wave vector. This is whyno acoustic resonance is observed at the second crossing off prec(μ0H) with fSAW, once the magnetization is aligned with 094401-4SURFACE-ACOUSTIC-W A VE-DRIVEN FERROMAGNETIC . . . PHYSICAL REVIEW B 90, 094401 (2014) the applied field ( Aχ|θ0=π/2=0). To check this point, we repeated the experiment with the field applied perpendicularto the sample (insets of Fig. 3). This time no resonance was observed, either in the attenuation changes or in the velocityvariations. A small, hysteretic kink (return sweep not shown)was observed at a field coinciding with the coercive field of thelayer, as already observed in Ref. [ 11]. This feature became undetectable above 20 K. Secondly, using the full depth dependence of the displace- ments results in a coupling of the u xanduzcomponents [ β terms in Eqs. ( 9) and ( 10)], contrary to simpler cases treated previously [ 11]. In fact, it is through the zdecay that c13and c33constants are modified by the magnetostrictive interaction; they would otherwise be left unchanged. Canceling the determinant of Eqs. ( 9) and ( 10) yields two solutions with the corresponding absorption coefficients β1,2 andx,zamplitude ratios Uz/Ux=ri(i=1,2; see Appendix, Sec. 4). As neither of these satisfy the normal boundary condi-tionσ xz|z=0=0 at the vacuum interface, a linear combination of these two solutions needs to be considered: ux=[Ux1exp(−β1z)+Ux2exp(−β2z)]ei(ωt−kx),(13) uz=[Uz1exp(−β1z)+Uz2exp(−β2z)]ei(ωt−kx).(14) The boundary conditions σxz|z=0=σzz|z=0=0 now lead to a new system, similar to Eqs. ( 9) and ( 10). Replacing ri,βiby their expressions and using ω/V r=k, its determinant eventually leads to Eq. ( 15). /parenleftbigg c44−ρω2 k2/parenrightbigg/bracketleftbigg c/prime 11c/prime 33−c/prime2 13−c/prime 33ρω2 k2/bracketrightbigg2 =c/prime 33c44/parenleftbigg c/prime 11−ρω2 k2/parenrightbigg/parenleftbigg ρω2 k2/parenrightbigg2 . (15) This implicit polynomial equation in km a yb es o l v e d numerically to yield the wave-vector solutions ksolin the presence of magnetostrictive interaction. There are threedistinct physical solutions to Eq. ( 15), but only the Rayleigh surface wave can be excited by our device [ 26]. In the absence of magnetostriction, the usual Rayleigh velocity [ 27] V r=ω ksol|Aχ=0=2852.2ms−1is recovered, very close to the crude experimental estimation made earlier. The amplitude of the transmitted SAW wave vector is proportional to exp[ −Im(ksol)l], and its phase is equal to Re(ksol)l. The relative variations are calculated with respect to the zero-field values. We can now plot the expected relativevariations of acoustic attenuation and velocity (e.g., at 40 K,Fig. 5) assuming we excite the IDTs at 549 MHz. In this calculation, we have also taken into account the A 4εterm. The procedure is identical to the one described above, but the ex-pressions are somewhat more cumbersome (see the Appendix,Sec. 5 for the corresponding effective elastic constants). V . DISCUSSION The variation of attenuation (Fig. 5, full black line) is monopolar and peaks at 88 mT, as expected from the simplecrossing of f prec(μ0H) with fSAW=ω/2π(Fig. 4). The relative variation of velocity (full red line) is bipolar, and cancels out when the amplitude variation is maximum. Bothcurves are quite asymmetric, plummeting to zero when the magnetization is aligned with the field (92 mT). Introducingas m a l l1 .2 ◦sample tilt in the ( x,z) plane with respect to the field direction pushes the saturation field away from theresonance field, restoring the symmetry of the resonance. Thistilt may have been introduced when gluing the sample. Itstrongly reduces the magnitude of the effect, almost by afactor of 20. The attenuation resonance fields thus obtained areslightly higher than without tilt. The higher-field bump of thevelocity variations disappears, making the resonance unipolarand at lower fields than the amplitude variations, as observedexperimentally. It is interesting to compare these results tothose of Dreher et al. [11], computed using a similar approach for an in-plane nickel thin film. Their closest comparableconfiguration is the one where the field is applied close tothe sample normal (hard axis configuration). Their simulations(last line of Fig. 8 in Ref. [ 11]) also show that a bipolar shape is expected for the relative velocity, as the sample is excitedcloser and closer to its resonance frequency. Their experiments,however, also seem to show more of a monopolar behavior,for fields close to the sample normal. Simulated attenuation and velocity variation resonance fields are now plotted along with the experimental ones inFig. 4(b) as a function of temperature. Their values are well reproduced, and so is their nonmonotonous temperaturevariation. The latter can be traced back to a sign inversionof the B 2/bardblterm with temperature, i.e., a swap between [110] and [1 −1 0] easy axes around 40 K. At high temperature, the match is less good, the simulation overestimating theresonance fields. This may be the signature of a slightmodification of the Curie temperature of the layer duringthe IDT deposition: The magnetostrictive coefficients wouldthen fall off faster with temperature than those estimated byFMR before the IDT deposition. This could also account forthe disappearing of the signal about 20 K below the Curietemperature (85 K, whereas T c=105 K). The resonance fields, as well as the fact that /Delta1V/V resonates at lower field than /Delta1/Gamma1 are overall well predicted. This is an indication that the data is reasonably well understoodby our simple model. One of the results of this approachis that the elastic constants are modified resonantly via themagnetoacoustic coupling [Eq. (12)]. In this respect, we pro-pose to reinterpret SAW-driven FMR as a resonant, dynamicform of the /Delta1E effect. This is a well-known effect [ 17,28] by which the Young modulus of a magnetostrictive materialchanges with applied field. Microscopically, the processesat play are generally the rotation of magnetization, ofteninvolving the rearrangement of magnetic domains. Although itis most often mentioned in static measurements [ 29], Ganguly et al. [23] had already identified the dynamic /Delta1E effect nature of the field-dependent velocity variations induced bya SAW on nickel. They had, however, demonstrated that itwas a nonresonant effect. In the case of SAW-driven FMR, wesuggest one interpret it as a resonant /Delta1E effect. Let us briefly comment on the influence of the depth decay of the SAW on the interaction with the magnetization: Asmentioned earlier, this resulted in a modification of, not onlyc 11, but also c13andc33. When not taken into account in the numerical calculation (e.g., at T=40 K), the peak of acoustic attenuation is found about 5 dB cm−1lower than otherwise, 094401-5L. THEVENARD et al. PHYSICAL REVIEW B 90, 094401 (2014) a sizable underestimation. The resonance field remains the same however, since c11,c13, and c33are all linear in Aχ, whose resonance field is solely given by the micromagneticparameters at the chosen temperature. Finally, we wish to address the quantitative agreement between predicted and measured effects. The magnetostrictiveconstants had to be reduced by a filling factor Fto best reproduce the amplitude of the effect since the magneticlayer occupies a small portion of the volume swept by theSAW: A 2ε/mapsto→FA 2ε,A4ε/mapsto→FA 4ε. This effective medium approximation is routinely used in other solid state physicssystems, such as the case of sparse quantum dots embeddedin a waveguide [ 30]. We converged to a value of F=0.10 to obtain good agreement between simulated and experimentalattenuation variations. The simulated velocity variations arethen, however, off by about an order of magnitude comparedto the experiment (Fig. 5). This disparity in quantitative agreement between the experimental and simulated phase shifthad already been observed in nickel [ 11]. We believe, however, that this filling factor has little physical meaning. First, wehave shown that not only F, but also the sample tilt play a great role in the amplitude of the effect, and this value is notknown experimentally. Secondly, the SAW amplitude is infact not uniform across the depth /Lambda1 SAW. To better reproduce quantitatively and qualitatively the shape and amplitude of theeffect a more complete multilayer approach using a transfermatrix formalism would clearly need to be adopted, as wasdone, for instance, in Ref. [ 23]. VI. CONCLUSION We have demonstrated the resonant excitation of magne- tization precession in (Ga,Mn)(As,P) by a surface acousticwave. Temperature-dependent measurements have clearlyshown that the magnitude of the effect and the positionof the resonance fields evolved together with temperaturedependence of the magnetostrictive coefficients. An analyticaldescription of a SAW traveling on a magnetostrictive cubicmedium was derived. It was evidenced that in that case,the elastic tensor coefficients are modified by a complexvalue depending on the magnetostrictive coefficients, the SAWfrequency, and the magnetization orientation (through thevalue of the applied field). Two of these results—the first evidence of SAW-induced magnetoelastic in (Ga,Mn)(As,P) and the prediction of res-onance fields—are important steps towards SAW-inducedprecessional magnetization switching in DFS. The generatedstrain ( ε max≈10−5) is for now about five times too small to obtain magnetization reversal, as sketched in the predictivediagram of Ref. [ 2]. This has been confirmed by the robust linearity of the observed effect versus acoustic power. The nextstep towards SAW-induced switching in DFS is therefore theoptimization of the amplitude of the generated strain waves,paired with the elaboration of higher frequency combs, in orderto work under smaller magnetic fields. ACKNOWLEDGMENTS This work was performed in the framework of the MANGAS and the SPINSAW projects (ANR 2010-BLANC-0424-02 and ANR 13-JS04-0001-01). We thank A. Lema ˆıtre (Laboratoire de Photonique et Nanostructures) for providingthe (Ga,Mn)(As,P) sample, and M. Bernard (INSP) for helpingus with the cryogenic setup. APPENDIX 1. Magnetization dynamics Following Dreher et al. [11], the (1,2,3) reference frame is defined by /vectorm3being aligned with the magnetization equilib- rium position ( θ0,φ0) and the following correspondence: mx=m1cosθ0cosϕ0−m2sinϕ0+m3sinθ0cosϕ0, my=m1cosθ0sinϕ0+m2cosϕ0+m3sinθ0sinϕ0, (A1) mz=−m1sinθ0+m3cosθ0. The susceptibility tensor defined in Eq. ( 8)i sg i v e nb y [χ]=1 D/parenleftBigg F22−F33+iαω γ−/parenleftbig F12−iω γ/parenrightbig −/parenleftbig F12+iω γ/parenrightbig F11−F33+iαω γ/parenrightBigg ,(A2) where Fij=∂2(Fmc+Fms,0) ∂mi∂mjand D=/parenleftbigg F11−F33+iαω γ/parenrightbigg/parenleftbigg F22−F33+iαω γ/parenrightbigg −F2 12−/parenleftbiggω γ/parenrightbigg2 . 2. Elastic coefficient tensor The elastic coefficient tensor being defined in the reference frame of a cubic material, we must rotate it by π/4f o r the particular case of a SAW traveling along [110]. Theequivalence with the usual elastic constants [ 27]C 0 ijis c11=1 2C0 11+1 2C0 12+C0 44, c12=1 2C0 11+1 2C0 12−C0 44, c13=C0 12, (A3) c33=C0 11, c44=C0 44, c66=1 2C0 11−1 2C0 12. Temperature variations of the elastic tensor have been neglected and (Ga,Mn)(As,P) elastic constants were assumedequal to those of GaAs. Note that since the medium is cubic,and not isotropic, the relationship C 0 12=C0 11−2C0 44is not verified. 3. Influence of the SiO 2/ZnO on the (Ga,Mn)(As,P) static strain Although GaAs is naturally piezoelectric, a SiO 2/ZnO bilayer was sputtered onto the magnetic layer to increase theamplitude of the SAW-generated strain. The silica underlayerwas required for good adhesion. An important question iswhether the high temperature (150 ◦C) deposition of the SiO 2/ZnO bilayer modifies the magnetic layer’s static strain. To check this, we performed room temperature high resolutionx-ray rocking curves around the (004) reflection at different 094401-6SURFACE-ACOUSTIC-W A VE-DRIVEN FERROMAGNETIC . . . PHYSICAL REVIEW B 90, 094401 (2014) steps of the bilayer deposition. The lattice mismatch of the reference (unpatterned) (Ga,Mn)(As,P) layer was around−0.152%, i.e., under tensile strain on GaAs. After the SiO 2deposition, the lattice mismatch dropped to −0.136%. However, the lattice mismatch of the layer after deposition ofthe full SiO 2/ZnO stack returned close to the reference value, around −0.161%, and remained unchanged after removal of the ZnO layer. The rocking curves also pointed to thepresence of a strain gradient extending into the GaAs substratesubsequently to the SiO 2/ZnO deposition. Given that the static strain of the magnetic layer seems to be affected by SiO 2/ZnO deposition, the FMR measurements of the magnetic anisotropyconstants were done on the (Ga,Mn)(As,P)/SiO 2/ZnO stackafter removal of the ZnO. We then used these values and the x-ray-diffraction-determined static strain to obtain themagnetostrictive coefficients of the layer, using the formulasgiven in Appendix Aof Ref. [ 2]. 4. Elastic wave equation This paragraph details solutions to the elastic wave equation when taking the displacement as uη=Uηe−βzexp[i(ωt−kx)] (η=x,y,z ). Inserting this expression into Eq. ( 7) leads to the system of Eqs. ( 9) and ( 10). Canceling this determi- nant leads to the following bisquared equation in q=β/k using the effective elastic tensor coefficients defined inEq. ( 12): q4+/bracketleftbig −c2 44−c/prime 11c/prime 33+(c/prime 13+c44)2/bracketrightbig +ρV2 r(c/prime 33+c44) c/prime 33c44q2+/parenleftbig c/prime 11−ρV2 r/parenrightbig/parenleftbig c44−ρV2 r/parenrightbig c/prime 33c44=0. (A4) This equation has two physical solutions, qi=βi/kwith Uz/Ux=ri(i=1,2) that verify q2 1+q2 2=/bracketleftbig c2 44+c/prime 11c/prime 33−(c/prime 13+c44)2/bracketrightbig −ρV2 r(c/prime 33+c44) c/prime 33c44, (A5) q2 1q2 2=/parenleftbig c/prime 11−ρV2 r/parenrightbig/parenleftbig c44−ρV2 r/parenrightbig c/prime 33c44, (A6) r1,2=iβ1,2k(c/prime 13+c44) k2c44−β2 1,2c/prime 33−ρω2. (A7) As neither of these satisfy the normal boundary condition σxz|z=0=0 at the vacuum interface, a linear combination of these two solutions needs to be considered, as furtherdeveloped in the text. 5. Solutions when taking into account the A4εterm At high temperatures ( T/greaterorequalslantTc/2),A4ε(cubic anisotropy) is routinely 10 smaller than A2ε(uniaxial anisotropy). At lowertemperatures, we rather have A2ε≈4–5A2ε. Following the same calculation as in the text but taking into account A4ε gives the following modified elastic constants: c13/mapsto→c/prime 13=c13+AξDB, c11/mapsto→c/prime 11=c11−AξD2, (A8) c33/mapsto→c/prime 33=c33−AξB2, where the field is applied along a [ ±110] axis and Aξ=Mssin2(2θ0)χ11, B=A2ε+A4ε 2[1+3cos(2 θ0)], (A9) D=B/2. The shape and position of the resonance remain globally unchanged, but the amplitude of the effect (on both therelative attenuation and the velocity variations) is stronglydiminished. [1] E. du Tremolet de la Lacheisserie, Magnetism: Fundamentals , 1st ed. (Springer, New York, 2006). [2] L. Thevenard, J.-Y . Duquesne, E. Peronne, H. J. von Bardeleben, H. Jaffres, S. Ruttala, J.-M. George, A.Lema ˆıtre, and C. Gourdon, Phys. Rev. B 87,144402 (2013 ). [3] O. Kovalenko, T. Pezeril, and V . V . Temnov, P h y s .R e v .L e t t . 110,266602 (2013 ). [4] W. Li, P. Dhagat, and A. Jander, IEEE Trans. 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Akimov, and M. Bayer, P h y s .R e v .B 85, 195324 (2012 ). 094401-7L. THEVENARD et al. PHYSICAL REVIEW B 90, 094401 (2014) [15] T. Dietl, H. Ohno, and F. Matsukura, P h y s .R e v .B 63,195205 (2001 ). [16] M. Cubukcu, H. J. von Bardeleben, K. Khazen, J. L. Cantin, O. Mauguin, L. Largeau, and A. Lema ˆıtre, P h y s .R e v .B 81, 041202(R) (2010 ). [17] E. Lee, Rep. Progr. Phys. 18,184(1955 ). [18] A. Lema ˆıtre, A. Miard, L. Travers, O. Mauguin, L. Largeau, C. Gourdon, V . Jeudy, M. Tran, and J.-M. George, Appl. Phys. Lett. 93,021123 (2008 ). [19] D. Royer and E. Dieulesaint, Elastic Waves in Solids I: Free and Guided Propagation , Advanced Texts in Physics (Springer, New York, 2000). [20] S. C. Masmanidis, H. X. Tang, E. B. Myers, M. Li, K. DeGreve, G. Vermeulen, W. V . Roy, and M. L. Roukes, Phys. Rev. Lett. 95,187206 (2005 ). [21] L. Landau, L. E. M, and L. Pitaevskii, Electrodynamics of Con- tinuous Media (Elsevier, Butterworth-Heinemann, Amsterdam, 1984). [22] This anisotropy has been attributed to the presence of a shear strain/epsilon1xyin the [100] reference frame [Sawicki et al. ,Phys. Rev. B71,121302 (R) ( 2005 )], which would imply that the SAW could have an action on it. However, since no experimentalevidence of this shear strain has ever been shown, we have not taken it into account. [23] A. K. Ganguly, K. L. Davis, D. Webb, and C. Vittoria, J. Appl. Phys. 47,2696 (1976 ). [24] N. Polzikova, S. Alekseev, I. Kotelyanskii, A. Raevskiy, and Y . Fetisov, J. Appl. Phys. 113,17C704 (2013 ). [25] L. Baselgia, M. Warden, F. Waldner, S. L. Hutton, J. E. Drumheller, Y . Q. He, P. E. Wigen, and M. Mary ˇsko, Phys. Rev. B 38,2237 (1988 ). [26] There are two other physical solutions. Each of them is the superposition of bulk quasilongitudinal and quasitransverseacoustic waves traveling away from the surface. 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PhysRevB.92.144424.pdf

PHYSICAL REVIEW B 92, 144424 (2015) Spin-orbit torque driven chiral magnetization reversal in ultrathin nanostructures N. Mikuszeit,1,2,3,*O. Boulle,1,2,3I. M. Miron,1,2,3K. Garello,4P. Gambardella,4G. Gaudin,1,2,3and L. D. Buda-Prejbeanu1,2,3 1Universit ´e Grenoble Alpes, INAC-SPINTEC, F-38000 Grenoble, France 2CNRS, INAC-SPINTEC, F-38000 Grenoble, France 3CEA, INAC-SPINTEC, F-38000 Grenoble, France 4Department of Materials, ETH Z ¨urich, H ¨onggerbergring 64, CH-8093 Z ¨urich, Switzerland (Received 7 August 2015; published 26 October 2015) We show that the spin-orbit torque induced magnetization switching in nanomagnets presenting Dzyaloshinskii- Moriya (DMI) interaction is governed by a chiral domain nucleation at the edges. The nucleation is inducedby the DMI and the applied in-plane magnetic field followed by domain-wall propagation. Our micromagneticsimulations show that the dc switching current can be defined as the edge nucleation current, which decreasesstrongly with increasing amplitude of the DMI. This description allows us to build a simple analytical modelto quantitatively predict the switching current. We find that domain nucleation occurs down to a lateral size of25 nm, defined by the length scale of the DMI, beyond which the reversal mechanism approaches a macrospinbehavior. The switching is deterministic and bipolar. DOI: 10.1103/PhysRevB.92.144424 PACS number(s): 75 .60.Jk,85.70.Ay,85.75.Dd I. INTRODUCTION The recent discovery that a current can switch the magnetization of a nanomagnet in ultrathin heavy-metal(HM)/ferromagnetic (FM) multilayers has opened a newpath to manipulate magnetization at the nanoscale [ 1]. The switching arises from structural inversion asymmetry and highspin coupling, resulting in a spin current from the HM into theFM. This novel switching mechanism has led to an innovativemagnetic memory concept, namely, the spin-orbit torquemagnetic random access memory (MRAM) [ 1–3], which combines large endurance, low power, and fast switching andthus appears to be a possible nonvolatile alternative for cachememory applications. Recently, Garello et al. [4] demonstrated deterministic magnetization switching by spin-orbit torque(SOT) in ultrathin Pt/Co/AlO x, as fast as 180 ps. These observations could not be explained within a simple macrospinapproach, suggesting a magnetization reversal mechanism bydomain nucleation and domain-wall (DW) propagation. Thefailure of the macrospin approach for quantitative descriptionis also underlined by the predicted switching current density,which is nearly one order of magnitude larger than experimen-tal ones [ 5–7]. Besides its fundamental importance, this lack of a proper quantitative modeling is an important issue for thedesign of logic and memory devices based on SOT switch-ing, which have so far considered a macrospin description[8–11]. The missing ingredient is the presence of anti- symmetric exchange interaction, i.e., Dzyaloshinskii-Moriyainteraction (DMI). This exchange tends to form states ofnoncollinear magnetization, promoting a homochiral N ´eel DW [ 12–14]. In the N ´eel configuration, a maximal SOT is applied on the DW [ 13,15–17], which explains the large current-induced DW velocity observed experimentally [ 1,17]. Moreover, the DMI can result in significant magnetizationtilting at the edges of magnetic structures, resulting, e.g.,in asymmetric field-induced domain nucleation [ 18,19]. The influence of the DMI on the magnetization pattern during *nikolai.mikuszeit@gmail.comSOT switching was recently pointed out in micromagneticsimulation studies [ 20–22], whereas recent experimental work [23] explained the SOT switching mechanism by the expansion of a magnetic bubble. Here, using micromagnetic simulations and analytic mod- eling, we show that the SOT-induced magnetization switchingin the presence of DMI is governed by domain nucleation onone edge followed by propagation to the opposite edge. Thisreversal process allows us to explain the ultrafast deterministicswitching observed experimentally. We systematically demon-strate that DMI leads to a large decrease of the switching current and of the switching time and thus strongly affects the reversal energy. On the basis of our micromagneticsimulations, we provide a simple analytical model, whichallows us to quantitatively predict the SOT switching currentin the presence of DMI. Finally, we address the evolution ofthe switching mechanism as the lateral dimension decreases,which is a key feature for the device scalability. II. REVERSAL MECHANISM The structures considered in this study are similar to the one used by Garello et al. [4]: a perpendicularly magnetized Co circular nanodot on top of a Pt stripe and cappedwith alumina. The DMI is included in the simulation usingthe expression of Ref. [ 13]. In addition to the standard micromagnetic energy density (which includes the exchange,magnetocrystalline anisotropy, Zeeman, and demagnetizingenergies), the current injected in the Pt layer leads to twoSOT terms in the the Landau-Lifshitz-Gilbert equation: thefieldlike T FL∝/vectorm×/vectoreyand the dampinglike TDL∝/vectorm×(/vectorm× /vectorey), where /vectoreyis the unit vector in the ydirection (see [ 24] for additional details). If not state otherwise, the externalapplied field is μ 0Happ=− 0.1 T, and the material parameters are [ 25] the saturation magnetization MS=1090 kA /m, the exchange constant Aex=1.0×10−11A/m, the perpendic- ular magnetic anisotropy constant Ku=1248 kJ /m3,t h e DMI amplitude D=2m J/m2, the Gilbert damping param- eterα=0.5, and the torques T0 FL=− 0.05 pTm2/A and T0 DL=+ 0.1p T m2/A. 1098-0121/2015/92(14)/144424(5) 144424-1 ©2015 American Physical SocietyN. MIKUSZEIT et al. PHYSICAL REVIEW B 92, 144424 (2015) The 3D micromagnetic simulations are performed using the solver MICRO3D [26] with a mesh size smaller than 1 .5n m .T h e initial magnetization state of the dot is the remanent state aftersaturation by a negative magnetic field ( −O z)a ss h o w nf o r 0p si nF i g . 1(d). In the presence of an applied magnetic field /vectorHappin the xdirection, magnetization dynamics is induced by a current pulse with a rise (and fall) time of 50 ps andvariable width and amplitude. Typical simulation results ofa 100-nm dot are presented in Fig. 1(a). Depending on the current amplitude, three regimes are identified: (1) For J app/lessorequalslant2.50×1012A/m2no magnetization switch- ing is observed. The SOT leads to a slight tilting of the mag-netization toward the plane of the dot, but the magnetizationrelaxes toward its initial equilibrium state after the pulse. (2) At intermediate current values (2 .60×10 12A/m2/lessorequalslant Japp/lessorequalslant3.70×1012A/m2) magnetization reversal occurs. The time evolution of the magnetization pattern in the dot [seeFig. 1(d)] reveals that, in contrast to recent interpretations [23], the magnetization reversal occurs by domain nucleation shortly after the pulse injection (100 ps), followed by fastDW propagation. The nucleation always occurs on the leftedge of the dot. Once nucleated, the DW propagates quicklythrough the dot and is expelled on the opposite edge. Theswitching time t 0, defined by /angbracketleftmz/angbracketright(t=t0)=0, decreases as Jappincreases; the increase of the slope of /angbracketleftmz/angbracketright(t) indicates that this is related to a faster DW propagation. As expected, the DWhas a N ´eel configuration due to the large DMI. The simulation highlights that the DW nucleation occurs for all current valueson the same edge in a deterministic way. Symmetrically, whenreversing the sign of the current, the reversal from the up tothe down state occurs on the opposite edge, i.e., the behavioris bipolar. (3) For higher currents ( J app/greaterorequalslant3.70×1012A/m2)t h e motion of the DW becomes turbulent (oscillatory), and thecoherence of the switching is destroyed. The magnetization reversal scheme can be explained in a simplified manner by considering the combined effect ofDMI, external magnetic field, and SOT but neglecting smallvariations of the demagnetizing field [ 27]. The DMI is too small to introduce a spin spiral but results in a magnetizationcanting at the dot edges [ 18,19,22]. The edge canting can be considered as effective field with spatial variation: on one side this field adds to the in-plane applied field, while it counteractsit on the other [see Fig. 1(b)]. This leads to an asymmetric tilting of the magnetization on both edges. Upon current injection the dampinglike torque emerges. Its effect can be interpreted as a rotating magnetic field of the form /vectorH DL∝Japp/vectorm×/vectorey[see Fig. 1(c)]. This leads to a rotation of the magnetization towards the film plane on one side and awayfrom the film plane on the other. Naturally, the current polarityis chosen such that the stronger tilted edge magnetization turnstowards the film plane. Above a critical current an instabilityoccurs, leading to domain nucleation and consecutive DWpropagation. It is clear that the current J c, required to introduce the instability, reduces with increasing DMI. This behavior isseen in Fig. 2(a), where J ctends to zero when D≈3.8m J/m2. Moreover, for Japp>J can increase of DMI decreases the switching time, as can be seen from Fig. 2(b). After expelling the DW on the opposite side, switching has occurred, and themore tilted edge appears on the opposite side. As the SOTrotates this side away from the film plane and is not sufficient to rotate the less tilted side into instability, the state is hencestable. It can easily be checked that this reversal scheme isin agreement with the hysteretic bipolar switching observedexperimentally when sweeping J appandHapp[1]. To understand these results better, we consider a simple analytical model which describes the reversal process in thepresence of both DMI and SOT. Using a Lagrangian approachand following Pizzini et al. [19], the strategy is, eventually, similar to the Stoner-Wohlfarth approach in a single domainparticle but using the energy functional per volume V E(θ) V=−Keffcos2θ−μ0MSHappsinθ−MSJappTDLθ, (1) where the effect of the SOT is introduced by the last term [24]. The equilibrium magnetization angle in the center θcis found by minimizing Eq. ( 1), while edge angle θeis found by solving [ E(θe)−E(θc)]/V=D2/(4A)[24]. For small SOT and/vectorHapp, two stable solutions for θeexist, corresponding to both sample edges. Above a threshold SOT one solutiondisappears, indicating that the magnetization on one edgeis unstable, i.e., domain nucleation occurs. Using numericalmethods, the critical current for nucleation J ccan be calculated easily as a function of D[see Fig. 2(a), black line]. Good agreement is obtained with micromagnetic simulation for adot diameter d=100 nm (circles). For Dtending to zero, the nucleation current tends to the critical current predicted bythe macrospin model J c=4.1×1012A/m2[6]. The absence of full quantitative agreement with micromagnetic simulationcan be attributed to variations of the demagnetizing tensor andvariations of the magnetization along the ydirection due to the curvature of the dot. Better agreement is obtained whenneglecting these effects in a quasi-one-dimensional simulation(square dots). Note that this nucleation current is actually thethreshold current for quasi-dc current pulse. III. CRITICAL CURRENT ANALYSIS In the following, we discuss the dynamics of the magne- tization switching. In Fig. 2(b) the switching time is shown as a function of Japp>J c. With increasing Jappthe switching time decreases rapidly as the DW velocity increases [ 16]. If Dis reduced, the DW propagation is slower, resulting in a larger switching time. In the inset we show Jappversus 1 /t0 forD=2m J/m2: a linear scaling is observed, in qualitative agreement with experiment [ 4]. Naturally, t0depends on the dot diameter. This is a key parameter for SOT applications. The evolution of the switchingtime vs the current density for varying dot sizes is shown inFig. 2(c). When decreasing the diameter from 100 down to 50 nm, a shift to shorter switching times is observed, whilea slightly higher onset current is found. Similar behavior isfound when decreasing the size down to 30 nm and furtherdown to 25 nm. It is, however, important to note that the lattertwo curves become identical for larger J app. Reducing the size down to 15 nm results in a dramatic increase of the thresholdcurrent density and deterministic switching is observed in onlya narrow current density region. Overall, one has indicationsfor three different size-dependent switching regimes. In thefirst regime the switching is covered by nucleation and 144424-2SPIN-ORBIT TORQUE DRIVEN CHIRAL MAGNETIZATION . . . PHYSICAL REVIEW B 92, 144424 (2015) Equilibrium μ0Happ=0,Japp=0 Initial State Happ<0,Japp=0 Switching Happ<0,Japp>0 Final State Happ<0,Japp=0switching HDLMleft McenterMright xz 0 ps 100 ps 150 ps 250 ps 350 ps 450 psHappJappxytime (ps)mz(a) (b) (c) (d)2.5×10122.6×10123.6×1012 3.7×1012 FIG. 1. (Color online) (a) Time evolution of the average out-of-plane magnetization for different applied current densities (variations in steps of 1011A/m2). The minimum current to trigger switching, i.e., the critical current, is highlighted in blue. The green curve indicates the threshold of turbulent behavior (see text). (b) Sketch of the magnetization configuration at different stages of the switching process. (c) Magnetization orientation in the center and at the left and right edges. The current-induced dampinglike torque (represented as effective field /vectorHDL) can only drive the left edge magnetization into instability, resulting in a nucleation at the left edge. (d) Snapshots of the magnetization configuration showing the reversal from down (black) to up (white) via domain-wall nucleation and propagation under an externally applied field of μ0H=0.1 T and a current density of 2 .6×1012A/m2. propagation of a DW, and the decrease of t0is mainly caused by a reduced distance for the DW to travel. In the secondregime the switching remains governed by DW propagation.The diameter, however, becomes comparable to approximatelytwice the value of ξ=2A/D≈10 nm, the characteristic length scale on which canting of the edge magnetization isobserved. In this situation the edge angle due to DMI differsfrom the ideal infinite case, and opposite edges are not com-pletely independent anymore (see Ref. [ 24]). While this does not cause coherent rotation yet, it affects the DW motion. Thecoherent regime is reached at diameters in the range of the DWwidth /Delta1=π√ A/K eff≈14 nm. This explains the significant change in switching behavior for the 15-nm dot. Note thatthe switching current at this size is close to the one pre-dicted by macrospin simulation (4 .1×10 12A/m2). It is worthmentioning that while the current density strongly increases with decreasing dot diameter, the current in the 3-nm-thick Ptstripe decreases almost linearly, as can be seen from Fig. 2(d). Therefore, the device exhibits favorable scaling behavior and,assuming a 1-k /Omega1resistance for the addressing transistor of a 30-nm dot, switching in about 300 ps, needs only 20 fJ for oneswitching event, which is significantly smaller than the energyfor perpendicular spin-transfer torque devices [ 28]. Naturally, the threshold current and switching time depend on several intrinsic as well as extrinsic parameters. Wehave studied in detail the influence of the applied field, thedamping constant, the strength of the fieldlike torque, andtemperature. The results are shown in Fig. 3. Variations in these parameters lead to quantitative changes of the nucleationcurrent as well as the switching time. In all cases this is (a) (b) (c) (d) FIG. 2. (Color online) (a) Critical current for destabilizing the system as a function of the DMI strength. (b) The relation between the critical current and the switching time t0for two different values of DMI. The inset shows the data for D=2m J/m2but in a transformed coordinate system Jappvst−1 0. (c) The switching time vs current for different dot diameters. (d) Critical current and current density for different dot sizes. The calculation of the current assumes a 3-nm-thick Pt line. 144424-3N. MIKUSZEIT et al. PHYSICAL REVIEW B 92, 144424 (2015) time (ps) time (ps)mz mzα=0.50 α=0.20 α=0.10 α=0.05Jc(1012A/m2)t0(ps) (a) (b) FIG. 3. (Color online) (a) Switching time as a function of the applied current density, varying intrinsic and extrinsic parameters.For the temperature case, the average t 0is plotted. The single event switching time is defined as before, while the average t0is defined as the time when the probability of stochastic switching reaches 90%. (b)Several switching graphs /angbracketleftm z/angbracketright(t) for varying damping at T=50 K andJapp=2.6×1012A/m2.F o rfi x e d αvariations are only due to temperature fluctuations. attributed mainly to changes in DW velocity; lower damping increases the wall velocity and so does an in-plane field,as it promotes and stabilizes a N ´eel-type wall. A negative fieldlike torque also stabilizes the DW, while a positive onedestabilizes it, therefore increasing the switching time. The edge nucleation/DW propagation mechanism, however, is notaffected. Most importantly, Fig. 3(a)shows that the mechanism of switching by nucleation and propagation is very robustagainst fluctuations due to temperature (see Ref. [ 24]f o rm o r e details). The temperature fluctuations strongly decrease thethreshold current [Fig. 3(b)]. Temperature effectively lowers the nucleation barrier, such that nucleation times get shorterand, consequently, the whole switching becomes faster. Ithas to be pointed out that the nucleation still takes place atthe same position on the dot edge, and the overall processremains bipolar with respect to field and current reversal. Thistemperature robustness, however, strongly relies on the largedamping, as can be seen from Fig. 3(b). With decreasing α an increasing tendency of oscillations is observed, such thatdeterministic switching cannot be guaranteed [ 6]. IV . CONCLUSIONS To conclude, we have studied the current-induced magneti- zation switching of a nanomagnet by spin-orbit torques in thepresence of Dzyaloshinskii-Moriya interaction. The criticalswitching current strongly decreases with increasing ampli-tude of DMI, and we provide a simple analytical model forthis dependency. This switching mechanism via chiral domainnucleation explains the deterministic switching observed ex-perimentally in ultrathin Pt /Co/AlO xeven for subnanosecond pulses. The switching is mainly introduced by the dampingliketorque, but the fieldlike torque cannot be neglected as itstrongly influences the switching time. Our systematic studyshows a change in the reversal mechanism below diameters of30 nm, while the switching remains deterministic and bipolar.However, at 0 K the operational window for current densitiesdecreases with decreasing dot diameter. The influence oftemperature on this technologically important limit will beinvestigated in the future. Most importantly, current scalabilityis maintained. Confirming the potential of SOT-MRAM forscalable fast nonvolatile memory application, our results willhelp in the design of devices based on this technology. ACKNOWLEDGMENTS This work was funded by the spOt project (318144) of the EC under the Seventh Framework Programme. [1] I. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V . Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P.Gambardella, Nature (London) 476,189(2011 ). [2] L. Liu, C.-F. Pai, Y . Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336,555(2012 ). [3] M. Cubukcu, O. Boulle, M. Drouard, K. Garello, C. Avci, I. Miron, J. Langer, B. Ocker, P. Gambardella, and G. Gaudin,Appl. Phys. Lett. 104,042406 (2014 ). [4] K. Garello, C. Avci, I. Miron, M. Baumgartner, A. Ghosh, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella, Appl. Phys. Lett.105,212402 (2014 ). [5] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, P h y s .R e v .L e t t . 109,096602 (2012 ).[6] K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, Appl. Phys. Lett.102,112410 (2013 ). [7] K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, Appl. Phys. Lett.104,072413 (2014 ). [8] Y . Kim, X. Fong, K.-W. Kwon, M.-C. Chen, and K. Roy, IEEE Trans. Electron Devices 62,561(2015 ). [9] K. Jabeur, G. Di Pendina, and G. Prenat, Electron. Lett. 50,585 (2014 ). [10] K. Jabeur, G. Di Pendina, G. Prenat, L. D. Buda- Prejbeanu, and B. Dieny, IEEE Trans. Magn. 50,1 (2014 ). [11] Z. Wang, W. Zhao, E. Deng, J.-O. Klein, and C. Chappert, J. Phys. D 48,065001 (2015 ). 144424-4SPIN-ORBIT TORQUE DRIVEN CHIRAL MAGNETIZATION . . . PHYSICAL REVIEW B 92, 144424 (2015) [12] G. Chen, J. Zhu, A. Quesada, J. Li, A. T. N’Diaye, Y . Huo, T. P. Ma, Y . Chen, H. Y . Kwon, C. Won, Z. Q. Qiu, A. K.Schmid, and Y . Z. Wu, P h y s .R e v .L e t t . 110,177204 (2013 ). [13] A. Thiaville, S. Rohart, E. Ju ´e, V . Cros, and A. Fert, Eur. Phys. Lett.100,57002 (2012 ). [14] J.-P. Tetienne, T. Hingant, L. J. Mart ´ınez, S. Rohart, A. Thiaville, L. H. Diez, K. Garcia, J.-P. Adam, J.-V . Kim, J.-F. Roch, I. M.Miron, G. Gaudin, L. Vila, B. Ocker, D. Ravelosona, and V .Jacques, Nat. Commun. 6,6733 (2015 ). [15] S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach, Nat. Mater. 12,611(2013 ). [16] O. Boulle, S. Rohart, L. D. Buda-Prejbeanu, E. Ju ´e, I. M. Miron, S. Pizzini, J. V ogel, G. Gaudin, and A. Thiaville, Phys. Rev. Lett. 111,217203 (2013 ). [17] K.-S. Ryu, L. Thomas, S.-H. Yang, and S. Parkin, Nat. Nanotechnol. 8,527(2013 ). [18] S. Rohart and A. Thiaville, Phys. Rev. B 88,184422 (2013 ). [19] S. Pizzini, J. V ogel, S. Rohart, L. D. Buda-Prejbeanu, E. Ju ´e, O. Boulle, I. M. Miron, C. K. Safeer, S. Auffret, G. Gaudin, and A.Thiaville, Phys. Rev. Lett. 113,047203 (2014 ). [20] N. Perez, E. Martinez, L. Torres, S.-H. Woo, S. Emori, and G. Beach, Appl. Phys. Lett. 104,092403 (2014 ).[21] G. Finocchio, M. Carpentieri, E. Martinez, and B. Azzerboni, Appl. Phys. Lett. 102,212410 (2013 ). [22] E. Martinez, L. Torres, N. Perez, M. A. Hernandez, V . Raposo, and S. Moretti, Sci. Rep. 5,10156 (2015 ). [23] O.-J. Lee, L.-Q. Liu, C.-F. Pai, Y . Li, H.-W. Tseng, P. G. Gowtham, J. P. Park, D. C. Ralph, and R. A. Buhrman,Phys. Rev. B 89,024418 (2014 ). [24] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.92.144424 for details on the equation of motion, a detailed elaboration of the Lagrange formalism andthe resulting analytical model, as well as additional data on sizeand temperature dependence. [25] K. Garello, I. Miron, C. Avci, F. Freimuth, Y . Mokrousov, S. Bl¨ugel, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8,587(2013 ). [26] L. D. Buda, I. Prejbeanu, U. Ebels, and K. Ounadjela, Comput. Mater. Sci. 24,181(2002 ). [27] S. Meckler, O. Pietzsch, N. Mikuszeit, and R. Wiesendanger, Phys. Rev. B 85,024420 (2012 ). [28] H. Liu, D. Bedau, J. Z. Sun, S. Mangin, E. E. 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PhysRevB.87.054406.pdf

PHYSICAL REVIEW B 87, 054406 (2013) Spin torque switching of an in-plane magnetized system in a thermally activated region Tomohiro Taniguchi,1Yasuhiro Utsumi,2Michael Marthaler,3Dmitri S. Golubev,4and Hiroshi Imamura1,* 1Spintronics Research Center, National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568, Japan 2Faculty of Engineering, Mie University, Tsu, Mie, 514-8507, Japan 3Institut f ¨ur Theoretische Festk ¨orperphysik, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany 4Institut f ¨ur Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany (Received 25 November 2012; published 8 February 2013) The current dependence of the exponent of the spin torque switching rate of an in-plane magnetized system was investigated by solving the Fokker-Planck equation with low temperature and small damping and currentapproximations. We derived the analytical expressions of the critical currents, I candI∗ c.A tIc, the initial state parallel to the easy axis becomes unstable while at I∗ c(/similarequal1.27Ic) the switching occurs without the thermal fluctuation. The current dependence of the exponent of the switching rate is well described by (1 −I/I∗ c)b, where the value of the exponent bis approximately unity for I/lessorequalslantIcwhile brapidly increases up to ∼2.2 with increasing current for Ic/lessorequalslantI/lessorequalslantI∗ c. The linear dependence for I/lessorequalslantIcagrees with the other works, while the nonlinear dependence for Ic/lessorequalslantI/lessorequalslantI∗ cwas newly found by the present work. The nonlinear dependence is important for analysis of the experimental results, because most experiments are performed in the current regionofI c/lessorequalslantI/lessorequalslantI∗ c. DOI: 10.1103/PhysRevB.87.054406 PACS number(s): 75 .78.−n, 85.75.−d, 75.60.Jk, 05.40.Jc I. INTRODUCTION Spin torque switching1,2of a magnetization in nanos- tructured ferromagnets has attracted much attention dueto its potential application to spintronics devices, such asspin random access memory (Spin RAM). 3,4An accurate estimation of the thermal stability, /Delta10, is very important for these applications, where the thermal stability is defined as theratio of the magnetic anisotropy energy to the temperature(k BT). For example, the retention time of the Spin RAM exponentially depends on the thermal stability.5 The thermal stability is experimentally determined by measuring the spin torque switching in the thermally activatedregion, 5,6and analyzing the switching probability7with the formula P=1−exp[−/integraltextt 0dt/primeν(t/prime)], where ν=fexp(−/Delta1) andfare the switching rate and the attempt frequency, re- spectively. Similar to many other nonequilibrium systems,8–10 the exponent of the switching rate can be written in the form, /Delta1=/Delta10/parenleftbigg 1−I Ic/parenrightbiggb , (1) is the energy barrier height of the switching divided by the temperature. In this paper, we call /Delta1the switching barrier. The thermal stability is defined as /Delta10. Here, IandIc are the current and the critical current of the spin torque switching at zero temperature. Equation (1)characterizes the switching in the thermally activated region, and is applicablefor|I|<|I c|. The exponent of the term 1 −I/I cin Eq. (1) is denoted as b. Equation (1)was analytically derived for the uniaxially anisotropic system by solving the Fokker-Planckequation, 11–14andb=2. Recently, it has also been confirmed by numerical simulations.15,16 It is difficult to derive the analytical formula of the switching barrier for an in-plane magnetized system, whichdoes not have axial symmetry due to the presence of twoanisotropic axes (an easy axis along the in-plane and a hard axisnormal to the plane). Previous analyses adopted b=1, which has been obtained by solving the Landau-Lifshitz-Gilbert(LLG) equation 17as well as the Fokker-Planck equation.18 The point of these works is that the effect of the spin torque on the switching barrier can be described by theeffective temperature 17T/(1−I/I c). However, the effective temperature approach is valid only for I/lessmuchIc17. The current dependence of the switching barrier in a relatively large currentregion ( I/similarequalI c) remains unclear, while such large current is applied to a ferromagnet in experiments to quickly observe the switching. These facts motivated us to study the switchingbarrier in the large current region. In Ref. 15, we numerically solved the LLG equation of the in-plane magnetized system,and found that the switching does not occur even if I>I c, although Ichas been assumed to be the critical current of the spin torque switching at zero temperature. We also found thatthe exponent bis larger than 2 by analyzing the numerical results with a phenomenological model of the switching. 19–21 In this paper, we develop an analytical theory of the spin torque switching of an in-plane magnetized system based onthe Fokker-Planck theory with WKB approximations, 8–10,22–26 where the temperature is assumed to be low, and the mag- nitudes of the damping and the spin torque are assumed tobe small. By assuming that the solution of the Fokker-Planckequation is given by W∝exp(−/Delta1), we find that the switching barrier is given by /Delta1=−V/integraldisplay Emax EmindEHsFs−αMF α αkBTMF α. (2) The physical meaning of Eq. (2)is as follows. The numerator in the integral arises from the drift terms of the Fokker-Planckequation, where the terms H sFsandαMF αare proportional to the work done by spin torque and the energy dissipation due tothe damping, respectively. On the other hand, the denominatorarises from the diffusion term of the Fokker-Planck equation,where the thermal fluctuation depends on the damping constantαand the temperature Tthrough the fluctuation-dissipation theorem. Then, the switching barrier is given by the integralof the ratio between the drift and the diffusion terms. The 054406-1 1098-0121/2013/87(5)/054406(9) ©2013 American Physical SocietyTANIGUCHI, UTSUMI, MARTHALER, GOLUBEV , AND IMAMURA PHYSICAL REVIEW B 87, 054406 (2013) boundaries of the integral, EminandEmax, are defined by the region of HsFs<α M F α, because the energy dissipation due to the damping overcomes the spin torque in this region, andthus, the energy absorption from the thermal bath is required toclimb up the switching barrier. The thermally activated regionis defined as the current region I/lessorequalslantI ∗ c, where I∗ csatisfies HsFs=αMF αat the saddle point of the energy map. The relation between Icin the conventional theory27andI∗ cis as follows. The initial state parallel to the easy axis becomesunstable at I=I c. However, the condition I>I cdoes not guarantee the switching. On the other hand, at I=I∗ c,t h e switching occurs without the thermal fluctuation. We derivethe analytical expression of I ∗ c, and find that I∗ c/similarequal1.27Ic. We also find that the current dependence of the switchingbarrier is well described by (1 −I/I ∗ c)b, where the value of the exponent bis approximately unity for the current I/lessorequalslantIc, while brapidly increases up to ∼2.2 with increasing current forIc/lessorequalslantI/lessorequalslantI∗ c, showing a good agreement with Refs. 17,18. The nonlinear dependence for Ic/lessorequalslantI/lessorequalslantI∗ cnewly found in this paper is important to evaluate the thermal stability becausemost experiments are performed in the current region ofI c/lessorequalslantI/lessorequalslantI∗ c. The paper is organized as follows. In Sec. II, we summarize the Fokker-Planck equation of the magnetization switching. InSec. IIIwe describe the details of the WKB approximation. Equations (22),(20), and (21) lead to the main results in the next sections. Readers who are interested in the applicationsof Eq. (2)can directly move to the next sections, Secs. IV and V. In Sec. IV, we apply the above formula to the uniaxially anisotropic system, and show that the presentformula reproduces the results b=2 derived in Refs. 11,13,14. In Sec. V, the switching barrier of an in-plane magnetized system is calculated. In Sec. VI, we compare the current results with our previous works. Section VII is devoted to the conclusion. II. FOKKER-PLANCK EQUATION FOR MAGNETIZATION SWITCHING The LLG equation is given by28–30 dm dt=−γm×H−γH sm×(np×m)+αm×dm dt, (3) where the gyromagnetic ratio and the Gilbert damping constant are denoted as γandα, respectively. m= (sinθcosϕ,sinθsinϕ,cosθ) and npare the unit vectors pointing to the directions of the magnetization of the free andpinned layers, respectively. H=−∂E/∂ (Mm) is the magnetic field, where MandEare the saturation magnetization and the magnetic energy density. For an in-plane magnetized system,Eis given by E=−MH K 2(m·ez)2+2πM2(m·ex)2 =−MH K 2z2+2πM2(1−z2) cos2ϕ, (4) where z=cosθ, andHKand 4πM are the uniaxial anisotropy field along the easy ( z) axis and the demagnetization along the hard ( x) axis, respectively. For the uniaxially anisotropicsystem discussed in Sec. IV, the demagnetization field should be absent. In Secs. IVand V, we assume that np=ez.T h e strength of the spin torque in the unit of the magnetic field, Hs, is given by Hs=¯hηI 2eMV, (5) where ηandVare the spin polarization of the current and the volume of the free layer, respectively. Although theexplicit form of the energy density is specified in Eq. (4),t h e extension of the following formula to the general system isstraightforward. Let us express Eq. (3)in terms of the canonical variables because the following formula is based on the canonicaltheory developed in Refs. 22–26. The magnetization dynamics without the spin torque and dissipation is described by thefollowing Lagrangian density, 31,32 L=−M γ˙ϕ(cosθ−1)−E. (6) The first term of Eq. (6)is the solid angle of the magnetization dynamics in the spin space, or equivalently, the Berry phase.The canonical coordinate is q=ϕ, and the momentum is defined as p=∂L/∂ ˙q=− (M/γ )(cosθ−1). In terms of the canonical variables ( q,p), the LLG Eq. (3)of the uniaxial and the in-plane magnetized system can be expressed as dq dt=1 1+α2∂E ∂p−αγg−1 (1+α2)M∂E ∂q+αγH s 1+α2, (7) dp dt=−1 1+α2∂E ∂q−αgM (1+α2)γ∂E ∂p+gMH s 1+α2, (8) where g=sin2θ. It should be noted that the spin torque term cannot be expressed as a gradient of the potential energy E,i n general, and can be regarded as a damping or antidamping. At a finite temperature, the random torque, −γm×h, should be added to the right-hand side of Eq. (3), where the components of the random field hsatisfy the fluctuation- dissipation theorem,7 /angbracketlefthi(t)hj(t/prime)/angbracketright=2D γ2δijδ(t−t/prime). (9) HereD=αγk BT/(MV) is the diffusion coefficient. Due to the random torque, the magnetization switching can beregarded as the two-dimensional Brownian motion of a pointparticle in the ( q,p) phase space. Let us introduce the distribution function of the magnetiza- tionWin the ( q,p) phase space. The Fokker-Planck equation is given by 7 ∂W ∂t=−∂ ∂qdq dtW+D 1+α2∂ ∂qg−1∂ ∂qW −∂ ∂pdp dtW+D 1+α2/parenleftbiggM γ/parenrightbigg2∂ ∂pg∂ ∂pW, (10) where dq/dt anddp/dt should be regarded as the right-hand sides of Eqs. (7)and (8). The terms proportional to D correspond to the diffusion terms while the others correspondto the drift terms. In equilibrium ( ∂W/∂t =0 andH s=0), the distribution function is identical to the Boltzmann distributionfunction ( ∝exp[−E/(k BT)]). 054406-2SPIN TORQUE SWITCHING OF AN IN-PLANE ... PHYSICAL REVIEW B 87, 054406 (2013) III. WKB APPROXIMATION In general, the distribution function determined by Eq. (10) depends on the two variables, ( q,p), and it is very difficult to solve Eq. (10) for an arbitrary system. Thus, we use the following two assumptions. First, the low temperature assumption corresponding to the WKB approximations in Refs. 8–10,22–26is employed. We assume that the distribution function takes the followingform, 22,24 W∝exp(−αS/D ). (11) In the zero temperature limit ( α/D/lessmuch1), Eq. (10) is reduced to33 ∂S ∂t=−dq dt∂S ∂q−αg−1/parenleftbigg∂S ∂q/parenrightbigg2 −dp dt∂S ∂p−αg/parenleftbiggM γ/parenrightbigg2/parenleftbigg∂S ∂p/parenrightbigg2 . (12) HereSand−∂S/∂t ≡Hcan be regarded as the effective action and the Hamiltonian density, respectively.24–26The corresponding Lagrangian density is then given by L=− ˙qλq−˙pλp−H, (13) where λq=−∂S/∂q andλp=−∂S/∂p conjugated to qand pare the counting variables.25The effective action is given by S=/integraltext dtL. Second, we utilize the fact that the Gilbert damping constant and the spin torque strength are small. The values of theGilbert damping constant αfor the conventional ferromagnetic materials 34such as Co, Fe, and Ni are on the order of 10−2. Also, the critical current of the spin torque switching27 is proportional to α|H|. Then, the switching time of the magnetization is much longer than the precession period τ, and the energy Eis approximately conserved during one period of the precession. Following Ref. 25, we introduce the new canonical variables ( E,s) accompanied by the new counting variables ( λE,λs). Then, the Lagrangian density is expressed as (see Appendix A) L=−dE dtλE−HE, (14) HE=λE∂E ∂q/bracketleftbiggαMH s 1+α2∂ ∂pm·np+αγ (1+α2)Mg−1∂E ∂q +γH s 1+α2g−1∂ ∂qm·np/bracketrightbigg −λE∂E ∂p/bracketleftbiggαMH s 1+α2∂ ∂qm·np −αM (1+α2)γg∂E ∂p−M2Hs (1+α2)γg∂ ∂pm·np/bracketrightbigg +λ2 Eα 1+α2g−1/parenleftbigg∂E ∂q/parenrightbigg2 +λ2 Eα 1+α2g/parenleftbiggM γ/parenrightbigg2/parenleftbigg∂E ∂p/parenrightbigg2 . (15) We perform the time average of Lover the constant energy orbit, L=/integraltextτ 0dtL/τ=− [M/(γτ)]/integraltext2π 0dϕL/(∂E/∂z ). Here we use the fact that the constant orbit isdetermined by the nonperturbative equation of motion,dϕ/dt =− (γ/M )(∂E/∂z ). The averaged Lagrangian density is expressed as L=−γ MdE dtλE−HE, (16) where we renormalize λEas (MλE/γ)→λEby which λE becomes a dimensionless variable. The effective Hamiltonian is given by HE=−λEγH s 1+α2Fs+λE(1+λE)αγM 1+α2Fα,(17) where FsandFαare given by Fs=1 τ/integraldisplay2π 0dϕ ∂E/∂z/bracketleftbigg (1−z2)/parenleftbigg∂E ∂z/parenrightbigg −α/parenleftbigg∂E ∂ϕ/parenrightbigg/bracketrightbigg ,(18) Fα=−1 τM2/integraldisplay2π 0dϕ ∂E/∂z/bracketleftBigg (1−z2)/parenleftbigg∂E ∂z/parenrightbigg2 +1 1−z2/parenleftbigg∂E ∂ϕ/parenrightbigg2/bracketrightBigg . (19) The term proportional to Hsin Eq. (17) describes the work done by the spin torque, while the term proportional to αλEin the second term of Eq. (17) describes the energy dissipation due to the Gilbert damping (Appendix B). The term proportional toαλ2 Earises from the thermal fluctuation. The switching barrier is obtained by integrating the La- grangian density along the switching path. It is sufficient tocalculate the integral along the optimal path 26in the low temperature limit, where the optimal path corresponds tothe switching path obtained by Eq. (3)without the thermal fluctuation. The optimal path 26can be found by solving HE=0. One of the solutions is given by E=−MH K/2, where z=± 1 and ϕis arbitrary, due to which Fs=Fα=0. The other two solutions of HE=0a r eg i v e nb y λE=0,λ∗ E, where λ∗ Eis given by λ∗ E=− 1+HsFs αMF α. (20) The equation of motion along λE=0 describes the drift in the energy space due to the competition between the spin torqueand the Gilbert damping at zero temperature. The energychange ˙E=−∂ HE/∂λEalongλE=0 is given by ˙E/vextendsingle/vextendsingle λE=0=γ(HsFs−αMF α) 1+α2. (21) On the other hand, λ∗ Ecorresponds to the time reversal path ofλE=0,26and thus, ˙E|λE=λ∗ E=− ˙E|λE=0. The switching barrier, /Delta1=αS/D , is then given by /Delta1=−V kBT/integraldisplayEmax EmindEλ∗ E, (22) which is identical to Eq. (2). The boundaries of the integral in Eq.(22) are determined as follows. According to its definition, Eq.(20),λ∗ Edepends on the work done by spin torque and the energy dissipation due to the damping (see also Appendix B). In the region λ∗ E>0, the spin torque overcomes the damping, and the magnetization can move from the initial state parallel tothe easy axis without thermal fluctuation. On the other hand,the damping exceeds the spin torque in the region λ ∗ E<0, 054406-3TANIGUCHI, UTSUMI, MARTHALER, GOLUBEV , AND IMAMURA PHYSICAL REVIEW B 87, 054406 (2013) energy, 2E/(MH K)-0.50 E I/Ic=0.2, 0.4, 0.6, 0.8, 1.0 0 -1.0-1.00.51.0 -0.8 -0.6 -0.4 -0.2(a) (b) * -0.50 E -1.00.51.0 * energy, 2E/(MH K)0 -1.0 -0.8 -0.6 -0.4 -0.2Emin EmaxE* E=0 FIG. 1. (Color online) (a) A typical dependence of λ∗ Eon the energy Efor the uniaxially anisotropic system. The energy is normalized by the factor MH K/2. The dotted line corresponds to λE=0. The switching barrier is obtained by integrating λ∗ EfromEmintoEmax, i.e., the shaded region. The lower boundary Eminis fixed to −MH K/2, while the upper boundary Emaxis located at −(MH K/2)(I/I c)2.( b )T h e dependence of λ∗ Efor the current I/I c=0.2,0.4,0.6,0.8, and 1 .0. and thus, the thermal fluctuation is required to achieve the switching. Then, the integral in Eq. (22) in the region λ∗ E<0 gives the switching barrier, i.e., the boundaries of the integralin Eq. (22) are those of λ ∗ E<0. In the latter sections, examples ofλ∗ Eare shown (see Figs. 1and5). It should be noted that the condition λ∗ E<0 is identical to HsFs<α M F αdiscussed after Eq. (2)because the energy dissipation due to the damping (∝−Fα) is always negative. Equation (22) with Eqs. (20) and(21) is the main result in this section, and enables us to calculate the current dependenceof the switching barrier in the next sections. Equation (22) is similar to Eq. (19) of Ref. 18, except for the additional condition on the integral range determined by Eq. (21).W e emphasize that this difference is crucial to obtain the exponentb, as shown in Secs. IVandV. IV . UNIAXIALLY ANISOTROPIC SYSTEM In this section, we calculate the switching barrier of the uniaxially anisotropic system, in which the energy densityis given by E=− (MH K/2) cos2θ.H e r e z=cosθcan be expressed in terms of EandHKasz=√−2E/(MH K). The metastable states of the magnetization locate at m=±ez, and the initial state is taken to be m=ez. The functions FsandFα [Eqs. (18) and(19)] of this system are given by Fs=γH Kz(1−z2), (23) Fα=γH2 K Mz2(1−z2), (24) respectively. The critical current Icof the uniaxially anisotropic system is given by Ic=2αeMV ¯hηHK. (25) The switching path, λ∗ Ein Eq. (20), is given by λ∗ E=− 1+ [I/(Icz)]. Figure 1(a) shows a typical dependence of λ∗ Eon the energy E. The region λ∗ E<0i sf r o m Emin=−MH K/2t o Emax=− (MH K/2)(I/I c)2. Figure 1(b) shows λ∗ Efor various currents. The upper boundary of λ∗ E<0,Emax, is zero at I=0, and approaches to −MH K/2 with increasing current. AtI=Ic,Emaxequals Emin. This behavior is particular forthe uniaxially anisotropic system, where the switching can be reduced to the one-dimensional problem due to the rotationalsymmetry along the zaxis. In this case, the effect of the spin torque can be described by the effective potential 11,12 Eeff=− (MH K/2) cos2θ+(MH s/α) cosθ. The effective po- tential has two minima at θ=0,πand one maximum at θ=cos−1(I/I c). The switching barrier, Eq. (22), is obtained by integrating the region of λ∗ E<0 [i.e., the shaded region of Fig. 1(a)], and is given by /Delta1=/Delta10/parenleftbigg 1−I Ic/parenrightbigg2 , (26) where /Delta10=MH K/(2kBT) is the thermal stability. Figure 2 shows the dependence of the switching barrier on the currentI/I c. The barrier height is normalized by /Delta10. The exponent of the term 1 −I/I cisb=2 in this system, which is consistent with Refs. 11,12. V . IN-PLANE MAGNETIZED SYSTEM In this section, we investigate the switching barrier of an in-plane magnetized system. The energy density of this system 1.0 0 current, I/I c0.20.40.60.8 1.0 0 0.2 0.4 0.6 0.8 FIG. 2. The dependence of the switching barrier of the uniaxially anisotropic system on the current I/I c. The barrier height is normalized by /Delta10. 054406-4SPIN TORQUE SWITCHING OF AN IN-PLANE ... PHYSICAL REVIEW B 87, 054406 (2013) 0 180 900180 90(a) hard axis easy axis(b) yx z FIG. 3. (Color online) (a) A typical energy map of an in-plane magnetized system. There are two minima ( E=−MH K/2) at θ=0,π, two maxima ( E=2πM2)a t(θ,ϕ)=(π/2,0),(π/2,π), and two saddle points ( E=0) at ( θ,ϕ)=(π/2,π/2),(π/2,3π/2). The blue and red regions correspond to the stable ( E< 0) and unstable (E> 0) regions, respectively. (b) A typical switching orbit in the in-plane magnetized system. The value of the parameters are M= 1000 emu /c.c.,HK=200 Oe, α=0.01, and T=20 K; see Ref. 15. is given by Eq. (4). The magnitudes of the demagnetization and the uniaxial anisotropy fields of the ferromagnetic materialsof conventional in-plane Spin RAM are on the order of 1 Tand 100 Oe 5, respectively. Thus, HK/(4πM) is on the order of 10−2. Figure 3(a) shows a typical energy map of an in-plane magnetized system. There are two minima of the energy(E=−MH K/2) atθ=0,πand two maxima ( E=2πM2) at (θ,ϕ)=(π/2,0),(π/2,π). Because of the large demag- netization field, the magnetization switches through one ofthe saddle points ( E=0) at (θ,ϕ)=(π/2,π/2),(π/2,3π/2). Figure 3(b) shows a typical switching orbit obtained by numer- ically solving the LLG equation. 15Starting from m=ez,t h e magnetization precesses around the easy axis, and graduallyapproaches the saddle point. Then, the magnetization passes close to the saddle point, and relaxes to the other stable state, m=−e z. Since the switching time is dominated by the time spent during the precession around the easy axis in which−MH K/2/lessorequalslantE/lessorequalslant0, it is sufficient to evaluate λ∗ Eof Eq. (22) in this energy region. The details of the calculation of the switching barrier are as follows. The variable zrelates to E,HK, and 4 πM as z=/radicalBigg 4πM cos2ϕ−2E/M 4πM cos2ϕ+HK. (27) Then, the functions FsandFα[Eqs. (18) and(19)]a r eg i v e n by Fs=2π(HK+2E/M ) τ√HK(HK+4πM), (28) Fα=4 τM/radicalBigg 4πM−2E/M HK/bracketleftBigg 2E MK/parenleftBigg/radicalBigg 4πM(HK+2E/M ) HK(4πM−2E/M )/parenrightBigg +HKE/parenleftBigg/radicalBigg 4πM(HK+2E/M ) HK(4πM−2E/M )/parenrightBigg/bracketrightBigg , (29)0.99981.0 01 0 2 0 3 0 4 0 5 0 time (ns)mz 0.8 0.61.0 01 0 2 0 3 0 4 0 5 0 time (ns) 01 0 2 0 3 0 4 0 5 0 time (ns) 01 0 2 0 3 0 4 0 5 0 time (ns)mz mz1.0 0mz1.0 0 -1.0mzmx my 1-1 01 0 -110 -1mzmx my 1-1 01 0 -110 -1 mzmx my 1-1 01 0 -110 -1 mzmx my 1-1 01 0 -110 -1(a) (b) (c) (d) FIG. 4. (Color online) The magnetization dynamics at zero temperature. The time evolution of mz(left) and the dynamic orbit (right) are shown. (a) At I=Ic. The magnetization oscillates around the initial state ( m=ez) with small amplitude. (b) and (c) AtI=r(I∗ c−Ic)+Ic[r=0.4 for (b) and r=0.8 for (c)]. The oscillation amplitude increases with increasing current. (d) At I=I∗ c, the switching occurs. where the precession period is given by τ=4 γ√HK(4πM−2E/M )K/parenleftBigg/radicalBigg 4πM(HK+2E/M ) HK(4πM−2E/M )/parenrightBigg . (30) The complete elliptic integrals of the first and second kinds are defined as K(k)=/integraltext1 0dy//radicalbig (1−k2y2)(1−y2) and E(k)=/integraltext1 0dy/radicalbig (1−k2y2)/(1−y2), respectively. In the limit of 4πM→0,τ,Fs, andFαare identical to those calculated for the uniaxially anisotropic system. It should be noted that λ∗ Eof Eq. (20) satisfies the following relations: lim E→−MH K/2λ∗ E=−/parenleftbigg 1−I Ic/parenrightbigg , (31) lim E→0λ∗ E=−/parenleftbigg 1−I I∗c/parenrightbigg , (32) 054406-5TANIGUCHI, UTSUMI, MARTHALER, GOLUBEV , AND IMAMURA PHYSICAL REVIEW B 87, 054406 (2013) energy, 2E/(MH K)-0.8-0.2 -0.4 -0.60 E r=0.2, 0.4, 0.6, 0.8, 1.0r=0.2, 0.4, 0.6, 0.8, 1.0 (I<Ic)(Ic<I<Ic*) 0 -1.0-1.00.20.4 -0.8 -0.6 -0.4 -0.2* energy, 2E/(MH K)-0.10E 0 -1.0-0.20.20.3 -0.8 -0.6 -0.4 -0.2*energy, 2E/(MH K)-0.8-0.2 -0.4 -0.60 E 0 -1.0-1.00.20.4 -0.8 -0.6 -0.4 -0.2* energy, 2E/(MH K)-0.1 -0.20 E 0 -1.00.1 -0.8 -0.6 -0.4 -0.2*Emin EmaxEmin Emax(a) (b) (I<Ic) (Ic<I<Ic*) (c) (d) 0.1I=Ic*E=0 E=0E* E* E* E=0at I=Ic E*at FIG. 5. (Color online) (a) Typical dependence of λ∗ Eof the in-plane magnetized system on the energy EforI<I c. The switching barrier is obtained by integrating λ∗ EfromEmin=−MH K/2t oEmax=0, i.e., the shaded region. (b) Typical dependence of λ∗ Eon the energy for Ic<I<I∗ c. The lower boundary of the integral, Emin, locates at −MH K/2<E min<0. (c)λ∗ EforI=IcandI=I∗ c. (d) The dependence of λ∗ Eon the various currents, I=rIcforI/lessorequalslantIcandI=r(I∗ c−Ic)+IcforIc<I/lessorequalslantI∗ c,w h e r e r=0.2,0.4,0.6,0.8, and 1 .0. where the critical currents IcandI∗ care, respectively, given by Ic=2αeMV ¯hη(HK+2πM). (33) I∗ c=4αeMV π¯hη/radicalbig 4πM (HK+4πM). (34) Below, we concentrate on the region of Ic<I∗ c(HK/(4πM)< 0.196). In the conventional ferromagnetic thin film, HKand 4πM are on the order of 100 Oe and 1 T, and thus, this condition is usually satisfied. For an infinite demagnetizationfield limit, I ∗ c/Ic/similarequal4/π=1.27. The physical meanings of IcandI∗ care as follows. In Fig. 4, we show the magnetization dynamics at the zero temperature.The parameters are M=1000 emu /c.c.,H K=200 Oe, γ=17.64 MHz /Oe,α=0.01,η=0.8, and V=π× 80×35×2.5n m3, respectively,15by which Ic=0.54 mA andI∗ c=0.68 mA, respectively. For I<I c, the damping overcomes the spin torque, and the initial state parallel tothe easy axis is stable. At I=I c, the initial state becomes unstable, and the magnetization oscillates around the easy axiswith a small constant amplitude [see Fig. 4(a)]. This has been already pointed out in Ref. 27, andI chas been considered to be the critical current of the magnetization switching. However,we emphasize that I=I cis not a critical point, because I=Icdoes not guarantee the switching.15AtIc<I<I∗ c, the oscillation amplitude increases with increasing current, asshown in Figs. 4(b) and4(c). The critical current is I ∗ c, over which the magnetization can switch its direction without thethermal fluctuation, as shown in Fig. 4(d).In Fig. 5, we show the dependence of λ ∗ Eon the energy for various currents. A typical λ∗ EforI<I cis shown in Fig. 5(a). Here, λ∗ Eis always negative because the damping exceeds the spin torque. The switching barrier is obtainedby integrating λ ∗ EfromEmin=−MH K/2t oEmax=0. On the other hand, Fig. 5(b) shows a typical λ∗ EforIc<I<I∗ c. From the initial state ( E=−MH K/2) to a certain energy Emin, the magnetization can move without the thermal fluctuationbecause the spin torque overcomes the damping ( λ ∗ E>0). At E=Emin, the magnetization dynamics is on a stable orbit, where the spin torque and the damping are balanced. The thermal fluctuation is required from EmintoEmaxto switch the magnetization direction. In Fig. 5(c),w es h o w λ∗ Efor I=IcandI=I∗ c.A tI=Ic,λ∗ E/lessorequalslant0, and λ∗ Eis zero at E=−MH K/2. On the other hand, at I=I∗ c,λ∗ E/greaterorequalslant0, and λ∗ E=0a tE=0. Figure 5(d) shows λ∗ Efor various currents. Figure 6shows the current dependence of the switching barrier obtained by numerically integrating the λ∗ Eshown in Fig. 5(d) [see Eq. (22)], in which the barrier height is normalized by /Delta10. The upper boundary of the integral range, Emax, is taken to be E=0. On the other hand, the lower boundary, Emin,i s−MH K/2f o r I/lessorequalslantIc, while it is determined by numerically solving λ∗ E=0f o rIc<I/lessorequalslantI∗ c. We find that the current dependence of the switching barrieris well described by (1 −I/I ∗ c)b(see also Appendix C). The dependence is approximately linear ( b/similarequal1) for I/lessorequalslantIc showing the consistence with Refs. 17,18. On the other hand, we find a nonlinear dependence for Ic<I/lessorequalslantI∗ c. The solid line in Fig. 7shows the dependence of the exponent bon 054406-6SPIN TORQUE SWITCHING OF AN IN-PLANE ... PHYSICAL REVIEW B 87, 054406 (2013) 1.0 0 current, I/I c*0.20.40.60.8 1.0 0 0.2 0.4 0.6 0.8 FIG. 6. The current dependence of the switching barrier normal- ized by /Delta10, i.e.,/Delta1//Delta1 0. The current magnitude is normalized by I∗ c. the current, where bis determined by the switching barrier shown in Fig. 6asb≡ln(/Delta1//Delta1 0)/ln(1−I/I∗ c). As shown, b slightly increases with increasing current for I<I c(/similarequal0.8I∗ c). ForIc/lessorequalslantI/lessorequalslantI∗ c,brapidly increases, and reaches b> 2 near I/lessorsimilarI∗ c. It should also be noted that bis not universal. The values of bwith different HKvalues are also shown in Fig. 7 by the dashed ( HK=500 Oe) and dotted ( HK=2250 Oe /similarequal 0.18×4πM) lines. As shown, the value of bdecreases with increasing HK. A further increase of HKbreaks the condition [Ic<I∗ c, or equivalently, HK/(4πM)<0.196], and beyond the scope of this paper. The nonlinear dependence of the switching barrier is important for an accurate estimation of the thermal stability. Typical experiments are performed in the large current region, I/lessorsimilarI∗ c, to quickly measure the switching. By applying a linear fit to Fig. 7, as done in the analysis of experiments,5,6the thermal stability would be significantly underestimated. VI. COMPARISON WITH PREVIOUS WORKS In Sec. III, we assume that the switching time from the initial state ( E=−MH K/2) to the saddle point ( E=0) is much longer than the precession period τ. Since the precession period [Eq. (30)] diverges at E=0, this assumptionexponent, b 1.0 current, I/I c*1.52.02.5 1.0 0 0.2 0.4 0.6 0.8HK=200, 500, 2250 (Oe) FIG. 7. The estimated value of the exponent bby analyzing Fig. 6 with a function ln( /Delta1//Delta1 0)/ln(1−I/I∗ c). The values of HKin the solid, dashed, and dotted lines are 200, 500, and 2250 Oe, respectively.0 current, I/I c*0.020.040.06 0.92 0.94 0.96 0.98 0.99 FIG. 8. The dots are the current dependence of the switching barrier in 0 .92I∗ c/lessorequalslantI/lessorequalslant0.99I∗ c. The solid line represents a fitting proportional to (1 −I/I∗ c)b/prime,w h e r e b/primeis assumed to be constant in this current region. is apparently violated in the vicinity of the saddle point. The condition under which our approximation is valid is τ|˙E|/vextendsingle/vextendsingle E=0/lessmuchMH K 2. (35) By using Eqs. (21),(28), and (29), we find that Eq. (35) can be expressed as 4πM/lessmuchHK 16α2. (36) The parameters used in Sec. Vsatisfy this condition. According to Eq. (36), the present formula does not work for an infinite demagnetization field limit, 4 πM/H K→∞ , where the switching occurs completely in-plane without precession.Then, the switching barrier is given by /Delta1 0(1−I/I c)2withb= 2 andIc=2αeMVH K/(¯hη), as shown in our previous work.12 Thus, the present work is valid for Eq. (36), while the previous work is valid for 4 πM/H K→∞ . The switching barrier in the intermediate region, HK/(16α2)/lessorsimilar4πM, remains unclear, and is beyond the scope of this paper. Let us also discuss the relation between the present work and one of our previous works in Ref. 15.I nR e f . 15,w e estimate that b∼3 for the in-plane magnetized system by numerically solving the LLG equation at various temperaturesand adopting the phenomenological model of the switching. 19 The parameters are identical to those used in Fig. 7.T h e switching currents at 0 .1 K and 20 K are 0 .67 mA ( /similarequal0.99I∗ c) and 0.62 mA ( /similarequal0.92I∗ c), respectively.35 The main difference between the present work and Refs. 15, 19is that, in Refs. 15,19the exponent bis assumed to be constant. In Fig. 8, the switching barrier shown in Fig. 6is fitted by a function proportional to (1 −I/I∗ c)b/prime, with a constant b/prime. The current range is from 0 .92I∗ cto 0.99I∗ c, according to Ref. 15. The obtained value of b/primeis 2.5, which is close to the estimated value ( b∼3) in Ref. 15. The difference may arise from the temperature dependence of /Delta1, or the current dependence of the attempt frequency neglected in the presentwork. VII. SUMMARY In summary, we have developed a theory of the spin torque switching of the in-plane magnetized system based on the 054406-7TANIGUCHI, UTSUMI, MARTHALER, GOLUBEV , AND IMAMURA PHYSICAL REVIEW B 87, 054406 (2013) Fokker-Planck theory with WKB approximation. We derived the analytical expressions of the critical currents, IcandI∗ c. The initial state parallel to the easy axis becomes unstableatI=I c, which has been derived in Ref. 27. On the other hand, at I=I∗ c(/similarequal1.27Ic), the switching occurs without the thermal fluctuation. We also find that the current dependenceof the switching barrier is well described by (1 −I/I ∗ c)b, where the value of the exponent bis approximately linear for the current I/lessorequalslantIc, while brapidly increases with increasing current for Ic/lessorequalslantI/lessorequalslantI∗ c. The nonlinear dependence for Ic/lessorequalslant I/lessorequalslantI∗ cis important for an accurate evaluation of the thermal stability because most experiments are performed in thecurrent region of I c/lessorequalslantI/lessorequalslantI∗ c.Note added. After we had submitted the manuscript, D. Pinna, A. D. Kent, and D. L.Stein informed us that they worked on a similar problemindependently. ACKNOWLEDGMENT The authors would like to acknowledge H. Kubota, H. Maehara, A. Emura, A. Fukushima, K. Yakushiji, T. Yorozu,H. Arai, K. Ando, S. Yuasa, S. Miwa, Y . Suzuki, H. Sukegawa,and S. Mitani for their valuable discussions. This work wassupported by Japan Society for the Promotion of ScienceKAKENHI Grant No. 23226001. APPENDIX A: DERIVATION OF EQS. (14)AND (15) First, we divide the Lagrangian density L,E q . (13),i n t o nonperturbative ( L0) and perturbative ( L1) parts. In terms of the canonical variables ( q,p), Eq. (3)can be expressed as dq dt=1 1+α2∂E ∂p−αMH s 1+α2∂ ∂pm·np−αγ (1+α2)Mg−1∂E ∂q −γH s 1+α2g−1∂ ∂qm·np, (A1) dp dt=−1 1+α2∂E ∂q+αMH s 1+α2∂ ∂qm·np−αM (1+α2)γg∂E ∂p −M2Hs (1+α2)γg∂ ∂pm·np. (A2) Equations (A1) and(A2) can be directly obtained from Eq. (3) for the general system by expressing Eq. (3)in terms of the spherical coordinate, ( θ,ϕ). By using the explicit forms of dq/dt anddp/dt inH [Eqs. (A1) and(A2) ], the nonperturbative Lagrangian density is given by L0=−λq/parenleftbiggdq dt−1 1+α2∂E ∂p/parenrightbigg −λp/parenleftbiggdp dt+1 1+α2∂E ∂q/parenrightbigg . (A3) On the other hand, the perturbative Lagrangian density is given by L1=−λq/bracketleftbiggαMH s 1+α2∂ ∂pm·np+αγ (1+α2)Mg−1∂E ∂q +γH s 1+α2g−1∂ ∂qm·np/bracketrightbigg +λp/bracketleftbiggαMH s 1+α2∂ ∂qm·np−αM (1+α2)γg∂E ∂p−M2Hs (1+α2)γg∂ ∂pm·np/bracketrightbigg −λ2 qα 1+α2g−1−λ2 pα 1+α2g/parenleftbiggM γ/parenrightbigg2 . (A4) The canonical transformation from ( q,p)t o( E,s)i s accompanied by the canonical transformation of the count-ing variables as λ q=(∂E/∂q )λE+(∂s/∂q )λsandλp= (∂E/∂p )λE+(∂s/∂p )λs. The nonperturbative Lagrangian is then given by L0=−λE(dE/dt )−λs[(ds/dt )−[s,H]P], where d/dt=(∂/∂q )(dq/dt )+(∂/∂p )(dp/dt ) and [,]Pis the Poisson bracket. In the small damping limit, scan be regarded as a physical time t. Since sis a conjugated variable of E ([s,H]P=1), the nonperturbative Lagrangian density is given by Eq. (14). Since the switching barrier is determined by L0 and is independent of λs,w es e t λs=0, according to Ref. 25. Then, HE=−L1is given by Eq. (15). APPENDIX B: WORK DONE BY SPIN TORQUE AND DAMPING Here we show that the functions FsandFαare proportional to the work done by the spin torque and damping. The time evolution of the magnetic energy, dE/dt = −MH·(dm/dt), is calculated by using Eq. (3)as dE dt=Ws+Wα, (B1) whereWsandWαare given by Ws=γMH s 1+α2[np·H−(m·np)(m·H)−αnp·(m×H)], (B2) Wα=−αγM 1+α2[H2−(m·H)2]. (B3) Here,Wsis the work done by spin torque while Wαis the energy dissipation due to the damping. It should be noted thatW αis always negative, while Wsis either positive or negative depending on the current direction. Let us consider the in-planemagnetized system as an example, where the magnetic field isgiven by H=(−4πMm x,0,HKmz). Then, the time averages ofWsandWαover one period of the precession around the easy axis are given by Ws=MH s 1+α2Fs, (B4) Wα=−αM2 1+α2Fα, (B5) respectively, where FsandFαare given by Eqs. (28) and (29), respectively. Thus, we can verify that FsandFαare proportional to the work done by the spin torque and thedamping, respectively. APPENDIX C: CURRENT DEPENDENCE OF THE SWITCHING BARRIER NEAR I∗ c Here, we derive an analytical formula of the switching barrier near I∗ c. For simplicity, we use the normalized variables k=HK/4πM,ε=E/4πM2, ands=Hs/4πM. First, let us derive the approximated form of λE[Eq. (20)] around ε∼0. 054406-8SPIN TORQUE SWITCHING OF AN IN-PLANE ... PHYSICAL REVIEW B 87, 054406 (2013) The Taylor expansions of HsFsτandαMF ατare, respectively, given by HsFsτ 4πM=2π(k+2ε)s√k(1+k), (C1) αMF ατ 4πM/similarequalα/parenleftbigg 4√ k+2(1−k)ε√ k/braceleftbigg 1−ln/bracketleftbigg −(1+k)ε 8k/bracketrightbigg/bracerightbigg/parenrightbigg , (C2) where εlnεterm appears and thus it is nonanalytic at /epsilon1=0. Then, the approximated λ∗ Eis given by λ∗ E/similarequal−/parenleftbigg 1−I I∗c/parenrightbigg +Iε 2I∗ck/braceleftbigg 3+k+(1−k)l n/bracketleftbigg −(1+k)ε 8k/bracketrightbigg/bracerightbigg . (C3)The energy Emin=4πM2ε0corresponding to the intersection ofλ∗ EandλE=0 can be obtained by solving the following self-consistent relation: ε0=/parenleftbigg 1−I I∗c/parenrightbigg2k(I∗ c/I) 3+k+(1−k)l n [−(1+k)ε0/(8k)].(C4) Up to the first order of Emin, the switching barrier is given by /Delta1/similarequal−EminV kBT/parenleftbigg 1−I I∗c/parenrightbigg , (C5) where Emin=4πM2ε0is determined by Eq. (C4) . Equation (C5) is only valid close to the switching current I∗ cwhere Eminlocates near E=0. *h-imamura@aist.go.jp 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, P h y s .R e v .B 54, 9353 (1996). 3S. Yuasa, J. Phys. Soc. Jpn. 77, 031001 (2008). 4Y . Suzuki and H. Kubota, J. Phys. Soc. Jpn. 77, 031002 (2008). 5S. Yakata, H. Kubota, T. Sugano, T. Seki, K. Yakushiji, A. 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Kent, Appl. Phys. Lett. 101, 262401 (2012); see also arXiv:1210.7675 and arXiv:1210.7682 . 17R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett. 92, 088302 (2004). 18D. M. Apalkov and P. B. Visscher, Phys. Rev. B 72, 180405 (2005).19T. Taniguchi and H. Imamura, J. Nanosci. Nanotechnol. 12, 7520 (2012). 20T. Taniguchi and H. Imamura, J. Appl. Phys. 111, 07C901 (2012). 21T. Taniguchi and H. Imamura, IEEE Trans. Magn. 48, 3803 (2012). 22E. Ben-Jacob, D. J. Bergman, B. J. Matkowsky, and Z. Schuss,Phys. Rev. A 26, 2805 (1982). 23R. S. Maier and D. L. Stein, P h y s .R e v .E 48, 931 (1993). 24V . N. Smelyanskiy, M. I. Dykman, and R. S. Maier, Phys. Rev. E 55, 2369 (1997). 25E. V . Sukhorukov and A. N. Jordan, Phys. Rev. Lett. 98, 136803 (2007). 26A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, 2011). 27J. Z. Sun, Phys. Rev. B 62, 570 (2000). 28L. Landau and E. Lifshits, Phys. Z. Sowjetunion 8, 153 (1935). 29E. M. Lifshitz and L. P. Pitaevskii, in Statistical Physics (Part 2) ,Course of Theoretical Physics , 1st ed., V ol. 9 (Butterworth- Heinemann, Oxford, 1980), Chap. 7. 30T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 31H.-B. Braun and D. Loss, Phys. Rev. B 53, 3237 (1996). 32S. Maekawa, ed., Concepts in Spin Electronics (Oxford Science Publications, Oxford, 2006). 33In our definition, the dimension of both SandDare taken to be 1/time, to make the notation simple. The dimension of Scan be changed to that of the action by multiplying the factor V/γ2. 34M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y . Ando, A. Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys. 45, 3889 (2006). 35The switching time shown in Fig. 4of Ref. 15is converted to the switching current by multiplying the current sweep rate κ. 054406-9

PhysRevB.102.245411.pdf

PHYSICAL REVIEW B 102, 245411 (2020) Dynamical and current-induced Dzyaloshinskii-Moriya interaction: Role for damping, gyromagnetism, and current-induced torques in noncollinear magnets Frank Freimuth ,1,2,*Stefan Blügel,1and Yuriy Mokrousov1,2 1Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany 2Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany (Received 17 August 2020; accepted 24 November 2020; published 9 December 2020) Both applied electric currents and magnetization dynamics modify the Dzyaloshinskii-Moriya interaction (DMI), which we call current-induced DMI (CIDMI) and dynamical DMI (DDMI), respectively. We report atheory of CIDMI and DDMI. The inverse of CIDMI consists in charge pumping by a time-dependent gradientof magnetization ∂ 2M(r,t)/∂r∂t, while the inverse of DDMI describes the torque generated by ∂2M(r,t)/∂r∂t. In noncollinear magnets, CIDMI and DDMI depend on the local magnetization direction. The resulting spatialgradients correspond to torques that need to be included into the theories of Gilbert damping, gyromagnetism,and current-induced torques (CITs) in order to satisfy the Onsager reciprocity relations. CIDMI is related tothe modification of orbital magnetism induced by magnetization dynamics, which we call dynamical orbitalmagnetism (DOM), and spatial gradients of DOM contribute to charge pumping. We present applications of thisformalism to the CITs and to the torque-torque correlation in textured Rashba ferromagnets. DOI: 10.1103/PhysRevB.102.245411 I. INTRODUCTION Since the Dzyaloshinskii-Moriya interaction (DMI) con- trols the magnetic texture of domain walls and skyrmions,methods to tune this chiral interaction by external means haveexciting prospects. Application of gate voltage [ 1–3]o rl a s e r pulses [ 4] are promising ways to modify DMI. Additionally, theory predicts that in magnetic trilayer structures the DMI inthe top magnetic layer can be controlled by the magnetizationdirection in the bottom magnetic layer [ 5]. Moreover, meth- ods to generate spin currents may be used to induce DMI,which is predicted by the relations between the two [ 6,7]. Recent experiments show that also electric currents modifyDMI in metallic magnets, which leads to large changes inthe domain-wall velocity [ 8,9]. However, a rigorous theoret- ical formalism for the investigation of current-induced DMI(CIDMI) in metallic magnets has been lacking so far, and thedevelopment of such a formalism is one goal of this paper. Recently, a Berry phase theory of DMI [ 6,10,11] has been developed, which formally resembles the modern theory oforbital magnetization [ 12–14]. Orbital magnetism is modi- fied by the application of an electric field, which is knownas the orbital magnetoelectric response [ 15]. In the case of insulators it is straightforward to derive the expressions forthe magnetoelectric response directly. However, in metals it ismuch easier to derive expressions instead for the inverse of themagnetoelectric response, i.e., for the generation of electriccurrents by time-dependent magnetic fields [ 16]. The inverse current-induced DMI (ICIDMI) consists in charge pumpingby time-dependent gradients of magnetization. Due to theanalogies between orbital magnetism and the Berry phase *Corresponding author: f.freimuth@fz-juelich.detheory of DMI one may expect that in metals it is convenientto obtain expressions for ICIDMI, which can then be usedto describe the CIDMI by exploiting the reciprocity betweenCIDMI and ICIDMI. We will show in this paper that this isindeed the case. In noncentrosymmetric ferromagnets spin-orbit interaction (SOI) generates torques on the magnetization—the so-calledspin-orbit torques (SOTs)—when an electric current is applied[17]. The Berry phase theory of DMI [ 6,10,11] establishes a relation to SOTs. The formal analogies between orbital mag-netism and DMI have been shown to be a very useful guidingprinciple in the development of the theory of SOTs driven byheat currents [ 18]. In particular, it is fruitful to consider the DMI coefficients as a spiralization, which is formally analo-gous to magnetization. In the theory of thermoelectric effectsin magnetic systems, the curl of magnetization describes abound current, which cannot be measured in transport ex-periments and needs to be subtracted from the Kubo linearresponse in order to obtain the measurable current [ 19–21]. Similarly, in the theory of the thermal spin-orbit torque spatialgradients of the DMI spiralization, which result from the tem-perature gradient together with the temperature dependence ofDMI, need to be subtracted in order to obtain the measurabletorque and to satisfy a Mott-like relation [ 10,18]. In non- collinear magnets the question arises whether gradients of thespiralization that are due to the magnetic texture correspond totorques like those from thermal gradients. We will show thatindeed the spatial gradients of CIDMI need to be included intothe theory of current-induced torques (CITs) in noncollinearmagnets in order to satisfy the Onsager reciprocity relations[22]. When the system is driven out of equilibrium by mag- netization dynamics rather than electric current one mayexpect DMI to be modified as well. The inverse effect of this 2469-9950/2020/102(24)/245411(21) 245411-1 ©2020 American Physical SocietyFREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) dynamical DMI (DDMI) consists in the generation of torques by time-dependent magnetization gradients. In noncollinearmagnets the DDMI spiralization varies in space. We willshow that the resulting gradient corresponds to a torque thatneeds to be considered in the theory of Gilbert damping andgyromagnetism in noncollinear magnets. This paper is structured as follows. In Sec. II A,w e give an overview of CIT in noncollinear magnets and intro-duce the notation. In Sec. II B, we describe the formalism used to calculate the response of electric current to time-dependent magnetization gradients. In Sec. II C,w es h o w that current-induced DMI (CIDMI) and electric current drivenby time-dependent magnetization gradients are reciprocal ef-fects. This allows us to obtain an expression for CIDMIbased on the formalism of Sec. II B. In Sec. II D,w ed i s - cuss that time-dependent magnetization gradients generateadditionally torques on the magnetization and show that theinverse effect consists in the modification of DMI by magne-tization dynamics, which we call dynamical DMI (DDMI).In Sec. II E, we demonstrate that magnetization dynamics induces orbital magnetism, which we call dynamical orbitalmagnetism (DOM) and show that DOM is related to CIDMI.In Sec. II F, we explain how the spatial gradients of CIDMI and DOM contribute to the direct and to the inverse CIT,respectively. In Sec. II G, we discuss how the spatial gradi- ents of DDMI contribute to the torque-torque correlation. InSec. II H, we complete the formalism used to calculate the CIT in noncollinear magnets by adding the chiral contribution ofthe torque-velocity correlation. In Sec. II I, we finalize the the- ory of the inverse CIT by adding the chiral contribution of thevelocity-torque correlation. In Sec. II J, we finish the compu- tational formalism of gyromagnetism and damping by addingthe chiral contribution of the torque-torque correlation and theresponse of the torque to the time-dependent magnetizationgradients. In Sec. III, we discuss the symmetry properties of the response to time-dependent magnetization gradients. InSec. IV A , we present the results for the chiral contributions to the direct and the inverse CIT in the Rashba model andshow that both the perturbation by the time-dependent mag-netization gradient and the spatial gradients of CIDMI andDOM need to be included to ensure that they are reciprocal. InSec. IV B , we present the results for the chiral contribution to the torque-torque correlation in the Rashba model and showthat both the perturbation by the time-dependent magneti-zation gradient and the spatial gradients of DDMI need tobe included to ensure that it satisfies the Onsager symmetryrelations. This paper ends with a summary in Sec. V. II. FORMALISM A. Direct and inverse current-induced torques in noncollinear magnets Even in collinear magnets the application of an electric fieldEgenerates a torque TCIT1on the magnetization when inversion symmetry is broken [ 17,23]: TCIT1 i=/summationdisplay jtij(ˆM)Ej, (1)where tij(ˆM) is the torkance tensor, which depends on the magnetization direction ˆM. This torque is called spin-orbit torque (SOT), but we denote it here CIT1, because it is onecontribution to the current-induced torques (CITs) in non-collinear magnets. Inversely, magnetization dynamics pumpsa charge current J ICIT1according to [ 24] JICIT1 i=/summationdisplay jtji(−ˆM)ˆej·/bracketleftbigg ˆM×∂ˆM ∂t/bracketrightbigg , (2) where ˆejis a unit vector that points into the jth spatial direction. Generally, JICIT1can be explained by the inverse spin-orbit torque [ 24] or the magnonic charge pumping [ 25]. We denote it here by ICIT1, because it is one contribution tothe inverse CIT in noncollinear magnets. In the special case ofmagnetic bilayers, one important mechanism responsible forJ ICIT1arises from the combination of spin pumping and the inverse spin Hall effect [ 26,27]. In noncollinear magnets, there is a second contribution to the CIT, which is proportional to the spatial derivatives ofmagnetization [ 28]: T CIT2 i=/summationdisplay jklχCIT2 ijklEjˆek·/bracketleftbigg ˆM×∂ˆM ∂rl/bracketrightbigg . (3) The description of noncollinearity by the derivatives ∂ˆM/∂rl is only applicable when the magnetization direction changes slowly in space like in magnetic skyrmions with large ra-dius and in wide magnetic domain walls. In order to treatnoncollinear magnets such as Mn 3Sn [29], where the mag- netization direction varies strongly on the scale of one unitcell, Eq. ( 3) needs to be modified, which is beyond the scope of the present paper. The adiabatic and the non-adiabatic[30] spin transfer torques are two important contributions toχ CIT2 ijkl, but the interplay between broken inversion sym- metry, SOI, and noncollinearity can lead to a large numberof additional mechanisms [ 22,31]. Similarly, the current pumped by magnetization dynamics contains a contributionthat is proportional to the spatial derivatives of magnetization[22,32,33]: J ICIT2 i=/summationdisplay jklχICIT2 ijkl ˆej·/bracketleftbigg ˆM×∂ˆM ∂t/bracketrightbigg ˆek·/bracketleftbigg ˆM×∂ˆM ∂rl/bracketrightbigg .(4) TCIT2 i andJICIT2 i can be considered as chiral contributions to the CIT and to the ICIT, respectively, because they distinguishbetween left- and right-handed spin spirals. Due to the reci-procity between direct and inverse CIT [ 22,24] the coefficients χ ICIT2 ijkl andχCIT2 jiklare related according to χICIT2 ijkl (ˆM)=χCIT2 jikl(−ˆM). (5) B. Response of electric current to time-dependent magnetization gradients In order to compute JICIT2based on the Kubo linear response formalism it is necessary to split it into two contri-butions, J ICIT2aandJICIT2b. While JICIT2ais obtained as linear response to the perturbation by a time-dependent magnetiza- tion gradient in a collinear ferromagnet, JICIT2bis obtained as 245411-2DYNAMICAL AND CURRENT-INDUCED … PHYSICAL REVIEW B 102, 245411 (2020) linear response to the perturbation by magnetization dynam- ics in a noncollinear ferromagnet. Therefore, as will becomeclear below, J ICIT2acan be expressed by a correlation function of two operators, because it describes the response of thecurrent to a time-dependent magnetization gradient: a time-dependent magnetization gradient is a single perturbation, which is described by a single perturbing operator. In contrast,J ICIT2binvolves the correlation of three operators, because it describes the response of the current to magnetization dy-namics in the presence of perturbation by noncollinearity.These are twoperturbations: one perturbation by the magne- tization dynamics and a second perturbation to describe thenoncollinearity. In the Kubo formalism the expressions for theresponse one the one hand to a time-dependent magnetizationgradient, which is described by a single perturbing opera-tor, and the response on the other hand to a time-dependentmagnetization in the presence of a magnetization gradient,which is described by two perturbing operators, are different.Therefore we split J ICIT2into these two contributions, which we call JICIT2aandJICIT2b. In the remainder of this section we discuss the calculation of the contribution JICIT2a.T h e contribution JICIT2bis discussed in section II Ibelow. JICIT2ais determined by the second derivative of magne- tization with respect to time and space variables and can bewritten as J ICIT2a i=/summationdisplay jkχICIT2a ijk∂2ˆMj ∂rk∂t. (6) A nonzero second derivative∂2ˆMj ∂rk∂tis what we refer to as a time-dependent magnetization gradient . We will show below that in special cases∂2ˆMj ∂rk∂tcan be expressed in terms of the products∂ˆMl ∂rk∂ˆMl ∂t, which will allow us to rewrite JICIT2a i in the form of Eq. ( 4) in the cases relevant for the chiral ICIT. How- ever, as will become clear below, Eq. ( 6) is the most general expression for the response to time-dependent magnetizationgradients, and it cannot generally be rewritten in the form ofEq. ( 4): This is only possible when it describes a contribution to the chiral ICIT. J ICIT2aoccurs in two different situations, which need to be distinguished. In one case, the magnetization gradient variesin time like sin( ωt) everywhere in space. An example is ˆM(r,t)=⎛ ⎝ηsin(q·r)s i n (ωt) 0 1⎞ ⎠, (7) where ηis the amplitude and the derivatives at t=0 and r=0 are ∂ˆM(r,t) ∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0=∂ˆM(r,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0=0( 8 ) and ∂2ˆM(r,t) ∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0=⎛ ⎝ηqiω 00⎞ ⎠. (9) In the other case, the magnetic texture varies like a propa- gating wave, i.e., proportional to sin( q·r−ωt). An exampleis given by ˆM(r,t)=⎛ ⎝ηsin(q·r−ωt) 0 1− η2 2sin2(q·r−ωt)⎞ ⎠, (10) where the derivatives at t=0 and r=0a r e ∂ˆM(r,t) ∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0=⎛ ⎝ηqi 0 0⎞ ⎠, (11) ∂ˆM(r,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0=⎛ ⎝−ηω 00⎞ ⎠, (12) and ∂ 2ˆM(r,t) ∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0=⎛ ⎝0 0 η2qiω⎞ ⎠. (13) In the latter example, Eq. ( 10), the second derivative, Eq. ( 13), is along the magnetization ˆM(r=0,t=0), while in the former example, Eq. ( 7), the second derivative, Eq. ( 9), is perpendicular to the magnetization when r=0 and t=0. We assume that the Hamiltonian is given by H(r,t)=−¯h2 2me/Delta1+V(r)+μBˆM(r,t)·σ/Omega1xc(r) +1 2ec2μBσ·[∇V(r)×v], (14) where the first term describes the kinetic energy, the sec- ond term is a scalar potential, /Omega1xc(r) in the third term is the exchange field, and the last term describes the spin-orbitinteraction. Around t=0 and r=0 we can decompose the Hamiltonian as H(r,t)=H 0+δH(r,t), where H0is ob- tained from H(r,t) by replacing ˆM(r,t)b y ˆM(r=0,t=0) and δH(r,t)=∂H0 ∂ˆMxηsin(q·r)s i n (ωt) =μB/Omega1xc(r)σxηsin(q·r)s i n (ωt) (15) in the case of the first example, Eq. ( 7). In the case of the second example, Eq. ( 10), δH(r,t)/similarequal∂H ∂ˆMxηsin(q·r−ωt) +∂H ∂ˆMzη2sin(q·r)s i n (ωt), (16) where for small randtonly the second term on the right- hand side contributes to∂2H(r,t) ∂rk∂t. We consider here only the time-dependence of the exchange field direction and ignorethe time-dependence of the exchange field magnitude /Omega1 xc(r) that is induced by the time-dependence of the exchange fielddirection. While the variation of the exchange field magnitudedrives currents and torques as well, as shown in Ref. [ 34], the variation of the exchange field magnitude is a small responseand therefore these secondary responses are suppressed inmagnitude when compared to the direct primary responsesof the current and torque to the variation in the exchangefield direction. We will use the perturbations Eqs. ( 15) and 245411-3FREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) (16) in order to compute the response of current and torque within the Kubo response formalism. An alternative approachfor the calculation of the response to time-dependent fields isvariational linear response, which has been applied to the spinsusceptibility by Savrasov [ 35]. The perturbation by the time-dependent gradient can be written as δH=∂H ∂ˆM·∂2ˆM ∂ri∂tsin(qiri) qisin(ωt) ω, (17) which turns into Eq. ( 15) when Eq. ( 9) is inserted. When Eq. ( 13) is inserted it turns into the second term in Eq. ( 16). In Appendix A, we derive the linear response to perturba- tions of the type of Eq. ( 17) and show that the corresponding coefficient χICIT2a ijk in Eq. ( 6) can be expressed as χICIT2a ijk=ie 4π¯h2/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr[viRvkRROjR+viRRvkROjR −viRROjRvkR−viRvkROjAA +viROjAvkAA+viROjAAvkA −viRvkRROjA−viRRvkROjA +viRROjAvkA+viAvkAOjAA −viAOjAvkAA−viAOjAAvkA], (18) where R=GR k(E) and A=GA k(E) are shorthands for the retarded and advanced Green’s functions, respectively, andO j=∂H/∂ˆMj.e>0 is the positive elementary charge. In the case of the perturbation of the type Eq. ( 7), the second derivative∂2ˆM ∂ri∂tis perpendicular to M. In this case it is convenient to rewrite Eq. ( 6)a s JICIT2a i=/summationdisplay jkχICIDMI ijk ˆej·/bracketleftbigg ˆM×∂2ˆM ∂rk∂t/bracketrightbigg , (19) where the coefficients χICIDMI ijk are given by χICIDMI ijk =ie 4π¯h2/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr[viRvkRRTjR+viRRvkRTjR −viRRTjRvkR−viRvkRTjAA +viRTjAvkAA+viRTjAAvkA −viRvkRRTjA−viRRvkRTjA +viRRTjAvkA+viAvkATjAA −viATjAvkAA−viATjAAvkA], (20) and T=ˆM×∂H ∂ˆM(21) is the torque operator. In Sec. II C, we will explain that χICIDMI ijk describes the inverse of current-induced DMI (ICIDMI). In the case of the perturbation of the type of Eq. ( 10), the second derivative∂2ˆMj ∂rk∂tmay be rewritten as product of the firstderivatives∂ˆMl ∂tand∂ˆMl ∂rk. This may be seen as follows: ∂H ∂ˆM·∂2ˆM ∂ri∂t =∂2H ∂t∂ri =∂ ∂t/bracketleftbigg/parenleftbigg ˆM×∂H ∂ˆM/parenrightbigg ·/parenleftbigg ˆM×∂ˆM ∂ri/parenrightbigg/bracketrightbigg =/bracketleftbigg/parenleftbigg∂ˆM ∂t×∂H ∂ˆM/parenrightbigg ·/parenleftbigg ˆM×∂ˆM ∂ri/parenrightbigg/bracketrightbigg =/bracketleftbigg/parenleftbigg/parenleftbigg ˆM×∂ˆM ∂t/parenrightbigg ׈M/parenrightbigg ×∂H ∂ˆM/bracketrightbigg ·/bracketleftbigg ˆM×∂ˆM ∂ri/bracketrightbigg =−/bracketleftbigg ˆM×∂ˆM ∂t/bracketrightbigg ·/bracketleftbigg ˆM×∂ˆM ∂ri/bracketrightbigg/bracketleftbigg ˆM·∂H ∂ˆM/bracketrightbigg =−∂ˆM ∂t·∂ˆM ∂ri/bracketleftbigg ˆM·∂H ∂ˆM/bracketrightbigg . (22) This expression is indeed satisfied by Eqs. ( 11)–(13): ∂ˆM ∂ri·∂ˆM ∂t=−∂2ˆM ∂ri∂t·ˆM (23) atr=0,t=0. Consequently, Eq. ( 6) can be rewritten as JICIT2a i=/summationdisplay jkχICIT2a ijk∂2ˆMj ∂rk∂t =−/summationdisplay jklχICIT2a ijk∂ˆMl ∂rk∂ˆMl ∂t[1−δjl] =/summationdisplay jklχICIT2a ijkl ˆej·/bracketleftbigg ˆM×∂ˆM ∂t/bracketrightbigg ˆek·/bracketleftbigg ˆM×∂ˆM ∂rl/bracketrightbigg ,(24) where χICIT2a ijkl=−/summationdisplay mχICIT2a iml [1−δjm]δjk. (25) Thus Eqs. ( 24) and ( 25) can be used to express JICIT2a i in the form of Eq. ( 4). C. Direct and inverse CIDMI Equation ( 20) describes the response of the electric cur- rent to time-dependent magnetization gradients of the typeEq. ( 15). The reciprocal process consists in the current- induced modification of DMI. This can be shown byexpressing the DMI coefficients as [ 10] D ij=1 V/summationdisplay nf(Ekn)/integraldisplay d3r(ψkn(r))∗Dijψkn(r) =1 V/summationdisplay nf(Ekn)/integraldisplay d3r(ψkn(r))∗Ti(r)rjψkn(r),(26) where we defined the DMI-operator Dij=Tirj.U s i n gt h e Kubo formalism the current-induced modification of DMImay be written as D CIDMI ij=/summationdisplay kχCIDMI kij Ek (27) 245411-4DYNAMICAL AND CURRENT-INDUCED … PHYSICAL REVIEW B 102, 245411 (2020) with χCIDMI kij=1 Vlim ω→0/bracketleftbigge ¯hωIm/angbracketleft/angbracketleftDij;vk/angbracketright/angbracketrightR(¯hω)/bracketrightbigg , (28) where /angbracketleft/angbracketleftDij;vk/angbracketright/angbracketrightR(¯hω)=−i/integraldisplay∞ 0dteiωt/angbracketleft[Dij(t),vk(0)]−/angbracketright(29) is the Fourier transform of a retarded function and Vis the volume of the unit cell. Since the position operator rin the DMI operator Dij= Tirjis not compatible with Bloch periodic boundary condi- tions, we do not use Eq. ( 28) for numerical calculations of CIDMI. However, it is convenient to use Eq. ( 28) in order to demonstrate the reciprocity between direct and inverseCIDMI. Inverse CIDMI (ICIDMI) describes the electric current that responds to the perturbation by a time-dependent magnetiza-tion gradient according to J ICIDMI k =/summationdisplay ijχICIDMI kij ˆei·/bracketleftbigg ˆM×∂2ˆM ∂t∂rj/bracketrightbigg . (30) The perturbation by a time-dependent magnetization gradient may be written as δH=−/summationdisplay jm·∂2ˆM ∂t∂rjrj/Omega1xc(r)sin(ωt) ω =/summationdisplay jT·/bracketleftbigg ˆM×∂2ˆM ∂t∂rj/bracketrightbigg rjsin(ωt) ω =/summationdisplay ijDijˆei·/bracketleftbigg ˆM×∂2ˆM ∂t∂rj/bracketrightbiggsin(ωt) ω. (31) Consequently, the coefficient χICIDMI kij is given by χICIDMI kij =1 Vlim ω→0/bracketleftBige ¯hωIm/angbracketleft/angbracketleftvk;Dij/angbracketright/angbracketrightR(¯hω)/bracketrightBig . (32) Using /angbracketleft/angbracketleftDij;vk/angbracketright/angbracketrightR(¯hω,ˆM)=− /angbracketleft /angbracketleft vk;Dij/angbracketright/angbracketrightR(¯hω,−ˆM), (33) we find that CIDMI and ICIDMI are related through the equa- tions χCIDMI kij (ˆM)=−χICIDMI kij (−ˆM). (34) In order to calculate CIDMI we use Eq. ( 20) for ICIDMI and then use Eq. ( 34) to obtain CIDMI. The perturbation Eq. ( 16) describes a different kind of time-dependent magnetization gradient, for which the recip-rocal effect consists in the modification of the expectationvalue/angbracketleftσ·ˆMr j/angbracketright. However, while the modification of /angbracketleftTirj/angbracketrightby an applied current can be measured [ 8,9] from the change of the DMI constant Dij, the quantity /angbracketleftσ·ˆMrj/angbracketrighthas not been considered so far in ferromagnets. In noncollinear magnetsthe quantity /angbracketleftσr j/angbracketrightcan be used to define spin toroidization [36]. Therefore, while the perturbation of the type of Eq. ( 15) is related to CIDMI and ICIDMI, which are both accessibleexperimentally [ 8,9], in the case of the perturbation of the type of Eq. ( 16) we expect that only the effect of driving current bythe time-dependent magnetization gradient is easily accessible experimentally, while its inverse effect is difficult to measure. D. Direct and inverse dynamical DMI Not only applied electric currents modify DMI, but also magnetization dynamics, which we call dynamical DMI(DDMI). DDMI can be expressed as D DDMI ij=/summationdisplay kχDDMI kij ˆek·/bracketleftbigg ˆM×∂ˆM ∂t/bracketrightbigg . (35) In Sec. II G, we will show that the spatial gradient of DDMI contributes to damping and gyromagnetism in noncollinearmagnets. The perturbation used to describe magnetizationdynamics is given by [ 24] δH=sin(ωt) ω/parenleftbigg ˆM×∂ˆM ∂t/parenrightbigg ·T. (36) Consequently, the coefficients χDDMI kij may be written as χDDMI kij=−1 Vlim ω→0/bracketleftbigg1 ¯hωIm/angbracketleft/angbracketleftDij;Tk/angbracketright/angbracketrightR(¯hω)/bracketrightbigg . (37) Since the position operator in Dijis not compatible with Bloch periodic boundary conditions, we do not use Eq. ( 37) for numerical calculations of DDMI, but instead we obtainit from its inverse effect, which consists in the generation oftorques on the magnetization due to time-dependent magneti-zation gradients. These torques can be written as T IDDMI k =/summationdisplay ijχIDDMI kij ˆei·/bracketleftbigg ˆM×∂2ˆM ∂t∂rj/bracketrightbigg , (38) where the coefficients χIDDMI kij are χIDDMI kij =1 Vlim ω→0/bracketleftbigg1 ¯hωIm/angbracketleft/angbracketleftTk;Dij/angbracketright/angbracketrightR(¯hω)/bracketrightbigg , (39) because the perturbation by the time-dependent gradient can be expressed in terms of Dijaccording to Eq. ( 31) and because the torque on the magnetization is described by −T[23]. Consequently, DDMI and IDDMI are related by χDDMI kij (ˆM)=−χIDDMI kij (−ˆM). (40) For numerical calculations of IDDMI, we use χIDDMI ijk =i 4π¯h2/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr[TiRvkRRTjR+TiRRvkRTjR −TiRRTjRvkR−TiRvkRTjAA +TiRTjAvkAA+TiRTjAAvkA −TiRvkRRTjA−TiRRvkRTjA +TiRRTjAvkA+TiAvkATjAA −TiATjAvkAA−TiATjAAvkA], (41) which is derived in Appendix A. In order to obtain DDMI we calculate IDDMI from Eq. ( 41) and use the reciprocity relation Eq. ( 40). 245411-5FREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) Equation ( 38) is valid for time-dependent magnetization gradients that lead to perturbations of the type of Eq. ( 15). Perturbations of the second type, Eq. ( 16), will induce torques on the magnetization as well. However, the inverse effect isdifficult to measure in that case, because it corresponds to themodification of the expectation value /angbracketleftσ·ˆMr j/angbracketrightby magnetiza- tion dynamics. Therefore, while in the case of Eq. ( 15), both direct and inverse response are expected to be measurable andcorrespond to IDDMI and DDMI, respectively, we expect thatin the case of Eq. ( 16) only the direct effect, i.e., the response of the torque to the perturbation, is easy to observe. E. Dynamical orbital magnetism (DOM) Magnetization dynamics does not only induce DMI, but also orbital magnetism, which we call dynamical orbital mag-netism (DOM). It can be written as M DOM ij=/summationdisplay kχDOM kijˆek·/bracketleftbigg ˆM×∂ˆM ∂t/bracketrightbigg , (42) where we introduced the notation MDOM ij=e V/angbracketleftvirj/angbracketrightDOM, (43) which defines a generalized orbital magnetization, such that MDOM i=1 2/summationdisplay jk/epsilon1ijkMDOM jk (44) corresponds to the usual definition of orbital magnetization. The coefficients χDOM kij are given by χDOM kij=−1 Vlim ω→0/bracketleftbigge ¯hωIm/angbracketleft/angbracketleftvirj;Tk/angbracketright/angbracketrightR(¯hω)/bracketrightbigg , (45) because the perturbation by magnetization dynamics is de- scribed by Eq. ( 36). We will discuss in Sec. II Fthat the spatial gradient of DOM contributes to the inverse CIT. Additionally,we will show below that DOM and CIDMI are related to eachother. In order to obtain an expression for DOM it is convenient to consider the inverse effect, i.e., the generation of a torque bythe application of a time-dependent magnetic field B(t) that acts only on the orbital degrees of freedom of the electronsand not on their spins. This torque can be written as T IDOM k=1 2/summationdisplay ijlχIDOM kij/epsilon1ijl∂Bl ∂t, (46) where χIDOM kij=−1 Vlim ω→0/bracketleftbigge ¯hωIm/angbracketleft/angbracketleftTk;virj/angbracketright/angbracketrightR(¯hω)/bracketrightbigg , (47) because the perturbation by the time-dependent magnetic field is given by δH=−e 2/summationdisplay ijk/epsilon1ijkvirj∂Bk ∂tsin(ωt) ω. (48) Therefore the coefficients of DOM and IDOM are related by χDOM kij(ˆM)=−χIDOM kij (−ˆM). (49)TABLE I. Relations between the inverse of the magnetization- dynamics induced orbital magnetism (IDOM) and inverse current-induced DMI (ICIDMI) in the 2d Rashba model when ˆMlies in the zxplane. The components of χ IDOM ijk [Eq. ( 50)] and χICIDMI ijk [Eq. ( 20)] are denoted by the three indices ( ijk). ICIDMI IDOM (211) (121) (121) (211) −(221) (221) (112) (112) −(212) (122) −(122) (212) (222) (222) (231) (321)(132) (312) −(232) (322) In Appendix A, we show that the coefficient χIDOM ijk can be expressed as χIDOM ijk=−ie 4π¯h2/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr[TiRvkRRvjR+TiRRvkRvjR −TiRRvjRvkR−TiRvkRvjAA +TiRvjAvkAA+TiRvjAAvkA −TiRvkRRvjA−TiRRvkRvjA +TiRRvjAvkA+TiAvkAvjAA −TiAvjAvkAA−TiAvjAAvkA]. (50) Equations ( 50) and ( 20) differ only in the positions of the two velocity operators and the torque operator between theGreen functions. As a consequence, IDOM are ICIDMI arerelated. In Tables IandII, we list the relations between IDOM and ICIDMI for the Rashba model Eq. ( 83). We will explain in Sec. IIIthat IDOM and ICIDMI are zero in the Rashba model when the magnetization is along the zdirection. Therefore we discuss in Table Ithe case where the magnetization lies in the xzplane, and in Table II, we discuss the case where the magnetization lies in the yzplane. According to Tables I andII, the relation between IDOM and ICIDMI is of the form TABLE II. Relations between IDOM and ICIDMI in the 2d Rashba model when ˆMlies in the yzplane. ICIDMI IDOM (111) (111) −(211) (121) −(121) (211) (221) (221) −(112) (112) (212) (122)(122) (212) −(131) (311) (231) (321)(132) (312) 245411-6DYNAMICAL AND CURRENT-INDUCED … PHYSICAL REVIEW B 102, 245411 (2020) χIDOM ijk=±χICIDMI jik . This is expected, because the index iin χIDOM ijk is connected to the torque operator, while the index j inχICIDMI ijk is connected to the torque operator. F. Contributions from CIDMI and DOM to direct and inverse CIT In electronic transport theory, the continuity equation de- termines the current only up to a curl field [ 37]. The curl of magnetization corresponds to a bound current that cannot bemeasured in electron transport experiments such that J=J Kubo−∇×M (51) has to be used to extract the transport current Jfrom the current JKuboobtained from the Kubo linear response. The subtraction of ∇×Mhas been shown to be important when calculating the thermoelectric response [ 37] and the anoma- lous Nernst effect [ 20]. Similarly, in the theory of the thermal spin-orbit torque [ 10,18] the gradients of the DMI spiraliza- tion have to be subtracted in order to obtain the measurabletorque: T i=TKubo i−/summationdisplay j∂ ∂rjDij, (52) where the spatial derivative of the spiralization arises from its temperature dependence and the temperature gradient. Since CIDMI and DOM depend on the magnetization di- rection, they vary spatially in noncollinear magnets. Similarto Eq. ( 52), the spatial derivatives of the current-induced spiralization need to be included into the theory of CIT.Additionally, the gradients of DOM correspond to currentsthat need to be considered in the theory of the inverse CIT,similar to Eq. ( 51). In Sec. IVwe explicitly show that Onsager reciprocity is violated if spatial gradients of DOM and CIDMIare not subtracted from the Kubo response expressions. Bytrial-and-error we find that the following subtractions are nec-essary to obtain response currents and torques that satisfy thisfundamental symmetry: J ICIT i=JKubo i−1 2/summationdisplay j∂ˆM ∂rj·∂MDOM ij ∂ˆM(53) and TCIT i=TKubo i−1 2/summationdisplay j∂ˆM ∂rj·∂DCIDMI ij ∂ˆM, (54) where JICIT iis the current driven by magnetization dynamics, andTCIT iis the current-induced torque. Interestingly, we find that also the diagonal elements MDOM ii are nonzero. This shows that the generalized definition Eq. ( 43) is necessary, because the diagonal elements MDOM ii do not contribute in the usual definition of Miaccording to Eq. ( 44). These differences in the symmetry properties be- tween equilibrium and nonequilibrium orbital magnetism canbe traced back to symmetry breaking by the perturbations.Also in the case of the spiralization tensor D ijthe nonequilib- rium correction δDijhas different symmetry properties than the equilibrium part (see Sec. III).The contribution of DOM to χICIT2 ijkl can be written as χICIT2c ijkl=−1 2ˆek·/bracketleftBigg ˆM×∂χDOM jil ∂ˆM/bracketrightBigg (55) and the contribution of CIDMI to χCIT2 ijkl is given by χCIT2b ijkl=−1 2ˆek·/bracketleftBigg ˆM×∂χCIDMI jil ∂ˆM/bracketrightBigg . (56) G. Contributions from DDMI to gyromagnetism and damping The response to magnetization dynamics that is described by the torque-torque correlation function consists of torquesthat are related to damping and gyromagnetism [ 24]. The chiral contribution to these torques can be written as T TT2 i=/summationdisplay jklχTT2 ijklˆej·/bracketleftbigg ˆM×∂ˆM ∂t/bracketrightbigg ˆek·/bracketleftbigg ˆM×∂ˆM ∂rl/bracketrightbigg ,(57) where the coefficients χTT2 ijklsatisfy the Onsager relations χTT2 ijkl(ˆM)=χTT2 jikl(−ˆM). (58) Since DDMI depends on the magnetization direction, it varies spatially in noncollinear magnets and the resultinggradients of DDMI contribute to the damping and to thegyromagnetic ratio: T TT i=TKubo i−1 2/summationdisplay j∂ˆM ∂rj·∂DDDMI ij ∂ˆM. (59) The resulting contribution of the spatial derivatives of DDMI to the coefficient χTT2 ijklis χTT2c ijkl=−1 2ˆek·/bracketleftBigg ˆM×∂χDDMI jil (ˆM) ∂ˆM/bracketrightBigg . (60) H. Current-induced torque (CIT) in noncollinear magnets The chiral contribution to CIT consists of the spatial gra- dient of CIDMI, χCIT2b ijkl in Eq. ( 56), and the Kubo linear response of the torque to the applied electric field in a non-collinear magnet, χ CIT2a ijkl : χCIT2 ijkl=χCIT2a ijkl+χCIT2b ijkl. (61) In order to determine χCIT2a ijkl , we assume that the magnetiza- tion direction ˆM(r) oscillates spatially as described by ˆM(r)=⎛ ⎝ηsin(q·r) 0 1⎞ ⎠1/radicalbig 1+η2sin2(q·r), (62) where we will take the limit q→0 at the end of the calcula- tion. Since the spatial derivative of the magnetization directionis ∂ˆM(r) ∂ri=⎛ ⎝ηqicos(q·r) 00⎞ ⎠+O(η 3), (63) the chiral contribution to the CIT oscillates spatially propor- tional to cos( q·r). In order to extract this spatially oscillating 245411-7FREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) contribution, we multiply with cos( q·r) and integrate over the unit cell. The resulting expression for χCIT2a ijkl is χCIT2a ijkl=−2e Vηlim q→0lim ω→0/bracketleftBigg 1 ql/integraldisplay cos(qlrl) ×Im/angbracketleft/angbracketleftTi(r);vj(r/prime)/angbracketright/angbracketrightR(¯hω) ¯hωd3rd3r/prime/bracketrightBigg , (64) where Vis the volume of the unit cell, and the retarded torque- velocity correlation function /angbracketleft/angbracketleftTi(r);vj(r/prime)/angbracketright/angbracketrightR(¯hω) needs to be evaluated in the presence of the perturbation δH=Tkηsin(q·r) (65) due to the noncollinearity (the index kin Eq. ( 65) needs to match the index kinχCIT2a ijkl ). In Appendix B, we show that χCIT2a ijkl can be written as χCIT2a ijkl=−2e ¯hIm/bracketleftbig W(surf) ijkl+W(sea) ijkl/bracketrightbig , (66) where W(surf) ijkl=1 4π¯h/integraldisplayddk (2π)d/integraldisplay dEf/prime(E) ×Tr/bracketleftbigg TiGR k(E)vlGR k(E)vjGA k(E)TkGA k(E) +TiGR k(E)vjGA k(E)vlGA k(E)TkGA k(E) −TiGR k(E)vjGA k(E)TkGA k(E)vlGA k(E) +¯h meδjlTiGR k(E)GA k(E)TkGA k(E)/bracketrightbigg (67) is a Fermi surface term ( f/prime(E)=df(E)/dE) and W(sea) ijkl=1 4π¯h2/integraldisplayddk (2π)d/integraldisplay dEf(E) ×/bracketleftBigg −Tr[TiRvlRRvjRTkR]−Tr[TiRvlRTkRRvjR] −Tr[TiRRvlRvjRTkR]−Tr[TiRRvjRvlRTkR] +Tr[TiRRvjRTkRvlR]+Tr[TiRRTkRvjRvlR] +Tr[TiRRTkRvlRvjR]−Tr[TiRRvlRTkRvjR] −Tr[TiRvlRRTkRvjR]+Tr[TiRTkRRvjRvlR] +Tr[TiRTkRRvlRvjR]+Tr[TiRTkRvlRRvjR] −¯h meδjlTr[TiRRRTkR]−¯h meδjlTr[TiAAATkA] −¯h meδjlTr[TiAATkAA]/bracketrightBigg (68) is a Fermi sea term. I. Inverse CIT in noncollinear magnets The chiral contribution JICIT2[see Eq. ( 4)] to the charge pumping is described by the coefficients χICIT2 ijkl=χICIT2a ijkl+χICIT2b ijkl+χICIT2c ijkl, (69)where χICIT2a ijkl describes the response to the time-dependent magnetization gradient [see Eqs. ( 18), (25), and ( 24)] and χICIT2c ijkl results from the spatial gradient of DOM [see Eq. ( 55)].χICIT2b ijkl describes the response to the perturbation by magnetization dynamics in a noncollinear magnet. In orderto derive an expression for χ ICIT2b ijkl , we assume that the mag- netization oscillates spatially as described by Eq. ( 62). Since the corresponding response oscillates spatially proportionalto cos( q·r), we multiply by cos( q·r) and integrate over the unit cell in order to extract χ ICIT2b ijkl from the retarded velocity- torque correlation function /angbracketleft/angbracketleftvi(r);Tj(r/prime)/angbracketright/angbracketrightR(¯hω), which is evaluated in the presence of the perturbation Eq. ( 65). We obtain χICIT2b ijkl=2e Vηlim q→0lim ω→0/bracketleftBigg 1 ql/integraldisplay cos(qlrl) ×Im/angbracketleft/angbracketleftvi(r);Tj(r/prime)/angbracketright/angbracketrightR(¯hω) ¯hωd3rd3r/prime/bracketrightBigg , (70) which can be written as (see Appendix B) χICIT2b ijkl=2e ¯hIm/bracketleftbig V(surf) ijkl+V(sea) ijkl/bracketrightbig , (71) where V(surf) ijkl=1 4π¯h/integraldisplayddk (2π)d/integraldisplay dEf/prime(E) ×Tr/bracketleftbig viGR k(E)vlGR k(E)TjGA k(E)TkGA k(E) +viGR k(E)TjGA k(E)vlGA k(E)TkGA k(E) −viGR k(E)TjGA k(E)TkGA k(E)vlGA k(E)/bracketrightbig (72) is the Fermi surface term and V(sea) ijkl=1 4π¯h2/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr[−Tr[viRvlRRTjRTkR]−Tr[viRvlRTkRRTjR] −Tr[viRRvlRTjRTkR]−Tr[viRRTjRvlRTkR] +Tr[viRRTjRTkRvlR]+Tr[viRRTkRTjRvlR] +Tr[viRRTkRvlRTjR]−Tr[viRRvlRTkRTjR] −Tr[viRvlRRTkRTjR]+Tr[viRTkRRTjRvlR] +Tr[viRTkRRvlRTjR]+Tr[viRTkRvlRRTjR]] (73) is the Fermi sea term. In Eq. ( 70), we use the Kubo formula to describe the response to magnetization dynamics combined with perturba-tion theory to include the effect of noncollinearity. Thereby,the time-dependent perturbation and the perturbation by themagnetization gradient are separated and perturbations of theform of Eqs. ( 15)o r( 16) are not automatically included. For example, the flat cycloidal spin spiral ˆM(x,t)=⎛ ⎝sin(qx−ωt) 0 cos(qx−ωt)⎞ ⎠ (74) 245411-8DYNAMICAL AND CURRENT-INDUCED … PHYSICAL REVIEW B 102, 245411 (2020) moving in xdirection with speed ω/qand the helical spin spiral ˆM(y,t)=⎛ ⎝sin(qy−ωt) 0 cos(qy−ωt)⎞ ⎠ (75) moving in ydirection with speed ω/qbehave like Eq. ( 10) when tandrare small. Thus, these moving domain walls correspond to the perturbation of the type of Eq. ( 10) and the resulting contribution JICIT2afrom the time-dependent mag- netization gradient is not described by Eq. ( 70) and needs to be added, which we do by adding χICIT2a ijkl in Eq. ( 69). J. Damping and gyromagnetism in noncollinear magnets The chiral contribution Eq. ( 57) to the torque-torque corre- lation function is expressed in terms of the coefficient χTT ijkl=χTT2a ijkl+χTT2b ijkl+χTT2c ijkl, (76) where χTT2c ijkl results from the spatial gradient of DDMI [see Eq. ( 60)],χTT2a ijkl describes the response to a time-dependent magnetization gradient in a collinear magnet, and χTT2b ijkl describes the response to magnetization dynamics in a non- collinear magnet. In order to derive an expression for χTT2b ijkl, we assume that the magnetization oscillates spatially according to Eq. ( 62). We multiply the retarded torque-torque correlation function/angbracketleft/angbracketleftT i(r);Tj(r/prime)/angbracketright/angbracketrightR(¯hω) with cos( qlrl) and integrate over the unit cell in order to extract the part of the response that variesspatially proportional to cos( q lrl). We obtain χTT2b ijkl=2 Vηlim ql→0lim ω→0/bracketleftBigg 1 ql/integraldisplay cos(qlrl) ×Im/angbracketleft/angbracketleftTi(r);Tj(r/prime)/angbracketright/angbracketrightR(¯hω) ¯hωd3rd3r/prime/bracketrightBigg . (77) In Appendix B, we discuss how to evaluate Eq. ( 77)i n first-order perturbation theory with respect to the perturbationEq. ( 65) and show that χ TT2b ijkl can be expressed as χTT2b ijkl=2 ¯hIm/bracketleftbig X(surf) ijkl+X(sea) ijkl/bracketrightbig , (78) where X(surf) ijkl=1 4π¯h/integraldisplayddk (2π)d/integraldisplay dEf/prime(E) ×Tr/bracketleftbig TiGR k(E)vlGR k(E)TjGA k(E)TkGA k(E) +TiGR k(E)TjGA k(E)vlGA k(E)TkGA k(E) −TiGR k(E)TjGA k(E)TkGA k(E)vlGA k(E)/bracketrightbig (79) is a Fermi surface term and X(sea) ijkl=1 4π¯h2/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr[−(TiRvlRRTjRTkR)−(TiRvlRTkRRTjR) −(TiRRvlRTjRTkR)−(TiRRTjRvlRTkR) +(TiRRTjRTkRvlR)+(TiRRTkRTjRvlR)TABLE III. Effect of mirror reflection Mxzat the xzplane, mirror reflection Myzat the yzplane, and c2 rotation around the zaxis. The magnetization Mand the torque Ttransform like axial vectors, while the current Jtransforms like a polar vector. Mx My Mz Jx Jy Tx Ty Tz Mxz−Mx My−Mz Jx−Jy−Tx Ty−Tz Myz Mx−My−Mz−Jx Jy Tx−Ty−Tz c2 −Mx−My Mz−Jx−Jy−Tx−Ty Tz +(TiRRTkRvlRTjR)−(TiRRvlRTkRTjR) −(TiRvlRRTkRTjR)+(TiRTkRRTjRvlR) +(TiRTkRRvlRTjR)+(TiRTkRvlRRTjR)] (80) is a Fermi sea term. The contribution χTT2a ijkl from the time-dependent gradients is given by χTT2a ijkl=−/summationdisplay mχTT2a iml[1−δjm]δjk, (81) where χTT2a iml=i 4π¯h2/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr[TiRvlRROmR+TiRRvlROmR −TiRROmRvlR−TiRvlROmAA +TiROmAvlAA+TiROmAAvlA −TiRvlRROmA−TiRRvlROmA +TiRROmAvlA+TiAvlAOmAA −TiAOmAvlAA−TiAOmAAvlA], (82) withOm=∂H/∂ˆMm(see Appendix A). III. SYMMETRY PROPERTIES In this section, we discuss the symmetry properties of CIDMI, DDMI, and DOM in the case of the magnetic Rashbamodel H k(r)=¯h2 2mek2+α(k׈ez)·σ+/Delta1V 2σ·ˆM(r). (83) Additionally, we discuss the symmetry properties of the cur- rents and torques induced by time-dependent magnetizationgradients of the form of Eq. ( 10). We consider mirror reflection M xzat the xzplane, mirror reflection Myzat the yzplane, and c2 rotation around the z axis. When /Delta1V=0 these operations leave Eq. ( 83) invariant, but when /Delta1V/negationslash=0 they modify the magnetization direction ˆM in Eq. ( 83), as shown in Table III. At the same time, these operations affect the torque Tand the current Jdriven by the time-dependent magnetization gradients (see Table III). In Tables IVandV,w es h o wh o w ˆM×∂ˆM/∂rkis affected by the symmetry operations. A flat cycloidal spin spiral with spins rotating in the xz plane is mapped by a c2 rotation around the zaxis onto the same spin spiral. Similarly, a flat helical spin spiral with spins 245411-9FREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) TABLE IV . Effect of symmetry operations on the magnetization gradients. Magnetization gradients are described by three indices ( ijk). The first index denotes the magnetization direction at r=0. The third index denotes the direction along which the magnetization changes. The second index denotes the direction of ∂ˆM/∂rkδrk. The direction of ˆM×∂ˆM/∂rkis specified by the number below the indices ( ijk). (1,2,1) (1,3,1) (2,1,1) (2,3,1) (3,1,1) (3,2,1) 3 −2 −31 2 −1 Mxz (−1,2,1) ( −1,−3,1) (2, −1,1) (2, −3,1) ( −3,−1,1) ( −3,2,1) −3 −23 −12 1 Myz (1,2,1) (1,3,1) ( −2,−1,1) ( −2,3,1) ( −3,−1,1) ( −3,2,1) 3 −2 −3 −12 1 c2 ( −1,2,1) ( −1,−3,1) ( −2,1,1) ( −2,−3,1) (3,1,1) (3,2,1) −3 −23 1 2 −1 rotating in the yzplane is mapped by a c2 rotation around thezaxis onto the same spin spiral. Therefore, when ˆM points in zdirection, a c2 rotation around the zaxis does not change ˆM×∂ˆM/∂ri, but it flips the in-plane current J and the in-plane components of the torque, TxandTy. Conse- quently, ˆM×∂2ˆM/∂ri∂tdoes not induce currents or torques, i.e., ICIDMI, CIDMI, IDDMI, and DDMI are zero, when ˆM points in zdirection. However, they become nonzero when the magnetization has an in-plane component (see Fig. 1). Similarly, IDOM vanishes when the magnetization points inzdirection: In that case Eq. ( 83) is invariant under the c2 rotation. A time-dependent magnetic field along zdirection is invariant under the c2 rotation as well. However, TxandTy change sign under the c2 rotation. Consequently, symmetry forbids IDOM in this case. However, when the magnetizationhas an in-plane component, IDOM and DOM become nonzero(see Fig. 2). That time-dependent magnetization gradients of the type of Eq. ( 7) do not induce in-plane currents and torques when ˆMpoints in zdirection can also be seen directly from Eq. ( 7): The c2 rotation transforms q→−qandM x→− Mx. Since sin(q·r)i so d di n r,E q .( 7) is invariant under c2 rota- tion, while the in-plane currents and torques induced bytime-dependent magnetization gradients change sign underc2 rotation. In contrast, Eq. ( 10) is not invariant under c2 rotation, because sin( q·r−ωt) is not odd in rfort>0. Consequently, time-dependent magnetization gradients of thetype of Eq. ( 10) induce currents and torques also when ˆM points locally into the zdirection. These currents and torques, which are described by Eqs. ( 24) and ( 82), respectively, need to be added to the chiral ICIT and the chiral torque-torquecorrelation. While CIDMI, DDMI, and DOM are zero whenthe magnetization points in zdirection, their gradients are not(see Figs. 1and2). Therefore the gradients of CIDMI, DOM, and DDMI contribute to CIT, to ICIT and to the torque-torquecorrelation, respectively, even when ˆMpoints locally into the zdirection. A. Symmetry properties of ICIDMI and IDDMI In the following, we discuss how Tables III–Vcan be used to analyze the symmetry of ICIDMI and IDDMI. Accordingto Eq. ( 19), the coefficient χ ICIDMI ijk describes the response of the current JICIT2a i to the time-dependent magnetization gradient ˆej·[ˆM×∂2ˆM ∂rk∂t]. Since ˆM×∂2ˆM ∂rk∂t=∂ ∂t[ˆM×∂ˆM ∂rk]f o r time-dependent magnetization gradients of the type Eq. ( 7)t h e symmetry properties of χICIDMI ijk follow from the transforma- tion behavior of ˆM×∂ˆM ∂rkandJunder symmetry operations. We consider the case with magnetization in xdirection. The component χICIDMI 132 describes the current in xdirection induced by the time-dependence of a cycloidal magnetizationgradient in ydirection (with spins rotating in the xyplane). M yzflips both ˆM×∂ˆM ∂yand Jx, but it preserves ˆM.Mzx preserves ˆM×∂ˆM ∂yandJx, but it flips ˆM. A c2 rotation around thezaxis flips ˆM×∂ˆM ∂y,ˆMandJx. Consequently, χICIDMI 132 (ˆM) is allowed by symmetry and it is even in ˆM. The compo- nentχICIDMI 122 describes the current in xdirection induced by the time-dependence of a helical magnetization gradient inydirection (with spins rotating in the xzplane). M yzflips ˆM×∂ˆM ∂yandJx, but it preserves ˆM.Mzxflips ˆM×∂ˆM ∂yand ˆM, but it preserves Jx. A c2 rotation around the zaxis flips Jxand ˆM, but it preserves ˆM×∂ˆM ∂y. Consequently, χICIDMI 122 is allowed by symmetry and it is odd in ˆM. The component χICIDMI 221 describes the current in ydirection induced by the TABLE V . Continuation of Table IV. (1,2,2) (1,3,2) (2,1,2) (2,3,2) (3,1,2) (3,2,2) 3 −2 −31 2 −1 Mxz (−1,−2,2) ( −1,3,2) (2,1,2) (2,3,2) ( −3,1,2) ( −3,−2,2) 32 −31 −2 −1 Myz (1,−2,2) (1, −3,2) ( −2,1,2) ( −2,−3,2) ( −3,1,2) ( −3,−2,2) −3231 −2 −1 c2 ( −1,2,2) ( −1,−3,2) ( −2,1,2) ( −2,−3,2) (3,1,2) (3,2,2) −3 −2 312 −1 245411-10DYNAMICAL AND CURRENT-INDUCED … PHYSICAL REVIEW B 102, 245411 (2020) FIG. 1. ICIDMI in a noncollinear magnet. (a) Arrows illustrate the magnetization direction. (b) Arrows illustrate the current Jy induced by a time-dependent magnetization gradient, which is de- scribed by χICIDMI 221 .W h e n ˆMpoints in zdirection, χICIDMI 221 andJyare zero. The sign of χICIDMI 221 and of Jychanges with the sign of Mx. time-dependence of a cycloidal magnetization gradient in x direction (with spins rotating in the xzplane). Mzxpreserves ˆM×∂ˆM ∂x, but it flips Jyand ˆM.Myzpreserves ˆM,Jy, and ˆM×∂ˆM ∂x. The c2 rotation around the zaxis preserves ˆM×∂ˆM ∂x, but it flips ˆMandJy. Consequently, χICIDMI 221 is allowed by sym- metry and it is odd in ˆM. The component χICIDMI 231 describes the current in ydirection induced by the time-dependence of a cycloidal magnetization gradient in xdirection (with spins rotating in the xyplane). Mzxflips ˆM×∂ˆM ∂x,ˆM, and Jy.Myz preserves ˆM×∂ˆM ∂x,ˆMand Jy. The c2 rotation around the zaxis flips ˆM×∂ˆM ∂x,Jy, and ˆM. Consequently, χICIDMI 231 is allowed by symmetry and it is even in ˆM. These properties are summarized in Table VI. Due to the relations between CIDMI and DOM (see Tables IandII), they can be used for DOM as well. When the magnetization liesat a general angle in the xzplane or in the yzplane several additional components of CIDMI and DOM are nonzero (seeTables IandII, respectively). Similarly, one can analyze the symmetry of DDMI. Ta- bleVIIlists the components of DDMI, χ DDMI ijk , which are allowed by symmetry when ˆMpoints in xdirection. FIG. 2. DOM in a noncollinear magnet. (a) Arrows illustrate the magnetization direction. (b) Arrows illustrate the orbital magnetiza-tion induced by magnetization dynamics (DOM). When ˆMpoints in zdirection, DOM is zero. The sign of DOM changes with the sign of M x.TABLE VI. Allowed components of χICIDMI ijk when ˆMpoints in xdirection. +components are even in ˆM, while −components are odd in ˆM. 132 122 221 231 + −−+ B. Response to time-dependent magnetization gradients of the second type (Eq. ( 10)) According to Eq. ( 13) the time-dependent magnetization gradient is along the magnetization. Therefore, in contrast to the discussion in Sec. III A , we cannot use ˆM×∂2ˆM ∂rk∂tin the symmetry analysis. Eqs. ( 24) and ( 25) show that χICIT2a ijjl de- scribes the response of JICIT2a i toˆej·[ˆM×∂ˆM ∂t]ˆej·[ˆM×∂ˆM ∂rl] while χICIT2a ijkl=0f o r j/negationslash=k. According to Eq. ( 23), the symmetry properties of [ ˆM×∂ˆM ∂t]·[ˆM×∂ˆM ∂rl] agree to the symmetry properties of ˆM·∂2ˆM ∂rl∂t. Therefore, in order to un- derstand the symmetry properties of χICIT2a ijjl we consider the transformation of Jand ˆM·∂2ˆM ∂rl∂tunder symmetry operations. We consider the case where ˆMpoints in zdirection. χICIT2a 1jj1 describes the current driven in xdirection, when the magne- tization varies in xdirection. Mxzflips ˆM, but preserves Jx and ˆM·∂2ˆM/(∂x∂t).Myzflips ˆM,Jx, and ˆM·∂2ˆM/(∂x∂t). c2 rotation flips ˆM·∂2ˆM/(∂x∂t) and Jx, but preserves ˆM. Consequently, χICIT2a 1jj1is allowed by symmetry and it is even inˆM. χICIT2a 2jj1describes the current flowing in ydirection, when magnetization varies in xdirection. Mxzflips ˆM and Jy, but preserves ˆM·∂2ˆM/(∂x∂t).Myzflips ˆM, and ˆM·∂2ˆM/(∂x∂t), but preserves Jy. c2 rotation flips ˆM· ∂2ˆM/(∂x∂t) and Jy, but preserves ˆM. Consequently, χICIT2a 2jj1 is allowed by symmetry and it is odd in ˆM. Similarly, one can show that χICIT2a 1jj2is odd in ˆMand that χICIT2a 2jj2is even in ˆM. Analogously, one can investigate the symmetry properties ofχTT2a ijjl. We find that χTT2a 1jj1andχTT2a 2jj2are odd in ˆM, while χTT2a 2jj1andχTT2a 1jj2are even in ˆM. IV . RESULTS In the following sections, we discuss the results for the direct and inverse chiral CIT and for the chiral torque-torque correlation in the two-dimensional (2d) Rashba model TABLE VII. Allowed components of χDDMI ijk when ˆMpoints in x direction. +components are even in ˆM, while −components are odd inˆM. 222 232 322 332 − ++− 245411-11FREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) -2 -1 0 1 2 Fermi ener gy [eV]-0.02-0.0100.010.020.030.040.05χijklCIT2 [ea0]2121 1121 2121 (gauge-field) 1121 (gauge-field) FIG. 3. Chiral CIT in the 1d Rashba model for cycloidal gradi- ents vs Fermi energy. General perturbation theory (solid lines) agrees to the gauge-field approach (dashed lines). Eq. ( 83), and in the one-dimensional (1d) Rashba model [ 38] Hkx(x)=¯h2 2mek2 x−αkxσy+/Delta1V 2σ·ˆM(x). (84) Additionally, we discuss the contributions of the time- dependent magnetization gradients, and of DDMI, DOM, andCIDMI to these effects. While vertex corrections to the chiral CIT and to the chiral torque-torque correlation are important in the Rashba model[38], the purpose of this work is to show the importance of the contributions from time-dependent magnetization gradients,DDMI, DOM and CIDMI. We therefore consider only theintrinsic contributions here, i.e., we set G R k(E)=¯h[E−Hk+i/Gamma1]−1, (85) where /Gamma1is a constant broadening, and we leave the study of vertex corrections for future work. The results shown in the following sections are obtained for the model parameters /Delta1V=1e V ,α=2 eV Å, and /Gamma1= 0.1R y=1.361 eV, when the magnetization points in zdirec- tion, i.e., ˆM=ˆez. The unit of χCIT2 ijkl is charge times length in the 1d case and charge in the 2d case. Therefore, in the 1dcase, we discuss the chiral torkance in units of ea 0, where a0 is Bohr’s radius. In the 2d case, we discuss the chiral torkance in units of e. The unit of χTT2 ijklis angular momentum in the 1d case and angular momentum per length in the 2d case.Therefore we discuss χ TT2 ijklin units of ¯ hin the 1d case, and in units of ¯ h/a0in the 2d case. A. Direct and inverse chiral CIT In Fig. 3, we show the chiral CIT as a function of the Fermi energy for cycloidal magnetization gradients in the 1d Rashbamodel. The components χ CIT2 2121 andχCIT2 1121 are labeled by 2121 and 1121, respectively. The component 2121 of CIT describesthe non-adiabatic torque, while the component 1121 describesthe adiabatic STT (modified by SOI). In the one-dimensionalRashba model, the contributions χ CIT2b 2121 andχCIT2b 1121 (Eq. ( 56)) from the CIDMI are zero when ˆM=ˆez(not shown in the-2 -1 0 1 2 Fermi ener gy [eV]-0.0200.020.04χijklICIT2 [ea0]1221 1121 χ1221ICIT2a FIG. 4. Chiral ICIT in the 1d Rashba model for cycloidal gradients vs Fermi energy. Dashed line: contribution from the time- dependent gradient. figure). For cycloidal spin spirals, it is possible to solve the 1d Rashba model by a gauge-field approach [ 38], which allows us to test the perturbation theory, Eq. ( 66). For comparison we show in Fig. 3, the results obtained from the gauge-field approach, which agree to the perturbation theory, Eq. ( 66). This demonstrates the validity of Eq. ( 66). In Fig. 4, we show the chiral ICIT in the 1d Rashba model. The components χICIT2 1221 andχICIT2 1121 are labeled by 1221 and 1121, respectively. The contribution χICIT2a 1221 from the time- dependent gradient is of the same order of magnitude as thetotalχ ICIT2 1221. Comparison of Figs. 3and 4shows that CIT and ICIT satisfy the reciprocity relations Eq. ( 5), that χCIT2 1121 is odd in ˆM, and that χCIT2 2121 is even in ˆM, i.e., χCIT2 2121= χICIT2 1221 andχCIT2 1121=−χICIT2 1121. The contribution χICIT2a 1221 from the time-dependent gradients is crucial to satisfy the reci-procity relations between χ CIT2 2121 andχICIT2 1221. In Figs. 5and6, we show the CIT and the ICIT, respec- tively, for helical gradients in the 1d Rashba model. Thecomponents χ CIT2 2111 andχCIT2 1111 are labeled 2111 and 1111, re- -2 -1 0 1 2 Fermi ener gy [eV]-0.04-0.0200.020.040.06χijklCIT2 [ea0]1111 2111 χ1111CIT2b χ2111CIT2b FIG. 5. Chiral CIT for helical gradients in the 1d Rashba model vs Fermi energy. Dashed lines: Contributions from CIDMI. 245411-12DYNAMICAL AND CURRENT-INDUCED … PHYSICAL REVIEW B 102, 245411 (2020) -2 -1 0 1 2 Fermi energy [eV]-0.0200.020.040.06χijklICIT2 [ea0]1111 1211 χ1111ICIT2a χ1111ICIT2c χ1211ICIT2c FIG. 6. Chiral ICIT for helical gradients in the 1d Rashba model vs Fermi energy. Dashed lines: Contributions from DOM. Dashed- dotted line: contribution from the time-dependent magnetizationgradient. spectively, in Fig. 5, while χICIT2 1211 andχICIT2 1111 are labeled 1211 and 1111, respectively, in Fig. 6. The contributions χCIT2b 2111 and χCIT2b 1111 from CIDMI are of the same order of magnitude as the total χCIT2 2111 andχCIT2 1111. Similarly, the contributions χICIT2c 1211 andχICIT2c 1111 from DOM are of the same order of magnitude as the total χICIT2 1211 andχICIT2 1111. Additionally, the contribution χICIT2a 1111 from the time-dependent gradient is substantial. Com- parison of Figs. 5and 6shows that CIT and ICIT satisfy the reciprocity relation Eq. ( 5), that χCIT2 2111 is odd in ˆM, and thatχCIT2 1111 is even in ˆM, i.e., χCIT2 1111=χICIT2 1111 andχCIT2 2111= −χICIT2 1211. These reciprocity relations between CIT and ICIT are only satisfied when CIDMI, DOM, and the response totime-dependent magnetization gradients are included. Addi-tionally, the comparison between Figs. 5and6shows that the contributions of CIDMI to CIT ( χ CIT2b 1111 andχCIT2b 2111 ) are related to the contributions of DOM to ICIT ( χICIT2c 1111 andχICIT2c 1211 ). These relations between DOM and ICIT are expected fromTable I. In Figs. 7and8, we show the CIT and the ICIT, respec- tively, for cycloidal gradients in the 2d Rashba model. In thiscase there are contributions from CIDMI and DOM in contrastto the 1d case with cycloidal gradients (Fig. 3). Comparison between Figs. 7and8shows that χ CIT2 1121 andχCIT2 2221 are odd inˆM, thatχCIT2 1221 andχCIT2 2121 are even in ˆM, and that CIT and ICIT satisfy the reciprocity relation Eq. ( 5) when the gradi- ents of CIDMI and DOM are included, i.e., χCIT2 1121=−χICIT2 1121, χCIT2 2221=−χICIT2 2221,χCIT2 1221=χICIT2 2121, andχCIT2 2121=χICIT2 1221.χCIT2 1121 describes the adiabatic STT with SOI, while χCIT2 2121 describes the non-adiabatic STT. Experimentally, it has been found thatCITs occur also when the electric field is applied parallelto domain-walls (i.e., perpendicular to the q-vector of spin spirals) [ 39]. In our calculations, the components χ CIT2 2221 and χCIT2 1221 describe such a case, where the applied electric field points in ydirection, while the magnetization direction varies with the xcoordinate. In Figs. 9and 10, we show the chiral CIT and ICIT, respectively, for helical gradients in the 2d Rashba model.The component χ CIT2 2111 describes the adiabatic STT with SOI-2 -1 0 1 2 Fermi ener gy [eV]-0.00200.0020.0040.006χijklCIT2 [e]1121 2221 1221 2121 χ2221CIT2b χ1221CIT2b χ2121CIT2b FIG. 7. Chiral CIT for cycloidal gradients in the 2d Rashba model vs Fermi energy. Dashed lines: contributions from CIDMI. and the component χCIT2 1111 describes the non-adiabatic STT. The components χCIT2 2211 andχCIT2 1211 describe the case when the applied electric field points in ydirection, i.e., perpendicu- lar to the direction along which the magnetization directionvaries. Comparison between Figs. 9and10shows that χ CIT2 1111 andχCIT2 2211 are even in ˆM, that χCIT2 1211 andχCIT2 2111 are odd in ˆMand that CIT and ICIT satisfy the reciprocity relation Eq. ( 5) when the gradients of CIDMI and DOM are included, i.e.,χCIT2 1111=χICIT2 1111,χCIT2 2211=χICIT2 2211,χCIT2 1211=−χICIT2 2111, and χCIT2 2111=−χICIT2 1211. B. Chiral torque-torque correlation In Fig. 11, we show the chiral contribution to the torque- torque correlation in the 1d Rashba model for cycloidalgradients. We compare the perturbation theory Eq. ( 78)p l u s -2 -1 0 1 2 Fermi energy [eV]-0.00200.0020.0040.006χijklICIT2 [e]1121 1221 2121 2221 χ2221ICIT2a χ1221ICIT2a χ2121ICIT2c χ1221ICIT2c χ2221ICIT2c FIG. 8. Chiral ICIT for cycloidal gradients in the 2d Rashba model vs Fermi energy. Dashed lines: contributions from DOM. Dashed-dotted lines: contributions from the time-dependent gradients. 245411-13FREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) -2 -1 0 1 2 Fermi ener gy [eV]-0.00200.0020.0040.006χijklCIT2 [e]2211 1111 1211 2111 χ2111CIT2b χ1211CIT2b χ2211CIT2b χ1111CIT2b FIG. 9. Chiral CIT for helical gradients in the 2d Rashba model vs Fermi energy. Dashed lines: contributions from CIDMI. Eq. ( 82) to the gauge-field approach from Ref. [ 38]. This com- parison shows that perturbation theory provides the correctanswer only when the contribution χ TT2a ijkl [Eq. ( 82)] from the time-dependent gradients is taken into account. The contribu-tionsχ TT2a 1221 andχTT2a 2221 from the time-dependent gradients are comparable in magnitude to the total values. In the 1d Rashbamodel, the DDMI-contribution in Eq. ( 60) is zero for cycloidal gradients (not shown in the figure). The components χ TT2 2121and χTT2 1221 describe the chiral gyromagnetism while the compo- nentsχTT2 1121andχTT2 2221describe the chiral damping [ 38,40,41]. The components χTT2 2121andχTT2 1221are odd in ˆMand they satisfy the Onsager relation Eq. ( 58), i.e., χTT2 2121=−χTT2 1221. In Fig. 12, we show the chiral contributions to the torque-torque correlation in the 1d Rashba model for heli-cal gradients. In contrast to the cycloidal gradients (Fig. 11) there are contributions from the spatial gradients of DDMI -2 -1 0 1 2 Fermi energy [eV]-0.004-0.00200.0020.0040.006χijklICIT2 [e]1111 1211 2111 2211 χ1111ICIT2a χ2221ICIT2a χ1111ICIT2c χ2111ICIT2c χ1211ICIT2c χ2211ICIT2c FIG. 10. Chiral ICIT for helical gradients in the 2d Rashba model vs Fermi energy. Dashed lines: contributions from DOM. Dashed- dotted lines: contributions from the time-dependent gradient.-2 -1 0 1 2 Fermi ener gy [eV]-0.00500.0050.01χijklTT2 [h_]2121 1221 2221 1121 χ1221TT2a χ2221TT2a 2121 (gf) 1221 (gf) 1121 (gf) 2221 (gf) FIG. 11. Chiral contribution to the torque-torque correlation for cycloidal gradients in the 1d Rashba model vs Fermi energy. Per- turbation theory (solid lines) agrees to the gauge-field (gf) approach(dotted lines). Dashed lines: contribution from the time-dependent gradient. [Eq. ( 60)] in this case. The Onsager relation Eq. ( 58)f o r the components χTT2 2111andχTT2 1211is satisfied only when these contributions from DDMI are taken into account, which areof the same order of magnitude as the total values. The com-ponents χ TT2 2111andχTT2 1211are even in ˆMand describe chiral damping, while the components χTT2 1111andχTT2 2211are odd in ˆM and describe chiral gyromagnetism. As a consequence of theOnsager relation Eq. ( 58) we obtain χ TT2 1111=χTT2 2211=0f o rt h e total components: Eq. ( 58) shows that diagonal components of the torque-torque correlation function are zero unless theyare even in ˆM. However, χ TT2a 1111,χTT2c 1111, andχTT2b 1111=−χTT2a 1111− χTT2c 1111 are individually nonzero. Interestingly, the off-diagonal components of the torque-torque correlation describe chiral -2 -1 0 1 2 Fermi ener gy [eV]-0.00500.0050.01χijklTT2 [h_]1111 2111 1211 2211 χ1111TT2c χ2111TT2c χ1211TT2c χ2211TT2c χ1111TT2a χ2111TT2a FIG. 12. Chiral contribution to the torque-torque correlation for helical gradients in the 1d Rashba model vs Fermi energy. Dashed lines: contributions from DDMI. Dashed-dotted lines: contributions from the time-dependent gradients. 245411-14DYNAMICAL AND CURRENT-INDUCED … PHYSICAL REVIEW B 102, 245411 (2020) -2 -1 0 1 2 Fermi energy [eV]-0.000500.00050.001χijklTT2 [h_/a0]1121 2121 1221 2221 χ1221TT2a χ2221TT2a χ2121TT2c χ1221TT2c FIG. 13. Chiral contribution to the torque-torque correlation for cycloidal gradients in the 2d Rashba model vs Fermi energy. Dashedlines: contributions from DDMI. Dashed-dotted lines: contributions from the time-dependent gradients. damping for helical gradients, while for cycloidal gradients the off-diagonal elements describe chiral gyromagnetism andthe diagonal elements describe chiral damping. In Fig. 13, we show the chiral contributions to the torque- torque correlation in the 2d Rashba model for cycloidalgradients. In contrast to the 1d Rashba model with cy-cloidal gradients (Fig. 11) the contributions from DDMI χ TT2c ijkl [Eq. ( 60)] are nonzero in this case. Without these contribu- tions from DDMI the Onsager relation ( 58)χTT2 2121=−χTT2 1221 is violated. The DDMI contribution is of the same order of magnitude as the total values. The components χTT2 2121andχTT2 1221 are odd in ˆMand describe chiral gyromagnetism, while the components χTT2 1121andχTT2 2221are even in ˆMand describe chiral damping. -2 -1 0 1 2 Fermi energy [eV]-0.000500.00050.001χijklTT2 [h_ /a0]1111 2111 1211 2211 χ1111TT2a χ2111TT2a χ1111TT2c χ1211TT2c χ2211TT2c FIG. 14. Chiral contribution to the torque-torque correlation for helical gradients in the 2d Rashba model vs Fermi energy. Dashed lines: contributions from DDMI. Dashed-dotted lines: contributions from the time-dependent gradients.In Fig. 14, we show the chiral contributions to the torque-torque correlation in the 2d Rashba model for heli-cal gradients. The components χ TT2 1211andχTT2 2111are even in ˆMand describe chiral damping, while the components χTT2 1111 andχTT2 2211are odd in ˆMand describe chiral gyromagnetism. The Onsager relation Eq. ( 58) requires χTT2 1111=χTT2 2211=0 and χTT2 2111=χTT2 1211. Without the contributions from DDMI these Onsager relations are violated. V . SUMMARY Finding ways to tune the Dzyaloshinskii-Moriya interac- tion (DMI) by external means, such as an applied electriccurrent, holds much promise for applications in whichDMI determines the magnetic texture of domain walls orskyrmions. In order to derive an expression for current-induced Dzyaloshinskii-Moriya interaction (CIDMI), we firstidentify its inverse effect: When magnetic textures vary asa function of time, electric currents are driven by variousmechanisms, which can be distinguished according to theirdifferent dependence on the time-derivative of magnetiza-tion,∂ˆM(r,t)/∂t, and on the spatial derivative ∂ˆM(r,t)/∂r: One group of effects is proportional to ∂ˆM(r,t)/∂t,a second group of effects is proportional to the product∂ˆM(r,t)/∂t∂ˆM(r,t)/∂r, and a third group is proportional to the second derivative ∂ 2ˆM(r,t)/∂r∂t. We show that the response of the electric current to the time-dependent mag-netization gradient ∂ 2ˆM(r,t)/∂r∂tcontais the inverse of CIDMI. We establish the reciprocity relation between in-verse and direct CIDMI and thereby obtain an expression forCIDMI. We find that CIDMI is related to the modification oforbital magnetism induced by magnetization dynamics, whichwe call dynamical orbital magnetism (DOM). We show thattorques are generated by time-dependent gradients of magne-tization as well. The inverse effect consists in the modificationof DMI by magnetization dynamics, which we call dynamicalDMI (DDMI). Additionally, we develop a formalism to calculate the chi- ral contributions to the direct and inverse current-inducedtorques (CITs) and to the torque-torque correlation innoncollinear magnets. We show that the response to time-dependent magnetization gradients contributes substantiallyto these effects and that the Onsager reciprocity relations areviolated when it is not taken into account. In noncollinearmagnets, CIDMI, DDMI, and DOM depend on the localmagnetization direction. We show that the resulting spatialgradients of CIDMI, DDMI and DOM have to be subtractedfrom the CIT, from the torque-torque correlation, and from theinverse CIT, respectively. We apply our formalism to study CITs and the torque- torque correlation in textured Rashba ferromagnets. We findthat the contribution of CIDMI to the chiral CIT is of the orderof magnitude of the total effect. Similarly, we find that thecontribution of DDMI to the chiral torque-torque correlationis of the order of magnitude of the total effect. ACKNOWLEDGMENTS We acknowledge financial support from Leibniz Collab- orative Excellence project OptiSPIN - Optical Control ofNanoscale Spin Textures. We acknowledge funding under 245411-15FREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) SPP 2137 “Skyrmionics” of the DFG. We gratefully acknowl- edge financial support from the European Research Council(ERC) under the European Union’s Horizon 2020 researchand innovation program (Grant No. 856538, project “3DMAGiC”). The work was also supported by the DeutscheForschungsgemeinschaft (DFG, German Research Founda-tion) - TRR 173 - 268565370 (project A11). We alsogratefully acknowledge the Jülich Supercomputing Centreand RWTH Aachen University for providing computationalresources under Project No. jiff40. APPENDIX A: RESPONSE TO TIME-DEPENDENT GRADIENTS In this Appendix, we derive Eqs. ( 18), (20), (41), and ( 82), which describe the response to time-dependent magnetizationgradients, and Eq. ( 50), which describes the response to time- dependent magnetic fields. We consider perturbations of theform δH(r,t)=Bb1 qωsin(q·r)s i n (ωt). (A1) When we set B=∂H ∂ˆMkand b=∂2ˆMk ∂ri∂t,E q .( A1) turns into Eq. ( 17), while when we set B=−eviand b=1 2/epsilon1ijk∂Bk ∂t, we obtain Eq. ( 48). We need to derive an expression for the response δA(r,t) of an observable Ato this perturbation, which varies in time like cos( ωt) and in space like cos( q·r), because∂2ˆM(r,t) ∂ri∂t∝cos(q·r) cos(ωt). Therefore we use the Kubo linear response formalism to obtain the coefficient χ in δA(r,t)=χcos(q·r) cos(ωt), (A2) which is given by χ=i ¯hqωV[/angbracketleft/angbracketleftAcos(q·r),Bsin(q·r)/angbracketright/angbracketrightR(¯hω) −/angbracketleft /angbracketleftAcos(q·r),Bsin(q·r)/angbracketright/angbracketrightR(−¯hω)], (A3) where /angbracketleft/angbracketleftAcos(q·r),Bsin(q·r)/angbracketright/angbracketrightR(¯hω) is the retarded func- tion at frequency ωandVis the volume of the unit cell. The operator Bsin(q·r) can be written as Bsin(q·r)=1 2i/summationdisplay knm/bracketleftbig B(1) knmc† k+nck−m−B(2) knmc† k−nck+m/bracketrightbig , (A4) where k+=k+q/2,k−=k−q/2,c† k+nis the creation op- erator of an electron in state |uk+n/angbracketright,ck−mis the annihilation operator of an electron in state |uk−m/angbracketright, B(1) knm=1 2/angbracketleftuk+n|[Bk++Bk−]|uk−m/angbracketright (A5) and B(2) knm=1 2/angbracketleftuk−n|[Bk++Bk−]|uk+m/angbracketright. (A6) Similarly, Acos(q·r)=1 2/summationdisplay knm/bracketleftbig A(1) knmc† k+nck−m+A(2) knmc† k−nck+m/bracketrightbig ,(A7)where A(1) knm=1 2/angbracketleftuk+n|[Ak++Ak−]|uk−m/angbracketright (A8) and A(2) knm=1 2/angbracketleftuk−n|[Ak++Ak−]|uk+m/angbracketright. (A9) It is convenient to obtain the retarded response function in Eq. ( A3) from the corresponding Matsubara function in imaginary time τ 1 V/angbracketleft/angbracketleftAcos(q·r),Bsin(q·r)/angbracketright/angbracketrightM(τ) =1 4i/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n/primem/prime/bracketleftbig A(1) knmB(2) kn/primem/primeZ(1) knmn/primem/prime(τ) −A(2) knmB(1) kn/primem/primeZ(2) knmn/primem/prime(τ)/bracketrightbig , (A10) where d=1,2 or 3 is the dimension, Z(1) knmn/primem/prime(τ)=/angbracketleftTτc† k+n(τ)ck−m(τ)c† k−n/prime(0)ck+m/prime(0)/angbracketright =−GM m/primen(k+,−τ)GM mn/prime(k−,τ), (A11) Z(2) knmn/primem/prime(τ)=/angbracketleftTτc† k−n(τ)ck+m(τ)c† k+n/prime(0)ck−m/prime(0)/angbracketright =−GM m/primen(k−,−τ)GM mn/prime(k+,τ), (A12) and GM mn/prime(k+,τ)=− /angbracketleft Tτck+m(τ)c† k+n/prime(0)/angbracketright (A13) is the single-particle Matsubara function. The Fourier trans- form of Eq. ( A10) is given by 1 V/angbracketleft/angbracketleftAcos(q·r),Bsin(q·r)/angbracketright/angbracketrightM(iEN) =i 4¯hβ/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n/primem/prime/summationdisplay p ×/bracketleftbig A(1) knmB(2) kn/primem/primeGM m/primen(k+,iEp)GM mn/prime(k−,iEp+iEN) −A(2) knmB(1) kn/primem/primeGM m/primen(k−,iEp)GM mn/prime(k+,iEp+iEN)/bracketrightbig , (A14) where EN=2πN/βandEp=(2p+1)π/β are bosonic and fermionic Matsubara energy points, respectively, and β= 1/(kBT) is the inverse temperature. In order to carry out the Matsubara summation over Ep,w e make use of 1 β/summationdisplay pGM mn/prime(iEp+iEN)GM m/primen(iEp) =i 2π/integraldisplay dE/primef(E/prime)GM mn/prime(E/prime+iEN)GM m/primen(E/prime+iδ) +i 2π/integraldisplay dE/primef(E/prime)GM mn/prime(E/prime+iδ)GM m/primen(E/prime−iEN) −i 2π/integraldisplay dE/primef(E/prime)GM mn/prime(E/prime+iEN)GM m/primen(E/prime−iδ) −i 2π/integraldisplay dE/primef(E/prime)GM mn/prime(E/prime−iδ)GM m/primen(E/prime−iEN), (A15) 245411-16DYNAMICAL AND CURRENT-INDUCED … PHYSICAL REVIEW B 102, 245411 (2020) where δis a positive infinitesimal. The retarded function /angbracketleft/angbracketleftAcos(q·r),Bsin(q·r)/angbracketright/angbracketrightR(ω) is obtained from the Mat- subara function /angbracketleft/angbracketleftAcos(q·r),Bsin(q·r)/angbracketright/angbracketrightM(iEN)b yt h e analytic continuation iEN→¯hωto real frequencies. The right-hand side of Eq. ( A15) has the following analytic con- tinuation to real frequencies: i 2π/integraldisplay dE/primef(E/prime)GR mn/prime(E/prime+¯hω)GR m/primen(E/prime) +i 2π/integraldisplay dE/primef(E/prime)GR mn/prime(E/prime)GA m/primen(E/prime−¯hω) −i 2π/integraldisplay dE/primef(E/prime)GR mn/prime(E/prime+¯hω)GA m/primen(E/prime) −i 2π/integraldisplay dE/primef(E/prime)GA mn/prime(E/prime)GA m/primen(E/prime−¯hω). (A16) Therefore we obtain χ=−i 8π¯h2qω/integraldisplayddk (2π)d[Zk(q,ω)−Zk(−q,ω) −Zk(q,−ω)+Zk(−q,−ω)], (A17) where Zk(q,ω)=/integraldisplay dE/primef(E/prime)Tr/bracketleftbig AkGR k−(E/prime+¯hω)BkGR k+(E/prime)/bracketrightbig +/integraldisplay dE/primef(E/prime)Tr/bracketleftbig AkGR k−(E/prime)BkGA k+(E/prime−¯hω)/bracketrightbig −/integraldisplay dE/primef(E/prime)Tr/bracketleftbig AkGR k−(E/prime+¯hω)BkGA k+(E/prime)/bracketrightbig −/integraldisplay dE/primef(E/prime)Tr/bracketleftbig AkGA k−(E/prime)BkGA k+(E/prime−¯hω)/bracketrightbig . (A18) We consider the limit lim q→0limω→0χ. In this limit, Eq. ( A17) may be rewritten as χ=−i 2π¯h2/integraldisplayddk (2π)d∂2Zk(q,ω) ∂q∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle q=ω=0. (A19) The frequency derivative of Zk(q,ω) is given by 1 ¯h∂Zk ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0=/integraldisplay dE/primef(E/prime)Tr/bracketleftBigg Ak∂GR k−(E/prime) ∂E/primeBkGR k+(E/prime)/bracketrightBigg −/integraldisplay dE/primef(E/prime)Tr/bracketleftBigg AkGR k−(E/prime)Bk∂GA k+(E/prime) ∂E/prime/bracketrightBigg −/integraldisplay dE/primef(E/prime)Tr/bracketleftBigg Ak∂GR k−(E/prime) ∂E/primeBkGA k+(E/prime)/bracketrightBigg +/integraldisplay dE/primef(E/prime)Tr/bracketleftBigg AkGA k−(E/prime)Bk∂GA k+(E/prime) ∂E/prime/bracketrightBigg . (A20)Using ∂GR(E)/∂E=−GR(E)GR(E)/¯h, we obtain ∂Zk ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0=−/integraldisplay dE/primef(E/prime)Tr/bracketleftbig AkGR k−GRk −BkGRk +/bracketrightbig +/integraldisplay dE/primef(E/prime)Tr/bracketleftbig AkGR k−BkGAk +GAk +/bracketrightbig +/integraldisplay dE/primef(E/prime)Tr/bracketleftbig AkGR k−GRk −BkGAk +/bracketrightbig −/integraldisplay dE/primef(E/prime)Tr/bracketleftbig AkGA k−BkGAk +GAk +/bracketrightbig .(A21) Making use of lim q→0∂GR k+ ∂q=1 2GR kv·q qGR k (A22) we finally obtain χ=−i 2π¯h2/integraldisplayddk (2π)dlim q→0lim ω→0∂2Z(q,ω) ∂q∂ω =−i 4π¯h2q q·/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr[AkRvRRBkR+AkRRvRBkR −AkRRBkRvR−AkRvRBkAA +AkRBkAvAA+AkRBkAAvA −AkRvRRBkA−AkRRvRBkA +AkRRBkAvA−AkAvABkAA −AkABkAvAA−AkABkAAvA], (A23) where we use the abbreviations R=GR k(E) and A=GA k(E). When we substitute B=∂H ∂ˆMj,A=−evi, and q=qkˆek,w e obtain Eq. ( 18). When we substitute B=Tj,A=−evi, and q=qkˆek, we obtain Eq. ( 20). When we substitute A=−Ti, B=Tj, and q=qkˆek, we obtain Eq. ( 41). When we substi- tuteB=−evj,A=−Ti, and q=qkˆek, we obtain Eq. ( 50). When we substitute B=∂H ∂ˆMj,A=−Ti, and q=qkˆek,w e obtain Eq. ( 82). APPENDIX B: PERTURBATION THEORY FOR THE CHIRAL CONTRIBUTIONS TO CIT AND TO THE TORQUE-TORQUE CORRELATION In this Appendix, we derive expressions for the retarded function /angbracketleft/angbracketleftAcos(q·r);C/angbracketright/angbracketrightR(¯hω)( B 1 ) within first-order perturbation theory with respect to the per- turbation δH=Bηsin(q·r), (B2) which may arise, e.g., from the spatial oscillation of the mag- netization direction. As usual, it is convenient to obtain theretarded response function from the corresponding Matsubarafunction /angbracketleft/angbracketleftcos(q·r)A;C/angbracketright/angbracketright M(τ)=− /angbracketleft Tτcos(q·r)A(τ)C(0)/angbracketright.(B3) 245411-17FREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) The starting point for the perturbative expansion is the equation −/angbracketleftTτcos(q·r)A(τ1)C(0)/angbracketright =−Tr[e−βHTτcos(q·r)A(τ1)C(0)] Tr[e−βH] =−Tr{e−βH0Tτ[Ucos(q·r)A(τ1)C(0)]} Tr[e−βH0U], (B4) where H0is the unperturbed Hamiltonian and we consider the first order in the perturbation δH: U(1)=−1 ¯h/integraldisplay¯hβ 0dτ1Tτ{eτ1H0/¯hδHe−τ1H0/¯h}. (B5) The essential difference between Eqs. ( A3) and ( B4)i st h a t in Eq. ( A3) the operator Benters together with the factor sin(q·r)s i n (ωt)[ s e eE q .( A1)], while in Eq. ( B4) only the factor sin( q·r) is connected to Bin Eq. ( B2), while the factor sin(ωt) is coupled to the additional operator C. We use Eqs. ( A4) and ( A7) in order to express Acos(q·r) andBsin(q·r) in terms of annihilation and creation opera- tors. In terms of the correlators Z(3) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1) =/angbracketleftTτc† k−n(τ)ck+m(τ)c† k+n/prime(τ1)ck−m/prime(τ1)c† k−n/prime/primeck−m/prime/prime/angbracketright,(B6) Z(4) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1) =/angbracketleftTτc† k−n(τ)ck+m(τ)c† k+n/prime(τ1)ck−m/prime(τ1)c† k+n/prime/primeck+m/prime/prime/angbracketright,(B7) Z(5) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1) =/angbracketleftTτc† k+n(τ)ck−m(τ)c† k−n/prime(τ1)ck+m/prime(τ1)c† k+n/prime/primeck+m/prime/prime/angbracketright,(B8) and Z(6) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1) =/angbracketleftTτc† k+n(τ)ck−m(τ)c† k−n/prime(τ1)ck+m/prime(τ1)c† k−n/prime/primeck−m/prime/prime/angbracketright.(B9) Equation ( B4) can be written as /angbracketleft/angbracketleftcos(q·r)A;C/angbracketright/angbracketrightM(τ1) =ηV 4i¯h/integraldisplayddk (2π)d/integraldisplay¯hβ 0dτ/summationdisplay nm/summationdisplay n/primem/prime/summationdisplay n/prime/primem/prime/prime ×/bracketleftbig −B(2) knmA(1) kn/primem/primeCk−n/prime/primem/prime/primeZ(3) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1) −B(2) knmA(1) kn/primem/primeCk+n/prime/primem/prime/primeZ(4) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1) +B(1) knmA(2) kn/primem/primeCk+n/prime/primem/prime/primeZ(5) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1) +B(1) knmA(2) kn/primem/primeCk−n/prime/primem/prime/primeZ(6) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1)/bracketrightbig (B10) within first-order perturbation theory, where we defined Ck−n/prime/primem/prime/prime=/angbracketleftuk−n/prime/prime|C|uk−m/prime/prime/angbracketrightandCk+n/prime/primem/prime/prime=/angbracketleftuk+n/prime/prime|C|uk+m/prime/prime/angbracketright. Note that Z(5)can be obtained from Z(3)by replacing k−byk+andk+byk−. Similarly, Z(6)can be obtained from Z(4)by replacing k−byk+andk+byk−. Therefore we write down only the equations for Z(3)andZ(4)in the following.Using Wick’s theorem, we find Z(3) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1) =−GM m/primen(k−,τ1−τ)GM mn/prime(k+,τ−τ1)GM m/prime/primen/prime/prime(k−,0) +GM mn/prime(k+,τ−τ1)GM m/prime/primen(k−,−τ)GM m/primen/prime/prime(k−,τ1) (B11) and Z(4) knmn/primem/primen/prime/primem/prime/prime(τ,τ 1) =−GM mn/prime(k+,τ−τ1)GM m/primen(k−,τ1−τ)GM m/prime/primen/prime/prime(k+,0) +GM mn/prime/prime(k+,τ)GM m/primen(k−,τ1−τ)GM m/prime/primen/prime(k+,−τ1).(B12) The Fourier transform /angbracketleft/angbracketleftcos(q·r)A;C/angbracketright/angbracketrightM(iEN) =/integraldisplay¯hβ 0dτ1ei ¯hENτ1/angbracketleft/angbracketleftcos(q·r)A;C/angbracketright/angbracketrightM(τ1) (B13) of Eq. ( B10) can be written as /angbracketleft/angbracketleftcos(q·r)A;C/angbracketright/angbracketrightM(iEN) =ηV 4i¯h/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n/primem/prime/summationdisplay n/prime/primem/prime/prime ×/bracketleftbig −B(2) knmA(1) kn/primem/primeCk−n/prime/primem/prime/primeZ(3a) knmn/primem/primen/prime/primem/prime/prime(iEN) −B(2) knmA(1) kn/primem/primeCk+n/prime/primem/prime/primeZ(4a) knmn/primem/primen/prime/primem/prime/prime(iEN) +B(1) knmA(2) kn/primem/primeCk+n/prime/primem/prime/primeZ(5a) knmn/primem/primen/prime/primem/prime/prime(iEN) +B(1) knmA(2) kn/primem/primeCk−n/prime/primem/prime/primeZ(6a) knmn/primem/primen/prime/primem/prime/prime(iEN)/bracketrightbig (B14) in terms of the integrals Z(3a) knmn/primem/primen/prime/primem/prime/prime(iEN) =/integraldisplay¯hβ 0dτ/integraldisplay¯hβ 0dτ1ei ¯hENτ1 ×GM mn/prime(k+,τ−τ1)GM m/prime/primen(k−,−τ)GM m/primen/prime/prime(k−,τ1) =1 ¯hβ/summationdisplay pGM k+mn/prime(iEp)GM k−m/prime/primen(iEp)GM k−m/primen/prime/prime(iEp+iEN) (B15) and Z(4a) knmn/primem/primen/prime/primem/prime/prime(iEN) =/integraldisplay¯hβ 0dτ/integraldisplay¯hβ 0dτ1ei ¯hENτ1 ×GM mn/prime/prime(k+,τ)GM m/primen(k−,τ1−τ)GM m/prime/primen/prime(k+,−τ1) =1 ¯hβ/summationdisplay pGM k+mn/prime/prime(iEp)GM k−m/primen(iEp)GM k+m/prime/primen/prime(iEp−iEN), (B16) where EN=2πN/βis a bosonic Matsubara energy point and we used GM(τ)=1 ¯hβ∞/summationdisplay p=−∞e−iEpτ/¯hGM(iEp), (B17) where Ep=(2p+1)π/β is a fermionic Matsubara point. AgainZ(5a)is obtained from Z(3a)by replacing k−byk+ andk+byk−andZ(6a)is obtained from Z(4a)in the same way. 245411-18DYNAMICAL AND CURRENT-INDUCED … PHYSICAL REVIEW B 102, 245411 (2020) Summation over Matsubara points Epin Eqs. ( B15) and (B16) and analytic continuation iEN→¯hωyields 2πi¯hZ(3a) knmn/primem/primen/prime/primem/prime/prime(¯hω) −/integraldisplay dEf(E)GR k+mn/prime(E)GR k−m/prime/primen(E)GR k−m/primen/prime/prime(E+¯hω) +/integraldisplay dEf(E)GA k+mn/prime(E)GA k−m/prime/primen(E)GR k−m/primen/prime/prime(E+¯hω) −/integraldisplay dEf(E)GA k+mn/prime(E−¯hω)GA k−m/prime/primen(E−¯hω)GR k−m/primen/prime/prime(E) +/integraldisplay dEf(E)GA k+mn/prime(E−¯hω)GA k−m/prime/primen(E−¯hω)GA k−m/primen/prime/prime(E) (B18) and 2πi¯hZ(4a) knmn/primem/primen/prime/primem/prime/prime(¯hω) −/integraldisplay dEf(E)GR k+mn/prime/prime(E)GR k−m/primen(E)GA k+m/prime/primen/prime(E−¯hω) +/integraldisplay dEf(E)GA k+mn/prime/prime(E)GA k−m/primen(E)GA k+m/prime/primen/prime(E−¯hω) −/integraldisplay dEf(E)GR k+mn/prime/prime(E+¯hω)GR k−m/primen(E+¯hω)GR k+m/prime/primen/prime(E) +/integraldisplay dEf(E)GR k+mn/prime/prime(E+¯hω)GR k−m/primen(E+¯hω)GA k+m/prime/primen/prime(E). (B19) In the next step, we take the limit ω→0[ s e eE q s .( 64), (70), and ( 77)]: −1 Vlim ω→0Im/angbracketleft/angbracketleftAcos(q·r);C/angbracketright/angbracketrightR(¯hω) ¯hω =η 4¯hIm[Y(3)+Y(4)−Y(5)−Y(6)], (B20) where we defined Y(3)=1 i¯h/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n/primem/prime/summationdisplay n/prime/primem/prime/primeB(2) knmA(1) kn/primem/primeCk−n/prime/primem/prime/prime ×∂Z(3a) knmn/primem/primen/prime/primem/prime/prime(¯hω) ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0, Y(4)=1 i¯h/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n/primem/prime/summationdisplay n/prime/primem/prime/primeB(2) knmA(1) kn/primem/primeCk+n/prime/primem/prime/prime ×∂Z(4a) knmn/primem/primen/prime/primem/prime/prime(¯hω) ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0, Y(5)=1 i¯h/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n/primem/prime/summationdisplay n/prime/primem/prime/primeB(1) knmA(2) kn/primem/primeCk+n/prime/primem/prime/prime ×∂Z(5a) knmn/primem/primen/prime/primem/prime/prime(¯hω) ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0, Y(6)=1 i¯h/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n/primem/prime/summationdisplay n/prime/primem/prime/primeB(1) knmA(2) kn/primem/primeCk−n/prime/primem/prime/prime ×∂Z(6a) knmn/primem/primen/prime/primem/prime/prime(¯hω) ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0, (B21)which can be expressed as Y(3)=Y(3a)+Y(3b)andY(4)= Y(4a)+Y(4b), where 2π¯hY(3a)=1 ¯h/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr/bracketleftbig AkGR k−(E)Ck−GA k−(E)BkGA k+(E)GA k+(E) +AkGR k−(E)GR k−(E)Ck−GA k−(E)BkGA k+(E) +AkGR k−(E)Ck−GA k−(E)GA k−(E)BkGA k+(E)/bracketrightbig =/integraldisplayddk (2π)d/integraldisplay dEf/prime(E) ×Tr/bracketleftbig AkGR k−(E)Ck−GA k−(E)BkGA k+(E)/bracketrightbig , (B22) and 2π¯hY(3b)=−1 ¯h/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr/bracketleftbig AkGA k−(E)Ck−GA k−(E)BkGA k+(E)GA k+(E) +AkGR k−(E)GR k−(E)Ck−GR k−(E)BkGR k+(E) +AkGA k−(E)Ck−GA k−(E)GA k−(E)BkGA k+(E)/bracketrightbig . (B23) Similarly, 2π¯hY(4a)=1 ¯h/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr/bracketleftbig AkGR k−(E)BkGR k+(E)Ck+GA k+(E)GA k+(E) −AkGR k−(E)GR k−(E)BkGR k+(E)Ck+GA k+(E) −AkGR k−(E)BkGR k+(E)GR k+(E)Ck+GA k+(E)/bracketrightbig =/integraldisplayddk (2π)d/integraldisplay dEf/prime(E) ×Tr/bracketleftbig AkGR k−(E)BkGR k+(E)Ck+GA k+(E)/bracketrightbig (B24) and 2π¯hY(4b)=−1 ¯h/integraldisplayddk (2π)d/integraldisplay dEf(E) ×Tr/bracketleftbig AkGA k−(E)BkGA k+(E)Ck+GA k+(E)GA k+(E) +AkGR k−(E)GR k−(E)BkGR k+(E)Ck+GR k+(E) +AkGR k−(E)BkGR k+(E)GR k+(E)Ck+GR k+(E)/bracketrightbig . (B25) We call Y(3a)andY(4a)Fermi surface terms and Y(3b)and Y(4b)Fermi sea terms. Again Y(5)is obtained from Y(3)by replacing k−byk+andk+byk−andY(6)is obtained from Y(4)in the same way. Finally, we take the limit q→0: /Lambda1=−2 ¯hVηIm lim q→0lim ω→0∂ ∂ω∂ ∂qi/angbracketleft/angbracketleftAcos(q·r);C/angbracketright/angbracketrightR(¯hω) =1 2¯hlim q→0∂ ∂qiIm[Y(3)+Y(4)−Y(5)−Y(6)] =1 2¯hIm[X(3)+X(4)−X(5)−X(6)], (B26) 245411-19FREIMUTH, BLÜGEL, AND MOKROUSOV PHYSICAL REVIEW B 102, 245411 (2020) where we defined X(j)=∂ ∂qi/vextendsingle/vextendsingle/vextendsingle/vextendsingle q=0Y(j)(B27) forj=3,4,5,6. Since Y(4)andY(6)are related by the inter- change of k−andk+it follows that X(6)=−X(4). Similarly, sinceY(3)andY(5)are related by the interchange of k−and k+it follows that X(5)=−X(3). Consequently, we need /Lambda1=1 ¯hIm[X(3a)+X(3b)+X(4a)+X(4b)], (B28) whereX(3a)andX(4a)are the Fermi surface terms and X(3b) andX(4b)are the Fermi sea terms. The Fermi surface terms are given by X(3a)=−1 4π¯h/integraldisplayddk (2π)d/integraldisplay dEf/prime(E) ×Tr/bracketleftbigg AkGR k(E)vkGR k(E)CkGA k(E)BkGA k(E) +AkGR k(E)CkGA k(E)vkGA k(E)BkGA k(E) −AkGR k(E)CkGA k(E)BkGA k(E)vkGA k(E) +AkGR k(E)∂Ck ∂kGA k(E)BkGA k(E)/bracketrightbigg (B29) and X(4a)=−[X(3a)]∗. (B30) The Fermi sea terms are given by X(3b)=−1 4π¯h2/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftbigg −(ARvRRCRBR) +(AACAABAvA)−(ARRvRCRBR) −(ARRCRvRBR)+(ARRCRBRvR) −(AAvACABAA)−(AACAvABAA) +(AACABAvAA)+(AACABAAvA) −(AAvACAABA)−(AACAvAABA) −(AACAAvABA)−/parenleftbigg ARR∂C ∂kRBR/parenrightbigg −/parenleftbigg AA∂C ∂kAABA/parenrightbigg −/parenleftbigg AA∂C ∂kABAA/parenrightbigg/bracketrightbigg (B31)and X(4b)=−[X(3b)]∗. (B32) In Eq. ( B31), we use the abbreviations R=GR k(E),A= GA k(E),A=Ak,B=Bk,C=Ck. It is important to note that Ck−andCk+depend on qthrough k−=k−q/2 and k+= k+q/2. The qderivative therefore generates the additional terms with ∂Ck/∂kin Eqs. ( B29) and ( B31). In contrast, Ak andBkdo not depend linearly on q. Equation ( B28) simplifies due to the relations Eqs. ( B30) and ( B32) as follows: /Lambda1=2 ¯hIm[X(3a)+X(3b)]. (B33) In order to obtain the expression for the chiral contribution to the torque-torque correlation we choose the operators asfollows: B→T k, A→−Ti, C→Tj, (B34) ∂C ∂k=0, v→vl. This leads to Eqs. ( 78)–(80)o ft h em a i nt e x t . In order to obtain the expression for the chiral contribution to the CIT, we set B→Tk, A→−Ti, C→− evj, ∂C ∂k→−e¯h mδjl, v→vl.(B35) This leads to Eqs. ( 66)–(68). In order to obtain the expression for the chiral contribution to the ICIT, we set B→Tk, A→− evi, C→Tj, ∂C ∂k→0, v→vl.(B36) This leads to Eqs. ( 71)–(73). [1] K. Nawaoka, S. Miwa, Y . Shiota, N. Mizuochi, and Y . Suzuki, Appl. Phys. Express 8, 063004 (2015) . [2] H. Yang, O. Boulle, V . Cros, A. Fert, and M. Chshiev, Sci. Rep. 8, 12356 (2018) . [3] T. Srivastava, M. Schott, R. Juge, V . K ˇrižáková, M. Belmeguenai, Y . 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PhysRevB.97.201411.pdf

PHYSICAL REVIEW B 97, 201411(R) (2018) Rapid Communications Parity-time symmetry breaking in spin chains Alexey Galda1,2and Valerii M. Vinokur2 1James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA 2Materials Science Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, Illinois 60439, USA (Received 28 January 2018; revised manuscript received 26 March 2018; published 29 May 2018) We investigate nonequilibrium phase transitions in classical Heisenberg spin chains associated with spontaneous breaking of parity-time ( PT) symmetry of the system under the action of Slonczewski spin-transfer torque (STT) modeled by an applied imaginary magnetic field. We reveal the STT-driven PT symmetry-breaking phase transition between the regimes of precessional and exponentially damped spin dynamics and show that itsseveral properties can be derived from the distribution of zeros of the system’s partition function, the approachfirst introduced by Yang and Lee for studying equilibrium phase transitions in Ising spin chains. The physicalinterpretation of imaginary magnetic field as describing the action of nonconservative forces opens the possibilityof direct observations of Lee-Yang zeros in nonequilibrium physical systems. DOI: 10.1103/PhysRevB.97.201411 Introduction. Phase transitions where physical systems experience nonanalytic changes of their properties are one of the most remarkable phenomena occurring in many-particle systems [ 1]. In 1952 Yang and Lee devised an approach that reveals a deep structure of singularities associated withphase transitions via investigation of points on the complexplane of physical parameters where the partition function ofa system vanishes (Lee-Yang zeros) [ 2,3]. Recently, the Lee- Yang description was generalized to nonequilibrium systems[4–8], and is now becoming a powerful tool promising to bring unified understanding of both equilibrium and nonequi-librium processes. The latter can be efficiently described in aframework of non-Hermitian Hamiltonian approach in whichnon-Hermiticity is proportional to the external bias [ 9,10]. Since the Lee-Yang approach rests on analytical continuationof the partition function to the complex plane of the controllingparameter, it opens an appealing opportunity for incorporatingthe Lee-Yang approach into a non-Hermitian scheme to enablea universal unified description of phase transitions in opendissipative systems. Here we meet the challenge and investigatenonequilibrium phase transitions in classical Heisenberg spinchains that offer an exemplary laboratory for both the Lee-Yang model and the non-Hermitian approach. We find thenonequilibrium phase transition associated with spontaneousbreaking of parity-time ( PT) symmetry of the system under the action of Slonczewski spin-transfer torque (STT) modeledby imaginary magnetic field. We relate the singularities of thisphase transition with the distribution of the Lee-Yang zeros. The pioneering Yang-Lee description of phase transitions [2,3] via the distribution of zeros of the partition function was achieved by going to the extended complex plane of theapplied magnetic field upon adding its imaginary component.The zeros were located on a unit circle in the complexfugacity plane, ξ=exp(−H/k BT), where His the applied transverse external magnetic field measured in energy units.Later, the Lee-Yang circle theorem has been generalized toHeisenberg models [ 11–17], ferromagnetic Ising models of arbitrary high spin [ 11,18–21], Heisenberg and general Isingmodels with multiple-spin interactions [ 22], and isotropic clas- sical spins of arbitrary dimensionality on a one-dimensional lattice [ 23,24]. Recently, the Lee-Yang zeros approach has emerged in nonequilibrium physics (see, e.g., Ref. [ 8]), where the Lee-Yang zeros of a partition function of trajectories describing the evolution of a stochastic process were studiedexperimentally in the nonequilibrium settings, where theycharacterize dynamic phase transitions occurring in a quantumsystem after a quench. In parallel, it has been discovered thatthe action of some nonconservative forces, such as Gilbertdamping and Slonczewski STT, on individual spins is equiva-lent to switching on an imaginary magnetic field [ 9,25]. Indeed, when considering the dynamics of a single classical spin,an imaginary magnetic field is necessary to account for theaction of nonconservative forces and dissipation. This bringsthe possibility for a direct observation of Lee-Yang zerosin nonequilibrium coupled spin systems. Altogether thesefindings suggest that incorporating Lee-Yang zeros descriptioninto the machinery of non-Hermitian quantum mechanics willcreate a powerful tool for quantitative description of out-of-equilibrium phase transitions. Here we explore this route. We build our approach on the non-Hermitian quantum mechanics of systems endowed with PT symmetry, devised by Bender and Boettcher [ 26,27]. They demonstrated that there exists a class of non-Hermitian but PT-symmetric Hamilto- nians whose energy spectrum is real as long as eigenstatesof the Hamiltonian are also eigenstates of the PT operator, but acquires an imaginary component as soon as the latter property is lost. The notion that PT symmetry describes open dissipative systems with “balanced loss and gain” [ 28] enables using non-Hermitian Hamiltonians for description of dynamicphase transitions between stationary and nonstationary dynam-ics of dissipative systems as transitions between the stateswith unbroken and broken PT symmetry of their eigenstates, correspondingly. Utilizing the theory of the STT-driven PT symmetry- breaking phase transition in single-spin systems [ 9,25], we investigate nonequilibrium phase transitions in the spin chain 2469-9950/2018/97(20)/201411(4) 201411-1 ©2018 American Physical SocietyALEXEY GALDA AND V ALERII M. VINOKUR PHYSICAL REVIEW B 97, 201411(R) (2018) which is a generic model for a broad variety of experimental systems. We consider a PT-symmetric classical Heisenberg spin chain: H=−J SN/summationdisplay k=1SkSk+1+hN/summationdisplay k=1Sk, (1) where Jand h=hˆx+iβˆyare dimensionless coupling strength (ferromagnetic when J> 0) and magnetic field, correspondingly, and |Sk|=S→∞ . The Hamiltonian ( 1)i s invariant under simultaneous parity ( Sy→−Sy) and time- reversal ( t→−t,i→−i) symmetries. There are 9 and 60 ways of defining parity lays (and every single one of them isright) symmetry on a lattice. We choose to define the parity op-erator Pas acting as a mirror reflection of the entire spin chain with respect to the xzplane: ( S x,Sy,Sz)→(Sx,−Sy,Sz). Dynamics. Our first step is to establish that the spin chain experiences nonequilibrium phase transition associatedwith the PT-symmetric STT term. To that end, we employ the SU(2) spin-coherent states [ 29,30] in the classical limit: |ζ/angbracketright=e ζˆS+|S,−S/angbracketright, where ˆS±≡ˆSx±iˆSy, andζ∈Cis the standard stereographic projection of the spin direction on a unitsphere, ζ=(s x+isy)/(1−sz), with sk≡Sk/S,S→∞ .ζ and ¯ζform a complex conjugate pair of stereographic projec- tion coordinates, and the expectation value of the Hamiltonian(1) in spin-coherent states is given by [ 31] H(ζ,¯ζ)=/angbracketleftζ|ˆH|ζ/angbracketright /angbracketleftζ|ζ/angbracketright. (2) Accordingly, the equations of motion for individual spins in stereographic coordinates are ˙ζk=i(1+|ζk|2)2 2S∂H ∂¯ζk,k=1,..., N, (3) which for the Hamiltonian ( 1) yields a system of coupled differential equations: ˙ζk(t)=−i(h+β) 2/parenleftbigg ζ2 k−h−β h+β/parenrightbigg +iJ(ζk−ζk+1)(1+ζk¯ζk+1) 1+|ζk+1|2 +iJ(ζk−ζk−1)(1+ζk¯ζk−1) 1+|ζk−1|2. (4) The first term on the right-hand side of Eq. ( 4) describes individual spin dynamics (see Ref. [ 9]), while the other two terms are responsible for interspin coupling in the chain. Numerical simulations of spin dynamics governed by Eq. ( 4) reveal two fundamentally different regimes. When |h|>|β|, the spin chain exhibits seemingly chaotic oscillating behavior, while for |h|/lessorequalslant|β|all spins saturate exponentially fast toward the stable fixed point ζ=ζ1=/radicalBig h+iβ h−iβin stereo- graphic projection coordinates. To show that a phase transition occurs at |h|=|β|, it is sufficient to consider the time evo- lution of the projection of the total spin on the zaxis,sz≡ 1/N/summationtextN k=1sz k, averaged over all possible initial conditions of allNspins, sk(0). As can be seen in Fig. 1,szis an oscillatory function of time when |h|>|β|(regime of unbroken PTFIG. 1. Divergence of the average period of oscillations. The points (red) represent numerical simulations of the period of oscil- lations of szas a function of applied imaginary magnetic field, β, averaged over 100 random initial conditions for N=4 spins, J=1, andh=1. The analytic dependence τ=2π(1−β2)−1/2is plotted as a straight line (blue). In the inset the exponential saturation of sz(t)i s shown in the regime of broken PT symmetry, for h=1a n dβ=1.1, sz(t)−sz 1∼exp(−/radicalbig β2−h2t). symmetry), and saturates exponentially quickly when the PT symmetry is broken, i.e., |h|/lessorequalslant|β|.T h e zprojection of the saturation direction is sz 1=−h/β. The dependence of the period of spin oscillations on the magnitude of imaginary magnetic field, τ=2π//radicalbig h2−β2,i s illustrated in Fig. 1, and is in agreement with the theorem that for every system with the exact (unbroken) PTsymmetry there exists a unitary similarity transformation mapping the system’snon-Hermitian Hamiltonian to a Hermitian one [ 32]. While the exact form of the equivalent Hermitian form of the spin chainHamiltonian ( 1) is unknown, for each individual spin the PT- symmetric Hamiltonian H (0)PT=(hˆx+iβˆy)Sin the regime of unbroken PT symmetry ( |h|/greaterorequalslant|β|) was shown to be equiv- alent to H/prime (0)=/radicalbig h2−β2ˆxS/prime[9], with the similarity trans- formation given by the Möbius transformation ζ/prime=/radicalBig h+β h−βζ in stereographic projection coordinates, which corresponds exactly to the oscillation period τ=2π(h2−β2)−1/2for individual spins. In the limit of weak interspin coupling, |J|/lessmuchh,β,t h es p i n chain is effectively uncoupled, with individual spin dynamicsdescribed by the single-spin solution with PT symmetry breaking at |β|=|h|[9,25]. In the opposite limit of very strong coupling, |J|/greatermuchh,β, spins exhibit correlated dynamics, yet they are not (anti-) ferromagnetically aligned, as could benaively expected. This is because the Hamiltonian ( 1) does not contain any nonconservative terms that can minimize theinterspin energy term directly. Shown in Fig. 2is the summary of our study of the dependence of the critical imaginary magnetic field, β c,o n the spin-spin interaction type and strength, J. We display the results of the numerical simulations of the spin chain dynamicswithN=4 and random initial conditions. We analyze two models, the isotropic Heisenberg model ( 1) and the anisotropic 201411-2PARITY-TIME SYMMETRY BREAKING IN SPIN CHAINS PHYSICAL REVIEW B 97, 201411(R) (2018) FIG. 2. Dependence of the critical imaginary magnetic field, βc, on the absolute magnitude of the interspin coupling, |J|, for classical isotropic (squares) and anisotropic (circles) Heisenberg models with periodic (red) and open (blue) boundary conditions for h=1. The model with anisotropic Sz-Szinteraction [see Eq. ( 5)] exhibits a nontrivial dependence of the critical imaginary magnetic field on the interaction strength, βc(|J|). Heisenberg model with S(z)-S(z)interaction: H(an)=−J SN/summationdisplay k=1Sz kSz k+1+hN/summationdisplay k=1Sk, (5) each with periodic and open boundary conditions. We find that in the model with isotropic spin-spin interaction, the criticalamplitude of imaginary magnetic field is independent of bothamplitude of the interaction, J, and the choice of boundary conditions. To conclude at this point, we find that the spin chain experiences the PTsymmetry-breaking transition manifesting as a transition between the stationary oscillating state (endowedwith the unbroken PT symmetry) and the PT symmetry- broken nonstationary state with each individual spin in the chain saturating in the direction ζ 1=−/radicalBig h−β h+βexponentially quickly, exhibiting the behavior identical to that of a single spin [ 9]. Thermodynamics. Now we turn to investigating the statis- tical physics of the linear Heisenberg spin chain described bythe Hamiltonian ( 1). The partition function is calculated as an integral over all possible orientations of Nspins on a 2-sphere: Z N=/integraldisplay ···/integraldisplayN/productdisplay k=1/parenleftbiggd/Omega1k 4π/parenrightbigg exp/parenleftbigg −H T/parenrightbigg , (6) where /Omega1kis the element of solid angle in the direction Sk for each of Ncoupled spins. We first analyze the partition function numerically, plotting it in Fig. 3for a chain of N=4 ferromagnetically coupled spins, as a function of applied realand imaginary magnetic fields, handβ, correspondingly. The partition function is real for all handβ, but only strictly positive in the parametric range of unbroken PT symmetry, |h|/greaterorequalslant|β|, where PT-symmetric systems are equivalent to Hermitian ones [ 32]. In the regime of broken PT symmetry, the par- tition function can assume negative values, and, accordingly,Lee-Yang zeros appear. Negative statistical weight becomes FIG. 3. Numerical result for the partition function of N=4f e r - romagnetically coupled spins ( J=1) with open boundary condition placed in magnetic field h=hˆx+iβˆy. Zeros of the partition function (Lee-Yang zeros) are only observed in the regime of broken PT symmetry, i.e., when |h|<|β|. The partition function takes strictly positive values in the regime of unbroken PT symmetry, |h|>|β|. possible when energy eigenvalues appear in complex conjugate pairs. This is related to the sign problem, which in turn is themanifestation of the general problem of complex weights ina statistical sum, arising naturally in Euclidean quantum fieldtheories with nonzero chemical potential [ 33–35]. At asymptotically low temperatures, T→0, the partition function ( 6) assumes the form lim T→0ZN=/bracketleftBigg sinh/parenleftbigJ T/parenrightbig /parenleftbigJ T/parenrightbig/bracketrightBiggN−1 sinc/parenleftBigg N/radicalbig β2−h2 T/parenrightBigg ,(7) where sinc( x)≡sin(x)/x, which is confirmed by our numer- ical calculations (see Fig. 4) and is in full agreement with the known results for classical Heisenberg spin chains with h=0 in the limit T→0, where the Lee-Yang zeros are uniformly FIG. 4. Partition function for N=4 spins with J=1a sa function of N/radicalbig β2−h2/T for different handβand a range of temperatures: T=1 (blue), 0.5 (orange), 0.2 (green), and 0.1 (red). The point shape represents different “slices” of the partition function:ath=0( c i r c l e s ) ,1 .5T(square), and 3 T(triangles). The dashed curve corresponds to the function sinc( N/radicalbig β2−h2/T). 201411-3ALEXEY GALDA AND V ALERII M. VINOKUR PHYSICAL REVIEW B 97, 201411(R) (2018) distributed along the unit circle in the complex fugacity plane, ξ=exp(−iβ/T )[13,17,22–24,36]. In order to study zeros of the partition function ( 6), we calculate the latter numerically as a function of real andimaginary applied magnetic fields, handβ, respectively (see Fig. 3). The behaviors of the partition function in the regimes of unbroken ( |h|>|β|) and broken ( |h|>|β|)PT symmetry of the Hamiltonian ( 1), differ fundamentally from each other. We find that in the regime of the broken PT symmetry, the partition function depends only on the product/radicalbig β2−h2rather than on handβseparately and independently. More generally, our results indicate that the following relation holds for all J andT: ZN(h,β)=ZN(0,/radicalbig β2−h2). (8) To demonstrate this, for N=4,J=1, andT=0.1,0.2,0.5,1, we calculate ZN(h,β) at several different values of hand plot them as a function of N/radicalbig β2−h2/Tin Fig. 4. As a result, we obtain a series of smooth curves that approach the zero-temperature result, Eq. ( 7). Conclusions. We showed that the PT symmetry-breaking phase transition in a system of coupled classical Heisenbergspins leads to a sharp transition from precessional to exponen-tially saturating dynamics, regardless of the initial conditionsof each spin. We found that in a system with the isotropicnearest-neighbor spin-spin interaction, the condition of the PT symmetry breaking does not depend on the coupling strength,for both open and periodic boundary conditions, which is notgenerally the case for anisotropic interactions. Furthermore, thePT symmetry-breaking condition is found to be indepen- dent of the length of the spin chain. The parametric regionofPT symmetry breaking (which is |β|>|h|for the spin chain Hamiltonian considered in this work) plays an importantrole for both dynamic and thermodynamic properties of thesystem of coupled classical Heisenberg spins. This proves thatthe notion of the imaginary fields describing nonconservativeforces, and, consequently, the PT symmetry in general, can be generalized to a much wider class of physical problemsthan considered before. The direct correspondence betweenthe actions of imaginary magnetic field on spin dynamics andSlonczewski STT [ 9] allows for an experimental verification of the PT symmetry-breaking phase transition in spin chainsby studying magnetization dynamics in a thin ferromagneticwire driven by spin Hall effect spin torque (see, e.g., [ 37,38]) generated by placing it on top of a thin nonmagnetic film ofa material with a sufficiently large spin Hall angle (e.g., W orTa). We generalized the Lee-Yang theorem for classical spin chains to PT-symmetric systems with real magnetic field applied perpendicular to the transverse imaginary magnetic field. The location of Lee-Yang zeros is modified according to Eq. ( 8) for nonzero applied real magnetic field along the xaxis, while still remaining on the imaginary axis for the transversefield applied in the ydirection. Acknowledgments. A.G. and V .M.V . were supported by the U.S. Department of Energy, Office of Science, MaterialsSciences and Engineering Division. [1] H. B. Callen, Thermodynamics and an Introduction to Thermo- statistics (Wiley, New York, 1985). [2] C. N. Yang and T. D. Lee, Phys. Rev. 87,404 (1952 ). [3] C. N. Yang and T. D. Lee, Phys. Rev. 87,410 (1952 ). [ 4 ] R .A .B l y t h ea n dM .RE v a n s , P h y s .R e v .L e t t . 89,080601 (2002 ). [5] I. Bena, M. Droz, and A. Lipowski, Int. J. Mod. Phys. B 19,4269 (2005 ). [6] M. Heyl, A. Polkovnikov, and S. Kehrein, P h y s .R e v .L e t t . 110, 135704 (2013 ). [7] J. M. Hickey, C. Flindt, and J. P. Garrahan, P h y s .R e v .E 90, 062128 (2014 ) [8] K. Brandner, V . F. Maisi, J. P. Pekola, J. P. Garrahan, and C. Flindt, Phys. Rev. Lett. 118,180601 (2017 ). [9] A. Galda and V . M. Vinokur, P h y s .R e v .B 94,020408(R) (2016 ). [10] V . Tripathi, A. Galda, H. Barman, and V . M. Vinokur, Phys. Rev. B94,041104 (2016 ). [11] M. Suzuki, Progr. Theor. Phys. 40,1246 (1968 ). [12] S. Katsura, Phys. Rev. 127,1508 (1962 ). [13] M. Suzuki, Progr. Theor. Phys. 41,1438 (1969 ). [14] C. Kawabata and M. Suzuki, J. Phys. Soc. Jpn. 28,16(1970 ). [15] H. F. Trotter, Prog. Amer. Math. Soc. 10,545 (1959 ). [16] O. J. Heilmann and E. H. Lieb, Phys. Rev. Lett 24,1412 (1970 ). [17] H. Kunz, Phys. Lett. A 32,311 (1970 ). [18] T. Asano, Progr. Theor. Phys. 40,1328 (1968 ). [19] T. Asano, J. Phys. Soc. Jpn. 25,1220 (1968 ).[20] M. Suzuki, J. Math. Phys. 9,2064 (1968 ). [21] R. B. Griffiths, J. Math. Phys. 10,1559 (1969 ). [22] M. Suzuki and M. E. Fisher, J. Math. Phys. 12,235 (1971 ). [23] M. E. Fisher, A m .J .P h y s . 32,343 (1964 ). [24] H. E. Stanley, Phys. Rev. 179,570 (1969 ). [25] A. Galda and V . M. Vinokur, Nat. Sci. Rep. 7,1168 (2017 ). [26] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80,5243 (1998 ). [27] C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40,2201 (1999 ). [28] A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38,L171 (2005 ). [29] E. H. Lieb, Commun. Math. Phys. 31,327 (1973 ). [30] M. Stone, K.-S. Park, and A. Garg, J. Math. Phys. 41,8025 (2000 ). [31] J. M. Radcliffe, J. Phys. A 4,313 (1971 ). [32] A. Mostafazadeh, J. Math. Phys. 36,7081 (2003 ). [33] M. G. Alford, A. Schmitt, K. Rajagopal, and T. Schäfer, Rev. Mod. Phys. 80,1455 (2008 ). [34] M. P. Lombardo, J. Phys. G 35,104019 (2008 ). [35] M. Stephanov, PoS LAT2006, 024 (2016).[36] G. S. Joyce, Phys. Rev. 155,478 (1967 ). [37] C. F. Pai, L. Liu, Y . Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 101,122404 (2012 ). [38] L. Liu, C. F. Pai, Y . Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336,555 (2012 ). 201411-4

PhysRevB.69.085209.pdf

Magnetization relaxation in Ga,MnAs ferromagnetic semiconductors Jairo Sinova,1T. Jungwirth,2,3X. Liu,4Y. Sasaki,4J. K. Furdyna,4,3W. A. Atkinson,5and A. H. MacDonald3 1Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA 2Institute of Physics ASCR, Cukrovarnicka ´10, 162 53 Praha 6, Czech Republic 3Department of Physics, University of Texas at Austin, Austin, Texas 78712-0264, USA 4Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA 5Department of Physics, Trent University, Ontario, Canada K9J 7B8 ~Received 19 August 2003; revised manuscript received 17 November 2003; published 27 February 2004 ! We describe a theory of Mn local-moment magnetization relaxation due to p-dkinetic-exchange coupling with the itinerant-spin subsystem in the ferromagnetic semiconductor ~Ga,Mn !As alloy. The theoretical Gilbert damping coefficient implied by this mechanism is calculated as a function of Mn-moment density, holeconcentration, and quasiparticle lifetime. Comparison with experimental ferromagnetic resonance data suggeststhat in annealed strongly metallic samples, p-dcoupling contributes significantly to the damping rate of the magnetization precession at low temperatures. By combining the theoretical Gilbert coefficient with the valuesof the magnetic anisotropy energy, we estimate that the typical critical current for spin-transfer magnetizationswitching in all-semiconductor trilayer devices can be as low as ;10 5Acm22. DOI: 10.1103/PhysRevB.69.085209 PACS number ~s!: 73.20.Mf, 73.40. 2c, 85.75. 2d I. INTRODUCTION The Gilbert coefficient describes the damping of small- angle magnetization precession and is one of the key param-eters that characterizes collective magnetization dynamics ina ferromagnet. Early theories of magnetization dynamics in transition metals viewed exchange coupling ( }Ss) between local moment d-shell spins Sand itinerant s-pband spins s as a key relaxation mechanism. 1We now recognize that this model needs to be elaborated for transition metals to accountfor the itinerant character of their d-electrons. Models of d-shell local moments that are exchange coupled to itinerant s-pband electrons have, however, been resurrected recently because they provide a good description of ferromagnetismin many diluted magnetic semiconductors ~DMS’s !,~Ga,M- n!As in particular. 2,3Exchange-coupling between local mo- ments and itinerant electrons should also contribute signifi-cantly to Gilbert damping in these new ferromagneticsystems. The elementary process for this damping mecha-nism is one in which a local-moment magnon is annihilatedby exchange interaction with a band electron that suffers aspin flip. This process cannot by itself change the total mag-netic moment since the exchange Hamiltonian commutes with the total spin S1s. Net relaxation of the magnetization requires another independent process in which the itinerantelectron spin relaxes through spin-orbit interactions. Recent experiments and ab initiocalculations 4have estab- lished that for doping levels up to several percent, a substi-tutional Mn impurity in GaAs introduces five strongly local-izeddelectrons and a delocalized hole in the As/Ga pband, and that ferromagnetic coupling between the S55/2 Mn mo- ments is mediated by the p-dkinetic exchange. Hence, we model 5the electronic structure of the free carriers by that of the host material, implicitly assuming a shallow acceptor pic- ture. The free-carrier quasiparticles are p-dexchange coupled to the local moments with strength3Jpd ’55 meVnm3, and have a finite life time that can be esti- mated perturbatively. This theoretical picture leads to an ac-curate description of many thermodynamic and transport properties of optimally annealed ~Ga,Mn !As samples, such as the measured transition temperature,6,7the anomalous Hall effect,3,8anisotropic magnetoresistance,8and magneto- optical properties.6,9Particularly important in justifying the present theory are results for the magnetocrystallineanisotropy, 10spin stiffness,11,12and Bloch domain width13 that all agree well with experiment. These parameters followfrom the long-wavelength limit of the theory of the Mn spin-wave dispersion, and reflect the retarded and nonlocal char-acter of the valence-band-hole mediated interactions betweenMn moments. 11The Gilbert damping of magnetization pre- cession discussed here is the aspect of this long-wavelengthcollective magnetization dynamics that is most directly de-pendent on valence-band spin-orbit coupling. In Sec. II of this paper, we present a fully microscopic theory of the kinetic-exchange contribution to the Gilbertcoefficient in DMS’s. By comparing the linear response pre-dicted by the classical phenomenological Landau-Lifshitz-Gilbert ~LLG!equation with microscopic linear-response theory, we identify the Gilbert coefficient with the dissipativepart of a susceptibility diagram. In Sec. III we present experimental ferromagnetic reso- nance ~FMR !data 12,14recorded as a function of temperature and external magnetic-field strength and orientation. Sincethe frequency dependence of the FMR linewidth is notavailable, 12,14we are unable to experimentally decouple in- homogeneous FMR broadening and the intrinsic Gilbert damping contributions to the linewidth to make a quantita-tive comparison with the theory. Nevertheless, the data indi-cate that the magnetic inhomogeneity contribution is largelysuppressed in the more metallic, annealed samples and thatmuch of the observed low-temperature FMR linewidth inthese samples can be explained by damping of the magneti- zation precession mediated through the p-dcoupling. By adding a spin-torque term 15to the LLG equation, we estimate in Sec. IV that the typical critical current for mag-netization switching due to spin-transfer torques in an all-PHYSICAL REVIEW B 69, 085209 ~2004! 0163-1829/2004/69 ~8!/085209 ~6!/$22.50 ©2004 The American Physical Society 69085209-1semiconductor trilayer device consisting of magnetically ‘‘soft’’ and ‘‘hard’’ DMS layers separated by a nonmagnetic spacer will be ;105Acm22. In metals, this spin-transfer effect is currently the focus of a considerable experimental16 and theoretical17research. Spin-transfer switching has not yet been demonstrated in all-semiconductor systems, but theeffect promises to have a richer phenomenology in this casebecause of the flexibility of semiconductor ferromagnet ma-terials, and because of the possibility of combining spin-transfer with other semiconductor circuit functionalities. 18,19 The relatively low critical currents we predict for semicon-ductors may also circumvent the incomplete magnetizationswitching encountered in metallic magnetic tunnel junctionsthat occurs due to the interference of strong self-field effectswith the spin-transfer torques. 20The paper concludes with a brief summary of our results. II. THEORY OF THE GILBERT DAMPING Semiclassical LLG linear response. The phenomenologi- cal LLG equation for collective magnetization dynamics is dM dt52gmB \M3Beff1a MM3dM dt, ~1! whereMis the local Mn-moment magnetization, Beff5 2]E/]Mis the effective magnetic field, gis the Lande ´ g-factor, mBis the Bohr magneton, and ais the phenomeno- logical Gilbert damping coefficient. Unless adepends strongly on the orientation of the magnetization21or if the magnetization is not fully aligned with the external staticmagnetic field, the Gilbert damping rate, observed in experi-ment through a frequency-dependent FMR linewidth, is in-dependent of the static field and of the details of magneticanisotropies present in the sample. 22This allows us to as- sume in this section a simple geometry in which the anisot-ropy fields are represented by a single, uniaxial anisotropyenergy density coefficient U. For small fluctuations of the Mn magnetization orientation around the easy axis, Eq. ~1! can be used to derive a phenomenological response functionof the magnetic system to a weak transverse field. For zeroexternal static magnetic field the corresponding inverse sus-ceptibility reads x215\ ~gmB!2NMnSSU˜2iav 2iv ivU˜2iavD, ~2! whereU˜5U/(\NMnS),NMn54x/alc3is the density of uni- formly distributed Mn moments in Ga 12xMnxAs (alcis the GaAs lattice constant !, and vis the frequency of the external rf field perturbation. Microscopic theory. We derive the zero-temperature quan- tum response function from our effective Hamiltonian theory and obtain a microscopic expression for aby equating the quantum-mechanical response function to the classical one in the uniform v!0 limit. We start by writing a quantum ana- log of Eq. ~1!using the linear-response theory,^Mx~r,t!&52i \E 2‘‘ dt8Edr8~^@Mx~r,t!, 2Mx~r8,t8!Bx~t8!#&1^@Mx~r,t!, 2My~r8,t8!By~t8!#&!u~t2t8!, ~3! which leads to the retarded transverse susceptibility: xi,jR~r,tur8,t8!5~gmB!2i \^@Si~r,t!,Sj~r8,t8!#&u~t2t8!. ~4! Herei5x,yandSi(r,t)5Mi(r,t)/(gmB) are the Mn trans- verse spin-density operators. To evaluate the correlation function ~4!we use the Holstein-Primakoff boson representation of the spin opera-tors assuming small fluctuations around the mean-field or- dered state, S 15bA2NMnSandS25b†A2NMnS@Sx5(S1 1S2)/2,Sy5(S12S2)/2i], and choosing the zˆdirection as the quantization axis. After integrating out the itinerant car-rier degrees of freedom within the coherent-state path-integral formalism of the many-body problem we obtain the partition function, Z5 *D@z¯z#exp(2S@z¯z#), with the action given to quadratic order in zandz¯~the complex numbers representing the bosonic degrees of freedom !by S@z¯z#51/bV( m,kz¯~k,Vm!~2iVm1P~iVm!!z~k,Vm!. ~5! In Eq. ~5!, the first term is the standard Berry’s phase and the second term is the itinerant carrier spin-polarization diagramfor the high-symmetry case where the cross terms of the formz¯z¯andzzvanish in S@z¯z#:11 P~iV!5NMnJpd2S 2bEd3k ~2p!3( m,a,bGa~ivm,k!Gb~ivm 1iV,k!u^fa~k!us1ufb~k!&u2. ~6! HereGa(z,k) is the single-particle band Green’s function and fa(k) are the band eigenstates. From the partition function we compute directly the imaginary time response functionsat finite Matsubara frequencies which after their correspond-ing analytic continuations yield xxxR~v!52~gmB!2NMnS 2\2Pret~v! v21id2Pret2~v!, xxyR~v!52i~gmB!2NMnS 2\2v v21id2Pret2~v!.~7! Here Pret~calculated below !describes mathematically the retarded interaction between the Mn bosonic degrees of free- dom due to the p-dkinetic exchange with valence-band holes. Connecting classical phenomenology and microscopic theory.Inverting the retarded susceptibility in Eq. ~7!for the uniform ( k50) precession mode, we obtainJAIRO SINOVA et al. PHYSICAL REVIEW B 69, 085209 ~2004! 085209-2xxxR21~v!5\Pret~v! ~gmB!2NMnS, xxyR21~v!52i\v ~gmB!2NMnS. ~8! Comparing Eqs. ~8!and~2!, we obtain the microscopic con- tribution to the Gilbert coefficient from kinetic-exchangecoupling: a52lim v!0@ImPret~v!/v#. ~9! Note that xxyR21is explicitly equal to the off-diagonal ele- ment of x21in Eq. ~2!and that, also consistently, the real part of Pret(v), in the limit of v!0, gives the magneto- crystalline contribution to the anisotropy energy U˜, as ex- plained in detail in Ref. 11. To compute Pret(v), we take into account the effects of disorder present in the system perturbatively by accountingfor the finite lifetime of band quasiparticles, which for sim- plicity we characterize by a single number G. The quasipar- ticle broadening Gwas chosen to be in the range estimated in previous detailed studies of transport properties of these sys-tems, which achieve good agreement with experiment. 8 The single-particle Green’s function for the valence-band quasiparticles is thus written as Ga(k,z)5*2‘‘dv8/ (2p)Aa(v8,k)/(z2v8) with a spectral function Aa(e,k) 5G/@(e2ea,k)21G2/4#. In the present case we take the valence-band electronic structure to be described by the six-band Kohn-Luttinger Hamiltonian in the presence of an ef- fective kinetic-exchange field h eff5JpdNMn^S&.10In a col- linear ferromagnetic state and zero-temperature ^S&5S, and we obtain a5lim v!0NMnJpd2S 4\vEd3k ~2p!3( a,bu^fa~k!us1ufb~k!&u2 3Ede 2pAa,k~e!Ab,k~e1\v!@f~e!2f~e1\v!# 5Jpdheff 4\Ed3k ~2p!3( a,bu^fa~k!us1ufb~k!&u2 3Aa,k~eF!Ab,k~eF!. ~10! In choosing a spin- and band-independent lifetime, we are implicitly appealing to the dominance of spin-independentCoulomb scattering off Mn acceptors and interstitials as thedominant 8scattering mechanism. Spin-orbit interactions en- ter through their presence in the intrinsic bands rather thanthrough spin-flip quasiparticle scattering events. In thismodel, the Gilbert coefficient in Eq. ~10!has intraband and interband contributions that have qualitatively different dis-order dependences, as illustrated in Fig. 1. Note that the ter-minology we use here recognizes that no band has spin char-acter sufficiently definite to justify the usual distinctionbetween majority and minority spin bands. The intrabandterm we refer to here would be spin-flip scattering within a given spin-split band in the more usual language and ispresent only because of intrinsic spin-orbit coupling in the host semiconductor bands.The intraband contribution to ais proportional to 1/ Gat small G, i.e., proportional to the con- ductivity rather than the resistivity, and would dominate thedamping if disorder were weak. The interband contribution to a, on the other hand, requires disorder to breach the wave-vector separation between different bands at the Fermi energy and is an increasing function of G. The overall de- pendence of the Gilbert coefficient on the sample’s mobilityis non-monotonic, as illustrated in Fig. 1, with the position ofthe minimum depending on both hole and Mn-moment den-sities, and on other parameters of the DMS system. The complexity and tunability of the Gilbert coefficient in DMS’s is illustrated in Fig. 2 where we plot aas a function of the hole density for G5150 and 50 meV and for Mn dopingx52–8%. These parameter values bracket the range typical for metallic ~Ga,Mn !As DMS’s. The theory curves in Fig. 2 predict that aincreases with increasing hole density because of the higher quasiparticle density of states at largerdensities. The dependence of the Gilbert coefficient on xis more complex. The prefactor h effin Eq. ~10!reflects the proportionality of the kinetic-exchange coupling to the Mnspin density and causes the Gilbert damping implied by thismechanism to decrease with decreasing xat low Mn doping. This behavior is clearly seen in Fig. 2 ~a!forx<6%. On the other hand, an opposite trend is predicted for higher Mn- moment densities where the effect of h effon the intraband and interband matrix elements in Eq. ~10!takes over. In that case, larger heffvalues lead to a reduced spin mixing in the quasiparticle bands and, therefore, to smaller Gilbert damp- ing rates. This implicit dependence of aonheffis more dramatic in higher-quality samples. We expect that the above kinetic-exchange mechanism of the Gilbert damping will dominate at low temperatureswhere other mechanisms such as magnon-magnon interac-tions vanish.At temperatures close to the Curie temperature,on the other hand, the contribution to magnetization preces-sion damping due to the kinetic-exchange coupling can be FIG. 1. Total Gilbert damping coefficient a, interband contribu- tionainter, and intraband contribution aintraas a function of quasi- particle lifetime broadening Gfor a carrier density of 0.5 nm23and x58%.MAGNETIZATION RELAXATION IN ~Ga,Mn !A s... P H Y SICAL REVIEW B 69, 085209 ~2004! 085209-3ignored. The argument is based on an approximation that combines our zero-temperature microscopic theory of awith a finite-temperature mean-field description of heff.10Within the mean-field model, heffis proportional to mean Mn spin- polarization ^S&whose temperature dependence is given by the Brillouin function with a temperature-dependent meanfield. 10The curves in Fig. 2 can therefore be approximately assigned also to a ~Ga,Mn !As DMS system where the effective-field value changes through the temperature-dependent average Mn-spin polarization rather than through the Mn-doping parameter x. Large values of h effcorrespond to low temperatures in this picture and, as seen from Fig. 2, the temperature dependence of ain this regime is quite com- plex and sensitive to details of the DMS sample structure. Generally, Fig. 2 suggests an initial increase of awith in- creasing temperature in samples with a large density of Mn moments, a nearly constant afor intermediate doping, and a suppression of ain samples with low Mn content. At high temperatures ~smallheff), the kinetic-exchange-driven Gil- bert damping rate will gradually decrease towards zero withincreasing temperature. III. EXPERIMENTAL FMR LINEWIDTH We now discuss our experimental FMR linewidth data recorded for the 120 nm thin Ga 12xMnxAs layer with x58% grown on GaAs ~001!substrate. The FMR measure- ments were carried out at 9.46 GHz, with the external dc magnetic field applied at different angles uwith respect to the growth axis ( u50 corresponds to the @001#crystal direc- tion and u590° to the @110#direction !. The Mn concentra- tion in the sample was estimated from x-ray-diffraction mea-surement of the lattice constant. The critical temperature in the as-grown ( T c565 K) and annealed ( Tc5110 K) samples were determined from superconducting quantum in-terference device magnetization measurements. A more de-tailed description of the sample properties and of our experi-mental setup can be found elsewhere. 14To analyze the measured peak-to-peak FMR linewidth DHpp, plotted in Fig. 3, we recall the following general relation between DHppanda:22,23 DHpp~v!5DHpp~0!12 A3v gmBa. ~11! Here DHpp(0) describes broadening due to sample inhomo- geneity which is assumed to be frequency independent23,24 but can depend on the dc field orientation. The second termin Eq. ~11!arises from the Gilbert damping term in the LLG equation ~1!, which gives a contribution to the FMR line- width which is linearly proportional to vand independent of the static magnetic-field direction, if the magnetization is aligned with the field24~and the dependence of aon the magnetization orientation can be neglected, as mentioned inthe preceding section 21!. This condition is satisfied in our sample since the FMR resonance field is larger than the mag-netic field at which saturation magnetizations for differentfield orientations coincide. 14 The strong dependence of the FMR linewidth on the field orientation in the as-grown sample suggests that magneticinhomogeneities in the ferromagnetic layer contribute sig-nificantly to the FMR broadening. As seen in both the main panel and the inset of Fig. 3, the angle dependence of DH pp becomes weaker in the annealed sample and also the overall magnitude of DHppis conspicuously reduced. This observa- tion is consistent with the improved quality of the sample ~as FIG. 2. Gilbert damping coefficient aas a function of carrier density for x52–8 % and for quasiparticle lifetime broadening of 150 meV ~a!and 50 meV ~b!. FIG. 3. Experimental peak-to-peak FMR linewidth in as-grown ~filled symbols !and annealed ~open symbols !Ga0.92Mn0.08As samples measured as a function of temperature for @001#and@110# dc magnetic-field orientations ~main plot !and as a function of the field angle at 4 K ~inset!.JAIRO SINOVA et al. PHYSICAL REVIEW B 69, 085209 ~2004! 085209-4indicated, e.g., by the enhanced Tc) and implies that the leading contribution to the FMR linewidth might in this casecome from the homogeneous ~Gilbert damping !broadening. The right yaxis in the main plot of Fig. 3 represents the experimental Gilbert coefficient obtained from the measured DH ppand from Eq. ~11!, assuming DHpp(0)50. Experi- mental low-temperature values of ain the annealed sample are around 0.03. As seen from Fig. 2, these values of the Gilbert coefficient can be fully explained by the p-dkinetic- exchange mechanism of the damping of magnetization pre-cession. However, because the density of Mn ions and theirdistribution in the lattice as well as the density of holes arenot precisely known in our sample, a fully quantitative com-parison between theory and experiment is not possible. Theresults suggest that experimental studies of the frequency- dependent FMR linewidth in high-quality DMS samples would be very valuable for understanding the complex be-havior of the Gilbert damping coefficient predicted in thetheoretical part of this paper, and in separating those effectsthat are arising from the inhomogeneity within the epitaxi-ally grown thin films. IV. CURRENT-INDUCED MAGNETIZATION SWITCHING Using theoretical values for the Gilbert coefficient and anisotropy energy,10we now estimate the critical current for the spin-transfer induced magnetization switching in a ferro-magnetic semiconductor multilayer structure. In general,spin-polarized perpendicular-to-plane currents in magneticmultilayer systems can transfer spin between magnetic layersand exert current-dependent torques. For a trilayer structure,arguments based on the conservation of the angular momen-tum suggest the following form of the torques on the twomagnetic layers: 15 t1(2)}Is eMVMˆ1(2)3~Mˆ13Mˆ2!, ~12! whereMˆ1(2)5M1(2)/MandIsis the spin-polarized electric current. The sign of the torque depends on the sign of thecurrent, so that magnetization vectors in the two magneticlayers can be aligned parallel or antiparallel by current flow-ing in one or the opposite direction. In a spin-valve structurewith one hard and one soft magnetic layer, switching occurswhen the torque in the soft layer overcomes the damping andthe anisotropy terms. There have been a series of theoretical papers 17aimed at the quantitative description of spin currents and their effectson magnetization switching in metallic spin-valve structures.The theories are based on a two-channel model ~spin up and spin down !and account for spin accumulation effects in the magnetic multilayers and spin-transfer effects due to reflec-tion at the ferromagnet/normal layer interface and due to theaveraging mechanism associated with rapid precession ofelectron spins after entering the ferromagnet. The two-channel model is not applicable for semiconductor valencebands with strong spin-orbit coupling, complicating thequantitative description of spin currents in DMS’s. Strongspin-orbit coupling leads to a reduced spin-coherence time.However, the exchange coupling between the Mn moments and hole carriers will make this time larger in ferromagneticthan in nonmagnetic p-type semiconductors. Experimentally, magnetic information can be transported by charge carriersin DMS multilayers, despite strong spin-orbit coupling, asdemonstrated, e.g., by the observation of the giant magne-toresistance effects. 25 For an order-of-magnitude estimate of the switching cur- rent in a ~Ga,Mn !As-based magnetic trilayer structure, we approximate the spin current as Is’I^s&, whereIis the elec- tric current and ^s&is the mean-field spin polarization of the itinerant holes in the ~Ga,Mn !As layers.10Adding the torque term ~12!to Eq. ~1!for the soft magnetic layer we obtain an effective damping rate D5(U˜a2I˜)/(11a2), where I˜ 5I^s&/(eNMnSV). An instability occurs at D50 and the corresponding critical current density for the magnetization switching is then given by jc5eUad/\^s&. Assuming a thickness d;10 nm for the soft ferromagnetic layer and typical parameters of a Ga 0.95Mn0.05As DMS, U ;1k Jm23,a’0.02, and ^s&’0.3, the critical current jc ;105Acm22. This estimate is two orders of magnitude smaller than critical switching currents characteristic of me-tallic spin-valves, 16,17primarily due to smaller saturation mo- ments and anisotropy energies in the DMS’s. Since the resis-tivities are only 2–3 orders of magnitude larger in DMS’sthan in metals, observation of this effect should be experi-mentally feasible in a ferromagnet/normal/ferromagnet semi-conductor spin-valve structure. 26 We note that achieving low critical currents is particularly important for magnetic tunneljunctions that are used in non- volatile memory devices. To avoid self-field effects, whichlead to a spin vortex state rather than to a complete reversaland therefore to a smaller giant magnetoresistance effect, 20 the in-plane diameter rof metallic spin-valve devices with critical current densities j;107Acm22must be of the order of;100 nm.16Magnetic tunnel junctions have resistances too high for applications when patterned to such small sizes. Since the Oersted field scales as ;rjand the critical currents for spin-torque-induced switching we predict are two orders of magnitude smaller in DMS’s than in metals, 1–10 mm size semiconductor tunnel junctions might still show suffi-ciently weak self-field effects and, therefore, a completecurrent-induced switching. V. SUMMARY In this paper we have studied magnetization precession damping in ferromagnetic semiconductor ~Ga,Mn !As alloys. We have attempted to employ theoretical analysis combinedwith existing experimental information to obtain the Gilbertdamping coefficient and to predict the scale of critical cur-rents for spin-transfer magnetization-reversal in these sys-tems. In spin-transfer induced reversal, damping of magneti-zation precession competes with current-induced spintorques and determines the scale of the current required toachieve reversal. Our theoretical analysis examines the mechanism that we expect to dominate at low temperatures in high-qualitysamples, due to the coupling of the d-level local moments toMAGNETIZATION RELAXATION IN ~Ga,Mn !A s... P H Y SICAL REVIEW B 69, 085209 ~2004! 085209-5valence-band holes. We derive an explicit expression for the Gilbert coefficient conventionally used to characterize damp-ing in experimental studies, by comparing microscopiclinear-response theory with the linear-response limit of thephenomenological Landau-Lifshitz-Gilbert equations, andstudy how the values predicted by this model for the Gilbertdamping coefficient depend on the hole density and on thesize of the mean-field exchange interaction experienced byvalence-band holes in the ferromagnetic state. We find that the magnitude predicted for the Gilbert coefficient, ;0.03, is consistent with experiment, but that the observed depen-dence on the external field and magnetization orientation islarger than can be accounted for by this mechanism. Theexperimental FMR linewidth appears to have an inhomoge-neous broadening contribution that is not included in ourtheoretical modeling developed for homogeneous bulk sys-tems. The uncertainty that presently exists in the relativeimportance of these two broadening mechanisms could bereduced by frequency-dependent FMR studies. In our view, the portion of the FMR linewidth broadening that is due to inhomogeneity should not, to a first approxi-mation, be included in assessing the competition betweenspin torques and spin-precession damping.We have thereforeused our theoretical value for damping in a homogeneoussystem to estimate the critical currents required for achieving magnetization reversal and obtained j c;105Acm22. ACKNOWLEDGMENTS The authors acknowledge helpful discussions with T. Dietl, Z. Frait, and H. Ohno.This work was supported in partby the Welch Foundation, the DOE under Grant No. DE-FG03-02ER45958, the Grant Agency of the Czech Republicunder Grant No. 202/02/0912, the Research Corporation un-der Grant No. CC5543, the DARPA SpinS Program, and bythe NSF-NIRT under Grant No. DMR-0210519. 1C. Kittel and A.H. Mitchell, Phys. Rev. 101, 1611 ~1956!; A.H. Mitchell, ibid.105, 1439 ~1957!. 2T. Dietl, Handbook on Semiconductors ~Elsevier, Amsterdam, 1994!. 3H. Ohno, J. Magn. Magn. Mater. 200,1 1 0 ~1999!. 4T.C. Schulthess, Bull.Am. Phys. Soc. K30,1~2003!; P.H. Deder- ichs, K. Sato, H. Katayama-Yoshida, and J. Kudrnovsky ´,ibid. S24,5~2003!. 5J. Ko¨nig, J. Schliemann, T. Jungwirth, and A.H. MacDonald, in Electronic Structure and Magnetism of Complex Materials , ed- ited by D.J. Singh and D.A. Papaconstantopoulos ~Springer- Verlag, Berlin, 2003 !. 6T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Ferrand, Sci- ence287, 1019 ~2000!. 7T. Jungwirth, J. Ko ¨nig, J. Sinova, J. Kuc ˇera, and A.H. Mac- Donald, Phys. Rev. B 66, 012402 ~2002!; K.W. Edmonds, K.Y. Wang, R.P. Campion, A.C. Neumann, C.T. Foxon, B.L. Gal-lagher, and P.C. Main, Appl. Phys. Lett. 81, 3010 ~2002!. 8T. Jungwirth, Jairo Sinova, K.Y. Wang, K.W. Edmonds, R.P. Campion, B.L. Gallagher, C.T. Foxon, Q. Niu, and A.H. Mac-Donald, Appl. Phys. Lett. 83, 320 ~2003!. 9Jairo Sinova, T. Jungwirth, S.-R. Eric Yang, J. Kuc ˇera, and A.H. MacDonald, Phys. Rev. B 66, 041202 ~2002!. 10T. Dietl, H. Ohno, and F. Matsukura, Phys. Rev. B 63, 195205 ~2001!; M. Abolfath, T. Jungwirth, J. Brum, and A.H. Mac- Donald,ibid.63, 054418 ~2001!. 11J. Ko¨nig, T. Jungwirth, and A.H. MacDonald, Phys. Rev. B 64, 184423 ~2001!. 12S.T.B. Goennenwein, T. Graf, T. Wassner, M.S. Brandt, M. Stutz- mann, J.B. Philipp, R. Gross, M. Krieger, K. Zu ¨rn, P. Ziemann, A. Koeder, S. Frank, W. Schoch, andA. Waag,Appl. Phys. Lett.82, 730 ~2003!. 13T. Shono,T. Hasegawa,T. Fukumura, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 77, 1363 ~2000!; T. Dietl, J. Ko ¨nig, and A.H. MacDonald, Phys. Rev. B 64, 241201 ~2001!. 14X. Liu, Y. Sasaki, and J.F. Furdyna, Phys. Rev. B 67, 205204~2003!. 15J.C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 ~1996!;L . Berger, Phys. Rev. B 54, 9353 ~1996!. 16E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, and R.A. Bu- hrman, Science 285, 867 ~1999!; M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, V. Tsoi, and P. Wyder, Nature ~London ! 409,4 6~2000!; W. Weber, S. Riesen, and H.C. Siegmann, Sci- ence291, 1015 ~2001!; F.J. Albert, N.C. Emley, E.B. Myers, D.C. Ralph, and R.A. Buhrman, Phys. Rev. Lett. 89, 226802 ~2002!. 17Ya.B. Bazaliy, B.A. Jones, and Shou-Cheng Zhang, Phys. Rev. B 57, R3213 ~1998!; J.C. Slonczewski, J. Magn. Magn. Mater. 195, L261 ~1999!; J.C. Slonczewski, cond-mat/0208207 ~unpub- lished !; M.D. Stiles and A. Zangwill, J. Appl. Phys. 91, 6812 ~2002!; Phys. Rev. B 66, 014407 ~2002!; S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 ~2002!. 18B.H. Lee, T. Jungwirth, and A.H. MacDonald, Phys. Rev. B 61, 15606 ~2000!; H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature ~London !408, 944~2000!. 19Byounghak Lee, T. Jungwirth, and A.H. MacDonald, Phys. Rev. B65, 193311 ~2002!; D. Chiba, M. Yamanouchi, F. Matsukura, and H. Ohno, Science 301, 943 ~2003!. 20Yaowen Liu, Zongzhi Zhang, P.P. Freitas, and J.L. Martins,Appl. Phys. Lett. 82, 2871 ~2003!. 21Stronger angle dependence of ais expected only in samples with large lattice-matching strains induced, e.g., by choosing a differ-ent III–V semiconductor as a substrate. 22S.V. Vonsovskii, Ferromagnetic Resonance ~Pergamon, Oxford, 1966!. 23F. Schreiber, J. Pflaum, Z. Frait, Th. Mu ¨hge, and J. Pelzl, Solid State Commun. 93, 965 ~1995!. 24W. Platow, A.N. Anisimov, G.L. Dunifer, M. Farle, and K. Bab- erschke, Phys. Rev. B 58, 5611 ~1998!. 25M. Tanaka and Y. Higo, Phys. Rev. Lett. 87, 026602 ~2001!. 26H. Ohno ~private communication !.JAIRO SINOVA et al. PHYSICAL REVIEW B 69, 085209 ~2004! 085209-6

PhysRevB.85.214413.pdf

PHYSICAL REVIEW B 85, 214413 (2012) Dispersion relations and low relaxation of spin waves in thin magnetic films L. V . Lutsev* A.F . Ioffe Physical-Technical Institute of Russian Academy of Sciences, 194021 St. Petersburg, Russia (Received 15 November 2011; revised manuscript received 13 May 2012; published 13 June 2012) We study spin excitations in thin magnetic films in the Heisenberg model with magnetic dipole and exchange interactions by the spin operator diagram technique and make comparison of their parameters with characteristicsof spin waves in thick films. Dispersion relations of spin waves in thin magnetic films (in two-dimensionalmagnetic monolayer and bilayer lattices) and the spin-wave resonance spectrum in N-layer structures are found. For thick magnetic films, spin excitations are determined by simultaneous solution of the generalized Landau-Lifshitz equations and the equation for the magnetostatic potential. Generalized Landau-Lifshitz equations arederived from first principles and have the integral (pseudodifferential) form. It is found that dispersion relationsof spin waves in monolayers and in bilayers differ from dispersion relations of spin waves in continuous thickmagnetic films. For normal magnetized ferromagnetic films, the spin-wave damping is calculated in the one-loopapproximation for a diagram expansion of the Green functions at low temperature. In thick magnetic films, themagnetic dipole interaction makes a major contribution to the relaxation of long-wavelength spin waves. Thinfilms have a region of the low relaxation of long-wavelength spin waves. In thin magnetic films, four-spin-waveprocesses take place and the exchange interaction makes a major contribution to the damping. It is found that thedamping of spin waves propagating in a magnetic monolayer is proportional to the quadratic dependence on thetemperature and is very low for spin waves with small wave vectors. DOI: 10.1103/PhysRevB.85.214413 PACS number(s): 75 .10.Jm, 75 .30.Ds I. INTRODUCTION Nanosized magnetic films are of great interest due to their perspective applications in spin-wave devices. At present,the most important spin-wave devices—microwave filters,delay lines, signal-to-noise enhancers, and optical signalprocessors—have been realized on the base of magneticfilms of microsized thickness. 1–3Nanosized films give us an opportunity to construct spin-wave devices of smallsizes and to design devices with new functional properties.Recently, new applications of spin waves have been proposedsuch as spin-wave computing, 4,5spin-wave filtering using width-modulated nanostrip waveguides,6and transmission of electrical signals by spin-wave interconversion in an insulatorgarnet Y 3Fe5O12(YIG) film based on the spin-Hall effect.7 Spin-wave logic elements have been done on the basis of aMach-Zehnder-type interferometer 6,8,9and can be realized on magnonic crystals.5Using nanosized magnetic films, we have a probability to construct an array of logic elements of smallsizes. Ferromagnetic monolayers, bilayers, and trilayers areof great interest for magnetic sensors and spin-wave devices.Spin excitations in these thin magnetic film structures aretheoretically investigated and are studied by the Brillouinlight-scattering method. 10–14 In order to design new spin-wave devices based on nano- sized magnetic films, it is necessary to determine dispersionrelations and damping of spin excitations in thin films. In thephenomenological model with the magnetic dipole interaction(MDI) and the exchange interaction, 15–18the magnetization dynamics in thick magnetic films is described by the Landau-Lifshitz equations, which are differential with respect to spatial variables. The differential form of equations is postulated. In this connection, the following question arises: is this form ofLandau-Lifshitz equations correct for thin nanosized films?Determination of the dispersion relations depends on theanswer of this question. In phenomenological models, the spin-wave damping is described by relaxation terms in Gilbert,Landau-Lifshitz, or Bloch forms. 18Properties of intrinsic relaxation processes are not taken into account in these terms and, therefore, the calculated spin-wave damping may beincorrect. The above-mentioned leads us to the main questionof the paper: what are the dispersion relations and dampingof spin waves in thin films and can they be derived fromfirst principles? In order to answer this question, we considergeneralized Landau-Lifshitz equations, spin excitations, and relaxation of spin waves in thin films in the framework of the Heisenberg model with the MDI and the exchange interaction.In the paper, we suppose that films are thin in two cases.(1) For the case, when we calculate dispersion relations ofspin waves, we say that an N-layer structure is thin, if Nis a low number (for example, monolayer, bilayer, trilayer). (2) Forthe case of relaxation processes a film is thin, if the spin-wave energy is less than energy gaps between spin-wave modes and, therefore, three-spin-wave processes are forbidden. The above-mentioned problems have not yet been investi- gated comprehensively. One of the cause of these problemsis the long-range action of the MDI. The spin-wave relax- ation and the spin-wave dynamics become dependent on the dimensions and shapes of ferromagnetic samples. In order toanalyze the Heisenberg model with the MDI and the exchangeinteraction, we use the spin operator diagram technique. 19–23 Advantages of the spin operator diagram technique give us the opportunity to calculate the spin-wave damping at high temperatures and obtain more exact relationships describingspin-wave scattering and excitations in comparison with meth-ods based on diagram techniques for creation and annihilationmagnon Bose operators. 24–32In Refs. 23and 33,t h es p i n operator diagram technique is generalized for models with arbitrary internal Lie-group dynamics. 214413-1 1098-0121/2012/85(21)/214413(17) ©2012 American Physical SocietyL. V . LUTSEV PHYSICAL REVIEW B 85, 214413 (2012) In Sec. II, we consider spin operator diagram technique for the Heisenberg model with the MDI and the exchangeinteraction. Spin-wave excitations are determined by polesof the Pmatrix: the matrix of the effective Green functions and interaction lines. On the basis of this diagram technique,dispersion relations of spin waves in a magnetic monolayerand in a bilayer and the spectrum of spin-wave resonancesin anN-layer structure are found (see Sec. III). It is found that dispersion relations of spin waves in monolayer andbilayer lattices differ from dispersion relations of spin wavesin continuous thick magnetic films. This difference is dueto the discreetness of the lattice. For the case when theMDI is equal or greater than the exchange interaction, forexample, for monolayer consisted of magnetic nanoparticleson the lattice, this difference becomes essential and is takeninto account. For thick magnetic films, it is more convenientto present the P-matrix-pole equation describing spin-wave excitations in the form of the Landau-Lifshitz equations andthe equation for the magnetostatic potential (see Sec. IV). Spin excitations are determined by simultaneous solutionof these equations. Landau-Lifshitz equations are integral(pseudodifferential) equations, but not differential ones withrespect to spatial variables. In the common case, the reductionof Landau-Lifshitz equations to differential equations withexchange boundary conditions is incorrect and their solutionsgive dispersion relations different from dispersion relationscalculated on the basis of integral (pseudodifferential) Landau-Lifshitz equations. The contradiction is removed, if the pinningparameter is equal to the spin-wave wave vector. In Sec. V, we consider spin-wave relaxation in thick and thin magneticfilms. In thick films, three-spin-wave processes take placeand the MDI makes a major contribution to the relaxationof long-wavelength spin waves. Thin films have a region oflow relaxation of long-wavelength spin waves. In this case,three-spin-wave processes are forbidden and the exchangeinteraction makes a major contribution to the relaxationprocess. II. HEISENBERG MODEL WITH MAGNETIC DIPOLE AND EXCHANGE INTERACTIONS A. Spin operator diagram technique Let us consider the Heisenberg model with the exchange interaction and the MDI on a lattice.22,23The exchange interaction is short ranged and the MDI is long ranged.Operators S ±=Sx±iSyandSzsatisfy the commutation relations: [Sz(/vector1),S+(/vector1/prime)]=S+(/vector1)δ/vector1/vector1/prime, [Sz(/vector1),S−(/vector1/prime)]=−S−(/vector1)δ/vector1/vector1/prime, [S+(/vector1),S−(/vector1/prime)]=2Sz(/vector1)δ/vector1/vector1/prime, where /vector1≡/vectorr1,/vector1/prime≡/vectorr1/primeis the abridged notation of lattice sites. The Hamiltonian of the Heisenberg model is H=−gμB/summationdisplay /vector1H(/vector1)Sz(/vector1)−gμB/summationdisplay /vector1hμ(/vector1)Sμ(/vector1) −1 2/summationdisplay /vector1,/vector1/primeJμν(/vector1−/vector1/prime)Sμ(/vector1)Sν(/vector1/prime), (1)where H(/vectorH/bardblOz) is the external magnetic field, hμis the auxiliary infinitesimal magnetic field, and μ=−,+,z.I ti s supposed that the summation in Eq. (1)and in all following relations is performed over all repeating indices μ,ν.T h es u m - mation is carried out over the lattice sites /vector1,/vector1/primein the volume V of the ferromagnetic sample. gandμBare the Land ´e factor and the Bohr magneton, respectively. Jμν(/vector1−/vector1/prime)=Jνμ(/vector1/prime−/vector1) is the interaction between spins, which is the sum of theexchange interaction I μνand the MDI: Jμν(/vector1−/vector1/prime)=Iμν(/vector1−/vector1/prime) −4π(gμB)2∇μ/Phi1(/vectorr−/vectorr/prime)∇/prime ν|/vectorr=/vector1,/vectorr/prime=/vector1/prime, (2) where the function /Phi1(/vectorr−/vectorr/prime) in the MDI term is determined by the equation /Delta1/Phi1(/vectorr−/vectorr/prime)=δ(/vectorr−/vectorr/prime), ∇μ={ ∇ −,∇+,∇z} =/braceleftbigg1 2/parenleftbigg∂ ∂x+i∂ ∂y/parenrightbigg ,1 2/parenleftbigg∂ ∂x−i∂ ∂y/parenrightbigg ,∂ ∂z/bracerightbigg . (3) In three-dimensional space, /Phi1(/vectorr−/vectorr/prime)=−1/4π|/vectorr−/vectorr/prime| and the MDI term in Hamiltonian (1)can be written as H(dip)=(gμB)2 2/summationdisplay /vector1,/vector1/prime/bracketleftbigg(/vectorS(/vector1),/vectorS(/vector1/prime)) |/vector1−/vector1/prime|3 −3(/vectorS(/vector1),/vector1−/vector1/prime)(/vectorS(/vector1/prime),/vector1−/vector1/prime) |/vector1−/vector1/prime|5/bracketrightbigg . For the following calculations of spin-wave dispersion rela- tions in magnetic films, we use a more convenient form of theMDI determined by relations (2)and(3). Spin excitations, interaction of spin waves, spin-wave relax- ation, and other parameters of excitations in the canonical spinensemble are determined by the generating functional 19,23,33,34 Z[h]=Sp exp[ −βH(h)] =∞/summationdisplay n=0/summationdisplay /vector1,...,/vectorn μ1,..., μ n/integraldisplayβ 0···/integraldisplayβ 0Qμ1...μn(/vector1,...,/vectorn,τ1,..., τ n) ×hμ1(/vector1,τ1)...h μn(/vectorn,τn)dτ1...dτ n, (4) where β=1/kT,kis the Boltzmann constant, and Tis the temperature, h={hμi}. Coefficients Qμ1...μnare proportional to the temperature Green function without vacuum loops: Gμ1...μn(/vector1,...,/vectorn,τ1,..., τ n) ≡/angbracketleft /angbracketleftTˆSμ1(/vector1,τ1)...ˆSμn(/vectorn,τn)/angbracketright/angbracketright =(βgμ B)−nZ−1 δnZ[h] δhμ1(/vector1,τ1)...δh μn(/vectorn,τn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle h→0, (5) where ˆSα(/vectorj,τ)=exp(τH)Sα(/vectorj)e x p (−τH) are the spin oper- ators in the Euclidean Heisenberg representation, τ∈[0,β]. Tis the τ-time ordering operator. Variable τis added in the auxiliary field hμin order to take into account Tordering. /angbracketleft/angbracketleft.../angbracketright/angbracketrightdenotes averaging of spin operators calculated with exp(−βH)/Sp exp( −βH). The symbol Sp denotes the trace. 214413-2DISPERSION RELATIONS AND LOW RELAXATION OF ... PHYSICAL REVIEW B 85, 214413 (2012) The frequency representation of the expansion (4)is more convenient for calculations. The Fourier transforms of Qμ1...μn are defined in terms of the Matsubara frequencies ω(1) m1= 2πm 1/¯hβ,..., ω(n) mn=2πmn/¯hβ35(m1,..., m nare integers): Qμ1...μn/parenleftbig/vector1,...,/vectorn,ω(1) m1,..., ω(n) mn/parenrightbig =/integraldisplayβ 0···/integraldisplayβ 0Qμ1...μn(/vector1,...,/vectorn,τ1,..., τ n) ×exp/bracketleftbig −i¯h/parenleftbig ω(1) m1τ1+···+ ω(n) mnτn/parenrightbig/bracketrightbig dτ1...dτ n.(6) The coefficients Qμ1...μncan be expanded with respect to the interaction Jμν(/vector1−/vector1/prime) [see Eq. (2)].19–23,33Each term of this expansion is represented by a diagram constructed ofpropagators, vertices, blocks and interaction lines. 1. Propagators. Spin propagators D ±(/vector1,/vector1/prime,ωm)=δ/vector1/vector1/prime p0±iβ¯hωm, (7) where p0=βgμ BH, are determined for the spin ensem- ble without any interaction between spins. The propagators D±(/vector1,/vector1/prime,ωm) are represented by directed lines in diagrams [see Fig. 1(a)]. The directions of arrows show the direction of growth of the frequency variable ωm. 2. Vertices. There are five types of vertices [see Fig. 1(b)]. Vertices aandbare the start and end points of propagators, respectively. In analytical expressions of diagrams the vertexacorresponds with the factor 2 and the vertex bwith the factor 1. The vertex cties three propagators and corresponds with the factor ( −1) in analytical expressions. The vertex dwith (a) (b) a b c d e (d)D (1,1 ,+m′ 1 1 D (1,1 ,-′ m 1 1′m m V (1-1 ,) = Jm (1-1 )=(0) 1 1m(c)′ ′ ′ ′ FIG. 1. (a) Propagators D±, (b) vertices, (c) block with isolated parts and (d) interaction lines V(0) μν.the factor 1 is defined as a single vertex. The vertex eties two propagators. The factor of the evertex is equal to ( −1). 3. Blocks. Blocks contain propagators and isolated vertices d[see Fig. 1(c)]. Propagators can be connected through vertices cande. In analytical expressions of the diagram expansion, each block corresponds with the block factorB [κ−1](p0), where κis the number of isolated parts in the block. The factor B[κ−1](p0) is expressed by partial derivatives of the Brillouin function BSfor the spin Swith respect top0: B(p0)=/angbracketleft /angbracketleftSz/angbracketright/angbracketright0=SBS(Sp0),B[n](p0)=S∂nBS(Sp0) ∂pn 0, (8) where /angbracketleft/angbracketleft.../angbracketright/angbracketright0denotes the statistical averaging performed over the states described by the Hamiltonian H(1)without the interaction Jμνbetween spins. BS(x)=(1+1/2S) coth[(1 + 1/2S)x]−(1/2S) coth( x/2S). 4. Interaction lines. The interaction line V(0) μν(/vector1−/vector1/prime,ωm)= βJμν(/vector1−/vector1/prime) connects two vertices in a diagram [see Fig. 1(d)]. The correspondence between the first index μof the interaction lineV(0) μνand the vertex type is the following. (1) If μ=−, then the left end point of V(0) −νis bound to the vertex a;( 2 ) ifμ=+, then this end point is bound to the vertices borc; and (3) if μ=z, then the end is bound to the vertices dore. The analogous correspondence is satisfied for the right end ν ofV(0) μν. Coefficients Qμ1...μnin the expansion (4)in the frequency representation (6)are the sum of Ntopologically nontrivial diagrams/summationtextN tQμ1...μn t (t=1,..., N ). The general form of the analytical expression of the diagram in the frequencyrepresentation is written as 19–23 Qμ1...μn t/parenleftbig/vector1,...,/vectorn,ω(1) m1,..., ω(n) mn/parenrightbig =(−1)L2maPk 2kk!/productdisplay lB[κl−1](p0)κl/productdisplay /vectori,/vectorj∈lδ/vectori/vectorj ×/summationdisplay /vector1/prime,.../vectork/prime /vector1/prime/prime.../vectork/prime/prime/summationdisplay miV(0) αγ/parenleftbig/vector1/prime−/vector1/prime/prime,ωm1/parenrightbig ×···× V(0) ρσ/parenleftbig/vectork/prime−/vectork/prime/prime,ωmk/parenrightbig ×ID/productdisplay /vectors,/vectors/primeD−/parenleftbig /vectors,/vectors/prime,ωms/parenrightbigIv/productdisplay vδ/parenleftBigg/summationdisplay r∈vβ¯hωmr/parenrightBigg , (9) where /vector1,...,/vectorn,ω(1) m1,..., ω(n) mnare the external lattice and frequency variables corresponded to the auxiliary fields hμi in the expansion (4).mais the number of avertices in a diagram. Lis the number of candevertices. Pkis the number of topological equivalent diagrams. 2 kis the number of vertices connected with kinteraction lines V(0) αγ...V(0) ρσ.T h e product/producttext lis performed over all blocks of a diagram. κlis the number of isolated parts in block l.T h et e r m/producttextκl /vectori,/vectorj∈lδ/vectori/vectorj denotes that all isolated parts in block lare determined on a single lattice site. IDis the number of propagators in a diagram. Ivis the number of vertices in a diagram./summationtext midenotes the summation performed over all inner frequency variables. The term/producttextIv vδ(/summationtext r∈vβ¯hωmr) gives the frequency 214413-3L. V . LUTSEV PHYSICAL REVIEW B 85, 214413 (2012) conservation in each vertex v, i.e., the sum of frequencies of propagators and interaction lines, which come in and go outfrom the vertex v, is equal to 0. The vertex dcan be connected with the single interaction line. In the analytical expression,this corresponds to the factor δ(β¯hω m). The lattice variables /vectorsand/vectors/primeof propagators D−can be inner or external. In the first case, end points of propagators are connected with the end points {/vector1/prime,/vector1/prime/prime,...,/vectork/prime,/vectork/prime/prime}of interaction lines V(0) αγ...V(0) ρσand the summation/summationtext /vector1/prime,.../vectork/prime /vector1/prime/prime.../vectork/prime/prime/summationtext miis performed. In the second case, end points of propagators are not connected with interaction lines. The first approximation of the diagram expansion (4)is the self-consistent field approximation in which the effective fieldacting on spins is derived and the self-consistent field H (c) μ induced by the neighboring spins is taken into account.19,22,23 This leads to the substitution p0→p=βgμ B|/vectorH+/vectorH(c)|in the propagator D−(7)and in the block factor B[κl−1](8)in the analytical expression (9). The self-consistent field is the sum of exchange and magnetic dipole self-consistent fields,H (c) μ=H(exch) μ+H(m) μ, where H(exch) μ (/vector1)=(gμB)−1/summationdisplay /vector1/primeIμν(/vector1−/vector1/prime)/angbracketleft/angbracketleftSν(/vector1/prime)/angbracketright/angbracketright H(m) μ(/vector1)=−4πgμ B∇μ/summationdisplay /vector1/prime/Phi1(/vectorr−/vectorr/prime)∇/prime ν/angbracketleft/angbracketleftSν(/vectorr/prime)/angbracketright/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectorr=/vector1 /vectorr/prime=/vector1/prime. (10) The second approximation of the expansion (4)is the approx- imation of the effective Green functions and interactions. Inthis approximation, the poles of the matrix of the effectiveGreen functions and interactions are determined and thedispersion curves are obtained. The next terms in the diagramexpansion determine the imaginary and real corrections tothe poles of the matrix of the effective Green functionsand interactions. The imaginary parts of the poles give therelaxation parameters of spin excitations and the real partsdetermine the corrections to the dispersion curves. In the nextsection, we consider the approximation of the effective Greenfunctions and interactions. B. Effective Green functions and interaction lines In the framework of this approximation, the matrix of the effective Green functions and effective interactions P= /bardblPAB(/vector1,/vector1/prime,ωm)/bardblis introduced.22,23We compose the Pmatrix from analytical expressions of connected diagrams with twoexternal sites. These sites are end points of propagators, singlevertices d, or end points of interaction lines. Accordingly, multi-indices A=(aμ),B=(bν) are the double indices, where μ,ν={ −,+,z}and indices a,bpoint out that A,B belong to a propagator or to a dvertex ( a,b=1), or belong to an interaction line ( a,b=2). The zero-order approximation P (0)of the Pmatrix is determined by the matrix of the bare interaction V(0)=/bardblV(0) μν(/vector1−/vector1/prime,ωm)/bardbland by the two-site Green functions (5)in the self-consistent-field approximationG(0)=/bardblG(0) μν/bardbl, given on a lattice site: P(0)=⎛ ⎜⎜⎜⎝/vextenddouble/vextenddoubleP(0) (1μ)(1ν)/vextenddouble/vextenddouble.../vextenddouble/vextenddoubleP(0) (1μ)(2ν)/vextenddouble/vextenddouble ··· ··· ··· /vextenddouble/vextenddoubleP(0) (2μ)(1ν)/vextenddouble/vextenddouble.../vextenddouble/vextenddoubleP(0) (2μ)(2ν)/vextenddouble/vextenddouble⎞ ⎟⎟⎟⎠ =⎛ ⎜⎜⎜⎝/vextenddouble/vextenddoubleG(0) μν/vextenddouble/vextenddouble...0 ··· ··· ··· 0.../vextenddouble/vextenddoubleV(0) μν/vextenddouble/vextenddouble⎞ ⎟⎟⎟⎠, where /vextenddouble/vextenddoubleG(0) μν/vextenddouble/vextenddouble=⎛ ⎜⎝0G(0) −+ 0 G(0) +− 00 00 G(0) zz⎞ ⎟⎠ =2B(p)⎛ ⎜⎜⎝0 D−(/vector1,/vector1/prime,ωm)0 D+(/vector1,/vector1/prime,ωm)0 0 00B[1](p) 2B(p)δ/vector1/vector1/primeδm0⎞ ⎟⎟⎠ (11) with the propagator (7)in which the substitution p0→p= βgμ B|/vectorH+/vectorH(c)|is performed. ThePmatrix is obtained by means of the summation of theP(0)matrix: the summation of all diagram chains consisted of the bare Green functions G(0) μνand the bare interaction lines V(0) μν(Fig. 2). These chains of propagators and interaction lines (a) G(0)==G+-(0) zz- +- + =G-+(0) =Gzz(0) P= G =(1 )(1 ) = + (b) P= V =(2 )(2 ) = + (c) P=(1 )(2 ) = P=(2 )(1 ) = FIG. 2. (a) Definition of the effective Green functions P(1μ)(1ν)= Gμνvia the bare two-site Green functions G(0) μν. (b) Definition of ef- fective interaction lines P(2μ)(2ν)=Vμν. (c) Definition of intersecting termsP(1μ)(2ν),P(2μ)(1ν). 214413-4DISPERSION RELATIONS AND LOW RELAXATION OF ... PHYSICAL REVIEW B 85, 214413 (2012) do not have any loop insertion. Analytical expressions of the considered diagrams can be written in accordance with relation(9). The summation gives equation of the Dyson type, which forms the relationship between P (0)- andP-matrices, P=P(0)+PσP(0), (12) where σ=⎛ ⎜⎜⎜⎝0...E ··· ··· ··· E...0⎞ ⎟⎟⎟⎠, E=/bardblδμν/bardblis the diagonal matrix. TheP-matrix consists of the two-site effective Green functions G=/bardblGμν/bardbl=G(0)(E−V(0)G(0))−1, where Gμν= P(1μ)(1ν), effective interactions V=/bardblVμν/bardbl= V(0)(E− G(0)V(0))−1, where Vμν=P(2μ)(2ν), and intersecting terms P(1μ)(2ν),P(2μ)(1ν)(see Fig. 2). The effective Green functions, effective interactions, and intersecting terms are denoted indiagrams by directed thick lines, empty lines, and composi-tions of the thick and empty lines, respectively. The Pmatrix determines the spectrum of quasiparticle excitations in the spinensemble. Spectrum relations for spin excitations are givenby the Pmatrix poles by zero eigenvalues of the operator 1−σP (0)or, equivalently, by E−V(0)G(0)under the analytical continuation iωm→ω+iεsignω, (13) δ(β¯hωm)=δm0→1 β¯h(ω+iεsignω)(ε→+ 0). Since zero eigenvalues of the operator E−V(0)G(0)may correspond to different eigenfunctions and can determine dif-ferent excitation modes, we introduce the spectral parameter λ for the eigenfunctions h (λ) μ(/vector1,ωm) of the operator E−V(0)G(0). The spectral parameter λcan be discrete or continuous. Taking into account the above mentioned, we get the equationdescribing spin-wave excitations: h (λ) μ(/vector1,ωm)−/summationdisplay /vector1/prime,/vector1/prime/prime,ν,ρV(0) μν(/vector1−/vector1/prime,ωm) ×G(0) νρ(/vector1/prime,/vector1/prime/prime,ωm)h(λ) ρ(/vector1/prime/prime,ωm)/vextendsingle/vextendsingle/vextendsingle/vextendsingle iωm→ω+iεsignω=0.(14) III. SPIN WA VES IN MAGNETIC FILMS A. Spin-wave equations for magnetic films Let us consider spin waves with the wave vector /vectorqin normal and in-plane magnetized films consisted of Nmonolayers at low temperature. Monolayers can be regarded as layersconsisting of ions with strong exchange interaction or layersconsisting of magnetic nanoparticles. In the second case, theexchange interaction between nanoparticles can reach lowvalues in comparison with the MDI. The external magneticfieldHis parallel to the zaxis. At low temperature, derivatives of the Brillouin function in B [n](p) in relation (8)tend to zero exponentially with decreasing temperature. Thus it followsthat diagrams containing blocks with isolated parts can bedropped, the Green function G (0) zzin relation (11) is negligible and only the Green functions G(0) −+,G(0) +−are taken into account in Eq. (14). Indices μ,νof interactions V(0) μνin Eq. (14) are {−,+}. We suppose that on monolayers, spins are placed on quadratic lattice sites with the lattice constant aand the spin orientation is parallel to the zaxis. The exchange interaction acts between neighboring spins and is isotropic between spins in monolayers, 2 I(mon) −+=2I(mon) +−=I(mon) zz=I0, and between neighboring layers, 2 I(lay) −+=2I(lay) +−=I(lay) zz=Id. As we consider spin waves in two-dimensional layers and films, it is necessary to discuss restrictions imposed bythe Mermin-Wagner theorem. 36The Mermin-Wagner theorem states that continuous symmetries cannot be spontaneouslybroken at finite temperature in systems with sufficientlyshort-range interactions in dimensions /lessorequalslant2. In accordance with the theorem, the isotropic spin Heisenberg model canbe neither ferromagnetic nor antiferromagnetic. The theoremextends to N-layer films: for any finite temperature and for any finite number of layers, a phase transition is ruled out. 37,38 In the case of the Heisenberg model with the Hamiltonian H(1), the Mermin-Wagner theorem is not applied: the O(3) rotational symmetry of the Hamiltonian His broken by the MDI and by the external magnetic field H. Therefore the two- dimensional layers and films considered below have nonzerofinite magnetization. We suppose that the magnitude of themagnetic field is sufficient to achieve magnetic saturation andto eliminate a domain structure. 1. Normal magnetized films In normal magnetized films, xandyaxes are in the monolayer plane and the zaxis is normal to monolayers. The magnetic field His normal to monolayers. The Fourier transform of the exchange interaction with respect to the longitudinal lattice variables /vector1xyis ¯I(/vectorq,1z−1/prime z)=/summationdisplay /vector1xy−/vector1/primexyI(/vector1xy−/vector1/prime xy,1z−1/prime z) ×exp[−i/vectorq(/vector1xy−/vector1/prime xy)] =¯I(0,1z−1/prime z)+2I0[cos(qxa) +cos(qya)]δ1z1/primez, (15) where /vector1xyand/vector1/prime xyare lattice sites in monolayers, 1 zand 1/primezarez positions of layers, /vectorq=(qx,qy) is the longitudinal wave vector in monolayers, and ¯I(0,1z−1/prime z) is the exchange interaction at/vectorq=0, which is equal to Idbetween spins of neighboring layers. The corresponding exchange part of the interaction line V(0) μν=V(exch) μν+V(dip) μν[see Fig. 1(d)]i s V(exch) μν (/vectorq,1z−1/prime z)=β¯I(/vectorq,1z−1/prime z)/2, (16) where μν=(−+),(+−). For indices μν=(−−) and ( ++), V(exch) μν=0. The MDI part V(dip) μνis determined by the Fourier 214413-5L. V . LUTSEV PHYSICAL REVIEW B 85, 214413 (2012) transform of Eq. (3): /parenleftbigg −q2+∂2 ∂z2/parenrightbigg /Phi1(/vectorq,z−z/prime)=S−1 aδ(z−z/prime) with the solution /Phi1(/vectorq,1z−1/prime z)=/Phi1(/vectorq,z−z/prime)|z=1z,z/prime=1/primez =−1 2qSaexp(−q|1z−1/prime z|), (17) where Sa=a2,q=| /vectorq|. According to the solution (17),t h e corresponding MDI part of the interaction line is V(dip) μν(/vectorq,1z−1/prime z)=−2πβ(gμB)2qμqν qSaexp(−q|1z−1/prime z|), (18) where μ,ν={ −,+}q−=1 2(qx+iqy),q +=1 2(qx−iqy). Taking into account relations (16) and(18),f r o mE q . (14),w e obtain equations for spin-wave modes with the wave vector /vectorqinN-layer magnetic films: h(λ) μ(/vectorq,1z,ωm)−/summationdisplay /vector1/primez/bracketleftbig V(0) μ−(/vectorq,1z−1/prime z,ωm) ×G(0) −+(1/prime z,1/prime z,ωm)h(λ) +(/vectorq,1/prime z,ωm)+V(0) μ+(/vectorq,1z−1/prime z,ωm) ×G(0) +−(1/prime z,1/prime z,ωm)h(λ) −(/vectorq,1/prime z,ωm)/bracketrightbig/vextendsingle/vextendsingle iωm→ω+iεsignω=0,(19) where G(0) −+ (+−)(1z,1/prime z,ωm)=2B(p)δ1z1/primez p±iβ¯hωm, λ=1,..., N is the mode number, V(0) μν(/vectorq,1z−1/prime z,ωm)= V(exch) μν (/vectorq,1z−1/prime z)+V(dip) μν(/vectorq,1z−1/prime z),μ,ν={ −,+}. Eigen- values of equations (19)give dispersion relations of spin waves in normal magnetized films. 2. In-plane magnetized films In in-plane magnetized films, xandzaxes are in the monolayer plane and the yaxis is normal to monolayers. The Fourier transform of the exchange interaction with respect to the longitudinal lattice variables /vector1xzis given by relation (15), where substitutions /vector1xy→/vector1xz,/vector1z→/vector1y, andqy→qzshould be done. The MDI part of the interaction is V(dip) −− (++)(/vectorq,1y−1/prime y)=πβ(gμB)2/parenleftbigg q2 x±2qx∂ ∂y+∂2 ∂y2/parenrightbigg /Phi1(/vectorq,y−y/prime)/vextendsingle/vextendsingle/vextendsingle/vextendsingle y=1y,y/prime=1/primey(20) V(dip) +−(/vectorq,1y−1/prime y)=V(dip) −+(/vectorq,1y−1/prime y)=πβ(gμB)2/parenleftbigg q2 x−∂2 ∂y2/parenrightbigg /Phi1(/vectorq,y−y/prime)/vextendsingle/vextendsingle/vextendsingle/vextendsingle y=1y,y/prime=1/primey, where /Phi1(/vectorq,y−y/prime)=−1 2qSaexp(−q|y−y/prime|), q=(q2 x+q2 z)1/2is the longitudinal wave vector. Taking into account relation (15) with the above mentioned substitutions and relation (20), from Eq. (14), we obtain equations for spin-wave modes in in-plane magnetized films analogous to Eq.(19), where the substitution /vector1z→/vector1yshould be done. In next sections, we find spin-wave dispersion relations forthe cases of monolayer and two-layer films and spin-waveresonance relations for the case of N-layer structures. B. Spin waves in magnetic monolayer 1. Normal magnetized monolayer films Dispersion relations of spin waves in normal magnetized monolayer lattice are determined by the determinant of Eq. (19) for variables h(1) −andh(1) +. Taking into account relations (16) and(18), we find ω2(/vectorq)=/Omega1(/vectorq)[/Omega1(/vectorq)+2πγσ mq], (21)where /Omega1(/vectorq)=γ(H+H(m))+2B(p)I0 ¯h[2−cos(qxa)−cos(qya)], γ=gμB/¯his the gyromagnetic ratio, H(m)=|/vectorH(m)|is the depolarizing magnetic field (10),σm=gμBB(p)/Sais the surface magnetic moment density, and q=(q2 x+q2 y)1/2.A s one can see from relation (21), in the monolayer lattice, spin waves have the one-mode character. In the next sections, we compare dispersion relations (21) with dispersion relations in thick magnetic films. Thereforewe calculate the dispersion curve for a monolayer filmwith parameters analogous to YIG films. YIG films have the magnetization 4 πM=4πgμ BB(p)/a3=1750 Oe and the exchange interaction constant α=B(p)I0a2/¯hγ4πM= 3.2×10−12cm2at room temperature.18Magnetic parameters of monolayer with /angbracketleft/angbracketleftSz/angbracketright/angbracketright0=B(p)=1/2 are analogous to YIG, if the lattice constant a=0.4 nm and the exchange interaction between neighboring spins I0=0.085 eV . Figure 3 presents the dispersion curve (21) of spin waves propagat- ing in the monolayer film. The spin-wave wave vector /vectorq is parallel to the xaxis ( qx=q,qy=0) and is in the range [0 ,π/a ]. Calculations have been done at the sum of 214413-6DISPERSION RELATIONS AND LOW RELAXATION OF ... PHYSICAL REVIEW B 85, 214413 (2012) H 0.0 1.0 2.0 3.0 aq0500100015002000Frequency ω/2π (GHz) FIG. 3. (Color online) Dispersion curve of spin waves propagat- ing in the normal magnetized monolayer film with quadratic lattice(a=0.4 nm) at the sum of magnetic fields H+H (m)=3 kOe. Exchange interaction I0is 0.085 eV . magnetic fields H+H(m)=3 kOe. For the given monolayer film, the exchange interaction makes a major contributionto the dispersion. The relatively weak MDI is significantfor the dispersion at small values of the wave vector q<q 0, where q0=¯hπγσ m B(p)I0a2=a 4α. Atq→0, the group velocity of spin waves is positive v=πγσ m. These spin waves are analogous to forward volume magnetostatic spin waves propagating in magneticfilms. 1–3,18 2. In-plane magnetized monolayer films Dispersion relations of spin waves in in-plane magnetized monolayers are determined by the determinant of Eq. (19) for the variables h(1) −andh(1) +with the substitution /vector1z→/vector1y. Taking into account relations (20), we obtain ω2(/vectorq)=[/Omega1(/vectorq)+/Omega1M−2πγσ mq]·/bracketleftbigg /Omega1(/vectorq)+2πγσ mq2 x q/bracketrightbigg , (22) where /Omega1(/vectorq)=γH+2B(p)I0 ¯h[2−cos(qxa)−cos(qza)], q=(q2 x+q2 z)1/2, and/Omega1M=4πγσ m/a. Spin waves propagat- ing along the xaxis (/vectorq⊥/vectorH,q=qx,qz=0) atq→0h a v e the positive group velocity v=πγσ m/Omega1M [/Omega1(0)(/Omega1(0)+/Omega1M)]1/2and, in this sense, are analogous to surface magnetostatic spin waves propagating in magneticfilms. 1–3,18In contrast with this, spin waves propagating along the zaxis ( /vectorq/bardbl/vectorH,q=qz,qx=0) atq→0 have the negative group velocity v=−πγσ m/Omega1(0)1/2 (/Omega1(0)+/Omega1M)1/2 and have features of backward volume magnetostatic spin waves. These backward spin waves propagate in the sector[−θ,θ], where sin θ=/Omega1(0)/[/Omega1(0)+/Omega1 M]. C. Spin waves in magnetic bilayer Let us consider spin waves in magnetized structures consisted of two monolayers of the quadratic lattice with thelattice constant a. The distance between layers is equal to d and the exchange interaction between spins of layers is I d. 1. Normal magnetized films Dispersion relations for two spin-wave modes in normal magnetized bilayer are determined by eigenvalues of Eq. (19) for variables h(1) −,h(1) +,h(2) −, andh(2) +and can be written as ω(1)2(/vectorq)=/Omega1(/vectorq){/Omega1(/vectorq)+2πγσ mq[1+exp(−qd)]}, ω(2)2(/vectorq)=/bracketleftbigg /Omega1(/vectorq)+2B(p)Id ¯h/bracketrightbigg/braceleftbigg /Omega1(/vectorq)+2B(p)Id ¯h +2πγσ mq[1−exp(−qd)]/bracerightbigg , (23) where q=(q2 x+q2 y)1/2and/Omega1(/vectorq) is defined in relation (21). For the first mode, spins in different layers change theirorientations in phase. In this case, spin waves of the firstmode correspond to spin waves in monolayer (21).A tq→ 0, the group velocity of spin waves v=2πγσ mis two times higher than the group velocity in monolayer. For thesecond mode, spins in different layers change orientations inantiphase and the energy of the spin wave with the givenlongitudinal wave vector qis higher than the energy of the spin wave of the first mode. For q→0, the spin-wave group velocity vtends to zero. Dispersion curves of spin waves determined by relations (23) are shown in Fig. 4. Spin waves propagate along the xaxis. Calculations have been done for the exchange interactions I 0=Id=0.085 eV and for the distance between layers d=a=0.4 nm at the sum of magnetic fields H+H(m)=3k O e . 2. In-plane magnetized films Dispersion relations of spin waves in in-plane magnetized bilayers are determined by eigenvalues of Eq. (19)for variables h(1) −,h(1) +,h(2) −, andh(2) +with the substitution /vector1z→/vector1y. Taking into account relations (20), for spin waves propagating along 214413-7L. V . LUTSEV PHYSICAL REVIEW B 85, 214413 (2012) IddH 0.0 1.0 2.0 3.0 aq0100020003000Frequency ω/2π (GHz) 12 FIG. 4. (Color online) Dispersion curve of spin waves propagat- ing in the normal magnetized bilayer with quadratic lattice ( a=0.4 nm) at the sum of magnetic fields H+H(m)=3 kOe. Exchange interactions, I0=Id=0.085 eV . The distance between monolayers dis equal to the lattice constant a. 1 and 2 are the first and the second modes of spin waves, respectively. thexaxis (/vectorq⊥/vectorH,q=qx,qz=0), we obtain ω(n)2(q)=/bracketleftbigg /Omega1(q)+B(p)Id ¯h/bracketrightbigg/bracketleftbigg /Omega1(q)+/Omega1M+B(p)Id ¯h/bracketrightbigg +/bracketleftbiggB(p)Id ¯h/bracketrightbigg2 +Q{/Omega1M−Q[1+2e x p (−2qd)]} ±/parenleftbigg/braceleftbigg/bracketleftbigg 2/Omega1(q)+2B(p)Id ¯h+/Omega1M/bracketrightbiggB(p)Id ¯h +Qexp(−qd)(2Q−/Omega1M)/bracerightbigg2 +4Q2exp(−2qd) ×/braceleftbigg Q2exp(−2qd)−/bracketleftbiggB(p)Id ¯h/bracketrightbigg2/bracerightbigg/parenrightbigg1/2 , (24)where Q=2πγσ mq,n=1,2 is the mode number, /Omega1(q) and /Omega1Mare defined in relation (22).A tq→0, the group velocity of the first mode v=2πγσ m/Omega1M {/Omega1(0)[/Omega1(0)+/Omega1M]}1/2 is two times higher than the group velocity of spin waves in monolayer. For the second mode, vtends to zero. For spin waves propagating along the zaxis (/vectorq/bardbl/vectorH,q=qz, qx=0) dispersion relations of two modes are ω(1)2(q)=/Omega1(q){/Omega1(q)+/Omega1M−2πγσ mq[1+exp(−qd)]}, ω(2)2(q)=/bracketleftbigg /Omega1(q)+2B(p)Id ¯h/bracketrightbigg/braceleftbigg /Omega1(q)+/Omega1M+2B(p)Id ¯h −2πγσ mq[1−exp(−qd)]/bracerightbigg . (25) For small wave vectors q, the group velocity of the first mode is negative and at q→0 is equal to v=−2πγσ m/Omega1(0)1/2 (/Omega1(0)+/Omega1M)1/2. The group velocity of the second mode tends to zero with the wave vector decrease. D. Spin-wave resonance in N-layer structure In this section, we consider a spin-wave resonance in a normal magnetized structure consisted of Nuniform mono- layer lattices with the exchange interaction Idbetween spins of layers. The distance between layers is equal to d.T h e spin-wave resonance is the limit case of a spin wave whenthe longitudinal wave vector q→0. Therefore the MDI terms V(dip) μν(/vectorq,1z−1/prime z)i nE q . (19) can be dropped and the equations with variables h(λ) +andh(λ) −are separated and eigenvalues are determined by the zero values of the determinant (we write the determinant D(+)for equations with the h(λ) +): D(+)=G(0)(1)...G(0)(N) det⎛ ⎜⎜⎜⎜⎜⎜⎜⎝[G (0)−1(1)−V(0)(11)] −V(0)(12) 0... −V(0)(21) [ G(0)−1(2)−V(0)(22)] −V(0)(23)... 0 −V(0)(32) [ G(0)−1(3)−V(0)(33)]... ··· ··· ··· ···⎞ ⎟⎟⎟⎟⎟⎟⎟⎠, where V (0)(kj) andG(0)(k) are the abridged notation of V(exch) +− (/vectorq,kz−jz,ωm)|/vectorq=0andG(0) −+(k,k,ω m)a tiωm→ω+iεsignω, respectively. ( k,j) are indices of layers. Taking into account that spins of outer layers ( k=1,N) interact with spins of one inner layer and spins of inner layers interact with spins of two layers and introducing the variable for inner layers in the determinantD (+), x=G(0)−1(k)−V(0)(kk) −V(0)(jk)=¯h B(p)Id[ω−γ(H+H(m))]−2(k/negationslash=1,N,j=k±1), 214413-8DISPERSION RELATIONS AND LOW RELAXATION OF ... PHYSICAL REVIEW B 85, 214413 (2012) we obtain that the spin-wave resonance spectrum is determined by roots of the polynomial RN(x)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(x+1 ) 100...0 0 1 x 10...0 0 01 x 1...0 0 00 1 x...0 0 ··· ··· ··· ··· ··· ··· ··· 0 000...x 1 0 000...1 ( x+1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=(x+1) 2PN−2(x)−2(x+1)PN−3(x)+PN−4(x)=0(N/greaterorequalslant2), where P−2(x)=−1,P−1(x)=0,P0(x)=1,PN(x)= xPN−1(x)−PN−2(x). Polynomial RN(x) hasNroots: x(n)=−2 cos/parenleftbiggπn N/parenrightbigg , where n=0,1,..., N −1. Taking into account the form of the roots x(n), we can introduce the transverse wave vector q(n) z=πn/Nd . Then the spin-wave resonance spectrum can be written as ω(n)=γ(H+H(m))+2B(p)Id ¯h/bracketleftbig 1−cos/parenleftbig q(n) zd/parenrightbig/bracketrightbig .(26) For the first mode ( n=0), spins in different layers change their orientations in phase. For the highest mode ( n=N−1), spins in different layers change orientations in antiphase andthe energy of spin-wave resonance is highest. Figure 5presents the spin-wave resonance spectrum (26) for the structure with N=40 layers. One can see that at low values of the transverse wave vector, the resonance spectrum is proportional to thequadratic dependence on q (n) z. d IdD 0.0 1.0 2.0 3.0 q d = n/N0.00.40.81.21.62.0 1- cos(q d)h( - )/2B(p)I(n) d(0) - (n) z(n) zωω FIG. 5. (Color online) Spin-wave resonance spectrum ω(n) (n=0,1,..., N −1) for the structure with N=40 layers. q(n) zis the transverse wave vector, dis the distance between layers, and Id is the exchange interaction between spins of layers.IV . LANDAU-LIFSHITZ EQUATIONS AND SPIN-WA VE EXCITATIONS IN THICK MAGNETIC FILMS A. Linearized Landau-Lifshitz equations Equations (14) and (19) describe spin-wave excitations. Solutions of these equations for magnetic samples of greatvolumes and for thick N-layer magnetic films with N/greatermuch1 become difficult, because determinants of Eqs. (14) and(19) have high orders. In order to overcome the difficulty and tofind the spin-wave spectrum for these samples, we deriveLandau-Lifshitz equations. 22,23Dispersion relations for spin excitations are determined by the P-matrix poles (12) that coincides with poles of the matrix Gof effective propagators. Accordingly, the dispersion relations can be derived from theeigenvalues of equation G=G (0)+G(V(exch)+V(dip))G(0), (27) where G(0)=/bardblG(0) μν/bardblis the matrix of bare propagators (11). Since the considered interaction is the sum of exchange andmagnetic dipole interactions, we can obtain the eigenvaluesand eigenfunctions of equation (27) by a two-step procedure. In the first stage, we perform the summation of diagrams, takeinto account the exchange interaction, and find the propagatormatrix G (1)=/bardblG(1) μν/bardbl G(1)=G(0)+G(0)V(exch)G(1). (28) In the second stage, the summation of diagrams with dipole interaction lines is performed. This gives the equation for thematrix Gof effective propagators expressed in terms of the matrix G (1): G=G(1)+GV(dip)G(1). (29) Thus the solution of Eq. (27), which determines the matrix G, is equivalent to the solution of Eqs. (28) and(29).A f t e rt h e performed two-step summation, Eq. (14) for eigenfunctions h(λ) μis written in the more convenient form h(λ) μ(/vector1,ωm)−/summationdisplay ρ,σ /vector1/prime/vector1/prime/primeV(dip) μρ(/vector1−/vector1/prime,ωm) ×G(1) ρσ(/vector1/prime,/vector1/prime/prime,ωm)h(λ) σ(/vector1/prime/prime,ωm)/vextendsingle/vextendsingle/vextendsingle/vextendsingle iωm→ω+iεsignω=0.(30) 214413-9L. V . LUTSEV PHYSICAL REVIEW B 85, 214413 (2012) The solution of simultaneous equations (28) and(30) gives the dispersion relations for spin excitations. These equationscan be reduced to linearized Landau-Lifshitz equations inthe generalized form and the equation for the magnetostaticpotential. In order to perform this transformation, one needsto make a transition to the retarded Green functions. Wetransform the matrix equation (28) to equations describing small variations of the magnetic moment density (or thevariable magnetization), m ν. The variable magnetization mν under the action of the magnetic field hν=¯hν, which is generated by the MDI V(dip), is given by the retarded Green functions, which are determined by the analytical continuedvalues of the propagator matrix G (1):39 mν(/vector1,ω)=β(gμB)2 Va/summationdisplay ρ,/vector1/primeG(1) νρ(/vector1,/vector1/prime,ωm)/vextendsingle/vextendsingle/vextendsingle/vextendsingle iωm→ω−iε¯hρ(/vector1/prime,ω), (31) where Vais the atomic volume. The analytical continuation iωm→ω−iεdefines the retarded Green functions. ¯hρ(/vector1,ω) is the field of the magnetic dipole-dipole interaction acting onspins. By multiplying matrix equation (28) byG (0)−1from the left and by ¯hρfrom the right, performing the analytical continu- ationiωm→ω−iε,δ(β¯hωm)→[β¯h(ω−iε)]−1and taking into account relation (31), we get the matrix equation (28) in the form of simultaneous equations: /summationdisplay ν,/vector1/prime/bracketleftbig G(0)−1 ρν(/vector1,/vector1/prime,ω)−βIρν(/vector1−/vector1/prime)/bracketrightbig mν(/vector1/prime,ω) =β(gμB)2 Va¯hρ(/vector1,ω). (32) For isotropic exchange interaction, 2 I−+=2I+−=Izz= I, equations (32) have the form ˆE±m±(/vector1,ω)=2γM(/vector1)¯h∓(/vector1,ω), (33) ˆEzmz(/vector1,ω)=B[1](p) B(p)γM(/vector1)¯hz(/vector1,ω), (34) where M(/vector1)=gμBB(p)/Vais the magnetic moment density at the low-temperature approximation. We say that the opera-tors ˆE ±and ˆEz, ˆE±m±(/vector1,ω)=[γ(H(/vector1)+H(m)(/vector1))±ω]m±(/vector1,ω) +B(p) ¯hVb/summationdisplay /vector1/prime/integraldisplay Vb[¯I(0)−¯I(/vectorq)] ×exp[i/vectorq(/vector1−/vector1/prime)]m±(/vector1/prime,ω)d3q and ˆEzmz(/vector1,ω)=ωmz(/vector1,ω)−B[1](p) ¯hVb/summationdisplay /vector1/prime/integraldisplay Vb¯I(/vectorq)e x p [i/vectorq(/vector1−/vector1/prime)] ×mz(/vector1/prime,ω)d3q, are Landau-Lifshitz operators. For a cubic crystal lattice, the Fourier transform of the exchange interaction with re- spect to the lattice variables is ¯I(/vectorq)=/summationtext /vector1I(/vector1) exp( −i/vectorq/vector1)= 2I0[cos(qxa)+cos(qya)+cos(qza)], where I0is the interac- tion between neighboring spins. The field H(m)(/vector1) is definedby relation (10) and depends on the magnetic moment density M(/vector1);Vb=(2π)3/Vais the volume of the first Brillouin zone. Equations (33) and (34) have the generalized form of the Landau-Lifshitz equations.18Solutions m±of Eq. (33)depend on temperature, because β=1/kT is contained in the variable pof the function B(p)(8), through which the magnetic moment density M(/vector1) is expressed. Equation (34) describes longitudinal variations of the variable magnetization under theinfluence of the field ¯h z. At low temperature, the derivative of the function B[1](p) tends to zero and the longitudinal variable magnetization mzis negligible. From the form of the magnetic dipole interaction in relations (2)and (3), it follows that the field hν=¯hν, which is generated by the MDI V(dip), in relation (31) is magnetostatic, i.e., it is expressed in terms of the magnetostaticpotential ϕ:¯h ν=− ∇ νϕ. We transform equation (30) to the equation for the magnetostatic potential ϕ(/vectorr,ω). Taking into account relation (31) and the explicit form of the magnetic dipole interaction in relations (2)and (3), performing the derivation ∇μ, the analytical continuation iωm→ω−iεand the summation of equation (30) over the index μ, we obtain the equation expressed in terms of ϕ,mν: −/Delta1ϕ(/vectorr,ω)+4π∇νmν(/vector1,ω)|/vector1→/vectorr=0. (35) Equation (35) gives the boundary conditions for the normal component of the field /vectorb=− ∇ ϕ+4π/vectorm, (/vectorb,/vectorn)|+∂V=(/vectorb,/vectorn)|−∂V, (36) where /vectornis the normal to the boundary, ∂V,+∂V, and−∂V denote different sides of the boundary. Thus, in considerationof the Landau-Lifshitz equations (33) and(34), the dispersion relations of spin excitations are given by the eigenvalues ofEq.(35). If the scale of the spatial distribution of the variable magnetization m ν(/vector1,ω) and the sample size are much greater than the lattice constant a, then the sum over the lattice variables/summationtext /vector1in Landau-Lifshitz operators ˆE±and ˆEzcan be converted into an integral over the sample volume V−1 a/integraltext d3r. In this approximation, we suppose that the film is continuousover the thickness and, therefore, one can use methods ofdifferential and integral calculus. Let us consider the case ofnormal and in-plane magnetized homogeneous films when thetemperature is low. Then we obtain that m z→0 and Eq. (34) is dropped. 1. Normal magnetized films The dispersion relations of spin waves in a normal mag- netized film with thickness Dare determined by Eqs. (33) and(35). Taking into account that the magnetic field H(m)in normal magnetized films is equal to −4πM,18we find the dispersion relations of spin waves: ω(n)2(/vectorq)=/Omega1(n)(/vectorq)/bracketleftbig /Omega1(n)(/vectorq)+/Omega1Mq2/slashbig q(n)2 0/bracketrightbig , (37) where n=1,2,3,... is the mode number, /vectorq=(qx,qy)i s the two-dimensional longitudinal wave vector of spin waves, 214413-10DISPERSION RELATIONS AND LOW RELAXATION OF ... PHYSICAL REVIEW B 85, 214413 (2012) q=| /vectorq|, /Omega1(n)(/vectorq)=γH−/Omega1M+2B(p)I0 ¯h ×/bracketleftbig 3−cos(qxa)−cos(qya)−cos/parenleftbig q(n) za/parenrightbig/bracketrightbig , /Omega1M=4πγM ,q(n) 0=(q2+q(n)2 z)1/2,q(n) zis the transverse vector. The magnetostatic potential over the thicknessz∈[−D/2,D/2] of the magnetic film is ϕ (n/vectorq)(x,y,z )=(2π)−1f(n)−1/2exp(iqxx+iqyy) ×cos/bracketleftbig q(n) zz+π(n−1)/2/bracketrightbig , (38) where f(n)=D/2+q/q(n)2 0. The boundary conditions (36) gives the relationship between the transverse q(n) zand the longitudinal qwave vectors: 2 cotq(n) zD=q(n) z q−q q(n) z. (39) For low values of qand for small mode numbers n, we can neglect the exchange term in the /Omega1(n)(/vectorq). Then, in this case, dispersion relations (37) correspond to dispersion relations of forward volume magnetostatic spin waves.1–3,18 2. In-plane magnetized films Let us consider the case, when xandzaxes are in the film plane and the y-axis is normal to the plane. The magnetic field/vectorHis parallel to the zaxis. Spin waves propagate along thexaxis. Dispersion relations of spin waves in an in-plane magnetized film with the thickness Dare determined by Eqs. (33) and(35) with boundary conditions (36). Taking into account that the magnetic field H(m)in in-plane magnetized films is equal to zero, we find the dispersion relations of surfacespin waves: ω (s)2(q)=/Omega12(q)+/Omega1(q)/Omega1M+/Omega12 M 4[1−exp(−2qD)],(40) where /Omega1(q)=γH+4B(p)I0 ¯h[1−cos(qa)], and dispersion relations of high spin-wave modes, ω(n)2(q)=/Omega1(n)(q)[/Omega1(n)(q)+/Omega1M], (41) where /Omega1(n)(q)=γH+2B(p)I0 ¯h/bracketleftbig 2−cos(qa)−cos/parenleftbig q(n) ya/parenrightbig/bracketrightbig , q(n) y=πn/D ,n=1,2,3,.... For low values of q, we can neglect the exchange term in the /Omega1(q) in relation (40).I nt h i s case, the dispersion relations correspond to dispersion relationsof Damon-Eshbach surface magnetostatic spin waves. 40 Let us consider the case when spin waves propagate along thezaxis (q=qz). Then the solution of Eqs. (33) and(35) gives the spin-wave dispersion relations ω(n)2(q)=/Omega1(n)(q)/bracketleftbigg /Omega1(n)(q)+/Omega1M−/Omega1Mq2 q(n)2 y+q2/bracketrightbigg ,(42) where /Omega1(n)(q) is defined in relation (41). The transverse wave vector q(n) yis determined by relation (39), where thesubstitution q(n) z→q(n) yshould be done. For low values ofqand for small mode numbers n, dispersion relations (42) correspond to dispersion relations of backward volume magnetostatic spin waves.1–3,18 B. Difference between dispersion relations of spin waves in monolayers, bilayers and in thick magnetic films We can single out the MDI part in the dispersion relations of spin waves in monolayers, bilayers, and in thick magneticfilms. Taking into account that for monolayers and for bilayers/Omega1 M=4πγσ m/a, we can write the dispersion relations (21), (23), and (37)of spin waves propagating in normal magnetized films in the form ω(n)2(/vectorq)=/Omega1(n)(/vectorq)[/Omega1(n)(/vectorq)+/Omega1Mη(n)(qD)], (43) where nis the mode number. η(n)(qD) is the function of qD, where for monolayers D=a, for bilayers D=2d=2a(we consider the case d=a), and in the case of thick films, Dis the thickness. /Omega1(n)(/vectorq) is defined in relation (37).T h eηfunction determines the action of the MDI. For spin waves propagating in in-plane magnetized films along the xaxis (/vectorq⊥/vectorH,q=qx,qz=0), dispersion relations (22)and(24)and surface spin-wave relation (40)can be written in the form ω2(q)=/Omega12(q)+/Omega1(q)/Omega1M+/Omega12 Mη, where for the case of monolayers and of thick films, ηis a function of qD. For bilayers, ηis a function of qD,/Omega1(q)//Omega1M, andB(p)I0//Omega1M./Omega1(q) is defined in relation (22). For spin waves propagating in in-plane magnetized films along the zaxis (/vectorq/bardbl/vectorH,q=qz,qx=0), dispersion relations (22),(25), and (42) have the form ω(n)2(q)=/Omega1(n)(q)[/Omega1(n)(q)+/Omega1M−/Omega1Mη(n)(qD)] with/Omega1(n)(q) defined in relation (41).T h e ηfunction for backward volume spin waves coincides with the ηfunction for forward waves in relation (43). For the first forward and backward spin-wave modes and for spin waves propagating in in-plane magnetized films alongthexaxis,ηfunctions are presented in Fig. 6. For in-plane magnetized bilayers, Fig. 6(b) shows the ηfunction of I 0=0 and of B(p)I0//Omega1M→∞ . For these cases, the ηfunction is independent on the variable /Omega1(q)//Omega1M. One can see that ηfunctions of spin waves in monolayers, bilayers, and in thick magnetic films are close for qD < 1. Thus, in order to calculate dispersion relations of spin waves in N-layer films ( N=1,2,...) consisted of monolayers for qD < 1, we can consider the N-layer film as continuous. For example, for a quadratic lattice monolayer with the lattice constant a, the parameters of this continuous film are the following: thethickness Dis equal to aand the volume magnetic moment density Mis determined by the surface magnetic moment density σ m,M=σm/a. Spin excitations in thin magnetic films, bilayers, and trilayers are calculated in Refs. 10–14in the continuous-film approximation for low wave vectors, qD/lessmuch1. In accordance with the above mentioned, in this case, the usageof the continuous-film approximation is correct. ForqD > 1, the difference between the ηfunction of monolayer and the ηfunction of continuous thick magnetic 214413-11L. V . LUTSEV PHYSICAL REVIEW B 85, 214413 (2012) (a) (b) FIG. 6. (Color online) (a) ηfunction characterizing the action of the MDI on dispersion curves of forward spin waves in normal magnetized films and backward spin waves in in-plane magnetizedfilms vs the normalized wave vector qD. 1 is a monolayer, 2 is a bilayer, and 3 is a thick film case. (b) ηfunction for surface spin waves in in-plane magnetized films. 1 is a monolayer, 2 is a bilayer[2a for I 0=0, and 2b for B(p)I0//Omega1M→∞ ], 3 is a thick film case. films is considerable. The difference is due to the discreetness of the lattice. If the exchange interaction is much greater thanthe MDI, the difference between dispersion relations of spinwaves in monolayers and in continuous thick magnetic filmsdetermined by ηfunctions is insignificant in comparison with the exchange interaction and can be dropped. But, when theMDI is equal or greater than the exchange interaction, thisdifference becomes essential and should be taken into account.It is important for the case of monolayer lattice with magneticnanoparticles on lattice sites. C. Exchange boundary conditions Let us consider the case when the size of a homogeneous film is much greater than the lattice constant aand the sum/summationtext /vector1 in operators ˆE±(33) can be converted into an integral over the sample volume V−1 a/integraltext d3r. The magnetic fields HandH(m) are homogeneous. If we restrict ourself to the second termin the Fourier transform of the exchange interaction ¯I(/vectorq)− ¯I(0)=−I0a2q2, then the operators ˆE±can be written in the pseudodifferential form of order 2:41 ˆE±m±(/vectorr,ω)=[γ(H+H(m))±ω]m±(/vectorr,ω) +4πγαM (2π)3/integraldisplay V/integraldisplay Vbq2exp[i/vectorq(/vectorr−/vectorr/prime)] ×m±(/vectorr/prime,ω)d3qd3r/prime, (44) where α=B(p)I0a2/¯hγ4πM is the exchange interaction constant, Vis the volume of the ferromagnetic sample. In Refs. 15–18,42–46, the pseudodifferential Landau- Lifshitz operators are reduced to the differential operators withrespect to spatial variables: ˆE ±(/vectorr,ω)=γ[H+H(m)−4παM/Delta1 ]±ω. (45) For solvability of Eq. (33) with differential Landau-Lifshitz operators (45), the exchange boundary conditions are imposed: ∂mν ∂/vectorn+ξmν/vextendsingle/vextendsingle/vextendsingle/vextendsingle ∂V=0, (46) where /vectornis the inward normal to the boundary ∂V and ξis the pinning parameter. As it is found in Appendix for the case of forward volume spin waves propagatingin a normal magnetized film, simultaneous equations (33) with operators (45) and with boundary conditions (46) and Eq.(35) with boundary conditions (36) have no solutions due to incompatibility of conditions (36) and (46). In order to evaluate the influence of the exchange boundary conditions onthe dispersion relations, we formally drop out the boundaryconditions (36). Then the exchange boundary conditions (46) give the relation for the transverse wave vector q (n) z(see Appendix): 2 cotq(n) zD=q(n) z ξ−ξ q(n) z, (47) where nis the mode number. Dispersion relations (37) of the first spin-wave mode propagating in the YIG film ofthe thickness D=0.5μm with 4 πM=1750 Oe, and α= 3.2×10 −12cm2at the applied magnetic field H=3000 Oe are shown in Fig. 7for the transverse wave vector q(1) z(47) with different pinning parameters ξ. In contrast with these curves, we show dispersion relations based on pseudodif-ferential operators (44) with the boundary conditions (36) (the curve A). One can see that there does not exist any constantpinning parameter ξat which the curve A calculated on the basis of relation (39) coincides with the curves calculated on the basis of the exchange boundary conditions. In order to overcome the contradiction based on simulta- neous solvability of relations (39) and(47), we should require that the pinning parameter ξ=q. Only in this case, the curve A calculated on the basis of equations with pseudodifferentialLandau-Lifshitz operators (44) coincides with the curves calculated on the basis of differential equations with theexchange boundary conditions (46). V . SPIN-WA VE RELAXATION In this section we answer the question: what is the value of spin-wave relaxation in the model with magnetic dipole 214413-12DISPERSION RELATIONS AND LOW RELAXATION OF ... PHYSICAL REVIEW B 85, 214413 (2012) 0.0 0.1 0.2 0.3 0.4 0.5 qD35004000450050005500Frequency ω/2π (MHz)21 3 4A FIG. 7. (Color online) Dispersion curves of the first spin-wave mode propagating in the YIG film of the thickness D=0.5μm with 4πM=1750 Oe, α=3.2×10−12cm2at the applied magnetic field H=3000 Oe. The curve A is calculated on the base of relation (39) for the case of pseudodifferential Landau-Lifshitz operators (44). Curves 1–4 are calculated for the case of differential Landau-Lifshitzoperators (45) on the base of relation (47) with different pinning parameters ξ.( 1 )ξD=0.01, (2) 0.1, (3) 1, and (4) 10. and exchange interactions derived from first principles? The answer depends on the ratio of the spin-wave energy tointervals between modes of the spin-wave spectrum and isdifferent for thick and for thin magnetic films. In thick films,the spin-wave energy is greater than energy gaps between modes and a three-spin-wave process takes place. If the exchange interaction is isotropic, it cannot induce three-magnon processes and, therefore, the MDI makes a majorcontribution to the relaxation. We consider the spin-wavedamping in thick films in the one-loop approximation. Inthin magnetic films (for example, in nanosized films), theenergy of long-wavelength spin waves is less than energy gapsbetween modes and three-spin-wave processes are forbidden. In this case, four-spin-wave processes take place, the exchange interaction makes a major contribution to the relaxation,and the spin-wave damping has lower values in comparisonwith the damping in thick films. We calculate the spin-waverelaxation for four-spin-wave processes in thin films forlong-wavelength spin waves in the two-loop approximation. A. Spin-wave relaxation in thick films The spin-wave relaxation induced by a three-spin-wave process in normal magnetized homogeneous ferromagneticfilms is considered in Refs. 22and23in the one-loop approxi- mation for spin waves with small longitudinal wave vectorsat low temperature. The relaxation is determined by self-energy diagram insertions /Sigma1 (1+)(1−)to the Pmatrix given by relation (12) (see Fig. 8). Damping of the j-mode excitation is q, j,m q,j,mj,j,q, )m= q,j,m q, j,m1 2B +(2B)21 (1+)(1-) FIG. 8. Self-energy diagrams in the one-loop approximation at low temperature. Bis determined by relation (8).defined by the imaginary part of the pole of the effective Green functions G−+=P(1−)(1+)with insertions /Sigma1(1+)(1−)under the analytical continuation (13): /Delta1(j)(/vectorq)=δω(j)(/vectorq) ω(j)(/vectorq)=2B(p)Va β¯hω(j)(/vectorq)Im/Sigma1(1+)(1−) ×(j,j,/vectorq,ωm)/vextendsingle/vextendsingle/vextendsingle/vextendsingle iωm→ω+iεsignω =Va 2β¯hω(j)Im/summationdisplay n,i,k/integraldisplay F(i)F(k)[¯P(1−)(1+)(i,−/vectorq1,−ωn) ׯP(2z)(2z)(k,/vectorq−/vectorq1,ωm−ωn) +1 8B(p)¯P(1−)(2z)(i,/vectorq1,ωn)¯P(2z)(1+) ×(k,/vectorq−/vectorq1,ωm−ωn)]N2(j,/vectorq;i,/vectorq1;k,/vectorq−/vectorq1) ×d2q1/vextendsingle/vextendsingle/vextendsingle/vextendsingle iωm→ω+iεsignω, (48) where ¯P(1−)(1+)(j,/vectorq,ωm)=2ρV2 a(/Omega1(j)+2η(j) −++iωm), ¯P(1−)(2z)(j,/vectorq,ωm)=−2η(j) +z(/Omega1(j)+iωm), ¯P(2z)(1+)(j,/vectorq,ωm)=−2η(j) z−(/Omega1(j)+iωm), ¯P(2z)(2z)(j,/vectorq,ωm)=F(j)−1βVa¯I(/vectorq)−ρ−1η(j) zz(/Omega1(j)2+iω2 m), F(j)=/parenleftbig ω(j)2+ω2 m/parenrightbig−1,ρ=B(p) β¯hVa, η(j) μν=/Omega1Mqμqν q(j)2 0(μ,ν=−,+,z), q±=1 2(qx∓iqy), ¯I(/vectorq)=2I0/bracketleftbig cos(qxa)+cos(qya)+cos/parenleftbig q(j) za/parenrightbig/bracketrightbig is the Fourier transform of the exchange interaction, N(j1,/vectorq1;j2,/vectorq2;j3,/vectorq3) =1 8πVa3/productdisplay k=11 f(jk)1/2/summationdisplay σ1,σ2,σ3sin/bracketleftbig/parenleftbig/summationtext3 k=1σkq(jk) z/parenrightbig D/2/bracketrightbig /summationtext3 k=1σkq(jk) z ×exp/bracketleftBigg i3/summationdisplay k=1σkπ(jk−1)/2/bracketrightBigg is the block factor in the representation of the functions (38),f(j)=D/2+q/q(j)2 0,σk=±1;/summationtext σ1,σ2,σ3denotes the summation over all sets {σ1,σ2,σ3}. The spin-wave frequency ω(j)and the transverse wave vector q(j) zare determined by relations (37) and(39), respectively. The damping /Delta1(j) increases directly proportionally to the temperature. Relation (48) describes relaxation of the spin-wave jmode caused by inelastic scattering on thermal excited spin-wavemodes. Relaxation occurs through the confluence of the j mode with the kmode to form the imode. From the explicit form of the block factor Nin relation (48),i tf o l l o w st h a t the confluence processes take place when the sum of modenumbers j+i+kis equal to an odd number. The confluence processes are induced by the MDI and are accompanied by 214413-13L. V . LUTSEV PHYSICAL REVIEW B 85, 214413 (2012) 0.0 0.1 0.2 0.3 0.4 0.5 qD1.0E-51.0E-41.0E-31.0E-21.0E-1Spin wave damping12 34 ∇(1) A FIG. 9. (Color online) Spin-wave damping /Delta1(1)=δω(1)/ω(1)of the first mode in normal magnetized YIG film with the magnetization4πM=1750 Oe and the exchange interaction constant α=3.2× 10 −12cm2atH=3000 Oe, T=300 K at different film thickness D. (1)D=500, (2) 300, (3) 200, (4) 120 nm. A is the low-relaxation region. transitions between thermal excited iandkmodes. Transitions take place when the equation ω(j)(/vectorq)=ω(i)(/vectorq(s))−ω(k)(/vectorq−/vectorq(s)) (49) has at least one solution /vectorq(s)for the given /vectorq,i,j,k.T h e existence of solutions /vectorq(s)of Eq. (49)depends on the thickness of the magnetic film. With decreasing film thickness D,t h e density of dispersion curves of modes on the plane ( ω,q) decreases and the frequency of the spacings between curvesincrease. The least frequency spacing occurs between thefirst (i=1) and the third ( k=3) modes. Figure 9shows the damping /Delta1 (1)of the first spin-wave mode versus the longitudinal wave vector qnormalized by the film thickness D at different film thicknesses. Calculations have been done fora YIG film with the magnetization 4 πM=1750 Oe and the exchange interaction constant α=3.2×10 −12cm2atH= 3000 Oe and T=300 K. One can see that for the YIG film with the thickness D=120 nm in the region qD < 0.14 the damping /Delta1(1)is equal to zero due to the absence of transitions between modes. Thus, in thin magnetic films, a low spin-waverelaxation region takes place. We define the low-relaxationregion as a region in the ( ω,q) space, where spin wave has no damping induced by three-spin-wave processes. For the givenjmode, this region appears when the excitation frequency ω (j)(/vectorq) is less than the difference ω(3)(/vectorq(s))−ω(1)(/vectorq−/vectorq(s)) at any values of the wave vector /vectorq(s). For the first mode ω(1) in the YIG film, the film thickness, when the low spin-wave relaxation region appears, is shown in Fig. 10atq→0. If ω(1)(0)<ω(3)(/vectorq(s))−ω(1)(/vectorq(s)), (50) the first mode has low values of the spin-wave damping /Delta1(1). Taking into account dispersion relations (37), from inequality (50), we can obtain an estimation of the characteristic thickness for the given frequency ω: D0=2π(α/Omega1M)1/2 [ω(ω+/Omega1M)]1/4.12 8 4 0 16 20050100150200Film thickness D( n m )0 Low relaxation region,q3 2 1 (1) Frequency /2 (GHz)(1)ω π FIG. 10. (Color online) Film thickness D0of YIG film versus the excitation frequency ω(1)(/vectorq)/2πof the first mode at the wave vector /vectorq→0. Low-relaxation region of the first spin-wave mode exists for YIG films with the thickness D<D 0. We say that a film is thin with respect to the relaxation process, if the film thickness D<D 0. In the next section, we consider the relaxation of spin waves in thin films. B. Relaxation in thin magnetic films What is the value of spin-wave damping in the low relaxation region in thin magnetic films? We consider four-spin-wave processes in the normal magnetized monolayer ofthe quadratic lattice with the lattice constant aat small lon- gitudinal wave vector values /vectorq=(q x,qy) at low temperature. As isotropy of the exchange interaction cannot forbid four-spin-wave processes and the value of the exchange interactionis much greater than the MDI, only the exchange interactionwill be taken into account in diagrams. We suppose that theexchange interaction acts between neighboring spins and isequal to I 0. In order to calculate self-energy diagram insertions to the effective Green functions in the two-loop approximation,we use the ladder expansion (see Fig. 11). At small values of wave vectors the bare /Gamma1 0vertex [see Fig. 11(a) ]i s /Gamma10(1,2; 3,4)≡/Gamma10(/vectork,/vectors+/vectorq−/vectork;/vectorq,/vectors) =β[¯I(/vectork−/vectorq)+¯I(/vectork−/vectors)−¯I(/vectors)−¯I(/vectorq)] =2βI0a2(/vectorq,/vectors), where 1 ,2; 3,4 is the abridged notation of two-dimensional wave vectors, which are variables of /Gamma10vertex; |/vectork|,|/vectorq|,|/vectors|/lessmuch a−1: ¯I(/vectorq)=/summationdisplay /vector1xy−/vector1/primexyI(/vector1xy−/vector1/prime xy)e x p [−i/vectorq(/vector1xy−/vector1/prime xy)] =2I0[cos(qxa)+cos(qya)]. The/Gamma1vertex in the ladder approximation [see Fig. 11(b) ]i s determined by the relationship /Gamma1(1,2; 3,4) ≡/Gamma1(/vectork,ω 1,/vectors+/vectorq−/vectork,ω 3+ω4−ω1;/vectorq,ω 3,/vectors,ω 4) =/Gamma10(/vectork,/vectors+/vectorq−/vectork;/vectorq,/vectors) 214413-14DISPERSION RELATIONS AND LOW RELAXATION OF ... PHYSICAL REVIEW B 85, 214413 (2012) (a) (1,2;3,4) =0=1 23 4+1 23 4+1 23 4+3 41 21 3 2 4 (b) 0 +0q,(q) = (1,2;3,4) =3 2 41 33 2 4 2 4s,(s)8B(p)21 (c) 0 +021(q, ) =(q) q,(q)q,(q)k,( ) s,(s)k, q,(q)q,(q)k+s-q,(k+s-q) 16B(p)21( ) FIG. 11. (Color online) (a) Bare /Gamma10vertex. (b) Ladder approxi- mation. (c) Self-energy diagram insertion. +1 8B2(p)Sb/summationdisplay ω(q) m/integraldisplay /Gamma10(/vectork,/vectors+/vectorq−/vectork;/vectorq/prime,/vectors+/vectorq−/vectorq/prime) ×G−+/parenleftbig/vectorq/prime,ω(q) m/parenrightbig G−+/parenleftbig /vectors+/vectorq−/vectorq/prime,ω3+ω4−ω(q) m/parenrightbig ×/Gamma1/parenleftbig/vectorq/prime,ω(q) m,/vectors+/vectorq−/vectorq/prime,ω3+ω4−ω(q) m;/vectorq,ω 3,/vectors,ω 4/parenrightbig d2q/prime, where G−+(/vectorq,ωm)=2B(p) β¯h(ω(/vectorq)−iωm) is the effective Green function determined by the Pmatrix (12),ω(/vectorq) is the frequency of spin excitations in monolayer (21), and Sbis the volume of the two-dimensional first Brillouin zone. The coefficient 1 /8B2(p) is due to the fact that the substitution of the bare Green function to effectiveones in diagrams are performed inside blocks. The self-energydiagram insertion [see Fig. 11(c) ] is given by /Pi1(/vectorq,ω (q) m)=1 2Sb/summationdisplay ω(k) n/integraldisplay /Gamma10(/vectorq,/vectork;/vectorq,/vectork)G−+/parenleftbig/vectork,ω(k) n/parenrightbig d2k +1 16B2(p)S2 b/summationdisplay ω(k) n,ω(s) l/integraldisplay/integraldisplay /Gamma10(/vectorq,/vectors+/vectork−/vectorq;/vectors,/vectork) ×G−+/parenleftbig −/vectors−/vectork+/vectorq,−ω(k) n−ω(s) l+ω(q) m/parenrightbig ×G−+/parenleftbig/vectork,ω(k) n)G−+/parenleftbig /vectors,ω(s) l/parenrightbig ×/Gamma1/parenleftbig /vectors,ω(s) l,/vectork,ω(k) n;/vectorq,ω(q) m,/vectors+/vectork−/vectorq, ω(k) n+ω(s) l−ω(q) m/parenrightbig d2kd2s. (51)The damping of spin-wave excitations at ω=ω(/vectorq)i s expressed by the imaginary part of the self-energy /Pi1(/vectorq,ω(q) m): /Delta1(/vectorq)=δω(/vectorq) ω(/vectorq)=Im/Pi1/parenleftbig /vectorq,ω(q) m/parenrightbig βω/vextendsingle/vextendsingle/vextendsingle/vextendsingle iω(q) m→ω+iεsignω.(52) Taking into account the self-energy /Pi1(/vectorq,ω(q) m)i nt h eB o r n approximation, namely, substituting /Gamma1→/Gamma10in relation (51), integrating over /vectorkand/vectors, and summing over the frequency variables ω(k) nandω(s) l, from equation (52) at ¯hω(/vectorq)<k T ,w e obtain /Delta1(/vectorq)=C(qa)2(kT)2 16πB2(p)I0ε(0), where C=1.12,kis the Boltzmann constant and ε(0)= ¯hγ(H+H(m)) is the Zeeman energy. In order to evaluate the damping of spin waves, we calculate /Delta1(/vectorq) for spin waves with the wavelength λ=5μm propagating in the monolayer film with the lattice constant a=0.4 nm and with the exchange interaction between neighboring spins I0=0.085 eV , B(p)= 1/2a tT=300 K. Then, taking into account that q=2π/λ, forε(0)/h=10 GHz, we obtain /Delta1(/vectorq)=4.28×10−6. Thus one can see that the damping of spin waves of small wavevectors is low. VI. CONCLUSIONS The results of the investigations can be summarized as follows. (1) Spin excitations in thin magnetic films inthe Heisenberg model with magnetic dipole and exchangeinteractions are studied by the spin operator diagram tech-nique. Dispersion relations of spin waves in two-dimensionalmagnetic monolayer and bilayer lattices and the spin-waveresonance spectrum in N-layer structures are obtained. It is found that dispersion relations of spin waves in monolayerand bilayer lattices differ from dispersion relations of spinwaves in continuous thick magnetic films. This differenceis due to the discreetness of the lattice. For the case whenthe magnetic dipole interaction is equal or greater than theexchange interaction, for example, for monolayer consisted ofmagnetic nanoparticles on the lattice, this difference becomesessential and is taken into account. (2) Generalized Landau-Lifshitz equations for thick mag- netic films, which are derived from first principles, have theintegral (pseudodifferential) form, but not differential one withrespect to spatial variables. Spin excitations are determinedby simultaneous solution of the Landau-Lifshitz equationsand the equation for the magnetostatic potential. It is foundthat the model based on differential Landau-Lifshitz equationswith exchange boundary conditions is contradictory. Thecontradiction is removed, if the pinning parameter ξis equal to the spin-wave wave vector q. (3) The magnetic dipole interaction makes a major con- tribution to the relaxation of long-wavelength spin waves inthick magnetic films. The spin-wave damping is determinedby diagrams in the one-loop approximation, which correspondto three-spin-wave processes. The three-spin-wave processesare accompanied by transitions between thermal excited spin-wave modes. The damping increases directly proportionally tothe temperature. 214413-15L. V . LUTSEV PHYSICAL REVIEW B 85, 214413 (2012) (4) Thin films have a region of low relaxation of long- wavelength spin waves. In thin magnetic films, the energy ofthese waves is less than energy gaps between spin-wave modes,therefore, three-spin-wave processes are forbidden, four-spin-wave processes take place and, as a result of this, the exchangeinteraction makes a major contribution to the relaxation. Itis found that the damping of spin waves propagating in amagnetic monolayer has the form of the quadratic dependenceon the temperature and is very low for spin waves with smallwave vectors. Low-damping spin waves can be observed in YIG films of nanometer thickness. Thin (nanosized) magnetic films canbe used in spin-wave devices. The low damping of long-wavelength spin waves gives us an opportunity to constructtunable narrow-band spin-wave filters with high quality at themicrowave frequency range. ACKNOWLEDGMENT This work was supported by the Russian Foundation for Basic Research, Grant 10-02-00516, and by the Ministry ofEducation and Science of the Russian Federation, Project2011-1.3-513-067-006. APPENDIX: SPIN-WA VE MODEL WITH EXCHANGE BOUNDARY CONDITIONS Let us consider forward volume spin waves propagating in a normal magnetized film homogeneous through the thickness D. The applied magnetic field /vectorHis parallel to the zaxis. In order to understand the role of the exchange boundaryconditions in Refs. 15and16, we consider Landau-Lifshitz equations (33) with differential operators (45) and with boundary conditions (46). The dispersion relations of spin waves are given by the eigenvalues of equation for themagnetostatic potential ϕ(35). Taking into account that the magnetic field of spin waves ¯h ν=− ∇ νϕ, after the Fourier transform with respect to the longitudinal variables xandy, we can write Landau-Lifshitz equations (33) in the form /bracketleftbigg /Omega1+α/Omega1M/parenleftbigg q2−∂2 ∂z2/parenrightbigg ±ω/bracketrightbigg m±=−i/Omega1Mq 4πϕ,(A1) where /Omega1=γ(H+H(m)) and/Omega1M=4πγM ,qis the longi- tudinal wave vector. Without loss of a generality, we supposethatq=q xandqy=0. The equation for the magnetostatic potential ϕ(35) is written as /parenleftbigg q2−∂2 ∂z2/parenrightbigg ϕ+2πiq(m++m−)=0. (A2) The boundary conditions (36) for the normal component of the field /vectorb=− ∇ ϕ+4π/vectormat boundaries z=−D/2 andD/2are reduced to the form ∂ ∂zϕ/vextendsingle/vextendsingle/vextendsingle/vextendsingle +∂V=∂ ∂zϕ/vextendsingle/vextendsingle/vextendsingle/vextendsingle −∂V, (A3) where +∂Vand−∂Vdenotes different sides of the boundary. The magnetostatic potential ϕis continuous in the boundary region ϕ|+∂V=ϕ|−∂V. (A4) The exchange boundary conditions (46) can be written as ∂m± ∂z+ξm±/vextendsingle/vextendsingle/vextendsingle/vextendsingle ∂V=0(z=−D/2) (A5) and −∂m± ∂z+ξm±/vextendsingle/vextendsingle/vextendsingle/vextendsingle ∂V=0(z=D/2), (A6) where ξis the arbitrary constant pinning parameter. In accordance with the form of Eqs. (A1) and(A2) , we find the magnetic moment density m±and the potential ϕover the film thickness z∈[−D/2,D/2] in the form m±(z)=A±exp(iqzz)+B±exp(−iqzz), (A7) ϕ(z)=Cexp(iqzz)+Dexp(−iqzz). (A8) Taking into account Eq. (A2) , the potential ϕin the external region of the film is given as ϕ(z)=Eexp(qz)(z<−D/2), (A9) ϕ(z)=Fexp(−qz)(z>D / 2). (A10) Relations (A7) ,(A8) ,(A9) , and (A10) contain eight unknown variables A±,B±,C,D,E, and F. In order to find these variables, we have eight equations: two equations (A3) at boundaries z=D/2 and −D/2, two equations (A4) atz= D/2 and −D/2, and four equations (A5) and (A6) .T h e substitution of ϕ(A8) ,(A9) , and (A10) in Eqs. (A3) and(A4) gives the relation for the transverse wave vector qz: 2 cotqzD=qz q−q qz. (A11) From Eqs. (A5) and(A6) and the form (A7) , we obtain the relation 2 cotqzD=qz ξ−ξ qz. (A12) Thus we have different relations (A11) and(A12) for deter- mination qz. In the common case, for given q,D, and the constant ξ, simultaneous solvability of Eqs. (A11) and(A12) is impossible and there is no solution of qz. 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PhysRevB.85.224408.pdf

PHYSICAL REVIEW B 85, 224408 (2012) Interference-mediated modulation of spin waves Sankha Subhra Mukherjee,1Jae Hyun Kwon,1Mahdi Jamali,1Masamitsu Hayashi,2and Hyunsoo Yang1,* 1Department of Electrical and Computer Engineering and NUSNNI-NanoCore, National University of Singapore, 117576, Singapore 2National Institute for Materials Science, Tsukuba 305-0047, Japan (Received 18 June 2011; revised manuscript received 16 April 2012; published 11 June 2012) The modulation of propagating spin wave amplitude in Ni 81Fe19(Py) films, resulting from constructive and destructive interference of spin wave, has been demonstrated. Spin waves were excited and detected inductivelyusing pulse-inductive time domain measurements. Two electrical impulses were used for launching two interferingGaussian spin wave packets in Py films. The applied bias magnetic field or the separation between two pulseswas used for tuning the amplitude of the resulting spin wave packets. This may thus be useful for spin wavebased low-power information transfer and processing. DOI: 10.1103/PhysRevB.85.224408 PACS number(s): 75 .30.Ds, 85 .75.−d I. INTRODUCTION Spin waves have been identified as promising candidates for information transfer,1,2quantum3and classical4–8information processing, control of THz dynamics,9and phase matching of spin-torque oscillators.10Spin waves form the basis for spin pumping11and have been used for the explanation12of the spin Seebeck effect.13Information transfer via spin waves does not suffer from phonon-mediated joule heating in the same wayas charge transfer in the diffusive transport regime. However,in metallic systems such as permalloy, attenuation of spinwaves is significant, and, hence for applications involving spinwaves in such systems, a method of spin wave amplificationwould be greatly beneficial. Previously, nonlinear parametricpumping 14has been used for amplifying spin wave signals. However, in these methods, significant care has to be taken tomake sure that the frequency of the pumping signal is preciselytwice that of the signal that needs to be amplified. This allowsonly a single frequency to be amplified at any time. Also, thecircuitry involved in the amplification process, such as an opendielectric resonator, may become prohibitively complicatedfor most applications. Furthermore, since the amplificationprocess is inherently nonlinear, extra spurious frequenciesare produced, which might adversely affect the usefulness ofthe spin waves in various applications, such as in spin wavecircuits. Recently, amplification has also been achieved bythermal-spin transfer torque in yttrium iron garnet (YIG). 15 Although this is a significant scientific demonstration, it isstill not the most practical method of achieving amplification. In this paper a method of spin wave amplitude modulation is presented by the linear superposition of spin waves. Resonantexcitation of spin dynamics has previously been exploited forreducing the power requirements of current-driven domainwall motion by Thomas et al. 16and for spin transfer torque (STT)-induced switching of magnetic tunnel junction (MTJ)devices by Garzon et al. 17We use interfering spin waves re- sulting from two closely spaced voltage impulses for the mod-ulation of the magnitude of the resultant spin wave packets. Al-though spin wave interference has been studied in theory 18and simulations,19,20demonstrated in optical measurements,21,22 and utilized for the generation of phase-shift keying signals23 previously, there is little study about quantitative time- domain electrical measurements of spin wave interference.We demonstrate how the applied bias magnetic field orthe interval between two adjacent pulses can be effectively used for the amplification and attenuation of spin wavesignals. II. EXPERIMENTAL DETAILS Figure 1(a) shows the optical micrograph of the device used for studying spin wave amplitude modulation. A 150 μm× 40μm×20 nm Ni 81Fe19(Py) strip was patterned on a Si/SiO 2 (100 nm) substrate. A 30-nm SiO 2layer was sputter deposited on top of the Py layer, and, subsequently, Ta (5 nm)/Au (85 nm)was sputter deposited and patterned into asymmetric coplanarstrips (ACPS). The distance between the source lines of theexcitation and detection ACPS is 10 μm. The width of the signal and ground arms of the ACPS is 10 μm and 30 μm, respectively, and the distance between the two is 5 μm. V oltage pulses applied at one of the waveguides launch a Gaussian spinwave packet 24,25and may be inductively detected by the other waveguide. V oltage pulses were applied by an Agilent 81134Apulse generator, and a Tektronix DPO 70604B real-timeoscilloscope was used for measuring the inductive voltagegenerated at the detection waveguide. A 20 dB low noiseamplifier was used for the amplification of the output signal.The output signals were averaged 10 000 times to improve thesignal-to-noise ratio. During the measurement, an out-of-planebias magnetic field ( H b) was applied. The signal obtained for no applied bias field is used as the background signal and issubtracted from the signals obtained at all other bias fieldsin order to obtain clean spin wave packets at each bias field.The frequencies of the resultant signals were calculated byfast Fourier transform (FFT) of the measured time domainsignal. To confirm that the measured signals were indeed spinwaves, the relationship between the frequency of the measuredsignals resulting from a single pulse excitation and the appliedbias field is shown in Fig. 1(b) and shows a distinct change with applied bias field, as has been shown in other reports onspin waves. 24,26The dependence is approximated by a second degree polynomial, shown by the red solid line in Fig. 1(b) . This frequency dependence of the spin waves with the biasmagnetic field is used for all subsequent calculations. Forthe study of the interference, Gaussian spin wave packetsgenerated from one and two pulse excitations have beenstudied. 224408-1 1098-0121/2012/85(22)/224408(6) ©2012 American Physical SocietyMUKHERJEE, KWON, JAMALI, HA Y ASHI, AND Y ANG PHYSICAL REVIEW B 85, 224408 (2012) III. SPIN WA VE INTERFERENCE MODEL The precession frequency ( ω), which has been previ- ously fitted with a second degree polynomial as men-tioned previously, wave vector ( k), and the group veloc- ityv g(dω/dk ) of the spin wave packets are a function ofHb. A Gaussian spin wave packet may be written as fG(t)=Aexp[−(t−tp)2/2σ2] cos(kx−ωt+φ), where A is the field- and position-dependent amplitude of the Gaussianwave packet, t pis the temporal position of the peak of the Gaussian wave packet, σis the field- and position-dependent standard deviation of the Gaussian wave packet, and φis the phase of the sinusoidal signal. The phase φis assumed to be constant in wave packets generated at different times.A signal excited at t 1may be written as fG(t−t1). Spin wave packets originating temporally close to one anotherinterfere linearly when the applied excitation is in the linearregime as f Tot(t)=fG(t)+fG(t−t1). When the phases of the sinusoidal components in the neighboring spin wave packetsmatch, the waves constructively interfere, while when they areout-of-phase, they destructively interfere. Thus, conditions ofconstructive and destructive interference may be obtained byfinding phase relationships between the sinusoidal parts ofthe wavefunction alone, temporarily neglecting the nonlinearGaussian dependencies. From simple trigonometric relations,it is possible to obtain the resultant interference proportionalityconstant as f Tot(t)∝|2 cos(0 .5ωt1)|. The relationship between the bias field and the frequency has been already measured andfitted in Fig. 1(b) . Thus, for the proper separation between two consecutive input pulses, one should be able to obtain bothconstructive as well as destructive interference over a range ofapplied bias field. In order to obtain destructive interference at2.5 GHz, for example, t 1should have a value of (2 πf t 1=π) 200 ps. However, due to the Gaussian envelopes, it is difficultto obtain an analytical expression for interference, and hencea numerical solution has been sought. In this work numericalsolutions have been compared with the measured data. IV . EXPERIMENTAL RESULTS AND ANALYSIS A. Experimental data at different bias fields A single-pulse excitation has a pulse width ( t0) of 100 ps (in the pulse mode) and a voltage of 2 V . A double-pulse excitationis two single-pulse excitations separated by a time period ( t 1) of 200 ps, created by combining two 100 ps signals from twochannels with a combiner. These are shown at the center ofFig. 1. Measured spin waves resulting from the single-pulse excitation at a bias field of −2.46 kOe is shown in Fig. 1(c) in red, and a simulated Gaussian wave packet is shown in blue.When two wave packets generated 200 ps apart interfere atthat particular field, they destructively interfere. The measuredvalue of this interference is shown in Fig. 1(d) by a green solid line. The result of a simulated interference between Gaussianpackets 200 ps apart is shown by a blue solid line in Fig. 1(d) . There is significant similarity between the simulated and themeasured signals. The simulated signal comprises two smallenvelopes and is zero at the center (marked by t c). This is the point at which the magnitudes of the Gaussian wave packetsexactly cancel each other and thus becomes zero. This pointis 100 ps from the center of either of the Gaussian wave FIG. 1. (Color online) (a) An optical micrograph of the device used for the inductive measurements of spin waves comprising a Py strip and ACPS patterned on top of it. (b) The FFT of resultant spinwaves as a function of applied bias field. Schematic representations of input excitations have been shown below (a) and (b). These pulses are not to scale. (c) Measured (red line) and simulated (blue line)signals for spin wave packets resulting from a single-pulse excitation at−2.46 kOe. (d) Measured (green line) and simulated (blue line) signals for spin wave packets resulting from a double-pulse excitationat−2.46 kOe, showing destructive interference. (e) Measured (red line) and simulated (blue line) signals for spin wave packets resulting from a single-pulse excitation at −3.5 kOe. (f) Measured (green line) and simulated (blue line) signals for spin wave packets resulting from a double-pulse excitation at −3.5 kOe, showing constructive interference. packets, leading the center of one of the wave packets and trailing the other. At t>t c, the spin wave packet launched at a later time has a larger amplitude than that launched earlier.Hence, the characteristics of the interference pattern beyondt care that of the spin wave packet launched later. Similarly, att<t c, the spin wave packet launched at an earlier time has the larger amplitude, and hence, the characteristics of theinterference pattern before t ccorrespond to that of the earlier spin wave. During destructive interference, the spin wavepackets are out-of-phase by πwith respect to one another. Therefore, as the characteristics of the resultant spin wavepacket after interference changes from one wave packet beforet cto another after tc, there is an abrupt phase change of πin the resultant wave packet at tc. The measured signal, shown by the green solid line in Fig. 1(d) , is characteristically similar to 224408-2INTERFERENCE-MEDIA TED MODULA TION OF SPIN W A VES PHYSICAL REVIEW B 85, 224408 (2012) the simulated signal. The two envelopes are separated at the center by complete destructive interference by a πphase shift, which is a clear indication of destructive interference. At−3.5 kOe, the resulting spin wave signal from a single- pulse excitation is shown by a red solid line in Fig. 1(e) , along with a simulated result for the same field shown by a solidblue line. At this field, the wave packets originating from thedouble-pulse excitation constructively interfere, and the resultis shown in Fig. 1(f). A blue solid line shows the simulated result, while a green solid line shows the measured data. Forconstructive interference, the wave packets are in-phase, and asa result, there is no abrupt phase change in the resultant signal.Furthermore, the total amplitude of the resultant interferenceis greater than that resulting from a single pulse. Unfortunately, simple Gaussian wave packets cannot be used for obtaining very accurate descriptions of the interfer-ence, especially in the low bias field regions. This is becausethe Gaussian pulses are created with rectangular pulses andare actually composed of two Gaussian wave packets, oneresulting from the rising edge of the pulse and another fromthe falling edge of the pulse, and hence the initial rise ofthe Gaussian wave packets is more abrupt than the trailingedges. 27A better description of the wave amplitudes at low fields may be obtained by taking the frequency transformof the two pulses directly. This gives additional insights intothe method in which constructive and destructive interferenceintensities may be calculated. It is known that the Fouriertransform of a rectangular pulse with a pulse width t 0isy1= sin(ωt0/2)/(ωt0/2). It is also known that the Fourier transform of two pulses separated from one another by t1isy3=y1×y2, where y2=[1+exp(−jωt 1)]. Calculated values of |y1|,|y2|, and|y3|are plotted as a function of frequency in Fig. 2,f o rt0= 100 ps and t1=200 ps. It is worth noting that the frequency characteristics of y1depend upon t0alone and that of y2depend upont1alone. Hence, effective independent control of both attenuation and amplification frequencies may be obtained. FIG. 2. (Color online) Calculated values of the frequency compo- nents of two similar pulses separated in time. The frequency compo- nents of a single pulse, y1(green dashed-dotted line); the frequency components of the two impulses separated one from the other, y2 (blue dashed line); and the product of the two, yielding the fre- quency components of two rectangular pulses separated from one another, y3(red solid line). FIG. 3. (Color online) (a) Contour plots of the spin wave signal and (b) the FFT of the time-domain signal due to a single-pulseexcitation. (c) Contour plots of the spin wave signal and (d) the FFT of the time-domain signal due to a double-pulse excitation. The scale bar is in mV . The contour plot of measured spin wave packets originating from a single 100-ps pulse is shown in Fig. 3(a) . The frequency of the measured signals increases with the magnetic field,and temporal widths between two subsequent peaks becomesmaller. In Fig. 3(b) the FFT of the time-domain signal shown in Fig. 3(a) is plotted. Spin waves originating from two 100 ps voltage pulses separated from each other by 100 ps(i.e.,t 1=200 ps) are plotted in Fig. 3(c) . The FFT of the time-domain signals resulting from two pulses is shown inFig. 3(d) . At bias fields above 3 kOe, one is clearly able to see an enhancement in the signal levels in comparisonwith spin wave signals arising due to the single pulseexcitation. B. Numerical analysis The magnitude mi(Hb) of the signal level originating from one- (i=1) and two-pulse ( i=2) excitations at a particular magnetic field is calculated as the difference between the max-imum and the minimum value of the measured signal at thatmagnetic field. This is plotted as a function of the bias field forsignals obtained for the single- and double-pulse excitationsin Fig. 4(a) . Note that the magnitude of the measured signal is dependent upon the excitation efficiency of the particularwaveguides that have been used, resulting in the change inthe intensity of m i(Hb), as shown. For bias fields less than 3 kOe, the magnitude of the spin wave packets due to the singlepulse is greater than that of double pulses. However, for biasfields between 3 kOe and 4.6 kOe, the magnitude of the spinwave packets due to double pulses constructively interfere,and the resultant magnitude become greater than that due toa single pulse. For comparing the effect of the interferencein the spin wave amplitude, the magnitude of the Gaussianwave packets originating from double-pulse excitations isnormalized by those originating from single-pulse excitations 224408-3MUKHERJEE, KWON, JAMALI, HA Y ASHI, AND Y ANG PHYSICAL REVIEW B 85, 224408 (2012) FIG. 4. (Color online) (a) The magnitude of a spin wave signal measured as the difference between the maximum and the minimumvalue of the signal at a particular bias field is plotted, using open squares for the single-pulse excitation and open circles for double- pulse excitations. (b) m 2/m 1(open squares), y3(thin blue line), and fTot(Hb) (thick red line) are plotted as a function of bias field. as [m2(Hb)/m 1(Hb)] and is plotted in Fig. 4(b) as open squares. In the same figure the result obtained from numericalanalysis is also plotted as a thick red solid line and shows areasonably good agreement with experiment. Figure 4shows that the magnitude of the signal due to interference changesregularly over the magnetic field. The largest increase in signalamplitude predicted by simulation is twofold. The value of|y 3|is also plotted as a function of the applied field as a thin blue solid line and, as discussed previously, is better atestimating the value of the interference at smaller bias fields.Measured signals are slightly larger than simulated signals atlarge values of bias fields, probably due to nonlinear mixing.This allows for the field-dependent control of the magnitudeof the spin wave signal and, hence, can be used as a spin wavemodulator. C. Experimental data at different pulse separations It is also important to note that the concept of the electrical modulation of spin waves using two subsequent pulses isgeneral and can be applied when the external bias field isapplied in another direction. For example, it is possible toapply an in-plane bias field along the signal line and as a resultobtain the surface-mode spin wave transport at much lowerfields. Neither is the bias field the sole parameter responsiblefor the generation of interference. The separation between twopulses is also a very effective way for tuning the modulationresulting from the interference. For demonstrating this phenomenon, a fixed in-plane bias field of 41 Oe is applied along the direction of the signal line,and the separation between two pulses is varied from −5t o 5 ns in steps of 20 ps. Both “unipolar” and “bipolar” pulses areused. Unipolar pulses are composed of two consecutive pulseshaving the same polarity, while bipolar pulses are composedof two consecutive pulses having opposite polarity. Schematic FIG. 5. (Color online) (a) The time-domain spin wave voltages measured for two bipolar input pulses separated from each other ( tδ) by−200 ps, 0 ns, and −5 ns. A schematic representation of unipolar and bipolar input pulses is shown. (b) The contour plot of measured spin wave signals as a function of tδclearly shows an interference pattern resulting from bipolar pulses. (c) and (d) represent the samemeasurements depicted in (a) and (b), respectively, resulting from unipolar pulses. (e) The amplitude of the measured spin wave signals is seen to be modulated from its value of 8.97 mV due to interferencefor both unipolar (blue/medium gray) and bipolar (red/dark gray) pulses. representations of both unipolar and bipolar pulses are shown in Fig. 5. Two 100 ps bipolar pulses separated from one another by 5 ns are applied to one of the ACPS and resultsin the generation of two Gaussian wave packets 5 ns apart, asshown in the lowest line plot in Fig. 5(a) . When the pulses are separated from each other by 0 ns, they destructively interfere,while when they are separated by 200 ps, they constructivelyinterfere. The contour plot of all measurements is shown inFig. 5(b) in the case of bipolar pulses. The interference of the two wave packets is clearly visible at the center of thecontour plot. Measurement data corresponding to unipolarpulses are shown in Figs. 5(c) and 5(d) , respectively. As can be seen, there is πphase shift in the interference output signals for unipolar pulses compared to the bipolar case.For example, they destructively (constructively) interfere forunipolar (bipolar) pulses for a t δof±200 ps. To quantify the modulation, the magnitude of the peak to peak amplitudeV Amp is plotted as a function of tδin Fig. 5(e) for bipolar (unipolar) pulses in red/dark gray (blue/medium gray). As canbe seen, the nominal amplitude of 8.97 mV , correspondingto a noninteracting wave packet, changes between 17.73 mVand 0.63 mV due to constructive and destructive interference,respectively. D. Micromagnetic simulations To better understand this behavior, we have performed micromagnetic simulations. The structure that we have usedin our simulations is 6 μm in length, 4.4 μm in width, and 20 nm in thickness to preserve the aspect ratio of lengthover width of the actual sample. The simulation cell size is10×10×20 nm 3and is made of Permalloy (Py), having a saturation magnetization ( MS) of 860 ×103A/m, an 224408-4INTERFERENCE-MEDIA TED MODULA TION OF SPIN W A VES PHYSICAL REVIEW B 85, 224408 (2012) FIG. 6. (Color online) (a) Simulated spin waves resulting from two bipolar pulses applied at a time difference ( tδ) of 200, 100, 0, −100, and −200 ps from one another, showing constructive and destructive interference patterns. (b) A contour plot of the interferenceof spin wave packets from two bipolar pulses as one of the inputs is shifted from the other by t δ. (c) Simulated spin waves resulting from two unipolar pulses applied at tδ. (d) A contour plot of the interference of spin wave packets from two unipolar pulses. The color bar is in arbitrary units. exchange stiffness ( Aex)o f1 . 3 ×10−11J/m, and a Gilbert damping constant ( α) of 0.01. We have used the object oriented micromagnetic framework (OOMMF) code for simulationsthat solves the Landau-Lifshitz-Gilbert (LLG) equation. 28,29 In order to generate spin waves, a pulse magnetic field with rise and fall times of 60 ps and a pulse width of 80 ps wasapplied to a 20 ×4400×20 nm 3volume at the center of the Permalloy film, and the spin waves were measured 1.5 μm away from the excitation source. A bias magnetic field of 200Oe was applied to the sample along the Permalloy width duringthe simulations. We have performed the simulations for two pulses with op- posite voltage polarities with different time intervals betweenthe two pulses. As can be seen in Fig. 6(a) ,f o r±100 ps time interval between the two pulses, we have constructiveinterference between the two spin wave packets generatedby the two pulses, while for ±200 ps one can observe destructive interference between these spin wave packets.When 2 πt δ·f=(2n)π, where tδis the time interval, fis thespin wave frequency, and nis an integer number, a destructive interference pattern results from the two spin wave packets.When 2 πt δ·f=(2n+1)π, a constructive interference is observed between the spin wave packets. In Fig. 6(b) we have simulated the spin wave profile for various time intervalsbetween the two pulses at a constant bias field of 200 Oe.Clear constructive and destructive interference patterns areobservable depending upon the phase difference between thetwo spin wave packets. Furthermore, for well separated pulses(t δ>1 ns), the two spin wave packets propagate independently from one another. We have also performed the simulationsfor two pulses with the same voltage polarities, as shownin Figs. 6(c) and 6(d) . In contrast to the case for pulses with opposite voltage polarities, constructive interference isobserved for 2 πt δf=(2n)π, while one can observe a destructive interference pattern when 2 πtδf=(2n+1)π, and is consistent with the measurement results. V . CONCLUSION The spin wave amplitude modulation either by controlling the bias field or the separation of two pulses has been electri-cally demonstrated using spin wave interference. Constructiveand destructive interference of spin wave has been utilized inPy films by the linear superposition of two spin waves. Bothnumerical calculation and micromagnetic simulations showgood agreement with the experimental data. The concept ofthe electrical modulation of spin waves using two successivepulses is general and can be applied to various spin wavemodes. This work lays the foundation for energy efficientinformation transfer as well as information processing inmagnonic systems. ACKNOWLEDGMENTS S.S.M. and J.H.K. contributed equally to this work. This work is supported by the Singapore National ResearchFoundation under CRP Award No. NRF-CRP 4-2008-06 andpartly supported by Grant-in-Aid for Scientific Research (No.22760015) from MEXT, Japan. *Corresponding author: eleyang@nus.edu.sg 1M. Murphy, S. Montangero, V . Giovannetti, and T. Calarco, Phys. Rev. A 82, 022318 (2010). 2Y . Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi,S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 3A. V . Gorshkov, J. Otterbach, E. Demler, M. Fleischhauer, and M. D. Lukin, Phys. Rev. Lett. 105, 060502 (2010). 4T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P . Kostylev, Appl. Phys. Lett. 92, 022505 (2008). 5K.-S. Lee and S.-K. Kim, J. Appl. Phys. 104, 053909 (2008). 6U.-H. 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PhysRevB.99.144411.pdf

PHYSICAL REVIEW B 99, 144411 (2019) Microscopic theory of spin transport at the interface between a superconductor and a ferromagnetic insulator T. Kato,1Y . Ohnuma,2M. Matsuo,2J. Rech,3T. Jonckheere,3and T. Martin3 1Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan 2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 3Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France (Received 8 January 2019; revised manuscript received 28 March 2019; published 15 April 2019) We theoretically investigate spin transport at the interface between a ferromagnetic insulator (FI) and a superconductor (SC). Considering a simple FI-SC interface model, we derive formulas for the spin currentand spin-current noise induced by microwave irradiation (spin pumping) or the temperature gradient (the spinSeebeck effect). We show how the superconducting coherence factor affects the temperature dependence of thespin current. We also calculate the spin-current noise in thermal equilibrium and in nonequilibrium states inducedby the spin pumping and compare them quantitatively for an yttrium iron garnet–NbN interface. DOI: 10.1103/PhysRevB.99.144411 I. INTRODUCTION Spin transport in hybrid systems composed of supercon- ductors (SCs) and ferromagnetic metals has been investigatedfor a long time [ 1–4]. In a superconductor, charge and spin imbalances may have different characteristic length scales due to spin-charge separation [ 5–7]. The interplay between superconductivity and magnetism also offers the potential for novel spintronic devices, in which fast logic operation can beperformed with minimum Joule heating [ 8]. One of the key ingredients there is the injection of spin-polarized carriers into SCs [ 7,9–12]. For conventional s-wave superconductors, spin injection is suppressed by opening a superconducting gap in the electronic spectrum. Thermally excited quasiparticles inSC, however, can carry a spin current over long distances, as spin excitations in SCs have long lifetimes [ 13–18]. There are several techniques for spin injection into SCs. Recently, it has been realized by taking advantage of thespin Seebeck effect (SSE) induced by a temperature gradient [19–25] or by applying a spin-pumping (SP) protocol using ferromagnetic resonance (FMR) under microwave irradiation[26–30]. The latter technique has successfully been used in experiments to realize spin injection from ferromagneticmetals into a SC [ 31–33]. These recent advances indicate a new path for spin injection into a wide class of SC materials.Remarkably, spin-current injection from a ferromagnetic in-sulator (FI) into a superconductor has also been performed re-cently [ 34], as revealed by the inverse spin Hall effect (ISHE) [35–37]. This last study opens up possible applications for novel superconducting spintronic devices using FI. In contrast to progress in experiments, the spin current at the FI-SC interface has been studied theoretically, to our knowledge, only by Inoue et al. [38]. They formulated the spin-pumping signal in terms of the local spin susceptibilityof the SC and showed that the signal is peaked below thetransition temperature due to the coherence factor in the BCStheory. In order to calculate the local spin susceptibility of theSC, they employed the Abrikosov-Gor’kov theory for dirtySCs taking spin diffusion into account [ 39–41]. The dynamic spin susceptibility thus obtained is, however, correct only forsmall wave numbers, whereas the local spin susceptibility,which involves all wave numbers, is dominated by the largewave number contribution [ 42] (for details, see Appendix A). Therefore, although their discovery of the coherence peak inspin transport is remarkable, their theory is expected to beinsufficient for a quantitative description of the spin-currentgeneration. In this paper, we consider a bilayer system composed of an s-wave singlet SC and a FI, as shown in Fig. 1. We formulate the spin current at the interface and study its temperaturedependence above and below the superconducting transitiontemperature. We also discuss the noise power of the pure spincurrent following the theory developed by three of the presentauthors and one collaborator [ 43] and estimate it using the experimental parameters for the yttrium iron garnet(YIG)–NbN interface [ 29,34]. This paper is organized as follows. We introduce the model for the FI-SC interface in Sec. IIand derive dynamic spin sus- ceptibilities in Sec. III. By using a second-order perturbative expansion with respect to the interface exchange coupling,we calculate the spin current and the spin-current noise inSecs. IVand V, respectively. It should be stressed that we evaluate the spin current just at the FI-SC interface. For ex-perimental detection, one needs a nanostructure for convertingthe spin current into an electronic response, a mechanismwhich depends, in general, on details of spin relaxation in thesuperconductor. We briefly discuss such a possible experimen-tal setup for detecting the spin current in Sec. VI. Finally, we summarize our results in Sec. VII. Detailed discussions of the impurity effect and the spin susceptibility of the SC are givenin Appendixes AandB, respectively. II. MODEL The system Hamiltonian is given by H=HSC+HFI+ Hex. The first term, HSC, describes a bulk SC and is given by 2469-9950/2019/99(14)/144411(8) 144411-1 ©2019 American Physical SocietyT. KATO et al. PHYSICAL REVIEW B 99, 144411 (2019) FIG. 1. Schematic picture of the FI-SC bilayer system. A spin current ISis generated in the SC by spin pumping using an external microwave irradiation or by the spin Seebeck effect induced by a temperature gradient ( TFI/negationslash=TSC). The large green arrow in the FI illustrates the magnetization, which can precess due to the applied microwave at frequency /Omega1. The arrows in the SC show an example of electron reflection at the interface, with a spin flip due to theexchange interaction. the mean-field Hamiltonian HSC=/summationdisplay k(c† k↑,c−k↓)/parenleftbiggξk/Delta1 /Delta1−ξk/parenrightbigg/parenleftBigg ck↑ c† −k↓/parenrightBigg , (1) where ckσ(c† kσ) is the annihilation (creation) operator of the electrons in the superconductors and ξkis the energy of conduction electrons measured from the chemical potential.The order parameter of the SC /Delta1is determined by the gap equation ln/parenleftbiggT Tc/parenrightbigg /Delta1=2πT/summationdisplay εn>0/parenleftBigg /Delta1/radicalbig ε2n+/Delta12−/Delta1 /epsilon1n/parenrightBigg , (2) where εn=(2n+1)πTis the Matsubara frequency and Tcis the SC transition temperature [ 38]. The second term HFIdescribes a bulk FI and is given by the Heisenberg model HFI=/summationdisplay /angbracketlefti,j/angbracketrightJijSi·Sj−¯hγhdc/summationdisplay iSz i −¯hγhac 2/summationdisplay i(e−i/Omega1tS− i+ei/Omega1tS+ i), (3) where Siis the localized spin at site iin the FI, Jijis the exchange interaction, hdcis a static magnetic field, hacand /Omega1are the amplitude and frequency of the applied microwave radiation, respectively, and γis the gyromagnetic ratio. Using the Holstein-Primakoff transformation [ 44] and employing the spin-wave approximation ( Sz j=S0−b† jbj,S+ j/similarequal(2S0)1/2bj), the Hamiltonian of the FI is rewritten as HFI/similarequalconst+/summationdisplay k¯hωkb† kbk −¯hγhac 2/radicalbig 2S0NF(e−i/Omega1tb† k=0+ei/Omega1tbk=0), (4) where ¯ hωkis the magnon dispersion, bkis the Fourier trans- form of bj,S0is the magnitude of the localized spin, and NF is the number of spins in the FI. For simplicity, we assumea parabolic dispersion ¯ hωk=Dk2+E0, where E0=¯hγhdcis the Zeeman energy. The last term in the system Hamiltonian, Hex, describes the exchange coupling at the interface. In this paper, we employ asimple model using the following tunneling Hamiltonian forspins: H ex=/summationdisplay k,q[Tk,qS+ ks− q+H.c.], (5) where Tk,qis the tunneling amplitude, S+ k=(2S0)1/2bk, and s− qis an operator defined as s− q:=/summationdisplay kc† k↓ck+q↑. (6) In what follows, we study the spin transport by considering a second-order perturbative expansion with respect to Hex. III. DYNAMIC SPIN SUSCEPTIBILITY In this section, we summarize the results for the dynamic spin susceptibilities for the unperturbed system, i.e., the de-coupled FI and SC, which are later used in the second-orderperturbation calculation of the spin current and spin-currentnoise. A. Retarded component We define the retarded components of the spin susceptibil- ity of the SC and the magnon propagator in the FI as χR(q,t):=i(¯hNS)−1θ(t)/angbracketleft[s+ q(t),s− q(0)]/angbracketright, (7) GR(k,t):=−i¯h−1θ(t)/angbracketleft[S+ k(t),S− k(0)]/angbracketright, (8) where NSis the number of unit cells in the SC. Their Fourier transformations are defined as χR(q,ω):=/integraldisplay∞ −∞dt eiωtχR(q,t), (9) GR(k,ω):=/integraldisplay∞ −∞dt eiωtGR(k,t). (10) We first consider the magnon propagator of the FI. By using the Holstein-Primakoff transformation [ 44] and em- ploying the spin-wave approximation [ S+ k/similarequal(2S0)1/2bk], the magnon propagator in the FI is calculated in the absence ofthe external field ( h ac=0) as GR(k,ω)=2S0/¯h ω−ωk+iαω, (11) where we have introduced the phenomenological dimension- less damping parameter α, which originates from the Gilbert damping. Next, we consider the dynamic spin susceptibility of the SC in the BCS theory. We define the local spin susceptibilityas χ R loc(ω):=1 NS/summationdisplay qχR(q,ω). (12) 144411-2MICROSCOPIC THEORY OF SPIN TRANSPORT AT THE … PHYSICAL REVIEW B 99, 144411 (2019) In the BCS theory, the local spin susceptibility is calculated as [45] ImχR loc(ω)=−πN(/epsilon1F)2/integraldisplay dE/bracketleftbigg 1+/Delta12 E(E+¯hω)/bracketrightbigg ×[f(E+¯hω)−f(E)]D(E)D(E+¯hω),(13) D(E)=|E|√ E2−/Delta12θ(E2−/Delta12), (14) where N(/epsilon1F) is the density of states per spin and per unit cell, f(E)=(eE/kBT+1)−1is the Fermi distribution func- tion, D(E) is the (normalized) density of states of quasi- particles, and θ(x) is the Heaviside step function (for a de- tailed derivation, see Appendix B). We note that the factor [1+/Delta12/E(E+¯hω)] in Eq. ( 13) is the so-called coherence factor, which produces singular behavior near the transitiontemperature [ 45]. For the normal metal ( /Delta1=0), the local spin susceptibility becomes Imχ R loc,/Delta1=0(ω)=πN(/epsilon1F)2¯hω. (15) B. Lesser component We define the lesser components of the spin susceptibilities for bulk SC and FI as χ<(q,t):=i(¯hNS)−1/angbracketlefts− q(0)s+ q(t)/angbracketright, (16) G<(k,t):=−i¯h−1/angbracketleftS− k(0)S+ k(t)/angbracketright. (17) Their Fourier transformations are defined as χ<(q,ω)=/integraldisplay∞ −∞dteiωtχ<(q,t), (18) G<(k,ω)=/integraldisplay∞ −∞dteiωtG<(k,t). (19) The lesser components include the information on the distri- bution function; we define the distribution functions as fSC(q,ω):=χ<(q,ω)/(2i)ImχR(q,ω), (20) fFI(k,ω):=G<(k,ω)/(2i)ImGR(k,ω). (21) In the setup of the SP, the SC is in equilibrium with the temperature T, whereas magnons in FI are excited by the external microwave irradiation. We split the Hamiltonian ofthe FI as H FI=H0+V, where H0=/summationdisplay k¯hωkb† kbk, (22) V=−h+ ac(t)b† 0−h− ac(t)b0, (23) h± ac(t)=¯hγhac 2/radicalbig 2S0NFe∓i/Omega1t. (24) While the perturbation Vdoes not change the retarded com- ponent of the dynamic spin susceptibility of FI, it does modifythe lesser component. The second-order perturbation withrespect to Vgives the correction: δG <(k,ω)=GR 0(k,ω)/Sigma1(k,ω)GA 0(k,ω), (25) /Sigma1(k,ω)=δk,0/integraldisplay dt(−i¯h−1)/angbracketlefth− ac(t)h+ ac(0)/angbracketrighteiωt, (26)where GR 0(k,ω) is the unperturbed spin susceptibility of FI. One can then straightforwardly obtain δfFI(k,ω)=δG<(k,ω)/(2i)ImGR 0(k,ω) =2πNFS0(γhac/2)2 αωδk,0δ(ω−/Omega1). (27) In the setup of the SSE, FI and SC are in equilibrium with temperatures TFIandTSC, respectively. Using their Lehmann representation, we can prove the relations [ 46,47] χ<(q,ω)=2iImχR(q,ω)nB(ω,TSC), (28) G<(k,ω)=2iImGR(k,ω)nB(ω,TFI), (29) where nB(ω,T) is the Bose distribution function defined as nB(ω,T)=1 e¯hω/kBT−1. (30) This result leads to the distribution functions of the FI and the SC [defined in Eqs. ( 20) and ( 21)] as fSC(q,ω)=nB(ω,TSC), (31) fFI(k,ω)=nB(ω,TFI). (32) IV . SPIN CURRENT A. Formulation The spin current at the SC-FI interface is defined by /angbracketleftˆIS/angbracketright, where /angbracketleft ···/angbracketright denotes the statistical average and ˆISis the operator for the spin current flowing from the SC to the FI,defined as ˆI S:=− ¯h∂t/parenleftbig sz tot/parenrightbig =i[sz tot,H], (33) sz tot:=1 2/summationdisplay k(c† k↑ck↑−c† k↓ck↓). (34) By substituting the expression for the system Hamiltonian, we obtain ˆIS=−i/summationdisplay k,q(Tk,qS+ ks− q−H.c.). (35) We consider the second-order perturbation calculation by taking HFI+HSCas an unperturbed Hamiltonian and Hexas a perturbation. The average of the spin-current operator iswritten as /angbracketleftˆI S/angbracketright=Re⎡ ⎣−2i/summationdisplay k,qTk,q/angbracketlefts− qS+ k/angbracketright⎤ ⎦ =lim t1,t2→0Re⎡ ⎣−2i/summationdisplay k,qTk,q/angbracketlefts− q(t2)S+ k(t1)/angbracketright⎤ ⎦, (36) where the average /angbracketleft ···/angbracketright is taken for the full Hamiltonian. By using the formal expression of perturbation expansion, the 144411-3T. KATO et al. PHYSICAL REVIEW B 99, 144411 (2019) time FIG. 2. The Keldysh contour C. spin current can be rewritten as [ 47,48] /angbracketleftˆIS/angbracketright=Re⎡ ⎣−2i/summationdisplay k,qTk,q/angbracketleftbigg TKs− q(τ2)S+ k(τ1) ×exp/parenleftbigg −i ¯h/integraldisplay CdτHex(τ)/parenrightbigg/angbracketrightbigg 0⎤ ⎦, (37) where the average /angbracketleft ···/angbracketright 0is now taken for the unperturbed Hamiltonian and TKis the time-ordering operator on the time variable τon the Keldysh contour C, which is composed of the forward path C+running from −∞ to∞and the backward path C−from∞to−∞ (see Fig. 2). We have put the time variables τ1andτ2on the contour C−andC+and have removed the limit operation for operator ordering. Expanding the exponential operator in Eq. ( 37) and keep- ing the lowest-order term with respect to Hex, we obtain /angbracketleftˆIS/angbracketright=−2 ¯h/integraldisplay CdτRe⎡ ⎣/summationdisplay k,q|Tk,q|2/angbracketleftTKs+ q(τ)s− q(τ2)/angbracketright0 ×/angbracketleftTKS+ k(τ1)S− k(τ)/angbracketright0/bracketrightBigg . (38) Using the real-time representation [ 46–48], we can rewrite the spin current in terms of the dynamic spin susceptibilities of FIand SC as /angbracketleftˆI S/angbracketright=− 2¯hRe/integraldisplay∞ −∞dt/summationdisplay k,q|Tk,q|2NS ×[χR(q,t)G<(k,−t)+χ<(q,t)GA(k,−t)],(39) where GA(k,t) is the advanced component. Using the defini- tions of the distribution functions and performing the Fouriertransformation for the dynamic spin susceptibilities, we obtain /angbracketleftˆI S/angbracketright=4¯h/integraldisplaydω 2π/summationdisplay k,q|Tk,q|2NSImχR(q,ω) ×[−ImGR(k,ω)][fFI(k,ω)−fSC(q,ω)].(40) Setting Tk,q=Tfor simplicity, we obtain /angbracketleftˆIS/angbracketright=¯hA/integraldisplayd(¯hω) 2π1 NSNF/summationdisplay k,qImχR(q,ω) ×[−ImGR(k,ω)][fFI(k,ω)−fSC(q,ω)], (41) where A=4|T|2N2 SNF/¯h. B. Spin pumping We first consider the case of spin pumping driven by microwave irradiation, keeping the same temperature for bothSC and FI. From Eq. ( 27), the difference of the distribution functions is given by fFI(k,ω)−fSC(q,ω)=2πNFS0(γhac/2)2 αωδk,0δ(ω−/Omega1). (42) The spin current generated by SP is then given by ISP S=¯hAg (/Omega1)I mχR loc(/Omega1), (43) g(/Omega1):=(γhacS0)2/2 (/Omega1−ω0)2+α2/Omega12, (44) where the local spin susceptibility χR loc(ω)i sg i v e nb y Eqs. ( 13) and ( 14) and ω0=γhdcis the angular frequency of the spin precession. For the normal-metal case ( /Delta1=0), we obtain for the spin current, using Eq. ( 15), ISP,N S=π¯hAg(/Omega1)N(/epsilon1F)2¯h/Omega1, (45) which is temperature independent for arbitrary values of /Omega1. We will use this expression as a normalization factor tocompare the results at finite /Delta1for various frequencies /Omega1. Before showing the results obtained in the superconducting case, we point out that in the low-frequency limit ( /Omega1→ 0), the expression for the spin current generated by SP issimilar to the one obtained when computing the nuclear spinresonance (NMR) signal [ 45]. It is known in the theory of the NMR measurement that the BCS singularity in the densityof states leads to a coherence peak below the SC transitiontemperature [ 49,50]. As a consequence, one can expect a similar coherence peak in the temperature dependence ofthe spin current at low frequency. However, the spin currentcontains more information than the NMR expression since/Omega1can be controlled arbitrarily up to high frequencies of the order of the transition temperature T c. In Fig. 3, we show the temperature dependence of the spin current induced by spin pumping. Here, the tempera-tures of both FI and SC are set to T, and the spin current is normalized by the value obtained for the normal-metalcase I SP,N S. For low excitation frequency /Omega1, the temperature dependence of ISP Sclearly shows a coherence peak below the SC transition temperature Tc, as expected. For ¯ h/Omega1< 2/Delta1(T= 0)/similarequal3.54kBTc, the spin current is strongly reduced at low temperatures [ kBT/lessmuch2/Delta1(T)] because spin-flip excitations in the SC are suppressed due to the energy gap 2 /Delta1in the one-electron excitation spectrum. As /Omega1increases, the coher- ence peak becomes insignificant, while a kink appears at thetemperature satisfying 2 /Delta1(T)=¯h/Omega1.F o r¯ h/Omega1> 2/Delta1(T=0), the spin current shows a plateau at low temperature corre-sponding to its zero-temperature value, ultimately recoveringthe normal-state value (dashed line) as ¯ h/Omega1is increased further. C. Spin Seebeck effect We now turn to the alternative technique for generating a spin current, namely, the spin Seebeck effect, which relies onthe presence of a temperature gradient between the FI and SClayers. Using Eqs. ( 31) and ( 32), the spin current induced by 144411-4MICROSCOPIC THEORY OF SPIN TRANSPORT AT THE … PHYSICAL REVIEW B 99, 144411 (2019) 0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.01.2ISSP(T) /ISSP,N T/Tc 3457.510(b)0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.51.01.5ISSP(T) /ISSP,N T/Tc0.1 0.5 11.5 2(a) FIG. 3. Temperature dependence of the spin current induced by spin pumping ISP S(T), normalized by the current obtained in the normal case ISP,N S(T), for different values of ¯ h/Omega1/Tc, as indicated near each curve. (a) shows ¯ h/Omega1/Tcfrom 0.1 to 2. (b) shows ¯ h/Omega1/Tcfrom 3.0 to 10. the spin Seebeck effect is given by ISSE S=¯hA/integraldisplayd(¯hω) 2πImχR loc(ω)/bracketleftbig −ImGR loc(ω)/bracketrightbig ×[nB(ω,TFI)−nB(ω,TSC)], (46) where GR loc(ω):=N−1 F/summationtext kGR(k,ω) is the local spin suscep- tibility in the FI. For simplicity, we consider the spin Seebeckeffect up to the linear term with respect to the temperaturedifference δT=T FI−TSC: ISSE S ISSE S,0=/integraldisplayEM E0dE D M(E)F(E)(E/2kBT)2 sinh2(E/2kBT), (47) F(E):=ImχR loc(E/¯h)/ImχR loc,/Delta1=0(E/¯h) =/integraldisplay∞ −∞dE/prime/bracketleftbigg 1+/Delta12 E/prime(E/prime+E)/bracketrightbigg ×/bracketleftbiggf(E/prime)−f(E/prime+E) E/bracketrightbigg D(E/prime)D(E/prime+E),(48) where T=TSC/similarequalTFIandISSE S,0=¯hAS 0kBδTN(/epsilon1F)2. The den- sity of states per site for a magnon is given by DM(E):=1 NF/summationdisplay kδ(E−¯hωk) =− (2πS0)−1ImGR loc(E/¯h), (49)0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.01.21.4ISSSE(T) /IS,0 T/Tc3 1 FIG. 4. Temperature dependence of the spin current in- duced by the spin Seebeck effect ISSE S, normalized by IS,0= ISSE S,0η(kBTc/EM)3/2, for SCs (solid lines) and normal metals (dashed lines) with EM/kBTc=∞ , 3, and 1 (as indicated near each curve), where EMis the high-energy cutoff of the magnon density of states. As we consider E0/lessmuchkBTc, the Zeeman energy E0is set to zero for simplicity. taking the limit α→0, and EM(/greatermuchE0) is the high-energy cut- off of the magnon dispersion relation, which is of the order ofthe exchange interaction in the FI. Under a uniform magneticfield, the local spin susceptibility is evaluated for the parabolic magnon dispersion as D M(E)=(3/2)(E−E0)1/2E−3/2 M.F o r normal metals ( /Delta1=0), the spin current at low temperatures (kBT/lessmuchEM) is given by ISSE S/ISSE S,0=η(kBT/EM)3/2, where η/similarequal6.69 is a numerical factor. In Fig. 4, we show the temperature dependence of ISSE S. The solid and dashed lines show ISSE Sfor the SC and the nor- mal metal ( /Delta1=0), respectively. For simplicity, the Zeeman energy is set to zero by assuming that it is much smaller thank BT. When EMis much larger than kBTc, the spin current monotonically decreases as the temperature is lowered. Belowthe transition temperature T c, the spin current at the FI-SC interface is suppressed due to the opening of the energy gapin the SC. When E Mis comparable to kBTc, the spin current shows a small maximum below Tcand saturates above Tc. V . SPIN-CURRENT NOISE The noise of the pure spin current has been studied for an interface between a FI and a nonmagnetic metal [ 43,51,52], as well as for several hybrid nanostructures [ 53–56]. It includes useful information on spin transport, as suggested from stud-ies of the (electronic) current noise [ 57]. In this section, we calculate the spin-current noise for the FI-SC interface. A. Formulation The noise power of the pure spin current is defined as [ 43] S:=lim T→∞1 T/integraldisplayT 0dt1/integraldisplayT 0dt21 2/angbracketleft{ˆIS(t1),ˆIS(t2)}/angbracketright, (50) where ˆIS(t):=eiHtISe−iHtand{A,B}=AB+BA.T h es p i n - current noise is calculated within the second-order perturba-tion calculation with respect to H exas S=¯h2/integraldisplay∞ −∞dω 2π/summationdisplay k,q|Tk,q|2NS[χ<(q,ω)G>(k,ω) +χ>(q,ω)G<(k,ω)]. (51) 144411-5T. KATO et al. PHYSICAL REVIEW B 99, 144411 (2019) Using Tk,q=T,E q s .( 20) and ( 21), and the relations χ>(q,ω)/(2i)ImχR(q,ω)=1+fSC(q,ω), (52) G>(k,ω)/(2i)ImGR(k,ω)=1+fFI(k,ω), (53) the spin-current noise is calculated as S=¯h2A/integraldisplayd(¯hω) 2π1 NFNS/summationdisplay k,q[−ImGR(k,ω)]ImχR(q,ω) ×[fSC(q,ω)[1+fFI(k,ω)] +[1+fSC(q,ω)]fFI(k,ω)]. (54) In the absence of both the external microwave excitation and the temperature gradient, the noise power is determinedby the equilibrium noise: S eq=2¯h2A/integraldisplay∞ −∞d(¯hω) 2πImχR loc(ω)[−ImGR loc(ω)] 4s i n h2(¯hω/2kBT).(55) Under the microwave radiation, the noise power is calculated from Eq. ( 42)a s S=Seq+SSP, (56) SSP=¯hcoth/parenleftbigg¯h/Omega1 2kBT/parenrightbigg ISP S, (57) where SSPis the nonequilibrium noise induced by spin pump- ing. While the nonequilibrium noise can similarly be inducedby SSE, we do not discuss it here as it requires us to considera large temperature gradient. B. Estimate As in the metal-FI bilayer system [ 43,51,52], the noise power of the pure spin current includes useful informationalso in the case of the SC-FI interface. At low temperatures(k BT/lessmuch¯h/Omega1), the ratio SSP/ISP Sapproaches ¯ h, indicating that each magnon excitation carries the angular momentum ¯ h.A t high temperatures ( kBT/greatermuch¯h/Omega1), this ratio becomes propor- tional to kBTdue to the nature of the Bose statistics. To illus- trate their temperature dependence, we estimate and comparethe noise powers, S eqandSSP, in realistic experiments. We use the parameters of the spin-pumping experiment for YIG[29]:α=6.7×10 −5,S0=16,hac=0.11 Oe, γ=1.76× 107Oe−1s−1, and/Omega1/2π=9.4 GHz. We consider NbN for the SC (Tc/similarequal10 K) and set D=532 meV Å2following Ref. [ 58]. Figure 5shows the results for the noise power, normalized byS0=SSP(T=0) for normal metals. For this parameter set, the nonequilibrium noise associated with spin pumping ismuch larger than the equilibrium noise. For both S eqandSSP, the temperature dependence peaks below the superconductingtransition temperature. VI. EXPERIMENTAL SETUP FOR DETECTION In the previous sections, we evaluated the spin current and its noise at the FI-SC interface. For their experimentaldetection, we need to consider a setup for converting the spinimbalance induced by the spin current into a charge signal.0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4020406080S/S0 T/TcSSP Seq(x100) FIG. 5. Temperature dependence of the equilibrium noise Seq and the nonequilibrium noise in the spin-pumping case SSPfor SCs (solid lines) and normal metals (dashed lines). The noise power is normalized by the nonequilibrium noise in the spin-pumping case S0 for the normal metals at T=0. For better visualization, data for the equilibrium noise have been multiplied by 100. There are several ways to perform such a spin-charge conver- sion. Here, we explain one possible way using the ISHE. Itwas theoretically predicted that such spin current flowing ina SC can be detected by the ISHE [ 35,36]. Indeed, a giant signal of the ISHE was recently observed by spin injectionfrom ferromagnetic metals into an s-wave superconductor NbN using the technique of the lateral spin valve [ 37]. Let us consider spin injection into a SC wire with a width w and a length 2 dfrom a FI at x=0 (see Fig. 6). By spin-orbit interaction in the SC, the spin current I Sis converted into a quasiparticle current IQin the direction perpendicular to both ISand the ordered spin in the FI S. This quasiparticle current induces a charge imbalance in the SC and produces a voltagebetween the two edges located at x=±d. The amplitude of the ISHE voltage depends on the spin relaxation in the SCas well as the spin Hall angle, so that the coefficient betweenthe spin current at the interface and the ISHE voltage is, ingeneral, temperature dependent. Here, we introduce a simpleformula employed in Ref. [ 37]: V ISHE=|e| ¯hISx w/bracketleftBigg aρxx 2f0(/Delta1)+b/parenleftbiggρxx 2f0(/Delta1)/parenrightbigg2/bracketrightBigg e−d/λQ,(58) f0(/Delta1)=1 e/Delta1/kBT+1. (59) This expression for the ISHE voltage has been derived assum- ing an extrinsic spin Hall effect due to spin-orbit scattering inthe SC. Here, λ Qis the charge relaxation length, aandbare coefficients determined by the strength of the skew scattering FI SC FIG. 6. A setup for detection of the spin current using the inverse spin Hall effect. 144411-6MICROSCOPIC THEORY OF SPIN TRANSPORT AT THE … PHYSICAL REVIEW B 99, 144411 (2019) and side jump, respectively, and ρxxis the resistivity of the SC. The correction due to nonuniform current distribution isrepresented by a shunting length x, which is determined by w, λ Q, and the shape of the junction [ 37]. Combining Eqs. ( 58) and ( 59) with careful determination of the parameters, we can obtain ISfrom the measurement of VISHE. In principle, the spin-current noise can also be measured within the same kindof setup via the fluctuations of V ISHE[43]. VII. SUMMARY In summary, we discussed the spin current and the spin- current noise for a bilayer system composed of a supercon-ductor and a ferromagnetic insulator. The spin current inducedby spin pumping has a maximum below the transition temper-ature when the pumping frequency /Omega1is much smaller than k BTc/¯h. As the ratio ¯ h/Omega1/kBTcincreases, the peak disappears, and the spin current at low temperatures is enhanced. Wealso discussed the spin current induced by the spin Seebeckeffect and the noise power of the pure spin current. Ourstudy provides a fundamental basis for the application ofspintronics using superconductors. Extension to spin injectionfrom antiferromagnetic insulators is left for a future problem[59–62]. ACKNOWLEDGMENTS The authors are grateful to S. Takei, Y . Niimi, Y .-C. Otani, K. Kobayashi, and T. Arakawa for useful discussions andcomments. This work is financially supported by KAKENHI(Grants No. 26103006, No. JP26220711, No. JP16H04023,and No. JP17H02927) from MEXT and JSPS, Japan. Thiswork has been supported by the Excellence Initiative of Aix-Marseille University (AMIDEX), a French “investissementsd’avenir” program. APPENDIX A: EFFECT OF IMPURITY SCATTERING Here, we explain that the diffusive behavior of conduc- tion electrons, which is taken into account in Ref. [ 38], can be neglected in the calculation of Im χR loc(ω) following Ref. [ 42]. We neglect the Coulomb interaction effect dis- cussed in Ref. [ 42] for simplicity. For a qualitative discussion, it is convenient to start with the interpolation formula (Eq. ( 6) in Ref. [ 42]) χR D(q,ω)/similarequalχ0(q,ω)Dq2 Dq2−iω, (A1) where q=|q|,χ0(q,ω) is the spin susceptibility per volume of the electron gas, D=vFl/3 is the diffusion constant, l= vFτis the mean free path, vFis the Fermi velocity, and τis the relaxation time. The leading behavior for small ωis(seeEq. (7) in Ref. [ 42]) Imχ(q,ω) ¯hω=N(/epsilon1F) ¯h/parenleftbiggπ 2vFq+1 Dq2/parenrightbigg (0<q<2kF), (A2) where kFis the Fermi wave number. Then, the local spin susceptibility is calculated as Imχloc(ω) ¯hω=/integraldisplayd3q (2π)3Imχ(q,ω) ¯hω =2πN(/epsilon1F)2/parenleftbigg1 2+3 πkFl/parenrightbigg . (A3) Since kFl/greatermuch1 for usual metals, the second term due to the diffusive Green’s function is usually a correction. Therefore,the leading contribution is obtained only by considering aclean system without impurities. For superconductors, a simi-lar discussion leads to the same conclusion. APPENDIX B: SPIN SUSCEPTIBILITY OF THE SC The dynamic spin susceptibility of the SC is calculated in the standard BCS theory as [ 45] χR(q,ω)=1 NS/summationdisplay k/summationdisplay λ=±1/summationdisplay λ/prime=±1/bracketleftbigg1 4+ξξ/prime+/Delta12 4EλE/prime λ/prime/bracketrightbigg ×f(E/prime λ/prime)−f(Eλ) ¯hω+iδ+Eλ−E/prime λ/prime, (B1) where ξ=ξk,ξ/prime=ξk+q,Eλ=λ/radicalbig /Delta12+ξ2,E/prime λ/prime= λ/prime/radicalbig /Delta12+ξ/prime2, and f(E)=[exp( E/kBT)+1]−1is the Fermi distribution function. For the normal state ( /Delta1=0), the spin susceptibility becomes χR(q,ω)=1 NS/summationdisplay kf(ξk+q)−f(ξk) ¯hω+iδ+ξk−ξk+q. 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PhysRevB.76.024418.pdf

Large-amplitude coherent spin waves excited by spin-polarized current in nanoscale spin valves I. N. Krivorotov Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA D. V. Berkov and N. L. Gorn Innovent Technology Development, Pruessingstraße 27B, D-07745 Jena, Germany N. C. Emley, J. C. Sankey, D. C. Ralph, and R. A. Buhrman Cornell University, Ithaca, New York 14853-2501, USA /H20849Received 17 March 2007; revised manuscript received 27 May 2007; published 13 July 2007 /H20850 We present spectral measurements of spin-wave excitations driven by direct spin-polarized current in a free layer of nanoscale Ir 20Mn80/Ni80Fe20/Cu/Ni 80Fe20spin valves. The measurements reveal that large-amplitude coherent spin-wave modes are excited over a wide range of bias current. The frequency of these excitationsexhibits a series of jumps as a function of current due to transitions between different localized nonlinearspin-wave modes of the Ni 80Fe20nanomagnet. We find that micromagnetic simulations employing the Landau- Lifshitz-Gilbert equation of motion augmented by the Slonczewski spin-torque term /H20849LLGS /H20850accurately de- scribe the frequency of the current-driven excitations including the mode transition behavior. However, LLGSsimulations give qualitatively incorrect predictions for the amplitude of excited spin waves as a function ofcurrent. DOI: 10.1103/PhysRevB.76.024418 PACS number /H20849s/H20850: 72.25.Ba, 75.75. /H11001a, 75.20. /H11002g I. INTRODUCTION The recent discovery of persistent current-driven excita- tions of magnetization in magnetic nanostructures1–13has created new opportunities for studies of magnetization dy-namics in extremely nonlinear regimes inaccessible withconventional techniques such as ferromagnetic resonance /H20849FMR /H20850. It was recently demonstrated 14that a spin-polarized current can excite motion of magnetization in metallic nano-magnets with precession cone angles over 30°, values farexceeding those achievable in typical FMR experiments per-formed on bulk and thin-film samples. There are two reasonswhy it is possible to have such large-amplitude current-driven motions of magnetization in nanomagnets: /H20849i/H20850sup- pression of Suhl instability processes 15,16due to quantization of the magnon spectrum in the nanomagnet17–28and /H20849ii/H20850ef- ficient amplification of spin waves by spin-transfer torquethat can act approximately as negative magnetic damping. 1,2 The possibility of exciting large-amplitude oscillations of magnetization in magnetic nanostructures by spin-polarizedcurrent provides a unique testing ground for theories of non-linear magnetization dynamics in ferromagnetic metals. 29–32 Most importantly, it gives an opportunity to test the validity of the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation for the de- scription of large-amplitude motion of magnetization. TheLLG equation is phenomenological in nature, and thus itsapplicability must be tested in every new type of experimen-tal situation. This equation has proved to be largely success-ful in the description of persistent small-angle magneticexcitations 33and transient large-angle magnetization dynamics34,35in thin films of ferromagnetic metals /H20849with some notable exceptions36/H20850. However, it is not known a pri- orithat the LLG equation is suitable for the quantitative description of a persistent magnetization precession withvery large amplitude. For example, the phenomenologicalGilbert damping term parametrized by a single constant inthe LLG equation may prove to be an approximation suitable for description of small-angle dynamics but not valid in gen-eral. Recently, large-angle persistent motion of magnetiza-tion was studied in thin films of Ni 80Fe20by time-resolved measurements, and a large increase of apparent damping wasobserved in the nonlinear regime. 37However, measurements of intrinsic damping in continuous ferromagnetic films areobscured by generation of parametrically excited spin wavesthat give rise to at least a large portion of the increaseddamping found in Ref. 37. This generation of parametrically pumped spin waves is expected to be suppressed in nanos-cale ferromagnets, 23and thus information on the amplitude dependence of intrinsic damping can, in principle, be ac-cessed. A number of recent models predict nontrivial angulardependence of damping 38,39and suggest how it may depend on the rate and amplitude of magnetization precession40in metallic magnetic nanostructures. These predictions remainlargely untested primarily due to the difficulty of excitingpersistent large-amplitude magnetization dynamics in nano-magnets. In this work we report a detailed comparison of experi- mentally measured spectra of current-driven magnetization oscillations in elliptical Py /H20849Py/H11013Ni 80Fe20/H20850nanoelements to the results of full-scale micromagnetic simulations for these structures and thus test the validity of the micromagneticLLG approach for the description of strongly nonlinear os-cillations of magnetization in magnetic nanostructures. Wefind that although simulations based on the LLG equationaugmented by the Slonczewski spin torque term 1/H20849the LLGS equation /H20850can successfully mimic many properties of current- driven magnetization dynamics such as the current depen-dence of the excitation frequency and abrupt frequencyjumps with increasing current, they qualitatively fail to re-produce the dependence of the amplitude of current-drivenspin waves as a function of current. Our results demonstratethe deficiencies of the current LLGS implementation for thePHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 1098-0121/2007/76 /H208492/H20850/024418 /H2084914/H20850 ©2007 The American Physical Society 024418-1description of spin-torque-driven magnetization dynamics and suggest the need for modification of this implementationfor a quantitative description of large-amplitude magnetiza-tion motion. We suggest that it may be necessary to introducea nonlinear dissipation or to consider effects of spin transferfrom lateral spin diffusion that are not contained in our cal-culation. II. EXPERIMENT A. Sample preparation and characterization The current-perpendicular-to-plane /H20849CPP /H20850nanopillar spin valves for our experiments are prepared by magnetron sput-tering of continuous magnetic multilayers onto an oxidizedSi wafer followed by a multistep nanofabrication process. 7 As a first step of the sample preparation process, a multilayerof Cu /H2084980 nm /H20850/Ir 20Mn 80/H208498n m /H20850Py/H208494n m /H20850/C u /H208498n m /H20850/Py /H208494nm /H20850/Cu/H2084920 nm /H20850/Pt/H2084930 nm /H20850is deposited onto a thermally oxidized Si /H20849100/H20850wafer by magnetron sputtering in a high- vacuum system with a base pressure of 2 10−8Torr. The 80-nm Cu layer is used as the bottom electrode of the CPPspin valve. The Pt capping layer is employed for protectionof the multilayer from oxidation during the nanopillar fabri-cation process. The multilayer is deposited at room tempera-ture in a magnetic field of approximately 500 Oe applied inthe plane of the sample and post-annealed at T=250 °C for 80 min in the same field. We use a subtractive process em-ploying e-beam lithography, photolithography, and etching of the multilayer in order to define nanoscale spin valves ofapproximately elliptical shape with major and minor axes of130 nm and 60 nm, respectively, and with Cu electrodesmaking contact with the top and bottom of the spin valve asshown in Fig. 1/H20849a/H20850. The role of the antiferromagnetic Ir 20Mn 80layer in the spin-valve structure is twofold: /H20849i/H20850to pin the direction of magnetization of the fixed Py nanomagnet at a nonzero anglewith respect to the easy axis of the free nanomagnet usingthe exchange bias effect as shown in Fig. 1/H20849b/H20850and /H20849ii/H20850to suppress current-driven excitations of magnetization in thefixed nanomagnet due to the giant enhancement of Gilbertdamping observed in exchange-biased ferromagnets. 41,42. The nominal direction of the exchange bias field set dur- ing the multilayer deposition and subsequent annealing is inthe plane of the sample at 45° with respect to the major axisof the ellipse. However, within a set of 40 samples we foundsignificant /H20849±35° /H20850sample-to-sample variations of the ex- change bias direction, as determined from the Stoner- Wohlfarth /H20849SW/H20850fitting procedure described below. These sample-to-sample variations of the exchange bias directionare not surprising in a magnetic nanostructure and may beattributed to finite-size effects 43–45as well as to resetting of the exchange bias direction due to sample heating that occursduring lithography and ion milling process employed to de-fine the nanopillar structure. In this paper we report experi-mental results for the most extensively studied sample al-though qualitatively similar results were obtained for theother samples from the set of 40. The quantitative differencesbetween the samples can be correlated with differences ofthe shapes of the hysteresis loop of resistance versus field,such as that shown in Fig. 1/H20849c/H20850, and ultimately to variations of the direction of the exchange bias field. Samples withsimilar resistance versus field hysteresis loops exhibit similarspectral properties of the current driven magnetization oscil-lations. All measurements reported in this paper were madeatT=4.2 K. We determine the direction and magnitude of the ex- change bias field for each nanopillar sample by fitting theStoner-Wohlfarth model to the experimental resistance–versus–magnetic-field hysteresis loop, such as that shown inFig.1/H20849c/H20850. For all measurements and simulations reported in this paper, we apply the external magnetic field in the planeof the sample at 45° with respect to the ellipse major axisand approximately perpendicular to the exchange bias direc-tion as shown in Fig. 1/H20849b/H20850. Stoner-Wohlfarth simulations5.85.855.95.956 -700 -350 0 350 700 Magnetic Field (Oe)R( O h m ) 5.755.855.956.056.15 -10 -5 0 5 10Current (mA)R( O h m )Py PyCuCu CuIrMn(a) (b) MP MFH (c) 5.75.96.16.3 -10 -5 0 5 10 Current (mA)dV/dI (Ohm)(e) (f)HEB θθθθp 0O e 680 Oe0O e 680 Oe00.20.40.60.81 -700 -350 0 350 700 Magnetic Field (Oe)DDDDR/DDDDRmax(d) FIG. 1. /H20849Color online /H20850/H20849a/H20850Schematic side view of the nanopillar spin valve used for studies of magnetization dynamics. /H20849b/H20850Sche- matic top view of the spin valve with approximate directions ofmagnetizations of the pinned, M P, and the free, MF, nanomagnets as well as the direction of positive external magnetic field Hand exchange bias field HEB./H20849c/H20850Experimentally measured resistance of the nanopillar as a function of the external magnetic field /H20849circles /H20850 and a macrospin Stoner-Wohlfarth fit to the data /H20849solid line /H20850with the parameters described in text. /H20849d/H20850Resistance versus field ob- tained from micromagnetic simulations using the giant magnetore-sistance /H20849GMR /H20850asymmetry parameter /H9273=0.5, the exchange bias field magnitude HEB=1600 Oe and its direction /H9258EB=30° obtained from the macrospin fit shown in /H20849c/H20850./H20849e/H20850Differential resistance of the sample as a function of bias current measured at H=0 Oe /H20849red/H20850 andH=680 Oe /H20849blue /H20850./H20849f/H20850dc resistance of the sample as a function of bias current obtained from the data in /H20849e/H20850by numerical integration.KRIVOROTOV et al. PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-2show that this choice of the bias field direction results in a weak dependence on the magnitude of external magneticfield for the equilibrium angle between magnetic moments ofthe free and pinned layers. According to Stoner-Wohlfarthsimulations, the equilibrium angle between magnetic mo-ments of the free and the pinned layer varies between 34°and 36° in the field range from 300 Oe to 1100 Oe. Thesolid line in Fig. 1/H20849c/H20850is a four-parameter Stoner-Wohlfarth fit to the data with the following fitting parameters: the ex-change bias field magnitude H EB, its direction /H9258EB, the mag- netoresistance /H20849MR/H20850asymmetry /H9273, and the MR magnitude /H9004R. The MR asymmetry parameter /H9273/H20849Refs. 46and47/H20850de- scribes a deviation of the angular dependence of the giantmagnetoresistance /H20849GMR /H20850from a simple cosine form: R/H20849 /H9258/H20850=R0+/H9004R1 − cos2/H20849/H9258/2/H20850 1+/H9273cos2/H20849/H9258/2/H20850. /H208491/H20850 Here/H9258is the angle between magnetic moments of the pinned and the free layers. The Stoner-Wohlfarth fit shown in Fig.1/H20849c/H20850yields H EB=1.6±0.5 kG, /H9258EB=/H2084930±6 /H20850°,/H9273=0.5±0.3, and/H9004R=0.161±0.007 /H9024. Two other parameters used in the Stoner-Wohlfarth simulations are the uniaxial shape aniso-tropy field H Kof the elliptical Py nanomagnets and the av- erage dipolar coupling field between the fixed and the pinnedlayers, H dip. The value of HK=600 Oe was obtained as the saturation field along the in-plane hard axis of the nanomag-net by employing micromagnetic simulations /H20849OOMMF /H20850. 48 The value of Hdip=80 Oe was obtained by numerical integra- tion of the dipolar coupling energy of the two uniformlymagnetized Py nanomagnets. The value of /H9273obtained from our fitting procedure is significantly less than that reportedfor a similar structure in Ref. 47/H20849 /H9273/H110152/H20850. The difference is probably due to the different values of the effective Py/Cu interfacial and Py bulk resistances in our spin valves, possi-bly due to interdiffusion of metallic layers of the spin valveduring the annealing process. 49 To test the validity of the Stoner-Wohlfarth approach for fitting the quasistatic MR hysteresis loop, we calculate theMR loop for this sample by employing full micromagneticsimulations 50with the values of HEB,/H9258EB, and /H9273obtained from the SW fit. Other input parameters for micromagneticsimulations were obtained by direct measurements. The satu-ration magnetization M Sof a 4-nm-thick Py film sandwiched between two Cu films and subjected to the same heat treat-ment as the spin valves under study was measured by super-conducting quantum interference device /H20849SQUID /H20850magne- tometry and was found to be M S=650 emu/cm3atT=5 K. The Gilbert damping parameter /H9261=0.025 /H20849needed for the dy- namic simulations described in Sec. III below /H20850for these samples was measured at T=7 K by a pump-probe technique described in Ref. 14. This value significantly exceeds that measured for Py nanomagnets at room temperature by spin-torque FMR spectroscopy /H20849/H9261=0.01 /H20850. 51A similar increase of the Gilbert damping parameter in nanoscale Py elements at low temperature was previously observed in Ref. 52and was attributed to exchange coupling of the nanomagnet to an an-tiferromagnetic oxide layer formed along the perimeter ofthe nanomagnet.The result of the full-scale micromagnetic simulation is shown in Fig. 1/H20849d/H20850. We find that the SW model is a reason- able approximation for the quasistatic hysteresis loop in thatthe coercivity predicted by micromagnetic simulation is/H1101180% of that given by the SW model and the shapes of the Stoner-Wohlfarth and micromagnetic hysteresis loops aresimilar. However, we could not obtain a quantitatively cor-rect fit of the measured GMR loop using full-scale micro-magnetic simulations. The main difference was that thesimulated loops were always narrower than the loop mea-sured experimentally /H20851Fig.1/H20849c/H20850/H20852: the difference between the left and right coercive fields for the loop shown in Fig. 1/H20849c/H20850 is/H9004H c=680 Oe while the maximum /H9004Hc=600 Oe was ob- tained for various directions of the exchange bias field thatwe have used in our simulations. Regarding this discrepancywe note that in full-scale simulations we do not have at ourdisposal the anisotropy field H Kand the dipolar coupling Hdipas adjustable parameters—the corresponding effective field contributions are calculated from the material saturationmagnetization and the sample geometry. Taking into accountthat the width of the GMR hysteresis loops varied substan- tially among different samples as discussed in Sec. IV, we donot consider this discrepancy as being qualitatively signifi-cant. The difference between the SW simulations and full-scale micromagnetic modeling means, first, that the fit pa-rameters obtained from the SW approximation should beconsidered not as exact values, but rather as reasonableguesses, and second, that some magnetic properties of thesystem under study /H20849e.g., surface anisotropy, sample shape imperfections, and the possible presence of antiferromag-netic oxides along the perimeter of the free layernanomagnet 52/H20850are still not included in our model. Because the main goal of this paper is the study of dynamic systemproperties, we postpone the discussion of this quite interest-ing problem to future publications. Figure 1/H20849e/H20850shows the measured differential resistance of the sample as a function of direct current flowing through thesample for H=0 Oe and for H=680 Oe. Positive current in this and subsequent figures corresponds to the flow of elec-trons from the free to the pinned layer. Figure 1/H20849f/H20850shows the dc sample resistance R=V/IforH=0 Oe and H=680 Oe as a function of direct current, obtained by numerical integra-tion of the differential resistance data in Fig. 1/H20849e/H20850. The qua- siparabolic increase of the resistance with increasing current/H20849most clearly seen for negative currents /H20850is due to a combi- nation of Ohmic heating 53and the Peltier effect in the nano- pillar junction.54The other features in the plots of dV/dI versus IandR/H20849I/H20850such as hysteretic switching of resistance at H=0 Oe or peaks in the differential resistance at H =680 Oe are due to changes of magnetic state of the nano-pillar. At fields below the coercive field of the free layer, weobserve current-induced hysteretic switching between thelow- and high-resistance states. For fields exceeding the co-ercive field, the time-averaged resistance of the sample R/H20849I/H20850 undergoes a transition from the low-resistance state to an intermediate-resistance state under the action of direct cur-rent as shown in Fig. 1/H20849f/H20850/H20851e.g., R/H20849680 Oe /H20850=R/H208490O e /H20850 −0.27/H9004RforI=10 mA /H20852. As we demonstrate below, this intermediate-resistance state is a state of persistent current-driven magnetization dynamics for the free nanomagnet.LARGE-AMPLITUDE COHERENT SPIN WAVES EXCITED … PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-3B. Measurements of current-driven oscillations of magnetization To measure the current-driven excitations of magnetiza- tion directly, we employ a spectroscopic technique developedin Ref. 10. Figure 2/H20849a/H20850schematically shows the measurement setup employed for detection of the current-driven excita-tions of magnetization. In this setup, direct current flowingperpendicular to the layers of the spin valve sample excitesmagnetization oscillations in the free Py nanomagnet, whichgive rise to a temporal variation in the resistance of the spinvalve due to the GMR effect, R/H20849t/H20850. Since the sample is cur- rent biased /H20849I dc/H20850, the temporal variation of the resistance gen- erates on ac voltage, V/H20849t/H20850=IdcR/H20849t/H20850. This ac voltage is ampli- fied with a microwave signal amplifier, and its spectral content is recorded with a spectrum analyzer. A spectrummeasured at zero-dc-bias current is subtracted from all spec-tra in order to eliminate a small background due to thermaland electronics noise. Figures 2/H20849b/H20850and2/H20849d/H20850show representative examples of typical spectra generated by the spin valve under direct cur-rent bias. The signals shown in Figs. 2/H20849b/H20850and2/H20849d/H20850are the normalized rms amplitude spectral density S/H20849f/H20850defined be- low. This quantity characterizes the amplitude, frequency, and coherence of oscillations of magnetization. To calculateS/H20849f/H20850, we start with the power spectral density measured with the spectrum analyzer, P an/H20849f/H20850. This quantity is corrected for frequency-dependent amplification and attenuation in the cir- cuit between the spectrum analyzer and the nanopillarsample in order to obtain the power spectral density P/H20849f/H20850of the signal emitted by the sample into a 50- /H9024transmission line. This latter quantity is used to calculate the rms voltagespectral density V/H20849f/H20850of the GMR signal due to oscillations of magnetization at the nanopillar as V/H20849f/H20850=/H20849R S+R0/H20850/H20881P/H20849f/H20850/R0.55 In this expression, R0=50/H9024is the characteristic impedance of all components of the microwave circuit shown in Fig.2/H20849a/H20850except for the nanopillar itself, and R S=26/H9024is the resistance of the nanopillar junction and leads. We define thenormalized rms amplitude spectral density S/H20849f/H20850as the rms voltage spectral density V/H20849f/H20850divided by the maximum rms GMR voltage signal amplitude /H20855V max/H20856 =/H20881/H20855/H20851/H20849I/H9004R/2/H20850sin/H20849/H9275t/H20850/H208522/H20856/2=I/H9004R/2/H208812 /H20849where /H9275=2/H9266f/H20850 achievable due to 360° uniform rotation of magnetization in the sample plane at a given current bias: S/H20849f/H20850=V/H20849f/H20850 /H20855Vmax/H20856=/H208818V/H20849f/H20850 I/H9004R. /H208492/H20850 The dimensionless integrated signal amplitude Sint, Sint=/H20881/H20885 0/H11009 S/H20849f/H208502df, /H208493/H20850 reaches its maximum value Sint=1//H208812 for the maximum pos- sible GMR voltage signal due to 360° uniform rotation ofmagnetization in the sample plane: V max/H20849t/H20850=I/H9004R 2sin/H208492/H9266ft/H20850. /H208494/H20850 The integrated signal amplitude Sintis a convenient di- mensionless scalar quantity that characterizes the amplitudeof magnetization precession. Its square is directly propor-tional to the integrated power emitted by the device. Thisdimensionless quantity is also convenient for comparison ofexperimental data to the results of micromagnetic simula-tions. A typical experimentally measured spectrum S/H20849f/H20850for our samples is characterized by a single frequency /H20851the funda- mental peak and higher harmonics such as that shown in Fig.2/H20849b/H20850/H20852. However, for some values of the bias current, two peaks that are not harmonically related to each other areobserved /H20851Fig.2/H20849d/H20850/H20852. Figure 2/H20849c/H20850shows a summary of spectra generated by the sample as a function of the direct current bias I dcmeasured at a fixed value of the applied magnetic field H=680 Oe. The most important features of these data are the following. /H20849i/H20850The frequency of the current-driven excitations de- creases with increasing current. This decrease of frequencywith increasing current can be explained as a nonlinear effectarising from the dependence of the frequency of precessingmagnetization on the precession amplitude. 10,29,30,56,57 /H20849ii/H20850The frequency of the current-driven excitations exhib- its downward jumps at I/H110153.7 mA and 4.85 mA. The current values at which the frequency jumps occur coincide with thepositions of the peaks in the plot of differential resistanceversus current /H20851Fig. 1/H20849e/H20850/H20852. A double-peak structure in the spectrum such as that shown in Fig. 2/H20849d/H20850is observed only for currents near frequency jumps, indicating that the apparentjumps are in fact nonhysteretic crossovers between two ex-citations with different frequencies. As the current is in-creased across the transition region, the emitted power is (a) (b) (d)(c) FIG. 2. /H20849Color online /H20850/H20849a/H20850Circuit schematic for measurements of magnetization dynamics driven by a direct current. /H20849b/H20850Normal- ized rms amplitude spectral density S/H20849f/H20850/H20849defined in text /H20850generated by the spin valve under a dc bias of 6.15 mA. /H20849c/H20850S/H20849f/H20850as a function of current for the nanopillar spin valve measured at H=680 Oe. /H20849d/H20850 S/H20849f/H20850at 3.7 mA and H=680 Oe.KRIVOROTOV et al. PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-4gradually transferred from the excitation with the higher fre- quency to the excitation with the lower frequency. We alsoobserve that the linewidths of the current-driven excitationsincrease in the current intervals where two excitationscoexist /H20851e.g., compare Figs. 2/H20849b/H20850and2/H20849d/H20850/H20852. In the current intervals where a single large-amplitude mode is excited,spectral lines as narrow as 10 MHz are observed whilefor currents in the mode transition regions spectral linesas wide as 250 MHz are found. The increase of the linewidthof the excitation indicates the decrease of its temporalcoherence. 57–60The linewidth increase is observed in all transition regions suggesting that the decrease of coherenceof the current-driven spin waves is induced by interactionbetween the two excited spin-wave modes. /H20849iii/H20850Modes with very low power visible only on the loga- rithmic amplitude scale of Fig. 2/H20849c/H20850are observed for currents above 3.7 mA. These modes are not harmonically related tothe dominant modes. Although these modes emit low inte-grated power, they may play an important role in determin-ing the coherence of the dominant spin-wave excitations. 61 III. NUMERICAL SIMULATIONS Full-scale micromagnetic simulations of the current- induced magnetization dynamics in the nanopillar describedabove were performed by solving the stochastic LLG equa-tion of motion for the magnetization M/H20849r/H20850using our com- mercially available simulation package MICROMAGUS /H20849Ref. 50/H20850supplemented by a spin injection module. Details of the simulation methodology are given in the Appendix. In thesesimulations, the spin torque in the LLGS simulations has theform/H9003=−f J/H20849/H9258/H20850/H20851M/H11003/H20851M/H11003p/H20852/H20852where the dimensionless spin- torque amplitude fJdepends on the angle /H9258between the mag- netization Mand the unit vector pof the polarization direc- tion of the electron magnetic moments /H20849in the spin-polarized current /H20850. We use, in general, the asymmetric angular depen- dence of the spin torque amplitude fJ/H20849/H9258/H20850, fJ/H20849/H9258/H20850=aJ·2/H90112 /H20849/H90112+1/H20850+/H20849/H90112−1/H20850cos/H9258, /H208495/H20850 given in Refs. 46and62, where /H9011is the asymmetry param- eter related to the GMR asymmetry parameter /H9273from Eq. /H208491/H20850 via/H90112=1+/H9273. As will be demonstrated below, variations in the direction of the spin current polarization p/H20849opposite to the magneti- zation of the pinned layer MP/H20850in the spin torque term /H9003=−fJ/H20849/H9258/H20850/H20851M/H11003/H20851M/H11003p/H20852/H20852can result in qualitative changes of magnetization dynamics and thus the orientation of pis a very important parameter of the problem under study.As explained in Sec. II, the direction of pcould not be determined quantitatively from the available experimentaldata. To understand the dependence of the magnetizationdynamics on the orientation of p, we first study magnetiza- tion dynamics for pdirected opposite to the exchange bias field extracted from the GMR hysteresis fit as described inSec. II /H20849i.e., the angle between pand the positive direction of the xaxis was set to /H9258p=150° /H20850. Then we perform two additional simulation sets for larger /H20849/H9258p=170° /H20850and smaller/H20849/H9258p=130° /H20850values of the equilibrium angle between magne- tization and current polarization to study the effect of the spin polarization direction on the current-driven dynamics. The decisive advantage of numerical simulations is the possibility to study and understand the influence of all rel-evant physical factors separately. For this reason we startfrom the “minimal model,” where the influence of the Oer-sted field and thermal fluctuations is neglected and the spintorque is assumed to be symmetric /H20851/H9011=1 in Eq. /H208495/H20850/H20852, and then switch on in succession all the factors listed above toanalyze their influence on the magnetization dynamics. Minimal model . Results for this model, for which the Oer- sted field and thermal fluctuations are not included and theasymmetry parameter /H9011=1/H20851f J/H20849/H9258/H20850=const /H20852, are presented in Fig.3/H20849a/H20850. For the spin orientation angle /H9258p=150° used for the first simulation series, the critical spin-torque value for the oscillation onset was found to be aJcr/H110150.308 /H208492/H20850. Using the simplest expression for the spin torque given, e.g., in Ref. 62, one can easily derive the relation between the reduced spin-torque amplitude aJand other device parameters as aJ=/H6036 2j /H20841e/H208411 dP1 MS2, /H208496/H20850 where eis the electron charge, jis the electric current den- sity, dis the thickness of a magnetic layer subject to a spin torque, and Pis the degree of spin polarization of the elec- trical current. Using the definition of the current density, j =I/Selem /H20849Iis the total current and Selemis the area of the nanopillar cross section /H20850, and substituting the values for the experimentally measured critical current Icr/H110152.7 mA and the threshold for the oscillation onset aJcr/H110150.3 found in simula- tions, we obtain that the polarization degree of the electronmagnetic moments is P/H110150.32. From the relation between the critical current I crand the critical spin torque amplitude aJcr, the proportionality factor /H9260between the spin-torque am- plitude aJused in the simulation and the experimental cur- rent strength I/H20849in mA /H20850is/H9260/H110150.11 /H20849mA−1/H20850/H20849whereby aJ=/H9260I/H20850 for this spin polarization direction /H20849/H9258p=150° /H20850. The simulated spectral lines are very narrow /H20849mostly /H11021100 MHz /H20850for all values of the spin-torque amplitude aJcr /H33355aJ/H333552.0, which means that for this simplest model a tran- sition to a quasichaotic regime similar to that found in Ref.63does not occur in the interval of currents studied. For this reason we show in Fig. 3/H20849a/H20850only the positions of the spectral maxima of the M zcomponent as a function of aJ/H20849red circles /H20850. In addition to the narrow lines, this minimal model also reproduces two other important qualitative features ofthe experimental results /H20851see Fig. 2/H20849c/H20850/H20852: a rapid decrease of the oscillation frequency with increasing current immedi-ately after the oscillation onset and /H20849ii/H20850downward frequency jumps at higher current values. The first feature, the rapid decrease of the oscillation fre- quency immediately after the oscillation onset, is a nonlineareffect due to the rapid growth of the oscillation amplitudewith increasing current. In the nonlinear regime, the fre-quency decreases with increasing amplitude because thelength of the precession orbit grows faster than the magneti-zation velocity. The corresponding effect was obtained ana-LARGE-AMPLITUDE COHERENT SPIN WAVES EXCITED … PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-5lytically in Refs. 29,30, and 57and observed numerically in our full-scale micromagnetic simulations63of the experi- ments published in Ref. 10 The second important observation, the existence of down- ward frequency jumps with increasing current, cannot yet beexplained using analytical theories, and such jumps are ab-sent in the macrospin description of current-driven magneti-zation dynamics. Spatially resolved spectral analysis of oursimulation data reveals that these jumps correspond to tran-sitions between strongly nonlinear oscillation modes /H20851see spatial maps of the magnetization oscillation power in Fig.3/H20849a/H20850/H20852. With each frequency jump, the mode becomes more localized but the oscillation power is still concentrated in onesingle-connected spatial region which has no node lines. Wediscuss these modes in more detail below when analyzing results for different current polarization directions. An ana-lytical theory of the nonlinear eigenmodes of a resonatorhaving the correct shape would be required to achieve a thor-ough understanding of this phenomenon. In the minimal model we also observe for the current strength a J/H110221.5 the so-called “out-of-plane” coherent pre- cession regime for which the magnetization acquires a non-zero time-average component perpendicular to the sampleplane. This regime is characterized by frequency increasingwith current and is well known from analytical considerationand numerical simulations. 64,65The out-of-plane regime was experimentally observed for a nanopillar sample with a2-nm-thick free Py layer and low critical current, I cr =1.4 mA, and thus relatively small Oersted field.66However, for the devices with a 4-nm-thick free layer that we study,this type of mode is an artifact of the minimal model /H20849due to the absence of the Oersted field /H20850and it was not observed experimentally. Effect of the Oersted field . The effect of the Oersted field is demonstrated in Fig. 3/H20849b/H20850where the dependences of the oscillation frequency on the spin-torque magnitude a Jare shown without /H20851red circles, identical to Fig. 3/H20849a/H20850/H20852and with the Oersted field /H20849green triangles /H20850. To compute the Oersted field, we have used the proportionality constant between thespin-torque magnitude a Jand experimental current value I /H20849in mA /H20850assuming that the simulated threshold value aJcr /H110150.31 corresponds to the experimentally measured critical current Icr/H110152.7 mA. The results shown in Fig. 3/H20849b/H20850demonstrate that the Oer- sted field has two major effects on magnetization dynamics.First, this field eliminates the out-of-plane precession: in-spection of magnetization trajectories shows that for all a J values they correspond to “in-plane” steady-state oscilla- tions. As a consequence, the Oersted field eliminates the up-ward frequency jump in the f/H20849a J/H20850dependence. The suppres- sion of the out-of-plane mode occurs because the Oersted field is a strongly inhomogeneous in-plane field that keepsmagnetization close to the plane of the sample. Second, the Oersted field shifts the frequency jumps to lower current values. This can be explained as follows: theOersted field is highly inhomogeneous, with its maximal val-ues at the edges of the elliptical element. For this reason itshould suppress magnetization oscillations at the elementedges, thus favoring spatial oscillation modes localized nearthe element center such as those shown in Fig. 3/H20849a/H20850. Hence the transition from the homogeneous mode to more localizedones should occur for lower currents when the Oersted fieldis taken into account. Effect of the spin-torque asymmetry . It can be seen di- rectly from Eq. /H208495/H20850that for /H9266/2/H11021/H9258p/H110213/H9266/2 the spin-torque magnitude in the case of positive GMR asymmetry /H20849/H9273/H110220, /H9011/H110221/H20850is larger than for the symmetric /H20849/H9273=0,/H9011=1/H20850case. This difference is expected to result in a decrease of thesteady-state precession frequency at a given current valuebecause larger spin torque results in larger amplitude of mag-netization oscillations. For the system studied in this paper,the GMR asymmetry is relatively low /H20849the Stoner-Wohlfarth fit of the quasistatic GMR curve gave the value /H9273=0.5 and /H90112=1.5 /H20850, so that the expected frequency decrease is quite FIG. 3. /H20849Color online /H20850/H20849a/H20850Dependence of the frequency of mag- netization oscillations, f, on spin-torque amplitude aJcalculated in the “minimal model” /H20849see text for details /H20850. Gray-scale maps repre- sent spatial distributions of the oscillation power for chosen aJval- ues/H20849bright corresponds to maximal oscillation power /H20850./H20849b/H20850Oersted field effect: f/H20849aJ/H20850without /H20849open circles /H20850and with /H20849open triangles /H20850 the Oersted field included in the simulations; arrows indicate thepositions of frequency jumps, and straight lines are guides to theeye. /H20849c/H20850Effect of the torque asymmetry: f/H20849a J/H20850for the symmetric /H20849/H9011=1.0, open triangles /H20850and asymmetric /H20849/H90112=1.5, solid triangles /H20850 spin torque.KRIVOROTOV et al. PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-6weak, but it is nevertheless clearly visible when comparing thef/H20849aJ/H20850dependences for symmetric and asymmetric cases in Fig. 3/H20849c/H20850. The frequency decrease is greatest in the low- current region where the dependence of the oscillation fre-quency on current is very steep. The asymmetric torque alsoleads to a small decrease of the threshold current for the oscillation onset, which becomes a Jcr/H110150.27. A more important effect of introducing the spin torque asymmetry is a shift of the frequency values where the tran-sitions between nonlinear eigenmodes of the system /H20849accom- panied by the frequency jumps as explained above /H20850take place. Even for the relatively low value /H9011 2=1.5 used in our simulations, this shift is significant /H20851see Fig. 3/H20849c/H20850/H20852. The rea- son, again, is that the asymmetric torque form gives a largerspin torque magnitude for a given current value, so that thetransitions to more localized modes occur earlier. Influence of thermal fluctuations . To take into account the influence of thermal fluctuations, we have first to estimatethe real temperature of the sample. Although the experimentswere performed at liquid helium temperature T=4.2 K, Joule heating of our multilayer nanoelement due to the direct cur-rent through the device was unavoidable. 53To estimate the maximal temperature of the nanoelement, we have measured/H20849i/H20850the temperature dependence of its resistance R 0/H20849T/H20850in the absence of any dc current by heating the whole setup and /H20849ii/H20850 the dependence of the resistance on current R0/H20849I/H20850measured at positive current and H=680 Oe as shown in Fig. 1/H20849f/H20850. The increase of the resistance with current is due both to theexcitation of coherent magnetization oscillations and Ohmicheating; therefore, an estimate of the sample temperaturefrom R 0/H20849I/H20850gives an upper bound on the temperature of the sample. Comparison of R0/H20849T/H20850andR0/H20849I/H20850shows that the nano- element temperature does not exceed T/H1101560 K for the high- est current I=10 mA used in the measurements. Taking into account that this maximal temperature is relatively low, wehave simply adopted a linear interpolation between the low-est temperature T=4 K for I=0 mA and T/H1101560 K for I /H1101510 mA /H20849with Iconverted into the reduced spin torque am- plitude a J/H20850for our simulations. The dependences of the excitation frequency on current f/H20849aJ/H20850forT=0 and T/H11008aJwith the proportionality factor /H9260 calculated as explained above are compared in Fig. 4/H20849a/H20850/H20849for both simulation sets the effect of the Oersted field and thespin torque asymmetry with /H9011 2=1.5 are included /H20850. It is clear that due to the relatively low temperatures, thermal fluctua-tions have a minor effect both on the oscillation frequencyand on the positions of the frequency jumps. Influence of the spin current polarization direction . The polarization direction /H9258pof the electron magnetic moments in the dc current is expected to be one of the most importantparameters of the problem. First, the onset threshold for os-cillations should depend strongly on this polarizationdirection. 56,67Second, the relative strength of the Oersted field /H20849with respect to the spin-torque magnitude aJ/H20850also should depend on /H9258p, because the Oersted field for different /H9258pis computed assuming that the threshold value aJcralways corresponds to the experimentally measured critical currentI cr/H110152.7 mA. Since /H9258pcould not be accurately determined from the fit of the quasistatic MR hysteresis loop /H20849see dis-cussion in Sec. II A /H20850, we have carried out additional series of simulation runs to study the effect of the spin current polar-ization direction on the magnetization dynamics. The results of these simulations are summarized in Fig. 5, where we show the dependences of the oscillation frequencyon the spin-torque magnitude f/H20849a J/H20850/H20851Fig. 5/H20849a/H20850/H20852and on the spin-torque magnitude normalized by the threshold value aJcr for the corresponding angle f/H20849aJ/aJcr/H20850/H20851Fig.5/H20849b/H20850/H20852. The dependence f/H20849aJ/H20850for/H9258p=150°—i.e., for the case which a detailed analysis has been presented above—is shown in this figure with open circles. For the increased polarization orientation angle /H9258p=170° /H20849open triangles /H20850the onset threshold for the magnetization dynamics decreases from aJcr/H20849/H9258p=150° /H20850/H110150.27 to aJcr/H20849/H9258p=170° /H20850/H110150.15 in a quali- tative agreement with Slonczewski’s prediction for the mac- rospin model /H20849Icr/H110111//H20841cos/H9258p/H20841/H20850 /H20851Ref. 1/H20852experimentally con- firmed in Ref. 67. The first frequency jump with increasing current is still present, but instead of the second jump weobserve a kink in the f/H20849a J/H20850curve /H20851see Fig. 5/H20849a/H20850/H20852. We note that(a) (b) (c) (d) FIG. 4. /H20849Color online /H20850/H20849a/H20850Effect of thermal fluctuations on the frequency of magnetization oscillations: f/H20849aJ/H20850forT=0/H20849triangles /H20850 and for T/H11008I/H11008aJ/H20849crosses /H20850. Simulated rms-amplitude spectral den- sities S/H20849f/H20850for oscillations /H20849b/H20850before the first frequency jump, /H20849c/H20850 between the first and second frequency jumps, and /H20849d/H20850after the second frequency jump. The linewidth of the peaks marked with /H9254 is below the resolution limit of our numerical simulations. See thetext for the detailed analysis of these spectra.LARGE-AMPLITUDE COHERENT SPIN WAVES EXCITED … PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-7the importance of the Oersted field relative to the spin-torque effect in this case is much larger than for /H9258p=150° for the following reason. The Oersted field is always computed as-suming that the critical value of the spin-torque magnitude a Jcrcorresponds to one and the same physical current value Icr/H110152.7 mA. This means that if aJcrdecreases, the same Oer- sted field corresponds to smaller aJvalues so that the impor- tance of the Oersted field effect increases relative to the spin-torque action. For a smaller polarization angle /H20849 /H9258p=135°, results are shown on Fig. 5with crosses /H20850the critical value of aJin- creases /H20851aJcr/H20849/H9258p=135° /H20850/H110150.66 /H20852, so that the influence of the Oersted field is weaker than for /H9258p=150°. This leads, in par- ticular, to the reappearance of the out-of-plane oscillationregime, which manifests itself in the increase of the oscilla-tion frequency with increasing a J. Recall that the out-of- plane precession regime was found for /H9258p=150° in the ab- sence of the Oersted field, but was suppressed by this field asexplained above /H20851see Fig. 2/H20849b/H20850/H20852. For the angle /H9258p=135° the Oersted field is not strong enough to eliminate this regimewhen the spin-torque magnitude increases. To compare magnetization dynamics for various spin po- larization angles we plot the frequency of oscillations for allthree values of /H9258pstudied as a function of spin-torque mag-nitude normalized to its critical value, aJ/aJcr/H20851Fig.5/H20849b/H20850/H20852. The most striking feature of the f/H20849aJ/aJcr/H20850curves for various angles /H9258pis that they all nearly collapse onto the universal f/H20849aJ/aJcr/H20850dependence for aJ/aJcrvalues up to aJ/aJcr/H110152. This region includes, in particular, the fast frequency decrease af- ter the oscillation onset /H20849see the discussion of this nonlinear effect above /H20850and the first frequency jump arising for all spin polarization directions at almost the same value of aJ/aJcr /H110151.5. These results clearly demonstrate that the initial nonlinear rapid frequency decrease and the first frequency jump areuniversal for the system under study, whereas further behav-ior of the magnetization dynamics /H20849in particular, the exis- tence of the second frequency jump /H20850are much more subtle features and thus may vary from sample to sample. The firstfrequency jump is always present because it marks the tran-sition from the homogeneous to a localized oscillation mode/H20851see Fig. 3/H20849a/H20850/H20852which is always accompanied by an abrupt change of the oscillation frequency. The next frequency jumpfor the situation when the Oersted field is neglected corre-sponds to the transition between the modes with different /H20849but symmetric /H20850localization patterns—before the second jump the mode is localized in the direction along the majorellipse axis only, whereas after this jump the new mode isconfined in both directions /H20851compare second and third maps in Fig. 3/H20849a/H20850/H20852. This latter transition is strongly disturbed by the Oersted field, which leads, in particular, to strongly asym-metric spatial mode patterns for localized modes. This mayeliminate the qualitative differences between modes with dif-ferent localization patterns that give rise to the frequencyjumps. To understand the degree of agreement that may be ex- pected between the simulation results and the experimentaldata for our samples, it is instructive to examine sample-to-sample variations of experimentally measured magnetizationdynamics for samples with different directions of the ex-change bias field. Figure 6shows resistance as a function of field /H20849left column /H20850and the corresponding dependence of the frequency of the current-driven excitations on current /H20849right column /H20850for three representative samples from the set of forty samples studied. These samples have different direc-tions of the exchange bias field as determined from theStoner-Wohlfarth fit of the hysteresis loop of resistance ver-sus field /H20851/H20849a/H20850,/H20849b/H20850 /H9258EB=22°, /H20849c/H20850,/H20849d/H20850/H9258EB=38°, /H20849e/H20850,/H20849f/H20850/H9258EB=48° /H20852. Figure 6along with Figs. 1and2illustrates typical sample- to-sample variations of the quasistatic hysteresis loop and thefrequency of current-driven excitations among nominallyidentical samples. While it is clear that there are significantsample-to-sample variations of the current dependence of thefrequency of excitations, the initial decrease of frequencywith current as well as downward frequency jumps is alwayspresent in these data. We note that there are correlations be-tween theoretically predicted trends in magnetization dynam-ics as a function of the exchange bias direction shown in Fig.5and experimental data such as those shown in Fig. 6.I n particular, we usually observe two frequency jumps forsamples with relatively large exchange bias angle /H20851 /H9258EB /H1140730°, e.g., Figs. 6/H20849e/H20850and6/H20849f/H20850/H20852and one jump for samples relatively small exchange bias angle /H20851/H9258EB/H1135130°, e.g., Figs. 6/H20849a/H20850and 6/H20849b/H20850/H20852. However, it is not always the case thatFIG. 5. /H20849Color online /H20850/H20849a/H20850Dependence of the frequency of os- cillations on spin torque amplitude aJfor different polarization angles of the spin current, /H9258p, as defined in Fig. 1/H20849b/H20850./H20849b/H20850The same frequencies plotted as functions of the normalized spin torque am-plitude a J/aJcr.KRIVOROTOV et al. PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-8samples with similar direction of exchange bias as deter- mined by the Stoner-Wohlfarth fitting procedure /H20851e.g., Figs. 1/H20849c/H20850and6/H20849c/H20850/H20852show identical dependence of frequency on current /H20851Figs. 2/H20849c/H20850and6/H20849d/H20850/H20852. We attribute these differences to sample shape imperfections. IV. DISCUSSION Having developed an understanding about how the vari- ous parameters influence the simulated dynamics, we canproceed with the analysis of magnetization oscillation spec-tra and a comparison with the experimentally observed mag-netoresistance power spectra. In Fig. 4/H20849a/H20850we display the dependence of the simulated spectral maximum frequencies on the spin-torque amplitudea J, in the presence of thermal fluctuations. These simulations take into account all the physical factors that are generallyincluded in a state-of-the-art micromagnetic model. Spectralamplitudes of magnetoresistance oscillations are displayed inFigs. 4/H20849b/H20850–4/H20849d/H20850. The spectra can be divided into the follow- ing three groups: /H20849i/H20850from the oscillation onset to the first frequency jump /H20851group A, Fig. 4/H20849b/H20850/H20852,/H20849ii/H20850from the first to the second frequency jump /H20851group B, Fig. 4/H20849c/H20850/H20852, and after the second frequency jump /H20851group C, Fig. 4/H20849d/H20850/H20852. For all groups, the frequency of the spectral maximum decreases monotoni-cally with the spin-torque magnitude a J. The dependencies of the linewidth and the integrated spectral power /H20851see Fig. 7/H20849b/H20850/H20852on the spin-torque magnitude require special discus- sion. For the first group—spectra from the oscillation onset to the first frequency jump—the linewidth for small aJis rela- tively large /H20849/H11011100 MHz /H20850due to a relatively large influenceof thermal fluctuations on small-amplitude motion of the magnetization.61The oscillation amplitude grows rapidly with increasing current /H20849compare spectra for aJ=0.30, 0.31, and 0.32 /H20850and the linewidth strongly decreases /H20849to/H1101120 MHz foraJ=0.32 /H20850, which is due to an increasing contribution from the spin-torque-driven dynamics resulting in the effectivesuppression of the influence of thermal fluctuations and thusin the decrease of the linewidth. 61,68When the current is increased further and approaches its value for the first jump,the contribution of the second nonlinear oscillation mode/H20849which will dominate the spectrum after the first frequency jump /H20850becomes visible, leading to line broadening and a de- crease of the maximal spectral amplitude /H20849see spectra for a J=0.34, 0.36 /H20850. After the first jump, the amplitude of magnetization pre- cession becomes large /H20849Sint/H110220.3/H20850and the relative influence of thermal fluctuations on the motion of magnetization be- comes small. For this reason, the linewidth for most spectraof the second group /H20849except for those close to the second frequency jump /H20850is extremely small. In fact, it is below the resolution limit of our simulations /H20849/H9004f min/H1101510 MHz /H20850, thus being in a good agreement with experimental observations. When approaching the current value of the second frequencyjump, the line width starts to increase again /H20849and the maxi- mal spectral power decreases /H20850due to the influence of the next nonlinear mode.(b) (a) (d) (c) (f) (e) FIG. 6. /H20849Color online /H20850/H20849a/H20850,/H20849c/H20850,/H20849e/H20850Examples of hysteresis loops of resistance versus field of different nominally identical samples./H20849b/H20850,/H20849d/H20850,/H20849f/H20850Frequency of persistent magnetization dynamics as a function of current for the samples in /H20849a/H20850,/H20849c/H20850,/H20849e/H20850measured at H =600 Oe. FIG. 7. /H20849Color online /H20850/H20849a/H20850Comparison of the experimentally measured /H20849solid triangles /H20850and simulated /H20849open circles /H20850dependence of the frequency of oscillations on current. /H20849b/H20850Experimentally mea- sured /H20849solid triangles /H20850and simulated /H20849open circles /H20850integrated rms- amplitude spectral density Sintas a function of current.LARGE-AMPLITUDE COHERENT SPIN WAVES EXCITED … PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-9For the last spectral group, the linewidth and the maximal value of the spectral power exhibit the same nonmonotonicbehavior. However, the line broadening for the large currentvalues /H20849a J/H110220.90 /H20850in this region is not due to the next incipi- ent frequency jump but due to the onset of spatially incoher- ent magnetization dynamics /H20849note that the maximal experi- mentally used current value Imax=10 mA corresponds to aJ /H110151.0/H20850. The line broadening for I/H110229.0 mA is also observed experimentally and is clearly visible in Fig. 2/H20849c/H20850. However, for values of aJgreater than the second frequency jump, the width of the simulated spectral peaks is substantially largerthan the width of spectral lines measured for correspondingcurrent values /H20849computed as I=a J//H9260/H20850. Another important dif- ference between experiment and the simulation results is thatthe narrowest spectral lines found in the simulations existbetween the first and the second frequency jumps, while thenarrowest lines observed experimentally occur after the sec-ond frequency jump. Before proceeding to a direct comparison with the experi- mental oscillation frequencies and amplitudes, we note an important difference between the magnetization dynamicsof the Py elliptical nanomagnet simulated in this paper andthat of the Co elliptical nanoelement studied in detail pre-viously in Ref. 63. For the Co element in Ref. 63the coher- ence of the magnetization oscillations was lost already forcurrents very close to the onset of the steady-state oscilla-tions, followed by a transition to a completely chaoticregime. 69In contrast to this behavior, magnetization dynam- ics of the Py element studied here remains nearly coherentup to current values several times larger than the criticalcurrent. This difference cannot be attributed to much lowertemperatures for which the experiment discussed here hasbeen performed /H20849compared to room temperatures used in Ref. 10/H20850, because the transition to the chaotic regime slightly above a crwas observed in Ref. 63already for simulations performed at T=0. The difference can also not be due to a slightly higher element thickness used here /H20849hPy=4 nm compared to hCo=3 nm in Ref. 63/H20850, because the much higher exchange constant of Co /H20849ACo=3/H1100310−6erg/cm; see Ref. 63for details /H20850when compared with the Py exchange /H20849APy=1.3/H1100310−6erg/cm /H20850should at least compensate this slightly larger thickness of the Py nanoelement. We argue that this important discrepancy in the behavior of the two quite similar systems studied here and in Ref. 63 is due to the very different character of the nonlinear mag-netization oscillation modes of these nanoelements. Whereasin Ref. 63several oscillation modes with a quite complicated localization patterns arose and coexisted when the oscillationamplitude increased /H20849see spatial maps in Figs. 1, 3, and 4 in Ref. 63/H20850, in this work we have found that for each given current value there is a single nonlinear eigenmode where theoscillating spins are confined in a localized area of the nano-magnet without any node lines between these oscillatingspins. It seems plausible that the transition to a quasichaoticbehavior from a single mode would be inhibited compared tothe case of several coexisting modes with different spatialprofiles. We believe that the physical reason for excitation ofa single mode in the case of substantially noncollinear mag-netizations of the pinned and the free layers is that spintorque is nearly spatially uniform. Indeed, in the case ofnominally collinear magnetizations, the direction and magni- tude of spin torque exerted on the free layer exhibits strongspatial variations due to spatially nonuniform magnetizationdirection predicted by micromagnetics. This results in localmagnetizations of the free and pinned layers making smallnegative angles with respect to each other in some parts ofthe sample and small positive angles in other parts of thesample. Since spin torque is proportional to the small anglebetween magnetizations of the free and pinned layers, thecase of nominally collinear magnetizations gives rise tostrongly spatially nonuniform spin torque. In the case of non-collinear magnetizations, small variations of the magnetiza-tion direction over the sample area result in small deviationsof spin-torque direction and magnitude from their averagevalues. A spatially nonuniform pattern of spin torque is morelikely to couple to multiple oscillation modes of the nano-magnet. In the case of nearly constant uniform torque, thecoupling to the longest-wavelength mode is expected to bethe strongest. Figure 7presents a direct comparison between experi- mental data and results of LLGS simulations. First, we showin Fig. 7/H20849a/H20850the current dependence of the magnetization os- cillation frequency as measured experimentally /H20849solid tri- angles /H20850and as obtained from micromagnetic simulations /H20849open circles /H20850. For plotting the simulation data as f/H20849I/H20850we have used the conversion from the spin torque amplitude a J to the current strength Iin the form I=aJ//H9260with the conver- sion factor /H9260/H110150.1 computed as explained above. The simu- lations reproduce the current dependence of the oscillationfrequency fairly well, except for the position of the secondfrequency jump, which occurs in simulations at a currentabout 20% higher than in the experiment. However, takinginto account that a nanomagnet with perfect edges was simu-lated and that the simulations did not contain any adjustableparameters /H20849except the conversion factor /H9260/H20850the agreement between simulations and experiment can be considered asvery satisfactory, as far as the oscillation frequency is con-cerned. The fact that micromagnetic LLGS simulations suc-cessfully reproduce the frequency jumps in the current de-pendence of the frequency of oscillations is a significantsuccess of the LLGS micromagnetic approach. Since thesejumps result from transitions between modes with differentdegrees of spatial localization, LLGS simulations in the mac-rospin approximation would not be able to describe suchtransitions. This shows that while LLGS simulations in themacrospin approximation 10,56are useful for an initial quali- tative understanding of many properties of current-drivenmagnetization dynamics such as the nonlinear shift of theoscillation frequency with current and the existence of dif-ferent types of nonlinear oscillation modes /H20849e.g., in-plane and out-of-plane precession modes 10,66/H20850, a quantitative descrip- tion of persistent current-driven dynamics requires a micro-magnetic approach. Despite the good agreement between experiment and simulations for the dependence of the oscillation frequencyon current, our micromagnetic simulations could not closelyreproduce the corresponding dependence of the oscillationamplitude on current. Figure 7/H20849b/H20850shows the experimentally measured /H20849solid triangles /H20850and simulated /H20849open circles /H20850inte- grated signal amplitudes S intas functions of the bias current.KRIVOROTOV et al. PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-10The general trend of the measured oscillation amplitude is to increase gradually with increasing current, together with aseries of dips at currents corresponding to the frequencyjumps shown in Fig. 2/H20849c/H20850. In contrast, the LLGS simulations predict a rapid increase of the oscillation amplitude justabove the critical current for the onset of oscillations, fol-lowed by a slow gradual decrease. Some minor anomalies onthe simulated S int/H20849I/H20850around the frequency jumps can be seen; however, they are far less pronounced than the corresponding experimentally observed dips. Taking into account the good agreement between simula- tions and experiment for the oscillation frequency, the dis-crepancy for the amplitude of oscillations is very surprisingand requires a detailed analysis. The failure of our LLGSsimulations to predict the correct dependence of the oscilla-tion amplitude on current indicates that the standard micro-magnetic LLGS approach for spin-torque-driven excitationsin nano-magnets requires modifications. Below we proposesome possible routes towards improvement of the theoreticaldescription of spin-torque-driven excitations in nanomag-nets. One possible way of solving this problem would be to introduce a nonlinear dissipation /H20851a dependence of the dissi- pation parameter /H9261on the rate of the magnetization change dm/dtin the form /H9261=/H9261 0/H208491+q1/H20849dm/dt/H208502+¯/H20850/H20852as suggested in Ref. 40. In making such an attempt, one should keep in mind that a too strong nonlinearity /H20849large values of the non- linear coefficient q1/H20850would destroy the good agreement be- tween simulated and measured oscillation frequencies, espe-cially for the initial part of the f/H20849a J/H20850dependence where the transition between linear and nonlinear oscillation regimes is observed. However, a moderate nonlinearity could weaklyaffect the oscillation frequency for small to moderate oscil-lation amplitudes /H20849small a J/H20850, while improving the coherence of the magnetization oscillations for large currents. /H20849If the dissipation coefficient /H9261increases with increasing dm/dt, then it should strongly suppress the short-wavelength excita-tions that lead to incoherent magnetization oscillations. /H20850In this way, one would obtain higher oscillation powers andnarrower linewidths for larger currents, thus improving theagreement between theory and experiment. Clearly, this sub-ject requires further investigation. Another possible way of reconciling theory and experi- ment for the current dependence of both frequency and am-plitude of the excited modes would be the generation of spin-wave modes that are more spatially nonuniform than thoseshown in Fig. 3/H20849a/H20850. Indeed, if only a part of magnetization of the nanomagnet moves with large amplitude /H20849e.g., edge modes /H20850, both a significant nonlinear shift of frequency and a relatively small average measured amplitude will be ob-served. Furthermore, the growth of the average amplitude ofsuch nonuniform spin-wave modes is likely to proceed via agradual spatial growth of the oscillating domain, whichshould give rise to a gradual increase of the measured am-plitude and result in a dependence of the amplitude on cur-rent similar to the experimentally observed dependenceshown in Fig. 7/H20849b/H20850. A possible mechanism leading to excita- tion of strongly spatially nonuniform modes is the instabilityof magnetization arising from lateral spin transport in spin-valve structures. 70–72A theoretical test of this scenario re-quires the development of a micromagnetic code that explic- itly treats magnetization dynamics coupled to spatiallynonuniform spin-dependent electrical transport, which is be-yond the scope of this work. Softening of the spin-wavespectrum by spin-polarized current 73,74could also be an im- portant factor to be taken into account for reconciling thetheory of current-driven excitations with the experimentalresults presented in this paper. V. CONCLUSIONS In conclusion, we have measured the spectral properties of current-driven excitations in nanoscale spin valves withnoncollinear magnetizations of the free and pinned ferromag-netic layers. We find that spin-polarized current in these de-vices excites a few coherent large-amplitude nonlinearmodes of magnetization oscillation in the free layer. Differ-ent modes are excited in different current intervals. We findthat the amplitude and the coherence of the current-drivenexcitations decrease in the current intervals where transitionsbetween these modes take place. We simulate the response ofmagnetization to spin-polarized current in our samples byemploying LLG micromagnetic simulations with a Sloncze-wski spin-torque term. 46These LLGS simulations capture a number of features of the experimental data: /H20849i/H20850the decrease of frequency of the excited oscillation modes with increasingcurrent, /H20849ii/H20850downward jumps of the frequency of excitations with increasing current resulting from transitions betweendifferent oscillation modes, and /H20849iii/H20850the high degree of co- herence /H20849narrow spectral linewidth /H20850of the excited modes. However, the LLGS simulations give qualitatively incorrectpredictions for the amplitude of the excited modes as a func-tion of current. Simulations predict rapid growth of the os-cillation amplitude above the threshold current for the onsetof spin-wave excitations, followed by a slow decrease of theamplitude. This is in sharp contrast to the more gradual in-crease of the oscillation amplitude with current observed inour experiment. Our results demonstrate that additional fac-tors possibly including nonlinear damping and/or lateral spintransport need to be taken into account for a quantitativedescription of large-amplitude magnetization dynamicsdriven by spin-polarized current in magnetic nanostructures. ACKNOWLEDGMENTS The authors thank J. Miltat, D. Mills, and A. Slavin for many useful discussions. This research was supported by theDeutsche Forschungsgemeinschaft /H20849DFG Grant No. BE 2464/4-1 /H20850, the Office of Naval Research, and the National Science Foundation’s Nanoscale Science and EngineeringCenters program through the Cornell Center for NanoscaleSystems. We also acknowledge NSF support through use ofthe Cornell Nanoscale Facility node of the National Nano-fabrication Infrastructure Network and the use of the facili-ties of the Cornell Center for Materials Research. APPENDIX: NUMERICAL SIMULATION METHODOLOGY In our full-scale micromagnetic simulations magnetiza- tion dynamics are simulated by solving the stochastic LLGLARGE-AMPLITUDE COHERENT SPIN WAVES EXCITED … PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-11equation of motion for the magnetization Miof each discreti- zation cell in the following form: dMi dt=−/H9253/H20851Mi/H11003/H20849Hidet+Hifl/H20850/H20852 −/H9253/H9261 MS/H20851Mi/H11003/H20851Mi/H11003/H20849Hidet+Hifl/H20850/H20852/H20852. /H20849A1/H20850 Here/H9253=/H92530//H208491+/H92612/H20850, where /H92530/H20849/H110220/H20850is the absolute value of the gyromagnetic ratio and /H9261is the reduced dissipation con- stant /H20849it is equal to the constant /H9251in the LLG equation writ- ten in the form M˙=−/H92530/H20851M/H11003H/H20852+/H20849/H9251/MS/H20850/H20851M/H11003M/H20852/H20850. The de- terministic effective field Hidetacting on the magnetization of theith cell includes all standard micromagnetic contributions /H20849external, anisotropy, exchange, and magnetodipolar interac- tion fields /H20850. In addition, this deterministic field includes the spin- torque effect in the following way. The spin torque in theSlonczewski formalism is taken into account adding the termG=−f J/H20849/H9258/H20850/H20851M/H11003/H20851M/H11003p/H20852/H20852to the equation of motion in the Gilbert form /H20849see, e.g., Ref. 56/H20850 dM dt=−/H92530/H20851M/H11003Heff/H20852+/H9251 MS/H20851M/H11003M˙/H20852 −/H92530fJ/H20849/H9258/H20850/H20851M/H11003/H20851M/H11003p/H20852/H20852, /H20849A2/H20850 where the dimensionless spin-torque amplitude fJdepends on the angle /H9258between the magnetization Mand the unit vector pof the polarization direction of the electron mag- netic moments /H20849in the spin-polarized current /H20850. From the com- putational point of view, this additional torque can be put into the effective field as HSTeff=Heff+fJ/H20851M/H11003p/H20852, after which Eq. /H20849A2/H20850can be converted to the numerically more conve- nient form /H20849A1/H20850in a standard way. We use the following asymmetric angular dependence of the amplitude fJ/H20849/H9258/H20850/H20849Refs. 46and62/H20850: fJ/H20849/H9258/H20850=aJ2/H90112 /H20849/H90112+1/H20850+/H20849/H90112−1/H20850cos/H9258. /H20849A3/H20850 Here aJgives the /H20849constant /H20850value of the spin-torque ampli- tude for the symmetric torque /H20849/H9011=1/H20850; the asymmetry param- eter/H9011can in principle be computed when the device con- figuration and various transport coefficients are known and isrelated to the GMR asymmetry parameter /H9273in Eq. /H208491/H20850via /H90112=1+/H9273. Expression /H20849A3/H20850is strictly valid only for sym- metrical spin valves46,62with identical ferromagnetic layers and identical top and bottom leads. The expression for fJ/H20849/H9258/H20850 in an asymmetric device is more complex62and involves effective resistances of the ferromagnetic layers and leads.However, we use a simplified expression /H20849A3/H20850forf J/H20849/H9258/H20850in our simulations for three reasons. First, we do not expect the spin-torque asymmetry of our device /H20849with respect to the above-mentioned effective resistances /H20850to be large because the thicknesses of two ferromagnetic layers of the spin valveare identical and the thicknesses of nonmagnetic leads arenot very different. Second, the spin diffusion length of Py/H20851/H110115n m /H20849Ref. 75/H20850/H20852is similar to the thickness of Py layers in our spin-valve structure, which substantially decreases theinfluence of the transport properties of the leads on spin torque. 62Third, Eq. /H20849A3/H20850can be considered as the simplest form /H20849apart from the form with fJ=const /H20850for studying the effect of the asymmetry of the spin-torque angular depen-dence on the magnetization dynamics. The more complexexpression derived in Ref. 62can be investigated after the effect of the simplest form of spin-torque asymmetry givenby Eq. /H20849A3/H20850is understood. The random fluctuation field H iflin Eq. /H20849A1/H20850represents the influence of thermal fluctuations and has standard /H9254-functional spatial and temporal correlation properties: /H20855H/H9264,ifl/H20856=0 , /H20855H/H9264,ifl/H208490/H20850H/H9274,jfl/H20849t/H20850/H20856=2D/H9254/H20849t/H20850/H9254ij/H9254/H9264/H9274 /H20849A4/H20850 /H20849i,jare the discretization cell indices; /H9264,/H9274=x,y,z/H20850, with the noise power Dproportional to the system temperature D =/H9261//H208491+/H92612/H20850/H20849kT//H9253/H9262/H20850; here, /H9262denotes the magnetic moment magnitude for a single discretization cell. The justification for using /H9254-correlated random noise for a finite-element ver- sion of an initially continuous system with interactions canbe found, e.g., in Ref. 76. The remaining simulation methodology is similar to that described in Ref. 63. We simulate spin-torque-driven excita- tions in the free ferromagnetic layer only. We neglect mag-netostatic and RKKY interactions between the free andpinned /H20851antiferromagnetic- /H20849AF-/H20850coupled /H20852Py layers. This approximation is justified because, first, the RKKY exchangecoupling via the thick /H20849h sp=8 nm /H20850Cu spacer is negligibly small and, second, the dipolar field acting on the free layer from the fixed one is on average /H20849/H1101180 Oe /H20850much smaller than the external field. The free layer /H20849130/H1100360/H110034n m3el- lipse /H20850is discretized into 50 /H1100324/H110031 cells; we checked that further refinement of the grid did not lead to any significantchanges in the results. Magnetic parameters of the free Pylayer used in simulations are: saturation magnetization M S =650 emu/cm3/H20849measured by SQUID magnetometry as ex- plained in Sec. II /H20850; exchange constant A=1.3/H1100310−6erg/cm /H20849standard value for Py /H20850; the random magnetocrystalline an- isotropy was neglected due to its low value /H20849Kcub=5 /H11003103erg/cm3/H20850for Py. The dissipation parameter is set to /H9261=0.025 /H20849see also Sec. II /H20850. As was shown above, the influence of the Oersted field HOeinduced by the current flowing through the spin valve may be very important. The calculation of this field is also ahighly nontrivial issue. In principle, its precise evaluationrequires the exact knowledge of the three-dimensional cur-rent distribution in the device itself and especially in adjacentelectrical contact layers, which is normally not available. Forthis reason the Oersted field is usually computed assumingthat the current is distributed homogeneously across thenanopillar cross section. Further, one of the following ap-proximations is used: /H20849i/H20850one assumes that H Oeis created by the infinitely long wire with the cross section correspondingto that of the nanopillar /H20849in our case the ellipse with l a/H11003lb =130/H1100360 nm2/H20850or/H20849ii/H20850the contribution to the Oersted field from the current inside the nanopillar itself only is included/H20849i.e.,H Oeis created by the piece of the wire with the length equal to the nanopillar height htot/H20850. Both approximations de- liver the same result for nanopillars with the height muchKRIVOROTOV et al. PHYSICAL REVIEW B 76, 024418 /H208492007 /H20850 024418-12larger than their characteristic cross-section size /H20851htot /H11271max /H20849la,lb/H20850/H20852, which is, however, very rarely the case for experiments performed up to now in the nanopillar geometry. In particular, in our situation the opposite inequality is true/H20849I a/H11022htot/H20850. Taking into account that the first approximation is also reasonably accurate for the system where the distribu- tion of currents in the nanopillar and adjacent leads is axiallysymmetric, and that the geometry of the electric contacts inour device is also highly symmetric, we have chosen the firstmethod to calculate H Oe. However, we point out once more that due to the importance of the influence of the Oerstedfield more precise methods for its calculation are highly de-sirable. Magnetization dynamics was simulated by integrating Eq. /H20849A1/H20850with the spin-torque term included using the addition-ally optimized Bulirsch-Stoer algorithm 77with the adaptive step-size control. /H20849The adaptive step-size control is especially important when the magnetization state significantly deviatesfrom a homogeneous one. /H20850For each current value /H20849each value of a Jin our formalism /H20850the dependence of the magne- tization on time for every discretization cell was saved forthe physical time interval /H9004t=400 ns. The spectral analysis of these magnetization “trajectories” was performed usingeither /H20849i/H20850the Lomb algorithm /H20849as described in Ref. 63/H20850espe- cially designed for nonevenly spaced sequences of time mo-ments as provided by the adaptive integration method or /H20849ii/H20850 interpolation of the “raw” results onto an evenly spaced tem-poral grid and usage of the standard fast Fourier transformroutines. 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PhysRevLett.120.097205.pdf

Magnon Valve Effect between Two Magnetic Insulators H. Wu, L. Huang, C. Fang, B. S. Yang, C. H. Wan, G. Q. Yu, J. F. Feng, H. X. Wei, and X. F. Han* Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China (Received 19 September 2017; revised manuscript received 10 December 2017; published 2 March 2018) The key physics of the spin valve involves spin-polarized conduction electrons propagating between two magnetic layers such that the device conductance is controlled by the relative magnetization orientation oftwo magnetic layers. Here, we report the effect of a magnon valve which is made of two ferromagnetic insulators (YIG) separated by a nonmagnetic spacer layer (Au). When a thermal gradient is applied perpendicular to the layers, the inverse spin Hall voltage output detected by a Pt bar placed on top of themagnon valve depends on the relative orientation of the magnetization of two YIG layers, indicating the magnon current induced by the spin Seebeck effect at one layer affects the magnon current in the other layer separated by Au. We interpret the magnon valve effect by the angular momentum conversion andpropagation between magnons in two YIG layers and conduction electrons in the Au layer. The temperature dependence of the magnon valve ratio shows approximately a power law, supporting the above magnon- electron spin conversion mechanism. This work opens a new class of valve structures beyond theconventional spin valves. DOI: 10.1103/PhysRevLett.120.097205 A spin valve, which comprises a nonmagnetic metallic [1,2] or insulating layer [3,4] sandwiched by two metallic ferromagnetic layers, has been widely adopted as an essential building block in spintronic devices such as magnetic reading heads of hard disk drives [5], magnetic random-access memory [6,7], and spin logic [8,9], etc. Information of a spin valve is encoded through the relativemagnetization orientation of two ferromagnetic layers.Although there are a variety of electric means to write aspin valve including spin-transfer torque [10,11] and spin- orbit torque [12,13] , the reading of the spin valve is exclusively based on giant magnetoresistance or tunnelmagnetoresistance in which spin-polarized electrons propa-gate from one magnetic layer to the other. Thus, a necessarycondition for a functional spin valve is that the magneticlayers must be metallic and possess a large electron spinpolarization. Fundamentally, the spin can be carried without electrons such as magnons, photons, neutrons, and so on; amongthem, magnons have attracted a great deal of interestrecently [14–17]. Magnons are quasiparticles of the spin wave, which represent the coherent collective excitation inmagnetic systems, and each quantized magnon carries aspin angular momentum of −ℏ. The wave property of magnons provides some unique features that are unavail-able in electron-based spintronic devices: Firstly, magnonsprovide long-distance spin information propagation with-out Joule heating, which could drastically reduce the powerconsumption of spintronic devices; secondly, the modula-tion of the phase parameter provides another degree offreedom to information processing, which could realize anon-Boolean logic operation. Moreover, the quantum property of magnons inspires some macroscopic quantum phenomena such as the spin superfluid [18], magnon Josephson effect [19], and so on. The concept of magnonics was proposed to study the magnon-based fundamental physics and potential applica-tions very recently [14]. As we know, transistors and spin valves act as the basic unit of semiconductor and spintronic devices, respectively. Therefore, in magnonic devices andcircuits, we need a basic building block —a magnon valve to accomplish functional information processing and datastorage. The typical magnon valve structure consists of twoferromagnetic layers that are separated by a space layer, andthe magnon valve effect means that the magnon currenttransmission coefficient could be controlled by the relativeorientation of two ferromagnetic layers. In the ideal case,all the magnon current could pass through the magnonvalve at the parallel magnetization state and be blocked atthe antiparallel magnetization state. Especially, the ferro- magnetic insulator (FMI)-based magnon valve is a prom- ising candidate, because the insulating property prohibitsany electron motion, and magnons become the sole spininformation carriers in FMI. Recent studies on the spin Seebeck effect (SSE) [20–25] and the magnon drag effect [26–29]have demonstrated that a magnon current in ferromagnetic insulators can be gen-erated by either a thermal gradient or an electron spininjection. More importantly, the magnon current can convertinto an electron spin current at the interface between the FMIand the nonmagnetic metal (NM) [26–29].A l s o ,s o m e theoretical works have investigated the magnon-mediatedPHYSICAL REVIEW LETTERS 120, 097205 (2018) Editors' Suggestion Featured in Physics 0031-9007 =18=120(9) =097205(6) 097205-1 © 2018 American Physical Societypure spin current transport and spin transfer torque between two FMI layers [30–33]. These studies lay the foundation of the experimental investigation of magnon valve structures inwhich the metallic magnetic layers are replaced by ferro- magnetic insulator layers. In this work, we propose to experimentally investigate spin transport in the magnon valve structure FMI/NM/FMI. When a thermal gradient is applied to the magnon valve, the magnon current in one FMI layer would be affected by themagnon current in the other FMI layer, mediated through the electron spin current in the NM layer. If one measures the magnon current across the magnon valve by depositinga heavy metal Pt layer, one would find that the inverse spin Hall effect (ISHE) [34,35] voltage depends on the relative orientation of the magnetization of the two FMI layers, i.e.,the magnon valve effect; see Fig. 1(a). We choose yttrium-iron-garnet (YIG) as both the top and bottom FMI layers. YIG is known for its low Gilbert damping factor ( α∼10 −4)[36] and wide band gap (Eg¼2.85eV)[37,38] . Magnon valve structures YIG ð40Þ/ AuðtAuÞ/YIG ð20Þ/Ptð10Þ(thickness in nanometers, from bottom to top) were deposited on 300-μmG d 3Ga5O12 (GGG) (111) substrates by an ultrahigh vacuum magnetronsputtering system (ULVAC MPS-4000-HC7 model), and the base pressure of the sputtering chamber was 1×10−6Pa. After deposition, an 800°C annealing in the air was carried out to improve the crystal structure of the YIG layers [39]. The multilayers were then patterned into the 100μm× 1000 μm stripe by a standard photolithography technique combined with Ar-ion etching, and then a 200 nm MgOinsulating layer and a 10 nm Au heating electrode (100μm× 1000 μm) were fabricated on top of the magnon valve to enable the longitudinal temperature gradient ∇Tvia on-chip Joule heating. The cross-sectional scanning andhigh-resolution transmission electron microscopy resultsof the YIG ð40Þ/Auð15Þ/YIG ð20Þ/Ptð10 nm Þmagnon valve structure were measured by a Tecnai G2 F20 S-TWINsystem. The magnetic field dependence of the magnetizationwas measured by a vibrating sample magnetometer (VSMEZ-9, MicroSense). All magnetotransport measurementswere performed in a physical property measurementsystem (PPMS-9T, Quantum Design) with a horizontalrotator option. In the YIG ð40Þ/Auð15Þ/YIG ð20Þ/Ptð10 nm Þmagnon valve structure, the well-defined epitaxial single crystal structure of YIG was formed on the GGG (111) surface, and the selected area electron diffraction (SAED) patterns showthat the YIG film was grown along the (111) direction, as shown in Fig. 1(b). From Fig. 1(c), both the bottom YIG/Au and the top Au/YIG interfaces are atomically sharp, whichpromises a reduced diffusive scattering and a higher rate of conversion between the magnon current and electron spin current at the interfaces. However, one would expect thecrystal quality of YIG deposited on Au to be worse than that on GGG. Such a difference is reflected in the coercive fields of the bottom and top YIG layers: The bottom YIG layer has a coercivity of 0.7 Oe, while the top YIG layer has more than one order of magnitude higher coercivity (47 Oe). Theseparation of the coercivity of the two YIG layers is necessary for generating an antiparallel configuration of the magnon valve structure as long as the coupling fieldfrom either magnetostatic coupling or indirect Ruderman- Kittel-Kasuya-Yosida exchange interaction is sufficiently weak. We have changed the thickness of the Au layer t Au from 2 to 15 nm and find that the coupling is weak enough for a well-separated magnetization reversal of the bottom and top YIG layers when tAuexceeds 6 nm (see Supplemental Material [40]). The temperature gradient ∇Tin the YIG ð40Þ/Auð15Þ/ YIGð20Þ/Ptð10 nm Þmagnon valve structure is created by applying a 20 mA electric current in the heating electrodeon the top of the magnon valve with a thick MgO spacerlayer in between. The temperature gradient would exist forboth the top and bottom YIG layers, generating a localmagnon current. Since the Pt layer is in contact with the topYIG layer, the ISHE voltage measured in Pt would beproportional to the total magnon current of the top YIGlayer. Aside from the magnon current associated with the FIG. 1. (a) Illustration of the magnon valve effect: When a temperature gradient is applied, the magnon current in the topYIG comes from two sources. One is generated by the presenceof the local temperature gradient, and the other is the magnoncurrent injected from the bottom YIG layer. If the magnetizationdirections of the YIG layers are parallel (antiparallel), these twomagnon currents are additive (subtractive). Since the ISHEvoltage measured by the Pt layer is proportional to the totalmagnon current through the top layer, a magnon valve effect isobserved. (b) The cross-sectional scanning transmission electronmicroscopy and selected area electron diffraction patterns of theGGG/YIG interface. (c) The cross-sectional high-resolutiontransmission electron microscopy of the YIG/Au/YIG region.The magnon valve structure measured in (b) and (c) isGGG/YIG ð40Þ/Auð15Þ/YIG ð20Þ/Ptð10 nm Þ.PHYSICAL REVIEW LETTERS 120, 097205 (2018) 097205-2local temperature gradient in the top YIG layer, the magnon current in the bottom YIG layer can flow intothe top YIG layer by first converting into an electron spincurrent in the Au layer and subsequently converting back tothe magnon current in the top YIG layer. Thus, depending on whether the magnetization of the two YIG layers is in the parallel or antiparallel state, the total magnon currentwould be the sum or difference of these two magnoncurrents. Figures 2(a)and2(b)show both the hysteresis loop (M-Hcurve) and the ISHE voltage-magnetic field loop (V ISHE-Hcurve), respectively. The M-Hloop illustrates a clear two-step magnetization reversal. Since the magneticmoment of the bottom YIG is made to be 2 times that of thetop YIG, the relatively sharp and large magnetization jumpat the smaller field indicates the magnetization reversal ofthe bottom YIG layer. For the V ISHE-Hloop, a clear difference is seen for the magnetization of two YIG layers in parallel and antiparallel states. If we defined a magnonvalve ratio MVR ¼ðV↑↑−V↓↑Þ/ðV↑↑þV↓↑Þ, where V↑↑ (V↓↑) is the measured ISHE voltage in Pt for the two YIG layers in the parallel (antiparallel) state, we found, for example, the MVR is 11% for a 15 nm Au interlayer. Next, a series of structures were designed to rule out possible artifacts of our observed effect. When the inter-layer Au in the YIG ð40Þ/Auð15Þ/YIG ð20Þ/Ptð10 nm Þ structure [Fig. 3(a)] was replaced by a 10-nm-thick insu- lating MgO layer, the magnon current from the bottom YIGwas completely blocked, and thus one would not expectany magnon current contribution from the bottom layer. Indeed, the V ISHE-Hloop, as seen in Fig. 3(b), only follows the magnetization of the top YIG layer, and the magneti-zation reversal of the bottom layer does not affect the ISHEvoltage. On the other hand, the sample with the bottomlayer only, YIG ð40Þ/Ptð10 nm Þ, displays a sharp transition within /C610Oe, as shown in Fig. 3(c), indicating a normal spin Seebeck signal for the bottom YIG layer, while for the Auð15Þ/YIG ð20Þ/Ptð10 nm Þsample, the inserted Au layer between GGG and YIG leads to a larger coercive filed withnonsquare hysteresis, as shown in Fig. 3(d). These con- trolled experiments support our proposed magnon valveeffect: The ISHE voltage depends on the relative orienta-tion of the magnetization of the two YIG layers. Another test of the magnon valve effect is to study the interlayer Au thickness dependence. When the thickness ofAu exceeds its spin diffusion length [41,42] , the magnon current in the bottom layer is unable to reach the top layer,and thus the magnon valve ratio diminishes. In Fig. 4(a),w e show the MVR- t Aurelation for tAufrom 6 to 15 nm. If the curve is fitted by a simple exponential decay function, we can obtain the spin diffusion length of Au (300 K) to be15.1 nm, which is close to the 12.6 nm from the spinpumping measurement [43]. Since the magnon valve effect comes from the magnon current propagating from the bottom to top YIG layers,mediated by the electron spin current of the spacer layer,the study of the temperature dependence would reveal theelectron-magnon spin conversion efficiency [26–29,44] at the two interfaces. Figure 4(b) shows the temperature dependence of the magnon valve ratio. As expected, the magnon valve ratio decreases as the temperature drops; this is consistent with magnon transport, in which the numberof magnon carriers is fewer at a lower temperature. A simple model can be used to quantitatively estimate the observed temperature dependence. The total magnon cur-rent in the top YIG layer comes from the local temperaturegradient as well as the magnon flow from the bottom layerto the top layer. In fact, the magnon current generatedby SSE would increase with increasing the thickness of YIG:ρ¼f ½coshðt FM/lmÞ−1/C138/½sinhðtFM/lmÞ/C138g, where ρis a factor that represents the effect of the finite YIG layer thickness, tFMis the thickness of YIG, and lmis the magnon diffusion length [45]. After considering the thickness dependence of SSE and using the 70 nm lmin previous work [45], the FIG. 2. The magnetic and magnon transport properties of the magnon valve. (a) Magnetization of the magnon valve structureGGG/YIG ð40Þ/Auð15Þ/YIG ð20Þ/Ptð10 nm Þas a function of the magnetic field applied in the plane of the layers. The arrowsindicate the magnetization directions of the two YIG layers.(b) The ISHE voltage in Pt as a function of the magnetic field forthe same magnon valve structure in the presence of the temper-ature gradient created by a 20 mA electric current applied at theheating electrode.PHYSICAL REVIEW LETTERS 120, 097205 (2018) 097205-3magnon current from the bottom YIG layer (40 nm) is 1.96 times of that from the top YIG layer (20 nm). The magnon current from the bottom layer would suffer three reduction factors to reach the top layer: the magnon-to-electron spinconversion rate at the bottom YIG/Au interface Gme, the electron-to-magnon spin conversion rate at the top Au/YIG interface Gem, and spin current loss in the Au layer. Since the ISHE voltage is proportional to the total magnon current FIG. 3. The ISHE voltage of different samples for controlled study. (a) –(d) Sample structures are marked in the figures, and the ISHE voltage is normalized by the resistance of the Pt detector to eliminate the sample-to-sample variation. FIG. 4. Thickness, temperature, magnetization direction, and heating current dependences of the magnon valve effect. (a) –(d) were measured in the GGG/YIG ð40Þ/Auð15Þ/YIG ð20Þ/Ptð10 nm Þsample. (a) The interlayer Au thickness dependence of magnon valve ratio MVR. The dashed line shows the exponential decay fitting curve. (b) The temperature dependence of MVR, and the dashed line shows theT5/2fitting curve. (c) The ISHE voltage as a function of the angle between the directions of the voltage probe and the in-plane magnetic field. The magnetic field (5 kOe) is large enough such that both YIG layers are in parallel with the magnetic field. The dashedline shows the sine fitting curve. (d) The heating current dependence of V ↑↑−V↓↓. The dashed line shows the parabolic fitting curve.PHYSICAL REVIEW LETTERS 120, 097205 (2018) 097205-4in the top YIG layer, we could write V↑↑ð↓↑Þ¼ a∇T½1/C61.96GmeGemeð−d/λÞ/C138, where ais the spin Seebeck coefficient, λanddare the spin diffusion length and the thickness of the Au layer, respectively, andwe neglect the magnon current decay in YIG layersin this formula. Thus, the magnon valve ratio is MVR ¼½ ðV ↑↑−V↓↑Þ/ðV↑↑þV↓↑Þ/C138 ¼ 1.96GmeGeme−d/λ. The spin conversion rates had been previously calculated [29]:GmeGem∝T5/2; apart from the offset signal, indeed the experimental data fit the T5/2temperature dependence very well, as seen in Fig. 4(b), which is consistent with the spin conversion theory. The offset signal is partly due to the on-chip heating that makes the temperature of the samplesignificantly higher (about 24 K) than the temperature of thecontrol. Further study is needed to map out the temperaturedependence of the magnon valve ratio. When a magnetic field of 5 kOe rotates in the sample plane such that the magnetizations of two YIG layers are parallel to the magnetic field, the ISHE voltage displays a perfect sine angular dependence, V ISHE∝sinα, as shown in Fig. 4(c), in agreement with the conventional spin Seebeck behavior [20–25]. The amplitude of the sine relation is proportional to the square of the heating current I, as shown in Fig. 4(d), indicating the temperature gradient ∇Tcreated by the on-chip heating scales as the heating power, as expected. In conclusion, we have fabricated the YIG/Au/YIG/Pt magnon valve structure and investigated thermally driven magnon current transport across the multilayers. The observed large magnon valve ratio supports the notion that the magnon current transmission between two mag- netic insulating layers mediated by a nonmagnetic metal has a high efficiency. The magnon valve ratio can be further improved via improving the spin conversion efficiency at the FMI/NM interface and optimizing the materials and thickness of FMI and space layers. Magnon valves could be used to manipulate the transmission coefficient of a magnon current, which has potential applications in magnon-based circuit, logic, memory,diode, transistor, waveguide and on-off switching devices etc. Utilizing magnetic insulators rather than magnetic metals for spintronic devices has superior advantages in terms of low energy consumption, and the present results open a door for fundamental research and device appli- cation beyond those based on conventional spin valve structures. We gratefully thank S. Zhang for enlightening discussions and theoretical help. This work was supported by theNational Key Research and Development Program ofChina [MOST, Grants No. 2016YFA0300802 andNo. 2017YFA0206200], the National Natural ScienceFoundation of China [NSFC, Grants No.11434014,No. 51620105004, and No. 11674373], and partially sup-ported by the Strategic Priority Research Program (B) [GrantNo. XDB07030200], the International Partnership Program (Grant No. 112111KYSB20170090), and the Key Research Program of Frontier Sciences (Grant No. QYZDJ-SSW- SLH016) of the Chinese Academy of Sciences (CAS). *Corresponding author. xfhan@iphy.ac.cn [1] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J.Chazelas, Phys. Rev. Lett. 61, 2472 (1988) . [2] G. Binasch, P. Grunberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989) . [3] T. Miyazaki and N. Tezuka, J. Magn. Magn. Mater. 139, L231 (1995) . [4] J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, Phys. Rev. Lett. 74, 3273 (1995) . 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PhysRevApplied.14.014037.pdf

PHYSICAL REVIEW APPLIED 14,014037 (2020) Nonlinear Control of Damping Constant by Electric Field in Ultrathin Ferromagnetic Films Bivas Rana ,1,*Collins Ashu Akosa ,1,2,†Katsuya Miura,3Hiromasa Takahashi,3Gen Tatara,1,4and YoshiChika Otani1,5 1RIKEN, Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan 2Department of Theoretical and Applied Physics, African University of Science and Technology (AUST), Km 10 Airport Road, Galadimawa, Abuja F.C.T, Nigeria 3Research and Development Group, Hitachi, Ltd., 1-280 Higashi-koigakubo, Kokubunji-shi, Tokyo 185-8601, Japan 4RIKEN Cluster for Pioneering Research (CPR), 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan 5Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan (Received 20 February 2020; revised 26 May 2020; accepted 19 June 2020; published 14 July 2020) The performances of many spintronic devices are governed by the damping constant and magnetic anisotropies of constituent materials. Spin-orbit coupling (SOC) plays a key role and is at the origins of these material parameters. Electric field control of magnetic anisotropy is in high demand for develop-ing energy-efficient nanoscale spintronics devices. Although electric field control of interfacial magnetic anisotropies is well studied and understood, the damping constant, on the other hand, is conventionally controlled by current-induced spin-orbit torque. Here, we use an alternative approach to demonstrate non- linear control of the damping constant in ultrathin ferromagnetic films by an electric field. We explicitly show that the presence of the Rashba SOC at a ferromagnet-insulator interface and the electric fielddependence of the Rashba coefficient may account for the observed nonlinear behavior. Furthermore, we show that engineering of the underlying and oxide material properties, i.e., bulk SOC and Rashba SOC, to tune the spin angular momentum relaxation pathways, can possibly increase the device functionalitysignificantly. DOI: 10.1103/PhysRevApplied.14.014037 I. INTRODUCTION Magnetic damping and anisotropy are the key material parameters that govern energy consumption, speed, and efficiency of many spintronic devices, such as magnetic random access memories, hard drives, nano-oscillators, sensors, logic gates, and transistors [ 1,2]. Higher or lower values of these parameters are desirable, depend- ing upon the requirement of the specific device applica- tion. For instance, magnetic elements with high magnetic anisotropies are required for the long-term thermal sta- bility of data in nanoscale storage devices, while a high damping constant is essential to reduce the data writing time in those devices. On the other hand, low damping and magnetic anisotropy are crucial in reducing the critical cur- rent densities for magnetization switching in data-storage devices [ 3,4] and for high-performance nano-oscillators and spin-wave- (SW) based magnonic devices [ 5–7]. It turns out that the quantum relativistic spin-orbit coupling *bivas.rana@riken.jp †collins.akosa@riken.jp(SOC) is at the origin of these physical parameters [ 5]. This is particularly intriguing because the damping and mag- netic anisotropy can be controlled via tuning the strength of SOC in the materials by material engineering [ 8] and/or external electric fields [ 9]. In particular, electric field con- trol of these parameters is in high demand for developing energy-efficient nanoscale spintronic devices. Recent studies on magnetic heterostructures composed of 3 dtransition metals (e.g., Co, Fe, and their alloys) and MOx(M=Mg, Al, Ta, Ru) show a high interfacial magnetic anisotropy (IMA) due to the hybridization of out- of-plane 3 dz2orbitals of Fe and out-of-plane 2 pzorbital of O[10,11]. In particular, (Co,Fe )B/MgO heterostructures, which constitute the basic building blocks in modern mem- ory devices, have attracted enormous interest, owing to their high tunneling magnetoresistance at room tempera- ture [ 12]. Interestingly, in addition to having a low damp- ing constant [ 13], the IMA in ultrathin (Co,Fe )Bfi l m s can be modulated by voltage, i.e., electric field, known as voltage-controlled magnetic anisotropy (VCMA), with- out the requirement for any charge current [ 14–16]. This VCMA is successfully used to control the coercive field 2331-7019/20/14(1)/014037(14) 014037-1 © 2020 American Physical SocietyBIVAS RANA et al. PHYS. REV. APPLIED 14,014037 (2020) [17]; magnetization [ 18]; magnetization switching [ 19]; magnetic phase [ 20]; domain-wall motion [ 21]; excitation [22], manipulation [ 23], and channeling of SWs [ 24]; and so on [ 25]. All of these reports show that VCMA is a promising tool for the development of all-electric-field- controlled SW-based devices, also known as magnonic devices [ 26]. It turns out that, although the VCMA phe- nomenon is well studied and understood, the study of voltage control of damping, on the other hand, is lacking, except for two experimental reports on metallic ferromag- nets [ 15] and a ferromagnetic semiconductor [ 27], showing linear modulation of damping with an electric field. In par- ticular, the nonlinear behavior of the damping constant has never been reported before. Here, the effect of applied electric field on the damp- ing constant of ultrathin (Co,Fe )B films is investigated by performing ferromagnetic resonance experiments. We demonstrate nonlinear control of damping in ultrathin ferromagnetic films by an electric field. In particular, we argue that an applied electric field modulates the Rashba spin-orbit coupling (RSOC) at the (Co,Fe )B/MgO interface, producing an electric-field-dependent additional angular momentum relaxation pathway, which may be the origin of our observed nonlinear behavior of the damp- ing constant. Furthermore, we show that the engineering of underlying material and oxide material properties can tune the Rashba coefficient, and hence, the spin angular momentum relaxation pathways, significantly. II. SAMPLE FABRICATION AND MEASUREMENT PROCEDURE The devices for our study are prepared in a multiple- step fabrication process. First, multilayer stacks are deposited on thermally oxidized Si (001)substrates by radiofrequency (rf) sputtering at room tempera- ture at a base pressure of about 10−8torr. The mul- tilayer stacks consist of the following layers from bottom to top: Ta (10)/Co20Fe60B20(t=2.2, 2.0, 1.8, 1.6)/MgO (2)/Al2O3(10), where the numbers in paren- theses indicate the nominal thicknesses of the corre- sponding layers in nanometers. The thick Al 2O3layer over MgO serves as a protecting layer to stop degra- dation of the MgO layer, and hence, (Co,Fe )B/MgO interface, due to the absorption of moisture. It also ensures almost perfect isolation of the top elec- trode from the bottom electrode [i.e., (Co,Fe )B] with- out tunneling any charge current across the junction. Some reference multilayers with stacking structures of Cu(10)/Co20Fe60B20(t=1.6)/MgO (2)/Al2O3(10),a n d Ta(10)/Co20Fe60B20(t=1.6)/Al 2O3(12) are also prepared. For simplicity, we refer to the abovementioned magnetic heterostructures as Ta /(Co,Fe )B(t)/MgO, Cu /(Co,Fe )B(t)/ MgO, and Ta /(Co,Fe )B(t)/Al 2O3, respectively, in the rest of the article. Before starting sample fabrication, thedeposited films are annealed at 280 °C in vacuum for 1 h under a perpendicular magnetic field of 600 mT. Anneal- ing is required to transform the as-deposited amorphous (Co,Fe )B films into crystalline (Co,Fe )B films with a bcc (001) structure. This helps to obtain the highest possible IMA with uniform distribution all over the film [ 28]. In the second step, rectangular structures with lateral dimensions of 200 ×12 µm2are prepared from the annealed multi- layer stacks by using maskless UV lithography followed by Ar+ion milling down to the substrate. In the third step, metal gate electrodes are prepared on top of the rectan- gular structures by maskless UV lithography followed by deposition of the Ti (5)/Au(100)layer by electron beam evaporation. In the same step, the contacts are also made at the edges of rectangular structures to measure inverse spin Hall effect (ISHE) signals. In the fourth step, the 180-nm-thick Al 2O3layer is deposited by rf magnetron sputtering everywhere, except on top of the contacts, to measure ISHE signals, and on micrometer-size rectangular areas on top of the metal gate electrodes. The uncov- ered areas on top of the metal gate electrodes allow us to make contacts for the application of the dc gate volt- age in the final stage of fabrication. Finally, contacts for the application of dc gate voltage and microwave anten- nae for the excitation of uniform ferromagnetic resonance (UFMR) are prepared by maskless UV lithography and deposition of the Ti (5)/Au(200)layer by electron beam evaporation. Notably, the microwave antennae are electri- cally isolated from rectangular (Co,Fe )B strips and from the top metal gate electrodes by the 180-nm-thick Al 2O3 layer. The microwave antennae are designed in such a way(widths of the signal and ground lines, W=12 µm; edge- to-edge separation between the arms, S=26 µm) that the generated Oersted field becomes almost uniform all over the rectangular structure, which is essential for exciting UFMR [Fig. 1(a)]. To ensure the uniformity of the Oersted field on the sample, the lateral dimension (12 µm) of the rectangular-shaped sample is designed to be significantly smaller than that of the edge-to-edge separation (26 µm) between the signal and ground lines, and the rectangular- shaped magnetic sample is placed at the center between signal and ground lines. We study the UFMR to characterize the IMA, VCMA coefficient β, and damping constants αof the (Co,Fe )B films used here. Figure 1(a) represents the schematic dia- gram of such a device and experimental setup for UFMR measurements. Figure 1(b) shows an expanded view of the rectangular multilayer structure and the mechanism for ISHE detection of the resonance signal under the applica- tion of a dc gate voltage (V G). A signal generator is used to send rf current ( Irf) through the micrometer-sized antenna surrounding the rectangular-shaped (200 ×12 µm2) mag- netic film [Fig. 1(a)]. This rf current through the antenna generates a microwave magnetic field ( hrf) perpendicular to the film plane. To obtain the largest possible signal 014037-2NONLINEAR CONTROL OF DAMPING . . . PHYS. REV. APPLIED 14,014037 (2020) (a) (b)(c) FIG. 1. (a) Schematic illustration of device structure and experimental setup for UFMR measurements. Radiofrequency current (Irf) is sent through a micrometer-sized antenna surrounding the rectangular-shaped multilayer film. Irf-induced Oersted field ( hrf) excites the UFMR in (Co,Fe )B film at resonance condition, as given by Eq. (2). (b) Schematic diagram of an expanded view of the rectangular multilayer structure and mechanism for ISHE detection of the resonance signal under application of dc gate voltage (VG). (c) Representative ISHE signal measured as a function of Hfrom Ta /(Co,Fe )B(2.0)/MgO film. Solid curves represent fitting with Eq.(1). detected through the ISHE, the magnetizations of the (Co,Fe )B films are set along the short axis of the rect- angular structures (along the yaxis) by applying a bias magnetic field ( H) from an electromagnet [ 29].His swept from−320 to +320 mT in 0.2–0.8 mT steps, while keep- ing the frequency of the rf current constant. The field stepsare decided by the line width of the measured UFMR signal. At the resonance condition [Eq. (2)], a signifi- cantly large pure spin current ( I S) is pumped from the (Co,Fe )B layer into the adjacent Ta (or Cu) layer [Fig. 1(b)].ISis converted into a transverse charge current ( IC) through the ISHE of Ta (or Cu), and the ensuing sig- nal is obtained by measuring the potential drop (VISHE) across the heterostructures by a nanovoltmeter. Although part of the pure spin current ISis also pumped towards the (Co,Fe )B/MgO interface and converted into the charge current ICby the inverse Edelstein effect, we do not show this in the schematic diagram for simplicity. To study the effect of voltage, i.e., electric field on IMA and damping constant, a dc gate voltage (VG)is applied across the top metal gate electrode and the (Co,Fe )B layer by using a dc power source. Here, positive VGmeans the top electrode has a positive potential with respect to the (Co,Fe )Bfi l m . The applied VGgenerates a uniform electric field EGat the (Co,Fe )B/MgO interface and modulates IMA and α. III. MEASUREMENT OF IMA AND DAMPING CONSTANT WITHOUT GATE VOLTAGE In Figure 1(c), a representative ISHE signal VISHEmea- sured from the Ta /(Co,Fe )B(2.0)/MgO film is shown. Therf power is set at 8 dBm (i.e., 6.3 mW), which is low enough to excite UFMR in the linear regime [see Fig. 10(a) of Appendix A]. Opposite signs of VISHE for oppo- site polarities of Hprove that the signal originates from spin pumping (SP) and the ISHE [ 30]. To extract values of the resonance field (H0), resonance line width, and signal amplitude, the ISHE signals are fitted with a mathematical expression, where VISHE is expressed as a linear combi- nation of symmetric and antisymmetric Lorentzian terms given by [ 23,31,32] VISHE=V0+Vs 1+(H−H0)2/σ2+Va(H−H0)/σ 1+(H−H0)2/σ2. (1) Here, V0is the dc background of VISHE;Vsand Vaare the weights of the symmetric and antisymmetric Lorentzian functions, respectively; and σis the half width at half maximum (HWHM) of the resonance spectrum. The near- perfect symmetric Lorentzian shape of the ISHE signal ensures that the UFMR is predominantly excited by the out-of-plane component of the rf Oersted field [ 29]. The solid curve in Fig. 1(c)represents the fitting with Eq. (1). To extract the values of IMA, the ISHE signals are measured for each (Co,Fe )B film for different values of microwave frequency f. In Figs. 2(a)–2(d), we show the ISHE signals measured for Ta /(Co,Fe )B(t=2.2, 2.0, 1.8, 1.6)/MgO films. Here, the ISHE signals are shown only for negative Hfor the sake of simplicity. The signal-to-noise ratios for all measured ISHE signals are high enough to extract the resonance fields and line widths. The resonance 014037-3BIVAS RANA et al. PHYS. REV. APPLIED 14,014037 (2020) (a) (b) (c) (d) FIG. 2. ISHE signals measured for (a) Ta /(Co,Fe )B(2.2)/MgO, (b) Ta /(Co,Fe )B(2.0)/MgO, (c) Ta /(Co,Fe )B(1.8)/MgO, and (d) Ta/(Co,Fe )B(1.6)/MgO films are represented for different values of microwave frequencies f. spectra are fitted (not shown) with Eq. (1)to determine the resonance fields for each value of f. After that, the resonance frequencies, fUFMR , are plotted as a function of magnetic field, H, and subsequently fitted with the Kittel formula [Fig. 3(a)], as follows [ 22]: fUFMR=/parenleftBigμ0γ 2π/parenrightBig {H[H+Ms−Hi(VG)]}1/2.( 2 ) Here, γis the gyromagnetic ratio, Msis the satura- tion magnetization, and Hiis the IMA field. We adopt γ=29.4 GHz T−1andμ0Ms=1.5 T from Refs. [ 22–24], while setting μ0Hias a free parameter for fitting. The extracted values of μ0Hiare 1.04 ±0.01, 1.15 ±0.01,1.30±0.01, and 1.47 ±0.01 T for Ta /(Co,Fe )B(t)/MgO films with t(Co,Fe)B =2.2, 2.0, 1.8, and 1.6 nm, respec- tively. The increment of μ0Hiwith the reduction of t(Co,Fe)B confirms the interfacial origin of magnetic anisotropy. As μ0Ms>μ 0Hifor all (Co,Fe )B films, the easy axis of magnetization lies in the plane of the films. This is fur- ther confirmed from the anomalous Hall effect (AHE) measurements (see Appendix Bfor more details). The line width (σ) of resonance spectra can be expressed in the following way [ 8]: σ=σ0+2πα γfUFMR .( 3 ) (a) (b) FIG. 3. (a) Plot of the resonance frequencies as a function of Hfor Ta/(Co,Fe )B(2.2, 2.0, 1.8, 1.6 )/MgO, Cu /(Co,Fe )B(1.6)/MgO, and Ta /(Co, Fe )B(1.6)/Al2O3films. Solid curves represent fittings with the Kittel formula [Eq. (2)]. (b) HWHMs of the resonance spectra measured for Ta /(Co,Fe )B(2.2, 2.0, 1.8, 1.6 )/MgO, Cu /(Co,Fe )B(1.6)/MgO, and Ta /(Co, Fe )B(1.6)/Al2O3films are plotted as a function of f. Solid lines represent linear fittings. 014037-4NONLINEAR CONTROL OF DAMPING . . . PHYS. REV. APPLIED 14,014037 (2020) Here, σ0is the frequency-independent line width, which originates from the inhomogeneous distribution of mag- netic properties of the ferromagnetic film, nonuniformity of the Oersted field hrf, and interfacial scattering. The second term is proportional to fUFMR and the damping constant α. The second term originates from the relax- ation of spin angular momentum through the intrinsic bulk SOC of the ferromagnetic film and through SP into the adjacent heavy metallic layer with a high SOC and through an interfacial RSOC. Hence, to evaluate α, the extracted values of HWHM of the resonance spectra are plotted as a function of fand fitted with a linear func- tion [Fig. 3(b)]. The values of αare then extracted from the slopes ( /Delta1) of the linear fittings by using the following expression [ 32]: α=γ 2π/Delta1.( 4 ) The extracted values of αfor Ta /(Co,Fe )B(t)/MgO films with t(Co,Fe)B =2.2, 2.0, 1.8, and 1.6 nm are 0.012 ± 0.001, 0.013 ±0.001, 0.016 ±0.001, and 0.022 ±0.001, respectively. The monotonic enhancement of αfor thin- ner films is due to the increment of the SP contribution [13,15,33], which will be shown in Sec. V.IV . V ARIATION OF IMA AND DAMPING CONSTANT WITH GATE VOLTAGE To evaluate the variation of the IMA and the damping constant with gate voltage (VG), the UFMR signals are measured from (Co,Fe )B films for various values of fand VG. The resonance spectra are fitted with Eq. (1)to deter- mine the resonance field and line width for each value of fand VG. Then, the resonance frequencies are plotted as a function of magnetic field, H, and values of HWHMs are plotted as a function of ffor each value of VGand subse- quently fitted with the Kittel formula [Eq. (2)] and a linear function to determine values of IMA and damping con- stant, respectively. Figures 4(a)–4(d) show some represen- tative resonance signals measured from Ta /(Co,Fe )B(2.2, 2.0, 1.8, 1.6)/MgO films for five values of VG.I ti s observed that the resonance fields are changed monoton- ically with VGfor all films due to the modulation of IMA by VG. In Figs. 5(a)–5(d), changes in the IMA field μ0Hi, i.e., d(μ0Hi), as a function of VGare plotted for four different Ta /(Co,Fe )B(t)/MgO films. It is observed that μ0Hivaries linearly with VGfor all thicknesses of (Co,Fe )Bfi l m s[ 14,15], and the slopes of the linear variation, i.e., the variation of μ0Hiper unit VGgives the value of the VCMA coefficient, β. The extracted values of βare(−4.7±0.2),(−5.7±0.1),(−6.8± 0.2),a n d(−7.1±0.1)mT V−1for t(Co,Fe)B =2.2, 2.0, (a) (b) (c) (d) FIG. 4. ISHE signals measured from (a) Ta /(Co,Fe )B(2.2)/MgO, (b) Ta /(Co,Fe )B(2.0)/MgO, (c) Ta /(Co,Fe )B(1.8)/MgO, and (d) Ta/(Co,Fe )B(1.6)/MgO films for five different values of VG. Solid curves represent the fittings with Eq. (1). 014037-5BIVAS RANA et al. PHYS. REV. APPLIED 14,014037 (2020) (a) (b) (c) (d) (e) (f) (g) (h) FIG. 5. Change in IMA fields, i.e., d(μ0Hi)with VGare plotted for Ta /(Co,Fe )B(t)/MgO films with t(Co, Fe)B=2.2 (a), 2.0 (b), 1.8 (c), and 1.6 nm (d). Solid lines represent linear fittings. Variation of αwith VGare plotted for Ta /(Co,Fe )B(t)/MgO films with t(Co, Fe)B=2.2 (e), 2.0 (f), 1.8 (g), and 1.6 nm (h). Solid curves represent fittings with Eq. (5). 1.8, and 1.6 nm, respectively. The linear variations of IMA of (Co,Fe )B films with the gate voltage were also reported previously [ 14,15]. This is related to the change in the number of electrons in the 3 dorbitals of the ferromagnet at the ferromagnet-insulator interface, which is shown to be linearly proportional to VG[34]. Therefore, the change in the magnetic moment, i.e., surface or interfacial mag- netization, and IMA should also be linearly proportional toVG[35,36]. The change in interfacial magnetization per unit VGis known as the surface magnetoelectric (ME) coefficient (β/prime). In Figs. 5(e)–5(h), the extracted values of αare plotted as a function of VG, which show that the variation of α with VGis gradually transformed from linear to nonlinear behavior as t(Co,Fe)B decreases from 2.2 to 1.6 nm. We note here that the resonance line width (HWHM) at f=0 GHz, i.e., frequency-independent line width (σ0)remains almost unchanged with VG(see Fig. 12of Appendix Cfor details). In a previous study, the linear variation of IMA and αwith VGwas reported for Ta /Ru/Ta/(Co, Fe )B/MgO/Al2O3 multilayers [ 15], showing negligible modulation of αby VGfor (Co,Fe )B films thicker than 1.5 nm. Interestingly, the variation of αwith VGis reported to be linear, even for thinner (Co,Fe )B films. Therefore, we note here that our observation of nonlinear behavior of αhas never been reported before. One possible reason behind this may be the different qualities of the deposited films and inter- faces, which may significantly affect the bulk SOC, RSOC, and hence spin angular momentum relaxation mechanisms. Another reason may be a different studied range of EG and different numbers of measured data points. Many data points in our study are suitable to demonstrate thenonlinear feature of α. From our experimental observation, we assume that the nonlinear variation of αwith VGcan be mathematically expressed by α(VG)=α0+α1VG+α2V2 G,( 5 ) where α0is the damping constant at VG=0V ,a n d α1and α2are the coefficients for the linear and quadratic variation ofαwith VG, respectively. The αversus VGdata points are fitted nicely with Eq. (5), as represented by the solid curves in Figs. 5(e)–5(h), and the extracted parameters from the fittings are plotted in Figs. 6(a)–6(c). It is observed that α0increases monotonically with the decrease of t(Co,Fe)B . α1also increases monotonically with the decrease of t(Co,Fe)B , with the exception of the Ta /(Co,Fe )B(1.6)/MgO film. Although α2is found to be negligibly small for Ta/(Co,Fe )B(2.2, 2.0)/MgO films, it increases signifi- cantly for thinner (Co,Fe )B films. Interestingly, α1and α2both have negative signs for all Ta /(Co,Fe )B(t)/MgO films. To elucidate the origin of this nonlinear behavior of damping, we develop a simplified theoretical model, as discussed in Sec. V. V . THEORETICAL ARGUMENT Our key assumption, based on previous studies [ 37–40], is that there is RSOC at the (Co,Fe )B/MgO interface that can be controlled by VG. When magnetization of the (Co,Fe )B layer is set into precession due to UFMR, it pumps a pure spin current (Is)towards both interfaces: Ta/(Co,Fe )B (heavy metal/ferromagnet, i.e., HM/FM) and (Co,Fe )B/MgO (ferromagnet/insulator, i.e., FM/I) 014037-6NONLINEAR CONTROL OF DAMPING . . . PHYS. REV. APPLIED 14,014037 (2020) (a) (b) (c)(d) FIG. 6. (a) Extracted values of α0(a),α1(b), and α2(c) for Ta /(Co,Fe )B(t)/MgO, Cu /(Co,Fe )B(1.6)/MgO, and Ta/(Co, Fe )B(1.6)/Al2O3films. (d) Schematic diagram showing the relaxation mechanism of spin angular momentum through (1) SP into an adjacent underlayer, such as Ta or Cu; (2) spin relaxation at the (Co,Fe )B/MgO Rashba interface and transmission of the relaxation effect to the Ta /(Co,Fe )Bo rC u /(Co, Fe )B interface due to long-range diffusive motion of electrons. [Fig. 6(d)]. We identify two pathways via which spin angu- lar momentum can be dissipated in our system. First, spin current pumped towards the Ta /(Co,Fe )B interface results in spin angular momentum dissipation via conventional SP [41–44] through the bulk SOC of Ta. The enhancement of αdue to SP into Ta can be expressed as [ 45] αTa/(Co,Fe)B eff=gμB 4πt(Co,Fe)B AG↑↓ Ms.( 6 ) Here, gis the Landé g-factor of an electron, μBis the Bohr magnetron, t(Co,Fe)B is the thickness of the (Co,Fe )B layer, Ais the area of the Ta /(Co,Fe )B interface, Msis the saturation magnetization, and G↑↓is the spin mixing conductance. There are two possible origins of voltage- dependent behavior of α: the saturation magnetization and the spin-mixing conductance. The saturation magnetiza- tion of an ultrathin ferromagnet is expected to depend upon the voltage, and theory predicts a linear depen- dence on VG[35,36], so that Mscan be written in the following way: Ms(VG)=Ms(0)+ΔMs=Ms(0)+β/primeVG.( 7 ) Here, Ms(0)is the saturation magnetization at VG=0 andβ/primeis the ME coefficient. The ME coefficient (β/prime)is defined as the change in saturation magnetization per unit of gate voltage VG, and hence, it is different from the VCMA coefficient (β), which is the change in IMA field or IMA energy per unit of VG. After substituting Eq. (7) into Eq. (6), it can be seen that the first relaxation chan- nel cannot explain the observed nonlinear behavior of the damping constant. The spin-mixing conductance is a parameter represent- ing the transparency of spin transmission through theHM/FM interface and is determined by the interface prop- erty at the atomic scale. Since the penetration depth of EGin the FM is very short ( ∼0.3 nm), it is expected that the spin current pumped into the Ta /(Co,Fe )B interface is well screened from the effect of VGand will have no direct effect on the atomic properties of the (Co,Fe )B/MgO interface. However, in the presence of spin-relaxation pro- cesses for the conduction electrons in the (Co,Fe )B/MgO interface, these relaxation effects will be transmitted to the Ta/(Co,Fe )B interface as a result of long-range diffusive motion of electrons. Here, the source of relaxation is the RSOC [ 46], which is known to exist at the (Co,Fe )B/MgO interface [ 37–40] and is sensitive to VG, since it arises from the breaking of inversion symmetry. This is the second relaxation channel of spin angular momentum, as repre- sented in Fig. 6(d). Although spin relaxation due to RSOC at a spatially separated interface has not been investigated, the relaxation effect due to the FM/HM interface itself, called the spin-memory-loss effect, was studied theoreti- cally in Ref. [ 47]. There, it was shown that RSOC reduced the spin-mixing conductance by a factor proportional to the second order of Rashba strength. Here, we assume that the spin memory loss at the FM/I interface causes qualitatively a similar (but maybe weakened) effect in spin-mixing conductance. So, the effective spin-mixing conductance, G↑↓ eff, of the Ta/(Co,Fe )B interface can be written as G↑↓ eff=G↑↓ 0/parenleftbigg 1−4me /planckover2pi12γ2 R EF/parenrightbigg ,( 8 ) where γRis the effective strength of the interfacial RSOC, i.e., Rashba coefficient acting on the HM/FM interface induced by the FM/I interface, EFis the Fermi energy, meis the effective mass of electrons, and /planckover2pi1is the reduced 014037-7BIVAS RANA et al. PHYS. REV. APPLIED 14,014037 (2020) Planck constant. The RSOC is modulated by the electric field induced by VG, and the dependence is expected to be linear, as the second-order effect cannot contribute by sym- metry, and, even if it exists, the effect would be negligibly small for the range of applied voltage. Thus, the Rashba coefficient, γR, can be expressed as a function of VGgiven by [37] γR≈γ0 R+γ1 RVG.( 9 ) Taking into account the two abovementioned voltage- dependent modulations, the effective mixing conductance becomes G↑↓ eff=G↑↓ 0/parenleftbigg 1−4me EF/planckover2pi12γ0 R2−8me EF/planckover2pi12γ0 Rγ1 RVG −4me EF/planckover2pi12γ1 R2VG2/parenrightbigg . (10) Thus, the enhancement of damping due to SP can be written as α(VG)≈α0+α1VG+α2V2 G.( 5 )Here α0=gμB 4πt(Co,Fe)B AG↑↓ 0 Ms(0)/parenleftbigg 1−4me EF/planckover2pi12γ0 R2/parenrightbigg , (11) α1=gμB 4πt(Co,Fe)B AG↑↓ 0 Ms(0) ×/bracketleftBigg 4me EF/planckover2pi12γ0 R2β/prime Ms(0)−8me EF/planckover2pi12γ0 Rγ1 R−β/prime Ms(0)/bracketrightBigg , (12) and α2=gμB 4πt(Co,Fe)B AG↑↓ 0 Ms(0)×/bracketleftbigg8me EF/planckover2pi12β/primeγ0 Rγ1 R Ms(0)−4me EF/planckover2pi12γ1 R2/bracketrightbigg . (13) α0increases monotonically with the decrease of t(Co,Fe)B , which may be due to the increment of SP into Ta /(Co,Fe )B and (Co,Fe )B/MgO interfaces. The monotonic incre- ment of α1with the decrease of t(Co,Fe)B may be due to the increment of the interfacial ME effect (β/prime), spin-flip back scattering from the (Co,Fe )B/MgO inter- face, and voltage control of γR. However, the defects and/or inhomogeneity present at the (Co,Fe )B/MgO inter- face may significantly affect the RSOC, its EGcontrol, (a) (b) (c) (d) FIG. 7. ISHE signals measured for (a) Cu /(Co,Fe )B(1.6)/MgO, and (b) Ta /(Co, Fe )B(1.6)/Al2O3films for different values of microwave frequencies f. ISHE signals measured for (c) Cu /(Co,Fe )B(1.6)/MgO, and (d) Ta /(Co, Fe )B(1.6)/Al2O3films plotted for five different values of VG. Solid curves represent fittings with Eq. (1). 014037-8NONLINEAR CONTROL OF DAMPING . . . PHYS. REV. APPLIED 14,014037 (2020) and corresponding loss of spin angular momentum. We estimate that the interfacial roughness of the Ta/(Co,Fe )B(1.6)/MgO film seems to be increased com- pared with other Ta /(Co,Fe )B(t)/MgO films, which is indirectly confirmed by extracting the resonance line width (HWHM) at f=0. This is achieved by plotting HWHM versus fdata points and extrapolating the lin- ear fitting line to f=0 [Fig. 3(b)]. Therefore, α1for the Ta/(Co,Fe )B(1.6)/MgO film is significantly smaller than that in the other films. The increment of α2for lower thick- nesses of (Co,Fe )B films may be due to the increment of the (Co,Fe )B/MgO interfacial contribution. We believe thatα2mainly originates from the spin angular momen- tum relaxation at the (Co,Fe )B/MgO Rashba interface. Notably, we do not observe a significant reduction of α2for the Ta /(Co,Fe )B(1.6)/MgO film, unlike α1. This may be because α2is not simply proportional to the Rashba coef- ficient (γ R). The value of α2is decided by the values of β/prime,γ0 R,a n dγ1 Rrelative to each other, as can be seen from Eq.(13). VI. STUDY OF REFERENCE SAMPLES Finally, we study two reference samples. In one sam- ple, the Ta layer is replaced by Cu and in another sample the MgO layer is replaced by Al 2O3, while keeping t(Co,Fe)B =1.6 nm. The purpose of studying the Cu/(Co,Fe )B(1.6)/MgO film is to verify the nonlin- ear dependence of αwith VGdue to SP at the(Co,Fe )B/MgO interface by suppressing spin relaxation through SP into Cu, whereas the purpose of studying the Ta /(Co, Fe )B(1.6)/Al2O3film is to verify the role of the (Co,Fe )B/MgO Rashba interface on the non- linear behavior of α. Figures 7(a) and 7(b) present ISHE signals measured for Cu /(Co,Fe )B(1.6)/MgO and Ta/(Co, Fe )B(1.6)/Al2O3films, respectively, for different values of microwave frequencies f. We notice that the ISHE signals measured for the Cu /(Co,Fe )B(1.6)/MgO film have the opposite sign to that of the Ta /(Co,Fe )B(t)/ MgO and Ta /(Co, Fe )B(1.6)/Al2O3films. Also, at the same rf power, the amplitude of VISHE for the Cu/(Co,Fe )B(1.6)/MgO, film is much smaller than that of the Ta /(Co,Fe )B(t)/MgO and Ta /(Co, Fe )B(1.6)/Al2O3 films. This is because Cu is known as a light mate- rial with a very weak bulk SOC. The spin Hall angle (SHA) of Cu has a much smaller and opposite sign value to the SHA of the heavy metal Ta. Fig- ures 7(c) and 7(d) present ISHE signals measured for Cu/(Co,Fe )B(1.6)/MgO, and Ta /(Co, Fe )B(1.6)/Al2O3 films for five different values of VG. The solid curves repre- sent fittings with Eq. (1). The extracted values of IMA and damping constants at VG=0f o rC u /(Co,Fe )B(1.6)/MgO and Ta /(Co, Fe )B(1.6)/Al2O3films are 1.15 ±0.01 T and 0.006 ±0.001 and 0.76 ±0.01 T and 0.020 ±0.001, respectively. Figures 8(a) and 8(b) show that the IMA varies linearly with VGfor both films with VCMA coefficients ( β)o f(−4.8±0.2)and(−0.5± 0.9)mT V−1, respectively. The lower value of βfor (a) (b) (c) (d)FIG. 8. Change in the IMA fields, i.e., d(μ0Hi), with VGare plotted for (a) Cu/(Co,Fe )B(1.6)/MgO, and (b) Ta /(Co, Fe )B(1.6)/Al2O3 films. Solid lines represent linear fittings. Variations of αwith VGare plotted for (c) Cu/(Co,Fe )B(1.6)/MgO, and (d) Ta/(Co, Fe )B(1.6)/Al2O3films. Solid curves represent fittings with Eq. (5). 014037-9BIVAS RANA et al. PHYS. REV. APPLIED 14,014037 (2020) (a) (b)FIG. 9. (a) Plot of damping constant αversus gate voltage VGfor Ta /(Co,Fe )B(1.6)/MgO film with opposite polarities of H. (b) Static I-Vchar- acteristic of a device withTa/(Co,Fe )B(1.6)/MgO film. Inset shows the experimental setup for the measurement ofstatic I-Vcharacteristics. the Cu /(Co,Fe )B(1.6)/MgO film than that of the Ta/(Co,Fe )B(1.6)/MgO film proves the important role of the Ta underlayer to enhance the value of β at the (Co,Fe )B/MgO interface. The negligible value ofβfor the Ta /(Co, Fe )B(1.6)/Al2O3film confirms the important role of the (Co,Fe )B/MgO Rashba interface on β. Interestingly, αvaries nonlinearly with VGfor the Cu /(Co,Fe )B(1.6)/MgO film [Fig. 8(c)], whereas αshows a linear dependence with VGfor the Ta/(Co, Fe )B(1.6)/Al2O3film [Fig. 8(d)]. This certainly proves our theory that only the (Co,Fe )B/MgO Rashba interface contributes to α2, i.e., the nonlinear behavior of damping with VGoriginates from the (Co,Fe )B/MgO Rashba interface. By fitting αversus VGdata points with Eq. (5),w e find that α0for the Cu /(Co,Fe )B(1.6)/MgO film is sig- nificantly reduced, while for the Ta /(Co, Fe )B(1.6)/Al2O3 filmα0is close to the Ta /(Co,Fe )B(1.6)/MgO film [see Fig. 6(a)], and the primary reason behind this is the much lower value of SHA for Cu compared with that of Ta. α1has relatively large value for the Cu/(Co,Fe )B(1.6)film, whereas it is significantly lower for the Ta /(Co, Fe )B(1.6)/Al2O3film compared with that of the Ta /(Co,Fe )B(1.6)/MgO film [see Fig. 6(b)]. One possible reason may be different values of β/prime, γ0 R,a n d γ1 Rfor (Co,Fe )B films with different under- layer and oxide layer materials. Interestingly, α2has a positive value for the Cu /(Co,Fe )B(1.6)/MgO film as opposed to that of the Ta /(Co,Fe )B(1.6)/MgO film [see Fig. 6(c)]. Asα2predominantly originates from the spin angular momentum relaxation at the (Co,Fe )B/MgO interface, we expect a negative sign of α2, simi- lar to that for Ta /(Co,Fe )B(1.6)/MgO. However, from Eq. (13), we find that the values of β/prime,γ0 R,a n d γ1 R, relative to each other, play a key role in defin- ing the sign and magnitude of α2. More studies arerequired to understand the detailed underlying mech- anism. Nevertheless, the observation of the nonlinear variation of αwith VGfor the Cu /(Co,Fe )B(1.6)/MgO film and the absence of a nonlinear variation of α with VGfor the Ta /(Co, Fe )B(1.6)/Al2O3film con- firms its origin from the (Co,Fe )B/MgO Rashba inter- face. We also measure the variation of αas a function of VG for the Ta /(Co,Fe )B(1.6)/MgO film for opposite polarities of bias magnetic field, H. Figure 9(a) shows that the variation of αwith VGis almost the same for opposite polarities of H. This confirms the absence of any spin- current-induced modulation of α(e.g., by dampinglike torque), which should be linear with charge current, i.e., VG, and should show the opposite trend for opposite polar- ities of H[31]. Modulation of αwith charge current (i.e., spin-polarized current) may be possible in the presence of a substantial charge current across the tunnel junc- tion while applying VG. However, the I-Vcharacteristics for our device [Fig. 9(b)] show that the tunnel current through the junction is too small to produce any noticeable change in α. This certainly confirms that the modula- tion of αis purely caused by the electric field. We also note that the static I-Vcharacteristics for other devices used for gate-voltage-dependent damping measurements are similar to the device with the Ta /(Co,Fe )B(1.6)/MgO film. VII. CONCLUSIONS We study the effect of electric field on αof ultrathin (Co,Fe )B films by performing UFMR through SP and ISHE techniques. We observe a nonlinear variation of α with VG, especially for ultrathin (Co,Fe )B films, in spite of a linear variation of IMA with VG. By using a the- oretical model, we explain that there are basically two 014037-10NONLINEAR CONTROL OF DAMPING . . . PHYS. REV. APPLIED 14,014037 (2020) channels for spin angular momentum relaxation. First, is SP into the adjacent HM and spin angular momen- tum relaxation by spin-flip scattering through the bulk SOC of the HM. The second channel is spin angular momentum relaxation by the RSOC at the (Co,Fe )B/MgO interface and transmission of the relaxation effect to the Ta/(Co,Fe )B interface due to long-range diffusive motion of electrons. This mechanism contributes to the linear and quadratic modulation of αby VG. We attribute that this may originate from the linear modulation of the Rashba coefficient by VG. This is verified by studying a reference sample made of the Cu /(Co, Fe )B(1.6)/MgO/Al2O3 heterostructure, where the spins are mostly allowed to relax at the (Co,Fe )B/MgO Rashba interface by sup- pressing relaxation through SP into the adjacent light metal. By studying another reference sample made of the Ta /(Co, Fe )B(1.6)/Al2O3heterostructure, we find that the absence of the (Co,Fe )B/MgO Rashba inter- face significantly reduces the IMA and VCMA coef- ficient and shows a linear variation of αwith VG. This confirms that the nonlinear variation of αwith VGoriginates from the relaxation of spin current at the (Co,Fe )B/MgO Rashba interface. Our study opens up a research direction to control the Rashba coefficient andαby engineering the underlayer and oxide material properties. ACKNOWLEDGMENTS The authors thank Y. Fukuma, F. Mahfouzi, and F. Ishii for fruitful discussions. This work is supported by a Grant-in-Aid for Scientific Research on Innovative Area, “Nano Spin Conversion Science” (Grant No. 26103002)and a Grant-in-Aid for Scientific Research on Innovative Areas (Grant No. 26103006) from the Ministry of Edu- cation, Culture, Sports, Science and Technology (MEXT) of Japan. B.R. acknowledges RIKEN Incentive Research Project Grant No. FY2019. C.A.A. and G.T. acknowledge support by a Grant-in-Aid for Exploratory Research (Grant No. 16K13853) and a Grant-in-Aid for Scientific Research (B) (Grant No. 17H02929) from the Japan Society for the Promotion of Science (JSPS). APPENDIX A: MICROW A VE POWER DEPENDENCE OF ISHE SIGNAL AMPLITUDE We measure ISHE signals for different values of applied microwave power ( Prf) for (Co,Fe )B films with Ta and Cu underlayers, i.e., for Ta /(Co,Fe )B(t)/MgO and Cu /(Co,Fe )B(1.6), films. In Fig. 10(a) , we plot the amplitudes of ISHE signals as a function of Prf for Ta /(Co,Fe )B(2.0, 1.6 )/MgO films. In both cases, the amplitudes of the ISHE signals vary linearly with Prf, within the range of our measurements. We also check that the resonance line width [as plotted in Fig. 10(b) ] does not change with Prfwithin this range. These behaviors confirm that the UFMRs are excited in the linear regime for all Ta /(Co,Fe )B(t)/MgO films of different thicknesses of (Co,Fe )B ,a tl e a s tu pt o1 0d B m( i . e . ,1 0m W )o f microwave power [ 16,29]. Therefore, we decide to use an optimum value of microwave power, Prf=6 . 4m W( i . e . , 8 dBm), for our study, which is well below the nonlinear regime of excitation, but high enough to get ISHE sig- nals with good signal to noise ratios. The expected value of the Oersted field ( hrf)a t Prf=6.4 mW is about 1.9 Oe (a) (c) (b) (d)FIG. 10. (a) Amplitudes of ISHE signals and (b)HMHM, measured for Ta/(Co,Fe )B(2.0, 1.6 )/MgO films, are plotted as a functionof microwave power P rf.V e r - tical dotted lines represent the microwave power (6.3 mW,i.e., 8 dBm) used for the mea- surements of ISHE signals for Ta/(Co,Fe )B(t)/MgO films. (c) Amplitudes of ISHE signals and (d) HMHM, measured for Cu/(Co,Fe )B(1.6)/MgO film, are plotted as a function of microwave power P rf. Verti- cal dotted lines represent themicrowave power (100 mW, i.e., 20 dBm) used for mea- surements of ISHE signals forCu/(Co,Fe )B(1.6)/MgO film. 014037-11BIVAS RANA et al. PHYS. REV. APPLIED 14,014037 (2020) (a) (b)FIG. 11. (a) Schematic diagram of device structure and experi- mental setup for measuring AHEsignals from (Co,Fe )Bfi l m s . (b) Measured AHE sig- nals (normalized) fromTa/(Co,Fe )B(t)/MgO films with t (Co, Fe)B=2.2, 2.0, 1.8, and 1.6 nm. [48]. In Fig. 10(c) , we plot the amplitudes of ISHE sig- nals as a function of Prffor the Cu /(Co,Fe )B(1.6)/MgO film. Also, in this case, the amplitudes of ISHE signals vary linearly with Prf, within the range of our measure- ments. We check that the resonance line width [as plotted in Fig. 10(d) ] does not change with Prfwithin this range. We also note that the amplitudes of ISHE signals for the Cu/(Co,Fe )B(1.6)/MgO film are much lower than the amplitudes of ISHE signals for the Ta /(Co,Fe )B(t)/MgO films, mainly due to the lower value of the spin Hall angle of Cu. Therefore, we decide to use a slightly higher value of microwave power, Prf=100 mW (i.e., 20 dBm), for the study of ISHE signals of the Cu /(Co,Fe )(1.6)film to obtain the required signal to noise ratio. A microwave power of Prf=63 mW (i.e., 18 dBm) is used to study ISHE signals of the Ta /(Co, Fe )B(1.6)/Al2O3film. The expected value of the Oersted field ( hrf)a t Prf=100 mW is about 7.4 Oe and at Prf=63 mW is about 5.9 Oe. APPENDIX B: AHE MEASUREMENT OF (Co,Fe)B FILMS We measure AHE signals for Ta /(Co,Fe )B(t)/MgO films with t(Co, Fe)B=2.2, 2.0, 1.8, and 1.6 nm. For thismeasurement, (Co,Fe )B films are designed to be like Hall bar structures with dimensions of 80 ×400 µm2, as shown in Fig. 11(a) . A dc current ( Idc) of 0.5 mA is sent through the Hall bar from a current source and the transverse Hall voltage ( VAHE) is measured by a nanovoltmeter, while sweeping the out-of-plane magnetic field ( Hz)f r o m −800 to+800 mT. Measured AHE signals as a function of Hz are plotted in Fig. 11(b) for t(Co, Fe)B=2.2, 2.0, 1.8, and 1.6 nm. The absence of a hysteresis loop and gradual switching of magnetization confirm the in-plane easy axis of magnetization. APPENDIX C: RESONANCE LINE WIDTH AT f=0 GHz AS A FUNCTION OF GATE VOLTAGE VG Figure 12shows that the resonance line width remains unchanged with VG.A s VGonly modulates the interfacial properties, therefore, we also do not expect any variation of the frequency-independent line width originating from the inhomogeneous distribution of magnetic properties of the ferromagnetic film and nonuniformity of the Oersted field, hrf. (a) (b) FIG. 12. 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PhysRevB.84.104429.pdf

PHYSICAL REVIEW B 84, 104429 (2011) Incomplete spin reorientation in yttrium orthoferrite J. Scola, Y . Dumont, and N. Keller Groupe d’ ´Etude de la Mati `ere Condens ´ee (GEMaC), UMR 8635 du CNRS, UVSQ, 45 Avenue des ´Etats-Unis, 78035 Versailles Cedex, France M. Vall ´ee and J.-G. Caputo INSA de Rouen, Math Laboratory, 76801 Saint- ´Etienne-du-Rouvray, France I. Sheikin Grenoble High Magnetic Field Laboratory, UPR 3228 du CNRS, 38042 Grenoble Cedex 9, France P. Lejay Institut N ´eel, CNRS and Universit ´e Joseph Fourier, Grenoble, France A. Pautrat Laboratoire CRISMAT, UMR 6508 du CNRS, ENSICAEN et Universit ´e de Caen, 6 Bd. Mar ´echal Juin, 14050 Caen, France (Received 30 November 2010; revised manuscript received 29 March 2011; published 15 September 2011) High-magnetic-field measurements of the magnetic moment of single crystals of yttrium orthoferrite were performed by torque and vibrating sample magnetometers. We investigated the magnetic states before and atthe end of the field-induced spin reorientation and compared them with the theoretical predictions given by amacrospin model. The model describes the spin reorientation for low magnetic fields well. For high magneticfields, the model predicts a 90 ◦spin rotation while the experiments indicate that the magnetic moment only rotates by 80◦forH=74 kOe and remains about 10◦out of a crystallographic axis, up to the highest measured field (280 kOe). This suggests that the initial magnetic interactions are altered by the strain induced by the spinreorientation, leading to a symmetry change. DOI: 10.1103/PhysRevB.84.104429 PACS number(s): 75 .30.Gw, 75 .47.Lx, 75 .50.Ee, 75 .40.Mg I. INTRODUCTION Rare-earth orthoferrites have a distorted perovskite struc- ture (space group D2h-16-Pbnm). Their low symmetry com-bined with spin-orbit coupling gives rise to antisymmetricexchange interactions as described by Dzyaloshinsky andMoriya (DM), 1–3in addition to the antiferromagnetic (AFM) exchange interaction. The DM interaction is responsible forthe canting angle (a few milliradians) between the two AFMsublattices and makes the compound a weak ferromagnet. Forall rare-earth metals except lanthanum and yttrium, the netmagnetization rotates by 90 ◦as the temperature varies between two characteristic temperatures; within this temperature range,the equilibrium magnetization can be found in any directionin the plane (010). This phenomenon, referred to as spinreorientation (SR), has been extensively studied in differentorthoferrites (see Ref. 4for an exhaustive review and Refs. 5 and6for more recent studies). SR was initially considered to be the consequence of a crystallographic phase transition,but it eventually turned out to be a purely magnetic phasetransition. 5,7Owing to the very strong magnetic anisotropy of orthoferrites, the rotation of the magnetization induces astrain on the ionic lattice. Investigations of such changes arecomplicated by the phononic thermal expansion that is mixedwith the magnetostriction during the SR. This difficulty can beovercome by inducing the SR by a magnetic field. As a fieldof tens of kilo-oersteds is applied along the AFM axis (i.e.,perpendicular to the weak ferromagnetic moment), the spinstructure undergoes a 90 ◦rotation in such a way that the weak ferromagnetic moment aligns with the applied field.8,9The field-induced SR at a fixed temperature was studied by neutrondiffraction by Koehler et al. on ErFeO 3and HoFeO 3up to 10 kOe.10Their results suggest structural domain transformations during the SR, but many details were lost because they studiedpowders. The problem can be simplified further by studyingyttrium orthoferrite [YFeO 3,TN=648 K (Ref. 3)]. The Y sites are not magnetic, so the magnetic description of the SRis limited to the iron lattice. Here we focus our investigations on the magnetic states before and after the SR by vectorial magnetization experimentsperformed on YFeO 3single crystals (Sec. II). By combining a torque magnetometer and vibrating sample magnetometer(VSM), we monitor the relative direction of the magneticmoment of YFeO 3during the SR. Particular attention is given to the magnetic state at very high fields where theSR is assumed to be completed (Sec. III). Then we compare our observations to a macrospin model using the commonlyused magnetic interactions for orthoferrites (Sec. IV). The numerical simulations of the Landau-Lifshitz equations wereperformed by a relaxation method. In Sec. V, we discuss the discrepancies between the experiments and the model. II. SAMPLE AND EXPERIMENTS A. Bulk single crystal A bulk single crystal was grown using the floating zone technique in a mirror furnace under air at ambient pressure.We used the very high quality precursors Y 2O35Nand Fe2O34N8. The single crystalline seed was tied with pure platinum wire and the crystal had no contact with any holderduring the growth, which rules out any pollution during the 104429-1 1098-0121/2011/84(10)/104429(6) ©2011 American Physical SocietyJ. SCOLA et al. PHYSICAL REVIEW B 84, 104429 (2011) FIG. 1. Laue picture of the YFeO 3single crystal. growth. Analysis by a scanning electron microscope with a field effect gun reveals that the only elements detectedwere Y , Fe, and O in the appropriate concentrations; noparasitic compounds were detected. This was confirmed bythe x-ray powder diffraction measurements carried out oncrushed single crystal mixed with pure silicon as standard. In the bulk, the refined lattice constants were a=5.2818 ◦A, b=5.5953◦A, and c=7.6054◦A. The mosaicity was estimated from a Laue picture to be less than 0.5◦(Fig. 1) The crystal was cut along the crystallographic faces into a1.90×1.96×1.99 mm 3cube and a 200 ×100×40μm3 nearly cuboid sample for the vibrating sample magnetometer (VSM) and torque magnetometer, respectively. B. Experiments Torque magnetometry informs on the magnetization vector and offers a very good resolution even at high field. Thesamples were rigidly glued on 125- μm-thick CuBe cantilevers. The small deflections of the cantilever caused by the magnetictorque T=m×H(mbeing the magnetic moment of the sam- ple and Hthe applied field) were detected by a high-resolution capacitance bridge. In the experimental configuration, theoutput signal is the xcomponent of Tas schematically represented in Fig. 2. Magnetization measurements were performed in 9-T and 14-T Quantum Design VSM’s. III. EXPERIMENTAL RESULTS A. Magnetization Figure 3presents the magnetization of the YFeO 3single crystal for Happlied along aandcatT=4.2K .W e duplicated the Jacobs et al. experiment9on a single crystal grown from state-of-the-art precursor purity. Comparisonswith the previous results are summarized in Table I. In contrast with the literature (e.g., Refs. 9and11), we observe that the magnetization remains smaller for H/bardblathan for H/bardblceven for high fields after the spin reorientation. This difference canbe due to either a smaller net magnetization or a finite angle FIG. 2. (Color online) Experimental geometry for torque mag- netometry measurements. Relative orientations of the sample crys- tallographic axes ( a,b,c) and the reference system of coordinates (ex,ey,ez) are given for the two experimental configurations. The deflection of the cantilever is symbolized by the curved arrow. In this drawing, β> 0a n dη> 0. between the magnetic moment and the applied field. The torque experiment will confirm the second explanation. B. Torque magnetometry We used a torque magnetometer in order to study the mag- netic moment direction at high field at room temperature. TheSR was probed in two distinct experimental configurations,sketched in Fig. 2and referred to as configurations I and II; (e x,ey,ez) represents the reference system of coordinates. In configuration I, the output signal is proportional to Tx H=cos(β)mc(H)−sin(β)ma(H), (1) where H=H(cosβez+sinβex). In our convention and for H> 0,mcis positive (negative) for βpositive (negative) while mais positive for all β. The torque was measured continuously while the magnetic field was swept up and down and is presented in Fig. 4. The angle β=6◦was obtained from the results of Fig. 5 FIG. 3. (Color online) Solid lines indicate magnetization mea- sured for two orientations of the magnetic field. +and×symbols are numerical simulations. Arrows indicate quantities discussed in the text.T=4.2K . 104429-2INCOMPLETE SPIN REORIENTATION IN YTTRIUM ... PHYSICAL REVIEW B 84, 104429 (2011) TABLE I. Summary of the experimental data and the deduced interaction fields in YFeO 3.M0is given in 10−5emu/cm3andχin 10−4emu/kOe. Also presented are the interaction fields obtained in Ref. 9atT=4.2K . T=4.2K T=300 K Ref. 9 M0 7.51 6.6 χ 5.38 5.79 H0(kOe) 70.0 74.3 74 φ0(mrad) 12.3 10.9 HD(kOe) 140 140 140 HE(kOe) 5700 5300 6400 H2(kOe) 1.3 1.3 1.2 H4(kOe) 0.50 0.44 0.52 as described below. The quantity Tx/Htakes its maximum value at low fields and decreases with increasing field. Theangle between Handmis initially 90 ◦and tends to decrease during the SR. As the field is applied, the magnetic momentmrotates from cforβ> 0(−cforβ< 0) toward a. In configuration II, the output signal is proportional to T x H=− cosηcosβmb(H)+sinηcosβmc(H) −sinβma(H), (2) FIG. 4. (Color online) Experimental (solid line) and simulated (dashed line) torque signal Tx/Hfor experimental configuration I (upper panel) and configuration II (lower panel). β=6◦,η=12◦, T=300 K.FIG. 5. (Color online) Torque signal Tx/Hmeasured for H/H= cosβez+sinβexwithβranging from 2◦to−0.3◦.β=5◦and 1◦ were also measured and not shown for clarity. β=6◦is shown in Fig.4, upper panel. T=300 K. where β=6◦as in configuration I and ηrepresents the misalignment of the sample when it was glued on thecantilever; using the experimental value of T x/Hat low field, one obtains η=12◦. The shape of the curve is similar to the one obtained in configuration 1, and is one order of magnitudesmaller due to the misalignment coefficients. This shows thatthe signal is dominated by the m candmacomponents of the magnetization, mbbeing negligible as expected. In other words, configuration II demonstrates that the SR takes placein the ( a,c) plane. It can be noticed that in this configuration, bothm candmaare positive and the signal converges to a negative value due to the negative sign of the third term.The cancellation of the experimental curve results from thecompensation of the last two terms of Eq. ( 2) and corresponds to a tilt angle of about 20 ◦of the weak ferromagnetic moment out of the aaxis. The quality of this estimation is poorer than the previous one because of the weakness of the signal and thefitting parameter η. In configuration I, a small misalignment angle analog to ηmay exist, but it was neglected since it introduces cos η≈1 coefficients for m aandmcand sin ηfor mb≈0. Figure 5gathers Tx/Hcurves measured for different values ofβ. The field directions were slightly changed around the ez direction (expected to coincide with the aaxis of the sample) in the ( ey,ez) plane. We choose β=0◦to be the angle at which the torque switches sign. Remarkably, the absolute value of the torque increases at the beginning of the SR for negative values of β: the angle between the directions of mandHis initially 90◦−βand as the field increases up to several teslas, it decreases, goingthrough 90 ◦thus maximizing the torque. This indicates once again that the net magnetic moment remains in the ( a,c) plane, at least at low fields. TheTx/Hsignal exhibits a plateau above 200 kOe. This may come from either a finite magnetic torque or from the forceapplied to the sample by the field gradient due to a residual 104429-3J. SCOLA et al. PHYSICAL REVIEW B 84, 104429 (2011) magnetic field inhomogeneity. However, this force should be identical in our two experimental configurations, which isclearly not the case (see Fig. 4). In addition it should not change sign with β(Fig. 5). Therefore, the measured plateau results from a m×Hterm. It follows that the plateau at a finite value clearly shows that the magnetic moment and theapplied field are never collinear for any β. Instead mforms a finite angle of 10 ◦withH. This value is of the order of the inaccuracy of the alignment of the crystallographic axes of thesample with respect to the reference system of coordinates.In order to determine if the plateau can originate from themisalignment of the sample within which mandaare collinear (/Gamma1 2configuration, as expected at high field), the effect of the different kinds of misalignment must be discussed; theconclusions drawn for experimental configuration I also holdfor configuration II. The first type of misalignment correspondsto a rotation of the sample around the zdirection and is represented by the angle ηin Fig. 2: it leaves the aaxis invariant and is thus irrelevant here. The second type arisesfrom a rotation around the xaxis and would only result in an offset in β. The third type of misalignment is a rotation around the yaxis and actually introduces a constant angle offset between Handain our experiment. This angle actually gives rise to a finite contribution to the torque but the y component of this term cannot be detected by our setup,and its xcomponent should cancel during the βscan, when Hbelongs to the ( a,b) plane. In any case, the measured torque would vanish like sin βif the magnetic moment and awere collinear. We conclude that unexpectedly, mis tilted away from the aaxis by 10 ◦.T h ev a l u eo f1 0◦agrees well with the ma(Ha)/mc(Hc)=cos 12◦ratio obtained by VSM at 14 T. During the SR, the net magnetic moment rotatesby 80 ◦rather than 90◦, and the final spin configuration is not/Gamma12. Abrupt changes of the torque signal are visible in Fig. 5. The jumps around 1 T correspond to the reversal of theferromagnetic moment, before the SR takes place: the fields at which the torque jumps fairly follow H r/sinβ, where Hris of the order of 1 kOe and refers to the reversal field for the ferromagnetic moment when His applied along thec. The curve at β=0◦is unique. It exhibits an irreversible large jump at 100 ±2 kOe that was observed several times: immediately after the first measurement and at the end of the β scan. However, this phenomenon takes place for a very narrowangle range |β|<0.3 ◦. The (quasi-) cancellation of the torque can have two relevant explanations: (i) the magnetic moment and the applied field are collinear everywhere in the sample, or (ii) the sample magnetization splits into two domains ofsimilar volumes within which the magnetic moment is tiltedby 10 ◦away from a. The narrow angle range for this jump to happen makes the first hypothesis quite unlikely. On the other hand, the irreversible and sharp features of the jumps matchthe behavior of a switch between metastable states. Figure 5 showed that there exist two possible spin configurations at theend of the SR, i.e., mbeing+10 ◦or−10◦away from ain the (a,c) plane. The 100-kOe jump appears for the field direction that is the closest to the mean direction between the two finalstates.We do not have enough information to determine if the small jumps at 20 and 35 kOe for β=2 ◦,1◦are switches of the tilt angle rather than late ferromagnetic reversals. IV . TWO-MACROSPIN MODEL In the the rest of the article, we will consider the origin of the magnetic state at high field. We will first discuss whetherour observations can be described by the current magneticmodel or if they reveal something new. The magnetic interactions in orthoferrites are commonly described by the following Hamiltonian: 3,9,11,12 H NgμBS=−HEm1·m2−HDeb·(m1×m2) −1 2H2/parenleftbig m2 1a+m2 2a/parenrightbig +1 4H4/summationdisplay α=a,b,ccα/parenleftbig m4 1α+m4 2α/parenrightbig −H·(m1+m2), (3) where gis the Lande factor; Nis the number of Fe3+ions (S= 5/2);HE,HD,H2, andH4are the AFM exchange interaction, the DM interaction, and the uniaxial and cubic anisotropyinteractions, respectively, expressed in field units; e a,b,care an orthogonal system of unit vectors based on the crystallographicaxes assuming an orthogonal structure; and m iare unit vectors pointing in the direction of a magnetic moment in the i th sublattice ( i=1,2). A. Analytical solutions Equilibrium states can be calculated analytically by min- imizing Eq. ( 3) if the magnetic field is applied either along eaorec. In these two particular cases, the magnetic moment remains in the ( ea,ec) plane axis as confirmed by the presented experimental results. Thus, the state of the system dependson only two variables—the polar angles θ 1,2. An additional assumption is required to obtain an analytical solution: wechoose c x=1 and cy,z=0. By doing this, the K4term loses its physical meaning of cubic anisotropy contributionand becomes a phenomenological correction. The interest ofintroducing such an artificial parameter is that it permits a fullyanalytical approach that will yield all the parameters of theproblem. We subsequently verified by numerical simulationsthat this stratagem makes no qualitative change in the observedquantities. We introduce/braceleftBigg α= 1 2(θ1+θ2−π) φ=1 2(θ1−θ2+π),(4) where θiare the polar coordinates of mion the unit sphere. αrepresents the angle of AFM with respect to eaandφ is the canting angle of the two sublattices with respect tothe AFM axis, respectively. Then, the net magnetizationM=Ngμ BS(m1+m2) in the particular cases H=Heaand H=Hecis, respectively, Ma(H)=NgμBSsinφsinα, (5) Mc(H)=NgμBSsinφ. (6) 104429-4INCOMPLETE SPIN REORIENTATION IN YTTRIUM ... PHYSICAL REVIEW B 84, 104429 (2011) TheHdependence of φandαis obtained by minimization of Eq. ( 3) as proposed by Jacobs et al. , sinφ=HD+H 2HE,∀H, (7) and⎧ ⎪⎨ ⎪⎩(2H2HE−4H4HE−H2)s i nα +4H4HEsin3α−HDH = 0,H <H 0 sinα =0,H >H 0.(8) Finally, the reorientation field H0is given by H0=−HD+/radicalBig H2 D+8H2HE 2. (9) B. Physical parameters The zero-field canting angle φ0can be evaluated from the experimental zero-field magnetization M0by Eq. ( 6). The obtained value of φ0in Eq. ( 7)f o rH=0 yields the HD/HEratio. For H> 0, the magnetization is described by Mc(H)=M0+χHwhere the slope χis the transverse AFM susceptibility. By identification of χin Eqs. ( 6) and ( 7), we getHEand thereby, HD. Introducing them into Eq. ( 9)g i v e s H2. Finally, H4is fitted to the experimental magnetization forH=Hea(H<H 0) using the previous parameters. The agreement with experimental data is illustrated by numericalsolutions (Fig. 3), which are identical to the analytical ones. The parameters are summarized in Table IforT=4.2 and 300 K; parameters from Ref. 9are also given for comparison. C. Numerical simulations The numerical minimization of Eq. ( 3) were calculated by a relaxation method. The time dependence of our system isruled by the Landau-Lifshitz-Gilbert equation, ∂m ∂t=−m×∂H ∂m−ε/parenleftbigg m×∂m ∂t/parenrightbigg , (10) where εis the damping parameter and is considered an arbitrary computational parameter. Equation ( 10) can be rewritten as (1+ε2)∂m ∂t=−m×∂H ∂m+εm×/parenleftbigg m×∂H ∂m/parenrightbigg .(11) The solutions of Eq. ( 11) converge toward the equilibrium state. The validity of the program was established by repro-ducing the analytical solutions (for Happlied along aandc) with excellent agreement. V . DISCUSSION The equilibrium magnetization curves calculated for H/bardbla andH/bardblcare plotted with the experimental data in Fig. 3.I n contradiction with the experimental results, the two curves areidentical at high field. This means that the magnetic interac-tions involved in the model of Eq. ( 3) yields a SR from /Gamma1 4at low field to /Gamma12above H0. The simulations also confirm that the misalignment is not responsible for the plateau at a finite valuein the torque signal. The magnetic torque was computed for the experimental conditions, i.e., including the misalignmentangle η=12 ◦: the applied magnetic field expressed H= cosβez+sinβeyin the reference system of coordinates cor- responds to H=cosβea−cosηsinβeb+sinηsinβec(with β=6◦) relative to the sample crystallographic directions. The simulations are presented in Fig. 4for configurations I and II. Similar to the magnetization, the theoretical calculationsagree very well with the experiment during the SR, butfail to reproduce the final state. In both configurations, thecalculated final states are essentially /Gamma1 2. This leads to the overestimation (underestimation) of ma(mc)i nE q s .( 1) and ( 2). To explain the failure of the model to describe the magnetic state after the SR, one may consider an extra interactionthat would not have been taken into account yet and whoseeffect can only be observed at high field. First, we introducedanisotropic exchange as suggested in Ref. 11. We did include anisotropic exchange either along the aaxis or along the caxis into the Hamiltonian expression and systematically varied itsreduced intensity from 10 −5to 10−2, with both signs. Second, we considered the effect of a more realistic DM interactionby taking into account four sublattices instead of only two.We used the model proposed by Herrmann 13that includes six Dzyaloshinsky-Moriya vectors corresponding to the anti-symmetric exchange coupling of each pair of sublattices andusing combinations of reduced values ranging from 0.01 h Dto 100hD. The calculation code was validated by the analytical solution as was previously done, using parameters matchingthe two-sublattice case. Third, we arbitrarily introduced asixth-order anisotropy term H 6=−1/6H6(m6 x1+m6 x2), with H6/Heranging from 0.01 h2to 100 h2. None of these three attempts could account for the incompleteness of the spinreorientation. For the sake of clarity, only the simplest modelis presented. The discrepancy between theory and experiment can be ascribed to a modification of the magnetic interactions duringthe SR. Structural changes, possibly associated with a symme-try change induced by magnetostriction, are likely to happenwith the SR. Given the very high anisotropy of the magneticinteractions, the rotation of the magnetization induces a strainon the ionic lattice. This strain has been evidenced throughthe macroscopic expansion of the samples (YFeO 314and ErFeO 37), but no information on the consequences of the strain on the crystallographic order have been published sofar. The field-induced SR at a fixed temperature was studiedby neutron diffraction by Koehler et al. on ErFeO 3and HoFeO 3 up to 10 kOe.10Their results suggest structural domain transformations during the SR but many details were lostsince they studied powders and they considered the structureas cubic to interpret the magnetic ordering. This hypothesisof a change of the magnetic interactions subsequent to asymmetry change of the crystal is coherent with the excellentagreement of the model at low field and its failure after theSR. Independent of its origin, the high-field magnetic state is noteworthy. The aaxis appears as a forbidden direction for the magnetization, whose direction remains tilted by 10 ◦away from the aaxis. This raises the question of the potential barrier separating the two metastable states. 104429-5J. SCOLA et al. PHYSICAL REVIEW B 84, 104429 (2011) VI. CONCLUSION We reported results on the magnetic-field-induced SR in single crystals of yttrium orthoferrite. By means of a torquemagnetometer under high magnetic field we showed that themagnetic state is not /Gamma1 2as predicted by theoretical models. Instead of rotating by 90◦under the effect of the applied field, the magnetic moment only rotates by 80◦, thus pointing about 10◦out of a crystallographic axis. These results point at the limitation of the description of the magnetic interactions thathave been considered so far in orthoferrites. However, the suc-cess of the existing model at low field suggests that its failure athigh field reveals a change of the symmetry of YFeO 3during the SR. ACKNOWLEDGMENTS We acknowledge the Transnational Access–Contrat n◦ 228043–Euromagnet II–Integrated Activities of the EuropeanCommission for granting the high-field experiments. Thiswork was supported by the region Ile-de-France, C’NanoNOV ATECS Project No. N ◦IF-08-1453/R. We thank the Centre de Ressources Informatiques de Haute-Normandie(CRIHAN) for the use of its computing resources. 1I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958). 2T. Moriya, Phys. Rev. 120, 91 (1960). 3D. Treves, Phys. Rev. 125, 1843 (1962). 4R. L. White, J. Appl. Phys. 40, 1061 (1969). 5L. T. Tsymbal, V . I. Kamenev, Y . B. Bazaliy, D. A. Khara, and P. E. Wigen, Phys. Rev. B 72, 052413 (2005). 6Y . B. Bazaliy, L. T. Tsymbal, G. N. Kakazei, V . I. Kamenev, and P. E. Wigen, P h y s .R e v .B 72, 174403 (2005). 7R. P. Chaudhury, B. Lorenz, C. W. Chu, Y . B. Bazaliy, and L. T. Tsymbal, J. Phys. Conference Series 150, 042014 (2009). 8J. Scola, W. Noun, E. Popova, A. Fouchet, Y . Dumont, N. Keller, P. Lejay, I. Sheikin, A. Demuer, and A. Pautrat, P h y s .R e v .B 81, 174409 (2010).9I. S. Jacobs, H. F. Burne, and L. M. Levinson, J. Appl. Phys. 42, 1631 (1971). 10W. C. Koehler, E. O. Wollan, and M. K. Wilkinson, Phys. Rev. 118, 58 (1960). 11C. W. Fairall and J. A. Cowen, Phys. Rev. B 2, 4636 (1970). 12G. Gorodetsky, S. Shtrikman, Y . Tenenbaum, and D. Treves, Phys. Rev.181, 823 (1969). 13G. F. Herrmann, Phys. Rev. 133, A1334 (1964). 14A. M. Kadomtseva, A. P. Agafonov, M. M. Lukina, V . N. Milov, A. S. Moskvin, V . A. Semenov, and E. V . Sinitsyn,Zh. Eksp. Teor. Fiz. 81, 700 (1981) [Sov. Phys. JETP 54, 374 (1981)]. 104429-6

PhysRevLett.111.087205.pdf

Evidence for a Magnetic Seebeck Effect Sylvain D. Brechet,1,*Francesco A. Vetro,1Elisa Papa,1Stewart E. Barnes,2and Jean-Philippe Ansermet1 1Institute of Condensed Matter Physics, Station 3, Ecole Polytechnique Fe ´de´rale de Lausanne—EPFL, CH-1015 Lausanne, Switzerland 2James L. Knight Physics Building, 1320 Campo Sano Avenue, University of Miami, Coral Gables, Florida 33124, USA (Received 5 June 2013; published 22 August 2013) The irreversible thermodynamics of a continuous medium with magnetic dipoles predicts that a temperature gradient in the presence of magnetization waves induces a magnetic induction field, whichis the magnetic analog of the Seebeck effect. This thermal gradient modulates the precession andrelaxation. The magnetic Seebeck effect implies that magnetization waves propagating in the direction of the temperature gradient and the external magnetic induction field are less attenuated, while magnetization waves propagating in the opposite direction are more attenuated. DOI: 10.1103/PhysRevLett.111.087205 PACS numbers: 75.76.+j, 76.50.+g The discovery of the spin Seebeck effects in ferromag- netic metals [ 1], in semiconductors [ 2], and in insulators [3] has generated much research for spin transport in ferromagnetic samples of macroscopic dimensions sub-jected to temperature gradients. The interplay of spin,charge, and heat transport defines the rich field known asspin caloritronics [ 4]. Prompted by these recent develop- ments, we established a formalism describing the irrevers- ible thermodynamics of a continuous medium withmagnetization [ 5]. In this Letter, we test a particular experimental predic- tion of this formalism on a yttrium iron garnet (YIG) slab.We argue that the thermodynamics of irreversible pro-cesses implies the existence of a magnetic counterpart tothe well-known Seebeck effect. We show how a thermally induced magnetic field modifies the Landau-Lifshitz equa- tion and provide experimental evidence for the magneticSeebeck effect by the propagation of magnetization wavesin thin crystals of YIG. The effect of a temperature gradienton the dynamics of the magnetization on a YIG slab withand without Pt stripes was investigated recently by Obryet al. [6], Cunha et al. [7], Silva et al. [8], Padro ´n- Herna ´ndez et al. [9,10], Jungfleisch et al. [11], and Lu et al. [12]. In general, irreversible thermodynamics predicts cou- plings between current and force densities. In Eq. (86) ofRef. [ 5], we identified the magnetization force term mrB. For an insulator like YIG, there is no charge current. Asexplained in detail in Ref. [ 5], the transport equation (94) of Ref. [ 5] implies that the magnetization force density MrB indinduced by a thermal force density /C0nkBrTis proportional and opposite to this force density, i.e., MrBind¼/C21nk BrT; (1) which corresponds to Eq. (155) of Ref. [ 5], where /C21> 0is a phenomenological dimensionless parameter, kBis Boltzmann’s constant, and n¼1:1/C21028m/C03is the Bohr magneton number density of YIG. Thethermodynamic formalism does not allow for a direct estimation of /C21. The numerical value of this parameter needs to be evaluated directly from the experimental data,as shown below. In the bulk of the sample, as shown in Ref. [ 5], the magnetization force density has the structure of aLorentz force density [ 13] expressed in terms of the mag- netic bound current density j M¼r/C2M[14] MrBind¼jM/C2Bind: (2) Thus, using vectorial identities, the phenomenological relations ( 1) and ( 2) imply that in the bulk of the system, the magnetic induction field Bind, induced by a uniform temperature gradient rTin the presence of a magnetic bound current density r/C2M, is given by, i.e., Bind¼"M/C2rT; (3) where the phenomenological vector "Mis given by "M¼/C0/C21nk Bðr/C2MÞ/C01: (4) By analogy with the Seebeck effect, we shall refer to this phenomenon as the magnetic Seebeck effect. The time evolution of the magnetization Mis given by the Landau-Lifschitz-Gilbert equation, i.e., _M¼/C13M/C2Beff/C0/C11 MSM/C2_M; (5) where /C13is the gyromagnetic ratio, /C11’10/C04is the Gilbert damping parameter of YIG [ 15], and MS¼1:4/C2 105Am/C01is the magnitude of the effective saturation magnetization of YIG at room temperature [ 16]. The ef- fective magnetic induction field Beffincludes the external fieldBext, the demagnetizing field Bdem, the anisotropy fieldBani, which behaves as an effective saturation mag- netization in the linear response [ 17], and finally a ther- mally induced field Bindgiven by the relation ( 3). The exchange field Bint[18] is negligible in the following, as we consider magnetostatic modes [ 19]. The demagnetizingPRL 111, 087205 (2013) PHYSICAL REVIEW LETTERSweek ending 23 AUGUST 2013 0031-9007 =13=111(8) =087205(5) 087205-1 /C2112013 American Physical SocietyfieldBdembreaks the spatial symmetry and generates an elliptic precession cone. After performing the linearresponse of the magnetization in the presence of a ther- mally induced field B ind, we shall describe how the demag- netizing field Bdemaffects the magnetic susceptibility. We found evidence for the magnetic Seebeck effect by exciting locally, at angular frequency !’2:74/C21010s/C01, the ferromagnetic resonance of a YIG slab of lengthL z¼10/C02m, width Ly¼2/C210/C03m, and thickness Lx¼2:5/C210/C05m, subjected to a temperature gradient as small as jrTj’2/C2103Km/C01generated by Peltier elements. The excitation field is applied on the slab usinga local antenna, as detailed in Ref. [ 20]. For signal trans- mission experiments, two antennas are used, set approx- imatively 8 mm apart, as shown in Fig. 1. Note that a similar setup for a gradient orthogonal to the YIG slabwas investigated recently [ 7]. For reasons explained below, these two setups can be expected to probe differentmechanisms. The external magnetic induction field B extapplied on the YIG film consists of a uniform and constant field B0and a small excitation field b¼bx^xþby^ylocally oscillating in a plane orthogonal to B0¼B0^z. In the limit of a small excitation field, i.e., in the linear limit, the magnetization fieldMconsists of a uniform and constant field MS¼MS^z and a response field m¼mx^xþmy^ylocally oscillating in a plane orthogonal to MSsuch that m/C28MS. The linear response of the magnetization to the excitation field,according to the time evolution equation ( 5), is given by _m¼/C13ðm/C2B 0þMS/C2B1Þ/C0/C11 MSMS/C2_m;(6) where the first-order magnetic induction field B1yields B1¼b/C0/C220ðkT/C1r/C01Þm; (7) and/C220is the magnetic permeability of vacuum and the thermal wave vector kT¼/C21nk B /C220M2 SrT: (8) To obtain the expressions ( 7) and ( 8), we used the linear vectorial identityðr/C2MÞ/C01/C2rT¼1 M2 Sðr/C01/C2mÞ/C2rT ¼1 M2 SðrT/C1r/C01Þm/C01 M2 SðrTÞr/C01m; wherer/C01/C1r¼1and the last term on the right-hand side vanishes since it averages out on a precession cycle. The vectorial time evolution equation ( 6) is written explicitly in Cartesian coordinates as _mx¼ð!0þ!MkT/C1r/C01Þmyþ/C11_my/C0!M/C22/C01 0by; _my¼/C0 ð !0þ!MkT/C1r/C01Þmx/C0/C11_mxþ!M/C22/C01 0bx;(9) where the angular frequencies !0and!Mare defined, respectively, as !0¼/C13B 0;! M¼/C13/C22 0MS: (10) In a stationary regime, the magnetic excitation field band the magnetization response mare oscillating at an angular frequency !, which is expressed in Fourier series as bx¼X kbkei½k/C1x/C0!tþð/C25=2Þ/C138;b y¼X kbkeiðk/C1x/C0!tÞ; mx¼X kmkei½k/C1x/C0!tþð/C25=2Þ/C138;m y¼X kmkeiðk/C1x/C0!tÞ; (11) where the eigenstates bkandmkare complex valued and dephased. The Cartesian components of the eigenmodes kx;y;zsat- isfy the boundary conditions of null mat the surface of the sample kx;y;z¼nx;y;z/C25 Lx;y;z; (12) where nx;y;z2N[20]. The eigenstates of the excitation field bkand the response field mkare related through the magnetic suscep- tibility /C31k, i.e., mk¼/C22/C01 0/C31kbk: (13) The time evolution equations ( 9), the definition ( 10), and the Fourier series ( 11) in the stationary regime imply that the magnetic susceptibility /C31kis given by /C31k¼/C01 /C10/C0/C100þið/C11/C10þkT/C1k/C01Þ; (14) where the dimensionless parameters /C10and/C100are, respec- tively, defined as /C10¼! !M; /C100¼!0 !M: (15) The demagnetizing field Bdem¼/C0/C220mx^xcauses the damping and the magnetic susceptibility /C31kxalong thexaxis to differ respectively from the damping and CuAntennae Peltier elementOscilloscopeRF pulse generatorAmplifierCrystal detector B0B1 YIG FIG. 1 (color online). Time-resolved transmission measure- ment of magnetization waves.PRL 111, 087205 (2013) PHYSICAL REVIEW LETTERSweek ending 23 AUGUST 2013 087205-2the magnetic susceptibility /C31kyalong the yaxis. The resonance frequencyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !0ð!0þ!MÞp is given by Kittel’s formula [ 21] to first order in /C11andkT. Thus, the magnetic susceptibilities /C31kx;yyield /C31kx;y¼/C01 /C10/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi /C100ð/C100þ1Þp þirx;yð/C11/C10þkT/C1k/C01Þ; (16) where rx;y>0are phenomenological damping scale fac- tors accounting for symmetry breaking. As shown by Cunha et al. in Fig. 1(a) of Ref. [ 7], the propagating modes of the magnetization waves in thebulk of YIG are magnetostatic backward volume modespropagating in the direction /C0k /C01. The expressions ( 8) and ( 16) for the magnetic susceptibilities and the thermal wave vector kTimply that the magnetization waves prop- agating from the cold to the hot side, i.e., kT/C1k/C01<0, are less attenuated by the temperature gradient and the mag- netization waves propagating from the hot to the cold sides,i.e.,k T/C1k/C01>0, are further attenuated. Thus, the opening angle of the precession cone of the magnetization mfor a magnetization wave propagating in the direction of the temperature gradient decreases lessthan the opening angle for a magnetization wave propagat- ing in the opposite direction, as shown in Fig. 2. This is confirmed experimentally by detecting induc- tively at one end of the sample the signal that resultsfrom an excitation pulse of 15 ns duration at the otherend. The signals obtained by sweeping the magnetic in-duction field B 0for the propagation of magnetization waves from the cold end to the hot end or from the hotend to the cold end are given in Fig. 3. Clearly, the waves propagating from the cold to the hot side appear to decayless rapidly than the waves propagating from the hot to thecold side. The time evolution of the signals for the waves propa- gating in the direction of the gradient or opposite to it isobtained by averaging the signals over the range of themagnetic induction field B 0displayed in Fig. 4. The signal is a convolution of kzmodes that have different group velocities and decay exponentially due to the damping.The peaks were identified in Ref. [ 22] as the result of the propagation of odd modes. Since the peaks of the trans-mitted signals are detected at the same time, the tempera-ture gradient does not affect significantly the k zmode group velocities. Moreover, from the logarithmic scalefor the signal in Fig. 4, a larger difference in attenuation between the signals for small k zmodes is inferred. This is in line with the theoretical prediction, made by Eq. ( 16), for the magnetic Seebeck effect to be proportional to k/C01z. Moreover, since the relative difference between the signalsis due to the temperature gradient, we can estimate therelative difference between the damping terms /C11/C10and Tm(0) B0 YIGHot to Coldm(τ) kT∝Tz yxm(0) B0 YIGCold to Hot kT∝m(τ)k-1k-1 k-1k-1 FIG. 2 (color online). Propagation of magnetization waves from the cold to the hot side (top) and vice versa (bottom). The cones describe the precession of the magnetization atexcitation mð0Þand at detection mð/C28Þ. The amount of damping depends on the relative orientation k Tof the temperature gradient with respect to the magnetization wave propagationdirection /C0k /C01. 86 84 82 80787674B0(mT) 8684 82 80 7876 74Hot to ColdHot to ColdB0(mT)-20 0 20 40 60 80 100 120 140 160Time (ns)Cold to HotCold to Hot FIG. 3 (color online). Transmitted signals from the cold to the hot side and from the hot to the cold side as a function of the magnetic field B0and of the detection time after a 15 ns pulsed excitation at 4.36 GHz. The lighter areas correspond to a largersignal.PRL 111, 087205 (2013) PHYSICAL REVIEW LETTERSweek ending 23 AUGUST 2013 087205-3kT/C1k/C01appearing in the expression ( 16) for the magnetic susceptibilities. Comparing the signals at t¼40 ns ,w e find that the dimensionless parameter /C21’6/C210/C07, which corresponds to a thermal damping ratio jkT/C1k/C01j=/C11/C10’ 0:3less that an order of magnitude below the self- oscillation threshold. The difference in attenuation between the signals is also shown on the ferromagnetic resonance (FMR) spectrumdetected 70 ns after the pulse and displayed in Fig. 5. The spectral linewidth /C240:2m T corresponds to inhomogene- ous broadening, since it is much larger than the homoge-neous linewidth /C24/C11B eff[23]. As is rightly pointed out in Ref. [ 6], the temperature dependence of the saturation magnetization affects theamplitude of the magnetization waves. However, sinceour experimental setup is sufficiently close to the self-oscillation threshold for a temperature gradient that issmall enough, we expect the dynamic contribution k T/C1 k/C01to be larger than the static contribution due to the temperature dependence of the saturation magnetization. Moreover, in contrast to the claim made in Ref. [ 6], Fig. 4 shows that magnetization waves can propagate with andagainst the temperature gradient and that the effect of the temperature is proportional to k/C01z. For a temperature gradient orthogonal to the YIG plane, Cunha et al. [7] showed that the temperature gradient affects the propagation of magnetization waves onlywhen Pt is deposited on the YIG slab. The effect isaccounted for by a model of spin injection and spin pump-ing, detailed by Ando et al. [24], at the interface between Pt and YIG. The quantitative analysis of the data is presentedin Ref. [ 8]. In Ref. [ 7], it is stated clearly that the effect does not occur in the absence of Pt on the surface. When Ptis removed in such a setup where k T/C1k/C01¼0, the mecha- nism invoked by Cunha et al. is not operative and our mechanism is not effective either. In summary, we point out that thermodynamics of irre- versible processes implies a coupling between heat currentand magnetization precession in a temperature gradient.This effect can be expressed by an induced magnetic fieldB indproportional to the applied temperature gradient. Thus, we suggest to refer to it as a magnetic Seebeckeffect, since it is the magnetic analog of the regularSeebeck effect. It is distinct from the magneto-Seebeck effect, which refers to a change in the Seebeck coefficient due to the magnetic response of nanostructures [ 25]. We analyze how the Landau-Lifshitz equation is modified andfind a contribution to the dissipation that is linear in rT. Hence, this effect can increase or decrease the damping,depending on the orientation of the wave vector of theexcited magnetostatic mode with respect to the tempera-ture gradient. If the temperature gradient could be madestrong enough, i.e., k T/C1k/C01>/C11/C10, then the damping would be negative and the magnetization would undergoself-oscillation. This would be analogous to the magneti-zation self-oscillation described in Chap. 7 of Ref. [ 26] and the heat equivalent of Berger’s spin amplification by simu-lated emission of radiation (SWASER) predicted forcharge-driven spin polarized currents [ 27]. We thank Franc ¸ois A. Reuse, Klaus Maschke, and Joseph Heremans for insightful comments and acknowl-edge the following funding agencies: Polish-SwissResearch Program NANOSPIN PSRP-045/2010 andDeutsche Forschungsgemeinschaft SS1538 SPINCAT,Grant No. AN762/1. *sylvain.brechet@epfl.ch [1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature (London) 455, 778 (2008) . [2] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nat. Mater. 9, 898 (2010) . [3] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nat. Mater. 9, 894 (2010) .kT·k-1<0 kT·k-1>0 Time (ns)0.46.0 0.60.81.02.04.0 -20 0 20 40 60 80 100 120 140 160 FIG. 4 (color online). Transmitted signal as a function of time after a 15 ns pulsed excitation at 4.36 GHz. kT·k-1<0 kT·k-1>0 0.51.01.52.0 74 76 78 80 82 84 86 B0 (mT) FIG. 5 (color online). FMR signal of a 15 ns pulsed excitation at 4.36 GHz detected after 70 ns, after baseline correction.PRL 111, 087205 (2013) PHYSICAL REVIEW LETTERSweek ending 23 AUGUST 2013 087205-4[4] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012) . [5] S. D. Brechet and J.-P. Ansermet, Eur. Phys. J. B 86, 318 (2013) . [6] B. Obry, V. I. Vasyuchka, A. V. Chumak, A. A. Serga, and B. Hillebrands, Appl. Phys. Lett. 101, 192406 (2012) . [7] R. O. Cunha, E. Padro ´n-Herna ´ndez, A. Azevedo, and S. M. Rezende, Phys. Rev. B 87, 184401 (2013) . [8] G. L. da Silva, L. H. Vilela-Leano, S. M. Rezende, and A. Azevedo, Appl. Phys. Lett. 102, 012401 (2013) . [9] E. Padro ´n-Herna ´ndez, A. Azevedo, and S. M. Rezende, J. Appl. Phys. 111, 07D504 (2012) . [10] E. Padro ´n-Herna ´ndez, A. Azevedo, and S. M. Rezende, Phys. Rev. Lett. 107, 197203 (2011) . [11] M. B. Jungfleisch, T. An, K. Ando, Y. Kajiwara, K. Uchida, V. I. Vasyuchka, A. V. Chumak, A. A. Serga, E.Saitoh, and B. Hillebrands, Appl. Phys. Lett. 102, 062417 (2013) . [12] L. Lu, Y. Sun, M. Jantz, and M. Wu, Phys. Rev. Lett. 108, 257202 (2012) . [13] F. A. Reuse, Electrodynamique (PPUR, Lausanne, 2012). [14] D. J. Griffiths, Introduction to Electrodynamics (Prentice- Hall, Upper Saddle River, 1999), 3rd ed. [15] H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J. Ferguson, and S. O. Demokritov, Nat. Mater. 10, 660 (2011) .[16] F. Boukchiche, T. Zhou, M. L. Berre, D. Vincent, B. Payet- Gervy, and F. Calmon, Proceedings of PIERS 2010 in Cambridge (MIT Press, Cambridge, MA, 2010), Vol. 1, p. 700. [17] J. A. Duncan, B. E. Storey, A. O. Tooke, and A. P. Cracknell, J. Phys. C 13, 2079 (1980) . [18] C. Kittel, Rev. Mod. Phys. 21, 541 (1949) . [19] A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D 43, 264002 (2010) . [20] E. Papa, S. E. Barnes, and J.-P. Ansermet, IEEE Trans. Magn. 49, 1055 (2013) . [21] C. Kittel, Introduction to Solid State Physics (Wiley, New York, 2004), 8th ed. [22] E. Padro ´n-Herna ´ndez, A. Azevedo, and S. M. Rezende, Appl. Phys. Lett. 99, 192511 (2011) . [23] S. V. Vonsovskii, Ferromagnetic Resonance (Pergamon, Oxford, 1966). [24] K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008) . [25] M. Walter, J. Walowski, V. Zbarsky, M. Munzenberg, M. Schafers, D. Ebke, G. Reiss, A. Thomas, P. Peretzki, M. Seibt, J. S. Moodera, M. Czerner, M. Bachmann, and C.Heiliger, Nat. Mater. 10, 742 (2011) . [26] S. E. Barnes, Spin Current , edited by S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura (Oxford UniversityPress, New York, 2012). [27] L. Berger, IEEE Trans. Magn. 34, 3837 (1998) .PRL 111, 087205 (2013) PHYSICAL REVIEW LETTERSweek ending 23 AUGUST 2013 087205-5

PhysRevLett.124.037202.pdf

Current-Induced Dynamics and Chaos of Antiferromagnetic Bimerons Laichuan Shen ,1,2Jing Xia ,1Xichao Zhang ,1Motohiko Ezawa ,3Oleg A. Tretiakov ,4 Xiaoxi Liu ,5Guoping Zhao,2,*and Yan Zhou1,† 1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China 2College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China 3Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan 4School of Physics, The University of New South Wales, Sydney 2052, Australia 5Department of Electrical and Computer Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan (Received 22 May 2019; revised manuscript received 1 November 2019; published 23 January 2020) A magnetic bimeron is a topologically nontrivial spin texture carrying an integer topological charge, which can be regarded as the counterpart of the skyrmion in easy-plane magnets. The controllable creation and manipulation of bimerons are crucial for practical applications based on topological spintextures. Here, we analytically and numerically study the dynamics of an antiferromagnetic bimeron driven by a spin current. Numerical simulations demonstrate that the spin current can create an isolated bimeron in the antiferromagnetic thin film via the dampinglike spin torque. The spin current can also effectively drive the antiferromagnetic bimeron without a transverse drift. The steady motion of an antiferromagnetic bimeron is analytically derived a nd is in good agreement with the simulation results. Also, we find that the alternating-current-induc ed motion of the antiferromagnetic bimeron can be described by the Duffing equation due to the presence of the nonlinear boundary-induced force. The associated chaotic behavior of the bimeron is analy zed in terms of the Lyapunov exponents. Our results demonstrate the inertial dynamics of an antiferroma gnetic bimeron, and may provide useful guidelines for building future bimeron-based spintronic devices. DOI: 10.1103/PhysRevLett.124.037202 Introduction. —Topologically protected magnetic tex- tures, such as magnetic skyrmions [1–7], have attracted a lot of attention because they have small size and can be used as nonvolatile information carriers in future spintronic devices [8–10]. The existence of magnetic skyrmions has been experimentally confirmed in many systems with bulk or interfacial Dzyaloshinskii-Moriya interaction (DMI) [2–5]. In addition, various topological structures, such as antiferromagnetic (AFM) skyrmions [11,12] , ferri- magnetic skyrmions [13], antiskyrmions [14], biskyrmions [15], bobbers [16], and bimerons [8,17 –30], are also current hot topics. In particular, a bimeron consists of two merons, which can be found in easy-plane magnets [8,18,20,28] , frustrated magnets [21], and magnets with anisotropic DMI [26]. The bimeron is a localized spin texture similar to a magnetic skyrmion, which can be constructed by rotating the spin direction of a skyrmion by 90°. Magnetic bimerons can also be used as informationcarriers for spintronic devices made of in-plane magnetized thin films [8,26,28] . On the other hand, AFM materials are promising for building advanced spintronic devices due to their zero stray fields and ultrafast spin dynamics [31–33]. Several theo- retical studies [11,12,34,35] predict that skyrmions may exist in AFM systems, which can be manipulated by spin currents [11,12] and magnetic fields [34]. Compared toferromagnetic (FM) skyrmions, AFM skyrmions do not show the skyrmion Hall effect [36,37] due to zero net Magnus force, so that they can move perfectly along thedriving force direction with ultrahigh speed [11,12,38,39] . Various methods have been proposed to control the AFM textures, such as using spin currents [40–42], magnetic anisotropy gradients [38], temperature gradients [34,43] , and spin waves [44]. For AFM systems, the motion equation of the AFM order parameter (N´ eel vector) is related to the second derivative with respect to time, whereas the FM Landau-Lifshitz-Gilbert (LLG) equation [45] is of first order [31]. Therefore, the dynamics of AFM spin textures are differ- ent from that of FM spin textures. For example, theoscillation frequency of AFM skyrmion-based spin torquenano-oscillators (STNOs) is higher than that of FM skyrmion-based STNOs as AFM skyrmions obey the inertial dynamics [46]. In addition, the motion equation of the systems, such as the LLG equation, is usually nonlinear, resulting in the dynamic behavior being com- plex or even chaotic [47,48] Note that, for the chaos, the nonlinearity is a necessary condition rather than a su ffi- cient condition, so that not all nonlinear systems will exhibit chaotic behavior. In nanoscale spintronic devices,spin torque oscillators are interesting candidates for chaotic systems [49–51], which are promising for variousPHYSICAL REVIEW LETTERS 124, 037202 (2020) 0031-9007 =20=124(3) =037202(6) 037202-1 © 2020 American Physical Societyapplications [52–54]. For the AFM bimeron, however, its dynamics induced by a spin current still remain elusive. In this Letter, we report the dynamics of an AFM bimeron induced by the spin current. Our theoretical and numerical results show that an isolated bimeron can be created and driven in the AFM thin film by spin currents. Furthermore, when an alternating current is applied to drive the AFM bimeron, the motion of the bimeron in a nanodisk can be described by the Duffing equation, which describes the oscillation of an object with a mass under the action of nonlinear restoring forces. The chaotic behavior associatedis also analyzed in terms of the Lyapunov exponents. Model and theory. —We consider a G-type AFM film with sublattice magnetization M 1andM2. By linearly combining the reduced magnetizations m1and m2 (mi¼Mi=MSwith the saturation magnetization MS), we obtain the staggered magnetization (or N´ eel vector) n¼ðm1−m2Þ=2and the total magnetization m¼ ðm1þm2Þ=2, where the former could be used to describe the AFM order, while the latter is related to the canting of magnetic moments. Here, we are interested in most realistic cases where the AFM exchange interaction is significantly strong, so that m2≪n2∼1[55,56] .mandnobey the following two coupled equations [40–42,57] : _n¼ðγf2−α_mÞ×nþT1;SOTþT1;STT; ð1aÞ _m¼ðγf1−α_nÞ×nþTnlþT2;SOTþT2;STT;ð1bÞ where γandαare the gyromagnetic ratio and the damping constant, respectively, and Tnl¼ðγf2−α_mÞ×mis the higher-order nonlinear term [40].T1;SOT¼γHdm×p×n and T2;SOT¼γHdn×p×nare dampinglike spin-orbit torques (SOTs), where pis the polarization vector and Hdrelates to the applied current density j, defined as Hd¼ jℏP=ð2μ0eMStzÞwith the reduced Planck constant ℏ, the spin polarization rate P, the vacuum permeability constant μ0, the elementary charge e, and the layer thickness tz. T1;STT¼γη∂xnandT2;STT¼γβ∂xn×nstand for spin- transfer torques (STTs) with the adiabatic (nonadiabatic) parameter η(β). In our simulations, η¼0.1βandβ¼Hdtz are adopted. f1¼−δE=μ0MSδnandf2¼−δE=μ0MSδm are the effective fields. From a classical Heisenberg Hamiltonian [57], the AFM energy Ecan be written as E¼RFdV, where F¼ðλ=2Þm2þLm·ð∂xnþ∂ynÞþ ðA=2Þ½ð∇nÞ2þ∂xn·∂yn/C138−ðK=2Þðn·neÞ2þwDwith the homogeneous exchange constant λ, parity-breaking con- stant L[41,44,57] , inhomogeneous exchange constant A, and magnetic anisotropy constant K.ne¼exstands for the direction of the anisotropy axis and wDis the DMI energy density, wD¼ðD=2Þ½nxð∂yny−∂xnzÞ−ny∂ynxþ nz∂xnx/C138with the DMI constant D[26,42,56,58] . Such a DMI energy can stabilize the bimeron, which can be induced at the antiferromagnet –heavy metal interface [26]. In addition, to form the bimeron, antiferromagnetswith in-plane easy-axis anisotropy, such as NiO [31], are favorable. Based on Eqs. (1a) and (1b), one can simulate the evolution of the staggered magnetization and also derive the steady motion equations for a rigid AFM bimeronby using Thiele (or the collective coordinate) approach[59–62](see Ref. [63] for details), written as a·M eff¼FαþFSOTþFSTT; ð2Þ where ais the acceleration, and Meffis the effective AFM bimeron mass, which is defined as μ2 0M2Stzd=γ2λwith the dissipative tensor d. The effective AFM texture mass Meff originates from the existence of two sublattices, [31]and it is intrinsic. The components dijof the dissipative tensor are dxx¼dyy¼d¼Rdxdy ð∂xn·∂xnÞanddxy¼dyx¼0.I n Eq.(2), the forces induced by the surrounding environment (e.g., the boundary effect) are not taken into account,F α¼−αμ0MStzv·d=γrepresents the dissipative force with the velocity v, and FSOTandFSTTare the forces induced by SOTs and STTs, respectively. Creation of an AFM bimeron by a spin current. — Creating an isolated AFM bimeron is essential for practical applications. Here, we employ a current to create an AFM bimeron via SOTs. As shown in Fig. 1, when a vertical current of j¼100MA=cm2is injected into the central circular region with a diameter of 30 nm, the N´ eel vector is continuously flipped, and then, a bimeronlike magnetic structure is formed. At t¼0.05ns, the current is turned off. Since the DMI energy density of the lower half of themagnetic texture has a positive value [63], the lower half of the magnetic texture is unfavorable and gradually recovers to the AFM ground state, while the upper half evolves into a metastable bimeron. The current-induced process from theAFM ground state to the metastable bimeron takes onlytens of picoseconds, as shown in Fig. 1. Such an ultrafast process also exists in the generation of the AFM skyrmions under the action of time-dependent magnetic fields [34], where the force induced by time-dependent magnetic fieldshas a similar form to that of dampinglike spin torques [60,67] . Similar to the AFM skyrmion, the AFM bimeron is a topologically protected magnetic texture with AFMtopological charge Q¼/C6 1, [see Fig. 1(i)] where the topological charge is defined as Q¼−ð1=4πÞRdxdy ½n· ð∂ xn×∂ynÞ/C138[12,18,68] . On the other hand, when the opposite DMI constant is adopted, the AFM bimeron is created in the lower plane (the result is given in Ref. [63]). In addition, for the creation of the AFM bimeron, increas-ing the injected region can effectively reduce the time and current density, and multiple bimerons will be generated when a small damping is adopted (see Ref. [63]). Note that the bimeron created here is symmetric, while it may deformunder the effect of thermal fluctuations [63]. Current-induced motion of an AFM bimeron. — Manipulating magnetic textures is indispensable inPHYSICAL REVIEW LETTERS 124, 037202 (2020) 037202-2information storage and logic devices. The current, which is a common method for manipulating magnetic materials, is employed to drive the AFM bimeron via SOTs and STTs. Taking the current density j¼5MA=cm2and the damp- ingα¼0.02, we simulate the motion of an AFM bimeron, where the initial state is a metastable AFM bimeron. Inorder to track the AFM bimeron, the guiding center ( r x,ry) of the bimeron is defined, described as ri¼R dxdy ½in·ð∂xn×∂ynÞ/C138R dxdy ½n·ð∂xn×∂ynÞ/C138;i ¼x; y; ð3Þ and the velocity vi¼_ri. As shown in Figs. 2(a)and2(b), considering the dampinglike SOTs, the steady motion speed reaches 725m=sa tt¼0.1ns, and the transmission path of the AFM bimeron is parallel to the racetrack, so thatthe fast-moving AFM bimeron will not be destroyed bytouching the racetrack edge due to the cancellation of the Magnus force. Therefore, in addition to the AFM sky- rmions, the AFM bimerons are also ideal informationcarriers in racetrack-type memory. Figure 2(c)shows the relation between the speed vand the damping α, where the speed of the AFM bimeron is inversely proportional to the damping constant for SOTs and STTs. In order to test the simulated speeds,we derived the steady motion speed from Eq. (2)(see Ref. [63] for details) v¼ π2RsγHd αd−γβ α; ð4Þ where Rsis the bimeron radius, which corresponds to the skyrmion radius. The first and second terms on the rightside of Eq. (4)are the SOT- and STT-induced speeds, respectively. We can see from Fig. 2(c)that the analytical speed given by Eq. (4)is in good agreement with the results of the numerical simulations. It is worth mentioning thatEq.(4)is also applicable to AFM skyrmions. Namely, the AFM bimeron and skyrmion have the same motion speed under the same driving force. Dynamics of the AFM bimeron induced by the alternating current. —Next, we discuss the forced oscil- lation of the AFM bimeron induced by the alternating current j¼j 0sinð2πftÞ, where j0andfare the amplitude and frequency of the applied currents. As shown inRef. [63], due to the harmonic current-induced driving force, the guiding center r xof the AFM bimeron exhibits a stable oscillation with amplitude r0∼11.64nm and phase difference φ∼89.14° between jandrx, where α¼0.002, f¼20GHz, and j0¼1MA=cm2are adopted. By chang- ing the frequency of the applied currents, the differentvalues of r 0andφare obtained by numerical simulations and are shown in Fig. 3, where three damping constants (α¼0.0015 , 0.002, and 0.003) are considered. We can see that the phase difference φbecomes larger with the increasing frequency, and interestingly, for the amplituder 0, there are current-induced resonance phenomena. To analyze such resonance phenomena, we return to Eq. (2) and focus on the motion in the xdirection, so that the Thiele equation becomes a scalar equation Meff ̈rxþα/C3_rxþFb¼FSOT;0sinð2πftÞ; ð5Þ0 100 20001230.00 0.05 0.100200400600800 vx vy t(ns)v(m/s) STT Numerical AnalyticalSOT Numerical Analyticalv(km/s) 1/t=0 . 5n s 0 200 400-50050(c) (b) y(nm) x(nm)(a) FIG. 2. (a) The evolution of the motion speed and (b) the top view for an AFM bimeron induced by the current via SOTs,where the polarization vector p¼−e y, the applied current density j¼5MA=cm2, and the damping α¼0.02. (c) The motion speed as a function of 1=αfor the AFM bimeron driven by the current j¼5MA=cm2via SOTs and STTs. Symbols are the results of the numerical simulations and lines are given by Eq. (4) with the numerical values of d∼15andRs∼7nm. 0.00 0.05 0.10 0.15 0.200.00.51.0t=0 . 0 0n s t=0 . 0 1n s Q j t(ns)Q(a) (b) (c) (d) (e) (i)(f) (g) (h)t=0 . 0 2n s t=0 . 0 4n s t=0 . 0 5n s t=0 . 0 6n s t=0 . 0 7n s t=0 . 2 0n s 050100 j(MA/cm2) FIG. 1. (a) –(h) The time evolution of the N´ eel vector induced by a spin-polarized current with the polarization vector p¼−ez, where the dampinglike spin-orbit torque (SOT) is taken intoaccount and the color represents the out-of-plane component ofthe N´ eel vector. (i) The evolution of the topological charge Qand the injected current density j. In our simulations, the current of j¼100MA=cm 2is injected in the central circular region with a diameter of 30 nm [see green lines in (a) –(d)] and we adopt the following parameters [12],A¼6.59pJ=m,K¼0.116MJ=m3, D¼0.6mJ=m2,MS¼376kA=m, λ¼150.9MJ=m3,L¼ 22.3mJ=m2,γ¼2.211×105m=ðAsÞ,α¼0.2, and P¼0.4. The mesh size of 1×1×1nm3is used to discretize the AFM film with the size 200×200×1nm3. (a)–(h) show the N´ eel vector in the 100×100nm2plane.PHYSICAL REVIEW LETTERS 124, 037202 (2020) 037202-3where α/C3¼αμ0MStzd=γand FSOT;0sinð2πftÞis the force induced by the alternating current with FSOT;0≈ π2Rsμ0HdMStz.Fbis the boundary-induced force, which can be described as Fb≈k1rxþk2r3xwith k1¼4.55× 10−6N=m and k2¼2×1010N=m3for the nanodisk with a diameter of 80 nm studied here (see Ref. [63]for details). Note that, for other nanodisks, the form of Fbmay change, resulting in other types of AFM-bimeron-based nonlinear oscillators, Since Fbcontains a cubic term, Eq. (5)is called the Duffing equation [47,66] , which describes a nonlinear system. Therefore, the AFM bimeron can be used as a Duffing oscillator, which is promising for various appli- cations, such as in weak signal detection [54,69] .W e assume that the solution of Eq. (5)satisfies this form rx≈r0sinð2πft−φÞ, and then substituting it into Eq. (5) gives the amplitude r0as r0¼FSOT;0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½k1þð3=4Þk2r2 0−Meffð2πfÞ2/C1382þð2πfα/C3Þ2p ;ð6Þ and the phase φ tanφ¼2πfα/C3 k1þð3=4Þk2r2 0−Meffð2πfÞ2; ð7Þ where sin3ð2πft−φÞ≈ð3=4Þsinð2πft−φÞhas been used. As shown in Fig. 3, the results given by Eqs. (6) and(7)are consistent with the numerical simulations for all damping constants. We can see from Eqs. (6)and(7)that the frequency response depends on the physical quantities of antiferromagnets, such as the damping and the effectivemass, so that they may be measured by applying alternating currents. It should be noted that, due to the existence of the nonlinear term ( k 2r3x), Eq. (6)indicates that an alternating current may induce multiple values of r0, resulting in the frequency response showing a jump phenomenon. For the nonlinear oscillator based on other types of AFM textures,such as the AFM skyrmion and domain wall, one can obtain a similar frequency response. If the nonlinear term and the damping are small, from Eq. (6), the resonance frequency is given by, fr¼1=ð2πÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k1=M effp , which equals to 16 GHz for the parameters used here. On the other hand, as mentioned earlier, r0∼11.64nm and φ∼89.14° for f¼20GHz. Equation (7)indicates that when fis equal to fπ=2¼1=ð2πÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðk1þ0.75k2r2 0Þ=M effp , φ¼90°. Taking r0¼11.64nm, fπ=2¼19.2GHz is obtained, which is consistent with the simulation result. In addition, for the case of k1¼0,k2¼0, and Meff¼0, i.e., there are no boundary effect and effective mass, Eq. (7) also gives the phase φ¼90°, which is independent of the damping and the frequency. For a nonlinear system, taking certain parameter values, it shows chaotic behavior. The Lyapunov exponents (LEs) are usually used to judge whether there is chaos, given as [48,64] LEi¼lim t→∞1 tlnkδxitk kδxi 0k; ð8Þ where kδxi 0kis the distance between two close trajectories at initial time, and kδxitkis the distance between the trajectories at time t. If the largest LE is positive, it means that two close trajectories will be separated. Therefore, a small initial error will increase rapidly, resulting in the evolution of rxbeing sensitive to initial conditions, and its value cannot be predicted for a long time, i.e., the AFMbimeron shows chaotic behavior. Based on Eq. (5),w e calculate the bifurcation diagram and LEs (see Ref. [63]for details), and the results are given in Fig. 4,showing that the periodic and chaotic windows appear at intervals . We find that a small damping αcan lead to the chaotic behavior. The sum of LEs, which equals to −αλγ=μ 0MS, agrees with the above result. On the other hand, the value of the damping αat the ith period-doubling bifurcation should05 10 15 20 2551015 05 10 15 20 25020406080100120r0(nm) f(GHz)Numerical Analytical 0.0015 0.002 0.003 (°) f(GHz)(a) (b) FIG. 3. (a) The amplitude r0and (b) phase φas functions of the frequency fof the alternating current [ j¼j0sinð2πftÞ], where the symbols are the results of our numerical simulations, whilethe lines are obtained from Eqs. (6)and(7). FIG. 4. (a) Calculated bifurcation diagram and (b) Lyapunov exponents (LEs) as functions of the damping constant α, where α1;2;3¼0.000 323 6 , 0.000 274 4, and 0.000 263 8. (c) Bifurca- tion diagram and (d) LEs as functions of the current density j.PHYSICAL REVIEW LETTERS 124, 037202 (2020) 037202-4satisfy the universal equation, i.e., the Feigenbaum constant δ¼limi→∞½ðαi−αi−1Þ=ðαiþ1−αiÞ/C138 ¼ 4.669.[48,64] For the case of Fig. 4(a),δ2is equal to 4.64, from which we estimate that chaos will occur at α∞¼0.0002609 .I n addition, the current density jis also of great importance to induce the occurrence of the chaos, as it can be easilytuned in experiment. Figures 4(c) and4(d) show that, for small currents, the system exhibits a periodic movement. With increasing currents, the period-doubling phenomenontakes place, and, then, the system shows chaotic behavior. It is worth mentioning that the chaotic behavior studied here is subject to the boundary-induced force F b, which depends on both the geometric and magnetic parameters.The effects of F bon chaos are discussed in Ref. [63]. Conclusions. —In summary, we have studied the dynam- ics of an isolated AFM bimeron induced by spin currents. We demonstrate that a spin current can create an isolatedbimeron in the AFM film, and drive the AFM bimeron at a speed of a few kilometers per second. Based on the Thiele approach, the steady motion speed is derived, which is ingood agreement with the simulation results. Also, we find that the AFM bimeron can be used as the Duffing oscillator. Furthermore, we study the chaotic behavior by calculatingthe Lyapunov exponents. Our results are useful for the understanding of bimeron physics in AFM systems and may provide guidelines for building spintronic devicesbased on bimerons. X. Z. acknowledges support by the Presidential Postdoctoral Fellowship of The Chinese University of Hong Kong, Shenzhen (CUHKSZ). M. E. acknowledgessupport by the Grants-in-Aid for Scientific Research from JSPS KAKENHI (Grants No. JP18H03676, No. JP17K05490, and No. JP15H05854) and also supportby CREST, JST (Grants No. JPMJCR16F1 and No. JPMJCR1874). O. A. T. acknowledges support by the Australian Research Council (Grant No. DP200101027), theCooperative Research Project Program at the ResearchInstitute of Electrical Communication, Tohoku University and by UNSW Science International Seed grant. X. L. acknowledges support by the Grants-in-Aid for ScientificResearch from JSPS KAKENHI (Grants No. 17K19074, No. 26600041, and No. 22360122). G. Z. acknowledges support by the National Natural Science Foundation of China(Grants No. 51771127, No. 51571126, and No. 51772004) of China, the Scientific Research Fund of Sichuan Provincial Education Department (Grants No. 18TD0010 andNo. 16CZ0006). Y. Z. acknowledges support by the President ’s Fund of CUHKSZ, Longgang Key Laboratory of Applied Spintronics, National Natural Science Foundationof China (Grants No. 11974298 and No. 61961136006),Shenzhen Fundamental Research Fund (Grant No. JCYJ20170410171958839), and Shenzhen Peacock Group Plan (Grant No. KQTD20180413181702403). L. S. and J. X. contributed equally to this work.*zhaogp@uestc.edu.cn †zhouyan@cuhk.edu.cn [1] U. K. Rößler, A. N. Bogdanov, and C. Pfleiderer, Nature (London) 442, 797 (2006) . [2] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013) . [3] G. Finocchio, F. Büttner, R. Tomasello, M. Carpentieri, and M. Kläui, J. Phys. D 49, 423001 (2016) . [4] A. Fert, N. Reyren, and V. Cros, Nat. Rev. Mater. 2, 17031 (2017) . [5] K. Everschor-Sitte, J. Masell, R. M. Reeve, and M. Kläui, J. Appl. Phys. 124, 240901 (2018) . [6] W. Kang, Y. Huang, X. Zhang, Y. Zhou, and W. Zhao, Proc. IEEE 104, 2040 (2016) . [7] Y. Zhou, Natl. Sci. Rev. 6, 210 (2019) . [8] X. Zhang, M. 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PhysRevLett.102.017001.pdf

Magnetic Moment Manipulation by a Josephson Current F. Konschelle and A. Buzdin * Condensed Matter Theory Group, CPMOH, Universite ´de Bordeaux and CNRS, F-33405 Talence, France (Received 9 October 2008; published 5 January 2009) We consider a Josephson junction where the weak link is formed by a noncentrosymmetric ferromag- net. In such a junction, the superconducting current acts as a direct driving force on the magnetic moment.We show that the ac Josephson effect generates a magnetic precession providing then a feedback to thecurrent. Magnetic dynamics result in several anomalies of current-phase relations (second harmonic,dissipative current) which are strongly enhanced near the ferromagnetic resonance frequency. DOI: 10.1103/PhysRevLett.102.017001 PACS numbers: 74.50.+r, 74.45.+c, 74.78.Fk, 76.50.+g Many interesting phenomena have been observed re- cently in the field of spintronics: the spin-dependent elec-tric current and inversely the current-dependentmagnetization orientation (see, for example, [ 1,2]). Moreover, it is well known that spin-orbit interactionmay be of primary importance for spintronics, namely,for systems using a two-dimensional electron gas [ 3]. In the superconductor/ferromagnet/superconductor (S/F/S) Josephson junctions, the spin-orbit interaction in a ferro-magnet without inversion symmetry provides a mechanismfor a direct (linear) coupling between the magnetic mo-ment and the superconducting current [ 4]. Similar anoma- lous properties have been predicted for Josephsonjunctions with a spin-polarized quantum point contact ina two-dimensional electron gas [ 5]. S/F/S junctions are known to reveal a transition to the /C25phase, where the superconducting phase difference ’in the ground state is equal to /C25[6]. However, the current-phase relation (CPR) in such a /C25-junction has a usual sinusoidal form I¼ I csin’, where the critical current Icdepends in a damped oscillatory manner on the modulus of the ferromagnetexchange field. In a noncentrosymmetric ferromagneticjunction, called hereafter ’ 0-junction, the time reversal symmetry is broken, and the CPR becomes I¼Icsinð’/C0 ’0Þ, where the phase shift ’0is proportional to the mag- netic moment perpendicular to the gradient of the asym-metric spin-orbit potential [ 4]. Therefore, manipulation of the internal magnetic moment can be achieved via thesuperconducting phase difference (i.e., by Josephsoncurrent). In the present work, we study theoretically the spin dynamics associated with such ’ 0-junctions. Though there is a lot of experimental progress in studying the staticproperties of S/F/S junctions, little is known about thespin-dynamics in S/F systems. Note here the pioneeringwork [ 7] where a sharpening of the ferromagnetic reso- nance was observed below the superconducting transitioninNb=Ni 80Fe20system. Theoretically, the single spin dy- namics interplay with a Josephson effect has been studied in [8–11]. More recently, the dynamically induced triplet proximity effect in S/F/S junctions was studied in [ 12,13],while the junctions with composite regions (including several F regions with different magnetization) were dis-cussed in [ 14,15]. Here, we consider a simple S/F/S ’ 0-junction in a low frequency regime "!J/C28Tc(!J¼ 2eV="being the Josephson angular frequency [ 16]), which allows us to use the quasistatic approach to treat the super-conducting subsystem in contrast with the case analyzed in [12,13]. We demonstrate that a dc superconducting current may produce a strong orientation effect on the F layermagnetic moment. More interestingly, the ac Josephsoneffect, i.e., applying a dc voltage Vto the ’ 0-junction, would produce current oscillations and consequently mag-netic precession. This precession may be monitored by theappearance of higher harmonics in CPR as well as a dccomponent of the current. In particular regimes, a total reversal of the magnetization could be observed. In the case of strong coupling between magnetic and supercon-ducting subsystems, complicated nonlinear dynamic re-gimes emerge. To demonstrate the unusual properties of the ’ 0-junction, we consider the case of an easy-axis magnetic anisotropy of the F material (see Fig. 1). Both the easy axis and gradient of the asymmetric spin-orbit potential nare along the z-axis. Note that suitable candidates for the F interlayer may be MnSi or FeGe. In these systems, the lack FIG. 1. Geometry of the considered ’0-junction. The ferro- magnetic easy-axis is directed along the z-axis, which is also the direction nof the gradient of the spin-orbit potential. The magnetization component Myis coupled with Josephson current through the phase shift term ’0/n:ðM^r/C9Þ, where /C9is the superconducting order parameter ( r/C9is along the x-axis in the system considered here).PRL 102, 017001 (2009) PHYSICAL REVIEW LETTERSweek ending 9 JANUARY 2009 0031-9007 =09=102(1) =017001(4) 017001-1 /C2112009 The American Physical Societyof inversion center comes from the crystalline structure, but the origin of broken-inversion symmetry may also beextrinsic, like in a situation near the surface of a thin F film. In the following, we completely disregard the magnetic induction. Indeed, the magnetic induction in the ( xy) plane is negligible for the thin F layer considered in this Letter,whereas the demagnetization factor cancels the internalinduction along the z-axis ( N¼1). The coupling between F and S subsystems due to the orbital effect has beenstudied in [ 17], and it appears to be very weak, and qua- dratic over magnetic moment Mfor the case when the flux ofMthrough the F layer is small in comparison with flux quantum /C8 0¼h=2e. The superconducting part of the energy of a ’0-junction is Esð’; ’ 0Þ¼EJ½1/C0cosð’/C0’0Þ/C138; (1) where EJ¼/C80Ic=2/C25is the Josephson energy, Icis the critical current, and ’0is proportional to the Mycompo- nent of the magnetic moment (see Fig. 1). Therefore, when the magnetic moment is oriented along the z-axis, we have the usual Josephson junction with ’0¼0. Assuming the ballistic limit we may estimate the characteristic Josephsonenergy as [ 6]/C8 0Ic=S/C24Tck2 Fsin‘=‘with ‘¼4hL="vF, where S,Landhare the section, the length and the exchange field of the F layer, respectively. The phase shift is ’0¼‘vso vFMy M0(2) where the parameter vso=vFcharacterizes the relative strength of the spin-orbit interaction [ 4]. Further on, we assume that vso=vF/C240:1. If the temperature is well below the Curie temperature, M0¼kMkcan be considered as a constant equal to the saturation magnetization of the F layer. The magnetic energy contribution is reduced to theanisotropy energy E M¼/C0KV 2/C18Mz M0/C192 ; (3) where Kis an anisotropy constant and Vis the volume of the F layer. Naturally, we may expect that the most interesting situ- ation corresponds to the case when the magnetic anisotropyenergy does not exceed too much the Josephson energy.From the measurements [ 18] on permalloy with very weak anisotropy, we may estimate K/C244/C210 /C05K/C1/C23A/C03.O n the other hand, typical value of Lin S/F/S junction is L/C24 10 nm andsin‘=‘/C241. Then, the ratio of the Josephson over magnetic energy would be EJ=EM/C24100 forTc/C24 10 K . Naturally, in the more realistic case of stronger anisotropy, this ratio would be smaller, but it is plausible to expect a great variety of regimes from EJ=EM/C281to EJ=EM/C291.Let us now consider the case when a constant current I<I cis applied to the ’0-junction. The total energy is (see, e.g., [ 16]): Etot¼/C0/C80 2/C25’IþEsð’; ’ 0ÞþEMð’0Þ; (4) and both the superconducting phase shift difference ’and the rotation of the magnetic moment My¼M0sin/C18(where /C18is the angle between the z-axis and the direction of M) are determined from the energy minimum conditions@ ’Etot¼@’0Etot¼0. It results in sin/C18¼I Ic/C0with /C0¼EJ KV‘vso vF; (5) which signifies that a superconducting current provokes the rotation of the magnetic moment Myin the ( yz) plane. Therefore, for small values of the rotation, /C18ðIÞdepen- dence is linear. In principle, the parameter /C0can be larger than one. In that case, when the condition I=Ic/C211=/C0is fulfilled, the magnetic moment will be oriented along they-axis. Therefore, applying a dc superconducting current switches the direction of the magnetization, whereas ap-plying an ac current on a ’ 0-junction could generate the precession of the magnetic moment. We briefly comment on the situation when the direction of the gradient of the spin-orbit potential is perpendicular(along y) to the easy axis z. To consider this case, we simply need to take ’ 0¼‘ðvso=vFÞcos/C18. The total energy (4) has two minima /C18¼ð0;/C25Þ, while applying the current removes the degeneracy between them. However, the en-ergy barrier exists for the switch from one minimum intoanother. This barrier may disappear if /C0>1and the cur- rent is large enough I>I c=/C0. In this regime, the super- conducting current would provoke the switching of themagnetization between one stable configuration /C18¼0 and another /C18¼/C25. This corresponds to the transitions of the junction between þ’ 0and/C0’0states. The readout of the state of the ’0-junction may be easily performed if it is a part of some SQUID-like circuit (the ’0-junction induces a shift of the diffraction pattern by ’0). In fact, the voltage-biased Josephson junction, and thus the ac Josephson effect, provides an ideal tool to studymagnetic dynamics in a ’ 0-junction. In such a case, the superconducting phase varies with time like ’ðtÞ¼!Jt [19]. If"!J/C28Tc, one can use the static value for the energy of the junction ( 4) considering ’ðtÞas an external potential. The magnetization dynamics are described bythe Landau-Lifshitz-Gilbert equation (LLG) [ 20] dM dt¼/C13M/C2Heffþ/C11 M0/C18 M/C2dM dt/C19 ; (6) whereHeff¼/C0/C14F=V/C14Mis the effective magnetic field applied to the compound, /C13the gyromagnetic ratio, and /C11 a phenomenological damping constant. The corresponding free energy F¼EsþEMyieldsPRL 102, 017001 (2009) PHYSICAL REVIEW LETTERSweek ending 9 JANUARY 2009 017001-2Heff¼K M0/C20 /C0 sin/C18 !Jt/C0rMy M0/C19 ^yþMz M0^z/C21 ; (7) where r¼‘vso=vF. Introducing mi¼Mi=M 0,/C28¼!Ft (!F¼/C13K=M2 0is the frequency of the ferromagnetic reso- nance) in LLG Eq. ( 6) leads to _mx¼mzð/C28Þmyð/C28Þ/C0/C0mzð/C28Þsinð!/C28/C0rmyÞ _my¼/C0mzð/C28Þmxð/C28Þ _mz¼/C0mxð/C28Þsinð!/C28/C0rmyÞ; (8) where !¼!J=!F. The generalization of Eq. ( 8) for/C11/C222 0is straightforward. One considers first the ‘‘weak cou- pling’’ regime /C0/C281when the Josephson energy EJis small in comparison with the magnetic energy EM. In this case, the magnetic moment precess around the z-axis. If the other components verify ðmx;myÞ/C28 1, then the Eqs. ( 8) may be linearized, and the corresponding solutions are mxðtÞ¼/C0!cos!Jt 1/C0!2and myðtÞ¼/C0/C0 sin!Jt 1/C0!2:(9) Near the resonance !J/C25!F, the conditions of lineariza- tion are violated, and it is necessary to take the dampinginto account. The precessing magnetic moment influencesthe current through the ’ 0-junction like I Ic¼sin!Jtþ/C0r 21 !2/C01sin2!Jtþ...; (10) i.e., in addition to the first harmonic oscillations, the cur- rent reveals higher harmonics contributions. The amplitudeof the harmonics increases near the resonance and changesits sign when ! J¼!F. Thus, monitoring the second harmonic oscillations of the current would reveal the dy-namics of the magnetic system. The damping plays an important role in the dynamics of the considered system. It results in a dc contribution to theJosephson current. Indeed, the corresponding expressionform yðtÞin the presence of damping becomes myðtÞ¼!þ/C0!/C0 rsin!Jtþ/C11/C0/C0/C11þ rcos!Jt;(11) where !/C6¼/C0r 2!/C61 /C10/C6and /C11/C6¼/C0r 2/C11 /C10/C6; (12) with /C10/C6¼ð!/C61Þ2þ/C112. It thus exhibits a damped resonance as the Josephson frequency is tuned to theferromagnetic one !!1. Moreover, the damping leads to the appearance of out of phase oscillations of m yðtÞ[term proportional to cos!Jtin Eq. ( 11)]. In the result, the current IðtÞ/C25Icsin!JtþIc!þ/C0!/C0 2sin2!Jt þIc/C11/C0/C0/C11þ 2cos2!JtþI0ð/C11Þ (13)acquires a dc component I0ð/C11Þ¼/C11/C0r 4/C181 /C10/C0/C01 /C10þ/C19 : (14) This dc current in the presence of a constant voltage V applied to the junction means a dissipative regime which can be easily detected. In some aspect, the peak of dc current near the resonance is reminiscent of the Shapirosteps effect in Josephson junctions under external r.f. fields.Note that the presence of the second harmonic in IðtÞ Eq. ( 13) should also lead to half-integer Shapiro steps in ’ 0-junctions [ 21]. The limit of the ‘‘strong coupling’’ /C0/C291(butr/C281) can also be treated analytically. In this case, my/C250and solutions of Eq. ( 8) yields mxðtÞ¼sin/C20/C0 !ð1/C0cos!JtÞ/C21 mzðtÞ¼cos/C20/C0 !ð1/C0cos!JtÞ/C21 ;(15) which are the equations of the magnetization reversal, a complete reversal being induced by /C0=! > /C25= 2. Strictly speaking, these solutions are not exact oscillatory functions in the sense that mzðtÞturns around the sphere center counterclockwise before reversing its rotation, and returnsto the position m zðt¼0Þ¼1clockwise, like a pendulum in a spherical potential [see Fig. 2(c)]. Finally, we have performed numerical studies of the nonlinear LLG Eq. ( 6) for some choices of the parame- ters when the analytical approaches fail. To check theconsistency of our numerical and analytical approaches, a) b) c) d) FIG. 2. Results of numerical analysis of the magnetic moment dynamics of the ’0-junction. (a) Reversal of mzfrom analytical expression Eq. ( 15) (dashed curve) and numerical one (normal curve). The other curves are related to the Mtrajectory: (b) in strong damping case (c) and (d) in the strong coupling regime revealing incomplete and complete magnetic moment reversal,respectively.PRL 102, 017001 (2009) PHYSICAL REVIEW LETTERSweek ending 9 JANUARY 2009 017001-3we present in Fig. 2(a) the corresponding mzðtÞdepen- dences for low-damping regimes. They clearly demon-strate the possibility of the magnetization reversal. InFigs. 2(b)–2(d), some trajectories of the magnetization vectors are presented for general coupling regimes. These results demonstrate that the magnetic dynamics of S/F/S’ 0-junction may be pretty complicated and strongly nonharmonic. If the ’0-junction is exposed to a microwave radiation at angular frequency !1, the physics that emerge are very rich. First, in addition to the Shapiro steps at !J¼n! 1, half-integer steps will appear. Second, the microwave magnetic field may also generate an additional magnetic precession with !1frequency. Depending on the parame- ters of ’0-junction and the amplitude of the microwave radiation, the main precession mechanism may be relatedeither to the Josephson current or the microwave radiation.In the last case, the magnetic spin-orbit coupling maysubstantially contribute to the amplitude of the Shapirosteps. Therefore, we could expect a dramatic increase of this amplitude at frequencies near the ferromagnetic reso- nance. When the influence of the microwave radiation andJosephson current on the precession is comparable, a verycomplicated regime may be observed. In the present work, we considered the case of the easy-axis magnetic anisotropy. If the ferromagnet presentsan easy-plane anisotropy, then qualitatively the mainconclusions of our article remain the same because the coupling between magnetism and superconductivity de- pends only on the M ycomponent. However, the detailed dynamics would be strongly affected by weak in-plane anisotropy. To summarize, we have demonstrated that S/F/S ’0-junctions provide the possibility to generate magnetic moment precession via Josephson current. In the regime ofstrong coupling between magnetization and current, mag-netic reversal may also occur. These effects have beenstudied analytically and numerically. We believe that thediscussed properties of the ’ 0-junctions could open inter- esting perspectives for its applications in spintronics.The authors are grateful to Z. Nussinov, J. Cayssol, M. Houzet, D. Gusakova, M. Roche, and D. Braithwaitefor useful discussions and comments. This work was sup- ported by the French ANR Grant No. ANR-07-NANO- 011: ELEC-EPR. *Also at Institut Universitaire de France . [1] I. Z ˇutic´, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [2] J. Hauptmann, J. Paaske, and P. Lindelof, Nature Phys. 4, 373 (2008). [3] R. Winkler, Spin-Orbit Coupling Effects in Two- Dimensional Electron and Hole Systems (Springer, New York, 2003). [4] A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008). [5] A. A. Reynoso et al. Phys. Rev. Lett. 101, 107001 (2008). [6] A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). [7] C. Bell et al. Phys. Rev. Lett. 100, 047002 (2008). [8] J.-X. Zhu and A. V. Balatsky, Phys. Rev. B 67, 174505 (2003). [9] L. Bulaevskii et al. , Phys. Rev. Lett. 92, 177001 (2004). [10] J.-X. Zhu et al. , Phys. Rev. Lett. 92, 107001 (2004). [11] Z. Nussinov et al. , Phys. Rev. B 71, 214520 (2005). [12] S. Takahashi et al. , Phys. Rev. Lett. 99, 057003 (2007). [13] M. Houzet, Phys. Rev. Lett. 101, 057009 (2008). [14] X. Waintal and P. W. Brouwer, Phys. Rev. B 65, 054407 (2002). [15] V. Braude and Y. M. Blanter, Phys. Rev. Lett. 100, 207001 (2008). [16] K. K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon and Breach Science Publishers, New York, 1986). [17] S. Hikino et al. , J. Phys. Soc. Jpn. 77, 053707 (2008). [18] A. Y. Rusanov et al. , Phys. Rev. Lett. 93, 057002 (2004). [19] B. D. Josephson, Superconductivity (in two volumes) (R. D. Parks, Marcel Dekker, , New York, 1968), Vol. 1, Chap. 9. [20] E. M. Lifshitz and L. P. Pitaevskii, Course of Theoretical Physics , Theory of the Condensed State Vol. 9 (Butterworth Heinemann, Oxford, 1991). [21] H. Sellier et al. , Phys. Rev. Lett. 92, 257005 (2004).PRL 102, 017001 (2009) PHYSICAL REVIEW LETTERSweek ending 9 JANUARY 2009 017001-4

PhysRevB.97.174404.pdf

PHYSICAL REVIEW B 97, 174404 (2018) Symmetry and localization properties of defect modes in magnonic superlattices R. A. Gallardo,1,2T. Schneider,3,4A. Roldán-Molina,5M. Langer,3,6A. S. Núñez,2,7K. Lenz,3J. Lindner,3and P. Landeros1,2 1Departamento de Física, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile 2Center for the Development of Nanoscience and Nanotechnology, 917-0124 Santiago, Chile 3Helmholtz-Zentrum Dresden–Rossendorf, Institut of Ion Beam Physics and Materials Research, Bautzner Landstrasse 400, 01328 Dresden, Germany 4Department of Physics, Technische Universität Chemnitz, Reichenhainer Strasse 70, 09126 Chemnitz, Germany 5Universidad de Aysén, Ovispo Vielmo 62, Coyhaique, Chile 6Paul Scherrer Institut, 5232 Villigen PSI, Switzerland 7Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, 8370415 Santiago, Chile (Received 21 December 2017; published 7 May 2018) Symmetry and localization properties of defect modes of a one-dimensional bicomponent magnonic superlattice are theoretically studied. The magnonic superlattice can be seen as a periodic array of nanostripes, where stripeswith different widths, termed defect stripes, are periodically introduced. By controlling the geometry of thedefect stripes, a transition from dispersive to practically flat spin-wave defect modes can be observed insidethe magnonic band gaps. It is shown that the spin-wave profile of the defect modes can be either symmetric orantisymmetric, depending on the geometry of the defect. Due to the localized character of the defect modes, aparticular magnonic superlattice is proposed wherein the excitation of both symmetric and antisymmetric flatmagnonic modes is enabled at the same time. Also, it is demonstrated that the relative frequency position ofthe antisymmetric mode inside the band gap does not significantly change with the application of an externalfield, while the symmetric modes move to the edges of the frequency band gaps. The results are complementedby numerical simulations, where excellent agreement is observed between the methods. The proposed theoryallows exploring different ways to control the dynamic properties of the defect modes in metamaterial magnonicsuperlattices, which can be useful for applications on multifunctional microwave devices operating over a broad frequency range. DOI: 10.1103/PhysRevB.97.174404 I. INTRODUCTION The dynamic properties of spin waves (SWs) in magnonic devices with artificial periodic modulation of the magnetic orgeometrical parameters have been a growing research areain recent years [ 1–10]. The magnetic metameterials called magnonic crystals (MCs) have been widely studied since their excitation spectrum presents magnonic band gaps (BGs), which can be controlled by external magnetic fields [ 11–13]. These systems can also be created by an artificial modulationof the magnetic properties [ 14–17] or by the modification of the film geometry [ 18–28]. The magnonic BGs are strongly dependent on the geometrical parameters of the periodic lattice,whose spatial range usually lies in hundreds of nanometers. Around these gaps, SWs can be excited in well-defined allowed frequency bands, where, depending on the wave vector, thewaves may have a standing or a propagating character. In par-ticular, the standing SWs occur at the borders of the Brillouinzones, and therefore, such waves can be excited only at somespecific wave vectors. This characteristic makes it difficult tochannel or guide the spin waves along specific regions, which is the key for applications in magnonic waveguides [ 10,13] and tunable narrow passband SW filters [ 29,30]. In the field of photonic crystals, it is well known that the incorporation of a local defect breaks the translationalsymmetry and electromagnetic modes can appear within theforbidden band gaps [ 31–34]. For instance, the addition of extra dielectric material in one of the unit cells gives riseto modes within the BGs that behave like a donor atom in a semiconductor, while the removal of dielectric material from the crystal produces acceptorlike modes [ 31]. This extra degree of freedom allows manipulating and controlling theproperties of light in dielectric metamaterials. Indeed, thedefect-induced phenomena in photonic crystals have beenapplied for controlling spontaneous light emission [ 35–37] and trapping optical pulses [ 38]. In analogy to photonic crystals, it has also been demonstrated that the controlled introduction ofperiodic defects in magnonic crystals induces defect modesinside the BGs that can beneficially enrich the SW bandstructure of the magnetic metamaterial [ 29,30]. Here, the periodic lattice induces translational symmetry to the SWs thatcan be broken by a controlled introduction of periodic defects.That is, a change in the periodic structure redefines the unit cellin the same way as in a crystal with a complex unit cell. Thissystem can be seen as a magnonic superlattice (MSL), which consists of a periodic array of magnonic supercells. The emergence of defect modes located within the band gap has been predicted [ 39–41] and was recently observed ex- perimentally in magnonic superlattices [ 29,30]. Multilayered ferromagnetic structures with variations in the magnetization,uniaxial anisotropy, and/or thicknesses have been theoreti-cally studied in backward volume (BV) geometry [ 39–41]. A 2469-9950/2018/97(17)/174404(8) 174404-1 ©2018 American Physical SocietyR. A. GALLARDO et al. PHYSICAL REVIEW B 97, 174404 (2018) theoretical analysis of short-wavelength perturbations in two- dimensional MCs with point defects was performed by Yanget al. [42–45], who investigated different configurations of the point defects. Defect-induced phenomena in one-dimensionalbicomponent MCs with structural defects were more recentlyinvestigated by Brillouin light scattering (BLS) measurementsand via numerical simulations [ 29,30], where arrays of 250- nm-wide permalloy (Py) stripes were fabricated in such a waythat every tenth wire is a defect wire with a larger widthranging from 300 to 500 nm. Since the recent experimentswere performed in the Damon-Eshbach (DE) geometry at smallwave vectors (around 10 μm −1), the dynamic dipolar contri- bution inevitably must be taken into account. Nevertheless,there is no theoretical description that considers the dynamicmagnetodipolar fields, and hence, the current measurementshave been compared only with micromagnetic and finite-element simulations [ 29,30,46]. Furthermore, the analysis of an arbitrary angle of the magnetization with respect to thesymmetry axes of the crystal has not been addressed so far.In addition, the evolution of the defect modes as a function ofthe external field has not been deeply explored either. Theseaspects have clearly hindered a complete study of the dynamic properties on MSL structures so far since there is no model available that considers a general way to introduce arbitraryarrays of periodic defects on the MC. In this paper, the symmetry and localization properties of defect modes within one-dimensional bicomponent magnonicsuperlattices are theoretically addressed and complementedwith micromagnetic simulations. It is shown that by controllingthe lattice parameter of the defect stripes, a transition fromslight to almost null dispersion of the defect modes is observed.In addition, by changing the width of the defect stripes thenature of the symmetry as well as the frequency of the defectmodes can be modified. It is also demonstrated that the externalfield can change the relative position of the symmetric modeswith respect to the BG, while the antisymmetric ones remain atthe same relative frequency position. The possibility of excitingboth symmetric and antisymmetric defect modes at the sametime is also proposed, which allows for observing the defectmodes in a straightforward way with ferromagnetic resonance(FMR) measurements. II. THEORETICAL DESCRIPTION By combining a defect-free lattice with a periodic array of stripes with different widths, a one-dimensional bicomponentmagnonic superlattice is formed, as shown in Fig. 1. Here, the lattice parameter of the defect-free crystal is a, while the lattice parameter of the defect stripes is νa. Here, ν=1,2,3,... is introduced to locate a defective wire at each νrepetition, allowing for a general description of the MSL. The dynamics of the magnetic system is described by the Landau-Lifshitz (LL) equation ˙M(r;t)=−γM(r;t)× H e(r;t), where γis the absolute value of the gyromagnetic ratio, M(r;t) is the magnetization, and He(r;t) is the effective field. For small magnetization deviations around the equilib-rium state, magnetization and effective field can be written asM(r;t)=M s(r)ˆZ+m(r;t) and He(r;t)=He0(r)+he(r;t), respectively. Here, Ms(r) is the saturation magnetization, ˆZ represents the equilibrium orientation of the magnetization, B xZ H X Y,yz δ δ A ϕ aνa ϕh FIG. 1. Geometry of the one-dimensional bicomponent magnonic superlattice composed of ferromagnetic materials A and B. The lattice parameter of the periodic array of nanostripes is a, while νa corresponds to the lattice parameter of the defect stripes, where ν represents the number of lattice repetitions that are necessary to form the magnonic superlattice. The width of the nanostripes (defects) is /lscript (/lscript+2δ). The spin waves are assumed to propagate in the zdirection, while the equilibrium magnetization (external field) forms an angle ϕ(ϕh) with the zaxis. The zoom denotes the unit cell of the MSL structure. andm(r;t)=mX(r;t)ˆX+mY(r;t)ˆYcorresponds to the dy- namic magnetization. In addition, He0(r) is the static part of the effective field, and he(r;t) is the time-dependent part. Now, assuming a harmonic time dependence, m(r;t)=m(r)eiωt, and neglecting the second-order terms in m(r), the LL equation can be written as i(ω/γ)mX(r)=−mY(r)He0 Z(r)+Ms(r)he Y(r)( 1 ) and i(ω/γ)mY(r)=mX(r)He0 Z(r)−Ms(r)he X(r), (2) withωbeing the angular frequency. Note that in Eqs. ( 1) and ( 2), the equilibrium conditions Ms(r)He0 Y(r)=0 and Ms(r)He0 X(r)=0 have been considered. Now, the effective field is given by He(r)=H+Hex(r)+Hd(r), where His the external field, Hex(r) is the exchange field, and Hd(r)i s the dipolar field. These fields are detailed in Appendix A. According to Bloch’s theorem, the dynamic magnetizationcomponents are expanded into Fourier series as m(r)=/summationtext Gm(G)ei(G+k)·r, where G=Gν nˆzdenotes the reciprocal lattice vector. Here, Gν n=(2π/νa )n, where nandνare integers. The saturation magnetization and exchange length arerespectively given by M s(r)=/summationtext GMs(G)eiG·randλex(r)=/summationtext Gλex(G)eiG·r. Here, it is assumed that the leading material contrasts are associated with the saturation magnetization andexchange stiffness. Nevertheless, a contrast in anisotropies orthe intrinsic damping can also modify the band structure of thespin waves as well as their relaxation time. A detailed analysisof these cases is beyond the scope of this paper, and in thefollowing the study will be limited to variations in M s(r) and 174404-2SYMMETRY AND LOCALIZATION PROPERTIES OF … PHYSICAL REVIEW B 97, 174404 (2018) f [GHz] kz ν = 5 ν = 10 Mode AS Mode S Mode S Mode AS δ=0 (f ) (g) (h) (i) (j) δ=−/2 δ=−/4 δ=/4 δ=/2 π/a 2π/a π/a 2π/a π/a 2π/a π/a 2π/a Mode AS Mode S Mode S First BG (a) (b) (c) (d) (e) π/a 2π/a π/a 2π/a π/a 2π/a π/a 2π/a π/a 2π/a First BG π/a 2π/a Mode AS FIG. 2. In (a)–(e) the dispersion of a superlattice as given by ν=5 is shown, while in (f)–(j) the case with ν=10 is depicted. The parameter δhas been varied from −/lscript/2u pt o /lscript/2 in such a way that the width ( /lscript+2δ) of the defects ranges from zero to a. The illustration above each plot schematically shows the unit cell of the magnonic superlattice structure, whereas the gray zone depicts the first band gap. In (f) and (j) the color code represents the numerical simulations, where the brightest color indicates a maximum of the response. This response is given in log scale and corresponds to δm=√ δm2 x+δm2 y+δm2 z,w h e r e δrefers to the subtraction of the ground state before the fast Fourier transform. λex(r), which provide information capable of reproducing the measured band structure [ 29,30] but not the lifetime of the modes. Now, by including the effective fields in Eqs. ( 1) and ( 2), the LL equation can be converted into the following eigenvalueproblem [ 1,5]: ˜Am G=i(ω/γ)mG, (3) where mT G=[mX(G1),..., m X(GN),mY(G1),..., m Y(GN)] is the eigenvector and ˜Ais given by ˜A=/parenleftBigg˜AXX ˜AXY ˜AYX ˜AYY/parenrightBigg . (4) By using standard numerical methods and a convergence test to check the reliability of the results, the eigenvalues andeigenvectors of Eq. ( 3) can be obtained. The matrix elements are given in Appendix A. III. MICROMAGNETIC SIMULATIONS Micromagnetic simulations were performed with the GPU- accelerated open-source code MUMAX3[47]. The bicomponent magnonic crystal was modeled as a 20 nm ×30 nm ×100μm stripe. Periodic boundary conditions were applied to regainthe thin-film nature of the system. The stripe was discretizedinto 4 ×1×16 384 cells, which results in a cell size of 5×30×6.1n m 3. The material parameters in the simulation were chosen as indicated in Sec. IV. In addition, a Gilbert damping value of 0.01 was chosen. Two kinds of simulationswere performed for the magnonic supercell. First, the SWdispersion relation was calculated by applying a sinc pulse intime and space [ 48]. In addition to the approach in Ref. [ 48], thesinc pulse was shifted in space by 30 .5 nm with respect to the unit cell to also excite the totally antisymmetric SW modes. Theresulting SW dispersion relations were obtained by performinga two-dimensional fast Fourier transform for every line of cellsin thezdirection. Furthermore, the FMR response of the system has been simulated. Therefore, the time evolution of the systemexcited by a sinc pulse in the time domain was recorded [ 49]. To excite the antisymmetric SW modes as well, an additionallinear offset was added. The SW frequencies were extracted asthe summation of the spatial fast Fourier transform in the timedomain within each cell. IV . RESULTS AND DISCUSSION To study the dynamic properties of the system, standard values of cobalt and permalloy are employed [ 29]. Namely, the magnetic properties of material A resemble those ofpermalloy (Ni 80Fe20), which are MA s=730 kA /m and AA ex= 1.1×10−11J/m. On the other hand, the magnetic properties of material B correspond to cobalt, i.e., MB s=1100 kA /m andAB ex=2.5×10−11J/m. Here, Aexis the exchange con- stant, and hence, λex=/radicalbig 2Aex/4πM2sis the exchange length. For both materials, an effective gyromagnetic ratio of γ= 0.0185556 GHz /G and thickness d=30 nm are used. Also, the lattice parameter of the defect-free crystal is a=500 nm, and its width is /lscript=250 nm. At 200 reciprocal lattice vectors, a convergence of the numerical solutions of Eq. ( 3) is reached. Figure 2shows the SW dispersion of MSLs created by ν=5 [Figs. 2(a)–2(e)] and ν=10 [Figs. 2(f)–2(j)]i nt h e Damon-Eshbach geometry at H=0. Here, the equilibrium magnetization is given by ϕ=π/2, and the SW propagation is along the zaxis. The parameter δhas been varied from −/lscript/2 174404-3R. A. GALLARDO et al. PHYSICAL REVIEW B 97, 174404 (2018) f[GHz] kzπ/a 2π/aν=3 ν=5 ν=1 0 π/a kzf[GHz]Af[GHz] ννAf(a)(b(((((((((((((((((((bbbbbbbbbbbbbbbbbbbbb) (ckzkk (cccccccccccccccccccccccc(cccccccccc) FIG. 3. (a) The antisymmetric defect mode evaluated at δ=/lscript/2 forν=3, 5, and 10. (b) Zoom of the dispersion around one boundary of the first Brillouin zone is shown. (c) The oscillation amplitude Af as a function of ν. up to/lscript/2 in such a way that the width ( /lscript+2δ) ranges from zero to a. In both cases, ν=5 and ν=10, a practically flat defect mode labeled as antisymmetric (AS) moves from thehigh-frequency region into the first BG when δ>0. As δ increases, this mode moves into the band gap and becomes localized around the center of the gap at δ=/lscript/2. Conversely, ifδ<0, the symmetric (S) mode at the low-frequency edge of the first BG enters into the magnonic BG and becomeslocalized close to the center of the gap at δ=−/lscript/2. Once both modes, S and AS, are located inside the BG, they arecharacterized by a nearly flat dispersion. Overall, one can seethat at higher values of νthe dispersion of the modes becomes flatter. Note that the case shown in Fig. 2(j)coincides with the system measured in Ref. [ 29]. Indeed, all parameters used in this paper are the same. Therefore, by comparing Fig. 3(b) ofRef. [ 29] with Fig. 2(j), one obtains excellent agreement be- tween them. Figures 2(f)and2(j)show a comparison between the micromagnetic simulations and the theoretical results.Overall, excellent agreement is reached between the methods,which corroborates the validity of the theoretical model. In thecaseν=10 depicted in Figs. 2(f)–2(j), it is clear that some defect modes also appear in the second BG. Nevertheless, thebehavior of these modes does not have a clear dependence onthe geometrical parameter of the modified stripe. For instance,atδ>0 they are localized within the second band gap, while atδ<0 the defect modes are localized around the second band-gap edges. In what follows, the results are focused onlyon the defect modes localized within the first band gap. On the other hand, it is possible to see that the defect modes always reveal a periodic dispersion with finite oscillationamplitude. Nonetheless, this amplitude decreases dramaticallyas the lattice parameter of the MSL νaincreases. This is depicted in Fig. 3(a), where the cases ν=3, 5, and 10 are shown. One can observe that the position of the defect modesis not significantly affected by ν; nevertheless, the oscillation amplitude A f[defined in Fig. 3(a)] and the number of peaks are clearly dependent on ν. Thus, at ν=10, for instance, the mode inside the BG seems to have no dispersion, which is inagreement with recent BLS experiments and micromagneticsimulations [ 29,30]. Figure 3(b) shows a zoom of Fig. 3(a) around one boundary of the first Brillouin zone ( k z=π/a), where a finite oscillation amplitude Afis observed. The behav- -5a-4a -3a -2a -a 0 a2a3a4a5a mX [arb. units] ](a) (b) z=3=5=1 0 δ=/2 δ=−/2ν ν ν -5a-4a -3a -2a -a 0 a2a3a4a5a FIG. 4. Defect mode for different values of ν. In (a) the case δ=−/lscript/2 is depicted, where the SW excitation exhibits a symmetric profile around the modified stripes, while in (b) an antisymmetric SW profile is observed for δ=/lscript/2. The vertical dashed (dot-dashed) line depicts the unit cell for ν=5(ν=3). ior ofAfas a function of νis illustrated in Fig. 3(c), where the oscillation amplitude decreases exponentially as νincreases. The spatial spin-wave profiles of the defect modes located within the first band gap, obtained from the in-plane dynamiccomponent m X, are depicted in Fig. 4forδ=±/lscript/2 and ν= 3, 5, and 10. The vertical dash-dotted (dashed) line depicts the unit cell for ν=3(ν=5). An important conclusion from Fig.4is that in addition to the reported antisymmetric defect states in Refs. [ 29,30], the MSL can be tuned by modifying the width of the defect stripe in such a way that the natureof the defect mode is either symmetric or antisymmetric. Forinstance, if δ<0, the mode is symmetric, as shown Fig. 4(a), whereas it is antisymmetric when δ>0 [see Fig. 4(b)]. Note that these symmetry properties are valid for other kinds ofmagnetic materials as long as the defect stripe corresponds tothe one with lower saturation magnetization since, if materialsA and B are exchanged, these symmetry properties are alsoreversed (not shown). On the other hand, unlike the defectmodes, where the excitation is mainly located in the defectzone, the branches at the band-gap edges show an extendedcharacter in such a way that these branches are excited in theentire unit supercell (see Ref. [ 29] for details). From Fig. 4it is easy to see that the SW profile of the defect modes decreases quickly as zincreases and this effect is enhanced as νincreases. Thus, for ν=10, the SW excitation is almost zero at z=±5a. This localization of the defect mode allows implementing the following: If the width of the fifthstripe (localized at z=±5a) in the lattice with ν=10 is geometrically modified, both the frequency and localization ofthe defect mode obviously should not change because the areaaround z=±5ais irrelevant for the dynamics of both S and AS modes. To corroborate this behavior, one may employ the caseν=10 with δ>0 in such a way to excite the AS mode and at the same time modify the width of the fifth stripe by changing/lscript→/lscript+2δ /prime(withδ/prime<0) in order to excite simultaneously both S and AS modes inside the magnonic band gap. Thecalculation of a superlattice with two alternating widths /lscript+2δ and/lscript+2δ /primefor each fifth stripe can be implemented in the theory by replacing the term cos( nπ)s i n(nπ/lscript/ 10a)i nE q .( B3) with cos( nπ)s i n[nπ(/lscript+2δ/prime)/10a] in such a way that δ modifies the width of the stripe located at z=j×10aand δ/primemodifies the width of the stripe in z=(j+1/2)10a, with j=0,1,2,3,....I nF i g . 5such a superlattice structure with two different defects characterized by δ=/lscript/2 andδ/prime=−/lscript/2 174404-4SYMMETRY AND LOCALIZATION PROPERTIES OF … PHYSICAL REVIEW B 97, 174404 (2018) f [GHz] mX [arb. units] -5a -4a -3a -2a -a 0 a2a3a4a5a z Mode AS Mode S Mode S Mode AS (a) M(b) kzπ/a 2π/a FIG. 5. In (a) a superlattice structure with ν=10,δ=/lscript/2a n d δ/prime=−/lscript/2 is depicted, while (b) shows the dynamic magnetization component mXof both S and AS modes. is shown. As mentioned above, neither the frequency nor the localization of S and AS modes is modified [see Figs. 2(f) and2(j)]. The interesting feature of this kind of system is that clearly uncoupled symmetric and antisymmetric defect modesmay be simultaneously excited and evolve from the upper andlower boundaries of the band gap, respectively, as δandδ /prime increase in magnitude. In Fig. 6(a) the evolution of the S and AS modes for a MSL with two different kinds of defects is shown as afunction of the magnitude of δandδ /prime,w h e r ei ti sa s s u m e d thatδ>0 and δ/prime<0. Figs. 6(b)–6(c) show the simulated and calculated dispersions for two specific values of δand |δ/prime|, 60 and 80 nm. Figures 6(d)–6(e) show the simulated and calculated dispersions for δ=|δ/prime|=100 and 125 nm. Note that there is a crossing point close to δ=|δ/prime|=115 nm where both S and AS modes have the same frequency. Then, forthe case δ=|δ /prime|=125 nm, the S mode has a slightly larger frequency than the AS mode, as opposed to the cases whereδ=|δ /prime|<115 nm. One can see in Figs. 5and6that two modes appear within the frequency BG. This is related tothe incorporation of two defect stripes in the unit supercellof the superlattice [see Fig. 5(b)]. Hence, it is expected that an arbitrary distribution of defect stripes in the unit supercell ofthe superlattice will induce a broad band consisting of multiplenearly flat modes within the band gap. The proposed MSL with two different defects would be especially interesting for FMR measurements since the natureof the external excitation in typical FMR setups allows us toexcite only the symmetric modes, and therefore, under specificconditions the S mode should be detected at k z=0. The applied field dependence of the S and AS modes is shown inFig. 7(a) at the FMR limit ( k z=0). Here, the low-frequency mode is plotted together with the S and AS modes, whereone notices that the symmetric mode is clearly influenced bythe field in such a way that at higher values of Hthe mode moves towards the high-frequency edge of the band gap (grayzone). Nevertheless, the AS mode remains almost in the samerelative frequency position with respect to the BG. Figures 7(b) and 7(c) show the SW profiles for the AS and S modes, respectively. Clearly, the profile of the AS modes remainsconstant as the external field increases, whereas the S mode isnotably modified. Since the dynamic part of the Zeeman energydensity can be expressed as /epsilon1 z=−(H/2Ms)(m2 X+m2 Y), it is expected that the AS modes will not have an additionaldynamic contribution in /epsilon1 zsince the term m2 X+m2 Yis not [nm] f [GHz] Mode S Mode AS (a) δ,|δ|kz f [GHz] f [GHz]δ=|δ|=6 0 δ=|δ|=8 0 δ=|δ|= 100 δ=|δ|= 125(b) (c) (d) (e) nm nm nm nmπ/a 2π/a π/a 2π/a π/a 2π/a π/a 2π/a kz FIG. 6. (a) Symmetric and antisymmetric modes as a function of δandδ/prime(note that for δ/primethe magnitude is plotted as δ/prime<0). (b)–(e) The simulated and theoretically calculated SW dispersion for some values of δand|δ/prime|. modified. Nevertheless, since for the symmetric mode the dynamic component of the magnetization mXchanges with the field, its frequency within the gap is influenced by theexternal field. Therefore, it is demonstrated that the symmetricmodes have a limited range of field where they can be observed,since when these modes reach the band-gap edges, they areextended along the crystal, and therefore, they can hardly bedetected [ 29]. On the other hand, once the antisymmetric mode is excited, it should be observable in a wider range of fields.The same behavior of the defect modes as a function of thefield value is valid for different magnitudes of δ(not shown). In the theoretical approach it has been assumed that the damping parameter is zero since this parameter does not sig-nificantly affect the band structure (real part of the frequency)of the modes. Nevertheless, the damping parameter is finiteand different in both materials in such a way that the lifetimeof the spin waves is dependent on the propagation direction ofthe waves [ 50,51]. Therefore, while the real part of the defect modes is nearly flat, the imaginary part should be dependenton the wave vector. Nonetheless, a quantitative analysis ofthe relaxation processes in magnonic superlattices is beyond 174404-5R. A. GALLARDO et al. PHYSICAL REVIEW B 97, 174404 (2018) z(b) (c)H [kOe]f[GHz] mX[arb. units] z(a) H = 0 H = 1 kOe H = 2 kOeLow-frequency mode -5a -4a -3a -2a -a 0 a2a3a4a5aMode ASMode SSimulation Theory -5a -4a -3a -2a -a 0 a2a3a4a5aAS S FIG. 7. (a) Evolution of the symmetric (dashed line) and anti- symmetric (solid line) modes as a function of the external field for kz=0. Here, the S mode is excited with δ=−/lscript/2, and the AS mode is excited with δ=/lscript/2. The gray zone depicts the first band gap, and the (blue) dotted line indicates the low-frequency FMR mode. (b) and (c) The SW profiles for the AS and S mode, respectively, clearly showing that only the S mode is influenced by the field. the scope of this study since here, the main focus is the band structure of magnonic superlattices. V . FINAL REMARKS The dynamic characteristics of one-dimensional bicompo- nent magnonic superlattices have been theoretically studiedby taking both the dipolar and exchange interactions intoaccount. Symmetry, localization, and the field-dependent prop-erties of the nearly flat defect modes have been theoreticallyaddressed and corroborated with micromagnetic simulations.It is found that by controlling the width of the modified stripe ofthe magnonic superlattice either symmetric or antisymmetricmodes can be excited. Also, by modifying the separationbetween defects, a transition from dispersive to practicallyflat spin-wave branches is observed inside the magnonic bandgaps. Due to the localization features of the defect modes,a system was proposed that consists of a superlattice withwide and narrow stripelike defects, where it is possible toobserve uncoupled symmetric and antisymmetric modes at thesame time. It was also demonstrated that the symmetric modeshave a limited range of fields where they can be observed,while the antisymmetric ones should be externally detectedin a wider range of external fields. The dynamic propertiesobserved in this work can be used to engineer the bandstructure of magnonic superlattice systems since the controlledintroduction of defects provides additional degrees of freedom,which can be of fundamental importance for technological applications in magnonic crystal-based devices. ACKNOWLEDGMENTS R.A.G. acknowledges financial support from FONDE- CYT Iniciacion Grant No. 11170736 and CONICYTPAI/ACADEMIA Grant No. 79140033. This work was alsosupported by FONDECYT 1161403 and 1150072, and theBasal Program for Centers of Excellence, Grant No. FB0807CEDENNA, CONICYT. T.S. acknowledges funding from theDeutsche Forschungsgemeinschaft (Grant No. GE1202/9-2)and funding from the In-ProTUC scholarship. M.L. acknowl-edges the funding from the Deutsche Forschungsgemeinschaft(Grant No. LE2443/5-1) as well as the European Union’sHorizon 2020 research and innovation programme under MarieSkłodowska Curie (Grant No. 701647). Funding from DAADGrant No. ALECHILE 57136331 and CONICYT PCCI140051are also highly acknowledged. APPENDIX A: EFFECTIVE FIELDS AND MATRIX ELEMENTS For the periodic structure shown in Fig. 1, the static exchange field is given by Hex0 Z(r)=−4π/summationdisplay G,G/primeG·(G/prime+G)Ms(G)[λex(G/prime)]2ei(G/prime+G)·r, (A1) where the other two static components are zero ( Hex0 X= Hex0 Y=0). On the other hand, the dynamic exchange com- ponents are hex X,Y(r)=− 4π/summationdisplay G,G/prime(G+k)·(G/prime+G+k)[λex(G/prime)]2 ×mX,Y(G)ei(G/prime+G+k)·r. (A2) According to Fig. 1, the external applied field is H0 Z= Hcos(ϕh−ϕ) andH0 X=Hsin(ϕh−ϕ), where ϕh(ϕ)i st h e angle between the external field (equilibrium magnetization)and the zaxis. On the other hand, the dynamic components of the dipolar field are h d Y(r)=−/summationdisplay GmY(G)ζ(G,k)ei(G+k)·r(A3) and hd X(r)=4π/summationdisplay GmX(G)ξ(G)2/bracketleftbiggζ(G,k)−1 |G+k|2/bracketrightbigg ei(G+k)·r.(A4) Here, it has been defined that ξ(G)=(Gν n+kz)s i nψand ζ(G,k)=2s i n h [ |G+k|d/2]e−|G+k|d/2 |G+k|d. (A5) Also, the Zcomponent of the static dipolar field is Hd0 Z(r)=−4π/summationdisplay GMs(G)χ(G)21−ζ(G,0) |G|2eiG·r,(A6) where χ(G)=Gν ncosϕ. 174404-6SYMMETRY AND LOCALIZATION PROPERTIES OF … PHYSICAL REVIEW B 97, 174404 (2018) By introducing the effective fields in the dynamic equation of motion, the submatrices in Eq. ( 4)a r eg i v e nb y AXX G,G/prime=AYY G,G/prime=0, (A7a) AXY G,G/prime=−Hcos(ϕ−ϕh)δG,G/prime+4πM s(G−G/prime)/bracketleftbigg χ(G−G/prime)21−ζ(G−G/prime,0) |G−G/prime|2−ζ(G/prime,k)/bracketrightbigg −4π/summationdisplay G/prime/primeMs(G−G/prime/prime)[(G/prime+k)·(G/prime/prime+k)−(G−G/prime/prime)·(G−G/prime)][λex(G/prime/prime−G/prime)]2, (A7b) AYX G,G/prime=Hcos(ϕ−ϕh)δG,G/prime−4πM s(G−G/prime)/braceleftbigg χ(G−G/prime)21−ζ(G−G/prime,0) |G−G/prime|2+ξ(G/prime)2/bracketleftbiggζ(G/prime,k)−1 |G/prime+k|2/bracketrightbigg/bracerightbigg +4π/summationdisplay G/prime/primeMs(G−G/prime/prime)[(G/prime+k)·(G/prime/prime+k)−(G−G/prime/prime)·(G−G/prime)][λex(G/prime/prime−G/prime)]2. (A7c) APPENDIX B: FOURIER COEFFICIENT OF ONE-DIMENSIONAL MAGNONIC SUPERLATTICES For a general one-dimensional superlattice, the Fourier coefficient of the saturation magnetization can be obtained by analyzing the one-dimensional periodic structure. Thus, according to Fig. 1, it is straightforward to see that Ms/parenleftbig Gν n/parenrightbig =1 2νa/bracketleftBigg MA s/integraldisplay−νa−/lscript 2 −νa 2e−iGν nzdz+MB s/integraldisplay−(ν−2)a+/lscript 2 −νa−/lscript 2e−iGν nzdz+MA s/integraldisplay−(ν−2)a−/lscript 2 −(ν−2)a+/lscript 2e−iGν nzdz + ···+ MA s/integraldisplay/lscript+2δ 2 −/lscript+2δ 2e−iGν nzdz+···+ MB s/integraldisplayνa−/lscript 2 (ν−2)a+/lscript 2e−iGν nzdz+MA s/integraldisplayνa 2 νa−/lscript 2e−iGν nzdz/bracketrightBigg (B1) ifνis an even number. On the other hand, if νis an odd number, the coefficient is calculated as Ms/parenleftbig Gν n/parenrightbig =1 2νa/bracketleftBigg MB s/integraldisplay−(ν−1)a+/lscript 2 −νa 2e−iGν nzdz+MA s/integraldisplay−(ν−1)a−/lscript 2 −(ν−1)a+/lscript 2e−iGν nzdz+MB s/integraldisplay−(ν−3)a+/lscript 2 −(ν−1)a−/lscript 2e−iGν nzdz + ···+ MA s/integraldisplay/lscript+2δ 2 −/lscript+2δ 2e−iGν nzdz+···+ MA s/integraldisplay (ν−1)a+/lscript 2 (ν−1)a−/lscript 2e−iGν nzdz+MB s/integraldisplayνa 2 (ν−1)a+/lscript 2e−iGν nzdz/bracketrightBigg . (B2) Then, by carrying out the appropriate integration of Eqs. ( B1) and ( B2), the result can be readily generalized as Ms/parenleftbig Gν n/parenrightbig =MB ssin(nπ) nπ+MA s−MB s nπ/braceleftbigg sin/bracketleftbiggnπ(/lscript+2δ) νa/bracketrightbigg +/Psi1(n,ν)s i n/parenleftbiggnπ/lscript νa/parenrightbigg/bracerightbigg , (B3) where /Psi1(n,ν)=cos(nπ) cos2/parenleftBig νπ 2/parenrightBig −2+2ν/summationdisplay j=1/braceleftbigg cos/bracketleftbigg(j−1)nπ ν/bracketrightbigg cos2/bracketleftBig (j+1)π 2/bracketrightBig cos2/bracketleftBig (ν+1)π 2/bracketrightBig +cos/bracketleftbigg(j−2)nπ ν/bracketrightbigg cos2/parenleftBig jπ 2/parenrightBig cos2/parenleftBig νπ 2/parenrightBig/bracerightbigg . (B4) Here, νrepresents the number of lattice repetitions that are necessary to form the MSL. A similar structure can be used for the exchange length λex(Gν n). Therefore, by choosing δandνany one-dimensional bicomponent magnonic superlattice can be modeled. [1] J. O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, and H. Puszkarski, Phys. Rev. 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PhysRevB.101.054419.pdf

PHYSICAL REVIEW B 101, 054419 (2020) Tailoring dual reversal modes by helicity control in ferromagnetic nanotubes H. D. Salinas ,1,*J. Restrepo,1,†and Òscar Iglesias2,‡ 1Grupo de Magnetismo y Simulación G +, Instituto de Física, Universidad de Antioquia, A.A. 1226, Medellín, Colombia 2Departament de Física de la Matèria Condensada and Institut de Nanociència i Nanotecnologia, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain (Received 10 April 2019; revised manuscript received 27 January 2020; accepted 28 January 2020; published 13 February 2020) We investigate the effects of the competition between exchange ( J) and dipolar ( D) interactions on the magnetization reversal mechanisms of ferromagnetic nanotubes. Using first atomistic Monte Carlo simulationsfor a model with Heisenberg spins on a cylindrical surface, we compute hysteresis loops for a wide range of theγ=D/Jparameter, characterizing the reversal behavior in terms of the cylindrical magnetization components along the tube length. For γ’s close to the value for which helical ( H) states are energetically favorable at zero applied field, we show that the hysteresis loops can occur in four different classes that are combinations of tworeversal modes with well-differentiated coercivities with probabilities that depend on the tube length and radius.This variety in the reversal modes is found to be linked to the metastability of the Hstates during the reversal that induces different paths followed along the energy landscape as the field is changed. We further demonstratethat reversal by either of the two modes can be induced by tailoring the nanotube initial state so circular stateswith equal or contrary chirality are formed at the ends, thus achieving low or high coercive fields at will withoutchanging γ. Finally, the results of additional micromagnetic simulations performed on tubes with a similar aspect ratio show that dual switching modes and its tailoring can also be observed in tubes of microscopic dimensions. DOI: 10.1103/PhysRevB.101.054419 I. INTRODUCTION Magnetic nanotubes have gained increasing interest in the last years from fundamental and technological standpointsdue to their double potential functionality arising from theircharacteristic inner and outer surfaces [ 1,2]. Recent advances in the experimental methods for fabrication of magnetic tubesand decoration techniques currently allow one to synthesizetubular and cylindric nanostructures by means of differentroutes [ 3–8] with a high degree of control on their com- position and geometry. The core-free aspect of the tubes ascompared to their filled counterparts or nanowires allows, inprinciple, a fast reversal process with a coercivity that can becontrolled by changing the shape factor or the aspect ratio.The variety of magnetic states that can be achieved in thesestructures have been exploited in sensors, logical devices [ 9], high-density magnetic memories [ 10], and even for magnetic hyperthermia and drug release [ 11–13]. In such systems, vortex ( V) and helical ( H) states have been predicted to occur theoretically by different methods[14–23]. In particular, vortex states are relevant for magnetic storage purposes since, being flux-closure configurations, theydo not produce stray fields, avoiding the consequent leakageof magnetic flux spreading outward from the tube. Advancesin imaging and characterization techniques have made pos- *hernan.salinas@udea.edu.co †johans.restrepo@udea.edu.co ‡oscar@ffn.ub.es; http://www.ffn.ub.es/oscar ; http://nanomagn.blogspot.comsible the direct visualization of magnetic configurations in individual nanotubes, confirming the formation of the above-mentioned VandHstates at different stages of the reversal process [ 24–29]. Several studies have reported field-driven magnetic-switching mechanisms of individual nanotubes hav-ing typical lengths in the range from 0.5 μm to some tens ofμm, radii from 100 −500 nm [ 30,31], and thicknesses from 10 −70 nm [ 27,28,32,33], with enough resolution to distinguish the nucleation of different states during reversaland the propagation of domain walls. Recent studies haveshown that it is even possible to register hysteresis loops ofa single-molecule magnet (SMM) [ 34]. Given the length scales of real tubes, theoretical under- standing of this subject has been gained mainly by us-ing micromagnetic simulations [ 35]. Apart from these, an- alytical calculations based on continuum approximations[16,17,36–41] have also succeeded in describing the main phenomenology. Other theoretical methods [ 42,43]o rs i m u - lation techniques [ 44–47] have also partially addressed these issues. However, most of the mentioned works focused on theinfluence of geometric parameters of the tubes or wires on thereversal mechanisms for a given material and the possibilityto explore the range of material parameters for which certainreversal modes can appear has not been exploited so far. Therefore, the objective of the present paper is to study how the competition between the two main competing en-ergies (given by the parameter γ=D/J, the ratio of dipo- lar (D) to exchange ( J) interactions between spins) may be used to tune or induce certain reversal modes when a mag-netic field is applied along the nanotube axis. Differentiatingfrom most of the above-mentioned works, we use both a 2469-9950/2020/101(5)/054419(12) 054419-1 ©2020 American Physical SocietySALINAS, RESTREPO, AND IGLESIAS PHYSICAL REVIEW B 101, 054419 (2020) micromagnetic approach as well as an atomistic Monte Carlo (MC) lattice model, performing simulations of hysteresisloops for certain representative geometric parameters of thetubes and considering values of γwithin a range that can be relevant to real materials. An atomistic approach allows us totackle finite-size effects, besides the accessibility to systemsat a nanoscopic level of a few nanometers where discretenessbecomes relevant, different from the micromagnetic approachwhere solution of the equations demands the considerationof a continuum model. Additionally, the consideration of arange of γvalues allows us to mimic a family of tubes with either different values of Jorμor the nearest-neighbor distance, which can in principle be tailor-made. The effect ofgeometrical parameters like the aspect ratio is also considered.We discover that for a certain range of γ’s, dual reversal modes of low and high coercivity appear as a consequenceof the metastability of helical and vortex spin configurationsformed during the first stages of the magnetization switching.The same phenomenology is also investigated for FeCo tubesat a micromagnetic level. In both cases, the chirality of helicaland vortex spin configurations is a key factor to understandthe two reversal mechanisms. The rest of the paper is organized as follows: In Sec. II,w e present the model and physical insights involved in the MCand micromagnetic simulations. In Sec. III, methodology and computational details are presented. In Sec. IV,w es t a r tb y analyzing the results of the MC simulated hysteresis loops,we show a procedure to tailor the different reversal modes,and end up presenting micromagnetic simulation results thatdemonstrate that a similar phenomenology can show up inFeCo tubes with sizes of tens of nanometeres. We finishwith a discussion about the relevance of these results as amechanism for coercivity enhancement and we present themain conclusions. II. MODEL A. Monte Carlo simulation As it concerns the MC simulation approach, a single- spin flip Metropolis dynamics and a three-dimensional (3D)classical Heisenberg Hamiltonian were implemented. TheHamiltonian reads as follows: H=E ex+Edip+EZ, (1) where Eexstands for short-range exchange coupling between 3D classical Heisenberg nearest-neighbors spins ( Sx,Sy,Sz), Eex=−/summationdisplay /angbracketlefti,j/angbracketrightJij/vectorSi·/vectorSj, (2) being Jij=J>0 the ferromagnetic (FM) exchange constant. This constant was fixed at 10 meV , a typical value of FMsystems. The second term E dipis the long-range magnetic dipolar interaction given by Edip=D/summationdisplay i<j/parenleftBigg /vectorSi·/vectorSj−3(/vectorSi·ˆrij)(/vectorSj·ˆrij) |/vectorrij|3/parenrightBigg , (3) where /vectorrijis the relative vector between iand jpositions and summation is over all the sets of pairs ( i,j), taking care tocount pair interactions once. Making explicit the fact that the pair distance vector can be considered as a multiple ofa minimum distance abetween nearest neighbors, which can eventually coincide with a lattice parameter, depending on thetype of system, we can define an atomistic dipolar couplingstrength parameter in energy units as D=μ 0μ2 4πa3, (4) withμ0andμbeing the vacuum permeability and the mag- netic moment per spin, respectively. Thus, we can define thedimensionless parameter γ=D/Jto quantify the degree of competition between long-range and short-range interactions.In this paper, γwas set to vary between 0.01 and 0.07, for which circular states ( VandH) can appear [ 23] and therefore it is expected that competition plays a relevant role on themagnetization reversal mechanisms. Finally, E Zis the Zeeman interaction of the spins with a uniform external field /vectorHapplied along the main axis of the tube, given by EZ=−/summationdisplay iμ/vectorSi·/vectorH. (5) B. Micromagnetics This approach was employed to simulate tubes of some tens of nanometers, i.e., one order of magnitude greaterthan those simulated atomistically via MC. Both zero tem-perature and finite-temperature micromagnetic simulationswere performed. For the latter, a thermal solver to theLandau-Lifshitz-Gilbert (LLG) equation was employed. Mag-netostatic, exchange, and Zeeman energies were consid-ered. The LLG equation, accounting for the magnetiza-tion dynamics of the system, was solved by using botha finite-difference method based on the object-oriented-micromagnetic-framework (OOMMF) [ 48], d/vectorM dt=− |γLL|/vectorM×/vectorHeff−|γLL|α M/vectorM×(/vectorM×/vectorHeff), where /vectorMis the magnetic moment per cell, γLLandαare the gyromagnetic ratio and the Gilbert damping parameter,respectively, whereas /vectorH effis an effective field that represents all the generalized forces acting on every magnetic moment.Temperature is included in the effective field through a fluctu-ating or stochastic term h flucfor which a Boltzmann distribu- tion is assumed for the spin ensemble and the stochastic fieldis modeled as white noise. The variance of the fluctuating fieldis given by σ 2 hfluc=α 1+α22kBT γLLMsV. In particular, we examined hysteresis loops and the time evo- lution of the magnetization at a temperature T=4 K. Loops were repeated for different values of the random numbergenerator used to generate the stochastic thermal fields. Weobtained also equilibrium magnetic configurations using thezero-temperature minimization driver. 054419-2TAILORING DUAL REVERSAL MODES BY HELICITY … PHYSICAL REVIEW B 101, 054419 (2020) III. METHODOLOGY AND COMPUTATIONAL DETAILS A. Monte Carlo simulation Single-wall nanotubes were modeled as shown in a pre- vious work [ 14] by rolling a square lattice along the (1,1) direction to get a zigzag-terminated 3D tube with symmetryaxis along the zdirection. We demonstrated in Ref. [ 23] that such spatial distribution of spins possesses a lower dipolarenergy than that of a columnar stacked realization. Tube dimensions are determined by the pairs ( N,N z),N being the number of spins per layer and Nzthe number of layers accounting for the height or length of the tubes.In particular, tubes having dimensions (8,20), (8,15), (8,14),and (8,8) were considered, with aspect ratios given by thequotient N z/N. Length and radius are given, respectively, by l=(Nz−1)√ 2a/2 and R=√ 2Na/(2π)[23]. We have conducted standard Metropolis MC simulations to obtain the hysteresis loops at a fixed low temperature T= 0.1J/kBby cycling the magnetic field in constant steps δH between 0.02 and 0.1, depending on the field region, whileperforming thermodynamic averages of several componentsof the magnetization and its rotational. Since thermodynamicstates in the hysteresis loops are not strictly equilibrium states,up to 100 different runs per loop were performed to carryout configurational averages with the corresponding error barscalculations. The maximum number of MC steps (5 ×10 3) and those discarded for thermalization (3 ×103) were kept fixed for all the hysteresis loops. As already proposed in our previous work [ 23], to ana- lyze the texture of noncollinear spin configurations duringreversal modes, more concretely VorHstates, we use also the vorticity order parameter defined as a discretized version of the curl of the magnetization /vectorρ=/vector∇×/vectorM, that quantifies the vorticity, helicity, or degree of circularity ofthe magnetic moments (similar to a toroidal moment [ 49]). Chirality can be characterized in terms of the sign of theazimuthal component of the magnetization m φ. Intermediate configurations attained during the reversal process of thehysteresis loops are tracked by expressing magnetic momentsin usual cylindrical coordinates [ 23] that are better suited to describe Vand Hstates. Thus, at every spin site /vectorr i= (Rcosϕi,Rsinϕi,zi), where Ris the tube radius and ϕiis the respective azimuthal space coordinate, a local referenceframe was considered so the spin vector components read /vectorS i= (Szi,Sφi,Sρi)=(cosθi,sinθisinφi,sinθicosφi)( s e eF i g . 1 for angle definitions). Therefore, a full characterization of spinconfigurations during the switching modes can be obtainedboth by plotting the average polar angle /angbracketleftθ/angbracketrightand the m φ component per layer for each zvalue along the entire length of the tube. B. Micromagnetic simulations To investigate both the hysteretic behavior of tubes with greater dimensions and their associated reversal modes, weconsider FeCo tubes having an internal diameter of 15 nm,an external one of 21 nm, and a height of 52.5 nm, with anaspect ratio very close to that of a (8,20) tube were considered.Simulations were performed considering magnetostatic, ex-change, and Zeeman energies with material parameters close FIG. 1. Local reference frame at a spin site where the magnetic moment is given by /vectorSi, showing the coordinates to describe magnetic configurations, with θandφthe polar and azimuthal spin angles, respectively. to that of FeCo, i.e., stiffness constant A=1.08×10−11J/m, saturation magnetization Ms=1.83×106A/m. The smallest mesh size was set at 1.5 nm, below the exchange length of thematerial. IV . RESULTS A. Hysteresis loops from MC simulations In our recent work [ 23], we established phase diagrams for the zero-field equilibrium configurations for nanotubes withcompeting exchange and dipolar interactions depending onthe value of γand geometric characteristics of the tubes. In particular, independently of the tube length and radius,at low γ, we found FM ground states along the tube axis while, for large enough γ,Vstates were found to become stable. Interestingly, for a range of γaround a critical value γ ⋆ that depends on the geometric parameters of the tubes, states Hwith helical order appear. Therefore, since it is possible to stabilize circular states in tubular structures from whicha switching mode can be initiated, it is expected that themagnetization reversal mechanisms of the nanotubes underthe application of a magnetic field depend on the γvalue. To study this, we first simulate low-temperature ( T=0.1J/k B) hysteresis loops for several values of γ, taking as a reference a (8,15) tube, for which we found γ⋆/similarequal0.035 [ 23]. The results of calculated hysteresis loops averaged over 30 runs, in whichthe initial random-number-generator seed was changed, areshown in Fig. 2for some selected values of γ. For low γvalues, when exchange interaction is dominant, the tubes behave as collinear ferromagnets reversing theirmagnetization coherently. Hysteresis loops exhibit a highdegree of squareness and appreciable but low values of thevorticity only around the coercive fields. On increasing γ,t h e coercive field diminishes as a consequence of the spin cantinginduced by dipolar interaction at the tube ends, where nucle-ation begins, which facilitates the spin reversal. Even thoughthe shape of the hysteresis loops in the range γ/lessorequalslant0.03 is qualitatively similar, suggesting that the reversal mechanisms 054419-3SALINAS, RESTREPO, AND IGLESIAS PHYSICAL REVIEW B 101, 054419 (2020) FIG. 2. Low-temperature hysteresis loops averaged over 30 dif- ferent runs for six different values of γfor a (8,15) tube with the field applied parallel along the tube axis. are preserved, microscopically some differences progressively emerge as indicated by the increasing height of the /angbracketleft|/vectorρ|/angbracketright peaks near the coercive fields in Fig. 3(a). This indicates the appearance of intermediate circular magnetic states duringthe switching process, close to the coercive field. For valuesofγ≈γ ⋆=0.035 for the (8,15) tube, the averaged loops become asymmetric and regions with considerable error barsappear progressively, pointing to variations in the inversionmodes from run to run. Bigger error bars are linked to ahigher metastability of the Hstates that can be formed at intermediate states of the reversal process for this range ofγ’s, as mentioned before. To elucidate the origin of these features, 100 additional different runs were performed forγ=0.035 by changing the initial seed each time. Results shown in Fig. 4reveal that all the hysteresis loops without exception, and under the same simulation conditions, fall intofour well-defined categories or paths with different proba-bilities of occurrence. A more detailed analysis allows usto identify the occurrence of two different switching modesalong the decreasing or increasing field branches, namely,one with lower coercivity ( Q1) and another with higher FIG. 3. Same as Fig. 2but for the average magnitude of the rotational of the magnetization /angbracketleft|/vectorρ|/angbracketright. coercivity ( Q2). Thus, the four classes of hysteresis loops can be categorized by combinations of them. Two classes ( Q1− Q1,Q2−Q2) correspond to symmetric loops [see Figs. 4(a) and4(d)], while for the other two ( Q1−Q2,Q2−Q1), cycles are asymmetric, resembling those found in exchange-biasedsystems. Their respective probabilities of occurrence are 86%(Q1−Q1), 11% ( Q1−Q2), 2% ( Q2−Q1), 1% ( Q2−Q2); while the total probabilities per mode or branch are 92.5% forQ1 and 7.5% for Q2. In the Q1 mode, the inversion is nucleated through the formation of Hstates with the same chirality at the tube ends (see snapshot 1 in Fig. 5, left column). Due to this, the magnetization switching proceeds in a completely coherentfashion through a gradual rotation of the angle θat both tube ends, reaching a state at remanence (snapshot 2) thatcorresponds to an almost perfect vortex. The formation of this Hstate and its gradual transition into a vortex is responsible for the low associated coercive field.This behavior is confirmed by the profiles shown in the upper 054419-4TAILORING DUAL REVERSAL MODES BY HELICITY … PHYSICAL REVIEW B 101, 054419 (2020) FIG. 4. The four possible paths followed when a (8,15) nanotube withγ=0.035 is submitted to a hysteresis loop starting from different seeds of the random number generator. The four scenariosare labeled according to the reversal modes followed along the decreasing-increasing field branches as (a) Q1−Q1, (b) Q1−Q2, (c)Q2−Q1, (d) Q2−Q2, respectively. left panels of Fig. 5, where for stage 2, the values /angbracketleftθ/angbracketright≈90◦, /angbracketleftmz/angbracketright≈0 and/angbracketleftmφ/angbracketright≈− 1 are attained. Unlike this, in the Q2 mode, reversal is started by the formation of Hstates having opposite chiralities at the tube FIG. 5. Magnetic configurations along the hysteresis loops for the different reversal modes displayed in Fig. 4(left and right columns correspond to panels (a) and (c) of that figure). Upper panels represent the height profiles of the quantities /angbracketleftθ/angbracketright,/angbracketleftmz/angbracketright,a n d( <mφ/angbracketright averaged per layer for the tube (8,15) and γ=0.035, whereas lower ones present snapshots of the spin configurations taken at points labeled in Fig. 4. FIG. 6. Dependence of the exchange (a) and dipolar (b) energies of the (8,15) nanotube on its magnetization for the configurationsattained along the decreasing field branch of the hysteresis loops shown in Figs. 4(a) and4(c) (green and blue symbols, respectively) corresponding to seeds 1 and 3, respectively, in the figure. ends as can be observed in snapshot 1 in Fig. 5(right column). As the reversal progresses, Hstates propagate by forming two domains with opposite directions connected by a domainwall (stage 2 in the right column panels of Fig. 5). This fact is confirmed by the positive and negative values of them φcomponent close to 1 and −1, respectively, as shown in the respective profiles of the right panels of Fig. 5.T h e confrontation of opposite chiralities in the central region ofthe tube makes the system magnetically harder and, therefore,a higher coercive field is obtained. The occurrence of different paths for reversal can be enlightened by studying the variation of the exchange anddipolar energies along the decreasing field branches of thehysteresis loops of Figs. 4(a) and4(c). Such variations are shown in Fig. 6as a function of the corresponding magnetiza- tion. For path Q1, while going from saturation to the coercive force, the dipolar energy decreases in a continuous fashion atthe expense of an increase in exchange energy, explaining theformation of a Vstate at a coercive field H c≈1.7T . O n t h e other hand, path Q2 is characterized by an excursion through 054419-5SALINAS, RESTREPO, AND IGLESIAS PHYSICAL REVIEW B 101, 054419 (2020) intermediate states with higher energies and this explains the lower probabilities of occurrence compared to those of pathQ1. Both in Q1 and Q2, reversal starts with identical decrease (increase) of the dipolar (exchange) energy. Differences be-tween paths initiate near remanence with a total average/angbracketleftm z/angbracketright≈0.8, from where the Q2 path drives the system through a state with /angbracketleftmz/angbracketright≈0.5, characterized by a local minimum in Edipabove the corresponding energy for the Q1 path. Near the coercive force, i.e., /angbracketleftmz/angbracketright=0, the Q2 path displays an abrupt decrease in both εexandεdipascribed to the formation of a domain wall connecting Hstates of opposite chirality. During switching, a confrontation of Hstates that propagate from the ends takes place in the central region of the tube. Oneof the Hstates leaves at the end where it was nucleated and at some point it is followed by the other Hstate (see snapshot 3 in the right column of Fig. 5). It is worth mentioning that when dipolar interactions dom- inate over exchange ( γ>γ ⋆), as can be seen in Fig. 2(b), the averaged loops become more tilted with higher closurefields. This is a consequence of the formation of almostperfect Vstates at remanence, which for this range of γ’s are the minimum energy configurations [ 23]. Two possible reversal modes are also found in this range, again dependingon the formation of states at the ends with equal or oppositechiralities in the early stages of reversal. However, now one ofthe modes ( Q1: low coercivity) has a negligible coercive field (not shown) and it corresponds to the formation of states forwhich vorticity varies monotonically with a maximum close to1. In contrast, the high coercivity mode ( Q2) is characterized by greater values of the coercive field compared to those oftheir low- γcounterparts. B. Tailoring reversal modes To verify the reproducibility of the paths observed in the Q1 and Q2 modes, we have examined the possibility to induce the magnetization reversal by either of the two modes ina controlled manner by preparing the tubes in two initialconfigurations near the remanence. Such as-prepared statesconsist in a central region where spins are aligned along thetube length ( /angbracketleftθ/angbracketright=0 ◦,/angbracketleftmz/angbracketright=1 and /angbracketleftmφ/angbracketright=0) and a helical order in the last two layers at both ends of the tube. Inone case, the two ends have the same chirality [ /angbracketleftθ/angbracketright=45 ◦, /angbracketleftmz/angbracketright=cos(45◦) and/angbracketleftmφ/angbracketright=− sin(45◦)], i.e., with the same sign of /angbracketleftmφ/angbracketrightat the ends, and in the other case, opposite chiralities are manufactured, i.e., ( /angbracketleftθ/angbracketright=45◦,/angbracketleftmz/angbracketright=cos(45◦) and/angbracketleftmφ/angbracketright=± sin(45◦)). Due to the cylindrical symmetry, both states result in the same total magnetization [ /angbracketleft|/vectorm|/angbracketright = /angbracketleftmz/angbracketright=0.92,/angbracketleftmx/angbracketright=/angbracketleft my/angbracketright=0] and the same energy ( /angbracketleftE/angbracketright= −4.5 meV per spin). These initial states were then allowed to evolve under the same conditions along the decreasing field branch of thehysteresis loop. Results are shown in Fig. 7and the respective initial as-prepared states along with some intermediate spinconfigurations during magnetization reversal are shown inFig. 8. Starting from the state with the same chirality at both ends, the system always evolves through the Q1 inversion mode of lower coercivity, i.e., P(Q1)=100%. However, starting from the configuration with opposite chiralities, the FIG. 7. Decreasing field branch of the hysteresis loops for a (8,15) tube with γ=0.035 for two states initially prepared with H states of the same ( Q1) or opposite ( Q2) chiralities at the tube ends. probability to follow the Q2 mode is P(Q2)=85%, whereas for the Q1 mode it is P(Q1)=15%. These features reveal the subtleties and complexity of the energy landscape and thedifferences in energy between Q1 and Q2 modes. Based on the results obtained, we have also investigated the effectthat the tube length plays in the observed metastability ofthe reversal modes without changing γ, i.e., for γ=0.035. For this purpose, additional simulations were performed for FIG. 8. Magnetic configurations during magnetization reversal when starting from an initial state Q1(Q2) prepared with the same (opposite) chirality at the ends of the tube. Upper panels represent the height profiles of the quantities /angbracketleftθ/angbracketright,/angbracketleftmz/angbracketright,a n d/angbracketleftmφ/angbracketrightaveraged per layer for the tube (8,15) and γ=0.035, whereas lower ones present snapshots of the spin configurations taken at the points labeled in Fig. 7. Profiles of the initial remanence prepared state are included for comparison. 054419-6TAILORING DUAL REVERSAL MODES BY HELICITY … PHYSICAL REVIEW B 101, 054419 (2020) FIG. 9. The four possible paths followed when a (8,20) nanotube withγ=0.035 is submitted to a hysteresis loop simulated starting from different seeds of the random number generator. The fourcases are named according to the reversal modes followed along the decreasing-increasing field branches: (a) Q1−Q1, (b) Q1−Q2, (c) Q2−Q1, (d) Q2−Q2. two other tubes, namely, (8,20) and (8,8) tubes. In the former, results evidence the same phenomenology of the (8,15) tube(see Figs. 9and10), although now the coercive field of the Q1 mode has increased whereas that of the Q2 mode has decreased compared to the respective values of the (8,15) tube. FIG. 10. Magnetic configurations along the hysteresis loops for the different reversal modes displayed in Fig. 9(left and right columns correspond to panels (a) and (c) of the figure). Upper panels represent the height profiles of the quantities /angbracketleftθ/angbracketright,/angbracketleftmz/angbracketright,a n d /angbracketleftmφ/angbracketright averaged per layer for the tube (8,20) and γ=0.035, whereas the lower ones are snapshots of the spin configurations taken at points labeled in Fig. 9.Moreover, differences can be observed in the probabilities of occurrence per category, which are now 31%( Q1−Q1), 26%( Q1−Q2), 26%( Q2−Q1), 17%( Q2−Q2) with total probabilities per single mode of P(Q1)=57% and P(Q2)= 43%. Notice that these last probabilities are closer eachother, with a trend to equiprobability, which contrasts withthe results for the (8,15) tube, where P(Q1)=92.5% and P(Q2)=7.5%. On the other hand, when performing the simulations with the two initial as-prepared states, i.e., with the same ( Q1) or opposite ( Q2) chiralities at the ends, by following the same procedure of the (8,15) tube, we now obtain a low-coercivereversal mode probability P(Q1)=100% when starting from remanence Q1 and a high-coercive reversal mode probability P(Q2)=100% when starting from remanence Q2. In this way, we have been able to demonstrate that, at least fornanotubes with a higher aspect ratio N z/N, under tuning of the remanence states Q1o r Q2, the coercive force can be completely tailored to a low or high value, respectively. Now, if we modify the degree of competition between energies, by tuning the γvalue to the respective γ⋆≈0.045 of a (8,20) tube, for which Hstates are energetically more favor- able in the zero-field magnetic-phase diagram [ 23], the total probabilities per single mode become almost equiprobablewith P(Q1)=52.5% and P(Q2)=47.5%. Moreover, those remanence states having opposite chiralities in the processesQ1−Q2 and Q2−Q1 result in having very different proba- bilities of occurrence if the number of layers is an odd number.This fact makes the remanence spin configuration asymmetricrelative to the field direction, i.e., the system is not invariantunder field reversal. This is the reason why for the (8,15) tubethe probabilities P(Q1−Q2)=11% and P(Q2−Q1)=2% were obtained. Contrary to this, if the number of layers is aneven number, the remanence spin configuration is symmetricand the probabilities P(Q1−Q2) and P(Q2−Q1) tend to be equal as indeed observed for the (8,20) tube. To verifythis issue, additional simulations were performed for a tube(8,14), very close to the (8,15) tube. In such a case, it turnsout that P(Q1−Q2)=P(Q2−Q1)=2%. In this respect, it is easy to notice that if the number of layers is odd, in a stateof opposite chiralities, there will be an unbalance betweenthe number of layers with a given chirality and the numberof layers with the opposite one. The asymmetry becomesmore relevant depending on the decreasing or increasing fielddirection. Such unbalance is negligible in scenarios of tubeswith an even number of layers. Regarding the (8,8) nanotube, hysteresis loops (averaged over 100 realizations) and the respective vorticity curves werealso studied, see Fig. 11. In this case, tilted hysteresis loops forγ=0.035 do not reveal the appearance of the two modes. Despite the existence of small regions in the cycle near satura- tion with a greater irreversibility, the associated coercivity isvery small, leading to the occurrence of one single ( Q1) mode. Here, the short length of the tube makes it unable to harborthe vortex domain wall of opposite chiralities, suppressingin this way the high-coercivity ( Q2) mode. Thus, only the low-coercivity ( Q1) mode subsists. These observations are in agreement with the fact that reversal modes in magnetic nanotubes are strongly dependenton geometrical factors, particularly the aspect ratios, as has 054419-7SALINAS, RESTREPO, AND IGLESIAS PHYSICAL REVIEW B 101, 054419 (2020) FIG. 11. Hysteresis loops of a (8,8) tube for two different values ofγ. The spin configuration corresponds to the single kind of remanence state with the same chiralities at the ends present in this system. been already published [ 30,31,36], so differences when the aspect ratios change are expected even if the same initial statesare considered. Finally, for a different competition degree,namely, γ=γ ⋆≈0.022, also one single Q1 mode subsists, but now the magnetization reversal occurs practically in onesingle step in a very narrow region in the vicinity of thecoercive field. Contrary to this, for γ=0.035, magnetization reversal occurs accompanied by circular states ( /angbracketleftρ z/angbracketright/negationslash=0) persistent along most of the cycle. C. Micromagnetic simulations To further investigate the possibility of observing the above-mentioned phenomenology in tubes of greater dimen-sions in the range of tens of nanometers, closer to those oftubes experimentally synthesized, a micromagnetic approachhas been conducted. For this purpose, tubes having the sameaspect ratio to that of the (8,20) tube considered in the MCsimulations have been modeled. We would like to stress that it is not possible to establish a one-to-one correspondence between an atomistic modelfor which we consider a layer of spins rolled to form amonoatomic thick tube with a micromagnetic model for whichthe tube thickness has to be finite. Therefore, even though theγparameter cannot be mapped exactly to a micromagnetic equivalent, we can have an estimation of the γparameter for the continuous model used in micromagnetics. To do so,we approximate the volume density of the energies E ex≈ A/a2and Emag≈μ0M2 s/2 and estimate their ratio as γ/similarequal Emag/Eex,abeing the lattice parameter. With the values for FeCo mentioned in Sec. II B,γ=0.016, while a value of γ close to the γ⋆≈0.045 of the (8,20) tube can be obtained by choosing, for example, Ms=2×106A/m,A=10×10−12 J/m and a=0.3 nm, which are within the typical values for ferromagnets. Micromagnetic calculations were performedfor these last values. In this approach, we have ensured first that, using the en- ergy minimization procedure via OOMMF at T=0, ground- FIG. 12. (a) Hysteresis loops simulated with OOMMF at T= 4 K starting from saturation for two different seeds for the randomnumber used to generate the stochastic thermal fields. Lower panels display the dependence of the total (b), exchange (c), dipolar (d), and Zeeman (e) energies of the nanotube on its magnetization forthe configurations attained along the decreasing field branch of the hysteresis loops shown in (a). state configurations similar to those obtained by MC are obtained for the values of Aand Msmentioned above. In particular, by starting from a FM configuration, results showthat, after some transient time, the value of the energy isstabilized when the magnetic configuration reaches the equiv-alent Hstate. Thus, once it was established that circular states are achievable in micromagnetic simulations, hysteresis loopswere computed. In analogy with the MC simulations, the hysteresis loops were repeated using different random seeds for the generationof the stochastic thermal fields at temperature T=4K . T h e intention is to give the system a chance to follow differentpaths through the energy landscape at each field stage, sodifferent metastable states, which can be very close in energybut have different microstates, can be reached along the hys-teresis loop. This is reasonable as long as a hysteresis loopis a nonequilibrium curve. Thus, as can be observed in thesimulated hysteresis loops of Fig. 12, two well-distinguished modes ( Q1 and Q2) for magnetization reversal with two different coercivities were also obtained, in analogy withthe ones obtained via MC for the (8,15) and (8,20) tubes. 054419-8TAILORING DUAL REVERSAL MODES BY HELICITY … PHYSICAL REVIEW B 101, 054419 (2020) FIG. 13. Snapshots of the spin configurations in the reversal modes Q1 (upper panel) and Q2 (lower panel) of the hysteresis loops obtained from micromagnetic simulations. Numerical labelscorrespond to those states shown in Fig. 12. Similarly, tracking of the spin configurations along the de- creasing field branches of the loops allows us to identifyboth switching modes in terms of a dual mechanism. TheQ1 mode (obtained for a particular random seed) follows an intermediate stage characterized by Hstates at the ends of the tube with the same chirality giving rise to a low coercivefield. In contrast, when a different random seed is used, theQ2 mode is observed, characterized by intermediate stages with helicities of opposite chirality that give rise to a highercoercive field. Snapshots of these two reversal modes areshown in Fig. 13. The variations of the energy contributions along the hys- teresis loops given in the lower panels of Fig. 12have a similar behavior to those obtained by MC simulations in Fig. 6. In particular, energies for both modes are almost identicalat high fields and they bifurcate near the remanence. Also,beyond that point, in both cases the mode with oppositechiralities makes an excursion to higher exchange energies[Fig. 12(c) ], altough in the MC case the variation is more progressive, in agreement with the smoother variation of themagnetization near the coercive field [compare Figs. 6(a)and 6(d) to Fig. 12(a) ]. A similar observation is applicable when FIG. 14. Magnetization reversal at T=4 K. Steps of 1 ps were considered for time evolution. H=0.9Hc. Error bars were computed over 50 runs starting from different random seeds. comparing dipolar energies in Fig. 6(b) to demagnetizing ones in Fig. 12(d) , although in the latter case the differences between both modes are not as pronounced as in the MCsimulations. Similar jumps in energy have also been reportedto occur during the reversal of vortices in micromagneticsimulations of nanodots [ 50]. D. Thermal effects To evaluate the thermal effects in the micromagnetic ap- proach upon the occurrence of the dual inversion modes, wehave examined the time evolution of the magnetization ata finite temperature T=4 K, and for an external applied field below and close to the largest coercive force, namely,H=0.9H c(coercive field of the Q2 mode), by starting from a saturated initial configuration for different seeds. Results,shown in Fig. 14, are the averages over 50 runs. As can be observed, the two magnetization reversal modes Q1 and Q2 are still present and they are well differentiated. The differences between the characteristic reversal times forboth modes are clearly linked to the different paths followedthrough the energy landscape. Of course, depending on tem-perature, differences regarding the probabilities of occurrenceper mode are expected to change with respect to the atomisticsimulations. V . DISCUSSION AND CONCLUSIONS We have shown that the occurrence of both helical ( H) and vortex ( V) states during the magnetization reversal of nanotubes is dictated by γ, the dipolar to exchange energy ratio. For a certain range of γ’s, the results of MC atomistic simulations have demonstrated that different reversal modescan occur along the hysteresis loop as a consequence of thehigh degree of metastability of the Hstates that facilitates different paths through the energy landscape when varyingthe magnetic field. In agreement with our results, recentexperimental works on individual magnetic nanotubes [ 32,33] have also evidenced that short FeCoB nanotubes (0.6 μm 054419-9SALINAS, RESTREPO, AND IGLESIAS PHYSICAL REVIEW B 101, 054419 (2020) long, 300 mn in diameter), with similar aspect ratios as the ones studied here, can be found in mixed states with endvortices of opposing or matching circulations depending onthe magnetic history or experiment repetition. Moreover, thecomparison of cantilever magnetometry [ 33] with micromag- netic simulations showed that reversal initiated with matchingvortices is correlated to lower energies and smoother energyvariations than for opposing vortices. It is worth mentioningalso that in Ref. [ 21] a high degree of data scattering in the two branches of the hysteresis loop as well as a smallexchange-bias-like effect in an individual pure Ni tube wasobserved, which we assume could be attributed to the differentreversal possibilities shown in this work. Our results allow us to conclude that the reversal modes initiated by vortices with same chirality have coercive fieldslower than the mode with opposite chiralities, since in thiscase the central region of the tube has to face the mergingof opposite directions of helicity due to the formation of adomain wall. Our proposal to induce the reversal through theQ1o r Q2 modes, based on controlling the initial chirality at the tube ends, would allow us to use a unique tube as a softor hard material without changing its composition. In fact,the experimental study of Ref. [ 33] has already suggested that control over relative chirality can be achieved introducingstructural asymmetries at the nanotube ends, but we haveshown that this might be achieved also without modifying thetube structure. Regarding vorticity control in magnetic tubes without modifying their structure, some previous works can be consid-ered as a step in such direction. Namely, (i) by using magneticfield pulses along the symmetry-axis of the tube [ 51], (ii) by means of the injection of electric current pulses along thesymmetry axis of the tube [ 52], and (iii) by injection of small electric currents flowing through the nanotube with a circularmagnetic field applied at the middle of the tube [ 53]a saw a yto stabilize a vortex state, which can eventually be used as a starting state for an inversion process. Finally, apart from the scenario of having clusters or nanoparticles grafted or attached to the surface of a tube as atemplate where dipolar interactions can gain importance, it isalso viable to pose as an alternative the use of SMMs, whichcan be assembled, by using coordination chemistry, into1D,2D, and 3D networks [ 54]. The experimental challenge would be to get a 2D network of SMMs in a cylindrical arrangementan estimate of the γvalues involved in this kind of systems can be assessed by considering, for instance, the FM Mn 4 molecular magnet [ 55] with an average magnetic moment between Mn+2and Mn+3of 4.15μB, an average distance of 3.31 Å, and a FM exchange interaction of 6.5 K. These values yieldγ=0.046, placing us in the range where dual reversal modes may appear. Although in this paper we have limited our study to tubes of reduced dimensions, we have given proof that our conclusionsare not peculiar to the range of sizes studied. Preliminaryresults to be shown in a forthcoming publication indicate thata similar phenomenology can be observed for longer or widertubes under certain conditions of competition according to γ. ACKNOWLEDGMENTS Ò.I. acknowledges financial support form the Spanish MINECO (No. MAT2015-68772 and No. PGC2018-097789-B-I00), Catalan DURSI (No. 2017SGR0598) and EuropeanUnion FEDER Funds (una manera de hacer Europa), alsoCSUC for supercomputer facilities. H.D.S and J.R. acknowl-edge financial support from the Colciencias Beca de Doctor-ados Nacionales, ConvocatoriaNo. 727, Project No. CODI-UdeA 2017-16253. J.R. acknowledges UdeA for the exclusivededication program. [1] M. Sta ˇno and O. Fruchart, Magnetic nanowires and nanotubes, inHandbook of Magnetic Materials , edited by E. Brück (Else- vier, the Netherlands, 2018), V ol. 27, Chap. 3, pp. 155–267. [2] Y . Ye and B. Geng, Magnetic Nanotubes: Synthesis, Properties, and Applications, Crit. Rev. Solid State Mater. Sci. 37,75 (2012 ). [3] K. Nielsch, F. J. Castaño, C. A. Ross, and R. Krishnan, Magnetic properties of template-synthesized cobalt polymercomposite nanotubes, J. Appl. Phys. 98,034318 (2005 ). 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PhysRevB.73.092416.pdf

Strategy to reduce minimal magnetization switching field for Stoner particles Z. Z. Sun and X. R. Wang Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China /H20849Received 7 July 2005; published 29 March 2006 /H20850 A strategy is proposed aimed at substantially reducing the minimal magnetization switching field for a Stoner particle. Unlike the normal method of applying a static magnetic field which must be larger than themagnetic anisotropy, a much weaker field, proportional to the damping constant in the weak damping regime,can be used to switch the magnetization from one state to another if the field is along the motion of themagnetization. The concept is to constantly supply energy to the particle from the time-dependent magneticfield to allow the particle to climb over the potential barrier between the initial and the target states. DOI: 10.1103/PhysRevB.73.092416 PACS number /H20849s/H20850: 75.60.Jk, 75.75. /H11001a, 85.70.Ay INTRODUCTION The recent advance in technology allows the fabrication of magnetic nanoparticles1,2that are potentially useful for high density information storage. A magnetic nanoparticle, inwhich the magnetic moments of all atoms are aligned in thesame direction, is called a Stoner particle. Manipulation of aStoner particle 3is of significant interest in information pro- cessing. Finding an effective way to switch the magnetiza-tion from one state to another requires a clear understandingof magnetization dynamics. One important issue in magneti-zation reversal of Stoner particles is the minimal switchingfield. This problem was first studied by Stoner andWohlfarth 4/H20849SW /H20850who showed that a field hlarger than the SW-limit hSWcan switch the magnetization from its initial state to the target value through a ringing effect.5–9However, recent theoretical and experimental studies8–12have shown that the minimal switching field can be smaller than the SWlimit. Most studies have assumed the magnetic field to betime independent. However, a very recent experiment 10has shown that a dramatic reduction of the minimal field is pos-sible by applying a small rf field pulse /H20849the decrease in the constant field is much larger than the amplitude of the rffield /H20850. In this study, it has been shown that a small time- dependent magnetic field can affect the magnetization of aStoner particle such that the magnetization can move upwardin its energy landscape against the dissipation effect. A con-sequence of this is that the minimal switching field is muchsmaller in comparison with the case of a time-independentmagnetic field. In the case where the field magnitude doesnot change but the direction is allowed to vary, it can also beshown that the minimal field is proportional to the dampingconstant at the weak damping limit. DYNAMICS OF MAGNETIZATION IN A MAGNETIC FIELD The magnetization M/H6023=m/H6023Msof a Stoner particle can be conveniently described by a polar angle /H9258and an azimuthal angle/H9278, shown in Fig. 1 /H20849a/H20850where m/H6023is the unit direction of the magnetization, and Msis the saturated magnetization of the particle. In /H9258-/H9278plane, each point corresponds to a par- ticular state of the magnetization. The evolution of a state isgoverned by the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation9,13 /H208491+/H92512/H20850dm/H6023 dt=−m/H6023/H11003h/H6023t−/H9251m/H6023/H11003/H20849m/H6023/H11003h/H6023t/H20850, /H208491/H20850 where tis in a unit of /H20849/H20841/H9253/H20841Ms/H20850−1, and the magnetic field is in the unit of Ms./H20841/H9253/H20841=2.21/H11003105/H20849rad/s /H20850//H20849A/m /H20850is the gyro- magnetic ratio and /H9251is a phenomenological dimensionless damping constant. The typical experimental values of /H925111 range from 0.037 to 0.22 for different Co films. The total field comes from an applied magnetic field h/H6023and the internal field h/H6023idue to the magnetic anisotropy. Let w/H20849m/H6023,h/H6023=0 /H20850be the magnetic energy density function. Then h/H6023t=−/H11612m/H6023w/H20849m/H6023,h/H6023/H20850//H92620 =h/H6023i+h/H6023, where /H92620=4/H9266/H1100310−7N/A2is the vacuum magnetic permeability. As shown in Fig. 1 /H20849a/H20850, the first term in the right-hand side /H20849rhs /H20850of Eq. /H208491/H20850describes a precession motion around the total field and the second term is the dampingmotion toward the field. The switching problem for a uniaxial Stoner particle is as follows: Before applying an external magnetic field, there aretwo stable fixed points /H20851denoted by Aand Bin Fig. 1 /H20849b/H20850/H20852 corresponding to magnetizations, say m /H60230/H20849point A/H20850and − m/H60230 /H20849point B/H20850along its easy axis. The shadowed areas in Fig. 1 /H20849b/H20850 denote basins Aand B. The system in basin A/H20849B/H20850will end up at state A/H20849B/H20850. Initially, the magnetization is m/H60230, and the goal FIG. 1. /H20849a/H20850Two motions of magnetization m/H6023:h/H6023tis the total magnetic field. − m/H6023/H11003h/H6023tand − m/H6023/H11003/H20849m/H6023/H11003h/H6023t/H20850describe the precession and dissipation motions, respectively. /H20849b/H20850The/H9258-/H9278phase plane. Two stable fixed points Aand Brepresent the initial and the target states, respectively. Two shadowed areas denote schematically basins of A and B. The solid curve L1 and dashed curve L2 illustrate two dif- ferent phase flows connected Aand B.PHYSICAL REVIEW B 73, 092416 /H208492006 /H20850 1098-0121/2006/73 /H208499/H20850/092416 /H208494/H20850/$23.00 ©2006 The American Physical Society 092416-1is to reverse the magnetization to − m/H60230by applying an exter- nal field as small as possible. The issue is what is the mini-mal field h cdefined as hc=max /H20853h/H20849t/H20850;"t/H20854for a given mag- netic anisotropy. TIME-DEPENDENT VERSUS TIME-INDEPENDENT MAGNETIC FIELD In order to show that the magnetization reversal in a time- dependent external magnetic field is qualitatively differentfrom that in a constant field, it is useful to look at the energychange rate. From Eq. /H208491/H20850, dw dt=−/H9251 1+/H92512/H20849m/H6023/H11003h/H6023t/H208502−m/H6023·h/H6023˙, /H208492/H20850 where h/H6023˙is the time derivative of h/H6023. If the external field is time independent, the second term on the rhs vanishes, andhence, the energy will always decrease. In other words, aconstant field is not an energy source. Conversely, a time-dependent field can provide energy to a particle. Accordingto Eq. /H208492/H20850, the second term on the rhs can be either positive or negative depending on the relative direction of m /H6023and h/H6023˙. This second term can even be larger than the first one so that the particle energy increases during its motion. − m/H6023·h/H6023˙is a maximum when m/H6023and h/H6023˙are in the opposite direction. From /H20841m/H6023/H20841=1, it is known that m/H6023and m/H6023˙are orthogonal to each other, which leads to m/H6023·m/H6023¨=−m/H6023˙·m/H6023˙. The second term on the rhs of Eq. /H208492/H20850is the maximum when h/H6023=h0m/H6023˙//H20841m/H6023˙/H20841for a fixed h0. Then, from Eqs. /H208491/H20850and /H208492/H20850, the maximal rate of energy increase is dw dt=/H20841m/H6023/H11003h/H6023t/H20841 /H208811+/H92512/H20873h0−/H9251 /H208811+/H92512/H20841m/H6023/H11003h/H6023t/H20841/H20874. /H208493/H20850 It should be highlighted that h/H6023is only well defined when m/H6023˙ /HS110050. Thus, in a numerical calculation, some numerical diffi- culties will exist when the system is near the extremes or thesaddle points. Special care must be taken at these points. STRATEGY A strategy based on Eq. /H208493/H20850, can be developed using a smaller switching field. The field of magnitude h0noncol- linear with the magnetization was applied to drive the systemout of its initial minimum. Noncollinear fields have beenproposed before in magnetization reversal. 14Fluctuations may also drive the system out of the minimum, but fluctua-tions are inefficient. When the system is out of the minimum and m /H6023˙/HS110050, a time-dependent field h/H6023=h0m/H6023˙//H20841m/H6023˙/H20841is applied such that w˙/H110220. The system will climb the energy landscape from the bottom. When the system energy is very close to thesaddle point, the field of magnitude h 0can be rotated to noncollinear with the magnetization, say /H9266/4 to the direction of the target state so that problems at m/H6023˙=0 are avoided and the system can move closer to the target state. When thesystem has overcome the potential barrier between the initial and target state and stays inside the basin of the target state,the field can be turned off or applied in the opposite direction to the motion of the magnetization, i.e., h/H6023=−h0m/H6023˙//H20841m/H6023˙/H20841.I nt h e first case, the system will reach the target state through theringing motion caused by the energy dissipation, often due tothe spin-lattice relaxation. In the second case, the system willmove faster toward the target state because both terms on therhs of Eq. /H208492/H20850will be negative, resulting in a faster energy release from the particle. The strategy is schematically illustrated in Fig. 2 /H20849a/H20850. The particle first spins out of its initial minimum by extractingenergy from the field, and then spins into the target state byboth energy dissipation and energy release /H20849to the field /H20850. Since the energy gain from the field is partially compensatedby the energy dissipation during the spinning-out processwhile both the field and the damping consume energy in thespinning-in motion, the particle moves out of its initial mini-mum slowly in comparison with its motion toward the targetstate. It can be readily seen in Fig. 2 /H20849a/H20850that the particle makes more turns around the left minimum and fewer aroundthe right minimum. A similar result was experimentally con-firmed in Ref. 10. It will be shown later that a /H20849linearly polarized /H20850rf-field used in Ref. 10 is not the optimum. In fact, a circularly polarizedlike microwave /H20849around 100 GHz for a Co film 11/H20850is enough to switch a magnetization. The present strategy should be compared with those of the SW and theprecessional pico-second magnetization reversal. 11,12As il- lustrated in Figs. 2 /H20849b/H20850and 2 /H20849c/H20850, the SW strategy is to apply a large enough field to destroy the minimum initial state so thatthe particle can end up in the target state /H20851Fig. 2 /H20849b/H20850/H20852.I nt h e precessional magnetization reversal, 9the field is applied in FIG. 2. /H20849Color online /H20850/H20849a/H20850Schematic illustration of the present strategy. The field along the motion of the magnetization providesthe energy to the particle initially in the left energy minimum suchthat the system spins out of the minimum. Then a field opposite tothe motion of the magnetization causes the system to spin into theright minimum. /H20849b/H20850The SW strategy: The target state /H20849left mini- mum /H20850becomes the only minimum when the magnetic field is larger than h SW. The system will roll down along the potential landscape and end up at the target state. /H20849c/H20850The strategy in the precessional magnetization reversal: Both the initial /H20849point A/H20850and the target /H20849B/H20850 states are not local minima under the reversal field. Due to themagnetization dynamics described by the LLG equation, the par-ticle will move along the trajectory denoted by the solid line on theenergy landscape. The field is switched off as soon as the particlearrives at B.BRIEF REPORTS PHYSICAL REVIEW B 73, 092416 /H208492006 /H20850 092416-2such a way that the energy of the initial state is larger than that at the saddle point between the initial and final states.When the particle moves down the energy landscape, it willpass through the saddle point and arrive at the target state.The magnetization switch is achieved if the field is switchedoff at this point /H20851Fig. 2 /H20849c/H20850/H20852. For simplicity, consider the case of an uniaxial magnetic anisotropy with the easy axis lying along the xaxis. The general form of w/H20849m /H6023,h/H6023/H20850can be written as w/H20849m/H6023,h/H6023/H20850=−/H92620/H208731 2kmx2+mxhx+myhy+mzhz/H20874, /H208494/H20850 where hx,hy, and hzare the applied magnetic fields along x, y, and zaxis, respectively. k/H110220 is the parameter measuring the strength of the anisotropy. RESULTS To find the minimal switching field for the uniaxial aniso- tropy of Eq. /H208494/H20850, it can be seen from Eq. /H208491/H20850, that m/H6023˙is linear in the magnetic field, and as illustrated in Fig. 1 /H20849a/H20850, each field generates two motions for m/H6023. The first is a precession around the field, and the second toward the field. Under the influ-ence of the internal field /H20849along the xaxis /H20850and of the applied field h/H6023=h0m/H6023˙//H20841m/H6023˙/H20841, the system evolves into a steady precession state for a small h0because the precession motion due to the applied field can exactly cancel the damping motion due tothe internal field. The net motion /H20849sum of precession around the internal field and damping motion due to the appliedfield /H20850is a precession around the xaxis /H20849easy axis /H20850. In this motion, the energy loss due to damping and the energy gainfrom the time-dependent external field are equal. The bal-ance equation is h 0−k/H9251cos/H9257sin/H9257=0 , /H208495/H20850 where /H9257is the angle between the magnetization and the x axis. The initial state is around /H9257=0. Any stable precession motion must be destroyed in order to push the system over the saddle point at /H9257=/H9266/2. Since Eq. /H208495/H20850has solutions only forh0/H33355/H9251k/2, the critical field is hc=/H9251k/2. /H208496/H20850 It is of interest to note that the minimal reversal field is proportional to the damping constant, and approach zerowhen the damping constant goes to zero irrespective of howlarge the magnetic anisotropy is. For an arbitrary magneticanisotropy, it may not be possible to find the analytical ex-pression for the minimal reversal field, and should thus usenumerical calculations. To demonstrate that this can indeedbe done numerically, a calculation for the magnetic aniso-tropy of Eq. /H208494/H20850has been performed. The result of the mini- mal reversal field versus damping constant /H9251is plotted in Fig. 3. For comparison, the minimal reversal field for a time-independent magnetic field laying at 135° from the xaxis has been plotted. As it was explained in Ref. 9, the minimalreversal field is smaller than the SW limit for a small damp-ing constant /H9251/H11021/H9251c/H20851which is 1 for the model given by Eq. /H208494/H20850/H20852and equals to the SW limit for /H9251/H11022/H9251c. It is clear that the FIG. 3. The minimal reversal field /H20849in unit k/2/H20850vs the damping constant. The diamond symbols are the numerical results of thepresent strategy for the uniaxial model of Eq. /H208494/H20850. The solid curve is the analytical results. For comparison, the dashed line is the mini-mal reversal field under a constant field 135° to the xaxis for the same magnetic anisotropy. FIG. 4. /H20849a/H20850The phase flow under the present strategy with k=2,/H9251=0.1, and h0=0.11. Just as illustrated in Fig. 2 /H20849a/H20850, the phase flow shows a slow spin-out motion near the initial state and a fastspin-in motion near the target state. /H20849b/H20850–/H20849d/H20850The time-dependent reversal field with the same parameter as that in /H20849a/H20850. Insets: The corresponding Fourier transforms.BRIEF REPORTS PHYSICAL REVIEW B 73, 092416 /H208492006 /H20850 092416-3present strategy is superior to that of SW or precessional reversal scheme only for /H9251/H110211, and is worse for larger /H9251. To explain the type of field to be used in the current strategy, the trajectory of the system is numerically calcu-lated and the time-dependent magnetic field is recorded. Theresults for k=2, /H9251=0.1, and h0=0.11/H11022hcare given in Fig. 4. Figure 4 /H20849a/H20850is the phase flow of the system starting from a point very close to the left minimum. As explained previ-ously, the particle moves many turns in the left half of thephase plane before it crosses the potential barrier /H20849the saddle point is on the middle line /H20850while it moves toward the right minimum /H20849the target state /H20850much faster /H20849with few turns /H20850. Fig- ures 4 /H20849b/H20850–4/H20849d/H20850are the corresponding time dependence of x, y, and zcomponents of the magnetic field. From these curves, it can be shown that h yand hzoscillate with time reflecting the spinning motion around minima. In general thespinning periods along different paths vary. Thus the time-dependent magnetic field contains many different frequen-cies as can be seen from the Fourier transform of h i/H20849t/H20850, i=x,y,zshown in the insets of Figs. 4 /H20849b/H20850–4/H20849d/H20850. For Co-film parameters of Ms=1.36/H11003106A/m,11the time unit is ap- proximately /H20849/H20841/H9253/H20841Ms/H20850−1=3.33 ps. Correspondingly, the field consists of circularly-polarized microwaves of about 100 GHz. DISCUSSION AND CONCLUSIONS Although the switching field in the present scheme is much smaller than that in the old ones, it is an experimentalchallenge to create a time-dependent magnetic field requiredby this strategy. A device that is sensitive to the motion of themagnetization may be needed such that a coil can be at- tached to generate the required field. It should be emphasizedthat the results are based on the LLG equation which doesnot include any quantum effects. Quantum effects may beimportant for small particles 15whose level spacings are com- parable with the energy quanta of the time-dependent field.In that case, a quantum version of LLG equation needs to bedeveloped, which is beyond the scope of the present work. In conclusion, a scheme is proposed to dramatically re- duce the magnetization reversal field based on the fact that atime-dependent magnetic field can be both energy source andenergy sink, depending on whether the field is parallel oranti-parallel to the motion of the magnetization. The idea isto constantly supply energy to a Stoner particle from thetime-dependent magnetic field to allow the particle to moveout of its initial minimum and to climb over the potentialbarrier. After the particle lands in the basin of the target state,the time-dependent field will act as an energy sink that con-stantly withdraw energy from the particle such that the par-ticle will accelerate to the target state. In a simple model withan uniaxial magnetic anisotropy, the conditions and the solu-tion of the steady precession motion in the present schemeforh 0/H11021hcwere also found. ACKNOWLEDGMENTS This work is supported by UGC, Hong Kong, through RGC CERG grants. X.R.W. would like to thank the hospi-tality of Laboratoire Pierre Aigrain, Ecole NormaleSupérieure. Discussion with G. Bastard, P. Tong, and Ke Xiais acknowledged. 1S. Sun, C. B. Murray, D. Weller, L. Folks, and A. Moser, Science 287, 1989 /H208492000 /H20850; D. Zitoun, M. Respaud, M.-C. Fromen, M. J. Casanove, P. Lecante, C. Amiens, and B. Chaudret, Phys. Rev.Lett. 89, 037203 /H208492002 /H20850. 2M. H. Pan, H. Liu, J. Z. Wang, J. F. Jia, Q. K. Xue, J. L. Li, S. Qin, U. M. Mirdaidov, X. R. Wang, J. T. Market, Z. Y. Zhang,and C. K. Shih, Nano Lett. 5,8 7 /H208492005 /H20850. 3Spin Dynamics in Confined Magnetic Structure sI&I I , edited by B. Hillebrands and K. Ounadjela /H20849Springer-Verlag, Berlin, 2001 /H20850. 4E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A 240, 599 /H208491948 /H20850. 5W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 /H208491997 /H20850. 6Y. Acremann, C. H. Back, M. Buess, O. Portmann, A. Vaterlaus, D. Pescia, and H. Melchior, Science 290, 492 /H208492000 /H20850. 7T. M. Crawford, T. J. Silva, C. W. Teplin, and C. T. Rogers, Appl. Phys. Lett. 74, 3386 /H208491999 /H20850. 8L. He and W. D. Doyle, J. Appl. Phys. 79, 6489 /H208491996 /H20850.9Z. Z. Sun and X. R. Wang, Phys. Rev. B 71, 174430 /H208492005 /H20850. 10C. Thirion, W. Wernsdorfer, and D. Mailly, Nat. Mater. 2, 524 /H208492003 /H20850. 11C. H. Back, D. Weller, J. Heidmann, D. Mauri, D. Guarisco, E. L. Garwin, and H. C. Siegmann, Phys. Rev. Lett. 81, 3251 /H208491998 /H20850; C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller,E. L. Garwin, and H. C. Siegmann, Science 285, 864 /H208491999 /H20850. 12H. W. Schumacher, C. Chappert, P. Crozat, R. C. Sousa, P. P. Freitas, J. Miltat, J. Fassbender, and B. Hillebrands, Phys. Rev.Lett. 90, 017201 /H208492003 /H20850; H. W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas, and J. Miltat, ibid. 90, 017204 /H208492003 /H20850. 13L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 /H208491953 /H20850; T. L. Gilbert, Phys. Rev. 100, 1243 /H208491955 /H20850. 14A. Brataas, Yu. V. Nazarov, and G. E. W. Bauer, Eur. Phys. J. B 22,9 9 /H208492001 /H20850; B. C. Choi, G. E. Ballentine, M. Belov, and M. R. Freeman, Phys. Rev. B 64, 144418 /H208492001 /H20850. 15X. R. Wang and X. C. Xie, Europhys. Lett. 38,5 5 /H208491997 /H20850;X .R . Wang, S. C. Ma, and X. C. Xie, ibid. 45, 368 /H208491999 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 73, 092416 /H208492006 /H20850 092416-4

PhysRevB.101.184418.pdf

PHYSICAL REVIEW B 101, 184418 (2020) Controlling domain wall oscillations in bent cylindrical magnetic wires R. Cacilhas ,C .I .L .d eA r a u j o , and V . L. Carvalho-Santos Departamento de Física, Universidade Federal de Viçosa, Avenida Peter Henry Rolfs s /n, 36570-000 Viçosa, Minas Gerais, Brazil R. Moreno Earth and Planetary Science, School of Geosciences, University of Edinburgh, Edinburgh EH9 3FE, United Kingdom O. Chubykalo-Fesenko Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain D. Altbir Departamento de Física, CEDENNA, Universidad de Santiago de Chile, Avenida Ecuador 3493, Santiago, Chile (Received 14 November 2019; revised manuscript received 14 March 2020; accepted 27 April 2020; published 18 May 2020) Magnetic cylindrical nanowires are promising candidates for future three-dimensional nanotechnology. Domain walls (DWs) in magnetic nanowires play the role of information carriers, and the development ofapplications requires proper description of their dynamics. Here we perform a detailed analytical and numericalanalysis of the DW motion along a bent magnetic nanowire under the action of tangential magnetic fields. Ourresults show that the DW velocity, precession, and oscillation frequencies can be controlled by the interplaybetween the curvature and the external magnetic field. Small magnetic fields induce a DW motion withoutprecession and oscillatory behavior, while higher magnetic fields yield a Walker breakdown regime, in which anoscillatory forward and backward DW motion is observed. Controlled DW motion under the Walker breakdownregime makes magnetic nanowires potential candidates for nanoscale microwave generation and sensing. DOI: 10.1103/PhysRevB.101.184418 I. INTRODUCTION Magnetic cylindrical nanowires (NWs) hold important promises for future three-dimensional (3D) nanotechnologies such as the nanoscale Internet of Things [ 1–3], a 3D com- plex platform of multiphysics interconnected devices [ 4] with different functionalities for information processing, commu-nication, and sensing. The importance of magnetic nanowires for these technologies comes from the fact that they can respond to many external stimuli such as electrical, magnetic, thermal, and stress excitations. Consequently, by changing their composition or geometries, they can be functionalizedfor different components in such hypothetical devices. A large number of possibilities opens the door for the use of magnetic nanowires as different components of nanoscale devices. In this line, we propose that bent cylindrical magnetic nanowires can be engineered for nanoscale communication technologies,similar to spin-torque nano-oscillators [ 5–9]. This statement is based on the fact that when a domain wall moves under the ac- tion of a magnetic field or electric current in these nanowires, it oscillates around and along the wire axis [ 10–12], and the frequency can be controlled by the nanowire geometry andexternal magnetic field. In magnetic nanowires, domain walls (DWs) play the role of information carriers. Depending on material parametersand geometrical constraints of the nanowire, the DW can beof the transverse or vortex (Bloch point) type [ 13–15]. Herewe will work with the first type, which appear in cylindrical nanowires of diameters /lessorsimilar20 nm and relatively low saturation magnetization such as in permalloy (FeNi). The cylindrical geometry of nanowires plays an important role. First and importantly, for technological applications, it has been predicted that domain walls in straight cylin-drical nanowires do not suffer from the so-called Walker breakdown phenomenon with a sudden drop in velocity due to internal instabilities, which happens in magnetic stripes [16–18]. Indeed, Yan et al. [19] showed that the DW motion along a straight magnetic nanowire with a cylindrical crosssection does not present the oscillation associated with the Walker breakdown and the DW velocity varies linearly as a function of the magnetic field even for high magnetic fields. Furthermore, the changes in the geometry of magnetic nanos- tructures lead to curvature-induced Dzyaloshinskii-Moriyainteractions (DMIs) [ 20,21], and thus, cylindrical nanowires are intrinsically magnetochiral systems [ 22]. Indeed, due to this curvature-induced DMI, intriguing geometrical effects were reported [ 23–27]. In the case of magnetic nanowires, it was shown that this effective DMI yields a DW pinning at themaximum of the curvature [ 28], and the average velocity of the DW increases as a function of the curvature [ 12]. Unfortunately, the presence of bends in cylindrical nanowires introduces back the Walker breakdown for the DWmotion [ 12]. This fact, however, may be useful since, at quite low applied fields, the DW performs an oscillatory motion 2469-9950/2020/101(18)/184418(8) 184418-1 ©2020 American Physical SocietyR. CACILHAS et al. PHYSICAL REVIEW B 101, 184418 (2020) FIG. 1. (a) Sketch of the adopted coordinate system. (b) The DW configuration in a bent nanowire and the magnetic field direction. with large amplitude along the nanowire, together with the precession around its axis. Similar behavior was describedin a theoretical analysis of the DW motion along a helicalnanowire induced by an electrical current [ 29]. This fact, to our mind, opens the possibility to generate an alternatingmagnetic field for microwave generation and sensing com-munications at the nanoscale. Since the oscillatory motionappears after the Walker breakdown, in this work, we lookfor its appearance. Namely, we consider the geometry of theazimuthal magnetic field in bent cylindrical nanowires, inanalogy to straight nanostripes, where the field acts along thenanowire axis. Experimentally, this field can be generated byan electric current passing through the center of curvature ofthe nanowire. Due to the above-described promises, the study of the in- fluence of the curvature on the DW motion in bent cylindricalnanowires becomes very relevant. Therefore, in this work, weperform a detailed analysis of the DW dynamics along bentnanowires described as toroidal sections. Through analyticaland numerical calculations, we investigate the DW motionin terms of position, phase, and velocity in bent nanowireswith different curvatures and under magnetic fields. The mainresults suggest that the periodic motion of the DW, charac-terizing a Walker regime, appears when the magnetic fieldintensity is above a small critical value. This critical value ofthe magnetic field depends on the curvature of the nanowire.Above the Walker field, the oscillatory motion of the DW,which can be controlled by external parameters, appears. II. THEORETICAL MODEL In our model, we consider a bent cylindrical nanowire, described as a toroidal section with length L, internal and external radii Randr, respectively, and toroidal and poloidal angles denoted by θandϕ(see Fig. 1). In this frame, the re- lation among the wire length, external radius, and the toroidalangle is given by L=2θ 0R. Therefore, R=L/2θ0, and then a straight wire can be obtained in the limit θ0→0, while the largest curvature, described as a half-torus section, is givenbyθ 0=π/2. For further analysis, we also define the NW curvature as K=R−1. The dynamics of a DW displacing along the bent wire will be described by the Landau-Lifshitz-Gilbert (LLG) equation ∂m ∂τ=m×δ/epsilon1 δm+αm×∂m ∂τ, (1)where m=M/Msis the normalized magnetization vector field, Msis the saturation magnetization, and τ=(γ0Ms)tand /epsilon1=E/(μ0M2 s) are the dimensionless time and normalized energy, respectively. μ0is the magnetic permeability, αis the dimensionless Gilbert damping parameter, and γ0=μ0|γ| is the gyromagnetic factor ( γ=− 1.76×1011rad/Ts). The magnetization mwill be parametrized in terms of spheri- cal coordinates lying on a curvilinear basis in the form ofm=cos/Theta1e θ+sin/Theta1cos/Phi1er+sin/Theta1sin/Phi1ez, where the di- rectional vectors ( er,eθ,ez) are related to the toroidal coordi- nate system presented in Fig. 1. This parametrization in terms of spherical angles allows us to rewrite the LLG equation as −sin/Theta1∂/Theta1 ∂τ=δ/epsilon1 δ/Phi1+αsin2/Theta1∂/Phi1 ∂τ(2) and sin/Theta1∂/Phi1 ∂τ=δ/epsilon1 δ/Theta1+α∂/Theta1 ∂τ. (3) Aiming to obtain the energetics and dynamics of a DW dis- placing along a nanowire, we propose that the magnetizationprofile of a head-to-head DW can be described by an ansatzusing the collective-variables approach [ 12], in the form /Theta1(θ,τ)=2a r c t a n/bracketleftbigg exp/parenleftbiggRθ−q(τ) /Delta1/parenrightbigg/bracketrightbigg ,/Phi1 (τ)=φ(τ), (4) where qandφrepresent the canonical conjugated pair of collective variables, i.e., the position and phase of the DWcenter. Additionally, /Delta1determines the DW width, which will be assumed to be constant and independent of the DW phase.Indeed, we have performed micromagnetic simulations to cal-culate the DW width as a function of φ, and we have observed that despite the DW width being a slave variable of the DWphase [ /Delta1≡/Delta1(φ)] and the curvature inducing a dependence of/Delta1onφ, this change is less than 3%. Moreover, although the magnetic field strength can change the DW width, performedmicromagnetic simulations have shown that this change doesnot produce significant qualitative or quantitative changesin the DW dynamics, as also observed in previous works[28–30]. In this context, the approximation of a rigid domain wall is valid, and a generalized DW model [ 31], in which the DW width can change, will produce small differences com-pared to our results. Additionally, it is worth noting that we areinterested in describing the DW dynamics when it displacesfar from the wire borders. Therefore, because /Delta1/lessmuchL, near the wire ends, the magnetization configuration consists of aquasitangential configuration, and therefore, cos /Theta1≈± 1a t θ=±θ 0. From these assumptions, the dynamics of the DW can be obtained by substituting the ansatz ( 4)i nE q s .( 2) and (3), leading to ˙q=1 2S∂/epsilon1 ∂φ+α/Delta1˙φ (5) and ˙φ=−1 2S∂/epsilon1 ∂q−α /Delta1˙q, (6) where the dot represents the derivative in relation to τ,S= πr2is the area of the circular cross section of the wire, and /epsilon1 184418-2CONTROLLING DOMAIN W ALL OSCILLATIONS IN BENT … PHYSICAL REVIEW B 101, 184418 (2020) is the DW energy. This energy can be obtained by adding the three contributions to the magnetic energy, that is, /epsilon1=SR/integraldisplayθ0 −θ0[ξx+ξd+ξz]dθ, (7) where ξx,ξd, andξzare, respectively, exchange, dipolar, and Zeeman energy densities. The exchange energy density normalized to M2 sis formally defined as ξx=/lscript2(∇m)2, where /lscript=/radicalbig A/(μ0M2s),Ais the exchange stiffness, and the gradient operator is given in termsof the curvilinear basis. In this context, the exchange energydensity can be written as ξ x=/lscript2 R2/bracketleftBigg/parenleftbigg∂/Theta1 ∂θ+cos/Phi1/parenrightbigg2 +/parenleftbigg sin/Theta1∂/Phi1 ∂θ−cos/Theta1sin/Phi1/parenrightbigg2/bracketrightBigg . (8) The dipolar energy density can be written as an easy- tangential anisotropy term, in the form ξd=−λcos2/Theta1, (9) where λ> 0 represents an easy-tangential anisotropy con- stant coming from magnetostatic contributions. It is worthnoting that since we are considering thin magnetic wires(r/lessmuch/lscript,R,L), the dependence of λonθandφcan be ne- glected, and then, the demagnetizing factor of the bent wirecan be given by the same value calculated for a cylindricalnanowire, i.e., λ=1/4[32–34]. Finally, since we are considering that the applied mag- netic field is pointing along the azimuthal direction, that is,H=He θ, the Zeeman energy density is given by ξz=−/lscript2 AMsH. (10) III. RESULTS AND DISCUSSION The integration of the three terms accounting for the total energy density can be the performed, and the results for ex-change and dipolar interactions were previously obtained (see Eqs. (A1) and (A2) of Ref. [ 12]). Additionally, the Zeeman energy is evaluated as E z=−HS/bracketleftBigg L−/Delta1ln/parenleftBigg 1+e−2q+L /Delta1 1+e−2q−L /Delta1/parenrightBigg/bracketrightBigg . (11) The analysis of the energy terms reveals that only the exchange energy presents a dependence on the DW phase insuch a way that the minimum exchange energy is obtainedforφ=π, and its maximum value is obtained for φ=0. This result evidences the existence of a curvature-induced DWphase selection [ 12,27,28]; that is, a head-to-head DW has minimum energy when it is pointing out of the bent. In our calculations we consider a permalloy wire whose exchange constant is A=1.3×10 −11J/m,μ 0Ms=1T,/Delta1= 30 nm (a typical value for a permalloy nanowire with a diam-eter of 20 nm), and we fix the damping constant as α=0.01. Simplified analytical expressions can be obtained for a bentnanowire with a small curvature, neglecting the terms of theorder of K 2in the energy calculations. In this context, we will assume that /Delta1/lessmuchLand that the DW is displacing far fromthe border of the nanowire, that is, q/lessorsimilarL/10. In this case, we obtained simplified expressions for energy contributions, i.e., Ex≈2π/lscript2SKcosφ, (12) Ed≈−λV, (13) and Ez≈− 2HSq. (14) In this case Eqs. ( 5) and ( 6) can be simplified to ˙φ=a+bsinφ (15) and ˙q=α/Delta1a−/Delta1 αbsinφ, (16) where a=H/(1+α2) and b=απ/lscript2K/[/Delta1(1+α2)]. The above equations form a dynamical system. It is clear that thereare no stationary points except in the case a=0, i.e., in the absence of external field. Importantly, Eq. ( 15) can be easily integrated to get solutions φ 1=2a r c t a n/bracketleftBigg −b a−√ a2−b2 acot/parenleftBigg√ a2−b2 2γt/parenrightBigg/bracketrightBigg , φ2=2a r c t a n/bracketleftBigg −b a−√ b2−a2 acoth/parenleftBigg√ b2−a2 2γt/parenrightBigg/bracketrightBigg . (17) In the above equations, φ1andφ2are the solutions for the cases a>banda<b, respectively. The DW position can be promptly obtained by substituting the above solutions inEq. ( 16) and performing the integration in t. Nevertheless, the obtained equation is cumbersome, and it will be omit-ted here. The analysis of Eqs. ( 17) reveals that for a>b, the DW presents a precessional motion with frequency ω=√ a2−b2/2. Therefore, the frequency of the DW precession depends on both the magnetic field and curvature. That is,diminishing curvature induces an increase in the frequency ofthe precession motion of the DW. On the other hand, if a< b, the DW phase asymptotically goes to φ 0=arcsin( −a/b) andq=/Delta1a(1+α2)t/α, and then due to the interplay be- tween curvature and magnetic field, the DW moves alongthe nanowire without precession, and its velocity is linearlyproportional to the magnetic field strength. In this context, themain aspects of the behavior of the DW phase and positioncan be observed in Fig. 2, where we present the DW phase and position as a function of time for the curvature K= 5×10 −3m−1and for different magnetic fields. The analysis of Fig. 2(a) reveals the dependence of the DW precession frequency on the magnetic field. That is, higher magneticfields induce precessional motions with higher frequencies.This behavior is evidenced in the DW translational motion,presented in Fig. 2(b), where the interplay between torques induced by curvature and magnetic field produce a nonmono-tonic behavior for the DW velocity. The critical field valuebetween the translational and oscillatory regimes is defined byH cr=απ /lscript2K//Delta1. This finding is very interesting because in the regime of small curvatures there is a Walker field linearly 184418-3R. CACILHAS et al. PHYSICAL REVIEW B 101, 184418 (2020) FIG. 2. (a) The DW phase as a function of the time for different magnetic fields. (b) The DW position for the same values of themagnetic field. In both plots, we have adopted K=5×10 −3m−1. proportional to the curvature. This result can be promptly compared with current-induced motion of DWs in helicoidalnanowires. Indeed, in Ref. [ 29] the authors show that in the limit of small curvatures, there is a critical current for whichthe Walker breakdown occurs. The velocity associated withthis critical current density is u c∝αK. The appearance of the Walker limit when the DW propagates along a bent wireis associated with curvature-induced anisotropy [ 35], which induces a preferential direction to the DW phase, generating acurvature-induced phase selection for the DW [ 28]. Finally, forb=0 (corresponding to a straight wire), Eq. ( 17)i s simplified to φ=γat, according to the expected DW rotation when it is displacing along a straight wire [ 19]. Next, we perform a complete analysis of the DW po- sition and phase as a function of the nanowire curvatureand magnetic field based on Eqs. ( 5) and ( 6). We start by analyzing a nanowire of L=1000 nm with two curvatures, K=π×10 6m−1(corresponding to φ0=π/2) and K= π/6×106m−1(φ0=π/12), subject to an external azimuthal magnetic field of H=3 mT. The main results are presented in Fig. 3, where we notice that in the two analyzed cases, the DW motion occurs in the Walker regime, that is, presents anoscillatory motion along the wire length. The oscillations ofboth DW position and phase occurring for higher curvatureslead to larger average DW velocity. Regarding results for aconstant (linear) magnetic field published in Ref. [ 12], we can assert that higher DW velocities are obtained when applyingan Oersted field. The influence of curvature in the DW dynamics for bent nanowires is shown in Fig. 4, where the DW position under the action of a 2-mT azimuthal magnetic field is represented.Specifically, we study three different curvatures: 2 π×10 6m−1 FIG. 3. (a) The DW position and (b) phase as a function of time under an azimuthal magnetic field of 3 mT for nanowire of L= 1000 nm with two curvature values, K=π×106m−1(φ0=π/2) andK=π/6×106m−1(φ0=π/12). [Figs. 4(a) and4(d)],π×106m−1[Figs. 4(b) and4(e)], and 5π×105m−1[Figs. 4(c) and 4(f)]. Under the condition of a half-torus geometry, L=π/K, the lengths corresponding to each case are L=500 nm, L=1μm, and L=2μm, respectively, which must be taken into account to understandwhen the domain wall exits from the end of the nanowire.Results clearly confirm that the higher the curvature is, thehigher the DW average velocity is. Furthermore, an excit-ing result from Fig. 4is the possibility of controlling DW oscillations along the nanowire axis in terms of curvature.Due to the increase in the exchange energy when the DWis pointing along the internal region of the bent, there isa fast dynamics in its phase in order to bring it back toφ=π(pointing outward the bent), resulting in DW oscillating behavior around NW. The analysis of Figs. 4(d)–4(f) reveals that the DW phase dynamics has a small dependence on thewire length (and, consequently, on its curvature). Indeed, itcan be observed that in all cases, after 5 ns, the DW phase is≈−π/2, and thus, the curvature has a small influence on the DW phase precession period. Finally, one can observe that forthe nanowire with L=500 nm (curvature 2 π×10 6m−1), the DW does not present the oscillatory behavior observed inthe other two cases. This result occurs because, in this case,the DW did not have sufficient time to change its phase fromzero to πbefore the arrival at the nanowire border. The influence of the magnetic field strength on the DW mo- tion is presented in Fig. 5for a nanowire with length L=2000 nm ( K=5π×10 5m−1). Again, starting from H=1.5m T both DW position and phase oscillate. It can be observedthat the frequency of the DW precession strongly depends 184418-4CONTROLLING DOMAIN W ALL OSCILLATIONS IN BENT … PHYSICAL REVIEW B 101, 184418 (2020) FIG. 4. The DW position as a function of time for nanowires with curvatures (a) 2 π×106m−1,( b )π×106m−1,a n d( c )5 π×105m−1 under the action of a magnetic field of 2 mT. The DW phase as a function of time for curvatures (d) 2 π×106m−1,( e )π×106m−1, and (f) 5π×105m−1. on the magnetic field and can be controlled by small fields. Generally, stronger magnetic fields produce higher torqueson the DW, and then the DW precesses faster. It can also benoticed that for magnetic fields above 1.5 mT, the smaller themagnetic field is, the faster the DW propagation is. However,by comparing the DW positions for H=0.5 mT and H= 1.5 mT, we can observe that this dependence is the opposite; that is, the DW is faster in the latter case. This fact evidencesthe existence of a Walker breakdown phenomenon for the DWdynamics when it is displaced along a bent wire. Applications using the DW motion along bent nanowires demand a complete understanding of the frequency and am-plitude of the DW oscillation. Indeed, during the domain wallmotion, the stray field in a fixed location in proximity to thenanostructure surface will either increase or decrease, generat-ing an electromotive force. The stray field generated by a DWdepends on the distance from the nanowire. It was shown thatthe stray field generated by a domain wall displacing alongstraight nanowires is on the order of 10 5A/m and decays to approximately 1 A /m at a distance of 150 nm [ 36]. Be- cause the frequency of precession and oscillation of the DWpropagating along a bent nanowire are the same, the previous results allow us to state that the curvature of the nanowire doesnot influence the oscillation frequency of the DW motion. Inthis context, we have analyzed the effects of the magneticfield strength and wire curvature on the amplitude of the DWoscillations and the influence of the magnetic field on theoscillation frequency. In Fig. 6(a), we show the amplitude of the DW oscillation as a function of the curvature for differentvalues of the magnetic field. We can notice that due to thecompetition between the torques produced by the exchange-driven curvature-induced anisotropy and the magnetic field,the amplitude of the DW oscillatory motion depends on both.Indeed, because the torque produced by the magnetic fieldinduces a faster precession motion, the amplitude of theDW oscillations decreases when the magnetic field increases.On the other hand, the increase in the curvature leads toan increase in the amplitude of the oscillations. This resultoccurs because the curvature-induced anisotropy generatesan extra torque, and the domain wall displaces longer dis-tances before performing a complete precession. For straightnanowires, the amplitude of the oscillations is zero, and the 184418-5R. CACILHAS et al. PHYSICAL REVIEW B 101, 184418 (2020) FIG. 5. (a)–(d) The DW position and phase for μH=0.5, 1.5, 2.0, and 3.0 mT, respectively. DW does not present the Walker behavior during its motion, as it should be [ 19]. In Fig. 6(b), we show the oscillation frequency as a function of the magnetic field, where one cannote that the oscillatory frequency varies linearly with themagnetic field. We also performed calculations of the average velocity as a function of the magnetic field for different curvatures,shown in Fig. 7. The average velocity is defined as /angbracketleftv/angbracketright=d/t, where dis the distance covered by the DW in t=5n s ,w h i c h is a proper time because, at this time, domain walls do notescape from the nanowire in any of the studied cases. TheNW curvature strongly influences the average DW velocity,and more curved nanowires present larger velocities. Thedependence on the magnetic field is highly nonlinear, and 184418-6CONTROLLING DOMAIN W ALL OSCILLATIONS IN BENT … PHYSICAL REVIEW B 101, 184418 (2020) FIG. 6. (a) The influence of magnetic field and curvature on the amplitude of the oscillations of the DW propagating along a bent nanowire. (b) The behavior of the oscillation frequency asa function of the magnetic field for a nanowire with curvature K=π×10 6m−1. the presence of the Walker breakdown is evident. A small magnetic field can control its occurrence. Compared to thecase of a constant nonazimuthal field considered in Ref. [ 12], the effect of curvature on the velocity is accentuated. IV . CONCLUSIONS In conclusion, bent magnetic cylindrical nanowires can present oscillatory behavior of domain wall dynamics, con-trolled by an external magnetic field. While in cylindricalstraight nanowires a DW rotates around the axis only by theaction of a constant magnetic field, the oscillations alongthe axis appear in bent nanowires due to the effect of theexchange interaction. Consequently, the Walker breakdownphenomenon is recovered. With aim of determining whenthis situation is happening, in this paper we have analyzedthe domain wall dynamics in bent wires, with a half-torus FIG. 7. DW average velocity as a function of the external mag- netic field for different curvatures. geometry, under the action of an azimuthal magnetic field. The obtained analytical results for nanowires with small curvaturerevealed that the oscillatory regime can appear depending onthe interplay between the curvature and magnetic field. For very small magnetic fields, the domain wall position does not present an oscillatory behavior, and it moves withvelocity linearly increasing with the field. Although nanowireswith larger curvatures present higher average velocities, theWalker limit appears, and a critical field value in which theDW velocity abruptly drops is observed. This drop is accom-panied by the domain wall oscillations along the nanowirelength, which are controlled by magnetic fields. It was alsoshown that the maximum DW average velocity increases as afunction of the curvature in such a way that the DW velocityfor a straight nanowire is recovered for small curvatures. Overall, the results allow the control of the conditions for domain wall dynamics and, particularly, its oscillatorybehavior. We believe that such control can be very useful fornanoscale communication applications such as those neededfor future technologies in the Internet of Things. ACKNOWLEDGMENTS In Brazil, this study was financed in part by the Coorde- nação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) - Finance Code 001. The authors also thankCNPq (Grants No. 401132/2016-1 and No. 302084/2019-3).In Scotland, we acknowledge the Natural Environment Re-search Council (Grant No. NE/S011978/1). In Chile, weacknowledge financial support from FONDECYT Grant No.1160198 and CONICYT through Centro Basal FB0807.In Spain, we acknowledge the financial support from theMinistry of Economy and Competitivity under Grant No.MAT2016-76824-C3-1-R. [1] Y . Zhan, Y . Mei, and L. Zhengac, J. Mater. Chem. C 2,1220 (2014 ). [2] H. Sun, M. Yin, W. Wei, J. Li, H. Wang, and X. Jin, Microsyst. Technol. 24,2853 (2018 )[3] L. Atzori, A. Iera, and G. Morabito, Comput. Networks 54, 2787 (2010 ). [4] A. Ferna ´ndez-Pacheco, R. Streubel, O. Fruchart, R. Hertel, P. F i s c h e r ,a n dR .P .C o w b u r n , Nat. Commum. 8,15756 (2017 ). 184418-7R. CACILHAS et al. PHYSICAL REVIEW B 101, 184418 (2020) [5] J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, P h y s .R e v .L e t t . 74,3273 (1995 ). [6] E. Bankowski, T. Meitzler, R. S. Khymyn, V . S. Tiberkevich, A. N. Slavin, and H. X. Tang, Appl. Phys. Lett. 107,122409 (2015 ). [7] M. Quinsat, F. Garcia-Sanchez, A. Jenkins, V . Tiberkevich, A. 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PhysRevB.99.134105.pdf

PHYSICAL REVIEW B 99, 134105 (2019) Strong anisotropy in strength and toughness in defective hexagonal boron nitride Tousif Ahmed, Allison Procak, Tengyuan Hao, and Zubaer M. Hossain* Laboratory of Mechanics & Physics of Heterogeneous Materials, Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716, USA (Received 9 September 2018; revised manuscript received 21 March 2019; published 17 April 2019) Using a combination of density functional theory calculations and molecular dynamics simulations we show that the strength and toughness of hexagonal boron nitride (hBN) containing isolated vacancy defectsare strongly anisotropic, regardless of the size of the defect core. The degree of anisotropy is preserved fora number of defect structures including monovacancy, tetravacancy, tridecavacancy, triheptacontavacancy, orheptatriacontavacancy defects. The chirality-dependent effects are strongly nonlinear and well characterized byclose-form mathematical equations indicating pronounced strength and toughness along the zigzag directioncompared to the strength and toughness along the armchair direction. Also, the size dependence of the strengthand toughness of the defective lattice shows an inverse relationship with the effective diameter of the defectcore. An atomistic analysis of the deformation fields reveals that nonuniformity in bond length, bond strain,and force distribution in the nonlinear regime of mechanical deformation surrounding the defect cores formsthe physical basis for the observed anisotropy. The anisotropic character of the lattice is governed primarily bythe nearest-neighbor covalent interactions (dominated by the first-nearest neighbors). Consequently, as soon asa set of bonds rupture at the defect core, the entire lattice undergoes catastrophic failure underscoring the brittlenature of the fracture state in hBN. Results also suggest that chirality-dependent elastic effects are dominatedby the third-order elastic modulus which stiffens the lattice at higher chiral angles, whereas the second- andfourth-order elastic moduli soften the lattice affecting the strength and toughness of the lattice in an intricatemanner. DOI: 10.1103/PhysRevB.99.134105 I. INTRODUCTION Hexagonal boron nitride (hBN) attracts special attention due to its inertness in chemical environments, its thermalstability at higher temperatures, its electrical insulating prop-erties, and more importantly its extraordinary compatibilitywith other two-dimensional materials (such as graphene)[1–5]. In single-layer hBN, observations of holes or vacancies of discrete sizes and shapes through high-resolution trans-mission electron microscopy have been widely reported inthe literature [ 6–10]. Not only is it inevitable from available scalable synthesis and growth processes that vacancy defectswill appear in hBN sheets, they can also presumably becontrolled in the future to design and engineer the effectivebehavior of hBN in a variety of applications. There is a widebody of literature that shows the formation mechanisms ofthese defects as well as their effects on electronic, optical,and chemical properties in hBN [ 11–15]. The mechanical behavior of pristine hBN has also been a subject matter ofactive research in the recent past [ 16–21]. Nonetheless the implications of vacancy defects on extreme mechanical prop-erties (such as strength, toughness, and higher-order elasticmoduli) along different chiral directions that regulate thedirectional mechanical and thermal response of a solid at finitedeformation remain mostly unexplored for hBN. As far as the linearly elastic behavior is concerned, it is now well known that hBN behaves as an isotropic media [ 18]. It is *zubaer@udel.eduhowever unclear if the isotropic character is preserved in the presence of defects or if the stiffness is improved or degradedby the vacancy defects due to the suppression of structuralfluctuations (as seen in graphene, which is phenomenologi-cally similar to hBN in terms of the mechanical behavior).Likewise there is a substantial lack of information on higher-order elastic moduli in hBN, and their dependence on materialdefects continues to puzzle scientists and engineers. Relevant to the extreme mechanical behavior of defective hBN, a number of polycrystalline or grain boundary struc-tures and domains with initial cracks have been investigated[22–30]. It has been revealed from first-principles simulations that the misorientation angle at the grain boundary adverselyaffects the ideal strength of the lattice [ 31]. Moreover, from atomistic simulations, grain size has been found to reduce thestrength of the material substantially [ 22,23], and temperature and extended defects have been attributed to causing the alter-ation of strength in polycrystalline hBN [ 24]. In the context of fracture, it is revealed that the criterion for propagation of anexisting crack in hBN is direction dependent [ 25–30], albeit the information about the condition for nucleation of the crackand its atomistic basis for loading along an arbitrary directionare yet to emerge. Apart from the extended defects, spatially confined smaller defects such as the vacancy defects (which are created byirradiation damage during the fabrication of the boron nitridemonolayer [ 6,10,32]) have been studied with a focus on understanding their formation energies and migration path-ways [ 12,33–35]. Confirmation of monovacancy as one of the stable defects in hBN motivates a detailed exploration of its 2469-9950/2019/99(13)/134105(17) 134105-1 ©2019 American Physical SocietyAHMED, PROCAK, HAO, AND HOSSAIN PHYSICAL REVIEW B 99, 134105 (2019) significance in altering the mechanical behavior [ 12]. Recent nanoscale experiments on defective graphene containing di-lute concentrations of monovacancies exhibit unusual defect-induced stiffening of the lattice [ 36–39], and the enhanced stiffening of graphene lattice is thought to originate fromthe suppression of structural fluctuations in the out-of-planedirection. Analogous studies however remain missing forhBN. The majority of existing findings on hBN are focused onits mechanical properties along the high-symmetry directions.They provide phenomenological insights on defect-inducedmechanical behavior, but the coupling between defects andchirality of the lattice remains less understood. In this paper we investigate the correlation between the chirality of the hBN lattice and its extreme mechanical proper-ties in the presence of a number of vacancy defect configura-tions. We focus on understanding the macroscopic mechanicalimplication of a collection of high-symmetry defects suchas the monovacancy, tetravacancy, tridecavacancy, heptatria-contavacancy, and triheptacontavacancy defects. These defectstructures have higher symmetry edge structures so that thenumber of defect configurations (for a given defect radius) isconsiderably less than that of other defect structures such asthe divacancy or trivacancy structures. To build a comprehen-sive understanding of the mechanical properties of defectivehBN, we considered ten different loading directions for eachof the defect structures and explore their effective elasticmoduli and extreme mechanical behavior over the linear andnonlinear regimes of mechanical deformation. Additionally,the condition of failure in defective hBN containing a vacancydefect core is explored for each of the ten chiral directions.Our results suggest the existence of strong coupling betweenchirality and mechanical properties of hBN, and the couplingis controlled by size-dependent nonuniform stress and defor-mation fields emanating from the defect cores. In the follow-ing sections we first outline our computational approach andthen discuss our findings from an atomistic viewpoint. II. COMPUTATIONAL DETAILS Ideally mechanical behavior of a solid is most accurately captured by first-principles simulations. However, owing tothe prohibitively large computational requirements, thesemethods are inapplicable for large systems containing morethan a few hundreds of atoms that are needed to determinethe mechanical properties of defective lattice along arbitrarychiral angles. As an alternative, molecular dynamics (MD)simulations with empirical potentials can be performed—andthere are several interatomic potentials available for hBNsuch as Tersoff [ 19] and ReaxFF [ 40]. These potentials are designed to capture the elastic and chemical behavior ofhBN. They reproduce experimentally measured elastic andchemical properties at equilibrium very well but are notalways directly applicable for investigating material proper-ties in the nonlinear regime of mechanical deformation, dueto the appearance of unphysical strain-hardening [ 30,41,42]. Although the strain-hardening feature can be corrected by ad hocmodification of the cutoff parameters of the potential (as shown in the Supplemental Material), there is no rigorousbasis for the modification.To address the limitations of existing computational tools in predicting the mechanical behavior of hBN over a widerange of deformation and defect conditions, we developed apotential in the Stillinger-Weber functional form and appliedan integrated approach containing two parts. The first partinvolves applying density functional thoery (DFT) simula-tions to determine the parameter set of the Stillinger-Weberinteratomic potential that reproduces DFT-generated stress-strain curves over the entire regime of mechanical defor-mation covering the linear and nonlinear mechanisms. Thesecond part involves applying MD simulations to explorethe implication of subnanometer scale vacancy defects oneffective mechanical properties of the lattice under a widerange of loading directions. We demonstrate in Secs. III A and III B that the effective nearest-neighbor interactions accounted for in the SW potential is sufficient to accurately determinethe strength and toughness properties of hBN—as long asthe potential is calibrated to reproduce the DFT-generatedequilibrium properties and strength of the material along atleast one crystallographic directions. The DFT calculations employed in this work are carried out by using the open source DFT-code named SIESTA [43]. Following the Troullier and Martins scheme [ 44] the core electrons are replaced by the norm-conserving pseudopo-tentials. For the exchange-correlation energy of the energyfunctional, we employ the generalized gradient approximation(GGA) [ 45]. The mesh cutoff is set to 500 Ry and the electronic temperature is set to 20 meV to obtain convergedelectron energy values. Prior to applying uniaxial deformationto the system quasistatically, the Broyden scheme is employedto equilibrate the electronic system with a force tolerance of0.01 eV /Å. The total energy is calculated by integrating over the Brillouin zone with a k-mesh of 4 ×4×6 Monkhorst- Pack grid. The convergence of total energy is achieved withan energy threshold of 0.1 meV /atom and a force tolerance of 0.04 eV /Å. The stress tensor over the lattice is computed using a virial scheme [ 46–48] in units of eV /Å 3. To transform the values in GPa taking into consideration the finite thicknesseffect of the hBN sheet, the stress components obtainedfrom DFT simulations are multiplied by 160.03 and a loadconversion factor (equal to the ratio between the box-lengthnormal to the hBN sheet and a thickness of 3.3 Å [ 49]). In the MD simulations, before applying deformation, the simulation box containing a single atomic layer of hBN isrelaxed in all directions using the conjugate gradient methodas implemented in LAMMPS [50]. The dimension of the sim- ulation box along the out-of-plane direction is kept fixedat 50 Å to avoid any interactions across the periodic repli-cas in that direction. During the deformation, the systemtemperature and pressure are set at 1 K and at 0 bar. Wechoose 1 K to get the simulation condition of the latticeas close to a static condition as possible (but yet considersome nonzero finite temperatures), to compare the results withthose of DFT under a similar physical platform. We alsoinvestigate the behavior of the lattice at 300 K to determine theinfluence of finite temperatures on the anisotropic mechanicalbehavior of the lattice. In our simulations, uniaxial stressdeformation is applied at a strain rate of 1 .0×10 8per second (or 1.0×10−4per picosecond), and the domain is allowed to relax in the lateral direction to maintain zero stress or 134105-2STRONG ANISOTROPY IN STRENGTH AND TOUGHNESS … PHYSICAL REVIEW B 99, 134105 (2019) the uniaxial stress condition of the lattice. This is done by applying an Andersen barostat in the NPT ensemble. Thedamping parameters used in the Nose-Hoover algorithm playa critical role in setting the physical behavior of the system.We find that with a time step of 1 fs, Tdamp =1 ps and Pdamp =1 ps yield reliable measurements of mechanical properties. III. RESULTS AND DISCUSSIONS Prior to conducting the simulations for defective hBN, we explore the mechanical behavior of the defect-free hBN forloading along the high-symmetry directions (the armchair andzigzag directions) by using the DFT-GGA calculations andMD simulations with two different interatomic potentials: theSW potential developed in this work and the most recentversion of the Tersoff potential. In Sec. III A we describe the potential developed here and in Sec. III B it is validated with DFT. Finally in Sec. III C we discuss the results on defective hBN obtained from the MD simulations with the SW potentialdeveloped here. A. Stillinger-Weber potential for hBN The SW potential models a solid as a collection of atoms bonded through the nearest-neighbor interactions only. Math-ematically the nearest-neighbor interactions are described bya two-body term, V 2(rij), and a three-body term, V3(θijk): V2=A/epsilon1/parenleftBigg Bσ4 r4 ij−1/parenrightBigg exp/parenleftbiggσ rij−rc/parenrightbigg , (1) V3=λ/epsilon1/parenleftbig cosθijk−cosθ0/parenrightbig2exp/parenleftbigg2γσ rij−rc/parenrightbigg , (2) where the first term is V2and the second term is V3;σand /epsilon1are length scale and energy parameters, respectively; rc is the cutoff distance beyond which the interaction potential is zero; rijis the distance between the atoms located at ri andrj;θ0is the angle between two bonds formed by atoms located at ri,rj, and rk; and A,B,λ,γ, andσare the SW parameters. We set /epsilon1=1 and rc=√ 3r0, the latter of which is the second-nearest-neighbor distance in the crystalline hBNlattice, as depicted in Fig. 1. FIG. 1. Interaction regime for an arbitrary atom ii ss h o w no nt h e left and the corresponding energy (denoted as V2) and force (denoted asF) diagrams are shown on the right as a function of the interatomic distance r. The shaded region indicates the total energy absorbed by a pair of B-N atoms in the bulk configuration over the strain window 0/lessorequalslant/epsilon1/lessorequalslant/epsilon1n,w h e r e /epsilon1nis the strain at which bond rupture is initiated.Any atoms entering rcduring any stage of dynamic evolution of the system are allowed to contribute to thetotal energy of the solid. In crystalline hBN the first-nearest-neighbor interactions involve interactions between the B andN atoms, and the second-nearest-neighbor interactions involveB-B and N-N interactions. To obtain the parameters of theSW potential we used an optimization scheme that has beensuccessfully applied for graphene [ 51] and SiC [ 52,53]. The details of the calculations and procedure are published else-where [ 51,54]. For brevity here we provide the key features of the framework. In fitting the parameters, we exploit five material prop- erties: the equilibrium lattice constant r 0, the bond force constant Kb, the bond-bending constant Kθ, the ideal strength of the lattice σmax, and its cohesive energy of Ec. These param- eters are obtained directly from DFT-GGA calculations. Toconnect the parameters with the material properties, we firstset the first derivative of the two-body potential to zero, thatis,dV 2/dr=0a tr=r0, to obtain an analytical expression for the parameter BasB=r5 0/[σ4r0+4σ3(r0−rc)2]. Next from the second derivative of V2atr=r0,o rd2V/dr2=Kb, we ob- tain an analytical expression for the parameter Aas a function ofr0,rc,σ, and Kb. Also, from the definition of the cohesive energy of the lattice we get a second expression for Afrom V2(r0)=Ecas a function of r0,rc,σ, and Ec. Equating these two equations (in conjunction with the expression for B)w e find a seventh-order polynomial that has only one real solutionforσ. We find this solution numerically after determining the values of r 0,rc,Kb, and Ecfrom DFT calculations. Once σ is known, we calculate Basr5 0/[σ4r0+4σ3(r0−rc)2] and A from the equation of V2(r0)=Ec. The parameter γis used to fit Kθ. Finally, the parameter λand an energy correction factor are used to calibrate the MD-generated stress-straincurve to the DFT-generated stress-strain curve of the latticeas well as the Poisson’s ratio of the solid. Consequently, inaddition to providing DFT-consistent elastic properties, thisapproach results in a reliable potential capable of reproduc-ing the DFT-consistent mechanical behavior over the entireregime of mechanical deformation up to the point of bondrupture. The approach results in the following SW parametersfor hBN: A=46.666 89, B=0.037 05, σ=2.705 00 Å, γ= 0.45,λ=41.000 00, r c=2.5 Å, and /epsilon1=1.0e V . The need for fitting the DFT-generated stress-strain curve up to the failure point is illustrated schematically inFig. 1. As evident, the lattice has a positive stiffness from AandBand a negative stiffness from BtoC; where A, B, and Cdenote the equilibrium state, the highest force state, and the interaction-free state of the solid underhydrostatic deformation of the lattice, respectively. Dueto negative-stiffness mediated instability of the solid andhigher-order nonlinear effects at r>r fm(whereat the bond force is the maximum), the physics of deformation is verydifferent on the two sides of r fm. As soon as the length of a bond exceeds r=rfm, the lattice experiences an unstable localized bond-rupture event, which leads to brittle fractureof the entire lattice. We therefore find that it is necessaryand sufficient for our SW potential to fit the DFT-generatedforcing behavior up to r=r fm. The validity of this potential is checked with DFT results and a second interatomic potential,the Tersoff potential with its most recent parameter set. 134105-3AHMED, PROCAK, HAO, AND HOSSAIN PHYSICAL REVIEW B 99, 134105 (2019) 0 0.1 0.2 0.3 0.4 Strain020406080100Stress (GPa)0o(MD-TER) 0o(MD-SW) 0o(DFT) 30o(MD-TER) 30o(MD-SW) 30o(DFT)AB C FIG. 2. Stress-strain behavior of hBN under uniaxial-stress de- formation: for loading along 0◦(armchair direction) and 30◦(zigzag direction). The linear regime covers the strain range of 0 </epsilon1/lessorequalslant 2%, marked by regime A, and beyond this regime the mechanical behavior is governed by nonlinear elasticity. The vertical blue and red lines indicate the onset of anisotropic behavior of the lattice in the MD-SW and DFT-GGA simulations, respectively. B. Stress-strain behavior of defect-free hBN To predict the stress-strain behavior of pristine hBN, we performed uniaxial deformation simulations along thearmchair and zigzag directions using DFT-GGA calculationsand classical MD simulations with our SW potential andthe recently updated Tersoff potential (named the BN-ExTePpotential) [ 19]. As shown in Fig. 2, there is a reasonable agreement for strain states below 15%. We notice three distinct regimes in the stress-strain re- sponse of hBN from the DFT-GGA and MD-SW results.They are identified by A,B, and C.R e g i m e Ais the linear regime covering 0 /lessorequalslant/epsilon1/lessorequalslant0.02. The DFT and MD results show identical behavior in regime A, indicating higher ac- curacy of the SW and Tersoff potentials in reproducing theDFT-generated mechanical behavior in the neighborhood ofthe equilibrium point. In regime Bthe normal stress along the armchair direction is higher than that along the zigzagdirection, indicating an anisotropic character of the lattice,although the maximum difference in their stress values isless than 1 GPa. This subtle feature is displayed by boththe DFT-GGA calculations and the MD-SW simulations, butnot by the MD-Tersoff simulations. Since the magnitude isless than 0.1% of the actual stress values, we can ignorethe associated anisotropic feature and consider regime Bas primarily an isotropic regime. It is noteworthy, however, thatthe MD-SW potential reproduced the DFT behavior. TheSW potential is a simple nearest-neighbor model, and it isdeveloped by calibrating the strength of the lattice along thearmchair direction only. Yet it has successfully reproducedthe precise details of the stress-strain response along a seconddirection in the nonlinear regime of mechanical deformation. In regime Cthe stresses along the zigzag direction are higher than those along the armchair direction. The differenceincreases dramatically with increasing strain. It is only regimeCwherein the lattice exhibits anisotropy in its mechanical behavior—and the chirality plays the central role in influenc-ing the stress-strain behavior of the lattice in this regime. OurMD-SW potential gives the transition strain between regimesBandCat/epsilon1=0.145, whereas in DFT the transition is at /epsilon1= 0.16. Nevertheless, they continue agreeing with each other up to the maximum stress point of the lattice. The MD-Tersoffon the other hand shows artificial stiffening of the lattice inregime Cand gives much higher strengths compared to DFT for each of the loading directions. Examination of the first-order elastic modulus indicates that it varies from 836.4 to 835.1 GPa from the armchair di-rection to the zigzag direction. The corresponding values fromDFT simulations are 814.3 and 811.4 GPa. Defect-free hBNlattice can therefore be assumed to be mostly isotropic, as faras the first-order elastic modulus is concerned. Nonetheless,the higher-order elastic moduli C 2,C3, and C4along the zigzag direction are around 21, 215, and 124% of the correspondingvalues along the armchair direction in DFT and 22, 170, and137% in MD-SW. From DFT and MD simulations results, itis thus clear that the higher-order elastic moduli in defect-freehBN are strongly anisotropic. The strength and toughness properties are also anisotropic. The strength values of hBN along the armchair and zigzagdirections from our DFT-GGA calculations are 86.6 and100.4 GPa respectively, which match with our MD-SW resultsof 87.1 and 99.0 GPa within 0.5 and 1.39%, respectively.While our MD-T calculations (performed with the potentialparameters obtained from the most recent version of Tersoff[19]) show unphysical stiffening of hBN, an ad hoc modi- fication of its cutoff function has been shown to reproducethe strength values as 102.35 and 113.23 GPa [ 28] along the armchair and zigzag directions, respectively. These valuesoverestimate the DFT-generated results by more than 21 and16%, respectively. In a recent work [ 54] we show that the cutoff function can be modified to correct the strength valuesobtained from the BN-ExTeP potential. The SW potential’s close quantitative agreement with the DFT results validates its applicability in modeling the elasticbehavior of hBN over the entire deformation history accu-rately. Furthermore, the close agreement between the DFT-GGA (which accounts for quantum effects) and the MD-SW(which accounts for the nearest neighbor effects only) resultshighlights an important point: the strength properties of hBNare governed mainly by the nearest-neighbor interactions.Higher-order nonlinear effects around r=r fmdo not affect the strength of hBN much, as reported for defect-free hBNlattice [ 54]. In this paper we find similar conclusions for defective hBN from a comparison of the results obtained fromthe SW potential for one defective configuration with thoseof the BN-ExTeP potential, the latter of which takes intoaccount the many-body effects in a rigorous manner. The mainobjective of this study is to investigate the extreme mechanicalbehavior of defective hBN, whose strength is much lowerthan the strength of the defect-free lattice. The majority of theatomic interactions are thus expected to involve much smallerinteratomic distances (such that r/lessmuchr fm) at which the DFT- GGA and MD-SW results show indistinguishable mechanicalbehavior. In the next section, we describe the mechanicalbehavior of defective hBN and compare the results with those 134105-4STRONG ANISOTROPY IN STRENGTH AND TOUGHNESS … PHYSICAL REVIEW B 99, 134105 (2019) of pristine hBN to identify the individual effects of various vacancy configurations. C. Mechanical behavior of defective hBN There are a variety of vacancy defect structures possible when the number of vacancies at the core is higher than afew (that requires application of statistical tools to extractan average response). Our focus here is to elucidate thesolitary effect of defect-core size on anisotropic mechanicalbehavior. To minimize any interference from other structuralor configurational sources of the defect cores, we created aselect set of defect configurations following three rules: (i) thedefect edge must not have atoms with more than one missingneighbor, (ii) all the defect structures should be C 3symmetric, and (iii) the defect core is an isolated defect such that itsinteraction with its periodic images is negligible. The result-ing structures with the five smallest diameters in ascendingorder are monovacancy ( V 1), tetravacancy ( V4), tridecavacancy (V13), heptatriacontavacancy ( V37), and triheptacontavacancy (V73) defects that represent 1, 4, 13, 37, and 73 missing atoms, respectively, at the defect core of diameter d. For each of these five defect configurations, we consider a set of ten differentchiral angles of the lattice: 0 ◦,5.2◦,9.82◦,1 3.89◦,1 7.4◦, 20.63◦,2 3.41◦,2 5.87◦,2 8.05◦, and 30◦. In total we have 50 atomic configurations—all of them have C3symmetry and similar defect edge structures. This helps us extract the defectsize effects only, with a minimal distraction from the edgestructure itself, which we explore as a separate work. Wefound these five configurations sufficient to obtain conclusiveoutcomes on defect-core-size-dependent anisotropic behavior.At the equilibrium state, the effective diameter of the defectcores at the equilibrium are 2 r 0for the V1configuration, 2√ 3r0forV4,4r0forV13,2√ 7r0forV37, and 2√ 13r0forV73, FIG. 3. (a) hBN lattice with different radial vectors considered to create the vacancy defect configurations for loading along the armchair direction. (b) Schematic showing a comparison of thedefect-core diameters. (c) Defect edge polygons created by connect- ing the locus of the nearest atoms from the defect core. (d) (From left to right): Monovacancy, tetravacancy, trideca-vacancy, heptatri-acontavacancy, and triheptacontavacancy denoting 1, 4, 13, 37, and 73 missing atoms at the core, respectively. Their effective radii are marked by red circles and the atoms removed are shown in red.where r0is the equilibrium bond length of the hBN lattice, as shown in Fig. 3, for loading along the armchair direction. To enable applying uniaxial deformation to a simulation cell with orthogonal lattice vectors, we construct rectan-gular domains bounded by two lattice vectors: T h=(n1+ m1/2)√ 3r0ˆx+(3m1r0/2)ˆyand Tv=(n2+m2/2)√ 3r0ˆx+ (3m2r0/2)ˆy, where n2=2m1+n1andm2=− (2n1+m1), andn1andm1are integer numbers of unit cells [ 55]. The do- mains are constructed large enough (22.6–39.1 nm along thelateral direction, 58.7–101.7 nm along the loading direction)to ward off any elastic interactions across the periodic imagesof the domain. The largest ratio between the defect size dand the lateral domain size L xis 0.06, which is sufficiently small for considering the defects as isolated defects. Considerationof ten different chiral angles and avoidance of elastic inter-actions make the simulation domains quite large (containingaround 300 000 atoms for the smallest domain) and beyondthe reach of DFT calculations. So we investigate the behaviorof the defective hBN lattice using the MD-SW calculationsonly. The robust accuracy of the potential in reproducing themechanical behavior of defect-free lattice over a large strainwindow leading up to bond rupture provides a reasonablebasis for relying on the outcomes of the MD simulations. 1. Constitutive behavior of defective hBN To determine the constitutive behavior of hBN (represent- ing the relationship between stress and applied strain) over alarge strain spectrum, we employ a constitutive equation thatis fourth order in strain and tenth order in the elasticity tensor.The explicit expression of the equation reads σ ij=Cijkl/epsilon1kl+Cijklm n/epsilon1kl/epsilon1mn+Cijklm n o p /epsilon1kl/epsilon1mn/epsilon1op +Cijklm n o p q r /epsilon1kl/epsilon1mn/epsilon1op/epsilon1qr, where /epsilon1ijis the second-rank strain tensor, Cijklis the fourth- rank elasticity tensor, Cijklm n is the sixth-rank elasticity tensor, Cijklm n o p is the eighth-rank elasticity tensor, and Cijklm n o p q r is the tenth-rank elasticity tensor. For loading along the y direction, this equation reduces to a fourth-order polynomialin strain: σ yy=Cyyyy/epsilon1yy+Cyyyyyy/epsilon12 yy+Cyyyyyyyy /epsilon13 yy+Cyyyyyyyyyy /epsilon14 yy. (3) Since each of the elastic constants appearing as the elasticity tensor components is a constant, we can use simpler sym-bols to represent the tensor components, such as C 1=Cyyyy, C2=Cyyyyyy ,C3=Cyyyyyyyy , and C4=Cyyyyyyyyyy . Substitution of these constants gives the constitutive behavior as σyy= C1/epsilon1yy+C2/epsilon12 yy+C3/epsilon13 yy+C4/epsilon14 yy. We determine the elastic con- stants by fitting a fourth-order polynomial to the stress-straindata obtained from the MD simulations. The fourth-orderpolynomial was needed to guarantee a best fit of the datapoints with a reasonable accuracy. Results show that the first-order elastic modulus C 1(or the Young’s modulus) changes monotonically with increasing the chiral angle, though themaximum change between the extremum of its values issufficiently low. For example, in the monovacancy configu-ration when the loading direction changes from the armchairdirection to the zigzag direction the first-order modulus C 1 134105-5AHMED, PROCAK, HAO, AND HOSSAIN PHYSICAL REVIEW B 99, 134105 (2019) 0 5 10 15 20 25 30 Chiral angle ( )867868869870871872E (GPa)MD data a+b +c2 fit a=867.415 b=0.019123c=0.0042336 0 5 10 15 20 25 30 Chiral angle ( )-2400-2350-2300-2250-2200-2150-2100C2 (GPa)MD data a+b +c2 fit a=-2100.3353 b=-4.0064c=-0.16116 0 5 10 15 20 25 30 Chiral angle ( )-4000-3000-2000-100001000C3 (GPa)MD data a+b +c2 fit a=-3274.6145 b=69.2852 c=2.1291 0 5 10 15 20 25 30 Chiral angle ( )0.511.522.5C4 (GPa)104 MD data a+b +c2 fit a=20343.5022 b=-181.6227 c=-8.6806 FIG. 4. Chirality-dependent elastic moduli C1or the Young’s modulus E,C2,C3,a n d C4of hBN lattice containing monovacancy under symmetry-breaking uniaxial deformation. The solid lines represent fitted curves with the polynomial Ci(θ)=ai+biθ+ciθ2,w h e r e ai,bi, andciare fitting parameters for the ith elastic modulus in units of GPa. Their values are shown in the plots. increases by less than (872–867.5) /867.5=0.5%. The higher- order moduli ( Ciwith i>1), however, vary substantially with increasing the chiral angle as exhibited in Fig. 4. They vary by more than 13, 133, and 75% for C2,C3, and C4, respectively. The chirality-dependent (denoted as θ-dependent) elastic behavior of all the elastic moduli is similar qualitatively forany of the defective configurations considered in this work.Their stress-strain curves as well as the θ-dependent behavior of the elastic moduli are provided in the Supplemental Ma-terial [ 56]. It is found that each and every elastic modulus is anisotropic in hBN, and their θ-dependent behavior is well fitted by a simple quadratic polynomial: C i(θ)=ai+biθ+ciθ2, (4) where Ciis the ith order elastic modulus and ai,bi, and ci are the fitting constants which denote the intensity and order of the θ-dependent elastic behavior. The general tendency for all the defective configurations is that the odd-order moduliC 1andC3increase with increasing chiral angle, whereas the even-order moduli C2andC4decrease with increasing chiral angle. Thus, at higher chiral angles the odd-order modulistiffen the lattice and the even-order moduli soften it. Thisphenomenological description raises an important question:which elastic modulus governs the θ-dependent anisotropic stress-strain response of the lattice? To find an answer tothat question, we explore the basis of the stress difference across different chiral angles at an arbitrary macroscopic load.Defining a quantity called the differential stress /Delta1σ that denotes the difference in stress along an arbitrary loadingdirection relative to the armchair direction, we determinethe effect of chirality on /Delta1σ(θ) from the elasticity equation shown in Eq. ( 4). This gives a simple expression for /Delta1σ(θ) applicable for any strain state of the lattice: /Delta1σ(θ)= 4/summationdisplay i=1/parenleftbig Cθ i−Czig i/parenrightbig /epsilon1i=4/summationdisplay i(biθ+ciθ2)/epsilon1i,(5) where Cθ i=ai+biθ+ciθ2=Czig i+biθ+ciθ2.F o rt h e monotonic increase in stress with increasing chiral angles (asseen in the stress-strain data given in the Supplemental Mate-rial [ 56]), the differential stress is the maximum when θis the maximum—and it occurs between the high-symmetry loadingdirections. Substituting θ=30 ◦and the fitting parameters de- scribed above, we get the maximum differential stress for themonovacancy configuration as /Delta1σ max=4.35/epsilon1−134.7/epsilon12+ 3.99×103/epsilon13−1.33×104/epsilon14. Plotting each of the four terms separately as a function of applied strain, their individualcontributions on the macroscopic stress-strain behavior can beeasily evaluated at each strain state, as illustrated in Fig. 5. 134105-6STRONG ANISOTROPY IN STRENGTH AND TOUGHNESS … PHYSICAL REVIEW B 99, 134105 (2019) 0 0.05 0.1 0.15 Strain-10-5051015maxC1 C22 C33 C44 stiffening softening 0 0.05 0.1 0.15 Strain-10-5051015maxC1+C33 C22+C44 stiffening softening FIG. 5. The first plot shows the contribution of the individual elastic modulus Cion the maximum differential stress /Delta1σmaxas a function of applied strain, and the second plot shows the contribution of the combined effects of the odd-order elastic moduli C1+C3and the even-order elastic moduli C2+C4on the maximum differential stress /Delta1σmaxas a function of applied strain. The stiffening regime /Delta1σmax>0 and the softening regime /Delta1σmax<0 are identified in the plots. It is evident that both C1andC3stiffen the lattice at higher chiral angles, but the effect of C1is negligible compared to that of C3. The third-order modulus C3thus plays the dominant role on the θ-dependent stiffening behavior of the lattice. On the other hand, the even-order moduli C2and C4, which have the softening effect on the lattice, exhibit a proportionate softening effect up to a strain level of /epsilon1=10%. Beyond this strain C4dominates the softening of the lattice. The combined stiffening effects of the odd-order moduli,however, dominate the combined softening effects of the even-order moduli at each deformation state of the solid, leadingto fracture. Also, the domination increases nonlinearly withapplied strain. Consequently the differential stress betweenthe highest-symmetry directions is positive and the latticeexperiences stiffening enhancement with higher chiral angles.Next we examine the role of defect configurations on theanisotropic character of the elastic moduli from the trend inthe fitting constants a i,bi, and cithat are presented in Table I for all the defective structures. It is clear that there are several notable features among the fitting constants. First, the behavior of the first-order modulusC 1is dominated by the constant afor the reason that |a|/greatermuch |10b|and|a|/greatermuch| 100c|, where the numerical prefactor implies the order of the associated strain. The domination of aonC1 indicates that the chirality effects are negligible for lower- order elastic modulus. Second, the odd-order elastic mod-uliC 1andC3decrease with increasing defect size, whereas even-order moduli C2andC4increase with increasing defect size. Third, all the elastic moduli are much more sensitiveto the defect-core size than to the chirality of the lattice,suggesting the possibility of defect engineering as a strongertool than chirality engineering in modulating the mechanicalproperties of hBN. Fourth, the size dependence of the elasticmodulus diminishes rapidly for the higher-order moduli, butthe lower-order elastic moduli remain relatively unaffected bydefect size. We believe that this higher sensitivity of higherorder moduli is a direct consequence of strength reduction ofthe lattice with increased defect size. It can nonetheless beinferred from the observations that the larger the defect size isthe smaller is the nonlinear effects are and thereby the smallerthe anisotropy is in higher-order moduli. If we continue in- creasing the defect size, we should eventually reach a situationwherein the elastic moduli of higher orders become chiralityindependent due to the annihilation of mechanical anisotropyby stronger defect-induced mechanical softening of thelattice. Despite the qualitative similarity of the θ-dependent elas- tic behavior of hBN among all the defective configurations,there is a sizable difference between the elastic behavior ofdefective hBN and that of defect-free hBN. Figure 6shows θ- dependent elastic moduli of defect-free hBN exposing a third- TABLE I. The fitting parameters a,b,a n d cof the θ-dependent elastic constant curves represented by a+bθ+cθ2. The symbols in the table, V0,V1,V4,V13,V37,a n d V73, denote monovacancy, tetravacancy, tridecavacancy, heptatriacontavacancy, and trihepta- contavacancy, respectively. Defect type Constant ab c V1 C1 863.2046 0.011317 0.0039176 V4 C1 856.4997 −0.061717 0.0072234 V13 C1 851.2594 0.051805 0.007772 V37 C1 788.5994 −0.032654 0.016783 V73 C1 780.606 −0.025685 0.018712 V1 C2 −4220.0742 −7.5992 −0.30966 V4 C2 −3458.5253 1.0036 −0.81506 V13 C2 −2783.09 −13.6035 −0.93121 V37 C2 1463.0349 −32.255 −2.3486 V73 C2 2674.8288 −26.0843 −3.3029 V1 C3 −3274.6145 69.2852 2.1291 V4 C3−10894.8181 −9.7134 8.0788 V13 C3−19069.5939 181.7205 9.432 V37 C3−53790.4269 309.9968 28.5057 V73 C3−74809.1936 120.69 50.1512 V1 C4 20343.5021 −181.6227 −8.6806 V4 C4 66192.3775 238.2367 −48.0632 V13 C4 125230.0457 −1271.5681 −56.0809 V37 C4 587560.4965 −678.0199 −427.4519 V73 C4 587560.4965 −678.0199 −427.4519 134105-7AHMED, PROCAK, HAO, AND HOSSAIN PHYSICAL REVIEW B 99, 134105 (2019) 0 5 10 15 20 25 30 Chiral angle ( )832832.5833833.5834834.5835E (GPa) MD data a+b +c2+d3 fita=831.7699 b=-0.024013c=0.011413d=-0.00023666 0 5 10 15 20 25 30 Chiral angle ( )-2000-1900-1800-1700-1600-1500C2 (GPa)MD data a+b +c2+d3 fit a=-1562.5643 b=1.2009c=-1.6403d=0.037463 0 5 10 15 20 25 30 Chiral angle ( )-2500-2000-1500-1000-50005001000C3 (GPa)MD data a+b +c2+d3 fit a=-2175.09 b=10.0782 c=10.9526 d=-0.27304 0 5 10 15 20 25 30 Chiral angle ( )-1000-500050010001500C4 (GPa)MD data a+b +c2+d3+e4 fit a=763.6531 b=174.1645 c=-49.9911 d=2.7925e=-0.043489 FIG. 6. Chirality-dependent variation in elastic moduli of different orders: C1or the Young’s modulus E,C2,C3,a n d C4. The MD data points are well fit by a second-order polynomial, Ci=a+bθ+cθ2+dθ3+eθ4,w h e r e a,b,c,d,a n d eare fitting parameters in units of GPa. Their values are shown in the plots. order dependence on the chiral angle for C1toC3and a fourth- order dependence for C4. Due to pronounced nonlinear effects near the fracture strain, a strong θ-dependent C4is observed in the defect-free lattice. Unlike the defective cases (whereinthe extremum occurs for loading along the high-symmetrydirections), in the defect-free lattice the minimum appearsat an intermediate chiral angle. Furthermore C 1is lower in the defect-free lattice compared to V1,V4, and V13, indicating an enhanced stiffening effect in the defective lattice, even atlower deformation. (This is consistent with experimental ob-servations of stiffening of defective lattice containing mono-vacancy [ 37,38]. Although that observation was for graphene, we anticipate similar behavior in hBN, considering a majorsimilarity between the stress-strain behavior of graphene andhBN.) We propose that the stiffening feature of defectivehBN originates from the highly stressed atoms at the defectcore and the associated localized buckling resistance in thecompressive regime that raises the ensemble-averaged stressstate of the lattice. Also, as the number of defects increasesat the core, the edge line increases its length and so does thepotential energy per atom, making the defect core softer thanthe rest of the domain. The larger the defect core is the smallerthe buckling resistance is and the higher the softening of thelattice is. With continued softening of the lattice, eventuallyit reaches a state for larger enough defect cores wherein theYoung’s modulus of the defective lattice becomes lower than that of the pristine hBN. With this analysis we conclude thatthe higher the number of missing atoms is at the defect corethe lower the dominance of the nonlinear effects is and sois the anisotropy. The overall θ-dependent behavior of the elastic moduli in defect-free hBN lattice is well represented by a fourth-orderpolynomial: C i=a+bθ+cθ2+dθ3+eθ4, (6) where a,b,c,d, and eare fitting parameters in units of GPa. The behavior of C1toC3is well characterized by the polynomial e=0, whereas for C4the fourth term is also needed. Though defective lattice shows reduced nonlinearityon the chiral angle, for all the structures we find that the latticeretains substantial anisotropy in elastic properties. This fea-ture can find critical applications in sensing and acoustics andcan help us understand the θ-dependent extreme mechanical behavior of the lattice that we explore next. 2. Anisotropy in strength and toughness In this section we develop a quantitative understanding of the role of chirality on toughness and strength in hBN.Strength is obtained as the maximum stress in the stress-strainresponse of the lattice and it is denoted as σ max, and toughness 134105-8STRONG ANISOTROPY IN STRENGTH AND TOUGHNESS … PHYSICAL REVIEW B 99, 134105 (2019) 0 5 10 15 20 25 30 Chiral angle (degree)0.9511.051.11.151.21.251.3Strength anisotropyNo vacancy Mono-vacancy Tetra-vacancy Trideca-vacancy heptatriconta-vacancy triheptaconta-vacancy 0 5 10 15 20 25 30 Chiral angle (degree)11.21.41.61.8Toughness anisotropyNo vacancy Mono-vacancy Tetra-vacancy Trideca-vacancy heptatriconta-vacancy triheptaconta-vacancy FIG. 7. Chiral-angle-dependent (the left figure) strength anisotropy χsand (the right figure) toughness anisotropy χtfor the five different defect configurations and for the defect-free lattice. is calculated by integrating the area under the stress strain curve and it is denoted as /Pi1c=/integraltext/epsilon1f 0σ(/epsilon1)d/epsilon1. Since the focus of this study is to evaluate the anisotropic character of thelattice, we normalize the strength and toughness values bythe corresponding values along the armchair direction (or atθ=0 ◦) and use the ratio as a measure of anisotropy. The expressions for strength and toughness anisotropy can bewritten as χ s(θ)=σmax(θ) σmax(0◦)andχt(θ)=/Pi1c(θ) /Pi1c(0◦),respectively . As displayed in Fig. 7, for all the defect cases the lattice preserves its anisotropic strength and toughness characteris-tics. The tetravacancy and monovacancy configurations showcloser θ-dependent behavior, and their anisotropy in strength and toughness is lower than that of the pristine hBN. Theremaining vacancy configurations display higher anisotropyin both toughness and strength compared to the pristine con-figuration. Furthermore, all of the defect structures exhibitsubstantially higher strength and toughness along the zigzagdirection. Their θ-dependent behavior is accurately described by a simple cubic polynomial of the following form: χ i=1+ai/parenleftbiggθ 30/parenrightbigg2 +bi/parenleftbiggθ 30/parenrightbigg3 , (7) where θis the chiral angle in degrees, χiis the anisotropic strength or toughness of the ith defect structure, and aiand biare the associated fitting parameters. Their values along with the strength and toughness along the armchair direction,denoted by σ max(0◦) and/Pi1c(0◦), are listed in Table II. With an increasing number of defects at the core the second-order chirality effect on strength or toughnessanisotropy has a tendency to go down. The defect structuredependence of the anisotropies does not seem to follow anyspecific pattern. Yet it is clear that the chirality dependenceof strength as well as toughness remains mostly unaffectedby the presence of defects. The anisotropy in strength is aslarge as 27%, which appears in the V 37andV73structures; and the maximum toughness anisotropy is around 80% displayedby the same defective structures. Moreover there is a trendof the anisotropies to saturate with increasing the defect-core diameter. Even though the anisotropic character remains unaltered by the defects, the individual values of both strengthand toughness, however, decrease with increasing core size,as illustrated by the decreasing magnitude of σ max(0◦) and /Pi1c(0◦) in Table II. Our discussion on θ-dependent elastic moduli sheds some light on the aforementioned anisotropic aspect of strengthand toughness properties in hBN. To build an atomistic un-derstanding, we map out bond deformation and stress fieldssurrounding the defect core and establish a correlation be-tween the anisotropic properties and the atomistic attributesof the defective structures. Figure 8shows bond deforma- tion and potential energy fields surrounding monovacancyand tridecavacancy defects in the 17 .5 ◦at the strain state of/epsilon1=0.28 (very close to the fracture point). The bond deformation field has two distinguishing field patterns: (i) itis heterogeneous around the core causing a nonuniform forcedistribution, and (ii) it has a macroscopic pattern over theentire domain. The nonuniform load distribution forms thebasis of the θ-dependent toughness and strength behavior of the lattice, and their patterns are unaffected by the size of TABLE II. The fitting parameters aiandbiof the chirality- dependent strength and toughness anisotropy. Defect type Property σmax(0◦) ab V0 χs 87.133 10.578 1.311 V1 χs 67.858 7.658 1.493 V4 χs 56.168 0.601 6.702 V13 χs 50.226 5.560 5.706 V37 χs 42.51 5.588 5.743 V73 χs 37.987 3.278 7.140 Defect type Property /Pi1c(0◦) ab V0 χt 10.38 0.210 5.948 V1 χt 4.162 0.825 0.830 V4 χt 2.472 0.165 0.813 V13 χt 1.898 0.447 0.838 V37 χt 1.313 0.340 0.662 V73 χt 1.01 0.447 0.838 134105-9AHMED, PROCAK, HAO, AND HOSSAIN PHYSICAL REVIEW B 99, 134105 (2019) FIG. 8. High symmetry vacancy configurations: monovacancy and tridecavacancy for a lattice with chirality 17 .5◦at the highest elastic energy density state at the onset of fracture. The bonds arecolored according to their lengths representing their strain states. The atoms are colored with the potential energy representing their energetic state. The dotted lines represent the missing bonds due tothe absence of the atoms at the defect center, and the circle marks the deformed core structure. The fracture plane in indicated by the inclined solid line intersecting the highly deformed bonds. Forcedistribution is shown on the right.the defect core. The load-carrying bonds can be classified into three distinct groups: Oa,Ob, and Ocbonds. Since the uniaxial stress deformation is applied along the ydirection for all the cases, the bond force is given by /vectorFb=/vectorF·ˆnb, where ˆ nb is the unit vector along the bond and Fis the applied force. The largest component acts along the Oadirection due to the smallest direction cosine and the smallest component actsalong the Obdirection—these angles are marked by θ minand θmax, respectively, in the schematic. The plane perpendicular to Oais the weakest plane, and failure initiation thus occurs along this plane. While there aremany such planes, the plane with the minimum nominal areaintersecting the least number of bonds is the most susceptibleone for undergoing fracture. That plane contains one missingbond in the V 1structure, two bonds in the V4structure (not shown in the figure), and three bonds in the V13structure. The bonds at the defect edge on these planes experiencethe highest force and they rupture first, leading the way formaterial separation along the θ ⊥direction, which is also the zigzag direction and at a right angle to the Oabonds. FIG. 9. Stress surface of σyyin hBN surrounding the tetravacancy defect for loading along (a) the armchair direction (0◦), (b) 13 .89◦with respect to the armchair direction, (d) 23 .41◦with respect to the armchair direction, and (d) the zigzag direction (0◦). All are at 5% macroscopic strain. (e) Nonuniform distribution of bond lengths at the core for the conditions shown in panels (a)–(d), respectively. The red dashed line indicates the fracture plane(s), and the intersection of the black dotted lines indicate the center of the defect core. 134105-10STRONG ANISOTROPY IN STRENGTH AND TOUGHNESS … PHYSICAL REVIEW B 99, 134105 (2019) For the situation when loading is applied along the arm- chair direction ( θmin=0◦)t h e Oabonds become parallel to the loading direction, and these bonds experience the highestforce with the direction cosine being equal to 1. For loadingalong the zigzag direction ( θ min=30◦)t h e OaandOcbonds lie at the same orientation angle relative to the loading direc-tion and they share the load equally, leaving the remainingObbonds to respond laterally only through the Poisson’s contraction. The equipartition happens only at θ min=30◦; the strength and toughness behavior of the lattice is thus themaximum when loaded along this direction. The presence ofdefects does not alter the directional force patterns discussedabove. As a result, without regard to the loading direction,the maximum toughness and strength always appear alongthe zigzag direction, and the minimum toughness and strengthalong the armchair direction. Furthermore, the atomic stress field displayed in Fig. 9 shows that for 0 <θ/lessorequalslant30 ◦the normal stress distribution around the defect breaks its symmetry relative to the sym-metry plane at y=0. For θ=0 ◦, the stress field retains its symmetry and distributes the stresses evenly around the defectcore. The θ-dependent symmetry breaking at 0 <θ/lessorequalslant30 ◦ and the symmetry preservation at θ=0◦are observed for all of the defect configurations, by virtue of identical symmetryof the respective defect structures. Though all the defectsare circular in an effective sense, edge-geometry breaks thestress-field symmetry surrounding the core. The asymmetricdistribution deviates from the continuum scale approximation of symmetric stress fields around a perfectly circular hole. The load-sharing by the bonds discussed earlier is directly related to the stress distribution such that any deviation fromthe symmetry pattern has a mechanical consequence on themacroscopic stress-strain behavior. The larger the chiral angleis the higher the asymmetry of the normal stress field isand the higher the asymmetry of the bond force is. As thechiral angle increases from 0 ◦, the asymmetry in the tensile stress field relative to the loading direction increases. Thisdistributes the load unevenly on the left- and right-hand sidesof the vertical midplane causing a macroscopic mechanicalanisotropy in the lattice. Thus, localized asymmetric stressfields configure and govern the macroscopic failure criteriaof the lattice, and thereby the anisotropic behavior of thelattice. The nonuniformity in the stress pattern is, however,undetectable from the macroscopic stress-strain data due tothe averaging of the stress fields over all the atoms. Also, thestress fields emanating from the defect core decay rapidly tothe nominal stress in the bulk within 1 nm from the defectedge, indicating stronger decay of the stress field than 1 /√ r. From the discussions above it is clear that defect size plays a critical role and it reduces the strength and toughness ofthe lattice, but it does not modulate the anisotropic behaviorsubstantially. To build a quantitative understanding of the roleof defect size, we plot the strength and toughness as a functionof defect diameter in Fig. 10, which represents the results for 0 0.05 0.1 0.15 0.2 Strain020406080100Stress (GPa)Pristine Monovacancy Tetravacancy Tridecavacancy Heptatriacontavacancy Triheptacontavacancy 0 0.05 0.1 0.15 0.2 0.25 Strain020406080100Stress (GPa)Pristine Monovacancy Tetravacancy Tridecavacancy Heptatriacontavacancy Triheptacontavacancy 02468 1 0 1 2 Defect radius (Angstrom) 30405060708090100Effective strength (GPa)armchair zigzag 02468 1 0 1 2 Defect radius (Angstrom) 05101520Effective toughness (Jm-2)armchair zigzag FIG. 10. The top panels show the stress-strain behavior of defective hBN for loading along the armchair and zigzag directions, respectively. The bottom panels show defect-size-dependent effective strength and toughness, where d=0 refers to the behavior of the defect-free lattice. 134105-11AHMED, PROCAK, HAO, AND HOSSAIN PHYSICAL REVIEW B 99, 134105 (2019) the vacancy configurations of different radii, for loading along the armchair and zigzag directions. With increasing defect size the effective toughness shows asymptotic nonlinear behavior, indicating the possibility ofsize invariance for sufficiently larger defect size. The sizesensitivity of effective strength is higher along the armchairdirection than along the zigzag direction. For any of the defectconfigurations the anisotropy in strength remains at around10 GPa; whereas the anisotropy in toughness is apparentlysize dependent—it reduces in magnitude from 7 Jm −2inpristine hBN to around 1.25 Jm−2in the triheptacontacva- cancy configuration. The difference in the maximum stresses between the tetravacancy and tridecavacancy defects is around12%, which is around half of the difference between themaximum stresses in the monovacancy and tetravacancy con-figurations. Further the maximum stress at the monovacancyedge is around 23 and 37% higher than that at the tetravacancyand tridecavacancy edges, respectively. The strength data points along either of the high-symmetry directions exhibit inverse diameter-dependent behavior of the 020 35yy (GPa) 1440 301260 25 10 8(a) 020 35yy (GPa) 1440 301260 25 10 8(b) 020 35yy (GPa) 1440 301260 25 10 8(c) 020 35yy (GPa) 1440 301260 25 10 8(d) 0 0.2 0.4 0.6 0.8 1 1.2 Defect core diameter (nm)2030405060Atomic max (GPa)MD data + d+ d2 fit =23.016 =35.3743 =-0.60723(e) 0 0.2 0.4 0.6 0.8 1 1.2 Defect core diameter (nm)22.4622.4822.522.5222.5422.56Average stress (GPa)MD data + d+ d2 fit =22.521 =0.17298 =-0.21552(f) FIG. 11. Symmetry-preserving stress surface of σyysurrounding the defects representing (a) tetravacancy, (b) tridecavacancy, (c) heptatri- acontavacancy, and (d) triheptacontavacancy at 5% macroscopic strain along the armchair direction. The color distribution refers to the stressrange of 0.0 to 60.0 GPa. The average stress is well captured by the color of the far-field stress in the stress surface plots. Panel (e) shows the variation in maximum atomic stress and panel (f) shows the average atomic stress for the V 1,V4,V13,V37,a n d V73configurations. They are well fit by first-order and second-order polynomials, respectively. 134105-12STRONG ANISOTROPY IN STRENGTH AND TOUGHNESS … PHYSICAL REVIEW B 99, 134105 (2019) following form: σmax(d,θ)=σ0(θ) 1+d/d0(θ), (8) where σ0=87.13 GPa and d0=8.1169 nm for loading along the armchair direction, and σ0=98.65 GPa and d0= 9.4877 nm for loading along the zigzag direction. The strength for pristine hBN can be obtained from Eq. ( 8) by setting d=0, which gives σmax(0,θ)=σ0(θ). The values of σ0(θ) are therefore the strength of the defect-free lattice along therespective directions. A first-order expansion of Eq. ( 8)g i v e s σ max(d,θ)≈[1−d/d0(θ)], which indicates a linear decrease of strength with increasing diameter for any of the chiralangles. Our MD data show a nonlinear decrease of strength—the behavior is well captured by (1 +d/d 0)−1w i t har a p i d decrease in values for smaller diameters and a slower decreasefor larger-diameter defect cores. The atomistic basis of the abovementioned reduction in strength is explored next by examining the atomic stress inthe lattice. It is found that the stress is maximum near theedge of the core and the maximum of the atomic stressesfor a given strain state increases with the defect-core size.For example, at /epsilon1=0.05 the maximum normal stress σ yy is 31.6, 38.8, 43.6, 51.5, and 57.5 GPa in the V1,V4,V13, V37, and V73configurations, respectively. The relationship between the maximum atomic stress and defect-core sizeis approximately linear. On the other hand, the ensemblearithmetic average stress decreases with the defect-core size.For the same configurations stated above, the average stress inthe configurations is 22.559, 22.5576, 22.5485, 22.5199, and22.476 GPa, respectively. The reduction in the average stressis tied to the softening of the lattice with increased defect core. Even though the softening effect reduces the average stress level in the lattice, the presence of larger defects makes it failearlier (since larger defects satisfy the bond rupture earlier)due to the rapid evolution of higher stress concentration atthe defect edges. The effective strength of defective hBN is thus controlled by the combined effect of softening and stressconcentration . The stress field surrounding the defect cores isshown in Fig. 11as a stress surface for a number of defect configurations for loading along the armchair direction. Eachsurface shows similarity in stress pattern, and the peaks ofthe stress surface exhibit the maximum atomic stresses andtheir locations. The maximum atomic stress and the ensembleaverage stress of the stress fields are plotted in the same fig-ure, showing approximately a monotonically increasing anddecreasing pattern for the former and the latter, respectively. Next analyzing size-dependent effective toughness of the defective lattice, we find that it has a stronger sensitivity tosize and obeys an exponential form: /Pi1 c(d,θ)=/Pi10(θ)+α(θ)e−d/d0(θ), (9) where /Pi10=0.739 and α=9.64 Jm−2, and d0=2.7563 nm along the armchair direction; and /Pi10=1.786 and α= 14.96 Jm−2, and d0=2.2548 nm along the zigzag direc- tion. The toughness reduction is much more sensitive to sizealong the zigzag direction. As d→∞ , under the situation ofd/W<1, effective toughness approaches a constant value equal to 2.0 Jm −2along the armchair direction and 3.0 Jm−2 along the zigzag direction. For holes of larger size than 10 nm, the toughness becomes independent of the defect size, as longas the size is small enough that the domain’s dimensionsand its dependence on defect size can be neglected. It isalso noted that the reduction in toughness is much strongercompared to the reduction in strength. While determiningthe atomistic basis of toughness variation is a difficult taskdue to the accumulative effects of all the elastic moduli andstrength that go into affecting the toughness of a solid, fromthe discussions on strength and elastic moduli variation itcan, however, be inferred that gradual insensitivity of higher-order elastic moduli for higher defect size and continuedreduction in strength can make the size-dependent toughnessapproach a slower variation for the lattice with larger defects.More detailed analysis is needed on this observation whichwe consider out of scope for this paper. Nonetheless sincethe results on anisotropy in strength and toughness of thedefective structures were obtained from the SW potential only,at 1 K we evaluate the accuracy of our results by determining 0 5 10 15 20 25 30 Chiral angle 11.021.041.061.081.11.121.14Strength anisotropySW BN-ExTeP 0 5 10 15 20 25 30 Chiral angle 11.11.21.31.4Toughness anisotropySW BN-ExTeP FIG. 12. Normalized strength and toughness at ten different chiral angles for the V4configuration, containing one tetravacancy in the domain. The strength and toughness values at a given chiral direction are normalized by the respective strength and toughness values along the armchair direction, which are 56.168 and 60.52 GPa with the SW and BN-ExTeP potentials, respectively, and the corresponding toughness values are 2.47 and 3.1 Jm−2, respectively. 134105-13AHMED, PROCAK, HAO, AND HOSSAIN PHYSICAL REVIEW B 99, 134105 (2019) FIG. 13. Comparison of atomic stress fields σyyobtained from the SW and BN-ExTeP (Tersoff) potential simulations for θ=17.4◦ at 8% macroscopic strain. The color bar represents stress in units of GPa. The uniaxial loading direction is shown by the double arrow. the results for one defective structure from the BN-ExTeP potential, which takes into account many-body effects in con-structing its bond-order term. Additionally, to assess the roleof temperature and examine its role, we conduct simulationswith both SW and BN-ExTeP potentials at 300 K. 3. Effects of interatomic potential To assess the validity of our SW results for the defec- tive structures we investigate the mechanical behavior of thedomain with a tetravacancy core, for ten different loadingdirections, using the BN-ExTeP potential and compare theresults with those obtained from the SW potential. As alreadypointed out in Sec. III B (in the context of Fig. 2), the BN- ExTeP potential in its original form shows stiffening of thelattice at higher deformation. Nevertheless, for a defectiveconfiguration, the domain fails at a much lower strain (com-pared to that in the pristine or defect-free configuration).The BN-ExTeP potential can thus be expected to providereasonable results for strength and toughness for the defective configurations. Results presented in Fig. 12show close agreement (within less than 0.1%) between the potentials on anisotropy instrength and toughness of the defective lattice. It should behighlighted here that the SW and BN-ExTeP potentials differsubstantially in terms of describing V 2andV3interactions, yet they produce consistent results for each of the ten differentloading directions. In the SW description of interatomic interactions there is only one set of equilibrium bond angles ( θ 0) and bond lengths ( r0), limiting the accessible local atomic structure at a defective site containing dangling bonds. The BN-ExTePpotential, on the other hand, has many-body effects accountedfor in its angle-dependent terms, allowing the possibility ofseveral local bond angles at a defective site. It is an extendedTersoff potential and it has an improved accounting of thechemical environment and bond order, on top of the usualTersoff bond-order terms. Nonetheless the close agreementbetween the potentials in reproducing the anisotropic char-acter of the lattice highlights an important point that theanisotropic behavior is strongly dominated by the nonlinearcovalent interactions of the lattice and the atomistic structureof the defect, as opposed to the many-body effects arisingfrom intricate bond-order terms. To determine if the difference between the potentials in accounting for the many-body effects has an effect on thestress fields (therefore the strength and toughness of thelattice) surrounding the defect core, we compare the atomicstress fields in the domain obtained from the SW and BN-ExTeP potentials at different strain states. We find consistentstress fields between the potentials, as illustrated in Fig. 13for 8% macroscopic strain state. The stress fields show consistentfeatures and highlight the possibility of negligible effects ofmany-body interactions. During deformation there is a localbuckling at the defective site but the large cohesive energybarrier in the lattice prohibits initiation of any dissipativeprocesses. The deformation state is found to be completelyreversible up to the point of bond rupture at the defect core. 0 5 10 15 20 25 30 Chiral angle 0.911.11.21.31.4Strength anisotropySW BN-ExTeP 0 5 10 15 20 25 30 Chiral angle 0.911.11.21.31.41.51.6Toughness anisotropySW BN-ExTeP FIG. 14. Chirality-dependent strength and toughness of defective hBN lattice containing a tetravacancy at 300 K. The strength and toughness values at a given chiral direction are normalized by the respective strength and toughness values along the armchair direction.The strengths of the lattice along the armchair direction are 50.13 and 38.56 GPa with the SW and BN-ExTeP potentials, respectively, and the corresponding toughness values are 2.02 and 1.61 Jm −2. 134105-14STRONG ANISOTROPY IN STRENGTH AND TOUGHNESS … PHYSICAL REVIEW B 99, 134105 (2019) FIG. 15. Deformation state of the hBN lattice at 300 K with (a) the SW potential and (b) the BN-ExTeP potential, at 30% macroscopic strain. Local morphology is evident in the insets in panels (a) and (b) indicating intact defect structures. The fracture states of the lattice areshown in panels (c) and (d), which are obtained from the SW and BN-ExTeP potentials, respectively. The anisotropic strength and toughness characteristics of the lattice are therefore governed primarily by the nonlinear elas-tic interactions surrounding the defect core. The results and discussions presented heretofore refer to the low-temperature behavior of the lattice, and both the SWand BN-ExTeP potentials consistently show that strength andtoughness are strongly retained by the lattice under defectiveconditions of the lattice. To evaluate the effect of temperatureon the observed anisotropy in strength and toughness, weexplore the behavior of one defective structure at 300 K andfind the potentials to exhibit similar anisotropic responses inregard to the anisotropy in strength and toughness, as depictedin Fig. 14. The lattice with both the potentials shows reduction in strength and toughness with increasing temperature, due tothermal softening of the lattice which is a widely knownphenomena for solids. With the SW potential the reductions instrength and toughness are 10 and 19%, respectively, whereaswith the BN-ExTeP potential they are 36 and 48%, respec-tively. In lieu of the differences in thermal softening behaviorof the SW vs BN-ExTeP potentials, they agree very well onthe degree of anisotropy in strength as well as well as tough-ness at 300 K. Furthermore, it is noted that the lattice formsa long-range ripple at 300 K, with both the potentials, thatarises from thermal fluctuations in the out-of-plane direction.Also, as demonstrated in Fig. 15, the fracture states show different fracture patterns and paths with the potentials. TheSW potential shows a clean brittle fracture pattern emanatingfrom the defect core, whereas the BN-ExTeP potential showsa non-brittlelike fracture (with a number of defects created atrandom locations in the lattice). To determine if the deformation is reversible or involves any plastic events propagating from the defect core, we applyuniaxial deformation to the defective lattice (up to the pointwhereat bond rupture takes place in the lattice from the core). We then unload the lattice at the same strain rate and monitorthe energetic and stress states of the lattice at different strainstates. We conduct this test at two different temperatures (1and 300 K) with the SW potential and find that the defor-mation is completely reversible, as illustrated in Fig. 16.T h e plastic events in the lattice do not kick off from the core evenat higher deformation. This accentuates the important pointthat, despite the presence of the defect core, the deformationof the lattice is governed by nonlinear elasticity mediated bystrongly covalent interatomic interactions. While our studies are focused on a few select numbers of C 3vacancy defects (with diameters less than 1 nm), the results are comprehensive enough to conclude that toughness andstrength anisotropy is hBN are retained by the defective lat- 0 0.02 0.04 0.06 0.08 Strain 0102030405060Stress (GPa)SW (loading at 1K) SW(loading at 300K) SW (unloading at 1K) SW(unloading at 300K) FIG. 16. Stress-strain behavior of a defective lattice ( V4withθ= 17.4◦) under one loading and unloading cycle at 1 and 300 K with the SW potential. 134105-15AHMED, PROCAK, HAO, AND HOSSAIN PHYSICAL REVIEW B 99, 134105 (2019) tice. The results motivate further exploration of longer length- scale effects (related to defect size and edge structures thatare currently under active investigation and will be reportedelsewhere). IV . CONCLUDING REMARKS This paper presents an atomistic understanding of elastic moduli, toughness, and strength in defective hBN containingisolated monovacancy, tetravacancy, tridecavacancy, trihepta-contavacancy, or heptatriacontavacancy defect structures forten different chiral angles spanning the high-symmetry load-ing directions (such as the armchair and zigzag directions)as well as the low-symmetry directions for the intermediatechiral angles. Results unambiguously show that all threehigher-order elastic moduli of hBN are anisotropic, and thefirst-order elastic modulus is isotropic only in the pristinelattice. The anisotropy in the first-order modulus is negligiblecompared to that in the higher-order moduli. It is also ob-served that the first-order modulus (denoting the stiffness ofthe lattice) increases in the presence of monovacancy, tetrava-cancy, or tridecavacancy. With the increasing defect size thelattice exhibits softening behavior. Moreover, the nonlinearityin chirality-dependent behavior reduces with defect size dueto the commensurate reduction in the strength of the lattice. In addition to the elastic properties, both strength and toughness are also strongly anisotropic in defective hBN, andit retains anisotropic strength and toughness over a rangeof vacancy defect configurations. It is also revealed that the anisotropic behavior is insensitive to the defect size, althoughthe individual values of toughness and strength reduce withincreasing defect size, following an inverse dependence oneffective diameter. A detailed atomistic deformation and stressfields unravel chirality-dependent nonuniformity in bond-length, bond-strain, and bond-force distributions as the primedetermining factor for governing the elastic and extreme be-havior of the defective lattice. With larger defect cores the me-chanical property degradation competes with the anisotropiceffect, and the mechanical degradation is stronger than theanisotropic effect. These observations underline the outstand-ing mechanical ability of hBN in preserving its anisotropyproperties even under defective and thermal conditions. Theclose-form mathematical relations that are developed in thiswork are expected to find critical applications in continuum-scale studies of larger-scale mechanical phenomena. Theresults are anticipated to motivate new materials researchand promote applications of hBN under extreme mechanicalconditions. 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[56] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.99.134105 for the chirality-dependent stress-strain curves and elastic moduli of the defective hBNlattice for five different defect configurations. 134105-17

PhysRevB.99.064418.pdf

PHYSICAL REVIEW B 99, 064418 (2019) Effects of lattice geometry on the dynamic properties of dipolar-coupled magnetic nanodisk arrays Sam D. Slöetjes,1,2,*Einar Digernes,1Christoph Klewe,2Padraic Shafer,2Q. Li,3M. Yang,3Z. Q. Qiu,3 Alpha T. N’Diaye,2Elke Arenholz,2Erik Folven,1and Jostein K. Grepstad1 1Department of Electronic Systems, NTNU, NO-7491, Trondheim, Norway 2Advanced Light Source, LBNL, Berkeley, California 94720, USA 3Department of Physics, UC Berkeley, Berkeley, California 94720, USA (Received 16 November 2018; revised manuscript received 1 February 2019; published 19 February 2019) We have studied the impact of lattice geometry on the dynamic properties of close-spaced arrays of circular nanomagnets, also known as magnonic crystals. To this end, we prepared 2D nanomagnet arrays with both squareand hexagonal lattice symmetries (300-nm disk diameter, 400-nm center-to-center distance) and performedbroadband ferromagnetic resonance (FMR) measurements. Micromagnetic simulations were used to interpretdistinct features of the measured resonance spectra. The FMR bias field was applied along two distinct principaldirections for each lattice, and a sample with well-separated, decoupled disks was measured for reference. Wefound that the interdisk dipolar coupling has a strong impact on the FMR for these 2D magnonic crystals.Distinctly different oscillation modes were found for the individual nanomagnets, dependent on lattice symmetryand direction of the bias field. Moreover, we find that spectral peak splitting from excitation of edge and centermodes, as well as the damping, depends on the lattice symmetry and the orientation of the bias field. Thesefindings demonstrate that lattice geometry has a strong influence on the excited spin-wave spectrum and is arelevant design parameter for spintronic devices. DOI: 10.1103/PhysRevB.99.064418 I. INTRODUCTION Magnonic crystals are metamaterials in which the magnetic properties vary periodically [ 1]. These analogs to photonic crystals offer unprecedented tunability of the magnetodynam-ics [2–4]. A special application of such systems is genera- tion of highly tunable microwaves from a nanoscale source[4]. Examples of magnonic crystals are magnetic thin films with nonmagnetic holes placed in a periodic fashion [ 5,6] or dipolar-coupled ferromagnetic nanodots [ 7], as discussed in this paper. The dynamics of single isolated magnetic dotshave been investigated extensively [ 8–11], whereas studies of closely spaced, dipolar-coupled systems are few [ 12–14]. The dynamic properties of magnonic crystals are still poorlyunderstood due to the high complexity of periodic dipolar-coupled systems [ 14]. Here, we present ferromagnetic reso- nance (FMR) studies of the spectral response of magnoniccrystals, dependent on their symmetry and the bias-fielddirection. II. METHODS The magnonic crystal samples were made from 15-nm permalloy (Py) films, deposited on a silicon substrate andcapped with a 2-nm aluminum oxide layer to serve as theoxidation barrier. Using electron-beam lithography, the filmswere patterned into arrays of disks with diameter d=300 nm and a center-to-center distance of 400 nm, shown in Figs. 1(a) and1(b). This results in an edge-to-edge spacing of 100 nm, *Corresponding author: sam.sloetjes@ntnu.noi.e., less than the disk diameter, and ensures sufficient dipole- dipole interaction to impact the dynamic properties of thearrays. The patterned area on each sample was 3 ×3m m 2 in order to ensure sufficient absorbed power in the FMR experiment. We performed broadband FMR measurements by acquir- ing FMR spectra at a constant microwave frequency whilesweeping the external field, μ 0H0, from 150 to 0 mT. The initial field of 150 mT ensures magnetic saturation of the disksfor the FMR experiments. In this setup, the sample is placedface down on a coplanar waveguide, where the microwaveradiation is applied at frequencies from 3 to 12 GHz in incre-ments of 1 GHz. The field derivative of the FMR absorptionintensity was measured using an rf diode and using a lock-inamplifier at an ac modulation field of 133 Hz. In order toprobe the anisotropy originating from intermagnet dipolarcoupling, we applied the field along two high-symmetry lat-tice directions for each nanodisk array. For the square lattice,we applied the field μ 0H0in the in-plane [10] and [11] directions, and for the hexagonal lattice the field was appliedin the [1 ¯2] and [ ¯10] directions; cf. Figs. 1(a) and1(b).I nt h e hexagonal lattice, these directions correspond to a direction ofnearest neighbors and a direction between nearest neighbors,respectively. Hysteresis loops measured using x-ray magneticcircular dichroism spectroscopy at beamline 6.3.1 at the Ad-vanced Light Source showed a remanent magnetization closeto saturation, confirming that the individual magnets remainin a monodomain state throughout this measurement range. Inorder to determine the spectral linewidths and resonance peakpositions, we fitted the resulting spectra with a superpositionof derivative and double-derivative Lorentzians [ 15]. 2469-9950/2019/99(6)/064418(7) 064418-1 ©2019 American Physical SocietySAM D. SLÖETJES et al. PHYSICAL REVIEW B 99, 064418 (2019) [10][11][10] [12] (a) (b) 7GHz 8GHzDifferential absorbtion (a.u.) 20 40 60 80 100 1209GHzSimulationField (mT) Field (mT)(c) (d)Field [mT]7GHz Field [mT]8GHzDifferential absorbtion (a.u.) 20 40 60 80 100 1209GHzExperiment FIG. 1. (a) SEM image of the magnonic crystal with (a) hexag- onal symmetry and (b) square symmetry. The disks are 300 nm in diameter with 100-nm separation. The lattice principal axes are indicated with red and blue arrows. (c) Measured and (d) simulatedFMR absorption spectra for frequencies of 7–9 GHz for the square array, with the applied field oriented in the [11] direction. For the micromagnetic simulations, we used the soft- ware package MUMAX 3, which solves the Landau-Lifshitz- Gilbert equation numerically for a given geometry [ 16]. The material parameters used to describe the magnetic proper-ties of the nanomagnets are standard values for Py, i.e.,exchange stiffness A ex=10 pJ/m, saturation magnetization MS=800 kA /m, and a crystalline anisotropy of 0 J /m3.T h e damping parameter was set to α=0.01, a typical value for patterned Py elements. The simulation cell size was opti- mized for mesh independence, to avoid anisotropies result- ing from projection of the circular nanodisks onto a squaregrid. However, the cells were large enough so that com-putation times were within reasonable bounds. The chosencells had in-plane dimensions of 2 .5n m×2.5n mf o rt h e hexagonal lattice and 1 .5n m×1.5 nm for the square lat- tice, both smaller than the magnetostatic exchange length, l S=√ 2Aex/μ0M2 S=4.98 nm. The simulated FMR spectra were obtained by exciting the magnetic moments with anin-plane field pulse in the direction perpendicular to H 0. The frequency spectra of the resulting oscillations are foundby Fourier transformation, using a procedure known as the“ringdown method” [ 17,18]. In order to obtain the full fre- quency spectra, this procedure is repeated for values of the fieldμ 0H0from 1 to 150 mT in steps of 1 mT. Magnetic nanodisks typically feature both edge and center modes [ 8], i.e., spatially inhomogeneous oscillations. Thus, the spatialdistribution of their amplitude is of interest. By applying theringdown method to every grid point m(x n,ym,t) to com- pute Fourier transforms ˜m(xn,ym,f), spatial amplitude and phase maps of the nanodisk oscillations were obtained, for applied bias fields μ0H0from 20 to 140 mT, in increments of 20 mT.III. RESULTS AND DISCUSSION Figures 1(c)and1(d) compare experimental and simulated (derivative) FMR absorption spectra for the square lattice,with H 0pointing in the [11] direction and excitation fre- quencies from 7 to 9 GHz. In this range of frequencies theresonance is split into two separate peaks, centered at differ-ent values of μ 0H0. The low-field peak retains a substantial amplitude throughout the frequency range, whereas the high-field peak broadens and rapidly attenuates with increasingfrequency. The simulated spectra show a similar splitting ofthe ferromagnetic absorption resonance around 7 GHz, andexcellent agreement between simulation and experiment isfound. We note there is an offset in peak position H FMR be- tween the simulated and experimental spectra, most likely dueto the measurements being carried out at finite temperature(T=295 K). The spectra were fitted with superpositions of first- and second-derivative Lorentzian functions [ 15,19]: dI FMR dH0∝cos(/epsilon1)2(H0−HFMR)/Delta1HHWHM/bracketleftbig /Delta1H2 HWHM+(H0−HFMR)2/bracketrightbig2 +sin(/epsilon1)/bracketleftbig /Delta1H2 HWHM −(H0−HFMR)2/bracketrightbig /bracketleftbig /Delta1H2 HWHM+(H0−HFMR)2/bracketrightbig2. Here, HFMRis the peak position, /Delta1HHWHM is the linewidth defined as the half width at half-maximum (HWHM), and /epsilon1is the mixing angle between the symmetric and the antisymmet-ric term. We employed a least-squares fitting method to derivethese fitting parameters. The resonance frequencies were sub-sequently plotted versus peak position H FMR,following the Kittel curve, which can be fitted using the Kittel equation[20]. Measured and simulated Kittel curves for an array of magnetically uncoupled nanodisks (300-nm disk diameter,800-nm center-to-center distance) are shown in Fig. 2.T h e Kittel curves for the square lattice, with the bias field H 0 applied along the [10] and [11] directions, are shown in Figs. 3(a)and3(b), respectively, and results for the hexagonal lattice are shown in Figs. 4(a) and4(b). The Kittel curves for a single disk and for the square lattice with H0oriented in the [10] direction show only one prominent branch. With H0 oriented in the [11] direction, two branches are obtained [cf. Fig.4(b)]. For the hexagonal lattice, the Kittel curves with H0 oriented along either the [1 ¯2] or [ ¯10] directions split into two branches. This splitting occurs at a higher bias field than forthe square lattice with H 0oriented in the [11] direction. In all cases, the simulated FMR spectra accurately repro- duce the spectral features of the experimental data, such as themode splitting. The corresponding spin-wave amplitude mapsfor the individual magnets are shown in the insets of Figs. 3 and4for the square and hexagonal lattice, respectively. The FMR spectra for the isolated disk and the square lattice withH 0in the [10] direction are nearly identical, differing only by a small offset in resonance frequency. For these systems, theresonances correspond chiefly to center modes throughoutthe entire frequency range (3–12 GHz), as can be seen fromthe insets in Fig. 3. A notable exception is the oscillation mode at an applied field of μ 0H0=60 mT, which shows a splitting in two spatial maxima rather than a single maximum. 064418-2EFFECTS OF LATTICE GEOMETRY ON THE DYNAMIC … PHYSICAL REVIEW B 99, 064418 (2019) FIG. 2. Upper graph shows the measured Kittel curve for a single disk. Insets show amplitude maps at fields of 20, 60, 100,and 140 mT, as determined by simulations. Lower graph shows corresponding contour plots for simulated absorption spectra. The noise at fields of μ 0H<5 mT is a result of some nanomagnets in the simulations having a flux-closure ground state.ForH0oriented in the [11] direction, we observe a much different resonance behavior. Here, the principal resonance splits into two branches for a field of approximately 40 mT[cf. Fig. 3(b)]. From simulations, we find that the resonance at low fields arises from a mode with maximal amplitudelocalized near the edges of the nanodisks, i.e., “edge modes”[cf. insets of Fig. 3(b)]. The high-field part of the Kittel curve features a high- and a low-frequency branch, corresponding toa center mode and an edge mode, respectively. The amplitude maps show that the high-field mode is not a pure edge mode, i.e., there is a finite but small oscillation amplitude for themagnetization throughout the disk. This oscillation is a stand-ing wave, with a wavelength of approximately the diameter ofthe magnetic disk. We find a similar behavior for the hexagonal lattice, where the main resonance feature splits into two peaks at an applied field of approximately 60 mT. The resonances beyond this bias field also feature a low-frequency edge mode and a high-frequency center mode, as can be seen in Fig. 4. However, in contrast to the square lattice with the field applied alongthe [11] direction, the resonance is a center mode in thelow-field range (i.e., for μ 0H0/lessorequalslant60 mT). The two applied field directions in the hexagonal lattice show little difference in the measured and simulated spectra (cf. Fig. 5), suggesting negligible magnetic anisotropy for the hexagonal lattice. Additional resonances can be observed in the simulated FMR data, such as the branch at approximately half thefrequency of the main mode, cf. Figs. 2–4. From the simulated amplitude maps, we find that this resonance corresponds to apure edge mode (i.e., zero amplitude at the center of the disk).This mode does not show up in the experimental FMR spectra, FIG. 3. Upper graphs show measured Kittel curves for the square lattice, with the field in the [10] direction (a) and the [11] direction (b). Blue solid dots and pink open dots represent center- and edge modes, respectively; insets show amplitude maps for applied fields of 20, 40, 60, and 80 mT. Lower graphs show corresponding contour plots for simulated absorption spectra, with insets showing magnetization directions. 064418-3SAM D. SLÖETJES et al. PHYSICAL REVIEW B 99, 064418 (2019) FIG. 4. Upper graphs show measured Kittel curves for the hexagonal lattice, with the field in the [1 ¯2] direction (a) and the [ ¯10] direction (b). Insets show amplitude maps for applied fields of 20, 40, 60, and 80 mT. Lower graphs show corresponding contour plots for simulatedabsorption spectra, with insets showing magnetization directions. most likely because it absorbs too little energy to be detected by our FMR setup. For the center-mode oscillations of the nanodisk magneti- zation, we can fit the Kittel equation for an ellipsoid [ 8,20]: f=γ 2π/radicalbig μ0H0+μ0Ha+μ0MS(Ny−Nx) ×/radicalbig μ0H0+μ0Ha+μ0MS(Nz−Nx). Here, Nx,y,zrepresent the demagnetization factors, Hais the anisotropy field, MSis the saturation magnetization, μ0 is the permeability of free space, and γis the gyromagnetic ratio. In our case, the disks are lying in the xyplane, and the bias field, H0, is applied in the xdirection. Because of the circular symmetry, we have Ny=Nxso that the term Ny−Nx vanishes, and we are left with f=γ 2π/radicalbig μ0H0+μ0Ha ×/radicalbig μ0H0+μ0Ha+μ0MS(Nz−Nx). (1) As the saturation magnetization for a blanket Py film has been determined by vibrating sample magnetometry, the de-magnetization factor difference ( N z−Nx) and the anisotropy fieldμ0Haare free fitting parameters. For the square-lattice edge mode measured with the bias field in the [11] direction, we take into account standingspin waves in the fitting procedure. The dispersion relationfor magnetostatic spin waves (i.e., spin waves with a largewavelength, virtually unaffected by the exchange energy) isgiven by [ 9,21,22] f(k)=γ 2π/radicalbig μ0H0×/radicalbig μ0H0+μ0MS·FD(k) (2)Here, FD(k) is a correction factor which arises from the dipole-dipole coupling between the spins, and kis the wave vector of the spin wave. Combining Eqs. ( 1) and ( 2), we obtain f(k)=γ 2π/radicalbig μ0H0+μ0Ha ×/radicalbig μ0H0+μ0Ha+μ0MS(Nz−Nx)·FD(k).(4) We fitted the Kittel equation for the center mode [Eq. ( 1)] to the relevant branch in Fig. 3(b) to obtain values for (Nz−Nx) and Ha. For the square lattice, this fitting resulted in anisotropy fields of μ0Ha=5 mT and μ0Ha=−2m Tf o rt h e field aligned along the [11] and [10] directions, respectively.Fitted values for the hexagonal lattice were μ 0Ha=1m T andμ0Ha=2 mT for the field aligned along the [1 ¯2] or [ ¯10] directions, respectively. This leaves FD(k) as the free fitting parameter in Eq. ( 4) for the edge mode. The best fit was found for FD(k)=0.9. Taking FD(k) to have the form valid for a blanket film, i.e., FD(k)=(1−e−kd)/kd, where dis the film thickness; the fitted wave vector kcorresponds to a spin- wave wavelength of λdip≈350 nm, which is approximately the disk diameter. This result is in close agreement with themicromagnetic simulations. The center-mode branches in Figs. 4(a) and4(b) have a difference in anisotropy field μ 0Haof 1 mT, which is within the error margin of ±1 mT. This finding suggests that the hexagonal lattice shows little to no anisotropy in the FMRresponse, which corresponds well to theoretical predictionsthat a hexagonal lattice of dipolar-coupled disks has a con-tinuous degeneracy with respect to the in-plane magnetizationdirection [ 23]. For the hexagonal lattice, the only distinct difference between the [1 ¯2] and the [ ¯10] orientations of H 0is 064418-4EFFECTS OF LATTICE GEOMETRY ON THE DYNAMIC … PHYSICAL REVIEW B 99, 064418 (2019) 2468 2468HMHWH]Tm[ 02468 1 0 1 2 Frequency [GHz]2468α=0.006 center mode α=0.006 center modeα=0.025 edge mode α=0.009 edge modeα=0.044 center mode H0 H0H0 H0 (a) (b) (c) FIG. 5. Absorption resonance linewidths, measured values, and fitted linear frequency dispersions (dashed lines) for the edge modes (open triangles) and center modes (open squares), respectively. Open circles in (b) and (c) denote mixed modes. (a)–(c) Data for differentlattice symmetries and bias-field orientations, as indicated in the insets. found in the amplitude maps for applied fields H0<60 mT. Figures 4(a) and 4(b) show that for μ0H0=20 mT, the oscillation mode features two amplitude maxima for H0in the [1¯2] orientation and one amplitude maximum only for H0in the [ ¯10] orientation. We have also investigated the damping of the oscillation modes for the different array symmetries and applied fieldorientations, shown in Fig. 5. The linewidth of the peaks in the FMR data is related to the damping αby μ 0/Delta1HHWHM (f)=μ0/Delta1H0 HWHM +2π γαf. (5) The quantity /Delta1H0 HWHM is the linewidth at zero frequency and is related to inhomogeneous broadening [ 8]. It has been previ- ously concluded that edge modes exhibit increased linewidth/Delta1H 0, as these modes are sensitive to edge imperfections from the lithography and lift-off processes [ 8,14]. The damping αis proportional to the slope of the linewidth plotted as a function of frequency and can be found fromEq. ( 5)a s α=γμ 0 2π∂ ∂f/Delta1HHWHM. (6)123 6789 Frequency [GHz]123HMHWH]Tm[ center modeedge modecenter modeedge modeα=0.01 α=0.01 α=0.01α=0.013H0 H0 FIG. 6. Simulated absorption resonance linewidths for the edge modes (triangles) and center modes (squares). Upper panel: square array with the bias field oriented in the [11] direction, lower panel: hexagonal array with the bias field oriented in the [1 ¯2] direction. We measure a linewidth frequency dependence with a splitting into two branches at some frequency (field) for alldipolar-coupled configurations except the square array withthe applied field oriented in the [10] direction, cf. Fig. 5.I n Fig. 5(a), we note that the resonance linewidth for the single disk increases monotonically with frequency in a mannersimilar to that observed for the square array with H 0oriented in the [10] direction. Fitting of Eq. ( 6) to the frequency disper- sion of the measured linewidths results in a damping constantofα=0.006±0.001, slightly less than that obtained for a blanket thin film of Py [ 8]. As center modes typically display low damping, this low value of αcorroborates the result from simulations that we only excite center modes in the geometriesshown in Fig. 5(a). When H 0is oriented in the [11] direction of the square lattice, we observe a significantly different damping behavior[cf. Fig. 5(b)]. The linewidth frequency dispersion then splits into two branches at a frequency of ∼6 GHz. The lower branch has a modest slope, corresponding to α=0.006± 0.002, whereas the upper branch has a steeper slope, cor- responding to a damping constant α=0.025±0.002. We attribute this difference in damping to a different resonancemode, with the upper branch corresponding to an edge modeand the lower branch to a center mode [ 8]. The resonance linewidth in the low-frequency regime ( f<6 GHz) shown by open circles in Fig. 5(b) pertains to a mixed (edge and center) oscillation mode, as seen in simulations [cf. Fig. 3(b)]. With the hexagonal lattice [Fig. 5(c)], the difference in linewidth for the two directions of H 0lies within the confines of the measurement uncertainty. The linewidth frequencydispersion splits into two branches at ∼8 GHz, i.e., at a higher frequency than that observed for the square latticewith H 0oriented in the [11] direction. The low-frequency part (f<8 GHz) of the linewidth dispersion (open circles) again 064418-5SAM D. SLÖETJES et al. PHYSICAL REVIEW B 99, 064418 (2019) pertains to a mixed oscillation mode (cf. Fig. 4). For f> 8 GHz, the center mode has a damping constant α=0.009± 0.003, which is higher than that of the isolated disk. The damping constant of the edge mode in the hexagonal latticeisα=0.044±0.004, almost twice the value obtained for the square lattice. We note from scanning electron microscopy(SEM) images of the nanomagnet arrays [cf. Figs. 1(a) and 1(b)] that there is negligible difference in edge roughness between the hexagonal and the square lattice, which indicatesthat the difference in damping between the two arrays can beattributed to the intermagnet coupling. Micromagnetic simu-lations show (cf. Fig. 6) that the damping of the edge mode exceeds that of the center mode for the hexagonal lattice,while the two modes are damped equally for the square latticewith the field oriented in the [11] direction. This difference indamping indicates that the edge mode is more sensitive to thelattice symmetry, which is plausible given that the magneticmoments at the edges of neighboring nanodisks are moreclosely separated ( ∼100 nm) than their centers ( ∼400 nm). In all cases, the measured damping of the edge mode isnoticeably higher than in simulations. This is most likely dueto edge roughness on the fabricated nanodisks. Numericalinvestigations on lattices with dipolar-coupled nanosphereshave been carried out previously by Mitsumata and Tomita[24], who also found that the damping is modified by dipole interactions between nanomagnets. IV . CONCLUSIONS Dipolar coupling between magnetic nanodisks in a lat- tice is found to have a profound impact on the dynamicresponse. This is seen from a distinct difference in the FMRspectra compared to a reference sample with uncoupled disks.Moreover, the lattice symmetry and the direction of the FMRbias field are found to promote different magnetic oscillationmodes in the individual disks. The field dependence of theresonance for a square lattice with the bias field imposed in the [10] direction shows a single dominant Kittel curveacross the full frequency range. A square lattice with thefield imposed in the [11] direction, however, shows a Kittelcurve splitting into edge- and center-mode branches for higherfrequencies. Such a splitting is also observed for a hexagonallattice, irrespective of the bias-field direction. Moreover, weobserve no anisotropy in dynamic response for this latticegeometry. The lattice symmetry and the orientation of thebias field are also found to impact the effective magneticdamping, α, with highest damping for the edge mode in the hexagonal lattice. The experimental results are corroboratedby micromagnetic simulations. The present findings will beof importance to the design and understanding of magnoniccrystals. ACKNOWLEDGMENTS We would like to thank Agne Ciuciulkaite and Jonathan Leliaert for stimulating discussions. S.D.S. acknowledgessupport from the ALS Doctoral Fellowship in Residence.The Advanced Light Source is supported by the Director,Office of Science, Office of Basic Energy Sciences, of theUS Department of Energy under Contract No. DE-AC02-05CH11231. Q.L., M.Y ., and Z.Q.Q. acknowledge supportfrom the US Department of Energy, Office of Science, Of-fice of Basic Energy Sciences, Materials Sciences and Engi-neering Division under Contract No. DEAC02-05-CH11231(van der Waals heterostructures program, KCWF16). Partialfunding was obtained from the Norwegian PhD Network onNanotechnology for Microsystems, which is sponsored bythe Research Council of Norway, Division for Science, underContract No. 221860/F60. The Research Council of Norwayis acknowledged for the support to the Norwegian Micro- andNano-Fabrication Facility, NorFab, Project No. 245963/F50. [1] A. Chumak, A. Serga, and B. Hillebrands, J. Phys. D: Appl. Phys. 50,244001 (2017 ). [2] M. Krawczyk and D. Grundler, J. Phys.: Condens. Matter 26, 123202 (2014 ). [3] V . V . Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43,264001 (2010 ). [4] S. Neusser and D. Grundler, Adv. Mater. 21,2927 (2009 ). [5] J. W. Kłos, M. Sokolovskyy, S. Mamica, and M. Krawczyk, J. Appl. Phys. 111,123910 (2012 ). [6] E. Wahlström, F. Macià, J. E. Boschker, Å. Monsen, P. N o r d b l a d ,R .M a t h i e u ,A .D .K e n t ,a n dT .T y b e l l , New J. Phys. 19,063002 (2017 ). [7] S. Tacchi, M. Madami, G. Gubbiotti, G. Carlotti, H. Tanigawa, T. Ono, and M. P. Kostylev, Phys. Rev. B 82,024401 (2010 ). [8] J. M. Shaw, T. J. Silva, M. L. Schneider, and R. D. McMichael, P h y s .R e v .B 79,184404 (2009 ). [9] Y . Huo, C. Zhou, L. Sun, S. T. Chui, and Y . Z. Wu, Phys. Rev. B 94,184421 (2016 ).[10] H. T. Nembach, J. M. Shaw, T. J. Silva, W. L. Johnson, S. A. Kim, R. D. McMichael, and P. Kabos, Phys. Rev. B 83,094427 (2011 ). [11] F. Guo, L. M. Belova, and R. D. McMichael, P h y s .R e v .L e t t . 110,017601 (2013 ). [12] S. Tacchi, G. Gubbiotti, M. Madami, and G. Carlotti, J. Phys.: Condens. Matter 29,073001 (2016 ). [13] M. Kostylev, R. Magaraggia, F. Y . Ogrin, E. Sirotkin, V . F. M e s c h e r y a k o v ,N .R o s s ,a n dR .L .S t a m p s , IEEE Trans. Magn. 44,2741 (2008 ). [14] N. Ross, M. Kostylev, and R. L. Stamps, J. Appl. Phys. 109, 013906 (2011 ). [15] G. Woltersdorf, Ph.D. thesis, Simon Fraser University, 2004.[16] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia- Sanchez, and B. Van Waeyenberge, AIP Adv. 4,107133 (2014 ). [17] V . Flovik, F. Macià, J. M. Hernàndez, R. Bru ˇcas, M. Hanson, and E. Wahlström, P h y s .R e v .B 92,104406 (2015 ). [18] R. D. McMichael and M. D. Stiles, J. Appl. Phys. 97,10J901 (2005 ). 064418-6EFFECTS OF LATTICE GEOMETRY ON THE DYNAMIC … PHYSICAL REVIEW B 99, 064418 (2019) [19] E. Montoya, T. McKinnon, A. Zamani, E. Girt, and B. Heinrich, J. Magn. Magn. Mater. 356,12(2014 ). [20] C. Kittel, Phys. Rev. 73,155(1948 ). [21] B. Kalinikos and A. Slavin, J. Phys. C: Solid State Phys. 19, 7013 (1986 ).[22] M. Grimsditch, G. K. Leaf, H. G. Kaper, D. A. Karpeev, and R. E. Camley, Phys. Rev. B 69,174428 (2004 ). [23] P. Politi and M. G. Pini, P h y s .R e v .B 66,214414 (2002 ). [24] C. Mitsumata and S. Tomita, Phys. Rev. B 84,174421 (2011 ). 064418-7

PhysRevLett.100.017207.pdf

Generation Linewidth of an Auto-Oscillator with a Nonlinear Frequency Shift: Spin-Torque Nano-Oscillator Joo-V on Kim * Institut d’Electronique Fondamentale, UMR CNRS 8622, Universite ´Paris-Sud, 91405 Orsay cedex, France Vasil Tiberkevich and Andrei N. Slavin Department of Physics, Oakland University, Rochester, Michigan 48309, USA (Received 8 March 2007; published 10 January 2008) It is shown that the generation linewidth of an auto-oscillator with a nonlinear frequency shift (i.e., an auto-oscillator in which frequency depends on the oscillation amplitude) is substantially larger than the linewidth of a conventional quasilinear auto-oscillator due to the renormalization of the phase noisecaused by the nonlinearity of the oscillation frequency. The developed theory, when applied to a spin- torque auto-oscillator, gives a good description of experimentally measured angular and temperature dependences of the linewidth. DOI: 10.1103/PhysRevLett.100.017207 PACS numbers: 85.75. /.0255d, 05.10.Gg, 05.40. /.0255a, 75.30.Ds It is well known that the linewidth /.02550of a passive oscillating circuit is determined by the ratio of its dissipa-tive element (e.g., resistance R) to its reactive element (e.g., inductance L):/.0255 0/.0136R=2L. When the oscillating circuit is connected to an active element (transistor, vac-uum tube, tunnel diode, etc.) and a source of a constant voltage (e.g., battery) the autogeneration of constant- amplitude oscillations at the resonance frequency of the oscillating circuit ( !/.01361=/.0129/.0129/.0129/.0129/.0129/.0129/.0129 LCp , whereCis the circuit capacitance) can take place [ 1,2]. The equilibrium ampli- tude of these auto-oscillations is determined by the dy- namic balance between the positive nonlinear damping of the oscillating system and negative nonlinear dampingintroduced into the system by the active element [ 1,2]. It is also well established that the generation linewidth /.0001!in a typical auto-oscillator is determined, for the most part, by thermal phase noise [see, e.g., Eq. (9.36) in [ 1]] and can be expressed in the following general form, /.0001!/.0136/.02550kBT E/.0133P/.0134; (1) wherekBis the Boltzmann constant, Tis the absolute temperature, E/.0133a/.0134/.0136/.0012P/.0136/.0012jaj2is the averaged energy of the auto-oscillation having the power P/.0136jaj2and complex amplitude a, and/.0012is the coefficient relating the averaged energy to the auto-oscillation power P.F o r example, in an auto-oscillator with a standard linear oscil- lating circuit, /.0012/.0136C=2, whereCis the capacitance of the oscillating circuit and ais the amplitude of the voltage on this capacitance. Equation ( 1) is rather general and is equally applicable to any type of conventional auto-oscillator (transistor, vacuum tube, tunnel diode, laser,etc.) in which the oscillation frequency is not stronglydependent on the amplitude, i.e., in the limit d!=dP!0. There exist, however, auto-oscillators for which the oscillation frequency exhibits a strong nonlinearity N/.0017d!=dP that is too large to be neglected. In such systems, one expects that even small fluctuations in the amplitude(or power) at steady state can give important contributionsto the phase noise. A pertinent example of present interest is the magnetic spin-torque nano-oscillator (STNO) [ 3–7], which consists of a nanosized metallic contact attached to amagnetic multilayer or a multilayered magnetic nanopillar.Direct electrical current passing through the multilayer canlead to a transfer of spin-angular momentum betweenmagnetic layers in the stack [ 3,4], which in turn creates an effective negative damping for the magnetization of thethinner (‘‘free’’) magnetic layer. This negative damping,analogous to the role played by an active element, can leadto self-sustained oscillations of magnetization in the free layer. The frequency of these auto-oscillations is deter- mined by the applied magnetic field, static magnetization,etc., and is, in general, close to the ferromagnetic reso-nance frequency, while the oscillation amplitude is deter-mined by the intrinsic nonlinearities of the system. In contrast to traditional (e.g., transistor) auto- oscillators, the frequency of the STNO strongly dependson the power of the magnetization precession P:!/.0133P/.0134/.0136 ! 0/.0135NP. The sign and magnitude of the nonlinear fre- quency shift coefficient Ndepend on the direction and magnitude of the bias magnetic field (see [ 6–9] for details) and can be varied over a range comparable to the oscilla- tion frequency itself. Thus, the classical result ( 1) cannot describe quantitatively the generation linewidth in STNO,and a new theory that explicitly takes into account thenonlinear frequency shift of the auto-oscillator isnecessary. In this Letter, we develop a theory to describe the generation linewidth in an auto-oscillator with nonlinearfrequency shift and show that this nonlinearity leads to asignificant linewidth broadening. The theory is then ap-plied to the STNO, and we demonstrate that the correctPRL 100, 017207 (2008)PHYSICAL REVIEW LETTERSweek ending 11 JANUARY 2008 0031-9007 =08=100(1)=017207(4) 017207-1 ©2008 The American Physical Societytreatment of such nonlinearities is essential for even the qualitative description of the nonlinear auto-oscillator. The general equation describing the time evolution of the oscillation amplitude ain a nonlinear auto-oscillator in the presence of noise can be written in the form @a @t/.0135i!/.0133P/.0134a/.0135/.0255/.0135/.0133P/.0134a/.0255/.0255/.0255/.0133P/.0134a/.0136fn/.0133t/.0134;(2) where!/.0133P/.0134is the nonlinearly shifted frequency of the excited oscillation mode, P/.0136jaj2,/.0255/.0135/.0133P/.0134is the natural positive damping of the oscillator, /.0255/.0255/.0133P/.0134is the effective negative damping introduced by an active element, andf n/.0133t/.0134is a random white Gaussian process that describes the influence of the thermal noise. The correlation function ofthis random noise can be written as hf n/.0133t/.0134f/.0003n/.0133t0/.0134i /.0136 2/.0255/.0135Pn/.0014/.0133t/.0255t0/.0134, where Pn/.0136kBT=/.0012 is the oscillator power at thermal equilibrium. The stationary solution of Eq. ( 2) in the absence of noise [fn/.0133t/.0134/.01360] can be easily obtained in the form a/.0133t/.0134/.0136/.0129/.0129/.0129/.0129/.0129/.0129 P0p e/.0255i!/.0133P0/.0134t/.0135i/.0030; (3) where the equilibrium oscillation power P0is determined by the condition /.0255/.0135/.0133P0/.0134/.0136/.0255/.0255/.0133P0/.0134and/.0030is a constant oscillation phase. Sufficiently far above the auto-oscillation threshold (i.e., forP0/.0029Pn) the solution of Eq. ( 2) with the noise term included will be similar to the noise-free solution Eq. ( 3)i n the sense that the oscillation amplitude will be close to themean value of/.0129/.0129/.0129/.0129/.0129/.0129P 0p, i.e.,ja/.0133t/.0134j /.0136/.0129/.0129/.0129/.0129/.0129/.0129P0p/.0135/.0014A/.0133t/.0134,j/.0014A/.0133t/.0134j2/.0028 P0, and the phase /.0030will be a slow function of time. Substituting the expression a/.0133t/.0134/.0136/.0137/.0129/.0129/.0129/.0129/.0129/.0129 P0p /.0135/.0014A/.0133t/.0134/.0138e/.0255i!/.0133P0/.0134t/.0135i/.0030/.0133t/.0134(4) fora/.0133t/.0134in Eq. ( 2), and retaining only the terms of the first order in /.0014A, we find equations for fluctuations of the amplitude @/.0014A @t/.01352/.0255effP0/.0014A/.0136Re/.0133~fn/.0133t/.0134e/.0255i/.0030/.0134 (5a) and phase @/.0030 @t/.01352N/.0129/.0129/.0129/.0129/.0129/.0129 P0p /.0014A/.01361/.0129/.0129/.0129/.0129/.0129/.0129P0pIm/.0133~fn/.0133t/.0134e/.0255i/.0030/.0134: (5b) Here/.0255effandNare the effective nonlinear damping and nonlinear frequency shift, respectively: /.0255eff/.0136d/.0255/.0135/.0133P/.0134 dP/.0255d/.0255/.0255/.0133P/.0134 dP;N/.0136d!/.0133P/.0134 dP;(6) where the derivatives are taken at P/.0136P0. In Eqs. ( 5a) and (5b)~fn/.0133t/.0134/.0136fn/.0133t/.0134ei!/.0133P0/.0134t. Note that the statistical proper- ties offn/.0133t/.0134and~fn/.0133t/.0134are identical. Therefore, the tilde will be omitted in the following text for simplicity. There is a significant qualitative difference between the behavior of the amplitude and the phase. Since the oscil-lation amplitude at steady state remains practically con-stant,jaj/.0025/.0129/.0129/.0129/.0129/.0129/.0129P 0p, the correlation function for the amplitude fluctuations KA/.0133/.0028/.0134/.0017h ja/.0133t/.0134jja/.0133t/.0135/.0028/.0134jiremains finite even if/.0028!1 , i.e.,KA!P0. Therefore, for large /.0028the behav- ior of the full correlation function K/.0133/.0028/.0134/.0017ha/.0133t/.0134a/.0003/.0133t/.0135/.0028/.0134i will be determined solely by the phase fluctuations, K/.0133/.0028/.0134/.0025P0hei/.0137/.0030/.0133t/.0134/.0255/.0030/.0133t/.0135/.0028/.0134/.0138iei!/.0133P0/.0134/.0028: (7) For the frequency linewidth of the auto-oscillation, we are interested only in the fluctuations taking place inside anarrow frequency region /.0001!/.0028/.0255 effP0, in which j@/.0014A=@tj/.0024/.0001!j/.0014Aj/.00282/.0255effP0j/.0014Aj. As such, the first (derivative) term in the left-hand side of Eq. ( 5a) can be neglected compared to the second term, and an explicitexpression for /.0014A/.0133t/.0134can be obtained, /.0014A/.01361 2/.0255effP0Re/.0133fne/.0255i/.0030/.0134: (8) Substituting this expression for /.0014A/.0133t/.0134in Eq. ( 5b) leads to a closed-form equation for the phase fluctuations /.0030/.0133t/.0134in the system, @/.0030 @t/.01361/.0129/.0129/.0129/.0129/.0129/.0129P0p/.0020 /.0255N /.0255effRe/.0133fne/.0255i/.0030/.0134/.0135Im/.0133fne/.0255i/.0030/.0134/.0021 ; /.01361/.0129/.0129/.0129/.0129/.0129/.0129P0p/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.0018N /.0255eff/.00192s Im/.0133fne/.0255i/.0011/.0255i/.0030/.0134;(9) where/.0011/.0136arctan/.0133N=/.0255eff/.0134. Equation ( 9) is formally identical to the equation for phase fluctuations in a system without a nonlinear fre- quency shift [see, e.g., second Eq. (9.8) in [ 1]], but with the increased noise level fn/.0133t/.0134!f0n/.0133t/.0134/.0136/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.0018N /.0255eff/.00192s e/.0255i/.0011fn/.0133t/.0134: (10) Application of the general methodology to compute auto- oscillator linewidths (see, e.g., Chap. 9 in [ 1]o r[ 10]) to Eq. ( 9) leads to the following expression for the Lorentzian linewidth of the auto-oscillator with a nonlinear frequencyshiftN, /.0001!/.0136/.02550/.0018kBT E0/.0019/.0020 1/.0135/.0018N /.0255eff/.00192/.0021 ; (11) where/.02550/.0136/.0255/.0135/.0133P0/.0134,E0/.0136hE/.0133a/.0134i /.0136/.0012P0is the average oscillator energy, and we have rewritten the ratio Pn=P0 askBT=E0. The comparison of the classical result ( 1) with the generalized Eq. ( 11) shows clearly that the nonlinear frequency shift in the auto-oscillator leads to a significantlinewidth broadening that is caused by effective renormal-ization of the phase noise ( 10) due to the frequency non- linearityN. The result in ( 11) is the principal result of this Letter and illustrates the fact that three key parameters determine the linewidth of an auto-oscillator with a nonlinear frequency shift. First, the relaxation rate of the oscillator /.0255 0deter-PRL 100, 017207 (2008)PHYSICAL REVIEW LETTERSweek ending 11 JANUARY 2008 017207-2mines the overall scale of the possible linewidth variations. Second, the generation linewidth is proportional to the ratio of the noise energy (which increases with tempera- ture) to the average energy of the auto-oscillation. Third,the ratio of the nonlinear frequency shift coefficient Nto the effective nonlinear damping /.0255 effgives a measure of the phase-noise renormalization due to amplitude fluctuations. It should be noted, that the importance of the nonlinear frequency shift for the STNO generation linewidth was explicitly pointed out in the pioneering paper [ 11], where experimental measurements and numerical calculations ofthe linewidth in a temperature interval were performed.However, the empirical linewidth expression [see Eq. (2) in [11]] and the numerical calculations performed in [ 11]g i v e the value of the linewidth that is about 1 order of magnitude larger than the experimentally measured one and a T 1=2 linewidth dependence on the temperature. We believe that both these results are caused by the approximation of a long correlation time ( /.0001!/.0029/.0255effP0) adopted in [ 11] that is not valid for typical STNO parameters. Another attempt to calculate the STNO linewidth was undertaken by one of the authors in [ 10], but the nonlinear frequency shift was neglected. The calculation resulted in an expression for the linewidth [see Eq. (28) in [ 10]] that can be cast in the classical form ( 1), where the constant /.0012is given by/.0012/.0136/.0133M0=/.0013/.0134!0Veff, where/.0013is the gyromagnetic ratio,!0is the oscillation frequency, and Veffis the effec- tive volume of the magnetic material of the free layerinvolved in the auto-oscillation [see Eq. (4) in [ 12]]. When compared to experiments, however, the result [ 10] underestimates the generation linewidth by 20–40 times. Now, it will be interesting to apply our new general result ( 11), where the frequency nonlinearity has been taken into account, to calculate the linewidth of a STNO. It has been shown previously [ 9,12,13] that the nonlinear oscillator equation ( 2) for the case of STNO can be derived from the Landau-Lifshitz-Gilbert equation with the Slonczewski term [ 3] describing the spin transfer torque. For the case of STNO the dimensionless complex ampli-tudeacan be defined as jaj 2/.0136/.0133M0/.0255Mz/.0134=2M0, where M0is the length of the magnetization vector in the free magnetic layer, and Mzis the projection of this vector on the equilibrium magnetization direction z(see [ 9] for de- tails), the negative damping caused by spin torque is givenby/.0255 /.0255/.0133P/.0134/.0136/.0027I/.01331/.0255P/.0134, whereIis the bias current and /.0027is the spin-polarization efficiency defined in Eq. (2) of [ 12], and the positive damping equals to /.0255/.0135/.0133P/.0134/.0136/.0255/.01331/.0135QP/.0134, where/.0255characterizes the oscillator equilibrium linewidth in the passive regime [see Eq. (31) in [ 9]] andQ>0is a phenomenological coefficient characterizing the nonline- arity of the positive damping (see [ 14] for details). The dependences of the STNO generation linewidth on the angle /.0018e, that the external bias magnetic field Hemakes with the plane of the STNO free layer, calculated using Eq. ( 11) forQ/.01363and typical parameters of STNO [ 6] areshown in Fig. 1. An important result that follows from Eq. ( 11) and Fig. 1is the prediction of a linewidth mini- mum that follows from a change in sign in the frequency shift [e.g., from ‘‘red’’ ( N<0) to ‘‘blue" ( N>0)] as the magnetization is tilted out of the film plane. Across thistransition the nonlinear frequency shift coefficient N passes through zero (see, e.g., Fig. 8 in [ 9]) at which one recovers the smallest value of the generation linewidth. In Fig. 2we directly compare the generation linewidth calculated using Eq. ( 11) with the results of experimental measurements of the temperature dependence of the STNO linewidth /.0001!/.0133T/.0134performed on the nanopillar devices no. 1 [Fig. 2(a)] and no. 2 [Fig. 2(b)] in Ref. [ 11] (see Fig. 2 in [11]), and with the angular dependence of the STNO linewidth /.0001!/.0133/.0018 e/.0134[Fig. 2(c)] experimentally measured on the nanocontact device in [ 15] (see Fig. 6 in [ 15]). Geometrical parameters of the nanopillar device [seeFigs. 2(a) and2(b)] were taken from Ref. [ 11] and it was assumed that the excited magnetization oscillation is pinned at the pillar lateral boundaries (see [ 16] for details); the Gilbert damping parameter /.0011 G/.01360:01, the nonlinear- ity parameter of positive damping Q/.01363, and the polar- ization efficiency "/.01360:4[see Eq. (2) of [ 12]] were assumed to be the same for both devices. Similarly, all 0.1110100Generation linewidth∆ω/2π (MHz) 0 3 06 09 0 Field angle θe (degree)11010010000 3 06 09 0 θe (degree)0500 Γ0 (MHz) (a) (b)σI = 1.0 GHz1.2 Tµ0He = 0.6 T 0.9 T 1.2 T 0.5 GHz FIG. 1. Generation linewidth as a function of applied field angle/.0018efor (a) three applied fields at constant /.0027I/.01361 GHz and (b) two bias currents at constant /.00220He/.01361:2T. Inset of (a): Equilibrium linewidth /.0255as a function of /.0018efor/.00220He/.0136 1:2T.PRL 100, 017207 (2008)PHYSICAL REVIEW LETTERSweek ending 11 JANUARY 2008 017207-3the parameters of the nanocontact device Fig. 2(c) were taken from [ 15]; current I/.01369m A and magnetic field /.00220He/.01360:9Tcorrespond to the center of the experimen- tally studied region, and the nonlinearity parameter ofpositive damping was again chosen to be equal to Q/.01363. As it is clear from Figs. 2(a) and 2(b), our simple analytical expression ( 11) gives a reasonably good estimate of the observed linewidths at different temperatures forboth nanopillar devices with the same parameters. Assuming that the parameters of the two devices areslightly different (which is possible due to different nano-patterning and different thicknesses of the ‘‘free’’ magneticlayer), one can obtain much better quantitative agreementwith the experiment [ 11]. It is also clear from Fig. 2(c) that the linewidth depen- dence on the bias field orientation calculated using ourresult Eq. ( 11) is in good quantitative agreement with the experimental results from Ref. [ 15]. In contrast to theclassical result ( 1), which predicts much narrower lines and a monotonic decrease in the linewidth as a function of/.0018 e, the renormalized phase-noise result ( 11) gives a rea- sonable qualitative and quantitative description of the ex- perimentally observed behavior, in particular, the linewidth minimum around /.0018e/.002580/.0014. In summary, we have developed a theory of the genera- tion linewidth of an auto-oscillator with a nonlinear fre-quency shift which generalizes the classical result ( 1). The additional nonlinearity in the oscillator frequency leads to a renormalization of the phase noise far above threshold. Applied to the particular case of a spin-torque nano-oscillator, the theory accounts for a number of character-istic, but previously unexplained, features observed in experiment: (i) general linewidth narrowing with increases in the bias current and oscillation amplitude (see Fig. 4 inRef. [ 15]), (ii) presence of a linewidth minimum as a function of the external magnetic field orientation (see Fig. 6 in Ref. [ 15]), and (iii) linear dependence of the linewidth on temperature. This work was in part supported by MURI Grant No. W911NF-04-1-0247 from the U.S. Department ofDefense, by Contract No. W56HZV-07-P-L612 from theU.S. Army TARDEC, RDECOM, by Grant No. ECCS- 0653901 from the U.S. National Science Foundation, and by the Oakland University Foundation. J. K. acknowledgessupport from the European Communities program ISTunder Contract No. IST-016939 TUNAMOS. *joo-von.kim@ief.u-psud.fr [1] A. Blaquiere, Nonlinear System Analysis (Academic, New York, 1966). [2] M. I. Rabinovich and D. I. Trubetskov, Oscillations and Waves in Linear and Nonlinear Systems (Kluwer, Dordrecht, 1989). [3] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [4] L. Berger, Phys. Rev. B 54, 9353 (1996). [5] M. Tsoi et al. , Phys. Rev. Lett. 80, 4281 (1998). [6] S. I. Kiselev et al. , Nature (London) 425, 380 (2003). [7] W. H. Rippard et al. , Phys. Rev. Lett. 92, 027201 (2004). [8] W. H. Rippard et al. , Phys. Rev. B 70, 100406(R) (2004). [9] A. N. Slavin and P. Kabos, IEEE Trans. Magn. 41, 1264 (2005). [10] J.-V . Kim, Phys. Rev. B 73, 174412 (2006). [11] J. C. Sankey et al. , Phys. Rev. B 72, 224427 (2005). [12] A. N. Slavin and V . S. Tiberkevich, Phys. Rev. B 74, 104401 (2006). [13] S. M. Rezende, F. M. de Aguiar, and A. Azevedo, Phys. Rev. Lett. 94, 037202 (2005). [14] V . S. Tiberkevich and A. N. Slavin, Phys. Rev. B 75, 014440 (2007). [15] W. H. Rippard et al. , Phys. Rev. B 74, 224409 (2006). [16] K. Yu. Guslienko and A. N. Slavin, Phys. Rev. B 72, 014463 (2005).FIG. 2 (color online). Generation linewidth of a spin-torqueauto-oscillator calculated from Eq. ( 11) (solid line) in compari- son with the temperature dependence of linewidth in a nanopillardevice no. 1 (a) and device no. 2 (b) measured in [ 11] (black dots) at/.0018 e/.01360, and (c) the angular dependence of the nano- contact STNO linewidth measured at a room temperature, taken from Fig. 6(a) in Ref. [ 15] (black dots). (a) The linewidth measured at the second harmonic of the signal /.0001!2/.01364/.0001!. The dashed line in (c) represents the multiplied by ten classical result for the oscillator linewidth calculated from Eq. ( 1).PRL 100, 017207 (2008)PHYSICAL REVIEW LETTERSweek ending 11 JANUARY 2008 017207-4

PhysRevB.80.104434.pdf

Emerging nonequilibrium bound state in spin-current–local-spin scattering Fatih Do ğan,1Lucian Covaci,1,2Wonkee Kim,1,3and Frank Marsiglio1 1Department of Physics, University of Alberta, Edmonton, Canada T6G 2J1 2Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada V6T 1Z1 3Department of Physics, University of Houston, Houston, Texas 77004, USA /H20849Received 29 May 2009; revised manuscript received 12 August 2009; published 25 September 2009 /H20850 Magnetization reversal is a well-studied problem with obvious applicability in computer hard drives. One can accomplish a magnetization reversal in at least one of two ways: application of a magnetic field or througha spin current. The latter is more amenable to a fully quantum-mechanical analysis. We formulate and solve theproblem whereby a spin current interacts with a ferromagnetic Heisenberg spin chain, to eventually reverse themagnetization of the chain. Spin flips are accomplished through both elastic and inelastic scattering. A conse-quence of the inelastic-scattering channel, when it is no longer energetically possible, is the occurrence of anonequilibrium bound state, which is an emergent property of the coupled local plus itinerant spin system. Forcertain definite parameter values the itinerant spin lingers near the local spins for some time, before eventuallyleaking out as an outwardly diffusing state. This phenomenon results in spin-flip dynamics and filteringproperties for this type of system. DOI: 10.1103/PhysRevB.80.104434 PACS number /H20849s/H20850: 03.65.Ud, 72.25.Ba, 75.10.Jm I. INTRODUCTION Most current computer hard drives utilize a technology for memory storage which requires a switching of states in-volving magnetized spin. This switching is accomplishedthrough the application of magnetic fields in appropriate di-rections. A theoretical understanding of this process is at-tained reasonably well through a classical description via theLandau-Lifshitz-Gilbert equations. 1,2These equations consti- tute a phenomenological description, since the requireddamping, whose analytical form is even under some debate, 3 has various possible origins. Just over a decade ago, however, theoretical proposals were made to accomplish magnetization switching throughspin transfer from applied spin currents to magnetized spins. 4,5A semiclassical description was used: the spin cur- rent was described by a plane wave while the magnetizedthin film that was to be flipped was described through aclassical magnetization vector. This problem became knownas the “spin-torque” problem; the incoming spin current ex-erts a torque on the local magnetization. It is noteworthy thatin this problem a phenomenological damping mechanism isnot required to torque the magnetization in the direction ofthe incoming spin current—whereas the use of a magneticfield leads only to precession unless some damping mecha-nism is introduced. The experimental observation of thespin-torque effect has met with some limited success. 6–9 Recently, a direct measurement of the spin-torque vector depending on the voltage has been made.10Furthermore, the results of this experiment imply that inelastic tunneling has an important effect on the spin-transfer torque. In fact, itappears that inelastic processes in the spin-flip scattering areinherent 11for ferromagnetic systems. In order to realize prac- tical applications of the spin-torque phenomenon, it is impor-tant to reduce the critical current required to reverse the mag-netization of ferromagnets. A couple of experiments 12,13have demonstrated experimental methodologies to decrease thecritical current. As another signature of spin transfer, spin-torque-induced magnetic vortex phenomena are also observed. 14–16 The semiclassical picture seems to work well in a practi- cal sense.17–21However, especially from a theoretical point of view, some aspects are missing. Ultimately, spin transferis a quantum-mechanical scattering problem, generally in-elastic, and so one would like to understand the spin-transferprocess in terms of excitations of the ferromagnet. Moreover,recent experimental work 22has focused on the impact of a spin current on cobalt nanoparticles with diameter less than 5nm, which can be used to examine the spin torque exerted onisolated nanoparticles. It has also been shown that it is ex-perimentally feasible to manufacture magnetic nanostruc-tures /H20849chains of 2–10 coupled atoms /H20850. 23In this case, only a fully quantum-mechanical description will suffice becausethe quantum nature of the spin operator representing the sta-tionary spins in the nanoparticle is significant. The scenario of an incoming /H20849electron /H20850spin, often mod- eled as a wave packet, whose spin degree of freedom iscoupled with local spins, has been advanced by a number ofworkers. 24–29The coupling between the incoming spin and the local spins is Kondo type while the local spins are them-selves ferromagnetically coupled via a Heisenberg exchangeinteraction. The model Hamiltonian is H=−t 0/H20858 /H20855i,j/H20856/H9268ci/H9268†cj/H9268−2/H20858 /H5129=1Ns J0/H9268/H5129·S/H5129−2/H20858 /H5129=1Ns−1 J1S/H5129·S/H5129+1, /H208491/H20850 where ci/H9268†creates an electron with spin /H9268at site i,S/H5129is a localized spin operator at site /H5129, and t0is the hopping ampli- tude between nearest-neighbor sites. The first term allows anelectron /H20849of either spin /H20850to propagate in a band that covers all space /H20849here in one dimension /H20850while the second term is re- sponsible for the Kondo-type interaction between the elec-tron and the local spins, with coupling constant J 0. This takes place over a finite chain of length Ns. Finally, the last term models the Heisenberg exchange interaction with strength J1PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 1098-0121/2009/80 /H2084910/H20850/104434 /H2084912/H20850 ©2009 The American Physical Society 104434-1between the local spins. For a ferromagnetic chain, J1/H110220. Note, moreover, that if so desired, both J0andJ1can depend on the position of the local spin within the finite chain. Fig-ure1shows a schematic of this model. The use of a wave packet to describe the incoming spin degree of freedom, and the subsequent “real-time” analysisof the scattering process allows us to examine the entire scat-tering process with very fine spatial and temporal resolution.While the present-day experimental capabilities do not quitematch this fine resolution, we anticipate that probing on thetime and length scales we use will be accessible in the nearfuture. In particular, in this work we identify a feature whichwe call a “nonequilibrium bound state” /H20849NEBS /H20850, whose char- acteristics would require careful experimental detection. Thisphenomenon results because of an inelastic-scattering pro-cess that is suppressed due to energy conservation. While ananalytical approach does reveal some of the properties of aNEBS, the numerical wave-packet calculations really allowus to see the nonequilibrium aspect of this phenomenon.Both calculations are presented here. This paper is organized as follows. In the following sec- tion we outline means by which we solve the time-dependentproblem. Some of our earlier work 25,27,30used straightfor- ward expansions in the basis states spanning the product Hil-bert space of electrons moving on a lattice and stationaryspins confined to a small portion of that same lattice. Thepresent work uses a different method; the exponentiatedHamiltonian operator is expanded in a series utilizingChebyshev polynomials. 31This allows us to easily generate large scale numerical results, as described in Sec. III.W e formulate the problem for an arbitrary number of stationaryspins /H20849in principle, representing a magnetized thin film, whose magnetization is being flipped /H20850but focus on two in- teracting stationary spins. This allows us to focus on thecharacteristic features of the larger system, including theNEBS, without the considerable complexity generated by themany scattering channels present when more than two sta-tionary spins are used. Snapshots of the propagating wavepacket reveal that in a particular region of parameter spacepart of the wave-packet “lingers” near the stationary spins.This feature is a signature of the NEBS. In Sec. IVwe develop an analytical approximation to de- scribe the same scattering process in the continuum limit. Apreliminary decomposition of the problem, into less familiarbut more useful basis states, allows us to readdress the nu- merical results of Sec. III. This analysis identifies the NEBS with the position-dependent amplitude of one of these basisstates. We further develop the analytical approximation toderive this amplitude, along with expectation values for theamount of spin flip expected. Thus, while we lose the trans-parency of the time-dependent /H20849i.e., nonequilibrium /H20850aspect of the problem, we clarify some of the physics of the bound-state part. In Sec. Vwe conclude with some discussion con- cerning experimental observation of this NEBS. II. THEORY We adopt the most straightforward approach to the scat- tering problem and study the time evolution of a wavepacket, defined, at t=0, as /H9272/H20849x/H20850=1 /H208812/H9266a2eik/H20849x−x0/H20850e−/H20849x−x0/H208502/2a2. /H208492/H20850 The calculation can take several routes at this stage. Consis- tent with the tight-binding formulation, Eq. /H208491/H20850, one can de- fine a Hilbert space /H20849with either open or periodic boundary conditions left of the wave packet and far to the right of thelocal spins /H20850, with typically hundred’s of lattice sites on which the itinerant spin /H20849hereafter referred to as the electron or electron spin /H20850can hop /H20849see Fig. 1for a schematic /H20850. One can diagonalize Eq. /H208491/H20850on this Hilbert space and find the com- plete spectrum of eigenstates and eigenvalues with whichone can construct the time evolution of the wavepacket. 26,27,30However, we find that the parameter regime and maximum possible size of the local-spin chain, for ex-ample, is severely restricted by computational expensewithin this approach. Instead we choose to solve the time dependence directly, using the formal solution /H9023/H20849x,t/H20850=e −iHˆt/H9272/H20849x/H20850. /H208493/H20850 A practical implementation of this solution is through the series expansion e−iHˆt=/H20858 nanYˆn, /H208494/H20850 where anare the coefficients of a complete orthonormal set of functions denoted by Yn. A very useful basis is provided by the Chebyshev polynomials, Tn/H20849x/H20850/H11013cos/H20849ncos−1x/H20850, with T0/H20849X/H20850=1, T1/H20849X/H20850=X, and Tn/H20849X/H20850=2XTn−1/H20849X/H20850−Tn−2/H20849X/H20850.31For this expansion to be useful, the argument X/H20849here, a matrix /H20850 is required to have norm less than unity so a scaled versionof the Hamiltonian is required /H20849accompanied by a scaled time variable /H20850, e −iHˆt=e−i/H20849Hˆ//H9254/H20850/H9254t=/H20858 n=−/H11009/H11009 an/H20849/H9254t/H20850Tn/H20873−Hˆ /H9254/H20874=/H20858 n=−/H11009/H11009 an/H20849y/H20850Tn/H20849x/H20850, /H208495/H20850 where y=/H9254tandx=−/H20849Hˆ /H9254/H20850. There are two reasons for choosing this particular basis. First, the coefficients an/H20849y/H20850can be written simply as32t0 t0 t0 t0 t0 t0 t0 S1S2 J1J0 J0σ FIG. 1. /H20849Color online /H20850A schematic of a lattice, on which an itinerant spin can hop /H20849with hopping parameter t0/H20850; it can interact with two stationary spins /H20851indicated by downward pointing /H20849red/H20850 arrows /H20852with coupling strength J0. The two stationary spins can interact with one another, with coupling strength J1.DOĞANet al. PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-2an/H20849y/H20850=1 /H9266/H20885 −11dx /H208811−x2Tn/H20849x/H20850e/H20849ixy/H20850=i/H20841n/H20841J/H20841n/H20841/H20849y/H20850, /H208496/H20850 where the Jn/H20849y/H20850are Bessel functions of the first kind. Second, these polynomials have a recursion relation that allows us touse a more compact calculation of the expansion of the ex-ponential of the Hamiltonian, T n+m/H20849x/H20850=2Tn/H20849x/H20850Tm/H20849x/H20850−T/H20841n−m/H20841/H20849x/H20850. /H208497/H20850 Using this equation we can rewrite the expansion up to a given order, N2as33 ei/H20849Hˆ//H9254/H20850/H9254t/H11061/H20858 0N2 aiTi=/H20858 0N bi0Ti+TN/H20875/H20858 1N bi1Ti+ ... +TN/H20873/H20858 1N bikTi+ ...+ TN/H20858 1N biN−1Ti/H20874.../H20876 /H208498/H20850 with bik=/H20858 j=0N−k /H20853mod /H20849j,2/H20850/H11569A/H20849j+k,k/H20850a/H20851/H20849j+k+1/H20850/H11569N−i/H20852+ mod /H20849j +1 , 2 /H20850/H11569A/H20849j+k,k/H20850a/H20851/H20849j+k/H20850/H11569N+i/H20852/H20854/H20849 9/H20850 and the matrix elements A/H20849i,j/H20850are defined by A/H20849i,j/H20850=/H20902A/H20849i−1 ,j/H20850+2/H11569A/H20849i−1 ,j−1/H20850mod /H20849i−j,2/H20850=0 −A/H20849i−1 ,j/H20850 mod /H20849i−j,2/H20850=1 0 i/H11021j/H20903 /H2084910/H20850 with A/H208490,0 /H20850=1. This formulation allows for an efficient evaluation of the time evolution of the wave function, such that large latticescan be studied, both for the electron spin and for the station-ary spin chain. III. NUMERICAL RESULTS A. Noninteracting stationary spins The result of a typical calculation is illustrated in Fig. 2. Here, we have used 1600 lattice sites, and, at t=0 we have “launched” a wave packet centered around site 700 with awidth given by a=30. The unit of length is the lattice spac- ing, which we take to be unity for convenience. In all ourfigures we also take t 0/H110131 as our energy scale. All our results will utilize an initial electron wave vector k=/H9266/2 so that no wave-packet broadening occurs.30The incoming electron spin has S=1 /2, and, in the calculations in this paper, the stationary spins have S=1 /2. A series of snapshots is shown as time progresses forward. Initially only the incoming elec-tron with a spin-up component is present, represented as aGaussian wave-packet /H20851shown as a solid /H20849red/H20850curve for the first time slice at the bottom /H20852. The initial conditions are such that all stationary spins /H20849not shown but situated at sites 800 and 801 /H20850have S z=−1 /2 and the incoming electron spin has Sz=1 /2. As time advances the electron spin interacts with thestationary spins and scatters. If there was only one stationary spin, the scattering would lead to 4 possibilities for the elec-tron wave packet: 25it can either be reflected or transmitted, with either spin up or spin down. With two /H20849or more /H20850inter- acting stationary spins, inelastic scattering is also possible.The choice of parameters in Fig. 2is such that the result is similar to that expected from a single spin /H20849J 1=0 here /H20850; after interacting with the local spins the wave packet both reflectsand transmits with both spin components. The scattering iselastic which means the associated wave vectors are /H11006 /H9266/2 so that no spreading of the wave packet occurs as timeprogresses /H20849there is some intrinsic spread because two neigh- boring scattering sites are involved /H20850. The “final state” of both the electron and the local spins is readily defined by waiting for a period of time after whichthe various electron components have separated a reasonabledistance from the local spins. This is clear from the figure/H20849the latest times shown clearly fulfill the above requirement /H20850 but we will encounter special parameter regimes where thisdefinition is not so clear, to be discussed later. B. “N” interacting stationary spins At the outset we wanted to understand how a /H20849macro- scopically /H20850long spin chain interacts with an incoming elec- tron spin to understand the effect of a spin current on amagnetic layer. With the technology discussed in Sec. IIfor treating the time evolution of a coupled electron-spin/local-spin system, the study of reasonably long spin chains is in-deed possible. However, the impact on the spin chain is suf-ficiently complex that this program was deemed overlyambitious for the present, even if we simply examine theimpact on the electron spin as it emerges from the spin chain.0.0000.0050.0100.0150.0200.0250.0300.0350.0400.0450.050 0 200 400 600 800 1000 1200 1400 160 0ni siteindexiTimespin up spin down FIG. 2. /H20849Color online /H20850Time evolution of an electron wave packet, interacting with two local spins /H20849located at sites 800 and 801 /H20850. For the electron spin we use a tight-binding model with nearest-neighbor hopping only; for reasons discussed in the text weusek= /H9266/2. For this figure the coupling with local spins is given by J0=2.0 t0and the coupling between local spins is set to zero /H20849J1 =0/H20850. The choice J1=0 causes the time evolution of the electron spin to closely resemble the one with a single local spin previously re-ported in Ref. 25. Subsequent time slices are displaced vertically for clarity.EMERGING NONEQUILIBRIUM BOUND STATE IN SPIN- … PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-3Looking at “long times” after the interaction, the complexity in a series of figures such that in Fig. 2for various values of J0andJ1is enormous. The summary of such a plot is shown in Fig. 3, where the value of the zcomponent of the electron spin is shown after interaction with a spin chain consisting of20 coupled S=1 /2 spins. As a function of the interaction parameters J 0and J1there are quite a number of visible ripplelike structures which no doubt are related to the exci-tations that are populated through the inelastic-scatteringchannels. This interpretation is reinforced by the observationthat, for smaller spin chains, the number of ripples is re-duced, as the number of possible internal excitations is re-duced. Slices for fixed values of J 0are illustrated in Fig. 3/H20849b/H20850 and again it is difficult to interpret all the various ripples. Forthis reason we focus, in the rest of this paper, on the simplersystem where there are only two coupled local spins.C. Two interacting stationary spins: Inelastic scattering We first examine the long-time behavior of the electron spin. Figure 4illustrates /H20849in a color plot /H20850thezcomponent of the electron spin once it has essentially left the vicinity of thetwo local spins, as a function of the Kondo coupling betweenelectron spin and each local spin, J 0, and the coupling be- tween local spins, J1. Curves are shown for the same quantity in Fig. 4/H20849b/H20850, for specific values of J0, as shown; these corre- spond to horizontal sweeps across the first plot. In Fig. 5 vertical sweeps across the first plot in Fig. 4are shown, along with the result for a single local spin.25The sweeps are plotted for extreme values of J1and avoid the complicated region characterized by a “trough” /H20849colored dark /H20850of signifi- cant spin flip rising upward to the right, and leaving the plotarea at /H20849J 1,J0/H20850/H11015/H208494,10 /H20850t0. This trough region will be dis- cussed in detail in Sec. III D . -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5 0 2 4 6 8 10σze J1J0=2 J0=5 J0=8 (b) (a) FIG. 3. /H20849Color online /H20850/H20849a/H20850The zcomponent of the electron spin long after the electron wave packet has interacted with the local spins, as a function of both electron-spin coupling J0and spin-spin interaction J1for 20 local spins. The outcome is sufficiently complicated that we will focus on the problem with only two local interacting spins hereafter. /H20849b/H20850Slices are plotted as a function of J1for various values of J0. As shown considerable complexity exists even in these plots. -0.4-0.3-0.2-0.10.00.10.20.30.40.5 0 1 2 3 4 5σze J1J0=2 J0=5 J0=8 (b) (a) FIG. 4. /H20849Color online /H20850/H20849a/H20850As in Fig. 3, the zcomponent of the electron spin long after the wave packet has interacted with the local spins, as a function of both electron-spin coupling J0and spin-spin interaction J1for two local spins. This plot is discussed extensively in the text. Note the horizontal band of strong spin flip /H20849dark colored /H20850centered around J0=2t0, broken only near J1/H110151.0t0. Smaller J1values result in independent behavior by the two localized spins while larger values of J1result in strongly coupled behavior by the two local spins. A prominent but very slight change occurs along the vertical line at J1=1t0, and a very obvious trough /H20849i.e., a valley as far as the zcomponent of the electron spin is concerned /H20850of spin flip occurs as shown /H20849in dark color /H20850sloping up toward the right and exiting the graph at /H20849J1,J0/H20850/H11015/H208494t0,10t0/H20850./H20849b/H20850Slices are plotted as a function of J1for various values of J0. For J0=5t0,8t0there is a definite valley corresponding to the dark trough just mentioned in the first plot while, for J0=2t0, the behavior is more complicated.DOĞANet al. PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-4A considerable amount of information is contained in Fig. 4. The horizontal band of strong spin flip /H20849dark /H20850centered around J0=2t0is further illustrated for specific values of J1 in Fig. 5, as a function of J0/H20851the dark horizontal band in Fig. 4/H20849a/H20850corresponds to the minima visible in Fig. 5/H20852. Whether or not the local spins are strongly coupled, the net effect on theelectron spin is similar, and in qualitative agreement withwhat happens when only a single localized spin is present 25 /H20851solid /H20849red/H20850curve in Fig. 5./H20852As already described for a single local spin,25–27,30the maximum spin flip occurs near J0=2t0; for very small values or very large values of J0the impact on the electron spin goes to zero. The reaction of the local spins does depend on the value of the coupling between local spins, as illustrated in Fig. 6, where the zcomponent of the two local spins are shown as a function of time for various values of J1. For zero coupling they react independently /H20849except the second local spin “sees” only part of the incoming electron spin because it has alreadyscattered and spin-flipped off the first /H20850while for low cou- pling some precession occurs. At high values of the coupling,the two local spins are essentially locked together. Referring again to Fig. 4, a subtle change occurs as J 1 passes through t0for all values of J0/H11022t0; this is more clearly seen in Fig. 4/H20849b/H20850, where a small rise occurs in the zcompo- nent of the electron spin as J1/t0crosses unity. For J0=2t0 the increase is considerable, followed by a peak and then a monotonically decaying result. This is in contrast to the othertwo curves which also show a minimum. In fact these twocurves are more “generic;” inspection of Fig. 4/H20849a/H20850shows that J 0=2t0passes right through the middle of the dark band which was discussed above. This region of the J0−J1phase diagram is fairly complicated—the three energy scales are allsimilar in size and no simple picture emerges. Focusing on the larger values of J 0, the small increase in thezcomponent of the electron spin shown in Fig. 4/H20849b/H20850/H20851alsovisible in Fig. 4/H20849a/H20850as a faint but abrupt break along the vertical line J1=t0/H20852can be understood as follows. First note that this increase signals a decrease in the spin-flip interac- tion. Recall that the electron spin is propagated with wavevector k= /H9266/2. This means that its kinetic energy is effec- tively 2 t0—the dispersion relation /H9280/H20849k/H20850=−2t0cos/H20849ka/H20850gives /H9280/H20849k=/H9266/2/H20850=0 but 2 t0is the energy with respect to the bottom of the band . Thus, the electron has a maximum energy 2 t0 that can be deposited into the local-spin system through the Kondo-type coupling J0. On the other hand, for a two spin system there is only one nonzero excitation energy—it isE ex=2J1—and this is essentially the spin-wave energy for a two spin system, as can be readily ascertained from the so-lution to the problem of two ferromagnetically coupledHeisenberg spins. 34ForJ1/H11022t0this mode of inelastic scatter- ing is no longer possible, so the amount of spin-flip scatter-ing decreases, as indicated in the figures. An explicit demonstration of this mode of scattering is provided in Fig. 7, where a series of snapshots of the electron wave packet is shown as a function of position. In contrast toFig. 2a second set of peaks is evident, all in the spin-flip channel /H20849i.e.,zcomponent of electron spin is −1 /2/H20850moving more slowly /H20849hence inelastic scattering /H20850both to the left /H20849re- flection /H20850and to the right /H20849transmission /H20850.A s J 1→t0the speed of this wave packet approaches zero /H20849so the extra wave pack- ets will appear almost vertically in a plot like Fig. 7/H20850. For more and more coupled local spins many more inelasticchannels are available for scattering, which in part explainsthe complexity in Fig. 3. D. Two interacting stationary spins: The NEBS The most striking feature in Fig. 4is the trough /H20849dark colored /H20850that extends upwards to the right and exits the graph at/H20849J1,J0/H20850/H11015/H208494,10 /H20850t0. This trough represents a domain in the coupling space in which the spin-flip interaction persistsmore than expected and is roughly associated with a “reso-nance” behavior. The evidence for this is very difficult to-0.4-0.3-0.2-0.10.00.10.20.30.40.5 0 1 2 3 4 5 6 7 8 9 10Electron Spin J01 site 2 sites J1=0 2 sites J1=10 1 site S=1 FIG. 5. /H20849Color online /H20850The zcomponent of the electron spin long after it has interacted with the local-spin system, as a functionof the Kondo coupling J 0. The solid /H20849red/H20850curve is the result for a single local spin with S=1 /2/H20849Ref.25/H20850. Note that the maximum spin flip occurs at an intermediate value of J0/H110152.3t0/H20849Ref. 25/H20850; when two local spins are present the result is similar, whether they arenoninteracting /H20849J 1=0/H20850or strongly interacting /H20849J1=10t0/H20850. As one would expect the degree to which the incoming electron can reverseits spin is much higher when interacting with more than one localspin.-0.5-0.4-0.3-0.2-0.10.00.10.2 10 20 30 40 50 60<Sz> timeJ1=0.0, Spin 1 J1=0.0, Spin 2 J1=0.3, Spin 1 J1=0.3, Spin 2 J1=1.4, Spin 1 J1=1.4, Spin 2 J1=10., Spin 1 J1=10., Spin 1 FIG. 6. /H20849Color online /H20850The zcomponent of the two local spins for different spin-spin interaction strengths J1, all for J0=2t0. For J1=0 the two spins are essentially independent of one another while forJ1=10t0the two local spins are locked together with the same value as a function of time.EMERGING NONEQUILIBRIUM BOUND STATE IN SPIN- … PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-5glean from the numerical calculations—we will have more to say based on analytical work to be presented in the nextsection. Nonetheless, examination of the numerical resultsfor a particular set of parameters on a logarithmic scaleshows an unusual feature, as illustrated in Fig. 8, for rela-tively high parameter values of electron-spin coupling, /H20849J 1,J0/H20850=/H208493.1,8 /H20850t0. On this scale the Gaussian wave packets are outside the displayed region at the latest times shown/H20849note that time progresses as one moves down from curve to curve, opposite to the progression shown in previous plots /H20850. The feature in question is the rather small peak located at thelocal-spin sites /H20849near site 800 and 801 /H20850that persists, albeit with strongly diminishing amplitude, for all times shown. This peak forms only for the spin-down component of theelectron; its amplitude decays away in both spin channels presumably through a diffusive process so eventually theelectron has scattered entirely. We refer to this state as aNEBS; this name will be further justified in the next section. In Fig. 9we show the various components of the local spins as a function of time, along with the electron spin. TheS xandSycomponents remain fixed at zero /H20849because of the initial conditions on these spins25/H20850while the Szcomponents flip partially and remain at the same value long after theflipping process has terminated. In the intermediate stages,however, they are notlocked together, and remarkably, the second spin flips before the first. This reversal of the ex- pected order of flipping occurs only for parameters in thetrough region; otherwise the local spin first encountered bythe incoming electron spin is the first to flip. While this phe-nomenon is clearly connected to the NEBS, we do not havea simple explanation for the spin-flip reversal. These results illustrate the variety of different behavior possible for the spin-flip scattering process as a function ofJ 0and J1. We now turn to an analytical approach to gain some further insight into the problem. IV . ANALYTICAL PLANE-WA VE APPROXIMATION A. A change in basis The problem of an incident spin represented as a plane- wave scattering off of an impurity with a contact Kondo-type00.010.020.030.040.05 0 200 400 600 800 1000 1200 1400 160 0ni siteindexiTimespin up spin down FIG. 7. /H20849Color online /H20850A series of snapshots of the electron wave packet, with both spin up /H20849solid, red curves /H20850and spin down /H20849dashed, green curves /H20850. Note that spin-down components are scattered both elastically /H20849same speed as incoming wave packet /H20850and inelastically /H20849slower speed, indicated by a more vertical profile on this plot /H20850. The scattering occurs off of two local spins, located at sites 800 and 801,ferromagnetically coupled with J 1/t0=0.8; we used J0/t0=2.0. 10-1810-1610-1410-1210-1010-810-610-410-2100 760 780 800 820 840ni site index iTimespin up spin down FIG. 8. /H20849Color online /H20850A series of snapshots of the electron wave packet, with both spin up /H20849solid, red curves /H20850and spin down /H20849dashed, green curves /H20850. Note that time progresses forward as one moves from curve to curve downward , and also note the logarithmic scale for the ordinate. By the last times shown the usual Gaussian wave-packet peaks have disappeared off to the sides; what remains, how-ever, is a small peak located near the local spins. We refer to this asa NEBS; justification for this name will come in the next section.Note that this small peak exists only in the flipped spin channel. Thescattering occurs off of two local spins, located at sites 800 and 801,ferromagnetically coupled with J 1/t0=3.1 and with a Kondo-type coupling J0/t0=8.0; with reference to Fig. 4these parameters place us in the middle of the dark colored trough of enhanced spin-flipscattering.-0.6-0.4-0.20.00.20.40.60.81.01.2 0 20 40 60 80 100 120 140 160 180spin up spin down electron spin S1xS2xS1yS2yS1zS2z FIG. 9. /H20849Color online /H20850TheSx,Sy, and Szcomponents of the two local spins for the parameter set discussed in the previous figure.Note that S xandSyremain equal to zero /H20849due to the initial condi- tions, as explained in Ref. 25/H20850while the Szcomponents change, although in reverse order than one would naively expect. This is aninstance where the classical notion of a “spin vector” that rotatesinto the direction of the spin current while maintaining a constantmagnitude is completely inapplicable.DOĞANet al. PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-6spin-spin interaction was solved analytically in Ref. 25.I n that problem we made use of the initial conditions and con-servation of angular momentum to simplify the problem.Here we do the same and utilize initial conditions such thattheS zcomponent of the incoming electron spin is +1 /2 while those of the two stationary spins are each −1 /2. The one-dimensional version, written in free space, has a Hamiltonian which can be written as H=−/H60362 2md2 dx2−2J0/H20851/H9268ˆ·Sˆ1/H9254/H20849x/H20850+/H9268ˆ·Sˆ2/H9254/H20849x−a/H20850/H20852−2J1Sˆ1·Sˆ2. /H2084911/H20850 The wave function for this problem consists of a spatial com- ponent which describes the electron-spin amplitude and aspin part which describes the spin state of the incoming elec-tron and the two stationary spins /H20849here located at positions x=0 and x=a/H20850. The Hilbert space concerning the spin de- grees of freedom has an overall size of 2 3=8 /H20849forS=1 /2 spins /H20850However, utilizing the conservation of total Szreduces this number to 3. As already stated, the initial state, in Diracnotation, is /H20841↑↓↓ /H20856, where the first arrow represents the zcom- ponent of the electron spin, and the next two arrows indicatethe respective zcomponents of the two local spins. Once the electron spin interacts with the local spins, two more spinstates are possible, /H20841↓↑↓ /H20856and /H20841↓↓↑ /H20856. In our numerical re- sults, we followed two separate routes: in cases with theinitial configuration as depicted here, we used this fact toreduce the Hilbert space to these three spin states, whichsped up the calculations considerably. Alternatively, whenthe initial configuration was not so straightforward /H20849and did not have a definite total S z, for example /H20850, we used all eight basis states. When we begin with an initial configuration such as /H20841↑↓↓ /H20856, we can combine these spin states into combinations with both good total Szandgood total Sto give rise to the following basis set:35 /H20841/H92741/H20856=1 /H208813/H20849/H20841↓↓↑ /H20856+/H20841↓↑↓ /H20856+/H20841↑↓↓ /H20856/H20850, /H2084912/H20850 /H20841/H92742/H20856=1 /H208816/H20849/H20841↓↓↑ /H20856+/H20841↓↑↓ /H20856−2/H20841↑↓↓ /H20856/H20850, /H2084913/H20850 /H20841/H92743/H20856=1 /H208812/H20849/H20841↓↓↑ /H20856−/H20841↓↑↓ /H20856/H20850. /H2084914/H20850 Writing the wave function as /H20841/H9274/H20849x/H20850/H20856=h/H20849x/H20850/H20841/H92741/H20856+f/H20849x/H20850/H20841/H92742/H20856+g/H20849x/H20850/H20841/H92743/H20856, /H2084915/H20850 then appropriate projection on to the spin basis states results in the three equations, −/H60362 2md2h dx2−2J0/H60362 4/H20851/H9254/H20849x/H20850+/H9254/H20849x−a/H20850/H20852h=/H9280elh, /H2084916/H20850−/H60362 2md2f dx2+J0/H60362 2/H20851/H9254/H20849x/H20850/H208492f−/H208813g/H20850+/H9254/H20849x−a/H20850/H208492f+/H208813g/H20850/H20852=/H9280elf, /H2084917/H20850 −/H60362 2md2g dx2−/H208813J0/H60362 2/H20851/H9254/H20849x/H20850−/H9254/H20849x−a/H20850/H20852f=/H20849/H9280el−2J1/H60362/H20850g, /H2084918/H20850 where /H9280elis the kinetic energy of the incoming electron. Note that the first equation results from the Stot=3/H6036/2 sector, and remains decoupled, while the second two are part of theS tot=/H6036/2Stotz=−/H6036/2 doublet. The state with spatial wave function g/H20849x/H20850, governed primarily by the third equation, ex- ists exclusively because of the possible inelastic-scatteringprocess. Equation /H2084914/H20850indicates that it contains only the spin-down component of the scattered electron, and, givenour initial conditions, exists only after scattering. It is “fu-eled” through the f/H20849x/H20850component, which, as Eq. /H2084913/H20850indi- cates, contains a component corresponding to the incomingelectron spin /H20849with /H9268z=+/H6036/2/H20850. That the g/H20849x/H20850component cor- responds to inelastic scattering is indicated by the eigenvalueon the right-hand side of Eq. /H2084918/H20850, with value /H9280el−2/H60362J1, which shows that an energy 2 /H60362J1is left behind in the form of a spin-wave excitation in the local-spin system, as ex-plained in the previous section. The first two equations, Eqs./H2084916/H20850and /H2084917/H20850, each have eigenvalue /H9280el, showing that the kinetic energy of the incoming electron is conserved /H20849elastic scattering /H20850. Note that this still results in spin-flip scattering; it is just that the two local spins are scattered by the sameamount so that they remain in their coupled ground state. B. A re-examination of the numerical solutions Equations /H2084916/H20850–/H2084918/H20850can be readily solved analytically and we will come to that solution shortly. However, alreadyEqs. /H2084912/H20850–/H2084914/H20850serve the important task of directing our at- tention to specific linear combinations of the spin states, asindicated. The numerical solutions presented in the previoussection were classified only according to the zcomponent of the electron spin. We now essentially replot those results, asseparate amplitudes h/H20849x/H20850,f/H20849x/H20850, and g/H20849x/H20850, in Fig. 10. Note that Eqs. /H2084916/H20850–/H2084918/H20850were derived for the continuum model de- fined by Eq. /H2084911/H20850; nonetheless the role of the various ampli- tudes, described at the end of Sec. IV A , applies equally well to the numerical results of the original tight-binding model. To demonstrate this, in Fig. 10/H20849a/H20850we plot the magnitude /H20841h/H20849x/H20850/H20841 2vs position for a number of time slices, for three dif- ferent values of J1. As predicted by Eq. /H2084916/H20850, there is no dependence on J1. It is important to note that these results still represent numerical solutions to the tight-binding modelpresented in the previous section; while we could have usedthe spin components as listed in Eqs. /H2084913/H20850and /H2084914/H20850as a basis set, these numerical solutions do not “use” the analyticalstructure of Eqs. /H2084916/H20850–/H2084918/H20850. Hence only one set of curves is visible /H20849forJ 1=1.4 t0/H20850as this set is identical to and covers entirely the sets corresponding to the other two values of J1. In contrast, the other two components, plotted in Figs. 10/H20849b/H20850and10/H20849c/H20850, are dependent on the value of J1. In bothEMERGING NONEQUILIBRIUM BOUND STATE IN SPIN- … PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-7cases the amplitudes of transmitted and reflected wave packet depend quantitatively on the value of J1. Note, more- over, that the amplitude g/H20849x/H20850has no “incoming” wave packet. As explained earlier this amplitude is generated en-tirely by the scattering process. Also note that for J 1/t0/H110211 /H20849i.e., J1/t0=0.8 in Fig. 10/H20850the slow moving piece belongs entirely to g/H20849x/H20850while the fast moving one belongs entirely to f/H20849x/H20850. To see the role of the gcomponent of the state more clearly, we separate the two local spins by 20 sites andproject out the gcomponent from the numerical solution, using Eq. /H2084915/H20850. In Fig. 11we show on a log scale the mag- nitude of the gcomponent, /H20841g/H20849x/H20850/H20841 2as a function of position; the two local spins are now located at sites 790 and 810. Theparameters /H20849J 1,J0/H20850=/H208493.1,8 /H20850t0/H20849solid curve /H20850situate the regime on the trough so apparent in Fig. 4whereas /H20849J1,J0/H20850 =/H208493.1,2 /H20850t0/H20849dashed curve /H20850puts one well away from the trough. This snapshot is taken at a time when theg-component amplitude is a maximum and it is clear that the gcomponent is almost two orders of magnitude larger on the trough /H20849solid curve /H20850than off /H20849dashed curve /H20850. A similar result holds for large values of J 0.C. Analytical solution An analytical solution of the problem with plane waves through Eqs. /H2084916/H20850–/H2084918/H20850is possible, though tedious. One de- fines three regions in space and defines the wave function ina piecewise continuous manner, as is done commonly in un-dergraduate physics texts. With k/H11013 /H208812m/H9280el//H60362and Q/H11013/H208812m/H208492/H60362J1−/H9280el/H20850//H60362, the wave function can be written as h/H20849x/H20850=/H9273/H20902hIeikx+uIe−ikx,x/H110210 hIIeikx+uIIe−ikx,0/H11021x/H11021a hIIIeikx, x/H11022a/H20903, /H2084919/H20850 f/H20849x/H20850=/H20902fIeikx+rIe−ikx,x/H110210 fIIeikx+rIIe−ikx,0/H11021x/H11021a fIIIeikx, x/H11022a/H20903, /H2084920/H20850 and0.0000.0050.0100.0150.0200.0250.030 0 200 400 600 800 1000 1200 1400 1600 site index|h(x)|2J1=0.0 J1=0.8 J1=1.4 0.0000.0050.0100.0150.0200.0250.030 0 200 400 600 800 1000 1200 1400 160 0 site index|f(x)|2J1=0.0 J1=0.8 J1=1.4 0.0000.0050.0100.0150.0200.0250.030 0 200 400 600 800 1000 1200 1400 1600 site index|g(x)|2J1=0.0 J1=0.8 J1=1.4(b) (a) (c) FIG. 10. /H20849Color online /H20850The time evolution of the magnitudes /H20849a/H20850/H20841h/H20849x/H20850/H208412,/H20849b/H20850/H20841f/H20849x/H20850/H208412, and /H20849c/H20850/H20841g/H20849x/H20850/H208412, as defined by the basis set, Eqs. /H2084912/H20850–/H2084914/H20850. These plots apply for J0=2t0and the values of J1indicated. Note that in /H20849a/H20850the plots are identical for all three values of J1,a s motivated by the structure of Eq. /H2084916/H20850.I n /H20849b/H20850and /H20849c/H20850differences are apparent; note that in /H20849c/H20850no amplitude is present before the time of scattering, and, furthermore, as one enters the trough region /H20849J1=1.4 t0/H20850/H20841g/H20849x/H20850/H208412consists of a single sharp peak near the local spins. In time this peak diffuses outwards but there is no wave-packet component.DOĞANet al. PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-8g/H20849x/H20850=/H20902gIeQx, x/H110210 gIIe−Qx+sIIeQx,0/H11021x/H11021a gIIIe−Qx, x/H11022a/H20903. /H2084921/H20850 Four conditions relate the various coefficients defining h/H20849x/H20850 to the incoming amplitude hIin Eq. /H2084919/H20850; similarly eight conditions determine the fandgcoefficients in terms of the incoming amplitude fI. The results for h/H20849x/H20850are standard and can be found in many undergraduate texts while, for fandg, the result is not standard but is nonetheless straightforward.Also note that we have written the wave function for themore relevant condition 2 /H6036 2J1/H11022/H9280el, in which case the func- tion g/H20849x/H20850is exponentially decaying; the alternative 2 /H60362J1 /H11021/H9280elis straightforward and gives a propagating wave solu- tion, with wave vector q=/H208812m/H20849/H9280el−2/H60362J1/H20850//H60362. This latter case corresponds to the situation whereby a spin-wave exci-tation is energetically allowed so that a spin-flipped wavepacket will emerge from the stationary spins at a reducedspeed, as we have already seen in the numerical solution inFig.7. When inelastic scattering is not allowed by energy con- siderations, it is not clear what will happen. Our intuitionsuggests that the stationary spins will respond as one and sothe spin-flip process will resemble that expected for scatter-ing from a single spin /H20851which, as we demonstrated earlier, is not so different from scattering off decoupled stationaryspins /H20849J 1=0/H20850/H20852. Inspection of Fig. 4shows that this is indeed the case, except for the trough region previously identified .I t is precisely in this regime that a peculiar enhancement ofspin-flip scattering occurs, which we now argue is connectedto the effective bound state /H20849NEBS /H20850defined by Eq. /H2084921/H20850. The solutions can readily be written down by using the definitions, /H9251/H11013J0/kand/H9252/H11013J0/Q/H20849the mass mis set equal to unity /H20850, and the terms v/H11013/H9251/H208531−3/H9252 4/H208491−e−Qaeika/H20850/H20854andu/H11013/H9251/H208531−3/H9252 4/H208491−e−Qae−ika/H20850/H20854appear often. Note that for real Qthese are complex conjugates of one another. However, these ex-pressions /H20849and the ones immediately following /H20850are valid for high electron kinetic energy as well, where /H9280el/H110222J1, and so it follows that Q=−iqwith qnow real, and uandvare no longer complex conjugates of one another. We find, for ex-ample, g II fI=/H208813 2/H92521+iv−iue2ika /H208491+iv/H208502+u2e2ika, /H2084922/H20850 with similar expressions for the other coefficients. To see how effective the spin-flip process is, we can calculate eitherthe expectation value of the electron spin, /H20855 /H9268z/H20856, or the spin torque, Nzx.20For the two local-spin system used here, the latter is given in terms of the former as Nzx=k/H208491/2−/H20855/H9268z/H20856/H20850.A s in the earlier numerical results, the quantity /H20855/H9268z/H20856will remain near 0.5 /H20849the initial electron-spin value /H20850if very little spin flip occurs whereas this quantity will deviate most from 0.5 /H20849and even become negative /H20850when significant spin flip occurs. Note that with the plane-wave solution given in Eqs./H2084919/H20850–/H2084921/H20850, the problem is no longer time dependent; one en- visions a continual influx of current /H20849this is f I/H20850while reflected and transmitted plane waves /H20849of both spin type /H20850take on “steady-state” values.36 The calculation of /H20855/H9268z/H20856is straightforward; we use an in- tegration region − L/H11021x/H11021+Land we allow L→/H11009. The plane- wave regions outside the local-spin region then dominate,and, for real values of Q, we obtain /H20855 /H9268z/H20856=1 18/H208535−4 /H208812R e /H20849hIIIfIII/H11569+uIrI/H11569/H20850/H20854, /H2084923/H20850 while, for pure imaginary values of Q, the expression for /H20855/H9268z/H20856is somewhat more complicated. In Fig. 12we show /H20849a/H20850/H20855/H9268z/H20856and /H20849b/H20850/H20841gII/H208412as a function of J1andJ0to emphasize the connection between the region /H20849described as a trough /H20850of enhanced spin-flip scattering and the NEBS. The range of both J1andJ0is considerably ex- tended compared with Fig. 4; nonetheless the qualitative similarities are striking; clearly the analytical solution cap-tures the essence of the numerical one. Furthermore, the ana-lytical approach has allowed us to make the association ofthe trough of enhanced spin-flip scattering with the NEBS.Quantitative details differ, in part because the numerical re-sults are based on a tight-binding model whereas the analyti-cal ones utilize a quadratic dispersion for the itinerant spin. Aspecific example is given in Fig. 13, where both /H20855 /H9268z/H20856and /H20841gII/H208412are plotted as a function of J0/H20849for a specific value of k and J1/H20850. Clearly the peaked region in /H20841gII/H208412/H20849near J0/H1101515/H20850 corresponds to the dip in /H20855/H9268z/H20856, showing strong evidence for the role of the NEBS in enhanced spin-flip scattering. Forlarge values of J 1/H11271/H9280el, Eq. /H2084922/H20850simplifies somewhat; we get /H20841gII/H208412=3 4/H20873J0 Q/H208742sin2ka+/H20849coska+2vsinka/H208502 1+4 v2/H20849coska+vsinka/H208502, /H2084924/H20850 where v/H11015J0 k/H208511−3 4J0 Q/H20852. Similar analytical expressions can be readily attained for all the coefficients but they are of limitedvalue.10-810-710-610-510-410-310-210-1100 780 785 790 795 800 805 810 815 820 site indexJ0=8 J0=2 FIG. 11. /H20849Color online /H20850A plot of /H20841g/H20849x/H20850/H208412/H20851see Eq. /H2084915/H20850/H20852vsx. This “snapshot” is taken immediately following the initial scattering ofthe electron spin with the two local spins, situated at sites 790 and810, i.e., they have been separated for clarity. We use J 1=3.1 t0so it is clear that for parameter values that fall on the trough /H20849J0=8t0/H20850the component of the wave function associated with inelastic scattering/H20851i.e.,g/H20849x/H20850/H20852is significantly enhanced /H20849almost 2 orders of magnitude /H20850 compared with the region away from the trough.EMERGING NONEQUILIBRIUM BOUND STATE IN SPIN- … PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-9The curve given by Eq. /H2084924/H20850is also plotted in Fig. 13, where it is seen to be very accurate /H20849in fact, it is fairly accu- rate all the way down to J1/H110152/H20850. The peak region in /H20841gII/H208412 follows roughly a dispersion relation J1/H11015/H9280el 2+9 64J02, /H2084925/H20850 and, as has been emphasized already, this corresponds to the region of most intense spin-flip scattering /H20849the “dark trough” region of previous figures /H20850. Thus, when J0andJ1are tuned to satisfy Eq. /H2084925/H20850we find an enhanced spin-flip process. D. Transmission and reflection amplitudes from the numerical solutions Having established the idea of a NEBS we once again re-examine the numerical solutions. In particular, one impor-tant property from the experimental point of view is that thestationary spins can act as a spin barrier. We have alreadyshown that a large electron-spin interaction /H20849J 0/H20850works as a high potential barrier for the incoming spin. In our frame-work, for instance, a large electron-spin interaction acts toprevent any spin-up component of the electron from trans-mitting through the stationary spin system. However, in theJ 0−J1phase space, at the onset of the trough described, for example, in Fig. 4, the transmitted component of the spin-up /H20849and spin-down /H20850electron is enhanced considerably; this is illustrated in the four plots shown in Fig. 14, where both transmitted and reflected intensities are plotted as a functionofJ 0andJ1. As is clearly evident in /H20849a/H20850and /H20849b/H20850, the trans- mission of both spin species is noticeably enhanced in thetrough region. Coincidentally the spin-up reflected compo-nent is decreased while the spin-down reflected componentshows an increase. The increase in the transmitted spin-upcomponent of the electron is not through “direct” transmis-sion. Rather it is achieved through the spin-flip interactionthat generates the component with amplitude gdiscussed in Sec. IV B . Recall that in this parameter regime this gcom- ponent does not exist outside the local spins; it first trans-forms into the component with amplitude f, which represents a propagating wave with both spin-up and spin-down spe-cies. These plots therefore reinforce the idea that the electrongoes through a two-step “virtual” spin-flip interaction /H20849cre- ation of the NEBS /H20850in the trough region. V . CONCLUSIONS We have modeled spin-current-induced spin torque in the quantum regime with a lattice, on which an itinerant spin/H20849constructed as a wave packet /H20850moves with a kinetic energy given by a tight-binding dispersion, to represent the spin cur-rent. Any number of ferromagnetically coupled spins canthen be flipped by repeating the process described here withmore itinerant electrons, i.e., a current. As described in Ref.26, this then requires a density-matrix description. We have (b) (a) FIG. 12. /H20849Color online /H20850/H20849a/H20850Plot of the expectation value of the zcomponent of the electron spin, /H20855/H9268z/H20856, as a function of the two coupling parameters, J1andJ0, based on the plane-wave solutions to this problem. The range of J1andJ0is considerably extended compared to Fig. 4to emphasize the presence of the trough /H20849shown in dark color /H20850that extends upward and to the right. In /H20849b/H20850we show a plot for 1 /2 −0.2 /H11569/H20841gII/H208412//H20841fI/H208412for the same parameters; the trough is immediately identifiable in this plot, which reinforces our contention that this region of enhanced spin-flip scattering is associated with the NEBS represented by gII/H20849a plot of sIIyields similar results /H20850. 0.00.10.20.30.40.5 0 5 10 15 20 25<σz>, |gII|2 J0k a=2 J1=3 0<σz> 0.2*|gII|2/|fI|2 FIG. 13. /H20849Color online /H20850/H20849a/H20850Plot of /H20855/H9268z/H20856/H20851solid /H20849red/H20850curve /H20852vsJ0 for specific values of J1and k, as indicated. Also shown is the coefficient /H20841gII/H208412/H20851solid /H20849green /H20850curve /H20852, which shows a peak precisely where /H20855/H9268z/H20856has a significant dip, indicative of enhanced spin-flip scattering. Also shown /H20849symbols /H20850is the result of an approximate analytical expression derived in the text. Agreement is extremelygood.DOĞANet al. PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-10focused on just two coupled local spins since this small sys- tem contains the essence of the processes we believe areresponsible for spin torque: /H20849i/H20850direct spin flip without inter- nal excitation of the local-spin system and /H20849ii/H20850spin flip through inelastic scattering, either real or virtual. The firstprocess exists even for a single local spin and has been ex-plored previously by us. The second process is the primarysubject of this paper, particularly in the regime where, ener-getically, the itinerant spin becomes momentarily bound inthe local system, a phenomenon which we have called theNEBS. The description here is for a one-dimensional systembut the NEBS should also be present in three dimensions. An analytical plane-wave approach, using a parabolic dis- persion for the itinerant spin, helps to elucidate the nature ofthe spin-flip processes. A scattering channel through which alocal-spin singlet /H20849i.e., /H20841 /H92743/H20856/H20850is generated is responsible for the enhanced spin-flip scattering along a trough in the /H20849J0,J1/H20850 phase diagram. This trough is reasonably well described inthe plane-wave approach by the relation J1=9J02/64+/H9280el/2. An experimental observation of the NEBS would be straightforward provided at least one of the parameters J0,J1, or/H9280elcan be tuned in a particular system. In this way the probability of spin flip can be monitored as a function ofparameter space and the NEBS would be identified by awell-defined region of enhanced spin flip, corresponding tothe trough in Fig. 4. One interesting consequence of our calculation is the pos- sibility of using the spin chain as an effective spin filter. Bytuning the parameters to correspond to the regimes of en-hanced spin flipping, the spin-up electrons will be flippedwhile the spin-down ones will be unaffected. This effect canbe achieved not only for the two-spin chain but also for thelonger chains, as shown in Fig. 3. The resonant trough pro- vides a controllable spin filter through the interspin couplingJ 1. 1See, for example, P. A. Rikvold, G. Brown, S. J. Mitchell, and M. A. Novotny, in Nanostructured Magnetic Materials and their Applications , edited by D. Shi, B. Aktas, L. Pust, and F. Mi- kailov, Springer Lecture Notes in Physics Vol. 593 /H20849Springer, Berlin, 2002 /H20850, p. 164.2M. R. Freeman and W. K. Hiebert, in Spin Dynamics in Confined Magnetic Structures I , edited by B. Hillebrands and K. Ounad- jela /H20849Springer, Berlin, 2002 /H20850. 3See, for example, T. L. Gilbert, IEEE Trans. Magn. 40, 3343 /H208492004 /H20850. 0.00.20.40.61.0 0.8 0 2 4 6 8 10 J1012345678910J0c)R↑ (b) (a) (c) (d) FIG. 14. /H20849Color online /H20850The /H20849a/H20850transmitted spin-up, /H20849b/H20850transmitted spin-down, /H20849c/H20850reflected spin-up, and /H20849d/H20850reflected spin-down intensities as a function of J0andJ1. These results are obtained with the same conditions as in Fig. 4. See the text for further description.EMERGING NONEQUILIBRIUM BOUND STATE IN SPIN- … PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-114J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850;J . Magn. Magn. Mater. 195, L261 /H208491999 /H20850. 5L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 6M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 /H208491998 /H20850; M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, V. Tsoi, and P. Wyder,Nature /H20849London /H20850406,4 6 /H208492000 /H20850. 7E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 /H208491999 /H20850. 8J.-E. Wegrowe, D. Kelly, T. Truong, Ph. Guittienne, and J.-Ph. Ansermet, Europhys. Lett. 56, 748 /H208492001 /H20850. 9S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850 425, 380 /H208492003 /H20850. 10J. C. Sankey, Y.-T. Cui, J. Z. Sun, J. C. Slonczewski, R. A. Buhrman, and D. C. Ralph, Nat. Phys. 4,6 7 /H208492008 /H20850. 11T. Balashov, A. F. Takacs, M. Dane, A. Ernst, P. Bruno, and W. Wulfhekel, Phys. Rev. B 78, 174404 /H208492008 /H20850. 12S. Mangin, Y. Henry, D. Ravelosona, J. A. Katine, and E. E. Fullerton, Appl. Phys. Lett. 94, 012502 /H208492009 /H20850. 13L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 94, 122508 /H208492009 /H20850. 14V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Nat.Phys. 3, 498 /H208492007 /H20850. 15G. Finocchio, O. Ozatay, L. Torres, R. A. Buhrman, D. C. Ralph, and B. Azzerboni, Phys. Rev. B 78, 174408 /H208492008 /H20850. 16J. P. Strachan, V. Chembrolu, Y. Acremann, X. W. Yu, A. A. Tulapurkar, T. Tyliszczak, J. A. Katine, M. J. Carey, M. R.Scheinfein, H. C. Siegmann, and J. Stohr, Phys. Rev. Lett. 100, 247201 /H208492008 /H20850. 17Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 /H208491998 /H20850. 18W. Kim and F. Marsiglio, Phys. Rev. B 69, 212406 /H208492004 /H20850. 19A. Brataas, Gergely Zaránd, Yaroslav Tserkovnyak, and Gerrit E. W. Bauer, Phys. Rev. Lett. 91, 166601 /H208492003 /H20850. 20W. Kim and F. Marsiglio, Can. J. Phys. 84, 507 /H208492006 /H20850.21See the many papers in J. Magn. Magn. Mater. 320, 1190 /H208492008 /H20850. 22X. J. Wang, H. Zou, and Y. Ji, Appl. Phys. Lett. 93, 162501 /H208492008 /H20850. 23C. F. Hirjibehedin, C. P. Lutz, and A. J. Heinrich, Science 312, 1021 /H208492006 /H20850. 24Y. Avishai and Y. Tokura, Phys. Rev. Lett. 87, 197203 /H208492001 /H20850. 25W. Kim and F. Marsiglio, Europhys. Lett. 69, 595 /H208492005 /H20850. 26W. Kim, L. Covaci, F. Do ğan, and F. Marsiglio, Europhys. Lett. 79, 67004 /H208492007 /H20850. 27F. Doğan, W. Kim, C. M. Blois, and F. Marsiglio, Phys. Rev. B 77, 195107 /H208492008 /H20850. 28F. Ciccarello, M. Paternostro, M. S. Kim, and G. M. Palma, Phys. Rev. Lett. 100, 150501 /H208492008 /H20850. 29F. Ciccarello, M. Paternostro, G. M. Palma, and M. Zarcone, arXiv:0812.0755 /H20849unpublished /H20850. 30W. Kim, L. Covaci, and F. Marsiglio, Phys. Rev. B 74, 205120 /H208492006 /H20850. 31A. Weiße and H. Fehske, Chebyshev Expansion Techniques , Lec- ture Notes in Physics Vol. 739, /H20849Springer, New York, 2008 /H20850,p . 545577. 32R. S. Dumont, S. Jain, and A. Bain, J. Chem. Phys. 106, 5928 /H208491997 /H20850. 33W. Liang, R. Baer, C. Saravanan, Y. Shao, A. T. Bell, and M. Head-Gordon, J. Comput. Phys. 194, 575 /H208492004 /H20850. 34N. W. Ashcroft and N. D. Mermin, Solid State Physics /H20849Thomson Learning, Toronto, 1976 /H20850, p. 680. 35L. Schiff Quantum Mechanics /H20849McGraw-Hill, New York, 1955 /H20850, p. 235. 36We find this description inferior to the wave-packet approach for which numerical results were earlier provided. This sentiment isshared by others, for example, T. Norsen, J. Lande, and S. B.McKagan, arXiv:0808.3566 /H20849unpublished /H20850. Nonetheless, as in undergraduate textbooks for simple barrier/well scattering prob-lems, this approach is useful because it allows for analyticalsolutions.DOĞANet al. PHYSICAL REVIEW B 80, 104434 /H208492009 /H20850 104434-12

PhysRevB.86.014418.pdf

PHYSICAL REVIEW B 86, 014418 (2012) Multiple synchronization attractors of serially connected spin-torque nanooscillators Dong Li,1Yan Zhou,2Bambi Hu,3Johan ˚Akerman,4and Changsong Zhou1,* 1Department of Physics, Centre for Nonlinear Studies and The Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex System (Hong Kong), Hong Kong Baptist University, Kowloon Tong, Hong Kong 2Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong 3Department of Physics, University of Houston, Houston, Texas 77204-5005, USA 4Department of Physics, University of Gothenburg, SE-412 96 Gothenburg, Sweden (Received 10 November 2011; revised manuscript received 11 June 2012; published 18 July 2012) Spin-torque nanooscillators (STNOs), which have both the common properties of nanosized oscillators (small size, tunable operating frequency) and some particular ones (wide operating range, easy on-chip integration,etc.), have received a great deal of attention due to their high potential in applications. Yet synchronizationof serially connected STNOs has been considered essential for applications. In this paper, we present findingsconcerning the following properties of synchronized serially connected STNOs: (i) multiple synchronizationattractors coexist, and the attracting basins are entangled in a complicated manner; (ii) these attractors havedifferent synchronized frequencies and output powers; and (iii) switching among these attractors can be inducedby a small noise, which causes a resonance peak in the power spectra to vanish. These characteristics can beunderstood using saddle-node bifurcations and have direct impact on laboratory experiments and the potentialapplications of STNO-based devices. DOI: 10.1103/PhysRevB.86.014418 PACS number(s): 85 .75.Bb, 05 .45.Xt, 75 .40.Gb, 62 .25.−g I. INTRODUCTION Various types of nanosized oscillators are currently attract- ing great interest. They lend themselves to many possibleapplications, thanks to their common properties, which includesmall size, tunable operating frequency, etc. 1–12Spin-torque nanooscillators (STNOs),1–6which are nanosized spintronic devices capable of microwave generation at frequencies inthe gigahertz-terahertz range with high quality factors, showgreat potential for wireless and radar communication. Thecollective behavior of an array of coupled STNOs is alsosignificant for many applications: For example, its magneticoscillation can be used as an information carrier in spin-basedlogic circuits 13–17and polychronous wave computation,18as well as serve as a model system for fundamental study.19More importantly, synchronization of coupled STNO arrays has beendemonstrated as a technique essential for potential microwavegeneration applications, as it leads to increased power andimproved signal purity and quality, 20–23in contrast with the very limited output power and relatively large linewidth ofas i n g l eS T N O . 19,24–26However, despite much experimental work in this direction, the dynamical characteristics of the syn-chronization states are still unclear. It is therefore imperativeto gain insight into the synchronization attractors of coupledSTNO networks in order to design and optimize STNO-basednovel spintronic radio-frequency devices for next-generationmicrowave applications. In this paper, we study the dynamical behaviors which arise in the synchronization region of serially connectedSTNO networks, where all the STNOs oscillate at a uniformsynchronized frequency. It is found that multiple attractorswith different phase shifts, synchronized frequencies, andresonance peak amplitudes can coexist in such a system acrossquite a broad range of system parameters. Both in-phase andantiphase synchronization can be observed in pairs of seriallyconnected STNOs. The attracting basins are entangled in acomplicated manner. A random switching between these twostates can be induced by means of a small noise, which causes the resonance peak in the power spectra to vanish.When the number of STNOs is greater than two, the attractorsbecome more complex. In contrast to the previously used Hopfbifurcation theory of coupled phase oscillators, 27,28we show that these results can be precisely understood by a model basedon saddle-node bifurcation theory. Our results are significant for the improved understanding of the nonlinear characteristics of such STNO devices, andhave a direct impact on experiments and applications involvingSTNO systems. We show that the synchronization of seriallyconnected STNOs does not necessary increase the resonancepeak, which was one of the original motivations for synchro-nizing STNOs. One of the common methods of identifyingsynchronization in such coupled nanosized high-frequencyoscillator systems is by measuring the resonance peak inexperimental power spectra, yet this becomes impracticalwhen the peak vanishes. In addition, these findings may havesome potential in STNO-based information processing anddigital computation applications. II. STNO MODEL In this work, we investigate an in-plane STNO model, as shown in Fig. 1(a). Since we focus on the interaction properties of electrically connected STNOs, the detailedmicromagnetics in each free layer is neglected and we adopta macrospin model to capture the main physics governing thefree layer magnetodynamics, where the unit vector of the freelayer magnetization mis described by the Landau-Lifshitz- Gilbert-Slonczewski equation, 29 dm dt=− |γ|m×Heff+αm×dm dt+|γ|βJm×(m×M), (1) where γis the gyromagnetic ratio, αis the Gilbert damp- ing parameter, and βcontains material parameters and 014418-1 1098-0121/2012/86(1)/014418(6) ©2012 American Physical SocietyLI, ZHOU, HU, ˚AKERMAN, AND ZHOU PHYSICAL REVIEW B 86, 014418 (2012) FIG. 1. (Color online) (a) Schematic of an in-plane STNO. The free layer magnetization mis separated from the fixed layer Mby a nonmagnetic layer. (b) Two serially connected STNOs. fundamental constants.20The electrical current Jis defined as positive when electrons flow from the fixed layer to thefree layer. The effective field H effcarries the contribution of an external applied magnetic field Ha, an anisotropy (easy axis) field Hkalong the xaxis, and a demagnetization (easy plane anisotropy) field Hdz≈4πMs, where Msis the saturation magnetization of the free layer material. We thusgetH eff=Haˆex+(Hkmxˆex-Hdzmzˆez)/|m|. In the case of a serial circuit [shown in Fig. 1(b)], the shared current in the STNObranch is20,21 J(t)=Jd/slashbigg/bracketleftbigg Ca−Cb/summationdisplay icosψi(t)/bracketrightbigg , (2) where ψiis the angle from Mtomof the ith STNO, and Ca andCbare constants. Such a macrospin model is valid when the spatial distribution of the magnetizations in the free layer isneglected and it has been successfully employed in interpretinga wide range of magnetic and more recent spin-torque relatedexperiments. 20,30,31 III. SYNCHRONIZATION ATTRACTORS In a serially connected STNO system, the parameter region for synchronization is predicted not to be large.32We therefore first investigate the synchronization of two identical STNOs.In the synchronization state, where the two STNOs havea common frequency ˜/Omega1and a time lag δ(phase shift), the current given by Eq. (2)should have the same value upon interchanging the two STNOs: J(t)| cosψ2(t)=cosψ1(t+δ)= J(t)|cosψ2(t)=cosψ1(t−δ). Therefore, both in-phase ( δ=0) and antiphase ( δ=T/2, half the period) synchronization are possi- ble. Our numerical simulations show that both are stable acrossa broad range of parameters, as shown in Fig. 2. This case is totally different from the typical coupled phase oscillators withthe form dψ 1,2/dt=/Omega11,2+Ksin (ψ2,1−ψ1,2). When /Omega11= FIG. 2. (Color online) The coexistence of in-phase (a) and (d) and antiphase (b) and (e) synchronizations of two identical STNOs. In the first row, the solid lines are the shared currents J, and the dashed lines are the time serials of cos ψifor the two STNOs. (d) and (e) The resonance peaks in the Fourier spectra of Jcan represent the resonance peaks in the power spectra in experiments. When a small noise is introduced, the system randomly switches between these two attractors, inducing the resonance peak to vanish, as shown in (f). Panel (g) shows the time serials with noise. In order to give a clear illustration of the switching phenomenon, we show a fragment of (g) in panel (c). 014418-2MULTIPLE SYNCHRONIZATION ATTRACTORS OF ... PHYSICAL REVIEW B 86, 014418 (2012) FIG. 3. (a) A section of the attracting basin of the antiphase (white) and in-phase (black) synchronization in the θ2−φ2plane in a pair of serially connected STNO system. (b) A section of the attracting basin of the antiphase synchronization (white) and desynchronization (black). θandφare, respectively, the polar and azimuth angles of m in a spherical coordinate. Both the initial θ1andφ1are equal to π/2. The system parameters of panels (a) and (b) are shown in Fig. 5, respectively. /Omega12, the attractor only depends on K. The mono-attractor will be either in-phase synchronization when K> 0 or antiphase synchronization when K< 0.33In Figs. 2(a) and2(b), one can also see that the in-phase and antiphase synchronizationscorrespond to the stable 1 : 1 and 2 : 1 lockings of an STNOto the shared current J(t), respectively. The fractional locking of an STNO to an external signal has also been reported inexperiments, 34,35and cannot be described by means of coupled phase oscillators.36 Because of the coexistence of these two attractors, the system may still exhibit different oscillations, despite beingsynchronized. From the Fourier spectra, one can observethat both the synchronized frequencies and the amplitudesare different. Through numerical simulation, we also foundthat the attracting basins of the two attractors are entangledin a complicated manner, as shown in Fig. 3(a).F o rt w o nonidentical STNOs, the entangled attracting basins are alsopresent, with some changes in the details. Furthermore, acrossa wide region of parameters, the attractors can be antiphaseeven if the initial conditions of these two STNOs are identical.In experiments it is therefore not an easy task to control theinitial conditions in order to favor either of the attractors. Thismay require some more advanced methods and sophisticatedtechnologies. The entangled attracting basins will have more influence on the experiments in the presence of noise perturbations, whichare inevitable in experiments. In this case, noise can induceswitching among these attractors, which is totally differentfrom the mono-attractor case in which noise only induces asmearing of the attractor. In this STNO system, switching canbe observed even with a very small noise, as shown in Figs. 2(c) and2(g). In this simulation, we simply add a Gaussian white noise term ξ(t)t oE q . (1)to refer to various kinds of stochastic factors which occur in experiments, including, but not limitedto, temperature. The standard deviation of ξ(t) is quite smaller compared to the deterministic terms, for example, in Figs. 2(c), 2(f), and 2(g),w eu s e /angbracketleftξ(t)ξ(s)/angbracketright=2×10 −4×γ 1+α2δ(t−s). Thus synchronization can be identified when measuringthe average frequencies of the two STNOs. However, theresonance peak in the Fourier spectra nearly vanishes. Thisswitching behavior does not have an intrinsic major frequency, as shown in Fig. 2(g). In such a nanosized high-frequency system, measurement of the resonance peak is commonly usedto identify the synchronization state from the power spectrain experiments. The result in Fig. 2(f) means that this type of synchronization will require other identification methods.Moreover, although these nontrivial characteristics may havepotential in some applications, if the original motivation ofsynchronizing is to increase the resonance power, this type ofsynchronization does not necessarily fulfil that goal. When more than two STNOs are serially coupled, the sit- uation becomes more complex. When an artificial time-delayDis introduced in J(t)=J d/[Ca−Cb/summationtext icosψi(t−D)] in order to enlarge the synchronization region,21,32further types of synchronization attractors will be observed. For example,when four STNOs are synchronized, we have observed oneattractor in which all four STNOs completely synchronize withuniform frequency and zero phase difference, another attractorin which the four STNOs are divided into two antiphasegroups, each of which consists of two in-phase STNOs,and even another in which three of the STNOs completelysynchronize with uniform frequency ˜/Omega1, while the fourth one fractionally synchronizes with the others at 3 4˜/Omega1, and so on. The attracting basins are entangled in a higher-dimensionalspace, which may lead to further impact on experiments andapplications. IV . BIFURCATION TO SYNCHRONIZATION In the following, we analyze the phenomena of the coexisting attractors, from the viewpoint of dynamical system.Starting with the two coupled nonidentical STNOs, weinvestigate the bifurcation process, where synchronizationis achieved by decreasing the difference between the twodevices. We set a parameter mismatch /Delta1H dzto characterize the difference, where Hdz1,2=Hdz0±/Delta1Hdz/2. If/Delta1Hdzdecreases to zero, the case reduces to the identical STNOs pair. Wehave always found the coexistence of in-phase and antiphasesynchronization attractors when the two STNOs are identical(i.e.,/Delta1H dz=0). However, when /Delta1Hdz/negationslash=0 for nonidentical STNOs pair, in-phase synchronization can emerge eitherbefore or after antiphase synchronization (i.e., the criticalvalue of /Delta1H dzfor the onset of in-phase synchronization can be either bigger or smaller than that of antiphase synchronization,depending on other system parameters). To study the bifurcation process, we construct a three- dimensional Poincar ´e section: The azimuth angle of the first device φ 1=2kπ, where kis an integer. The desynchronization (or synchronization) state, which is a two-dimensional torus(or one-dimensional line) in this system, exhibits a one-dimensional loop (or zero-dimensional point) on this section.To be clear, in the rest of this paper, if we just plot two ofthe total three dimensions of the Poincar ´e section, we preserve the two-dimensional desynchronization and one-dimensionalsynchronization states; otherwise, if we plot one dimension ofthe Poincar ´e section, we construct another cross section again and thus the desynchronization state will exhibit as two pointsand the synchronization state as one point. As we mentioned, there are two cases in this system, depending on the system parameters. In the first case, as shown 014418-3LI, ZHOU, HU, ˚AKERMAN, AND ZHOU PHYSICAL REVIEW B 86, 014418 (2012) FIG. 4. (Color online) The two cases of bifurcation process: (a) and (c) the in-phase synchronization is achieved first when decreasing /Delta1Hdz, and (b) and (d) the antiphase synchronization is achieved first, where (a) and (c) /Delta1J=3.75 mA and (b) and (d) /Delta1J=1.65 mA, respectively. A Poincar ´e section is constructed as the azimuth angle of the first device φ1=2kπ. Its projection on the plane θ1−θ2can be seen in panels (c) and (d), where the three attractors are of different system parameters: (c) /Delta1Hdz=0.0048 T (near the onset of the in-phase synchronization where the critical value /Delta1Hdzc1≈0.0047 T) for the desynchronization state (green limit cycle), /Delta1Hdz=0.0047 T for the in-phase state (red circle) and its transient process (orange diamonds connected by solid lines), and /Delta1Hdz=0.0042 T (near the onset of the antiphase synchronization) for the antiphase state (black square); (d) /Delta1Hdz=0.002225 T for the coexistence of desynchronization (green limit cycle) and antiphase synchronization (black square) and /Delta1Hdz=0.0022 T (near the onset of the in-phase synchronization) for the in-phase state (red circle). In panels (a) and (b), we demonstrate the bifurcation processes when decreasing /Delta1Hdz. The equilateral and inverted triangles are the maximum and minimum values of θ1of the limit cycle on the Poincar ´e section, and the red circles and black squares are the in-phase (red circle) and antiphase (black square) attractors, respectively. In the inset of panel (b), we enlarge the region around the bifurcation points, and in the inset of panel (c), we zoom in the region where the system evolves into the in-phase synchronization attractor. in Fig. 4(a), when the parameter difference /Delta1Hdzis decreased, the desynchronization attractor immediately disappears ac-companied by the emergence of a synchronization attractor at acritical parameter /Delta1H dzc1. With further investigation, we know that this synchronization state is the in-phase one. However,with further decreasing /Delta1H dzc1to another critical parameter /Delta1Hdzc2, a new synchronization state (antiphase) can emerge. Meanwhile, the in-phase synchronization state still exists. At the first bifurcation point /Delta1Hdzc1, the emergence of the in-phase synchronization is on the disappeared limit cycle,shown in Fig. 4(c), consistent with a saddle-node homoclinic bifurcation (a special case of saddle-node bifurcation, where astable node-type solution and an unstable saddle-type solutionemerge together on the limit cycle, so that the limit cycleturns into two heteroclinic orbits). In order to further checkthe bifurcation process, we plot a certain range of transientpoints in Fig. 4(c). It is clearly seen that the system evolves along a trajectory near the disappeared limit cycle towardsthe in-phase synchronization state, consistent with the birthof the heteroclinic orbits. These heteroclinic orbits and thesaddles bring some complexity to the manifolds of the system,facilitating the complicated manner of the entangled attractingbasins when the antiphase synchronization emerges, after the system reaches the second bifurcation point /Delta1H dzc2.T h e attracting basin of the antiphase synchronization increasesfrom zero to a certain portion of the whole phase space until/Delta1H dz=0. The onset of antiphase synchronization is therefore a typical saddle-node bifurcation. On the other hand, there can be the second case, where the antiphase synchronization is achieved firstly when theparameter difference /Delta1H dzis decreased, as shown in Fig. 4(b). However, the desynchronization attractor still exists, such thatwe can get the coexistence of the antiphase synchronizationand desynchronization in some regions of system parameters,as shown in Fig. 4(d). This phenomenon is quite similar to the case of synchronization in the injected STNO system. 37–39 One example of the attracting basins of this type of coexisting attractors is shown in Fig. 3(b), which also exhibits the similar pattern to that of the injected STNO system [compared toFig. 8(C) in Ref. 39]. In one part of the phase space [the lower right corner in Fig. 3(b)], the system is always attracted to the desynchronization attractor, whereas in the remaining part theattracted basins are entangled together. The reason is thatthe desynchronization orbit is a two-dimensional tori in the 014418-4MULTIPLE SYNCHRONIZATION ATTRACTORS OF ... PHYSICAL REVIEW B 86, 014418 (2012) FIG. 5. (Color online) The bifurcation process in achieving syn- chronization for different shared currents. [Case I (correspondingly II)]The onset of in-phase synchronization occurs before (or after)that of antiphase synchronization when the difference between the two STNO devices is reduced. /Delta1J denotes the amplitude of the shared currents, which stands for the coupling strength between thetwo STNOs. The two points AandBshow the two parameter points for the states, respectively, as shown in Figs. 3(a)and3(b). phase space which separates the whole phase space into two parts. When the antiphase synchronization is achieved viaa saddle-node bifurcation, its attracting basin cannot spreadthrough the tori. In a general case, the fate of the toriusually develops in one of two ways. In the first situation,it disappears via a homoclinic bifurcation. It should benoted that this type of bifurcation does not exist for oursystem of a chain of electrically coupled STNOs. However,synchronization in the injected STNO system is an exampleof this case. In the second situation, the tori disappears itselfvia a bifurcation process, resulting in the stabilizing of theother synchronization attractor. The attracting basins of the twosynchronization attractors are entangled throughout nearly thewhole space. We find that the serially connected STNO systemstudied in this work always belongs to this situation (i.e.,when the parameter difference /Delta1H dzis further decreased, in- phase synchronization is achieved via saddle-node homoclinicbifurcation), as shown in Figs. 4(b) and4(d). Actually, for both cases, the onsets of these two types of bifurcation processes do not show dependence on each other.But it should be noted that both of them will happen when/Delta1H dzdecreases, so that we always find the coexistence of the two synchronization attractors when two identical STNOsare serially connected. The whole picture of the bifurcation processes is shown in Fig. 5, where the two aforementioned cases can be easily found. To sum up, the synchronization of serially connected STNOs system is achieved via a saddle-node bifurcation. Atype of characteristic of the saddle-node bifurcation is thatone synchronization attractor emerges together with anotherunstable orbit, rather than the desynchronization or othersynchronization attractors necessarily disappearing, so thatthe coexistence of multiple synchronization attractors is acommon phenomenon. 37,38,40–44Furthermore, the manifolds in the phase space may be very complicated if heteroclinicor homoclinic orbits emerge together with the saddle-nodebifurcation. As is shown in this work, the attracting basins areentangled together, which have serious impacts on experimentsand applications of the system. V . CONCLUSIONS In a serially connected STNO system, we find that synchro- nization takes place via saddle-node bifurcation across quite awide range of system parameters, resulting in the coexistenceof multiple synchronization attractors. These coexistent attrac-tors show different oscillatory frequencies and amplitudes. Avery small noise can make the resonance peak vanish. Theseresults have great significance for experiments and applica-tions. For example, they pose a challenge to the commonmethod of experimentally identifying synchronization in suchnanosized high-frequency systems by measuring the resonancepeak. Thus new methods to identify synchronization will berequired; meanwhile, this type of synchronization does notnecessarily increase the resonance peak, which was one of theoriginal motivations for synchronizing STNOs. These effectson experiments and applications should be taken into seriousconsideration in similar nanosized high-frequency systems. ACKNOWLEDGMENTS This work is supported by Hong Kong Baptist University and conducted using the resources of the High PerformanceCluster Computing Centre, Hong Kong Baptist University,which receives funding from Research Grant Council, Univer-sity Grant Committee of the HKSAR, and HongKong BaptistUniversity. J. ˚A. acknowledges support from the Swedish Research Council, the Swedish Foundation for StrategicResearch (Future Research Leader Programme), and the G ¨oran Gustafsson Foundation. J. ˚A. is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knutand Alice Wallenberg Foundation. *cszhou@hkbu.edu.hk 1J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000). 2D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 321, 2508 (2009).3M. R. Pufall, W. H. Rippard, S. Kaka, T. J. Silva, and S. E. Russek,Appl. Phys. Lett. 86, 082506 (2005). 4T. J. Silva and W. H. Rippard, J. Magn. Magn. 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PhysRevB.85.075206.pdf

PHYSICAL REVIEW B 85, 075206 (2012) Hole spin relaxation and coefficients in Landau-Lifshitz-Gilbert equation in ferromagnetic (Ga,Mn)As K. Shen and M. W. Wu* Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Received 22 September 2011; revised manuscript received 15 January 2012; published 9 February 2012) We investigate the temperature dependence of the coefficients in the Landau-Lifshitz-Gilbert equation in ferromagnetic GaMnAs by employing the Zener model. We first calculate the hole spin relaxation time basedon the microscopic kinetic equation. We find that the hole spin relaxation time is typically several tens offemtoseconds and can present a nonmonotonic temperature dependence due to the variation of the interband spinmixing, influenced by the temperature-related Zeeman splitting. With the hole spin relaxation time, we are ableto calculate the coefficients in the Landau-Lifshitz-Gilbert equation, such as the Gilbert damping, nonadiabaticspin torque, spin stiffness, and vertical spin stiffness coefficients. We find that the nonadiabatic spin torquecoefficient βis around 0 .1−0.3 at low temperature, which is consistent with the experiment [J.-P. Adam et al. , Phys. Rev. B 80, 193204 (2009) ]. As the temperature increases, βmonotonically increases. We show that the Gilbert damping coefficient αincreases with temperature below the Curie temperature, showing good agreement with the experiments [J. Sinova et al. ,P h y s .R e v .B 69, 085209 (2004) ;Kh. Khazen et al. ,ibid. 78, 195210 (2008) ]. Moreover, we also calculate the temperature dependences of the spin stiffness and vertical spin stiffness. DOI: 10.1103/PhysRevB.85.075206 PACS number(s): 72 .25.Rb, 75 .50.Pp, 72 .25.Dc, 75 .30.Gw I. INTRODUCTION The ferromagnetic semiconductor, GaMnAs, has been pro- posed to be a promising candidate to realize all-semiconductorspintronic devices, 1,2where the existence of the ferromagnetic phase in the heavily doped sample sustains seamless spin injec-tion and detection in normal nonmagnetic semiconductors. 3,4 One important issue for such applications lies in the efficiency of the manipulation of the macroscopic magnetization, whichrelies on properties of the magnetization dynamics. Theoret-ically, the magnetization dynamics can be described by theextended Landau-Lifshitz-Gilbert (LLG) equation, 5–11 ˙n=−γn×(Heff+h)+αn×˙n−(1−βn×)(vs·∇)n −γ Mdn×/parenleftbig Ass−Av ssn×/parenrightbig ∇2n, (1) with nandMdstanding for the direction and magnitude of the magnetization, respectively. Heffis the effective magnetic field and/or the external field. The stochastic magnetic fieldhis included to describe the thermal spin fluctuations at finite temperature. 11,12The second term on the right-hand side of the equation is the Gilbert damping torque, with α denoting the damping coefficient.5,6The third one describes the spin-transfer torque induced by the spin current vs.7,8As reported, the out-of-plane contribution of the spin-transfertorque, measured by the nonadiabatic torque coefficient β, can significantly ease the domain wall motion. 7,8In Eq. ( 1), the spin stiffness and vertical spin stiffness coefficients areevaluated by A ssandAv ss, respectively, which are essentially important for the static structure of the magnetic domainwall. 10Therefore, for a thorough understanding of properties of the magnetization dynamics, the exact values of the abovecoefficients are required. In the past decade, the Gilbert damping and nonadiabatic torque coefficients have been derived via many microscopicapproaches, such as the Boltzmann equation, 13diagrammaticcalculation,14,15Fermi-surface breathing model,16–18and ki- netic spin Bloch equations.10,19According to these works, the spin lifetime of the carriers was found to be critical to both α andβ. However, to the best of our knowledge, the microscopic calculation of the hole spin lifetime in ferromagnetic GaMnAsis still absent in the literature, which prevents the determinationof the values of αandβfrom the analytical formulas. Alternatively, Sinova et al. 20identified the Gilbert damping from the susceptibility diagram of the linear-response theoryand calculated αas function of the quasiparticle lifetime and the hole density. A similar microscopic calculation onβwas later given by Garate et al. 21In those papers, the quasiparticle lifetime was also treated as a parameter instead ofexplicit calculation. Although there have been various methodsdeveloped to determine the spin relaxation time in differentsystems in the past decade, 22–32the accurate calculation of the hole spin and/or quasiparticle lifetime in ferromagneticGaMnAs is still lacking due to the complex band structureof the valence bands. In the present paper, we employ themicroscopic kinetic equation to calculate the spin lifetimeof the hole gas and then evaluate αandβin ferromagnetic GaMnAs. For the velocity of the domain wall motion dueto the spin current, the ratio β/α is an important parameter, which has attracted much attention. 14,21,33Recently, a huge ratio (∼100) in nanowire was predicted from the calculation of the scattering matrix by Hals et al.33By calculating α andβ, we are able to supply detailed information on this interesting ratio in bulk material. Moreover, the peak-to-peakferromagnetic resonance measurement revealed pronouncedtemperature and sample preparation dependences of theGilbert damping coefficient. 20,34,35For example, in annealed samples, αcan present an increase in the vicinity of the Curie temperature,20,34which, to the best of our knowledge, has not been studied theoretically in the literature. Here, weexpect to uncover the underlying physics of these features.In addition, the nonadiabatic torque coefficient βin GaMnAs 075206-1 1098-0121/2012/85(7)/075206(9) ©2012 American Physical SocietyK. SHEN AND M. W. WU PHYSICAL REVIEW B 85, 075206 (2012) has been experimentally determined from the domain wall motion and quite different values were reported by differentgroups, from 0.01 to 0.36, 36,37which need to be verified also by the microscopic calculation. Moreover, to the best of ourknowledge, the temperature dependence of βhas not been studied theoretically. We will also address this issue in thepresent paper. In the literature, the spin stiffness in GaMnAs was studied by K ¨onig et al. , 38,39who found that Assincreases with hole density due to the stronger carrier-mediated interactionbetween magnetic ions, i.e., A ss=Nh/(4m∗), with Nhandm∗ being the density and effective mass of hole gas, separately. However, as shown in our previous paper, the stiffness shouldbe modified as A ss∼Nh/[4m∗(1+β2)] in ferromagnetic GaMnAs with a finite β.10As a result, Assas well as the vertical spin stiffness Av ss=βA ssmay show a temperature dependence introduced by β. This is also a goal of the present paper. For a microscopic investigation of the hole dynamics, the valence-band structure is required for the description of theoccupied carrier states. In the literature, the Zener model 40 based on the mean-field theory has been widely used foritinerant holes in GaMnAs, 41–44where the valence bands split due to the mean-field p-dexchange interaction. In the present paper, we utilize this model to calculate the band structure withthe effective Mn concentration from the experimental valueof the low-temperature saturate magnetization in GaMnAs.The thermal effect on the band structure is introduced via thetemperature dependence of the magnetization following theBrillouin function. Then we obtain the hole spin relaxationtime by numerically solving the microscopic kinetic equationswith the relevant hole-impurity and hole-phonon scatterings.The carrier-carrier scattering is neglected here by consideringthe strongly degenerate distribution of the hole gas below theCurie temperature, where this scattering mechanism is signifi-cantly suppressed by the Pauli blocking. 23We find that the hole spin relaxation time decreases (increases) with increasing tem-perature in the small (large) Zeeman splitting regime, whichmainly results from the variation of the interband spin mixing.Then we study the temperature dependence of the coefficientsin the LLG equation, i.e., α,β,A ss, andAv ss,b yu s i n gt h e analytical formulas derived in our previous papers.10,19We find that βincreases with increasing temperature and can exceed one in the vicinity of the critical point, resulting ininteresting behaviors of other coefficients. For example, αcan present an interesting nonmonotonic temperature dependencewith the crossover occurring at β∼1. Specifically, αincreases with temperature in the low-temperature regime, which isconsistent with the experiments. Near the Curie temperature,an opposite temperature dependence of αis predicted. Similar nonmonotonic behavior is also predicted in the temperaturedependence of A v ss. Our results of βandAssalso show good agreement with the experiments. This paper is organized as follows. In Sec. II,w es e t up our model and lay out the formalism. Then we showthe band structure from the Zener model and the holespin relaxation time from microscopic kinetic equations inSec. III. The temperature dependence of the Gilbert damping, nonadiabatic spin torque, spin stiffness, and vertical spinstiffness coefficients are also shown in this section. Finally,we summarize in Sec. IV.II. MODEL AND FORMALISM In thesp-dmodel, the Hamiltonian of hole gas in GaMnAs is given by44 H=Hp+Hpd, (2) withHpdescribing the itinerant holes. Hpdis the sp-d exchange coupling. By assuming that the momentum kis still a good quantum number for itinerant hole states, one employsthe Zener model and utilizes the k·pperturbation Hamiltonian to describe the valence band states. Specifically, we take theeight-band Kane Hamiltonian H K(k)( R e f . 45) in the present paper. The sp-dexchange interaction reads Hpd=−1 N0V/summationdisplay l/summationdisplay mm/primekJmm/prime exSl·/angbracketleftmk|ˆJ|m/primek/angbracketrightc† mkcm/primek,(3) withN0andVstanding for the density of cation sites and the volume, respectively. The cation density N0=2.22×1022 cm−3. The eight-band spin operator can be written as ˆJ= (1 2σ)⊕J3/2⊕J1/2, where1 2σ,J3/2, and J1/2represent the total angular momentum operators of the conduction band, /Gamma18 valence band, and /Gamma17valence band, respectively. Jmm/prime exstands for the matrix element of the exchange coupling, with {m}and {m/prime}being the basis defined as the eigenstates of the angular momentum operators ˆJ. The summation of lis through all localized Mn spins Sl(atrl). Then we treat the localized Mn spin in a mean-field approximation and obtain ¯Hpd=−xeff/angbracketleftS/angbracketright·/parenleftBigg/summationdisplay mm/primekJmm/prime ex/angbracketleftmk|ˆJ|m/primek/angbracketrightc† mkcm/primek/parenrightBigg , (4) where /angbracketleftS/angbracketrightrepresents the average spin polarization of Mn atoms with uncompensated doping density NMn=xeffN0. Obviously, ¯Hpdcan be reduced into three blocks as ˆJ, i.e., ¯Hmm/prime pd(k)=/Delta1mmn·/angbracketleftmk|ˆJ|m/primek/angbracketrightwith the Zeeman splitting of them-band /Delta1mm=−xeffSdJmm exM(T)/M(0). Here, nis the direction of /angbracketleftS/angbracketright. For a manganese ion, the total spin Sd=5/2. The temperature-dependent spontaneous magnetization M(T) can be obtained from the following equation of the Brillouinfunction 46 BSd(y)=y(Sd+1)T/(3SdTc), (5) where y=3SdTcM(T)/[(Sd+1)TM(0)], with Tcbeing the Curie temperature. Here, BSd(y)=2Sd+1 2Sdcoth(2Sd+1 2Sdy)− 1 2Sdcoth(1 2Sdy). The Schr ¨odinger equation of the single-particle Hamilto- nian is then written as [HK(k)+¯Hpd(k)]|μ,k/angbracketright=Eμk|μ,k/angbracketright. (6) One obtains the band structure and wave functions from the diagonalization scheme. In the presence of a finite Zeemansplitting, the structure of the valence bands deviates fromthe parabolic dispersion and becomes strongly anisotropic,as we will show in the next section. Moreover, the valencebands at the Fermi surface are well separated in ferromagneticGaMnAs because of the high hole density ( >10 20cm−3) and Zeeman splitting, suggesting that the Fermi golden rule canbe used to calculate the lifetime of the quasiparticle states. Forexample, the contribution of the hole-impurity scattering due 075206-2HOLE SPIN RELAXATION AND COEFFICIENTS IN ... PHYSICAL REVIEW B 85, 075206 (2012) to negatively charged scattering centers on the μth-band state with energy /epsilon1can be expressed by /bracketleftbig τhi μ,p(/epsilon1)/bracketrightbig−1=2π/summationdisplay νni Dμ(/epsilon1)/integraldisplayd3k (2π)3/integraldisplayd3q (2π)3δ(/epsilon1−/epsilon1μk) ×δ(/epsilon1μk−/epsilon1νq)U2 k−q|/angbracketleftμk|νq/angbracketright|2f(/epsilon1μk)[1−f(/epsilon1νq)], (7) where Dμ(/epsilon1) stands for the density of states of the μth band. f(/epsilon1μk) satisfies the Fermi distribution in the equi- librium state. The hole-impurity scattering matrix elementU 2 q=Z2e4/[κ0(q2+κ2)]2withZ=1.κ0andκdenote the static dielectric constant and the screening constant underthe random-phase approximation, 47respectively. A similar expression can also be obtained for the hole-phonon scattering. However, it is very complicated to carry out the multifold integrals in Eq. ( 7) numerically for an anisotropic dispersion. Also the lifetime of the quasiparticle is not equivalent tothe spin lifetime of the whole system, which is requiredto calculate the LLG coefficients according to our previouspapers. 10,19Therefore, we extend our kinetic spin Bloch equation approach22to the current system to study the relaxation of the total spin polarization as follows. By taking into account the finite separation between different bands, one neglects the interband coherence and focuses on the carrierdynamics of the nonequilibrium population. The microscopickinetic equation is then given by ∂ tnμ,k=∂tnμ,k|hi+∂tnμ,k|hp, (8) withnμ,kbeing the carrier occupation factor at the μth band with momentum k. The first and second terms on the right-hand side stand for the hole-impurity and hole-phonon scatterings,respectively. Their expressions can be written as ∂ tnμ,k|hi=− 2πni/summationdisplay ν,k/primeU2 k−k/prime(nμk−nνk/prime)|/angbracketleftμk|νk/prime/angbracketright|2 ×δ(Eμk−Eνk/prime)( 9 ) and ∂tnμ,k|hp=− 2π/summationdisplay λ,±,ν,k/prime/vextendsingle/vextendsingleMλ k−k/prime/vextendsingle/vextendsingle2δ(Eνk/prime−Eμk±ωλ,q) ×[N± λ,q(1−nνk/prime)nμk−N∓ λ,qnνk/prime(1−nμk)] ×|/angbracketleftμk|νk/prime/angbracketright|2, (10) withN± λ,q=[exp(ωλ,q/kBT)−1]−1+1 2±1 2. The details of the hole-phonon scattering elements |Mλ q|2can be found in Refs. 23,48, and 49. Here, we assume that almost all the Mn atoms are correlated together to contribute to thecollective magnetization and only a few random Mn spinsexist, therefore the direct spin-flip collision due to the exchangeinteraction with random Mn spins is neglected. From aninitial condition with a small nonequilibrium distribution, thetemporal evolution of the hole spin polarization is carried outby J(t)=1 Nh/summationdisplay μ,k/angbracketleftμk|ˆJ|μk/angbracketrightnμ,k(t), (11) from the numerical solution of Eq. ( 8). The hole spin relaxation time can be extracted from the exponential fitting of Jwith respect to time. One further calculates the concerned coefficients such as α,β,Ass, andAv ss. III. NUMERICAL RESULTS In the Zener model, the sp-dexchange interaction constants Jmm ex are important parameters for the band structure. In the experimental works, the p-dexchange-coupling constant Jpp exwas reported to vary from −1t o2 .5 eV , depending on the doping density.50–52In ferromagnetic samples, Jpp ex is believed to be negative, which was demonstrated by theoretical estimation Jpp ex≈− 0.3e V .53In our calculation, the antiferromagnetic p-dinteraction Jpp exis chosen to be −0.5o r −1.0 eV . The ferromagnetic s-dexchange-coupling constant is taken to be Jss ex=0.2e V .44 Another important quantity for determining the Zeeman splitting is the macroscopic magnetization or the effectiveconcentration of the Mn atoms. As deduced from the low-temperature saturate magnetization, only around 50% Mnatoms can contribute to the ferromagnetic magnetization,which has been recognized as the influence of the compen-sation effect due to the deep donors (e.g., As antisites) orthe formation of sixfold-coordinated centers defect Mn 6As (Ref. 54). As only the uncompensated Mn atoms can supply holes and contribute to the ferromagnetic magnetization,55 one can also estimate the total hole density from the saturatemagnetization. 56However, the density of the itinerant hole can be smaller than the effective Mn concentration because of thelocalized effect in such disordered material. It was reported thatthe hole density is only 15%–30% of the total concentrationof the Mn atoms. 54 In our calculation, the magnetization lies along the prin- ciple axis chosen as the [001] direction.44The conventional parameters are mainly taken from those of GaAs in Refs. 57 and58. Other sample-dependent parameters such as the Curie temperature and effective Mn concentration are picked upfrom the experimental works. 20,34,36,56For samples A, B, and E (C), only the saturate magnetization at 4 (104) K wasgiven in the references. Nevertheless, one can extrapolatethe zero-temperature magnetization M sfrom Eq. ( 5). The effective Mn concentrations listed in Table Iare derived TABLE I. The parameters obtained from the experi- ments for different samples: A: Ga 0.93Mn 0.07As/Ga 0.902In0.098As. B: Ga 0.93Mn 0.07As/GaAs. C: Ga 0.93Mn 0.07As/Ga 1−yInyAs. D: Ga0.92Mn 0.08As. E: Ga 0.896Mn 0.104As0.93P0.07.Msstands for the saturate magnetization at zero temperature M(0). Tc Ms NMn (K) (emu cm−3)( 1 020cm−3) Aa130 38 8 Ba157 47 10 Cb114 33 6.9 Dc110 Ed139 53.5 11.5 aReference 34. bReference 36. cReference 20. dReference 56. 075206-3K. SHEN AND M. W. WU PHYSICAL REVIEW B 85, 075206 (2012) fromNMn=Ms/(gμBSd). It is clear to see that all of these effective Mn concentrations are much smaller than the dopingdensity ( /greaterorequalslant1.5×10 21cm−3) due to the compensation effect as discussed above. Since the saturate magnetization of sampleD is unavailable, we treat the effective Mn concentration as aparameter in this case. Moreover, since the exact values of theitinerant hole densities are unclear in such strongly disorderedsamples, we treat them as parameters. Two typical values arechosen in our numerical calculation, i.e., N h=3×1020and 5×1020cm−3. The effective impurity density is taken to be equal to the itinerant hole density. For a numerical calculation of the hole spin dynamics, the momentum space is partitioned into blocks. Comparedto the isotropic parabolic dispersion, the band structurein ferromagnetic GaMnAs is much more complex, as wementioned above [refer to Figs. 1(b) and 4]. Therefore, we need to extend the partition scheme used in isotropic parabolicdispersion 59into the anisotropic case. In our scheme, the radial partition is still carried out with respect to the equal-energyshells, while the angular partition is done by following Ref. 59. In contrast to the isotropic case, the number of states inone block is generally different from that in another block,even if both of them are on the same equal-energy shell. Wecalculate the number of states of each block from its volumein momentum space. A. Density of states By solving Eq. ( 5), one obtains the magnetization at finite temperature M(T) and the corresponding Zeeman energy /Delta1pp. From Fig. 1(a), it is seen that the Zeeman energy is tens of meV at low temperature and decreases sharply near theCurie temperature due to the decrease of the magnetization.To show the anisotropic nonparabolic feature of the bandstructure in the presence of the Zeeman splitting, we illustratethe valence bands along the [001] and [111] directions inFig. 1(b), which are obtained from Eq. ( 6)a tT/T c=0.1 forNMn=8×1020cm−3. In this case, the Zeeman splitting /Delta1pp=45 meV . The Fermi levels for the hole densities Nh=3×1020and 5 ×1020cm−3are shown in the figure. One can see that all of the four upper bands can be occupiedand the effective mass approximation obviously breaks down. By integrating over the volume of each equal-energy shell, one obtains the density of states (DOS) of each band asfunction of energy in Figs. 1(c) and 1(d). Here the energy is defined in the hole picture so that the sign of the energyis opposite to that in Fig. 1(b). It is seen that the DOS of the upper two bands are much larger than those of the otherbands, regardless of the magnitude of the Zeeman splitting.ForT/T c=0.99, the systems approach the paramagnetic phase and the nonparabolic effect is still clearly seen fromthe DOS in Fig. 1(d), especially in the high-energy regime. Moreover, the pronounced discrepancy of the DOS for thetwo heavy-hole bands suggests the finite splitting betweenthem. We find that these features are closely connected withthe anisotropy of the valence bands, corresponding to theLuttinger parameters γ 2/negationslash=γ3in GaAs.60In our calculation, we take γ1=6.85,γ2=2.1, and γ3=2.9f r o mR e f . 58.A s a comparison, we apply a spherical approximation ( γ1=6.85 andγ2=γ3=¯γ=2.5) and find that the two heavy-hole 0 10 20 30 40 50 60 0.2 0.4 0.6 0.8 1Δpp (meV)T/Tc (a) 6.9×1020 cm-3 8×1020 cm-3 1×1021 cm-3 -0.4-0.3-0.2-0.1 0 0.1-0.1 0 0.1 0.2 E (eV)k (2π/a ) [111] [001] (b) 0 51015 -0.1 -0.05 0 0.05 0.1 0.15 0.2DOS (1020/eVcm3) E (eV)T = 0.1 TcNMn = 8×1020 cm-3 (c) -0.05 0 0.05 0.1 0.15 0.2 0 5 10 15 DOS (1020/eVcm3) E (eV)T = 0.99 Tc (d) FIG. 1. (Color online) (a) Zeeman energy as function of temper- ature for three effective Mn concentrations. (b) The valence-bandstructure along the [001] and [111] directions with /Delta1 pp=45 meV . The blue dashed curve illustrates the Fermi level for the hole density Nh=3×1020cm−3, while the green dotted curve gives Nh=5×1020cm−3. The density of states as a function of energy at (c) T/T c=0.1a n d( d ) T/T c=0.99 for the uncompensated Mn density NMn=8×1021cm−3. In (d), the blue dashed curve stands for the upper heavy-hole band from the spherical approx- imation, and the corresponding DOS from the analytical formula (√ 2E[√m∗/(2π¯h)]3) is given as the green dotted curve. Here, Jpp ex=− 0.5e V . bands become approximately degenerate.23In Fig. 1(d),w e plot the DOS of the upper heavy-hole band for γ2=γ3as the blue dashed curve, which shows good agreement with thatfrom the analytical expression with m ∗=m0/(γ1−2¯γ). This suggests the good precision of our numerical scheme. B. Hole spin relaxation Here we investigate the hole spin dynamics by numerically solving the microscopic kinetic equation, i.e., Eq. ( 8). By taking into account the equilibrium hole spin polarization, wefit the temporal evolution of the total spin polarization alongthe [001] direction by J z(t)=J0 z+J/prime ze−t/τs, (12) where J0 zandJ/prime zcorrespond to the equilibrium and nonequi- librium spin polarizations, respectively. τsis the hole spin relaxation time. In all the cases of the present paper, the equilibrium hole spin polarization for a fixed hole density is found to beapproximately linearly dependent on the Zeeman splitting. InFig. 2,J 0 zin samples A and B (similar behavior for others) are shown. One notices that the average spin polarization becomessmaller with an increase of the hole density, reflecting the largeinterband mixing for the states in the high-energy regime. 075206-4HOLE SPIN RELAXATION AND COEFFICIENTS IN ... PHYSICAL REVIEW B 85, 075206 (2012) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 10 20 30 40 50 60Equilibrium Hole Polarization Δpp (meV)A: Nh=3×1020 cm-3 5×1020 cm-3 B: Nh=3×1020 cm-3 5×1020 cm-3 FIG. 2. (Color online) The equilibrium hole spin polarization J0 z as function of the Zeeman splitting for samples A and B. Here, Jpp ex=− 0.5e V . The temperature dependence of the hole spin relaxation time in samples A and B with Jpp ex=− 0.5e Vi ss h o w n in Fig. 3(a), where the spin relaxation time monotonically decreases with increasing temperature. This feature can beunderstood from the enhancement of the interband mixingas the Zeeman splitting decreases (shown below). 61To gain a complete picture of the role of the Zeeman splitting on the holespin relaxation in ferromagnetic GaMnAs, we also carry outthe calculation with the exchange constant J pp ex=− 1e V .44,50 Very interestingly, one finds that the hole spin relaxation time at low temperature increases with increasing temperature,resulting in a nonmonotonic temperature dependence of thehole spin relaxation time in sample B. The results in this caseare shown as solid curves in Fig. 3(b),w h e r ew ea l s op l o t the Zeeman splitting dependence of the hole spin relaxationtime as dashed curves. It is seen that the hole spin relaxationtime for the hole density N h=3×1020cm−3first increases with increasing temperature (alternatively speaking, decreas-ing Zeeman splitting) and starts to decrease at around 0 .8T c, where the Zeeman splitting /Delta1pp=70 meV . To understand this feature, we show the typical band structure in the increase(decrease) regime of the hole relaxation time at T/T c=0.4 (0.99), corresponding to /Delta1pp=105 (16.7) meV , in the inset at the left (right) upper corner. The Fermi levels of the holedensity 3 ×10 20cm−3are labeled by blue dashed curves. One finds that the carrier occupations in the increase and decreaseregimes are quite different. Specifically, all of the four upperbands are occupied in the decrease regime while only threevalence bands are relevant in the increase regime. One may naturally expect that the increase regime origi- nates from the contribution of the fourth band via the inclusionof the additional scattering channels or the modification of thescreening. However, we rule out this possibility through thecomputation with the fourth band artificially excluded, wherethe results are qualitatively the same as those in Fig. 3(b). Moreover, the variations of the screening and the equilibriumdistribution at finite temperature are also demonstrated tobe irrelevant to the present nonmonotonic dependence byour calculation (not shown here). Therefore, the interestingfeature has to be attributed to the variations of the banddistortion and spin mixing due to the exchange interaction. 30 40 50 60 70 80 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1τs (fs) T/Tc(a)Jexpp = -0.5 eV A: Nh=3×1020 cm-3 5×1020 cm-3 B: Nh=3×1020 cm-3 5×1020 cm-3 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 0 20 40 60 80 100 120τs (fs) T/TcΔpp (meV) (b) Jexpp = -1 eV0.4 Tc 0.99 Tc B: Nh=3×1020 cm-3 5×1020 cm-3 FIG. 3. (Color online) (a) Spin relaxation time as a function of temperature with Jpp ex=− 0.5 eV for samples A and B. (b) Spin relaxation time as a function of temperature and Zeeman splitting obtained from the calculation with Jpp ex=− 1 eV for sample B. The inset at the left (right) upper corner illustrates the band structure from the [001] direction to the [111] direction [refer to Fig. 1(b)]f o r T/T c=0.4 (0.99) and /Delta1pp=105 (16.7) meV . The Fermi levels of Nh=3×1020and 5×1020cm−3are shown as the blue dashed and green dotted curves, respectively, in the insets. This is supported by our numerical calculation, where the nonmonotonic behavior disappears once the effect of theinterband mixing is excluded by removing the wave-functionoverlaps |/angbracketleftμk|νk /prime/angbracketright|2in Eqs. ( 9) and ( 10) (not shown here). For a qualitative understanding of the nonmonotonic temperature dependence of the hole spin relaxation time,we plot the Fermi surface in the k x-kz(ky=0) and kx-ky (kz=0) planes at Nh=3×1020cm−3in Fig. 4. We choose typical Zeeman splittings in the increase regime ( /Delta1pp= 105 meV), the decrease regime ( /Delta1pp=16.7 meV), and also the crossover regime ( /Delta1pp=70 meV). One notices that the Fermi surfaces in Figs. 4(a) and 4(d) are composed of three closed curves, meaning that only three bands are occupiedfor/Delta1 pp=105 meV [see also the inset of Fig. 3(b)]. For the others with smaller Zeeman splittings, all of the four upperbands are occupied. The spin expectation of each state atthe Fermi surface is represented by the color coding. Notethat the spin expectation of the innermost band for /Delta1 pp=70 meV is close to −1.5 [see Figs. 4(b) and 4(e)], suggesting that this band is the spin-down heavy-hole band and themixing of other spin components in this band is marginal. 075206-5K. SHEN AND M. W. WU PHYSICAL REVIEW B 85, 075206 (2012) -1.5-1-0.5 0 0.5 1 1.5ξΔpp = 105 meV 70 meV 16.7 meV (a)-0.2-0.10.00.10.2kz (2π/a) -1.5-1-0.5 0 0.5 1 1.5 (b) -1.5-1-0.5 0 0.5 1 1.5 (c) -1.5-1-0.5 0 0.5 1 1.5 (d) -0.2 -0.1 0.0 0.1 0.2-0.2-0.10.00.10.2ky (2π/a) -1.5-1-0.5 0 0.5 1 1.5 (e) -0.2 -0.1 0.0 0.1 0.2 kx (2π/a)-1.5-1-0.5 0 0.5 1 1.5 (f) -0.2 -0.1 0.0 0.1 0.2 FIG. 4. (Color online) The Fermi surface in the kx-kz(ky= 0) and kx-ky(kz=0) planes with (a), (d) /Delta1pp=105 meV , (b), (e) 70 meV , and (c), (f) 16.7 meV . The color coding represents thespin expectation of each state, ξ=/angbracketleftμ|J z|μ/angbracketright. Here, Nh=3×1020 cm−3. Therefore, the spin-flip scattering related to this band is weak and cannot result in an increase of the hole spin relaxation timementioned above. By comparing the results with /Delta1 pp=105 and 70 meV , one notices that the spin expectation of the Fermisurface of the outermost band is insensitive to the Zeemansplitting. Therefore, this band cannot be the reason also for theincrease regime. Moreover, for the second and third bandsin Figs. 4(a) and 4(b), from the comparable color coding between the two figures in this regime [see also Figs. 4(d) and 4(e) withk z=0], one finds that the spin expectation for the states with small kzis also insensitive to the Zeeman splitting. However, for the states with large kz, the spin expectation of the spin-down states ( ξ< 0) approaches a large magnitude ( −1.5) with decreasing Zeeman splitting, suggesting the decrease ofthe mixing from the spin-up states. As a result, the interbandspin-flip scattering from/to these states becomes weak and thehole spin relaxation time increases. In the decrease regime ofthe hole spin relaxation time, Figs. 4(c) and4(f)show that the two outer (inner) bands approach each other, leading to a strongand anisotropic spin mixing. Therefore, the spin-flip scatteringbecomes more efficient in this regime and the spin relaxationtime decreases. One may suppose that the nonmonotonictemperature dependence of the hole spin relaxation timecan also arise from the variation of the shape of the Fermisurface, according to Fig. 4. However, this variation itself is not the key to the nonmonotonic behavior, because thecalculation with this effect but without band mixing cannotrecover the nonmonotonic feature as mentioned above. For thehole density N h=5×1020cm−3, the structures of the Fermi surface at /Delta1pp=105 meV are similar to those in Figs. 4(b) and4(e). This explains the absence of the increase regime for this density in Fig. 3(b). Moreover, we should point out that the increase regime of the hole spin relaxation time in sample A for Jpp ex=− 1e V is much narrower than that in sample B. The reason liesin the fact of a lower effective Mn density in sample A,leading to a smaller maximal Zeeman splitting ∼90 meV , only slightly larger than the crossover value of 70 meV at N h=3× 1020cm−3.As a summary of this section, we find different temperature dependences of the hole spin relaxation time due to thedifferent values of the effective Mn concentration, hole density,and exchange-coupling constant J pp ex. In the case with a large coupling constant and a high effective Mn concentration,the interband spin mixing can result in a nonmonotonictemperature dependence of the hole spin relaxation time.Our results suggest a possible way to estimate the exchange-coupling constant with knowledge of the itinerant hole density,i.e., by measuring the temperature dependence of the hole spinrelaxation time. Alternatively, the discrepancy between thehole relaxation time from different hole densities in Fig. 3(b) suggests that one can also estimate the itinerant hole density ifthe exchange-coupling constant has been measured from othermethods. C. Gilbert damping and nonadiabatic torque coefficients Facilitated by the knowledge of the hole spin relaxation time, we can calculate the coefficients in the LLG equation.According to our previous papers, 10,19the Gilbert damping and nonadiabatic spin torque coefficients can be expressed as α=Jh/[NMn|/angbracketleftS/angbracketright|(β+1/β)] (13) and β=1/(2τs/Delta1pp), (14) respectively. In Eq. ( 13),Jhrepresents the total equilibrium spin polarization of the itinerant hole gas, i.e., Jh=NhJ0 z, withJ0 zbeing the one defined in Eq. ( 12) in our paper. The average spin polarization of a single Mn ion is given by |/angbracketleftS/angbracketright| = SdM(T)/M(0). From Eqs. ( 13) and ( 14), one finds that the hole spin relaxation time, together with the Zeeman spittingand hole spin polarization, can influence the Gilbert dampingand nonadiabatic torque coefficients. In Figs. 5(a),5(c), and 5(e), the nonadiabatic spin torque coefficients βin samples A–C are plotted as functions of temperature. Our results in sample C show good agreementwith the experimental data in Fig. 5(e). 36At low temperature, the value of βis around 0.1–0.3, which is also comparable with the previous theoretical calculation.21Very interestingly, one finds that βsharply increases when the temperature approaches the Curie temperature. This can be easily understood fromthe pronounced decreases of the spin relaxation time and theZeeman splitting in this regime [see Figs. 1(a) and 3]. By comparing the results with different values of the exchange-coupling constant, one finds that βfromJ pp ex=− 1e Vi s generally about one half of that obtained from Jpp ex=− 0.5e V because of the larger Zeeman splitting. Moreover, one noticesthat the nonmonotonic temperature dependence of the holespin relaxation time in Fig. 3(b) is not reflected in βdue to a marked decrease of the Zeeman splitting with increasingtemperature. In all cases, the values of βcan exceed 1 very near the Curie temperature. The results of the Gilbert damping coefficient from Eq. ( 13) are shown as curves in Figs. 5(b),5(d), and 5(f).T h e dots in these figures are the reported experimental datafrom the ferromagnetic resonance along different magnetic-field orientations. 34Both the magnitude and the temperature dependence of our results agree well with the experimental 075206-6HOLE SPIN RELAXATION AND COEFFICIENTS IN ... PHYSICAL REVIEW B 85, 075206 (2012) 0 0.5 1 1.5 2 2.5 3 20 40 60 80 100 120β T (K)Sample A(a)-0.5 eV, Nh=3×1020 cm-3 5×1020 cm-3 -1.0 eV, Nh=3×1020 cm-3 5×1020 cm-3 0 0.01 0.02 0.03 0.04 20 40 60 80 100 120α T (K)(b) 0 0.5 1 1.5 2 2.5 3 20 40 60 80 100 120 140 160β T (K)Sample B(c) 0 0.01 0.02 0.03 0.04 20 40 60 80 100 120 140 160α T (K)(d) 0 0.5 1 1.5 2 2.5 3 20 40 60 80 100 120β T (K)Sample C(e) 0 0.01 0.02 0.03 0.04 0.05 20 40 60 80 100 120α T (K)(f) FIG. 5. (Color online) βandαas functions of temperature with Jpp ex=− 0.5a n d −1.0 eV in samples A–C. In (b) and (d), the dots represent the experimental data from the ferromagnetic resonance measurement for [001] (brown solid upper triangles), [110] (orange solid circles), [100] (green open squares), and [1-10] (black open lower triangles) dc magnetic-field orientations (Ref. 34). The brown solid square in (e) stands for the experimental result from the domain wall motion measurement (Ref. 36). data. From Fig. 2, one can conclude that the prefactor in Eq. ( 13),Jh/(NMn|/angbracketleftS/angbracketright|), is almost independent of temperature. Therefore, the temperature dependence of αmainly results from the nonadiabatic spin torque coefficient β. Specifically, α is insensitive to the temperature in the low-temperature regimeand it gradually increases with increasing temperature dueto the increase of β. Moreover, we predict that αbegins to decrease with increasing temperature once βexceeds 1. This crossover lying at β≈1 can be expected from Eq. ( 13). By comparing the results with different values of J pp ex, one finds that the value of αis robust against the exchange-coupling constant in the low-temperature regime. In this regime, β/lessmuch1 and one can simplify the expression of the Gilbert dampingcoefficient as α≈N hJ0 z/(NMnSdτs/Delta1pp). Since the total hole spin polarization is proportional to the Zeeman splitting (seeFig. 2) and τ sis only weakly dependent on the Zeeman splitting (see Fig. 3) in this regime, the increase of Jpp ex does not show a significant effect on α. However, at high temperature, the scenario is quite different. For example,one has the maximum of the Gilbert damping coefficientα m≈NhJ0 z/(2NMn|/angbracketleftS/angbracketright|)∝Jpp exatβ=1. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 20 40 60 80 100β T (K)Sample D(a)-0.5 eV, Nh=3×1020 cm-3 5×1020 cm-3 -1.0 eV, Nh=3×1020 cm-3 5×1020 cm-3 0 0.02 0.04 0.06 0.08 0.1 20 40 60 80 100α T (K)(b) FIG. 6. (Color online) βandαas functions of temperature by taking NMn=5×1020cm−3withJpp ex=− 0.5a n d −1.0e V in sample D. The dots are from the ferromagnetic resonancemeasurement (Ref. 20) for [001] (brown solid upper triangles) and [110] (orange solid circles) dc magnetic-field orientations. Since the effective Mn concentration of sample D is unavailable as mentioned above, here we take NMn=5× 1020cm−3. The results are plotted in Fig. 6. It is seen that the Gilbert damping coefficients from our calculationwithJpp ex=− 1 eV agree very well with the experiment. As reported, the damping coefficient in this sample is muchlarger ( ∼0.1) before annealing. 20The large Gilbert damping coefficient in the as-grown sample may result from the directspin-flip scattering between the holes and the random Mnspins, existing in low-quality samples. In the presence ofthis additional spin-flip channel, the hole spin relaxation timebecomes shorter and results in an enhancement of αand β(forβ< 1). Moreover, in the low-temperature regime, a decrease of the Gilbert damping coefficient was observed byincreasing temperature, 20which is absent in our results. This may originate from the complicated localization or correlationeffects in such a disordered situation. The quantitativelymicroscopic study in this case is beyond the scope of thepresent paper. In addition, one notices that βin Ref. 37was determined to be around 0.01, which is one order of magnitude smaller than our result. The reason is because of the incorrect parameter used in that work, as pointed out by Adam et al. 36 D. Spin stiffness and vertical spin stiffness In this section, we calculate the spin stiffness and ver- tical spin stiffness coefficients according to our previousderivation 10 Ass=Nh/[4m∗(1+β2)] (15) and Av ss=Nhβ/[4m∗(1+β2)]. (16) Since the effective mass m∗is a rough description for the anisotropic valence bands in the presence of a large Zeemansplitting, it is difficult to obtain an accurate value of thestiffness coefficients from these formulas. Nevertheless, onecan still estimate these coefficients with the effective masstaken as a parameter. By fitting the DOS of the occupied holestates, we find m ∗≈m0, which is consistent with previous work.44The results are plotted in Fig. 7. The sudden decrease ofAssoriginates from an increase of βin the vicinity of the 075206-7K. SHEN AND M. W. WU PHYSICAL REVIEW B 85, 075206 (2012) 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140Ass(v) (pJ/m) T (K)Nh=3×1020 cm-3, 1.0m0 1×1020 cm-3, 1.0m0 1×1020 cm-3, 0.5m0 FIG. 7. (Color online) Spin stiffness (vertical spin stiffness) coefficient as a function of temperature is plotted as curves with (without) symbols. The calculation is carried out with Jpp ex=− 0.5e V in sample E. The effective mass is taken to be 1.0(0.5) m0as labeled in the figure. The brown solid (from the period of the domains) andopen (from the hysteresis cycle) squares are the experimental data of spin stiffness from Ref. 56. Curie temperature (see Fig. 5). Our results are comparable with the previous theoretical work from the six-band model.39 As a comparison, we take m∗=0.5m0, which is widely used to describe the heavy hole in the low-energy regime in theabsence of Zeeman splitting. 62The spin stiffness becomes two times larger. Moreover, Av ssis found to present a nonmonotonic behavior in the temperature dependence, as predicted byEq. ( 16). In Fig. 7, we also plot the experimental data of the spin stiff- ness coefficient from Ref. 56. It is seen that these values of A ss are comparable with our results and show a decrease as the tem- perature increases. However, one notices that the experimentaldata is more sensitive to the temperature, especially for thosedetermined from the domain period in the low-temperatureregime. This may originate from strong anisotropic interbandmixing and inhomogeneity in the real material. In Ref. 10, we have shown that the vertical spin stiffness can lead to the magnetization rotated around the easy axis within the domain wall structure by /Delta1ϕ=(/radicalbig 1+β2−1)/βin the absence of a demagnetization field. For β=1,/Delta1ϕ≈0.13π, while /Delta1ϕ=β/2→0f o rβ/lessmuch1. As illustrated above, βis always larger than 0.1. Therefore, the vertical spin stiffness canpresent observable modifications in the domain wall structurein the GaMnAs system. 10Finally, we should emphasize that our calculation is based on the mean-field theory, where the fluctuations are neglected.For temperatures very close to the Curie temperature, thethermal spin fluctuations can strongly influence the magne-tization dynamics and hence our predictions, e.g., the peaksof the Gilbert damping and vertical spin stiffness. An explicitinclusion of the spin fluctuations is beyond the scope of thepresent paper. IV . SUMMARY In summary, we theoretically investigate the temperature dependence of the LLG coefficients in ferromagnetic GaM-nAs, based on the microscopic calculation of the hole spinrelaxation time. In our calculation, we employ the Zener model with the band structure carried out by diagonalizing the 8 ×8 Kane Hamiltonian together with the Zeeman energy due tothesp-dexchange interaction. We find that the hole spin relaxation time can present different temperature dependences,depending on the effective Mn concentration, hole density,and exchange-coupling constant. In the case with a high Mnconcentration and a large exchange-coupling constant, thehole spin relaxation time can be nonmonotonically dependenton temperature, resulting from the different interband spinmixings in the large and small Zeeman splitting regimes.These features are proposed to be for the estimation of theexchange-coupling constant or itinerant hole density. By sub-stituting the hole relaxation time, we calculate the temperaturedependence of the Gilbert damping, nonadiabatic spin torque,spin stiffness, and vertical spin stiffness coefficients. We obtaina nonadiabatic spin torque coefficient around 0 .1–0.3a tl o w temperature, which is consistent with the experiment. As thetemperature increases, this coefficient shows a monotonicincrease. In the low-temperature regime, the Gilbert dampingcoefficient increases with temperature, which shows goodagreement with the experiments. We find that the spin stiffnessand vertical spin stiffness can be also markedly influenced bythe temperature. ACKNOWLEDGEMENT This work was supported by the National Basic Research Program of China under Grant No. 2012CB922002 and theNational Natural Science Foundation of China under GrantNo. 10725417. *mwwu@ustc.edu.cn 1H. Ohno, Science 281, 951 (1998). 2T. Jungwirth, J. Sinova, J. Ma ˇsek, J. Ku ˇcera, and A. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006). 3M. G. 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PhysRevLett.122.117205.pdf

Barkhausen Noise from Precessional Domain Wall Motion Touko Herranen1and Lasse Laurson1,2,* 1Helsinki Institute of Physics, Department of Applied Physics, Aalto University, P.O. Box 11100, FI-00076 Aalto, Espoo, Finland 2Computational Physics Laboratory, Tampere University, P.O. Box 692, FI-33014 Tampere, Finland (Received 8 June 2018; revised manuscript received 28 January 2019; published 21 March 2019) The jerky dynamics of domain walls driven by applied magnetic fields in disordered ferromagnets —the Barkhausen effect —is a paradigmatic example of crackling noise. We study Barkhausen noise in disordered Pt =Co=Pt thin films due to precessional motion of domain walls using full micromagnetic simulations, allowing for a detailed description of the domain wall internal structure. In this regime the domain walls contain topological defects known as Bloch lines which repeatedly nucleate, propagate, andannihilate within the domain wall during the Barkhausen jumps. In addition to bursts of domain wall propagation, the in-plane Bloch line dynamics within the domain wall exhibits crackling noise and constitutes the majority of the overall spin rotation activity. DOI: 10.1103/PhysRevLett.122.117205 Understanding the bursty crackling noise response of elastic objects in random media —domain walls (DWs) [1], cracks [2], fluids fronts invading porous media [3], etc.—to slowly varying external forces is one of the main problems of statistical physics of materials. An important example is given by the magnetic field driven dynamics of DWs in disordered ferromagnets, where they respond to a slowly changing external magnetic field by exhibiting a sequence of discrete jumps with a power-law size distribution [1,4]. This phenomenon, known as the Barkhausen effect [5], has been studied extensively, and a fairly well-established picture of the possible universality classes of the avalanche dynamics, using the language of critical phenomena, is emerging [1,4]. Magnetic DWs constitute a unique system exhibiting crackling noise since the driving field may, in addition to pushing the wall forward, excite internal degrees of free- dom within the DW [6]. This effect is well known especially in the nanowire geometry —important for the proposed spintronics devices such as the racetrack memory [7]—where the onset of precession of the DW magnetiza- tion above a threshold field leads to an abrupt drop in the DW propagation velocity (the Walker breakdown [8]), and hence to a nonmonotonic driving field versus DW velocity relation [9]; these features are well captured by the so-called 1dmodels [10]. In wider strips or thin films, the excitations of the DW internal magnetization accompanying the velocity drop cannot be described by precession of an individual mag- netic moment. Instead, one needs to consider the nuclea- tion, propagation, and annihilation of topological defects known as Bloch lines (BLs) within the DW [11–13]. BLs, i.e., transition regions separating different chiralities of the DW, have been studied in the context of bubble materialssince the 1970s [13]. Their role in the physics of the Barkhausen effect needs to be studied. The typical models of Barkhausen noise, such as elastic interfaces in random media [4,14] , scalar field models [15], or the random field Ising model (RFIM) [16–18], exclude BLs by construction. Here, we focus on understanding the consequences of the presence of BLs within DWs on the jerky DW motionthrough a disordered thin ferromagnetic film. To this end, we study field-driven DW dynamics considering as a test system a 0.5-nm-thick Co film within a Pt =Co=Pt multi- layer [19] with perpendicular magnetic anisotropy (PMA) by micromagnetic simulations, able to fully capture the DW internal structure. By tuning the strength of quenched disorder, we match the DW velocity versus applied field curve to the experimental one reported in Ref. [19]. This leads to a depinning field well above the Walker field of the corresponding disorder-free system. Hence, when applying a driving scheme corresponding to a quasistatic constant imposed DW velocity, the resulting Barkhausen jumps take place within the precessional regime. We find that in addition to avalanches of DW propaga- tion, also the in-plane BL magnetization dynamics within the DW exhibits crackling noise, and is responsible for the majority of the overall spin rotation activity during the Barkhausen jumps; the latter dynamics is not directly observable in typical experiments (magneto-optical imag- ing[20] or inductive recording [21]). The DW can locally move backwards, so it does not obey the Middleton no- passing theorem [22]. Functional renormalization group calculations [23]crucially depend on this property, but we find that in linelike DWs, BLs do not change the scaling picture of avalanches if one looks at measures related to DW displacement. Remarkably, simple scaling relations applicable to short-range elastic strings in random mediaPHYSICAL REVIEW LETTERS 122, 117205 (2019) 0031-9007 =19=122(11) =117205(5) 117205-1 © 2019 American Physical Societyremain valid in the much more complex scenario we consider here. In our micromagnetic simulations of the DW dynamics, the Landau-Lifshitz-Gilbert (LLG) equation, ∂m=∂t¼ γHeff×mþαm×∂m=∂t, describing the time evolution of the magnetization m¼M=MS, is solved using the MUMAX3software [24]. In the LLG equation, γis the gyromagnetic ratio, αthe Gilbert damping parameter, and Heffthe effective field, with contributions due to exchange, anisotropy, Zeeman, and demagnetizing energies. The simu- lated magnetic material is a 0.5-nm-thick Co film in a Pt=Co=Pt multilayer with PMA. Micromagnetic parameters for the material are exchange stiffness Aex¼1.4× 10−11J=m, saturation magnetization MS¼9.1×105A=m, uniaxial anisotropy Ku¼8.4×105J=m3, and damping parameter α¼0.27; these have been experimentally deter- m i n e di nR e f . [19]. The resulting DW width parameter is ΔDW¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aex=K0p ≈7nm, where K0¼Ku−1 2μ0M2 Sis the effective anisotropy. The system size is fixed to Lx¼1024 nm, Ly¼4096 nm, and Lz¼0.5nm. The simulation cell dimensions are Δx¼Δy¼2nm and Δz¼0.5nm. In every simulation the DW, separating domains oriented along /C6z, is initialized along the þy direction as a Bloch wall with the DW magnetization in theþydirection. Periodic boundary conditions are used in the ydirection to avoid boundary effects. The LLG equation is then solved using the Dormand-Prince solver (RK45) with an adaptive time step. For thin films with thicknesses of only a few atoms, a natural source of disorder [25] is given by thickness fluctuations of the film. Thus, for simulations of disorderedfilms, the sample is divided into “grains ”of linear size 20 nm (defining the disorder correlation length) by Voronoi tessellation, each grain having a normally distributed random thickness t G¼hþNð0;1Þrh, with rthe relative magnitude of the grain-to-grain thickness variations and h the mean thickness of the sample. These thickness fluctua-tions are then modeled using an approach proposed inRef. [26], by modulating the saturation magnetization and anisotropy constant according to M G S¼MStG=hand KGu¼Kuh=tG. We start by considering the response of a Bloch DW to a constant Bextalong the þzdirection; this leads to DW motion in the þxdirection. Our algorithm solves the spatially averaged DW velocity vDWby determining the local DW position along the DW as XðyÞ¼min yjmzðxÞj, withmzðxÞinterpolated across the minimum. By scanning different values of the thickness fluctuations r, we found thatr¼0.03produces a similar vDWðBextÞbehavior as in the finite temperature experiments of Ref. [19] for the 0.5-nm-thick sample in the range of 0 –30 mT. Because of thermal rounding of the depinning transition [27] in experiments of Ref. [19], this value of rshould be interpreted as a lower limit. The resulting vDWðBextÞcurveis shown in Fig. 1, along with the corresponding curve from the disorder-free system. The depinning field of roughly15 mT due to the quenched pinning field exceeds the Walker threshold of 2.5 mT of the nondisordered system, thus suggesting that the experiment of Ref. [19]is operating in the precessional regime. We then proceed to address the main problem of this Letter, i.e., how Barkhausen noise is affected by the presence of BLs. To this end, we consider the system with r¼0.03, and a simulation protocol involving a moving simulation window where the DW center of mass is always kept within one discretization cell from the center of the simulation window, using the ext_centerWall function ofM UMAX3with a modified tolerance. This minimizes effects due to demagnetizing fields that may slow down the DW during avalanches. To “reintroduce ”this feature in a controllable fashion, we utilize a driving protocol analo-gous to the quasistatic limit of the constant velocity drive, where the driving field B extis decreased during avalanches (i.e., when vDW>vth DW¼0.1m=s) as _Bext¼−kjvDWj, with k¼0.18mT=nm chosen to adjust the avalanche cutoff to be such that the lateral extent of the largestavalanches is smaller than L y, in order to avoid finite size effects. In between avalanches (i.e., when vDW<0.1m=s), Bextis ramped up at a rate _Bext¼0.037mT=ns until the next avalanche is triggered. The latter rate is chosen to get well-separated avalanches in time, while at the same time avoiding excessively long waiting times between ava-lanches. This leads to a B extðtÞ, which after an initial transient fluctuates in the vicinity of the depinning field. To characterize the bursty DW dynamics, in addition to the “standard ”DW velocity vDW, we study different measures of the rate of spin rotation (or “activity ”) associated with the DW dynamics. To study the dynamics FIG. 1. vDWas a function of Bextin a perfect strip and in a disordered system where the disorder strength rhas been tuned to roughly match the vDWðBextÞcurve with the experimental one of Ref. [19]; the disorder-induced depinning field exceeds the Walker field of the perfect strip. Inset: Example snapshot of arough DW containing BLs in the disordered system withB ext¼17mT.PHYSICAL REVIEW LETTERS 122, 117205 (2019) 117205-2of the internal degrees of freedom of the DW, we consider separately contributions from in-plane and out-of-plane spin rotation, defined as AxyðtÞ¼P i∈B_ϕi·jmi;xyjand AzðtÞ¼P i∈B_θi, respectively, where ϕiand θiare the spherical coordinate angles of the magnetization vector mi in the ith discretization cell. The sums are taken over a band Bextending 20 discretization cells around the DW on both sides, moving with the DW. The multiplication by jmi;xyjin Axyis included to consider only contributions originating from inside of the DW. Figure 2(a) shows examples of DW magnetization configurations in between successive avalanches, defined by thresholding the vDWðtÞsignal with vth DW¼0.1m=s. To quickly reach the stationary avalanche regime, we use 15 mT as the initial field. Notice how the initially straight Bloch DW (green) is quickly transformed into a roughinterface with a large number of BLs, visible in Fig. 2(a)as abrupt changes of color along the DW; see also Movie 1 in Supplemental Material [28]. Figures 2(b)–2(d) show the corresponding v DWðtÞ, AxyðtÞ, and−AzðtÞsignals, respectively; notice that AzðtÞ has a minus sign to compensate for the fact that Bextalong þztends to decrease θi. In addition to the fact that all three signals exhibit the characteristic bursty appearance of acrackling noise signal, we observe two main points. (i)v DW, as well as the two activity signals AxyðtÞand −AzðtÞ, may momentarily have negative values; this indi- cates that the DW center of mass is moving against the direction imposed by Bext, and hence the DW does not respect the Middleton theorem [22]. (ii) While the appear- ance of the three signals is quite similar, AxyðtÞhas a significantly larger magnitude than AzðtÞ: We findhAxy=Azi≈1.7, showing that in relative terms the BL activity within the DW is more pronounced during ava- lanches than the overall propagation of the DW. Comparing the distribution PðϕinitialÞof the local in-plane magnetiza- tion angle ϕinitial of the DW segments from which an avalanche is triggered to that of the angle ϕDWof all DW segments (Fig. 3) suggests that the avalanche triggering process is not affected by the local DW structure. To analyze the statistical properties of the Barkhausen avalanches, we consider 200 realizations of the three signals discussed above. Denoting the signal by VðtÞ, the avalanche size is defined as SV¼RT 0½VðtÞ−Vth/C138dt, where Vthis the threshold level used to define the avalanches; the integral is over a time interval T(the avalanche duration) during which the signal stays contin- uously above Vth. We consider separately the three cases where VðtÞisvDWðtÞ,AzðtÞ,o r AxyðtÞ. Figures 4(a) and4(b) show the distributions PðSAzÞandPðSAxyÞfor different threshold values ( AthzandAthxy, respectively); the corresponding avalanche duration distributions PðTAzÞand PðTAxyÞare shown in Figs. 4(c) and4(d), respectively. Insets of Figs. 4(a)and4(b) show the distributions PðSvÞ and PðTvÞextracted from the vDW signal using vth DW¼0.1m=s. All the distributions can be well described by a power law terminated by a large-avalanche cutoff. Solid lines in Fig. 4 show fits of PðSVÞ¼S−τS Vexp½−ðSV=S/C3 VÞβ/C138,w h e r e τSis a scaling exponent, βparametrizes the shape of the cutoff, andS/C3 Vis a cutoff avalanche size (avalanche durations follow a similar scaling form). We find τS¼1.1/C60.1 and τT¼1.2/C60.1, respectively; i.e., close to the values expected for the quenched Edwards-Wilkinson (QEW) equation, ∂hðx; tÞ=∂t¼ν∇2hðx; tÞþηðx; hÞþFext, describing a short-range elastic string hðx; tÞdriven by an (a)(b) (c) (d) FIG. 2. (a) An example of a sequence of DW magnetization configurations in between successive avalanches (as defined bythresholding the v DWsignal); the DW is moving to the þx direction. The corresponding crackling noise signals, with (b) theDW velocity v DWðtÞ, (c) the in-plane activity AxyðtÞ, and (d) the out-of-plane activity −AzðtÞ.FIG. 3. Distribution of the in-plane magnetization angle ϕinitial of the DW segments where avalanches are initiated versus the corresponding distribution of ϕDWfor all DW segments. The two distributions look almost identical, suggesting that the presenceor absence of BLs within the DW is not important for theavalanche triggering process.PHYSICAL REVIEW LETTERS 122, 117205 (2019) 117205-3external force Fextin a quenched random medium η[29]. The value of the τSexponent is also close to that found very recently for “creep avalanches ”[30], and to that describing avalanches in the central hysteresis loop in a 2D RFIM with a built-in DW [31]. The cutoff avalanche size and duration depend on the imposed threshold level, but appear to saturate to a value set by the “demagnetizing factor ”kin the limit of a low threshold. Figure 5shows the scaling of the average avalanche size as a function of duration, hSvðTÞiin Fig. 5(a), hSAzðTÞiin Fig. 5(b),a n d hSAxyðTÞiin Fig. 5(c).T h e exponent γdescribing the scaling as hSvðTÞi∼Tγ[and similarly for hSAzðTÞiandhSAxyðTÞi] is found to be thresh- old dependent, in analogy to recent observations forpropagating crack lines [32] and the RFIM [33],w i t ht h e γvalue close to 1.6 expected for the QEW equation in the limit of zero threshold [2]approximately recovered for low thresholds [insets of Figs. 5(a) and5(c)]. Thus, our exponent values satisfy within error bars the scaling rela- tion γ¼ðτT−1Þ=ðτS−1Þ. Hence, we have shown how DWs with a dynamical internal structure consisting of BLs generate Barkhausen noise in disordered thin films with PMA. One of the uniquefeatures of this system is the large relative magnitude of the internal, in-plane bursty spin rotation activity within the DW, which in our case actually exceeds that of the out-of-planespin rotations contributing to DW displacement. We have103104105 SAz[rad]10-810-610-410-2P(SAz) Azth = 3 × 1011 rad/s Azth = 5 × 1011 rad/s Azth = 1 × 1012 rad/s τS = 1 × 05 104105 SAxy [rad]10-810-610-4P(SAxy) Axyth = 2.5 × 1011 rad/s Axyth = 1.0 × 1012 rad/s Axyth = 2.0 × 1012 rad/s τS = 1.1510-910-8 Sv [m]106109P(Sv) τS = 1.09 10-910-810-7 TAz[s]104106108P(TAz) Azth = 3 × 1011 rad/s Azth = 5 × 1011 rad/s Azth = 1 × 1012 rad/s τT = 1 × 17 10-910-810-7 TAxy [s]106108P(TAxy) Axyth = 2.5 × 1011 rad/s Axyth = 1.0 × 1012 rad/s Axyth = 2.0 × 1012 rad/s τT = 1.2610-810-7 Tv [s]106109P(Tv) τT = 1.26(a) (b) (c) (d) FIG. 4. Distributions of the avalanche sizes obtained by thresholding (a) the AzðtÞsignal and (b) the AxyðtÞsignal. The corresponding avalanche duration distributions are shown in (c) and (d), respectively. Different threshold values ( AthzandAthxy, respectively) considered are indicated in the legends. The insets in (a) and (c) show the corresponding avalanche size and duration distributions computed fromthev DWðtÞsignal using vth DW¼0.1m=s. Solid lines correspond to fits of power laws terminated by a large-avalanche cutoff (see text), while the dashed lines show the fitted power-law exponent in each case. (a) (b) (c) × × × ×× × FIG. 5. Scaling of the average avalanche size as a function of duration for different threshold values: (a) hSvðTÞi, (b) hSAzðTÞi, and (c)hSAxyðTÞi. The insets in (a) and (c) illustrate the threshold-dependent nature of the exponent γcharacterizing the size versus duration scaling.PHYSICAL REVIEW LETTERS 122, 117205 (2019) 117205-4demonstrated that this internal dynamics within the DW leads to a violation of the Middleton no-passing theorem. It is quite remarkable that the scaling exponents describ- ing the Barkhausen jumps cannot be distinguished from those expected for the much simpler QEW equation. Theavalanche triggerings appear not to be correlated with the internal structure of the DW. Thus, commonly used simple models based on describing DWs as elastic interfaces,neglecting Bloch line dynamics by construction, seem to be capturing correctly the large-scale critical dynamics of the system. This may be rationalized by noticing that Blochlines, being localized N´ eel wall-like segments within the Bloch DW, produce dipolar stray fields decaying as 1=r 3in real space. For 1dinterfaces, such interactions are short- ranged, and hence are not expected to change the univer- sality class of the avalanche dynamics from that of systems with purely local elasticity. In higher dimensions dipolarinteractions are long-ranged, so we expect that the internal dynamics of the DWs will have important consequences; the role of Bloch lines in the case of 3dmagnets with 2d DWs should be addressed in future studies. Another important future avenue of research of great current interest would be to extend the present study to thin films withDzyaloshinskii-Moriya interactions [34,35] . This work has been supported by the Academy of Finland through an Academy Research Fellowship (L. L., Project No. 268302). We acknowledge the computa- tional resources provided by the Aalto University School ofScience “Science-IT ”project, as well as those provided by CSC (Finland). *lasse.laurson@tuni.fi [1] G. Durin and S. Zapperi, in The Science of Hysteresis , edited by G. Bertotti and I. Mayergoyz (Academic, Amsterdam,2006). [2] L. Laurson, X. Illa, S. Santucci, K. T. Tallakstad, K. J. Måløy, and M. J. Alava, Nat. Commun. 4, 2927 (2013) . [3] M. Rost, L. Laurson, M. 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E. Ferrero, A. B. Kolton, J. Curiale, V. Jeudy, and S. Bustingorry, Phys. Rev. E 97, 062122 (2018) . [16] F. J. P´ erez-Reche and E. Vives, Phys. Rev. B 70, 214422 (2004) . [17] A. Mughal, L. Laurson, G. Durin, and S. Zapperi, IEEE Trans. Magn. 46, 228 (2010) . [18] D. Spasojevi ć, S. Jani ćević, and M. Kne žević,Phys. Rev. E 84, 051119 (2011) . [19] P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier, J. Ferr´ e, V. Baltz, B. Rodmacq, B. Dieny, and R. L. Stamps, Phys. Rev. Lett. 99, 217208 (2007) . [20] D.-H. Kim, S.-B. Choe, and S.-C. Shin, Phys. Rev. Lett. 90, 087203 (2003) . [21] S. Papanikolaou, F. Bohn, R. L. Sommer, G. Durin, S. Zapperi, and J. P. Sethna, Nat. Phys. 7, 316 (2011) . [22] A. A. Middleton, Phys. Rev. Lett. 68, 670 (1992) . [23] P. Le Doussal, K. J. Wiese, and P. Chauve, Phys. Rev. B 66, 174201 (2002) . [24] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv. 4, 107133 (2014) . [25] J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson, G. Durin, L. Dupr´ e, and B. Van Waeyenberge, J. Appl. Phys. 115, 17D102 (2014) . [26] S. Moretti, M. Voto, and E. Martinez, Phys. Rev. B 96, 054433 (2017) . [27] S. Bustingorry, A. Kolton, and T. Giamarchi, Europhys. Lett. 81, 26005 (2008) . [28] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.122.117205 for a movie illustrating the bursty dynamics of a DW containingBloch lines. [29] A. Rosso, P. Le Doussal, and K. J. Wiese, Phys. Rev. B 80, 144204 (2009) . [30] M. P. Grassi, A. B. Kolton, V. Jeudy, A. Mougin, S. Bustingorry, and J. Curiale, Phys. Rev. B 98, 224201 (2018) . [31] B. Tadi ć,Physica (Amsterdam) 493A , 330 (2018) . [32] S. Jani ćević, L. Laurson, K. J. Måløy, S. Santucci, and M. J. Alava, Phys. Rev. Lett. 117, 230601 (2016) . [33] S. Jani ćević, D. Jovkovi ć, L. Laurson, and D. Spasojevi ć, Sci. Rep. 8, 2571 (2018) . [34] A. Thiaville, S. Rohart, É. Ju´ e, V. Cros, and A. Fert, Europhys. Lett. 100, 57002 (2012) . [35] Y. Yoshimura, K.-J. Kim, T. Taniguchi, T. Tono, K. Ueda, R. Hiramatsu, T. Moriyama, K. Yamada, Y. Nakatani, and T.Ono, Nat. Phys. 12, 157 (2016) .PHYSICAL REVIEW LETTERS 122, 117205 (2019) 117205-5

PhysRevApplied.13.044036.pdf

PHYSICAL REVIEW APPLIED 13,044036 (2020) Parametric Amplification of Magnons in Synthetic Antiferromagnets A. Kamimaki,1,2S. Iihama,2,3K.Z. Suzuki,2,3N. Yoshinaga,2,4and S. Mizukami2,3,5, * 1Department of Applied Physics, Tohoku University, Aoba 6-6-05, Sendai, 980-8579, Japan 2WPI AIMR, Tohoku University, Katahira 2-1-1, Sendai, 980-8577, Japan 3CSRN, Tohoku University, Sendai, 980-8577, Japan 4MathAM-OIL, AIST, Sendai, 980-8577, Japan 5CSIS (CRC), Tohoku University, Sendai, 980-8577, Japan (Received 13 November 2019; revised manuscript received 31 January 2020; accepted 18 March 2020; published 14 April 2020) We demonstrate the parametric amplification of an acoustic magnon mode induced by an optical magnon mode in synthetic antiferromagnets, which was achieved by using the all-optical pump-probe time-resolved magneto-optical Kerr effect. The acoustic and optical modes with low and high frequen-cies, respectively, are clearly observed under different field directions and pump-laser powers. For a relatively high laser power, the acoustic mode shows a temporal increase in amplitude in the time domain; this is observed when the acoustic mode frequency is approximately half the frequency of the opticalmode. Correspondingly, we also observe a large enhancement in the spectral density of the acoustic mode in the frequency domain. These data are supported by a numerical simulation using a macrospin model; furthermore, the optical mode amplitude threshold for achieving the acoustic mode amplificationis also discussed. The parametric effect in synthetic antiferromagnets demonstrated here can be applied to nanoscale parametric amplifiers and oscillators of magnons, which are the building blocks for spintronic and magnonic computing beyond von Neumann architectures. DOI: 10.1103/PhysRevApplied.13.044036 I. INTRODUCTION Exploration of energy-efficient spontaneous oscillation of magnetization and/or magnon amplification is of funda- mental and technological interest for utilization in numer- ous applications, such as neuromorphic architectures [ 1] and magnonic majority logic gates [ 2]. Such nonlinear dynamics are typically approached with use of an electric- current-induced spin torque [ 3–7] and a nonlinear magnon process such as three-magnon splitting [ 8,9]. The latter process was first explored in Suhl’s studies of the second- order instability in ferromagnets (FMs) [ 10,11]. In this phenomenon, two spatially nonuniform magnons with cer- tain energies /planckover2pi1ω/2 and momentums ±/planckover2pi1kare formed by the annihilation of one spatially uniform (/planckover2pi1k=0) magnon with energy /planckover2pi1ω; however, this is observed only when the amplitude of the latter magnon is larger than a certain threshold [ 10]. Thus far, such parametric excitation and amplification of ferromagnetic magnons has been widely investigated in ferromagnets of various shapes, such as slabs, thin films, strips, and disks [ 12–18]. In this article, we report the parametric amplification of a magnon in a synthetic antiferromagnet (AFM). A synthetic AFM is a stacked layer of FM 1/nonmagnet (NM)/FM 2, *shigemi.mizukami.a7@tohoku.ac.jpwhich is a useful structure for antiferromagnetic spintron- ics [19]. The synthetic AFMs investigated in this study have easy-plane magnetic anisotropy due to the large mag- netization Bsin each ferromagnetic layer. The antiparallel state of the two magnetization unit vectors m1and m2 for FM 1and FM 2is stabilized by an effective magnetic field Bexdue to interlayer coupling via the NM. Because a synthetic AFM has two magnetizations, it exhibits acous- tic and optical magnon modes, as studied before [ 20,21]. When a relatively weak external magnetic field B0is applied along the in-plane xdirection [Figs. 1(a)and1(b)], it induces a canted state with a net magnetization vector m≡(m1+m2)/2 parallel to the xaxis at equilibrium. mprecesses about the xaxis in the acoustic mode with a lower frequency [Fig. 1(a)] and oscillates along the xaxis in the optical mode with a higher frequency [Fig. 1(b)]. These modes are analogous to the ferromagnetic and anti- ferromagnetic resonances, which arise from the in-phase and out-of-phase precessions of m1and m2, respectively. These linear dynamics in the canted state were examined using coupled macrospin equations of mand n≡(m1− m2)/2[21]. Herein we demonstrate that amplification via nonlinear interaction of these two spatially uniform acous- tic and optical magnon modes is possible, which enables us to realize magnon amplification using synthetic AFMs on the nanoscale. 2331-7019/20/13(4)/044036(15) 044036-1 © 2020 American Physical SocietyA. KAMIMAKI et al. PHYS. REV. APPLIED 13,044036 (2020) (a) (b) FIG. 1. (a) Acoustic and (b) optical modes for a synthetic AFM. The zaxis is parallel to the normal to the film. An external magnetic field B0is applied parallel to the xaxis. m1(m2)i s the magnetization unit vector for ferromagnetic layer 1 (2), and m≡(m1+m2)/2.mx0is the xcomponent of mat equilibrium. II. SIMPLE THEORETICAL ANALYSIS To gain insight into the physics of the parametric ampli- fication in this study, we approximately derive the equa- tions beyond the linear regime in the large- Bslimit and under the application of an in-plane magnetic field B0,a s shown in Fig. 1. We consider the free energy Ffor the synthetic AFMs expressed as F=2/summationdisplay j=1/bracketleftbigg −MsB0x·mj+1 2MsBs/parenleftbig mj·z/parenrightbig2/bracketrightbigg +Jex dFMm1·m2,( 1 ) where Ms=μ−1 0Bs,Jex(greater than 0) is the antiferro- magnetic interlayer exchange coupling constant, and dFM is the magnetic layer thickness. Bexis given as Bex= Jex/MsdFM. In the polar coordinate system, the following equations corresponding to the Landau-Lifshitz-Gilbert equations are generally obtained [ 22,23]: dθj dt=−γ1 Mssinθj∂F ∂φ j−γα01 Ms∂F ∂θ j,( 2 ) sinθjdφj dt=γ Ms∂F ∂θ j−γα01 Mssinθj∂F ∂φ j,( 3 ) with mj=(cosφjsinθj,s i nφjsinθj,c o sθj). Here γis the gyromagnetic ratio and α0is the Gilbert damping constant for the FM. We consider the case where the easy-plane magnetic anisotropy is very strong, B0<2Bex/lessmuch Bs.T h e n we obtain coupled second-order differential equations from Eqs. (1)–(3)with the approximation of θj/similarequalπ/2 using a method similar to that in Ref. [ 22]: d2φm dt2=−γ2B0Bscosφnsinφm−α0γBsdφm dt,( 4 )d2φn dt2=−γ2B0Bscosφmsinφn +γ2BexBssin 2φn−α0γBsdφn dt,( 5 ) where we introduce the variables φm=(φ1+φ2)/2a n d φn=(φ1−φ2)/2. We consider the Taylor series of the small deviations around the equilibrium values, which are expressed as φm(t)/similarequalδφ m(t)andφn(t)/similarequalφn0+δφ n(t). Then the following equations for the acoustic and optical modes are obtained from Eqs. (4)and(5)with the lowest order of the nonlinear cross (mode-mixing) term for the acoustic mode (see Appendix Afor details): d2δmy dt2=−ω2 ac/parenleftbig 1+m−1 x0δmx/parenrightbig δmy−2 τacdδmy dt,( 6 ) d2δmx dt2=−ω2 opδmx−2 τopdδmx dt,( 7 ) respectively. Some quantities are transformed into those in x-y-zcoordinates with use of the following defini- tions: mx0=cosφn0,ny0=sinφn0,δmx=−δφ nsinφn0, andδmy=δφ mcosφn0. Here the equilibrium angle of the canted state φn0is determined by the relation cos φn0≡ B0/2Bex. The angular frequencies of the two modes are expressed as ωac=2πfac=mx0γ/radicalbig 2BexBs,( 8 ) ωop=2πfop=ny0γ/radicalbig 2BexBs.( 9 ) Finally, the relaxation times of the two modes are expressed as 1/τac=1/τop=α0γBs/2. (10) The higher-order terms in Eqs. (6)and(7)are disregarded because we discuss an initial process of the parametric amplification of the acoustic mode by the optical mode in this study. When the optical mode amplitude δmxis small enough, the nonlinear cross term δmxδmyin Eq. (6)is negligi- ble and the two modes are decoupled. The value of mx0 increases as the magnetic field increases, which implies that the canted state gradually changes to a parallel state [Fig. 1(a)]. This change causes a linear increase in the acoustic mode frequency ωac[Eq. (8)], as shown in Fig.2. On the other hand, the optical mode frequency ωop decreases as the magnetic field increases via a decrease in ny0[Eq. (9)], as shown in Fig. 2. In the large- Bslimit, the relaxation times for the two modes are identical [Eq. (10)]; however, they are generally different. When the optical mode amplitude δmxis not small enough, the nonlinear cross term δmxδmyin Eq. (6)starts 044036-2PARAMETRIC AMPLIFICATION OF MAGNONS... PHYS. REV. APPLIED 13,044036 (2020) 0 0.2 0.4 0.6 0.8 1.0012Reduced frequency Reduced magnetic fieldωop ωacωop= 2ωac FIG. 2. Theoretical frequencies of the acoustic mode ωacand the optical mode ωopas a function of the magnetic field B0for the in-plane-magnetized synthetic AFM in the large- Bslimit. The data for the acoustic and optical modes are calculated with Eqs. (8)and(9), respectively, with Bs/2Bex/similarequal2.2. The frequency and magnetic field are reduced by γ2Bexand 2 Bex, respec- tively. The dashed line denotes the magnetic field, on which the condition of ωop=2ωacis satisfied. playing an important role. Indeed, Eq. (6)for the acous- tic mode is the Mathieu equation that predicts a parametric instability. Therefore, Eqs. (6)and(7)predict a parametric instability of the acoustic mode that is induced by the opti- cal mode. This is the mathematical analog of a suspended pendulum with periodically changing length [ 24]. Here we assume that a certain magnetic field is applied so as to sat- isfy the condition of ωop=2ωac, as denoted in Fig. 2by the dashed line. We also consider that the optical mode is continuously excited by an external torque, and then δmx is expressed as δmx=δmx0cos 2ωact, (11) where δmx0is the oscillation amplitude of the optical mode. Substituting Eq. (11) into Eq. (6), one can find the solution for the oscillation amplitude of the acoustic mode δmy0, which is proportional to the exponential factor [ 24] exp/bracketleftbig/parenleftbig −1/τac+ωacm−1 x0δmx0/4/parenrightbig t/bracketrightbig . (12) Equation (12) indicates that the relaxation time of the acoustic mode apparently increases and becomes negative as the optical mode amplitude δmx0increases. This appar- ent negative relaxation time corresponds to the acous- tic mode instability or amplification, which occurs when the optical mode amplitude δmx0overcomes a threshold determined by the following relation: 1/τac−ωacm−1 x0δmx0/4≤0. (13)In transient dynamics, the optical mode is not continu- ously excited and its amplitude decays exponentially, such as exp (−t/τop). Even in this case, Eqs. (12) and(13) are approximately valid, because the decay of the optical mode amplitude is much slower than its oscillation frequency. Thus, the amplification may be observed for some duration during which the optical mode amplitude δmx0is larger than the threshold. The acoustic mode grows exponentially in the initial process [Eq. (12)] but eventually its amplitude is saturated by higher-order nonlinear terms, which is not described above. It should also be noted that the above- mentioned parametric amplification is unique to synthetic AFMs, because the nonlinear cross term in Eq. (6)vanishes when the exchange coupling Jexis of the ferromagnetic type (see Appendix Afor details). Although the basic physics of the parametric amplifi- cation in this study is understood as mentioned above, the theoretical expressions described in this section are derived for in-plane-magnetized synthetic AFMs in the large- Bs limit. Thus, we also describe a more-general case that is based on the macrospin models for synthetic AFMs to achieve the simulation and analysis of the experimental data, as found in Appendixes BandC, respectively. III. EXPERIMENTAL METHODS The dynamics are investigated with an all-optical pump- probe technique in which transient dynamics are induced by the pump pulse and detected by the delayed probe pulse via the magneto-optical Kerr effect [ 25–28]. We use the standard optical technique with a Ti:sapphire pulsed laser and a regenerative amplifier, as used in pre- vious studies [ 29,30]. The output laser wavelength, pulse width, and repetition rate are approximately 800 nm, approximately 120 fs, and 1 kHz, respectively. The spot diameter of the pump (probe) laser is approxi- mately 1.1 mm (0.12 mm.) The incident angle of the probe beam is approximately 5◦, and the measured Kerr rotation angle is proportional to the zcomponent of the magnetization vectors [Fig. 3(a)]. The film stacking is Ta(3n m)/Fe60Co20B20(3n m)/Ru(0.4 nm )/Fe60Co20B20 (3n m)/Ta(3n m), which is deposited by magnetron sput- tering on thermally oxidized Si substrates. Bs/similarequal1.57 T and Bexof approximately 0.4 T for the sample are evaluated from magnetization measurements at room temperature using vibrating-sample magnetometers. The magnetiza- tions for the sample are also measured with a physical property measurement system (PPMS) at 300–400 K. To achieve parametric amplification conditions while maintaining a sufficiently large signal-to-noise ratio for both modes, we measure the dynamics while systemati-cally varying the magnetic field angle θ Bunder fixed B0 of 0.32 T for different pump-laser powers P[Fig. 3(a)]. In this case, the out-of-plane component of B0is sig- nificantly smaller than the out-of-plane saturation field 044036-3A. KAMIMAKI et al. PHYS. REV. APPLIED 13,044036 (2020) Bsat=2Bex+Bsfor the synthetic AFM in this study, so the magnetizations for the synthetic AFM are nearly in the film plane, as depicted in Fig. 3(a). Consequently, the magnetic field B0in the relations for the mode frequencies given by Eqs. (8)and(9)is approximately regarded as its in-plane component; that is, B0sinθB(see Appendix Cfor details). By changing θB, we vary the mode frequencies via the change in the in-plane component of the magnetic field, as discussed in Sec. II. Therefore, the condition ωop=2ωac is satisfied at a certain angle θB. IV . EXPERIMENTAL RESULTS AND DISCUSSION Figures 3(b)–3(m) display the time-domain data recorded at typical values of θBfor various values of P. The laser-induced changes in the Kerr rotation angle /Delta1φ Kin the figures are normalized at the saturation value of the polar Kerr rotation angle φK0. For most of the angles, the data show damped periodic changes with dif- ferent frequencies [Figs. 3(b)–3(e) and3(j)–3(m) ]. These are the weighted sums of the acoustic and optical modes that are excited by the pump pulse and subsequently decay with damping. On the other hand, for the data at angles of approximately 45◦–60◦, the oscillation ampli- tudes show nonmonotonic decays [Figs. 3(f)–3(i)]. In this case, the amplitude first increases within 0.1–0.2 ns and then decreases for 10 mW at 45◦[Fig. 3(i)]. This behavior becomes less pronounced with decreasing P [Figs. 3(f)–3(h)]. Figures 4(a)–4(d) show the spectral densities (SDs) of|/Delta1φ K/φ K0|in the frequency fdomain as a function of the angle θBfor different pump-laser powers P.T h e SDs are obtained with use of a fast Fourier transformof the time-domain data for /Delta1φ K/φ K0. The bright and less-intense curves are attributed to the acoustic and opti- cal modes, respectively. The experimental θBvariations offacand fopare qualitatively consistent with being pro- portional to sin θBand show no significant change in ny0 under relatively small B0, as described in Secs. IIandIII. Figures 4(e)–4(h) and4(i)–4(l)display the absolute detun- ing|fop−2fac|and SDs obtained from the two mode peak positions and heights in the frequency-domain data, respectively. The condition fop/similarequal2facis achieved at a cer- tain magnetic field angle, and we define this angle as θc, as shown in Figs. 4(e)–4(h).A t fop/similarequal2fac, the SDs for the acoustic mode increase remarkably for the 7- and 10-mW data, which correspond to the time-domain data shown in Figs. 3(h) and3(i), respectively. To understand the amplitude behavior of the two modes in the time and frequency domains, we simulate the dynamics using a numerical integration of the coupled macrospin equations for mand nin the thin-NM limit [ 21] (see also Appendix B): dm dt=−γm×[B0−Bs(m·z)z]+γBs(n·z)n×z +α0/parenleftbigg m×dm dt+n×dn dt/parenrightbigg , (14) dn dt=−γn×[B0−Bs(m·z)z−2Bexm]+γBs(n·z)m ×z+(α0+αsp)/parenleftbigg m×dn dt+n×dm dt/parenrightbigg −αsp/bracketleftbigg1 m2m·/parenleftbigg n×dm dt/parenrightbigg m+1 n2n·/parenleftbigg m×dn dt/parenrightbigg n/bracketrightbigg , (15) ΔφK/ φK0(%) Δt(ns) Δt(ns) Δt(ns) Δt(ns)(b) (f) (j)(c) (g) (k)(d) (h) (l)(e) (i) (m)ΔφK/ φK0(%) ΔφK/ φK0(%)(a)P= 3 mW 5 mW 7 mW 10 mW θB= 75° 60° 45° 70° 55° 40°70° 55° 40° 60° 45° 30° m1 m2 Probe laser FIG. 3. (a) The coordinate system for the optical measurement. The field angle θBis measured relative to the film normal ( zaxis). (b)–(m) Normalized laser-induced change in the Kerr rotation angle /Delta1φ K/φ K0versus delay time /Delta1tmeasured at typical θBvalues for different pump-laser powers P. 044036-4PARAMETRIC AMPLIFICATION OF MAGNONS... PHYS. REV. APPLIED 13,044036 (2020) |ΔφK/φK0| (10–8/Hz0.5)f(GHz)|ΔφK/φK0| (10–8/Hz0.5) |ΔφK/φK0| (10–8/Hz0.5) ac ac ac ac op op op opac ac acop opop acop|ΔφK/φK0| (10–8/Hz0. 5) θc θc θc θc 0 20 40 60 80012|ΔϕK/ϕK0| (10–8/Hz0.5) θB (deg)0 20 40 60 800102030|fop–2fac| (GHz) 0 20 40 60 80012345 θB (deg)0 20 40 60 800102030 0 20 40 60 800246 θB (deg)0 20 40 60 800102030 0 20 40 60 800510 θB (deg)0 20 40 60 800102030(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) FIG. 4. (a)–(d) SDs of the laser-induced change in the Kerr rotation angle /Delta1φ K/φ K0as functions of the frequency fand the field angle θB.θBvariation of (e)–(h) the absolute detuning |fop−2fac|and (i)–(l) the SDs for the acoustic (ac; circles) and optical (op; squares) modes. The solid and open symbols denote the experimental and simulated data, respectively. (a),(e),(i) for Pof 3 mW; (b),(f),(j) for 5 mW; (c),(g),(k) for 7 mW; and (d),(h),(l) for 10 mW. θcin (e)–(h) denotes the field angle θBat which fopis equal to 2 fac. where αspis the additional Gilbert damping constant due to the spin-pumping effect [ 31]. The laser-induced changes in the effective fields are phenomenologically modeled, as explained below. Ultrafast demagnetization and recov- ery [ 26,32] are taken into account with use of the relation Bs(t)=Bs0−/Delta1Bsc(t), with the mean value Bs0and reduc- tion at the zero delay /Delta1Bs. The temporal change used in Ref. [ 33] is approximated as c(t)=(1−c0)e−t/τ+c0√t/t0+1(t≥0), (16) where the first and second terms represent rapid and slow recoveries, respectively. These recoveries are character- ized by the amplitude constant c0(≤1), picosecond relax- ation time τ, and relatively long heat-diffusion time t0.T h e initial femtosecond-to-subpicosecond change is simplified as a step function. Bexalso decreases with rising temper- ature [ 34,35], so the relation Bex(t)=Bex0−/Delta1Bexc(t)is also used similarly to its use in previous studies [ 36,37].Here Bex0and/Delta1Bexare defined similarly to Bs(t). Our simulation focuses on the dynamics on a timescale of hundreds of picoseconds and is irrelevant to the details of microscopic modeling [ 26,38,39], which takes into account the femtosecond-to-picosecond dynamics of elec- tron, spin, and lattice temperatures. The observed quantity is computed with use of the relation /Delta1φ K φK0=/bracketleftbigg 1−/Delta1Bs Bs0c(t)/bracketrightbigg [mz(t)+rmnnz(t)]−mz0, (17) where rmn(much less than 1) is the ratio of the Kerr rotation angle of nzand mz, which stems from the two magnetic layers located at different depths [ 40] (see also Appendix Dfor details). The data obtained in the simulation using reasonable parameters (see Table Iand its legend) are similar to the experimental data [Figs. 4(i)–4(l)]. The broad θBvariations of the simulated SDs originate from the θBvariation of mx0,mz0, and the magnitude of the laser-induced torques. 044036-5A. KAMIMAKI et al. PHYS. REV. APPLIED 13,044036 (2020) TABLE I. The parameters used in the simulation. We use γ/2π=29.5 GHz /T,Bs0=1.57 T, α0=0.01,αsp=0.01,τ= 1.5 ps, t0=30 ps, and rmn=−0.065 for any Pvalues. /Delta1Bs/Bs0, c0,τ,a n d t0are obtained from the time-domain data below the 20-ps regime (see Appendix E).γ,α0,a n dαspare evalu- ated with use of the experimental data at 3 mW (see AppendixF).B ex0is determined by our fitting the detuning values in Figs. 4(e)–4(h)./Delta1Bexand rmnare obtained by our fitting the SDs in Figs. 4(i)–4(l). P(mW) Bex0(T) /Delta1Bex/Bex0 /Delta1Bs/Bs0 c0 3 0.355 0.075 0.028 0.58 5 0.350 0.12 0.042 0.597 0.340 0.16 0.055 0.61 10 0.325 0.29 0.11 0.66 The sharp peaks of the simulated SDs near θcare caused by the nonlinear mode coupling. The slight change in the simulated θcwith increasing Pis due to the reduction of fopvia that of Bex0(Table I)[ s e eE q . (9)]. This Bex0 reduction indicates a rise in the time-averaged temperature under laser illumination for higher P. A similar reduction inBexis observed at a temperature slightly higher than room temperature in our magnetization measurements (see Appendix G). Figures 5(a)–5(d) and 5(e)–5(h) show the simulated time-domain data in the vicinity of θcat different pow- ers Pfor the laser-induced change in the Kerr rotation ΔφK/ φ(%) Δt(ns) Δt(ns)ΔφK/ φ(%) ΔφK/ φ(%) ΔφK/ φK0 K0 (%) 102 δ mx (a) (b) (c) (d)(e) (f) (g) (h)102 δ mx 102 δ mx 102 δ mxK0 K0 FIG. 5. Simulated time-domain data. Laser-induced change in the Kerr rotation angle /Delta1φ K/φ K0for (a) 57.5◦at 3 mW, (b) 55.0◦ at 5 mW, (c) 52.5◦at 7 mW, and (d) 47.5◦at 10 mW. (e)–(h) The corresponding optical mode amplitude δmx. The dashed lines in (f),(g),(h) denote the threshold δmx0,c. The arrows denote increases in the amplitude.angle /Delta1φ K/φ K0and the corresponding optical mode δmx (≡mx−mx0), respectively. Initial increases in the ampli- tude of /Delta1φ K/φ K0on the negative sides are observed in Figs. 5(c)and5(d), as indicated by arrows, which well cap- ture the experimental observations in Figs. 3(h) and3(i). As discussed at the end of Sec. II, such initial increases can be observed for some duration where the optical mode amplitude δmx0is larger than the threshold. To evaluate the threshold, we extend the nonlinear theory described in Sec. IIto the more-general case with the weak mag- netic field applied obliquely (see Appendix Cfor details). Then we obtain approximate analytical expressions for the threshold as well as the mode frequencies and relaxation times. The threshold in the present case is derived from the instability condition at fop=2fac, given by δmx0,c=1 1−p−24mx0 ωacτac, (18) with the acoustic mode frequency ωac=2πfac=pγB0sinθB (19) and its relaxation time: 1/τac=α0γBex[m2 x0+p2(1−m2 z0)]. (20) Here mx0and mz0are the xand zcomponents of mat equilibrium, respectively [Fig. 3(a)], mx0=(B0/2Bex)sinθB, (21) mz0=p−2(B0/2Bex)cosθB, (22) and pis the acoustic mode ellipticity, p=(1+Bs/2Bex)1/2. (23) Equations (19)–(21) become identical to the correspond- ing equations described in Sec. IIwhen we set the large- Bs limit and θB=90◦in Eqs. (19)–(21). We estimate the threshold δmx0,c/similarequal0.054 from Eqs. (18)–(23) with the parameters in this study, as shown in Figs. 5(f)–5(h) by the dashed lines. Initial increases in the amplitude of /Delta1φ K/φ K0[Figs. 5(c) and5(d)] occur whenδmxmoves across the threshold [Figs. 5(g) and5(h)]. Therefore, the initial temporal increase in the amplitude [Figs. 3(h) and 3(i)] and the sharp peaks of the SD [Figs. 4(k) and4(l)] for the experimental data at 7 and 10 mW are evidence of parametric amplification of the acoustic mode. The optical mode amplitude in the sim- ulated 3- and 5-mW data is smaller than the threshold[Figs. 5(e) and5(f)] and, therefore, the small increase in the simulated SDs in the vicinity of θ c[Figs. 4(i)and4(j)] may be understood as an effective increase in the acous- tic mode relaxation time due to nonlinear interactions that 044036-6PARAMETRIC AMPLIFICATION OF MAGNONS... PHYS. REV. APPLIED 13,044036 (2020) is not large enough to overcome the threshold. A more- quantitative discussion of the threshold requires further investigations that are beyond the scope of this study. Finally, it is worthwhile commenting on the possibility of observing the amplification in synthetic AFMs by means of the conventional radio-frequency (rf) technique. In the three-magnon-splitting or subsidiary absorption experi- ments in ferromagnets, as described at the beginning of Sec. I, the rf magnetic field is applied perpendicular to the direction of the static magnetic field. When the magnitude of the rf magnetic field exceeds the threshold, parametric instability of ferromagnetic magnons occurs. This thresh- old depends on the sizes, shapes, and material parameters of ferromagnets. Typical values for the threshold of the rf magnetic field were reported as approximately 0.1–0.6 mT for thick permalloy films [ 12] and approximately 1–3 mT in strip-shaped permalloy films [ 17]. In the case of syn- thetic AFMs, the optical mode can be excited when the rf magnetic field is applied parallel to the direction of the static magnetic field, so-called longitudinal pumping [ 20]. The typical threshold of the rf magnetic field for the syn- thetic AFMs is evaluated as about 1.5 mT for the present synthetic AFM sample when the optical mode is reso- nantly excited by longitudinal pumping (see Appendix H for details). This threshold is comparable to the above- mentioned value in strip-shaped permalloy films [ 17]. Thus, the parametric amplification in synthetic AFMs can be observed with a rf setup similar to that in Ref. [ 17]. V . SUMMARY AND OUTLOOK We investigate the dynamics of acoustic and optical magnon modes in synthetic AFMs with the all-optical pump-probe technique. We clearly observe acoustic and optical modes with low and high frequencies for different field directions and pump-laser powers. We find that the acoustic mode amplitudes temporally increase in the time domain at a relatively high laser power Pand fop/similarequal2fac. Correspondingly, large enhancements in the SDs are also observed in the frequency domain. These are character- istics of parametric amplification of the acoustic magnon mode induced by the optical magnon mode. The numer- ical simulation using the macrospin model well explain the experimental frequency-domain and time-domain data, and we also discuss the threshold necessary to realize the amplification. As mentioned in Sec. I, a parametric amplifier or oscillator is one of the fundamental building blocks to develop computers beyond von Neumann architectures. One interesting device concept is the magnonic majority logic gate, in which the phases of ferromagnetic magnonsare used as the information encoded into the inputs and the parametric amplifier of ferromagnetic magnons is used for nonlinear bistable phase elements [ 2]. This is a magnonic parametron in a modern sense, being analogousto the parametron using ferrite cores invented in the past [41]. Similarly, one can design a synthetic-AFM-based magnonic parametron by utilizing the parametric ampli- fication of the acoustic magnon mode. Synthetic AFMs show relatively strong nonreciprocity in magnon prop- agation [ 42–45], so the synthetic-AFM-based magnonic parametron could have an input or output isolation better than that of the ferromagnet-based one. In addition, our study naturally suggests that synthetic AFMs also work as parametric nano-oscillators. Nano-oscillators play a major role in some types of modern reservoir computing. Such computing architecture is being extensively studied using nanoscale dc-biased spin-torque oscillators (STOs) [ 1,46] and large numbers of arrays of nano-STOs [ 47]. The auto- oscillation of the acoustic modes in synthetic AFMs can be driven by the wireless rf magnetic field via optical mode pumping. Thus, synthetic-AFM-based oscillator arrays are free from complicated wiring issues and simple and low- cost integrations would be possible. The above-mentioned threshold of the rf magnetic field can be reduced by a fac- tor of about 100 in the case of synthetic AFMs composed of low-damping magnets with α 0of about 0.001 or less [48–52]. Therefore, an energy-efficient nanoamplifier or nano-oscillator of magnons is feasible. ACKNOWLEDGMENTS A.K. acknowledges the Graduate Program in Spin- tronics (GP-Spin) at Tohoku University A.K. and S.M. thank K. Saito for his technical support. S.M. thanks H. Kurebayashi and Y. Nozaki for valuable discussions. This work was supported in part by KAKENHI (Grants No. 26103004 and No. 19K15430), CSRN, and ATP of WPI-AIMR. APPENDIX A: SOME DETAILS OF THE ANALYSIS IN SEC. II In Eqs. (4)and(5), we consider the Taylor series of the small deviations around the equilibrium values, which are expressed as φm(t)/similarequalδφ m(t)andφn(t)/similarequalφn0+δφ n(t). Then the following equations are obtained with the lowest order of the nonlinear cross (mode-mixing) terms: d2δφ m dt2=−γ2B0Bscosφn0/parenleftbigg 1−sinφn0 cosφn0δφ n/parenrightbigg δφ m −α0γBsdδφ m dt, (A1) d2δφn dt2=−γ22BexBssin2φn0/parenleftbigg 1−1 2cos2φn0 sin2φn0δφ2 m/parenrightbigg δφ n +1 2γ2B0Bssinφn0δφ2 m−α0γBsdδφ n dt. (A2) By transforming some quantities into those in the x-y-z coordinates, as performed in Sec. II, we obtain Eqs. (6) 044036-7A. KAMIMAKI et al. PHYS. REV. APPLIED 13,044036 (2020) and(7), as follows: d2δmy dt2=−ω2 ac/parenleftbig 1+m−1 x0δmx/parenrightbig δmy−2 τacdδmy dt, (A3) d2δmx dt2=−ω2 op/parenleftbigg 1−1 2n−2 y0δm2 y/parenrightbigg δmx−1 2ω2 acn2y0m−3 x0δm2 y −2 τopdδmx dt, (A4) /similarequal−ω2 opδmx−2 τopdδmx dt. (A5) Here we use the expressions for ωac,ωop,τac,a n dτopas in Eqs. (8)–(10).I nE q . (A4) , the terms δm2 yandδmxδm2 yare negligible when we discuss an initial process of parametric amplification of the acoustic mode by the optical mode in this study. The nonlinear cross term in Eq. (A1) vanishes for the case of ferromagnetic coupling because sin φn0=0, as mentioned in Sec. II. APPENDIX B: COUPLED MACROSPIN EQUATIONS WITH THE SPIN-PUMPING EFFECT USED IN THIS STUDY We consider the case where an external magnetic field B0is applied in an arbitrary direction. As described in Sec. II, the free energy Fis expressed as F=2/summationdisplay j=1/bracketleftbigg −MsB0u·mj+1 2MsBs/parenleftbig mj·z/parenrightbig2/bracketrightbigg +Jex dFMm1·m2,( B 1 ) where uis the unit vector for the direction of the exter- nal magnetic field. The Landau-Lifshitz-Gilbert equation for the jth ferromagnetic layer without the spin-pumping effect is expressed as dmj dt=−γmj×Bj+α0mj×dmj dt(j=1, 2).( B 2 ) Here the effective magnetic field Bjis given as Bj= −M−1 s∂F/∂mj.F r o mE q s . (B1) and(B2), the two coupled equations for mand nwithout the spin-pumping effect are derived as dm dt=−/Omega1Lm×u+/Omega1B[(m·z)m×z+(n·z)n×z] +α0/parenleftbigg m×dm dt+n×dn dt/parenrightbigg ,( B 3 )dn dt=−/Omega1Ln×u+/Omega1B[(m·z)n×z+(n·z)m×z] +2/Omega1exn×m+α0/parenleftbigg m×dn dt+n×dm dt/parenrightbigg ,( B 4 ) where we define /Omega1L=γB0,/Omega1B=γBs,a n d/Omega1ex=γBex. When the spin-pumping effect is taken into account, extra torque terms are added to Eqs. (B3) and(B4) [21]: Tm=αm/parenleftbigg m×dm dt+n×dn dt/parenrightbigg +2αmη/bracketleftbigg1 1−η(m2−n2)m·/parenleftbigg n×dn dt/parenrightbigg m +1 1+η(m2−n2)n·/parenleftbigg m×dm dt/parenrightbigg n/bracketrightbigg , Tn=αn/parenleftbigg m×dn dt+n×dm dt/parenrightbigg −2αnη/bracketleftbigg1 1+η(m2−n2)m·/parenleftbigg n×dm dt/parenrightbigg m +1 1−η(m2−n2)n·/parenleftbigg m×dn dt/parenrightbigg n/bracketrightbigg , where some quantities are expressed as αm=α1gr 1+grcoth(dNM/2λ), αn=α1gr 1+grtanh(dNM/2λ), η=gr sinh(dNM/λ)+grcosh(dNM/λ), α1=γ1 MsdFM/parenleftbigg/planckover2pi1 2e/parenrightbigg2 (ρλ)−1, gr=2ρλGr. Here dNMis the thickness of the nonmagnetic layer, λis the spin-diffusion length, ηis the back-flow efficiency of the spin current, ρis the resistivity of the nonmagnetic layer, and Gris the real part of the mixing conductance for the FM/NM interface. In our case, we take the limit ofdNM/λ→0 because λin the case where the NM is Ru is larger than dNMof 0.4 nm in this study [ 53]. Then, the additional damping αmis ignored, and ηis taken as unity. Consequently, we obtain Tm=0, and Tnis expressed as Tn=αsp/parenleftbigg m×dn dt+n×dm dt/parenrightbigg −αsp ×/bracketleftbigg1 m2m·/parenleftbigg n×dm dt/parenrightbigg m+1 n2n·/parenleftbigg m×dn dt/parenrightbigg n/bracketrightbigg , (B5) 044036-8PARAMETRIC AMPLIFICATION OF MAGNONS... PHYS. REV. APPLIED 13,044036 (2020) where we use m2+n2=1 and define the additional damp- ingαspas αsp=γ1 MsdFM/parenleftbigg/planckover2pi1 2e/parenrightbigg2 2Gr. The value of αspis independent of the resistivity of the NMρin this case. Equations (B3) and(B4) with Eq. (B5) correspond to Eqs. (14) and(15), respectively. APPENDIX C: THEORETICAL ANALYSIS FOR THE CASE OF APPLICATION OF A RELATIVELY WEAK OBLIQUE MAGNETIC FIELD We consider the case where a relatively weak mag- netic field is applied in the x-zplane. To obtain some approximate analytical expressions, the vectors mand n are expressed as the sum of the equilibrium and time- dependent parts: m(t)=m0+δm(t)and n(t)=n0+ δn(t). Substituting these quantities into Eqs. (14) and(15) and using the equilibrium conditions, we obtain mand n at equilibrium, which are expressed as m0=mx0x+mz0z and n0=ny0y. Each component is given as follows: mx0=B0 2Bexux,( C 1 ) mz0=B0 Bs+2Bexuz,( C 2 ) ny0=/radicalBig 1−m2 x0−m2 z0,( C 3 ) where ux=sinθBand uz=cosθBwith the magnetic field angle θBwith respect to the film normal, and the rela- tion m2+n2=1 is used. Subsequently, from the equation of the first order and second order for δ, we obtain the following six coupled equations: dδmx dt=/Omega1Bny0δnz−2/Omega1exmz0δmy−α0mz0dδmy dt +α0ny0dδnz dt,( C 4 ) dδmy dt=−(/Omega1Lux+/Omega1Bmx0)δmz+2/Omega1exmz0δmx −α0mx0dδmz dt+α0mz0dδmx dt−/Omega1Bδmxδmz, (C5) dδmz dt=/Omega1Luxδmy+α0mx0dδmy dt−α0ny0dδnx dt,( C 6 ) dδnx dt=(/Omega1B+2/Omega1ex)ny0δmz−/parenleftbig α0+αsp/parenrightbig mz0dδny dt +/parenleftbig α0+αsp/parenrightbig ny0dδmz dt,( C 7 )dδny dt=−/Omega1Bmx0δnz−/parenleftbig α0+αsp/parenrightbig mx0dδnz dt +/parenleftbig α0+αsp/parenrightbig mz0dδnx dt,( C 8 ) dδnz dt=−2/Omega1exny0δmx+/parenleftbig α0+αsp/parenrightbig mx0dδny dt −/parenleftbig α0+αsp/parenrightbig ny0dδmx dt.( C 9 ) Here we use Eqs. (C1) and (C2) and the definitions of /Omega1L=γB0,/Omega1B=γBs,a n d /Omega1ex=γBex, as defined in Appendix B.I nE q s . (C4)–(C9), we take only the term −/Omega1Bδmxδmzas a leading nonlinear term, according to the analysis in Appendix A.I nE q s . (C4)–(C9), we drop the last term of Eq. (15) because this term is negligibly small in this study. In this study, mz0is on the order of 0.1, which is smaller than mx0and ny0for most of the θBvalues. Thus, we solve Eqs. (C4)–(C9) in a perturbative way: δm(t)/similarequalδm(0)(t)+mz0δm(1)(t)+m2 z0δm(2)(t)+··· , δn(t)/similarequalδn(0)(t)+mz0δn(1)(t)+m2 z0δn(2)(t)+··· . Substituting δm(t)andδn(t)into Eqs. (C4)–(C9),w e decouple the six coupled equations [Eqs. (C4)–(C9)] into two sets of three zeroth-order coupled equations for each acoustic and optical mode, except for the nonlinear cross term: dδm(0) x dt=/Omega1Bny0δn(0) z+α0ny0dδn(0) z dt, (C10) dδn(0) y dt=−/Omega1Bmx0δn(0) z−/parenleftbig α0+αsp/parenrightbig mx0dδn(0) z dt, (C11) dδn(0) z dt=−2/Omega1exny0δm(0) x+/parenleftbig α0+αsp/parenrightbig mx0dδn(0) y dt −/parenleftbig α0+αsp/parenrightbig ny0dδm(0) x dt(C12) for the optical mode, and dδm(0) y dt=−(/Omega1Lux+/Omega1Bmx0)δm(0) z−/Omega1Bδm(0) xδm(0) z −α0mx0dδm(0) z dt, (C13) dδm(0) z dt=/Omega1Luxδm(0) y+α0mx0dδm(0) y dt−α0ny0dδn(0) x dt, (C14) dδn(0) x dt=(/Omega1B+2/Omega1ex)ny0δm(0) z+/parenleftbig α0+αsp/parenrightbig ny0dδm(0) z dt (C15) 044036-9A. KAMIMAKI et al. PHYS. REV. APPLIED 13,044036 (2020) for the acoustic mode. Finally, we obtain two coupled second-order differential equations for the optical and acoustic modes from Eqs. (C10) –(C12) and(C13) –(C15) , respectively, with the condition that α0andαspare much smaller than unity: d2δm(0) x dt2=−ω2 opδm(0) x−2 τopdδm(0) x dt, (C16) d2δm(0) z dt2=−ω2 ac/parenleftbig 1+/epsilon1δm(0) x/parenrightbig δm(0) z −2 τacdδm(0) z dt. (C17) Here the acoustic and optical mode angular frequencies are expressed as ωac=/radicalbig /Omega1Lux(/Omega1Lux+/Omega1Bmx0), (C18) ωop=ny0/radicalbig 2/Omega1ex/Omega1B. (C19) The relaxation times of the acoustic and optical modes are expressed as 1 τac=1 2α0/bracketleftBig mx0(2/Omega1Lux+/Omega1Bmx0)+n2 y0(/Omega1B+2/Omega1ex)/bracketrightBig , (C20) 1 τop=1 2α0/bracketleftBig /Omega1Bm2 x0+(/Omega1B+2/Omega1ex)n2 y0/bracketrightBig +1 2αsp/Omega1B/parenleftBig m2 x0+n2 y0/parenrightBig . (C21) The coupling constant of the acoustic and optical modes /epsilon1 in Eq. (C17) is given as /epsilon1=/Omega1Lux/Omega1Bω−2 ac. (C22) Consequently, the following relation is obtained from Eq.(C17) in a way similar to the description in Sec. II: 1 τac−1 4ωac/epsilon1δmx0≤0. (C23) From Eq. (C23) , the threshold is given as δmx0,c=4 ωacτac/epsilon1. (C24) We note that Eqs. (18),(19),(20),(21),a n d (22) are rewrit- ten using Eqs. (C24) ,(C18) ,(C20) ,(C1),a n d (C2) with ellipticity p[Eq. (23)], respectively. The ellipticity pofthe acoustic mode in the linear regime without damping is derived from Eqs. (C13) and(C18) as p=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleδm (0) y δm(0) z/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/Omega1 Lux+/Omega1Bmx0 ωac=/radicalBigg 1+Bs 2Bex. (C25) In the large- Bslimit,δm(0) yis much larger than δm(0) zfrom Eq.(C25) , so the acoustic mode can be approximated as described in Sec. II. The first-order equations are written as dδm(1) y dt=−(/Omega1Lux+/Omega1Bmx0)δm(1) z+2/Omega1exδm(0) x+··· , dδm(1) z dt=/Omega1Luxδm(1) y+··· . These equations mean that the optical mode is hybridized into the acoustic mode as the first-order correction: δmz/similarequalmz02/Omega1ex/Omega1Lux ω2ac−ω2opδm(0) x. Thus, the optical mode has a zcomponent under the appli- cation of an oblique magnetic field, the amplitude of which is proportional to mz0. The perturbation corrections for the frequency, the relaxation time, etc. are also inversely proportional to the mode separation in their frequencies ωop−ωac. Therefore, as an approximation these correc- tions are ignored here. This is because B0is small enough and no frequency degeneracies for the acoustic and optical modes are observed in this study. APPENDIX D: SOME DETAILS OF THE MAGNETO-OPTICAL KERR EFFECT The polar magneto-optical Kerr rotation angle φKin the synthetic AFMs is the weighted sum of the angles from each ferromagnetic layer. This is proportional to the zcomponent of the magnetization: φK(t)=a1(t)m1z(t)+a2(t)m2z(t), (D1) where aj(j=1, 2) is a proportionality factor that depends not only on the optical and magneto-optical parameters of jth ferromagnetic layer but also on those of other layers [54]. By introducing the quantities am=(a1+a2)/2a n d an=(a1−a2)/2, we rewrite Eq. (D1) as φK(t)=2am(t)[mz(t)+rmnnz(t)]. (D2) Here we approximate rmn=an/amas a constant. In the conventional manner, Eq. (D2) is expressed as φK(t)=2bB s(t)[mz(t)+rmnnz(t)], (D3) with the constant proportionality factor b. The laser- induced change in the Kerr rotation angle /Delta1φ Kin the 044036-10PARAMETRIC AMPLIFICATION OF MAGNONS... PHYS. REV. APPLIED 13,044036 (2020) experiments is expressed as /Delta1φ K(t)=φK(t)−φK(−0) =2bB s(t)[mz(t)+rmnnz(t)] −2bB s0(mz0+rmnnz0). (D4) Using the relations φK0=2bB s0and nz0(−0)=0, we obtain the expression /Delta1φ K(t) φK0=Bs(t) Bs0[mz(t)+rmnnz(t)]−mz0. (D5) We finally obtain Eq. (17) by substituting Bs(t)=Bs0− /Delta1Bsc(t)into Eq. (D5) . The very small factor rmnoriginates from the small difference in the magneto-optical Kerr rotation angle between FM 1and FM 2located at different depths [40]. To examine the validity of the rmnvalue of −0.065 used in the simulations (see the legend for Table I), we perform a numerical calculation of rmn using the standard magneto-optical theory for multi- layer films [ 54,55]. We consider the model stacking of Si substrate /SiO 2(100)/Ta(3n m)/Fe60Co20B20(3n m)/Ru (0.4 nm )/Fe60Co20B20(3)/Ta(1.5 nm )/Ta-O(1.5 nm )/air, where we take into account the oxidation of the Ta cap- ping layer. The literature values of the complex refractive indices for the materials at a light wavelength of 800 nm are used, as shown in Table II. The value of the com- plex magneto-optical constant Qfor Fe 60Co20B20at a light wavelength of 800 nm is unknown. Therefore, we evalu- ate rmnusing several values ( −Q/prime=0.035−0.020, Q/prime/prime= 0.005−0.014) expected from the values of the complex magneto-optical constant Qfor those alloys measured at a light wavelength of approximately 630 nm [ 56]. The val- ues of rmnare evaluated as 0.04–0.05, meaning that the Kerr rotation angle of the top Fe 60Co20B20layer is slightly larger than that of the bottom Fe 60Co20B20layer. This value is in rough agreement with the rmnvalue used in the simulations. The sign of rmndepends on the definition of the positions of FM 1and FM 2(i.e., top or bottom) in the macrospin model. Thus, it is irrelevant to the discussion in this study. APPENDIX E: THE PARAMETERS RELATED TO THE DEMAGNETIZATION AND RECOVERY Figure 6shows the time-domain data near zero delay for the synthetic AFM sample measured with a magnetic field of approximately 2 T applied perpendicular to the film plane with different pump-laser powers P. The rapid change of /Delta1φ Kat/Delta1tof about 0 ps is due to the ultrafast demagnetization induced by the pump laser. Subsequently, /Delta1φ Kshows relatively slow change. The curves in Fig. 6 show the calculated data for c(t)obtained with Eq. (16) and are fitted to the experimental data. Here we assumethatτand t0are approximately independent of P.F r o m this fitting we obtain τ,t0,a n d c0, as shown in Table Iand its legend. From Eqs. (17),(22),a n d (23), we obtain the relation for the demagnetization and the magnetic field angle: /Delta1φ K(t=0) φK0=−/Delta1Bs Bs0mz0=−ηBcosθB.( E 1 ) Here we define the factor ηB, which is expressed as ηB=/Delta1Bs Bs0B0 Bs0+2Bex0.( E 2 ) Using Eq. (E1), we obtain the ηBvalue from the experi- mental data for the θBvariation of /Delta1φ K/φ K0at zero delay measured at 0.32 T for each laser power P. Then we extract TABLE II. Refractive indices for the materials used in the cal- culation of the magneto-optical effect for the synthetic AFMsample in this study. Material Refractive index Reference Ta2O5 2.10+i0.00128 [ 57] Ta 1.11 +i3.48 [ 58] Ru 3.95 +i5.34 at 633 nm [ 59] Fe60Co20B20 3.40+i4.30 [ 60] SiO 2 1.45 [ 61] Si 3.68 +i0.0054 [ 62] 00.51.0 (a) 00.51.0 (b) 00.51.0 (c) 0 10 2000.51.0Normalized ΔφK Δt (ps)(d) FIG. 6. Time-domain normalized values of the laser-induced change in the Kerr rotation angle /Delta1φ Knear zero delay with a pump-laser power Po f( a )1 0m W ,( b )7m W ,( c )5m W ,a n d( d ) 3 mW. The data are measured with a magnetic field of approx- imately 2 T applied perpendicular to the film plane. The curvesrepresent the calculated data fitted to the experimental data. 044036-11A. KAMIMAKI et al. PHYS. REV. APPLIED 13,044036 (2020) value of /Delta1Bs/Bs0from the ηBvalue using the values for Bex0for each laser power P,Bs0,a n d B0that are input into the simulation. APPENDIX F: ANALYSIS OF THE EXPERIMENTAL DATA OBTAINED AT 3 MW IN THE TIME AND FREQUENCY DOMAINS Figure 7shows the typical time-domain and frequency- domain data measured at a pump-laser power Pof 3 mW. As described earlier, the measured time-domain data are normalized by the saturation Kerr rotation angle. Sub- sequently, we obtain the SD in the frequency domain using the fast Fourier transform with zero padding [ 63]. In the absence of both nonlinear effects and large tem- poral changes in the magnetic moment values, the data can be analyzed by the summation of the following func- tions for the two modes with a proper offset function, as conventionally performed: ∝e−t/τisin(2πfi+φi) (F1) for the time-domain data /Delta1φ K(t)/φ K0,a n d ∝/bracketleftbig (2π)2(f−fi)2+τ−2 i/bracketrightbig−1/2(F2) for the frequency-domain data |/Delta1φ K(f)/φ K0|. Here fiis the frequency, τiis the relaxation time, and φiis the phase for the acoustic mode (i=ac)or the optical mode (i= op). The results of the fittings using Eqs. (F1) and(F2) are shown in Figs. 7(a) and7(b), respectively by the solid curves. 10 20 3000.20.40.60.8 f (GHz)|Δϕ/ϕK0 K | (10–8/Hz0.5)(a) (b)0 0.2 0.4 0.6 0.8–0.4–0.200.20.4ΔϕK / ϕK0 (%) Δt (ns)(a FIG. 7. Typical data obtained with a pump-laser power Pof 3 mW at a magnetic field angle θBof 75◦in the time domain (a) and the frequency domain (b). Solid curves represent the theo-retical data and are fitted to the experimental data. The data near zero delay are removed in (a).Figure 8displays the frequencies fand relaxation times τfor the acoustic and optical modes as a function of the magnetic field angle θB. Here we show the data obtained from the fitting to both the time-domain data and the frequency-domain data. The data for the θBdependence offac,fop,a n d1 /τacobtained from the time-domain data are nearly equal to those obtained from the frequency- domain data, as shown in Figs. 8(a) and8(b). However, there are relatively large uncertainties in the data for the θBdependence of 1 /τop, as shown in Fig. 8(c). This is because the zcomponent of the amplitude for the optical mode observed in our measurement is much smaller than that for the acoustic mode. We also show the data calcu- lated using Eqs. (C18) –(C21) with the parameters given in Table Iand its legend. The theoretical data for facare in accordance with the experimental data for most of the θBvalues [Fig. 8(a)]. The experimental 1 /τopvalues are 024681/τ (grad/s) θB (deg)0 20 40 60 80024680 20 40 60 80010203040f (GHz) (c) (b)(a) Optical mode Optical modeAcoustic modeAcoustic mode FIG. 8. Dependence of (a) the mode frequencies fand relax- ation time τon the magnetic field angle θBfor the (b) acoustic mode and the (c) optical mode evaluated with the time-domain data (solid circles) and the frequency-domain data (open cir-cles) measured at a pump-laser power Pof 3 mW. Solid curves represent the theoretically calculated data. 044036-12PARAMETRIC AMPLIFICATION OF MAGNONS... PHYS. REV. APPLIED 13,044036 (2020) approximated by the theoretical values [Fig. 8(c)]. The experimental data for fopand 1/τacslightly and system- atically deviate from the theoretical values at θB/similarequal0◦,a s seen in Fig. 8(a) and8(b), respectively. The increase in the experimental 1 /τacvalues at θB/similarequal0◦[Fig. 8(b)]m a y stem from some magnetic inhomogeneities in the present sample, because a similar behavior is observed in the magnetization dynamics in ferromagnets as an increase of the ferromagnetic resonance linewidth in a low-frequency regime [ 63]. On the other hand, the inconsistency between the experimental and theoretical data for the fopvalues [Fig. 8(a)] is not yet clear. This is not due to the approx- imation accuracy of Eq. (C19) since the exact theoretical values of fopshow a difference of only a few percent from the values calculated with Eq. (C19) (not shown here). We believe this discrepancy for fopwill not significantly influ- ence the discussion of the parametric amplification in this study, since we focus on the data at θBof about 40◦–60◦, where the theoretical values are not very different from the experimental ones. The fitting using Eq. (F1) did not work well for the data measured at a magnetic field angle θBnearθcwith higher pump-laser powers P. This is because the ampli- tudes of the optical mode are much smaller than those of the acoustic mode with larger PatθB/similarequalθc. In addition, the temporal change of the measured Kerr rotation angles for the acoustic mode are very different from Eq. (F1) at larger P, as seen in Figs. 3(g)–3(i), because of the effect of the parametric amplification on its amplitude. APPENDIX G: TEMPERATURE DEPENDENCE OF THE INTERLAYER COUPLING Figure 9shows several values for the effective mag- netic field due to the interlayer coupling Bexin the syn- thetic AFM sample in this study obtained at different temperatures, which are evaluated from the out-of-plane 200 300 400 50000.20.40.6Bex (T) T (K) FIG. 9. Effective magnetic field due to the interlayer coupling Bexas a function of the temperature Tin the synthetic AFM sample in this study.saturation magnetic field Bsatwith the relation Bsat= 2Bex+Bs. The large reduction of Bexwith increasing tem- perature is similar to that reported in Ref. [ 35]. We also confirm that the temperature dependence of Bsis negligible in this temperature range. APPENDIX H: THE PARAMETRIC AMPLIFICATION IN rf EXPERIMENTS We consider the in-plane-magnetized synthetic AFM under static magnetic field B0x(i.e.,θB=90◦)a n dr f microwave magnetic field δbx(t)x; namely, longitudinal pumping [ 20]. By adding the external torque due to the rf field γny0δbx(t)to Eq. (C12) , we obtain the equation corresponding to Eq. (C16) : d2δmx dt2=−ω2 opδmx−2 τopdδmx dt+/Omega1Bn2 y0γδbx(t). (H1) By substituting the sinusoidal rf magnetic field with its frequency tuned to that of the optical mode, δbx(t)= δbx0sinωopt, we obtain the sinusoidal solution of the optical mode, such as Eq. (11),f r o mE q . (H1) , whose amplitude δmx0is expressed as δmx0=/parenleftbigg2 τopωop/parenrightbigg−1 /Omega1Bn2 y0γδbx0, (H2) where δbx0is the amplitude of the rf magnetic field. With the condition ωop=2ωac, we obtain the threshold of the rf-magnetic-field amplitude δbx0,cfor the parametric amplification of the acoustic mode by substituting Eq. (H2) into Eq. (18): δbx0,c=/Delta1ω op/Delta1ω ac γ/Omega1B4 1−p−2mx0 n2 y0, (H3) where /Delta1ω ac=2/τacis the full linewidth of the rf absorp- tion for the acoustic mode and /Delta1ω op=2/τopis that for the optical mode in the frequency domain. 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PhysRevE.71.026120.pdf

Hysteresis multicycles in nanomagnet arrays J. M. Deutsch, Trieu Mai, and Onuttom Narayan Department of Physics, University of California, Santa Cruz, California 95064, USA sReceived 14 September 2004; published 23 February 2005 d We predict two physical effects in arrays of single-domain nanomagnets by performing simulations using a realistic model Hamiltonian and physical parameters. First, we find hysteretic multicycles for such nanomag-nets. The simulation uses continuous spin dynamics through the Landau-Lifshitz-Gilbert sLLG dequation. In some regions of parameter space, the probability of finding a multicycle is as high as ,0.6. We find that systems with larger and more anisotropic nanomagnets tend to display more multicycles. Our results alsodemonstrate the importance of disorder and frustration for multicycle behavior. Second, we show that there isa fundamental difference between the more realistic vector LLG equation and scalar models of hysteresis, suchas Ising models. In the latter case spin and external field inversion symmetry is obeyed, but in the former it isdestroyed by the dynamics, with important experimental implications. DOI: 10.1103/PhysRevE.71.026120 PACS number ssd: 05.50. 1q, 82.40.Bj, 75.60.Ej, 61.46. 1w I. INTRODUCTION Hysteresis in magnets f1,2gis a paradigm for all history dependent behavior in nature f3–5g. In addition, hysteresis is the cornerstone of the magnetic data storage industry, and ofgreat technological importance f6g. Understanding the full possibilities of magnetic hysteresis is thus important for bothfundamental insights and practical implications. Recently, we have shown f7gthat spin glasses can exhibit stable “multicycle” hysteresis loops, in that when the exter-nal magnetic field is cycled adiabatically sover a range that does not reach saturation d, the magnetization returns to itself afterm.1 cycles of the magnetic field.This behavior should be experimentally observable in spin glass nanoparticles atlow temperature. Thus simple one-cycle hysteresis loops, al-though ubiquitous and generally assumed to be universal, areonly part of a much richer phenomenon. In our previous work f7g, we used the standard Edwards- Anderson spin glass Hamiltonian f8g, with Ising spins and nearest neighbor interactions in three dimensions. Zero tem-perature dynamics or Monte Carlo dynamics at low tempera-ture were used. Starting from saturation, the magnetic fieldwas lowered adiabatically, and then cycled repeatedly over asuitably chosen range.When the system reached steady state,the order of the multicycle sas defined in the previous para- graph dwas measured.Whether a multicycle is present, and if so, its order, depended on the realization of randomness,varying from one system to another. There were two weaknesses in the previous work. First, the systems considered were susceptible to thermal noisemeaning that very low temperatures were needed to preventfluctuations from destroying the periodicity. Second, themodel considered was an Ising model.Although this has welldefined equilibrium statistics, it is not clear that it adequatelyrepresents the dynamics. In this paper, we overcome the problems of the previous paragraph by examining a different system as a candidate formulticycle hysteresis behavior: an array of magnetic nano-particles. With current technology, it is standard to fabricatesuch magnetic arrays according to specification, and a widevariety has been studied experimentally f9gand these haveimportant applications for magnetic storage technology, so called “patterned media” f10g. For our purposes, these sys- tems offer many experimental advantages over the spin glass nanoparticles that we considered earlier. One can build “de-signer arrays” with optimized parameters that maximize mul-ticycles and select them in a predictable manner. Spatiallyresolved measurements are possible, unlike for the spin glasscase, making experimental observations of multicycles morepractical. The relatively large size and shape of these alsogreatly reduce the effects of thermal noise swhich is one important reason that they are useful in disk drive recording d. Finally, the ability to address small regions of the array in-stead of only applying a uniform external magnetic field, andthe sensitivity of multicycle phenomena to relatively smallchanges in system parameters, may open up the possibility ofusing these systems for computation. Through numerical simulations, we demonstrate that mul- ticycles can be seen in an array of pillars made of ferromag-netic material, coupled to each other through dipolar forces,arranged in square and triangular lattices. The external mag-netic field is applied perpendicular to the lattice, i.e., parallelto the axes of the pillars; to be specific, we will refer to thisas the vertical or zdirection. The existence of multicycles is robust, persisting over a large range of system parameters. We also examine the importance of frustration and disor- der in achieving multicycles, a question that has been ofgreat interest in spin glass research f8g. We find that when a square lattice is used instead of a triangular one, multicyclesare not as likely to be seen. As discussed later in this paper,the magnetization in the pillars prefers to be approximatelyvertical, so that the dipolar coupling between the pillars isantiferromagnetic. For a square lattice, the dipolar forces be-tweennearestneighborsarenotfrustrated,unlikethecasefora triangular lattice. sThere is still some frustration because of further neighbor interactions. dFrom this result, we conjec- ture that frustration plays an important role for the existenceof multicycles. However, it has been shown f11gthat systems with different coordination numbers have qualitativelydifferent hysteresis loops regardless of the amount offrustration.PHYSICAL REVIEW E 71, 026120 s2005 d 1539-3755/2005/71 s2d/026120 s7d/$23.00 ©2005 The American Physical Society 026120-1For the spin glass system considered earlier, disorder was explicitly present through the random bond strengths. For thenanoparticle array, although the bond strengths are not ran-dom sunless the spacings between the pillars are varied d, there is crystalline anisotropy, arising from the fact that themagnetization prefers to align itself in a specific directionrelative to the crystal axes. Because of the way in which thepillars are grown, the orientation of the crystal axes is differ-ent in each pillar, and random. Even if the crystalline aniso-tropy energy is small compared to the dipolar sand other d energies, we find that it is sufficient to cause multicycles inan otherwise regular triangular lattice. The order of the mul-ticycle for a specific sample depends on the orientation of thecrystalline axes in its pillars. Although in this paper we con-sider only the case of randomly oriented crystal axes, if thepillars could be grown with the orientations specified, itwould be possible to make arrays whose hysteresis loops aremulticycles of desired order. Regardless of the source, it is desirable to have somein- homogeneity in the model to see multicycles. Without this,as the external magnetic field is reduced from saturation, thesequence in which the magnetic moments of the pillarschanges depends strongly on thermal noise, and repeatablemulticycles are not obtained for any given sample. The model used for the dynamics in our numerical simu- lations is discussed in detail in the next section; the magneticmoment of each pillar is treated as a single “Heisenbergspin,” i.e., with its orientation as a continuous variable, withcontinuous time dynamics. This is in contrast to the earlierspin glass work, with Ising spins and discrete sevent driven d dynamics. Even though the dynamics are continuous, as dis-cussed in the next section there is a shape anisotropy energyfor tall pillars that causes the magnetization to be nearlyvertical and to jump from up to down sor vice versa das the external field is changed. This jump can trigger instabilitiesin other pillars, forming an avalanche. We believe that inorder for multicycles to be seen, it is essential for the inter-action between pillars to be sufficiently strong to cause ava-lanches; in the extreme case, when the pillars are indepen-dent, it is clear that a one-cycle hysteresis loop would beseen. However, avalanches are not sufficient to produce dis- order: for disordered nearest neighbor Ising ferromagnets,the phenomenon of return point memory sRPM df12,13 gcan be proved, precluding multicycles. This leads us to speculatethat frustration is needed. This is the first paper, as far as we are aware, that studies adiabatic hysteresis loops in magnetic systems using themore fundamental Landau-Lifshitz-Gilbert sLLG dequations rather than simplified relaxational dynamics. Experiencefrom critical phenomena might lead one to believe that thedifference between this approach and previous work wouldbe trivial. However there is an important physical differencethat we believe has been overlooked. LLG dynamics destroysymmetry under global spin flip, even though the Hamil-tonian is symmetric under this operation. This mechanismfor the asymmetry is impossible for scalar models such asIsing models. This result, which will be discussed further inthe next section of this paper, has significant experimentalimplications f14g. In the next section of this paper, the dynamical equation used in the numerical simulations is introduced, and variousterms in the model Hamiltonian are calculated. Details of the numerics are given in Sec. III, and the results thereof arepresented in Sec. IV. II. CLASSICAL SPIN DYNAMICS AND THE MODEL HAMILTONIAN Microscopically, the evolution of classical spins is de- scribed by the Landau-Lifshitz-Gilbert equation of motionf15g. The LLG equation is the simplest equation describing micromagnetic dynamics which contains a reactive term anda dissipative term: ds dt=−g1s3B−g2s3ss3Bd, s1d wheresis a microscopic spin, Bis the local effective field, g1is a precession coefficient, and g2is a damping coeffi- cient. The effective field is B=−]H/]s+z, where His the Hamiltonian and zrepresents the effect of thermal noise. Terms in the Hamiltonian will be discussed and computedlater in this section. The reactive term of the LLG equation describes the pre- cession of the spin about its local field, with the angle be-tween the two remaining constant. sThe coefficient of the reactive term g1will be set to unity throughout this paper unless otherwise noted. dThe dissipative term aligns the spin with its local effective field. The cross products in the dissi-pative term ensure that only the tangential component of thefield causes damping, since the length of scannot change. The relaxation time is inversely related to the damping coef-ficient g2. Reasonable approximations for g2are difficult to obtain, but it will be shown that the hysteresis multicyclephenomenon studied in this paper is present for a large rangeof g2. With current technology, nanomagnetic pillars that are ap- proximately 50 nm wide and 100 nm tall can be made offerromagnetic materials such as nickel f9g. For such small pillars, it is found that the ferromagnetic coupling betweenthe atoms dominates the antiferromagnetic dipolar interac-tions. Thus the entire pillar consists of a single magneticdomain. Using the lattice constant of nickel, each pillar holdsapproximately 10 7atoms, allowing us to treat the pillar as a continuous magnetic medium. Edge effects such as splayingnear the boundaries are neglected, and the pillar is treated asa saturated nanomagnet with uniform magnetization. Eachsingle-domain nanomagnet can be viewed as a single degreeof freedom: a magnetic moment of fixed magnitude, whoseorientation represents the direction of the magnetization f16g. The time evolution of this magnetic moment has the same structure as the micromagnetic LLG equation: a reactive part− g1s3B, and a dissipative part − g2s3ss3Bd, although g2 is different from its microscopic value. sHenceforth swill denote a unit vector in the direction of the magnetic momentof a pillar, rather than an individual spin. dAs before, the field Bis given by − ]H/]s; the large number of spins evolving in unison in each pillar allows the thermal noise zto be ne- glected. To complete the specification of the dynamics of the mag- netic moments through Eq. s1d, the Hamiltonian has to beDEUTSCH, MAI, AND NARAYAN PHYSICAL REVIEW E 71, 026120 s2005 d 026120-2calculated in terms of the magnetic moment of each pillar. The various terms in the Hamiltonian are discussed in thefollowing paragraphs. First, the geometry of the pillars introduces a shape aniso- tropy term in the Hamiltonian: H SA=−dzo isz,i2, s2d wheredzis a constant to be calculated in the next paragraph andsz,iis thezcomponent of the magnetic moment of the ith pillar. Shape anisotropy energy is present because of the di-polar interactions between the individual spins within a pil-lar. Qualitatively, if the magnetization of a tall skinny pillaris vertical, the spins are predominantly lined up “head totoe.” This configuration has a lower energy than when themagnetization is horizontal, in which case the spins are pre-dominantly side by side. For a short wide disk, the effect isclearly reversed. For the case of tall pillars, the shape anisotropy reduces the magnetic moment to an almost Ising-like variable, thatcansapproximately dpoint only up or down. The dynamics of anisotropic and isotropic spins are qualitatively different,with avalanche phenomena more likely to occur in theformer than the latter. As mentioned earlier, we believe thatavalanchesarenecessaryforhystereticmulticycles.Notethateven when the shape anisotropy is large, we evolve eachmagnetic moment according to Eq. s1drather than as an Ising variable, i.e., with an orientation that evolves continuouslywith time, although the shape anisotropy causes rapid transi-tions from up to down states. Deriving the form of H SAand the value of dzrequires solving a magnetostatic problem. The energy of the field dueto the microscopic spins in a single pillar is W =s1/2 m0ded3xuBsxdu2, whereBsxdis the magnetic field at x due to the spins. Through Ampere’s law and vector calculus manipulations, we can rewrite this in terms of the magneti-zationMsxd.Foruniformmagnetization,theresultcan supto an additive constant dbe converted to a squadruple dintegral over two surfaces, similar to electrostatics, with the self-energy of the magnetic surface “charge” to be calculated. Forcylindrical pillars, which we consider in the rest of this pa-per, if the magnetization has magnitude M 0and makes an angle uto the vertical, and Randhare the radius and height of the pillars, the final result is W=m0M02R3FSh RDcos2u s3d up to an additive constant independent of u. Comparing with Eq.s2d, we see that dz=−m0M02R3Fsh/Rd, whereFsh/Rdis a function of the aspect ratio that can be evaluated numerically. WithM0equal to the saturation magnetization for nickel, 4.843105A/m f9g, the values of dzfor different sized pil- lars are given in Table I. If the pillars are ellipsoidal insteadof cylindrical, d zcan be obtained analytically instead of nu- merically f9,16g. Asecond form of anisotropy energy is caused by the crys- tal structure of Ni.As mentioned earlier, the crystal axes givea preferential direction to the magnetization, independent ofthe shape of the pillar. Nickel has a face-centered cubic structure which tends to align spins in the f111gdirection. If ax,i,ay,i, and az,iare the direction cosines of the magnetiza- tion of the ith pillar to its sx,y,zdcrystal axes, the crystal anisotropy energy can be expanded in powers of a. The first two terms are f9,16g HCA=o i−K1 2sax,i4+ay,i4+az,i4d+K2ax,i2ay,i2az,i2,s4d where an additive constant has been dropped. The material parameters K1andK2can be obtained by multiplying the experimentally obtained energy densities by the volume ofthe pillars. Crystal anisotropy energy densities have approxi-mate values of −5 310 3and −2 3103J/m3forK1/Vand K2/Vrespectively f9g. Athough the ratio of K2/K1is,0.4, the first term dominates, since it has two fewer powers of a. In addition to the shape and crystal anisotropies that affect each pillar by itself, there is dipolar coupling between pillars.Microscopically, this is similar to the shape anisotropy en-ergy, except that it arises from interactions between spins ondifferent pillars. The resultant interaction energy is of theform H dip=o i,jÞisi·Asrijd·sj. s5d Asrijdis a second-rank tensor that depends only on the sepa- rationofthepillars.Theelementsof Asrijdaredeterminedby numerically solving integrals similar to the integrals for the shape anisotropy energy. The last term in the Hamiltonian is due to the external magnetic field, which we take to be in the zdirection. The form of this term is the conventional one, Hext=−Beosz,i. Hysteresis occurs as Beis varied adiabatically, with the sys- tem evolving according to the LLG equation. In summary, the full Hamiltonian has four terms: shape anisotropy, crystalline anisotropy, external field, and dipole-dipole interaction. The first two are properties of the pillarsindividually, the external field term is the term that is adia-batically changed to observe hysteresis, and the dipole termis an interaction between pillars:TABLE I. Calculated shape anisotropy coefficients dzfor pillars with radius R=30 nm and different aspect ratios h/R. The ratios between crystalline anisotropy coefficients and dzare also given. h/Rd zs10−18Jdu K1/dzuu K2/dzu 0.5 −2.866 0.0740 0.0296 1 −2.689 0.1577 0.06312 0.769 1.102 0.4413 5.754 0.2211 0.08854 11.30 0.1501 0.06005 17.11 0.1240 0.0496 10 47.41 0.0894 0.0358HYSTERESIS MULTICYCLES IN NANOMAGNET ARRAYS PHYSICAL REVIEW E 71, 026120 s2005 d 026120-3H=o iS−dzsz,i2−K1 2sax,i4+ay,i4+az,i4d+K2ax,i2ay,i2az,i2 −Besz,i+o jÞisi·Asrijd·sjD. s6d In the numerics, the actual values of the coefficients in the Hamiltonian are inconsequential, and only the ratios of termsare relevant. Table I shows the results of the calculationsdescribed above for d z,K1, andK2. Evidently, for the dimen- sions of the nanomagnetic pillars of interest, the shape an-isotropy term is larger than the crystalline anisotropy. Thedipolar coupling is also small compared to d zfor the lattice spacings of interest. In the simulations, all energies are nor-malized to d z.Although dzis dominant for pillars with aspect ratios of interest, the other terms must be included in theHamiltonian because they affect the dynamics qualitatively.Without the dipole term, each pillar would be isolated. Thecrystalline anisotropy term introduces quenched randomnessin the system, and determines the order in which the s i’s flip when the magnetic field is changed; its importance has beendiscussed toward the end of Sec. I. Thus the Hamiltonian ofEq.s6dhas all the important terms that have to be kept. As mentioned in Sec. I, although the Hamiltonian of Eq. s6dis invariant if all the s i’s are flipped salong with the ex- ternal magnetic field d, the dynamics of Eq. s1dare not. In Eq. s1d, under spin and external field reversal, the left hand side and the dissipative term on the right hand side change sign,but the reactive term does not. Therefore the spin inversionsymmetry, although relevant to equilibrium static properties,does not apply to the nonequilibrium dynamics appropriatefor hysteresis. In particular, the two branches of the majorhysteresis loop are not complementary to each other. III. NUMERICS As mentioned earlier, the pillars are modeled as single degrees of freedom which follow the LLG equation of mo-tion. The effective field for each magnetic moment in theLLG equation is the “spin” derivative of the Hamiltonian ofthe previous section. Pillars are placed on a two-dimensionaltriangular lattice, to maximize the frustration of the dipolarbonds. All systems studied are 4 34 lattices with open boundary conditions. Unlike simulations of conventionalcondensed matter systems with Os10 23dparticles, the array size chosen here is not an approximation because arrays could be fabricated with an arbitrary number of pillars. Thesize and boundary condition dependence of the phenomenawe observe may have interesting features; this is left forfuture work. The positions of the pillars are ixˆ+jsxˆ/2 +yˆ ˛3/2dwithi,j=0,...,3. The orientation of the crystallo- graphic axes is separately and randomly chosen for each pil- lar. Depending on the choice of these random orientations, asample can have multicycles of various orders m, or a simple hysteresis loop si.e.,m=1d. Square lattices are also consid- ered. The dimensions of the cylinders and the separation be- tween them, the external field range, and the damping coef-ficient g2are input parameters. These are used to calculatedz,K1,K2, and the elements of Asrijd. All the input param- eters can be adjusted to maximize the occurrence of multi- cycles, except g2. Since g2is a property of the material, but is unknown, we make sure that the results reported here arevalid over a wide range of g2: essentially the entire s0,‘d range for multicycles, and 0.0005 ,g2,50 for asymmetric major hysteresis loops, which should include the experimen-tally appropriate value. For instance, in NiFe films, g2/g1is measured to be 0.013 f17g. As long as the ratio g2/g1is finite, the major hysteresis loop will be asymmetric, althoughthe asymmetry will become small as g2!‘and the dynam- ics are effectively Ising-like. Numerical modeling of the adiabatic field variation is straightforward. The external field is lowered or raised by asmall field step dBe. To optimize speed, the field step dBeis adjusted adaptively, since a small step is required during ava-lanches. The effective field is then calculated and the systemevolves by a small time step dtwith this field. This time evolution is repeated, without changing Be. Numerical inte- gration of the LLG equation is implemented using the fourth-order Runge-Kutta algorithm. Once the system “settles” to astationary state, the external field is changed again. Waitingfor the system to reach a stationary state is equivalent tovaryingB emore slowly than all the dynamics of the system, i.e., adiabatically. The time scale of an avalanche is pre-sumed to be very short; therefore during an avalanche, dBeis adjusted to be extremely small to maintain adiabatic change. The requirement for settling is that the configuration after the evolution by a time step dtis essentially the same as the configuration before. In practice, some numerical tolerance isallowed, and the initial and final configurations must differ by less than this tolerance. The sums dsx,i2+dsy,i2+dsz,i2for eachiandoidsz,imust all be less than 10−11in one time step for the system to be considered stationary. The results re-ported here are insensitive to a reduction of dBe,dt,o rt h e tolerance, and therefore represent adiabatic field variationwith continuous time dynamics. Starting from a large positive B esso that all the pillars are magnetized upward dthe external field is lowered and cycled adiabatically over a range f−Bemax,Bemaxg. The configuration hsijis compared at Bemaxafter each cycle. If hsijis the same after every moccurrences of Be=Bemax, the system is in an m-cycle. Similar to the condition for settling, the configura- tions match up to a tolerance; we have verified in numerouscases that the tolerance does not introduce spurious multi-cycles. A tolerance of 10 −4for each component of the mag- netization was found to be sufficient. Initially, the systemundergoes a transient period of a few cycles of B ebefore reaching a limit cycle. Figure 1 shows the major hysteresis loop for a sample realization of randomness. Since all the pillars are magne- tized vertically each time Be=±Bemax=±‘,mis trivially equal to 1. However, one can see the avalanching dynamicscharacteristic of this system, and the fact—discussedearlier—that the ascending and descending branches of themajor loop are not complementary. Figure 2 shows a hyster-esis minor loop with a two-cycle. IV. RESULTS Using an algorithm that performs the operations of the previous section, we search through a large number of real-DEUTSCH, MAI, AND NARAYAN PHYSICAL REVIEW E 71, 026120 s2005 d 026120-4izations of randomness to find regions in parameter space where the probability of finding multicycles is high. The pa-rameters in the model are the radius Rand height hof the pillars,B emax, and the damping coefficient g2. If the lattice spacing,R, andhare all scaled by a factor l, all the terms in the Hamiltonian are scaled by l3, which does not affect the ratios of the terms.Accordingly, the lattice spacing can be setto 100 nm without loss of generality. Given a set of param-eters, the algorithm determines the periodicity mfor a given realization of randomness. By classifying the periodicity fora large number of realizations, we obtained the approximateprobability for finding an m-cycle as a function of m. The easiest parameter to vary experimentally is B emax.A s mentioned in the previous section, when Bemaxis too large or too small, multicycles will not be present. We find that mul- ticycles can be roughly optimally found when Bemaxap- proaches the saturation field Besatbut not greater. Besatis dif- ferent for every realization of randomness; therefore theoptimal field can only be determined by scanning over vari- ous values of Bemax. The range of Bemaxwhere the occurrence of multicycles is appreciable depends on the pillar dimen-sions. For R=30 nm and h=180 nm, the probability of find- ing multicycles when B emaxis optimal is ,2–3 times the probability when Bemaxis,15% smaller than the optimal field. In general, for systems in which the probability of find- ing multicycles is small, the range of Bemaxwhere the prob- ability is nonzero is narrow. The narrow range in Bemaxis not an obstacle to finding multicycles due to the ease of tuningthis parameter experimentally. The damping coefficient of the LLG equation, g2, cannot be easily calculated. In fact, different experimental environ-ments could allow for a large range of g2. Because of our inability to obtain a reasonable and realistic approximationfor g2, we run searches for a wide range of g2srelative to g1d. The results show that multicycles exist for very small g2 to essentially infinite damping. sThe large damping limit is implemented by setting g1=0 and keeping g2finite. dSmall values of g2tend to give more multicycles, as one might expect: the probability of finding multicycles increases by afactor of ,1.5 when g2is reduced from g1to,0.1g1. Un- expectedly, the multicycle probability also seems to increaseslightly when g2is larger than g1.When the dynamical equa- tion is strictly dissipative sg1=0d, the multicycle probability is comparable to the probability when g2*g1. There are no clear trends in the distribution of mwhen g2is changed. Because of the difficulty in calculating g2, we conserva- tively set g2to a value where the probability of finding mul- ticycles is approximately minimal. As mentioned in the pre- ceding paragraph, this occurs when g2<g1. WithBemaxat its optimal value, and g2=g1=1, the probabilities of finding multicycles for systems with pillars of different Randhare found. The different terms in the Hamiltonian scale differ-ently asRandhare varied. Table II shows results for sys- tems of the same aspect ratio sh/R=5dand different radii. The pillar radius cannot be larger than ,45 nm with a lattice spacing of 100 nm. From Table II, one can see that systemsof pillars with larger radii generally display multicycles moreoften. The occurrence of multicycles depends largely on the as- pect ratio of the pillars. Avalanches tend to occur only forpillars where h/Ris large; accordingly, systems with disk- FIG. 1. Major hysteresis loop ssolid curve df o ra4 34 triangular lattice of pillars. The steps demonstrate avalanching dynamics. Inorder to see that the ascending and descending branches of the loopare not complementary, the dashed curve shows the same hysteresisloop, with M!−MandB!−B; the solid and dashed curves clearly do not coincide. FIG. 2. A hysteresis two-cycle, starting at Be=−Bemax. The solid curve is a hysteresis loop after one cycle of the external field. Thedashed curve is the hysteresis loop after the second cycle. Anothersweep of the external field would retrace the solid curve, indicatingthat this particular realization of randomness undergoes a two-cycle.B emaxis less than the saturation field Besat.sIfBemaxwere increased beyondBesat, no multicycles would be found. Thus the two-cycle shown in this figure is a minor loop. dTABLE II. Approximate probabilities of finding an m-cycle for systems with pillars of different radii. The lattice spacing is 100 nm and the aspect ratio h/Ris 5. RsnmdPm=2Pm=3Pm=4Pm.4Pm.1 1 0 00000 20 0.08 0.02 0 0 0.125 0.14 0.12 0.02 0.04 0.3230 0.2 0.06 0.04 0.16 0.4635 0.22 0.22 0.04 0.12 0.640 0.22 0.12 0.1 0.18 0.6245 0.28 0.12 0 0.10 0.50HYSTERESIS MULTICYCLES IN NANOMAGNET ARRAYS PHYSICAL REVIEW E 71, 026120 s2005 d 026120-5shaped pillars, i.e., with negative dz, do not display multi- cycles. In fact, even when dzis positive, multicycles are only found for a sufficiently large shape anisotropy energy. TableIII shows numerical results for systems of pillars with a30 nm radius and various aspect ratios. Multicycles are onlylikely to be found for systems with long pillars. No multi-cycles are found for disk-shaped pillars as expected. Systems with pillar vacancies are also studied. Vacancies are implemented by introducing a finite probability for a pil-lar to be missing at every lattice site. One might expect thatthese vacancies would introduce more randomness in thesystem, thereby increasing the number of multicycles. Forpillars with R=30 nm and h=150 nm, the probability of finding a multicycle is ,0.46 without any vacancies. When the probability of having a vacancy at a site is smalls,0.2d, the number of multicycles drops by about 40%. When the vacancy probability increases to *0.5, the prob- ability of finding multicycles decreases to less than 0.1. Thisdecrease in probability could be due to the decrease in thenumber of pillars. We conclude that, contrary to what onemight expect, random vacancies do not increase the probabil-ity of multicycles. The probability of finding a multicycle is significantly less on a square lattice than on a triangular lattice; for a pillararray arranged in a square lattice, the probability of finding amulticycle is approximately half of the corresponding prob-ability for a triangular lattice. We speculate that this differ-ence in probabilities could be due to the different amount offrustration between the configurations. If a square lattice isused instead of a triangular one, the dipolar couplings be-tween nearest neighbor pillars are not frustrated. In a check-erboard pattern, all nearest neighbor bonds would be satis-fied, but the next nearest neighbor bonds sand certain neighbors further apart dwould not be satisfied. Thus, if frus- tration is important for the existence of multicycles, multi-cycles are expected to be much less probable on a squarelattice. A possible mechanism for increasing frustration and dis- order is to introduce random ferromagnetic couplings be-tween pillars. These bonds could be manufactured by build-ing one large pillar instead of two small ones. The largepillar would have two domains that are coupled ferromag- netically. By randomly assigning the bonds with some prob-ability, more disorder can be introduced in addition to thatfrom the crystalline anisotropy. We did not study how theseferromagnetic bonds affect the number of multicycles. One interesting question is whether exact return point memory f12gsurvives when it is extended beyond the ran- dom field Ising model with purely ferromagnetic interactionsto continuous time vector models. To answer that, we use theLLG model with nearest neighbor random ferromagnetic in-teractions and the same crystalline and shape anisotropymodel that is used above swithout dipole coupling d. We ap- ply a random field and search for violations of RPM withdifferent random seeds. We start at high fields and go to aminimum field of −1.7 and record the spin configuration.Then we go up to a field 1.7 and back down to −1.7. We findthat for a 4 34 square lattice of spins, of the order of 1% of systemsviolateRPM because the initial and final minimum configurations and total magnetization are substantially dif-ferent. Violation of RPM is seen both with and without theprecessional term in Eq. s1d. This shows that it is not possible to extend the proof of RPM to continuous vectormodels. V. CONCLUSIONS In this paper, we have investigated the feasibility of ob- serving multicycles and noncomplementary hysteresis loopsin a candidate experimental system: that of cylindrical mag-netic nanopillars arranged on a lattice. We have performedrealistic numerical simulations of this system by calculatingthe magnetic interactions between the pillars and then em-ploying continuous spin dynamics and the Landau-Lifshitz-Gilbert equation to obtain their time evolution. Using physi-cally appropriate parameters, we have shown that there isoften multicycle hysteretic behavior, i.e., a periodic adiabaticexternal magnetic field causes a subharmonic steady stateresponse in the magnetization. Because systems of this kindare currently the subject of much experimental investigation,we believe that it would be fruitful to attempt to observe theunusual behavior predicted here. We have also shown that, even though the Hamiltonian is invariant under spin and external field reversal, the dynamicsare not, so that the ascending and descending branches of themajor hysteresis loop are not complementary. This result re-quires both a precessional and a relaxational term in the dy-namics, emphasizing the importance of both. In particular,since it is impossible to include precession for Ising models,they are—despite their ubiquitousness for strongly aniso-tropic magnets—qualitatively inadequate in certain respects.Further implications of the noncomplementary nature ofLLG dynamics will be studied elsewhere f14gand in future research. ACKNOWLEDGMENTS We thank John Donohue, Holger Schmidt, Mark Sherwin, and Larry Sorensen for very useful discussions.TABLE III. Approximate probabilities of finding an m-cycle for systems with pillars of different aspect ratios h/R. The lattice spac- ing is 100 nm and the radius is fixed at 30 nm. h/RP m=2Pm=3Pm=4Pm.4Pm.1 0.5 0 0 0 0 0 1 000003 0.16 0.04 0.02 0.12 0.344 0.12 0.08 0.02 0.16 0.385 0.2 0.06 0.04 0.16 0.466 0.16 0.1 0.06 0.16 0.48 10 0.24 0.2 0.06 0.1 0.6DEUTSCH, MAI, AND NARAYAN PHYSICAL REVIEW E 71, 026120 s2005 d 026120-6f1gC. P. Steinmetz, Trans. Am. Inst. Electr. Eng. 9,3s1892 d. f2gH. Barkhausen, Z. Phys. 20, 401 s1919 d. f3gJ. Katz, J. Phys. Colloid Chem. 53, 1166 s1949 d;P .H .E m m e t t and M. Cines, ibid.51, 1248 s1947 d. f4gZ. Z. Wang and N. P. Ong, Phys. Rev. B 34, 5967 s1986 d. f5gJ. Ortin and L. Delaey, Int. J. Non-Linear Mech. 37, 1275 s2002 d, and references therein; J. Ortin, J. Appl. Phys. 71, 1454 s1992 d. f6gMagnetic Recording Technology , 2nd ed., edited by C. D. Mee and E. D. Daniel sMcGraw-Hill Professional, New York, 1996 d. f7gJ. M. Deutsch and O. Narayan, Phys. Rev. Lett. 91, 200601 s2003 d. f8gK. H. Fischer and J. A. Hertz, Spin Glasses sCambridge Uni- versity Press, Cambridge, U.K., 1991 d, and references therein. f9gM. C. Abraham, H. Schmidt, T. A. Savas, H. I. Smith, C. A. Rose, and R. J. Ram, J. Appl. Phys. 89, 5661 s2001 d;H . Schmidt and R. J. Ram, ibid.89, 507 s2001 d.f10gS. H. Charap, P. L. Lu, and Y. He, IEEE Trans. Magn. 33, 978 s1997 d; H. N. Bertram, H. Zhou, and R. Gustafson, ibid.34, 1845 s1998 d; D. Weller and A. Moser, ibid.35, 4423 s1999 d. f11gS. Sabhapandit, D. Dhar, and P. Shukla, Phys. Rev. Lett. 88, 197202 s2002 d. f12gJ. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts, and J. D. Shore, Phys. Rev. Lett. 70, 3347 s1993 d. f13gJ. A. Barker, D. E. Schreiber, B. G. Huth, and D. H. Everett, Proc. R. Soc. London, Ser. A 386, 251 s1983 d; J. Ortin, J. Appl. Phys. 71, 1454 s1992 d. f14gM. S. Pierce et al., Phys. Rev. Lett. 90, 175502 s2003 d;94 017202 s2005 d. f15gF. H. de Leeuw, R. van den Doel, and U. Enz, Rep. Prog. Phys. 43, 689 s1980 d. f16gE. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. Lon- don, Ser. A 240, 599 s1948 d. f17gG. M. Sandler et al., J. Appl. Phys. 85, 5080 s1999 d.HYSTERESIS MULTICYCLES IN NANOMAGNET ARRAYS PHYSICAL REVIEW E 71, 026120 s2005 d 026120-7

PhysRevB.103.165122.pdf

PHYSICAL REVIEW B 103, 165122 (2021) Low Gilbert damping in epitaxial thin films of the nodal-line semimetal D03-Fe 3Ga Shoya Sakamoto ,1,*Tomoya Higo ,1,2Shingo Tamaru ,3Hitoshi Kubota ,3Kay Yakushiji ,2,3 Satoru Nakatsuji ,1,2,4,5and Shinji Miwa1,2,5,† 1The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan 2CREST, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan 3National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Emerging Computing Technologies, Tsukuba, Ibaraki 305-8568, Japan 4Department of Physics, University of Tokyo, Hongo, Bunkyo, Tokyo 113-0033, Japan 5Trans-scale Quantum Science Institute, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan (Received 12 January 2021; accepted 30 March 2021; published 20 April 2021) D03-ordered Fe 3Ga has been recently reported to show a giant anomalous Nernst effect due to a large Berry curvature along the interconnected nodal lines concentrated near the Fermi level. This peculiar bandstructure may be advantageous for spintronics device applications because one can expect a strong responseto a gate-voltage application. This study evaluates the Gilbert damping constants, a key parameter governingthe magnetization dynamics of epitaxial thin films of D0 3-Fe 3Ga and bcc-Fe by ferromagnetic resonance measurements. We find that the Gilbert damping constant of D03-Fe 3Ga [(6 .0±0.2)×10−3] is relatively low and comparable to that of Fe [(2 .3±0.2)×10−3]. The low Gilbert damping suggests that the interconnected nodal lines near the Fermi level do not hinder magnetization dynamics, making D03-Fe 3Ga even more attractive as a building block of spintronics devices. DOI: 10.1103/PhysRevB.103.165122 I. INTRODUCTION In recent years, there has been growing interest in physical phenomena associated with the nonvanishing Berry curvatureof Bloch states [ 1]. The anomalous Hall effect (AHE) and anomalous Nernst effect (ANE) are typical examples: AHEresults from the summation of the Berry curvature of all occu-pied states [ 1,2], and ANE results from the Berry curvature of bands near the Fermi level [ 1,3]. Indeed, exceptionally large AHE and ANE were observed in ferromagnetic Weylsemimetals such as Co 2MnGa [ 4,5] and Co 3Sn2S2[6,7] and also in antiferromagnetic Weyl semimetals such as Mn 3Sn [8,9] and Mn 3Ge [10,11], where there is claimed to be a large Berry curvature around the Weyl nodes. Although not a Weyl semimetal, D03-ordered Fe 3Ga has been recently reported to show a giant ANE of 6 μV/K [12], close to the highest values reported thus far and 20 times stronger than that of α-Fe.D03-Fe 3Ga consists of only two elements and has a simple cubic structure, as shown inFig. 1(a), and therefore, it is relatively easy to fabricate even in a thin-film form. These features make D0 3-Fe 3Ga promis- ing for the realization of low-cost and flexible thermoelectricgenerators. The origin of the giant ANE was attributed to thelarge density of states (DOS) and Berry curvature along thequasi-two-dimensional (2D) network of gapped nodal lines(referred to as nodal web). The nodal web consists of twonearly flat valence and conduction bands touching each other *Corresponding author: shoya.sakamoto@issp.u-tokyo.ac.jp †Corresponding author: miwa@issp.u-tokyo.ac.jpthat acquire a large Berry curvature when the spin-orbit in-teraction gaps the intersections. The large density of statesand Berry curvature at the Fermi level may be useful forspintronics device applications because a gate voltage may ef-fectively modulate the Berry-curvature-associated response aswell as magnetic anisotropy. In this respect, characterizing themagnetic damping, which governs magnetization dynamics, isof great importance [ 13,14]. It is worth mentioning that Fe-Ga alloys (galfenol) also show a sizable ANE [ 15] and are famous for their large magnetostriction [ 16–19]. These aspects may also be advantageous for applications. In the present study, we perform ferromagnetic resonance (FMR) measurements on epitaxial thin films of D0 3-Fe 3Ga and bcc Fe to characterize their magnetic damping. The in-trinsic Gilbert damping constant of Fe 3Ga is deduced to be (6.0±0.2)×10−3, which is relatively low and comparable to the Fe value of (2 .3±0.2)×10−3. The low Gilbert damping constant makes D03-Fe 3Ga even more attractive for device applications that involve magnetization switching. II. EXPERIMENT AND CALCULATION Fe and Fe 3Ga thin films were grown on MgO(001) sub- strates by molecular beam epitaxy (MBE) and dc magnetronsputtering, respectively. The sample structures are schemat-ically shown in Fig. 1(b). The substrates were annealed at 800 ◦C for 10 min before the deposition. For the Fe thin film, a 5-nm-thick MgO seed layer was first grown at a rate of0.1 Å/s, and a 50-nm-thick Fe layer was grown at a rate of 0.2 Å /s. The sample was annealed at 350 ◦C for 30 min and covered by a 5-nm-thick MgO layer deposited at a rate 2469-9950/2021/103(16)/165122(5) 165122-1 ©2021 American Physical SocietySHOYA SAKAMOTO et al. PHYSICAL REVIEW B 103, 165122 (2021) (a) (c)(b) (d) (e) Intensity[arb.unit] 30282624 2θ[degree](111) Fe3Ga(f) (e)(LogIntensity[arb.unit] 80604020 2θ[degree] (004)(002) MgOsubstrateFe3GaMgO(001) substratecappinglayer Fe(001) 50nm Fe[110]MgO(001) substrateFe3Ga(001) 50nmcappinglayer Fe3Ga[110] Fe Ga FIG. 1. (a) Crystal structure of D03-Fe 3Ga drawn using VESTA [20]. (b) Thin-film structures. (c) and (d) In situ reflection high- energy electron diffraction images of the annealed Fe and Fe 3Ga surfaces taken along the Fe [110] and Fe 3Ga [110] (or MgO [100]) direction. (e) Out-of-plane 2 θ/ω x-ray diffraction (XRD) patterns of the Fe 3Ga thin film and an MgO (001) substrate. (f) Fe 3Ga (111) superlattice peak is detected by an 2 θ/ω scan, indicating the D03 ordering of Fe 3Ga. of 0.1 Å /s. For the Fe 3Ga thin film, a 50-nm-thick Fe 3Ga layer was directly grown on an MgO substrate at a rate of0.4 Å/s and annealed at 500 ◦C for 30 min. The 2-nm-thick MgO and 5-nm-thick AlO xcapping layers were grown at a rate of 0.1 Å /s. This Fe 3Ga growth procedure is the same as in our previous report [ 12]: We used the same apparatuses, the same growth rate, and the same annealing temperature andtime. Note that only the Fe 3Ga layer was grown by sputtering and the other layers were grown by MBE. Both sputteringand MBE chambers are connected, and all the depositionwas conducted without exposing the samples to air. The basepressures of the sputtering and MBE chambers were below5×10 −7and 3×10−8Pa, respectively. We conducted FMR measurements using a vector network analyzer and a coplanar waveguide (CPW). The width andthe thickness of the center conductor were 200 and 12 μm, respectively, and the gap between the center conductor andthe ground was 60 μm. The 20- μm-thick polyester tape was inserted between the CPW and the samples. Unlike a conven-tional measurement setup, the magnetic field was modulatedby±0.48 mT at each magnetic field H, and the magnetic-field derivative of the transmission coefficient S 21, namely /Delta1S21= S21(H+0.48 mT) −S21(H−0.48 mT), was measured. This field-modulation technique gives background-free FMRspectra and enables precise estimation of FMR peak positions and peak widths even when the signals are weak [ 21]. Note that it is confirmed that the field-modulation technique yieldsessentially the same FMR peak positions and widths as theconventional field-sweep technique. All the FMR measure-ments were performed at room temperature. Magnetic fieldsup to∼2 T were applied to the [100], [110], and [001] direc- tions of Fe 3Ga and Fe. To discuss the relationship between the DOS and the Gilbert damping constants of Fe and D03-Fe 3Ga, we per- formed density-functional-theory (DFT) calculations underthe Perdew-Burke-Ernzerhof generalized gradient approxi-mation using the WIEN 2Kpackage [ 22]. The experimental lattice constants of 2.87 and 5.80 Å [ 12]w e r eu s e df o r Fe and D03-Fe 3Ga, respectively. A spin-orbit interaction was included. Brillouin-zone integration was performed on a17×17×17k-point mesh. III. RESULTS Figures 1(c) and 1(d) show the in situ reflection high- energy electron diffraction (RHEED) images of the annealedFe and Fe 3Ga surfaces. The sharp streak patterns guarantee the epitaxial growth of Fe and Fe 3Ga layers. In addition to the main intense streaks, the RHEED image of Fe 3Ga shows weak streaks in between marked by the red arrows in Fig. 1(d). We attribute the weak streaks to the superlattice diffractionfrom the√ 2×√ 2 Ga arrangement on the D03-Fe 3Ga sur- face. The crystallinity of the Fe 3Ga thin film is also evaluated by performing x-ray diffraction (XRD) measurements. Theout-of-plane XRD 2 θ/ω pattern [Fig. 1(e)] does not show any unknown second-phase peaks but only shows the Fe 3Ga (002) and (004) peaks. This second-phase-free XRD patterntogether with the streaky RHEED image indicate the single-crystalline epitaxial growth of the Fe 3Ga layer. Note that an XRD φscan (not shown) exhibits a fourfold symmetric pattern accordingly. The D03ordering is confirmed by the 2θ/ω XRD pattern [Fig. 1(f)] that exhibits the D03-specific (111) peak. From the XRD (111) and (004) [or (002)] peaks,the in-plane and out-of-plane lattice constants are estimatedto be 5.788 and 5.761 Å, respectively. The longer in-planelattice constant is consistent with the fact that there is tensileepitaxial strain from the MgO substrate caused by ∼3% lattice mismatch. Figures 2(a) and2(b) show typical FMR spectra of the Fe and Fe 3Ga thin films, respectively. Here, an FMR spectrum refers to /Delta1S21as a function of magnetic field ( H) at a fixed frequency. In order to deduce the FMR peak position ( Hr) and peak width ( /Delta1H), we fit the following function to each spectrum [ 23,24], /Delta1S21=S(H+0.48 mT) −S(H−0.48 mT) , (1) S(H)=A H2r−H(H−i/Delta1H)eiφ. (2) Here, Arepresents the amplitude of the signals, and eiφ accounts for a phase shift. Equation ( 2) can be approxi- mated as the summation of the Lorentzian and antisymmetricLorentzian functions when /Delta1H/lessmuchH r, and/Delta1Hcorresponds to the full width at half maximum of the Lorentzian 165122-2LOW GILBERT DAMPING IN EPITAXIAL THIN FILMS … PHYSICAL REVIEW B 103, 165122 (2021) 25 20 1510 50Frequency[GHz] 0.40.20 μ0Hr[T]FeH||[100] H||[110](c) (e) (f) 2.01.825 20 1510 50Frequency[GHz] 0.40.20 μ0Hr[T]H||[001]Fe3Ga H||[100]H||[110](d) 1.7 1.6 MagneticField[T]5GHz 9GHz 13GHzFe3GaH||[001] (b) Real Imaginary0.30.20.1 MagneticField[T]19GHz 23GHz 27GHzFeH||[100] (a) Real Imaginary8 6420μ0ΔH[mT] 2520151050 Frequency[GHz]α=0.0023(2)[110] [001][100]Fe α=0.0024(2) 50 40 30 20 10 0μ0ΔH[mT] 2520151050 Frequency[GHz]α=0.0060(2)[110] [001][100]Fe3GaΔS21[a.u.]ΔS21[a.u.] ΔS21[a.u.] FIG. 2. (a) and (b) Typical ferromagnetic resonance (FMR) spectra of the Fe and Fe 3Ga thin films. The solid curves are the fit of Eqs. ( 1) and ( 2). (c) and (d) Resonance frequency plotted against resonance magnetic fields. The solid curves are the fit of Eqs. ( 4)–(6). (e) and (f) FMR peak width as a function of the resonance frequency. The data are fitted by a linear function (solid lines), the slope of which represents theintrinsic Gilbert damping constant. component. The fitted curves reproduce the experimental spectra very well, as shown in Figs. 2(a) and2(b) by solid curves. Figures 2(c) and2(d) show the deduced FMR peak po- sitions for the Fe and Fe 3Ga thin films as a function of microwave frequency. We did not perform out-of-plane FMRmeasurements for the Fe thin film because the saturation mag-netization was too large for the present experimental setup. In thin-film samples, the crystal symmetry is lowered from cubic to tetragonal because of the epitaxial strain from thesubstrate. Under the tetragonal symmetry, the total energy ofthe system can be written as [ 25] E=μ 0MsH[cosθcosθH+sinθsinθHcos(φ−φH)] −1 2μ0M2 ssin2θ−K2⊥cos2θ −1 2K4⊥cos4θ−1 2K4/bardbl1 4(3+cos 4φ)s i n4θ, (3) where μ0is the vacuum permeability, and Msis the saturation magnetization. θ(θH) denotes the tilt angle between the [001] direction and the magnetization (magnetic-field) direction,andφ(φ H) denotes the azimuth angle of the magnetization (magnetic field) with respect to the [100] direction. K2⊥,K4⊥, andK4/bardblrepresent the out-of-plane uniaxial, out-of-plane cu- bic, and in-plane cubic magnetocrystalline anisotropy energy,respectively. Based on Eq. ( 3), the relationship between the resonance magnetic field and frequency can be written as[25,26] f [100] r=μ0γ 2π/radicalbig (Hr+H4/bardbl)(Hr+Meff+H4/bardbl), (4) f[110] r=μ0γ 2π/radicalbig (Hr−H4/bardbl)(Hr+Meff+H4/bardbl/2),(5) f[001] r=μ0γ 2π(Hr−Meff+H4⊥). (6)Here, frdenotes the resonance frequency with its superscript representing the magnetic-field direction. γis the gyromag- netic ratio expressed as γ=gμB/¯h, where gis the Landé g-factor, μBis the Bohr magneton, and ¯ his the reduced Planck constant. Meffis the effective saturation magnetization defined asMeff=Ms−H2⊥.H2⊥,H4/bardbl, and H4⊥are the anisotropy fields defined as Hi=2Ki/μ0Ms. Table Isummarizes g-factors, Meff, and H4/bardbldeduced by fitting Eqs. ( 4)–(6) to the data. Here, Mefffor the Fe 3Ga [001] direction is obtained assuming H4⊥=H4/bardbl∼−15.5m T .N o t e thatMeff/similarequalMs(orH2⊥/lessmuchMs) because almost the same in- plane and out-of-plane lattice constants would not inducesignificant uniaxial magnetic anisotropy. In fact, the obtainedM effof∼1.43 T is in good agreement with the saturation magnetization of bulk D03-Fe 3Ga at 300 K, 1.42 T [ 12]. We find that the g-factors of Fe and Fe 3Ga are almost the same within error bars. On the other hand, the in-plane cubic TABLE I. The results of the fitting using Eqs. ( 4)–(7):g-factors (g), in-plane cubic anisotropy fields ( H4||), effective saturation mag- netizations ( Meff), intrinsic Gilbert damping constants ( α), and the inhomogeneous line broadening ( /Delta1H0). The numbers in parentheses refer to the fitting errors. Fe Fe 3Ga [100] [110] [100] [110] [001] g 2.07(4) 2.05(3) 2.08(3) 2.08(4) 2.05(1) μ0H4||(mT) 62(2) 60(1) −20(3) −11(4) μ0Meff(T) 2.15(9) 2.18(8) 1.46(6) 1.40(8) 1.43(2) α×10−32.3(2) 2.4(2) 6.9(4) 7.4(9) 6.0(2) μ0/Delta1H0(mT) 2.8(2) 2.9(2) 33(1) 22(3) 2.5(1) 165122-3SHOYA SAKAMOTO et al. PHYSICAL REVIEW B 103, 165122 (2021) magnetic anisotropy K4/bardblof Fe 3Ga is about seven times weaker than that of Fe, and the sign is opposite. The [110] easy axisseems intrinsic to the D0 3-ordered Fe 3Ga because the previ- ous FMR study reported that disordered Fe 81Ga19thin films have a sizable positive (or [100] easy axis) H4/bardblof 75–100 mT [27]. This difference in magnetic anisotropy indicates that the D03atomic ordering significantly alters the electronic band structure of Fe-Ga alloys and induces the giant ANE as aresult [ 12]. Figures 2(e) and2(f)show the deduced FMR peak widths plotted against the resonance frequency. Here, the slope of thedata represents the intrinsic Gilbert damping constant α, while theyintercept ( /Delta1H 0) represents extrinsic linewidth broad- ening caused by inhomogeneous (resonance) magnetic-fielddistribution. That is, μ 0/Delta1H=μ0/Delta1H0+4πα γfr. (7) The linear fitting with Eq. ( 7) yields the Fe Gilbert damping constant of α=0.0023, which is as low as the previously reported α∼0.0023 [ 28,29]. This agreement implies that the radiative damping [ 30] is not significant in the present study. The linear relationship does not hold for the Fe 3Ga data taken with in-plane magnetic fields: The data show a steepincrease (with a hump) at low frequencies. We attributethis nonlinear behavior to the two-magnon scattering (TMS)[31–33], where a uniform-precession magnon with k=0 scatters into another energetically degenerate magnon withk/negationslash=0. The TMS occurs when some inhomogeneity, which would be related to the surface roughness or the grain structurein the present case, breaks translational symmetry and makesthe crystal momentum ( k) nonconserved. The measurements with out-of-plane magnetic fields do not suffer from the TMS as there is no k/negationslash=0 magnon to scatter into [ 34]. The intrinsic Gilbert damping of the D0 3-Fe 3Ga thin film is thus estimated to be 0.0060(2). The intrinsic Gilbertdamping constants ( α) and extrinsic linewidth broadening (/Delta1H 0)o fF ea n d D03-Fe 3Ga are listed in Table I. The extrinsic linewidth broadening for the Fe 3Ga [001] direction is deduced to be 2.5 mT, which is as small as the Fe values. This smallextrinsic broadening suggests that there is no significant inho-mogeneity in the resonance magnetic field distribution. The Gilbert damping constant, or the relaxation fre- quency G=αγM s, can be roughly approximated as G∼ ξ2D(EF)[32], where ξandD(EF) represent the spin-orbit coupling strength and the DOS at the Fermi level, respectively.Figures 3(a) and3(b) show the calculated spin-resolved DOS of Fe and D0 3-Fe 3Ga, respectively. The DOS of D03-Fe 3Ga is markedly different from that of Fe and show sharp flat-band features. Figure 3(c) shows the total DOS of Fe and D0 3-Fe 3Ga per unit-cell volume of bcc Fe. The total DOS ofD03-Fe 3Ga at the Fermi level relative to that of Fe [DFe3Ga(EF)/DFe(EF)] falls within the range of 0.7–1.7 with a Fermi-level shift of ±130 meV . Assuming the same ξfor both Fe3Ga and Fe, the ratio GFe3Ga/GFealso becomes 0.7–1.7, which is in fair agreement with the experimental value of 1.7.This agreement suggests that the Ga inclusion does not sig-nificantly increase the spin-orbit coupling strength probablybecause Ga is nonmagnetic and does not have states near theFermi level. To be more precise, the ratio ξ Fe3Ga/ξFeshould be3 210 -1-2DOS[states/eV/f.u.] -4 -2 0 2 E-EF[eV]Fe (a)15 10 50 -5 -10DOS[states/eV/f.u.] -4 -2 0 E-EF[eV]D03 Fe3Ga GaFeTotal(b) 2 5 43210DOS[states/eV/VFe] -2 -1 0 1 E-EF[eV]TotalDOS Fe D03-Fe3Ga(c) FIG. 3. (a) and (b) Spin-resolved densities of states (DOS) of Fe andD03-Fe 3Ga per formula unit. The positive and negative signs represent the majority-spin and minority-spin states, respectively.(c) Total DOS of Fe and D0 3-Fe 3Ga per unit-cell volume of bcc Fe (VFe). in the range of 1–1.6. Note that the magnetostrictive responses of Fe 3Ga may partially contribute to the enhanced damping. Although the intrinsic Gilbert damping constant of D03-Fe 3Ga is larger than that of Fe, it is still low compared to those of typical ferromagnets: 0.0005–0.003 for FeCo alloys[28], 0.004–0.006 for CoFeB [ 35–37],∼0.007 for Permal- loy [ 38,39], and 0.0065 for disordered A2-Fe 81Ga19[27]. In other words, the nodal web with a large Berry curvature nearthe Fermi level does not hinder the magnetization dynamicsmuch. The low Gilbert damping may be useful for spintronicsdevice applications because one may be able to efficientlyswitch magnetization by exploiting the peculiar band structureofD0 3-Fe 3Ga. For an application, it is important to realize low absolute damping. Thus, while the TMS would be less of anissue when devices are made smaller than the magnon wave-length, the sample growth process might have to be furtheroptimized to suppress the TMS. IV . CONCLUSION We have investigated the magnetization dynamics of the epitaxial thin films of the nodal-line semimetal D03-Fe 3Ga by ferromagnetic resonance measurements. We have deducedthe intrinsic Gilbert damping constant of D0 3-Fe 3Ga to be (6.0±0.2)×10−3, which is larger than the Fe Gilbert damp- ing constant of (2 .3±0.2)×10−3but as low as other typical ferromagnets. The low Gilbert damping and the peculiar bandstructure with a large Berry curvature make D0 3-Fe 3Ga attrac- tive as a building block of spintronics devices. ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI (No. JP18H03880, No. JP19H00650, No. JP20K15158), JSTCREST (JPMJCR18T3), JST-Mirai Program (JPMJMI20A1),the Murata Science Foundation, and the Spintronics ResearchNetwork of Japan (Spin-RNJ). 165122-4LOW GILBERT DAMPING IN EPITAXIAL THIN FILMS … PHYSICAL REVIEW B 103, 165122 (2021) [1] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010) . [2] N. Nagaosa, J. Sinova, S. Onoda, A. H. 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PhysRevB.93.014432.pdf

PHYSICAL REVIEW B 93, 014432 (2016) Spin-orbit interaction enhancement in permalloy thin films by Pt doping A. Hrabec,1,*F. J. T. Gonc ¸alves,2,†C. S. Spencer,1E. Arenholz,3A. T. N’Diaye,3R. L. Stamps,2and Christopher H. Marrows1,‡ 1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 2SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom 3Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 22 October 2015; revised manuscript received 7 December 2015; published 21 January 2016) The spin-orbit interaction is an inherent part of magnetism, which links up the independent world of spins to the atomic lattice, thus controlling many functional properties of magnetic materials. In the widely used 3 d transition metal ferromagnetic films, the spin-orbit interaction is relatively weak, due to low atomic number.Here we show that it is possible to enhance and tune the spin-orbit interaction by adding 5 dplatinum dopants into permalloy (Ni 81Fe19) thin films by a cosputtering technique. This is achieved without significant changes of the magnetic properties, due to the vicinity of Pt to meeting the Stoner criterion for the ferromagnetic state.The spin-orbit interaction is investigated by means of transport measurements (the anisotropic magnetoresistanceand anomalous Hall effect), ferromagnetic resonance measurements to determine the Gilbert damping, as wellas by measuring the x-ray magnetic circular dichroism at the L 3andL2x-ray absorption edges to reveal the ratio of orbital to spin magnetic moments. It is shown that the effective spin-orbit interaction increases with Ptconcentration within the 0%–10% Pt concentration range in a way that is consistent with theoretical expectationsfor all four measurements. DOI: 10.1103/PhysRevB.93.014432 I. INTRODUCTION The spin-orbit interaction (SOI) is the underlying effect for many phenomena in magnetism, since it connects twoindependent worlds: that of the orbital angular momentumL, which is closely connected to the atomic lattice, and the electron spin magnetic angular momentum S, a quantity that otherwise exists on its own in the world of quantummechanics. The SOI is often expressed as ξS·L, where ξ is the SOI constant. Since the SOI is strongly influenced bythe nuclei, large effects occur when a heavy element with alarge nuclear charge, such as Pt or Au, is involved. Since the3dferromagnets—Fe, Co, Ni, and their alloys—are relatively light, the SOI is comparatively weak in these conventionalferromagnets. Therefore, a combination of a heavy elementwith a ferromagnet is one of the possible ways to drive astronger SOI within a conventional ferromagnet. For instance,the physics of thin magnetic films has recently attracted a lot ofattention due to new emerging phenomena when a heavy metalis brought in contact with a thin ferromagnet. In such a waymagnetic moments sitting at the surface of the ferromagnetexperience the broken spatial symmetry and consequently giverise to new interfacial interactions which can have crucialimpact on those surface states [ 1–4]. The other way to enhance the SOI in the ferromagnet is to dope it with a heavy element, ashas been demonstrated in doped magnetic semiconductors [ 5]. Since the SOI affects a vast number of magnetic properties,many of which are important for various nanomagnetic orspintronic technologies, it would be convenient to tailorits strength and observe its impact on properties such as *Present address: Laboratoire de Physique des Solides, CNRS, Universit ´es Paris-Sud et Paris-Saclay, 91405 Orsay Cedex, France. †Present address: Graduate School of Science, Hiroshima Univer- sity, Higashi-Hiroshima, Hiroshima 739-8526, Japan. ‡c.h.marrows@leeds.ac.ukanisotropic magnetoresistance (AMR) [ 6], anomalous Hall effect (AHE) [ 7], magnetization damping phenomena [ 8,9], or different contributions to the torque acting on a magneticdomain wall by a field[ 10] or a spin-polarized current [ 11]. Permalloy (Py =Ni 81Fe19) is a widely used material due to its unique properties combining a small magnetocrystallineanisotropy and a negligible magnetostriction constant. Besidesthese properties it has also high relative permeability, theproperty that initially attracted attention to it [ 12]. It has thus become an interesting material for many aspects ofnanomagnetism and spintronics research. The effect of 3 d,4d, and 5dtransition metal doping of Py on the Gilbert damping has been systematically studied showing a relation to Hund’srules [ 13,14]. Here we report an investigation of the influence of Pt doping on the SOI of Py in the range of Pt concentrationsup to 10%, where the magnetic properties of the ferromagnetare maintained. We have probed the SOI strength usingmagnetotransport (AMR and AHE), measurements of theGilbert damping αthrough ferromagnetic linewidth measure- ments, and x-ray magnetic circular dichroism (XMCD), whichhave all provided consistent results. This cross-correlation ofthe different effects of changing the SOI strength gives acomprehensive overview of the interplay between the SOIstrength and various observable properties. Thus we haveshown that the SOI can be controlled by the Pt concentrationin Py in order to fine-tune functional properties. II. EXPERIMENT A. Thin film growth and characterization The magnetic films were deposited by cosputtering using Py and Py90Pt10targets at base pressure of 10−8Torr and Ar pressure of 5 ×10−3Torr. The exact balance between the rates from the two targets determined the final overall Pt dopinglevel in the film which is deduced from the stoichiometryof the targets. 20 nm thick films of Py, Py 97.5Pt2.5,P y95Pt5, 2469-9950/2016/93(1)/014432(6) 014432-1 ©2016 American Physical SocietyA. HRABEC et al. PHYSICAL REVIEW B 93, 014432 (2016) FIG. 1. Exchange stiffness Aas a function of xin PyxPt1−x20 nm thick films. Blue and red data points correspond to results for Gdand V doped Py which were extracted from Ref. [ 15]a n dR e f .[ 16], respectively. The black dashed line corresponds to a linear fit to the data for Pt. The inset shows temperature dependent normalizedmagnetization for different films with corresponding fits of the T 3/2 Bloch law. Py92.5Pt7.5, and Py90Pt10compositions have been sputtered directly onto thermally oxidized silicon substrates. Because ofthe cosputtering method, the relative uncertainties in the filmstoichiometries are small. Deposition rates were about 1 ˚A/s and the film thicknesses were calibrated by low-angle x-rayreflectometry. The films are uncapped apart from the set ofsamples used for synchrotron measurements where a layer ofAl (2 nm) was used to protect the surface. No magnetic fieldwas applied during the deposition to minimize parasitic effectsof uniaxial anisotropy in our experiments. The magnetic properties were characterized with vibrating sample magnetometry (VSM) where the magnetization ofPyM Py=660±20 kA /m and of Py90Pt10MPyPt=650± 20 kA /m, unchanged to within the uncertainty. The magne- tization of all the other films was the same to within thisuncertainty. The inset of Fig. 1shows a VSM data set of normalized magnetization M sas a function of temperature Tfor various levels of Pt doping. All the curves are seen to be very similar. In order to obtain the exchange stiffnessconstant A, the experimental points are fitted using the Bloch lawM s=1−cT3/2from which the exchange stiffness can be determined by using the formula A=4.22×108kB/c2/3.T h e results of doing so are shown in Fig. 1. The change of Ais very small in comparison with previously reported suppression of A in Py doped with Gd [ 15]o rV[ 16]. Such a small variation of A can be attributed to the fact that Pt tends to be easily polarizedby the surrounding ferromagnetic atoms, since it is very closeto satisfying the Stoner criterion as a pure element, and there-fore it does not significantly affect the ferromagnetic state [ 17]. B. Magnetotransport To determine the impact of Pt doping on the SOI, two types of magnetotransport measurements, directly connectedto the SOI, were carried out: AHE and AMR. In order toFIG. 2. Anomalous Hall effect at room temperature. (a) Hall resistance RHat room temperature in films with various Pt content as a function of out-of-plane magnetic field. (b) Anomalous Hall resistivity as a function of Pt content. The red dashed curve representsa linear fit to the data. The error bars are smaller than the data points. The inset shows the dependence of the anomalous Hall conductivity on the longitudinal conductivity in the case of x=10%. measure the AHE, a set of films was deposited with a Hall bar structure, with the cross feature of size 50 μm×50μm, using a shadow mask deposition technique. The Hall resistivity ρAH was measured at room temperature by a standard four-probe method with the field normal to the sample plane. The roomtemperature Hall hysteresis loops are shown in Fig. 2(a). The transverse resistivity ρ xyis described by an empirical formula ρxy=RoB+RsMz, (1) where Rois the ordinary Hall resistivity coefficient and Rs is the anomalous Hall resistivity coefficient [ 7]. Unlike Rs, the mechanism of ordinary Hall effect is well understoodandR odepends only on the inverse density of carriers, therefore it is found to be small in metals. The anomalous Hallresistivity, ρ A xy=RsMzwas obtained by the usual method of extrapolating the high field Hall resistivity data back to zerofield. The dependence of ρ A xyonxis shown in Fig. 2(b), where a linear dependence on xis clearly evident, with ρA xyroughly quadrupling when 10% Pt is added to Py. It is now generally believed that the AHE comprises three contributions, each with different underlying physics: theintrinsic mechanism, which arises due to the SOI causing Berrycurvature of the electron bands; and two extrinsic contributions 014432-2SPIN-ORBIT INTERACTION ENHANCEMENT IN . . . PHYSICAL REVIEW B 93, 014432 (2016) arising from the skew and sidejump scattering mechanisms [ 7]. The knowledge of how the longitudinal σxxand anomalous HallσA xyconductivities scale with each other allows some distinction between these mechanisms. Here, σA xyincreases as the temperature is varied for x=10% in a manner that is rather linear in x, as can be seen from the data in the inset of Fig. 2(b), which is consistent with the skew scattering mechanism butnot the intrinsic or sidejump scattering. Considering the room temperature data across the series of samples with different x, we find that σ xxlies in the range (2 .3±0.2)×104(/Omega1cm)−1for them all, with little discernible systematic dependence on x. This is reasonable since in sputtered films of a transition metal solid solutionlike Py there is already strong disorder and some additionalPt impurity atoms will hardly affect the overall scatteringrate. It is therefore reasonable to consider that the strongerSOI induced by the presence of the Pt gives rise to skewscattering that has a larger “skew angle,” albeit at a similaroverall scattering rate, leading to a higher anomalous Hallconductivity, suggesting the proportionality σ A xy∝ξ, where ξrises linearly in xas the introduction of Pt increases the overall SOI strength from the low initial level found in undopedPy. While this proportionality is certainly not to be expectedin general [ 18], if the SOI strength is rather small, then the spin-conserving transitions will dominate and the scaling ofρ A xywith the SOI strength ξshould be dominated by a linear term [ 19]. Turning to the AMR, this effect is driven mainly by the probability of s-dscattering, leading to a dependence of the resistivity on the relative orientation of the magnetization M and electric current I. It is defined as ( R⊥−R/bardbl)/R⊥, where R⊥andR/bardblare the resistances in I/bardblMandI⊥Mconfigurations, respectively. AMR data for undoped Py is shown in the insetof Fig. 3. The value of 2.6% compares well to other data on thin Py films [ 20]. The AMR dependence on the Pt content xwas extracted from a series of such measurements across FIG. 3. AMR as a function of Pt content. The blue curve represents a fit to the AMR data as described in the text. The inset shows an example of AMR measurements in I/bardblM(red) and I⊥M (black) configurations for an undoped Py film.the set of samples, and is displayed in Fig. 3, showing that the AMR increases with increasing Pt content up to 5.8% forPy 90Pt10. This enhancement easily reaches the Py bulk value and compares well with the large AMR ratios of 6%–7% inCo 70Fe30[6]. It does not appear to be linear in x. Campbell et al. proposed a model for the AMR mechanism in strong ferromagnets that works well for Ni-based alloys[21]. The maximum AMR is of the order of 3 4(ξ/E ex)2(ζ−1), where Eexis the exchange splitting and the parameter ζis derived from the residual resistivity, which depends on thetype of impurity and is independent of impurity concentrationand temperatures well below T c. The AMR therefore scales quadratically with the SOI. Based on the arguments givenabove with regard to the AHE, we assume the relationship ξ(x)=ξ Py+kx, (2) where ξPyis the SOI of pure Py, and kis a scaling constant. The blue curve in Fig. 3represents a fit of the square of Eq. ( 2) to the data, motivated by the fact that the variation in Eexwith xis likely to be very small, given the small variation in A shown in Fig. 1. The good agreement between the fit and the data demonstrates that the increase of the AMR is consistentwith such a model. C. Ferromagnetic resonance The SOI is also closely linked to the magnetization dynamics, since it gives rise to dissipation and damping. Oncethe magnetization Mis excited from the equilibrium state, its motion is described by the Landau-Lifshitz-Gilbert equation: dM dt=−γM×Beff+α M/parenleftbigg M×dM dt/parenrightbigg , (3) where γ=|gμB//planckover2pi1|is the gyromagnetic ratio, /planckover2pi1is the reduced Planck constant, and μBis the Bohr magneton. Beside the first precessional term expressing an infinite magnetization rotationaround the effective field B eff, the equation also includes a second term representing energy dissipation, allowing themagnetization to relax into the direction of the effectivemagnetic field. This term is purely phenomenological andit controls the rate at which the magnetization reaches itsequilibrium. Its strength is given by the Gilbert dampingparameter α. Because this process does not conserve the spin there is an obvious connection to the SOI. Althoughthis phenomenon is at the heart the magnetization dynamics,its exact mechanism is not fully understood. A commonlycited theory is the Kambersk ´y Fermi-surface breathing model, in which αis calculated based on the SOI-induced spin-flip scattering rate as well as on the ordinary scattering [ 22] α∼μ 2 BD(EF)(/Delta1g)2 τγM s, (4) where τis the electron momentum scattering time, D(EF)i s the density of states at the Fermi level, and the change in g factor is expressed as /Delta1g=g−2. The Gilbert damping can be experimentally measured for example by the time resolved magneto-optical Kerr effecttechnique [ 23], or by domain wall velocity measurements [ 24], but most commonly by measuring linewidths in ferromagneticresonance (FMR) although significant discrepancies can arise 014432-3A. HRABEC et al. PHYSICAL REVIEW B 93, 014432 (2016) FIG. 4. Ferromagnetic resonance measurements. (a) Absorption /Delta1S 21for different frequencies given in GHz as a function of applied magnetic field, from which the FWHM /Delta1H can be obtained. (b) The inset depicts /Delta1H as a function of a fixed frequency for different Pt contents. The damping parameter αis calculated by using Eq. ( 5)a n d the results are displayed in the main plot of this panel. The red dashed curve represents a quadratic fit to the data. due to film inhomogeneities [ 25]. To determine α,w eh a v e employed a vector network analyzer FMR method, where theresonance line shape is measured through the relative variationof the forward transmission parameter S 21of a two-port microstrip circuit. This quantity is measured as a functionof the frequency and external applied field. Examples ofsuch absorption peaks for different frequencies are displayedin Fig. 4(a) for a permalloy film. One can clearly see that these peaks broaden with the increasing frequency fin the expected manner. The damping αwas obtained from fits of the expression /Delta1H=/Delta1H 0+4πα μ0γf, (5) where /Delta1H is the absorption full width at half maximum (FWHM) and /Delta1H 0is the inhomogeneous contribution to the linewidth [ 26]. The/Delta1H as a function of frequency for different Py1−xPtxfilms is shown in inset of Fig. 4(b) and the resulting αvalues are plotted as a function of Pt content xin Fig. 4(b). The damping increases from 0 .0095±0.0005 for pure Py up to 0.0141±0.0002 for Py90Pt10. The relationship between α andxdoes not appear to be linear. The dashed line representsa quadratic fit to the data, which will be discussed in more detail below. D. X-ray magnetic circular dichroism In the light of the relationship between the damping constant and the gfactor in the Kambersk ´y model [Eq. ( 4) ] ,i ti s desirable to gain information about the g-factor in our films. In the case of 3 dtransition metals, the orbital magnetic moment is almost completely quenched by the surrounding crystal fields,and so the gfactor is very close to the free-electron value of 2. The deviation of the gfactor from the free-electron value is proportional to ξ//Delta1, where /Delta1is the ground state–first excited state splitting of the corresponding 3 dion, and therefore reflects the strength of the SOI. For small orbital contributionsit can be expressed as [ 27] g=2/parenleftbiggμ L μS+1/parenrightbigg , (6) where μLandμSare the orbital and spin magnetic moment, respectively. This formula can be extended to alloys andcompounds [ 27], and the effective gfactor g effin the case of Py reads geff=(0.81MNi+0.19MFe)/slashbigg/parenleftbigg0.81MNi gNi(x)+0.19MFe gFe(x)/parenrightbigg , (7) where gNi(x) andgFe(x) are the Pt concentration dependent g factors of Ni and Fe obtained by Eq. ( 6) andMNi=0.69μB andMFe=2.28μBare the magnetizations per atom of Ni and Fe, respectively [ 28]. In this model we suppose that the magnetizations of Fe and Ni sublattices remain unchangedas Pt is introduced. This assumption is based on the totalinsensitivity of the magnetization of Py 1−xPtxto the value of xreported above. It is possible to straightforwardly measure the ratio μL/μS using the x-ray magnetic circular dichroism (XMCD) tech- nique. We did so here with measurements that were performedat beamline BL6.3.1.1 at the Advanced Light Source. Due tothe surface sensitivity of this technique, the magnetic layersfor this part of the study were capped by 2 nm of Al, whichforms a self-limiting oxide. The total electron yield intensitiesμ +andμ−around the L3andL2edges for Fe (690–760 eV) and Ni (820–920 eV) were measured in 30◦grazing incidence with the sample saturated in a positive or negative magneticfield. The x-ray absorption spectroscopy (XAS) and XMCD(μ +−μ−) spectra that were measured are plotted in Fig. 5(a). By using the XMCD sum rules [ 29], these spectra were used to calculate the ratio of orbital magnetic moment μLand the spin magnetic moment μSfor both Fe and Ni according to the formula μL μS=2q 9p−6q, (8) where pis the integral under the L3XMCD peak and qis the integral under the L2+L3peaks. The results are plotted as a function of xin Fig. 5(b).T h eμL/μSvalues measured in pure permalloy correspond to the data measured previously [ 30]. The behavior of the μL/μSratios for Fe and Ni as Pt is added are rather different though. The ratio for Fe rises smoothly and 014432-4SPIN-ORBIT INTERACTION ENHANCEMENT IN . . . PHYSICAL REVIEW B 93, 014432 (2016) FIG. 5. (a) XAS spectra for right μ+and left μ−circularly polarized x rays in pure Py film. The XMCD contrast is obtained by subtracting these two intensities. The XMCD data is mutually offset for clarity. (b) Ratio of spin and orbital magnetic momentsand as a function of Pt doping extracted from Fe (black) and Ni (red) absorption edges. (c) Calculated g effas a function of Pt concentration. The blue dashed line corresponds to linear fit to the data. /Delta1gis obtained from FMR data by using Eq. ( 4). The black dashed line corresponds to linear fit of the data.linearly, whilst that for Ni fluctuates around a constant value without any discernible pattern. The reason for this is not clearat present, and we hope that this intriguing result will stimulatefuture studies in this area, both experimental and theoretical. Inserting these values into Eq. ( 6), the gfactors for the Ni and Fe magnetic sublattices were obtained. These werecombined according to Eq. ( 7) to give the effective gfactor g eff, and the results are plotted in Fig. 5(c) as a function of Pt doping x. One can see that gefffactor increases with the Pt concentration, signaling the SOI enhancement. This lineartrend of μ L/μSwithx—partly masked by the fluctuations arising from the Ni results—justifies, through the Kambersk ´y model [Eq. ( 4)], the quadratic-in- xfit to the αdata in Fig. 4(b). The fact that geffwould show a much smoother trend with x if we neglected the Ni and only treated the clear variation inμ L/μSfor the Fe suggests that a more sophisticated theoretical treatment, perhaps based on first principles calculations, couldyield a better formula to use than that in Eq. ( 7) to treat cases such as this. The gfactor in Fig. 5(c)obtained by the XMCD technique shows a good agreement with /Delta1gobtained from the FMR data calculated by Eq. ( 4). III. CONCLUSIONS In our experimental work we show that heavy element doping of permalloy with Pt can lead to a significant enhance-ment of the SOI. This was proved by a series of transportmeasurements, element specific XMCD observations, andby the magnetization dynamics behavior, which showed anenhancement of the Gilbert damping. The spin-orbit interac-tionξincreases linearly with Pt concentration xwithin the 0%–10% Pt concentration range in way that is consistent withtheoretical expectations across all four measurements. Thisgives a comprehensive overview of the way in which changesin the SOI strength affect the interplay of different observablemagnetic properties, many of which are of technologicalimportance. Data associated with this work are available from the Research Data Leeds repository [ 31] under a CC-BY license. ACKNOWLEDGMENTS This work was supported by the EPSRC (Grants No. EP/I011668/1, No. EP/M024423/1, No. EP/I013520/1, andNo. EP/J000337/1). The Advanced Light Source is supportedby the Director, Office of Science, Office of Basic EnergySciences, of the U.S. Department of Energy under ContractNo. DE-AC02-05CH11231. We would like to thank G. Tatarafor the discussions that led to the suggestion for theseexperiments, and Y . 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PhysRevLett.107.066603.pdf

Ab Initio Calculation of the Gilbert Damping Parameter via the Linear Response Formalism H. Ebert, S. Mankovsky, and D. Ko ¨dderitzsch University of Munich, Department of Chemistry, Butenandtstrasse 5-13, D-81377 Munich, Germany P. J. Kelly Faculty of Science and Technology and MESAþInstitute for Nanotechnology, University of Twente, P .O. Box 217, 7500 AE Enschede, The Netherlands (Received 1 March 2011; published 2 August 2011) A Kubo-Greenwood-like equation for the Gilbert damping parameter /C11is presented that is based on the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn-Rostoker bandstructure method in combination with coherent potential approximation alloy theory allows it to beapplied to a wide range of situations. This is demonstrated with results obtained for the bcc alloy systemFe 1/C0xCoxas well as for a series of alloys of Permalloy with 5dtransition metals. To account for the thermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. The corresponding calculations for Ni correctly describe the rapid change of /C11when small amounts of substitutional Cu are introduced. DOI: 10.1103/PhysRevLett.107.066603 PACS numbers: 72.25.Rb, 71.20.Be, 71.70.Ej, 75.78. /C0n The magnetization dynamics that is relevant for the performance of any type of magnetic device is in generalgoverned by damping. In most cases the magnetizationdynamics can be modeled successfully by means of theLandau-Lifshitz-Gilbert (LLG) equation [ 1] that accounts for damping in a phenomenological way. The possibility tocalculate the corresponding damping parameter from firstprinciples would open the perspective of optimizing mate- rials for devices and has therefore motivated extensive theoretical work in the past. This led among others toKambersky’s breathing Fermi surface (BFS) [ 2] and torque-correlation models (TCM) [ 3], that in principle provide a firm basis for numerical investigations basedon electronic structure calculations [ 4,5]. The spin-orbit coupling that is seen as a key factor in transferring energyfrom the magnetization to the electronic degrees of free- dom is explicitly included in these models. Most ab initio results have been obtained for the BFS model though thetorque-correlation model makes fewer approximations[4,6]. In particular, it in principle describes the physical processes responsible for Gilbert damping over a widerange of temperatures as well as chemical (alloy) disorder.However, in practice, like many other models it depends ona relaxation time parameter /C28that describes the rate of transfer due to the various types of possible scattering mechanisms. This weak point could be removed recentlyby Brataas et al. [7] who described the Gilbert damping by means of scattering theory. This development suppliedthe formal basis for the first parameter-free investigationson disordered alloys for which the dominant scatteringmechanism is potential scattering caused by chemical dis-order [ 8] or temperature induced structure disorder [ 9]. As pointed out by Brataas et al. [7], their approach is completely equivalent to a formulation in terms of thelinear response or Kubo formalism. The latter route is taken in this communication that presents a Kubo-Greenwood-like expression for the Gilbert damping pa-rameter. Application of the scheme to disordered alloysdemonstrates that this approach is indeed fully equivalentto the scattering theory formulation of Brataas et al. [7]. In addition a scheme is introduced to deal with the tempera-ture dependence of the Gilbert damping parameter. Following Brataas et al. [7], the starting point of our scheme is the Landau-Lifshitz-Gilbert (LLG) equation for the time derivative of the magnetization ~M: 1 /C13d~M d/C28¼/C0 ~M/C2~Heffþ~M/C2/C20~Gð~MÞ /C132M2sd~M d/C28/C21 ; (1) where Msis the saturation magnetization, /C13the gyromag- netic ratio, and ~Gthe Gilbert damping tensor. Accordingly, the time derivative of the magnetic energy is given by _Emag¼~Heff/C1d~M d/C28¼1 /C132_~m½~Gð~mÞ_~m/C138 (2) in terms of the normalized magnetization ~m¼~M=M s.O n the other hand, the energy dissipation of the electronic system _Edis¼hd^H d/C28iis determined by the underlying Hamiltonian ^Hð/C28Þ. Expanding the normalized magnetiza- tion ~mð/C28Þ, that determines the time dependence of ^Hð/C28Þ about its equilibrium value, ~mð/C28Þ¼~m0þ~uð/C28Þ, one has ^H¼^H0ð~m0ÞþX /C22~u/C22@ @~u/C22^Hð~m0Þ: (3) Using the linear response formalism, _Ediscan be written as [7]PRL 107, 066603 (2011) PHYSICAL REVIEW LETTERSweek ending 5 AUGUST 2011 0031-9007 =11=107(6) =066603(4) 066603-1 /C2112011 American Physical Society_Edis¼/C0/C25@X ii0X /C22/C23_u/C22_u/C23/C28 ci/C12/C12/C12/C12/C12/C12/C12/C12@^H @u/C22/C12/C12/C12/C12/C12/C12/C12/C12ci0/C29/C28 ci0/C12/C12/C12/C12/C12/C12/C12/C12@^H @u/C23/C12/C12/C12/C12/C12/C12/C12/C12ci/C29 /C2/C14ðEF/C0EiÞ/C14ðEF/C0Ei0Þ; (4) where EFis the Fermi energy and the sums run over all eigenstates /C11of the system. Identifying _Emag¼_Edis, one gets an explicit expression for the Gilbert damping tensor ~Gor equivalently for the damping parameter /C11¼ ~G=ð/C13MsÞ[7]. In full analogy to electric transport [ 10], the sum over eigenstates jciimay be expressed in terms of the retarded single-particle Green’s functionImG þðEFÞ¼/C0 /C25P ijciihcij/C14ðEF/C0EiÞ. This leads for the parameter /C11to a Kubo-Greenwood-like equation /C11/C22/C23¼/C0@/C13 /C25M sTrace/C28@^H @u/C22ImGþðEFÞ@^H @u/C23ImGþðEFÞ/C29 c (5) with h...icindicating a configurational average in case of a disordered system (see below). Identifying T/C22¼@^H=@u /C22 with the component of the magnetic torque operator^~I along the direction ~n, such that ^I~n¼@^H=@~uð~n/C2~uÞ¼ @^H=@u /C22ð~n/C2~uÞ/C22this expression obviously gives the pa- rameter /C11in terms of a torque-torque correlation function. However, in contrast to the conventional TCM the elec-tronic structure is not represented in terms of Bloch states but using the retarded electronic Green’s function giving the present approach much more flexibility. The most reliable way to account for spin-orbit coupling as the source of Gilbert damping is to evaluate Eq. ( 5) using a fully relativistic Hamiltonian within the frameworkof local spin density formalism (LSDA) [ 11]: ^H¼c~/C11~pþ/C12mc 2þVð~rÞþ/C12~/C27~mBð~rÞ: (6) Here/C11iand/C12are the standard Dirac matrices and ~pis the relativistic momentum operator [ 12]. The functions Vand Bare the spin-averaged and spin-dependent parts, respec- tively, of the LSDA potential. Equation ( 6) implies for the T/C22operator occurring in Eq. ( 5) the expression T/C22¼@ @u/C22^H¼/C12B/C27 /C22: (7) The Green’s function Gþin Eq. ( 5) can be obtained in a very efficient way by using the spin-polarized relativisticversion of multiple scattering theory [ 11] that allows us to treat magnetic solids: G þð~rn;~r0m;EÞ¼X /C3/C30Zn /C3ð~rn;EÞ/C28nm /C3/C30ðEÞZm/C2 /C30ð~r0m;EÞ /C0X /C3Zn /C3ð~r<;EÞJn/C2 /C30ð~r>;EÞ/C14nm: (8) Here coordinates ~rnreferred to the center of cell n have been used with j~r<j¼minðj~rnj;j~r0njÞand j~r>j¼ maxðj~rnj;j~r0njÞ. The four-component wave functionsZn /C3ð~r; EÞðJn /C3ð~r; EÞÞare regular (irregular) solutions to the single-site Dirac equation for site nand/C28nm /C3/C30ðEÞis the so- called scattering path operator that transfers an electronic wave coming in at site minto a wave going out from site n with all possible intermediate scattering events accountedfor coherently. Using matrix notation, this leads to the following ex- pression for the damping parameter: /C11 /C22/C22¼g /C25/C22 totX nTrace hT0/C22~/C280nTn/C22~/C28n0ic (9) with the gfactor 2ð1þ/C22orb=/C22spinÞin terms of the spin and orbital moments, /C22spinand/C22orb, respectively, the total magnetic moment /C22tot¼/C22spinþ/C22orb, and ~/C280n /C3/C30¼ 1 2ið/C280n /C3/C30/C0/C280n /C30/C3Þand with the energy argument EFomitted. The matrix elements of the torque operator, Tn/C22, are identical to those occurring in the context of exchangecoupling [ 13] and can be expressed in terms of the spin- dependent part Bof the electronic potential with matrix elements: Tn/C22 /C30/C3¼Z d3rZn/C2 /C30ð~rÞ½/C12/C27/C22Bxcð~rÞ/C138Zn /C3ð~rÞ: (10) As indicated above, the expressions in Eqs. ( 5)–(10) can be applied straightforwardly to disordered alloys. In thiscase the brackets h...i cindicate the necessary configura- tional average. This can be done by describing in a first step the underlying electronic structure (for T¼0K) on the basis of the coherent potential approximation (CPA) alloytheory. In the next step the configurational average inEq. ( 5) is taken following the scheme worked out by Butler [ 10] when dealing with the electrical conductivity atT¼0K or residual resistivity, respectively, of disor- dered alloys. This implies, in particular, that so-calledvertex corrections of the type hT /C22ImGþT/C23ImGþic/C0 hT/C22ImGþichT/C23ImGþicthat account for scattering-in processes in the language of the Boltzmann transport formalism are properly accounted for. Thermal vibrations as a source of electron scattering can in principle be accounted for by a generalization ofEqs. ( 5)–(10) to finite temperatures and by including the electron-phonon self-energy /C6 el-phwhen calculating the Green’s function Gþ. Here we restrict ourselves to elastic scattering processes by using a quasistatic representationof the thermal displacements of the atoms from theirequilibrium positions. We introduce an alloy-analogymodel to average over a discrete set of displacementsthat is chosen to reproduce the thermal root mean square average displacementffiffiffiffiffiffiffiffiffiffiffi hu 2iTp for a given temperature T. This was chosen according to hu2iT¼1 43h2 /C252mk/C2D½/C8ð/C2D=TÞ /C2D=Tþ1 4/C138 with /C8ð/C2D=TÞthe Debye function, hthe Planck constant, kthe Boltzmann constant, and /C2Dthe Debye temperature [14]. Ignoring the zero temperature term 1=4and assuming a frozen potential for the atoms, the situation can be dealtPRL 107, 066603 (2011) PHYSICAL REVIEW LETTERSweek ending 5 AUGUST 2011 066603-2with in full analogy to the treatment of disordered alloys described above. The approach described above has been applied to the ferromagnetic 3d-transition metal alloy systems bcc Fe1/C0xCox, fcc Fe1/C0xNix, and fcc Co1/C0xNix. Figure 1shows as an example results for bcc Fe1/C0xCoxforx/C200:7. The calculated damping parameter /C11ðxÞforT¼0Kis found to be in very good agreement with the results based on thescattering theory approach [ 8] demonstrating numerically the equivalence of the two approaches. An indispensablerequirement to achieving this agreement is to include thevertex corrections mentioned above. In fact, ignoring them leads in some cases to completely unphysical results. To check the reliability of the standard CPA, that implies asingle-site approximation when performing the configura-tional average, we performed calculations on the basis ofthe nonlocal CPA [ 15]. Using a four-atom cluster led to practically the same results as the CPA except for the very dilute case. As found before for fcc Fe 1/C0xNix[8] the theoretical results for /C11reproduce the concentration de- pendence of the experimental data quite well but are foundto be too low (see below). As suggested by Eq. ( 9) the variation of /C11ðxÞwith concentration xmay reflect to some extent the variation of the average magnetic moment of the alloy, /C22 tot. Because the moments and spin-orbit coupling strength do not differ very much for Fe and Co, thevariation of /C11ðxÞshould be determined in the concentrated regime primarily by the electronic structure at the Fermi energy E F. As Fig. 1shows, there is indeed a close corre- lation with the density of states nðEFÞthat may be seen as a measure for the number of available relaxation channels. While the scattering and linear response approach are completely equivalent when dealing with bulk alloys thelatter allows us to perform the necessary configurationaveraging in a much more efficient way. This allows usto study with moderate effort the influence of varying the alloy composition on the damping parameter /C11.Corresponding work has been done, in particular, using Permalloy as a starting material and adding transition metals (TM) [ 16] or rare earth metals [ 17]. If we use the present scheme to study the effect of substituting Fe and Ni atoms in Permalloy with a 5dTM, we find an increase of /C11 nearly linear with the 5dTM content, just as in experiment [16]. The total damping for 10% 5dTM content shown in Fig. 2(top) varies roughly parabolically over the 5dTM series. In contrast to the Fe 1/C0xCoxalloys considered above, there is now an S-like variation of the moments /C225d spinover the series (Fig. 2, bottom), characteristic of 5dimpurities in the pure hosts Fe and Ni [ 18,19]. In spite of this behavior of/C225d spinthe variation of /C11ðxÞseems again to be correlated with the density of states n5dðEFÞ(Fig. 2bottom). Again the trend of the experimental data is well reproduced by the calculated values that are, however, somewhat too low. One possible reason for the discrepancy between the theoretical and experimental results shown in Figs. 1and 2might be the neglect of the influence of finite tempera- tures. This can be included as indicated above to account for the thermal displacement of the atoms in a quasistateway by performing a configurational average over the displacements using the CPA. This leads even for pure systems to a scattering mechanism and this way to a finite value for /C11. Corresponding results for pure Ni are given in Fig.3that show in full accordance with experiment a rapid decrease of /C11with increasing temperature until a regime with a weak variation of /C11withTis reached. This behavior is commonly interpreted as a transition from conductivity- like to resistivitylike behavior reflecting the dominance of intra- and interband transition, respectively [ 4], that is related to the increase of the broadening of electron energy bands caused by the increase of scattering events with temperature. Adding even less than 1 at. % Cu to Ni strongly reduces the conductivitylike behavior at low tem- peratures while leaving the high-temperature behavior es-sentially unchanged. A further increase of the Cu content leads to the impurity-scattering processes responsible for 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 concentration xCo0123456α(x) x10-3Expt Theory (CPA), bcc Theory (NL CPA)Fe-Co n(EF) n(EF) (sts./Ry) 102030405060 0 FIG. 1 (color online). Gilbert damping parameter for bcc Fe1/C0xCoxas a function of Co concentration: full circles—the present results within CPA; empty circles—within nonlocal CPA (NL CPA); and full diamonds—experimental data byOogane [ 20].02468α x10-2 Ta W Re Os Ir Pt Au-0.300.30.6mspin5d (µB) n5d(EF) 5d spin moment 061218 n5d(EF) (sts./Ry) FIG. 2 (color online). Top: Gilbert damping parameter /C11for Py/5dTM systems with 10% 5dTM content in comparison with experiment [ 16]; bottom: spin magnetic moment m5d spinand density of states nðEFÞat the Fermi energy of the 5dcomponent in Py/ 5dTM systems with 10% 5dTM content.PRL 107, 066603 (2011) PHYSICAL REVIEW LETTERSweek ending 5 AUGUST 2011 066603-3band broadening dominating /C11. This effect completely suppresses the conductivitylike behavior in the low-temperature regime because of the increase of scatteringevents due to chemical disorder. Again this is fully in linewith the experimental data, providing a straightforward explanation for their peculiar variation with temperature and composition. From the results obtained for Ni one may conclude that thermal lattice displacements are only partly responsiblefor the finding that the damping parameters obtained for Pydoped with the 5dTM series, and Fe 1/C0xCoxare somewhat low compared with experiment. This indicates that addi-tional relaxation mechanisms like magnon scattering con- tribute. Again, these can be included at least in a quasistatic way by adopting the point of view of a disordered localmoment picture. This implies scattering due to randomtemperature-dependent fluctuations of the spin momentsthat can also be dealt with using the CPA. In summary, a formulation for the Gilbert damping parameter /C11in terms of a torque-torque-correlation func- tion was derived that led to a Kubo-Greenwood-like equation. The scheme was implemented using the fully relativistic Korringa-Kohn- Rostoker band structuremethod in combination with the CPA alloy theory. Thisallows us to account for various types of scattering mecha-nisms in a parameter-free way. Corresponding applicationsto disordered transition metal alloys led to very goodagreement with results based on the scattering theoryapproach of Brataas et al. demonstrating the equivalence of both approaches. The flexibility and numerical efficiency of the present scheme was demonstrated by astudy on a series of Permalloy- 5dTM systems. To inves- tigate the influence of finite temperatures on /C11, a so-called alloy-analogy model was introduced that deals with the thermal displacement of atoms in a quasistatic manner.Applications to pure Ni gave results in very good agree-ment with experiment and, in particular, reproduced thedramatic change of /C11when Cu is added to Ni. The authors would like to thank the DFG for financial support within the SFB 689 ‘‘Spinpha ¨nomene in redu- zierten Dimensionen’’ and within project Eb154/23 forfinancial support. P. J. K acknowledges support by EUFP7 ICT Grant No. 251759 MACALO. [1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004) . [2] V. Kambersky, Can. J. Phys. 48, 2906 (1970) . [3] V. Kambersky, Czech. J. Phys. 26, 1366 (1976) . [4] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007) . [5] M. Fa ¨hnle and D. Steiauf, Phys. Rev. B 73, 184427 (2006) . [6] V. Kambersky, Phys. Rev. B 76, 134416 (2007) . [7] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 101, 037207 (2008) . [8] A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010) . [9] Y. Liu, A. A. Starikov, Z. Yuan, and P. J. Kelly, Phys. Rev. B84, 014412 (2011) . [10] W. H. Butler, Phys. Rev. B 31, 3260 (1985) . [11] H. Ebert, in Electronic Structure and Physical Properties of Solids , edited by H. Dreysse ´, Lecture Notes in Physics Vol. 535 (Springer, Berlin, 2000), p. 191. [12] M. E. Rose, Relativistic Electron Theory (Wiley, New York, 1961). [13] H. Ebert and S. Mankovsky, Phys. Rev. B 79, 045209 (2009) . [14] E. M. Gololobov, E. L. Mager, Z. V. Mezhevich, and L. K. Pan, Phys. Status Solidi (b) 119, K139 (1983) . [15] D. Ko ¨dderitzsch, H. Ebert, D. A. Rowlands, and A. Ernst, New J. Phys. 9, 81 (2007) . [16] J. O. Rantschler, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F. Egelhoff, B. B. Maranville, D. Pulugurtha, A. P. Chen, and L. M. Connors, J. Appl. Phys. 101, 033911 (2007) . [17] G. Woltersdorf, M. Kiessling, G. Meyer, J.-U. Thiele, and C. H. Back, Phys. Rev. Lett. 102, 257602 (2009) . [18] B. Drittler, N. Stefanou, S. Blu ¨gel, R. Zeller, and P. H. Dederichs, Phys. Rev. B 40, 8203 (1989) . [19] N. Stefanou, A. Oswald, R. Zeller, and P. H. Dederichs, Phys. Rev. B 35, 6911 (1987) . [20] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys. 45, 3889 (2006) . [21] S. M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974) .00.050.10.15α(T)expt: pure Ni theory: pure Ni 00.050.10.15α(T)expt: Ni + 0.17 wt.%Cu theory: Ni + 0.2 at.%Cu 0 100 200 300 400 500 Temperature (K)00.050.10.15α(T)expt: Ni + 5 wt.%Cu theory: Ni + 5 at.%Cu FIG. 3 (color online). Temperature variation of Gilbert damp- ing of pure Ni and Ni with Cu impurities: present theoretical results vs experiment [ 21].PRL 107, 066603 (2011) PHYSICAL REVIEW LETTERSweek ending 5 AUGUST 2011 066603-4

PhysRevLett.53.2497.pdf

VOLUME 53,NUMBER 26 PHYSICAL REVIEW LETTERS 24DECEMBER 1984 Landau-Lifshitz Equation ofFerromagnetism: ExactTreatment oftheGilbertDamping M.Lakshmanan Department ofPhysics, Bharathidasan University, Tiruchirapalli 620-023,India,andDepartment of Physics, KyotoUniversity, Kyoto606,Japan and K.Nakamura Fukuoka InstituteofTechnology, Higashi ku,Fu-kuoka81102-Ja,pan (Received 24September 1984) IntheLandau-Lifshitz equation whichdescribes theevolution ofspinfieldsinnonequi- libriumcontinuum ferromagnets, bystereographic projection oftheunitsphereofspinonto acomplex plane,itisshownthattheeffectoftheLandau-Lifshitz-Gilbert damping termisa mererescaling oftimebyacomplex constant. Consequently, foranygivenundamped motionofspatially regularand/orirregular spinstructures, thenatureofthedamping canbe analyzed exactly inasimplified manner. PACSnumbers: 75.30.Ds,03.40.Kf,75.10.Hk TheLandau-Lifshitz (LL)equation' which describes theevolution ofspinfieldsincontinuum ferromagnets bearsafundamental roleinthe understanding ofnonequilibrium magnetism, just astheNavier-Stokes equation doesinthatoffluid dynamics. Inthecontextofnonlinear dynamics, it isbeingrealized thattheLLequation possesses fas- cinating geometrical properties andthatitsspecial versions without damping in(1+1)dimensions are completely integrable solitonsystems.47Herewe wishtoshowtheastonishing factthattheeffectof Landau-Lifshitz-Gilbert orsimply theGilbert,damping' isjustarescaling ofthetimevariable t byacomplex constant, sothatforeverygivensolu- tionoftheundamped LLequation inanydimen- siontheexactsolutionofthefullydamped version canbegivenstraightforwardly. Thewayinwhich wedemonstrate thisresultisbyprojecting theunit sphereofspinS(r,t)stereographically ontoacom- plexplaneofto(r,t)andthenrewriting theLL equation intermsofthelattervariable. Itiswellknownthatthenormalized formofthe Landau-Lifshitz equation fortheferromagnetic spin systemisoftheform' t1S(r,t)/Qt=SxF«—XSx(SxF,«)=SxF,«+ whereS=—(S",S»,S')andS=1,and A,isadimen- sionless Gilbert damping parameter. InEq.(1)the effective fieldFefftypically contains contributions fromexchange interaction, crystalline anisotropy, magnetostatic self-energy, external magnetic fields, thermal fluctuations, etc.Equation (1)canalsobe written inthealternative Gilbertformas BS(r,t)/Bt=SxF,«X(SxBS/Bt).—(2) OnecaneasilycheckthatEqs.(1)and(2)are identical towithinaconstant scalingfactor1+A.of thetimevariable t.Wewilluseboththeformsin Eqs.(1)and(2)inthefollowing analysis. Tobeginwithweconsider atypicalformofF,ff, corresponding toauniaxial anisotropic Heisenberg ferromagnet inexternal magnetic fields: F,«=VS—2A(Sn)n+p,B, n=(0,0,1),(3) where Aistheanisotropy parameter (A)0,easy&~F.«—(SFc«)S~. plane; A(0,easyaxis),p,isthegyromagnetic ra- tioinBohrmagnetons, andB=B(t)istheexternal magnetic field.Itmaybenotedthatintheun- damped caseEq.(1)or(2)forEq.(3)maybe derived startingfromafieldHamiltonian H=dr—,O'S+23Sn—2pBS, withsuitable Poisson brackets forthespin fields.248 Intheundamped case(A.=O)recentinvestiga- tionshaveestablished thatthe(1+1)-dimensional versionoftheLLequation forthepureisotropic case[A=0andB=OinEq.(3)]iscompletely integrable andisequivalent toanonlinear Schrodinger equation, andthattheelementary ex- citations areenvelope solitons andmagnons. Also forA&056andB=(O,O,BL)itisanintegrable solitonic system. Wewishtoconsider nowtheef- fectofthedamping termsproportional toPinEq.1984TheAmerican Physical Society 2497 VOLUME 53,NUMBER 26 PHYSICAL REVIEW LETTERS 24DECEMBER 1984 (1)or(2)ontheundamped spinmotion. Tradi- tionaltreatments ofEq.(1)inpolarcoordinates tendtomixuptheevolutions ofthetwoanglesina complicated wayandsothedamping istreatedonly approximately. Eventheother geometrical parametrizations whichprovedtobesuccessful in theundamped casetendtocomplicate thetreat-,mentofdamping. Inthefollowing, however, we showthattheparametrization ofthespinfieldin termsofastereographic variable simplifies the structure ofEq.(1)orEq.(2)drastically. Wetherefore project theunitsphereofspin S(r,t)=1stereographically ontoacomplex vari- ableco(r,t)9: ()S"+iS» (1+S')' 2Redo(r,t) (1+QJQj)S»()21m&v(r,t)S,()(1—o)o)) (1+QJQI) (1+rdcu)(4a) (4b) Thenthederivatives areeasilyseentobe S,"=BS"/Bt=I'[o),(1—co')+(o,"(I—o)')], S»=—ir[o),(1+(u')—cu,'(I+tu')], Sg=2I(QJgQJ+Q)QJg), and '7'S"=I'[(1—~')'7'~+(1—m')'7'o)"] —2r't'[2(~+ ~')7~'7~'+~'(I—~')(7~)'+~(1—~')(7~')'], v'S»=ir[(1+—N')v'N(1+')—v''] +2ir'/'[2(~ —~')7~'7~'+~'(I+~')(7~)'—~(1+~')(7~')'], S=—2Ii[cu(1+6)Q))7Qi+QJ(l+ (Uco)7QJ +2(l—o)o)')V(u'7m—2~"'(7~)' —2m'('7m")'],(Sa) (5b) (Sc) (6a) (6b) (6c) whereI'=(1+co~') 2.Wethenreexpress theindividual components ofEq.(1)forthespecific formof F,rrinEq.(3).Thexcomponent ofEq.(1)withB=0(forBA0,seebelow)reads riS"/rit=(SM'S' S"7'S»)—2AS»S'+ X[—7'S"—(S72S—2A(Sn)']S"]. (7) Byuseofthestereographic transformation inEqs.(4)-(6) andthefactthat S'7S=—47''7CU/(I+QÃd) Eq.(7)canberewritten as (1—(o'2)G(tu,o)')—(1—(o2)G'(o),(u')=0, where G(cu,o)')=i(l+o)co')(u,+(1—iA)[(I+o)o)')'72m)—2o)'(7co) 2+2Ao)(I—o)o)')]. Similarly theothertwocomponents ofEq.(1)become —i(1+a)') G((u,co')—i(1+o)')G'(tu,(o')=0 and 2QJG(QJ, QJ)20)G(QJ,Cd)=0.(9) (10) (12) Consistency ofEqs.(9)-(12) thenobviously impliesG(co,co')=0andG'(co,co')=0sothattheevolution equation forthestereographic variable co(r,t)inthepresence ofdamping becomes i(1+o)co")co,+(1—iA.)[(1+coo)')7o)—2o)'(7o) )+2Acu(1—o)co')]=0, (13) 2498 VOLUME 53,NUMBER 26 PHYSICAL REVIEW LETTERS 24DECEMBER 1984 anditscomplex conjugate. Ontheotherhand,redefining thetimevariable weobtain i(1+tutu')cu, +[(I+tutu')'vr cu—2''Vcu 'vrcu+2Atu(1 —tutu")]=0,(14) (15) whichisexactlythesameastheundamped evolution equation forcoendowed herewiththescaledtime~. Thusforeverysolution intheA.=0case,wehavethecorresponding solution inthedamped (A.A0)case justwiththerescaling inEq.(14)ofthetimeparameter. Thecorresponding damped spinfieldS(r,t)can thenbeconstructed simplyfromEq.(4). Itcanbeeasilyobserved thattheeffectofmagnetic fieldsp,BonF,«inEq.(3)doesnotaltertheabove facteither. Weverifythattheeffectofalongitudinal fieldB(t)=(0,0,B(t))istoaddaterm—p,BL(1+tutu')cu tothetermsproportional to(1—iA.)inEq.(13)andtheeffectofatransverse field B(t)=(B(t),0,0)istoaddafactor —,'pB(1—ik)(1+tucu')(1 —tu)tothelefthandsideofEq.(13). Moregenerally, weclaimthatforagivenarbitraryF,ff,ouraboveassertion istrue.Thereasonforthiscan beelucidated bymanipulation oftheGilbertforminEq.(2):Thetimederivative termsinEq.(2)canbe summarized instereographic coordinates as "riS/r)t+)t(SxtiS/t)t)=[(1+iX)/(I+ tutu")2][(1—tu'z)et—i(1+tu'2)e2—2cue3]cu,+cc., (16) wheretheunitorthonormal vectorse;,i=1,2,3, defineS=S"e&+S~e2+S'e3andc.c.standsfor complex conjugate inEq.(16).Sincetheremaining terminEq.(2),i.e.,SxFeffincludes no dependent term,thespinevolution equation inEq. (2)rewritten instereographic coordinates isob- tainedfromitsundamped version bymultiplying thetimevariable bythefactor(1+i'.)'inthe latter.Combining thisfactwiththealreadyexisting scaledifference1+)tzbetween Eqs.(1)and(2),we onceagainbutinamoregeneral wayarriveatthe earlier conclusion thattheeffectofLandau- Lifshitz-Gilbert damping isjustarescaling oftime bythefactor 1—iXoftheundamped spinmotion. Thissimplification makesfeasible asystematic studyonpattern-forming transitions andkineticsof topological singularities innonequilibrium mag- nets,'wheretheGilbert damping aswellasdriv- ingmagnetic fieldsplayanessential role. Oneoftheauthors(M.L.)wishestoacknowledge thehospitality ofProfessor H.Hasegawa duringhis stayatKyoto.Thisworkwassupported inpartby theJapanSocietyforPromotion ofScience. L.D.Landau andE.M.Lifshitz, Phys.Z. Sowjetunion 8,153(1935), reproduced inCollected Pa persofL.D.Landau, editedbyD.terHaar(Pergamon, NewYork,1965),p.101. 2A.I.Akhiezer, V.G.Baryakhtar, andS.V.Peletmin- skii,SpinWaves(North-Holland, Amsterdam, 1968);F.H.DeLeeuw,R.vandenDoel,andU.Enz,Rep. Prog.Phys.43,689(1980). 3A.P.Malozemoff andJ.C.Slonczewski, Magnetic Domain WallsinBubbleMaterials (Academic, NewYork, 1979);J.C.Slonczewski, inPhysicsofDefects, editedby R.Balian,M.Kleman, andJ.P.Poirier,LesHouches Summer SchoolProceedings Vol.35(North-Holland, Amsterdam, 1981). 4K.Nakamura andT.Sasada, Phys.Lett.48A,321 (1974); M.Lakshmanan, Th.W.Ruijgrok, andC.J. Thompson, Physica (Utrecht) 84A,577(1976); M.Lakshmanan, Phys.Lett.61A,53(1977);L.A. Takhtajan, Phys.Lett.64A,235(1977). 5K.Nakamura andT.Sasada,J.Phys.C15,L915 (1982);A.KunduandO.Pashaev,J.Phys.C16,L585 (1983). sY.U.L.Rodin,Physica(Utrecht) 11D,90(1984). 7D.C.Mattis, TheTheoryofMagnetismI(Springer, Berlin,1980),p.198. 8M.Lakshmanan andM.Daniel, Phys.Rev.B24, 6751(1981);M.Daniel andM.Lakshmanan, Physica (Utrecht) 120A,125(1983);M.Daniel,"Nonlinear Ex- citations intheHeisenberg Ferromagnetic SpinChain," Ph.D.thesis,University ofMadras, 1983(unpublished). sM.Lakshmanan andM.Daniel, Physica (Utrecht) 107A,533(1981). Thestereographic projection method wasalsoeffective infindingtheinstanton solution ina nonlinear o-model.SeeA.A.Belavin andA.M.Pol- yakov,Pis'maZh.Eksp.Tear.Fiz.22,503(1975)JETP Lett.22,245(1975)]. ~OSimilar phenomena inRayleigh-Benard convection arebeinganalyzed. See,forexample, M.C.Crossand A.C.Newell, Physica(Utrecht) 10D,299(1984). 2499

PhysRevB.82.020407.pdf

Hysteretic synchronization of nonlinear spin-torque oscillators Phillip Tabor Department of Physics, West Virginia University, Morgantown, West Virginia 26506, USA Vasil Tiberkevich and Andrei Slavin Department of Physics, Oakland University, Rochester, Michigan 48309, USA Sergei Urazhdin Department of Physics, West Virginia University, Morgantown, West Virginia 26506, USA /H20849Received 24 June 2010; published 23 July 2010 /H20850 We report the observation of hysteretic synchronization of point-contact spin-torque nano-oscillators by a microwave magnetic field. The hysteresis is asymmetric with respect to the frequency detuning of the drivingsignal and appears in the region of a strong dependence of the oscillation frequency on the bias current.Theoretical analysis shows that hysteretic synchronization occurs when the width of the synchronization range,enhanced by the oscillator’s nonlinearity, becomes comparable to the dissipation rate, while the observedasymmetry is a consequence of the nonlinear dependence of frequency on the bias current. Hysteretic synchro-nization is a general property of strongly nonlinear oscillators, and can therefore be expected to affect thedynamics of a variety of physical systems. DOI: 10.1103/PhysRevB.82.020407 PACS number /H20849s/H20850: 76.50. /H11001g, 05.45.Xt, 75.75.Jn, 85.70.Ec Among the auto-oscillating systems, the class of nonlinear /H20849or nonisochronous /H20850auto-oscillators is characterized by a strong dependence of their frequency /H92750on the oscillation power p.1–3When such an oscillator is driven by a periodic force, its nonlinearity can enhance the range of synchroniza-tion. This effect results from the reduction in the detuningbetween /H92750and the frequency /H9275eof the driving force, caused by the variations of p.2,3 The nonlinearity can also have a qualitative effect on the dynamical properties of the oscillator. In quasilinear auto- oscillating systems, synchronization can be described byAdler’s phase equation 2,4and the synchronization transition is interpreted as the simultaneous creation of a stable nodeand a saddle point, which is a nonhysteretic process. In con-trast, for nonisochronous oscillators such as magnetic spin-torque nano-oscillators /H20849STNO /H20850, 5–8analysis that includes variations of both the phase and the amplitude indicates thatperiodic and quasiperiodic states can remain simultaneouslystable, resulting in hysteretic synchronization. 9This pre- dicted but previously unobserved effect is qualitatively dif-ferent from the well-known hysteretic transition between dif-ferent synchronization regimes, 2which is caused by the coexistence of multiple stable synchronized states and doesnot require nonlinearity. Understanding the synchronizationproperties of STNO may be important for the application ofSTNO arrays as microwave sources with enhanced genera-tion characteristics. 7,10–16 Here, we report the observation of hysteretic synchroni- zation of a point-contact STNO to an external periodic signalprovided by a large microwave magnetic field, enabled by adevice geometry incorporating a nanoscale microwave an-tenna. The hysteresis is observed in the region where theauto-oscillation frequency exhibits a strong dependence onthe bias current, confirming the correlation of this phenom-enon with the nonlinear properties of the oscillator. The hys-teretic synchronization is asymmetric , i.e., at each value ofthe bias current the hysteresis generally occurs only at the lower or the upper frequency boundary of the synchroniza-tion range, and in this respect is different from the symmetrichysteresis predicted in Ref. 9. Our analysis based on a ge- neric model of a nonlinear auto-oscillator suggests that the phenomenon of hysteretic synchronization is not specific toSTNO, but is rather a general property of nonlinear auto- oscillators, independent of their physical nature. Devices with structure Cu/H2084940/H20850Py/H208493.5/H20850Cu/H208498/H20850Co 70Fe30/H2084910/H20850Cu/H2084960/H20850 were fabricated on sapphire substrates patterned into copla- nar striplines, by a procedure described elsewhere.17All thicknesses are in nanometers, Py=Ni 80Fe20. The polarizing CoFe layer was patterned into a 100 nm /H1100350 nm nanopillar while the free Py /H208493.5/H20850layer was left extended with dimen- sions of several micrometers /H20851Fig.1/H20849a/H20850/H20852, thus forming a de- vice geometry similar to the magnetic point contact studiedin Ref. 8. Magnetic oscillations of the Py layer in the point- contact area were induced by a dc bias current I 0/H110220 flowing from CoFe. We also performed measurements in a morecommon device geometry with a nanopatterned free layerbut have not observed hysteresis in these devices, likely dueto the smaller nonlinearity and stronger effects of fluctua-tions that generally suppress hysteretic phenomena. In all the previous measurements of synchronization in STNO, the external driving signal was provided by an accurrent superimposed on the dc bias current. 10,18,11In con- trast, in our experiments the driving signal was provided by amicrowave magnetic field, enabling us to achieve signifi-cantly stronger driving. The microwave field was generatedby a 300-nm-wide and 250-nm-thick Cu microstrip fabri-cated on top of the STNO, and electrically isolated from it bya SiO 2/H2084950/H20850layer /H20851Fig.1/H20849a/H20850/H20852. The microstrip was oriented at 45° with respect to the nanopillar easy axis. To calibrate thePHYSICAL REVIEW B 82, 020407 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 1098-0121/2010/82 /H208492/H20850/020407 /H208494/H20850 ©2010 The American Physical Society 020407-1Oersted field heproduced by the microstrip, we compared the dependence of the auto-oscillation frequency f0on the dc bias field H0to the dependence of f0on the dc current iapp applied to the microstrip with H0oriented perpendicular to the direction of the microstrip, yielding he/H20849Oe/H20850=9.0 iapp /H20849mA/H20850. The frequency-dependent microwave losses were de- termined by utilizing a separately fabricated transmissionline with the same microstrip geometry. These losses weretaken into account when determining the microwave Oerstedfield generated by the microstrip. Loss-adjusted ac currentsof up to 4 mA rms were applied to the microstrip withoutnoticeable heating, producing h eof up to 36 Oe rms at the location of the STNO. Microwave signals applied to the microstrip produced parasitic currents in STNO in addition to the Oersted field. Inour measurements, the parasitic coupling did not exceed−25 dB, inducing microwave currents of less than 12 /H9262A rms at the largest driving amplitude. As demonstrated below,the effects of these parasitic currents on the oscillation aresmaller than those of the microwave field. All the measure-ments were performed at 5 K, at a dc bias fieldH 0=1.1 kOe. The reported results were confirmed for two devices. Magnetic static and dynamical characteristics of STNO were determined by measurements of magnetoresistance andauto-oscillation spectra. The auto-oscillation spectra had aLorentzian lineshape with a typical full width at half maxi-mum of 4 MHz, consistent with the effects of thermalbroadening 3,19,20/H20851Fig. 1/H20849b/H20850/H20852. Above the oscillation onset at bias current It=2.0 mA, the frequency f0of the auto- oscillation was initially approximately constant. AtI 0/H110222.75 mA, it exhibited a region of strong redshifting with a gradually decreasing slope df0/dI0/H20851Fig. 1/H20849c/H20850/H20852. This red- shifting region is correlated with the increase in the slope inthe dependence of the emitted power on the bias current /H20851Fig. 1/H20849d/H20850/H20852. A similar transition from a nearly constant frequency to a strongly redshifting behavior was observed in the otherstudied device. In both samples, hysteresis was observed inthe vicinity of this strong frequency redshift. Parasitic coupling produced peaks at f=f ein spectro- scopic measurements of driven oscillations, regardless of the oscillation regime. Therefore, the transition to the synchro-nized regime was identified as an abrupt disappearance of the f 0/H11015const. line of unlocked oscillation /H20849Fig.2/H20850. Note that the frequency pulling, which is a general feature of synchroniza-tion in linear oscillators, is negligible in the data of Fig. 2. The frequency at the lower boundary of the detuning intervalin the synchronized regime is larger by 0.31 GHz for f e scanned up /H20851Fig.2/H20849a/H20850/H20852than for fescanned down /H20851Fig.2/H20849b/H20850/H20852, demonstrating hysteretic synchronization. The synchronization range and the hysteresis were signifi- cantly larger for H0/H11036hethan for H0/H20648he, confirming the dominance of the effects of the microwave field over theparasitically induced microwave current /H20851Figs. 3/H20849a/H20850and 3/H20849b/H20850/H20852. A similar dependence of hysteresis on the field orien- tation was also observed in the other tested device. Thesedata explain why hysteresis has not been previously ob-served in measurements of synchronization induced by a mi-crowave current. Indeed, in a typical geometry of spin trans-fer devices, the current polarization is nearly collinear withthe static magnetization of the free layer. In this case, thesynchronization effects are similar to those of h e/H20648H0in Fig. 3/H20849b/H20850, producing only a small hysteresis. A driving mi- crowave current with polarization perpendicular to the mag- netization of the free layer would likely produce a large hys-FIG. 1. /H20849a/H20850Schematic of the device including an STNO and a microstrip line generating a synchronizing Oersted field. /H20849b/H20850Ex- ample of auto-oscillation spectrum at I0=3 mA /H20849symbols /H20850and Lorentzian fitting with full width at half maximum of 4 MHz /H20849solid curve /H20850./H20849c/H20850Symbols: auto-oscillation frequency vs bias current. Curve: our model fitting, as described in the text. /H20849d/H20850Emitted power under the oscillation peak vs bias current.FIG. 2. /H20849Color online /H20850Dependence of the oscillation frequency f of STNO on the driving frequency fe, demonstrating hysteretic syn- chronization to the external signal: /H20849a/H20850increasing feand /H20849b/H20850decreas- ingfe,a t I0=2.7 mA. The bias field H0is perpendicular to the microwave driving field he=18 Oe. Arrows show the direction of the scan and dashed vertical lines show the limits of the hystereticsynchronization range; a double arrow shows the width of the hys-teresis interval.TABOR et al. PHYSICAL REVIEW B 82, 020407 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 020407-2teresis, similar to our geometry with he/H11036H0. However, such a geometry would not allow excitation of auto-oscillationsvia the spin transfer effect. 21In addition, Figs. 3/H20849a/H20850and3/H20849b/H20850 demonstrate that hysteresis appears only above a certainthreshold driving amplitude, which may not have beenachieved in the published measurements of synchronizationby microwave current. Measurements of the dependence of the synchronization onI 0indicate that the hysteretic behaviors are correlated with the nonlinear characteristics of the auto-oscillation. At asmall driving field h e=8 Oe /H20851Fig. 3/H20849c/H20850/H20852, the hysteresis ap- pears at the lower boundary of the synchronization range at2.65 mA /H11349I 0/H113492.8 mA, i.e., close to the region where the auto-oscillation /H20849dashed curve /H20850exhibits the largest frequency redshift. A small hysteresis is also observed at the upperboundary of the synchronization range at I 0=2.8–2.85 mA. At a larger driving amplitude he=32 Oe /H20851Fig.3/H20849d/H20850/H20852, the syn- chronization interval and the magnitude of hysteresis in-crease and hysteretic synchronization is observed over alarger range of bias currents. The hysteresis at the upperboundary of the synchronization interval remains small, butoccurs over a broad range of I 0from 2.85 to 3.2 mA. The observed asymmetry of hysteresis was confirmed in bothtested devices for several values of the bias field. Our experimental results confirm the prediction of syn- chronization hysteresis made by Bonin et al. 9The primary distinction between the predicted behaviors and our results isthe observation of a strong asymmetry with respect to thesign of frequency detuning. To understand the origin of theasymmetry, we performed numerical simulations of synchro-nization taking into account the strong nonlinear dependence of the auto-oscillation frequency on the bias current. Oursimulations were based on the nonlinear auto-oscillatormodel 3for the complex oscillation amplitude c/H20849t/H20850 dc dt+i/H92750/H20849p/H20850c+/H20851/H90030−/H9268I0/H208491−p/H20850/H20852c=/H9253hee−i/H9275et. /H208491/H20850 Here, /H9253is the gyromagnetic ratio, p=/H20841c/H208412is the dimen- sionless precession power, /H92750/H20849p/H20850is the auto-oscillation fre- quency, /H90030is the natural positive magnetic damping, which for simplicity is assumed independent of p. The parameter /H9268 was defined by Eq. /H208494b/H20850in Ref. 3, such that /H9268I0/H208491−p/H20850is the negative damping induced by the spin transfer. The right-hand side of Eq. /H208491/H20850represents the action of the driving mi- crowave magnetic field of amplitude h eand frequency /H9275e. For simplicity, we neglect the ellipticity of the magnetizationprecession. This assumption is equivalent to an effectiverenormalization of the driving amplitude h eand does not qualitatively modify the results. The driving field is orientedorthogonal to the magnetization precession axis to model theexperimental geometry exhibiting the largest hysteresis. The coefficient /H9268in the linear relationship between the current and the spin torque is determined predominantly bythe spin-polarization efficiency of the polarizing layer. Wechose /H9268=2.0 ns−1mA−1, which corresponds to the dimen- sionless spin-polarization efficiency /H9255/H112290.4. The damping rate/H90030is determined by a combination of the Gilbert damp- ing /H20849/H9251/H112290.01 /H20850and radiative damping due to spin wave propagation away from the area of the nanocontact. In oursimulations, we chose /H9003 0=4.0 ns−1to reproduce the experimental value of the auto-oscillation onset currentI t=/H90030//H9268=2.0 mA. The dependence of the power of the sta- tionary auto-oscillation on the bias current was calculatedfrom Eq. /H208491/H20850asp 0=/H20849I0−It/H20850/I0. The adequacy of our model for the analysis of hysteretic synchronization was initially verified by assuming a linearrelationship between the oscillation frequency and bias cur-rent, similar to the model of Bonin et al. In this approxima- tion, our simulations yielded symmetric hysteresis consistentwith the results of Ref. 9. To model the effect of the actual nonlinear properties of our devices, we analyzed the experi-mental relationship /H20851Fig. 1/H20849c/H20850/H20852between of the auto- oscillation frequency and the bias current, yielding the de-pendence /H92750/H20849p/H20850of the frequency on the power of the oscillator, without any additional assumptions. We fitted thisdependence by /H92750/H20849p/H20850=54.5+ /H208491.16−7.85 p/H20850/H208511+tanh /H2084938.5p −11.3 /H20850/H20852, where /H92750is expressed in ns−1, which provides an excellent approximation for the experimental data /H20851Fig.1/H20849c/H20850/H20852. Figure 4/H20849a/H20850shows an example of the simulated synchro- nization at I0=2.4 mA, he=30 Oe, for two opposite direc- tions of the driving frequency sweep. The simulation exhibitsa significant hysteresis only at the lower synchronizationboundary. In addition, the unlocked oscillation frequency re-mains constant up to the synchronization transition, i.e., thefrequency pulling is negligible. Both of these features are inremarkable agreement with the experiment /H20849Fig.2/H20850. The simulated dependence of synchronization on the bias current is consistent with the most prominent features of theexperimental data, as illustrated in Fig. 4/H20849b/H20850forh e=30 Oe.FIG. 3. /H20849a/H20850Dependence of the synchronization boundaries on the rms amplitude heof the driving microwave magnetic field, for H0/H11036heatI0=2.7 mA. Crosses: upper synchronization boundary forfescanned up, dots: upper synchronization boundary for fe scanned down, open circles: lower synchronization boundary for fe scanned up, and solid circles: lower synchronization boundary for fescanned down. /H20849b/H20850Same as /H20849a/H20850, for H0/H20648he./H20849c/H20850Dependence of the synchronization boundaries on the bias current I0,a the=8 Oe per- pendicular to H0. Dashed curves show the auto-oscillation fre- quency. /H20849d/H20850Same as /H20849c/H20850,a the=32 Oe.HYSTERETIC SYNCHRONIZATION OF NONLINEAR SPIN- … PHYSICAL REVIEW B 82, 020407 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 020407-3In our simulations, the synchronization hysteresis appeared athe/H11022ht/H1101515 Oe, when the width of the synchronization interval, which is proportional to heand is enhanced by the oscillator’s nonlinearity,2,3becomes comparable to the auto- oscillator dissipation rate /H90030. Additional simulations per- formed for different values of /H90030showed that the hysteresis generally appears at driving field amplitudes exceeding thethreshold value determined by /H9004f e/H11015/H90030//H208492/H9266/H20850. Thus, mea- surements of the synchronization range for he/H11015htcan be used to determine the dissipation rate /H90030of a nonlinear auto- oscillator. Athe=30 Oe above the excitation threshold /H20851Fig. 4/H20849b/H20850/H20852, the simulated synchronization exhibits a large hysteresis atthe lower synchronization boundary, at currents below the point of the maximum slope /H20841d/H92750/H20849p/H20850/dp/H20841. A small hysteresis also appears at the upper synchronization boundary over abroad range of the bias current, above the point of the maxi-mum slope /H20841d /H92750/H20849p/H20850/dp/H20841. These features are in excellent agreement with the experimentally observed behaviors, con-firming that the observed asymmetry of the hysteresis iscaused by the strong nonlinearity of the dependence of theauto-oscillation frequency on the oscillation power. In summary, we have demonstrated the predicted hyster- etic synchronization of a nonlinear magnetic nano-oscillatorby a microwave magnetic field. Our observations indicatethat the hysteresis is correlated with the region of large non-linear red frequency shift, linking the phenomenon of hyster-esis to the nonlinearity of the oscillator. Simulations basedon the model of strongly nonlinear auto-oscillator, takinginto account the observed dependence of frequency on thebias current, yielded good semiquantitative agreement withthe most salient features of the experiment. The subtle de-pendence of hysteretic synchronization on the nonlinearproperties of the oscillator can be utilized to extract impor-tant information about the dynamical characteristics of spintorque nano-oscillators, as well as other strongly nonlinearoscillating systems. This work was supported by NSF under Grants No. DMR-0747609 and No. ECCS-0653901, the Research Cor-poration, U.S. Army TARDEC, RDECOM /H20849Contract No. W56HZV-09-P-L564 /H20850, and Oakland University Foundation. 1A. Blaquiere, Nonlinear System Analysis /H20849Academic, New York, 1966 /H20850. 2A. Pikovsky, M. Rosenbblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences /H20849Cambridge Univer- sity Press, New York, 2001 /H20850. 3A. Slavin and V. Tiberkevich, IEEE Trans. Magn. 45, 1875 /H208492009 /H20850. 4R. Adler, Proc. IRE 34, 351 /H208491946 /H20850. 5J. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 6L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 7S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850 425, 380 /H208492003 /H20850. 8W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 /H208492004 /H20850. 9R. Bonin, G. Bertotti, C. Serpico, I. D. Mayergoyz, and M. d’Aquino, Eur. Phys. J. B 68, 221 /H208492009 /H20850. 10W. H. Rippard, M. R. Pufall, S. Kaka, T. J. Silva, S. E. Russek, and J. A. Katine, Phys. Rev. Lett. 95, 067203 /H208492005 /H20850. 11B. Georges, J. Grollier, M. Darques, V. Cros, C. Deranlot, B. Marcilhac, G. Faini, and A. Fert, Phys. Rev. Lett. 101, 017201 /H208492008 /H20850.12S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature /H20849London /H20850437, 389 /H208492005 /H20850. 13F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Nature /H20849London /H20850437, 393 /H208492005 /H20850. 14D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 /H208492008 /H20850. 15D. V. Berkov and J. Miltat, J. Magn. Magn. Mater. 320, 1238 /H208492008 /H20850. 16T. J. Silva and W. H. Rippard, J. Magn. Magn. Mater. 320, 1260 /H208492008 /H20850. 17S. Urazhdin and P. Tabor, J. Appl. Phys. 105, 066105 /H208492009 /H20850. 18J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 96, 227601 /H208492006 /H20850. 19J. C. Sankey, I. N. Krivorotov, S. I. Kiselev, P. M. Braganca, N. C. Emley, R. A. Buhrman, and D. C. Ralph, Phys. Rev. B 72, 224427 /H208492005 /H20850. 20V. S. Tiberkevich A. N. Slavin, and Joo-Von Kim, Phys. Rev. B 78, 092401 /H208492008 /H20850. 21F. B. Mancoff, R. W. Dave, N. D. Rizzo, T. C. Eschrich, B. N. Engel, and S. Tehrani, Appl. Phys. Lett. 83, 1596 /H208492003 /H20850.FIG. 4. /H20849Color online /H20850/H20849a/H20850Numerical simulations of the STNO synchronization with increasing /H20849blue solid curve /H20850and decreasing /H20849red dashed curve /H20850driving frequency featI0=2.4 mA, he=30 Oe. /H20849b/H20850Calculated dependence of the synchronization limits on I0for he=30 Oe. Black dotted curve shows the free running frequency.TABOR et al. PHYSICAL REVIEW B 82, 020407 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 020407-4

PhysRevApplied.9.064014.pdf

Nonreciprocal Surface Acoustic Waves in Multilayers with Magnetoelastic and Interfacial Dzyaloshinskii-Moriya Interactions Roman Verba,1Ivan Lisenkov,2,3,*Ilya Krivorotov,4Vasil Tiberkevich,5and Andrei Slavin5 1Institute of Magnetism, Kyiv 03680, Ukraine 2School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, Oregon 97333, USA 3Kotelnikov Institute of Radioengineering and Electronics, Russian Academy of Sciencies, Moscow 125009, Russia 4Department of Physics and Astronomy, University of California, Irvine, California 92697, USA 5Department of Physics, Oakland University, Rochester, Michigan 48309, USA (Received 29 December 2017; revised manuscript received 13 April 2018; published 12 June 2018) Surface acoustic waves (SAWs) propagating in a piezoelectric substrate covered with a thin ferromagnetic –heavy-metal bilayer are found to exhibit a substantial degree of nonreciprocity, i.e., the frequencies of these waves are nondegenerate with respect to the inversion of the SAW propagation direction. The simultaneous action of the magnetoelastic interaction in the ferromagnetic layer and theinterfacial Dzyaloshinskii-Moriya interaction in the ferromagnetic –heavy-metal interface results in the openings of magnetoelastic band gaps in the SAW spectrum, and the frequency position of these band gaps is different for opposite SAW propagation directions. The band-gap widths and the frequency separationbetween them can be controlled by a proper selection of the magnetization angle and the thickness of the ferromagnetic layer. Using numerical simulations, we demonstrate that the isolation between SAWs propagating in opposite directions in such a system can exceed the direct SAW propagation losses by morethan 1 order of magnitude. DOI: 10.1103/PhysRevApplied.9.064014 I. INTRODUCTION Surface acoustic wave (SAW) transmission lines, based on high-quality piezoelectric single crystals, found appli-cations as frequency filters, sensors, and other signal processing devices [1–4]. SAWs have very low propagation losses at frequencies ranging from a megahertz to severalgigahertz. They can be excited with a very high efficiency in piezoelectric crystals, and the use of unidirectional transducers [3,4] can reduce the insertion losses of SAW transmission lines to just several decibels. Moreover, typical propagation speeds (and therefore wavelengths) of SAWs in crystals are several orders of magnitude lessthan the speed of electromagnetic waves, thus allowing a miniaturization of SAW signal processing devices com- pared to their electromagnetic counterparts. A typical frequency spectrum ω kof a SAW is reciprocal, i.e., it is degenerate for SAWs having opposite wave vectorskand−k:ω k¼ω−k. This degeneracy is a result of a fundamental time-reversal symmetry in the laws of mechanics. However, frequency nonreciprocity (when ωk≠ω−k) is extremely important for applications: it allows us to isolate signals traveling in opposite directions [5,6].From a practical point of view, a good isolator should demonstrate high rejection; i.e., it should block most of the power traveling in one direction (say, from port 2 to port 1). Simultaneously, it should have low insertion loss, i.e.transmit nearly all of the power traveling in the oppositedirection (from port 1 to port 2). It is known thatSAW-based devices demonstrate very low transmissionlosses—and therefore should be useful as isolators —if a nonreciprocal propagation SAW propagation in these devices can be demonstrated. Unfortunately, a nonreciprocal propagation of a SAW is not easy to achieve. Thus far, the nonreciprocal propagation of a SAW has been found in devices with moving or rotating elements [7,8], where the effect of the summation of velocities of sound and moving media was used. Analternative way to achieve acoustic nonreciprocity is to usenonlinear effects in high-power acoustic waves, where theacoustic wave loss or gain are power dependent [9–12]. Unfortunately, neither of these ways have led to the development of practical nonreciprocal devices based onacoustic waves. In contrast to acoustic waves, the frequency nonreciproc- ity of spin waves (SWs) propagating in ferromagnetic mediais not an exotic phenomenon. The SW nonreciprocity isa consequence of the intrinsic breaking of time-reversal *ivan.lisenkov@phystech.eduPHYSICAL REVIEW APPLIED 9,064014 (2018) 2331-7019 =18=9(6)=064014(11) 064014-1 © 2018 American Physical Societysymmetry in magnetized magnetic materials, where mag- netization precesses only clockwise around its equilibriumdirection. The frequency nonreciprocity for SWs can be achieved in multiple ways, e.g., by nonsymmetric boundary conditions [13–15], by patterning of a ferromagnet [16–18], or by a bulk or interfacial Dzyaloshinskii-Moriya interaction [19–22]. It should be noted that the application of the SWs themselves for the development of compact nonreciprocalmicrowave devices is a challenging task, as the magnetic field bias is needed, and the problem of a relatively high SW propagation losses should be solved. Fortunately, owing to the magnetoelastic interaction in magnetostrictive materials, the SWs and acoustic waves can interact with each other, and SWs can act as a “source ” of nonreciprocity for acoustic waves. The nonreciprocity of magnetoelastic waves [23–25], as well as the magnetoe- lastic interaction itself [26–28], have been studied for a long time in the case of bulk samples. The bulk materials, however, typically do not show good acoustic, magnetic,magnetoelastic, and piezoelectric properties simultane- ously, which hinders their practical applications in nonre- ciprocal devices. For example, many papers have beendevoted to the study of magnetoelastic waves in yttrium iron garnet (YIG), which has nice magnetic properties, but the magnetoelastic interaction in YIG is weak, and thismaterial has no piezoelectricity at all. At the same time, metallic ferromagnets (such as Ni, Co, and Fe) have rather large magnetostriction (3 to 4 orders larger than in YIG),but prohibitively bad acoustic properties. Rather promising recent experiments [29–32] have demonstrated that the propagation of SAWs in piezoelectricsubstrates can be controlled by a thin magnetic layer placed atop the substrate. The use of such heterostructures allows one to combine in a single device a high-quality piezo-electric substrate (like LiNbO 3) and a ferromagnet with large magnetostriction (e.g., Ni). Moreover, it has been shown that SAWs propagating in a LiNbO 3substrate covered by a thin Ni film indeed demonstrate some degree of nonreciprocity [32], although the observed nonreciproc- ity effect was small. In such a case, the SWs are generally reciprocal, and the small transmission nonreciprocity comes from the slightly different widths of the magne-toelastic band gaps having the same central frequency in the SAW spectrum for waves with opposite wave vectors. As a result, the nonreciprocal transmission appears on abackground of large propagation (insertion) losses. In this work, we propose a way to substantially enhance the nonreciprocal properties of SAW in piezoelectric-ferromagnetic heterostructures using the materials with the interfacial Dzyaloshinskii-Moriya interaction (IDMI). We show that the IDMI results in a nondegeneracy of thecentral frequencies of the magnetoelastic band gaps with respect to the inversion of the SAW propagation direction. Since the central frequencies of the band gaps are differentfor the two counterpropagating waves, a wave traveling in one direction falls within the band gap, while the wavetraveling in the opposite direction does not “feel”the band gap at all. Therefore, the damping of the wave propagating in one direction is tremendously increased, while that forthe wave propagating in the opposite direction is practically unaffected, resulting in the simultaneously high isolation and low insertion losses. In our numerical simulations,we use the parameters of a transmission line based on a LiNbO 3substrate covered by a thin Ni =Pt bilayer, and we show that, using a high-quality Ni film, one can achievean isolation of up to 45 dB with insertion losses of about 20 dB. The article is organized as follows. In Sec. II, we present a general formalism for the magnetoelastic coupling between the linear SWs and SAWs in the framework of a perturbation theory. Then we consider the conditions forthe appearance of nonreciprocal magnetoelastic band gapsin the wave spectrum (Sec. III B), and the ways for the optimization of the nonreciprocal properties (Sec. III C). Finally, Sec. III D is devoted to the calculation of the SAW line transmission characteristics in the presence of the IDMI and the SAW coupling to SWs. II. THEORY OF WEAKLY COUPLED LINEAR MAGNETOELASTIC WAVES In this section, we revisit the theory of magnetoelastic interactions in ferromagnetic samples, and we develop an analytical formalism for magnetoelastic coupling between the spin waves and acoustic waves suitable for the systemshaving arbitrary wave profiles (e.g., suitable for surface magnetoelastic waves), limiting ourselves to the case of linear coupling between SWs and SAWs. The dynamics of the magnetoelastic waves is governed by the coupled Landau-Lifshitz equation for SWs and elasticmechanical equations [2,5] for acoustic waves. Simulations solution of these equations is complicated and is often possible only numerically [33–36].H o w e v e r ,i na l m o s ta l l of the practically important situations, the magnetostriction isweak in comparison to the other interactions in a ferromag- net, which allows us to consider the magnetoelastic inter- action in the framework of perturbation theory. For the consideration of linear excitations, the magneti- zation vector Mcan be represented as a sum of static and dynamic components, Mðr;tÞ¼M s½μðrÞþmðr;tÞ/C138, where Msis the saturation magnetization, μis the unit vector pointing in the direction of the static magnetization, and m is a dimensionless dynamic magnetization ( jmj≪1). Then the equations describing the coupled magnetoelastic dynamics can be written as 1 γˆJ·dmðr;tÞ dt−Z ˆΩ·mðr0;tÞdr0¼bmeðr;tÞ;ð1ÞROMAN VERBA et al. PHYS. REV. APPLIED 9,064014 (2018) 064014-2ρ∂2 ∂t2ξiðr;tÞ−cijln∂2 ∂xj∂xlξnðr;tÞ¼fme iðr;tÞ:ð2Þ Here, ˆJ¼ˆe·μis the operator of the angular momentum, ˆe is the Levi-Civit` a antisymmetric tensor, ˆΩ¼ ˆΩðr;r0Þis the operator of magnetic interactions (see Refs. [37–39]for additional details), ξðr;tÞis the elastic displacement, ρis the density, and cijlmrepresents the components of the elastic stiffness tensor. The magnetoelastic coupling is given by the terms on the right-hand side of the equation, where bmeðr;tÞis the effective magnetic field generated by the acoustic deformations via the inverse magnetostriction,andf meis the effective force generated by the magnetiza- tion dynamics and acting on the sample via the direct magnetostriction effect. In Eq. (1), we skip another cou- pling term of the form mðμ·bmeÞ, as it is of the second order of smallness and cannot result in a linear coupling between the waves. In Eq. (2)and below, the repeating indices ði; j; l; m Þ¼ðx; y; z Þare assumed to be summed. The magnetoelastic coupling can be obtained from the following magnetoelastic energy density Wme¼1 M2sbijlnuijMlMn; ð3Þ where ˆbis the tensor of magnetostriction [40]and ˆuðr;tÞis the tensor defining the strain created by the displacementξðr;tÞ[2] u ij¼1 2/C18∂ξi ∂rjþ∂ξj ∂ri/C19 : ð4Þ The values of the field bmeand the force fmecan be calculated using Eq. (3)as bmenðr;tÞ¼−∂Wme ∂M≈−2 Msbijlnuijðr;tÞμlðrÞ;ð5Þ fme iðr;tÞ¼∂ ∂xi∂ ∂uijWme≈2∂ ∂xjbijlnμlðrÞmnðr;tÞ; ð6Þ where we leave only the terms that are linear in the dynamic magnetization mor displacement ξ, as they are responsible for the linear coupling between the waves. The other terms,corresponding, e.g., to the parametric coupling between the waves, are disregarded. Equations (1)and(2)can be solved within a standard framework of an eigenmode expansion: mðr;tÞ¼X νcνðtÞmνðrÞþc:c:; ð7Þ ξðr;tÞ¼X λqλðtÞξλðrÞþc:c:; ð8Þwhere mνðrÞandξλðrÞare the profiles of the linear SWs and acoustic modes, while cνðtÞandqλðtÞare the unknown complex amplitudes of the eigenmodes. The spatial profilesof the eigenmodes and their eigenfrequencies, ω νand ˜ωλ, respectively, are the solutions of Eqs. (1)and(2)with zero right-hand-side parts in the form mðr;tÞ¼mνðrÞe−iωνt, while ξðr;tÞ¼ξλðrÞe−i˜ωλt. The linear SW modes satisfy the following orthogonality relation [37,38] Ms γZ m/C3 ν0ðrÞ·μðrÞ×mνðrÞdr¼−iAνδν;ν0; ð9Þ where Aν>0is the spin-wave normalization constant having the dimensionality of action [41]. A similar ortho- gonality condition can be written for the acoustic modes [2,42] 2ωλZ ρðrÞξ/C3 λðrÞ·ξλ0ðrÞdr¼Qλδλλ0; ð10Þ where Qλ>0is a positive normalization constant having the same dimensionality as Aν. Substituting the expansion equations (7)and (8)for mðr;tÞand ξðr;tÞinto Eqs. (1)and (2)and using the orthogonality relations, we get the following final equations for the amplitudes of the coupled spin and acoustic waves: dcν dtþiωνcνþΓνcν¼iX λffiffiffiffiffiffi Qλ Aνs κν;λqλ; dqλ dtþi˜ωλqλþ˜Γλqλ¼iX νffiffiffiffiffiffi Aν Qλs κ/C3 ν;λcν; ð11Þ where we also introduce in a common way [8,38] the damping rates of the spin and acoustic modes Γνand ˜Γλ, respectively. The coupling coefficient is equal to κν;λ¼2ffiffiffiffiffiffiffiffiffiffiffiAνQλpZ μðrÞ·½ˆb·ˆuλðrÞ/C138·m/C3νðrÞdr:ð12Þ This expression is the central result of the above-developed theory. The coupling coefficient can be calculated for the arbitrary spatial profiles of the acoustic and SW modes.The exact profiles of the SW and acoustic modes, as well as the mode eigenfrequencies ω νand ˜ωλ, in simple cases can be found analytically or, otherwise, can be extracted fromnumeric simulations. In the case of propagating waves, characterized by a wave vector k, the solution of Eq. (11) can be easily obtained. In the equations above, we change m ν→ mkðρÞeik·randξλ→ξk0ðρÞeik0·r, where ρis two-dimensional radius vector, perpendicular to the wave propagation direction, defined by k. Then Eq. (12) is transformed toNONRECIPROCAL SURFACE ACOUSTIC WAVES IN … PHYS. REV. APPLIED 9,064014 (2018) 064014-3κk;k0¼2ffiffiffiffiffiffiffiffiffiffiffiffiAkQk0pZ μðrÞ·½ˆb·ˆuk0ðρÞ/C138·m/C3 kðρÞeiðk0−kÞ·rdr: ð13Þ It is clear that, in the case when the static magnetization is uniform along the wave propagation direction, the expo-nent under the integral gives a zero integration result untilk≠k 0. Therefore, in this case, the spin and acoustic waves can interact only if they have the same wave vector k. Note that the length Lof the sample in the wave propagation direction, which appears after the integration in Eq. (13),i s canceled out by the same term in the normalizationconstants of the SW and the SAW. The dispersion relationfor the interacting waves can be written as ω k¼ωSWþωAW 2/C6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/C18ωSW−ωAW 2/C192 þjκkj2s ;ð14Þ where ωSW¼ωSW;kandωAW¼ωAW;kare the dispersion relations of the noninteracting SWs and acoustic waves, respectively. III. NONRECIPROCAL SURFACE MAGNETOELASTIC WAVES A. Spin-wave modes in a ferromagnetic film with the IDMI In this section, we apply the above-presented general theory to the study of the surface magnetoelastic waves in amagnetic-nonmagnetic heterostructure. A sketch of a con-sidered heterostructure is shown in Fig. 1. It consists of a nonmagnetic substrate which supports the propagation of aSAW. Typically, this substrate is a piezoelectric singlecrystal, like LiNbO 3, LiTaO 3or quartz. The piezoelectric substrate is covered with a thin ferromagnetic film having alarge magnetostriction (e.g., Ni), and then by a thin heavy-metal layer (typically Pt), which induces the IDMI at theferromagnetic –heavy-metal interface. The ferromagnetic layer is biased by an external magnetic field B eapplied in the film plane at the angle ϕwith respect to the wave propagation direction. The value of the bias field should besufficient to saturate the ferromagnetic film in its plane,thus overcoming the effect of the surface perpendicularmagnetic anisotropy, which can exist at the ferromagnetic – heavy-metal interface. Since the IDMI is an interface effect, the thickness of a ferromagnetic film necessary to produce a significant SW nonreciprocity should be sufficiently small. As has becomeclear from the results of our numerical simulations, the ferromagnetic film thickness should not exceed several tens of nanometers. In this case, we can use the assumption of auniform SW profile across the thickness of a ferromagnetic layer, m k∉fðzÞ. The dispersion of SWs propagating along thexaxis (see Fig. 1) can be expressed as [20,43] ωSW;k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΩzzΩIPp −ωM˜Dksinϕ; ð15Þ where ΩIP¼ωHþωM½λ2exk2þfðktÞsin2ϕ/C138; Ωzz¼ωH−ωanþωM½λ2exk2þ1−fðktÞ/C138:ð16Þ In these equations, ωH¼γBe,ωM¼γμ0Ms,a n d ωan¼ 2γKs=ðMstFMÞ, where Ksis the constant of the surface perpendicular anisotropy, k¼kx,fðxÞ¼1−ð1−e−jxjÞ=jxj is a function describing the dynamic demagnetization, andthe effect of the IDMI is described by the expression ˜D¼2Db=ðμ 0M2stFMÞ, where Dis the IDMI constant, tFM is a thickness of the ferromagnetic layer, and bis the thickness of an atomic monolayer of the ferromagnet [21,44] .F r o mE q . (15), it is clear that the SW dispersion is nonreciprocal: ωSW;k≠ωSW;−kifϕ≠0;π. Because of the symmetry of the effective field, produced by the IDMI, the vector structure of the SW mode does not depend on the IDMI [43], and it can be expressed as mk¼½−mIPsinϕ;mIPcosϕ;i m z/C138, where mIPis the in-plane dynamic component of magnetization, and the relation between the magnetization dynamic components is mz=mIP¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΩIP=Ωzzp . B. Analysis of the coupling of surface acoustic waves with spin waves The calculation of a dispersion relation and mode profile of SAWs in a layered structure consisting of a piezoelectricsubstrate and a metallic layer is, in itself, not a simple task.Thus, to simplify our analytical analysis, we use severalapproximations. First, we consider the substrate as isotropicand nonpiezoelectric and use the Poisson ratio as an adjustable parameter, as is often done in analytical calcu- lations [1]. Within this approximation, it is not possible to answer the question on the SAW stability, but it is possibleto adequately describe the profile of the SAW mode and,therefore, to evaluate the main characteristics of themagnetoelastic coupling of the SAWs and SWs. Second, we neglect the influence of the thin metallic layer on the SAW properties. In general, shear acoustic waves in sputtered metals are slower than the shear acousticwaves in piezoelectric single crystals. Thus, in such a FIG. 1. A layout of the heterostructure under study, a piezo- electric substrate covered by a ferromagnetic –heavy-metal bilayer.ROMAN VERBA et al. PHYS. REV. APPLIED 9,064014 (2018) 064014-4system, the substrate is loaded by the metal layer, and the surface acoustic wave does not have any cutoff wave numbers [2,45,46] . In reality, the thickness of the ferro- magnetic layer is on the order of tNi≈10nm, and the layer of the heavy metal can be as thin as 2 to 3 nm because thefurther increase of the heavy-metal thickness does not affect the strength of the IDMI [47]. The SAW wave number for the considered range of frequencies (1 –5 GHz) is on the order of k x<10μm−1. Thus, we can work in an approximation in which kxðtNiþtPtÞ≪1, and we assume that the SAW is only weakly affected mechanically by the bilayer. Therefore, we can use the wave dispersion and thewave mode profiles calculated for a free substrate [2,45] .I t should be noted that the above-presented formalism (Sec. II) remains valid if one considers the exact values of both the acoustic field distribution and the SAWdispersion. This property could be useful in the futurefor making more-accurate calculations of the coupling parameter κ. Taking into account the above-described approximations, we consider a Rayleigh surface acoustic wave [48]. The in- plane component of the displacement perpendicular to the SAW propagation direction is absent, ξ y¼0, so the strain tensor components uxy¼uyz¼uyy¼0. The oscillations in thexandzdirections are shifted in phase by π=2, resulting in an effective rotation in the medium over an elliptic trajectory. The only nonzero components of the SAW strain tensor areuxx,uzz, and uxz. At the surface ( z¼0in Fig. 1), the off-diagonal strain component vanishes, uxzðz¼0Þ¼0, while the components uxxanduzzremain nonzero. The dispersion of the SAW is linear, ωSAW ;k¼cSAWjkj, where cSAW is the SAW velocity. The magnetoelastic coupling tensor ˆbin the case of a cubic crystal (Ni, Fe, or Co) has only two independentcomponents [5]:b iiii¼b1andbijij¼b2(fori≠j), while all of the other components are zero (in the case of an isotropic media b1¼b2). Noting the symmetry of the magnetoelastic tensor, we calculate the coupling coefficient κkfor a SW in the ferromagnet and a Rayleigh SAW using Eq.(12): κk¼2tFMffiffiffiffiffiffiffiffiffiffiffiAkQkp ½−b1¯uxx;km/C3 IP;ksinϕþb2¯uxz;km/C3 z;k/C138cosϕ; ð17Þ where ¯uijrepresents the strain components, averaged over the ferromagnetic film thickness, and, in the definition of the normalization constants AkandQk[Eqs. (9) and(10)], the integration over the volume is replaced by the integration over the zcoordinate. It is clear from Eq. (17) that the magnetoelastic inter- action between the SW and SAW vanishes for ϕ¼π=2, while this angle corresponds to the maximum IDMI- induced SW nonreciprocity; see Eq. (15). As was pointed out earlier, at the free surface, the strain componentuxz¼0, so the averaged value j¯uxzj≪j¯uxxj, and the coupling coefficient is determined mainly by the first term in brackets in Eq. (17). Therefore, the coupling coefficient is approximately proportional to the function κk∼sin2ϕ, which reaches its maximum at ϕ¼π=4. Consequently, the maximal coupling of SW and SAW is realized for the magnetization angle close to ϕ¼π=4. This feature has been already observed in Refs. [29,32] . We also note that the SW eigenmode does not change with the reversal of the propagation direction m¼m−k.A t the same time, the SAW strain tensor transforms asu xx;−k¼−uxx;k,uxz;−k¼uxz;k[48]. Therefore, the cou- pling between the SW and the SAW is nonreciprocal even without the IDMI, κk≠κ−k(ifϕ≠0;π=2), and this non- reciprocity becomes more pronounced for thicker ferro-magnetic film due to an increase in the ¯u xzcomponent. Unequal coupling results in different propagation losses ofSAW in opposite directions, which was observed in Ref. [32]. However, to achieve a good isolation, while maintaining a low insertion loss, one should havejκ kj≪jκ−kj. A simple analysis from Eq. (17) reveals that this requirement leads to the requirement on the staindistribution that ¯u xx≈¯uxz, which can be realized if ferro- magnetic film thickness becomes of the order of the SAW penetration depth. For metallic layers, this requirement isdifficult to fulfill because a thick ferromagnetic layersignificantly affects the mechanical properties of the sub-strate and increases the acoustic loss. However, this regime may possibly be implemented in dielectric single-crystal ferromagnets, such as YIG. C. Wave spectrum and magnetoelastic band gaps For our numerical example demonstrating nonreciprocal surface magnetoelastic waves, we choose a LiNbO 3=Ni=Pt heterostructure. LiNbO 3=Ni heterostructures have already been fabricated and studied in Refs. [29,32] , and they have demonstrated good magnetoelastic coupling. LiNbO 3is one of the best piezoelectric materials supporting SAW propagation in a frequency range of up to 10 GHz [49], while Ni shows large magnetostriction, and the combina-tion Ni =Pt gives the largest IDMI among the studied combinations of Ni with other heavy metals. We use the Ycut of LiNbO 3having the density ρ¼4650 kg=m3as a substrate, and the SAW propagates along the Zaxis. The substrate had the following material parameters: longi-tudinal and transversal sound velocities c l¼7350 m=s and ct¼3600 m=s[50], and the corresponding SAW velocity iscSAW¼3361 m=s. For the Ni layer, we use the following parameters: saturation magnetization μ0Ms¼0.66T, exchange stiffness A¼9.5×10−12J=m3(λex¼7.4nm), sur- face perpendicular anisotropy energy Ks¼6×10−4J=m2, gfactor g¼2.21, magnetoelastic coupling coefficients b1¼9.38MJ=m3,b2¼10MJ=m3[51,52] . The IDMI energy at the Pt-Ni interface is equal toNONRECIPROCAL SURFACE ACOUSTIC WAVES IN … PHYS. REV. APPLIED 9,064014 (2018) 064014-5D¼−2.7×10−3J=m2, while the lattice constant is b¼0.352nm[53]. An example of the spectra of SW and SAW in the heterostructure is shown in Fig. 2. By selection of the magnitude of Beand the angle ϕof the bias magnetic field, it is possible to achieve a crossing between the spectra of the noninteracting SW and SAW in a desirable frequency range. The interaction between the SAW and the SWleads to the opening of band gaps in the spectrum of amagnetoelastic surface wave. The widths of the band gapsare determined by the coupling coefficient κ k:Δω¼ 2πΔf¼2jκkj. Since the SW spectrum is nonreciprocal, the crossing of the SW dispersion curves with the SAWspectrum takes place at different points, and the magne-toelastic band gaps open at different frequencies and wave numbers for waves propagating in opposite directions. This feature is clearly visible in Fig. 2(b), where the central frequencies of the band gaps are shifted by 170 MHz with respect to each other. Therefore, within the frequency range of one of the band gaps, the SAW propagating in onedirection is strongly coupled to the SW, forming a slow anddissipative magnetoelastic wave, while the wave travelingin the opposite direction is almost unaffected by themagnetoelastic interaction. This property exists due onlyto the SW frequency nonreciprocity induced by the IDMI inour case. The widths of the band gaps and the separation between their central frequencies depend on the thickness of the ferromagnetic layer. The width Δfof a band gap increases with the thickness of the Ni layer t Nibecause the coupling coefficient between the SW and the SAW is proportional tot Ni; see Eq. (17). However, because of the interfacial nature of the IDMI, the nonreciprocity of the SWs —and, there- fore, the frequency separation between the band gapscorresponding to the opposite propagation directions — decreases with an increase in Ni thickness. These tendenciesare clearly illustrated in Fig. 3, where the positions and the widths of the band gaps are plotted as functions of t Ni.T h e bias field at each value of tNiis chosen in such a way that the average frequency position of the band gaps is kept constant [3 GHz in Fig. 3(a) and 5 GHz in Fig. 3(b), respectively]. For applications, it is desirable to have thewidest possible band gaps, which, however, should be well separated from one another, at least by a frequency interval on the order of the band-gap width. Thus, there is an optimumrange of ferromagnetic film thickness in which it is possible to achieve the best nonreciprocal properties of the magne- toelastic surface waves propagating in opposite directions.For example, in the above-described heterostructureLiNbO 3=Ni=Pt, the optimum thickness of the Ni layer is tNi≈8to 9 nm for average frequencies of both 3 and 5 GHz (see Fig. 2). For higher average frequencies, the optimum Ni thickness remains almost the same, at least up to a frequency of 10 GHz, at which the SAW excitation and propagation in LiNbO 3were observed experimentally in Ref. [49]. It should be noted that the crossing and hybridization of the dispersion curves of the SW and the SAW at any (a) (b) FIG. 2. (a) Spectra of surface magnetoelastic waves in the LiNbO 3=Ni=Pt heterostructure that, away from the points of wave hybridization, look like independent crossing spectra of theSAW and the SW, respectively. (b) Close-up of the spectra nearthe hybridization points [marked by dashed rectangles in (a)],where the magnetoelastic band gaps are clearly seen. Ni thick-ness, t Ni¼10nm; magnetization angle, ϕ¼π=4; bias field, Be¼41mT. (a) (b) FIG. 3. Positions and widths of the magnetoelastic band gaps in a spectrum of surface magnetoelastic waves for opposite propa-gation directions. (a) Average frequency of 3 GHz. (b) Averagefrequency of 5 GHz. The magnetization angle is optimum,ϕ¼π=4.ROMAN VERBA et al. PHYS. REV. APPLIED 9,064014 (2018) 064014-6desirable frequency cannot always be satisfied for the optimum magnetization angle of ϕ¼π=4. For example, fortNi>19nm, the crossing cannot be achieved at a frequency below 3 GHz for any value of the bias field.This property is related to the increase of SW group velocity at k→0taking place with the increase of film thickness. A solution of this problem is to use a smaller magnetization angle, ϕ<π=4, which decreases the SW group velocity, and even changes its sign for sin 2ϕ<ωH= ðωHþωM−ωanÞ. At such a magnetization angle, one can achieve the formation of magnetoelastic band gaps at almost any desirable frequency. However, the widths ofthe band gaps, as well as the separations between them, become smaller [see Eqs. (15)and(17)]. This feature limits the applicability of the IDMI-induced nonreciprocity of surface magnetoelastic waves in a relatively low-frequency range (below 2 GHz). D. Transmission characteristics In this section, we consider how the appearance of the magnetoelastic band gaps affects the transmission character- istics of a SAW line. In general, the appearance of the bandgaps leads to a variation in the wave group velocity v gr¼ ∂ωk=∂k(the slope of the dispersion curve), and to a variation in the wave damping rate in the vicinity of the band gaps. Both of these factors contribute to the variation of a trans- mission rate in a magnetoacoustic transmission line. It should be noted that common methods of SAW transmission calculations (see, e.g., Refs. [1,54] ) are not applicable in our case. These methods use the assumption of a negligibly small resonance linewidth so that the wave group velocity and the efficiency of interdigital transducers(IDTs) can be calculated locally, at the point k¼kðωÞ. This assumption is natural for SAWs, which typically have a very small linewidth (for example, for LiNbO 3, this linewidth is only 500 kHz at the 5-GHz frequency [55]). However, the SW damping rate —and, consequently, the damping rate of magnetoelastic waves in the vicinity of the band gaps —can be comparable to (or larger than) the band- gap width. In such a case, the nonresonant wave excitation becomes important, and one should integrate contributions from all of the excited waves within the resonance line. To calculate the transmission characteristics, we need to introduce into Eq. (11) an external harmonic force which describes the excitation of SAWs by an IDT. The appli-cation to an IDT of a microwave voltage VðtÞ¼V ine−iωtof the frequency ωresults in the appearance of a mechanical force in the LiNbO 3substrate, and this force has a certain spatial profile which depends on the IDT geometry. The efficiency of the coupling of IDT to a SAW having a certainwave vector k¼ke xcan be decomposed into two terms. The first of these terms is the normalized Fourier transform Fkof the force spatial profile in the xdirection, which determines the kdependence of the excitation efficiency. This term is often approximated by a functionFk¼sinc½πNfðk−k0Þ=k/C138, where Nfis the number of fingers in the IDT and ω0¼cSAWk0is the central frequency of the IDT [54]. The second term describes all of the other effects: piezoelectric coupling, overlap of the mechanical force with SAWs (in the zdirection), etc. A detailed consid- eration of this second term lies beyond the scope of thisarticle, and the influence of this second term is described below by a coefficient C 1. The coefficient C1can also be k dependent, but this dependence is much weaker than that ofthe term F kand therefore is neglected below. Thus, the excitation force, which appears on the right-hand side of the equation for the SAW amplitudes qkin the system equation (11), is expressed as feðtÞ¼C1VinFke−iωt. The solution of Eq. (11) with the excitation term gives the amplitudes of the excited SAWs qkin the form qk¼−iC1VinFkω−ðωSW−iΓSWÞ ðω−ω1Þðω−ω2Þ; ð18Þ where ω1;2¼ωSW−iΓSWþωSAW−iΓSAW 2 /C6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/C20ðωSW−iΓSWÞ−ðωSAW−iΓSAWÞ 2/C212 þjκkj2s ; ð19Þ and the obtained frequencies of the coupled waves are complex because damping is taken into account. At the receiving IDT, the displacement created by a SAW ξðxÞ¼ð 1=2πÞRξkqkeikxdkis transformed into the output voltage via the piezoelectric effect. Similar to the excitation efficiency, the efficiency of detection of a SAW having thewave vector kcan be decomposed into two terms and then, similar to the description of the efficiency at the input IDT, represented as C 2Fk. The total output microwave voltage is obtained by the integration over all of the SAW wave vectors, Vout¼ðC2=2πÞRFkqkeikLdk, where Lis the dis- tance between the input and output IDTs. Thus, the transmission parameters S12andS21, which are defined as ratios of the output voltage to the input one for the two opposite directions of the signal propagation (from port 1 to port 2, and vice versa) are equal to S12;21¼C1C2 2πiZω−ðωSW−iΓSWÞ ðω−ω1Þðω−ω2ÞF2 ke/C6ikLdk; ð20Þ where S12differs from S21by the sign in front of the length Lof the SAW line, and both IDTs are assumed to be the same. In a general case, this expression cannot be further simplified because the widths of the magnetoelastic band gaps, the SW damping rate, and the characteristic width ofthe function F kcan be on the same order of magnitude. InNONRECIPROCAL SURFACE ACOUSTIC WAVES IN … PHYS. REV. APPLIED 9,064014 (2018) 064014-7the limiting case of the absence of magnetoelastic coupling and a sufficiently wide spectrum of the IDT (i.e., in the case where the range of variation of the function Fkis much larger than ΓSAW=cSAW), Eq. (20) simplifies to the form S12;21¼ðC1C2=cSAWÞexp½−ΓSAWL=c SAW/C138. The calcula- tion of the coefficients C1andC2requires an accurate accounting of the piezoelectric coupling and impedance matching between the SAW line and the external circuit and lies beyond the scope of this article. Below, we use thenormalization C 1C2=cSAW¼1; i.e., we consider only the effects of the propagation losses of magnetoelastic waves and the spatial spectra of IDTs, given by Fk. As was pointed out above, the widths of the magne- toelastic band gaps Δω¼2jκkjare several orders of magnitude larger than the SAW linewidth, ΓSAW≪Δω, and, at the same time, the SW linewidth is typically larger than the width of the magnetoacoustic band gap,Γ SW>Δω. An example of the transmission characteristics calculated for this case is given in Fig. 4. For these calculations, we use the thickness of a polycrystalline Ni layer of tNi¼5nm, which is smaller than the optimum thickness, in order to demonstrate that a significantnonreciprocity of the transmission characteristic cannot be achieved only in the unique optimum case. We choose acentral frequency of the band gaps of 5 GHz, and the Gilbert damping parameter of the nickel layer is chosen to beα G¼0.045, which is a typical value for polycrystalline Ni films [51]. For these parameters, the spectral widths areΓSAW=ð2πÞ¼500kHz,ΓSW=ð2πÞ¼360MHz, and Δω¼2jκkj=ð2πÞ¼60MHz. The parameters of the SAW line transmission character- istics can be adjusted by the selection of the IDT centralfrequency f 0and the number of IDT fingers. For example, if the IDT central frequency f0lies between the magne- toelastic band gaps and the spectrum of the IDT is wideenough to cover both band gaps (a small number of fingers), then the transmission characteristic contains two nonreciprocal bands where the transmission reaches a maximum value at different frequencies for the opposite wave propagation directions [see Fig. 4(a)]. By contrast, if f 0lies within one of the band gaps and the IDT spectrum is narrow (a large number of fingers), there is one main unidirectional transmission band, as shown in Fig. 4(b). The isolation in both cases is close to 10 dB, while the propagation losses at the transmission maximum do not exceed 10 dB. We note that these values of the isolation are much larger than the ones that were observed for a single Ni film on a LiNbO 3substrate (without Pt) [32], and they can be easily measured and, possibly, used in applications. The isolation of the SAW line with magnetoelastic coupling can be substantially improved if a high-qualityferromagnetic film is used. As one can show from Eq. (19), the damping of the magnetoelastic waves at the crossing point depends on the relative values of the magnetoelasticcoupling coefficient jκ kjand the SW linewidth ΓSW.I f jκkj>ðΓSWþΓSAWÞ=2, the damping rate of the hybrid- ized waves is Γ1;2¼ðΓSW−ΓSAWÞ=2. Otherwise, for jκkj<ðΓSWþΓSAWÞ=2, the damping rate of the magneto- acoustic waves is equal to Γ1;2¼ðΓSWþΓSAWÞ=2/C6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðΓSW−ΓSAWÞ2=4−jκkj2p , i.e., the damping rate of one of the hybridized magnetoacoustic waves decreases with the increase of ΓSW, and, in the limit ΓSW≫jκkj,i ti s reduced to Γ1→ΓSAW. Naturally, this low-damping hybridized wave makes a dominant contribution to the signal transmission rate, and the isolation in the trans- mission line decreases in spite of the fact that the signal frequency lies within a magnetoacoustic band gap. Thus, to maximize the influence of the magnetoelastic coupling on the signal transmission, the SW damping rate should be ΓSW<2jκkj, while the use of ferromagnetic materials with high magnetic damping leaves SAW trans-mission almost unaffected. For the studied heterostructure LiNbO 3=Ni=Pt, such an optimum case can be realized if high-quality monocrystal-line Ni film is used. In our numerical example illustrated in Fig.5, we use a high-quality Ni film with a Gilbert damping constant of α G¼0.014[56]. Here, we choose a Ni layer(a) (b) FIG. 4. Transmission characteristics of the SAW lines with a polycrystalline Ni film ( αG¼0.045) having different IDTs for the opposite directions of wave propagation [ S12(the dashed line) andS21(the solid line)]. (a) IDT1 with central frequency f0¼5GHz, number of fingers Nf¼10. (b) IDT2 with f0¼5.16GHz, Nf¼20. The thickness of the Ni layer is tNi¼5nm, the SAW line length L¼250μm, and the bias magnetic field Be¼82mT is applied at the angle ϕ¼π=4to the line axis.ROMAN VERBA et al. PHYS. REV. APPLIED 9,064014 (2018) 064014-8thickness of tNi¼9nm, for which 2κk=ð2πÞ¼105MHz andΓSW¼104MHz. As is clear from Fig. 5, the isolation in this case is increased remarkably, up to 45 dB, and this isolation also exists in a rather wide frequency band. It follows from Eq. (20) that the maximum achievable isolation in such a SAW transmission line is on the order of S12−S21∼exp½ðΓmin−ΓSAWÞL=c SAW/C138, where Γmin¼ min Im ½ω1;2/C138is the smallest damping rate of the hybridized magnetoelastic waves. This maximum isolation is achieved at the frequency in the center of one of the magnetoelastic band gaps, provided that the excitation spectra of the used IDT is sufficiently narrow compared to the band-gap width.The desired IDT bandwidth can be achieved using an IDT with a sufficiently large number of “fingers ”N f. As was pointed out previously, if the magnetoelastic coupling is relatively weak, jκkj<ðΓSWþΓSAWÞ=2,w eg e t Γmin¼ðΓSWþΓSAWÞ=2−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðΓSW−ΓSAWÞ2=4−jκkj2p .I n this case, a weak damping of one of the hybridized magne- toelastic waves strongly limits the maximum achievableisolation, and an increase of the number N fof the IDT fingers does not lead to a significant improvement in the isolation. For example, for the parameters used in Fig. 4(b), the increase of Nffrom Nf¼20toNf¼100gives only 1 dB of the isolation enhancement. By contrast, in the optimum case of a strong coupling and a high-quality ferromagnetic layer, whenjκ kj>ðΓSWþΓSAWÞ=2, the maximum isolation can be very high, so the signal level in one direction can be below the level of a thermal noise. For example, it follows fromFig. 5that, at the frequency 5.09 GHz, the isolation is 84 dB and can be made even higher for the increased number N fof IDT fingers. A large number of fingers, however, leads to a severe limitation in the frequency band of the transmitted signal.Thus, we show here that, by using a well-known acoustic and magnetic material such as LiNbO 3in combination with a Ni film covered by a thin layer of Pt, it is possible toachieve the transmission of hybridized magnetoacoustic waves with quite a large level of nonreciprocal isolation. The characteristics of such a nonreciprocal magnetoacous-tic isolator could be further improved by using ferromag-netic materials with lower damping. Promising materials for this purpose could be CoFe alloys, which, at a certain composition, show ultralow magnetic damping of α G¼ 0.0014 [57]. Unfortunately, the magnetoelastic properties and the DMI of these alloys have not been studied yet. IV. CONCLUSION In this work, we present a general theory of linear magnetoelastic coupling between the spin and acoustic waves propagating in an arbitrary magnetic-nonmagneticlayered structure and having arbitrary mode profiles. The developed theory uses the relative weakness of the mag- netoelastic interaction and reduces the problem to astandard form of equations for coupled oscillators. The theory provides a simple method for the calculation of the magnetoelastic wave dispersion and the damping param-eters of the coupled waves, as well as for the determinationof the condition of the nonzero magnetoelastic interaction between the acoustic and spin waves. Using the developed theory, we demonstrate that the SW nonreciprocity induced by the IDMI can be “transferred ” to the “hybridized SAWs ”existing in piezoelectric – ferromagnetic –heavy-metal heterostructures. The magne- toelastic interaction results in the appearance of band gaps in the spectra of magnetoelastic surface waves, and,because of the IDMI-induced SW nonreciprocity, these band gaps exist at different frequency and wave-number positions for the opposite wave propagation directions. Thewidths of these band gaps and the frequency separation between them can be optimized by a proper selection of the in-plane magnetization angle ( ϕ≈π=4relative to the direc- tion of the SAW propagation) and the thickness of aferromagnetic layer (about 8 to 9 nm for the studied LiNbO 3=Ni=Pt heterostructure), while the central frequency of the band gaps can be tuned by varying the magnitude of thebias magnetic field. We demonstrate in this paper that the transmission characteristics of the surface magnetoelastic waves can be substantially nonreciprocal while having relatively low direct insertion losses. Our calculations show that, forLiNbO 3covered by a think Ni/Pt layer, it is possible to achieve an isolation of 10 –45 dB while maintaining the SAW propagation losses below 10 –20 dB. The isolation could be further improved by a selection of a proper ferromagnetic material having large values for the magnetostriction and the IDMI, but low magneticlosses.FIG. 5. Transmission characteristics of a SAW line with a monocrystalline Ni film ( αG¼0.014) for the opposite directions of wave propagation [ S12(the dashed line) and S21(the solid line)]. The thickness of the Ni layer is tNi¼9nm, the SAW line length L¼250μm, the IDT central frequency f0¼5.095GHz, the number of fingers Nf¼30, and the bias magnetic field Be¼ 45mT is applied at the angle ϕ¼π=4to the line axis.NONRECIPROCAL SURFACE ACOUSTIC WAVES IN … PHYS. REV. APPLIED 9,064014 (2018) 064014-9ACKNOWLEDGMENTS This work was supported in part by a grant from the Center for NanoFerroic Devices (CNFD) and theNanoelectronics Research Initiative (NRI), by Grants No. EFMA-1641989 and No. ECCS-1708982 from the U.S. NSF, and by the DARPA M3IC grant under ContractNo. W911-17-C-0031. R. V. acknowledges support fromthe Ministry of Education and Science of Ukraine (ProjectNo. 0118U004007). I. L. acknowledges support from DARPA SPAR (Grant No. HR0011-17-2-2005) and the Russian Science Foundation (Project No. 14-19-00760). [1] C. Campbell, Surface Acoustic Wave Devices and Their Signal Processing Applications (Academic Press, New York, 1989). [2] B. A. Auld, Acoustic Fields and Waves in Solids (Krieger Publishing, Malabar, FL, 1990). [3] D. Morgan, Surface Acoustic Wave Filters: With Applica- tions to Electronic Communications and Signal Processing , 2nd ed., Studies in Electrical and Electronic Engineering(Academic Press, London, 2010). [4] E. A. Ash, A. 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APPLIED 9,064014 (2018) 064014-11

PhysRevB.101.014433.pdf

PHYSICAL REVIEW B 101, 014433 (2020) Dynamics of domain-wall motion driven by spin-orbit torque in antiferromagnets Luis Sánchez-Tejerina,1Vito Puliafito,2Pedram Khalili Amiri,3Mario Carpentieri,1and Giovanni Finocchio4 1Dipartimento di Ingegneria Elettrica e dell’Informazione, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy 2Dipartimento di Ingegneria, Università di Messina, C.da Di Dio s/n, 98166 Messina, Italy 3Department of Electrical and Computer Engineering, Northwestern University, 633 Clark St, 60208 Evanston, Illinois, USA 4Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F . Stagno d’Alcontres 31, 98166 Messina, Italy (Received 28 March 2019; revised manuscript received 29 November 2019; published 22 January 2020) The excitation of ultrafast dynamics in antiferromagnetic materials is an appealing feature for the realization of spintronic devices. Several experiments have shown that static and dynamic behaviors of the antiferromagneticorder are strictly related to the stabilization of multidomain states and the manipulation of their domainwalls (DWs). Hence, a full micromagnetic framework should be used as a comprehensive theoretical toolfor a quantitative understanding of those experimental findings. This model is used to perform numericalexperiments to study the antiferromagnetic DW motion driven by the spin-orbit torque. We have derivedsimplified expressions for the DW width and velocity that exhibit a very good agreement with the numericalcalculations in a wide range of parameters. Additionally, we have found that a mechanism limiting the maximumapplicable current in an antiferromagnetic racetrack memory is the continuous domain nucleation from its edges,which is qualitatively different from what observed in the ferromagnetic case. DOI: 10.1103/PhysRevB.101.014433 I. INTRODUCTION The nucleation and manipulation of ferromagnetic (FM) domain walls (DWs) have attracted a lot of attention in recentyears due to the promising results for the development ofspintronic devices such as racetrack memories [ 1,2], mem- ristors [ 3–5], and sensors [ 6]. Nevertheless, the FM DW velocity, which is a key metric for evaluating the performanceof those devices, driven by an external field drops beyonda certain field threshold (Walker breakdown) [ 7], while it saturates when an electric current is used as a driving force[2–8]. Recent experiments in synthetic antiferromagnets have demonstrated that the DW velocity can be as large as 750 m /s [9] and does not saturate within the applicable current ranges [10]. Ferrimagnetic DWs can also reach high velocities at the angular momentum compensation point as well [ 11,12]. In addition, it has been predicted that the velocity of DWs inantiferromagnets (AFM) should reach tens of km /s and it is limited by the group velocity of spin waves [ 13–15]. Here, we will focus on this latter category of materials due to their intriguing properties (absence of stray fields and low magnetic susceptibilities) [ 13,15–19] and potential importance either from a technological point of view, design of high-speeddevices, and better scaling in storage devices, and from a fun-damental point of view to study the statics and the dynamics ofmultidomain states. Out of equilibrium, the antiferromagneticorder exhibits relaxation processes at ps time scale [ 20–22]. This THz dynamics makes those materials also appealing for the development of ultrafast spintronic devices [ 19]. In particular, on the path towards antiferromagnetic spintronics,AFM domains can play the same role as the FM ones beingthe information carriers. The writing process can be achievedemploying laser pulses [ 23] or spin-orbit-torques (SOT) [24,25], the manipulation by using alternating magnetic fields [13,23] or SOT, and the detection can be performed using one of the readout mechanism already observed experimentallysuch as tunneling anisotropic magnetoresistance, anisotropicmagnetoresistance, or spin Hall magnetoresistance [ 26–28]. From a numerical point of view, antiferromagnetic dy- namics can be described by atomistic micromagnetic models[29], or at mesoscopic scale by a continuous micromagnetic framework that has proven to be very powerful for its abil-ity to reproduce experimental observations in FM materials[30]. These models are based on the numerical solution of two Landau-Lifshitz-Gilbert (LLG) equations, each of themdescribing one of the two sublattices of the antiferromagnet,strongly coupled through the exchange interactions. However,in the continuous formulation derived from atomistic modelsthe exchange interactions are characterized by homogeneous,inter- and intralattice inhomogeneous terms at least [ 31]. Here, we perform an ideal numerical experiment to study therole of each of those exchange terms in the DW stability anddynamics [ 22]. In particular, we find that the homogeneous interlattice exchange does not affect the DW velocity and itsrole is limited to the stabilization of the antiferromagneticorder. On the other hand, the DW velocity as a function of ei-ther interlattice or intralattice inhomogeneous exchange fieldfollows a square root dependence. We have derived simplified expressions for both DW size and velocity exhibiting a good agreement with numerical calculations that can be used fora fast exploration of DW statics and dynamics in a largespace of material parameters. We also have found that themechanism limiting the maximum applicable current is thenucleation of domains from the edges originating by boundary 2469-9950/2020/101(1)/014433(10) 014433-1 ©2020 American Physical SocietyLUIS SÁNCHEZ-TEJERINA et al. PHYSICAL REVIEW B 101, 014433 (2020) FIG. 1. (a) A schematic of the device under investigation charac- terized by antiferromagnetic material / heavy metal bilayer, with the indication of the Cartesian coordinate reference system and the de-vice dimensions. The panel also includes the directions of the current density, J, and the spin polarization, p, and an example of the cubic discretization mesh (finite difference scheme) with a 2 nm side usedin this work to study the AFM. (b). Example of one computational cell with the magnetization vectors of the two sublattices m 1and m2. (c)–(e) Description of the three different exchange interactions included in this study, (c) inhomogeneous intralattice A11=A22,( d ) homogeneous interlattice A0acting on the same computational cell, and (e) inhomogeneous interlattice A12=A21both acting on the neighbors. Here, we consider the six neighbors for the computation of the inhomogeneous exchange terms indicated in (c) and (e). conditions imposed by the Dzyaloshinskii-Moriya interaction. In general, the continuous micromagnetic framework shouldbe used for the qualitative understanding of recent switchingexperiments on antiferromagnetic devices, tens of microns insize, involving multiple domain states and memristive behav-ior [3–5]. The paper is organized as follows. In Sec. IIthe micromagnetic framework is described. Section IIIdiscusses the steps to derive the one-dimensional formulation. Resultsand conclusions are discussed in Secs. IVandV, respectively. II. MICROMAGNETIC MODEL Device description. Figure 1shows a schematic of the system under investigation. It is a thin slab of an insulatingAFM with perpendicular anisotropy, having lateral dimen-sions of 400 ×200×2n m 3, on top of a heavy metal (HM) (e.g., Pt, Ta). A Cartesian coordinate system is introduced[see Fig. 1(a)] with the zaxis being the out-of-plane direction, while the xandyaxes are related to the length and the width of the device, respectively. The electric current is applied alongthexdirection and flows in the HM layer, because of the SOT [32–34], a spin density along the ydirection at the interface HM/AFM is accumulated. Model description. Within the micromagnetic approach the AFM order is described by means of the magnetizationsof two sublattices ( m 1andm2) strongly antiferromagnetic coupled by the exchange interaction. We consider a finitedifference discretization scheme (see Fig. 1) with the value ofm 1andm2reflecting the average magnetization of the spins within the same discretization cell. The AFM dynamicsdriven by the current can be described by the following LLG-Slonczewski equations [ 35–37]: dm 1 dt=−γ0m1×Heff,1+αm1×dm1 dt+τSH,1 (1) dm2 dt=−γ0m2×Heff,2+αm2×dm2 dt+τSH,2, where γ0is the gyromagnetic ratio and αis the Gilbert damping parameter, while τSH,i=−γ0HSHmi×(mi×p) (2) is the antidamping SOT mainly due to the spin-Hall effect originating from a current density Jflowing through the HM [32–34], with the amplitude given by HSH=¯hθSH 2etμ0MSJ.I nt h e last expression, ¯ h,θSH,e<0,t,μ0are the reduced Planck’s constant, the spin Hall angle, the electron charge, the AFMfilm thickness, and the vacuum permeability, respectively.The saturation magnetization is equal in both sublatticesM S1=MS2=MS.p=z×jis the direction of the spin Hall polarization (see Fig. 1),jbeing the unit vector of the cur- rent density direction. Additionally, Heff,1andHeff,2are the effective fields for the first and second sublattice, respectively[38]. They include the uniaxial anisotropy, the demagnetizing term, and the interfacial Dzyaloshinskii-Moriya interaction(iDMI) contribution [ 8,39]. The exchange energy density can be written as ε exch=A11(∇m1)2+A11(∇m2)2−4A0 a2m1m2 +A12(∇m1)(∇m2), (3) giving an exchange field with three contributions H1,exch=2A11 μ0Ms∇2m1+4A0 a2μ0Msm2+A12 μ0Ms∇2m2, (4) H2,exch=2A11 μ0Ms∇2m2+4A0 a2μ0Msm1+A12 μ0Ms∇2m1, where ais the lattice constant. In Eq. ( 3), the first term, A11> 0, is the inhomogeneous intralattice contribution [Fig. 1(c)], the second one, A0<0, is the homogeneous interlattice [Fig. 1(d)], and the third, A12<0, is the inhomogeneous inter- lattice contribution [Fig. 1(e)]. The demagnetizing field is cal- culated by solving the magnetostatic problem [ 40] for the total magnetization MS1+MS2where MSi=MSmi. Our scheme is based on a field-based approach, so we compute directlythe effective field rather than derive it from the energy density[41]. The antiferromagnetic material has been discretized into cubic cells with a side of 2 nm [Fig. 1(b)]. The following ma- terial parameters have been used [ 11,22,42]: lattice constant a=0.35 nm, saturation magnetization M S=0.4M A/m, uniaxial anisotropy constant Ku=64 kJ/m3,zbeing its easy axis, spin Hall angle θSH=0.044, Gilbert damping α=0.1, and gyromagnetic ratio γ0=0.221 Mm /As. The expressions for the iDMI field are HDMI,1=−2D μ0MS(uz(∇·m1)−∇m1,z), HDMI,2=−2D μ0MS(uz(∇·m2)−∇m2,z), where the iDMI pa- rameter D=0.11 mJ/m2anduzis the unit zvector. In or- der to investigate the role of exchange fields in statics and 014433-2DYNAMICS OF DOMAIN-WALL MOTION DRIVEN BY … PHYSICAL REVIEW B 101, 014433 (2020) dynamics, the exchange constants range from few pJ /mt of e w tens of pJ /m. Boundary conditions. At the edges, the iDMI boundary conditions [ 43], are determined by the fields HDMI,1S= D μ0MS(m1×(n×uz)) and HDMI,2S=D μ0MS(m2×(n×uz)) where nis the normal vector to the edge as in the case of the FM. In addition, it necessary to take into account thecontribution from the interlayer exchange field, therefore,the boundary conditions for the ith sublattice are given by the relation 2A 11∂nmi+A12mi×(∂nmj×mi)+Dmi×(n×uz)=0, (5) where nis the unit vector perpendicular to the edge [ 44]. At the right edge, we have m(Nx+1,cy,cz)i,x=m(Nx,cy,cz)i,x−ξ/parenleftbigg m(Nx,cy,cz)i,z −A12 2A11m(Nx,cy,cz)j,z/parenrightbigg , m(Nx+1,cy,cz)i,y=m(Nx,cy,cz)i,y, m(Nx+1,cy,cz)i,z=m(Nx,cy,cz)i,z+ξ/parenleftbigg m(Nx,cy,cz)i,x −A12 2A11m(Nx,cy,cz)j,x/parenrightbigg , (6) where the couple iand jcan be the following values, i,j= (1,2) or, i,j=(2,1) and ξ=D/Delta1x 2A11(1−(A12/2A11)2).Nxis the number of computational cells along the xdirection, /Delta1xthe lateral cell size, and cy,czrefer to an arbitrary cell along yand zdirections. Similar expressions are also valid for the other edges. For the special case A12=−2A11,t h eE q .( 6) becomes m(Nx+1,cy,cz)i,x=m(Nx,cy,cz)i,x−D 4A11m(Nx,cy,cz)i,z, m(Nx+1,cy,cz)i,y=m(Nx,cy,cz)i,y, (7) m(Nx+1,cy,cz)i,z=m(Nx,cy,cz)i,z+D 4A11m(Nx,cy,cz)i,x, which is essentially the boundary condition of the ferromag- netic case for an exchange two times larger. Domain wall stability. All the simulations were per- formed considering an antiferromagnetic Néel DW type as aground state that are stabilized by the iDMI. The equilibriumconfiguration has been computed by solving the equations { m1×heff,1=0 m2×heff,2=0with a residual of 10−9. Figure 2(a) shows a snapshot (the color is related to the out-of-plane component) of a typical ground state of the two sublattices. III. ANALYTICAL DERIVATION OF DOMAIN WALL VELOCITY AND WIDTH The derivation of the simplified expressions for the DW velocity and width is based on the one-dimensional approx-imation (only the spatial dependence along the xdirection is considered) and assuming that the magnetization profilecan be described by the Walker ansatz as reported in Eq. ( 8) [(see Fig. 2(c) for a comparison with the micromagnetic(a) 2 y1m or 2l (b)x yq -200 -100 0 100 2001D M x(c) position (nm)2m 2x,1= x 20=1mor 2ml1+ 11zm2zlor q 2 FIG. 2. (a) A snapshot of an antiferromagnetic Néel DW (the color is related to the out-of-plane component of the magnetization as indicated in the color bar), where its position, q, and its size, /Delta1, are also indicated. (b) Definition of the Néel order parameter l,t h e magnetization of the first sublattice m1and the magnetization of the second sublattice m2used in this work. /Phi11and/Phi12are the angles ofm1andm2vectors with respect to the x-axis. (c) An example o ft h eD Wp r o fi l e( zcomponent of the magnetization m1,z) for the first sublattice as computed from micromagnetic simulations (emptycircles) compared with the Walker ansatz (solid line) considering the parameters listed in Sec. II,a n df o r A 11=15 pJ/ma n d A12= 0p J/m.qand/Delta1are also indicated for comparison with (a). profile]. θ(t,x)=2a r c t a n/parenleftbigg exp/parenleftbigg Qx−q(t) /Delta1(t)/parenrightbigg/parenrightbigg , (8) /Phi1i(t,x)=/Phi1i(t), where q,/Delta1, and /Phi11,/Phi12are the DW position, width, and sublattice internal angles, as defined in Figs. 2(a) and2(b). Q=±1 allows distinguishing between an up-down transition (Q=1) or a down-up transition ( Q=−1). Simplified model. Analytical models for the description of DW dynamics in AFM have been already derived. See, forexample, the Appendix for the σmodel where the dynamics can be written in term of Néel order parameter l=m 1− m2and the small magnetization m=m1+m2[13,14,35,42]. Here, we develop a generalization of the previous modelswhere (1) the ldependence of the homogeneous exchange and themdependence of the anisotropy are taken into account, (2)A 12andA11are independent parameters, (3) /Phi11and/Phi12 are free to evolve independently, and (4) the DW width is a dynamical variable /Delta11(t)=/Delta12(t)=/Delta1(t). Within these hypotheses, Eq. ( 1) in spherical coordinates reads ˙θi=−1 Lsinθiδu δϕi−αsinθi˙ϕi+1 LhSHcosθisinϕi sinθi˙ϕi=1 Lδu δθi+α˙θi+1 LhSHcosϕi, (9) i=1,2 where L−1=γ0/μ0MS,hSH=¯hθSH 2etJ. It is possible to com- pute a surface energy density from the integral of the energydensity along xand taking the Walker ansatz of Eq. ( 8), by making the hypothesis q 1=q2=q,Q1=−Q2. Within this 014433-3LUIS SÁNCHEZ-TEJERINA et al. PHYSICAL REVIEW B 101, 014433 (2020) assumption, the surface energy density is σ=4A11 /Delta1−2A12 /Delta1/parenleftbigg2−cos(/Phi11−/Phi12) 3/parenrightbigg −2hexch/Delta1(cos(/Phi11−/Phi12)+1)+πQD(cos/Phi11−cos/Phi12) +4Ku/Delta1+μ0/Delta1MS2Nx(cos/Phi11+cos/Phi12)2+μ0/Delta1MS2Ny(sin/Phi11+sin/Phi12)2, (10) where hexch=4A0 a2andNk(k=x,y,z) are the demagnetizing factors associated with the DW corresponding to a prism having as size the strip width, the strip thickness, and 2 /Delta1. We wish to stress one more time that, differently from the σmodel, two different values for the intralattice and the interlattice inhomogeneous exchanges are considered. It is possible to link the surfaceenergy density with the dynamic variables of the system through the LLG-Slonczewski Eq. ( 9), giving the relations between the variational derivatives and the partial derivatives of the surface energy density with respect to q,/Delta1,/Phi1 1,and/Phi12. These relations lead to a set of differential equations describing the dynamics of the DW ˙q /Delta1=/bracketleftbig απ 2hSH(cos/Phi11−cos/Phi12)−π 2hD(sin/Phi11+sin/Phi12)+2(hexch/prime+hexch)s i n/Phi1d+Hd1+Hd2/bracketrightbig (1+α2)LT, ˙/Delta1 /Delta1=−12 π2αLT/bracketleftbigg 2Ku−2A11 /Delta12+hexch/prime(2−cos(/Phi11−/Phi12))−hexch(cos(/Phi11−/Phi12)+1)+Hd/prime/bracketrightbigg , αL˙/Phi11=Hd1+π 2hDsin/Phi11−(hexch/prime+hexch)s i n/Phi1d+L˙q /Delta1, αL˙/Phi12=Hd2−π 2hDsin/Phi12+(hexch/prime+hexch)s i n/Phi1d−L˙q /Delta1, (11) Hdi=−μ0M2 S/parenleftbigg (Nx−Ny)sin 2/Phi1i 2+(Nxsin/Phi1icos/Phi1j−Nysin/Phi1jcos/Phi1i)/parenrightbigg , Hd/prime=1 2μ0M2 S(Nx(cos/Phi11+cos/Phi12)2+Ny(sin/Phi11+sin/Phi12)2), i/negationslash=j,/Phi1 d=/Phi11−/Phi12,LT=2L, (12) hexch/prime=A12/3/Delta12andhD=πQD/2/Delta1. The values of the two in-plane angles /Phi11and/Phi12are given by a trade-off be- tween the torque exerted by the spin Hall effect (SHE), whichtends to align the in-plane magnetization for each sublatticealong the same direction, and the antiferromagnetic exchangeenergy that has a minimum for /Phi1 1=π+/Phi12. Additionally, once the values for /Phi11and/Phi12are reached, the DW width also acquires a stationary value /Delta1=/radicalBigg 2A11−A12/parenleftbig2−cos(/Phi11(J)−/Phi12(J)) 3/parenrightbig 2Ku−hexch(cos(/Phi11(J)−/Phi12(J))+1)+Hd/prime(J). (13) At these stationary conditions, taking the difference betweenthe two last equations of the system Eq. ( 11), the expression for the DW velocity reads ˙q=Q/Delta1(J)π 2hSH LTα(cos/Phi11(J)−cos/Phi12(J)) (14) and, differently from Eq. ( A6), the velocity depends on the stationary values of /Phi11,/Phi12, and /Delta1(all are a function of the applied current J). As compared to Eq. ( A6), a first qualitative difference is that a saturation velocity is expectedfor large currents due to the transformation from Néel to Bloch(/Phi1 1→±π/2 and /Phi12→±π/2 depending on the current sign) domain wall similar to what is found in the ferromag-netic counterpart. Additionally, a decrement on the velocitywith respect to the linear behavior is also expected due to thecontraction of the DW width (note that A 0<0s o−hexch>0) However, for the parameters used in this work, a deviationfrom the linear behavior below the 0 .2% is expected for a homogeneous interlattice exchange of A0=0.5p J/m and a current density J=1T A/m2while is still below the 15% for a current density of J=10 TA/m2. Higher homogeneous interlattice exchanges would fit better with the linear behavior,and therefore these discrepancies are not easily observedexperimentally. The expression for the static DW width /Delta1is also derived from Eq. ( 13) taking into account that at equilibrium, no current is applied and /Phi1 1=0,π, /Phi1 2=π,0 so that the DW width reads /Delta1e=/radicalBigg 2A11−A12 2Ku. (15) This formula is a generalization of the expression for the DWs in FM [ 45] and it is a key result of this work. At equilibrium there are no misalignments between the two sublattices and,consequently, the static /Delta1 edoes not depend on the homoge- neous exchange. On the other hand, the two inhomogeneousexchange terms and the anisotropy have a key role in thedetermination of the DW size. The DW width /Delta1 ealso does not depend on the iDMI parameter being its energy equal toσ iDMI=πQD(cos/Phi11−cos/Phi12). Full numerical simulations confirm this finding, showing that the /Delta1change is less than 1.5% while changing the iDMI parameter from 0 .1m J/m2to 0.5m J/m2. 014433-4DYNAMICS OF DOMAIN-WALL MOTION DRIVEN BY … PHYSICAL REVIEW B 101, 014433 (2020) FIG. 3. DW width /Delta1as a function of (a) the intralattice in- homogeneous exchange, A11, for three values of the interlattice inhomogeneous exchange ( A12=0,−5,−10 pJ/m), and (b) the in- terlattice inhomogeneous exchange, A12, for four different values of the inhomogeneous intralattice exchange ( A11=2,6,10,15 pJ/m). In both figures, the symbols stand for micromagnetic simulations and the solid lines are computed with Eq. ( 15).IV . RESULTS AND DISCUSSIONS Static properties. First of all, we have studied the static properties of the DW width by comparing calculations frommicromagnetic simulations with Eq. ( 15). The results show a good agreement in a wide range of parameters. As anexample, Figs. 3(a) and3(b) summarize some of those com- parisons. The square root dependence emerges when oneof the two inhomogeneous exchange terms is zero [blackline in Fig. 3(a)]. The good agreement between numerical calculations and Eq. ( 15) confirms the lack of misalignments between both sublattices. Larger values of the inhomogeneousexchange contributions increase the DW width and, therefore,the minimum domain size. Dynamic properties: DW velocity. Figure 4compares the DW velocity ˙ qcomputed by numerically solving Eq. ( 11) with the one obtained from full micromagnetic simulations for awide range of the exchange parameters (see figure caption).The agreement between micromagnetic simulations and one-dimensional calculations is very good with slight differencesat very high current density J>=7T A/m 2. At such a high current density the domains themselves acquire a non-negligible in-plane component as can be seen inFigs. 5(a) and5(b),s oE q .( 8) is no longer valid. Differently from the FM case, here the linear behavior of the DW velocityis maintained at larger currents due to the stabilization role forthe Néel configuration of the homogeneous exchange, anal-ogously to the Ruderman-Kittel-Kasuya-Yosida interactionin the case of synthetic antiferromagnets [ 10]. Even though the proposed model allows for misalignments of the in-planecomponents of the two sublattice magnetizations, no signif-icant misalignments are observed for realistic parameters.Nevertheless, it is possible to get the condition for which this A11= 2 pJ/m A11= 6 pJ/m A11= 15 pJ/m FIG. 4. DW velocity ˙ qas a function of the applied current. In all the panels we have used three values of the inhomogeneous interlattice exchange A12=0,−5,−10 pJ/m. For the inhomogeneous intralattice exchange, the values are A11=2p J/m for (a) and (b), A11=6p J/m for (c) and (d), and A11=15 pJ/m for (e) and (f). The homogeneous interlattice exchange is A0=−2p J/m for (a), (c), and (e) and A0=−15 pJ/m for (b), (d), and (f). Lines are calculated by numerically solving Eq. ( 11) while dots are from full micromagnetic simulations. 014433-5LUIS SÁNCHEZ-TEJERINA et al. PHYSICAL REVIEW B 101, 014433 (2020) FIG. 5. Snapshots of the first sublattice magnetization from µM for (a) equilibrium state and (b) under a high current density(7 TA/m 2). In the latter, both domains acquire a non-negligible in- plane component affecting the reliability of the simplified models. behavior is kept. To do that, we set the stationary conditions, ˙/Phi11=˙/Phi12=˙/Delta1//Delta1=0. The sum of the dynamic equations for the in-plane angles give us the condition sin /Phi11=sin/Phi12.I t can be checked that /Phi11=/Phi12is unstable so /Phi11=π−/Phi12. Because of this relation the third equation from Eq. ( 11) can be rewritten as tan/Phi11=πhSH 2α(hexch/prime+hexch) π 2hD/(hexch/prime+hexch)+2 cos/Phi11. (16) From Eq. ( 16), it is trivial to demonstrate that if hSH/lessmuch α|hexch/prime+hexch|/Phi11=0,π, /Phi1 2=π−/Phi11, the DW width remains the one at equilibrium [see Eq. ( 15)] and the linear behavior for the DW velocity is recovered [see Eq. ( 14)]. Dynamic properties: DW nucleation. We observe that it exists a maximum current density, Jth=8T A/m2forA11= 2pJ/m(I≈160 mA), that can be applied without leading to the nucleation of other domains at the edges. The Supplemen-tal Material (Movie 1) [ 46] shows these dynamics achieved for J=9T A/m 2(approximately I≈180 mA). This DW nucle- ation from the edge driven by the current, already observed inFM [ 43], is determined by the iDMI boundary conditions (in fact simulations without those boundary conditions show noDW nucleation, see Supplemental Material (Movie 2) [ 46]). In AFM, this mechanism is more efficient due to the stabilizationof the xcomponent of the magnetization. In other words, in FM the magnetization at the edge rotates towards the y direction reducing the SOT, but in AFM this rotation doesnot take place because of the antiferromagnetic exchange.A systematic study of the domain nucleation at the edge asa function of the iDMI parameter, D, for different intralat- tice and interlattice inhomogeneous exchange interactions,A 11and A12, respectively, is summarized in Fig. 6.T h e x component of the sublattice magnetization in absolute value(the same for both) is displayed in Fig. 6(a). It increases as a function of Dand, on the other hand, decreases as a function of A 11and|A12|. This tilting originates a torque at the edge (roughly proportional to the tilting angle) promotingthe nucleation of the domain with the opposite sign of z.As a consequence, the minimum threshold current density J THA12=-5 pJ/mA12=-10 pJ/m A12=0 pJ/m FIG. 6. (a), (c) and (e) xcomponent of the sublattice magnetiza- tion, responsible of the nucleation of domains at the edge. (b), (d), and (f) threshold current density needed to nucleate new domains asa function of the iDMI parameter, D, for different inhomogeneous intralattice parameters, A 11. Bottom panels stand for a inhomoge- neous interlattice parameters A12=0p J/m, middle panels for A12= −5p J/m, and top panels for A12=−10 pJ/m. for domain nucleation decreases as a function of D[see data summarized in Fig. 6(b)]. As the current increases, the DWs also acquire a slight curvature (see Supplemental Material (Movie 1) [ 46]) due to the smaller torques at the edges caused by the reduction ofthexcomponent and the increase of the ycomponent of the magnetization. We conclude that, in antiferromagnetic racetrack memo- ries, the domain nucleation from the edges is the mechanismlimiting the maximum velocity of an AFM DW, at least with-out changing the numbers of DWs and hence the informationcontent of the racetrack itself. Dynamic properties: Role of exchange contributions. In this section, we show the results of a systematic study ofthe DW velocity as a function of the different exchangeinteractions. Figures 7(a)–7(c) summarize a comparison (full micromagnetic and the generalized simplified model) for acurrent density J=1T A/m 2, and a good agreement is ob- served in a wide range of parameters. The solid lines are fromthe generalized simplified model calculations while the dotsindicate the full numerical computations. A main result is thatthe DW velocity is insensitive to the homogeneous exchangeat low currents [Fig. 7(a)], provided it is large enough to avoid misalignments between the magnetization of the two sublat-tices. On the other hand, the DW velocity is a square rootfunction of both inhomogeneous terms, trend originated bythe proportionality with the DW width /Delta1[yellow dashed lines in Figs. 7(b) and7(c)], see Eq. ( 14). This demonstrates the in- homogeneous terms modify the DW width parameter withoutchanging the DW structure in the stationary state determinedby the two in-plane angles /Phi1 1and/Phi12. Since the DW velocity 014433-6DYNAMICS OF DOMAIN-WALL MOTION DRIVEN BY … PHYSICAL REVIEW B 101, 014433 (2020) -15 -12 -9 -6 -3 0 30.20.40.60.8 A11=10 pJ/m 1D MA11=6 pJ/m 1D MA11=2 pJ/m 1D M A11=15 pJ/m 1D Mq (km/s) A0 (pJ/m) 369 1 2 1 50.20.40.60.8q (km/s) A11 (pJ/m)A0=-2 pJ/m 1D M A0=-6 pJ/m 1D M A0=-10 pJ/m 1D M sqrt fittingA0=-15 pJ/m 1D M -12 -10 -8 -6 -4 -2 00.00.20.40.60.8 A11=2 pJ/m 1D M A11=6 pJ/m 1D M A11=10 pJ/m 1D M A11=15 pJ/m 1D M sqrt fittingq (km/s) A12 (pJ/m)(a) (b) (c) FIG. 7. DW velocity computed for a current density J= 1T A/m2as a function of the exchange interactions, (a) depen- dence on the homogeneous interlattice coefficient for different values of A11=2,6,10,15 pJ/m, (b) dependence on the inho- mogeneous intralattice coefficient for different values of A0= −2,−6,−10,−15 pJ/m,and (c) dependence on the inhomo- geneous interlattice coefficient for different values of A11= 2,6,10,15 pJ/m.is proportional to the DW width [see Eq. ( 14)], the induced increase of the DW width leads to a larger DW velocity. Forlarger currents, the DW velocity is expected to depend on thehomogeneous exchange by the DW width dependence on thiscontribution [see Eq. ( 13), which predicts a decreasing trend when the two sublattice magnetizations align]. Nevertheless,we are well below the condition h SH/lessmuchα(hexch/prime+hexch) and we only observe a small dependence for the largest currentsconsidered in this work, J th=7T A/m2(not show here). Dynamic properties. Interlattice damping. Finally, we have considered the role of an interlattice damping parameter α12 [47], which enters in the LLG of Eq. ( 1) in the following way: ˙m1=−m1×heff,1+τ1+α11m1×˙m1+α12m1×˙m2 ˙m2=−m2×heff,2+τ2+α22m2×˙m2+α21m2×˙m1, (17) where the intralattice damping parameter has been renamedfor a clearer description ( α=α 11=α22). In spherical coordi- nates, Eq. ( 17) becomes ˙θi=−1 Lsinθiδu δϕi−α11sinθi˙ϕi −α12cos(ϕi+ϕj)s i nθi˙ϕj+1 LhSHcosθisinϕi sinθi˙ϕi=1 Lδu δθi+α11˙θi+1 LhSHcosϕi −α12(sin2θicosϕi˙ϕj−cos2θisinϕi˙θj)s i nϕj.(18) Making the same assumption as in the previous case (Walker ansatz, q1=q2=q, and /Delta11=/Delta12=/Delta1). Equation ( 11)i s now α11˙q /Delta1+(1+α12sin/Phi12cos/Phi11)˙/Phi11 2−(1+α12sin/Phi12cos/Phi11)˙/Phi12 2=π 2hSH LT(cos/Phi11−cos/Phi12) −LT/parenleftbigg/parenleftbiggπ2 6α11−α12 2sin/Phi11sin/Phi12/parenrightbigg˙/Delta1 /Delta1+2 3α12(sin/Phi12cos/Phi11˙/Phi12+sin/Phi11cos/Phi12˙/Phi11)/parenrightbigg =4Ku−4A11 /Delta12+2A12 /Delta12/parenleftbigg2−cos(/Phi11−/Phi12) 3/parenrightbigg −2hexch(cos(/Phi11−/Phi12)+1)+2Hd/prime, LT/Delta1/bracketleftbigg Q˙q /Delta1−α11˙/Phi11−α12cos(/Phi11+/Phi12)˙/Phi12/bracketrightbigg =−QπDsin/Phi11+2hexch/Delta1sin/Phi1d, LT/Delta1/bracketleftbigg −Q˙q /Delta1−α11˙/Phi12−α12cos(/Phi11+/Phi12)˙/Phi11/bracketrightbigg =QπDsin/Phi12−2hexch/Delta1sin/Phi1d. (19) Because the considerations made previously to derive Eq. ( 16) are still valid, we can consider that /Delta1is constant and equal tothe equilibrium value for low currents. Then we can omit the second equation. A fast exploration of the system including 014433-7LUIS SÁNCHEZ-TEJERINA et al. PHYSICAL REVIEW B 101, 014433 (2020) the interlattice damping parameter (not shown here), with α12=0.01,0.05,0.09,0.0999, shows no changes on the stationary values. Moreover, we observe that the interlatticedamping only affects the terms that are zero at the stationaryregime, so no changes are expected for the stationary DW ve-locity. Nevertheless, we observe small changes in the transientregime even for low currents so these new terms could beimportant when considering other conditions, such as AFMoscillators [ 18,21,22]. V . CONCLUSIONS Velocities up to a few km/s for antiferromagnetic domain walls have been predicted making antiferromagnets a testbedmaterial for the development of ultrafast racetrack memoriesand THz spintronic devices. Here, we have extended theresults of previous works on this topic, by deriving a gener-alized expression for DW width and velocity that has beenbenchmarked with continuous micromagnetic simulations ina wide range of parameters. A systematic study of the role ofdifferent exchange interactions shows a DW velocity indepen-dent of the homogeneous interlattice exchange at low currents,and with a square root dependence on both inhomogeneousexchanges, i.e., intralattice and interlattice. This dependenceis inherited from the behavior of the DW width, which ispredicted to decrease at high currents due to the homogeneousinterlattice exchange. Finally, we show that the domain wallvelocity in an antiferromagnetic racetrack memory will belimited by the nucleation of new domains at the edges ofthe system, due to the iDMI boundary conditions which, forexample in racetrack memories, can change the content ofstored information. Therefore, it should be noticed that evena small iDMI parameter is needed to promote the Néel typewall, and large Dvalues are undesirable as they would lead to lower threshold currents. On the contrary, high inhomo-geneous exchange interaction would increase the thresholdcurrent, and also the DW width, promoting higher DW veloc-ities. Nevertheless, the larger DW width would increase theminimum domain size and then decrease the storage density.The analytical approach employed here can be used as astarting point for the development of a one-dimensional modelfor the description of DW motion in ferrimagnets. ACKNOWLEDGMENTS G.F. and M.C. would like to acknowledge the con- tribution of the COST Action CA17123 “Ultrafast opto-magneto-electronics for nondissipative information technol-ogy”. P.K.A. acknowledges support by a grant from the USNational Science Foundation, Division of Electrical, Com-munications and Cyber Systems (No. NSF ECCS-1853879).This work was also supported by PETASPIN. The authorsalso acknowledge N. Kioussis for the discussion regarding theinterlattice damping. APPENDIX: SIGMA MODEL The DW width /Delta1, the DW position q, and the in-plane angle of the magnetization of the Néel order parameter, /Phi1 for the σmodel are defined in Figs. 2(a) and2(b). Equation(1) can be rewritten in terms of the Néel order parameter l=m1−m2and the small magnetization m=m1+m2as [13,35,42] ˙m=−γ0(m×Hm+l×Hl)+α 2(m×˙m+l×˙l) −γ0HSH 2(m×(m×p)+l×(l×p)), (A1a) ˙l=−γ0(l×Hm+m×Hl)+α 2(l×˙m+m×˙l) −γ0HSH 2(l×(m×p)+m×(l×p)), (A1b) where the dot convention for the time derivative has been adopted and HmandHlare the effective fields with respect tomandl. We start with a simplified formulation, where the energy density uhas the following expression: u=A11(∇l)2−A0 a2m2−Ku 2l2 z+D 2(mz(∇·m)−(m·∇)mz +lz(∇·l)−(l·∇)lz). (A2) The expression derived in Ref. [ 38] is obtained neglecting the ldependence of the homogeneous exchange and mdepen- dence of the uniaxial anisotropy has been neglected, assumingthat−A 12=2A11.F r o mE q .( A1b) it is possible to determine mas a function of lconsidering the anisotropy term and the spatial derivatives are much smaller than the other terms [ 38], neglecting dissipative terms [ 35], and by taking into account thatl×(m×l)=ml2≈4m. Inserting this expression in Eq. ( A1a), the dynamics of ldoes not depend on m. Thus, Eq. ( A1a) in spherical coordinates for lreads ¨θ−c2θ/prime/prime+sinθcosθ(c2ϕ/prime/prime−˙ϕ2)+b2sinθcosθ +d2sin2θsinϕϕ/prime =+2γ0αHexch˙θ+2γ2 0HSHHexchcosϕ, d dt(sin2θ˙ϕ)−c2d dx(sin2θϕ/prime)−d2sin2θsinϕθ/prime =−2γ0αHexchsin2θ˙ϕ+2γ2 0HSHHexchsinθcosθsinϕ, (A3) with c2=−(2γ0 μ0Ms)22A0 a22A11,b2=−(2γ0 μ0Ms)22A0 a2Ku,d2= −(2γ0 μ0Ms)22A0 a2D,Hexch=2A0 a2μ0Msand/primestanding for the xpartial derivative. At equilibrium, one exact solution of the system of equations ( A3) is the Walker ansatz [ 48], which describes an approximate DW profile: θ(t,x)=2a r c t a n/parenleftbigg exp/parenleftbigg Qx−q(t) /Delta1/parenrightbigg/parenrightbigg , (A4) /Phi1(t,x)=/Phi1(t), where q, and/Delta1are again the DW position and width and now /Phi1stands for the in-plane angle of the Néel order parameter, as have been defined in Figs. 2(a) and2(b).Q=±1a l l o w s distinguishing between an up-down transition ( Q=1) or a down-up transition ( Q=−1). Under the hypothesis that 014433-8DYNAMICS OF DOMAIN-WALL MOTION DRIVEN BY … PHYSICAL REVIEW B 101, 014433 (2020) Eq. ( A4) is still valid for moving DWs, it is possible to derive a couple of equations for qand/Phi1which, at stationary conditions ( ¨ q=¨/Phi1=˙/Phi1=0), transform into ˙q=Qπ 2γ0/Delta1HSH αcos/Phi1 HDsin/Phi1=0⇒/Phi1=0,π (A5)where HD=D/μ0MS. 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PhysRevB.98.184424.pdf

PHYSICAL REVIEW B 98, 184424 (2018) Theory of chiral effects in magnetic textures with spin-orbit coupling C. A. Akosa,1,2,*A. Takeuchi,3Z. Yuan,4and G. Tatara1,5 1RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan 2Department of Theoretical and Applied Physics, African University of Science and Technology (AUST), Km 10 Airport Road, Galadimawa, Abuja F .C.T, Nigeria 3Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 252-5258, Japan 4The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 Beijing, China 5RIKEN Cluster for Pioneering Research (CPR), 2-1 Hirosawa, Wako, Saitama, 351-0198 Japan (Received 19 September 2018; revised manuscript received 29 October 2018; published 21 November 2018) We present a theoretical study of two-dimensional spatially and temporally varying magnetic textures in the presence of spin-orbit coupling (SOC) of both the Rashba and Dresselhaus types. We show that the effectivegauge field due to these SOCs, contributes to the dissipative and reactive spin torques in exactly the same wayas in electromagnetism. Our calculations reveal that Rashba (Dresselhaus) SOC induces a chiral dissipationin interfacial (bulk) inversion asymmetric magnetic materials. Furthermore, we show that in addition to chiraldissipation α c, these SOCs also produce a chiral renormalization of the gyromagnetic ratio ˜ γc,a n ds h o wt h a t the latter is intrinsically linked to the former via a simple relation αc=(τ/τ ex)˜γc,w h e r e τexandτare the exchange time and the electron relaxation time, respectively. Finally, we propose a theoretical scheme basedon the Scattering theory to calculate and investigate the properties of damping in chiral magnets. Our findingsshould in principle provide a guide for material engineering of effects related to chiral dynamics in magnetictextures with SOC. DOI: 10.1103/PhysRevB.98.184424 I. INTRODUCTION The recent years have witnessed a surge in research in nanoscale magneto-electronics that focuses on the utilizationof the spin degree of freedom of electrons in combination withits charge, to create new functionalities and devices such asmagnetic random access memories, hard drives, and sensors[1,2]. The performance of these devices strongly depends on the dissipation of magnetization dynamics. The latter detectsthe energy required, the speed and efficiency at which thesedevices operate. As a result, the qualitative estimation ofdamping in magnetic materials is, in principle, indispensablefor piloting and designing alternative materials for differentspintronics applications. Over the past years, several microscopic theories of mag- netization dissipation in which SOC is the mediating interac-tion via which angular momentum (and energy) is dissipatedby the precessing magnetization [ 3–5] have been proposed. Recent theories have highlighted the important role that thes-dinteraction between the local magnetization and the spins of itinerant electrons play in the dynamics of magnetization[6]. Indeed, it has been shown that the interaction between a nonuniform precessing magnetization and spins of itinerantelectrons give rise to a nonlocal contribution to the Gilbertdamping [ 7,8]. A class of magnetic materials that have attracted enormous research interest owing to their offer of enhanced device per-formances such as low threshold current density and ultrafast *collins.akosa@riken.jpcurrent-induced domain wall motion [ 9] are chiral magnets common in materials with SOC and broken inversion symme-try. It was recently pointed out that magnetization dampingin these materials include a chiral contribution [ 10–14]. Even though this prediction is appealing towards the realizationof ultralow damping, little information is known about therelative strength of the chiral with respect to the nonchiralcontributions to the damping. Furthermore, the nature ofthe SOC in chiral magnets determines the type of magnetictexture that can be stabilized in the system. Indeed, it has beenshown that an effective chiral energy, i.e., Dzyaloshinskii-Moriya interaction can be derived from a microscopic modelof electrons moving in a magnetic texture in the presence ofSOC [ 15,16]. This chiral energy has been shown to stabilize Néel (Bloch) domain walls in systems with Rashba (Dressel-haus) SOC as a result of interfacial (bulk) inversion symmetrybreaking. In this study, we present a theoretical study of an interplay of Rashba and Dresselhaus SOCs in two-dimensional chiralmagnets with spatially and temporally varying magnetization.We propose schemes based on the Green’s function formalismand the scattering theory to qualitatively calculate the chiraldamping and chiral renormalization of the gyromagnetic ratioinherent in these materials. We show that just as in the casefor chiral energy, these SOCs induce a chiral damping ( α c) and chiral renormalization of the gyromagnetic ratio ( ˜ γc) that are intrinsically linked via αc=(τ/τ ex)˜γc, where τexand τare the exchange time and the electron relaxation time, respectively. Finally, we elucidate the nature and propertiesof both the chiral and nonchiral contributions to damping inthese materials. 2469-9950/2018/98(18)/184424(10) 184424-1 ©2018 American Physical SocietyC. A. AKOSA, A. TAKEUCHI, Z. YUAN, AND G. TATARA PHYSICAL REVIEW B 98, 184424 (2018) This work is organized as follows. In Sec. II, we introduce the theoretical model based on the Green’s function formalismemployed to calculate the spin torque induced by a spatiallyand temporally varying magnetization in the presence ofRashba SOC and Dresselhaus SOC. In Sec. III, we study the corresponding current-induced dynamics in the presenceof the torques calculated in the preceding section to obtainanalytical expressions and estimates of the chiral damping andchiral renormalization of the gyromagnetic ratio. In Sec. IV, we provide a scheme based on the Scattering theory to calcu-late the chiral damping contribution. This scheme is appliedin Sec. Vvia a tight-binding model to numerically compute and investigate the properties of chiral damping and thusverify our theoretical model. Finally, in Sec. VI, we provide a summary of the main results in this work. II. THEORETICAL MODEL In this section, we outline the theoretical framework em- ployed to calculate the spin torque induced by a spatiallyand temporally varying magnetization in the presence of bothRashba SOC and Dresselhaus SOC. The calculated torquesare classified into dissipative orreactive based on whether they are oddoreven under time reversal symmetry. The dissi- pative torques contribute to a damping that is proportional tothe first-order derivative of the magnetization and hence chiral by nature [ 10–14]. The reactive torques contribute to the renormalization of the gyromagnetic ratio, which is also chiral [13,14]. Our considerations are based on a two-dimensional inversion asymmetric magnet with spatially and temporarilyvarying magnetization m(r,t) described by the Hamiltonian ˆH=ˆp 2 2m+Jm(r,t)·ˆσ+ˆHso, (1) where mis the mass of electron, ˆpis the momentum operator, Jis the s-dexchange coupling between the local moment m and the electrons with spin represented by the vector of Paulimatrices ˆσ. The third term on the right-hand side of Eq. ( 1) represents an interplay of Rashba SOC due to interfacialinversion symmetry breaking [ 17] and Dresselhaus SOC due to bulk inversion symmetry breaking [ 18] given as ˆH R=βR ¯h(ˆσyˆpx−ˆσxˆpy) (2a) and ˆHD=βD ¯h(ˆσxˆpx−ˆσyˆpy), (2b) of strength βRandβD, respectively. In the case of mag- netic textures, the exchange term in Eq. ( 1) includes off- diagonal terms. This term is diagonalized via a local unitarytransformation in the spin space ˆU(r,t)=n(r,t)·ˆσ, i.e., ˆU †(m·σ)ˆU=σz, where n=(cosφsinθ 2,sinφsinθ 2,cosθ 2) [19,20]. In the transformed space [rotating frame with the spin quantization axis along m(r,t)], the electrons see a background of a uniform ferromagnetic state that is coupled to the corresponding spin gauge fields due to (i) the textureAμ sand (ii) the SOC Aμ so, given as Aμ s=−i¯h 2eTr[ ˆσμˆU†∇ˆU]=¯h e(m×∇m)μ(3a)and Aμ so=m e¯hλν soRμν, (3b) respectively, where λμ sois given as ⎛ ⎜⎝λx so,xλy so,xλz so,x λx so,yλy so,yλz so,y λx so,zλy so,zλz so,z⎞ ⎟⎠=⎛ ⎜⎝βD βR 0 −βR−βD0 00 0⎞ ⎟⎠ (4) andRμνare components of the rotation matrix given by Rμν=2nμnν−δμν. (5) Furthermore, this unitary transformation modifies spin- dependent observables such as the spin torque and thenonequilibrium spin density of itinerant electrons. In partic-ular, the nonequilibrium spin density in the transformed ( ˜s) and original ( s) frames transforms as s μ=Rμν˜sν. (6) We recall that the presence of nonequilibrium spin density s regardless of its source in a magnetic system, exerts a torqueTon the local magnetization mgiven as T=Ja 2 0 ¯hm×s, (7) where a0is the lattice constant and ¯ his the reduced Planck’s constant. Therefore, to calculate the spin torque on the localmagnetization, it suffices to calculate ˜s. In this study, we focus on the time-varying magnetization as the primary sourceof˜s. We treat the interaction between the spin gauge fields A μ=Aμ so+Aμ sand the background conduction electrons in the transformed frame to be weak; this allows us to apply theperturbation theory to calculate ˜s. In particular, we consider the adiabatic limit in which the spins of electrons follow thedirection of the local magnetization, and calculate ˜svia the Green’s function approach [ 21], in which the spin gauge fields A μare treated perturbatively (see Appendix Afor details). Since this work focuses on chiral effects, we consider onlyup to first order in the spin gauge fields due to SOC. Therelevant contributions to the spin torque induced by the time-dependent texture is calculated using Eq. ( 7) as (see Appendix Afor details) T=C 1(∂tA|| so·∇)m+C2m×(∂tA|| so·∇)m +C3[(m×∂tm)·(A⊥,μ so·∇)m]eμ +C4/bracketleftbig (m×∂tm)·/parenleftbig λμ so·∇/parenrightbig m/bracketrightbig eμ+Tnl, (8) where the in-plane andout-of-plane components of the SOC- induced spin gauge fields are given as A|| so=λμ somμandA⊥,μ so=/epsilon1μνcmνλc so. (9) The last term on the right-hand side of Eq. ( 8) represents other contributions to the torque given as Tnl=(C5+C6m×)(A|| so·∇)∂tm. (10) In domain walls, even though Tnlis locally finite, it vanishes upon the integration over space. The torque prefactors in 184424-2THEORY OF CHIRAL EFFECTS IN MAGNETIC TEXTURES … PHYSICAL REVIEW B 98, 184424 (2018) Eqs. ( 8) and ( 10)a r eg i v e na s C1=−1 4πma2 0 ¯h2εFJ2(J2−η2) η(J2+η2)2, (11a) C2=1 2πma2 0 ¯h2εFJ3 (J2+η2)2, (11b) C3=−1 4πma2 0 ¯h2εFJ2(J2+3η2) η(J2+η2)2, (11c) C4=−1 2πma2 0 ¯h2εFJη2 (J2+η2)2, (11d) C5=−1 πma2 0 ¯h2εFJ2η (J2+η2)2, (11e) C6=−1 2πma2 0 ¯h2εFJ(J2−η2) (J2+η2)2, (11f) where η=¯h/2τ,τbeing the elastic relaxation time of con- duction electrons. Notice from Eq. ( 11) that C1andC3/greatermuchC2 andC4and thus dissipative torque effects are dominant over the reactive torque effects in chiral domain walls. Observe that Eq. ( 8) includes torque terms that are both dissipative ( ∝C1andC3) and reactive ( ∝C2andC4) based on their symmetry under time reversal. Interestingly, Eq. ( 8), which constitutes one of the main result of this study, showsthat in the presence of relaxation [ 22], the first two terms of the torque takes the same form of the adiabatic ( ∝C 1) and the nonadiabatic ( ∝C2) spin transfer torque proportional to (E·∇)mandm×(E·∇)m, respectively [ 6,23], where Eis the applied electric field expressed in terms of the electromagnetic vector potential A(i.e.,E=−∂tA). In fact, our result indicates that the effective gauge field of anyorigin contributes to the torque in exactly the same way as theelectromagnetic gauge field. Even though this is as expectedfrom symmetry point of view, what is significant is that thespin transfer torque arising from the gauge field due to spin- orbit interaction, indeed, has a nature of a damping torque, asthe gauge field is linear in magnetization. III. CURRENT-INDUCED CHIRAL MAGNETIZATION DYNAMICS The previous section was devoted to establishing the nature of the spin torque that itinerant electrons exert on the localmagnetization as a result of a time-dependent backgroundmagnetization in the presence of SOC. In this section, weprovide analytic expressions and a qualitative estimate of thechiral contribution to both the damping and the gyromagneticratio. To achieve this, we investigate the influence on dynam-ics of chiral domain walls via the incorporation of Eq. ( 8)i n t o the equation of motion of the magnetization described by theextended Landau-Lifshiftz-Gilbert (LLG) equation ∂ tm=−γm×Heff+α0m×∂tm+T, (12) where for completeness we have included the phenomenolog- ical Gilbert damping with constant α0,γis the gyromagnetic ratio, Heff=−1 μ0Ms∂E ∂mis the effective field, Eis the energy density, Msis the saturation magnetization, and μ0is the permeability of free space. We consider a one-dimensionalWalker domain wall with magnetization parametrized by the domain wall center Xcand tilt angle φ, and given in spheri- cal coordinates as m=(cosφsinθ,sinφsinθ,cosθ), where θ(x)=2t a n−1(e x p(sx−Xc λdw)),s=+1(−1) for ↑↓(↓↑) do- main wall, φ=φ(t), and λdwis the width of the wall. The dynamics of the wall is given by coupled equations (1+s¯γc)∂tφ+α0s∂tXc λdw=−/Gamma1θ (13a) and (α0+sαc)∂tφ−s∂tXc λdw=/Gamma1φ, (13b) where /Gamma1θ(φ)=1 2γ/integraldisplay+∞ −∞Heff·eθ(φ)dx, (14) eθ=(cosφcosθ,sinφcosθ,−sinθ) and eφ=(−sinφ, cosφ,0). The terms αcand ¯γcin Eq. ( 13) represent the chiral damping and chiral renormalization of the gyromagnetic ratioand given as α c=πnFβso 4¯hλdwτ 1+τ2ex/τ2cos(φ+φso) (15a) and ¯γc=πnFβso 4¯hλdwτex 1+τ2ex/τ2cos(φ+φso), (15b) respectively, where nF=νa2 0εFis the number of conduction electrons at the Fermi level, βso=/radicalBig β2 R+β2 D (16a) and φso=tan−1(βD/βR) (16b) characterizes the strength of the effective SOC present in the material. Equations ( 15a) and ( 15b) constitute one of the main results of this work, from which we infer that (i) chiraldamping and chiral renormalization of the gyromagnetic ra-tio are Fermi-surface effects since they are proportional tothe number of available conduction electrons at the Fermilevel. (ii) The chiral damping constant is proportional to theelastic relaxation time of electrons (i.e. α c∝τ), which is well described by the SOC mediated breathing Fermi surface mechanism for magnetization relaxation [ 24–26]. It is worthy to note here that the source of electron relaxation can be fromscattering with impurities or the domain wall itself and henceτshould, in principle, depend on the domain wall width λ dw and therefore makes the dependence of the αcon the domain wall width a bit subtle. (iii) The chiral renormalization ofthe gyromagnetic ratio is inversely proportional to exchangestrength (i.e., ˜ γ c∝1/J) since τex=¯h/2Jand, therefore, is more significant in weak ferromagnets. (iv) The chiraldamping and gyromagnetic ratio renormalization are relatedvia α c=(τ/τ ex)˜γc. (17) 184424-3C. A. AKOSA, A. TAKEUCHI, Z. YUAN, AND G. TATARA PHYSICAL REVIEW B 98, 184424 (2018) This simple relation provides a means by which one effect can be deduced with the knowledge of the other. It turns out thatsimilar correspondence has been established by Kim et. al. [27], in the context of texture-induced intrinsic nonadiabatic- ity in the absence of SOC. For a realistic estimate of theseeffects, we consider typical material parameters such as β so= 2×10−11eV m, τ=1×10−14s,τex=1×10−15s,λdw= 10 nm, and nF=1, from which we obtain αc=3×10−2and γc=3×10−3. In general, for real ferromagnetic materials, τex/τ/lessmuch1, therefore from Eq. ( 17), it is expected that in chiral magnets, chiral damping constitutes the dominant mechanismthat detects the dynamics of chiral domain walls [ 11,12]. Now that we have established the analytical form of thedissipative torque given by Eq. ( 8), and the corresponding estimate of the chiral damping and chiral gyromagnetic ratiogiven by Eqs. ( 15a) and ( 15b), respectively, in what follows, we use the well established scattering theory of magnetizationdissipation based on the conservation of energy [ 28,30]t o compliment our analytical calculations and propose a schemeto numerically compute the damping in chiral magnets. IV . MAGNETIZATION DAMPING FROM THE SCATTERING THEORY In what follows, we compliment our analytical treatment of the preceding sections by providing a scheme based on thescattering theory of magnetization damping to calculate thenonchiral and chiral damping [and hence the chiral renormal-ization of the gyromagnetic ratio by virtue of Eq. ( 17)]. We focus on dissipative torque terms in Eq. ( 8) and neglect the chiral renormalization of the gyromagnetic ratio (i.e., torqueterms that are even under time reversal symmetry). However,notice that effects associated with the chiral renormalizationof the gyromagnetic ratio can be straightforwardly inferredfrom our calculations via Eq. ( 17), which establishes a simple relation between chiral damping and chiral gyromagnetic ratiorenormalization due to SOC. The dynamics of magnetizationis well described by the extended Landau-Lifshiftz-Gilbertequation given by ∂ tm=−γm×Heff+α0m×∂tm+Tdp, (18) Tdpis the dissipative contribution to the torque given in Eq. ( 8). Again, we consider a one-dimensional Walker domain wall parametrized by the domain wall center Xc=Xc(t) and tilt angle φ=φ(t). Furthermore, since the scattering theory of magnetization dissipation is based on the conservation ofenergy, we first calculate the rate of change of the magneticenergy density from Eq. ( 18)a s dE dt=−μ0Ms γ(α0∂tm+Tdp×m)·∂tm, (19) where the negative sign shows that energy is lost by the magnetic system. Notice that the right-hand side of Eq. ( 19) is bilinear in ∂tmand can therefore be rewritten in the form dE dt≡Do(∂tφ)2+Dm∂tφ∂tXc+Di(∂tXc)2, (20) where Do,Di, and Dmrepresents the out-of-plane ,in-plane , andmixdissipation, respectively. The substitution of Eq. ( 8)into Eq. ( 19) yields Do=μ0Ms γ(α0+sαcsinθ)s i n2θ, (21a) Dm=−μ0Ms γ˜αccosθsin3θ, (21b) Di=μ0Ms γα0 λ2 dwsin2θ, (21c) where αcis the chiral damping defined in Eq. ( 15a) and ˜αcrepresents aπ 2-phase shift in φofαc[i.e., ˜αc(φ)= αc(φ−π/2)]. Interestingly, SOC induces in addition to the in-plane and out-of-plane damping, a mixterm Dmwhich is locally finite as shown in Eq. ( 21b). Even though, in principle, the spatial integration of Dmvanishes, the nonequilibrium dynamics of the magnetization might result to a finite value and hencerenormalizes the overall contribution of the chiral damping.However, such corrections are expected to be small and,hence, we neglect this effect in the rest of this study. The totalrate of energy loss by the magnetic system with cross sectionalareaAis given as dE dt=A/integraldisplay+∞ −∞dE dtdx. (22) Following the representation of Eq. ( 20), Eq. ( 22) can be rewritten in the form dE dt=Do(∂tφ)2+Di(∂tXc)2, (23) where Do(i)=A/integraldisplay+∞ −∞Do(i)dx, (24) and after performing the integration, we obtain Do=2μ0MsAλdw γ(α0+sαc) (25a) and Di=2μ0MsAλdw γλ2 dwα0. (25b) The application of the scattering theory of magnetization dissipation, in which the magnetic system is considered to beat a constant temperature and the energy loss by the magneticsystem is equal to the total energy pumped into the system,yields [ 28–32] dE dt=¯h 4πTr/parenleftbiggdS dtdS† dt/parenrightbigg , (26) where Sis the scattering matrix at the Fermi energy. Further- more, since S=S(m), we have that S=S(Xc(t),φ(t)) and therefore Eq. ( 26) is transformed into dE dt=Ao(∂tφ)2+Ai(∂tXc)2, (27) where Ao=¯h 4Tr/parenleftbigg∂S ∂φ∂S† ∂φ/parenrightbigg (28a) 184424-4THEORY OF CHIRAL EFFECTS IN MAGNETIC TEXTURES … PHYSICAL REVIEW B 98, 184424 (2018) and Ai=¯h 4Tr/parenleftbigg∂S ∂Xc∂S† ∂Xc/parenrightbigg (28b) are proportional to the out-of-plane and in-plane contribution to damping, respectively. Next, comparing Eqs. ( 23) and ( 27), we have that Do=Ao=¯h 4Tr/parenleftbigg∂S ∂φ∂S† ∂φ/parenrightbigg (29a) and Di=Ai=¯h 4Tr/parenleftbigg∂S ∂Xc∂S† ∂Xc/parenrightbigg . (29b) Finally, we obtain the expression of the out-of-plane damping using Eq. ( 25) and Eq. ( 29)a s α0+sαc=CTr/parenleftbigg∂S ∂φ∂S† ∂φ/parenrightbigg , (30) where C=γ¯h 8μ0MsAλdw. (31) Equation ( 30) provides a very transparent way to extract both thenonchiral andchiral contribution of the damping. Indeed, sinces=±1f o r↑↓(↓↑) domain walls, the nonchiral and chiral contribution of damping can be computed as α0=C 2Tr/parenleftBigg ∂S↑↓ ∂φ∂S† ↑↓ ∂φ+∂S↓↑ ∂φ∂S† ↓↑ ∂φ/parenrightBigg (32a) and αc=C 2Tr/parenleftBigg ∂S↑↓ ∂φ∂S† ↑↓ ∂φ−∂S↓↑ ∂φ∂S† ↓↑ ∂φ/parenrightBigg , (32b) respectively. Therefore the calculation of nonchiral, chiral, and by extension chiral renormalization of the gyromagneticratio requires the knowledge of the derivative of the scatteringmatrix with respect to the domain wall tilt angle φ. The deriva- tion of a close form analytic expressions of the scattering ma-trix in the presence of SOC is nontrivial even though asymp-totic expressions have been derived in the limits k Fλdw/greatermuch1 [34] and kFλdw/lessmuch1[35–37], where kFis the Fermi wave number. Therefore, in the following section, we calculatethese damping contributions by numerically computing thederivatives of the scattering matrix and its conjugate withrespect to the tilt angle φof a domain wall to ascertain the correctness of the theoretical treatment presented above. V . NUMERICAL RESULTS In this section, we follow the procedure outlined in the preceding section and numerically compute the nonchiral andchiral contributions to the damping. To achieve this, we con-sider a two-dimensional tight-binding model on a square lat-tice with lattice constant a 0. In our calculations, we consider a scattering region of size 1001 ×101a2 0to ensure that a domain wall of width λdw=15a0is fully relaxed into a ferromagnet at the contact with the left and right leads. The scattering matrix FIG. 1. Shows the φdependence of the nonchiral (dash lines) andchiral (solid lines) contribution to the damping in the presence of different SOC. Results show that αcis SOC-driven and proportional to the Fermi energy evident in the smaller amplitude for (a) εF= −4.5tcompared to (b) εF=−3.2t. In all calculations with SOC, βso=0.02t. and its derivatives are calculated with the help of KWANT pack- age [ 33] from which the nonchiral and chiral contributions of the damping are extracted based on Eqs. ( 32a) and ( 32b), respectively. Furthermore, in all our calculations, we consideran exchange constant of J=−2t/3 and an on-site energy ε i=0. The damping parameters are calculated based on the material parameters Ms=8×105Am−1,a0=0.35 nm. Our numerical results of the φdependence of the nonchiral (dash lines) and chiral (solid lines) contributions to the damp- ing in the presence of different SOC for different transportenergies, ε F=−4.5tin Fig. 1(a)andεF=−3.2tin Fig. 1(b), are in good agreement with our analytical predictions given byEq. ( 15a). Indeed, the relative increase in the strength of α cin Fig.1(b)compared to Fig. 1(a)shows that the effect is a Fermi energy effect, i.e., ∝εF. Furthermore, in the absence of SOC, i.e.,βso=0 (green curves), αc=0, and α0is a constant. In the presence of SOC, we considered three interesting cases:(i)β so=βR, i.e.,βD=0 (red curves), and from Eq. ( 16b), φso=0, therefore, αc∝cosφ. (ii) βso=βD, i.e., βR=0 (blue curves) similarly, φso=π/2, therefore, αc∝sinφ. (iii) βR=βD(black curves) and using similar arguments, φso= π/4, therefore, αc∝cos(φ+π/4). It is worth mentioning here that in the presence of SOC, the nonchiral damping α0 shows a small oscillatory ∝cos2(φ+φso) as a result of small SOC-induced anisotropic magnetoresistance. The completedescription of the φdependence of α cpresented here should, in principle, provide a guide for material engineering ofeffects related to damping in chiral magnets. The validityof our analytical model is strengthened with the result ofthe investigation of the dependence of chiral damping onthe strength of the SOC. Indeed, Figs. 2(a) and2(b) show that (i) the nonchiral damping α∝β 2 so[24], (ii) the chiral damping αc∝βso, and (iii) chiral and nonchiral damping are Fermi energy effects, i.e., ∝εF. This is, again, in agreement with our analytical prediction of Eq. ( 15a). Observe that the Dresselhaus SOC, which stabilizes Bloch walls in materialswith bulk inversion symmetry breaking, affects chiral damp-ing in these materials in exactly the same way that the RashbaSOC, which favors Néel in materials with interfacial inversionsymmetry breaking interaction, affects the chiral dampingin these materials. Furthermore, Dresselhaus (Rashba) SOCinduces no chiral contribution Néel (Bloch) as a result of the 184424-5C. A. AKOSA, A. TAKEUCHI, Z. YUAN, AND G. TATARA PHYSICAL REVIEW B 98, 184424 (2018) FIG. 2. Dependence of chiral and nonchiral damping on the strength of the SOC for (a) εF=−4.5tand (b) εF=−3.2t.N o t i c e that the blue and red curves as well as the green and black curves are superimposed showing that the Dresselhaus SOC influences thedamping in Bloch walls ( φ=π/2) exactly the same way Rashba SOC influences it in Néel walls ( φ=0). sinφ(cosφ) symmetry of the chiral damping. Therefore our work shows that the symmetry of the SOC-induced chiraldamping is inherited from the symmetry of the materials. VI. CONCLUSIONS We have carried out a detailed theoretical investigation of the nature of spin torque and the corresponding dynamics gen-erated by two-dimensional, spatially and temporally varyingchiral magnetic textures in the presence of both Dresselhausand Rashba SOCs. We employed the Green’s function formal-ism to derive expressions for the nonequilibrium spin densityand hence the spin torque generated by a spatially and tem-porally varying chiral magnetic textures in which the gaugefield induced by these SOCs is treated perturbatively. Ourresult indicates that the effective gauge field associated withthese SOCs, and by extension of any origin, contributes to thetorque in exactly the same way as the electromagnetic gaugefield. In order to investigate the impact these torques have onthe dynamics of chiral magnets, we then incorporated thecalculated torques into the LLG equation that governs the dy-namics of the magnetization and derived analytic expressionsfor both the chiral damping α cand the chiral renormalization of the gyromagnetic ratio ˜ γcand show that αc=(τ/τ ex)˜γc, where τexandτare the exchange and electron relaxation times, respectively. Furthermore, we propose a theoreticalscheme based on the scattering matrix formalism to calculateand investigate the properties of damping in chiral magnets.Our findings should, in principle, provide a guide for materialengineering of effects related to damping in chiral magnets. ACKNOWLEDGMENTS G.T. acknowledges financial support from Grant-in-Aid for Exploratory Research (No. 16K13853), Grant-in-Aid for Sci-entific Research (B) (No. 17H02929) from Japan Society forthe Promotion of Science (JSPS ), Grant-in-Aid for ScientificResearch on Innovative Areas (No. 26103006) from the Min-istry of Education, Culture, Sports, Science and Technology(MEXT), Japan and the Graduate School Materials Science inMainz (DFG GSC 266). A.T. acknowledges financial supportfrom Grant-in-Aid for Scientific Research (No. 17H02924)from JSPS. Z.Y . acknowledges financial support from the National Natural Science Foundation of China (Grants No.61774018 and No. 11734004), the Recruitment Program ofGlobal Youth Experts, and the Fundamental Research Fundsfor the Central Universities (Grant No. 2018EYT03). C.A.A.thanks A. Abbout and Y . Yamane for useful discussions. APPENDIX: NONEQUILIBRIUM SPIN DENSITY CALCULATION In this section, we present a detailed calculation of the nonequilibrium spin density induced by a time-varying mag-netization. To calculate the nonequilibrium spin density ˜s,w e treat the spin gauge fields A μ=Aμ s+Aμ soperturbatively in the adiabatic limit of slow dynamics (¯ h/Omega1/lessmuchεF) and smooth variation of the magnetization ( q/lessmuchkF), where /Omega1,q, and kFare the frequency, the wave number, and the Fermi wave number, respectively. To simplify notation and render ouranalysis trackable, we define the Green’s functions g k,ω=1 2/summationdisplay σ=±(1+σσz)gk,ω,σ, (A1a) gr k,ω,σ=1 ¯hω−εk+εF+σJ+iη, (A1b) such that gr k,ω,σ=(ga k,ω,σ)∗andη=¯h/2τ, where τis the elastic relaxation time of conduction electrons. The nonequi-librium spin density is defined up to linear order in /Omega1as ˜s μ(q,t)=e¯h2 2πm/summationdisplay k,q/prime,q/prime/prime∂tAν i(q/prime,t)Tr/bracketleftbigg kiˆσμgr/parenleftbigg k+q 2,k+q/prime/prime 2/parenrightbigg ׈σνga/parenleftbigg k+q/prime/prime−q/prime 2,k−q 2/parenrightbigg/bracketrightbigg +e2¯h 2πm/summationdisplay k,q/prime∂t/bracketleftbig Aν s,i(q/prime,t)Aν so,i(q−q/prime,t)/bracketrightbig ×Tr/parenleftbig ˆσμgr kga k/parenrightbig , (A2) where gr(a)(k,k/prime) is the retarded (advanced) Green’s func- tion represented by gr(a)(k,k)≡gr(a) k=(1/2)/summationtext σ=±(1+ σσz)gr(a) k,σ, with gr k,σ=(ga k,σ)∗=1/(−εk+εF−σJ+iη). The dominant contributions are linear in qandλ, and they are depicted in Fig. 3. )b( )a( As Aso ~s FIG. 3. Diagrammatic representation of the nonequilibrium spin density ˜s. The solid, wavy, and dashed lines represent the Green’s function, spin gauge potential As, and gauge potential due to SOC Aso, respectively. (a) First order and (b) second order in Acontribu- tions to the nonequilibrium spin density. 184424-6THEORY OF CHIRAL EFFECTS IN MAGNETIC TEXTURES … PHYSICAL REVIEW B 98, 184424 (2018) 1. First order in A Up to first order in A, the diagram that contributes to the nonequilibrium spin density is given by Fig. 3(a), from which the components of the spin density are computed as ˜sμ(q,t)=e¯h2 2πm∂tAν so,i(q,t)/summationdisplay kTr/parenleftbig kiˆσμgr k+q 2ˆσνga k−q 2/parenrightbig /similarequal−ie¯h4 2πm2qj∂tAν so,i(q,t)/summationdisplay σ=±/summationdisplay kkikj/braceleftbig δμzδνzIm/bracketleftbig gr k,σ/parenleftbig ga k,σ/parenrightbig2/bracketrightbig +[δμx(δνxIm+σδνyRe)+δμy(δνyIm−σδνxRe)]gr k,−σ/parenleftbig ga k,σ/parenrightbig2/bracerightbig . (A3) 2. Second order in A For completeness we also calculated the second order in Acontribution to the nonequilibrium spin density as depicted in Fig.3(b) as ˜sμ(q,t)=e2¯h3 2πm2/summationdisplay q/prime/bracketleftbig ∂tAν s,i(q/prime,t)Ao so,j(q−q/prime,t)+∂tAν so,i(q/prime,t)Ao s,j(q−q/prime,t)/bracketrightbig ×/summationdisplay kTr/parenleftbig kikjˆσμgr kˆσνga kˆσoga k+kikjˆσμgr kˆσogr kˆσνga k/parenrightbig +e2¯h 2πm/summationdisplay q/prime∂t/bracketleftbig Aν s,i(q/prime,t)Aν so,j(q−q/prime,t)/bracketrightbig/summationdisplay kTr/parenleftbig ˆσμgr kga k/parenrightbig =e2¯h3 πm2/summationdisplay q/prime/bracketleftbig ∂tAν s,i(q/prime,t)Ao so,j(q−q/prime,t)+∂tAν so,i(q/prime,t)Ao s,j(q−q/prime,t)/bracketrightbig ×/summationdisplay σ=±σ/summationdisplay kkikj/parenleftbig −δμz/braceleftbig (δνo−δνzδoz)Re/bracketleftbig gr k,σ/parenleftbig ga k,σ/parenrightbig2−gr k,σga k,−σga k,σ/bracketrightbig −σ/epsilon1νozIm/parenleftbig gr k,σga k,−σga k,σ/parenrightbig/bracerightbig +δμx/bracketleftbig δνz(δoxRe+σδoyIm)gr k,σga k,−σga k,σ+δoz(δνxRe−σδνyIm)gr k,−σ/parenleftbig ga k,σ/parenrightbig2/bracketrightbig +δμy/bracketleftbig δνz(δoyRe−σδoxIm)gr k,σga k,−σga k,σ+δoz(δνyRe+σδνxIm)gr k,−σ/parenleftbig ga k,σ/parenrightbig2/bracketrightbig/parenrightbig . (A4) The dominant contributions of the xandycomponents of the nonequilibrium spin density, ˜sxand˜sy, are obtained as ˜sx=−e¯h2 2πm∂i∂tAν so,i/summationdisplay σ=±(δνxIm+σδνyRe)C1,σ+e2¯h πm/parenleftbig ∂tAν s,iAo so,i+∂tAν so,iAo s,i/parenrightbig ×/summationdisplay σ=±σ[δνz(δoxRe+σδoyIm)C2,σ+δoz(δνxRe−σδνyIm)C1,σ], (A5) ˜sy=−e¯h2 2πm∂i∂tAν so,i/summationdisplay σ=±(δνyIm−σδνxRe)C1,σ+e2¯h πm/parenleftbig ∂tAν s,iAo so,i+∂tAν so,iAo s,i/parenrightbig ×/summationdisplay σ=±σ[δνz(δoyRe−σδoxIm)C2,σ+δoz(δνyRe+σδνxIm)C1,σ], (A6) where C1(2),σare calculated as C1,σ=/summationdisplay kεkgr k,−σ/parenleftbig ga k,σ/parenrightbig2 /similarequal−ν/integraldisplay∞ −∞dεε (ε−εF−σJ−iη)(ε−εF+σJ+iη)2 =−iπν 2εF+σJ+iη (σJ+iη)2 =−πν 2(J2+η2)2[η(J2+η2+2σεFJ)+iεF(J2−η2)+iσJ(J2+η2)], (A7) 184424-7C. A. AKOSA, A. TAKEUCHI, Z. YUAN, AND G. TATARA PHYSICAL REVIEW B 98, 184424 (2018) C2,σ=/summationdisplay kεkgr k,σga k,−σga k,σ /similarequal−ν/integraldisplay∞ −∞dεε (ε−εF+σJ−iη)(ε−εF−σJ+iη)(ε−εF+σJ+iη) =πν 2ηεF−σJ+iη σJ−iη =−πν 2η(J2+η2)(J2+η2−σεFJ−iεFη). (A8) The effective magnetic field H∗ effdue to this nonequilibrium spin density is given by H∗μ eff=−Ja2 0 γ¯h(Rμx˜sx+Rμy˜sy) =−eJa2 0 πγ¯hλν so,i/summationdisplay σ=±/braceleftbigg 2mν∂t/bracketleftbig/parenleftbig RμxAx s,i+RμyAy s,i/parenrightbig σReC1,σ−/parenleftbig RμxAy s,i−RμyAx s,i/parenrightbig ImC1,σ/bracketrightbig +∂tmν/bracketleftbig/parenleftbig RμxAx s,i+RμyAy s,i/parenrightbig σRe(C1,σ+C2,σ)−/parenleftbig RμxAy s,i−RμyAx s,i/parenrightbig Im(C1,σ−C2,σ)/bracketrightbig −/parenleftbig ∂tAz s,i+∂iAz s,t/parenrightbig [(RμxRνx+RμyRνy)σRe(C1,σ−C2,σ)−(RμxRνy−RμyRνx)Im(C1,σ+C2,σ)] +4e ¯hmμmν/bracketleftbig/parenleftbig Ax s,iAy s,t−Ay s,iAx s,t/parenrightbig σReC1,σ−/parenleftbig Ax s,iAx s,t+Ay s,iAy s,t/parenrightbig ImC1,σ/bracketrightbig +4e ¯hmνAz s,t/bracketleftbig/parenleftbig RμxAy s,i−RμyAx s,i/parenrightbig σReC1,σ+/parenleftbig RμxAx s,i+RμyAy s,i/parenrightbig ImC1,σ/bracketrightbig +∂iAz s,t[(RμxRνx+RμyRνy)σRe(C1,σ−C2,σ)−(RμxRνy−RμyRνx)Im(C1,σ+C2,σ)]/bracerightbigg . (A9) Here we used the relations ∂tRμx=−2e ¯h/parenleftbig Ay s,tmμ−Az s,tRμy/parenrightbig , (A10) ∂iRμx=2e ¯h/parenleftbig Ay s,imμ−Az s,iRμy/parenrightbig , (A11) ∂tRμy=2e ¯h/parenleftbig Ax s,tmμ−Az s,tRμx/parenrightbig , (A12) ∂iRμy=−2e ¯h/parenleftbig Ax s,imμ−Az s,iRμx/parenrightbig . (A13) To make our calculation tractable and simplify notation, we define constants Cias C1=−Ja2 0 2πRe/summationdisplay σ=±σ(C1,σ+C2,σ)=−Ja2 0 2πm ¯h2εFJ(J2−η2) 2η(J2+η2)2, (A14) C2=−Ja2 0 2πIm/summationdisplay σ=±(C1,σ−C2,σ)=Ja2 0 2πm ¯h2εFJ2 (J2+η2)2, (A15) C3=−Ja2 0 2πRe/summationdisplay σ=±σ(C1,σ−C2,σ)=−Ja2 0 2πm ¯h2εFJ(J2+3η2) 2η(J2+η2)2, (A16) C4=−Ja2 0 2πIm/summationdisplay σ=±(C1,σ+C2,σ)=−Ja2 0 2πm ¯h2εFη2 (J2+η2)2, (A17) C5=−Ja2 0 2πRe/summationdisplay σ=±σC 1,σ=−Ja2 0 2πm ¯h2εFηJ (J2+η2)2, (A18) C6=−Ja2 0 2πIm/summationdisplay σ=±C1,σ=−Ja2 0 2πm ¯h2εF(J2−η2) 2(J2+η2)2. (A19) 184424-8THEORY OF CHIRAL EFFECTS IN MAGNETIC TEXTURES … PHYSICAL REVIEW B 98, 184424 (2018) From which we obtain H∗μ eff=−λν so,i γ/braceleftbigg −2mν∂t[C5(m×∂im)μ−C6∂imμ]+∂tmν[C1(m×∂im)μ−C2∂imμ] +∂im·(m×∂tm)[C3(δμν−mμmν)−C4/epsilon1μνomo]+2mμmν[C5∂im·(m×∂tm)+C6∂im·∂tm] −4e ¯hmνAz s,t[C5∂imμ+C6(m×∂im)μ]−2e ¯h∂iAz s,t[C3(δμν−mμmν)−C4/epsilon1μνomo]/bracerightbigg . (A20) (Note that the last two terms proportional to Az s,tand∂iAz s,tare expected to cancel out for gauge invariance with the other contributions we do not consider here.) In the above calculation, we used the following relations for spin gauge field As: RμxAx s,i+RμyAy s,i=−¯h 2e(m×∂im)μ, (A21) RμxAy s,i−RμyAx s,i=−¯h 2e∂imμ, (A22) ∂tAz s,i+∂iAz s,t=¯h 2e∂im·(m×∂tm), (A23) RμxRνx+RμyRνy=δμν−mμmν, (A24) RμxRνy−RμyRνx=/epsilon1μνomo, (A25) Ax s,iAy s,t−Ay s,iAx s,t=/parenleftbigg¯h 2e/parenrightbigg2 ∂im·(m×∂tm), (A26) Ax s,iAx s,t+Ay s,iAy s,t=−/parenleftbigg¯h 2e/parenrightbigg2 ∂im·∂tm. 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PhysRevB.104.014403.pdf

PHYSICAL REVIEW B 104, 014403 (2021) Dynamic magnetoelastic boundary conditions and the pumping of phonons Takuma Sato ( /ZdZ1148/ZdZ15120/ZdZ6143/ZdZ11796),1Weichao Yu ( /ZdZ1157/ZdZ1099/ZdZ17073),1Simon Streib ,2and Gerrit E. W. Bauer1,3,4 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden 3WPI-AIMR, Tohoku University, Sendai 980-8577, Japan 4Zernike Institute for Advanced Materials, Groningen University, Groningen, The Netherlands (Received 14 April 2021; accepted 10 June 2021; published 6 July 2021) We derive boundary conditions at the interfaces of magnetoelastic heterostructures under ferromagnetic reso- nance for arbitrary magnetization directions and interface shapes. We apply our formalism to magnet |nonmagnet bilayers and magnetic grains embedded in a nonmagnetic thin film, revealing a nontrivial magnetization angledependence of acoustic phonon pumping. DOI: 10.1103/PhysRevB.104.014403 I. INTRODUCTION The functionalities explored by spintronics may lead to novel information technologies that take advantage of the spindegrees of freedom. The current-induced spin transfer by elec-trons [ 1] and magnons [ 2,3] can, e.g., be used for nonvolatile memories and magnetic logic devices. Electromagnetic fields [4,5] and lattice vibrations [ 6–13] may also carry spin. The orbital and spin angular momentum of the deformation fieldsof continuous isotropic acoustic media with SO(3) rotationalsymmetry derive from Noether’s theorem [ 7,10]. The phonon spin is the angular-momentum contribution that does not de- pend on the origin of the coordinate system [ 10,12,13]. Magnetic anisotropy and magnetoelasticity in magnetic materials couple the phonon spin with the magnetization [ 14]. Interfaces between magnets (M) and nonmagnets (NM) play crucial roles in spintronics. An interesting material for the“spin mechanics” extension of spintronics are thin films ofmagnetic insulator yttrium iron garnet (YIG) grown on theparamagnet gadolinium gallium garnet (GGG), which areboth of very high acoustic quality. The magnetization dynam-ics in YIG emits phonons into the GGG ( phonon pumping ) [15]. In YIG |GGG|YIG phononic spin valves, the magnetic layers communicate by the exchange of phonons over mil-limeters [ 16,17], much larger than the propagation distance of diffuse magnon spin currents in YIG [ 18]. The Landau-Lifshitz-Gilbert (LLG) equation governs the magnetization dynamics and the elastic equation of mo-tion (EOM) that of the underlying lattice. They are coupledby effective forces and fields, which are functional cross- derivatives of the total energy [ 19–23]. This approach is ap- propriate in the GHz frequency regime in which wavelengthsfar exceed the lattice constants. Ferromagnetic resonance(FMR) excites the uniform precession (the Kittel mode) forwhich effective forces and torques in the bulk cancel out toa large extent. Dynamical magnetoelastic stresses at surfacesand interfaces of the magnet, however, are a source of phonons [24] and its generation is governed by boundary conditions (BCs). Comstock and LeCraw [ 24] formulated the BCs for planar M|NM interfaces with magnetization normal to the plane.Tiersten [ 25,26] addressed BCs of general structures such as sketched in Fig. 1in the framework of nonlinear contin- uum mechanics, but the practical consequences of spin-latticecoupling are difficult to distill from the heavy mathematics.Here, we address the BCs in linear system, clearly separatingelastic and magnetoelastic effect, generalizing Ref. [ 24]t o arbitrary interface geometries and directions of the macrospindynamics, including shear and pressure waves. We interpretthe BCs in terms of physically appealing conservation lawsfor linear- and angular-momentum currents. The formalismcan handle different material combinations; here we focus onYIG|GGG and Galfenol |GaAs. We illustrate the formalism by calculating the FMR spectra and phonon pumping for twogeometries: M |NM bilayers and M grains embedded in NM thin film. The model leads to analytic expressions of FMR andphonon pumping in the planer systems, with a good agreementwith the experiments by An et al. [16] when magnetization is normal to the interface, and we predict a magnetizationangle-dependence that reveals generation of pressure waves.By the curvilinear BCs, the dynamics of magnetic grains, onthe other hand, emits a nontrivial distribution of phonon spincurrents. This article is organized as follows. In Sec. II we intro- duce and simplify Tiersten’s formalism [ 26] and define the linear- and angular-momentum phonon currents. Sections III andIVdeal with phonon pumping from planar and curvilinear interfaces, respectively. In Sec. Vwe discuss applications and justify the approximations. In Sec. VIwe summarize our results and give an outlook. II. FORMALISM A. Variational principle We consider the M-NM composite system in Fig. 1with NM|vacuum surface A, M |vacuum surface B, and NM |M interface C. Sound waves in elastic media with frequenciesup to tens of GHz have wavelengths much longer than thelattice constants and continuum theory applies. We disregardthe effect of global rotations on magnons and phonons [ 27], assuming the total system size to be macroscopic. We focus 2469-9950/2021/104(1)/014403(15) 014403-1 ©2021 American Physical SocietySATO, YU, STREIB, AND BAUER PHYSICAL REVIEW B 104, 014403 (2021) FIG. 1. The displacement (deformation) vector fields of lattice u and magnetization mdetermine the state of magnet (M)-nonmagnet (NM) composite systems studied in this paper. The boundaries con- sist of NM |vacuum surface A (black), M |vacuum surface B (blue), and NM |M interface C (red). on the linear regime in which the material (Lagrangian) and spatial (Eulerian) coordinates coincide [ 28] and denote them byr, while u(r,t) is the displacement vector at time tof a volume element with equilibrium position r. M and NM are assumed bonded, with identical displacement on both sidesclose to the interface C in Fig. 1: u| NM=u|Mon interface C . (1) The magnetization vector field is normalized by its modu- lusMsasm(r,t)=M(r,t)/Ms. Throughout, Greek letters α,β,... denote spatial coordinates and we adopt the sum- mation convention over repeated indices. We compartmentalize the Lagrangian density as L=/braceleftbiggKel−Uel in NM Kel−Uel+Kmag−Umag−Usurf−Ume in M, (2) where the kinetic energy Kel(r)i sρ˙u(r)2/2 in NM and ˜ρ˙u(r)2/2 in M. The mass densities ρ,˜ρand constitu- tive parameters are taken to be constant in each material,but allowed to differ. The spin kinetic term reads K mag= Ms/(−γ)˙φ(cosθ−1), where −γ,θandφare the gyromag- netic ratio and the polar- and azimuthal angles, respectively[29,30]. We consider elastic, magnetic, and magnetoelastic energy densities that depend linearly on the deformations,magnetization, and their derivatives U el=Uel(∂βuα), (3a) Umag=Umag(mα,∂βmα), (3b) Ume=Ume(∂βuα,mα,∂βmα). (3c) The deformation gradients ∂βuα=εαβ+ωαβconsist of strains εαβ=(∂βuα+∂αuβ)/2 and rotational deformations ωαβ=(∂βuα−∂αuβ)/2. The coupling of strain and magne- tization is the “magnetoelastic coupling” [ 14,19]. Rotational deformations tilt the anisotropy axis, leading to the mag- netorotation coupling [31–34]. The general form Eq. ( 3c) includes both interactions but we still refer to it as mag- netoelastic coupling (MEC) in the following. The surface anisotropy energy Usurf=μ0M·Hsurf/2 arises from crys- talline and magnetodipolar effective fields Hsurf at the boundaries in Figs. 1(b) and 1(c) [19]. The classical action functional reads S[u,m]=/integraldisplayT 0dt/integraldisplay M+NMd3rL(˙uα,∂βuα,mα,∂βmα).(4)We can derive the governing equations by the principle of least action. We assume a strong ferromagnet with constantmodulus of the magnetization vector: δS+δ/integraldisplay T 0dt/bracketleftbigg/integraldisplay Md3rλ1|m|2+/integraldisplay A+BdSλ2|m|2/bracketrightbigg =0.(5) The Lagrange multipliers λ1,2enables independent variations of the three magnetization components {δmα}. In the absence of external forces at surfaces and interfaces, the variationsat the boundaries must be taken into account when mini-mizing the action. For mathematical convenience, we imposeδu α(r,0)=δuα(r,T)=δmα(r,0)=δmα(r,T)=0, which does not affect the results as long as the time Tis sufficiently large. Equation ( 5) yields the following EOMs and BCs for the lattice and magnetization [ 26]: ρ¨uα=∂σαβ el ∂rβin NM , (6a) ˜ρ¨uα=∂σαβ el ∂rβ+∂σαβ me ∂rβin M, (6b) σα el·n=0o n A , (7a) /parenleftbig σα el+σα me/parenrightbig ·n=0o n B , (7b) /parenleftbig σα el|M+σα me/parenrightbig ·n=σα el|NM·n on C, (7c) ˙m=−γμ 0m×(Hmag+Hme)i n M , (8) and /epsilon1αβγmβ/parenleftbig1 2μ0MsHγ surf+Xγ·n/parenrightbig =0( 9 ) on boundaries to M, where σαβ el=∂Uel ∂(∂βuα),σαβ me=∂Ume ∂(∂βuα), (10a) Hmag≡−1 μ0Ms/bracketleftbigg∂Umag ∂m−∂ν∂Umag ∂(∂νm)/bracketrightbigg , (10b) Hme≡−1 μ0Ms/bracketleftbigg∂Ume ∂m−∂ν∂Ume ∂(∂νm)/bracketrightbigg , (10c) Xαβ≡∂Umag ∂(∂βmα)+∂Ume ∂(∂βmα), (10d) the vectors of tensor components are defined as Aα≡ (Aαx,Aαy,Aαz)T, and nis a surface normal. In Appendix A we discuss the definition of the stress tensor Eq. ( 10a). MEC causes a nonvanishing magnetoelastic stress tensor σmeand forces Fα me=∂βσαβ mein Eq. ( 6b). A uniform magne- tization appears to not affect the dynamics, but comes intoplay via the BCs [ 19,24], as seen in Eqs. ( 7) that ensures the continuity of stress across the interfaces and boundaries.Equations ( 7) contain all surface and interface stresses and uniquely determines the solution to Eqs. ( 6)[35]. In the ab- sence of MEC ( σ α me=0 ) ,E q s .( 7) reduce to the free- and bonded-boundary conditions in the ordinary theory of elas-ticity [ 36]. Akhiezer [ 37] and Tiersten [ 26] derived the BCs at the boundaries B and C more than half a century ago. How-ever, these authors did not separate magnetoelastic and elasticcontributions, which is helpful for practical implementations 014403-2DYNAMIC MAGNETOELASTIC BOUNDARY CONDITIONS … PHYSICAL REVIEW B 104, 014403 (2021) TABLE I. Material parameters used in this paper. ρ(kg/m3) ct(m/s) c/lscript(m/s)ηel/(2π) (MHz) αG μ0Ms(T) K1(MJ/m3) b1(MJ/m3)b2(MJ/m3) YIG 5170a3843a7209a0.35 9 ×10−5b0.172b−6.10×10−4c0.348a0.696a GGG 7070d3568d6411d0.35b Galfenol 7800e4.0×103e5.0×103f1 0.017g1.59e3.3×10−2g-15.6h-6.19e GaAs 5317i3.34×103i4.73×103i1j aReference [ 19]. bReference [ 16]. cReference [ 47]. dReference [ 48]. eReference [ 23]. fCalculated from C44=1.23×1011Pa and C11=1.96×1011Pa [49]. gThin film value [ 50]. hBulk single crystal value [ 51]. iRef. [ 52]. For propagation along /angbracketleft100/angbracketrightand at 300 K. jIn the absence of experimental data at 3–10 GHz, we average the room-temperature results at 1.03 GHz [ 53][ηel/(2π)∼0.1 MHz] and 56 GHz [ 54](∼60 MHz). and physical understanding. Equations ( 7) hold for arbitrarily curved interfaces, magnetization direction, and MEC energy.For a bilayer system with macrospin magnetization normal orparallel to the plane, Eq. ( 7c) reduces to MEC-BCs involving only pure shear waves [ 15,17,21,24]. Magnetization dissipation can be taken into account in Eq. ( 8) by adding a phenomenological viscous damping torque to arrive at the LLG equation: ˙m=−γμ 0m×(Hmag+Hme)+αGm×˙m, (11) where αGis the Gilbert damping constant. Ultrasonic atten- uation in solids arises from thermoelasticity, phonon-phononinteractions, and defects and depends on frequency and tem-perature [ 38]. Here we focus on a small frequency range and room temperature and model the attenuation by additionaldamping forces −ρη el˙uαon the right-hand side of Eq. ( 6) with frequency-independent attenuation per unit length, for whichwe adopt the parameters from the MHz-GHz experiments (seeTable I). Both types of phenomenological damping, α Gand ηel, cause a loss of angular momentum that can in a micro- scopic description be accounted for by a transfer of angularmomentum to global rotations [ 27] or to the environment that supports the sample [ 12]. The magnetization obeys the BC ( 9). In the long- wavelength limit the exchange contribution to the MECvanishes, i.e., ∂U me/∂(∂βmα)=0. When, on the other hand, the surface anisotropy is small compared to exchange in-teraction, Eq. ( 9) simplifies to “free” BC, i.e., vanishing magnetization gradient at the boundaries. B. Linear-momentum current According to Streib et al. [15] the BCs for the bilayer with perpendicular and in-plane magnetization reflect conservationof the linear-momentum current at the interface. Here weextend this notion for arbitrary shape of the boundaries andmagnetization directions. Newton’s Eqs. ( 6) are equivalent to the conservation law of linear momentum, dp α dt=− divjα p, (12)where jα p(r)≡/braceleftbigg−σα el(r) r∈NM −σα el(r)−σα me(r)r∈M(13) is the (outward) linear-momentum current density tensor with units [N m−2]=[kg m /ss−1m−2] (linear-momentum flux per unit area). The index αdenotes Cartesian component of the linear momentum, whereas the vector is the current flowdirection. The minus signs in Eq. ( 13) indicate that the stress is a force exerted on the volume by its surrounding partsof the body [ 39] equivalent to an incoming flow of linear momentum. Our central result Eq. ( 7) and assumption Eq. ( 1) are therefore equivalent to the continuity of linear-momentumcurrent and displacement: j α p·n=0 on outer surfaces A and B , u|NM=u|M jα p|NM·n=jα p|M·n/bracerightBigg on interface C. (14) C. Angular-momentum currents Akhiezer [ 37] and Kamra et al. [21] derived the BCs by considering energy flux and energy conservation (integral ofmotion) but did not address the angular momentum. Here, wefind that the linear-momentum current introduced in Sec. II B is closely related to the magnon-phonon angular-momentumcurrent across M |NM interfaces. Let us consider a volume element of a deformed elastic magnet located at the vector sum of its equilibrium positionand displacement, r+u. When |u|/lessmuch|r|, the volume integral of physical quantities over a deformed body and equilibriumbody with volume Vis the same. The motion of a volume element in continuous media may acquire Newtonian angularmomentum J ph=Lph+Sph, where [ 9,10,12,40] Lph=/integraldisplay Vd3rr×p,Sph=/integraldisplay Vd3ru×p. (15) The first integral expresses a global rotation of the body, depends on the choice of the origin, and vanishes for elasticplane waves with finite wavelengths [ 12]. The second may 014403-3SATO, YU, STREIB, AND BAUER PHYSICAL REVIEW B 104, 014403 (2021) FIG. 2. (a) Sketch of a planar M |NM bilayer with magnetization angle θmand film thicknesses Land d.( b ,c ) θm-dependence of the amplitudes of the dynamical magnetoelastic interface stress Eq. ( 29) induced by pure FMR for different magnets with material parameters from Table I. Magnetization aligns the applied static fields of (b) μ0H0=0.3609 T and (c) 1.6 T and is excited with the same perpendicular microwave intensity. be interpreted as “phonon spin,” which is caused by unidi- rectional rotations of mass particles around their equilibriumpositions [ 40].J phis generated not only by the magne- torotation coupling in the volume [ 10,40] but also by the angular-momentum current through the boundaries, since ac-cording to Eq. ( 12): ˙L α ph=−/integraldisplay ∂Vjα L·ndS+/integraldisplay VTα Ld3r, (16a) ˙Sα ph=−/integraldisplay ∂Vjα S·ndS+/integraldisplay VTα Sd3r, (16b) where ∂Vis the surface and jα L=/epsilon1αβγrβjγ p,Tα L=/epsilon1αβγjγβ p, (17a) jα S=/epsilon1αβγuβjγ p,Tα S=/epsilon1αβγ∂νuβjγν p. (17b) The angular-momentum current densities jα L,Sare in units (angular momentum) /(area)/(time). The surface integrals in Eqs. ( 16) represent the angular-momentum transfer across the boundaries that has not been considered in the bulk theory[40]. For levitating (non)magnets with stress-free boundary, the surface integrals vanishes owing to the BC Eq. ( 14).T α L,S are torque densities. Tα Linduces a rigid rotation of the body and therefore does not involve local deformations. On theother hand, T α Sexerts local torques by elastic deformations and hence depends on the derivatives of the mechanical dis-placement. In NM, T α Lvanishes due to the symmetry of the elastic stress tensor, while Tα Smay be disregarded since it is quadratic in the strain. In M, Tα Lis finite due to the antisym- metric part of the magnetoelastic stress tensor [see Eq. ( A2) in Appendix A], which actuates a rigid rotation [ 40]. In the examples illustrated in Secs. IIIandIV, however, the antisym- metric part (i.e., magnetorotation coupling) is negligible andangular momenta are mostly supplied by currents across the boundaries. In Sec. IVwe calculate the phonon spin current density jα Semitted from a magnetic disk. III. APPLICATIONS: BILAYERS As a first example, we consider a flat interface between a magnetic film of thickness dand a nonmagnetic substrate of thickness L[Fig. 2(a)]. The governing equations derived in Sec. IIallows us to compute the FMR signals as a function of layer thicknesses and magnetization orientation, therebymicroscopically modeling the published experiments withperpendicular magnetization [ 16]. A. Model We take the zaxis normal to the interface and assume translational symmetry in the x-yplane. We consider cubic lattices, whose elastic energy is given by three elastic stiffnessconstants C ijand the strains [ 41]: Uel=C11 2/parenleftbig ε2 xx+ε2 yy+ε2 zz/parenrightbig +C12/parenleftbig εyyεzz+εzzεxx+εxxεyy/parenrightbig +2C44/parenleftbig ε2 yz+ε2 zx+ε2 xy/parenrightbig . (18) Substitution into the first of Eq. ( 10a) reproduces Hooke’s law. In YIG and GGG, C11−C12/similarequal2C44and the sound velocities are virtually isotropic [ 19], but in general anisotropic in sin- gle cubic crystals. We adopt the crystallographic orientatione z/bardbl/angbracketleft100/angbracketright, so the transverse and longitudinal velocities of the ultrasounds propagating in the zdirection read ct=√C44/ρ andc/lscript=√C11/ρ, respectively. 014403-4DYNAMIC MAGNETOELASTIC BOUNDARY CONDITIONS … PHYSICAL REVIEW B 104, 014403 (2021) The magnetic energy density consists of the Zeeman cou- pling UZ=−μ0Msm·Hextand the shape and crystalline anisotropy energies [ 42] UA=1 2μ0M2 smTNm+K1(m×n0)2 =K1+/parenleftbig1 2μ0M2 s−K1/parenrightbig (m·ˆez)2, (19) where μ0is the vacuum permeability, N=diag(0 ,0,1) is the demagnetization tensor of thin films, and K1the uniaxial crystalline anisotropy constant. Here we adopt a perpendicu-lar anisotropy axis n 0/bardblˆez. We disregard surface anisotropies which could pin the magnetization at the boundaries [ 43–45]. The uniform equilibrium magnetization is assumed to be par-allel to a strong enough applied magnetic field H 0. The Kittel mode is excited by a weak AC magnetic field transverse to themagnetization. The equilibrium magnetization lies in the z-xplane [Fig. 2(a)] at an angle θ mwith the zaxis. The external field and magnetization consist of static and dynamical components Hext(t)=Ry(θm)⎛ ⎝h/bardbl(t) h⊥(t) H0⎞ ⎠, (20) m(r,t)/similarequalm(t)/similarequalRy(θm)⎛ ⎝m/bardbl(t) m⊥(t) 1⎞ ⎠, (21) where Ry=⎛ ⎝cosθm 0s i n θm 01 0 −sinθm 0 cos θm⎞ ⎠ (22) is the rotation matrix around the yaxis. In Eq. ( 21), we have assumed homogeneous and small magnetization amplitudesm /bardbl,⊥/lessmuch1. We expand the continuum MEC energy to linear order in strain and rotation tensor elements, Ume=mαmβ[bαβεαβ+Kαβωαβ], (23) where bαβ=δαβb1+(1−δαβ)b2and b1,2are the MEC parameters. The magnetorotation coupling arises from therotation of the hard anisotropy axis ˆe z→ˆez+δn, where δn=(∇×u)׈ez/2=(ωxz,ωyz,0) [15,33]. From Eq. ( 19) we derive Kxz=Kyz=− K1+μ0M2 s/2,Kzx=Kzy=K1− μ0M2 s/2, while the other components vanish. B. Magnetization dynamics We derive from Eqs. ( 19) and ( 23) the anisotropy and effective fields: γμ 0HA=−γ Ms∂UA ∂m=− (ωM−ωK)mzez, (24) γμ 0Hme=−γ Ms∂Ume ∂m=−ΩmeRy⎛ ⎝m/bardbl m⊥ 1⎞ ⎠, (25) where ωK=γ2K1/Ms,ωM=γμ 0Ms and Ωαβ me= 2γ Ms[bαβεαβ+Kαβωαβ] are angular frequencies. From Eqs. ( 11), (21), (24), and ( 25) we obtain the linearizedLLG equation in frequency domain: /parenleftbigg m/bardbl m⊥/parenrightbigg (ω)=χFMR(ω,θ m)/bracketleftbigg/parenleftbiggh/bardbl h⊥/parenrightbigg −1 γμ 0/parenleftbigg /Omega1/prime13 me /Omega1/prime23 me/parenrightbigg/bracketrightbigg (ω), (26) where Ω/prime me=R−1 yΩmeRyrepresents the MEC, whereas the susceptibility tensor χFMR(ω,θ m)=γμ 0 /Delta1FMR/parenleftbiggω0 11−iω iωω0 22/parenrightbigg (27) governs the pure FMR. The determinant /Delta1FMR= ω0 11ω0 22−ω2and the matrix elements ω0 11=ωH−(ωM− ωK) cos2θm−iαGω, ω0 22=ωH−(ωM−ωK) cos 2 θm− iαGω.F o rθm/negationslash=0◦the magnetization precession is elliptic. The “tickle” field [ 46]i nE q .( 26) is induced by the lattice strains and rotations. For the present case u(r,ω)=u(z,ω) and /Omega1/prime13 me=ωc∂zuxcos 2θm−ω/lscript c∂zuzsin 2θm, /Omega1/prime23 me=ωc∂zuycosθm,(28) where ωc=ωM/2+γ(b2−K1)/Msandω/lscript c=γb1/Ms parametrize the magnetostriction and magnetorotation coupling. C. Magnetoelastic surface stresses Magnetization precession at frequency ωinduces ac sur- face stresses on the x−yplane. In frequency space, Eq. ( 10a) reduces to ⎛ ⎜⎝σxz me σyz me σzz me⎞ ⎟⎠(ω)=⎛ ⎜⎝/parenleftbig b2−K1+1 2μ0M2 s/parenrightbig m/bardbl(ω) cos 2 θm/parenleftbig b2−K1+1 2μ0M2 s/parenrightbig m⊥(ω) cosθm −b1m/bardbl(ω)s i n2θm⎞ ⎟⎠,(29) where we discarded static ( ω=0) as well as higher order terms in the transverse magnetization, implying that the stress(not the strain) adiabatically follows the magnetization preces-sion [ 55]. For YIG, K 1/b2=0.0009, ( μ0M2 s/2)/b2=0.02, sob2dominates [Eq. ( 29)]. In iron gallium alloy (Fe 0.81Ga0.19, Galfenol), K1/b2=0.005, (μ0M2 s/2)/b2=0.1, so the mag- netorotation coupling due to dipolar anisotropy may becomesignificant. The angular dependencies of the magnetoelastic stresses Eq. ( 29) in YIG and Galfenol are plotted in Figs. 2(b) and2(c), respectively, for the Kittel mode excited by the pure FMR(27). The shear stresses σ xz meandσyz meare maximal for θm=0◦. While the former remains finite at θm=90◦, the latter van- ishes, leading to less efficient pumping of linearly polarizedphonons [ 15]. The pressure force vanishes at θ m=0◦and 90◦, but at intermediate angles pumps longitudinal phonons, as discussed in the next subsection. For fixed microwave in-tensity, the Kittel mode amplitude in Galfenol is smaller due tothe large Gilbert damping, which reduces the magnetoelasticstresses despite its large MEC parameters. D. 1D phonon pumping The phonon pumping problem derived in Sec. IIcan be solved analytically by a plane-wave ansatz. We first solve theelastic EOM and substitute it into the BCs to find the relation 014403-5SATO, YU, STREIB, AND BAUER PHYSICAL REVIEW B 104, 014403 (2021) between the elastic wave amplitudes and magnetization. The strain effective field in Eq. ( 26) is then proportional to the magnetization and affects the magnetic response. The imag-inary part of the magnetic susceptibility is proportional to themicrowave power absorption. The EOM ( 12) in M for sound waves propagating normal to the film separates into three modes, two degenerate trans-verse ( α=x,y) and a longitudinal, which solve ∂ 2uα ∂z2+ω2 ˜c2 t/parenleftBig 1+i˜ηel ω/parenrightBig uα=0, (30a) ∂2uz ∂z2+ω2 ˜c2 /lscript/parenleftBig 1+i˜ηel ω/parenrightBig uz=0. (30b) Similar equations without tilde hold in NM. The sound velocities and phenomenological ultrasonic attenuation pa-rameters are summarized in Table I, where we assume the same attenuation parameter for the magnetic film and the sub-strate. The characteristic attenuation length of the TA modesisδ=c t/ηel=1.7 mm in GGG and 0.5 mm in GaAs. The general solution to Eq. ( 30)i s uα(z,ω)=˜Aαei˜k/prime tz+˜Bαe−i˜k/prime tz, ˜k/prime t=ω ˜ct/radicalbigg 1+iηel ω≈˜kt+i˜κt, (31) where ˜kt=ω/˜ctand the damping parameter ˜ κt=˜ηel/2˜ct. The BCs Eq. ( 14) read for α=x,y,z, uα(0−)=uα(0+) σαz el(−d)+σαz me(−d)=0 σαz el(0−)+σαz me(0−)=σαz el(0+) σαz el(L)=0, (32) which should be used with Eq. ( 29). Since the Kittel mode feels only the spatial average of the effective field, we aver-age the strain in Eq. ( 28) over the film thickness and rewrite it in terms of transverse magnetization using Eq. ( 32) (see Appendix B), such that /parenleftbigg m /bardbl m⊥/parenrightbigg (ω)=χtot(ω,θ m)/parenleftbigg h/bardbl h⊥/parenrightbigg (ω). (33) The susceptibility tensor χtot(ω,θ m)=γμ 0 /Delta1(ω,θ m)/parenleftbigg ω11−iω iωω 22/parenrightbigg , (34) where /Delta1(ω,θ m)=ω11ω22−ω2, includes the coupling to the lattice. The matrix elements ω11=ω0 11−g(ω) cos2θm andω22=ω0 22−g(ω) cos22θm−g/lscript(ω)s i n22θmare shifted by the complex coupling strengths (in units of angularfrequency) g(ω)=M s γd˜ρ˜ctω2 c ω+i˜ηel/2F(ω), (35a) g/lscript(ω)=Ms γd˜ρ˜c/lscript/parenleftbig ω/lscript c/parenrightbig2 ω+i˜ηel/2F/lscript(ω), (35b) and depend on magnetization orientation. The real and imagi- nary part of Eq. ( 35) modify the anisotropy fields and dampingtorques, respectively. F(ω) and F/lscript(ω) are complex function of system geometry and material parameters (Appendix B). The coupling strengths in our microscopic theory includesthe effect of acoustic damping and scales as ∼ω −1, while in the simple coupled oscillator models [ 16,56] the coupling is independent of the damping and tends to ∼ω−1/2. These differences may be important when a wider frequency rangeis of interest and the frequency dependence of ˜ η elbecomes significant [ 38,57]. For the out-of-plane ( θm=0◦) configura- tionχ11 tot=χ22 totinduces circular precession, whereas for other angles the precession is elliptic. The microwave power ab-sorption P abs(ω,θ m)∝Im(hTm)=Imχ11 toth2 /bardbl+Imχ22 toth2 ⊥ (36) is the observable in FMR experiments. The absorption spectrum Eq. ( 36) contains the resonances sketched in Fig. 3. Figure 3(c1) illustrates the FMR frequency dependence on magnetic anisotropies, external field, and mag-netization orientation. The resonance Fig. 3(c2) occurs whenthe magnetoelastic surface stresses acting in the opposite di-rections on the two magnetic surfaces excite odd acousticwaves, which requires that the magnetic film thickness fulfillsthestress-matching condition ˜k td=π(2m−1) ( m=1,2,... ). (37) Under this condition the lattice displacement and the ad- ditional FMR broadening are maximized [ 15,17]. These resonances are very broad for YIG |GGG because of the strong coupling at the interface. In Fig. 3(c3), standing sound wavesform when sin˜k tdcosktL+ρct ˜ρ˜ctcos˜ktdsinktL=0. (38) The acoustic impedance mismatch ρct/(˜ρ˜ct)=1.27 between YIG and GGG or 0.87 between Galfenol and GaAs (Table I) is not important at GHz frequencies. The acoustic resonancefrequencies then simplify to f nt=nt 2(d/˜ct+L/ct)(nt=1,2,... ). (39) The same equation holds for the pressure waves by replacing the transverse by the longitudinal sound velocities. When thefilm thickness exceeds the sound attenuation length δthe back and forth reflected waves cannot interfere anymore andthe discrete spectrum is smeared out into a continuous one,which is the regime considered by Streib et al. [15]. The phase matching of the acoustic waves reflected by the twoboundaries z=− dand z=0 may also enhance the phonon pumping: 2˜k td=π(2s−1) (s=1,2,... ), (40) w h i c hw ec a l l thin-film interference condition [Fig. 3(c4)]. We first focus on the normal ( θm=0◦) configuration, in which the FMR excites only the transverse acoustic (TA)modes (see Fig. 2). In Fig. 3(a) we plot the FMR spectrum [Eq. ( 36)] of a YIG |GGG bilayer system with large thick- ness L+d=5( m m ) >δ / 2. The external field of 0.3609 T leads to the f FMR=5.129 GHz. The horizontal axis is nor- malized by half of the TA mode wavelength at fFMR,˜λ/2= ˜ct/2fFMR=375 nm. When varying dwhile keeping L+d constant we observe phonon-pumping increased linewidths at 014403-6DYNAMIC MAGNETOELASTIC BOUNDARY CONDITIONS … PHYSICAL REVIEW B 104, 014403 (2021) FIG. 3. Phonon pumping in a YIG |GGG bilayer. (a, b) FMR absorption spectrum [Eq. ( 36)] under a perpendicular static field μ0H0= 0.3609 (T) that pulls the magnetization fully out of plane ( θm=0◦) as a function of the frequency of the microwave and magnetic film thickness dfor bilayer thickness of (a) 5 mm and (b) 0.5 mm, normalized by half of the TA phonon wavelength at the FMR frequency. The red arrow indicates the film thickness d/(0.5˜λ)=0.53 in the experiments [ 16]. White lines labeled by the mode index mindicate the stress-matching conditions [Eq. ( 37)]. The ladder on the ordinate in (b) marks the eigenfrequencies [Eq. ( 39)] of the TA standing waves with mode numbers from nt=1430 (bottom) to 1443 (top) [see also panels (d) and (e)]. (c) Schematic of the resonances in M |NM bilayer. Green curves represent acoustic waves. Panel (d) shows the absorption fine structure in the vicinity of the FMR frequency as a function of applied static field for normal to and in-plane magnetization, respectively ( d=200 nm, L=0.5 mm). (e) The corresponding plots for Galfenol |GaAs bilayers. The contour color scales are normalized by the maximum values. the resonances indicated by vertical white lines [the resonance labeled (c2) in Fig. 3]. On the other hand, for thicknesses d=˜λ,2˜λ,... , the magnetic precession is out of phase with the phonons and the coupling is suppressed. When L+d=0.5( m m ) <δ / 2, clear standing waves form by wave interference and the phonon spectrum is dis-crete [Fig. 3(b)]. In addition to the resonance in Fig. 3(c2), equidistant satellite peaks appear at the acoustic eigenfre-quencies in Fig. 3(c3) indicated by the white ladder on theordinate. The effect of sound waves decreases for higher-order stress-matching conditions m=2,3,... because of the interference with the Kittel mode, but may couple strongerto higher order perpendicular spin wave modes (R. Schlitz,private communication). We observe clear avoided crossingsof the FMR with the acoustic resonance frequencies whenthe three resonance conditions are simultaneously fulfilled.In Fig. 3(b) we observed dips when the f FMR=5.129 GHz is tuned to the acoustic mode n=1438 and m=1,2,3. In Fig. 3(d) we adopt the layer thicknesses of the sample used by Anet al. [16][ r e da r r o wi nF i g . 3(b)] and sweep the external field for the out-of-plane (left) and in-plane (right) magnetiza-tions. The Kittel mode in the in-plane configuration is shiftedto lower fields because of the thin-film shape anisotropy. Theanticrossing in the left panel indicates strong coupling be-tween the magnetic and elastic excitations, as observed [ 16]. Our estimates of the Kittel frequencies are slightly shiftedfrom observed ones, which we tentatively attribute to residualanisotropies not included in our model. The sample thicknessin the experiment deviates from the optimal stress-matchingcondition m=1, but the strong coupling is still achievedbecause of the broadness of the m=1 resonance, which is effective for a wider range of the YIG thickness 0 .5/lessorsimilar d/(0.5˜λ)/lessorsimilar1.5. Note that the thin-film interference condi- tion labeled (c4) in Fig. 3[Eq. ( 40)] favors the half-integers d/(0.5˜λ)=0.5,1.5,... . On the right panel we again observe regularly spaced anticrossings, suggesting strong coupling be-tween the Kittel and TA modes. The gap is, however, smallercompared to the normal configuration due to the absence ofone of the transverse stresses, as shown in Fig. 2(σ yz me=0f o r θm=90◦). Physically, the phonon pumping is less efficient because the emitted sound waves are now linearly polarized[15]. The magnetoelastic stress does not rotate but oscillates and no net angular momentum is pumped into the NM. SinceGalfenol and GaAs have larger elastic damping, we choosefor Fig. 3(e) a thin NM film with L=0.15 mm ( <δ / 2= 0.26 mm). The spectra are broad due to the large Gilbert damping of Galfenol, yet exhibiting interaction with discreteTA modes for the θ m=0◦configuration (left). When θm= 90◦(right), they are not resolved because the large thin-film shape anisotropy significantly confines the precession withinthe film, suppressing σ xz me[see Eq. ( 29) and Fig. 2(c)]. The longitudinal waves interact with the dynamic magnetization when 0◦<θ m<90◦. The maximum coupling is not universal but depends on the material parameters, foundatθ m=30◦for YIG |GGG and at θm=9◦for Galfenol |GaAs (Fig. 2). Figures 4(a) and4(c) shows the resonance conditions and Figs. 4(b) and4(d) the FMR spectra at the magnetic film thickness indicated with the red arrows on the left. Figure 4(b) exhibits LA mode anticrossings. TA phonon features aresuppressed because the thickness d∼513 nm of the m l=1 014403-7SATO, YU, STREIB, AND BAUER PHYSICAL REVIEW B 104, 014403 (2021) FIG. 4. Coupling to longitudinal phonons in different bilayers. (a,c) Acoustic eigenfrequencies (TA in gray and LA in red) and LA stress- matching condition (green) against the magnetic film thickness. TA stress-matching is not achieved in the plotted parameter space. FMRspectra on the right panels are for the film thicknesses indicated by the red arrows on the left panels and normalized (b) by the same factor as in Fig. 3and (d) by the maximum value. resonance lies in the middle between mt=1 and 2 resonances, so the TA mode destructively interfere withthe Kittel mode [see Fig. 3(b)]. The selective coupling is possible in YIG |GGG bilayers because the transverse sound velocity ˜ c tin YIG is about a half of ˜ c/lscript. The anticrossing gap is smaller, however, since it is governed by the longitudinalMEC parameter b 1∼0.5b2. Figure 4(d) exhibits avoided crossings with not only TA but also LA phonons. At thesimultaneous crossing with n t=320 and nl=226 modes the gap is large. We conclude that the Kittel mode can coupleto the faster pressure waves for appropriate magnetizationorientation and film thickness. IV . APPLICATIONS: MAGNETIC GRAIN Next we apply our formalism to a thin magnetic disk of radius aembedded in a nonmagnetic film. The extension to, e.g., a spherical magnetic grain in a nonmagnetic matrix isstraightforward. We focus on the regimes in which the filmthickness is much smaller than the wavelengths of sound atfrequencies up to several GHz, so the deformation is constantover the film thickness (in z-direction). We consider an in- finitely extended medium, which means that emitted wavesare not coming back. The MEC-BCs on the top and bottomsurfaces of the film only rigidly shift the spectra and aretherefore disregarded. The dipolar interaction of thin magnetic films favors in- plane magnetization. We define the xaxis along an in-plane external static magnetic field and introduce a cylindrical coor-dinate system: x=Rcosφandy=Rsinφ. Elastic waves are excited coherently by uniform magnetization precession /parenleftbigg m y mz/parenrightbigg =χFMR(ω,π/ 2)/parenleftbigg hy hz/parenrightbigg , (41) where the susceptibility tensor is defined in Eq. ( 27) and hy,zare the magnetic fields of an applied microwave field atfrequency ω. The effect of strain in Eq. ( 26) represents the back-action from the lattice and lead to the anticrossing spec-tra in Sec. III. Here we focus on the propagation of pumped sound waves and disregard the higher-order self-consistenttickle fields in Eq. ( 41). The displacement field in terms of scalar and vector displacement potentials read [ 36] u(R,φ,t)=∇/Phi1(R,φ,t)+∇×/Xi1(R,φ,t). (42) Equation ( 12) separates into dilatation and shear motions with wave equations (in the absence of acoustic damping) ∇ 2/Phi1=¨/Phi1/c2 /lscript, ∇2/Xi1=¨/Xi1/c2 t. (43) The in- and out-of-plane motions decouple, which in the cylindrical coordinate system leads to independent pair ofsolutions ( /Phi1,/Xi1 z) and ( /Xi1R,/Xi1φ), respectively, given by the Bessel functions multiplied by sinusoidal angular dependence(Appendix C): /Phi1=/braceleftBigg˜A /lscriptJ2(˜k/lscriptR)s i n2φ A/lscriptH(1) 2(k/lscriptR)s i n2φ, (44a) /Xi1z=/braceleftBigg˜AtJ2(˜ktR) cos 2 φ AtH(1) 2(ktR) cos 2 φ, (44b) /Xi1R=/braceleftBigg˜CJ0(˜ktR)s i nφ CH(1) 0(ktR)s i nφ, (44c) /Xi1φ=/braceleftBigg˜CJ0(˜ktR) cosφ CH(1) 0(ktR) cosφ. (44d) 014403-8DYNAMIC MAGNETOELASTIC BOUNDARY CONDITIONS … PHYSICAL REVIEW B 104, 014403 (2021) Here the upper rows hold for 0 <R<a(in M) and the second fora<R(in NM). Jnand H(1) nare the Bessel and Hankel functions of the first kind, respectively. For this model, theBCs Eq. ( 14) become u α(a−0,φ)=uα(a+0,φ), σαR el(a−0,φ)+σαR me(a−0,φ)=σαR el(a+0,φ),(45)where φ∈[0,2π],α=R,φ,z,and ⎛ ⎜⎝σRR me σφR me σzR me⎞ ⎟⎠=⎛ ⎜⎝b2mysin 2φ b2mycos 2φ b2mzcosφ⎞ ⎟⎠. (46) The angular dependence in Eq. ( 44) follows from the con- straint that Eq. ( 45) holds for arbitrary φ.E q .( 45) then yields two decoupled matrix equations for the coefficients: ⎛ ⎜⎜⎜⎜⎝˜A /lscript A/lscript ˜At At⎞ ⎟⎟⎟⎟⎠=b 2a2my⎛ ⎜⎜⎜⎜⎝˜k /lscriptaJ/prime 2(˜k/lscripta)−k/lscriptaH(1)/prime 2(k/lscripta)−2J2(˜kta)2 H(1) 2(kta) 2J2(˜k/lscripta)−2H(1) 2(k/lscripta)−˜ktaJ/prime 2(˜kta)ktaH(1)/prime 2(kta) −˜MR /lscript MR /lscript˜MR t −MR t −˜Mφ /lscriptMφ /lscript−˜Mφ t Mφ t⎞ ⎟⎟⎟⎟⎠−1 ⎛ ⎜⎝0 0 11⎞ ⎟⎠, (47a) /parenleftbigg˜C C/parenrightbigg =b 2a2mz/parenleftbigg˜ktaJ/prime 0(˜kta) −ktaH(1)/prime 0(kta) −˜μ(˜kta)2J/prime/prime 0(˜kta)μ(kta)2H(1)/prime/prime 0(kta)/parenrightbigg−1/parenleftbigg 0 1/parenrightbigg , (47b) of in- and out-of-plane oscillations, respectively. The matrix elements ˜MR /lscript=2˜μ(˜k/lscripta)2J/prime/prime 2(˜k/lscripta)−˜λ(˜k/lscripta)2J2(˜k/lscripta),˜MR t=4˜μ[˜ktaJ/prime 2(˜kta)−J2(˜kta)], ˜Mφ /lscript=4˜μ[˜k/lscriptaJ/prime 2(˜k/lscripta)−J2(˜k/lscripta)], ˜Mφ t=˜μ[2˜ktaJ/prime 2(˜kta)+(˜kta)2J2(˜kta)],(48) depend on frequency. The other components are given by removing tildes and replacing J2with H(1) 2. Equations ( 47) and ( 41) determine the phonon pumping. We consider a YIG nano-disk embedded in a GGG thinfilm. For the in-plane static field μ 0H0=0.3609 T and trans- verse microwave field hy=hz=5A/m, the magnetization amplitudes are |my(ω)|≈0.05 and |mz(ω)|≈0.04 at the FMR frequency√ωH(ωH+ωM−ωK)/(2π)=12.6 GHz. The disk radius a=76.2 nm satisfies the first stress-matching condition of the TA modes at this frequency. The numericalsolutions of Eqs. ( 47) for the displacement field at resonance are depicted in Fig. 5(a). The anisotropy of the magnetoelastic stress [Eq. ( 46)] generates the observed angular patterns. The symmetry and location of the nodes do not depend on thematerial parameters. The magnetization precession Eq. ( 41) introduces a phase shift of π/2 between u φand uz, i.e., ex- cites rotational lattice motion in the vicinity of the xaxis. In the absence of damping, the elastic energy of the wavefront ∝/integraltext2π 0u(R,φ,ω )2dφ, decays as 1 /Rin the far field region by geometrical spreading, so uzdecreases as 1 /√ R. The ampli- tudes of uRanduφare coupled according to Eq. ( 47a), thereby oscillating as a function of Rby exchanging energy during propagation. The linear-momentum current jαR pfollows the angle dependence of uα(not shown). In Fig. 5(b) we plot the associated phonon spin current density in cylindrical coordinates (see Sec. II C), jRR S=− uzjφR p+uφjzR p, jφR S=uzjRR p−uRjzR p, jzR S=− uφjRR p+uRjφR p,(49)where the indices αandβinjαβ Srefer to the phonon spin polarization and current direction, respectively. The plots inFig. 5(b) represent the currents leaving the magnet in radial directions that vanish at the nodes of the respective displace-ment components, along which the sound waves are linearlypolarized. Even though the displacement fields oscillate intime with the FMR frequency, j RR Sis a DC current, which is ensured by the π/2 phase shift and similar wavelengths be- tween uφanduz. The amplitude are extreme along the xaxis, transporting phonon spins /bardblˆeRin the forward angles −π/4< φ<π / 4 and spins /bardbl−ˆeRin the backward 3 π/4<φ< 5π/4 directions. The other two components are AC currents carrieddominantly by pressure waves /Phi1. V . DISCUSSION The stress tensors and the BCs derived in Sec. IIare applicable to a wide range of materials of any crystal sym-metry and MEC as well as geometries. We focus in Secs. III and IVon the Kittel mode excited by FMR in the presence of uniaxial crystalline and dipolar shape anisotropies. Wecan adopt MEC generated, e.g., by exchange [ 19]o ri n t e r - facial Dzyaloshinskii-Moriya interaction [ 34,58] to address phonon pumping by spin waves and magnetorotation couplingin basically any material combination with ferromagnets. Inprinciple, the analysis can also be extended to antiferromag-nets [ 59–63], but the MEC energy and its parameters are less established. The magnon frequencies in antiferromagnets aretypically higher than that of ultrasound [ 19,62] and magnon- phonon hybridization requires application of large magneticfields [ 63]. Large MEC coefficients and large magnetiza- tion amplitudes are important for phonon pumping. Efficient 014403-9SATO, YU, STREIB, AND BAUER PHYSICAL REVIEW B 104, 014403 (2021) FIG. 5. Spatial profile of the (a) displacement components and (b) phonon spin currents at the FMR frequency 12.6 GHz emitted from a YIG disk of radius a=76.2 nm at the center (white). Magnetization precesses around the positive x-axis. Illustrations in panel (b), left, indicate the phonon spin orientation and propagation directions. phonon pumping does not necessarily require matching of wave numbers as demonstrated in Secs. IIIand IV, which unlocks the possibility of magnon-phonon strong couplingaway from the intersections of their dispersion branches. Moreresearch is required to identify the best material combinationsfor an optimal sound generation by magnetization dynamics. The BCs derived here allows deploying textbook knowl- edge of elastic waves [ 35,36,64] to address various boundary shapes. In Sec. IIIwe computed the phonon pumping in bilayers. M |NM|M phononic spin valves [ 16] can be cal- culated by attaching another YIG layer to the free surfaceof GGG. The transmission and reflection of sound wavesin the opposite sandwich, i.e., a thin magnetic film insertedin an infinite nonmagnetic matrix [ 22], is a simple exten- sion of our model as a function of magnetization angle. OurBCs can also address the energy partition between surfaceand bulk modes [ 65–67]. A magnetic stripline attached to a nonmagnetic substrate [ 16,68,69] excites not only the bulk phonons addressed here but also surface modes. Nonplanarstructures such as acoustic whispering gallery modes around asphere [ 70–73], or 3D ferromagnetic nanoparticles embedded in nonmagnetic media, as well as evanescent acoustic wavesat interfaces with metamaterials [ 11], are within the scope of our formalism. Multiple magnets in a nonmagnetic matrixindirectly coupled via ultrasounds is a playground to studycollective magnonic excitations, viz. a phononic extension ofspin cavitronics [ 74,75]. A large magnetostriction constant should lead to efficient phonon pumping, but may also induce a static deformation inthe ground state that depends on the magnetization direction.For example, a magnetization m=(1,0,0) in the circular YIG disk discussed in Sec. IVgenerates static magnetoelasticstresses ⎛ ⎝σ RR me σφR me σzR me⎞ ⎠ static=⎛ ⎝b1cos2φ −b1sinφcosφ 0⎞ ⎠, (50) that compresses the disk in the xdirection via the BC, ( σαR el+ σαR me)|M=σαR el|NM. This is consistent to the conventional static magnetostriction [ 42]. Other components of the magnetoelas- tic stress are finite as well, but the strains vanish by symmetry.The correction Eq. ( 50) modify the phonon dispersion and the elastic constants [ 76]. The correction σ me∼105Pa is, however, much smaller than the Lamé parameters ∼1011Pa in YIG, justifying that we disregard this effect in Secs. III andIV. In Sec. IVwe discuss phonon pumping in elastically isotropic GGG, where the angular pattern solely originatesfrom the anisotropy of σ me. In other single crystals, the elastic anisotropy may further affect the angular dependence. We focus here on microwave absorption experiments such as carried out by An et al. [16]. However, all experiments sen- sitive to the magnon polarons in YIG, such as pump and probespectroscopy [ 77,78], local and non-local spin Seebeck effect [79,80], and Brillouin light scattering [ 81] are affected by the phonon pumping into GGG substrates and can in principle testour results. VI. SUMMARY We present the BCs of magnet-nonmagnet composite systems for arbitrary interface geometries, magnetization ori-entation, and magnetoelastic interactions, with a focus onthe consequences of MEC. Our formalism is tuned to FMR 014403-10DYNAMIC MAGNETOELASTIC BOUNDARY CONDITIONS … PHYSICAL REVIEW B 104, 014403 (2021) conditions or small magnets, in which spin waves have little effect on the lattice and boundary dynamics becomes impor-tant. This natural extension of continuum mechanics allowstransfer of knowledge from ultrasonics for a better under-standing of the spintronics with phonons. The phonon pumping scheme formulated here allows us to magnetically activate phonon modes in nonmagnets. Mag-netic elements and fields may therefore be a tool to studyquantum ground states of phonons and phononic computing[82]. The magnetization-angle- and geometry dependence of phonon pumping may be useful for engineering magnon-photon-phonon hybrids [ 75,83,84]. A formulation of the dynamics of magnetic core /shell type nanoparticles levitated in traps may require extension of our BCs to include effects ofrigid body rotations [ 27,85] and oscillations [ 86]. ACKNOWLEDGMENTS We thank Kei Yamamoto for fruitful discussions and shar- ing his insights. The work was supported by JSPS KAKENHIGrants No. 20K14369 and No. 19H00645. S.S. acknowledgesfinancial support from the Knut and Alice Wallenberg Foun-dation through Grant No. 2018.0060. APPENDIX A: DEFINITION OF STRESS TENSORS Our definition of the stress tensor Eq. ( 10a) deviates from the conventional definition σαβ old=∂U/∂εαβ, where εis the strain tensor. The latter only holds for infinite media withvanishing surface stresses [ 39], i.e., disregarding surfaces and boundaries. The derivation of σ oldalso postulates the symmetry of the stress tensor to ensure angular-momentumconservation [ 36,39]. In the presence of spin-lattice coupling, magnetization can be a source of angular momentum eitherin the bulk or at the boundaries, rendering the stress tensorasymmetric [ 25,26,40], which is not reflected in σ old.T h e stress tensors defined here from the Lagrangian overcomethese issues [ 25,26,37,40]. We can clarify the relation between σandσ oldby rewriting Eq. ( 10a) in terms of strain and rotation tensors, εαβ(r,t)=1 2(∂βuα+∂αuβ), ωαβ(r,t)=1 2(∂βuα−∂αuβ),(A1) i.e., by switching from the nine variables {∂βuα} to the set of independent tensor elements,{ε xx,εyy,εzz,εyz,εzx,εxy,ωyz,ωzx,ωxy}. The first six components determines the elastic potential energy Eq. ( 3a) [39], whereas the coupling Eq. ( 3c) in general depends on all nine elements. The chain rule leads to σel=⎛ ⎜⎝∂ ∂εxx1 2∂ ∂εxy1 2∂ ∂εzx 1 2∂ ∂εxy∂ ∂εyy1 2∂ ∂εyz 1 2∂ ∂εzx1 2∂ ∂εyz∂ ∂εzz⎞ ⎟⎠Uel, (A2a) σme=⎛ ⎜⎝∂ ∂εxx1 2/parenleftbig∂ ∂εxy+∂ ∂ωxy/parenrightbig1 2/parenleftbig∂ ∂εzx−∂ ∂ωzx/parenrightbig 1 2/parenleftbig∂ ∂εxy−∂ ∂ωxy/parenrightbig∂ ∂εyy1 2/parenleftbig∂ ∂εyz+∂ ∂ωyz/parenrightbig 1 2/parenleftbig∂ ∂εzx+∂ ∂ωzx/parenrightbig1 2/parenleftbig∂ ∂εyz−∂ ∂ωyz/parenrightbig∂ ∂εzz⎞ ⎟⎠Ume. (A2b)When calculating stress from σold, differentiations with re- spect to off-diagonal strain components yield twice the correctvalue, as pointed out in Ref. [ 39]. This is because ε αβ↔εβα andωαβ↔−ωβαin the energy densities are not independent. Not taking care of the degrees of freedom of the strain tensorleads to a relation inconsistent with Eq. ( A2a): σαβ el=∂Uel ∂(∂βuα)=∂ενη ∂(∂βuα)∂Uel ∂ενη =1 2(δνβδηα+δηβδνα)∂Uel ∂ενη =∂Uel ∂εαβ, (A3) where the summation over ν,η doubly adds the off-diagonal elements. In our expression ( A2), in contrast, the factor 1 /2 appropriately compensates the doubled values in the off-diagonal elements. Equation ( A2a) implies that in the linear regime the elastic stress tensor is symmetric for any crys-tals whose elastic energy has the form Eq. ( 3a). Equation (A2b) reveals that the symmetric part of the magnetoelas- tic stress arises from magnetostriction (coupling to strain),whereas its antisymmetric part originates from the magne-torotation coupling. This implies that σ mecan differ from the conventional form ∂Ume/∂εαβwhen the magnetorotation coupling is relevant. In YIG [ 19] and other ferromagnets such as Galfenol [ 50,51], iron, and nickel [ 42] the effects of crystalline anisotropy are orders of magnitude smaller thanthat from the magnetostriction. However, the magnetorotationcoupling is significant in CoFeB or Ni /Ag films that are thinner than acoustic wavelengths [ 34,87]. APPENDIX B: A VERAGE STRAIN IN 1D PROBLEM The linearized LLG reads [Eq. ( 26)] /parenleftbigg m/bardbl m⊥/parenrightbigg (ω)=χFMR(ω,θ m)/bracketleftbigg/parenleftbigg h/bardbl h⊥/parenrightbigg −1 γμ 0/parenleftbigg /Omega1/prime13 me /Omega1/prime23 me/parenrightbigg/bracketrightbigg (ω), (B1) where χFMR reflects the purely magnetic response. We first compute the average strain in the magnetic film induced byMEC-BCs. For the one-dimensional problem, the componentsof the tensor Ω/prime mereads /Omega1/prime13 me=ωc∂zuxcos 2θm−ω/lscript c∂zuzsin 2θm, /Omega1/prime23 me=ωc∂zuycosθm,(B2) where ωc=ωM/2+γ(b2−K1)/Msandω/lscript c=γb1/Ms parametrize the magnetostriction and magnetorotation coupling. The BCs determine the relation between thecomplex acoustic wave amplitudes and magnetization. Wethen find the average strain in the magnet from Eq. ( 31) and write it in terms of magnetoelastic stress: u α(0,ω)−uα(−d,ω) d=−σαz me(θm) d˜ρ˜ctF(ω) ω+i˜ηel/2, uz(0,ω)−uz(−d,ω) d=−σzz me(θm) d˜ρ˜c/lscriptF/lscript(ω) ω+i˜ηel/2,(B3) 014403-11SATO, YU, STREIB, AND BAUER PHYSICAL REVIEW B 104, 014403 (2021) where α=x,yand F(ω)=F3+iF4 F1+iF2,F/lscript(ω)=F/lscript 3+iF/lscript 4 F/lscript 1+iF/lscript 2. (B4) The real-valued functions are defined as F1=−ω(β11sin˜ktdcosktL+β22cos˜ktdsinktL) +˜ηel 2(β/prime 21cos˜ktdcosktL−β/prime 12sin˜ktdsinktL), F2=−ω(β21cos˜ktdcosktL−β12sin˜ktdsinktL) −˜ηel 2(β/prime 11sin˜ktdcosktL+β/prime 22cos˜ktdsinktL), F3=ω[(˜CC+β11) cos ˜ktdcosktL −(˜SS+β22)s i n ˜ktdsinktL−2CcosktL] +˜ηel 2[(˜SC+β/prime 21)s i n ˜ktdcosktL +(˜CS+β/prime 12) cos ˜ktdsinktL−2SsinktL], F4=−ω[(˜SC+β21)s i n ˜ktdcosktL +(˜CS+β12) cos ˜ktdsinktL−2SsinktL] +˜ηel 2[(˜CC+β/prime 11) cos ˜ktdcosktL−(˜SS +β/prime 22)s i n ˜ktdsinktL−2CcosktL], where β11=˜CC+ρct ˜ρ˜ct˜SS,β 12=˜CS+ρct ˜ρ˜ct˜SC, β21=˜SC+ρct ˜ρ˜ct˜CS,β 22=˜SS+ρct ˜ρ˜ct˜CC.(B5) C=coshκtLandS=sinhκtLrepresent the wave attenuation in NM, while ˜C=cosh ˜κtdand ˜S=sinh ˜κtdquantify the attenuation in M. β/prime ijare given by multiplying ηel/˜ηelto the impedance ratios. F/lscript(ω)i nE q .( B3) is defined with corre- sponding longitudinal parameters. Replacing the strains in Eq. ( B2) with the average ( B3), and substituting the result into Eq. ( B1)g i v eE q .( 33)i n the main text. In the limit of vanishing acoustic damping ˜ηel,ηel→0,βis diagonal, F2,F4→0, and consequently the coupling strength [Eq. ( 35a)] becomes real, i.e., magnetization damping is not enhanced by phonon pumping. The theory byStreib et al. [15] corresponds to the case ˜ η el=0,ηel/negationslash=0 and L→∞ . APPENDIX C: ELASTIC WA VES IN A DISK The Helmholtz relation between the displacement vector and potentials in cylindrical coordinates reads [ 36] uR=∂/Phi1 ∂R+1 R∂/Xi1 z ∂φ−∂/Xi1φ ∂z, uφ=1 R∂/Phi1 ∂φ−∂/Xi1 z ∂R+∂/Xi1 R ∂z, uz=1 R/bracketleftbigg∂ ∂R(R/Xi1φ)−∂/Xi1 R ∂φ/bracketrightbigg +∂/Phi1 ∂z,(C1)where the zderivatives vanish for thin films. The choice of the gauge div /Xi1=ψ(r,t), is a constraint on the four components (/Phi1,/Xi1 α) so that the both sides of Eq. ( C1) have the same de- grees of freedom [ 36]. While ψ(r,t)=0 is suitable for planar configurations, we chose a different one for the present systemas discussed below. The elastic strain tensor components read[36] ε RR=∂uR ∂R, (C2a) εφφ=1 R∂uφ ∂φ+uR R, (C2b) εφz=1 2R∂uz ∂φ, (C2c) εzR=1 2∂uz ∂R, (C2d) εRφ=1 2/parenleftbigg1 R∂uR ∂φ+∂uφ ∂R−uφ R/parenrightbigg . (C2e) The stress tensors transform as σ=R−1 zσcarRz, where σcaris the tensor in Cartesian coordinates and the matrix Rzrotates the axes around the zaxis, Rz(φ)=⎛ ⎝cosφ−sinφ 0 sinφ cosφ 0 00 1⎞ ⎠. (C3) The stress-strain relation takes the same form as in Carte- sian coordinate systems since, using the property of the tracetr[ε car]=tr[R−1 zεcarRz], σαβ el=/parenleftbig R−1 zσel,carRz/parenrightbig αβ =λδαβtr[εcar]+2μ/parenleftbig R−1 zεcarRz/parenrightbig αβ =λδαβtr[ε]+2μεαβ, (C4) forα,β∈{R,φ,z}. We then write the elastic stress within NM in terms of displacement potentials: σRR el=/parenleftbig 2μ∂2 R−λk2 /lscript/parenrightbig /Phi1+2μ R∂φ/parenleftbigg ∂R−1 R/parenrightbigg /Xi1z,(C5a) σφR el=μ/parenleftbigg2 R∂R+k2 t/parenrightbigg /Xi1z+2μ R∂φ/parenleftbigg ∂R−1 R/parenrightbigg /Phi1,(C5b) σzR el=μ/bracketleftbigg −∂φ R/parenleftbigg ∂R−1 R/parenrightbigg /Xi1R +/parenleftbigg ∂2 R+1 R∂R−1 R2/parenrightbigg /Xi1φ/bracketrightbigg . (C5c) The magnetoelastic stress tensor transforms analogously as σme=R−1 zσme,carRz. The components relevant for the BCs are, to linear order in transverse magnetization and for ω> 0, ⎛ ⎝σRR me σφR me σzR me⎞ ⎠=⎛ ⎝b2mysin 2φ b2mycos 2φ b2mzcosφ⎞ ⎠. (C6) Magnetoelastic stress on a circular boundary does not depend on the longitudinal coupling b1. In frequency domain the 014403-12DYNAMIC MAGNETOELASTIC BOUNDARY CONDITIONS … PHYSICAL REVIEW B 104, 014403 (2021) EOM for the deformation potentials Eq. ( 43) become /bracketleftbigg ∂2 R+1 R∂R+1 R2∂2 φ+k2 /lscript/bracketrightbigg /Phi1=0, /bracketleftbigg ∂2 R+1 R∂R+1 R2∂2 φ+k2 t/bracketrightbigg /Xi1z=0, /bracketleftbigg ∂2 R+1 R∂R+k2 t+1 R2/parenleftbig ∂2 φ−1/parenrightbig/bracketrightbigg /Xi1R−2 R2∂/Xi1φ ∂φ=0, /bracketleftbigg ∂2 R+1 R∂R+k2 t+1 R2/parenleftbig ∂2 φ−1/parenrightbig/bracketrightbigg /Xi1φ+2 R2∂/Xi1 R ∂φ=0,(C7) where k/lscript=ωc/lscript,kt=ω/ct. Equations ( C5), (C7), and ( C1) with∂z=0 confirm that the in-plane and out-of-plane dy- namics decouple. By the separation of variables /Phi1(R,φ)= f(R)X(φ) and /Xi1α(R,φ)=fα(R)Xα(φ), the first two equa- tions in Eq. ( C7) reduce to the Bessel differential equations for the radial component, whose solutions are given either bya linear superposition of the Bessel function of the first andsecond kind, AJ m(kR)+BYm(kR), or by a combination of the Hankel function of the first and second kind, AH(1) m(kR)+ BH(2) m(kR). Since the amplitudes must be finite in M and backward waves represented by H(2) mare not excited in the infinite NM, f(R)=/braceleftBigg˜A/lscriptJm(˜k/lscriptR) A/lscriptH(1) m(k/lscriptR), (C8) fz(R)=/braceleftBigg˜AtJn(˜ktR) AtH(1) n(ktR), (C9) where the first cases are for M (0 <R<a) and the second for NM ( a<R).X(φ) is then a superposition of cos mφand sinmφandXz(φ) a superposition of cos nφand sin nφ, where m,nare integers. It follows from Eqs. ( C5) and ( C6) that the BCs Eq. ( 45) in the main text hold for all φonly if X,dXz dφ∝sin 2φ, (C10) dX dφ,Xz∝cos 2φ, (C11)and thus we may assume X(φ)=sin 2φ and Xz=cos 2φ. We next consider out-of-plane oscillations. If the BCs are to hold for arbitrary φ, then Xφand dXR/dφmust be proportional to cos φ, allowing us to assume XR=sinφ and Xφ=cosφ. Out-of-plane dynamics in Eq. 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PhysRevB.79.054424.pdf

Spin and charge pumping in magnetic tunnel junctions with precessing magnetization: A nonequilibrium Green function approach Son-Hsien Chen,1,2,*Ching-Ray Chang,2,†John Q. Xiao,1and Branislav K. Nikoli ć1 1Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716-2570, USA 2Department of Physics, National Taiwan University, Taipei 10617, Taiwan /H20849Received 21 November 2008; published 20 February 2009 /H20850 We study spin and charge currents pumped by precessing magnetization of a single ferromagnetic layer within F/H20841I/H20841NorF/H20841I/H20841F/H20849F-ferromagnet; I-insulator; N-normal metal /H20850multilayers of nanoscale thickness attached to two normal-metal electrodes with no applied bias voltage between them. Both simple one-dimensionalmodel, consisting of a single precessing spin and a potential barrier as the “sample,” and realistic three-dimensional devices are investigated. In the rotating reference frame, where the magnetization appears to bestatic, these junctions are mapped onto a four-terminal dc circuit whose effectively half-metallic ferromagneticelectrodes are biased by the frequency /H6036 /H9275/eof microwave radiation driving magnetization precession at the ferromagnetic resonance /H20849FMR /H20850conditions. We show that pumped spin current in F/H20841I/H20841Fjunctions, diminished behind the tunnel barrier and increased in the opposite direction, is filtered into charge current by the secondFlayer to generate dc pumping voltage of the order of /H110111 /H9262V/H20849at FMR frequency /H1101110 GHz /H20850in an open circuit. In F/H20841I/H20841Ndevices, several orders of magnitude smaller charge current and the corresponding dc voltage appear concomitantly with the pumped spin current due to barrier induced asymmetry in the transmissioncoefficients connecting the four electrodes in the rotating-frame picture of pumping. DOI: 10.1103/PhysRevB.79.054424 PACS number /H20849s/H20850: 76.50. /H11001g, 72.15.Gd, 72.25.Mk, 72.25.Ba I. INTRODUCTION The pursuit of “second generation” spintronic devices1 has largely been focused on harnessing coherent spin states and their dynamics in metals and semiconductors. This re-quires to maintain and control spin orientations transverse toexternally applied or internal magnetic fields. The salient ex-ample of phenomena involving both coherent spins and theirtime evolution is the spin-transfer torque where spin current of large enough density injected into a ferromagnetic layereither switches its magnetization from one static configura-tion to another or generates a dynamical situation withsteady-state precessing magnetization. 2In the reciprocal ef- fect, termed spin pumping because it occurs in setups with- out applied bias voltage,3microwave driven precessing mag- netization of a single ferromagnetic layer under the FMRconditions emits pure spin current /H20849not accompanied by any net charge flux 4/H20850into adjacent normal-metal layers. In the conventional picture of spin pumping,3F/H20841Ninterface pumps spin current in both directions,5so that its magnitude is de- termined by the interfacial parameters which govern trans-port of spins that are noncollinear to the magnetization di-rection at the interface. 2,3 The spin current emitted from the Flayer with moving magnetization has been observed6–8in early experiments only indirectly as an enhancement of Gilbert damping8of magnetization dynamics in inhomogeneous structures due tothe presence of F/H20841Ninterfaces and fast relaxation of pumped spins in good “spin sink” 3Nlayers which ultimately leads to a loss of the angular momentum.9Very recently it has been converted10into the conventionally measurable voltage sig- nals through the inverse spin Hall effect /H20849where longitudinal spin current injected into a metal with spin-orbit couplingsgenerates transverse voltage between lateral edges of thesample 4/H20850. Another electrical scheme is based on N1/H20841F/H20841N2multilayers11where different voltages develop at different F/H20841Niinterfaces due to backflow spin current /H20849driven by the spin accumulation in Nilayers built up by directly pumped spin current /H20850, which is detected by the precessing Flayer itself.12These experiments suggest that spin pumping de- vices could be exploited as generators3,7of elusive pure spin currents,4where spin current injected from Finto adjacent N layers carries fast precessing spins in gigahertz range of fre-quencies offering new functionality for metal spintronics. 8 They can also be used to probe important aspects of spindynamics in thin Flayers. 13 Unlike spin-transfer torque that has been demonstrated in F/H20841I/H20841Fmagnetic tunnel junctions /H20849MTJ /H20850,2it has been consid- ered that low transparent interfaces would completely screenthe interfacial spin pumping effect /H20849as observed in some experiments 14/H20850, unless the tunnel barrier has nontrivial mag- netic properties.15Thus, recent surprising measurements16,17 of large voltage signals of the order of /H110111/H9262V/H20849at FMR frequencies f/H112292 GHz and precession cone angles /H9258/H1122910°/H20850 in microwave driven F/H20841I/H20841Nand F/H20841I/H20841Ftunnel junctions, as opposed to /H1101110 nV pumping signals11inN1/H20841F/H20841N2multilay- ers, have attracted considerable theoretical attention.18–20 Nevertheless, the puzzle of unexpectedly large magnitude of the observed dc pumping voltages persists: /H20849i/H20850the scattering18approach to transport of noninteracting quasipar- ticles through defect-free epitaxial F/H20841I/H20841FMTJ finds /H110111n V signals at FMR frequency f=2 GHz and precession cone angle /H9258=10°; /H20849ii/H20850the tunneling Hamiltonian approach19for clean F/H20841I/H20841FMTJ and the same fand/H9258parameters sets the maximum dc pumping voltage at /H110110.01/H9262V in parallel and /H110111/H9262V in antiparallel configuration of two Felectrodes; and /H20849iii/H20850the tunneling Hamiltonian approach combined with semiclassical modeling of the interplay of spin diffusion andself-consistent screening around interfaces in F/H20841I/H20841FandF/H20841I/H20841N junctions involves too many unknown phenomenological pa-PHYSICAL REVIEW B 79, 054424 /H208492009 /H20850 1098-0121/2009/79 /H208495/H20850/054424 /H208498/H20850 ©2009 The American Physical Society 054424-1rameters, thereby offering only a wide range of possible pumping voltages for both of these junctions.20 Here we address the problem of spin pumping and in- duced voltages by high-frequency magnetization dynamicsinF/H20841I/H20841FandF/H20841I/H20841Njunctions within the framework of non- equilibrium Green function /H20849NEGF /H20850formalism. 21,22We note that NEGFs have been utilized before to study spin23,24and charge25pumping by time-dependent fields acting on finite- size paramagnetic devices attached to electrodes held at thesame electrochemical potential. Since NEGF formalismtakes as an input a microscopic Hamiltonian, it makes itpossible to include, in a controlled fashion, the fullgeometry 26of experimental devices /H20849such as the finite thick- ness of F,I, and Nlayers, down to few atomic monolayers, which play an important role in the magnetoresistance27and spin-transfer torque28of crystalline MTJs /H20850, the properties of the insulating barrier /H20849including disorder effects /H20850, as well as the interactions responsible for spin-flip processes in F. The NEGF formalism also makes it easy to take into account ab initio input27–30on the F/H20841Iinterface electronic and magnetic structure and the self-consistently developed nonequilibriumspin and charge distributions around it. Furthermore, NEGF approach yields a remarkably trans- parent physical picture of pumping in ferromagnetic multi-layered systems. For example, in the simplest model ofpumping, generated by a single spin precessing with fre- quency /H9275in Fig. 1/H20849a/H20850, the NEGF rotated into the rest frame of the spin maps the original laboratory-frame device onto adc circuit in Fig. 1/H20849b/H20850. The central sample of this circuit, which contains time-independent spin interactions, is at-tached to four electrodes that allow only one spin species topropagate through them and are, therefore, labeled by L-left, R-right, spin- ↑, and spin- ↓. These four electrodes are biased by the voltage /H6036 /H9275/e, so that spin- ↓electrons flow from elec- trodes at higher electrochemical potential and precess insidethe sample due to spin-dependent interactions to be able toenter into electrodes at a lower electrochemical potential asspin- ↑states. Thus, this picture reduces the quantitative analysis of spin and charge pumping by precessing spins tomultiterminal Landauer-Büttiker-type formulas for spin-resolved charge currents as encountered in, e.g., the mesos-copic spin Hall effect. 22,31 The paper is organized as follows. In Sec. IIwe exploit the physical picture of pumping provided by Fig. 1/H20849b/H20850to analyze local spin and charge currents flowing away from the single precessing spin toward the neighboring sites alongthe tight-binding chain in one dimension /H208491D/H20850. This frame- work is extended to total pumped currents and associated voltages in three-dimensional /H208493D/H20850multilayered structures, such as F/H20841N/H20841F,F/H20841I/H20841F, and F/H20841I/H20841N, in Sec. III. We conclude in Sec. IV. Our principal results—pumped spin and charge cur- rents in 1D model and voltage signals in F/H20841I/H20841Fand F/H20841I/H20841N junctions—are shown in Figs. 2and5, respectively. II. NEGF APPROACH TO SPIN AND CHARGE PUMPING BY A SINGLE PRECESSING SPIN IN ONE DIMENSION The toy 1D model in Fig. 1/H20849a/H20850encodes most of the essen- tial physics of pumping by precessing spins while making itpossible to obtain analytical solution for the magnitude ofpumped currents. For simplicity, we start from the often em-ployed in spin-transfer torque 32and spin pumping19studies Stoner-type Hamiltonian,33 Hˆlab/H20849t/H20850=/H20858 r,/H9268,/H9268/H11032/H20873/H9255r/H9254/H9268/H9268/H11032−/H9004r 2mr/H20849t/H20850·/H9268ˆ/H9268/H9268/H11032/H20874cˆr/H9268†cˆr/H9268/H11032 −/H9253/H20858 /H20855rr/H11032/H20856/H9268cˆr/H9268†cˆr/H11032/H9268, /H208491/H20850 in the local orbital basis suited for NEGF calculations.21,22Its time dependence stems from the unit vector m/H20849t/H20850along the local magnetization direction, which is assumed to be spa-tially uniform and steadily precessing around the zaxis with a constant cone angle /H9258. The operators cˆr/H9268†/H20849cˆr/H9268/H20850create /H20849an- nihilate /H20850electron with spin /H9268at site r, and/H9253is the nearest- neighbor hopping. The coupling of itinerant electrons to col-lective magnetic dynamics is described through the material- dependent exchange potential /H9004 r, where /H9268ˆ=/H20849/H9268ˆx,/H9268ˆy,/H9268ˆz/H20850is the vector of the Pauli matrices and /H9268ˆi/H9268/H9268/H11032denotes the Pauli- matrix elements. The on-site potential /H9255raccounts for the presence of the barrier /H20851such as /H9255r=/H9255Ion the second site of the sample in Fig. 1/H20849a/H20850/H20852, disorder, external electric field, and it can also be used to shift the band bottom conveniently. Thesample is attached to two semi-infinite ideal /H20849spin and charge interaction free /H20850electrodes, which terminate in macroscopic reservoirs held at the same electrochemical potential /H9262p =EFwhere EFis the Fermi energy. The fundamental objects21of the NEGF formalism are the retarded FIG. 1. /H20849Color online /H20850/H20849a/H20850The 1D model of spin pumping where the sample consisting of two sites, one hosting the single spin ro-tating with frequency /H9275and the other one hosting the potential barrier of height /H9255I, is attached to two semi-infinite tight-binding chains /H20849/H9253is the hopping parameter /H20850playing the role of electrodes with no applied bias voltage between them. In the rotating referenceframe the spin is static and the device /H20849a/H20850is mapped into the four- terminal dc circuit in panel /H20849b/H20850whose electrodes have electrochemi- cal potential shifted by /H11006/H6036 /H9275/2 with respect to the equilibrium Fermi level EFof unbiased electrodes in the laboratory frame.CHEN et al. PHYSICAL REVIEW B 79, 054424 /H208492009 /H20850 054424-2Grr/H11032r,/H9268/H9268/H11032/H20849t,t/H11032/H20850=−i /H6036/H9008/H20849t−t/H11032/H20850/H20855/H20853cˆr/H9268/H20849t/H20850,cˆr/H11032/H9268/H11032†/H20849t/H11032/H20850/H20854/H20856, /H208492/H20850 and the lesser Grr/H11032/H11021,/H9268/H9268/H11032/H20849t,t/H11032/H20850=i /H6036/H20855cˆr/H11032/H9268/H11032†/H20849t/H11032/H20850cˆr/H9268/H20849t/H20850/H20856, /H208493/H20850 Green functions /H20849/H20855¯/H20856denotes the nonequilibrium statistical average21/H20850which describe the density of available quantum states and how electrons occupy those states, respectively.Since nonequilibrium problems are not time-translation in-variant, these Green functions depend on two time variablesseparately. However, the cumbersome double time depen-dence of NEGF in general pumping problems 25can be eliminated24for the special case of time-dependent potential caused by precessing magnetization using the compensatingrotation 34of the system described by the unitary transforma- tion Uˆ=ei/H9275/H9268ˆzt/2/H20849for magnetization precessing counterclock-wise /H20850. Thus, the Hamiltonian in the rotating frame19 Hˆrot=UˆHˆlab/H20849t/H20850Uˆ†−i/H6036Uˆ/H11509 /H11509tUˆ†=Hˆlab/H208490/H20850−/H6036/H9275 2/H9268ˆz, /H208494/H20850 is time independent. The term /H6036/H9275/H9268ˆz/2, which appears uni- formly in the Hamiltonian of the sample or Nelectrodes, will spin-split the bands of the Nelectrodes. This yields a rotating-frame picture of pumping based on the four-terminaldevice in Fig. 1/H20849b/H20850. The device in Fig. 1/H20849b/H20850guides us in setting up the NEGF equations for the description of currents flowing between itsfour electrodes, labeled by p, /H9268/H20849p=L,Rand/H9268=↑,↓/H20850, which are biased by the voltage /H6036/H9275/e. The electrodes behave effec- tively as the half-metallic ferromagnets, emitting, or absorb-ing only one spin species. The rotating-frame Green func-tions Gr/H20849E/H20850=/H20851E−Hrot−/H9018r/H20849E/H20850/H20852−1, /H208495/H20850 and G/H11021/H20849E/H20850=Gr/H20849E/H20850/H9018/H11021/H20849E/H20850Ga/H20849E/H20850, /H208496/H20850 depend on /H9270=t−t/H11032, or energy Eafter the time difference /H9270is Fourier transformed. Here the advanced Green function is Ga/H20849E/H20850=/H20851Gr/H20849E/H20850/H20852†, and Hrotis the matrix representing Hˆrotin the local-orbital basis. The retarded self-energy matrix /H9018r/H20849E/H20850=/H20858p,/H9268/H9018pr,/H9268/H20849E/H20850is the sum of self-energies introduced by the interaction with the leads which determine escape ratesof spin- /H9268electron into the electrodes p,/H9268in Fig. 1/H20849b/H20850. For interacting systems /H9018r/H20849E/H20850would also contain electron-electron and electron-phonon contributions, whilefor noninteracting systems, described by Hamiltonian /H208494/H20850, the lesser self-energy is expressed in terms of /H9018pr,/H9268/H20849E/H20850as /H9018/H11021/H20849E/H20850=/H20858 p,/H9268ifp/H9268/H20849E/H20850/H9003p/H9268/H20849E/H20850. /H208497/H20850 The level broadening matrix /H9003p/H9268/H20849E/H20850=−2I m /H9018pr/H20873E+/H9268/H6036/H9275 2/H20874, /H208498/H20850 is obtained from the usual self-energy matrices21/H9018pr/H20849E/H20850of semi-infinite leads in the laboratory frame with their energyargument being shifted by /H9268/H6036/H9275/2 to take into account the “bias voltage” in accord with Fig. 1/H20849b/H20850. The distribution function of electrons in the four electrodes of the rotating-frame dc circuit is given by f p/H9268/H20849E/H20850=1 exp/H20851/H20849E−EF+/H9268/H6036/H9275/2/H20850/kT/H20852+1, /H208499/H20850 where /H9268=+ for spin- ↑and/H9268=− for spin- ↓. Since the device is not biased in the laboratory frame, the shifted Fermi func-tion in Eq. /H208499/H20850is uniquely specified by the polarization ↑or↓ of the electrode, so that we remove the lead label pfrom it in the equations below. The basic transport quantity for the rotating-frame dc cir- cuit is the spin-resolved bond charge current 22carrying spin-/H9268electrons from site rto neighboring site r/H11032(b)(a) FIG. 2. The /H20849a/H20850Sz-spin currents and /H20849b/H20850charge currents pumped by a single precessing spin as a function of the potential barrier onthe second site of the sample in 1D model shown in Fig. 1/H20849a/H20850. The parameters of the model are: f= /H9275/2/H9266=20 GHz; /H9258=10°, /H9004/EF =0.85, and electrons in the macroscopic reservoirs to which the electrodes are attached have the Fermi energy EF=2/H9253.SPIN AND CHARGE PUMPING IN MAGNETIC TUNNEL … PHYSICAL REVIEW B 79, 054424 /H208492009 /H20850 054424-3Jrr/H11032/H9268=e/H9253 /H6036/H20885 −/H11009/H11009 dE/H20851G¯ r/H11032r/H11021,/H9268/H9268/H20849E/H20850−G¯ rr/H11032/H11021,/H9268/H9268/H20849E/H20850/H20852. /H2084910/H20850 This gives spin Jrr/H11032S=Jrr/H11032↑−Jrr/H11032↓, /H2084911/H20850 and charge Jrr/H11032=Jrr/H11032↑+Jrr/H11032↓, /H2084912/H20850 bond currents flowing between neighboring sites.22Equation /H2084910/H20850can be evaluated analytically for 1D model assuming that for small enough /H6036/H9275/H11270EFwe can use f↓/H20849E/H20850−f↑/H20849E/H20850 =/H6036/H9275/H9254/H20849E−EF/H20850at zero temperature. Such “adiabatic approximation”24is analogous to linear-response limit in conventional transport calculations for devices biased bysmall voltage difference. The S zcomponent of the bond spin current between the precessing site and its nearest neighbor in the sample, asillustrated in Fig. 1/H20849a/H20850, is given by J rr/H11032Sz=/H6036/H9275sin2/H9258/H92532/H90042/H20849Im/H90181D/H208502 8/H9266/H20841R/H208412 /H11003/H208514/H20849/H92532+/H9255I2/H20850+4/H20841/H90181D/H208412−8/H9255IRe/H90181D/H20852, /H2084913/H20850 where the terms O/H20851/H20849/H6036/H9275/H208502/H20852are neglected. Here /H90181D=/H20849EF −2/H9253−/H20881/H20849EF−2/H9253/H208502−4/H92532/H20850/2 is the self-energy of 1D semi- infinite lead /H20849i.e., tight-binding chain /H20850and R=/H208494/H90181D2 −/H90042/H20850/H20849/H90181D−/H9255I/H208502/4+/H208494/H9255I/H90181D−4/H90181D2/H20850/H92532/2+/H92534. In deriving Eq. /H2084913/H20850, we assume uniform band bottom, so that /H9255I/H21739/H9255I −/H20849/H9004cos/H9258/H20850/2 and EF/H21739EF+/H20849/H9004cos/H9258/H20850/2 is used to plot Fig. 1. The expression in Eq. /H2084913/H20850reproduces all major features of the scattering approach3to adiabatic /H20849/H6036/H9275/H11270/H9004 /H20850regime of spin pumping by F/H20841Ninterface in 3D multilayers: /H20849i/H20850the pure spin current carrying Szspins is proportional to /H6036/H9275and sin2/H9258;/H20849ii/H20850Szcomponent Jrr/H11032Szof the spin current tensor is time independent in both rotating and laboratory frames; and /H20849iii/H20850SxandSycomponents of the pumped spin current oscil- late harmonically with time in the laboratory frame. More-over, when potential barrier /H9255 Iis introduced into the sample, we find in Fig. 2/H20849a/H20850that spin current on the right decays with increasing /H9255Iwhile pumped spin current flowing on the left increases to about twice the value of the sum Jrr/H11032Sz,left+Jrr/H11032Sz,right of the left and right spin currents pumped symmetrically in the absence of the barrier. This effect can clarify the origin ofpossible Gilbert damping enhancement in realistic MTJ de-vices consisting of N/H20841F/H20841I/H20841F/H20841Nmultilayers, rather than infinite Felectrodes, where angular-momentum loss develops due to increasing spin pumping into the left Nelectrode even when the insulating barrier Isuppresses spin pumping on the right side of the junction. Figure 2/H20849b/H20850demonstrates that nonzero potential /H9255 I/HS110050 also leads to concomitant pumping of a tiny charge currentinto the right electrode, which is several order of magnitudesmaller than the pumped spin current. Unlike pumping ofspins which is linear in frequency, such pumped charge cur-rent scales as /H11011 /H92752. We discuss its origin in Sec. IIIby ana- lyzing total charge current in the Nterminals expressed interms of the transmission coefficients between the fully spin- polarized electrodes in the rotating frame. III. NEGF APPROACH TO SPIN AND CHARGE PUMPING IN 3D F/H20870N/H20870F,F/H20870I/H20870F, AND F/H20870I/H20870NMULTILAYERS We extend this analysis to a 3D MTJ shown in Fig. 3 which consists32of infinite planes of F,N, and Imaterials modeled on a simple-cubic tight-binding lattice with singles-orbital per site using Hamiltonian /H208491/H20850. The effective dc cir- cuit /H20851Fig. 1/H20849b/H20850/H20852in the rotating frame makes it easy to write the expression for the total charge current in the left and right Nelectrodes. For example, spin- ↓electrons can flow from L↓ lead at higher electrochemical potential into R↑ lead at the lower electrochemical potential. They enter R↑ lead as spin- ↑ electrons with probability determined by the precession in- side the sample since Hˆrotcontains terms proportional to /H9268ˆx for which the injected spin states /H20841↓/H20856from L↓ lead /H20849polarized along the zaxis /H20850are not the eigenstates. Since pumped charge current is necessarily conserved, as exemplified by Fig. 2/H20849b/H20850, we arbitrarily select the right elec- trode /H20849current flowing into the electrode is assumed to be positive /H20850to find its explicit expression in terms of the mul- titerminal Landauer-Büttiker formulas31for spin-resolved quantum transport, I=e h/H20885 −/H11009/H11009 dE/H20853TRL↑↓/H20851f↓/H20849E/H20850−f↑/H20849E/H20850/H20852−TLR↑↓/H20851f↓/H20849E/H20850−f↑/H20849E/H20850/H20852/H20854. /H2084914/H20850 Here the transmission coefficients Tpp/H11032/H9268/H9268/H11032determine the prob- ability for /H9268/H11032electrons injected through lead p/H11032to emerge in electrode pas spin- /H9268electrons. They can be computed from the NEGF-based formula21 FIG. 3. /H20849Color online /H20850The magnetic tunnel junction with pre- cessing magnetization in the left Flayer is modeled on a simple- cubic tight-binding lattice. The thicknesses of the ferromagnetic /H20849F, F/H11032/H20850and thin insulating /H20849I/H20850layers are measured using the number of atomic monolayers dF,dF/H11032, and dI, respectively. The P and AP configurations of MTJ correspond to the magnetization of the rightFlayer being parallel or antiparallel to the zaxis around which spatially uniform magnetization of the left Flayer steadily pre- cesses with a constant cone angle /H9258.CHEN et al. PHYSICAL REVIEW B 79, 054424 /H208492009 /H20850 054424-4Tpp/H11032/H9268/H9268/H11032=T r /H20853/H9003p/H9268Gpp/H11032r,/H9268/H9268/H11032/H9003p/H11032/H9268/H11032/H20851Gpp/H11032r,/H9268/H9268/H11032/H20852†/H20854, /H2084915/H20850 which is written here in the spin-resolved form. The block Gpp/H11032r,/H9268/H9268/H11032of the retarded Green function matrix consists of those matrix elements which connect the layer of the sample at- tached to lead p/H11032to the layer of the sample attached to lead p. In general, the spin current is not conserved, as illustrated by Fig. 2/H20849a/H20850, and we choose to compute it in the left N electrode ILS=e h/H20885 −/H11009/H11009 dE/H20853TLR↑↓/H20851f↓/H20849E/H20850−f↑/H20849E/H20850/H20852+TRL↑↓/H20851f↓/H20849E/H20850−f↑/H20849E/H20850/H20852 +2TLL↑↓/H20851f↓/H20849E/H20850−f↑/H20849E/H20850/H20852/H20854. /H2084916/H20850 The expressions Eqs. /H2084914/H20850and /H2084916/H20850for total currents are equivalent to the sum of all bond charge Jrr/H11032or bond spin Jrr/H11032Scurrents, respectively, where summation is performed over the pairs of sites within the electrode at a chosen cross section.22By the same token, the analytical expression /H20851Eq. /H2084913/H20850/H20852for the spin current is already equivalent to the result obtained from Eq. /H2084916/H20850since no summation is necessary for the cross section consisting of a single site. The pumped charge current in multilayers with the second analyzing Flayer originates from spin filtering by the static magnetization of the analyzing Flayer of current pumped toward the right. That is, we find IRSz=0,ILSz/HS110050 and I/HS110050i n such systems. In junctions with a single precessing Flayer the pumped spin current is pure ifILSz=IRSz/HS110050 and I=0. The possibility of nonzero pumped charge current even in junc-tions with only one Flayer whose magnetization is precess- ing, as exemplified by Fig. 2/H20849b/H20850andF/H20841I/H20841Njunctions in gen- eral /H20851see Figs. 5/H20849c/H20850and5/H20849d/H20850/H20852, is explained by Eq. /H2084914/H20850as the consequence of the asymmetry in transmission coefficients T RL↑↓−TLR↑↓/HS110050 when arbitrary potential /H9255I/HS110050 is introduced in one of the layers.The transmission coefficients can also explain the unex- pectedly large enhancement of ILSzorJmm /H11032Sz,leftin Fig. 2. As the barrier height /H9255Iincreases, TLR↑↓and TRL↑↓diminish to very small value while 2 TLL↑↓increases to about four times its value at/H9255I=0 due to quantum interferences effects on the left side of the device /H20849quantum interferences were also found to en- hance pumped spin current when coherent backscatteringfrom disorder occurs in finite-size conductors at paramag- netic resonance 24/H20850.A t/H9255I=0,TLR↑↓=TRL↑↓=TLL↑↓so that spin cur- rents of the same magnitude are pumped in both directionssymmetrically. The pumped charge current is translated into dc voltage in open circuits via V pump=I G/H20849/H9258/H20850, /H2084917/H20850 where G/H20849/H9258/H20850is the conductance of F/H20841I/H20841F/H20849orF/H20841I/H20841Nwhen the second Flayers is removed /H20850junction sketched in Fig. 3 whose first Flayer has its static magnetization tilted by an angle /H9258away from the zaxis and the linear-response bias voltage is applied between the Nelectrodes in the laboratory frame. The quantity G/H20849/H9258/H20850=2e2TRL/hcan also be computed via the standard NEGF formula21as in Eq. /H2084915/H20850but for total TRL, rather than spin-resolved, transmission coefficient ex- pressed in terms of the retarded Green function and self-energies in the laboratory frame. The largest voltage signal of spin pumping is expected in high quality epitaxial Fe /H20841MgO /H20841Fe tunnel junctions. 19To mimic their huge tunneling magnetoresistance /H20849TMR /H20850, while using the simple single-orbital tight-binding Hamiltonian /H208491/H20850, we adopt the same parameters employed in Ref. 18:EF =4.5 eV, /H9004/EF=0.85, /H9253=1.0 eV, and the barrier height measured relative to the Fermi energy Ub=/H20849/H9255I−EF/H20850is Ub/EF=0.25. The band bottom is aligned across all layers of the junction with the bottom of the band for majority spins inF/H20849similarly to Ref. 18/H20850. The “optimistic” TMR ratio for this junction with d I=5 monolayers of the insulating material is TMR= /H20849RAP−RP/H20850/RP/H112293900 %, which is close to ab initio computed zero-bias TMR /H112293700 % for defect-free Fe/H20841MgO /H20841Fe MTJ containing five MgO layers.30 In the coherent limit of tunneling,30applicable to ideal crystalline structures without any defect scattering, the in-plane wave vector k /H20648=/H20849ky,kz/H20850is conserved and all NEGF quantities depend on it. This requires to integrate Tpp/H11032/H9268/H9268/H11032/H20849E,k/H20648/H20850 in Eq. /H2084914/H20850and /H2084916/H20850over the two-dimensional /H208492D/H20850Brillouin zone /H20849BZ/H20850. Thus, in the adiabatic limit and at zero tempera- ture we use the following formulas to obtain the charge cur-rent: I=e /H9275 2/H9266/H20885 BZdk/H20648/H20849TRL↑↓−TLR↑↓/H20850, /H2084918/H20850 and the spin current in the left lead ILS=e/H9275 2/H9266/H20885 BZdk/H20648/H20849TLR↑↓+TRL↑↓+2TLL↑↓/H20850, /H2084919/H20850 pumped by magnetization precessing at frequency /H9275. The computational algorithm for this integration can beFIG. 4. /H20849Color online /H20850The dc pumping voltage in F/H20841N/H20841Fmulti- layers attached to two semi-infinite Nelectrodes as the function of the thickness of Flayer whose magnetization is precessing with cone angle /H9258=10° at frequency f=/H9275/2/H9266=20 GHz. The parameters describing the multilayer are EF=4.5 eV, /H9004/EF=0.85 /H20849in both F layers /H20850, and/H9253=1.0 eV.SPIN AND CHARGE PUMPING IN MAGNETIC TUNNEL … PHYSICAL REVIEW B 79, 054424 /H208492009 /H20850 054424-5substantially accelerated by transforming the 2D planar mo- mentum integral into a single integral over the in-plane ki-netic energy /H20885 −/H9266/a/H9266/a/H20885 −/H9266/a/H9266/a dkydkzTpp/H11032/H9268/H9268/H11032/H20849E,k/H20648/H20850=/H208732/H9266 a/H208742/H20885 −/H11009/H11009 d/H9255yz/H92672D/H20849/H9255yz/H20850 /H11003Tpp/H11032/H9268/H9268/H11032/H20849E,/H9255yz/H20850, /H2084920/H20850 where we utilize the two-dimensional density of states /H92672D/H20849/H9255yz/H20850for a square lattice and the fact that Tpp/H11032/H9268/H9268/H11032depends on k/H20648through the in-plane kinetic energy /H9255yz. In the case of nearest-neighbor hopping on a square lattice, the kinetic en-ergy within a monolayer is given by /H9255 yz=4/H9253−2/H9253/H20851cos/H20849kya/H20850 +cos /H20849kza/H20850/H20852, where ais the lattice spacing. The effect of the in-plane kinetic energy is equivalent to an increase in theon-site potential /H9255 r/H21739/H9255r+/H9255yz. To provide reference values for understanding the magni- tude of pumping voltages in tunnel junctions, as well as toconnect our theory to a “standard model” of interfacial spinpumping provided by the scattering theory, 3,5,8we first com- pute the dc voltage Vpump generated in F/H20841N/H20841Fmultilayers. The chosen cone angle /H9258=10° and FMR frequency f =/H9275/2/H9266=20 GHz are within the range of typical values en- countered in experiments11where the results in Figs. 4and5can easily be rescaled for other values of these two param- eters using the general /H11008sin2/H9258and/H11008/H9275dependence in Eq. /H2084913/H20850. Figure 4demonstrates that pumping involves only a thin layer of Fmaterial around the F/H20841Ninterface. However, while in the scattering theory3adiabatic pumping develops over the atomistically short ferromagnetic coherence length/H11011/H6036 vF//H9004, which in our junction is /H6036vF//H9004/H11229a, we find that pumping in Fig. 5involves about five monolayers of the ferromagnetic material. Here we assume that the magnitudeof pumped current generated on this length scale is notaffected 19by spin-relaxation processes /H20851not included in Hamiltonian /H208491/H20850/H20852that typically occur on a much longer length scale.35The pumped voltages in both P /H20849parallel /H20850and AP /H20849antiparallel /H20850configurations are below the maximum3,18 expected voltage Vpump/H11021/H6036/H9275/H1101583/H9262V/H20849for the explanation of P and AP junction setups in the context of pumping byprecessing magnetization, see Fig. 3/H20850. The dc pumping voltage for tunnel junctions is shown in Fig.5. Although the presence of the potential barrier within I layer of F/H20841I/H20841Fjunction increases the resistance of the junc- tion in Eq. /H2084917/H20850, the pumped charge current decreases faster so that V pump decreases with increasing barrier height Ub.I n contrast to the scattering result of Ref. 18where Vpump in- creases with increasing Ubfor all thicknesses of the Ilayer, we find in Fig. 5/H20849b/H20850such increase only if the Ilayer consists of a single monolayer. The large difference between VpumpPFIG. 5. /H20849Color online /H20850The dc pumping voltage in /H20851/H20849a/H20850and /H20849b/H20850/H20852F/H20841I/H20841Fand /H20851/H20849c/H20850and /H20849d/H20850/H20852F/H20841I/H20841Nmultilayers attached to two semi-infinite N electrodes as the function of the barrier height Ub/H20849measured relative to the Fermi energy /H20850. The magnetization of the left Flayer is precessing with cone angle /H9258=10° at frequency f=/H9275/2/H9266=20 GHz. The parameters describing the multilayer are EF=4.5 eV, /H9004/EF=0.85 /H20849in both F layers for F/H20841I/H20841Fjunction /H20850, and/H9253=1.0 eV.CHEN et al. PHYSICAL REVIEW B 79, 054424 /H208492009 /H20850 054424-6andVpumpAPconfiguration stems from huge TMR ratio for this junction while the magnitude of pumped charge current re-mains virtually the same for both P and AP configurations. These results are quite close to /H110111 /H9262V for VpumpAP −VpumpPobserved in MTJs with Al 2O3barriers, and an order of magnitude larger voltages in MTJs with MgO barriers.17 The experimentally observed17change in sign of Vpump de- pending on the type of the barrier /H20849Al2O3vs MgO /H20850or its thickness might be related to a difference in sign betweenFigs. 5/H20849a/H20850and5/H20849b/H20850. We also compute the pumped spin cur- rents in the left I LSz=0.012 e/hand right IRSz=0 electrodes, which do not depend on /H9255IorUbin the range shown in Fig. 5. Analogously to pumped charge current of 1D model in Fig. 2/H20849b/H20850, we find nonzero charge current and corresponding dc pumping voltage in F/H20841I/H20841Njunctions shown in Figs. 5/H20849c/H20850 and5/H20849d/H20850. Nevertheless, Vpump of the order of /H1101110 pV are way to small to explain recent experiments on F/H20841I/H20841N junctions16where Vpump /H112291/H9262V is measured at frequencies of the applied rf field in the range f=2–3 GHz and the precession cone angle /H9258=10° –17° tuned by the microwave input power. IV. CONCLUDING REMARKS In conclusion, we have demonstrated that pumping of spin and charge currents by the precessing magnetization ofa ferromagnetic layer within various multilayer setups con-sisting of F,N, and Ilayers of nanoscale thicknesses can be understood within the framework of NEGF rotated into theframe moving with the magnetization as a simple four-terminal dc circuit problem, as illustrated by Fig. 1/H20849b/H20850. The four leads of this circuit are labeled as: L ↑ ,L↓ ,R↑ , and R↓ /H20849i.e., they act as half-metallic ferromagnetic electrodes /H20850. They are biased by the voltage difference /H6036/H9275/eeffectively emerging between the electrodes of opposite polarization. Our formal-ism provides a transparent physical picture of how: /H20849i/H20850single precessing spin pumps pure spin current symmetrically /H20849in the absence of any barriers /H20850toward the left and the right in 1D; /H20849ii/H20850pumped spin currents are suppressed by the tunnel barrier in one direction and enhanced in the opposite direc-tion beyond naïve sum of currents before the introduction ofthe barrier; /H20849iii/H20850pumped spin currents develop over few monolayers of Fmaterial in 3D junctions; and /H20849iv/H20850pumped spin currents become filtered by the second Flayer with static magnetization which converts them into charge currentand the corresponding dc pumping voltage in open circuits.Our physical picture of spin and charge pumping in MTJswith time-dependent magnetization suggests that these set-ups can serve as a sensitive probe of MTJ parameters, suchas the properties of the tunnel barrier and damping param-eters. The pumping voltages in N/H20841F/H20841I/H20841F/H20841Ntunnel junctions of the order of /H110111 /H9262V at FMR frequencies /H1101110 GHz could ex-plain some of the recent measurements of large voltage sig- nals in microwave driven MTJs under the FMR conditions.17 They are much larger than /H1101110 nV signal /H20849at FMR frequen- cies /H1101110 GHz /H20850recently predicted by the scattering theory18 for MTJs with similar TMR, but whose infinite Felectrodes are assumed to have strong spin-flip scattering leading to avanishing spin accumulation in F. 18The spin-flip scattering can easily be introduced in Hamiltonian /H208491/H20850via spin-orbit /H20849SO/H20850coupling terms36whose strength is tuned to match ex- perimental values for spin-diffusion length.35Nevertheless, here we use simpler Hamiltonian following assumptionssimilar to Ref. 19—typical spin-relaxation lengths 35are much longer than the length scale /H20849illustrated by Fig. 4/H20850over which pumping develops so that it does not affect thestrength of pumped currents. On the other hand, computationof realistic patterns of spin accumulation 20throughout the device requires to consider balance between transport andrelaxation processes. Also, the NEGF formalisms developedhere, with spin-diffusion length vs selected layer thicknesstuned via microscopic SO scattering terms, can tackle com-plicated spin pumping multilayer setups involving Ilayers where conventional approaches are not applicable /H20849because of spin accumulation not being well defined in aninsulator /H20850. 26 Although we do find nonzero charge current in N/H20841F/H20841I/H20841N multilayers when potential barrier is introduced in the devicethrough the Ilayer, the voltage signal /H1101110 pV is several orders of magnitude smaller than /H110111 /H9262V observed in ex- periments on such devices.16Also, this charge current is pro- portional to /H20849/H6036/H9275/H208502, rather than /H6036/H9275for spin and charge cur- rents in N/H20841F/H20841I/H20841F/H20841Njunctions or experimentally observed dc voltage signal in N/H20841F/H20841I/H20841Njunctions.16Its origin is in asym- metry of transmission coefficients connecting the four elec-trodes of the dc circuit in the rotating reference frame. Whilethe magnitude of measured voltages remains a puzzle for avariety of approaches 18–20utilized very recently to address some of the aspects of the experiment in Ref. 16, we believe that combining NEGF approach to spin pumping outlinedhere with the density functional theory /H20849DFT /H20850to take into account nonequilibrium self-consistent spin and charge den-sities /H20849akin to NEGF-DFT approach 29to spin-transfer torque in spin valves and MTJs /H20850could be capable of addressing this problem. ACKNOWLEDGMENTS We thank G. E. W. Bauer, T. Moriyama, and Y . Tserk- ovnyak for illuminating discussions. This work was sup-ported by DOE Grant No. DE-FG02-07ER46374 through theCenter for Spintronics and Biodetection at the University ofDelaware. S.-H. Chen and C.-R. 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PhysRevB.90.094402.pdf

PHYSICAL REVIEW B 90, 094402 (2014) Temperature-dependent ferromagnetic resonance via the Landau-Lifshitz-Bloch equation: Application to FePt T. A. Ostler and M. O. A. Ellis Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom D. Hinzke and U. Nowak Fachbereich Physik, Universit ¨at Konstanz, D-78457 Konstanz, Germany (Received 4 July 2014; revised manuscript received 14 August 2014; published 2 September 2014) Using the Landau-Lifshitz-Bloch (LLB) equation for ferromagnetic materials, we derive analytic expressions for temperature-dependent absorption spectra as probed by ferromagnetic resonance. By analyzing the resultingexpressions, we can predict the variation of the resonance frequency and damping with temperature and couplingto the thermal bath. We base our calculations on the technologically relevant L1 0FePt, parametrized from atomistic spin dynamics simulations, with the Hamiltonian mapped from ab initio parameters. By constructing a multimacrospin model based on the LLB equation and exploiting GPU acceleration, we extend the study toinvestigate the effects on the damping and resonance frequency in μm-sized structures. DOI: 10.1103/PhysRevB.90.094402 PACS number(s): 75 .70.−i,75.50.Bb,75.50.Vv,75.78.−n I. INTRODUCTION The magnetic properties of ferromagnetic structures such as thin films, nanowires, and nanoparticles have been studiedextensively both experimentally [ 1,2] and theoretically [ 3,4]. The interest in these particles is driven by fundamentalfeatures on the one hand and technological perspectives onthe other [ 5–7]. Ferromagnetic resonance (FMR), which has been applied with great success to thin ferromagnetic filmsin the past [ 8], can be used to measure important material properties, such as the damping, gyromagnetic ratio, andanisotropy constant. The temperature dependence of theseproperties for large or complex structures is often difficultto predict using analytical treatments, especially when tem-perature effects are included [ 4,9]. As well as being difficult to calculate analytically, temperature-dependent calculations of(for example) FMR can be slow to converge. The convergencecan become particularly troublesome if thermal fluctuationsare accounted for. A specific motivation for this work is theinterest in L1 0FePt materials, which is a promising candidate for ultrahigh density magnetic recording [ 10,11]. The ability to tune magnetic properties such as the damping is important, for example, in devices based on spin-transfertorque where a low damping of a free layer is essential forreducing the power consumption and can affect the signal-to-noise ratio [ 12]. In some cases, such as in giant magnetoresis- tive (GMR) read sensors, high damping is preferred to improvethermal stability [ 13]. For technologies based on heat-assisted magnetic recording (HAMR), understanding temperature effects and fluctuationsin strongly anisotropic materials will be crucially important.In this paper, we present analytical and numerical calculationsof the material properties of strongly anisotropic materials atelevated temperatures. We do so by utilizing the formalismof the Landau-Lifshitz-Bloch (LLB) equation of motion forferromagnetic particles, which has an intrinsic temperaturedependence via various input functions. There are a numberof different approaches to calculating FMR and introducingtemperature effects. The work of Usadel [ 14] utilizes an ap- proach based on the Landau-Lifshitz-Gilbert (LLG) equationfor nanoparticles whereby ensembles of atomic spins are treated as a single macrospin, in the same manner as theLandau-Lifshitz-Bloch equation. However, the work presentedin Ref. [ 14] does not take into account the contraction of the magnetization length. As the LLG model does not takeinto account the longitudinal relaxation of the magnetization,which becomes important at elevated temperatures, there is arequirement to use an approach such as the LLB to correctlydescribe the temperature-dependent properties as we approachthe Curie temperature. Other approaches for numerical determination of FMR properties in systems where exchange between macrospinsis important, such as thin films or granular media, includingmicromagnetic simulations such as that of Ref. [ 3]. In Ref. [ 3] the study focuses on granular media with the exchange be- tween macrospins within a grain and between grains taken into account. The use of this kind of micromagnetic model is wellaccepted at temperatures clearly below the Curie temperaturewhere the magnitude of the magnetization is determinedby the temperature. However, at higher temperatures thesusceptibility increases and due to thermal fluctuations themagnetization locally cannot be regarded as constant [ 15]. The use of the LLB model is of greatest importance when the susceptibility begins to increase and small variations intemperature result in large changes in magnetization (aroundT/greaterorsimilar3T C/4) [16]. In the first part of the paper, we present the derivation of the temperature-dependent analytic expression for the powerabsorbed by the particle. This analytic expression allows us to look at the effect of temperature on FMR curves for single-domain particles. The temperature-dependent inputfunctions that enter into the LLB formalism have beenparametrized from atomistic spin dynamics with the exchangeparameters calculated directly from ab initio calculations [ 17]. We have tested the expressions with a single-spin andmultispin (with exchange) LLB numerical model, by showing a number of resonance curves at different temperatures against the derived expressions (without demagnetizing fields).The analytic expressions for the damping and resonance 1098-0121/2014/90(9)/094402(11) 094402-1 ©2014 American Physical SocietyT. A. OSTLER, M. O. A. ELLIS, D. HINZKE, AND U. NOW AK PHYSICAL REVIEW B 90, 094402 (2014) frequency show the overall trend of the temperature-dependent behavior. In the second half of the paper, we extend the scope of our analysis using a multimacrospin model based onthe LLB formalism with large number of exchange-coupledmacrospins. We present numerical calculations of FMR intwo-dimensional (2D) and three-dimensional (3D) structureswith the inclusion of demagnetizing effects and (stochastic)thermal fluctuations. Specifically, we have investigated theeffects of the anisotropy constant and film thickness andanisotropy on the measured damping in out-of-plane filmsat high temperatures. Our findings show that, depending onthickness or anisotropy, there is a competition between the de-magnetizing and anisotropy energy that can modify the damp-ing significantly. We have implemented this large-scale modelon the CUDA GPU platform so that even with the inclusionof the stochastic thermal terms, it is possible to obtain goodaveraging of the FMR power spectra. There are limited experimental ferromagnetic studies of chemically ordered FePt due to its large magnetocrystallineanisotropy [ 18]. However, it is possible to perform so-called optical FMR with the use of laser pulses [ 19]. In a theoretical work by Butera [ 20], the resonance spectra were calculated using a computational model for disordered nanoparticles ofFePt. This study showed that the measured damping dependedstrongly on the amount of disorder. To our knowledge, thereare no systematic studies on the temperature dependence of theproperties such as damping due to the limited fields in typicalFMR setups; our results provide insight into this complexissue. II. LANDAU-LIFSHITZ-BLOCH EQUATION The LLB equation for magnetic macrospins describes the time evolution of an ensemble of atomic spins and allowsfor relaxation of the magnitude of the magnetization. Theequation was originally derived by Garanin [ 21] within a mean-field approximation from the classical Fokker-Planckequation for atomistic spins interacting with a heat bath.The resulting LLB equation has been shown to be ableto describe linear domain walls, a domain-wall type withnonconstant magnetization length. These results are consistentwith measurements of the domain-wall mobility in yttrium irongarnet (YIG) crystals close to the Curie point ( T c)[22] and with atomistic simulations [ 23]. Furthermore, the predictions for the longitudinal and transverse relaxation times havebeen successfully compared with atomistic simulations [ 24]. Consequently, we use this equation in the following forthe thermodynamic simulations of macrospins. The useof the LLB formalism has the advantage over traditionalmicromagnetics that it automatically allows for changes inthe modulus of the magnetization. In theory, it is indeedpossible to calculate temperature-dependent FMR using theatomistic spin dynamics (ASD) model, however, such anapproach would be extremely computationally expensive. Thiscomputational expense in the ASD model arises because,for FMR calculations, large system sizes are required toreduce the effects of thermal noise. While large systemsare possible to calculate, the FMR calculations also requireaveraging over many cycles of the driving field, up to hundredsof nanoseconds. These two restrictions combined means that this method is not suitable, even with GPU accelera-tion or a (for example, MPI) distributed memory solution[25]. A further challenge for accurate calculation of magnetic properties is the accounting of the long-ranged exchange inmaterials such as FePt. Through proper parametrization of theLLB equation [ 17], one can account for such long-ranged interactions in the so-called multiscale approach [ 17]. Via this multiscale approach we can then bridge the gap betweenelectronic-structure calculations to large-scale (of the orderof micrometres) calculations of material properties. With thisin mind, the LLB model is then ideally placed to describetemperature-dependent ferromagnetic resonance. The LLB equation, without the stochastic term, can be written in the form ˙m=−γ[m×H eff]+γα/bardbl m2(m·Heff)m −γα⊥ m2[m×[m×Heff]]. (1) Aside from the usual precession and relaxation terms, the LLB equation contains another term which controls longitudinalrelaxation [second term in Eq. ( 1)]. Hence, mis a spin polarization which is not assumed to be of constant length andeven its equilibrium value m e(T) is temperature dependent. The value of mis equal to the ratio of the magnetization of the macrospin normalized by the magnetization at saturation(M/M sV).α/bardblandα⊥are dimensionless longitudinal and transverse damping parameters (defined below) and γis the gyromagnetic ratio taken to be the free-electron value. Thetransverse damping parameter in this equation is related towhat is usually measured in experiments (the Gilbert dampingα g) by the expression αg=α⊥ m. (2) The LLB equation is valid for finite temperatures and even above Tc, although the damping parameters and effective fields are different in the two regions. Throughout this paper, weare only interested in the case T/lessorequalslantT cwith the damping parameters [ 21]α/bardbl=2λT 3Tcandα⊥=λ(1−T 3Tc). The single- particle free energy (without demagnetizing fields) is givenby f=−BM 0 smz+M0 s 2˜χ⊥/parenleftbig m2 x+m2 y/parenrightbig +M0 s 8˜χ/bardblm2e/parenleftbig m2−m2 e/parenrightbig2, (3) and the effective fields Heff=−1 Ms0δf δmgiven by [ 21] Heff=B+HA+1 2˜χ/bardbl/parenleftbigg 1−m2 m2e/parenrightbigg m, (4) where Brepresents an external magnetic field and the anisotropy field HAis given by HA=− (mxex+myey)/˜χ⊥. (5) Here, the susceptibilities ˜ χlare defined by ˜ χl=∂ml/∂H l, where Hlis thel=/bardbl,⊥. In these equations, λis a microscopic parameter which characterizes the coupling of the individual, 094402-2TEMPERATURE-DEPENDENT FERROMAGNETIC RESONANCE . . . PHYSICAL REVIEW B 90, 094402 (2014) atomistic spins with the heat bath. The anisotropy field HA [Eq. ( 5)] defines a hard axis in the xandyplanes, essentially giving a uniaxial anisotropy in the zdirection. This allows the anisotropy field to be defined in terms of the transversesusceptibility [ 17](χ ⊥) and gives the correct scaling of the anisotropy [ 26]. For the purpose of testing the model, we use a thermal bath coupling constant of λ=0.05, consistent with Ref. [ 18]. There are differing values of the damping constant in the literature,for example, for granular FePt Becker et al. measured a damping constant of 0.1 using an optical FMR technique,whereas Alvarez et al. found a value of 0.055 using standard FMR in a broad frequency range [ 18]. It should be pointed out here that while λis a coupling to the thermal bath equivalent to that used in atomistic spin dynamics, it is assumed to betemperature independent. At this point, we should take some time to define the different constants related to the damping and their differences.The parameters λ,α ⊥,α/bardbl, andαgcorrespond to the thermal bath coupling, the temperature-dependent transverse andlongitudinal damping parameters, and the damping parameterthat one would measure experimentally, respectively. Thethermal bath coupling is temperature independent and is aphenomenological parameter that is the same as that usedin atomistic spin dynamics. The transverse and longitudinaldamping parameters that enter into the LLB equation definethe relaxation rates of the transverse and longitudinal magne-tization components. Finally, the parameter α gis equal to the perpendicular damping ( α⊥) that enters into the equation of motion, divided by the magnetization and is what one wouldfind in an FMR measurement from the linewidth. For the application of this equation, one has to know a priori the spontaneous equilibrium magnetization m e(T) and the perpendicular [ ˜ χ⊥(T)] and parallel [ ˜ χ/bardbl(T)] susceptibilities. In this work, these are calculated separately from a Langevindynamics simulation of an atomistic spin model, however,it is possible to calculate these properties from mean-fieldcalculations [ 27]. We use a model for FePt which was introduced earlier and which is meanwhile well establishedin the literature [ 26,28–30]. Since this model was derived from first principles, a direct link is made from spin-dependentdensity functional theory calculations, via a spin model, to ourmacrospin simulations. The calculation of these parameters isdiscussed in more detail in Ref. [ 17]. III. ANALYTIC SOLUTION FOR THE FMR ABSORBED POWER SPECTRUM P( ω) The focus of this section is on the derivation of an analytical solution for the power spectrum P(ω) using the LLB equation for a single macrospin. The power P(ω) absorbed in an FMR experiment is given by [ 4] P(ω)=/angbracketleftbigg M·∂B ∂t/angbracketrightbigg =−ω 2π/integraldisplay2π/ω 0MSVmx˙Bxdt, (6) where Vis the volume of the macrospin, Mis the magnetiza- tion (MsVm), and ωis the frequency of the driving field. The right-hand side of Eq. ( 6) assumes that the time-varying field is applied in the xdirection with the static applied field in z.T h e time dependence of the xcomponent of the magnetizationcan be derived from Eq. ( 1). Using the assumptions that m2is constant and mxas well as myare small leads to the approximation mz≈m. Under this assumption, Eq. ( 1) can be written in linearized form. Together with the linearized formof the effective field, the solution of the resonance frequencyω 0and transverse relaxation time τcan be obtained (for full details see Appendix A): ω0(T)=γ/parenleftbigg Bz+m(T) ˜χ⊥(T)/parenrightbigg , (7) τ(T)=m(T) λ/bracketleftbig/parenleftbig 1−T 3Tc/parenrightbig ω0(T)−2 3γT TcHz eff(T)/bracketrightbig. (8) Here, m=me+˜χ/bardblBzis an approximation written to first order of the susceptibility for the purposes of the analyticcalculation. In the zero-temperature case under the conditionsthatα=α ⊥,α/bardbl=0, and m=me=1,ω0andτare the same as for the Landau-Lifshitz-Gilbert (LLG) equation ω0= γ(Bz+1 ˜χ⊥) andτ=1 λω0. The analysis of Eqs. ( 7) and ( 8) shows that there is little variation of the measured damping αgwith the applied field as one would expect [ 31]. Also, at low temperature, as expected, the measured damping is equal to the input coupling to thethermal bath λ. The temperature dependence of α gshows that (for a chosen value of λ) there is an increase with temperature, diverging at the Curie point. In a recent paper [ 31], the measured damping as a function of applied field (up to 7 T)was shown to be almost independent of temperature. In thesame study, the damping was measured at two temperatures:170 and 290 K. Between these two temperatures the dampingwas shown to be around 0.1 with a slight increase as one wouldexpect. Figure 1shows the analysis of Eqs. ( 7) and ( 8)f o rt h e physical input parameters for FePt. In the figure the measureddamping is calculated as α g=1/ω0τand is shown to increase with temperature, diverging at the Curie point. The contoursshow lines of constant damping explicitly. This demonstratesthat if one assumes no temperature dependence of the thermalbath coupling λ, the measured damping will not be constant. 0.50.4 0.3 0.2 αg=0.1 αg0.75 0.500.250.00λ0.20 0.15 0.100.050.00 T[ K ]600 500 400 300 200 100 0λ T[K] αg FIG. 1. (Color online) Analytically derived Gilbert damping as a function of temperature and the intrinsic coupling to the thermal bathλ, valid for a single macrospin without demagnetizing effects. For each value of λthe damping is shown to increase with temperature consistent with other works [ 24,31]. The lines are contours of constant measured damping. 094402-3T. A. OSTLER, M. O. A. ELLIS, D. HINZKE, AND U. NOW AK PHYSICAL REVIEW B 90, 094402 (2014) The value of λs h o w no nt h e yaxis of Fig. 1is a temperature-independent parameter. As mentioned above, thisparameter is equivalent to the coupling/damping parameterused in atomistic spin dynamics models. The general approachfor atomistic spin dynamics models is to use a constant valueofλwhich governs the rate of energy transfer to the bath [ 32]. The overall damping measured in the atomistic model isdetermined by this rate of energy transfer but is also affectedby the presence of spin-wave interactions in the system. Themeasured damping in atomistic spin dynamics is larger thanthe coupling to the thermal bath (at elevated temperatures) dueto spin-wave broadening. The LLB equation for a single spin contains the parallel and perpendicular damping constants ( α ⊥andα/bardbl). These values depend on λ(the microscopic coupling to the bath) and intrinsically give a temperature-dependent damping thatwas derived via the Fokker-Planck equation for the interactingatomistic spins [ 21]. The solution of the resulting inhomogeneous differential equation ( A4) combined with Eq. ( 6) leads us to the analytic solution for the power absorbed during ferromagnetic reso-nance as a function of the frequency of the driving field: P(ω,T )=M sVω2 4γα⊥B2 0 1 τ2+(ω−ω0)2, (9) where the temperature dependence comes from ω0andτ[see Eqs. ( 7) and ( 8)] and B0is the amplitude of the driving field. In the zero-temperature case, this solution reduces to that fromthe LLG equation. In Fig. 2(shown and discussed below) the temperature dependence of the analytic solution enters via thetemperature-dependent input functions of the LLB equation.The FMR equation given by Eq. ( 9) (and shown analytically in Fig. 2) uses the functions for FePt that were presented in Ref. [ 17], however, similar functions could be calculated using mean-field theory [ 27]. The analytic solution given by Eq. ( 9) can be compared to the numerical results, by integration of the LLB equation andusing Eq. ( 6). By applying an alternating driving field in the x direction and numerically averaging Eq. ( 6) until convergence we can compare the results of a single spin to the analyticexpression. For FePt, there is a strong uniaxial exchangeanisotropy, therefore, in the absence of any static applied field 650K500K300K100K0K ω/γ[T]P/M sV[ T2s] 25 20 15 10 5 01.2×10−4 8.0×10−5 4.0×10−5 0 FIG. 2. (Color online) Power spectrum vs frequency in a 1-T applied field. The data points are from LLB simulations for a singlemacrospin and the solid lines are given by Eq. ( 9).we still see a very strong FMR line for single-domain particles. Throughout the calculations we use a driving field amplitude(B 0) of 0.005 T and a static applied field ( Bz)o f1T .T h eC u r i e temperature for our system was assumed to be 660 K [ 17], the saturation magnetization used was M s=1 047 785 JT−1m−3. The anisotropy field at 0 K is equal to 1/ ˜ χ⊥(0 K) and is equal to 15.69 T, i.e., a value of the anisotropy constantK(viaH A=2K/Ms)o f8 .2×106Jm−3. The temperature dependence of the transverse susceptibility via Eq. ( 5) scales the anisotropy with M2.1as shown in Ref. [ 26]. In the second part of the paper, we have scaled the anisotropy constant at 0 Kby a given amount to give a different value of the anisotropyconstant used. The reason for using a static applied field and varying the frequency of the driving field is for computational efficiency. Inthe second part of this work, we have simulated a large numberof macrospins coupled via exchange and magnetostatics,which is computationally very expensive. To obtain goodaverages of the absorbed power during FMR we require alarge number of cycles of the driving field. The use of drivingfrequencies around 10–60 GHz would drastically increase thesimulation time, particularly for the low-temperature (high-resonance-field) simulations. Therefore, the simulation timefor higher frequencies is lower, increasing as it is reduced. Theuse of a higher-frequency driving field would overcome thecomputational problem, however, a large static applied field(particularly at low temperature) would be required to drivethe system to resonance. Both the high frequency and the highfield would be very difficult to obtain experimentally. Theexpression for the FMR power [Eq. ( 9)] can also be presented in the form P(B z). We have shown the analytic curves for this representation in Appendix B, although the quantities derived, such as damping, from the curves in either representationshould be consistent. We integrate the LLB equation using the Heun numerical scheme with a 5-fs time step. The input functions [ m e(T), χ⊥,/bardbl(T), andA(T)] that were used for FePt [as used in Eq. ( 9)] were calculated from atomistic spin dynamics [ 17] and the functional forms are polynomial fits [ 33]. The exact functions can be found in Ref. [ 33], specifically on pages 143 and 144 and are the same as those in Figs. 1, 2, and 3 of Ref. [ 17]. Figure 2shows the calculated absorbed power as a function of frequency for a range of temperatures using the single-spinLLB model. As we can see from Fig. 2, there is a large decrease in the resonance frequency, given by Eq. ( 7) ,w h i c hw ew o u l d expect to occur because of the decrease in the anisotropy field.The analytic solution agrees perfectly with the numeric model,except as we approach the Curie temperature. This is becausein the analytical treatment we approximate the magnetizationin a field B zto depend on the parallel susceptibility ( m=me+ ˜χ/bardblBz) which diverges as we approach the Curie temperature. This point has been discussed in Appendix Aand is an error in the analytic treatment only, not in the form of the LLBequation. The reduction in the anisotropy with temperature shown in Fig. 2is represented by a reduction in the resonance frequency in the P(ω) representation. As we have shown in Appendix B,i nt h e P(B z) representation, the resonance field increases with temperature. Both of these representations areconsistent with the expected decrease in temperature and are in 094402-4TEMPERATURE-DEPENDENT FERROMAGNETIC RESONANCE . . . PHYSICAL REVIEW B 90, 094402 (2014) qualitative agreement with other works, for example, the experimental works of Schulz and Baberschke [ 34] for the case of perpendicular films with the field applied perpendicular tothe film. The work of Antoniak [ 35] on FePt nanoparticles, as well as the theoretical work of Usadel [ 14] using an LLG-based model for (dipole) interacting nanoparticles, show a similarincrease in the resonance field with temperature (reduction inthe resonance frequency). IV . MULTIMACROSPIN NUMERICAL RESULTS In the following section, we introduce the stochastic LLB equation that takes into account thermal fluctuations. As wellas the normal terms in the LLB described by Eq. ( 4), we also include exchange coupling between the macrospins andthe magnetostatic fields. The LLB equation with stochasticthermal terms included is written for each spin iin the form ˙m i=−γ/bracketleftbig mi×Hi eff/bracketrightbig +ζi,/bardbl−γα⊥ m2 i/bracketleftbig mi×/bracketleftbig mi ×/parenleftbig Hi eff+ζi,⊥/parenrightbig/bracketrightbig/bracketrightbig +γα/bardbl m2 i/parenleftbig mi·Hi eff/parenrightbig mi. (10) The stochastic fields ζi,⊥andζi,/bardblhave zero mean and the variance [ 16] /angbracketleftbig ζη i,⊥(0)ζθ j,⊥(t)/angbracketrightbig =2kBT(α⊥−α/bardbl) |γ|MsVα2 ⊥δijδηθδ(t), (11)/angbracketleftbig ζη i,/bardbl(0)ζθ j,/bardbl(t)/angbracketrightbig =2|γ|kBTα/bardbl MsVδijδηθδ(t), where /bardblis the additive noise, and ηandθrepresent the Cartesian components. As well as the stochastic field, theexchange is also included in the form H i ex=A(T) m2e2 M0s/Delta12/summationdisplay j∈neigh( i)(mj−mi), (12) where A(T) is the exchange stiffness, /Delta1is the cell length, and M0 sis the saturation magnetization. It should be pointed out that the inclusion of the stochastic term into the LLB equationleads to a slightly reduced T Cas compared to the LLB without the stochastic term [ 16]. Figure 3shows the power spectrum as a function of fre- quency for multimacrospin calculations (coupled by exchange)for a system size of (100 nm) 3with a unit-cell discretization of (6.25 nm)3(i.e., 16 ×16×16 macrospins), though we have checked unit-cell sizes down to (3.125 nm)3(32×32×32 macrospins), i.e., below the typical domain-wall size of 4–6 nm. The solid lines are the analytical solution ( 9). The macrospin lattice is represented as a simple cubic arrangement with only nearest-neighbor interaction taken into account andis the same for all simulations involving many macrospins. As discussed in the Introduction, we have also introduced into our model demagnetizing effects to extend the analyticstudy to more realistic materials. We have taken the approachof that of Lopez-Diaz et al. used in the GPMAGNET soft- ware [ 36]. In this approach, we write the magnetostatic field in a (cubic) cell i(Hd,i)a s Hd,i=−Ms/summationdisplay jN(ri−rj)·mj, (13)650K600K500K400K300K200K100K50K ω/γ[T]P/M sV[ T2s] 20 15 10 5 01.4×10−4 1.2×10−4 1.0×10−4 8.0×10−5 6.0×10−5 4.0×10−5 2.0×10−5 0 FIG. 3. (Color online) Power spectrum vs frequency in a 1-T applied field. The data points are from LLB simulations for many exchange-coupled macrospins including the stochastic fields and exchange and the solid lines are given by Eq. ( 9) (no magnetostatic fields are included here). where Nis the 3 ×3 symmetric demagnetizing tensor. The sum runs over all cells at positions ri,j. The demagnetizing tensor is given by N(ri−rj)=1 4π/contintegraldisplay Si/contintegraldisplay SjdSi·dS/prime j |r−r/prime|, (14) Si(Sj) are the surface of cell i(j), respectively, randr/primeare the points on the surface iandj. This sum requires a summation from all cells and requires integration over each of the surfacesiandj, making it extremely computationally expensive. If one were to perform the integration ( 14) numerically for each surface of each cell, the calculation is extremely timeconsuming and converges very slowly with the number ofmesh points on each surface. To that end, we have employedthe method of Newell [ 37], whereby the surface integration of the cubes is calculated analytically as in the OOMMF code [ 38]. Some further details can be found in Appendix C. V . FERROMAGNETIC RESONANCE IN THIN FILMS OF FEPT In this section, we present calculations of thin films of FePt. We begin by looking at the effect of temperature onthe damping of 2-nm thin films using the stochastic formof the LLB equation with demagnetizing fields [Eq. ( 13)]. We compare this to the results for the single-spin analyticresults. The thin films show a large increase in the predicteddamping over the analytic results due to the inclusion of thedemagnetizing term as we approach the Curie temperature.The thickness dependence of the films is also calculated usingthe multispin model, showing that at low temperatures thereis little variation in damping with film thickness though, attemperatures approaching the Curie point, there is a largereduction with increasing thickness. For the thin films of FePt where we have included the magnetostatic interactions into the system, the analytic formof the resonance curves can no longer be fitted to the data.We therefore extract the damping by fitting the following 094402-5T. A. OSTLER, M. O. A. ELLIS, D. HINZKE, AND U. NOW AK PHYSICAL REVIEW B 90, 094402 (2014) ˜αFi lm αBulk T[ K ]600 300 02.00 1.501.00T=600KT=450KT=300KT=150K ω /γ[T]P/M sV[ T2s] 30 25 20 15 10 5 01.2×10−4 8.0×10−5 4.0×10−5 0 FIG. 4. (Color online) Ferromagnetic resonance curves in thin films of FePt for a range of temperatures below the Curie temperature. The points here are simulated data and the lines are the fits to Eq. ( 15). The inset shows the ratio of the damping as measured in our 2D film to the damping calculated analytically for a single macrospin. For low temperatures, the two are equivalent, however, at higher temperaturesthere is an enhanced damping in the thin films due to the effect of the magnetostatic field. expression to the FMR curves: P(ω)=P0ω2 (ω˜αg)2+(ω−˜ω0)2, (15) where ˜ αg,P0, and ˜ω0are free-fitting parameters and we use the tilde to distinguish the resonance frequency and damping fromthe analytically derived values. The use of this fitting procedureallows us to compare with experimental observations as thiswould be the kind of analysis required to extract the dampingparameter ( α g). For the single-spin calculations (as shown in Sec. III), we have verified that the use of this expression recovers the analytic value of the damping αg. By systematically varying the anisotropy we have shown that this increase in damping occurs when the demagnetizingfield dominates over the anisotropy term. Finally, this modifi-cation in the damping is shown to affect the switching timesas we transition from one regime to another. Thexandydimensions of the thin films in this section are 0.4μm×0.4μm. The zdimension is initially one cell (2 nm) thick, i.e., a 2D film. Our cell discretization is 2 nm ×2n m× 2 nm, below the domain-wall width. We apply the fields in thesame orientation as discussed above. The resonance curvesare shown on Fig. 4for a range of temperatures for the 2D (2-nm-thick) film. From each FMR curve we have used afitting procedure, as in Fig. 2, to calculate the damping in the 2D structures (solid lines). The inset of Fig. 4shows then the ratio of the damping that we calculate for the 2D structures tothe analytically derived damping for single-domain particlesin Sec. III. In the low-temperature limit, this ratio is consistent with the analytic solution for a single macrospin (i.e., it is 1);for high temperature, however, the damping is increased as thedemagnetizing effects start to dominate over the anisotropy. Next, we consider the effects of film thickness on the damping and resonance frequency. We increase the thickness300 K500 K600 K Thickness [nm]αg 18 14 10 6 20.13 0.11 0.09 0.070.0518 14 10 6 214.0 12.0 10.0 8.06.0 4.0 Thickness˜ω0/γ˜αg [nm] FIG. 5. (Color online) Damping as a function of film thickness for a range of temperature. In the low-temperature regime, there isa slight increasing damping as a function of thickness. As the Curie temperature is approached, there is a large decrease in the damping with film thickness. The inset shows the variation of the resonancefrequency with thickness. The resonance frequency shows an overall increase over all temperatures due to the decrease in the effective magnetostatic field. of the film from 2 to 20 nm (1 to 10 cells) and calculate the ferromagnetic resonance curve for each thickness (a maximumof 400 000 cells for around 100 ns). The resulting FMR curveswere again analyzed to extract the damping and resonancefrequencies. Figure 5shows the variation of the damping and resonance frequency as a function of the thickness of the thin film.The largest variation in the damping is shown close tothe Curie temperature (blue square, dotted line). For T= 500 K, there is a small increase in the damping with filmthickness when going from 2 to 4 nm. After 4 nm, the curveshows little variation, consistent with the T=300 K (red circles, dotted-dashed line) line. The variation in the damping,with film thickness, close to the Curie point will have a largeeffect on the magnetization dynamics in heat-assisted magneticrecording (for which FePt is a promising candidate), thatoperates at elevated temperatures. The elevated temperaturesallow for the reduction in the anisotropy so that the fieldgenerated by the write head of a hard disk drive (around 1–2 T)is sufficient to reverse the magnetisation. This reduction indamping for thick layers of FePt would lead to longer switchingtimes (as we show in the following), limiting the write times. In Ref. [ 39], Liu et al. showed that the damping in a magnetic tunnel junction consisting of a FeCoB free layerincreased with decreasing thickness. The mechanism was saidto be caused by spin pumping and nonlocal background effects.Our results, while not calculated for FeCoB, show that it isnot required to invoke a mechanism via spin pumping butcan arise due to an interplay between the anisotropy and thedemagnetizing fields. As well as looking at the effect of the film thickness on the damping parameter, we have also performed a systematicvariation of the anisotropy constant. In FePt, the anisotropycan be modified, for example, by inducing lattice distortion 094402-6TEMPERATURE-DEPENDENT FERROMAGNETIC RESONANCE . . . PHYSICAL REVIEW B 90, 094402 (2014) T=500KT=400KT=300K Anisotropy Energy [J/m3]˜αg 8×1066×1064×1062×10600.075 0.070 0.0650.0600.0550.050 FIG. 6. (Color online) Dependence on the damping in FePt for a range of anisotropy constants for three values of temperature (300 K, blue circle points; 400 K, green triangle points; and 500 K, red square points). In the lower anisotropy range the damping increases,consistent with the results of Fig. 5. The lines are fits to exponential decays to give a guide to the eye. or chemical disorder [ 40]. For the 2-nm-thick films, we have calculated the FMR spectra at three different temperatures(300, 400, and 500 K) for a range of anisotropy values belowthe bulk value (vertical dashed gray line in Fig. 5). From these calculations we have measured the effective dampingparameters using the method described above. The overalltrend shows a decrease in the measured damping, the result ofwhich is shown in Fig. 6. The overall trend in Fig. 6shows a decrease in the damping when the anisotropy becomes dominant over the demagnetizing field, consistent with the results of Fig. 5. Figure 7shows the calculated switching times for four temperatures (610, 620, 630, and 640 K) as a function ofthe thickness of the film. To calculate the switching times,we equilibrated the system at the temperature shown in thefigure, we then applied a field with a step function to 2 T toreverse the magnetization in the zdirection. The switching times were then averaged over 25 runs per point. The errors inthe switching times are quite small, so 25 runs seems to be asufficient number to take a good average. The thickness dependence of the switching times shown in Fig. 7is consistent with the calculations of the damping presented in Fig. 5. As the thickness is increased, there is an observed decrease in the damping which leads to the reducedswitching times seen in Fig. 7. It should be pointed out that the field that we apply is not sufficient to switch the magnetizationbelow around 610 K, consistent with Ref. [ 41]. The large reduction in the switching time seen for the T=640 K case is due to the fact that with the inclusion of the stochastic termthere is a slight reduction in the Curie temperature as shownin Ref. [ 16]. VI. CONCLUSION We have derived, using the LLB formalism, an analytic solution to the power frequency spectrum for nanometer-sized,single-domain ferromagnets during ferromagnetic resonance.Using the technologically relevant FePt, this analytic solutionT=640 KT=630 KT=620 KT=610 K 20 18 16 14 12 10 8 6 4 2140 120100 80 60 40 20 0Switching Times [ps] Thickness [nm] FIG. 7. (Color online) Switching times for thin films of FePt of differing thicknesses for a range of temperatures. Consistent with the result of Fig. 5, the reduction in the damping with increasing film thickness leads to an increase in the switching time. A Heaviside stepfunction of 2 T was applied to the field to reverse the magnetization after equilibration and the runs were averaged over 25 realizations of the random number seed. With the inclusion of the stochastic termthere is a reduced T Cso the T=640 K line is already above the transition temperature. agrees well with numerical simulations of both single-spin and exchange-coupled multispin calculations including thestochastic thermal term. Analysis of the resulting FMR expressions for a single macrospin show that the analytically derived damping isconsistent with those of extended thin films up to quite hightemperatures. At temperatures approaching T C, the anisotropy decreases more rapidly than the magnetization. This leadsto a region where demagnetizing field dominates over theanisotropy in the thin films. This means that our analyticexpressions for thin films of magnetically soft materialswould not hold, however, the analysis is still valid for singlemacrospins (or small structures) of soft materials. We have extended the calculations to include the thermal stochastic term and demagnetizing effects to explore theeffect this plays on the temperature-dependent ferromagneticresonance curves. By calculating FMR spectra as a functionof film thickness, we have shown that there is an increaseddamping for thinner films due to the interplay between thedemagnetizing fields and the anisotropy. For the thinner films,there is more of a tendency for the magnetization to want to liein plane due to the demagnetizing field. For highly anisotropicmaterials (shown here for FePt), this effect is more dominantat elevated temperatures. We have verified that this increasein damping can be explained by a change in the dominance ofthe demagnetizing energy by varying the anisotropy constantfor the thin films. As the anisotropy constant is decreased thedamping increases, consistent with the results of varying thefilm thickness. Finally, we have shown that this reduction in the damp- ing has an effect on the switching times. This conclusioncould have important consequences for heat-assisted magneticrecording, which operates at elevated temperatures, and 094402-7T. A. OSTLER, M. O. A. ELLIS, D. HINZKE, AND U. NOW AK PHYSICAL REVIEW B 90, 094402 (2014) requires sufficiently thick grains to have sufficient material for good readback of the magnetic signal. ACKNOWLEDGMENTS This work was supported by the European Commission under Contract No. 281043, FemtoSpin . The financial support of the Advanced Storage Technology Consortium is gratefullyacknowledged. The authors also thank R. W. Chantrell forhelpful discussions. APPENDIX A: DETAILS OF ANALYTIC DERIV ATION OF P( ω) This section gives some more detail regarding the derivation of the key equations discussed in Sec. III. The linearized equations of motion for the LLB equation ( 1) are written as ˙mx≈−γ/parenleftbig myHz eff−mHy eff/parenrightbig +γ(α/bardbl−α⊥) m/parenleftbig mxHz eff/parenrightbig +γα⊥ m/parenleftbig mHx eff/parenrightbig , ˙my≈−γ/parenleftbig mHx eff−mxHz eff/parenrightbig +γ(α/bardbl−α⊥) m/parenleftbig myHz eff/parenrightbig (A1) +γα⊥ m(mHy eff), ˙mz=0, with the linearized effective fields then written as Hx,y eff=Bx,y−mx,y ˜χ⊥+1 2˜χ/bardbl/parenleftbigg 1−m2 m2e/parenrightbigg mx,y, (A2) Hz eff=Bz+1 2˜χ/bardbl/parenleftbigg 1−m2 m2e/parenrightbigg m. In equilibrium, the zcomponent of the effective field vanishes, Hz eff=0. Using the linearized form of m,m=me(1+/Delta1m) as well as m2=m2 e(1+2/Delta1m), we arrive at an expression for thezcomponent of the applied magnetic field Bz−me/Delta1m+me/Delta1m2 ˜χ/bardbl=0. Using the linearized form of this equation Bz−me/Delta1m ˜χ/bardbl=0 as well as the approximation /Delta1m=(m−me)/m e,w eh a v e an approximation for mduring FMR that is both field and temperature dependent: m(T,Bz)=˜χ/bardbl(T)Bz+me(T). (A3) As is discussed in the main text, the approximation ( A3) leads to errors in the analytic treatment if the resonancecurve is calculated in an applied field. This is due to thefact that the susceptibility diverges as we approach the Curietemperature. This does not occur in the numerical simulationsand is only a problem in the analytic calculations due to theabove approximation ( A3). In order to calculate the resonance frequency ( ω 0)a sw e l la s the transverse relaxation time ( τ) for the power spectrum P(ω), one has to solve the linearized LLB equation [see Eq. ( A1)]. Using the notation ˜m=mx+imyand ˜Heff=Hx eff+iHy effleads to the differential equation ˙˜m γ=˜m/parenleftbigg i+α/bardbl−α⊥ m/parenrightbigg Hz eff+m/parenleftbiggα⊥ m−i/parenrightbigg ˜Heff. As can be easily seen from Eq. ( A2),˜Heffis also ˜m dependent. Writing the effective field as ˜Heff=˜B+A˜m, with A=−1 ˜χ⊥+1 2˜χ/bardbl(1−m2 m2e) and ˜B=Bx+iBywe arrive at an inhomogeneous differential equation ˙˜m γ=˜m/bracketleftbigg/parenleftbigg i+α/bardbl−α⊥ m/parenrightbigg Hz eff/bracketrightbigg +˜m/bracketleftbigg m/parenleftbiggα⊥ m−i/parenrightbigg A/bracketrightbigg +m/parenleftbiggα⊥ m−i/parenrightbigg ˜B. (A4) In the first step, we solve the homogeneous part of the differential equation ( A4), using the ansatz ˜mhom(t)=exp (ωt) whose solution leads to the expressions for ω0andτ: ω0=γ/parenleftbigg Bz+m ˜χ⊥/parenrightbigg , (A5) τ=m λ/bracketleftbig/parenleftbig 1−T 3Tc/parenrightbig ω0−2 3γT TcHz eff/bracketrightbig. (A6) In the next step, we solve the inhomogeneous differential equation ( A4) under the assumption that the applied magnetic field Bhas the form B=[B0exp(iωt),0,Bz], where B0/lessmuchBz. These lead to the following simplification of the right-handside of Eq. ( A4): m/parenleftbiggα ⊥ m−i/parenrightbigg ˜B=m/parenleftbiggα⊥ m−i/parenrightbigg B0exp(iωt). (A7) Using the ansatz ˜m(t)=u(t)˜mhom(t) where u(t)i sg i v e nb y u(t)=/integraldisplayt t0m/parenleftbigα⊥ m−i/parenrightbig B0exp(iωt) exp/parenleftbig −t τ/parenrightbig exp(iω0t)dt, and assuming t0=0 andt→∞ ,E q .( A4) has the solution ˜m(t)=/parenleftbig −i+α⊥ m/parenrightbig γmB 0/bracketleftbig1 τ−i(ω−ω0)/bracketrightbig 1 τ2+(ω−ω0)2exp(iωt). From this general solution, mxcan easily be derived mx=γmB 0 1 τ2+(ω−ω0)2/bracketleftbigg/parenleftbiggα⊥ τm−(ω−ω0)/parenrightbigg cos(ωt) +/parenleftbigg1 τ+α⊥ m(ω−ω0)/parenrightbigg sin(ωt)/bracketrightbigg , (A8) and substituted into the definition for the power spectrum P(ω) [see Eq. ( 6)]. This leads to the analytic solution for the power spectrum P(ω): P(ω)=MsVω2 4γα⊥B2 0 1 τ2+(ω−ω0)2. (A9) As we can see from Eq. ( A9), the analytic solution for the absorbed power depends on the magnetization, which inturn depends on the longitudinal susceptibility. As mentionedabove, we approximate the magnetization in the presenceof an applied field [Eq. ( A3)] in terms of the zero-field susceptibility. Therefore, Eq. ( A3) is only strictly correct in the zero-field limit. Away from the critical temperature, the 094402-8TEMPERATURE-DEPENDENT FERROMAGNETIC RESONANCE . . . PHYSICAL REVIEW B 90, 094402 (2014) Bz= 10TBz=b5TBz=b0TLines: me+˜χBzPoints: Numerical Data T[K]m[Red.] 660 620 580 540 5001.0 0.8 0.6 0.40.20.0 FIG. 8. (Color online) Equilibrium magnetization vs tempera- ture in different applied fields. The (red) solid line represents the zero-field equilibrium magnetization megained from atomistic FePt simulations, a fit to which defines the input function me(T)[17]. The dashed (blue) and dotted (black) lines represent expression ( A3) for different applied fields. The symbols represent the equilibrium magnetization in the presence of an applied field Bz=0,5,10 T, from the numerical simulations of a single macrospin without demagnetizing, stochastic, or exchange fields. zero-field susceptibility is small, therefore, in this limit the approximation holds. As we approach the critical temperature,the susceptibility diverges as we approach the phase transition.This means that our analytic expression shows a deviation fromthe numerically calculated result. A plot of the magnetization as a function of temperature using Eq. ( A3) and data from numerical simulations can be seen in Fig. 8. For small values of the applied field, this error reduces as the susceptibility is defined for small changes in theapplied field. Figure 8shows the equilibrium magnetization (red solid curve), initially calculated from atomistic spin dynamicssimulations [ 17], which is used as an input to the numeric simulation. As well as the equilibrium magnetization, the fig-ure also shows the magnetization as a function of temperaturein 5- and 10-T applied fields, which is of course not zeroat the (zero-field) Curie temperature. The dashed and dottedline is the analytic solution to the magnetization (also in 5-and 10-T fields), diverging across the Curie temperature. Aswe can see, the magnetization in an applied field from theanalytic expression shows a diverging behavior as we approachthe Curie temperature because of the diverging susceptibility,whereas the numerical simulations (points) show no suchdivergence. APPENDIX B: ANALYTIC FERROMAGNETIC RESONANCE CURVES FOR FIXED FREQUENCY As was pointed out in the main text, the experimentally more easily accessible measurement involves keeping the driv-ing frequency fixed (usually around 10–60 GHz) and varyingthe applied magnetic field until the resonance condition is met.The representation that we have used in our calculations andthe analytic solutions that we have shown in the main paperkeeps the applied field constant at 1.0 T in the zdirection and varies the frequency. The reason for using this representation600K ( ×3)500K300K100K0K Bz[T]P/M sV[ T2s] 16141210864204×10−7 3×10−7 2×10−7 1×10−7 0 FIG. 9. (Color online) Power absorbed during ferromagnetic res- onance for a fixed frequency of the driving field of 504.2 GHz as a function of the applied field Bz. The curves are given by the analytic solution [Eq. ( 9)] and are shown for a range of temperatures. There is an increasing field required to drive the system to resonance. Note that the T=600 K curve has been scaled for clarity as shown in the legend. is that for the case of a large number of macrospins coupled by exchange and demagnetizing fields, the variation of thefrequency is computationally more efficient. Therefore, to beconsistent between our results and the subsequent analysiswe also presented the analytic solutions and single macrospincalculations (which would be easily calculated with a fixedfrequency) in the same representation [ P(ω)]. It should be pointed out that experimentally measuring FMR for FePt is quite difficult in general (as was pointedout in Ref. [ 18]) it has not been possible (to our knowledge) to measure FMR in ordered L1 0FePt due to its high anisotropy, particularly at low temperatures. As was also shown inRef. [ 19], the resonance frequency is in the hundreds of GHz regime for FePt with a high degree of ordering. In Fig. 9, we have shown Eq. ( 9) for a fixed frequency and varied the applied field, essentially giving us P(B z). Due to the extremely large magnetocrystalline anisotropy in FePt, the use of drivingfrequencies of 10–60 GHz would show negative resonancefields (the field required to drive the system to resonance).For our single macrospin analytic approximation at 0 K, thefrequency required to give zero resonance field would be justbelow 450 GHz. Figure 9shows the results of the analytic solution [Eq. ( 9)] for a fixed value of the driving frequency ( ω=504.2 GHz) as a function of the applied field. The data are shown for a rangeof temperatures and show that as the temperature is increased,the value of the field required to drive the system to resonanceincreases consistent with a reduction in the anisotropy asseen in Refs. [ 14,34]. For room-temperature values of the temperature, the field required to drive the system to resonanceis quite large (around 6 T). Note that the value of the drivingfrequency of 504.2 GHz was chosen so that a reasonablepositive resonance field was required to drive the system toresonance at lower temperatures. The data of Fig. 9would give a qualitatively similar result for a lower frequency butwith a shifted set of resonance fields shifted in the negativefield range. 094402-9T. A. OSTLER, M. O. A. ELLIS, D. HINZKE, AND U. NOW AK PHYSICAL REVIEW B 90, 094402 (2014) APPENDIX C: MAGNETOSTATIC FIELDS For efficient calculation of the magnetostatic fields we write the convolution ( 13)a s Hη d,i=/summationdisplay θ,jWηθ ijmθ j, (C1) where the greek symbols η,θagain denote Cartesian compo- nentsx,y,z and latin symbols i,jdenote the lattice sites. Wηθ ij are interaction matrices which only depend on the structure of the material (cubic in this work). Since we are consideringa translationally invariant lattice, one can apply the discreteconvolution theorem and calculate the fields in Fourier space: H η d,k=/summationdisplay θWηθ kmθ k. (C2) It should be pointed out here that we have absorbed the prefactor Msinto the interaction matrix Wηθ ij. Furthermore, to write the fields in terms of units of Tesla to be consistentwith the form of the fields above, we have multiplied Eq. ( 13)byμ 0. The Fourier transform of the interaction matrix only has to be performed once and thus stored in memory. There are a number of conditions that must be met in order to utilize the convolution theorem. In terms of signalprocessing theory, the interaction matrix is seen as the responsefunction and the magnetization data is the signal. We shouldnote that there are two conditions that must be satisfied toutilize the convolution theorem. The first is that the signal(spin system) must be periodic in space. 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PhysRevB.92.180401.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 92, 180401(R) (2015) Noncircular skyrmion and its anisotropic response in thin films of chiral magnets under a tilted magnetic field Shi-Zeng Lin*and Avadh Saxena Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 17 August 2015; published 3 November 2015) We study the equilibrium and dynamical properties of skyrmions in thin films of chiral magnets with oblique magnetic field. The shape of an individual skyrmion is noncircular and the skyrmion density decreases with thetilt angle from the normal of films. As a result, the interaction between two skyrmions depends on the relativeangle between them in addition to their separation. The triangular lattice of skyrmions under a perpendicularmagnetic field is distorted into a centered rectangular lattice for a tilted magnetic field. For a low skyrmiondensity, skyrmions form a chainlike structure. The dynamical response of the noncircular skyrmions depends onthe direction of external currents. DOI: 10.1103/PhysRevB.92.180401 PACS number(s): 75 .70.Kw,75.10.Hk,75.70.Ak A magnetic skyrmion is a topologically protected spin texture, which has been observed in magnets without inversionsymmetry recently such as MnSi and FeGe [ 1–5]. A skyrmion is characterized by a topological charge Q= 1 4π/integraltext dr2S· (∂xS×∂yS)=± 1 with S(r) being a unit vector describing the direction of the spin. The typical size of a skyrmion isabout 5 to 100 nm and skyrmions form a triangular lattice. Inbulk crystal, the skyrmion lattice is stabilized in a small regionclose to the critical temperature in the temperature-magneticfield phase diagram [ 1]. Skyrmions are found to be much more stable in thin films [ 2,3]. Skyrmions respond to various external stimuli, such as magnetic field, electric current, andtemperature gradient. One extremely attractive feature ofskyrmions is that they can be depinned by a low currentdensity of the order of 10 6A/m2, which is 5 to 6 orders of magnitude smaller than that for magnetic domain walls[6–8]. Moreover, the conduction electrons in a metal interact with skyrmions and acquire a Berry phase, which producesan emergent electromagnetic field acting on these electrons.This gives rise to the topological Hall effect which has beenobserved experimentally [ 9–11]. Skyrmions in insulators can be driven by a temperature gradient [ 12–14] or electric field [15,16]. For their unique physical properties, skyrmions are believed to be a prime candidate for the next generationspintronic devices [ 17,18]. It is crucial to tailor the skyrmion structure to optimize the desired functionalities. For instance, to achieve high densitymemory utilizing skyrmions, one needs to have skyrmions withsize in the nanometer range. The size of the skyrmion can becontrolled by spin anisotropy or external magnetic fields, whilethe density of skyrmions can be tuned by external magneticfields. Even the chirality of the skyrmion can be tuned by thesign of the Dzyaloshinskii-Moriya (DM) interaction throughchemical substitution [ 19]. The skyrmion in these cases has circular shape and the response is isotropic. It is of fundamentalinterest and of relevance for applications whether there existnoncircular skyrmions with an anisotropic interaction betweenthem. We note that such noncircular skyrmions were observedexperimentally in strained crystals, where the DM vector *szl@lanl.govbecomes anisotropic [ 20]. Here we propose a simple way to stabilize noncircular skyrmions in thin films of chiral magnetby tilted magnetic fields. Skyrmions are more stable in thin films by suppressing the competing conical phase when a magnetic field is appliedperpendicular to the thin film. When the magnetic field istilted from the normal, the skyrmion phase is suppressed,which causes the decrease of the topological Hall resistivityas was measured recently in experiments [ 21]. When the field is parallel to the film, the skyrmion phase is suppressedcompletely and the conical phase is stabilized. Meanwhilethe skyrmion shape is distorted for a tilted magnetic fieldbecause the region with spin parallel to the in-plane componentof the field grows while the region with the opposite spinshrinks. Because of the distortion of skyrmion shape, thepairwise interaction between two skyrmions also becomesanisotropic, i.e., the interaction energy depends on the relativeangle between the two skyrmions. The resulting skyrmionlattice is no longer a triangular lattice. For a low densityof skyrmions, chains of skyrmions are stabilized because ofthe anisotropic interaction. At last, we will show that thedynamical response of skyrmions to an external current drivealso becomes anisotropic. We consider the following Hamiltonian for a classical spin S with |S|=1 in two dimensions ( x-yplane) [ 22] H=−J/summationdisplay /angbracketlefti,j/angbracketrightSi·Sj−D/summationdisplay i,μ=x,y(Si×Si+μ·eμ)−H·/summationdisplay iSi, (1) where Jis the exchange interaction between the nearest neighbor spins, Dis the DM vector along the bond due to the breaking of inversion symmetry, and His the external magnetic field with a tilt angle φfrom the zaxis, i.e., Hx=Hsinφ andHz=Hcosφ.H e r e eμwithμ=x, y is a unit vector in the xandydirection, respectively. The magnetic dipolar interaction was neglected because its strength is much weakerthan the interactions in Eq. ( 1). Equation ( 1) can reproduce satisfactorily the measured phase diagram in experiments. Toobtain the skyrmion configuration at zero temperature T= 0, we anneal the system using the Landau-Lifshitz-Gilbert 1098-0121/2015/92(18)/180401(5) 180401-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS SHI-ZENG LIN AND A V ADH SAXENA PHYSICAL REVIEW B 92, 180401(R) (2015) FIG. 1. (Color online) Skyrmion becomes noncircular when the magnetic field in the x-zplane is tilted away from the normal of the film. The tilt angle φis (a)φ=0◦,( b )φ=40◦,( c )φ=60◦,a n d( d ) φ=66◦. The color denotes the spin component along the zdirection, while the arrows represent the in-plane spin component. Inset in (a) is a schematic view of the setup. Here H=0.8D2/Jand the plotted area is 20×20J2a2/D2, with abeing the lattice constant of the spin system. equation with the Slonczewski spin-transfer torque term [ 23] ∂tSi=−γSi×(Heff+˜H)+αSi×∂tSi −/planckover2pi1γ 2eJextSi×(Sj×Si). (2) The effective field is Heff≡−δH/δSiand ˜His the Gaussian noisy field. Here αis the Gilbert damping coefficient and γis the gyromagnetic ratio. To discuss the skyrmion dynamics,we have also introduced the spin current J extto describe the adiabatic spin-transfer torque between the conductionelectrons and localized spins. Let us first consider a single skyrmion in the tilted magnetic field. The spin configuration of a skyrmion as a function of tiltangleφis presented in Fig. 1. For a perpendicular magnetic field, the skyrmion is centrosymmetric. As the magnetic field deviates from the normal of the film, the skyrmion elongates along the direction perpendicular to the in-plane componentof the magnetic field due to the Zeeman interaction term.For a tilt angle φ> 66 ◦, a clear coexistence phase of the skyrmion and the conical state can be seen. Such a skyrmionremains metastable even for φ=90 ◦when we tilt the field continuously towards the in-plane direction, because of thetopological protection. For a noncircular skyrmion, the interaction between skyrmions becomes anisotropic. To calculate the pairwiseinteraction, let us first define the center of mass of a skyrmionas its topological center R=1 4π/integraldisplay dr2S·(∂xS×∂yS)r. (3) Here we have used the continuum approximation Si→S(r), which is valid when D/J/lessmuch1. The interaction energy between two skyrmions U(R1−R2) depends on R 1−R2, in contrast to the |R1−R2|dependence for circular skyrmions. We then calculate U(R1−R2) by fixing spins of a skyrmion around its topological center |r−R1|<rcin order to pin the skyrmion at a desired position. The results for U(R1−R2) are shown in Fig.2. The interaction is mediated by exchange of magnons be- tween two skyrmions and is repulsive. For circular skyrmionsat a perpendicular magnetic field ( φ=0) the interaction is isotropic and is given by U(R 1−R2)=U(|R1−R2|)∼ K0(|R1−R2|/ξ) for a large separation |R1−R2|/greatermuchJa/D . Here the length scale ξis related to the magnon gap, ais thelattice constant of the spin system, and K0(x) is the modified Bessel function. For a tilted magnetic field, the interactionbecomes anisotropic. The repulsion between skyrmions in thedirection where they are elongated is stronger than that in theother directions. This can be best seen by looking at l 2/l1, where l1(l2) is the vector along the principal axes connecting the origin and the energy contour at U=− 0.7974D2/J.H e r e magnetic field has the component in the l1direction. The ratiol2/l1increases with the tilt angle φ, meaning that the interaction becomes more anisotropic for a larger tilt angle. Such an anisotropic interaction between skyrmions has profound consequences for the equilibrium configuration ofskyrmion lattice. The triangular lattice is distorted into acentered rectangular lattice (space group p6mm toc2mm transition), i.e., the skyrmion lattice constant in the directionwith a stronger repulsion becomes larger than that in theother directions. This can be seen from the spin structurefactor /angbracketleftS z(−q)Sz(q)/angbracketrightdisplayed in Fig. 3(a). Still we have six dominant Bragg peaks and higher order peaks for the skyrmionlattice. The ratio l 2/l1is larger than 1 because of the distortion due to the anisotropic interaction. In the coexisting phase FIG. 2. (Color online) Pair interaction U(R1−R2) between two skyrmions at different tilt angle of the magnetic field: (a) φ=0◦, (b)φ=45◦,a n d φ=60◦. Here l1andl2are two vectors along the principal axes connecting the origin and the energy contour at U=− 0.7974D2/J. As indicated by the ratio l2/l1>1, the pair interaction becomes anisotropic when the magnetic field is tilted.HereH=0.8D 2/J,D=0.2J,T=0, and rc=Ja/D . We restrict the pair interaction in the region |R1−R2|/greaterorequalslant3Ja/D because the method to pin skyrmions by fixing spins in the skyrmion core becomesinvalid when they are close to each other. 180401-2RAPID COMMUNICATIONS NONCIRCULAR SKYRMION AND ITS ANISOTROPIC . . . PHYSICAL REVIEW B 92, 180401(R) (2015) FIG. 3. (Color online) (a) Spin structure factor /angbracketleftSz(−q)Sz(q)/angbracketrightplotted in the logarithmic scale at H=0.336J,D=0.7265J,T=0.05J, andφ=45◦obtained by Monte Carlo simulations of model Eq. ( 1) with L=60a. The skyrmions form a centered rectangular lattice with l1=0.0642×2π/a andl2=0.09016 ×2π/a. (b) A skyrmion chain state obtained by numerical annealing of model Eq. ( 2)t oT=0a t H=0.7D2/J,D=0.2J,φ=48◦,a n dL=60Ja/D . The color denotes the spin component along zdirection, while the arrows represent the in-plane spin component. of skyrmions and ferromagnetic state with a low skyrmion density, as shown in Fig. 3(b), skyrmions arrange themselves into a chain along the direction with a weaker repulsion. Wenote that such skyrmion chains have also been observed innanowires due to the geometry confinement [ 24].Figures 4(a)–4(c) summarize the equilibrium phase dia- gram of model Eq. ( 1) at different tilt angles φ.F o ral o w tilt angle, we have a magnetic spiral with the ordering wavevector qalong the [11] direction at low magnetic fields. For intermediate magnetic fields, the skyrmion lattice was 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5 Spiral with q||[11]Field-polarized ferromagnetic phase Skyrmion latticeH [J] T [J]0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5 Field-polarized ferromagnetic phase Spiral with q||[11]H [J] T [J]Skyrmion lattice 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5 Spiral with q||[10]H [J] T [J]Spiral with q||[11]Field-polarized ferromagnetic phase 03 0 6 0 9 00.0000.0060.0120.018Skyrmion density φ(a) φφ φ=0° (b) =45° (c) =60° (d) FIG. 4. (Color online) (a)–(c) Temperature-magnetic field ( T-H) phase diagram of the model Eq. ( 1) at different tilt angles of the applied magnetic field obtained by Monte Carlo simulations. Here D=0.7265Jand the system size is L=60a. The phase boundary is obtained by checking the spin structure factor and spin configuration. (d) Skyrmion density as a function of tilt angle obtained by numerical annealing ofthe model from the paramagnetic state to T=0a tH=0.7D 2/JandD=0.2J. 180401-3RAPID COMMUNICATIONS SHI-ZENG LIN AND A V ADH SAXENA PHYSICAL REVIEW B 92, 180401(R) (2015) stabilized. The skyrmion lattice is distorted into a centered rectangular lattice for an oblique magnetic field. At high fieldswe have a field-polarized ferromagnetic state. The skyrmionlattice phase shrinks with the tilt angle, and it disappears atφ≈60 ◦. In this case the spiral with qalong the [11] direction transits into the conical phase with qparallel to the in-plane component of the magnetic field direction. (Here it is alongthexdirection.) The skyrmion density as a function of the tilt angleφatT=0 obtained by numerical annealing of Eq. ( 2) is presented in Fig. 4(d). The skyrmion density decreases and vanishes completely at φ≈60 ◦. The threshold tilt angle where skyrmion density vanishes decreases with temperature.The decrease of the skyrmion density as evidenced from thetopological Hall resistivity as a function of tilt angle wasmeasured in Mn 0.96Fe0.04Si thin films recently in Ref. [ 21], where the topological Hall resistivity vanishes at φ≈40◦at T=20 K. Finally, let us discuss the equation of motion for the noncentrosymmetric skyrmion. It is more convenient to adoptthe continuum approximation here. We follow Thiele’s col-lective coordinate approach [ 25] by treating a skyrmion as a rigid object, i.e., S(r,t)=S(r−vt). In this rigid skyrmion approximation, ∂ tS(r,t)≈− (v·∇)SandS×Heff=0. After properly integrating out the internal degrees of freedom forskyrmions, we obtain the equation of motion for skyrmions asparticles [ 26–28]f r o mE q .( 2), αη ijvj−Gij(Jext,j+vj)=0. (4) Here i,j=x,y and summation over repeated indices is assumed. The tensor Gand the form factor tensor ηare given by Gij=1 4π/integraldisplay dr2S·(∂iS×∂jS)=/parenleftbigg 0−1 10/parenrightbigg , (5) ηij=1 4π/integraldisplay dr2∂iS·∂jS. (6) For a noncircular skyrmion, ηxx/negationslash=ηyyandηxy/negationslash=0. We compute numerically ηijfor skyrmions in Fig. 1at different φ, and the results are shown in Fig. 5. While ηxy≈0 andηxx is almost independent on φ,ηyyincreases rapidly with φwhen the conical phase starts to appear. Note that yis the direction along which skyrmions are elongated. For a current in the x direction, the skyrmion acquires a velocity in the ydirection because of the damping. The Hall angle θxof the skyrmion motion is defined as tan(θx)=vy/vx=−αηxx/(1+αηyy). (7) For a current in the ydirection, the Hall angle is given by tan(θy)=−vx/vy=−αηyy/(1−αηxx). (8)0 1 32 63 95 26 50.00.51.01.52.02.53.0 φxx xy yy y/x FIG. 5. (Color online) Skyrmion form factors defined in Eq. ( 6) and the ratio of the Hall angle defined in Eqs. ( 7)a n d( 8)f o re x t e r n a l currents in the xandydirections. Here α=0.1,H=0.8D2/J, T=0, and D=0.2J. As shown in Fig. 5, the response of skyrmions to current becomes anisotropic, i.e., the Hall angle depends on thedirection of the current, because of the noncentrosymmetricnature of the skyrmion in tilted magnetic fields. To summarize, we have studied the equilibrium phase and dynamics of skyrmions with an oblique magnetic field.When the magnetic fields are tilted away from the normalof magnetic films, the skyrmion phase is less favorable andthe skyrmion density decreases. Meanwhile, the shape ofa skyrmion becomes noncentrosymmetric, which resultsin an anisotropic interaction between skyrmions. Thisanisotropic interaction stabilizes a centered rectangular latticeof skyrmions. In the low skyrmion density regime, a chain ofskyrmions can be stabilized in the ferromagnetic background.Note that skyrmion chains have been observed in a confinedgeometry recently [ 24]. 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PhysRevB.102.104413.pdf

PHYSICAL REVIEW B 102, 104413 (2020) Shapiro steps and nonlinear skyrmion Hall angles for dc and ac driven skyrmions on a two-dimensional periodic substrate N. P. Vizarim ,1,2C. Reichhardt,1P. A. Venegas ,3and C. J. O. Reichhardt1 1Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2POSMAT—Programa de Pós-Graduação em Ciência e Tecnologia de Materiais, Faculdade de Ciências, Universidade Estadual Paulista—UNESP, Bauru, SP , CP 473, 17033-360, Brazil 3Departamento de Física, Faculdade de Ciências, Universidade Estadual Paulista—UNESP, Bauru, SP , CP 473, 17033-360, Brazil (Received 9 March 2020; revised 27 June 2020; accepted 26 August 2020; published 10 September 2020) For an overdamped particle moving over a two-dimensional periodic substrate under combined dc and ac drives, a series of steps can appear in the velocity force curves that are known as Shapiro steps. Here we showthat for skyrmions driven over a two-dimensional periodic obstacle array with a dc drive and an ac drive thatis either parallel or perpendicular to the dc drive, the system exhibits numerous transverse and longitudinalsynchronization dynamics due to the Magnus force. These phenomena originate in interactions between twodifferent types of phase-locking effects: Shapiro steps and directional locking. In some cases, the skyrmion Hallangle is constant but longitudinal Shapiro steps appear, while in other regimes the skyrmion Hall angle caneither increase or decrease with increasing dc drive during the phase locking as the skyrmion locks to differentsymmetry directions of the obstacle lattice. For a transverse ac drive, we find that strong Hall angle overshootscan occur in certain locked phases where the skyrmion is moving at an angle that is considerably larger than theintrinsic Hall angle. For the strongest Magnus force, the phase-locking effects are reduced and there are largerregions of disordered dynamics. We show that the skyrmion Hall angle can be controlled by fixing the dc driveand changing the amplitude of the ac drive. DOI: 10.1103/PhysRevB.102.104413 I. INTRODUCTION Systems with multiple interacting frequencies are known to exhibit various nonlinear dynamical effects such as syn-chronization or phase locking [ 1,2]. Such phenomena arise across a wide range of fields ranging from coupled pendula [3] to biological systems [ 4]. One of the simplest examples of a system that can exhibit phase locking is an overdampedparticle on a periodic substrate under a combined dc and acdrive, where there can be resonances between the ac drivingfrequency and the frequency of the oscillations generated bythe motion of the particle over the periodic substrate. Theseresonance effects create a series of steps in the velocity force curves since the particle remains locked to a specific velocity over an interval of the external drive to remain in the resonantstate. One of the first systems where such resonant steps wereobserved was Josephson junctions, where so-called Shapirosteps appear in the current-voltage response [ 5,6]. Many sys- tems that exhibit phase locking can be described as effectivelyone-dimensional, and locking dynamics have been studied for Josephson junction arrays [ 7], incommensurate sliding charge density waves [ 8,9], vortices in type-II superconductors with one-dimensional (1D) [ 10–12] and two-dimensional (2D) pe- riodic substrates [ 13,14], driven Frenkel-Kontorova models [15], frictional systems [ 16], and colloids moving over 1D periodic substrates [ 17–19]. Even in the 1D case, a variety of additional phenomena such as fractional locking can arise when additional nonlinear effects come into play.Particles moving over a periodic 2D substrate exhibit many of the same phase locking effects as 1D systems, but theadditional degrees of freedom available in 2D make it possibleto align the ac drive perpendicular to the dc drive. In thiscase, new phase-locking effects that are distinct from Shapirosteps can appear that are known as transverse phase locking,in which the step widths generally grow with increasing acamplitude [ 20,21] rather than oscillating with increasing ac amplitude as in Shapiro steps. For 2D substrates, it is alsopossible to have biharmonic ac drives applied both paralleland perpendicular to the dc drive, which generate a circularmotion of the driven particle. Here, an increasing dc driveproduces chiral scattering effects that result in phase lockedregions in which the particle motion is both transverse andlongitudinal to the dc drive direction [ 22,23]. In most of the above systems, the dynamics is overdamped; however, in some situations nondissipative effects such asinertia can arise [ 24]. Another type of nondissipative effect is a gyro-coupling or Magnus force, which creates velocitycomponents that are perpendicular to the forces experiencedby the particle. In a 1D system, a Magnus force has no effect;however, in 2D systems it can strongly modify the dynam-ics. Magnus forces can be significant or even dominatingfor skyrmions in chiral magnets [ 25–28], where the ratio of the Magnus force to the damping term can vary from 0.1to 10. Skyrmions can interact with pinning sites, be set intomotion readily with an applied current, and exhibit depinningthresholds [ 27,29–34]. One of the most prominent effects of 2469-9950/2020/102(10)/104413(13) 104413-1 ©2020 American Physical SocietyN. P. VIZARIM et al. PHYSICAL REVIEW B 102, 104413 (2020) the Magnus force is that the skyrmions move at an angle with respect to the applied driving force, which is known as theskyrmion Hall angle θ sk[27,35,36], as has been observed in simulations [ 37–40] and experiments [ 41,42]. The Magnus force strongly modifies the interaction of the skyrmion witha substrate by creating spiraling motions of skyrmions thatare in a trapping potential [ 42–48]. The pinning or defects produce a strong drive dependence of the skyrmion Hall angle,which starts off near zero just at depinning and increaseswith increasing skyrmion velocity before saturating to theintrinsic value at high drives. This effect was first observed insimulations of skyrmions interacting with periodic or randomdisorder [ 38–40,44,49,50] and was then found in experiments [41,42,51–54]. The drive dependence arises due to a side jump effect when the skyrmion scatters off a pinning site [ 38,44]. For random disorder, θ skincreases smoothly with increasing drive; however, for a periodic substrate a guiding effect occurswhich causes the skyrmion motion to become directionallylocked to specific symmetry directions of the substrate overa range of drives, producing a quantized skyrmion Hall angle[38,55]. There are a number of proposals on how to create localized skyrmion pinning sites, which can be attractive orrepulsive [ 56–59], and there are now experimental realizations of skyrmion phases in periodic substrates [ 60] and supercon- ducting vortices interacting with skyrmions [ 61], making it feasible to create tailored 2D pinning arrays of attractive orrepulsive obstacles with which skyrmions can interact. Skyrmions moving over a 2D substrate under dc and ac drives are expected to exhibit a variety of new synchronizationeffects not observed in overdamped systems. In numericalwork examining dc and ac driven skyrmions in 2D systemsmoving over a periodic 1D substrate, Shapiro steps appearedfor the velocity in both the longitudinal and transverse direc-tions [ 62]. In an overdamped 2D system with a 1D substrate, phase locking occurs only when the dc drive, ac drive, andsubstrate periodicity direction are all aligned. The inclusionof the Magnus force allows any combination of the ac and dcdrive directions to produce some form of phase locking, andalso generates new effects such as Shapiro spikes in the veloc-ity force curves, which are distinct from Shapiro steps [ 63]. It is even possible for absolute transverse mobility to appear inwhich the skyrmion moves at 90 ◦to the driving direction, as well as negative mobility in which the net skyrmion motion isin the direction opposite to the applied drive [ 63], or ratchet effects [ 64,65]. The Magnus force opens entirely new aspects of nonlinear dynamics, and skyrmions moving over periodicsubstrates can serve to provide experimental realizations ofsuch dynamics. These results suggest that skyrmion motioncan be controlled by combining a periodic substrate arraywith different driving protocols, which can be important forapplications [ 66]. In this paper, we extend our previous results on dc and ac driven skyrmions on a 1D periodic array [ 62,63,67]t ot h e case of skyrmions interacting with a 2D periodic array ofobstacles, where the ac drive can be applied either parallel orperpendicular to the dc drive direction. We find two dominanteffects. The first is directional locking, which is similar tothat found previously for purely dc driven skyrmions on a2D substrate [ 39,55,68]. The second is Shapiro steps similar to those found for skyrmions and vortices under ac and dc Y XFDFac,x (b) Y XFD Fac,y(a) FIG. 1. A schematic of the system, which consists of a square array of obstacles (red circles) modeled as repulsive Gaussian scat-tering sites. The black line is the trajectory of a skyrmion which is subjected to both damping and a Magnus term as well as a dc drive F Dapplied along the xdirection. An additional ac drive is applied either (a) along the xdirection, Fac,x, or (b) along the ydirection, Fac,y. drives on a 1D periodic substrate [ 62]. These two effects can interfere with each other. In some cases, we find a constantskyrmion Hall angle accompanied by steps in both the paralleland perpendicular velocity components, while in other cases,a series of steps in the skyrmion Hall angle coexists withregimes in which the skyrmion Hall angle is nonmonotonicand either increases, decreases, or reverses sign as a functionof the applied drive. We also show that for fixed dc driving,the skyrmion Hall angle can be controlled by changing theamplitude of the ac drive. We note that the results presentedhere are distinct from those described in Ref. [ 68], in which there is only a dc drive but no ac driving. The steps in thevelocity force curves produced by pure dc driving are duesolely to a directional locking effect and are not the same asthe Shapiro step phase locking which we observe here. II. SIMULATION We consider a two-dimensional L×Lsystem with pe- riodic boundary conditions and model a single skyrmionmoving over a square obstacle array with lattice constant a. In Fig. 1, we show a schematic of the system highlighting the obstacles and the skyrmion trajectory for a dc drive F D applied along the xdirection. We apply an additional ac drive along the xdirection, as shown in Fig. 1(a), or along the y direction, as illustrated in Fig. 1(b). In the presence of only a dc drive, a series of directional locking steps appear due to thevelocity dependence of the skyrmion Hall effect [ 68]. We use a particle-based model for skyrmions interacting with disorder [ 31,38,39,68,69]. The equation of motion for skyrmion iis α dvi+αmˆz×vi=Fobs+FD+Fac(1) The first term αdon the left is the damping term, and the second term αmis the Magnus force, which produces veloci- ties that are perpendicular to the net force experienced by theskyrmion. Unless otherwise noted, we normalize the dampingand Magnus coefficients so that α 2 d+α2 m=1. 104413-2SHAPIRO STEPS AND NONLINEAR SKYRMION HALL … PHYSICAL REVIEW B 102, 104413 (2020) The first term on the right, Fobs, represents the interaction between the skyrmion and the obstacles. The potential en- ergy of this interaction has a Gaussian form Uo=Coe−(rio/ao)2, where Cois the strength of the obstacle potential, riois the distance between skyrmion iand obstacle o, and aois the ob- stacle radius. The force between an obstacle and the skyrmion is given by Fo i=−∇Uo=−Forioe−(rio/ao)2/hatwiderio, where Fo= 2Uo/a2 o. For computational efficiency, we place a cutoff on the obstacle interaction at rio=2.0 since the interaction becomes negligible beyond this distance. We set the obstacle density toρ o=0.093/ξ2and the obstacle radius to a0=0.65. The dc drive is represented by the term FD=FD/hatwidex. We increase the dc drive in small increments of δFD=0.001, and we wait 105 simulation time steps between increments to ensure that the system has reached a steady state. The ac drive has the formF ac=Asin(ωt)ˆαfor driving in the x(ˆα=ˆx)o r y(ˆα=ˆy) direction. Here Ais the amplitude of the ac drive and the drive frequency is ω=2×10−4inverse simulation steps. We measure the skyrmion velocity parallel, /angbracketleftV/bardbl/angbracketright=/angbracketleftvi·ˆx/angbracketright, and perpendicular, /angbracketleftV⊥/angbracketright=/angbracketleftvi·ˆy/angbracketright, to the dc drive. In the absence of any obstacles, the skyrmion moves at the intrinsic skyrmionHall angle, θ int sk=arctan( αm/αd) . We can also quantify the dynamics by measuring R=/angbracketleftV⊥/angbracketright//angbracketleftV/bardbl/angbracketright, where the skyrmion Hall angle is given by θsk=arctan( R). In this paper, we employ the rigid particle or Thiele equa- tion representation of the skyrmion dynamics, which neglectsthe internal modes or distortions of the skyrmion that canarise when the skyrmion interacts with a defect. The drivein our model is phenomenological and could represent theapplication of a current, but there are other possible ways todrive skyrmions such as via the excitation of internal modes[70,71]. In Sec. VI, we suggest some possible effects that could be produced by inclusion of internal skyrmion modes. III. DC AND AC DRIVE IN THE SAME DIRECTION We first consider the case where the ac drive is applied along the same direction as the dc drive, Fac=Asin(ωt)ˆx. For an overdamped particle moving over a periodic array, thisdrive configuration produces Shapiro steps in the velocity-force curves, and the motion is strictly in the drive direction,giving a Hall angle of zero. As shown in Ref. [ 68], when F D is increased under zero ac driving, a series of jumps in the velocity-force curves appear that are associated with differentlocking directions for the skyrmion motion, and are accompa-nied by jumps in θ sk. This is a result of the pinning-induced velocity dependence of the skyrmion Hall angle, as previouslystudied for skyrmions moving over a periodic pinning or ob-stacle array [ 38,68]. When a finite ac drive of A=0.5 is applied along the xdi- rection in the same system, the behavior changes as illustratedin Fig. 2.I nF i g . 2(a),w ep l o t /angbracketleftV ||/angbracketrightand/angbracketleftV⊥/angbracketrightversus FD, while in Fig. 2(b) we show the corresponding θskversus FDcurve. The skyrmion motion is initially locked along the xdirection forFD<0.65, and above this drive /angbracketleftV⊥/angbracketrightbegins to increase in a series of steps. The skyrmion Hall angle is nonmonotonicbetween the steps. Above the first step in /angbracketleftV ⊥/angbracketright, the Hall angle is close to θsk=−12.5◦, and decreases in magnitude with increasing drive to θsk=−8◦before increasing in magnitudeFIG. 2. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vs dc driving force FDfor a system with αm/αd=0.45 under a finite ac drive with A=0.5 applied along the xdirection. (b) The corresponding θskvsFD. There are windows of drive over which the magnitude of θskdecreases with increasing FD. again. This pattern repeats several times until, at high drives, θsksaturates to θsk=24◦, a value close to the pin-free intrinsic skyrmion Hall angle. At the higher drives, the steps in thevelocity force curves also become smoother. The decreasesin magnitude of the skyrmion Hall angle with increasing F D have not been observed for skyrmions interacting with random pinning. In Figs. 3(a)and3(b), we zoom in on the range 0 <FD< 0.6 for the two velocity components and θskin the system from Fig. 2.H e r e θsk=0.0 and /angbracketleftV⊥/angbracketright=0, indicating that the motion is locked along xdirection; however, a set of phase locking steps still appear in /angbracketleftV||/angbracketright. These are Shapiro 0.00.20.4 0.0 0.2 0.4 0.6-4-2020.00.40.8 0.8 0.9 1.0-14-12-10<V||>, <V⊥> (d)(c) (b)θsk FD(a) <V||>, <V⊥> θsk FD FIG. 3. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vsFDfor the system in Fig. 2withαm/αd=0.45 and xdirection ac driving of A=0.5 zoomed in on the range 0 .0<FD<0.6. (b) The corresponding θskvsFDshowing Shapiro steps. (c) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) for the same system over the range 0.7<FD<1.2. (d) The corresponding θskvsFD, showing a regime of decreasing skyrmion Hall angle magnitude. 104413-3N. P. VIZARIM et al. PHYSICAL REVIEW B 102, 104413 (2020) steps, which also occur in the overdamped limit. The steps correspond to windows of drive over which /angbracketleftV||/angbracketrightis locked to a constant value. In contrast, the directional locking found inthe absence of an ac drive in Ref. [ 68] is not associated with constant velocity steps but instead is accompanied by dips andcusps in the velocities. Figures 3(c)and3(d)show the curves from Fig. 2over the interval 0 .7<F D<1.2, where we find two new features. The first is that /angbracketleftV⊥/angbracketrighthas a fixed finite value, indicating that the particle is moving at an angle to the dc drive. The second isthat the series of steps which appear in /angbracketleftV ||/angbracketrightare correlated with steps in θsk, which is decreasing in magnitude as FDincreases. This indicates that the velocity is increasing in the xdirection but remains constant in the ydirection and the different phase locking steps are associated with decreases in the magnitudeof the skyrmion Hall angle. Near F D=1.2i nF i g . 2, there is a substantial jump in θskto a larger magnitude which coincides w i t haj u m pt oan e ws t e pi n /angbracketleftV⊥/angbracketright. The results in Figs. 2and3show that the phase-locking behavior found in Fig. 2is actually a mixture of two different types of locking. The first is the Shapiro-step phase lockingassociated with the matching of the ac drive frequency or itshigher harmonics to the increasing frequency of the skyrmionvelocity oscillations caused by the periodic collisions withthe obstacles under an increasing dc drive. This locking isassociated with a θ skvalue that is either constant or increasing in magnitude. The second is the directional locking whichoccurs even in the absence of an ac drive, as shown in previousworks [ 38,55,68]. These two locking phenomena can interact with each other to create regions where the magnitude of theskyrmion Hall angle is either constant or decreasing with driveinstead of increasing with drive. We note that directional lock-ing effects for a particle moving over a periodic substrate canalso occur for overdamped systems such as vortices in type-IIsuperconductors moving over 2D pinning arrays [ 72,73] and colloids moving over optical traps [ 73–78] or periodic sub- strates [ 79,80]; however, in those systems the direction of the drive with respect to the substrate must be varied, whereas forthe skyrmions, the velocity dependence of the skyrmion Hallangle changes the direction of motion even when the drivingdirection is fixed [ 38,68]. In Fig. 4(a), we plot the skyrmion trajectories for the system in Fig. 2atF D=0.3 where the skyrmion motion is locked in the xdirection with θsk=0◦. We note that in the absence of an ac drive, there is still a single locking step atθ sk=0◦as described in Ref. [ 68] where the orbit has a similar appearance; however, inclusion of the ac drive can producemultiple phase locking steps even for motion that remainslocked along the xdirection. At F D=0.85 in Fig. 4(b),t h e skyrmion has a finite displacement in the negative ydirection and it traverses five obstacles in the xdirection for every one obstacle in the ydirection, giving a ratio of R=1/5 andθsk=arctan(1 /5)=−11.3◦.I nF i g . 4(c)atFD=1.0, the velocity in the ydirection is unchanged but the skyrmion Hall angle has a smaller magnitude of −8.1◦, and the skyrmion moves seven lattice constants in xand one lattice constant inyduring a single ac drive period. Figure 4(d) shows the trajectories in the same system at FD=2.0, where θskis close toθsk=−22◦. Here the system is not on a locking step and the trajectories are more disordered. FIG. 4. Skyrmion trajectory (black line) and obstacle locations (red circles) for the system in Fig. 2withαm/αd=0.45 and x direction ac driving with A=0.5. (a) FD=0.3 where the motion is locked in the xdirection. (b) FD=0.85 where there is finite motion along ywithθsk=−11.3◦.( c ) FD=1.0, where θsk=−8.1◦.( d ) FD=2.0, where θsk=−22◦. In Fig. 5(a),w ep l o t /angbracketleftV||/angbracketrightand/angbracketleftV⊥/angbracketrightversus FDfor a system with xdirection ac driving of magnitude A=0.5a si nF i g . 2 but with αm/αd=1.0, where the intrinsic Hall angle is θint sk= 45◦. Figure 5(b)shows the corresponding measured θskwhich has only two values, with θsk=0◦at small drives followed by a jump to the intrinsic value θsk=−45◦, indicating that there are no intermediate directional locking phases. Once thesystem is locked to −45 ◦, a series of Shapiro steps still appear in both the parallel and perpendicular velocities in Fig. 5(a) that do not correspond to changes in θsk. This shows that it is possible for Shapiro steps to occur even when the systemmotion is fixed along a locking angle. On the Shapiro steps inFig.5, the skyrmion trajectories are much more ordered, while forF D=0.1 the trajectories are less ordered. In general, we find that if the ratio αm/αdproduces an intrinsic skyrmion Hall angle that gives a ratio of ytoxmotion that is close to 1/4,1/3,1/2, or 1, which correspond to strong symmetry di- rections of the substrate lattice, the system locks permanentlyto this symmetry direction even for very low drives, and stepsin the velocity appear that are a signature of Shapiro stepsinstead of directional locking steps. In Fig. 6(a),w ep l o t /angbracketleftV ||/angbracketrightand/angbracketleftV⊥/angbracketrightversus FDfor a system with xdirection ac driving of magnitude A=0.5a si nF i g . 2 but with αm/αd=1.732, giving an intrinsic Hall angle of θsk=60◦. Figure 6(b) shows the corresponding θskversus FD. Here the system is directionally locked to θsk=45◦, but there is still a series of steps in the velocities at lowF Ddespite the fact that the Hall angle is constant in this regime. For FD>0.6, a series of steps appear in θskas the system switches between different locking steps. The largerincreases in the magnitude of θ skare followed by regions in 104413-4SHAPIRO STEPS AND NONLINEAR SKYRMION HALL … PHYSICAL REVIEW B 102, 104413 (2020) FIG. 5. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vsFDfor a system with xdirection ac driving of magnitude A=0.5 atαm/αd=1.0, where the intrinsic Hall angle is θint sk=45◦.( b )T h e corresponding θskvsFD. Here the motion is locked to 45◦but there is still a series of Shapiro steps that are not associated with a changingskyrmion Hall angle. which the magnitude of θskdecreases by a smaller amount, and at large FD,θskgradually approaches the intrinsic value. In Fig. 7(a) we illustrate the skyrmion trajectory for the system in Fig. 6atFD=0.5 where the skyrmion is in the FIG. 6. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vsFDfor a system with xdirection ac driving of magnitude A=0.5 atαm/αd=1.732. (b) The corresponding θskvsFD.FIG. 7. Skyrmion trajectory (black line) and obstacle locations (red circles) for the system in Fig. 6withαm/αd=1.732 and x direction ac driving of magnitude A=0.5. (a) FD=0.5, where the system is locked to θsk=−45◦.( b ) FD=1.0. (c) FD=1.3. (d) FD=1.75 with θsk=−57.5◦. θsk=−45◦directional locking regime, while in Fig. 7(b) we show the FD=1.0 state where the skyrmion is locked to an angle close to θsk=−50◦.A t FD=1.3i nF i g . 7(c),t h e skyrmion is moving in an alternating fashion. In Fig. 7(d) at FD=1.75,θsk=−57.5◦. For increasing Magnus force, the dynamics become in- creasingly disordered, weakening both the directional lockingand the Shapiro steps. Figure 8(a)shows /angbracketleftV ||/angbracketrightand/angbracketleftV⊥/angbracketrightversus FDfor a system with xdirection ac driving at A=0.5, as in Fig. 6,b u tf o r αm/αd=9.962, where the intrinsic Hall angle is θsk=84.3◦. Figure 8(b) shows the corresponding θskversus FD. In this case there are only small steps in the velocity force curves that are associated with steps in θsk, which has an average value near θsk=−82.5◦. For these higher Magnus forces, the skyrmion starts to perform fullor partial loops around the obstacles, as shown in Fig. 8(c) atF D=0.33. In Fig. 8(d),a t FD=0.5, the system is in a disordered phase. For FD>1.5, the locking regimes are lost and θskgradually approaches the intrinsic Hall angle value. IV . AC DRIVING IN THE TRANSVERSE DIRECTION We next consider the case where the ac drive is applied along the ydirection, transverse to the dc drive. In Fig. 9(a), we plot the velocity components versus FDand in Fig. 9(b) we show the corresponding θskversus FDfor a system withαm/αd=0.45 and A=0.5. The features in the velocity curves are more steplike, rather than the cusplike shapes foundforxdirection ac driving in Fig. 2, and in general there are more locking regions which are associated with both direc-tional locking and the ac phase locking. Another interesting 104413-5N. P. VIZARIM et al. PHYSICAL REVIEW B 102, 104413 (2020)Y X(d) (c) Y X FIG. 8. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vsFDfor a system with xdirection ac driving of magnitude A=0.5 atαm/αd=9.962. (b) The corresponding θskvsFD. (c) Skyrmion trajectory (black line) and obstacle locations (red circles) at FD= 0.33, where loop orbits appear. (d) Skyrmion trajectory and obstacle locations at FD=0.5, where the orbits are disordered. feature is that near FD=0.15, there is a window of locking toθsk=−45◦, which is considerably larger in magnitude than the intrinsic skyrmion Hall angle of θsk=24.2◦. We call this a Hall angle overshoot. As FDincreases, θskundergoes a number of oscillations until it reaches a saturation near theintrinsic value at high F D.I nF i g s . 10(a) –10(c) ,w ep l o t /angbracketleftV||/angbracketright, /angbracketleftV⊥/angbracketright, and θskversus FDfor the system in Fig. 9over the interval 0 .4<FD<0.85. There are sudden jumps both up and down in θsk. Additionally, there are regions where /angbracketleftV⊥/angbracketright remains constant but steps appear in /angbracketleftV||/angbracketrightthat are associated with jumps in θsk. In the interval 1 .35<FD<1.75 shown in Figs. 10(d) –10(f) , there are regions where the velocity can decrease with increasing FD. In Fig. 11, we show some of the representative skyrmion orbits for the system in Fig. 9.A tFD=0.1i nF i g . 11(a) ,t h e motion is locked in the xdirection, and the skyrmion executes a zigzag pattern. In Fig. 11(b) ,a tFD=0.16, the motion is locked to θsk=−45◦. Figure 11(c) shows the trajectory at FD=0.26, where the skyrmion moves at a much smallerFIG. 9. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) for a system with αm/αd=0.45 and ydirection ac driving of magni- tude A=0.5. (b) The corresponding θskvsFD. We find more steps than for the same system with ac driving in the xdirection (Fig. 2). angle of θsk≈−18.4◦.I nF i g . 11(d) ,a tFD=0.44, we still findθsk=18.4◦but the orbit shape has changed, with the skyrmion moving 3 ain the xdirection and ain the ydirection during each ac drive cycle. At FD=0.61 in Fig. 11(e) ,t h e motion is along θsk=−26.6◦, and in Fig. 11(f) atFD=1.4, θsk=−18.4◦, indicating that the system has returned to the 1/3 locking region. The orbit differs from that shown in Fig.11(d) , indicating that R=1/3 locking can occur in sev- eral different ways. In Fig. 12(a) , we plot the velocity curves versus FDand in Fig. 12(b) we show the corresponding θskversus FDfor a system with αm/αd=1.732, where there are again a series of steps at which θskincreases or decreases. Locking occurs in several regimes and the system jumps in and out of theθ sk=−45◦locked state since the 45◦locking is a particularly 0.300.450.600.75 -0.25-0.20-0.15-0.10 0.4 0.5 0.6 0.7 0.8-30-25-20-151.21.41.6 -0.50-0.45-0.40 1.4 1.5 1.6 1.7-24-21-18-15<V||> <V
>θsk FD <V||><V
> (f)(b)(a) θsk FD(d) (e) (c) FIG. 10. (a) /angbracketleftV||/angbracketright,( b )/angbracketleftV⊥/angbracketright,a n d( c ) θskvsFDfor the system in Fig.9withαm/αd=0.45,A=0.5, and ydirection ac driving shown over the interval 0 .4<FD<0.85. (d) /angbracketleftV||/angbracketright,( e )/angbracketleftV⊥/angbracketright,a n d( f ) θskvs FDfor the same system over the interval 1 .35<FD<1.75. 104413-6SHAPIRO STEPS AND NONLINEAR SKYRMION HALL … PHYSICAL REVIEW B 102, 104413 (2020) FIG. 11. Skyrmion trajectory (black line) and obstacle locations (red circles) for the system in Fig. 9withαm/αd=0.45,A=0.5, andydirection ac driving. (a) At FD=0.1 the motion is aligned with the xdirection. (b) At FD=0.16, the motion is locked to θsk= −45◦. (c) At FD=0.26, the motion is at a smaller angle of θsk= −18.4◦.( d ) FD=0.44. (e) FD=0.61. (f) FD=1.4. strong symmetry direction of the square obstacle lattice. In Figs. 13(a) and13(b) we plot the skyrmion trajectories for the system in Fig. 12atFD=0.15 in the −45◦locking regime and at FD=0.185, where the skyrmions move at a lower magnitude angle of θsk=−33.7◦.I nF i g . 13(c) atFD=0.3, the system jumps to a new −45◦locking phase with a braiding patten, and in Fig. 13(d) atFD=0.43, the motion is along θsk=−56.3◦. For higher values of αm/αd, we again observe extended regions in which the trajectories are disordered, and thephase-locking phenomena is generally reduced. In Fig. 14, we show the velocities and skyrmion Hall angle versus F D for a system with A=0.5 and ydirection ac driving as in Fig.12but for αm/αd=9.962. There are a number of smaller steps, particularly in the range 0 .35<FD<1.0, along with one larger step near FD=2.5. The trajectories become in- creasingly aligned with the ydirection on the steps as FD increases. For ac driving in the ydirection, there is an interplay between three types of phase locking. These are the ShapiroFIG. 12. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vsFDfor a system with αm/αd=1.732, A=0.5, and ydirection ac driving. (b) The corresponding θskvsFD. steps, the directional locking, and the transverse phase- locking effect. This is the reason that there are a larger numberof steps in the velocity and skyrmion Hall angle curves com-pared to ac driving in the xdirection. FIG. 13. Skyrmion trajectory (black line) and obstacle locations (red circles) for the system in Fig. 12withαm/αd=1.732, A=0.5, andydirection ac driving. (a) At FD=0.15, the motion is locked toθsk=−45◦.( b )A t FD=0.175, the motion locks to θsk=−33◦. (c) At FD=0.3, the motion is along θsk=−45◦.( d )A t FD=0.43, the motion locks to θsk=−56.3◦. 104413-7N. P. VIZARIM et al. PHYSICAL REVIEW B 102, 104413 (2020) FIG. 14. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vsFDfor a system with αm/αd=9.962, A=0.5, and ydirection ac driving. (b) The corresponding θskvsFD. Hall angle reversal In most cases, we have shown that although the skyrmion Hall angle increases or decreases with drive, it maintains thesame sign. Under certain circumstances, however, we find re-gions in which the skyrmion Hall angle changes from positiveto negative. This effect is generally associated with windowsof disordered motion at smaller F Dwhere the skyrmion is jumping among different orbits. In Fig. 15,w ep l o t /angbracketleftV||/angbracketright,/angbracketleftV⊥/angbracketright, andθskversus FDfor a system with ydirection ac driving atαm/αd=1.0. If the ac driving were in the xdirection, this ratio of the Magnus to damping terms would produce aconstant skyrmion Hall angle of θ sk=45◦with only Shapiro steps. When the ac driving is along the ydirection, however, a variety of locking regions appear that are associated withjumps both up and down in /angbracketleftV ||/angbracketrightand/angbracketleftV⊥/angbracketright. Jumps also oc- cur in θskamong the values θsk=−45◦,−38.65◦,−36.87◦, −33.6◦, and−26.56◦. The corresponding velocity ratios are /angbracketleftV⊥/angbracketright//angbracketleftV||/angbracketright=1, 4/5, 4/3, 2/3, and 1 /2, respectively. At higher drives, θskdecreases in magnitude to angles smaller than 45◦. Meanwhile, for FD<0.1 there are several regions in which /angbracketleftV||/angbracketrightand/angbracketleftV⊥/angbracketrightare both finite but positive, which produces a positive skyrmion Hall angle of θsk≈10◦.T h em o t i o ni nt h i s regime is illustrated in Figs. 16(a) and16(b) atFD=0.045, where the motion is locked along x, and at FD=0.065, where the skyrmion is jumping intermittently in the positive ydirec- tion. Figure 16(c) shows the locking phase with θsk=−45◦at FD=0.2, and in Fig. 16(d) atFD=0.55,θsk=−33.7◦.I ti s possible that by varying other parameters such as the size ofthe obstacles, clear regions of skyrmion Hall angle reversalswill also emerge, but the results above indicate that suchreversal effects can arise for skyrmion motion on periodicsubstrates.FIG. 15. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vsFDfor a system with αm/αd=1.0,A=0.5, and ydirection ac driving. (b) The corresponding θskvsFD.F o r FD<0.1, there are regions in which both velocity components are positive, giving apositive skyrmion Hall angle. V . CHANGING AC AMPLITUDE We next consider the case of a fixed dc drive of FD=1.0 and changing ac drive amplitude Ain a system with xdirection ac driving at αm/αd=0.45. In Fig. 17(a) we plot /angbracketleftV||/angbracketrightand /angbracketleftV⊥/angbracketrightversus Aand in Fig. 17(b) we show the corresponding θsk versus A. When A=0.0, the skyrmion motion is locked along thexdirection, giving θsk=0. As Aincreases, /angbracketleftV||/angbracketrightremains fairly constant due to the fixed value of FD, but small cusps are present which are correlated with a series of increasingsteps in /angbracketleftV ⊥/angbracketright. The steps in /angbracketleftV⊥/angbracketrightproduce a series of steps in θsk=arctan( R)a tR=0, 1/10, 1/6, 1/5, and a small step near 1 /4. There are extended steps for R=1/3, 3/7, and 1/2. In general, we find that the magnitude of the Hall angle increases with increasing A.I nF i g . 18(a) , we illustrate the trajectories for the system in Figs. 17(a) and17(b) atA=0.2 on the R=1/10 locking step, where the skyrmion moves 10 a in the xdirection and ain the ydirection during each ac drive cycle. Figure 18(b) shows the same system on the R=1/3 step at A=1.0, where the orbit jumps between two different paths to produce the 1 /3 ratio. In Figs. 17(c) and17(d) , we plot the velocities and θsk versus Afor the same system as in Figs. 17(a) and17(b) but for ac driving in the ydirection. Here the Hall angle is initially zero since the skyrmion motion is locked along the x direction. The velocities and skyrmion Hall angle increase anddecrease in a series of jumps as Ais varied. In Fig. 18(c) ,w e illustrate the trajectories at A=0.34 along a step on which the skyrmion moves 11 ainxand 4 ainyduring every ac cycle. At A=1.0i nF i g . 18(d) , there is a more complicated orbit along a step where the skyrmion moves 11 ainxand 5 ainyper ac cycle. We find similar behavior for higher values of α m/αd. 104413-8SHAPIRO STEPS AND NONLINEAR SKYRMION HALL … PHYSICAL REVIEW B 102, 104413 (2020) FIG. 16. Skyrmion trajectory (black line) and obstacle locations (red circles) for the system in Fig. 15withαm/αd=1.0,A=0.5, andydirection ac driving. (a) At FD=0.05, the motion is locked along x.( b )A t FD=0.065, the skyrmion is also translating along the positive ydirection, giving a positive skyrmion Hall angle. (c) At FD=0.2, there is locking at θsk=−45◦.( d )A t FD=0.55, there is locking at θsk=−33.7◦. These results indicate that the Hall angle can be controlled by varying A. VI. DISCUSSION In the locked phases, the skyrmions perform quantized motion along the xand/orydirections. This suggests that 0.00.51.0 0.0 0.5 1.0 1.5 2.0-30-20-100100.00.51.0 0.0 0.5 1.0 1.5 2.0-30-20-10010 (b)<V||>, <V⊥> (a)θsk A <V||>, <V⊥> (d)(c)θsk A FIG. 17. (a) /angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vs ac drive amplitude Afor a system with αm/αd=0.45, fixed FD=1.0, and xdirection ac driving. (b) The corresponding θskvs FD. Here the magnitude of the skyrmion Hall angle increases in a series of steps with increasing A.( c )/angbracketleftV||/angbracketright(upper black curve) and /angbracketleftV⊥/angbracketright(lower red curve) vs Afor the same system with ydirection ac driving. (d) The corresponding θskvsFD. Here the Hall angle increases in magnitude in a series of steps with increasing A. FIG. 18. Skyrmion trajectory (black line) and obstacle locations (red circles) for the system in Figs. 17(a) and17(b) withαm/αd= 1.0,FD=1.0, and xdirection ac driving. (a) At A=0.2, the skyrmion moves 10 ain the xdirection and ain the y direction during each ac cycle. (b) An R=1/3 step at A=1.0. (c), (d) The same system from Figs. 17(c) and17(d) with ydirection driving at (c) A=0.34 and (d) A=1.0. ac drives could be used to control skyrmion motion in dif- ferent types of devices [ 66]. Such controlled motion could be applied to more complex geometries such as rows ofpinning or different tailored geometries. We expect that sim-ilar results would appear in triangular arrays of obstacles,where the dominant directional locking angles are 30 ◦and 60◦. Future areas to address include the role of tempera- ture, where thermal effects could strongly affect the transitionpoints or jumps between different locking phases and couldalso produce thermal creep [ 81]. At higher temperatures, the phase-locking effects would gradually wash away. Wefind that the locking effects are most prominent for systemswith repulsive obstacles, but if attractive obstacles or pin-ning sites are used instead, the locking effects persist but aresmaller for both the directional locking [ 68] and the Shapiro steps. We model the skyrmions as point particles; however, actual skyrmions often have additional internal modes of motion.These modes could be excited at much higher frequencieswhere they could induce additional locking frequencies. Sucheffects could be explored more fully with continuum basedsimulations [ 82], which can also capture skyrmion creation and annihilation, distortion of the skyrmions by the obsta-cles, and different types of driving. For example, skyrmionscan be driven by inducing changes in the skyrmion shapeusing parametric driving or oscillating magnetic fields. Theskyrmion motion arises from the competition between thefrequencies of the drive and those of the internal skyrmion 104413-9N. P. VIZARIM et al. PHYSICAL REVIEW B 102, 104413 (2020) modes. In this case, an ac drive could be generated by the wiggling motion of the skyrmions in time-dependent fields[70,71]. One could imagine situations in which the periodicity of the wiggling motion could match with the periodicity of anunderlying periodic substrate to create new types of phase-locking effects that are distinct from what we study in thispaper. Internal modes have been shown to produce skyrmion or ratchet motion in the absence of a substrate [ 64]. If a periodic substrate were added, this motion could become quantizedor could lock to different substrate symmetry directions.Different types of skyrmions, such as antiskyrmions or an-tiferromagnetic skyrmions, can exhibit distinct dynamics. In antiferromagnetic skyrmions, the Hall angle is small or zero, so the dynamics could be the same as overdamped systemsand just exhibit Shapiro steps; however, internal modes couldstill be excited. For antiskyrmions, the dynamics could beeven richer since they could depend on the orientation of theskyrmion with respect to the applied driving force. Anotherinteresting effect to study would be the competition between the pinning array and naturally occurring defects of the type that can arise in granular films [ 48]. In this paper, we have focused on the motion of a sin- gle isolated skyrmion. If multiple interacting skyrmions arepresent, additional locking effects could arise as a result ofemergent soliton dynamics, which would be most pronouncedjust outside of rational filling fractions of 1 /2o r1/1[83,84], where the filling fraction indicates the ratio of the numberof skyrmions to the number of obstacles or pinning sites.At commensurate fillings, the skyrmion-skyrmion interactionsshould cancel and the dynamics should be similar to the singleskyrmion case. Although our results are focused on skyrmions, similar effects could arise for particles in effectively 2D systemswhere gyroscopic forces can arise, including active spinners[85–88] or charged particles in magnetic fields moving over periodic substrates [ 89–91]. Another system with similar dy- namics is classical charges interacting with periodic substratesor superlattices, where various dynamical commensurationeffects and phase-locking phenomena can occur [ 89,92,93]. In this case, when a magnetic field is applied, the charges canalso exhibit a Hall angle. It would be interesting to study theresulting electron pinball systems [ 89] under combined dc and ac driving, where we expect effects similar to those which weobserve in our system to occur.VII. SUMMARY We have numerically examined a skyrmion moving over a 2D periodic array of obstacles under a dc drive with an addi-tional ac drive applied either parallel or perpendicular to thedc driving direction. We find that the Magnus force inducesnew types of dynamical locking effects that are not observedfor overdamped systems with 2D periodic substrates. Whenthe ac and dc drives are parallel, the skyrmion exhibits bothShapiro steps similar to those observed in the overdampedcase as well as directional locking in which the skyrmionmotion locks to different symmetry directions of the substrate.The locking is associated with steps or cusps in the velocitiesas well as changes in the skyrmion Hall angle. Under strictlydc driving, the skyrmion Hall angle changes monotonicallywith drive, but when ac driving is added, the skyrmion Hallangle can both increase and decrease along the locking steps.For certain ratios of the Magnus force to the damping term,we find that even though the skyrmion Hall angle is fixed in aparticular direction of motion, Shapiro steps still appear in thevelocity force curves either parallel or perpendicular to thedc drive. At high drives, the skyrmion Hall angle graduallyapproaches the intrinsic value and shows oscillations as afunction of increasing drive. In general, cusps in the velocityforce curve are indicative of directional locking, while stepsindicate that Shapiro steps are occurring. When the ac drive isperpendicular to the dc drive, we generally find an even largernumber of steps in the velocity force curves and the skyrmionHall angle. It is also possible to observe Hall angle overshootsin which the skyrmion Hall angle locks to a value that is muchlarger than the intrinsic value. When the dc drive amplitude isfixed, steps in the Skyrmion Hall angle can occur as functionof changing ac drive amplitude. For higher Magnus forces,we generally find that the steps are reduced and there areincreased regions of disordered flow. ACKNOWLEDGMENTS This work was supported by the US Department of En- ergy through the Los Alamos National Laboratory. LosAlamos National Laboratory is operated by Triad NationalSecurity, LLC, for the National Nuclear Security Admin-istration of the US Department of Energy (Contract No.892333218NCA000001). N.P.V . acknowledges funding fromFundação de Amparo à Pesquisa do Estado de São Paulo -FAPESP (Grant No. 2018 /13198-7). [1] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences , 1 (Cambridge University Press, Cambridge, 2001). [2] E. 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PhysRevB.73.020403.pdf

Current-induced magnetic vortex motion by spin-transfer torque Junya Shibata * RIKEN-FRS, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan and CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan Yoshinobu Nakatani University of Electro-communications, Chofu, 182-8585, Tokyo, Japan Gen Tatara Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo, 192-0397, Japan and PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan Hiroshi Kohno Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan Yoshichika Otani Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan; RIKEN-FRS, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan; and CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan /H20849Received 25 August 2005; published 4 January 2006 /H20850 We investigate the dynamics of a magnetic vortex driven by spin-transfer torque due to spin current in the adiabatic case. The vortex core represented by collective coordinate experiences a transverse force proportionalto the product of spin current and gyrovector, which can be interpreted as the geometric force determined bytopological charges. We show that this force is just a reaction force of Lorentz-type force from the spin currentof conduction electrons. Based on our analyses, we propose analytically and numerically a possible experimentto check the vortex displacement by spin current in the case of single magnetic nanodot. DOI: 10.1103/PhysRevB.73.020403 PACS number /H20849s/H20850: 75.70.Kw, 72.25.Ba, 85.75. /H11002d Manipulation of nanoscale magnetization by electric cur- rent is one of the most attractive subjects in both basic phys-ics and technological applications. After its theoreticalprediction, 1,2it has been widely recognized that spin- polarized current /H20849spin current /H20850plays a crucial role in mag- netization dynamics. The spin current exerts a torque on lo-calized spins by transferring the spin angular momenta ofelectrons through the exchange interaction between conduc-tion electrons and localized spins, which is called the spin-transfer torque. The key understanding of the effect is thatthe spin current favors magnetic configurations with spatialgradient, or more precisely, with finite Berry-phase curvaturealong the current. Such spatial gradient by spin current in-deed gives rise to the motion of domain wall, 3,4spin-wave instability in a uniform ferromagnet,5–7and domain nucleation.8 Recent experiments9,10and numerical simulation11for current-induced domain wall motion have shown that there is a vortexlike configuration in magnetic nanowire. Also, mag-netic vortices in nanodots have drawn much attention sincethe magnetic force microscopy /H20849MFM /H20850observation of a vor- tex core. 12However, an effective force on the vortex and its dynamics by spin-transfer torque due to the spin current havenot been clarified. In this Rapid Communication, we present a microscopic theory of vortex dynamics in the presence of spin current byusing the collective coordinate method. In the adiabatic ap-proximation, we derive an effective force exerted on the vor-tex core due to spin-transfer torque. It is shown that the vortex core experiences a transverse force, which compen-sates the Magnus-type force derived from the so-calledBerry’s phase term. This specific force is topologically in-variant, which is characterized by topological charges. Un-like the case of domain wall, 4we show that there is no threshold current to induce the vortex motion in the absenceof an external pinning. It is of great interest to the vortex-based devices for application. To verify the existence ofcurrent-induced transverse force on the vortex, we propose apossible experiment for the current-induced vortex displace-ment in a single magnetic nanodot. We consider the Lagrangian of the localized spins in the continuum approximation. The spins are assumed to have aneasy plane taken to be the x-yplane, and are described by the Lagrangian L S=/H6036S/H20885d3x a3/H9278˙/H20849cos/H9258−1 /H20850−HS, /H208491/H20850 HS=S2 2/H20885d3x a3/H20853J/H20849/H11612n/H208502+K/H11036cos2/H9258/H20854, /H208492/H20850 where S/H20849x,t/H20850=Sn/H20849x,t/H20850represents the localized spin vector with unit vector n=sin/H9258cos/H9278ex+sin/H9258sin/H9278ey+cos/H9258ez, and the magnitude of spin S;ei/H20849i=x,y,z/H20850are unit vectors of Cartesian frame. The Jand K/H11036are, respectively, the ex-PHYSICAL REVIEW B 73, 020403 /H20849R/H20850/H208492006 /H20850RAPID COMMUNICATIONS 1098-0121/2006/73 /H208492/H20850/020403 /H208494/H20850/$23.00 ©2006 The American Physical Society 020403-1change and the hard-axis anisotropy constants and ais the lattice constant. The first term of the right-hand side in Eq./H208491/H20850is the so-called Berry-phase term, which in general deter- mines the dynamical property of the localized spins. Let us denote the spin configuration of a vortex centered at the origin by a vector field n V/H20849x/H20850with unit modulus. As a vortex profile, we take an out-of-plane vortex; nV/H20849x→0/H20850=pez, where p= ±1 is the polarization, which re- fers to the spin direction of the vortex core center and n/H20849/H20841x/H20841/H11271/H9254V/H20850=cos /H20849q/H9272+C/H9266/2/H20850ex+sin /H20849q/H9272+C/H9266/2/H20850ey, where /H9254V is the vortex core radius, /H9272=tan−1/H20849y/x/H20850,q=±1,±2,..., is the vorticity, which describes the number of windings of the spin vector projected on the order-parameter space and C=±1 i s the chirality, which refers to the counterclockwise /H20849C=1 /H20850or the clockwise /H20849C=−1 /H20850rotational direction of the spin in the plane. We here introduce a collective coordinate X/H20849t/H20850=X/H20849t/H20850ex+Y/H20849t/H20850ey, which represents the vortex core center, and assume that a moving vortex can be written as n/H20849x,t/H20850=nV/H20851x−X/H20849t/H20850/H20852at least as a first approximation; that is, ignoring the spin-wave excitation. Substituting this into Eq. /H208491/H20850, we obtain the Lagrangian for the collective coordinate as LV=1 2G·/H20849X˙/H11003X/H20850−U/H20849X/H20850. /H208493/H20850 HereGis the gyrovector defined by G=ez/H6036S/H20885d3x a3n·/H20849/H11509xn/H11003/H11509yn/H20850=/H6036S a32/H9266Lpqez, /H208494/H20850 with Lbeing the thickness of the system, and U/H20849X/H20850is a potential energy of a vortex evaluated from the Hamiltonian HS. The gyrovector Gis topologically invariant correspond- ing to polarization pand vorticity qand the number covering the mapping space D. In the case of nV/H20849x/H20850, this mapping number is 1/2 in units of surface area 4 /H9266. The first term of the right-hand side in Eq. /H208493/H20850, which comes from the Berry-phase term, represents that Xand Y are essentially canonically conjugate each other. This term provides a transverse force − G/H11003X˙on the moving vortex, the so-called Magnus force, perpendicular both to the gyrovectorand to the vortex velocity, whose term has been derived anddiscussed by many workers 15–19in various systems. Let us investigate the force acting on the vortex by spin current of conduction electrons. The Lagrangian of the elec-trons is given by L el0=/H20885d3xc†/H20849x,t/H20850/H20877i/H6036/H11509 /H11509t+/H60362 2m/H116122/H20878c/H20849x,t/H20850−Hsd, /H208495/H20850 where c/H20849c†/H20850is anihilation /H20849creation /H20850operator of conduction electrons. The last term Hsdrepresents the exchange interac- tion between localized spins and conduction electrons givenbyH sd=−/H9004/H20848d3xn·/H20849c†/H9268c/H20850x, where 2 /H9004is the energy splitting, and/H9268is a Pauli-matrix vector. Here we perform a local gauge transformation4,13in electron spin space so that the quantization axis is parallel to the localized spins n/H20849x,t/H20850at each point of space and time; c/H20849x,t/H20850=U/H20849x,t/H20850a/H20849x,t/H20850, where a/H20849x,t/H20850is the two-component electron operator in the rotated frame, and U/H20849x,t/H20850=m/H20849x,t/H20850·/H9268is an SU /H208492/H20850matrix withm=sin /H20849/H9258/2/H20850cos/H9278ex+sin /H20849/H9258/2/H20850sin/H9278ey+cos /H20849/H9258/2/H20850ez. The La- grangian is now given by Lel=/H20885d3xa†/H20875i/H6036/H20849/H115090+iA0/H20850+/H60362 2m/H20849/H11509i+iAi/H208502/H20876a +/H9004/H20885d3xa†/H9268za, /H208496/H20850 where A/H9263=A/H9263·/H9268=−iU†/H11509/H9263U/H20849/H9263=0,x,y,z/H20850is an SU /H208492/H20850gauge field determined by the time and spatial derivative of the localized spins. For slowly varying magnetic configurations,the electron spins can mostly follow adiabatically the local-ized spins. This is justified for the condition k F/H9261/H112711, where kFis the Fermi wave number of conduction electrons, and /H9261 is the characteristic length scale of the spin texture of thelocalized spins. In this adiabatic approximation, taking theexpectation value of L elfor the current-carrying nonequilib- rium state, we can obtain the following interaction Hamil-tonian of the first-order contribution to the localized spins; 8 HST=/H20885d3x/H6036 2ejs·/H11612/H9278·/H208491 − cos /H9258/H20850, /H208497/H20850 where jsis the spin-current density, which is written by using the distribution function fk/H9268=/H20855ak/H9268†ak/H9268/H20856in the rotated frame specifying the current-carrying nonequilibrium state as js=/H208491/V/H20850/H20858k,/H9268/H9268/H20849/H6036k/m/H20850fk/H9268. As seen from Eq. /H208497/H20850, the spin cur- rent favors a finite Berry-phase curvature along the current. Indeed, the Hamiltonian HSTleads to the spin-transfer torque in the case of domain wall.8 From Eq. /H208497/H20850, we can derive the effective Hamiltonian represented by the collective coordinate for the vortex core.Taking the variation /H9254n=−/H11509jnV/H20849x−X/H20850/H9254Xj, where repeated roman indices imply sum over the in-plane spatial direction j=x,y, we obtain /H9254HST=/H6036S a32/H9266Lpq /H20849vs,x/H9254Y−vs,y/H9254X/H20850, /H208498/H20850 where vs=/H20849a3/2eS/H20850jsrepresents the drift velocity of electron spins. By integrating Eq. /H208498/H20850, we obtain HST=G·/H20849vs/H11003X/H20850. /H208499/H20850 Thus, a force acting on the vortex core is given by FST=−/H11509HST /H11509X=−G/H11003vs. /H2084910/H20850 This current-induced transverse force has been previously derived by Berger14based on a phenomenological treatment in the case of Bloch line. Here we have derived this forcefrom microscopic theory. It is noted that this force does notdepend on the chirality C= ±1 of the vortex in contrast to a force produced by a magnetic field. Since it is hard to controlthe chirality, this fact would great advantage in application.A microscopic derivation of a general relation between forceand torque in the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation is presented in Ref. 20. Before proceeding, we here briefly remark that this force can be interpreted as a reaction force of a Lorentz-type forceSHIBATA et al. PHYSICAL REVIEW B 73, 020403 /H20849R/H20850/H208492006 /H20850RAPID COMMUNICATIONS 020403-2from the spin current of conduction electrons. The interac- tion Hamiltonian, Eq. /H208497/H20850, can be rewritten by using the the U/H208491/H20850gauge field, A/H20849x/H20850=/H20849/H6036/2e/H20850/H11612/H9278/H20849x/H20850/H20851cos/H9258/H20849x/H20850−1 /H20852, interact- ing with the spin-current density. Thus the magnetic field B/H20849x/H20850is given by B=/H20849/H11509xAy−/H11509yAx/H20850ez=−/H6036 2en·/H20849/H11509xn/H11003/H11509yn/H20850ez, /H2084911/H20850 which corresponds to the so-called topological field.21It is noted that the SU /H208492/H20850field intensity, F/H9262/H9263=/H11509/H9262A/H9263−/H11509/H9263A/H9262 −i/H20851A/H9262,A/H9263/H20852, vanishes, since the original Lagrangian Lel0in Eq. /H208495/H20850does not include the local gauge field. The finite magnetic field is a consequence of the projection from SU /H208492/H20850to the U/H208491/H20850by taking the adiabatic approximation. Under this mag- netic field, the spin current of the conduction electrons mayexperience the following Lorentz-type force as: F L=/H20885d3xjs/H11003B=G/H11003vs=−FST, /H2084912/H20850 which can be interpreted as the reaction force acting on the vortex. Let us derive the equation of motion for the collective coordinate of vortex in the presence of spin current based onthe Euler-Lagrange equation, d dt/H11509L /H11509X˙−/H11509L /H11509X=−/H11509W /H11509X˙, /H2084913/H20850 where L=LV−HSTis the total Lagrangian, Wis the so-called dissipation function given by W=/H9251/H6036S 2/H20885d3x a3n˙2/H20849x,t/H20850=/H9251 2DX˙2, /H2084914/H20850 with/H9251being the Gilbert damping constant. The constant D=/H6036S a3L/H20885 Ddxdy /H20853/H20849/H11509i/H9258/H208502+ sin2/H9258/H20849/H11509i/H9278/H208502/H20854, /H2084915/H20850 generally includes a factor ln /H20849RV//H9254V/H20850, where RVis the sys- tem size. Here we assume that system has rotational invari- ance along the zaxis. The concrete expression in Eq. /H2084913/H20850is given by G/H11003/H20849vs−X˙/H20850=−/H11509U/H20849X/H20850 /H11509X−/H9251DX˙. /H2084916/H20850 This is the equation of motion for the vortex dynamics in the presence of spin current. If the right-hand side in Eq. /H2084916/H20850is absent, we obtain X˙=vs, where the vortex core moves along the spin current perpendicular to the transverse force FST. This situation can be seen from the current-induced domain wall motion.4,8On the other hand, the damping term − /H9251DX˙ acts as a deviation from the orbital direction of the moving vortex along the current. Importantly, there is no intrinsicpinning in the dynamics of vortex unlike the case of a do-main wall. This leads to a vanishing threshold current for thevortex motion in the absence of an external pinning. This isbecause, in the translationally invariant system, there is nopinning on Xand Y. This is in contrast with the case of thedomain wall, where /H92780is pinned by the hard-axis magnetic anisotropy even in the translationally invariant system.4Thus vortex-based devices would have great advantages in low-current operations. To verify the existence of current-induced force on the vortex, we propose the vortex displacement by spin currentin a single magnetic nanodot, 22where an out-of-plane vortex with vorticity q=1 is stabilized. We assume the electric cur- rent is uniform in the nanodot, and flowing in the positive x direction; that is, vs=/H20849a3/2eS/H20850js=vsex. We assume the full spin polarization of the current, P=1, for simplicity /H20849Fig. 1 /H20850. The potential energy U/H20849X/H20850is modeled by a harmonic one U/H20849X/H20850=/H9260X2/2, where /H9260is a force constant. In Ref. 23, /H9260is evaluated in detail, which depends on the aspect ratio g=L/R, where Ris the dot radius. From Eq. /H2084916/H20850, the equa- tion of motion is given by /H208491+i/H9251˜/H20850Z˙=−i/H9275Z+vs, /H2084917/H20850 where Z=X+iY,/H9251˜=/H9251D/G, and/H9275=/H9260/G. For an initial con- dition Z/H208490/H20850=0, the solution is given by Z/H20849t/H20850=ivs /H9275/H20877exp/H20873−i/H9275t 1+i/H9251˜/H20874−1/H20878. /H2084918/H20850 Thus the vortex center exhibits a spiral motion, whose rota- tional direction depends on the sign of the core polarization p=±1 /H20851Fig. 2 /H20849a/H20850/H20852. The final displacement of the vortex core is perpendicular to the current direction and given by FIG. 1. Schematic illustration of an experimental setup for current-induced vortex displacement. The topological charges arechosen to be p=1, q=1, and C=1 in the above vortex. FIG. 2. /H20849a/H20850Spiral motion of vortex center under spin current obtained from the analytical calculations. We took /H9251˜=0.02. /H20849b/H20850Nu- merical results of the time evolution of the volume-averaged mag-netization components.CURRENT-INDUCED MAGNETIC VORTEX MOTION BY … PHYSICAL REVIEW B 73, 020403 /H20849R/H20850/H208492006 /H20850RAPID COMMUNICATIONS 020403-3/H9254Y=−Gvs//H9260, which also depends on p, where the transverse force − G/H11003vsbalances the restoring force − /H9260X. This result is consistent with the recent experimental one in Ref. 10, wherea distorted vortex wall was shifted to the diagonal directionand stopped moving. For comparison, the current-induced dynamics of a vortex was calculated by micromagnetic numerical simulationsbased on the LLG equation with spin-current terms, /H11509M /H11509t=−/H92530M/H11003Heff+/H9251 MsM/H11003/H11509M /H11509t−vs·/H11612M, /H2084919/H20850 where M/H20849x,t/H20850is the magnetization vector, /H92530is the gyromag- netic constant, Heffis the effective magnetic field including the exchange and demagnetizing field, and Msis the sartura- tion magnetization. The last term represents the spin-transfertorque.5–8The sample is divided into identical cells, in each of which magnetization is assumed to be constant. The di-mension of the cells is 4 /H110034/H11003hnm 3with dot thickness h =10,20,30 nm. The dot radius is taken to beR=500 nm. The computational material parameters are typical for permalloy: M s=1.0 T /H20849Ms//H92620=8.0/H11003105A/m /H20850, the exchange stiffness constant A=1.0/H1100310−11J/m, and /H92530=1.8/H11003105m/A s. We take /H9251=0.02. The programming code is based on those of Refs. 11 and 24. Figure 2 /H20849b/H20850shows the time evolution of volume-averaged magnetization, which exhibits the spiral motion of vortexcore. It is noted that the rotational direction is opposite to thecase of analytical result because of the replacement/H6036S/a 3→−Ms//H92530. Figure 3 shows that vortex displacement as a function of the spin-current density in various aspectratios g=L/R. The numerical results are in good agreement with the analytical ones for small vortex displacement. It isfound that the smaller gis more advantageous to the vortex displacement. In conclusion, we have clarified the transverse force on the vortex and its dynamics by spin-transfer torque due to thespin current in the adiabatic regime. We have proposed ana-lytically and numerically a possible experiment for the vor-tex displacement by spin current in the case of single mag-netic nanodot. Finally, this vortex displacement may affect amagnetoresistance, which is probable to be detected, for ex-ample, by using planar Hall effect. 25 The authors would like to thank T. Ishida and T. Kimura for valuable discussion. *Electronic address: jshibata@riken.jp 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3L. Berger, J. Appl. Phys. 71, 2721 /H208491992 /H20850; E. Salhi and L. Berger, ibid. 73, 6405 /H208491993 /H20850. 4G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 /H208492004 /H20850. 5Ya. B. Bazaliy, B. A. Jones, and Shou-Cheng Zhang, Phys. Rev. B57, R3213 /H208491998 /H20850. 6J. Fernández-Rossier, M. Braun, A. S. Núñez, and A. H. Mac- Donald, Phys. Rev. B 69, 174412 /H208492004 /H20850. 7Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 /H208492004 /H20850. 8J. Shibata, G. Tatara, and H. Kohno, Phys. Rev. Lett. 94, 076601 /H208492005 /H20850. 9A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 /H208492004 /H20850. 10M. Kläui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A. C. Bland, G. Faini, U. Rüdiger, C. A. F. Vaz, L. Vila, and C. V ouille, Phys.Rev. Lett. 95, 026601 /H208492005 /H20850. 11A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys. Lett. 69, 990 /H208492005 /H20850. 12T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono,Science 289, 930 /H208492000 /H20850. 13G. Tatara and H. Fukuyama, Phys. Rev. Lett. 72, 772 /H208491994 /H20850;J . Phys. Soc. Jpn. 63, 2538 /H208491994 /H20850. 14L. Berger, Phys. Rev. B 33, 1572 /H208491986 /H20850. 15A. A. Thiele, Phys. Rev. Lett. 30, 230 /H208491973 /H20850. 16G. V olovik, JETP Lett. 44, 185 /H208491986 /H20850. 17P. Ao and D. J. Thouless, Phys. Rev. Lett. 70, 2158 /H208491993 /H20850. 18M. Stone, Phys. Rev. B 53, 16573 /H208491996 /H20850. 19H. Kuratsuji and H. Yabu, J. Phys. A 29, 6505 /H208491996 /H20850;ibid. 31, L61 /H208491998 /H20850. 20H. Kohno, G. Tatara, and J. Shibata /H20849unpublished /H20850. 21P. Bruno, V . K. Dugaev, and M. Taillefumier, Phys. Rev. Lett. 93, 096806 /H208492004 /H20850. 22P. Vavassori, M. Grimsditch, V . Metlushko, N. Zaluzec, and B. Llic, Appl. Phys. Lett. 86, 072507 /H208492005 /H20850. 23K. Yu. Guslienko, B. A. Ivanov, V . Novosad, Y . Otani, H. Shima, and K. Fukamichi, J. Appl. Phys. 91, 8037 /H208492002 /H20850. 24Y . Nakatani, A. Thiaville, and J. Miltat, Nat. Mater. 2, 521 /H208492003 /H20850. 25T. Ishida, T. Kimura, and Y . Otani, cond-mat/0511040 /H20849unpub- lished /H20850. FIG. 3. Displacement of the vortex center as a function of the spin-current density jsin various aspect ratios g. The solid lines and dots represent the analytical and numerical results, respectively.SHIBATA et al. PHYSICAL REVIEW B 73, 020403 /H20849R/H20850/H208492006 /H20850RAPID COMMUNICATIONS 020403-4

PhysRevB.102.214406.pdf

PHYSICAL REVIEW B 102, 214406 (2020) Derivation of magnetic inertial effects from the classical mechanics of a circular current loop Stefano Giordano* Institute of Electronics, Microelectronics and Nanotechnology - UMR 8520, Université de Lille, CNRS, Centrale Lille, ISEN, Université de Valenciennes, LIA LICS /LEMAC, F-59000 Lille, France Pierre-Michel Déjardin† Laboratoire de Mathématiques et de Physiques, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, F-66860 Perpignan, France (Received 2 October 2020; revised 20 November 2020; accepted 20 November 2020; published 7 December 2020) The dynamical equation of a single magnetic moment constituted by a rigid circular current loop is derived from the mechanical Lagrange equations of motion, introducing the Lorentz force and the damping process,described by a well-defined dissipative mechanism. It is demonstrated that magnetic inertial effects arisenaturally by simple mechanical considerations and superimpose onto the Gilbert original dynamical equation.The comparison with models proposed in the recent literature is drawn and discussed. DOI: 10.1103/PhysRevB.102.214406 I. INTRODUCTION The so-called Landau-Lifshitz-Gilbert (LLG) equation de- scribes the orientational dynamics of a single magnetic dipoleor the magnetization distribution in a given ferromagneticmaterial [ 1–4]. The first version of this equation was proposed by Landau and Lifshitz in 1935 to study the magnetic perme-ability dispersion in ferromagnetic bodies [ 5]: d/vectorM dt=γ/vectorM∧/vectorB−γα M/vectorM∧(/vectorM∧/vectorB). (1) Here /vectorMis the dipole moment, γis the gyromagnetic ratio, αis the damping coefficient, and /vectorBis the effective magnetic induction. This equation was modified in 1955 by Gilbert todescribe the behavior of materials with large damping [ 6,7]: d/vectorM dt=γ 1+α2/bracketleftBig /vectorM∧/vectorB−α M/vectorM∧(/vectorM∧/vectorB)/bracketrightBig . (2) While the two equations exhibit the same mathematical form, they have a slightly different behavior, especially for largeenough values of α. However, this second-order difference cannot be appreciated with experimental measurements andthe choice between the models must be made on the basis oftheoretical arguments [ 8–13]. The problem is still the subject of debate and different opinions can be found in the literature.For instance, Ref. [ 12] is in favor of the Gilbert version since it corresponds to an isotropic damping action, while Ref. [ 13] opts for the Landau-Lifshitz equation being coherent withirreversible thermodynamics. Today, the LLG equation is very important since it de- scribes the magnetization dynamics in several systems anddevices of crucial technological importance [ 14,15]. A clas- sical application concerns the modeling of the switching *stefano.giordano@univ-lille.fr †dejardip@univ-perp.frstate in memory elements based on ferromagnetic materi-als [16,17]. More generally, magnetoelectroelastic structures [18–21], typically composed of piezoelectric and magnetoe- lastic subsystems, are largely investigated, being promisingprototypes for the reduction of the energetic consumption indata storage and elaboration systems [ 22–24]. One of the most important features of these systems is the stability of the stored information over long times. Thismeans that thermal fluctuations must not alter the informa- tion recorded in the memory elements [ 25,26]. To perform the analysis of this problem one can use the extension ofthe LLG equation with the additional term representing thethermal noise, as proposed by Brown in his pioneering works[27–30]. This generalization converts the LLG equation into a stochastic differential equation, or Langevin equation, whichcan be typically studied through the Fokker-Planck formal-ism [ 31,32]. Several applications of this methodology can be found in the recent literature [ 33–37]. The theory of the magnetization dynamics is also of cru- cial importance to study the movement of a domain wall inferromagnetic materials. The first important analytical resultis given by the Walker solution describing, under simpleassumptions, the one-dimensional steady-state motion of adomain wall in a uniform magnetic field [ 38]. Because of the many advantages such as reliability, fast operation, andlow power consumption, devices based on domain walls arewidely seen as promising tools for various applications, in-cluding data storage, sensing, and logic [ 39–43]. Also in this context, the question of whether Landau-Lifshitz dampingor Gilbert damping provides the more natural description ofdissipative magnetization dynamics has been reopened [ 44]. It has been suggested that the Gilbert damping term is moreadapted, showing the purely energy dissipative property alsoin the presence of nonconservative fields (e.g., with spin-transfer torques) [ 45]. This state of affairs motivates the present readdressing of the magnetization dynamics. We propose here an origi-nal derivation of the corresponding equations based on the 2469-9950/2020/102(21)/214406(13) 214406-1 ©2020 American Physical SocietyGIORDANO AND DÉJARDIN PHYSICAL REVIEW B 102, 214406 (2020) explicit consideration of the magnetic dipole structure as a charge distribution (first discrete and then continuous) rotatingaround a given axis with an intrinsic angular frequency (circu-lar current loop). We first develop a model with a fixed angularvelocity ωof charges. This hypothesis allows for the use of the Lagrange formalism for rheonomic systems, where thetwo-variables Lagrangian function explicitly depends on time(via the angular frequency ω). Then, in order to obtain a more symmetric formalism, we introduce a second model with anarbitrarily varying angular velocity of the charges. In thiscase the system is scleronomous, but with three generalizedcoordinates. In both cases, while the dipole structure is in-variant during the dynamics, the dipole orientation changes inresponse to the external actions. Within these approaches, theLorentz and the damping forces can be naturally introducedwithout the need to specify a scalar magnetic potential anda Rayleigh dissipation function. This direct method allowstherefore to obtain the explicit Lorentz contribution due to thedipole reorientation (always neglected) and a precise descrip-tion of the damping force, defined by an explicit dissipationmechanism. Moreover, the introduction of the realistic mag-netic dipole structure naturally generates in the final equationsthe inertial terms, corresponding to the second derivatives ofthe orientation angles with respect to the time. It is importantto remark that these inertial terms and the Lorentz terms dueto the reorientation are both of the order of 1 /ωand therefore they are often neglected in practical applications. Neverthe-less, the possible use of magnetic fields with extremely highfrequencies (at pico- and femtosecond time scales) has re-cently generated a wide interest for the generalization of theLLG equation with inertial effects [ 46–56]. In this context, our approach yields two generalized sets of second-order differen-tial equations for the magnetization dynamics, including theinertial effects and the Lorentz terms due to the reorientation.To get a simplified description, the concept of ideal dipole,corresponding to an infinitely small size, an infinitely largeelectric current, and a finite dipole moment, can be introducedthrough the limit ω→∞ . In this condition, our result reduces to a set of first-order differential equations in perfect agree-ment with Eq. ( 2). This clearly explain why the LLG equation is a first-order differential equation while the Lagrange equa-tions are second-order differential equations, coherently to theNewton law. Moreover, this analysis shows that the Gilbertform of the damping is more adapted than the Landau-Lifshitzform to describe dissipation in ferromagnetic materials. It is interesting to underline that even Gilbert himself rec- ognized that he was not able to conceive a purely mechanicalsystem with a behavior described by his magnetic preces-sional equation. Indeed, the inertia tensor of such a systemwould have only one nonzero principal moment [ 6,7]. As discussed below, this point is clarified by the models pro-posed here, where three positive moments of inertia can beidentified. This problem was noticed in particular by We-growe and co-workers who, in a long series of papers [ 46,48– 51], following previous experiments regarding ultrafast mag- netization switching [ 57], proposed to complete the Gilbert equation by including inertial terms in their derivation. To thisaim, they introduced a true inertial tensor and constructed aLagrangian for which kinetic energy is that of a symmetrictop with one point fixed and the potential energy consistsof the ferromagnetic one. They further utilized the Rayleigh dissipation function proposed by Gilbert [ 6,7] and supported by Brown [ 58,59] in order to account for damped precession. In combining these concepts with the gyromagnetic relationlinking the angular momentum of the top and the magneticmoment, they were able to demonstrate that Eq. ( 2) is comple- mented by extra terms, the importance of which rises in timescales which are shorter than that of gyromagnetic precessionby orders of magnitude. At last, very recently, experimentalevidence of such (resonant) inertial effects were achieved[56], and were found to occur at a probing frequency in the terahertz region, involving characteristic time scales that can-not be described by the classical Gilbert equation. Now, thederivation of Wegrowe et al. [48], although strongly indicating a close analogy between a magnetic dipole and a precessingand nutating symmetric top, does not allow to definitively con-clude regarding this analogy, which is nevertheless extremelyimportant for a qualitative understanding. Our analysis allowsfor a thorough explanation of this issue and better justifies theequation proposed by Wegrowe and co-workers. To summarize, by means of our approach, we clarify the following points concerning the dynamic equation for themagnetization reorientation: (i) We prove that a purely classical mechanics model is able to reproduce the LLG behavior with three positive mo-ments of inertia. (ii) This model supports the Gilbert damping term against the Landau-Lifshitz counterpart. (iii) We show that the effect of the externally applied mag- netic field can be directly described by the general Lorentzforce, without the necessity to introduce a scalar magneticpotential. (iv) We are able to precisely define the damping mech- anism without resting on the Rayleigh function, which isconvenient to use but hides the real dissipative process. (v) The proposed models automatically yield the terms describing the Lorentz force due to the reorientation of themagnetic dipole, a phenomenon always neglected. (vi) Importantly, the models here developed naturally lead to the inertial terms that must be added to the classical LLGequation to describe the specific response at pico- and fem-tosecond time scales. (vii) In particular, the second model, when properly ap- proximated, gives exactly the dynamic equation proposed byWegrowe and co-workers [ 46,48–51]. (viii) We provide evidence that the frequency response based on the two variants of our models and on the We-growe equation is exactly the same and show the classicalferromagnetic resonance together with the inertial or nutationresonance. We believe that these points are important to give a clearer picture of the reorientation process of the magnetization vec-tor and to get a better understanding of its underlying physics. The paper is structured as follows. In Sec. II, we develop the model based on a circular current loop, where the electriccharges rotate at constant angular velocity. Then, in Sec. III, we propose an alternative approach, with an arbitrarily vary-ing angular velocity of the charges. This second approachshows a more elegant symmetry and allows to better explainthe dynamic equation proposed by Wegrowe and co-workers. 214406-2DERIV ATION OF MAGNETIC INERTIAL EFFECTS FROM … PHYSICAL REVIEW B 102, 214406 (2020) FIG. 1. Physical and geometrical description of the system. (a) Discrete distribution of charge rotating around the center of the system with angular frequency ω. (b) Position and velocity vectors of the ith particle with charge q0and mass m0, and definition of the orthonormal bases (/vectore1,/vectore2,/vectore3). (c) Definition of the precession and nutation angles ϕandϑ, and representation of the orthonormal bases ( /vectorλ,/vectorμ,/vectorn) rigidly joined with the magnetic dipole. Finally, in Sec. IVwe analyze the frequency response ob- tained through the proposed models and with the Wegroweequation. II. DERIVATION OF GILBERT’S EQUATION FROM THE MOTION OF A CIRCULAR CURRENT LOOP WITH FIXED CENTER MODEL A magnetic dipole is typically considered as an elementary electric current flowing in a circular loop [ 60,61]. Here, to follow this idea, we initially consider a very simple magneticdipole structure characterized by a sequence of Nequally spaced material points (having mass m 0and charge q0) rotat- ing on a circular trajectory of radius Rwith angular frequency ω. The angular spacing between the points is given by 2 π/N [see Fig. 1(a)]. The resulting dipole moment Mcan be ob- tained through the classical expression M=IS, where I= Nq0ω 2π=qω 2πis the effective electric current in the loop, q= Nq0is the total charge, and S=πR2is its surface. The dipole moment is therefore given by M=1 2qωR2. Also, the total mass can be defined as m=Nm 0. In order to obtain a contin- uous system for the magnetic dipole, we analyze the limitingcase with N→∞ ,q 0→0, and m0→0 while keeping a finite value of qand m. It is also important to define the classical gyromagnetic ratio γ=q 2m, playing an important role in the dynamics of the dipole (according to the laws ofclassical physics, it is the ratio of the magnetic moment to theangular momentum). We suppose that the discrete structure ofthe dipole is invariant (i.e., q 0,m0,N,R, andωare fixed in our process), whereas the dipole plane is subject to arbitraryreorientations around the geometrical center of the system.From the point of view of the analytical mechanics, this is arheonomic system since the mechanical constraint implicatesthe time as an explicit variable [ 62]. It means that the material points have a preexisting orbital motion, independent of thedipole reorientation. It must be admitted that this is an op-erative assumption, which is difficult to be justified from anenergetic point of view. Indeed, in our model, the origin of thepower necessary to maintain the motion of the charges in theloop is not explained. Of course, the model we are going topresent is useful to describe the dipole moment reorientationbut is not able to discuss the origin of the spin behavior ofmagnetized matter. One difficulty comes from the fact that ourmodel is based on classical physics whereas the origin of spinand of the magnetic dipole must be actually discussed withinquantum physics. Indeed, the effective electric current in theloop corresponds, at the atomic scale, to electron spin, nucleonspin, and electron orbital motions within the atom structure, and all these phenomena can be only explained through quan-tum mechanics. Therefore, the energetics of spins or dipolesis beyond the scope of the present paper. Anyway, to better explain the system geometry, we observe that the plane where the dipole is confined can be character-ized by its unit normal vector /vectorn, /vectorn=(cosϕsinϑ,sinϕsinϑ,cosϑ), (3) where ϕandϑare the precession and nutation angles, respec- tively [see Figs. 1(b)and1(c)]. Our aim is to find the equations governing the time evolution of ϕandϑbased on the external actions applied to the magnetic dipole. On the moving planeof the dipole, we can define a couple of unit vectors such thatthey coincide with /vectore 1and/vectore2whenϕ=ϑ=0. We have /vectorλ=∂/vectorn ∂ϑ=(cosϕcosϑ,sinϕcosϑ,−sinϑ), (4) /vectorμ=1 sinϑ∂/vectorn ∂ϕ=(−sinϕ,cosϕ,0). (5) Therefore, ( /vectorλ,/vectorμ,/vectorn) represents an orthonormal basis rigidly bound to the loop [see Fig. 1(c)]. Now, we can introduce the motion /vectorri(t)o ft h e ith point charge q0as follows: /vectorri(t)=/vectorλRcos(ωt+pi)+/vectorμRsin(ωt+pi), (6) which manifestly shows the rheonomic character of the con- straint since the time is explicit within the terms cos( ωt+pi) and sin( ωt+pi). Here, pi=2π N(i−1) with i=1,..., N.O f course, from the trajectory /vectorri(t), we can also define the veloc- ity vector /vectorvi(t)=d/vectorri dt[see Fig. 1(b)]. Within the Lagrangian mechanics, ϕ(t) andϑ(t) assume the role of generalized co- ordinates. By adopting this formalism, the motion equationscan be written as [ 62] d dt∂T ∂˙ϑ−∂T ∂ϑ=Qϑwith Qϑ=N/summationdisplay i=1/vectorFi·∂/vectorri ∂ϑ, (7) d dt∂T ∂˙ϕ−∂T ∂ϕ=Qϕwith Qϕ=N/summationdisplay i=1/vectorFi·∂/vectorri ∂ϕ, (8) where T=1 2m0/summationtextN i=1/vectorvi·/vectorviis the kinetic energy of the parti- cle system, /vectorFiis the total force applied to the ith particle, and (Qϑ,Qϕ) are the so-called generalized forces [ 62]. We discuss below the physical contributions to /vectorFi. We remark that Eqs. ( 7) and ( 8) represent the most general formulation of Lagrangian mechanics where both conservative and dissipative actions 214406-3GIORDANO AND DÉJARDIN PHYSICAL REVIEW B 102, 214406 (2020) can be envisaged [ 62]. Now, to obtain the kinetic energy, we develop the velocity vectors as follows: /vectorvi(t)=˙/vectorλRcos(ωt+pi)−/vectorλωRsin(ωt+pi) +˙/vectorμRsin(ωt+pi)+/vectorμωRcos(ωt+pi).(9) From the orthonormality properties /vectorλ·/vectorμ=0,/vectorλ·/vectorλ=1, and /vectorμ·/vectorμ=1, we get by differentiation˙/vectorλ·/vectorμ+/vectorλ·˙/vectorμ=0,˙/vectorλ·/vectorλ= 0, and ˙/vectorμ·/vectorμ=0. Hence, we simply obtain /vectorvi(t)·/vectorvi(t)=˙/vectorλ·˙/vectorλR2cos2(ωt+pi)+˙/vectorμ·˙/vectorμR2sin2(ωt+pi) +ω2R2+2˙/vectorλ·˙/vectorμR2cos(ωt+pi)s i n (ωt+pi) −2/vectorλ·˙/vectorμωR2. (10) To further simplify this expression we can use the relationships ˙/vectorλ·˙/vectorλ=˙ϑ2+˙ϕ2cos2ϑ, (11) ˙/vectorμ·˙/vectorμ=˙ϕ2, (12) /vectorλ·˙/vectorμ=− ˙ϕcosϑ, (13) ˙/vectorλ·˙/vectorμ=˙ϕ˙ϑsinϑ. (14) We eventually get the kinetic energy in the final form T=1 2m0R2N/summationdisplay i=1[˙ϑ2cos2(ωt+pi)+˙ϕ2+ω2 −˙ϕ2sin2ϑcos2(ωt+pi)+2ω˙ϕcosϑ +2˙ϕ˙ϑsinϑcos(ωt+pi)s i n (ωt+pi)]. (15) Now, we can apply the continuous limit to this result. To do this, we use the relation m0=m/Nand we transform the sum into an integral over p(i.e., over the distribution of charge and mass) by observing that dp/similarequal2π/N. Hence, we get T=1 2mR21 2π/integraldisplay2π 0[˙ϑ2cos2(ωt+p)+˙ϕ2+ω2 −˙ϕ2sin2ϑcos2(ωt+p)+2ω˙ϕcosϑ +2˙ϕ˙ϑsinϑcos(ωt+p)s i n (ωt+p)]dp. (16) A straightforward integration delivers T=1 2mR2/parenleftbig1 2˙ϑ2+˙ϕ2+ω2−1 2˙ϕ2sin2ϑ+2ω˙ϕcosϑ/parenrightbig =1 4mR2(˙ϑ2+˙ϕ2sin2ϑ)+1 2mR2(ω+˙ϕcosϑ)2.(17) This expression allows us to calculate the left-hand side of the Lagrange equations for our system as follows: d dt∂T ∂˙ϑ−∂T ∂ϑ=mωR2/bracketleftbigg ˙ϕsinϑ+1 2ω(¨ϑ+˙ϕ2sinϑcosϑ)/bracketrightbigg , (18) d dt∂T ∂˙ϕ−∂T ∂ϕ=mωR2/bracketleftbigg −˙ϑsinϑ+1 2ω ×(2 ¨ϕ−¨ϕsin2ϑ−2˙ϑ˙ϕsinϑcosϑ)/bracketrightbigg ,(19)FIG. 2. Instantaneous decomposition of the charge velocity /vectorvi in the two components tangent ( /vectorvi,/bardbl) and perpendicular ( /vectorvi,⊥)t o the instantaneous position of the dipole circle. The perpendicular component is crucial for the introduction of the damping force. While /Pi1represents the dipole plane, /Sigma1is the sphere circumscribed to the dipole. where we separated the terms proportional to ωfrom the others. We discuss this separation below. In order to com-plete the Lagrange equations of motion, we need to elaboratethe generalized forces Q ϑand Qϕ. To do this, we assume that the charged particles are subjected to two kinds offorces. First, we consider the Lorentz force /vectorF i,L=q0/vectorvi∧/vectorBgener- ated by an external magnetic induction /vectorB=(Bx,By,Bz). Of course, this is an effective magnetic induction representingmany real factors: the Zeeman effect induced by an appliedmagnetic field, the demagnetization effect generated by themagnetic field created by the magnetization itself, the ex-change effect depending of the gradients of the magnetization,the magnetic anisotropy effect induced by the crystallinestructure of the materials, and the magnetoelastic effectsgenerated by the interaction of magnetic and elastic fields.Typically, all these contributions are summed up through aneffective energy function, which can be derived with respectto the magnetization to give the effective magnetic induction /vectorBapplied to the dipole [ 58,59]. Second, we introduce a damping force describing the effec- tive viscous drag acting opposite to the reorientation motionof the dipole plane /Pi1. We precisely describe the dissipation mechanism as follows. At a given time t,w eh a v eag i v e n orientation of /Pi1identified by /vectornor, equivalently, by ϕandϑ (see Fig. 2). At that time t, each charge velocity /vectorv ican be decomposed in the two components tangent ( /vectorvi,/bardbl) and perpen- dicular ( /vectorvi,⊥) to the instantaneous position of the dipole circle (see Fig. 2). Indeed, while each particle velocity is tangent to the sphere /Sigma1circumscribed to the dipole, it is not tangent to the dipole circle because it is in motion ( ˙ϑ/negationslash=0 and ˙ ϕ/negationslash=0). Now, since we want to describe the damping of the dipolemotion and not of the particle motion, we apply a drag forceopposite to the perpendicular component of each instanta-neous particle velocity. Hence, we define the damping force /vectorF i,D=−k0/vectorvi,⊥=−k0(/vectorvi·/vectorn)/vectorn, where /vectornis defined in Eq. ( 3). This is a phenomenological approach able to effectivelyrepresent all the microscopic processes responsible for theoverall dipole damped motion. The two forces can be summed 214406-4DERIV ATION OF MAGNETIC INERTIAL EFFECTS FROM … PHYSICAL REVIEW B 102, 214406 (2020) to give /vectorFi=/vectorFi,L+/vectorFi,D. In conclusion, we have to calculate the following contributions to the generalized forces: Qϑ,L=q0N/summationdisplay i=1/vectorvi∧/vectorB·∂/vectorri ∂ϑ, (20) Qϑ,D=−k0N/summationdisplay i=1(/vectorvi·/vectorn)/parenleftbigg /vectorn·∂/vectorri ∂ϑ/parenrightbigg , (21) Qϕ,L=q0N/summationdisplay i=1/vectorvi∧/vectorB·∂/vectorri ∂ϕ, (22) Qϕ,D=−k0N/summationdisplay i=1(/vectorvi·/vectorn)/parenleftbigg /vectorn·∂/vectorri ∂ϕ/parenrightbigg , (23) which can be summed to give Qϑ=Qϑ,L+Qϑ,D, (24) Qϕ=Qϕ,L+Qϕ,D. (25) While the exact expressions of the quantities in Eqs. ( 20)–(23) are given in the Appendix, we perform here the continuouslimit of these generalized forces. By recalling that q 0=q/Nand that dp/similarequal2π/Nin Eqs. ( A1) and ( A2), a straightforward integration, as before, yields the following results for thecomponents related to the Lorentz force: Q ϑ,L=qωR2 2/vectorλ·/vectorB+q˙ϕR2 2cosϑ/vectorλ·/vectorB, (26) Qϕ,L=qωR2 2sinϑ/vectorμ·/vectorB−q˙ϑR2 2cosϑ/vectorλ·/vectorB, (27) where we used the definitions of /vectorλand/vectorμgiven in Eqs. ( 4) and ( 5), respectively. Concerning the damping terms given in Eqs. ( A3) and ( A4), we define k=NK 0and we get the continuous limit as Qϑ,D=−1 2kR2˙ϑ, (28) Qϕ,D=−1 2kR2˙ϕsin2ϑ. (29) We have now all the explicit terms to write down the Lagrange equations for the dipole time evolution. The left-hand sidesare summarized in Eqs. ( 18) and ( 19) while the right-hand sides are given in Eqs. ( 26)–(29), which can be summed as in Eqs. ( 24) and ( 25). Hence, Eqs. ( 7) and ( 8) can be finally written as mωR2/bracketleftbigg ˙ϕsinϑ+1 2ω(¨ϑ+˙ϕ2sinϑcosϑ)/bracketrightbigg =qωR2 2/bracketleftbigg /vectorλ·/vectorB+˙ϕ ωcosϑ/vectorλ·/vectorB/bracketrightbigg −1 2kR2˙ϑ, (30) mωR2/bracketleftbigg −˙ϑsinϑ+1 2ω(2 ¨ϕ−¨ϕsin2ϑ−2˙ϑ˙ϕsinϑcosϑ)/bracketrightbigg =qωR2 2/bracketleftbigg sinϑ/vectorμ·/vectorB−˙ϑ ωcosϑ/vectorλ·/vectorB/bracketrightbigg −1 2kR2˙ϕsin2ϑ. (31) By straightforward simplifications, we get ˙ϕsinϑ+1 2ω(¨ϑ+˙ϕ2sinϑcosϑ)=γ/bracketleftbigg /vectorλ·/vectorB+˙ϕ ωcosϑ/vectorλ·/vectorB/bracketrightbigg −α˙ϑ, (32) −˙ϑsinϑ+1 2ω(2 ¨ϕ−¨ϕsin2ϑ−2˙ϑ˙ϕsinϑcosϑ)=γ/bracketleftbigg sinϑ/vectorμ·/vectorB−˙ϑ ωcosϑ/vectorλ·/vectorB/bracketrightbigg −α˙ϕsin2ϑ, (33) where we introduced the gyromagnetic ratio γ=q 2mand the Gilbert damping coefficient α=k 2mω. This is the main achievement of the present section and represents the set ofdynamical equations for the reorientation of the magneticdipole subjected to external magnetic field and damping.These equations have been obtained without any form ofapproximation, starting from the basic assumptions reportedabove. We further observe that three parameters γ,α, andω completely control these dynamical process. It is important to observe that the obtained equations con- tain some terms that are proportional to 1 /ω. These terms can be explained as follows. The terms proportional to 1 /ωin the left-hand sides of Eqs. ( 32) and ( 33) are responsible for the inertial behavior of the magnetic dipole and are indeed relatedto the second derivatives of precession and nutation angles ϕ andϑ. On the other hand, the terms proportional to 1 /ωin the right-hand sides of Eqs. ( 32) and ( 33) represent the Lorentz force generated by the reorientation of the dipole plane /Pi1. Indeed, such a reorientation produces a charge velocity notrelated to ωbut rather to ˙ ϕand ˙ϑ. All these kinds of terms can be typically neglected for the real microscopic dipole or,equivalently, for the so-called ideal dipole. An ideal magnetic dipole is indeed characterized by ω→∞ andR→0, but with a finite value of M= 1 2qωR2. It means that we have a magnetic dipole with an infinitely small size and an infinitelylarge electric current, so that we have a finite dipole moment.In other words, to deal with an ideal dipole, we have tosuppose that ω/greatermuch˙ϑandω/greatermuch˙ϕ. Equivalently, the intrinsic rotation of the charged particles is much faster than the reori-entation process of the dipole plane. Concerning the dampingprocess, the ideal dipole is characterized by the limiting valuesk→∞ andω→∞ , performed by taking a finite value for the damping coefficient α= k 2mω. The meaning of the ideal dipole approximation charac- terized by ω→∞ can be appreciated by considering the paradigmatic magnetic dipole constituted by a hydrogen Bohratom with one electron and one proton. In this case, we haveq=e=1.6×10 −19C,m=me=9.1×10−31kg, and the dipole radius Rcoincides with the Bohr radius a0given by a0=4π/epsilon10¯h2 mee2/similarequal0.5×10−10m. (34) 214406-5GIORDANO AND DÉJARDIN PHYSICAL REVIEW B 102, 214406 (2020) Moreover, the Bohr theory allows the determination of the electron orbital velocity as ve=e2 4π/epsilon10¯h/similarequal2.1×106ms−1, (35) also corresponding to the fine-structure constant ( ∼1/137) times the speed of light in vacuum, c=299 792 458 m /s. The Bohr radius and the electron velocity can be used to directlycalculate the angular frequency as follows: ω=v e a0=mee4 (4π/epsilon10)2¯h3/similarequal4.2×1016s−1. (36) Therefore, we observe that for this magnetic dipole, ωas- sumes a very large value, confirming the validity of the idealdipole hypothesis. To conclude, we can also determine thedipole moment of the electron rotation as M=ev ea0 2=e¯h 2me/similarequal9.2×10−24Am2, (37) corresponding to the so-called Bohr magneton. We can state that, for such a system, the ideal dipole approximation is validif the frequency fof the applied magnetic induction /vectorBis much lower than ω/(2π)∼10 16s−1. Consequently, the inertial ef- fect in the magnetization reorientation can be appreciated onlywith very large frequencies of the applied magnetic field. Weremark that neglecting the terms of the order 1 /ωin Eqs. ( 32) and ( 33) transforms the second-order Lagrange equations in a set of first-order differential equations. This is coherent withthe classical forms of the LLG equation, as generally used inmicromagnetism [ 27,28]. As discussed in the Introduction, the problem of the inertial effect in the dynamics of magnetizationhas been investigated in recent literature [ 46–56]. In these works, an evolution equation for the magnetization has beenproposed. However, it is not completely consistent with ourEqs. ( 32) and ( 33). Indeed, in the previously proposed equa- tion, the terms corresponding to the Lorentz force generatedby the reorientation dynamics [our 1 /ωterms in the right-hand sides of Eqs. ( 32) and ( 33)] have been completely neglected and the purely inertial terms are similar but not coincidingwith ours. The origin of the differences between our approachand previous works is due to the fact that in Refs. [ 46–56]t h e intrinsic rotational motion of the charge defining the magneticdipole is not considered as a basic assumption and thereforethe inertial and Lorentz forces are introduced in a differentway. An alternative approach useful to better draw a compar-ison with the equation proposed by Wegrowe and co-workersis discussed in the next section. Anyway, if we neglect the terms of the order of 1 /ωin Eqs. ( 32) and ( 33), we get the simplified relations ˙ϕsinϑ=γ/vectorλ·/vectorB−α˙ϑ, (38) ˙ϑ=−γ/vectorμ·/vectorB+αsinϑ˙ϕ. (39) This is a first-order system of differential equations, which is not written in normal form. To obtain its normal form,we can substitute ˙ϑfrom Eq. ( 39) into Eq. ( 38) and, recip- rocally, ˙ ϕsinϑfrom Eq. ( 38) into Eq. ( 39). This procedureeventually yields ˙ϕsinϑ =γ 1+α2(/vectorλ·/vectorB+α/vectorμ·/vectorB), (40) ˙ϑ=γ 1+α2(α/vectorλ·/vectorB−/vectorμ·/vectorB). (41) These polar forms of the equations for the magnetization dynamics have been largely used in different applications[63,64]. To conclude, it is not difficult to prove that Eqs. ( 38) and ( 39) are equivalent to the first form (implicit) of the LLG equation, d/vectorM dt=γ/vectorM∧/vectorB−α M/vectorM∧d/vectorM dt, (42) while Eqs. ( 40) and ( 41) are equivalent to the second form (explicit) of the LLG equation, d/vectorM dt=γ 1+α2/bracketleftbigg /vectorM∧/vectorB−α M/vectorM∧(/vectorM∧/vectorB)/bracketrightbigg . (43) To directly prove these equivalences, it is sufficient to consider that/vectorM=M/vectornand use the definition of /vectorngiven in Eq. ( 3). From this result we deduce that the Gilbert damping processappears to be more adapted to describe the magnetizationdynamics than the Landau-Lifshitz counterpart since it hasbeen obtained from a purely mechanical model. Notice thatwith the above formulation, it is difficult to close the equationsof motion for /vectornaccounting for inertial effects when ω→∞ , as all second-order derivatives are, at first glance, wiped outby such a limiting process. Thus, the purpose of the nextsection is to demonstrate that, in the same limit ω→∞ ,a closed-form equation for /vectorncan be found that includes second- order derivatives, and therefore of magnetic inertial effectssimilar with those which have been experimentally evidencedrecently [ 56]. III. MAGNETIC INERTIA CORRECTED GILBERT EQUATION FROM THE CIRCULAR LOOP MODEL The main equations derived in the previous section, namely, Eqs. ( 32) and ( 33), are able to describe all effects produced by an external magnetic field on a magnetic dipole.However, as alluded to in the previous section their form isnot symmetric and it is difficult, if not impossible, to obtaina dynamic equation written only in terms of the vector /vectorn or/vectorM, as expected to get a generalization of the classical LLG equation. So, we describe here an alternative approacheventually yielding a more symmetric formalism and giving arigorous justification of the Wegrowe equation. As before, we consider the moving frame defined by (/vectorn,/vectorλ,/vectorμ), constituting a convenient basis which is rigidly bound with the rotating loop. Now, the rotational frequencyof the charges is arbitrarily varying and we can write theirpositions as /vectorr i(t)=/vectorλRcos(ψ(t)+pi)+/vectorμRsin(ψ(t)+pi),(44) where ˙ψrepresents the arbitrary angular velocity of the charges. Notice that here, the constraints are holonomicand time independent. However, this does not fundamentallychange the basic analysis conducted in the previous section. 214406-6DERIV ATION OF MAGNETIC INERTIAL EFFECTS FROM … PHYSICAL REVIEW B 102, 214406 (2020) The kinetic energy of the system is in the form T=mR2 4(˙ϑ2+˙ϕ2sin2ϑ)+mR2 2(˙ψ+˙ϕcosϑ)2,(45) which is consistent with Eq. ( 17) by replacing ˙ψbyω in Eq. ( 45). Here again, we introduced m=Nm 0and we performed the continuum limit, in exactly the same fash-ion as discussed in the previous section. Written in thisform, the kinetic energy is identical to that of a symmetrictop with one point fixed, with principal moments of inertiaI 1=I2=mR2/2 and I3=mR2[62]. This point definitively shows that the classical mechanics is able to mimic themagnetization dynamics with three positive moments of in-ertia. Now, the generalized coordinates are three in number,namely, ( q 1,q2,q3)=(ϕ,ϑ,ψ ) as it must for a symmetric top with one point fixed, and the generalized velocities are (˙q1,˙q2,˙q3)=(˙ϕ,˙ϑ,˙ψ). Before writing the equations of mo- tion, we also write the total force to which the charges aresubjected. This is given by the Lorentz force combined withthe damping force, /vectorF i=/vectorFi,L+/vectorFi,D=q0/vectorvi∧/vectorB−k0(/vectorvi·/vectorn)/vectorn. (46) Then, we write the Lagrange equations as [ 62] d dt∂T ∂˙qk−∂T ∂qk=N/summationdisplay i=1/vectorFi·∂/vectorri ∂qk,k=1,2,3. (47) By introducing the gyromagnetic ratio γ=q 2m=q0 2m0,t h e apparent damping constant k=Nk0, and using the discrete to continuous limit to evaluate the sums in Eq. ( 47), we explicitly obtain the Lagrange equations d dt(2 ˙ϕ−˙ϕsin2ϑ+2˙ψcosϑ) =2γ˙ψsinϑ/vectorμ·/vectorB−2γ˙ϑcosϑ/vectorλ·/vectorB−k m˙ϕsin2ϑ,(48) ¨ϑ+˙ϕsinϑ(˙ϕcosϑ+2˙ψ) =2γ˙ψ/vectorλ·/vectorB+2γ˙ϕcosϑ/vectorλ·/vectorB−k m˙ϑ, (49) and d dt(˙ψ+˙ϕcosϑ)=−γ˙ϑ/vectorλ·/vectorB−γ˙ϕsinϑ/vectorμ·/vectorB.(50) We set now /Omega1=˙ψ+˙ϕcosϑ. (51) From Eq. ( 50), we may write γ˙ϑ/vectorλ·/vectorB=−γ˙ϕsinϑ/vectorμ·/vectorB−˙/Omega1, (52) and this result can be substituted in the first Lagrange equation given in Eq. ( 48). After straightforward algebra we get ¨ϕsinϑ+2˙ϑ˙ϕcosϑ−2/Omega1˙ϑ=−k m˙ϕsinϑ+2γ/Omega1/vectorμ·/vectorB, (53) ¨ϑ−˙ϕ2sinϑcosϑ+2/Omega1˙ϕsinϑ=−k m˙ϑ+2γ/Omega1/vectorλ·/vectorB,(54)while we also have Eq. ( 50), viz., ˙/Omega1=−γ˙ϑ/vectorλ·/vectorB−γ˙ϕsinϑ/vectorμ·/vectorB. (55) In order to handle the inertial terms, we consider now the vector /vectorJdefined by /vectorJ=/vectorn∧d2/vectorn dt2 =(¨ϑ−˙ϕ2sinϑcosϑ)/vectorμ−(¨ϕsinϑ+2˙ϑ˙ϕcosϑ)/vectorλ =Jμ/vectorμ+Jλ/vectorλ. (56) Then, we may write Eqs. ( 53) and ( 54)a s −Jλ−2/Omega1˙ϑ=−k m˙ϕsinϑ+2γ/Omega1/vectorμ·/vectorB, (57) Jμ+2/Omega1˙ϕsinϑ=−k m˙ϑ+2γ/Omega1/vectorλ·/vectorB. (58) This form exhibits a complete symmetry and can be further developed as follows: −Jλ−2˙ψ/parenleftbigg 1+˙ϕcosϑ ˙ψ/parenrightbigg ˙ϑ =−k m˙ϕsinϑ+2γ˙ψ/parenleftbigg 1+˙ϕcosϑ ˙ψ/parenrightbigg /vectorμ·/vectorB,(59) Jμ+2˙ψ/parenleftbigg 1+˙ϕcosϑ ˙ψ/parenrightbigg ˙ϕsinϑ =−k m˙ϑ+2γ˙ψ/parenleftbigg 1+˙ϕcosϑ ˙ψ/parenrightbigg /vectorλ·/vectorB. (60) These equations must be combined with Eq. ( 50). Indeed, we remark that Eqs. ( 59) and ( 60) completely describe the motion of/vectorn, the unit normal to the loop, if the dynamics of ψis known. However, for the description of the magnetic momentdynamics, we can consider the value of ˙ψlarge (with respect to˙ϑand ˙ϕ) and constant since the modulus of the magnetic moment and the damping coefficient should be considered asconstant parameters. This is accomplished if ˙ψ=ω, (61) where ωis a constant (the same considered in the previous section). This choice is actually legitimate since the dampingforce does not play any role in Eq. ( 50). With this hypothesis, the motion of /vectornis governed by the couple of equations −J λ−2ω/parenleftbigg 1+˙ϕcosϑ ω/parenrightbigg ˙ϑ =−k m˙ϕsinϑ+2γω/parenleftbigg 1+˙ϕcosϑ ω/parenrightbigg /vectorμ·/vectorB,(62) Jμ+2ω/parenleftbigg 1+˙ϕcosϑ ω/parenrightbigg ˙ϕsinϑ =−k m˙ϑ+2γω/parenleftbigg 1+˙ϕcosϑ ω/parenrightbigg /vectorλ·/vectorB, (63) where ωis a constant representing the angular frequency of the charge rotation (nutation frequency). The set of Eqs. ( 62) and ( 63) represents our second proposed model for the mag- netization dynamics. Its form is more elegant and symmetricthan the one given in Eqs. ( 32) and ( 33). Moreover, it allows 214406-7GIORDANO AND DÉJARDIN PHYSICAL REVIEW B 102, 214406 (2020) to draw a comparison with the dynamic equation recently proposed by Wegrowe and co-workers. Actually, a further simplification can be introduced by as- suming that ˙ϕcosϑ/lessmuchωor ˙ϕ/lessmuchω. (64) The last two equations become −Jλ−2ω˙ϑ=−k m˙ϕsinϑ+2γω/vectorμ·/vectorB, (65) Jμ+2ω˙ϕsinϑ=−k m˙ϑ+2γω/vectorλ·/vectorB. (66) Finally, redefining the dimensionless damping constant α=k 2mω, (67) and introducing the time constant τby τ=1 2ω, (68) we easily obtain −τJλ−˙ϑ=−α˙ϕsinϑ+γ/vectorμ·/vectorB, (69) τJμ+˙ϕsinϑ=−α˙ϑ+γ/vectorλ·/vectorB, (70) or, equivalently, the equation for /vectornin the form d/vectorn dt=γ/vectorn∧/vectorB−α/vectorn∧d/vectorn dt−τ/vectorn∧d2/vectorn dt2, (71) which is equivalent to Eqs. ( 65) and ( 66) and represents the equation of motion for the magnetic dipole proposed in recentliterature by Wegrowe and co-workers [ 46–51]. Of course, in thenoninertial limit defined by ω→∞ orτ→0, we obtain again the simplified form d/vectorn dt=γ/vectorn∧/vectorB−α/vectorn∧d/vectorn dt, (72) which is the Gilbert equation for the dynamics of the mag- netization direction. As a conclusion, we can state that theequation of Wegrowe and co-workers can be obtained as anapproximation (with ˙ ϕ/lessmuchω) of the exact equations of motion governing the dynamics of a circular current loop, in turngiven by that of a symmetric top with one point fixed withwell-identified moments of inertia. IV . FREQUENCY RESPONSE We investigate now the characteristic frequency response corresponding to the proposed models, since it is the featuretypically investigated with standard experimental approaches.It means that we apply a uniform and constant bias field /vectorB 0 to the magnetic dipole, with an additive time-varying small perturbation δ/vectorB, and we observe the resulting dipole motion. To simplify the notation, we define the vector /vectorx=(ϕ,ϑ), describing the magnetization orientation. In response to theapplied field, we can observe a preferential fixed directionidentified by /vectorx 0=(ϕ0,ϑ0), perturbed by a small time-varying quantity δ/vectorx. In previous sections, we discussed three different versions of the equations describing the noninertial dynamicsof magnetization: (i) Eqs. ( 32) and ( 33), obtained by consid- ering a uniformly rotating distribution of charge; (ii) Eqs. ( 62) and ( 63), obtained through a symmetric top with one point fixed; and (iii) Eqs. ( 69) and ( 70), which represent a simpli- fication of the second form for high values of ω(coinciding with the equation proposed by Wegrowe and co-workers).These three sets of equations can be cast into the followinggeneral form: f 1(/vectorx,˙/vectorx,¨/vectorx,/vectorB)=0, (73) f2(/vectorx,˙/vectorx,¨/vectorx,/vectorB)=0, (74) where f1andf2are suitable functions representing any of the three models above. We describe here an ad hoc procedure of linearization for this arbitrary system of differential equa-tions. To begin, we can substitute the assumed hypotheses /vectorx=/vectorx 0+δ/vectorxand/vectorB=/vectorB0+δ/vectorBin Eqs. ( 73) and ( 74), eventu- ally obtaining f1(/vectorx0+δ/vectorx,δ˙/vectorx,δ¨/vectorx,/vectorB0+δ/vectorB)=0, (75) f2(/vectorx0+δ/vectorx,δ˙/vectorx,δ¨/vectorx,/vectorB0+δ/vectorB)=0. (76) Since the applied perturbation δ/vectorBand the resulting perturba- tionδ/vectorxare supposed to be small with respect to /vectorB0and/vectorx0, respectively, we can develop previous equations to the firstorder as follows: f 1(/vectorx0,0,0,/vectorB0)+∂f1 ∂/vectorx·δ/vectorx+∂f1 ∂˙/vectorx·δ˙/vectorx +∂f1 ∂¨/vectorx·δ¨/vectorx+∂f1 ∂/vectorB·δ/vectorB=0, (77) f2(/vectorx0,0,0,/vectorB0)+∂f2 ∂/vectorx·δ/vectorx+∂f2 ∂˙/vectorx·δ˙/vectorx +∂f2 ∂¨/vectorx·δ¨/vectorx+∂f2 ∂/vectorB·δ/vectorB=0, (78) where the partial derivatives are calculated for /vectorx=/vectorx0, ˙/vectorx=0,¨/vectorx=0, and /vectorB=/vectorB0. Now, we clearly have that f1(/vectorx0,0,0,/vectorB0)=0 and f2(/vectorx0,0,0,/vectorB0)=0, since /vectorx0is the magnetization direction induced by /vectorB0when the perturbations are not applied. To make this procedure more effective, wesuppose that the perturbation of the applied magnetic induc-tion is given by the sinusoidal oscillation δ/vectorB=Re{/vectorbe i/Omega1Bt}, (79) where /Omega1Bis the angular frequency and /vectorbis the corresponding complex amplitude (phasor). Here Re {z}stands for the real part of the complex number z. Of course, also the angle perturbation follows a similar time evolution δ/vectorx=Re{/vectoraei/Omega1Bt}, (80) with the same angular frequency and where /vectorais its complex amplitude (phasor). It means that we are in asinusoidal steady-state regime. While the applied phasor /vectorb=(b x,by,bz)∈C3is known, the resulting phasor /vectora= (aϕ,aϑ)∈C2is unknown and it can be determined as follows. 214406-8DERIV ATION OF MAGNETIC INERTIAL EFFECTS FROM … PHYSICAL REVIEW B 102, 214406 (2020) By using Eqs. ( 79) and ( 80)i nE q s .( 77) and ( 78), we easily obtain /parenleftbigg∂f1 ∂/vectorx+i/Omega1B∂f1 ∂˙/vectorx−/Omega12 B∂f1 ∂¨/vectorx/parenrightbigg ·/vectora+∂f1 ∂/vectorB·/vectorb=0,(81) /parenleftbigg∂f2 ∂/vectorx+i/Omega1B∂f2 ∂˙/vectorx−/Omega12 B∂f2 ∂¨/vectorx/parenrightbigg ·/vectora+∂f2 ∂/vectorB·/vectorb=0,(82) which is a system of two linear equations in the two unknown components of the vector /vectora. This vector /vectoracan be simply obtained by calculating all the partial derivatives needed inEqs. ( 81) and ( 82), starting from the mathematical expressions off 1and f2, and by solving the linear system. To simplify the calculation, we fix /vectorB0=(B,0,0); i.e., we suppose the bias field is applied to the xdirection of the reference frame. Of course, this assumption does not limit the generality ofthe following achievements. The interesting point is that weget exactly the same result for the three models proposedand discussed previously. It means that Eqs. ( 32) and ( 33), Eqs. ( 62) and ( 63), and Eqs. ( 69) and ( 70) yield the same vector /vectoragiven by a ϕ=2ωγ/parenleftbig 2i/Omega1Bωbz−2byγBω−2iby/Omega1Bαω+by/Omega12 B/parenrightbig D, (83) aϑ=2ωγ/parenleftbig 2i/Omega1Bωby+2bzγBω+2ibz/Omega1Bαω−bz/Omega12 B/parenrightbig D, (84) where D=4/Omega12 Bω2−4γ2B2ω2−8iγBω2/Omega1Bα+4γBω/Omega12 B +4/Omega12 Bα2ω2+4i/Omega13 Bαω−/Omega14 B, (85) which is a fourth degree polynomial in the applied angular frequency /Omega1B. The quantities aϑandaϑare used as follows to obtain the fluctuations of the direction /vectorn. To begin, we can write /vectorn= /vectorn0+δ/vectorn, where /vectorn0is identified by /vectorx0=(ϕ0,ϑ0). Concerning the perturbation we can assume that δ/vectorn=Re{/vectorνei/Omega1Bt}, (86) where /vectorνis the complex amplitude associated to δ/vectorn. Then, a simple use of Eq. ( 3) leads to the first-order relations νx=− sinϕ0sinϑ0aϕ+cosϕ0cosϑ0aϑ, (87) νy=cosϕ0sinϑ0aϕ+sinϕ0cosϑ0aϑ, (88) νz=− sinϑ0aϑ. (89) Therefore, we obtain from Eqs. ( 83) and ( 84) the follow- ing simplified expressions based on the assumption /vectorB0= (B,0,0): νx=0, (90) νy=aϕ, (91) νz=−aϑ. (92)10 11 12 13 log10ΩB-3-2-10123B log10|νz by| log10|νz bz| FIG. 3. Frequency response of the system with a varying value of the applied magnetic induction B. We adopted the parameters γ= 1.76×1011s−1T−1;B=0.1, 0.2, 0.3, 0.4, and 0.5 T; α=0.1; and ω=1×1012s−1. The arrow indicates the increasing values of B. We study the behavior of νzwhen only by/negationslash=0 and when only bz/negationslash=0 and we eventually get νz by/vextendsingle/vextendsingle/vextendsingle/vextendsingle bx=bz=0=−4iγω2/Omega1B D, (93) νz bz/vextendsingle/vextendsingle/vextendsingle/vextendsingle bx=by=0=−2ωγ/parenleftbig 2γBω+2i/Omega1Bαω−/Omega12 B/parenrightbig D,(94) whereDis the polynomial defined in Eq. ( 85). These results represent the frequency response of the system and they areshown in Figs. 3–5. In all plots we can see a first reso- nance that can be identified with the classical ferromagneticresonance and a second resonance that can be ascribed tothe inertial effects taken into consideration in our models.Indeed, in Fig. 3, we can observe that only the first res- onance frequency is shifted with an increasing polarizingfield B, which is the classical behavior of the ferromagnetic 10 11 12 13 log10ΩB-3-2-10123 log10|νz bz|log10|νz by| ω FIG. 4. Frequency response of the system with a varying value of the intrinsic frequency ω. We adopted the parameters γ=1.76× 1011s−1T−1,B=0.25 T,α=0.1, and ω=1.6×1011,2.5×1011, 4×1011,6.3×1011,a n d1 ×1012s−1. The arrow indicates the in- creasing values of ω. 214406-9GIORDANO AND DÉJARDIN PHYSICAL REVIEW B 102, 214406 (2020) 10 11 12 13 log10ΩB-3-2-10123log10|νz by| log10|νz bz|α α FIG. 5. Frequency response of the system with a varying value of the damping factor α. We adopted the parameters γ=1.76×1011 s−1T−1;B=0.25 T;α=0.01, 0.06, 0.11, 0.16, and 0.21; and ω= 1×1012s−1. The arrow indicates the increasing values of α. resonance. Moreover, from Fig. 4, we deduce that only the second resonance is shifted with an increasing value of ω, which is the characteristic frequency describing the inertialeffects. Finally, in Fig. 5, we can observe the effect of the damping factor on the resonance behavior and we concludethat a smaller damping induces a sharper resonance mecha-nism while a larger damping produces a smoother resonanceresponse. This is true for both the ferromagnetic and the in-ertial resonances. The existence of the second resonance peakdue to the inertial effect has been experimentally confirmedin Ref. [ 56], where it has been observed in ferromagnetic thin films at a frequency of approximately 0.6 THz. The fact that the three studied models exhibit exactly the same frequency response means that the mathematicaldifferences among them do not generate different physicalbehaviors. This is true, at least, for the results concerningthe resonance behavior of the frequency response. A fur-ther analysis should be conducted in order to compare thecomplete time evolution of the magnetization for the threemodels with experimental data. We leave this point to furtherinvestigations. V . CONCLUSIONS In this work we readdressed the problem of mimicking the dynamics of a magnetic dipole subjected to a dampingforce and an external magnetic field from purely classicalconcepts. While the classical approaches are based on theLandau-Lifshitz equation and on its Gilbert refinement, recentexperimental and theoretical investigations have shown theneed to extend these theories to include inertial effects. Tothis aim, we propose here to consider a magnetic dipole as acircular current loop and we obtain its quantitative descriptionthrough the Lagrangian mechanics. It is important to place this dipole structure in the context of previous approaches. The idea of using the mechanical anal-ogy between a magnetic dipole and a spinning top has beenefficiently developed by Gilbert in order to derive the equationthat bears his name [ 6,7]. The corresponding dipole can becalled a Gilbert magnetic dipole in order to be distinguished from the Ampère magnetic dipole . This latter is defined by a loop in which the electric current is confined (see, e.g.,Ref. [ 51]). This dichotomy has been proposed by Griffiths [65], who proved that the two dipoles are equivalent in the subrelativistic regime of the electromagnetism. The dipolestructure proposed here is a sort of intermediate version ofthe two systems above. Indeed, we exploited the current loopof the Ampère dipole combined with the possibility to rotateits plane through external actions, as in the Gilbert case. Inthis sense, the electromagnetic and the mechanical behaviorare coupled to eventually obtain the dynamic equation withthe inertial effects. While the Ampère magnetic dipole isclassically used to determine the magnetic field produced bya dipole (by defining the dipole moment M=IS), here we use the current loop to evaluate the forces applied from anexternal magnetic field to the dipole itself. Since our dipole isfree to rotate, these forces produce the reorientation, whosedynamics can be studied by the classical mechanical laws.We can also remark that the idea of merging the Gilbert andAmpère visions can open new perspectives concerning thefull electromagnetic and mechanical analysis of the problem(based on the Maxwell and Lagrange equations). As a matterof fact, the Lagrangian function for the system can be adoptedin the context of the electrodynamics in order to study themost general time-dependent situation. Here, we followed two different lines. In the first one, we supposed a constant angular frequency for the electriccharges rotating in the loop and we dealt with a rheonomicsystem with two degrees of freedom. In the second one,we supposed an arbitrary angular frequency for the chargesby obtaining a holonomic and time-independent system withthree degrees of freedom. In both cases we first introduceda discrete distribution of charges and we performed the limittowards a continuous structure in a second step. The effect ofthe magnetic field is directly introduced through the Lorentzforce without using the magnetic scalar potential. Moreover,the dissipative process is defined by a specific damping forcewithout the need to introduce a Rayleigh dissipation function,but rather asking for this phenomenological damping forcenot to brake the orbital motion of the charges inside the loop(so that the current intensity inside the loop is maintainedconstant). This point allows a better understanding of thedissipative mechanism with a clearer definition of the forcethat opposes the orientation of the magnetic dipole in anexternally applied magnetic field. Importantly, the proposedmodels naturally lead to extra terms with respect to the classi-cal LLG equation, characterizing two important phenomena:(i) the inertial effect that can be observed for high values of thefrequency of the applied magnetic field and (ii) the effect ofthe Lorentz force generated by the reorientation of the dipoleplane, which is usually neglected in previous models. Whilethe first proposed model, with constant angular frequencyof the charges, contains these two terms, its mathematicalform is not symmetric and it is therefore difficult to drawa comparison with the results of the recent literature. Forthis reason, we introduced a refined treatment, with an arbi-trary angular frequency, and we obtained a more elegant andsymmetric form for the dynamical equations. Moreover, itsapproximation, obtained for a reasonably slow reorientation 214406-10DERIV ATION OF MAGNETIC INERTIAL EFFECTS FROM … PHYSICAL REVIEW B 102, 214406 (2020) motion with respect to the rotation of charges, is found to be coinciding with the equation recently proposed by Wegroweand co-workers. Our analysis represents therefore an inde-pendent derivation of this equation. The importance of theinertial effect can be appreciated by taking into account thefrequency response of the dipole system. We provided evi-dence that both the first and second model proposed, and alsothe Wegrowe equation, lead to exactly the same mathematicalform of the frequency response. On the one hand, this provesthat the differences between the mathematical details of thethree models have no observable consequences on the physi-cal response of the system. On the other hand, the frequencyresponse is characterized by two resonance phenomena: whilethe first represents the classical ferromagnetic resonance, thesecond is induced by inertial effects. It is important to notethat this second resonance (between 10 11and 1012Hz) has been observed experimentally only recently [ 56]. If we con- sider a sufficiently low applied magnetic field frequency, theproposed models can be approximated by neglecting inertial effects and provide Gilbert’s equation as a result. Therefore,we can also state that a purely mechanical approach to theproblem of the dynamics of a magnetic dipole gives a strongindication that the damping process is better represented bythe Gilbert assumption than the Landau-Lifshitz counterpart.Finally, we clarified that the motion of a symmetric top witha fixed point, and with three positive moments of inertia, iscoherent with the LLG dynamics of a magnetic dipole. APPENDIX: GENERALIZED FORCES We show here the complete expressions of the generalized forces defined in Eqs. ( 20)–(23), which can be summed as in Eqs. ( 24) and ( 25). The results shown here concern the case of a discrete distribution of charge within the loop. Concerningthe Lorentz force, a long but straightforward calculation leadsto the expressions Qϑ,L=q0ωR2N/summationdisplay i=1[−Bzsinϑcos2(ωt+pi)+Bycosϑsinϕcos2(ωt+pi)+Bycosϕsin(ωt+pi) cos(ωt+pi) +Bxcosϑcosϕcos2(ωt+pi)−Bxsinϕsin(ωt+pi) cos(ωt+pi)]+q0˙ϕR2N/summationdisplay i=1[−Bzsinϑcosϑcos2(ωt+pi) +Bycos2ϑsinϕcos2(ωt+pi)+Bycosϕcosϑsin(ωt+pi) cos(ωt+pi)+Bxcos2ϑcosϕcos2(ωt+pi) −Bxsinϕcosϑsin(ωt+pi) cos(ωt+pi)], (A1) Qϕ,L=q0ωR2N/summationdisplay i=1[−Bzsin2ϑsin(ωt+pi) cos(ωt+pi)+Bysinϑcosϕsin2(ωt+pi)+Bycosϑsinϑsinϕsin(ωt+pi) ×cos(ωt+pi)−Bxsinϑsinϕsin2(ωt+pi)+Bxcosϑsinϑcosϕsin(ωt+pi) cos(ωt+pi)], +q0˙ϑR2N/summationdisplay i=1[Bzsinϑcosϑcos2(ωt+pi)−Bycos2ϑsinϕcos2(ωt+pi)−Bycosϑcosϕsin(ωt+pi) cos(ωt+pi) −Bxcos2ϑcosϕcos2(ωt+pi)+Bxcosϑsinϕsin(ωt+pi) cos(ωt+pi)], (A2) where, as in the main text, we separated the terms proportional to ωfrom the others. 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PhysRevB.91.174112.pdf

PHYSICAL REVIEW B 91, 174112 (2015) Understanding the martensitic phase transition of Ni 2(Mn 1−xFex)Ga magnetic shape-memory alloys from theoretical calculations Chun-Mei Li,1,2,*Qing-Miao Hu,2Rui Yang,2B¨orje Johansson,3,4,5and Levente Vitos3,4,6 1College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China 2Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China 3Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden 4Condensed Matter Theory Group, Physics Department, Uppsala University, P .O. Box 516, SE-75120 Uppsala, Sweden 5School of Physics and Optoelectronic Technology and College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China 6Research Institute for Solid State Physics and Optics, Budapest H-1525, P .O. Box 49, Hungary (Received 6 January 2014; revised manuscript received 11 May 2015; published 26 May 2015) By using first principles in combination with atomistic spin dynamics calculational methods, we determine the temperature-dependent free energies of the L21-a n dL10-Ni 2(Mn 1−xFex)Ga (0/lessorequalslantx/lessorequalslant1), including phonon vibrational and magnetic energies. The x-dependent martensitic phase transformation (MPT) temperature ( TM) and the Curie temperature ( TC) are well represented. It is found that, the abnormal nonmonotonic behavior ofTM∼xmainly originates from the phonon vibrational free energy. The magnetic energy, which does not change the trend of TMagainst xbut decreases the driving force of the MPT, is indispensable as well to get reasonable TMvalues. This insight provides a good understanding of the physical mechanisms driving the MPT of Ni 2(Mn 1−xFex)Ga, and promotes the improvement of their magnetic shape memory properties. DOI: 10.1103/PhysRevB.91.174112 PACS number(s): 62 .20.fg,31.15.es,64.70.kd,75.50.Cc I. INTRODUCTION Ni2(Mn 1−xFex)Ga quaternary magnetic shape-memory al- loys have drawn much attention in recent years. The Fe dopingimproves the poor ductility of Ni 2MnGa [ 1]. It increases the Curie temperature ( TC) and consequently enhances the low output stress (less than 2 MPa) of Ni 2MnGa during the marten- sitic phase transformation (MPT) [ 2]. The Ni 2(Mn 1−xFex)Ga family is a good candidate of the alloys possessing theunique magnetic shape-memory and magnetocaloric ef-fects. They are hoped to possess more numerous technicalapplications [ 2–4]. The MPT in Ni 2MnGa-based alloys generally from the high temperature L21structure (austenite) to the low temperature L10one (martensite) [ 5] is highly composition dependent. From an energetic point of view, all changes in the en-ergy of an alloy system owe their origin to the electronicconsiderations [ 6,7]. The electron-to-atom ( e/a) ratio was thus proposed to be a powerful factor [ 8,9]. In many alloys it has been shown to vary in a systematic manner with anumber of thermophysical properties, such as the formation ofdefect phases [ 10], stacking fault energy [ 11], superconducting transition temperature [ 12], elastic constants [ 13], and so on. In most of the Ni 2MnGa-based alloys, the coarse-grained general rule is that the large e/a ratio corresponds to the high MPT temperature ( TM)[14]. However, it is interesting that there is a nonmonotonic behavior of the TMversus xin Ni 2(Mn 1−xFex)Ga [ 15]. For x/greaterorequalslant0.7, the TMindeed increases with xore/a, corresponding to the above established relationship of TM∼e/a. Whereas, for x/lessorequalslant0.7, the TM decreases with increasing x, i.e., with increasing e/a, which *Corresponding author: cmli@imr.ac.cnresults in the opposite trend of TM∼e/a. The reason for this inconsistency has not yet been found. There are several other plausible quantities to connect the composition with TMof Ni 2MnGa-based alloys. The large electronic energy difference ( /Delta1E el) between the L21andL10 phases corresponds to the high TMin Ni 2+xMnGa 1−x[16]. The decrease of the lattice parameter ( a) of the austenite, or the increase of the tetragonality ( |c/a−1|) of the martensite generally results in the increase of TMin many Ni 2MnGa- based alloys [ 17–19]. The shear elastic constant of the austenite [C/prime=1 2(C11−C12)], relying on the details of the electronic structure of the system, was confirmed to be a better predictorof the composition-dependent T Mthane/ain our published papers [ 20–23]: the softer the C/prime, the higher the TM. However, the opposite trends of the above established TM∼/Delta1E el, TM∼a,TM∼|c/a−1|, and TM∼C/primewere found also in Ni2(Mn 1−xFex)Ga with x/lessorequalslant0.7[22,23]. The true story of the abnormal behavior of TM∼xin this group of alloys is so far still a mystery. In this paper, we compute separately the contributions of the free energies of Ni 2(Mn 1−xFex)Ga (0 /lessorequalslantx/lessorequalslant1) with bothL21andL10phases. Different from the previous work [ 24], the phonon vibrational free-energy part is calculated with the Debye model instead of the quantum theory withinthe quasiharmonic approximation. The magnetic momentat finite temperature is evaluated with first-principles andatomistic spin dynamics methods but not with some empiricalapproximations. As a result, we show a good representationof the composition-dependent T MandTCvalues of this group of alloys. The nonmonotonic trend of TM∼xis confirmed mainly originating from the temperature-dependent phononvibrational free-energy term. The magnetic energy which doesnot influence the trend of T M∼xbut depresses the occurrence of the MPT is indispensable as well to produce reasonable TM values. 1098-0121/2015/91(17)/174112(5) 174112-1 ©2015 American Physical SocietyLI, HU, YANG, JOHANSSON, AND VITOS PHYSICAL REVIEW B 91, 174112 (2015) II. METHODOLOGY A. Calculational tool Based on density functional theory within the generalized- gradient approximation (GGA) described by Perdew, Burke,and Ernzerhof (PBE) [ 25], the present energy and electronic structure calculations are performed with the exact muffin-tinorbitals (EMTO) method [ 26–28], in combination with the coherent potential approximation (CPA) [ 29]. Here, the EMTO basis sets include s,p,d, and fcomponents, and the scalar-relativistic and soft-core approximations are adopted.The overlapping potential spheres ( R mt) are optimized by RNi mt=0.95RWSandRNi WS=1.10RWSfor Ni atoms, where RWSis the average Wigner-Seitz radius. For atoms on Mn/Ga sublattices, the usual setup RMn/Ga mt=RWSandRMn/Ga WS=RWS are adopted. The Brillouin zone is sampled by a 13 ×13×13 uniform k-point mesh. The temperature-dependent magnetism is evaluated with the Uppsala Atomistic Spin Dynamics (UppASD) pro-gram [ 30–34]. Within this method, the itinerant electron system is mapped to an effective classical Heisenberg model, H=−1 2/summationdisplay i/negationslash=jJijmi·mj, (1) where Jijare the interatomic exchange interactions, and the indices iandjare 1, 2, and 3, representing the Mn, Ni, and Fe atoms. The miis the magnetic moment of atom i, the motion of which is described using the Landau-Lifshitz-Gilbert (LLG)equation [ 30,31], ∂m i ∂t=−γmi×[Bi+bi(t)]−γα mmi×{mi×[Bi+bi(t)]}. (2) In this expression, Bi=−∂H ∂miis the so-called effective field experienced by each atom i.γis the gyromagnetic ratio. bi(t) is a stochastic magnetic field with a Gaussian distributionwith respect to temperature T. The impact of b i(t)o nt h e dynamics of the magnetic system is weighted by the dampingparameter α, which eventually brings the system into thermal equilibrium. With the solved m ifrom Eq. ( 2), the magnetic moment of the system ( M) is then calculated from M=1 N/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftBigg/summationdisplay imx,i/parenrightBigg2 +/parenleftBigg/summationdisplay imy,i/parenrightBigg2 +/parenleftBigg/summationdisplay imz,i/parenrightBigg2 .(3) In our calculations, the periodic box size is kept to 10×10×10 unit cells. The time step for solving the above differential equation ( 2)i s1 0−16s, and the number of the time steps used is 10 000. The αis set to 0.01. The Jij includes the exchange interactions between the atoms within the tenth nearest neighbors. They are calculated at 0 K using themagnetic force theorem [ 35] implemented in the EMTO-CPA program [ 26]. B. Helmholtz free energy The Helmholtz free energy ( F) is decomposed as F=Eel+Fvib+Emag−TSmix. (4)Here, the chemical mixing entropy, −TSmix= 1 4kBT/summationtext4 i=1[xilnxi+(1−xi)ln(1−xi)], is taken into account. For Ni 2MnGa at 300 K, the Helmholtz free energy difference ( /Delta1F) between the L21andL10phases is tested merely 0.01 mRy contributed by the electronic entropy term(−T/Delta1 S el, with /Delta1S elbeing the electronic entropy difference between the two phases). At the same temperature, thecontribution of the magnetic entropy ( −T/Delta1 S mag, with /Delta1S mag being the magnetic entropy difference between the two phases) to the /Delta1F of the alloy is less than 0.02 mRy, even when the alloy is described with the paramagnetic (PM) state,i.e., fully disordered local magnetic (DLM) picture [ 36]. Thus, both the −T/Delta1 S eland−T/Delta1 S magterms may be neglected when we evaluate the relative stability of the L21to the L10 phase of Ni 2(Mn 1−xFex)Ga. The vibrational free energy at finite temperature, Fvib,i s calculated with Fvib=−kBT[D(/Theta1D/T)−3ln(1−e−/Theta1D/T)]+9 8kB/Theta1D, (5) where /Theta1DandD(/Theta1D/T) correspond to the Debye temperature and the Debye function [ 37–39], respectively. The magnetic excitations gradually transform the fer- romagnetic magnetic moment into the DLM one over atemperature interval of T C. During this process, with the partially disordered local moment (PDLM) model [ 40], the Ni 2(Mn 1−xFex)Ga system may be described as a (Ni↑ 2Mn↑1−xFe↑ x)1−y(Ni↓ 2Mn↓1−xFe↓ x)yGa pseudoalloy. Here, the yis named magnetic disordering degree, which is connected with the temperature-dependent normalized magnetic moment(M/M 0≈1−2y, with MandM0the magnetic moments at finite temperature and 0 K, respectively). For the sake ofsimplicity, the magnetic energy, E mag, may be defined as the electronic total energy difference between the PDLM systemand the ordered one ( y=0) with the fixed equilibrium volume. In the present work, we first calculate the E magversus yas well as the M/M 0against Tdirectly with the EMTO-CPA in combination with the UppASD methods, and then withthe obtained E mag∼yandM/M 0∼T,t h e Emag∼Tis estimated by means of M/M 0≈1−2y. FIG. 1. (Color online) Free-energy difference between the L21 andL10phases ( /Delta1F≈/Delta1E el+/Delta1F vib)o fN i 2(Mn 1−xFex)Ga (0 /lessorequalslant x/lessorequalslant1) as functions of Tandx. 174112-2UNDERSTANDING THE MARTENSITIC PHASE . . . PHYSICAL REVIEW B 91, 174112 (2015) FIG. 2. (Color online) Normalized magnetic moment ( M/M 0) versus T(a) and the magnetic energy difference ( /Delta1E mag) between theL21andL10phases against the M/M 0( b )o fN i 2(Mn 1−xFex)Ga (0/lessorequalslantx/lessorequalslant1). III. RESULTS AND DISCUSSION A. Contribution of the phonon vibration Figure 1shows the free-energy difference ( /Delta1F≈/Delta1E el+ /Delta1F vib) between the L21andL10phases, including the /Delta1E el and the phonon vibrational free-energy difference ( /Delta1F vib). At 0 K, the /Delta1F is almost the same with the /Delta1E eland it increases with x, indicating that with the Fe doping the Eel term tends to favor the relative stability of the L10phase to the L21one. This results that the TMincreases with x, opposite to the experimental trend of TM∼xbelow x=0.7[15]. Further, adding the /Delta1F vibterm at finite temperature, the /Delta1F of each composition decreases with increasing T. Above room temperature, the phonon vibration term even varies the trend of/Delta1Fagainst x.T h e/Delta1Ffirst decreases, whereas above x=0.6 it still increases with x. Correspondingly, the estimated T M from/Delta1F=0 decreases with increasing xwhen x/lessorequalslant0.6, whereas for x/greaterorequalslant0.6, it increases with x, which is roughly in line with the nonmonotonic trend of the experimental TM against x[15]. It is worth noting that in Fig. 1, these evaluated critical temperature TMvalues are much larger than the experimental ones [ 5,15,41]. For example, when x=0 the obtained TM presents to be about 450 K, which is even more than two times the experimental one (around 202 K) [ 5]. To get reasonable TMvalues, another negative contribution to the /Delta1Fstill needs to be considered at finite temperature.B. Contribution of the magnetism Combining the EMTO-CPA method with the UppASD method, we calculate the magnetic moment Mat temperatures from 0 to 500 K with intervals of 50 K. In Fig. 2(a), the obtained M/M 0with respect to Tis shown for each composition x.F r o m the critical temperature where the M/M 0suddenly drops close to zero, we could first test the TCvalue for each x. Listed in Table I, our theoretical TCturns out to be a little smaller than the experimental one in the same composition [ 15,41]. Nevertheless, with increasing xfrom 0 to 1, the present TC increases from about 340 to 420 K, which is in line with the experimental result that it goes up from 360 to 430 K inthis composition range [ 15,41]. This coincidence confirms the accuracy of the UppASD calculations and thus ensures thereliability of our theoretical data. In Fig. 2(b), the calculated magnetic energy difference between the austenite and martensite ( /Delta1E mag) is shown against theM/M 0with intervals of M/M 0=0.2(y=0.1). In each composition, the /Delta1E magis negative and it increases with the M/M 0. When M/M 0/greaterorequalslant0.4, the increase of the /Delta1E magwith theM/M 0seems to get faster and faster with the Fe doping. Comparing Figs. 2(a) and2(b), it could be approximated that theM/M 0decreases linearly with increasing T, whereas the /Delta1E magincreases linearly with the M/M 0for 0.6/lessorequalslantM/M 0/lessorequalslant 1, where the corresponding temperature is not more than200 K, i.e., below the experimental T Mvalues [ 15]. Therefore, forT/lessorequalslant200 K, we may get the slopes of the linear fitting of M/M 0∼T(d(M/M 0) dT)a sw e l la s /Delta1E mag∼M/M 0(d(/Delta1E mag) d(M/M 0)), and then get the variation of the /Delta1E magagainst T(d(/Delta1E mag) dT). Shown in Table I, with increasing x,t h ed(M/M 0) dTis almost the same, whereas thed(/Delta1E mag) d(M/M 0)increases. Thed(/Delta1E mag) dT, with negative value, thus decreases with the Fe doping. By means of the obtainedd(/Delta1E mag) dTvalues above, the /Delta1F is then recalculated with /Delta1F≈/Delta1E el+/Delta1E mag, and shown as a function of Tandxin Fig. 3. With fixed concentration x,t h e/Delta1F decreases with increasing T, indicating that the /Delta1E magterm prefers the relative stability of the L21phase to theL10one at finite temperature. It is directly opposite to the contribution of the /Delta1E elterm to the relative stability of the two phases. Nevertheless, it seems that the /Delta1E magterm could not vary the trend of /Delta1F∼x.T h e/Delta1F as well as the /Delta1E el always increases with xat each temperature corresponding to /Delta1F/lessorequalslant0. This results in that the estimated TMfrom/Delta1F=0 increases monotonically with x. Comparing Figs. 1and3,t h e TABLE I. The present and experimental Curie temperatures ( TTheor. C andTExpt. C, in K), together with the estimated slopes of M/M 0∼T (d(M/M 0) dT,i n1 / K ) , /Delta1E mag∼M/M 0(d(/Delta1Emag) d(M/M 0),i n1 0−3mRy), and /Delta1E mag∼T(d(/Delta1Emag) dT,i n1 0−3mRy/K) of Ni 2(Mn 1−xFex)Ga (0 /lessorequalslantx/lessorequalslant1). TExpt. C are cited from Refs. [ 15]a n d[ 41]. xTTheor. C TExpt. Cd(M/M 0) dTd(/Delta1Emag) d(M/M 0)d(/Delta1Emag) dT 0.0 340 360 −1.46±0.18 0.66 ±0.01 −0.96±0.15 0.2 355 408 −1.31±0.11 0.95 ±0.06 −1.24±0.19 0.4 360 422 −1.41±0.15 1.09 ±0.07 −1.54±0.27 0.6 385 425 −1.43±0.16 1.15 ±0.09 −1.64±0.33 0.8 405 −1.37±0.13 1.21 ±0.08 −1.66±0.28 1.0 420 430 −1.35±0.08 1.25 ±0.06 −1.69±0.18 174112-3LI, HU, YANG, JOHANSSON, AND VITOS PHYSICAL REVIEW B 91, 174112 (2015) FIG. 3. (Color online) Free-energy difference between the L21 andL10phases ( /Delta1F≈/Delta1E el+/Delta1E mag)o fN i 2(Mn 1−xFex)Ga (0 /lessorequalslant x/lessorequalslant1) as functions of Tandx. nonmonotonic behavior of TM∼xis confirmed originating from the /Delta1F vibbut not the /Delta1E magterm. The latter one, nevertheless, decreases the driving force of the MPT andcontributes to lowering the T Mof each alloy. C. Estimation of the TM Including both the phonon vibration and the magnetic excitations outlined above, we finally get the /Delta1Fwith Eq. ( 4). For comparison, the relative free energies of the L21andL10 phases are calculated as well in the present work by Frel= F−1 4[2ENi+(1−x)EMn+xEFe+EGa], where ENi,EMn, EFe, andEGaare the total energies per atom of Ni, Mn, Fe, and Ga in a hypothetical fcc lattice. Depicted in Figs. 4(a) and4(b), respectively, the Frelof the two phases decrease with increasing Tbut increase with x, meaning that the two phases become more and more stable with increasing T, whereas they become more and more unstable with the Fe doping.Around experimental T M=125 K ∼200 K, in comparison with the L10phase, the increase of Frelwithxin the L21 phase may be relatively slower for 0 /lessorequalslantx/lessorequalslant0.6 but faster for 0.6/lessorequalslantx/lessorequalslant1. Therefore, in Fig. 4(c),t h e/Delta1F decreases with increasing x/lessorequalslant0.6 but increases with x/greaterorequalslant0.6 above 125 K. Correspondingly, the obtained TMvalues from /Delta1F=0fi r s t decrease and then increase with xin the figure, which is in line with the nonmonotonic trend of experimental TM∼x[15]. Moreover, they are greatly lowered in comparison with thoseestimated in Fig. 1, where only the phonon vibration term is considered. Depicted in Fig. 4(d),f o rN i 2MnGa, the present TMvalue is 191 K, corresponding to the reported theoretical data(175 K) [ 24] as well as the experimental one (202 K) [ 5]. For Ni 2FeGa, our TMis about 142 K, in good agreement with the experimental result (145 K) [ 41]. The xdependence of the TMis consistent with that measured by Kikuchi et al. [15]. In the present work, the abnormal behavior ofTM∼xin Ni 2(Mn 1−xFex)Ga is well represented with both composition- and temperature-dependent /Delta1F. Including the composition dependence but neglecting the temperatureeffects, the previous established empirical relationships ofFIG. 4. (Color online) Relative free energies ( Frel)o ft h e L21 phase (a), L10phase (b), and their difference /Delta1F (c) of Ni2(Mn 1−xFex)Ga (0/lessorequalslantx/lessorequalslant1) as functions of Tandx, together with the predicted TMvalues in comparison with the available experimental and theoretical data from Refs. [ 15,24,41]( d ) . TM∼e/a,TM∼/Delta1E el,TM∼a,TM∼|c/a−1|, andTM∼ C/primemay thus fail to account for the TM∼xof the studied alloys. IV . CONCLUSION Using the first-principles EMTO-CPA in combination with the UppASD methods, we have determined thetemperature-dependent free energies of the L2 1- and L10-Ni 2(Mn 1−xFex)Ga (0 /lessorequalslantx/lessorequalslant1), by taking into account the phonon vibration and the magnetic excitations. It is foundthat, the present theoretical T MandTCvalues well represent the phase diagram of the group of alloys. The abnormalnonmonotonic behavior of T M∼xis mainly dominated by theFvibterm. The Emag, which does not influence the trend ofTMagainst xbut depresses the occurrence of the MPT, is indispensable as well to predict the TMvalues. ACKNOWLEDGMENTS The authors acknowledge the financial support from the NSFC under Grants No. 51171187, No. 51271181, andNo. 51301176, and the MoST of China under Grant No.2014CB644001. 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PhysRevApplied.12.014043.pdf

PHYSICAL REVIEW APPLIED 12,014043 (2019) Tunability of Domain Structure and Magnonic Spectra in Antidot Arrays of Heusler Alloy Sougata Mallick,1Sucheta Mondal,2Takeshi Seki,3,4Sourav Sahoo,2Thomas Forrest,5 Francesco Maccherozzi,5Zhenchao Wen,3,4,†Saswati Barman,6Anjan Barman,2Koki Takanashi,3,4 and Subhankar Bedanta1,* 1Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni 752050, Odisha, India 2Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India 3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 5Diamond Light Source Ltd., Diamond House, Didcot, Oxfordshire OX11 0DE, United Kingdom 6Institute of Engineering and Management, Sector V , Salt Lake, Kolkata 700091, India (Received 10 January 2019; revised manuscript received 28 April 2019; published 24 July 2019) Materials suitable for magnonic crystals demand low magnetic damping and long spin-wave propa- gation distance. In this context Co-based Heusler compounds are ideal candidates for magnonic based applications. In this work, antidot arrays (with different shapes) of epitaxial Co 2Fe0.4Mn 0.6Si Heusler- alloy thin films are prepared using e-beam lithography and sputtering technique. Magneto-optic Kerr effect(MOKE) and ferromagnetic resonance analysis confirm the presence of dominant cubic and moderate uni- axial magnetic anisotropies in the thin film. Domain imaging via x-ray photoemission electron microscopy on the antidot arrays reveals chainlike switching or correlated bigger domains for different antidot shapes.Time-resolved MOKE microscopy is performed to study the precessional dynamics and magnonic modes of the antidots with different shapes. We show that the optically induced spin-wave spectra in such antidot arrays can be tuned by changing the shape of the holes. The variation in internal-field profiles, pinningenergy barrier, and anisotropy modifies the spin-wave spectra dramatically within the antidot arrays with different shapes. We further show that by combining the magnetocrystalline anisotropy with the shape anisotropy, an extra degree of freedom can be achieved to control the magnonic modes in such antidotlattices. DOI: 10.1103/PhysRevApplied.12.014043 I. INTRODUCTION Magnon spintronics has emerged as a future poten- tial candidate for alternative computing and data-storage technology due to several advantages viz. highly efficient wave-based computing, applicability in devices of dimen- sions down to approximately 10 nm, operation frequency varying from sub-GHz to THz range, room-temperature transport of spin information without generating Joule heating, etc [ 1]. Further, the spin waves (SWs) also find their applications in on-chip communication systems due to their wavelength being one order shorter than that of the electromagnetic waves, which enables them to minimize the data-processing bits [ 1]. In recent years, magnonic crystals have been rigorously studied for the *sbedanta@niser.ac.in †Present address: National Institute for Materials Science, Tsukuba 305-0047, Japan.propagation and confinement of SWs. There are two kinds of interactions present in magnonic crystals such as short- ranged exchange and long-ranged dipolar interactions. In the large wave-vector ( k) limit (i.e., short wavelength of SW), the interaction is primarily exchange dominated. Due to such strong exchange interaction (nearest-neighbor Heisenberg exchange interaction), the atomic spins remain parallel to each other in the ground state. However, in the case of a ferromagnetic thin film, the moments align themselves in the film plane under an in-plane bias mag- netic field. Such modes traveling in the film plane usually possess a long wavelength (hundreds of nm to several μm), which is significantly larger than the interatomic dis- tance. In such a low klimit, the exchange interaction is weak and dipolar interactions dominate. Further, for SWs with relatively higher kvalues both interactions become non-negligible, which is known as the dipole-exchange SW modes. The interactions can be controlled to tune the dispersion relation in such systems. 2331-7019/19/12(1)/014043(9) 014043-1 © 2019 American Physical SocietySOUGATA MALLICK et al. PHYS. REV. APPLIED 12,014043 (2019) Further two-dimensional magnonic crystals can be broadly subdivided into two parts: dot and antidot lat- tice arrays [ 2]. Magnetic antidot lattice (MAL) arrays are an arrangement of periodic holes in a continuous thin- film system [ 3]. A major advantage of MAL arrays over dot arrays is the miniaturization of their dimension not being restricted by the superparamagnetic limit to the bit size [ 4]. Additionally, MAL arrays with well-defined peri- odic holes, provide precise control over the magnetization reversal, relaxation, and domain structure in comparison to its thin-film counterparts [ 5–9]. Recently, several other aspects of MALs viz. magnetotransport, ferromagnetic res- onance, geometric coercivity scaling, magnetoresistance, domain structure with varying shapes of the holes have been reported [ 10–15]. Further, the MALs are superior to the dot arrays as magnonic crystals because of their larger SW propagation velocity (viz. steeper dispersion) [ 16,17]. The edges of the holes in the antidot arrays quantize the SW modes and modulate internal magnetic field periodi- cally due to the demagnetizing effect [ 16]. Over the last decade, several works have been reported on the control of SW dynamics in MAL arrays by varying the antidot architecture [ 18–24]. The most desired characteristics of a material for magnonic based applications are low magnetic damp- ing, high saturation magnetization, high Curie temper- ature, long SW propagation distance, etc [ 1]. Among various materials reported to date, yttrium iron garnet (YIG) possesses remarkably low magnetic damping of approximately 0.0002 and high SW propagation distance of approximately 22.5 μm[1]. However there are certain drawbacks of YIG such as its large structure and low satu- ration magnetization (approximately 0.14 ×106A/m) [1]. On the other hand, permalloy (NiFe) has been used for SW detection due to their low damping (approxi- mately 0.008) and relatively high saturation magnetization (approximately 0.80 ×106A/m). However, the SW prop- agation distance is rather short (approximately 3.9 μm) in permalloy thin films [ 1]. In this context, Co-based Heusler compounds are promising candidates for magnonic based applications because of their high saturation magnetization (approximately 1.00 ×106A/m), low magnetic damping (approximately 0.003), and moderately high SW prop- agation distance (approximately 10.1 μm) [ 1,25–28]. It has been shown that due to the lower density of states at the Fermi level in a one-spin channel, the spin-flip scattering gets reduced leading to such a remarkable low damping in Co-based Heusler compounds [ 29]. Recently, it has been observed that low magnetic damping down to approximately 0.0045 can be achieved from epitax- ial Co-based Heusler alloy thin films deposited on Crbuffer layer [ 30]. It has been reported that the Co-based Heusler compounds comprise of growth-induced uniaxial magnetic anisotropy and cubic anisotropy, which can be tuned by varying the thickness as well as the choice ofseed layers [ 28,30]. Hence, such antidots of Heusler-alloy thin films provide additional degrees of freedom (over con- ventional permalloy films) to control the SWs and may be useful for future applications. The combination of magne- tocrystalline and shape anisotropy to tune the SW spectra is unexplored. In addition, there are few reports on anti- dot arrays with the anisotropic structures like triangular and diamond-shaped holes, which may significantly mod- ify the internal-field distribution in the vicinity of the holes leading to further modifications of the SW spectra of such systems [ 16,31,32]. In this present work, we choose Co 2Fe0.4Mn 0.6Si (CFMS) thin films and their antidot arrays for the inves- tigation of SW dynamics. To the best of our knowledge, there are no reports yet for the study of magnetization dynamics in antidot arrays of similar materials. We show that under the influence of both magnetocrystalline and shape anisotropy, the magnonic spectra can be tuned with formation of various modes by varying the shape of the antidots. We also explore the possibility of tuning the domain structure in such antidot arrays depending on the available magnetic area and anisotropy distribution. II. EXPERIMENTAL DETAILS Epitaxial thin films of Cr(20 nm)/CFMS(25 nm)/ Al(3 nm) are deposited at room temperature on MgO(100) substrates in an ultrahigh vacuum-compatible magnetron- sputtering chamber with a base pressure of approximately 1.5×10−9mbar. Prior to deposition, the MgO(100) sub- strate is annealed at 600◦C for 15 min for surface recon- struction and removal of impurity from the surface. The 20-nm-thick Cr seed layer is deposited on MgO(100) at a rate of approximately 0.03 nm /sa t1 . 2 ×10−3mbar. After the deposition of Cr, it is annealed in situ at 700◦Cf o r 1 h to form a flat surface. The Cr layer is placed between MgO and CFMS to reduce the magnetic damping of the system [ 30]. Further, the CFMS layer is deposited at a rate of approximately 0.02 nm /sa t1 . 2 ×10−3mbar. Next, the thin-film heterostructure was postannealed in situ at 500◦C for 15 min to promote the B2 (random position of Fe, Mn, and Si, with respect to Co) and L21(completely ordered state) ordering of CFMS. Finally, a 3-nm-thick capping layer of Al is deposited at approximately 0.04 nm /sa t 1.2×10−3mbar to avoid oxidation of the magnetic layer. The CFMS layer shares the following epitaxial relation- ship with MgO: CFMS(001)[110] /bardblMgO(001)[100] [ 33]. Microfabrication of the MAL arrays with different shapes (circular, square, triangular, and diamond) and feature size of 200 nm is performed using e-beam lithography and Ar ion milling. See Supplemental Material for details ofthe microfabrication technique of the MAL arrays [ 34]. The surface structural quality of the films is investigated in situ using reflection high-energy electron diffraction (RHEED). The x-ray diffraction (XRD) measurement is 014043-2DOMAIN STRUCTURE AND MAGNONIC SPECTRA... PHYS. REV. APPLIED 12,014043 (2019) performed ex situ to determine the crystalline quality and atomic site ordering in the film. Saturation magnetization (MS) of the film is extracted from the room-temperature M-Hmeasurement using vibrating sample magnetometry (VSM). High-resolution domain imaging on the nanodi- mensional MAL arrays is performed along the easy axis by x-ray photoemission electron microscopy (XPEEM) at the I06 nanoscience beamline, Diamond Light Source, UK. The nature of anisotropy is extracted from the angle- dependent hysteresis loops measured using a magneto- optic Kerr effect (MOKE) magnetometer with a microsize laser spot. To quantify the growth-induced anisotropy, angle-dependent ferromagnetic resonance (FMR) mea- surement is performed using a Phase FMR spec- trometer manufactured by NanoOsc AB, Sweden. The time-resolved precessional dynamics is measured using an all-optical time-resoled MOKE microscope setup on two- color collinear pump-probe geometry at applied magnetic fields significantly higher than the saturation fields of the samples [ 35]. See Supplemental Material for details of the time-resolved MOKE measurements [ 34]. The experi- mentally observed SW spectra are qualitatively reproduced using micromagnetic simulation (OOMMF) [ 36]. The sim- ulation area is taken as 1600 ×1600×25 nm3for an array of 4 ×4 holes (this ensures the feature size of 200 nm). The cell size for the simulations are considered to be 4×4×25 nm3, which ensures the presence of only one shell along the thickness of the sample. Although dis- cretization of the sample thickness would reveal additional information about the mode variation continuously along the thickness of the film, the resulting simulation time would be computationally challenging. Hence, we com- promise on the small variation along the thickness to focus on the in-plane configurations of the SW modes. We use the following material parameters: γ=2.14×105m/As, MS=9.2×105A/m, K2=3.4×102J/m3,K4=1.17× 103J/m3,α=0.006, and exchange stiffness (A)=1.75× 10−11J/m. The values of γ,α,K2,a n d K4are extracted from the FMR fitting, whereas Ais taken from Ref. [ 30]. Two-dimensional periodic boundary condition (2D PBC) is used for approximating the large sample area. It should be noted that although the simulation qualitatively repro- duces the experimental observation, there is, however, quantitative disagreement due to the limitations in the simulation, viz., edge roughness of the holes, statistical dif- ference in the hole structures, etc [ 37]. Further, the exper- iments are performed at ambient temperature whereas the simulations do not consider any effect arising from the tem- perature. The power and phase profiles of the SW modes are calculated by a home-built code, Dotmag [ 38]. III. RESULTS AND DISCUSSION The parent thin film is denoted as TF, whereas the circular, square, triangular, and diamond-shaped antidotsare defined as CA, SA, TA, and DA, respectively. See Table S1 within the Supplemental Material for a detailed sample description [ 34]. Figures 1(a) and1(b) show the in situ RHEED patterns for TF with electron-beam inci- dence parallel to MgO [110] (CFMS [100]) and MgO [100] (CFMS [110]) directions, respectively. The well-defined reflected spots in the RHEED patterns ensure the formation of epitaxial CFMS film on MgO with the relation: CFMS [110] /bardblCr [110] /bardblMgO [100]. The formation of streak lines in the RHEED pattern indicates the flat film surface. Superlattice streaks are also observed in TF indicating the chemical ordering of CFMS into either B2o r L21structure. Figure 1(c) shows the XRD pattern obtained for TF in theθ−2θgeometry. MgO(200) corresponds to the most intense peak whereas the peak marked by ∗appears from (200) diffraction of the MgO substrate arising from the Cu−Kβsource. Both CFMS (200) superlattice and CFMS (400) fundamental diffractions are visible, which is consis- tent with the results of the RHEED observation. The CFMS (200) superlattice peak corresponds to the formation of B2o r L21structure. However, it is difficult to quantita- tively determine the strength of the B2o r L21ordering individually from the θ−2θgeometry. Figure 1(d) shows the M-Hloop measured using VSM at room tempera- ture for TF while the external magnetic field is applied along the easy axis (CFMS [ ¯110] direction). See Fig. S1 within the Supplemental Material for the orientation of the easy axis and applied field direction with respect to the crystallographic axes of the MgO substrate and CFMS thin film [ 34]. Square hysteresis loop with approximately 100% remanence is observed. The extracted value of MS is 920 ×103A/m. The value of the coercive field ( μ0HC) for TF along CFMS [ ¯110] is 1.7 mT. (a) (b) (c) (d) FIG. 1. (a),(b) RHEED images for TF along CFMS [100] and CFMS [110] directions. (c) XRD pattern for TF showing the CFMS, Cr, and MgO peaks. The peak denoted by ∗appearing around 38◦arises from the (200) diffraction of the MgO substrate with Cu −Kβsources. (d) M-Hcurve for TF measured by VSM at room temperature along the easy, i.e., CFMS [ ¯110] direction. 014043-3SOUGATA MALLICK et al. PHYS. REV. APPLIED 12,014043 (2019) Angle-dependent MOKE measurements reveal that TF exhibits the presence of cubic anisotropy with the easy axes along 0◦,9 0◦, 180◦, and 270◦. See Fig. S2(a) within the Supplemental Material for angular-dependent MOKE data [ 34]. It further reveals the presence of an additional uniaxial magnetic anisotropy. The uniaxial anisotropy can be introduced in a film due to several reasons, viz., oblique angular deposition, anisotropic strain relaxation, miscut in the substrate, interfacial roughness, interfa- cial alloy formation, growth on a stepped substrate, etc [39–42]. The strength of uniaxial and cubic anisotropies are extracted by fitting the angular-dependent FMR data [see Fig. S2(b) within the Supplemental Mate- rial for angular-dependent FMR data [ 34]] with the two-anisotropy Kittel equation, which under small-angle approximation can be represented as f=γ/2π/braceleftbigg/bracketleftbigg H+2K2 MScos2φ−4K4 MScos4φ/bracketrightbigg ×/bracketleftbigg H+4πMS+2K2 MScos2φ−K4 MS(3+cos4φ)/bracketrightbigg/bracerightbigg1/2 , (1) where γis the gyromagnetic ratio, φis the angle between the easy axis and applied magnetic field ( H),K2is the uni- axial anisotropy constant, and K4is the cubic anisotropy constant. The value of MSis obtained from the VSM measurement and used here to calculate K2and K4.T h e fitted values of uniaxial and cubic anisotropy constants are 0.34 ×103, and 1.2 ×103J/m3, respectively. Hence, it is concluded that a noticeable contribution of uniax- ial anisotropy is present in TF, which is approximately 28% of the dominant cubic anisotropy. We further extract the damping constant (α)=0.0056 from the frequency- dependent FMR measurements. The parameters extracted from the FMR measurements are used in OOMMF simu- lation, which is discussed in the later part of the paper. Figure 2shows the hysteresis loops measured using micro-MOKE along CFMS [ ¯110] direction for (a)–(d) CA, SA, TA, and DA, respectively. The insets of Figs. 2(a)–2(d) show the scanning electron microscopy (SEM) images of the corresponding antidot samples. The values of HCare 7.3, 9.5, 5.5, and 9.4 mT, for CA, SA, TA, and DA, respectively. The values of HSfor CA, SA, TA, and DA are 9.0, 10.1, 6.2, and 10.3 mT, respectively. The values of HSare significantly higher for all the anti- dot samples in comparison to their thin-film counterpart. This is expected since the introduction of periodic holes pin the magnetic domains and do not allow the domain walls (DWs) to propagate through them leading to mag-netic hardening. The variation in the strength of H Cand HS between the antidot samples can be explained by the active area (total magnetic area in the antidot samples, which is calculated by subtracting the total area covered by all the (a) (b) (c) (d) FIG. 2. Hysteresis loops measured using micro-MOKE along CFMS [ ¯110] direction at room temperature for CA, SA, TA, and DA, shown in (a)–(d), respectively. The insets show the corresponding SEM images of the antidot samples. holes from the total area) of the samples. The available active area in the triangular antidot is always higher than that in the diamond-shaped antidot with the same feature size and lattice geometry. Hence, the DWs require higher energy to avoid the holes and propagate in a zigzag path for DA in comparison to that in TA. Figure 3(a) shows the SEM image of CA. Fig- ures 3(b)–3(f)show the domain images for CA measured using XPEEM at field pulses of −40.6, 5.1, 8.1, 9.1, and 40.6 mT, respectively. The x-ray magnetic circular dichorism (XMCD) measurements are performed at the L3edge of Fe. The direction of applied magnetic field as well as the sensitivity direction of XPEEM is set along CFMS [ ¯110] direction. To saturate the sample, initially a high magnetic field pulse ( −40.6 mT) is applied and subsequently set to zero to observe the remanent mag- netic state [Fig. 3(b)]. Further, gradually field pulses in the reverse direction are applied followed by removing the field and taking the images at remanence states. The val- ues of the reverse-field pulses are mentioned above for Figs. 3(b)–3(f). The presence of the periodic holes in the film act as nucleation centers and the domains remain pinned in between two successive holes. The domains propagate in a zigzag path avoiding these holes. This leads to formation of a chainlike domain structure [ 43,44]. Fur- ther, the DWs propagate in the transverse direction with the enhancement of applied-field pulses to complete the rever- sal. The domain width in CA near nucleation [Fig. 3(c)] and coercive fields [Fig. 3(d)] are 0.37 and 1.07 μm, respectively. The circular shape of the antidot itself does not contribute to any shape anisotropy but the lattice sym- metry of the two-dimensional magnonic crystal contributes to a shape anisotropy [ 19,45]. This, in conjunction with the 014043-4DOMAIN STRUCTURE AND MAGNONIC SPECTRA... PHYS. REV. APPLIED 12,014043 (2019) (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) FIG. 3. (a) SEM image of CA (circular antidot). (b)–(f) Domain images in CA measured by XPEEM (XMCD at the Fe L3edge) at magnetic field pulses of −40.56, 5.07, 8.11, 9.13, and 40.56 mT, respectively. (g) SEM image of TA (triangular antidot). (h)–(l) Domain images in TA at field pulses of −40.56, 4.26, 4.46, 5.07, and 40.56 mT, respectively. Scale bars for the XPEEM images are the same for both samples, which are shown in (b),(h). The field is applied along the EA (CFMS [ ¯110]) and the direction is shown in (l). magnetocrystalline anisotropy of the CFMS Heusler-alloy thin film, determines its resultant magnetic anisotropy. Since during the XPEEM measurements, the external mag- netic field is applied along CFMS [ ¯110], the domains align in a chainlike structure [Fig. 3(d)]. Nevertheless, with enhancement of the Zeeman energy (external mag- netic field), the domain walls are forced to move in the transverse direction before coalescing with the neighbor- ing domains to complete the reversal. Figure 3(g) shows the SEM image of TA whereas (h)–(l) show the domain images measured at −40.6, 4.3, 4.5, 5.1, and 40.6 mT, respectively. In contrast to the chainlike domain formation in CA, the domains are correlated and bigger in TA. The domain width in TA near nucleation [Fig. 3(i)] and coer- cive fields [Fig. 3(j)] are 0.44 and 3.65 μm, respectively. The triangular holes in TA are not isotropic in nature like the circular ones in CA. Hence, the local-field distribu-tion as well as the anisotropy is different in the vicinity of the holes in the triangular antidot. This modifies the overall anisotropy nature of the parent film. Hence the DWs propagate along both the field as well as transversedirection leading to elevation in size of the domains. The enhancement of the domain size can be further explained by comparing the availability of the active area in CA, and TA. The active area in CA is 71.7% of the total sample area, whereas the same for TA is 84.4%. Hence, due to the availability of larger magnetic area in sample TA, the DWs have more freedom to expand laterally during the reversal. From the above discussion, we conclude that by varying the shape of the holes in the antidot array, domain engineering can be accomplished. Figures 4(a)–4(d) show the SW spectra for CA, SA, TA, and DA, respectively, at an applied field of 153 mT along CFMS [ ¯110]. It should be noted that the applied field is significantly higher than the saturation field so as to employ precession of the magnetization under its sat- uration state. The SW spectra is obtained by taking FFT of the background-subtracted time-resolved Kerr rotation data. See Fig. S3 within the Supplemental Material for the details about obtaining the SW spectra from the Kerr (a) (b) (c) (d)(e) (f) (g) (h) FIG. 4. (a)–(d) SW spectra obtained at μ0H=153 mT from the experimental time-resoled Kerr rotation data for CA, SA, TA, and DA, respectively. (e)–(h) Simulated SW spectra for CA, SA,TA, and DA, respectively. The numbers corresponding to differ- ent modes are shown in the individual images. The images in the insets of (a)–(d) and (e)–(h) show the shape of a single hole inthe antidot array obtained from SEM images and bitmap images considered in the simulation, respectively. 014043-5SOUGATA MALLICK et al. PHYS. REV. APPLIED 12,014043 (2019) rotation in TF [ 34]. It can be observed from Figs. 4(a) and4(b) that there are a total of three modes in the case of CA and SA. The gap between consecutive modes for these two lattices has a slight variation within the spectral width of 9.90 and 12.25 GHz for CA and SA, respectively. In TA, the modes undergo a significant blueshift as opposed to those in CA and SA, and the frequency gaps between con- secutive modes also reduce. The spectral width of TA also reduces to 5.94 GHz. In contrast, in DA, four modes appear with a similar blueshift in frequencies like TA and the gaps between consecutive modes become narrower with respect to CA and SA. The spectral width of DA is 8.91 GHz. Figures 4(e)–4(h) show the simulated SW spectra for CA, SA, TA, and DA, respectively. The simulated spec- tra agree qualitatively with the experimental results. In the simulated spectra, the spectral widths for CA, SA, TA, and DA are 9.65, 13.36, 3.96, and 7.43 GHz, respectively. The spectral width decreases due to the compression of the resonant modes in frequency domain for TA and DA. There are a few low-power modes present in the simu- lated spectra for TA and DA, which are not well resolved in the experiment. The remarkable difference in the mode profile between the different antidots arises due to the dif- ference in their internal-field distribution, which leads to formation of extended and localized modes [ 31]. Addition- ally, asymmetry introduced from the fabrication processimposes further differences in the internal-field profiles in the antidot arrays. Nevertheless, we try to address this issue by considering roundish edges during simulations [see insets of Figs. 4(f)–4(h)] for square, triangular, and diamond-shaped antidots. Indeed, the simulation results are closer to the experimental findings when round corners [see insets of Figs. 4(f)–4(h)] are considered in compari- son to the sharp corners of the holes. Previous reports on such structures have considered zero magnetocrystalline anisotropy for the constituent films [ 31,32]. However, in our study both cubic and uniaxial anisotropies along with the shape anisotropy play important roles in determining the total anisotropy in the antidot samples. In CA and SA, the symmetric structure of the holes does not lead toany additional shape anisotropy along with the anisotropy of the parent thin film. However, the anisotropy is mod- ified in the case of the triangular and diamond-shaped antidots. This leads to the appearance of high-frequency modes. Similar reasons can be corroborated to the absence of low-frequency mode in TA and DA. To understand the nature of the precessional modes, we further simulate the power and phase profiles of the modes using a code described in Ref. [ 38]. Figure 5shows the simulated power maps for the precessional modes for CA, SA, TA, and DA. The phase profiles are shown inside the dashed boxes at the top right corner of each image. See (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m)FIG. 5. Simulated power maps for different precessional modes as shown in Figs. 4(e)–4(h) for cir- cular, square, triangular, and dia-mond antidots, respectively. Phase profiles are shown inside the dashed box at the top right cornerof the images. The color map for the power distribution is shown at the bottom of the image. The respective mode numbers as per Fig. 4is shown in the right side of the image. The quantization numbers are calculated from the profiles within the blue-coloredboxed regions as shown in (g),(j). An external magnetic field of 153 mT is applied along the easy axis(CFMS [ ¯110]) as shown in the figure. 014043-6DOMAIN STRUCTURE AND MAGNONIC SPECTRA... PHYS. REV. APPLIED 12,014043 (2019) (a) (b) (c)FIG. 6. (a) Magnetostatic field distribution of different antidot lattices is shown. The line scansofycomponent (CFMS [ ¯110]) of the magnetostatic field (b) across the antidots [orange dashed linein (a)] and (c) along the chan- nel between the two columns of antidots [green dotted line in (a)].The magnified view of the min- ima in the line scan is shown at the inset of (b). The hollowboxes in (c) represent regions I and II as highlighted in (a). The color bar is presented at the bot-tom of (a). Fig. S4 within the Supplemental Material for the detailed phase map [ 34]. Figures 5(a)and5(b) show low-frequency edge modes (mode no. 1) in CA and SA, respectively. From the mode profile, it can be observed that this is a symmetric mode where the maximum power is stored at the vicinity of the circular and square holes in CA, and SA, respectively. This low-frequency symmetric edge mode is absent in TA and DA due to the modification of internal- field profile. Mode 1 for TA, DA and mode 2 for CA, SA are extended modes, which flow in Damon-Eshbach (DE) geometry through the channels in between the holes [Figs. 5(c) and 5(d)]. The frequencies of the extended modes are slightly reduced in comparison with TF because of modulation by the antidots and varying demagnetized regions in the vicinity of the holes. These modes can also be characterized as quantized modes in backward vol- ume (BV)-like geometry with quantization number q=1. Higher-frequency modes in these lattices are the standing SW modes appearing within the potential barrier between two consecutive holes acting like a cavity resonator. Mode 2 for TA and DA are localized modes in BV-like geometry with quantization number q=4. Mode 3 [Figs. 5(i)–5(l)] is also a localized mode with q=5 for CA, SA, and TA. However, for DA mode 3 and mode 4 are mixed modes where the mode quantization numbers are difficult to assign. This is attributed to the effect of the shape of the diamond antidots in modifying the anisotropy, internal- field profiles, and pinning-energy landscapes in the system. To have a deeper understanding about the magneto- static field distributions of the antidot lattices with different shapes, we perform further simulations using LLG micro- magnetic simulator [ 46]. Figure 6(a) shows the contour plots of the field distribution and internal spin configura- tion for a bias magnetic field, μ0H=153 mT along CFMS [¯110]. Due to the shape-dependent demagnetizing effects, the contours exhibit nonuniform distribution around theantidot edges. For CA and SA, the lines of force at the vicinity of vertical edges of the antidot are denser than that for DA and TA, which prompts the appearance of edge mode for CA and SA. To quantify the magnetostatic field, we take a line scan across the antidots and along the chan- nel between two columns of antidots. The variation of y component (along CFMS [ ¯110]) of magnetic field with distance is plotted in Figs. 6(b) and6(c). The line scans reflect the imprint of the spatial modification in the mag- netostatic field distribution due to different antidot shapes. The magnitude of demagnetizing field inside the antidots is about 40 mT, which is nearly invariant with antidot shape [Fig. 6(b)]. However, the minima in the magneto- static field values show asymmetry for TA [shown in the inset of Fig. 6(b)] due to strong interactions between the unsaturated spins residing at the vertices of the triangular antidot. When we analyze the magnetostatic field inside the channels between the columns of antidot, the situa- tion becomes more interesting. Here, we observe alternate maxima and minima, with the maxima occurring at posi- tions between a pair of nearest-neighbor antidots (region I), while the minima occurring in the continuous-film region (region II) [Fig. 6(c)]. This is further strengthened by the observation of increased density of magnetic field lines in region I as opposed to region II. This effect is most promi- nent in DA and that probably leads to the appearance of additional high-frequency mode in this lattice. IV . CONCLUSION In summary, we study the domain engineering as well as SW dynamics in antidot arrays of Heusler-based CFMSthin films. Epitaxial CFMS thin films are grown on MgO substrates with highly ordered B2o r L2 1phase. MOKE and FMR measurements confirm the formation of cubic anisotropy due to epitaxial growth of the film along with 014043-7SOUGATA MALLICK et al. PHYS. REV. APPLIED 12,014043 (2019) a moderate uniaxial anisotropy. The quantification of the anisotropy is performed using angle-dependent FMR mea- surements, which shows that the uniaxial anisotropy is almost approximately 28% of the cubic anisotropy. We show that the domain structure changes from chainlike fea- tures to wide correlated domains by changing the shape of the holes from circular to triangular. This behavior is explained in terms of availability of the active area and the anisotropy distribution in the vicinity of the holes in different antidot structures. The continuous thin film shows the presence of a uniform precessional mode in the FFT spectra of the time-resolved Kerr rotation data. However, introduction of periodic holes leads to the for- mation of extended and quantized modes in the antidots. The isotropic circular and square antidots show the pres- ence of two dominant modes (higher frequency) along with a low-frequency mode having relatively low power. Appearance of extra modes reduces the frequency gaps in the triangular and diamond-shaped antidots. The extended DE mode along with other quantized modes are present in all the arrays. However, the sharp corners of the holes modify the demagnetizing field in the vicinity of the anti- dots in triangular and diamond-shaped antidots. Further, the intrinsic magnetocrystalline and the shape anisotropy are not aligned in the case of triangular and diamond antidots. Due to its complicated structure and local-field distribution another mixed mode appears at high frequency in the diamond-shaped antidot, which further reduces the frequency gap, which is suitable for SW-based devices. This observed tunability of the SW spectra by varying the shape of the antidots in CFMS thin films having low αand high MSmight have significant impact in future applica- tions based on magnonic filters, splitters, other magnonic devices, etc. ACKNOWLEDGMENTS S.M. and S.B. thank Department of Atomic Energy (DAE), Department of Science and Technology-Science and Engineering Research Board (DST-SERB) (project no. SB/S2/CMP-107/2013), Govt. of India, for providing the funding to carry out the research. We also thank Inter- national Collaboration Center of the Institute for Materi- als Research (ICC-IMR), Tohuoku University, Japan, for providing the funding for visit of S.M. for sample fabri- cation, RHEED, XRD, micro-MOKE, and VSM measure- ments. We acknowledge Diamond Light Source for time on I06 under proposal SI16582 for XPEEM measurements. We further thank Visitor, Associates, and Students’ Pro- gramme (VASP), of S. N. 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PhysRevLett.124.087702.pdf

Disorder Dependence of Interface Spin Memory Loss Kriti Gupta ,1Rien J. H. Wesselink,1Ruixi Liu,2Zhe Yuan ,2,*and Paul J. Kelly1,2,† 1Faculty of Science and Technology and MESA+Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands 2The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 Beijing, China (Received 11 November 2019; revised manuscript received 17 January 2020; accepted 29 January 2020; published 27 February 2020) The discontinuity of a spin-current through an interface caused by spin-orbit coupling is characterized by the spin memory loss (SML) parameter δ. We use first-principles scattering theory and a recently developed local current scheme to study the SML for Au jPt, Au jPd, Py jPt, and Co jPt interfaces. We find a minimal temperature dependence for nonmagnetic interfaces and a strong dependence for interfaces involving ferromagnets that we attribute to the spin disorder. The SML is larger for Co jPt than for Py jPt because the interface is more abrupt. Lattice mismatch and interface alloying strongly enhance the SML that is larger for a AujPt than for a Au jPd interface. The effect of the proximity-induced magnetization of Pt is negligible. DOI: 10.1103/PhysRevLett.124.087702 With the discovery of the giant magnetoresistance effect in magnetic multilayers [1,2], it was recognized that interfaces play a key role in spin transport phenomena.In semiclassical formulations [3–5]of transport, these appear as discrete resistances and the description of the transport of electrons through a multilayer requires a resistivity ρfor each material as well as a resistance R I for each interface. Bulk resistivities are readily measured, interface resistances much less so. Magnetic materialsrequire spin-dependent bulk resistivities ρ σand interface resistances Rσ. Because spin is not conserved, describing its transport additionally requires a spin-flip diffusionlength (SDL) l sfin each material, as well as its counterpart for each interface, the spin memory loss (SML) parameterδ. Determining l sfrequires distinguishing interface and bulk contributions. Because doing so is nontrivial, theinterface contribution had been largely neglected and values of l sfreported over the last decade for well-studied materials like Pt span an order of magnitude [6–8]. Almost everything we know about interface parameters is from current-perpendicular-to-the-plane (CPP) magneto-resistance experiments [5,6,9] interpreted using the semi- classical Valet-Fert (VF) model [4]. Though these experiments are relatively simple to interpret, they arerestricted to low temperatures as they require supercon-ducting leads [5], while calculations have so far only addressed ballistic interfaces [10–15]. Because the vast majority of experimental studies in spintronics is carriedout at room temperature, there is a need to know howtransport parameters, in particular, those describing inter-faces, behave at finite temperatures. This need is accentuated by the huge interest in recent years [7,16] in the spin Hall effect (SHE) [17–19], whereby a longitudinal charge current drives a transverse spincurrent in nonmagnetic materials, and in its inverse, the inverse SHE (ISHE). Determination of the spin Hall angle(SHA) Θ sHthat measures the efficiency of the SHE is intimately connected with the SDL and, because an interfaceis always involved, with the SML [20]. When use is made of spin pumping and the ISHE [21–24]or the SHE and spin- transfer torque [25], the interface in question is between ferromagnetic (FM) and nonmagnetic (NM) materials. Whenthe nonlocal spin-injection method is used [26,27] ,t w o interfaces are involved: an FM jNM interface to create a spin accumulation and an NM jNM 0interface to detect it. Progress has been made by recognizing that bulk parameters like lsf andΘsHare very sample dependent and that the SML plays a key role in determining their values [20,28 –30]. Recent studies suggesting that measurements of the SHA mayactually be dominated by interface effects [20,28,31,32] are stimulating attempts to tailor these [33–37]. This makes it crucial to have a way to independently determine interface parameters. We recently described aformalism to evaluate local charge and spin currents [8] from the solutions of fully relativistic quantum mechanicalscattering calculations [38] that include temperature- induced lattice and spin disorder [39,40] . This yielded a layer-resolved description of spin currents propagatingthrough atomic layers of thermally disordered Pt and Pythat allowed us to unambiguously determine bulk transportproperties [41]. By focusing on spin currents, we can straightforwardly evaluate all the parameters entering theValet-Fert semiclassical formalism [4]that is universally used to interpret experiment [5]. In this Letter, we focus on interface transport properties and study realistic interfaces between thermally disorderedmaterials. Typical structures used in experimental studies ofthe SHE contain a heavy NM metal with strong spin-orbitPHYSICAL REVIEW LETTERS 124, 087702 (2020) Editors' Suggestion 0031-9007 =20=124(8) =087702(7) 087702-1 © 2020 American Physical Societycoupling (SOC) and a 3dtransition metal (TM) or TM alloy ferromagnet [7,20 –25,42] . We will study (i) Au jPt and AujPd interfaces to shed light on the role of SOC and roughness at interfaces involving heavy TMs and (ii) Py jPt and Co jPt interfaces to examine the role of the FM magnetization and disorder in determining interface param- eters, as well as the temperature dependence of all these. Method.—We begin by solving the VF equations ana- lytically for the spin accumulation μsiðzÞand spin current jsiðzÞin a metallic multilayer. The solution in each region i involves two coefficients Aiand Bithat are determined by appropriate boundary conditions [4]. For an NM jNM0 system, we will require that jsð0Þ¼1at the left-lead jNM interface, corresponding to injecting a fully spin polarized current from the left lead, and that jsð∞Þ¼0, requiring the NM0material to be much thicker than its lsfvalue. The interface (I) is initially considered as a bulklike material with resistivity ρI,S D L lI≡lI sf, and thickness tso that at the NM jI and I jNM0interfaces the spin accumulation and spin current are continuous [43,44] . We then eliminate the coefficients AIand BIand take the limit t→0, thereby defining the areal interface resistance ARIand the SML parameter δas ARI¼lim t→0ρItand δ¼lim t→0t=lI: ð1Þ We finally express δas js;NMðzIÞ js;NM0ðzIÞ¼cosh δþρNM0lNM0 ARIδsinh δ ð2Þ in terms of jsiðzIÞ, the spin current at the interface zIon thei¼NM and NM0sides as well as ρNM0,lNM0≡lNM0 sf, and RI. The relationship of the SML to the spin-current discontinuity js;NMðzIÞ−js;NM0ðzIÞis nontrivial. AujPtinterface. —We illustrate our methodology in Fig. 1for a Au jPt interface between “room-temperature ” Au and Pt, in which a Gaussian distribution of atomicdisplacements in a 7×7lateral supercell was used to reproduce the experimentally observed resistivities of each bulk material at T¼300K,ρ Au¼2.3μΩcm and ρPt¼ 10.7μΩcm[45], for which lAu∼80nm and lPt∼5.25/C6 0.05nm[8]. The empty gray circles in Fig. 1represent jsðzÞ, obtained [46]from the results of quantum mechani- cal scattering calculations [38] for a Au jPt bilayer when a fully polarized spin current was injected into the bilayer from the left Au lead. The lattice constant of fcc Au is initially chosen to be that of Pt ( a0 Au¼aPt¼3.923Å), which does not affect the Au electronic structure qualita- tively. The figure also shows the VF solutions in Au (blue curve) and Pt (red curve) found by fitting jsðzÞfar from the interface. The initial spatial decay of jsðzÞis determined bylAu, the rapid decay in the vicinity of the interface is described in the semiclassical VF framework by the inter- face discontinuity and, after this abrupt decay, the spincurrent that survives in Pt decays to zero on a length scale described by lPt. By fitting jsðzÞto the solution of the VF equation, we obtain values of js;AuðzIÞandjs;PtðzIÞ. From the Landauer expression for the conductance in terms of the transmission matrices, ARI¼0.54/C60.03fΩm2is directly determined, leaving just δas the only unknown parameter. Using a numerical root finder to solve (2), we find δ¼0.62/C60.03, where the error bar is evaluated from the uncertainties in the other parameters. The bulk parameters ρPtand1=lPtare known to increase linearly with temperature [31,40,45,47] , but virtually nothing is known about the temperature dependence ofinterface parameters. We therefore calculate AR AujPtandδ at 200 and 400 K and plot the results in Fig. 1(inset). Within the error bars of the calculations, both parametersremain constant between 200 and 400 K. The temperatureindependence that we observe for δis in agreement with the results of a very recent CPP-magnetoresistance experiment for a Cu jPt interface that shows δto be nearly constant over the temperature range 0 –300 K [48]. Interface mixing. —Unlike the sharp interfaces between bulk Au and Pt we have considered so far, experimentalinterfaces are believed to comprise a few intermixed layers.To study the effect of interface mixing, we insert Natomic layers of Au 50Pt50random alloy at the interface of the lattice-matched Au jPt bilayer [49]. The results for the spin current jsðzÞand corresponding values of δat 300 K are-10 -5 0 5 10 15 20 25 30 z (nm)0.00.20.40.60.81.0js T = 300 K ARI(fm2) Temperature (K)ARI 200 300 4000.8 0.4 0.00.6 0.3 0.0tP|uAtP|uA tP|uAtP|uA|uAdP|uA Au|PdPt Au Au (300 K)P AuAu AuAAA (30(((3330000000K)KKK)K)K)Pt AAuAAAAAAAAAAAAAAuuu FIG. 1. A fully polarized spin-current jsinjected at 300 K from the left Au lead into a Au ð50ÞjPtð140Þbilayer sandwiched between Au leads decays exponentially in Au and in Pt; thenumbers in brackets denote the number of atomic layers. Thesolid lines indicate fits for j sðzÞin individual layers using solutions of the VF equations. (Top inset) Schematic of thescattering region. (Middle and bottom insets) Temperature dependence of the interface parameters δ(yellow) and R I(purple) for a Au jPt interface (circles). The corresponding parameters for AujPd at 300 K are included (diamonds).PHYSICAL REVIEW LETTERS 124, 087702 (2020) 087702-2shown in Fig. 2forN¼0, 2, and 4. When the spin current from bulk Au approaches the mixed interface layers(yellow for N¼2, green for N¼4), then compared to the sharp interface, j sðzÞdecreases more and δincreases rapidly with increasing N(inset). Electron scattering at a commensurable and clean Au jPt interface only involves Bloch states with equal kk, but intermixing (and thermal disorder) break momentum conservation and allow kk→k0 kscattering. The higher scattering rate results in a higher spin-flipping probability and hence a larger δfor the intermixed interfaces. Moreover, conduction electronsinjected into Au are only weakly affected by SOC untilthey enter Pt, where as dstates they become very susceptible to the large SOC. The interatomic mixing effectively increases the region where conduction electrons experience large SOC and therefore increases the SML.The interface resistance AR Ialso increases monotonically as the disordered region increases in thickness, suggestingthat δ∝AR I,s o ρIlI∼const. Lattice mismatch. —To study the effect of kk→k0 k scattering on its own, we reexamine the sharp (111) AujPt interface where both Au and Pt have their equilib- rium bulk volumes, aAu¼4.078 and aPt¼3.923Å. A (111) oriented 5×5unit cell of Au matches to a (111) oriented 3ffiffi ffi 3p ×3ffiffi ffi 3p unit cell of Pt to better than 0.02%. For this fully relaxed Au jPt geometry, we repeat our calculations at 300 K and obtain δ¼0.81/C60.05comparedtoδ¼0.62/C60.03with commensurable Au. This calcu- lation indicates that the kk→k0 kscattering does indeed lead to an increase of the interface SML. Our finding is in agreement with calculations for a Cu jPd interface using the ansatz of Schep et al. [10,50] , which indicated that δ increases on going from a sharp to a rough interface [14]. AujPd.—To examine the effect of changing the strength of the SOC, we apply the procedures described above to a commensurable Au jPd interface, choosing a0 Au¼aPd¼ 3.891Å. Corresponding to the experimental resistivity of Pd at 300 K, ρPd¼10.8μΩcm[45], we find lPd¼7.06/C6 0.02nm and a value of ARAujPd¼0.81/C60.05fΩm2, which is much larger than the value 0.54/C60.03fΩm2 found for Au jPt (inset Fig. 1, bottom panel). By substitut- ing all the input parameters and their uncertainties into (2), we extract a value of δAujPd¼0.43/C60.02(Fig. 1, inset). Compared to Au jPd, the larger SOC in Pt leads to a larger value of δfor Au jPt. Our results for ARIandδfor Au jPd interfaces are in good agreement with theoretical estimatesmade by Belashchenko et al. [14], combining the ansatz of Schep et al. [10] with calculations for ballistic Cu jPd interfaces. Cu and Au have very similar electronic struc- tures and the very different SOC of the filled 3dand5d states below the Fermi level is not expected to play a major role. FMjPtinterfaces. —We developed an analogous pro- cedure to study FM jNM interfaces. Compared to the NMjNM 0case, two additional parameters enter: spin asym- metry parameters β¼ðρ↓−ρ↑Þ=ðρ↓þρ↑Þfor the bulk FM andγ¼ðR↓−R↑Þ=ðR↓þR↑Þfor the interface. To avoid interfaces between a lead and Py or Co, we considered symmetric NM jFMjNM scattering geometries and studied them by passing an unpolarized charge current throughthem. The appropriate boundary conditions are that both the spin accumulation and spin current vanish at z¼/C6∞and the analysis results in two implicit equations, containing thediscontinuity in the spin current at the FM jNM interface, as described by j s;FMðzIÞand js;PtðzIÞ,a sw e l la st h ee i g h t transport parameters ρNM,lNM sf,ρFM,lFM sf,βFM,RI,δ,a n d γ. Figure 3(a)illustrates the spin current jsðzÞthat we calculate for a Pt jPyjPt trilayer at 300 K. The five bulk parameters are determined independently as well as ARIobtained from the Landauer formula, leaving us with two equations and two unknowns, δandγ,t ob ed e t e r m i n e d . This procedure was applied to Py jPt and Co jPt interfaces assuming completely relaxed geometries and 8×8inter- face unit cells of (111) Py or Co matched to 2ffiffiffiffiffi 13p ×2ffiffiffiffiffi 13p interface unit cells of Pt. Thermal lattice and spin disorder were taken into account as described in Refs. [8,38 –40].ρPt andlPtwere already determined above and the appropriate corresponding calculations were performed for bulk Py and Co [40,51] . The temperature dependence of the three interface parameters that we extract for Py jPt and Co jPt interfaces- 2 02468 1 0 z (nm)0.00.20.40.60.81.0jsAu|AuPt(2)|Pt 0.0 0.5 1.0 ARI(fm2)0.00.51.0Au|AuPt(4)|PtAu|Pt N=0N=2N=4T=300 K Au Pt FIG. 2. A fully polarized spin current jsðzÞis injected into a AujPt bilayer with a sharp interface (vertical black line), two layers of Au 50Pt50interface (yellow shaded region), and four layers of Au 50Pt50interface (green shaded region) between them. The calculated spin currents jsðzÞfor the three cases are shown as gray circles, yellow diamonds, and green squares, respectively.The solid blue line indicates a fit to the VF equation in Au. Thesolid, dashed, and dotted red lines indicate fits to the VF equationin Pt for Au jPt, Au jAu 50Pt50ð2ÞjPt, and Au jAu50Pt50ð4ÞjPt, respectively. (Inset) δvsARIforN¼0, 2, and 4 interface layers of mixed Au 50Pt50.PHYSICAL REVIEW LETTERS 124, 087702 (2020) 087702-3is summarized in Figs. 3(b)–3(d).ARIand γare seen to decrease monotonically with temperature for both inter-faces. γis found to vary in a small range of /C60.15for Py jPt and/C60.03for Co jPt.δ, the main focus of our interest, decreases monotonically and substantially withtemperature for both interfaces. δ CojPtis larger than δPyjPt for all temperatures in the range 200 –500 K. This temper- ature dependence contrasts starkly with the temperatureindependence we found for Au jPt. We speculate that it is the variation of the spin disorder associated with the FM magnetization that affects the interface parameters most. Consistent with this is our finding that δandAR Iare larger for Co jPt than for Py jPt. With a higher Curie temperature, Co is more ordered at any given temperature than Py. To test this hypothesis, we repeated the T¼200, 300, and 400 K calculations for Py jPt keeping the atomic spins ordered and including only lattice disorder in Py. Theresults for the three interface parameters with only latticedisorder are included in Fig. 3(open circles, dotted lines) for comparison. With only lattice disorder included, wefind that the Py jPt interface parameters decrease much more slowly with temperature. This weak variation can beattributed to the lattice disorder, but the decrease is muchsmaller compared to that brought about by spin disorder. Inthe low-temperature limit, we also expect δto be smaller for PyjPt because this interface is less abrupt than Co jPt, owing to Py ’s intrinsic disorder. SOC-induced interface splittings are smeared out by alloy disorder in Py compared to Co,leading to smaller δ. Finally, we found that proximity- induced magnetization of Pt by Co or Py has no effect onthe interface parameters within the error bars of thecalculations; see the left inset to Fig. 3(a). Summary. —First-principles scattering theory and a recently developed local current scheme have been usedto study how spin currents propagate through interfacesbetween two nonmagnetic (Au jPt and Au jPd) materials and between a ferromagnetic and a nonmagnetic (Py jPt and CojPt) material at finite temperatures. By extracting values ofδ,R I,a n d γ, we could study how δdepends on various properties of the interfaces and temperature. For nonmag- netic interfaces, we found that δandARIremain unchanged over a wide range of temperature and found values of ARI that are in remarkably good agreement [50]with an ansatz introduced more than twenty years ago by Schep et al [10]. δAujPtwas found to be larger than δAujPdowing to the larger SOC in Pt, indicating a direct link between the magnitude of δand SOC strength of NM metals. An incommensurable Au jPt interface with relaxed Au and Pt lattices has a substantially larger δthan the lattice-matched interface. Mixing at an interface also leads to larger values ofδ.T h u s ,t om i n i m i z e δ, lattice-matched and clean interfaces should be targeted in experiments to avoid momentum-nonconserving scattering of conduction electrons. FMjPt interface parameters decrease strongly with increasing temperature. This dependence stems directlyfrom the magnetization of the FM. Co is a stronger FMthan Py and we find that AR CojPtandδCojPtare larger than ARPyjPtandδPyjPtfor all temperatures. By turning off spin disorder in Py jPt, the variation of interface parameters with temperature becomes negligible.100 200 300 400 500 Temperature (K)0.00.51.0 100 500 3000.51.0 100 300 500 Temperature (K)0.60.81.0ARI(f-m2) 100 300 500 Temperature (K)-0.20.00.2Lattice disorder only Lattice disorder only Lattice disorder only(b) (c) (d)z (nm)jszI js,Pt(zI)js,Py(zI) Pt Pt PyT = 300 K 0.00.20.40.60.8 02 0 4 0 -40 -20(a) CoPt PyPtPyPt PyPt CoPt CoPtPyPt PyPtPyPt-10 -5 0 z (nm)00.20.40.6jsExcluded IncludedInduced moment in Pt Pt Py FIG. 3. (a) Open circles: spin current jsðzÞthrough a Pt jPyjPt trilayer calculated for T¼300K. The solid blue (orange) curve is a fit to the VF equations in bulk Pt (Py). These fits areextrapolated to the interface z Ito obtain the values js;PtðzIÞand js;PyðzIÞ, shown in detail in the right inset. (Left inset) The spin current with (red) and without (blue) proximity-induced momentsin Pt. (b) δfor Py jPt (circles, solid lines) and Co jPt (diamonds, dashed lines) plotted as a function of temperature. (Inset) δfor PyjPt compared with results with only lattice disorder in Py jPt (open circles, dotted lines). (c),(d) Interface parameters AR Iandγ for Py jPt (circles, solid and dotted lines) and Co jPt (diamonds, dashed lines) plotted as a function of temperature. The dottedlines show the results for Py jPt with only lattice disorder.PHYSICAL REVIEW LETTERS 124, 087702 (2020) 087702-4This work was financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek ”(NWO) through the research program of the former “Stichting voor Fundamenteel Onderzoek der Materie, ”(NWO-I, formerly FOM) and through the use of supercomputer facilities of NWO “Exacte Wetenschappen ”(Physical Sciences). K. G. acknowledges funding from the Shell-NWO/FOMComputational Sciences for Energy Research Ph.D. program (CSER-PhD; No. i32; Project No. 13CSER059) and is grateful to Yi Liu for help in starting this work and to S.Wildeman for helpful discussions. The work was also supported by the Royal Netherlands Academy of Arts and Sciences (KNAW). Work in Beijing was supportedby the National Natural Science Foundation of China (Grant No. 61774018), the Recruitment Program of Global Youth Experts, and the Fundamental Research Funds for theCentral Universities (Grant No. 2018EYT03). *zyuan@bnu.edu.cn †P.J.Kelly@utwente.nl [1] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Giant Magnetoresistance of ð001ÞFe=ð001ÞCr Magnetic Superlattices, Phys. Rev. Lett. 61, 2472 (1988) . [2] G. Binasch, P. Grünberg, F. Saurenbach, and W. 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PhysRevB.93.060401.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 93, 060401(R) (2016) Dynamics of Dirac strings and monopolelike excitations in chiral magnets under a current drive Shi-Zeng Lin*and Avadh Saxena Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 19 January 2015; revised manuscript received 25 January 2016; published 10 February 2016) Skyrmion lines in metallic chiral magnets carry an emergent magnetic field experienced by the conduction electrons. The inflow and outflow of this field across a closed surface is not necessarily equal, thus it allows for theexistence of emergent monopoles. One example is a segment of skyrmion line inside a crystal, where a monopoleand antimonopole pair is connected by the emergent magnetic flux line. This is a realization of Dirac stringlikeexcitations. Here we study the dynamics of monopoles in chiral magnets under an electric current. We showthat in the process of creation of skyrmion lines, skyrmion line segments are first created via the proliferation ofmonopoles and antimonopoles. Then these line segments join and span the whole system through the annihilationof monopoles. The skyrmion lines are destroyed via the proliferation of monopoles and antimonopoles at highcurrents, resulting in a chiral liquid phase. We also propose to create the monopoles in a controlled way byapplying an inhomogeneous current to a crystal. Remarkably, an electric field component in the magnetic fielddirection proportional to the current squared in the low current region is induced by the motion of distortedskyrmion lines, in addition to the Hall and longitudinal voltage. The existence of monopoles can be inferred fromtransport or imaging measurements. DOI: 10.1103/PhysRevB.93.060401 Skyrmion in magnets is a stable mesoscopic topological excitation where spins wrap the surface of a sphere once [ 1,2]. Skyrmions have been observed in magnetic materials, suchas B20 chiral magnets without inversion symmetry [ 3–5], multiferroics [ 6,7], and even centrosymmetric compounds [8,9]. These findings suggest that skyrmions are ubiquitous in magnetic systems. The typical size of skyrmions is about10 nm and skyrmions form a triangular lattice. Skyrmions canbe driven by electric current [ 10–12], temperature gradient [13–15], and electric field gradient [ 16,17]. Moreover, the threshold current that makes skyrmions mobile is five to sixorders of magnitude smaller than that for a magnetic domainwall [ 10–12]. Thus, skyrmions are promising candidates for spintronics applications and recently they have attractedsignificant attention [ 18,19]. In metals, the spin of the conduction electron interacts with the localized magnetic moment associated with skyrmionsthrough the Hund’s coupling. In the strong-coupling limit,the spin of a conduction electron changes adiabatically whenit moves around a skyrmion and it gains a Berry phasewhich can be described by an effective magnetic field. Theemergent magnetic field is related to the skyrmion topologicalcharge, B E=/planckover2pi1c/epsilon1ijkn·(∂jn×∂kn)/(2e), and is extremely strong due to the small size of the skyrmion. Here nis a unit vector representing the direction of localized spin and/epsilon1 ijkis the totally antisymmetric matrix with i,j,k=x,y,z . A skyrmion line in a three-dimensional (3D) metal thus canbe regarded as an emergent magnetic flux line with quantumflux/Phi1 0=hc/e . It was demonstrated experimentally that two skyrmion lines can merge and become a single line. Thisimplies an emergent magnetic monopole with flux /Phi1 0at the merging point, which can be verified by integrating BEover a closed surface around the merging point [ 20]. A monopole and antimonopole pair can also be created by thermal fluctuations *szl@lanl.gov[21]. The emergent magnetic flux of the skyrmion and hence the monopole is an effective description for the conductionelectrons [ 22], thus these concepts cannot be applied to insulators. While magnetic monopoles as elementary particleshave never been found experimentally, quasiparticles resem-bling monopoles have been identified in several condensed-matter systems, such as spin ice [ 23] and chiral magnets discussed here as well as Bose-Einstein condensates [ 24]. A unique feature of monopoles in skyrmion systems is thatthey can be driven by electric currents. The monopolesin chiral magnets have attracted considerable interestrecently [ 25–27]. In a thin film, a skyrmion becomes a pancakelike object and no monopole is allowed to appear because the system is uniform along the direction perpendicular to the film. It was demonstrated theoretically that skyrmions can be createddynamically by applying a strong current to a fully spinpolarized state in chiral magnetic thin films [ 28]. This is because the current induces a Doppler shift in the magnonspectrum [ 29,30] and renders the spin polarized state unstable when the spectrum becomes gapless, after which skyrmions are nucleated dynamically. When the current increases further, skyrmions are destroyed and the system evolves into a chiralliquid with strong spatial and temporal fluctuations in spinchirality. The creation and destruction of skyrmions by current can also occur in 3D, where the emergent monopoles playan important role. A skyrmion line percolating the whole system (connecting surfaces of the system) cannot be created instantly. Instead, segments of skyrmion lines are created. Atthe ends of these lines, ∇·B E=±ρm, i.e., there is a monopole and antimonopole pair connected by a skyrmion line, asschematically shown in Fig. 1(a). This is a realization of Dirac string in condensed matter systems. These segments then jointogether to form skyrmion lines spanning the whole systemthrough the annihilation of monopoles and antimonopoles. In the course of skyrmion destruction, skyrmion lines break into segments and monopole and antimonopole pairs are created 2469-9950/2016/93(6)/060401(5) 060401-1 ©2016 American Physical SocietyRAPID COMMUNICATIONS SHI-ZENG LIN AND A V ADH SAXENA PHYSICAL REVIEW B 93, 060401(R) (2016) FIG. 1. (a) Schematic view of a skyrmion segment as a Dirac stringlike excitation, where there is a monopole and antimonopole pair at the two ends. (b) Straight skyrmion line lattice. (c) Snapshot of spin configuration after the magnonic instability of spin polarized state. Skyrmion segments are created dynamically associated with monopole and antimonopole excitations. The monopole and antimonopole annihilate ordisappear at the surfaces, through the merging of the skyrmion segments. Finally the system reaches a state with straight skyrmion line lattice, and they move as a whole. (d) Chiral liquid at a high current when skyrmion lines are destroyed. Color at three surfaces represents n z(red for nz=1 and blue for nz=− 1) and the vector field at the top surface denotes the nxandnycomponents. dynamically by current. These physical pictures are borne out by our numerical simulations below. We consider a 3D chiral magnet described by the following Hamiltonian in the continuum limit because the skyrmionsize is much bigger than the underlying lattice parameter[1,2,31–33]: H=/integraldisplay dr 3/bracketleftbiggJex 2(∇n)2−Ha·n/bracketrightbigg +HDM, (1) HDM=D/integraldisplay dr3[nx∂ynz−ny∂xnz+nz(∂xny−∂ynx)], (2) where Jexis the exchange coupling and HDM is the Dzyaloshinskii-Moriya (DM) interaction due to the absenceof inversion symmetry [ 34–36]. Here we consider that the DM vector Dis along the bond direction in the x-yplane and the component along the zdirection is zero, D x=Dy=D andDz=0[37]. The field is along the zdirection, Ha=Haˆz with a unit vector ˆz, and the skyrmion lines are aligned in the same direction. In 3D systems, skyrmions are stable only in asmall phase region in magnetic-field–temperature parameterspace [ 3,38]. Here we consider a metastable skyrmion line lattice at zero temperature which can be obtained throughfield cooling [ 20]. The dynamics of spins follows the Landau-Lifshitz-Gilbert equation ∂ tn=/planckover2pi1γ 2e(J·∇)n−γn×Heff+αn×∂tn, (3) where γis the gyromagnetic ratio, αis the Gilbert damping constant, Heff≡−δH/δnis the effective magnetic field, and Jis the spin polarized current. The current is in the xdirection in our simulations. In simulations, we calculate the skyrmion density in the x-yplaneρ(r)=n·(∂xn×∂yn)/(4π), or the BE zcomponent. The emergent electric field induced by skyrmion motion isE E=/planckover2pi1n·(∇n×∂tn)/(2e). The system size is 40 ×40× 40 (Jex/D)3. The system is discretized into a cubic meshwith periodic boundary conditions in most cases. To count the monopole/antimonopole number, we split each surfaceof a unit cubic cell into two triangles, and calculate thesolid angle /Omega1 isubtended by the three spins niin a tri- angle, tan( /Omega1i/2)=n1·(n2×n3)[1+n1·n2+n2·n3+n3· n1]−1[20]. The monopole number of this unit cell is obtained by summing the solid angle for these 12 triangles on the surface of the unit cubic cell, i.e., nM=(4π)−1/summationtext12 i=1/Omega1i, which is quantized, nM=± 1,0. We first study the nucleation of skyrmions in the ferromag- netic state due to the magnon instability induced by current.The magnon dispersion in the fully spin polarized state is givenby [28] /Omega1(k)=/planckover2pi1γ 2eJ·k+γ(1+iα) α2+1(Ha+Jexk2). (4) The term J·kaccounts for the magnonic Doppler shift. The magnon gap for J=0 is due to the external field and it becomes gapless at a threshold current Jm= 4e√HaJex/[/planckover2pi1(α2+1)], signaling an instability. The averaged skyrmion and monopole density in the stationary state as afunction of current is displayed in Fig. 2(a). Right after the instability, skyrmion segments are nucleated accompanyingthe creation of monopoles and antimonopoles [see Fig. 1(c)]. In Fig. 2(b), the skyrmion and monopole densities as a function of time are shown when the instability is triggered.Short skyrmion segments appear with a high monopoledensity. During the relaxation, the length of skyrmion segmentincreases accompanying an increase in skyrmion density.Once these segments meet, the monopole and antimonopoleannihilate resulting in longer skyrmion segments. This processrepeats until the skyrmion lines span the whole system when nomonopole and antimonopole is left. Meanwhile the skyrmionlines become straight to minimize the line tension. Themonopole density decreases exponentially and the typical timescale for this process is about 300 J ex/(γD2). In the end, straight skyrmion line lattice driven by current moves as awhole [see Fig. 1(b)]. 060401-2RAPID COMMUNICATIONS DYNAMICS OF DIRAC STRINGS AND MONOPOLELIKE . . . PHYSICAL REVIEW B 93, 060401(R) (2016) FIG. 2. (a) Skyrmion and monopole density versus current in the stationary state for Ha=0.8D2/Jex.A tJ=3.84De//planckover2pi1, skyrmion segments are created after the instability of the ferromagnetic state. (b) Time dependence of skyrmion and monopole density when the instability is triggered at J=3.7De//planckover2pi1att=0f o rHa=0.6D2/Jex. Monopoles and antimonopoles are created after the instability but they finally disappear in the stationary state, resulting in a straight skyrmion line lattice. (c) Dynamics of skyrmion destruction at a high current J=10.0De//planckover2pi1andHa=0.5D2/Jex, where monopoles and antimonopoles are nucleated. (d) Distribution of monopoles and antimonopoles in the chiral liquid phase at J=8.64De//planckover2pi1andHa=0.8D2/Jex. The skyrmion density increases stepwise with current. For a low skyrmion density, the system is close to a ferromagneticstate and one would expect a similar magnonic instability asthat in Eq. ( 4). For Jclose to J m, the skyrmion density is low and the magnetic instability persists at the same current.In this case skyrmion segments do not relax into straightlines. When the skyrmion density becomes high, the instabilitycurrent increases. The magnonic instability disappears whenthe skyrmion lattice is reached at a high skyrmion density foran intermediate J. This accounts for the stepwise increase of skyrmion density. In these steps, no monopole exists becausethe skyrmion line lattice spans the whole system. At a high current, the skyrmion density decreases sharply indicating destruction of skyrmions [see Fig. 2(a)]. The system evolves into a chiral liquid displayed in Fig. 1(d), where spin chirality fluctuates strongly in space and time. In the courseof skyrmion destruction, skyrmion lines break into segmentswith a monopole and an antimonopole at ends (similar to theDirac strings). In other words, skyrmion lines are destroyeddue to the proliferation of monopole-antimonopole pairs in this dynamic phase transition. The time evolution of monopole and skyrmion density is presented in Fig. 2(c). Skyrmion density decreases while the monopole density increases. In the chiralliquid, both densities fluctuate justifying the liquid nature ofthe final state. These fluctuations are induced dynamicallyby current. As shown in Fig. 2(d), the monopoles and antimonopoles are distributed randomly in space, and can be regarded as a monopole gas. The dynamical phase transitions associated with the skyrmion creation and destruction are ofthe first order because there is a strong hysteresis [ 28]. Both transitions can be detected from current-voltage characteristicsbecause the emergent electric field depends on the skyrmiondensity. We next discuss the controlled creation of monopoles and antimonopoles by driving part of the skyrmion lines. For ease of discussion, let us consider the case that current only flows at the bottom surface of the crystal. The skyrmion lines nearthe surface are driven by the Lorentz force due to the current,Jδ(z), and start to move. This causes distortion of the lines and drags the skyrmion lines inside the crystal to move due to theelastic energy of the skyrmion lines [see Fig. 3(b)]. For a small current, the elastic force can balance the different Lorentzforces acting on the surface and middle of the skyrmion line, and the skyrmion lines move as a whole. For an elasticskyrmion line lattice, the equation of motion for the lattice canbe described by a displacement vector u(z)=[u x(z),uy(z),0] [39,40], 4π γ[ηv(z)+v(z)׈z]=/Phi10 cˆz×Jδ(z)+/lscript 2∂2 zu(z), (5) where v=[∂tux,∂tuy,0] is the skyrmion velocity, and η/lessmuch1 is the dimensionless viscosity due to the Gilbert dampingfor localized spins and the Ohmic dissipation by conductionelectrons. Here /lscriptis the tilt modulus of the skyrmion lattice and it becomes skyrmion self-energy per unit length for a singleline. From Eq. ( 5), the velocity component parallel to the current is v /bardbl=−γ/planckover2pi1J/[2e(1+η2)Lz] and that perpendicular to the current is v⊥=γ/planckover2pi1ηJ/[2e(1+η2)Lz] with Lzthe system size along the zdirection. For a small current, the equation of motion is the same as that for a single skyrmion inthin films. The displacement field is u∝Jz 2. However, for a strong current, the skyrmion line breaks. The skyrmion segments at the surface and inside the crystaltravel with different velocities. This decoupled motion of theskyrmion segments is possible because the monopole andantimonopole pairs are allowed for the emergent magneticflux line associated with the skyrmion. The above picture is confirmed by our simulations. We consider a T-shaped crystal shown in the inset of Fig. 3(a). The current is injected through the bottom of the crystal. Theskyrmion lines produce an emergent electromagnetic fieldwhich affects the motion of conduction electrons. However,as shown in the Supplemental Material [ 41], the emergent electric field is much smaller than the real electric field inducedby the injected current, and we neglect the emergent electricfield in the calculation of current distribution. The currentdistribution is obtained by solving ∇ 2V=0, subject to the proper boundary conditions, with Vbeing the voltage [ 41]. As displayed in Fig. 3(a), below a threshold current, the electric fields at the top and bottom are the same, indicating motion ofskyrmion lines as a whole. Beyond the threshold current, theskyrmion segments at the bottom surface move at a highervelocity. At the same time, monopoles and antimonopolesare created which are confined at the bottom surface [see 060401-3RAPID COMMUNICATIONS SHI-ZENG LIN AND A V ADH SAXENA PHYSICAL REVIEW B 93, 060401(R) (2016) FIG. 3. (a) Emergent electric fields at the top and bottom surfaces of the crystal. At a small current, EEat the opposite surfaces is the same indicating an elastic motion of skyrmion lines. Beyond a threshold current, EEat the bottom surface under a current drive is much bigger than the top one, implying the breaking of skyrmion lines. The inset is a sketch showing how to inject current at the bottom surface and thecalculated current distribution [ 41]. (b) Snapshot of spin configuration after the breaking of skyrmion lines. (c) Monopole and antimonopole density versus current. (d) Distribution of monopoles and antimonopoles after the segmentation of skyrmion lines. In this calculation, we have used open boundary conditions in the zdirection. Here H a=0.5D2/JexandJ=0.26De//planckover2pi1for (b) and (d). The system size is Lz=2Lx=2Ly=100Jex/D. Figs. 3(c) and3(d)]. At a large current, the skyrmion density near the bottom surface decreases due to the dynamicaldestruction of skyrmions. This causes a rapid increase ofantimonopole density. Meanwhile the skyrmion density inthe current-free region does not change, which results in ahigher density of antimonopoles than that of monopoles [seeFig. 3(c)]. The electric field along the magnetic field direction E E z is particularly interesting. Let us first consider EE zdue to the stretching of skyrmion lines by a surface current. Theelectric field is given by E E=v×BE/c. For a straight skyrmion line, EE z=0 because BE=BEˆzonly has the zcomponent. When the skyrmion lines are distorted ac- cording to Eq. ( 5),BEacquires an in-plane component, BE=BE[∂zux,∂zuy,1]//radicalbig (∂zux)2+(∂zuy)2+1 and EE zis nonzero, as shown in Fig. 4(a). The induced electric field EE z∝J2increases nonlinearly with current in the elastic region and it suddenly decreases when the skyrmion linebreaks, after which skyrmion segments become mostly alignedwith the magnetic field. In the chiral liquid phase, E E zfluctuates around zero due to the fluctuations of the density of monopolesand antimonopoles [see Fig. 4(b)]. Therefore E E zprovides a clear transport signature of the existence of monopoles. FIG. 4. (a) Emergent electric field along the magnetic field direction EE zwhen skyrmion lines are stretched by a current at the bottom surface in a T-shaped crystal in Fig. 3(a). (b) Fluctuations of EE zas a function of time in the chiral liquid phase at J=10.0De//planckover2pi1. HereHa=0.5D2/Jex.We remark that the nonuniform current distribution can also produce a local real electric field Ezalong the magnetic field direction. However, Ezaveraged over the crystal van- ishes because there is no net current in the magnetic fielddirection. The existence of monopoles can also be inferred from imag- ing measurements, such as with a magnetic force microscope[20]. For instance, one may first apply a strong current to drive the system into the chiral liquid phase. Then the current isremoved and the system can be trapped in a metastable statewith monopoles. By visualizing the skyrmion density at theopposite surface, one can extract the information about themonopoles inside the crystal. If we start with a skyrmion line lattice in the ground state and drive it, the magnon instability due to the Doppler shiftis absent. At a high current, we still have the destruction ofskyrmion lines in nonequilibrium similar to those in Fig. 2(a) via the proliferation of monopoles and antimonopoles. To summarize, we show that the emergent monopoles play an important role in the skyrmion line creation and destructionprocess in the presence of an electric current. The currentdensity required for the monopole proliferation is of the orderof 10 12A/m2for typical material parameters. The monopoles can also be created in a controlled way by applying aninhomogeneous current to the crystal. Specifically, we predictan induced electric field component along the magnetic fielddirection, in addition to the Hall and longitudinal electricfields. The authors are indebted to Achim Rosch and Jiadong Zang for helpful discussions. The authors also thank CristianD. Batista for helpful suggestions and a critical readingof the manuscript. Computer resources for numerical cal-culations were supported by the Institutional ComputingProgram at LANL. This work was carried out under theauspices of the National Nuclear Security Administrationof the US DOE at LANL under Contract No. DE-AC52-06NA25396 and was supported by the LANL LDRD-DRProgram. 060401-4RAPID COMMUNICATIONS DYNAMICS OF DIRAC STRINGS AND MONOPOLELIKE . . . PHYSICAL REVIEW B 93, 060401(R) (2016) [1] A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP 68, 101 (1989). [ 2 ] U .K .R ¨oßler, A. N. Bogdanov, and C. Pfleiderer, Nature (London) 442,797 (2006 ). [3] S. M ¨uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. B ¨oni, Science 323,915 (2009 ). [4] X. Z. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. H. Han, Y . Matsui, N. Nagaosa, and Y . Tokura, Nature (London) 465,901 (2010 ). [5] X. Z. Yu, N. Kanazawa, Y . Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y . Matsui, and Y . Tokura, Nat. Mater. 10,106 (2011 ). [6] S. Seki, X. Z. 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PhysRevB.102.224308.pdf

PHYSICAL REVIEW B 102, 224308 (2020) Coherent control in ferromagnets driven by microwave radiation and spin polarized current Marina Brik,*Nirel Bernstein, and Amir Capua† Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (Received 13 September 2020; revised 1 December 2020; accepted 2 December 2020; published 21 December 2020) Coherent control is a method used to manipulate the state of matter using oscillatory electromagnetic radiation which interacts with the state in a nonequilibrium manner before the steady precessional state is reached.It is commonly applied in quantum processing applications. This technique is interesting in the context offerromagnetic materials because of the ability to combine it with spintronics for the purpose of fundamental spintransport research, low-power information processing, and potentially future quantum bit (Qubit) applications.In this work we address the theoretical grounds of coherent manipulation in practical ferromagnetic systems. Westudy the electromagnetic radiation-driven interaction that is enhanced in the presence of spin-polarized currentand map the conditions that allow coherent manipulation for which Rabi oscillations take place. The role of themagnetic anisotropy field is shown to act as an additional oscillatory driving field. We discuss the Gilbert lossesin the context of effective coherence decay rates and show that it is possible to control these rates by applicationof a static spin current. The case of coherent manipulation using oscillatory spin current that is free of radiation isdiscussed as well. Our work paves the way towards spin current amplification as well as radiation-free coherentcontrol schemes that may potentially lead to novel Qubits that are robust and scalable. DOI: 10.1103/PhysRevB.102.224308 I. INTRODUCTION Coherent control is a method of controlling dynamical pro- cesses using electromagnetic (EM) radiation that translates adynamical system from one state to another. At its basis standsa nonadiabatic process which means that the system losesor gains energy during the interaction with the radiation. Asopposed to the nonadiabatic process, the ferromagnetic reso-nance (FMR) experiment is an adiabatic process: A harmonicstimulus drives the system in steady state and the energystored in the magnetic medium is constant over time. Simi-larly, the pump-probe type free-induction decay experiment inferromagnets (FM) is also not considered a nonadiabatic inter-action because the driving EM radiation is absent. However,the perturbed FMR interaction is a nonadiabatic process. Studies of spin dynamics in magnetic media have been mainly carried out under either the adiabatic regime (e.g.,Refs. [ 1–3]), or the free-induction decay regime (e.g., Refs. [ 4–7]). These regimes have played an increasingly important role in understanding spin transport processes inatomically engineered solid-state devices and key funda-mental phenomena have been explored, e.g., spin angularmomentum losses [ 8–10], the spin Hall effect (SHE) [ 11–15], the anomalous Hall effect [ 16], motion of magnetic domains [17–20], the spin transfer torques (STT) [ 21–23], and more. The third dynamical regime of the nonadiabatic interaction has so far received little attention in the context of FM systems[24]. It can be considered as a hybrid dynamical regime which is a combination of the microwave-driven steady-state inter- *marina.brik@mail.huji.ac.il †amir.capua@mail.huji.ac.ilaction [ 1–3] and the impulse stimulated free-induction decay [4–7] so that the EM radiation and the magnetization state are not in equilibrium and excessive energy is transferred backand forth between the two before the steady precessional stateis reached. When the energy is exchanged periodically, Rabioscillations arise that are characterized by the Rabi frequencyand are the basis for coherent control similarly to a two-levelsystem [ 25]. The analogy to quantum two-level system stems from the fact that the observables of angular momentum intwo-level systems obey the classical equations of motion.In fact, Rabi’s original theory [ 24] which was derived for a nuclear two-level spin system that is driven by an oscillatoryfield under the nonadiabatic interaction can be equally appliedto an isolated electron or equivalently to a FM in the absenceof the Gilbert losses and anisotropies under the macrospinlimit. Furthermore, similarly to the two-level system, in thenonadiabatic interaction coherence plays a role in the sensethat the outcome of the interaction is dependent on the initialphase relations between the magnetization state and the os-cillatory torque [ 26]. Hence, a great deal of insight into the quantum world is gained from studies of the spin ensembles. Because of the large gyromagnetic ratio of the electron, exploration of the nonadiabatic regime in FM systems oftenrequires fast electronics and/or synchronization circuitry ca-pable of operating in the GHz range. Hence, experimentalstudies of the nonadiabatic regime in magnetic solid-statesystems is usually more cumbersome. In the work byKarenowska (Ref. [ 27]), a spatial nonequilibrium energy ex- change was demonstrated between counterpropagating spinwaves in yttrium iron garnets using artificial magnonic crys-tals. In these experiments a periodic spatial modulationfulfilled the role of the oscillatory signal whereas the effectwas recognized to be valuable for signal processing purposes. 2469-9950/2020/102(22)/224308(10) 224308-1 ©2020 American Physical SocietyMARINA BRIK, NIREL BERNSTEIN, AND AMIR CAPUA PHYSICAL REVIEW B 102, 224308 (2020) At the quantum limit, coherent control of single artificial magnetic spins was demonstrated using a scanning tunnel-ing microscope [ 28]. In Ref. [ 28] magnetic Ti atoms were excited using microwaves to induce Rabi oscillations whileinitialization of the atoms was achieved by passing a DCspin current through the atom. This study was carried outin the time domain and the magnetization state was read outmagnetoresistively. Recently, we have demonstrated a hybrid time-frequency domain method to excite the nonadiabatic regime in magneticmedia which we can describe as the pump-probe opticallysensed ferromagnetic resonance [ 29]. In this method, Rabi oscillations were excited in a few Å-thick film of a CoFeBferromagnet and in the presence of rf radiation following aperturbation by an intense ultrashort demagnetizing opticalpulse. These experiments revealed a frequency chirp whichwas controllable by the static magnetic field and that themicrowave field induced coherence in the inhomogeneously(IH)-broadened spin ensemble. Moreover, the experimentsshowed that according to Gilbert’s damping theory the in-trinsic relaxation times were tunable by proper choice of theexternal magnetic field and when taken long enough theyeventually initiated a resonant spin mode locking of the sys-tem. In the present work we provide the theoretical grounds for the nonadiabatic regime in a FM. We consider a sys-tem being driven into precession by microwave EM radiationand spin current. We compute the equations of motion forsmall deviations away from the steady precessional state.The deviations nutate at the Rabi frequency with a decayingamplitude. The frequency and decay rates of these nutationsin a variety of circumstances are derived. To that end wecombine the formalism of the adiabatic interaction in FMsand approaches that are commonly used to describe two-statequantum systems. We compare between the representation ofthe spin-lattice and transverse spin-polarization decay times,T 1andT∗ 2, used to describe quantum coherent phenomena and the Gilbert damping constant and IH broadening arising fromGilbert’s damping theory. The influence of the injection of DCspin current on the effective coherence and decay times andthe role of the magnetic anisotropy is studied. In addition,we present the case of using the AC spin current to drivethe nonadiabatic dynamics instead of EM microwaves. Wediscuss the limitations of using AC STT as a driving force,and clarify its distinct nature compared to the ordinary rf EMfield case. Our work is presented as follows: We start by introducing the general conditions for observing Rabi oscillations in amagnetic system that is driven by a rf EM field. Next, weinclude the anisotropy fields and examine the case of the filmhaving a perpendicular magnetic anisotropy (PMA) which isrelevant for practical applications. We then add to our modela DC SHE. Specifically, we look into the influence of theinjection of spin current on the overdamped interaction forwhich the magnetization simply decays and Rabi oscillationsare not observable. Finally, the nonadiabatic interaction isstudied in the presence of a driving oscillatory STT generatedby the SHE.II. MODEL AND RESULTS A. Model framework: Rabi oscillations in FMs Our analysis is carried out under the framework of the macrospin approximation. To that end we start with theLandau-Lifshitz-Gilbert (LLG) equation for the magneti-zation, /vectorM /prime, in the presence of the effective field, /vectorH/prime eff. Throughout the paper we use a prime to indicate variables inthe lab frame of reference and the unprimed variables referto the rotating system of coordinates. In the lab frame ofreference d/vectorM /prime dt=−γ(/vectorM/prime×/vectorH/prime eff)+α Ms/parenleftbigg /vectorM/prime×d/vectorM/prime dt/parenrightbigg , (1) in which Msis the magnetization saturation, αis the Gilbert damping parameter, and γis the gyromagnetic ratio. In spher- ical coordinates the LLG equation converts to ˙θ/prime=γH/prime ϕ sinθ/prime˙ϕ/prime=−γH/prime θ, (2) where H/prime θand H/prime ϕare the polar and azimuthal components of the effective field, respectively. In order to study the nonadiabatic interaction, we convert Eq. ( 2) to a coordinate system rotating about the ˆ z/primeaxis at the driving angular frequency, ω. In spherical coordinates, this corresponds to the substitutions θ/prime=θandϕ/prime=ϕ+ωt.F o l - lowing linearization, we express the solution of Eq. ( 2)i nt h e rotating frame by θ(t)=θ0+/Delta1θ(t) andϕ(t)=ϕ0+/Delta1ϕ(t) with [/Delta1θ(t),/Delta1ϕ (t)] being small deviations from equilibrium (θ0,ϕ0). (θ0,ϕ0) indicate the coordinates that the mag- netization decays towards in the rotating frame and not theenergy minimum in the lab frame as in conventional FMRmodels. Alternatively, in the lab frame, these coordinates cor-respond to the coordinates of the steady precessional state (upto a constant phase difference). /Delta1θ(t) and /Delta1ϕ(t) are then expressed by their phasors /Delta1θ=/Delta1θ iexp(−i/Omega1t) and /Delta1ϕ= /Delta1ϕ iexp(−i/Omega1t), with /Delta1θiand/Delta1ϕ ibeing constants of the prob- lem that are determined by the initial conditions. The complexfrequency, /Omega1, consists of the real part responsible for the os- cillatory component known as the generalized Rabi frequency,/Omega1 G R, and of the imaginary part responsible for the dampening of the response, indicated by the decay rate, /Gamma1, according to/Omega1=−i/Gamma1+/Omega1G R. Rabi oscillations are generally observable when the decay time is longer than the Rabi cycle, namely,when/Gamma1</Omega1 G Rand the response becomes underdamped. When /Gamma1>/Omega1G Rthe response becomes overdamped and the magnetic moment in the rotating frame decays exponentially towards(θ 0,ϕ0) without oscillating. Finally, /Omega1G Ris given by /Omega1G R=/radicalbig /Omega12σ−/Gamma12,where /Omega1σand/Gamma1are obtained by satisfying the secular equation /Omega12+i2/Omega1/Gamma1−/Omega12 σ=0 in the usual manner. B. Fundamental interaction: EM-driven dynamics 1. Rabi frequency and linewidth We first examine the microwave magnetic field-driven in- teraction that will also serve as a reference case. The externalmagnetic field of magnitude H 0is chosen in the ˆ z/primedirection 224308-2COHERENT CONTROL IN FERROMAGNETS DRIVEN … PHYSICAL REVIEW B 102, 224308 (2020) while the oscillatory driving field of amplitude hrfis applied along the ˆ x/primeaxis. In the rotating frame under the rotating wave approximation, Eq. ( 2) becomes ˙θ=−1 2γhrfsinϕ−αsinθ(˙ϕ+ω) sinθ˙ϕ=γ/parenleftbigg H0−ω γ/parenrightbigg sinθ −1 2γhrfcosθcosϕ+α˙θ. (3) (θ0,ϕ 0) can be inferred from the equilibrium conditions ˙θ= ˙ϕ=0, while the time-dependent part of Eq. ( 3)g i v e s /Delta1˙θ=−1 2γhrfcosϕ0/Delta1ϕ −αsinθ0/Delta1˙ϕ−αωcosθ0/Delta1θ sinθ0/Delta1˙ϕ=γ/parenleftbigg H0−ω γ/parenrightbigg cosθ0/Delta1θ +1 2γhrfcosθ0sinϕ0/Delta1ϕ +1 2γhrfsinθ0cosϕ0/Delta1θ+α/Delta1˙θ. (4) The set of Eq. ( 4) describes the conventional problem of a two-level system [ 30] with the difference that the spin angular momentum losses are incorporated through Gilbert’s dampingtheory [ 31]. The Gilbert damping in the LLG equation has a rigorous physical origin. It originates from a Rayleigh frictionprocess that is included to model losses such as those medi-ated by the spin-orbit and exchange interactions. Hence, theenergy dissipation rate in our model is inherently dependenton numerous parameters of the problem with the most criticalof them being the frequency of the precessional motion andconsequently the external magnetic field [ 29]. In contrast, in the Bloch-Bloembergen formalism the losses are incorporatedthrough T 1andT∗ 2and are generally independent of the ef- fective field of the problem. The IH broadening that arisesfrom variations in local anisotropy fields can be added in our model by taking variations in the effective bias field to the first order [ 32]. Figure 1highlights the differences between the two models. To calculate the Rabi flopping frequency, /Omega1 σand/Gamma1can be determined from Eq. ( 5): /Omega1σ=γ/braceleftbigg1 (α2+1)/parenleftbigg/parenleftbiggαω γ/parenrightbigg2 cos2θ0 +/parenleftbigg H0−ω γ/parenrightbigg2 +/parenleftbigg1 2hrf/parenrightbigg2 cos2ϕ0/parenrightbigg/bracerightbigg1/2 , /Gamma1=γα 2(α2+1)/parenleftbigg2ω γcosθ0+/parenleftbigg H0−ω γ/parenrightbigg cosθ0 +1 2hrfsinθ0cosϕ0+1 2hrfcosϕ0 sinθ0/parenrightbigg . (5) On resonance ( H0−ω γ)=0 and the solutions for (θ0,ϕ 0) require θ0=+90◦orϕ0=−90◦. The solution θ0= +90 and ϕ0=−90◦corresponds to αω=1 2γhrf, which in- dicates the transition from the overdamped to underdampeddynamics. FIG. 1. Geometrical representation of the damping and relax- ation torques and the IH broadening of Bloch-Bloembergen and Gilbert pictures. Blue arrows represent the lattice and transverse relaxation torques, T1andT∗ 2, respectively. /vectorH/prime effis the effective mag- netic field. The Gilbert damping torque is indicated by the orange arrow and the IH broadening, /Delta1/vectorH/prime IH, is modeled as variations in /vectorH/prime eff. In the underdamped regime in which Rabi oscilla- tions are observable, αω <1 2γhrf,θ0=90◦, and ϕ0= −arcsin(2 αω/γ hrf), resulting in /Omega1G R=γ/radicalBigg/parenleftbigg1 2hrf/parenrightbigg2 cos2(ϕ0)/parenleftbigg1 (α2+1)−α2 (α2+1)2/parenrightbigg , /Gamma1=α 2(α2+1)γhrfcos(ϕ0), (6) and when αω >1 2γhrf,ϕ0=−90◦and θ0= arcsin( γhrf/2αω) and the response is overdamped with /Omega1G R=γ/radicalBigg/parenleftbiggαω γ/parenrightbigg2 cos2(θ0)/parenleftbigg1 (α2+1)−1 (α2+1)2/parenrightbigg , /Gamma1=αω (α2+1)cos(θ0), (7) The calculated results are presented in Fig. 2. The given geometry is shown in Fig. 2(a). Figure 2(b) illustrates /Omega1G R and/Gamma1forα=0.01 and 10 GHz on resonance as a function of the normalized fieldγhrf 2αω.The data resemble closely the dependence of the resonance frequency of a FM on the appliedfield when the external magnetic field is applied perpendicu-larly to the easy axis. In this case the quantity 1 2hrffulfills the same role as the static external field,αω γplays the role of the effective anisotropy field, and the easy axis is the rotation axis, ˆz/prime. This effective anisotropy field arises from the projection of the Gilbert damping torque into the rotating frame andis hence dependent on both αandω,a n ds i m i l a r l yt oa n actual anisotropy field the torque that arises from it dependson the angle between the magnetization and the easy axis.Forαω > 1 2γhrf,/Gamma1rapidly increases with decreasing hrfand becomes more than two orders of magnitude greater than /Omega1G R as shown in the inset so that Rabi oscillations are not obtained. When αω <1 2γhrfthis behavior abruptly changes and /Omega1G R becomes much greater than /Gamma1with increasing hrfgiving rise 224308-3MARINA BRIK, NIREL BERNSTEIN, AND AMIR CAPUA PHYSICAL REVIEW B 102, 224308 (2020) FIG. 2. (a) Geometry of the rf EM-driven dynamics. (b) /Omega1G R,/Gamma1for zero detuning. Inset shows closeup of /Omega1G Ratγhrf/2αω < 1. (c)/Omega1G R−/Gamma1 as a function of the detuning. The red zone indicates the overdamped region, while the blue zone is the underdamped region. (d) Normalized and shifted temporal responses as a function of H0calculated numerically for hrf=20 Oe. The slight high-frequency modulation observed as a background arises from the counter-rotating terms that are neglected in the model. Results are presented for α=0.01 and 10 GHz. to Rabi nutations. Figure 2(c)illustrates the difference /Omega1G R−/Gamma1 as a function of the detuning, ( H0−ω γ), and the normalized field from which the oscillatory nature can be determined. Starting from ( H0−ω γ)=∼30 Oe the behavior is always oscillatory irrespective of hrf. Figure 2(d)shows a typical tem- poral response for various H0values for which the response is overdamped at resonance and away from resonance becomesoscillatory. 2. Interpretation of the Gilbert damping torque in the rotating frame In the rotating frame of reference, the damping torque can be interpreted in a comprehensive manner providing furtherinsight to the nonadiabatic interaction. In Cartesian coordi-nates Eq. ( 1) transforms to d/vectorM dt=−γ/vectorM×/parenleftbigg/parenleftbigg /vectorH0−/vectorω γ/parenrightbigg +/vectorhrf/parenrightbigg +α Ms/parenleftbigg /vectorM×δ/vectorM δt/parenrightbigg −α Ms/vectorM×(/vectorM×/vectorω), (8) where /vectorωis the vector (0 ,0,ω) and /vectorhrfis the rf field. The first term on the right-hand side of Eq. ( 8) is the effective field ( /vectorH0−/vectorω γ)+/vectorhrfwhich /vectorMprimarily precesses about. The second term on the right-hand side of Eq. ( 8) is identical to the Gilbert damping term in the LLG equation and is responsiblefor the decay of the magnetic field towards the effective field.The third term, − α Ms/vectorM×(/vectorM×/vectorω), does not appear in theLLG equation in the lab frame. It behaves as a nonconserving torque that has the form of the antidamping STT term of theLandau-Lifshitz-Gilbert-Slonczewski (LLGS) equation. Thisterm gives rise to the effective field α Ms(/vectorM×/vectorω γ) and scales withαω. In steady state,δ/vectorM δt=0, it balances the primary field ( /vectorH0−/vectorω γ)+/vectorhrfand causes the system to decay towards a new steady state different than the one dictated solely by the primary field. Hence, this torque can be used as an additionalcontrol in a coherent manipulation scheme. Before steadystate is reached its contribution to /Omega1 σand hence also to /Omega1G Ris readily seen in Eq. ( 5) where it appears as an additional term in the Euclidean norm of the fields consisting of /vectorhrf,(/vectorH0−/vectorω γ), andα/vectorω/γ that eventually determine the Rabi frequency. This STT-like torque can be enhanced by increasing the drivingfrequency which is analogous to increasing the spin currentin the LLGS equation. 3. Large-angle nonadiabatic interaction The analytical model addresses small deviations from steady precessional state. For large deviations, the nona-diabatic response becomes nonlinear, giving rise to thegeneration of higher harmonics. We examine this nonlinearitynumerically [ 33–35]. A typical representative temporal re- sponse in the lab frame of reference is presented in Fig. 3at 10 GHz and h rfof 90 Oe on resonance (following the experiments of Ref. [ 29]). In this example the magnetization was initialized to the ˆ z/primedirection and traversed the full swing towards the 224308-4COHERENT CONTROL IN FERROMAGNETS DRIVEN … PHYSICAL REVIEW B 102, 224308 (2020) FIG. 3. Large signal response calculated numerically. (a) Re- sponse calculated at 10 GHz and α=0.01. (b) Same simulation presented in (a) at later times. (c) Same conditions as in (a) but with α=0.001. −ˆz/primedirection. Figure 3(a) shows the response for αof 0.01. The figure depicts M/prime zwhich is proportional to the energy of the system and M/prime y. A nonlinear response consisting of higher harmonics is readily seen as well as an asymmetric behaviorasM /prime zevolves, namely, as energy is absorbed or emitted. The “down” transition is slower than the “up” transition forwhich /vectorM /primealigns with /vectorH/prime 0. This is also readily seen on the M/prime y component which stretches or compresses in time depending on the up or down transition. At later times, as the responsefurther decays and /vectorM /primeprecesses at small angles near the steady state, the asymmetry vanishes and a harmonic responseis revealed [Fig. 3(b)]. This behavior is highly dependent on the Gilbert damping as shown in Fig. 3(c). When αis reduced to a value of 0.001 the nonlinearity vanishes and M /prime zoscillates at a single frequency according to the analytical model. Theeffect of the Gilbert damping on the nonlinear nature of theresponse is understood by examining the acting torques. Thetorque arising from the applied magnetic fields, namely rf and DC fields, d/vectorM/prime dtapp., can be decomposed into two components: a tangential component,d/vectorM/prime dt/bardbl, responsible for the primary longi- tudinal precessional motion, and a transverse component,d/vectorM/prime dt⊥, responsible for the “downward”/“upward” transition of /vectorM/prime. When /vectorM/primeshifts towards −ˆz/prime, the transverse component,d/vectorM/prime dt⊥, is balanced by the Gilbert damping torque and the transitionoccurs at a slower rate. Likewise, when /vectorM /primeshifts towards ˆ z/prime the transverse component,d/vectorM/prime dt⊥, is enhanced by the Gilbert damping torque. Therefore, the Gilbert torque is responsiblefor the asymmetry in the upward/downward transition rates. FIG. 4. (a) Geometry of the rf EM-driven dynamics in a PMA sample. (b) Temporal Mzresponses as a function of hrfcalculated numerically for H0=ω γ(quasiresonance), effective anisotropy field of 120 Oe, and α=10−4. Finally, as /vectorM/primefurther decays towards steady precessional state the applied torque is primarily tangential and the transverse torque componentd/vectorM/prime dt⊥is negligible, resulting in a harmonic response. This behavior is more prominent the greater αis [Figs. 3(a)and3(c)]. Hence, changes in αcan be readily seen on the nonlinear nonadiabatic response, thereby providing anadditional way to investigate the loss mechanisms. 4. Inclusion of magnetic anisotropy Practical magnetic systems exhibit magnetic anisotropy fields such as demagnetization and/or crystalline anisotropies.We focus on the case of a sample having PMA such as thegeometry studied in Ref. [ 29] which is usually more important for technological purposes and allows high density and lowercrosstalk between devices in practical applications. The modeled geometry is illustrated in Fig. 4(a). The easy axis of magnetization is set along ˆ y /prime. The effective anisotropy isHKeff=2Ku Ms−4πMs, where Kuis the crystalline anisotropy constant. The analysis was carried out under the conditionH 0>HKefffor which the precession takes place around the ˆz/primeaxis in the lab frame. Under these conditions the problem has a closed form analytical solution. Hence, Eq. ( 4) becomes /Delta1˙θ=−1 2γhrfcosϕ0/Delta1ϕ−αsinθ0/Delta1˙ϕ−αωcosθ0/Delta1θ sinθ0/Delta1˙ϕ=−γHKeff 2cos 2θ0/Delta1θ+γ/parenleftbigg H0−ω γ/parenrightbigg cosθ0/Delta1θ +1 2γhrfcosθ0sinϕ0/Delta1ϕ +1 2γhrfsinθ0cosϕ0/Delta1θ+α/Delta1˙θ (9) while ( θ0,ϕ0) is found as before. 224308-5MARINA BRIK, NIREL BERNSTEIN, AND AMIR CAPUA PHYSICAL REVIEW B 102, 224308 (2020) The solutions for /Omega1σand/Gamma1are /Omega1σ=γ/braceleftbigg1 (α2+1)/parenleftbigg/parenleftbiggαω γ/parenrightbigg2 cos2θ0−1 2hrf1 2HKeffcos 2θ0cosϕ0 sinθ0+1 2hrf/parenleftbigg H0−ω γ/parenrightbiggcosθ0cosϕ0 sinθ0+/parenleftbigg1 2hrf/parenrightbigg2 cos2ϕ0/parenrightbigg/bracerightbigg1/2 , /Gamma1=γα 2(α2+1)/parenleftbigg2ω γcosθ0−HKeff 2cos 2θ0+/parenleftbigg H0−ω γ/parenrightbigg cosθ0+1 2hrfsinθ0cosϕ0+1 2hrfcosϕ0 sinθ0/parenrightbigg . (10) For small α,s m a l l hrf, and quasiresonance conditions (H0−ω γ)=0,ϕ0=−180◦and the condition for θ0is sinθ0=hrf/HKeff. Substituting the above conditions into the /Omega1σterm in Eq. ( 10), we get /Omega1G R=γ/radicalBigg/parenleftBigHKeff 2/parenrightBig2 −/parenleftbigghrf 2/parenrightbigg2 . (11) Equation ( 11) has the familiar form of resonance frequency for the PMA case where the external field is applied per-pendicular to the easy axis. Thus, in the rotating frame h rf takes the role of DC field applied perpendicularly to the easy axis. In comparison with the PMA case [ 36],1 2HKeffappears as an effective anisotropy field in the rotating frame. Thedependence of /Omega1 G Ronhrfin Eq. ( 11) is fundamentally different from its general dependence in standard two-level systems.Here/Omega1 G Rdecreases with increasing hrfwhereas in conven- tional two-level systems /Omega1G Rincreases with the driving-field amplitude. In conventional two-level systems the driving-fieldamplitude determines the rate at which the occupation prob-abilities evolve, therefore /Omega1 G Rgenerally increases with hrf. In contrast, in Eq. ( 11)/Omega1G Rdecreases with hrfand is due to the role played by the anisotropy field which effectivelyacts as an additional oscillatory driving field. Therefore, theobserved response deviates from the conventional two-levelsystem behavior until h rfreaches the limit of1 2HKeff.T h e numerical model shows this behavior as well. The temporalresponses presented in Fig. 4(b) show that /Omega1G Rdecreases as hrf increases up to 60 Oe (1 2HKeff). Above 60 Oe hrfovercomes the anisotropy field and /Omega1G Rincreases with hrfas expected. C. Interaction in the presence of DC spin current From a technological point of view, a static STT may play an important role in the nonadiabatic interaction in magneticsystems because it can be used to actively tune the decayrates according to the LLGS equation [ 22,23]. Specifically, STT can be used to extend the coherence time of the sys-tem making the FM system a versatile platform for coherentcontrol schemes. In the model derived hereon we considerthe antidamping-like STT and assume that the spin currentis generated by the SHE in a heavy metal-FM bilayer [ 37]. Hence, a DC charge current of magnitude J DC cis applied along the ˆx/primedirection and generates a spin current density Jsˆy/primewith spin angular momentum aligning in the ˆ z/primedirection. Figure 5 illustrates the modeled geometry. The presence of spin current introduces the torqueγHSHE,DC Ms(/vectorM/prime×(/vectorM/prime׈s/prime)) into Eq. ( 1). Here ˆ s/primeis a unit vector in the direction of the injected spin an- gular momentum and HSHE,DCis the SHE parameter defined byHSHE,DC=¯hθSHJDC c 2eMstFMwhere ¯ his the reduced Planck constant, eis the electron charge, θSHis the spin Hall angle (SHA), and tFMis the thickness of the FM layer into which the spin current is injected. With these substitutions Eq. ( 4) becomes /Delta1˙θ=−1 2γhrfcosϕ0/Delta1ϕ−αsinθ0/Delta1˙ϕ−(αω−γHSHE,DC)cosθ0/Delta1θ sinθ0/Delta1˙ϕ=γ/parenleftbigg H0−ω γ/parenrightbigg cosθ0/Delta1θ+1 2γhrfcosθ0sinϕ0/Delta1ϕ+1 2γhrfsinθ0cosϕ0/Delta1θ+α/Delta1˙θ, (12) resulting in /Omega1σand/Delta1/Omega1: /Omega1σ=γ/braceleftbigg1 (α2+1)/parenleftbigg/parenleftbigg HSHE,DC−αω γ/parenrightbigg2 cos2θ0+/parenleftbigg H0−ω γ/parenrightbigg2 +/parenleftbigg1 2hrf/parenrightbigg2 cos2ϕ0/parenrightbigg/bracerightbigg1/2 /Gamma1=γ1 2(α2+1)/parenleftbigg 2/parenleftbiggαω γ−HSHE,DC/parenrightbigg cosθ0+α/parenleftbigg H0−ω γ/parenrightbigg cosθ0+α1 2hrfsinθ0cosϕ0+α1 2hrfcosϕ0 sinθ0/parenrightbigg . (13) Equation ( 13) shows that the spin current compensates the Gilbert damping term according to the differenceαω γ− HSHE,DC. Equation ( 13) reveals that for the critical case of HSHE,DC=αω γwhich may be achieved in realistic systems having SHA of 0.15, e.g., Pt, W [ 12,13] the response is al- ways underdamped, namely, /Gamma1</Omega1G R, and Rabi oscillations appear irrespective of the magnitude of hrf. Obviously, theseoscillations still decay as /Gamma1/negationslash=0. To examine the influence of the injected spin current on the existence of Rabi os-cillations and the damping rate we explored the interactionunder resonance conditions. Figure 6illustrates the temporal responses from which the transition between the overdampedand underdamped regimes for various charge current levels isseen. The figure shows M zas a function of hrfwithα=0.01, 224308-6COHERENT CONTROL IN FERROMAGNETS DRIVEN … PHYSICAL REVIEW B 102, 224308 (2020) FIG. 5. Geometry of the EM rf-driven dynamics in the presence of DC spin current indicated by /vectorJ/primeDC cin the lab frame. Ms=300 emu /cm3,θSH=0.15,and tFM=11.5 Å corre- sponding to Ref. [ 37]. Figure 6(a) presents the case with no spin current. It is seen that the transition between the over-damped and underdamped responses occurs at h rf=71 Oe above which Rabi oscillations take place. When the DCcurrent is increased the threshold reduces up to the pointofH SHE,DC=αω γ[Fig. 6(c)] in which the oscillations are obtained for any value of hrf. As the DC current is further in- creased, the threshold in hrfincreases again [Figs. 6(d)–6(f)]. This behavior stems from the fact that when α<γHSHE,DC ωthe term 2 γ(αω γ−HSHE,DC) cosθ0in Eq. ( 13) adds a positive contribution to the damping rate. Thus, at high DC currents the Rabi oscillations eventually become overdamped. Mostimportantly, Fig. 6shows that when Rabi oscillations takeplace the coherence times can be tuned by the DC spin current. This is seen from the varying decay rates as marked by theguiding red dashed lines in Figs. 6(a)–6(c). It is seen that as J DC cincreases to the critical value (2 .5×106A/cm2in our case) the coherence times extend. D. AC STT-driven nonadiabatic dynamics Inclusion of AC charge current creates an AC STT which serves as an alternative driving force. A driving force of thiskind is advantageous over the rf-driven case for scalabilitypurposes since it does not require a radiating microantennabut only physical contact to the device. However, the ACSTT driving force has a different nature compared to theordinary Zeeman oscillatory magnetic field. We include the AC STT by replacing /vectorh /prime rfin Eq. ( 1) with an AC charge current density /vectorJ/primeAC c=J0cos(ωt)ˆy/prime. The AC charge current is converted by the SHE to an AC spin current density Jsˆz/prime having ˆ s/primein the ˆ x/primedirection which introduce a STT term ofγhSHE,AC Ms(/vectorM/prime×(/vectorM/prime׈s/prime)) in Eq. ( 1).hSHE,ACis the SHE parameter as in Sec. II C, that refers now to an AC current. The geometry is presented in Fig. 7(a).T h e/vectorJ/primeAC cdirection was chosen such that ˆ s/primeis orthogonal to the static magnetization equilibrium vector in the lab frame /vectorM/prime 0. Following lineariza- tion, the AC STT in the lab frame equals approximately γhSHE,AC Ms(/vectorM/prime 0×(/vectorM/prime 0׈s/prime)). Thus, if /vectorM/prime 0and ˆs/primeare collinear, the AC STT vanishes. The time-dependent equations become /Delta1˙θ=1 2γhSHE,ACcosθ0sinϕ0/Delta1ϕ+1 2γhSHE,ACsinθ0cosϕ0/Delta1θ−αsinθ0/Delta1˙ϕ−αωcosθ0/Delta1θ , sinθ0/Delta1˙ϕ=γ/parenleftbigg H0−ω γ/parenrightbigg cosθ0/Delta1θ+1 2γhSHE,ACcosϕ0/Delta1ϕ+α/Delta1˙θ, (14) and/Omega1σand/Gamma1for this case are given by /Omega1σ=γ/braceleftbigg1 (α2+1)/parenleftbigg/parenleftbiggαω γ/parenrightbigg2 +/parenleftbigg H0−ω γ/parenrightbigg2 cos2θ0+/parenleftbigg1 2hSHE,AC/parenrightbigg2 cos2ϕ0/parenrightbigg/bracerightbigg1/2 /Gamma1=γ1 2(α2+1)/parenleftbiggαω γcosθ0+2α/parenleftbigg H0−ω γ/parenrightbigg cosθ0−1 2hSHE,ACsinθ0cosϕ0−1 2hSHE,ACcosϕ0 sinθ0/parenrightbigg . (15) FIG. 6. Temporal responses as a function of hrfcalculated numerically for JDC clevels of 0 A /cm2to 8×106A/cm2[(a) to (f)]. The red dashed guiding lines indicate the varying decay rates. 224308-7MARINA BRIK, NIREL BERNSTEIN, AND AMIR CAPUA PHYSICAL REVIEW B 102, 224308 (2020) FIG. 7. FM system driven by AC STT. (a) Geometry of the AC STT driven nonadiabatic dynamics. (b) Response on resonance ( H0−ω γ)= 0 for AC charge current density /vectorJ/primeAC cwith amplitude of J0=1×106A/cm2andα=0.001. (c) Normalized and shifted temporal responses as a function of H0calculated numerically for the same J0value as in (b) and α=0.005. To understand the general behavior of the solution we assume very small αand resonance conditions. From Eq. ( 15) it is seen that /Gamma1takes a nonzero value even when α→0; thus, the AC STT contributes to the decay term. From steadystate we find ( θ 0,ϕ0)=(90◦,180◦). Substituting ( θ0,ϕ0) into Eq. ( 15) we get /Omega1G R=0. Hence, the response is over- damped [Fig. 7(b)] regardless of the hSHE,ACvalue even in the absence of damping in contrast to the hrf-driven case. It can be further verified that the overdamped response persists inthe vicinity of resonance as long as |(H 0−ω γ)|<1 2hSHE,AC. This distinctly different behavior of the AC STT compared to the rf magnetic field case can be understood by observing Eq. ( 8) in the rotating frame in which /vectorhrfappears in the primary torque term −γ/vectorM×((/vectorH0−/vectorω γ)+/vectorhrf). In contrast, when the system is driven solely by AC STT the primary torque vanishes on resonance, leaving only the AC damp-inglike STT. For this reason, an additional DC STT cannotexcite Rabi oscillations under resonance conditions, but onlychange the steady state. Away from resonance conditions andfor|(H 0−ω γ)|>1 2hSHE,ACRabi oscillations are observable.When losses are included in addition, a DC STT applied in the geometry of Sec. II C affects the losses in the same manner as in the /vectorhrfcase where it extends the coherence time as long as HSHE,DC/lessorequalslantαω γ. When HSHE,DC>αω γ, the coherence time decreases and for relatively high HSHE,DCvalues the oscillations are totally suppressed. In the current analysis, thefieldlike term of the AC STT was neglected since in manymaterial systems it is much smaller than the dampinglike term.However, when the fieldlike term is not negligible, it canexcite Rabi oscillations even on resonance because its form isidentical to the /vectorh rftorque and hence appears in the primary precessional torque term of Eq. ( 8). Figure 7(c) shows the temporal responses of Mzas a function of H0. It is readily seen that despite the differences between the /vectorhrfand the AC STT cases especially on resonance, the AC STT-driven dynamics qualitatively behave in the same manner as the /vectorhrf-driven interaction [Fig. 2(d)]. Finally, when the magnetic anisotropy is included, it can be verified that for the same geometry introduced earlier(Sec. II B 4 ), Eq. ( 15) takes the form /Omega1σ=γ/braceleftbigg1 (α2+1)/parenleftbigg/parenleftbiggαω γ/parenrightbigg2 −1 2hSHE,AC/parenleftbigg H0−ω γ/parenrightbiggcos2θ0sinϕ0 sinθ0 +1 2hSHE,AC1 2Hkeffcos 2θ0cosθ0sinϕ0 sinθ0+/parenleftbigg1 2hSHE,AC/parenrightbigg2 cos2ϕ0/parenrightbigg/bracerightbigg1/2 /Gamma1=γ1 2(α2+1)/parenleftbiggαω γcosθ0+α/parenleftbigg H0−ω γ/parenrightbigg cosθ0−α1 2Hkeffcos 2θ0 −α1 2hSHE,ACcosθ0sinϕ0 sinθ0−1 2hSHE,ACsinθ0cosϕ0−1 2hSHE,ACcosϕ0 sinθ0/parenrightbigg . (16) Forα→0 and resonance conditions the steady-state equations give ϕ0=90◦and the condition for θ0is sin 2θ0=hSHE,AC/0.5HKeff,forhSHE,AC<HKeff. Inserting these conditions into Eq. ( 16), we get /Omega1G R=/radicalbig /Omega12σ−/Gamma12=/radicalBig 1 2γhSHE,AC1 2γHkeffcos 2θ0cosθ0 sinθ0. Thus, when the anisotropy fields are included, /Omega1G R/negationslash=0 and Rabi oscillations take place on resonance (quasiresonance).III. SUMMARY In this work we examined the nonadiabatic interaction which is the basis for coherent control schemes in mag-netic materials and relied on a hybrid two-level/adiabaticinteraction in FMs formalism. We explored the ordinary nona-diabatic interaction driven by rf field and mapped the condi-tions for reaching the Rabi oscillations for which coherent 224308-8COHERENT CONTROL IN FERROMAGNETS DRIVEN … PHYSICAL REVIEW B 102, 224308 (2020) control is made possible. We studied the energy transfer rates and showed that at large angles of precession and largeαvalues the absorption and emission rates become highly nonsymmetric. Furthermore, this nonlinear nonadiabatic re-sponse provided an additional way to investigate the lossmechanisms. We demonstrated that it is possible to controlthe effective coherence time by the injection of DC currentand explored the nonadiabatic interaction in a system drivenby an alternative driving source, namely, the AC STT, andconcluded that there are no Rabi oscillations on resonance, aslong as the AC STT fieldlike term is negligible. However, it ispossible to get on-resonance Rabi oscillations if the fieldlike term is non-negligible and this can motivate the search formagnetic materials that possess significant STT fieldlike term.Extensions of our work include complementing the existingexperimental work to fully map the nonadiabatic regime inFM systems as well as to discuss a truly coherent controlscheme that relays on the principles outlined here as well ascoherent spin current amplification schemes (to be discussedin a follow-up paper). Further into the future, STT can beutilized as a versatile platform for coherent control schemesto be used in the manipulation of Qubits. [1] S. S. 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PhysRevB.94.134421.pdf

PHYSICAL REVIEW B 94, 134421 (2016) Spatial symmetry of spin pumping and inverse spin Hall effect in the Pt /Y3Fe5O12system Hengan Zhou,1Xiaolong Fan,1,*Li Ma,2Qihan Zhang,1Lei Cui,1Shiming Zhou,2Y. S . G u i ,3C.-M. Hu,3and Desheng Xue1,† 1The Key Lab for Magnetism and Magnetic Materials of Ministry of Education, Lanzhou University, Lanzhou 730000, China 2Shanghai Key Laboratory of Special Artificial Microstructure and Pohl Institute of Solid State Physics and School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 3Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2 (Received 17 July 2016; published 17 October 2016) The spatial symmetry of spin pumping, i.e., the voltage amplitude of the inverse spin Hall effect varying with the orientation of static magnetization and the polarization direction of a dynamic magnetic field, has beensystematically studied both theoretically and experimentally. This symmetry is quite important because it is theunique foundation and criterion for identifying the spin pumping signal from other contributions to the voltageand for unveiling the underlying physics of spin pumping and pure spin current. As a straightforward proof, wefound the spin mixing conductance to be out-of-plane anisotropic, which comes from an elaborate study of thesymmetry of a Pt/YIG bilayer after examining every effect that may contribute to the angular dependence of spinpumping; the spin Hall angle, however, is a constant that is independent of the polarization direction of the spincurrent. DOI: 10.1103/PhysRevB.94.134421 I. INTRODUCTION Spin pumping is a fundamental approach for generating pure spin currents. The precessing spins in ferromagnets(FMs) act as peristaltic pumps, adiabatically injecting spinsout of the FM layer into the adjacent normal metal (NM)without charge transport [ 1–5]. Therefore, spin pumping can be understood as the inverse effect of spin torque and providesaccess to the physics of spin dynamics, damping, and spincurrent [ 6–8]. However, due to the inverse spin Hall effect (ISHE), the pumped spin current is subsequently convertedinto a detectable charge current with a resonance feature in theNM [ 9–11], which has made spin pumping a standard method for determining spin-related parameters, such as spin diffusionlength, spin mixing conductance, and spin Hall angle [ 12–17]. The spatial symmetry of spin pumping, i.e., the ISHE volt- age amplitude varying with the orientation of magnetizationand polarization direction of the dynamic magnetic field, isthe key feature of the spin pumping effect [ 18–20]. This symmetry is quite important because it is the only criterionfor distinguishing spin pumping from other ferromagneticresonance (FMR) related electrical effects [ 21–25], and it is also the foundation for discovering physics for pure spincurrents [ 26–28]. For example, large discrepancies exist among the different spin pumping experiments conductedover the past decade to quantify the magnitude of θ SH [4,13,23,25,29–31], and the difficulty arises from the tangled dc voltage signal of spin pumping and spin rectification. Thelatter is a nonlinear coupling between conducting electronsand local spin through magnetoresistance effects [ 25,32–35]. Until recently, this problem has been solved by Bai et al. [21,22], who proposed a universal method for separating them in NM/FM devices based on the difference in thespatial symmetries of spin pumping and spin rectification. Onthe other hand, there is a debate about whether exchange- *fanxiaolong@lzu.edu.cn †xueds@lzu.edu.cnmediated or polaron hopping dominates the spin transportmechanism in organic semiconductors. Watanabe et al. [27] hypothesized that the discrepancy between the experimentalspatial symmetry of spin pumping and theoretical predicationsis due to the Hanle effect, which is proof of polaron hopping.However, Jiang et al. [28] considered that such discrepancy arises from the not well-controlled dynamic magnetic fieldin Watanabe’s device, and they performed straightforwardcontrolled measurements to exclude the Hanle effect, whichsupported the exchange-mediated mechanism rather thanhopping. Thus, comprehending the spatial symmetry of spinpumping is necessary and important. In this paper, we systematically studied the spatial sym- metry of spin pumping associated with the details of spindynamics controlled by dynamic and static magnetic fields.We first present the theoretical results showing that the basicsymmetry comes from ISHE and the dynamic torque that actson the magnetization, and other details of spin dynamics, suchas the ellipticity of magnetization precession and resonancelinewidth, are considered as corrections to the basic symmetry.Experiments were conducted using two types of devices, inwhich the spin pumping is excited in Pt/YIG strips withdifferent dynamic magnetic fields. The experimental resultsare highly consistent with the theoretical expectations ifan out-of-plane anisotropy of spin mixing conductance isconsidered. II. THEORY OF SPIN PUMPING VOLTAGE Figure 1presents a schematic illustration of the inverse spin Hall effect induced by spin pumping in a NM/FMbilayer. Two coordinate systems are used: one is xyz for the device geometry, in which a static magnetic field His applied to cause the magnetization to rotate either in-plane(xy) or out-of-plane ( yzorxz), a dynamic magnetic field is applied either in-plane along the yaxis or out-of-plane along thezaxis, and the voltage signal is measured along the x axis. The other is x /primey/primez/primefor the magnetization Mwith the equilibrium direction along x/prime; the unit vector of dynamic 2469-9950/2016/94(13)/134421(11) 134421-1 ©2016 American Physical SocietyZHOU, FAN, MA, ZHANG, CUI, ZHOU, GUI, HU, AND XUE PHYSICAL REVIEW B 94, 134421 (2016) FIG. 1. (a) Schematic illustration of the inverse spin Hall effect induced by spin pumping in an FM/NM bilayer system. (b)–(d) show the device coordinate systems ( xyz) and magnetization coordinate systems ( x/primey/primez/prime) for magnetic fields applied in the xy,yz,a n dxz planes, respectively. magnetization is m=(1,my/primee−iωt,mz/primee−iωt),my/primeandmz/primeare the complex amplitude of m, andωis the angular frequency of magnetization precession. In an FM/NM bilayer system, a spin current is pumped out of an FM layer into NM via FMR, as shown in Fig. 1(a).T h e dccomponent of spin current density j0 sthrough the interface is given by [ 18,23,36] j0 sσ=−ω 2π/integraldisplay 2π ω 0/planckover2pi1 4πg↑↓/bracketleftbigg m×dm dt/bracketrightbigg dt =−/planckover2pi1 4πωg↑↓Im(m∗ y/primemz/prime)ˆx/prime, (1) where σis the polarization direction of dcspin current, which is opposite to ˆx/prime;g↑↓is the real part of spin mixing conductance [37]; and /planckover2pi1is the Dirac constant. Due to spin relaxation and diffusion in normal metals, the spin current density exponentially decays along the zdirection, which can be calculated by solving the spin diffusion equationtogether with boundary conditions of j s(0)=j0 sandjs(tN)= 0, as follows [ 10–13] js(z)=j0 ssinh[(tN−z)/λsd] sinh(tN/λsd),(2) where λsdis the specific spin diffusion length and tNis the thickness of the NM layer. The strong spin-orbit interactionin normal metals allows transferring the spin current j s(z)i n t o charge current jc(z)v i aI S H Ea s[ 4,38,39] jc(z)=θSH2e /planckover2pi1js(z)×σ, (3) where θSHis the spin Hall angle. After integrating the charge current along the zaxis in the thickness region, the effective charge current due to ISHE associated with spin pumping isgiven by j c=/integraltexttN 0jc(z)dz tN=−θSH2e /planckover2pi1λsd tNtanh/parenleftbiggtN 2λsd/parenrightbigg j0 s׈x/prime.(4) Because it is the voltage signal along ˆxthat will be detected in the following experiments, Vx=LNEx=LNjc·ˆx/σN, where LNandσNare the length and conductivity of the NM,respectively. Together with Eqs. ( 1) and ( 4), and noting that the direction of j0 sisˆz, the voltage signal due to spin pumping is given by Vx=−θSHeλsdLNω 2πσNtNtanh/parenleftbiggtN 2λsd/parenrightbigg g↑↓Im(m∗ y/primemz/prime)(ˆz׈x/prime·ˆx). (5) Based on the linear solution to the Landau-Lifshitz-Gilbert (LLG) equation with spin pumping, the Im( m∗ y/primemz/prime)t e r ma s a function of applied magnetic field at a certain frequency isgiven by Im(m ∗ y/primemz/prime)=Ph2 eff /Delta1H2L(H), (6) where Pis a correction factor due to the ellipticity of the magnetization precession; for a perfect circle precessionP=1/4[40] and Eq. ( 6) degrades to sin 2θc[4], where θc is the open angle of magnetization precession; heffis the dynamic magnetic field that drives magnetization precession;/Delta1H=α effω/γ is the effective resonance linewidth; γis the gyromagnetic ratio; αeffis the effective damping parameter, which includes the intrinsic Gilbert damping αGin FMs and the additional damping αspdue to spin pumping, i.e., αeff=αG+ αsp;L(H)=/Delta1H2/[(H−Hr)2+/Delta1H2] is a dimensionless symmetric Lorentz lineshape [ 41,42]; and Hris the resonance field. The derivation of Eq. ( 6) can be found in Appendix 2, Eqs. ( A1)–(A7). Based on Eqs. ( 5) and ( 6), the spatial symmetry of the spin pumping signal primarily results from the direction ofthe dynamic magnetic field h effand the mixed product of ˆz׈x/prime·ˆx. To highlight the angular dependence, we define Cx=−θSHeλsdLNω 2πσNtNtanh(tN/2λsd) as an angular-independent parameter that only depends on the frequency, the intrinsicproperties and the geometry of the NM. In the following, thespatial symmetry of the spin pumping signal will be discussedfor three cases. In the configuration with magnetization rotating in the xy plane [Fig. 1(b)],ˆz׈x /prime·ˆx=sinαM(αMis the angle between stable magnetization and the xaxis). Because the stable magnetization is along ˆx/prime, only the components of the dynamic magnetic field that are perpendicular to ˆx/primewill contribute to dynamic magnetization. In the following experiments,the dynamic magnetic field was generated by using on-chipmicrowave devices, and the directions are fixed either along ˆyorˆz.F o rt h e h zdevice, heff=hz, while for the hydevice, heff=hycosαM. Therefore, the spin pumping voltages for the xyplane configuration are given as Vxy x,hz=Cxg↑↓Pxy zh2 z /Delta1H2sinαML(H), Vxy x,hy=Cxg↑↓Pxy yh2 y /Delta1H2cos2αMsinαML(H), (7) where Pxy zandPxy yare the ellipticity correction factors of the hzandhydevices, which are constants for the in-plane case as discussed in Eq. ( A6) of Appendix 1. When the applied magnetic field rotates in the yzplane [Fig. 1(c)],ˆz׈x/prime·ˆx=sinβM(βMis the angle of Mwith respect to the zaxis in the yzplane). The effective dynamic magnetic field acting on magnetization is heff=hzsinβM 134421-2SPATIAL SYMMETRY OF SPIN PUMPING AND INVERSE . . . PHYSICAL REVIEW B 94, 134421 (2016) FIG. 2. Spatial symmetry of spin pumping signal (ISHE voltage). The three rows correspond to magnetization rotating in the xy,yz,a n d xzplanes. Each column corresponds to a certain device setup: The voltage can be detected either along the longitudinal direction [columns (a) to (c)] or the transversal direction [columns (d) to (f)], where the double-headed arrows indicate the polarization direction of the dynamic magnetic field. (heff=hycosβM)f o rt h e hz(hy) device. Moreover, consider- ing the misalignment between MandHdue to the demagneti- zation field of the film, the Lorentz lineshape L(H)i nE q .( 6) should be replaced by L/prime(H)=/Delta1H/prime2/[(H−Hr)2+/Delta1H/prime2] with/Delta1H/prime=/Delta1H/ cos(β−βM). The spin pumping voltage for the yzplane configuration can be obtained as Vyz x,hz=Cxg↑↓Pyz yh2 z /Delta1H2sin3βML/prime(H), Vyz x,hy=Cxg↑↓Pyz yh2 y /Delta1H2cos2βMsinβML/prime(H). (8) According to the discussion in Appendix 4, the influence ofPyz z(Pyz y) on the overall trend of angular-dependent spin pumping voltage can be ignored. Thus, in the yzconfiguration, the calculated spatially resolved spin pumping voltage showssin 3βMand cos2βMsinβMtendencies in the hzandhy devices. For the case of the applied magnetic field rotating in the xzplane [Fig. 1(d)], no signal can be detected. Because the polarization direction of the pumped spin current σ=− ˆx/primelies in thexzplane, the charge accumulation due to ISHE ( ˆz׈x/prime) is along the yaxis. It directly follows that the detected voltage signal along the xaxis is zero ( ˆz׈x/prime·ˆx=0). In Fig. 2, we summarize the spatial symmetries of the voltage due to spin pumping associated with ISHE, andwe only highlight the dominated trigonometric functions; the influences of P,/Delta1H, and g↑↓on the symmetries are temporarily ignored. All the angular dependencies have beentheoretically studied in Appendices 1–3. In the followingexperiments, we test the cases shown in Figs. 2(a1) –2(a3) and 2(b1) –2(b3) ; the case of Fig. 2(e1) has been tested in Ref. [ 43]. III. DEVICE FABRICATION The FM/NM bilayer that we used is an 80-nm-thick ferrimagnetic Y 3Fe5O12(YIG) film capped with an 18-nm- thick Pt layer. The single-crystal YIG films were grown on(111)-oriented single crystalline Gd 3Ga5O12(GGG) substrates via pulsed laser deposition with a base pressure greaterthan 1 ×10 −6Pa [44]. After deposition, the samples were post-annealed at 810◦C for four hours. Then, the Pt film was deposited on top of the YIG by magnetron sputtering. Themagnetic hysteresis loops of the YIG film were measured withan in-plane magnetic field Happlied along ˆxand ˆyusing a vibrating sample magnetometer (VSM; EV9, MicroSense,Westwood, MA, USA), as shown in Fig. 3(b). The saturation magnetization M s=2.4 kOe, and the magnetic anisotropy fieldHk≈5 Oe. Figure 3(c) shows the cavity FMR spectra (ω/2π=9 GHz) of a YIG/Pt bilayer and of a pure YIG film. The spectral width for the YIG film is clearly enhanced bythe adjacent Pt layer, which is direct evidence of the spin 134421-3ZHOU, FAN, MA, ZHANG, CUI, ZHOU, GUI, HU, AND XUE PHYSICAL REVIEW B 94, 134421 (2016) -1.0-0.50.00.51.0M/Ms -30 -15 0 15 30 H (Oe)α=0ο α=90ο1.0 0.5 0.0 -0.5 -1.0Intensity (a.u.) -20 -10 010 20 H-Hr (Oe) YIG Pt/YIGPt YIG Cu GGG GGGV Vx yz Hα SGG SG G(a) (b) (c) FIG. 3. (a) Sketch of our devices and measurement setup. (b) Magnetic hysteresis loops of the YIG/Pt bilayer. (c) Cavity FMRspectra of the YIG film before and after the deposition of Pt. pumping effect. Assuming that the extrinsic contribution to the linewidth is the same for both YIG/Pt and YIG, the valuesofg ↑↓=3.8×1018m−2can be calculated from the linewidth enhancement via [ 36,45,46] g↑↓=4πγM sdYIG gμ Bω(/Delta1H Pt/YIG−/Delta1H YIG), (9) where γ/2π=2.8 GHz/kOe is the gyromagnetic ratio, which will be determined later, g=2 is the Lande factor, μBis the Bohr magneton, and dYIG=80 nm is the thickness of YIG. To obtain the spatial symmetry of spin pumping under the excitation of a well-controlled dynamic magnetic field, twotypes of devices were fabricated in which YIG/Pt bilayer stripsare coupled with the co-planer waveguide (CPW) in differentways. As shown in Fig. 3(a), two YIG/Pt bilayer strips with a length of 2 mm and width of 20 μm were patterned using ion-milling; then, a CPW with ground-signal-ground (G-S-G)lines was made from Cu with a thickness of 150 nm usingmagnetron sputtering and the lift-off method. The YIG/Ptstrips were placed in the gaps between S and G such thatthe dynamic magnetic field would act perpendicularly on theYIG/Pt strip when the microwave signal from the microwavegenerator (E8257D Agilent) is feeding into the CPW, i.e., ah zdevice [ 33,41]. In the other device, the YIG/Pt bilayer was patterned in the shape of a 2 mm ×50μm strip. Together with two Cu strips deposited on each side, a CPW of 50 /Omega1was created for impedance matching. The devices were attached toa printed circuit board, connected with microwave connectorand electrodes by using wire bonding. By feeding a microwavesignal into the CPW, an rfcurrent along the xaxis will emerge in the Pt layer, which would simultaneously induce an dynamicmagnetic field h ythat can push the magnetic moment of YIG into precession, i.e., a hydevice. Note that all the devices and the YIG/Pt films were made from one sample to eliminate anyquality differences between different YIG films.-40040V (μV) 1.76 1.72 1.68 1.64 H (kOe)80 40 0 -40 -80V (μV) 1.76 1.72 1.68 1.64 H (kOe) 12 10 8 6 4ω/2π (GHz) 3.0 2.5 2.0 1.5 1.0 Hr (kOe) 5.5 5.0 4.5 4.0 3.5ΔH (Oe) 12 10 8 6 4 ω/2π (GHz)hy hz 45o 225o45o 225o Ms =2.34 kOe γ/2π =2.8 GHz/kOe αG=0.0007 (a) (b) (c) (d) hz hy FIG. 4. (a) and (b) are typical resonance spectra measured at αM=45◦and 225◦forhzandhydevices. (c) The symbols represent the resonance positions of FMR, and the red curve is a fitting line based on the Kittle equation. (d) Symbols are the frequency- dependent linewidth, accompanied by a linear fitting curve. IV . RESULTS AND DISCUSSION Figures 4(a) and4(b) show typical voltage spectra measured onhzandhydevices with the applied magnetic field in the xy plane at αM=45◦and 225◦. The FMR was driven by feeding a microwave signal with a frequency of ω/2π=7.5 GHz and power of 15 dBm. The two curves in each figure show thesame symmetric lineshape but opposite polarities, which isthe typical feature of a spin pumping signal. The magneticfield-dependent voltage can be well fitted with V(H)=V sp/Delta1H2 (H−Hr)2+/Delta1H2, (10) which is a simplified expression of Eq. ( 7). The fitting gives rise to the values of the voltage amplitude Vsp, linewidth /Delta1H, and resonance position Hr. Frequency-dependent V(H) curves were measured on both devices at α=45◦. Figure 4(c) shows the frequency-dependent resonance positions Hr.T h e resonance position shifts toward the high-field range withincreasing frequency, which can be well fitted by usingthe Kittle equation ω=γ√ (M0+Hr)Hr, where γ/2π= 2.8 GHz /kOe is the gyromagnetic ratio of YIG and M0= 2.34 kOe is the effective demagnetization field, which isclose to the saturation magnetization of YIG films determinedby VSM. The data shown in Fig. 4(d) are/Delta1H increasing linearly with frequency, from which a reasonable small Gilbertdamping parameter α G=0.0007 and extrinsic contribution to linewidth /Delta1H 0=2.5 Oe were determined. Based on the 134421-4SPATIAL SYMMETRY OF SPIN PUMPING AND INVERSE . . . PHYSICAL REVIEW B 94, 134421 (2016) -50050Vsp (μV) 360 270 180 90 0 αM (degrees)1000 800 600 400 200 0V (μV) -0.08 -0.04 0.00 0.04 0.08 H (kOe) -1000100Vsp (μV) 360 270 180 90 0 αM (degrees)800 600 400 200 0V (μV) -0.08 -0.04 0.00 0.04 0.08 H (kOe) 4.8 4.4 4.0ΔH (Oe) 1.72 1.71 1.70Hr (kOe) 360 270 180 90 0 αM (degrees)0°360° step45° 0°360° step45° hy hz Hk=6.5±0.5 Oe(a) (b) (c) (d) (e) (f) hz hy FIG. 5. In-plane V(H) curves measured at different αfor (a) hzand (b) hydevices. (c) and (d) are angular-dependent resonance amplitudes, accompanied with fitting curves based on Eq. ( 7). (e) Angular-dependent linewidths for both devices, which are in-plane isotropic. (f) Angular-dependent resonance positions, which have been fitted using a −cos 2αM-like function. data shown in Fig. 4, it is clear that both devices ( hzand hy) have identical properties, such as magnetization dynamic parameters and g↑↓, which is important for later discussions. The angular-dependent V(H) curves were measured in the xyplane. Figures 5(a) and 5(b) show the original spectra measured at different αM(α=αMbecause the in-plane anisotropy field Hkis far smaller than Hr, which will be proved later). Distinct periodicity of the resonance amplitudecan be observed, in which the voltage signal vanished atn×180 ◦(n×90◦)f o rt h e hz(hy) device. After fitting each curve with Eq. ( 10), the voltage amplitudes Vspare plotted in Figs. 5(c) and 5(d) as a function of αM, and they are quite similar to the theoretical curves shown in Figs. 2(a1) and 2(b1) , respectively. To examine the influences of linewidth and in-plane anisotropy on the angular dependence, /Delta1H andHr are plotted in Figs. 5(e) and5(f). For both devices, the values of /Delta1H are essentially identical and independent of αM.T h eHr shown in Fig. 5(f)presents a −cos 2αM-dependent oscillation and can be well fitted with Hr=ω2 γ2Meff−Hkcos 2αM. The fit provides a positive value of Hk=6.5±0.5 Oe, which meansthat the in-plane anisotropy of YIG is very weak, and the misalignment between HandMcan be ignored because Hkis far smaller than the resonance field of 1700 Oe. After excluding the influences of linewidth and in-plane anisotropy on the angular dependence and with the ellipticitycorrection factors Pxy zandPxy ybeing constant based on Eq. ( A6) in Appendix 1, the data shown in Figs. 5(c) and 5(d) can be simply fitted with sin αMand cos2αMsinαM according to Eq. ( 7). The simple trigonometrical fitting works quite well for the hzdevice [Fig. 5(c)], but there is a slight discrepancy between the data and fitting curve for the hy device around 90◦and 270◦[Fig. 5(d)]. The reason for this discrepancy may arise from a small ˆxand/or ˆzcomponents of the dynamic magnetic field induced by the CPW such that theycan drive magnetization precession when Mis parallel to ˆy. Nevertheless, the dominated component of h effishybecause the symbols shown in Fig. 5(d) are essentially consistent with a cos2αMsinαMcurve. We measured the angular-resolved V(H) curves out-of- plane on both hzandhydevices. The magnetic field H was applied in the yzplane with an angle βwith respect toˆz. Because of the relatively strong demagnetization field HDin the YIG films, the misalignment between MandH has to be considered to obtain the angular-dependent spinpumping predicted by Eq. ( 8). This task was performed using the static equilibrium condition of magnetization when FMR occurs: M×(H r+HD)=0. This equation was solved in the xyz system with Hr=Hr(0,−sinβ,cosβ),HD= −M0(0,0,cosβM), and M=M0(0,−sinβM,cosβM), and βMcan be determined via 2Hrsin(β−βM)+Mssin 2βM=0. (11) After determining the resonance position Hras a function of β [Fig. 6(a)], the relation between βMandβwas obtained as the data shown in the inset of Fig. 6(a). The insets in Figs. 6(b) and 6(c) show the γM-dependent Vspof the hzandhydevice with magnetization rotating in the xzplane. The Vspis zero, which is consistent with the theoretical prediction shown in Figs. 2 (a3) and 2(b3). The symbols shown in Figs. 6(b) and6(c) are Vspas a function of βMfor the hzandhydevices, respectively. The figures also show the sin3βMand cos2βMsinβMcurves predicted using Eq. ( 8). The general angular dependencies of the experimental results are essentially consistent with thecurves; however, the discrepancies between them are obvious.Therefore, we need to consider other influences on the detailsof the angular dependence. Based on Eq. ( 8), there are three parameters that may distort the trigonometric function, i.e., ellipticity correction factorP yz y, linewidth /Delta1H, and spin mixing conductance g↑↓.T h e influence of Pyz ycan be ignored, which has been discussed in Appendix 4. The angular-dependent /Delta1H is shown in Fig. 7(a), which presents distinct anisotropy. The maximum value of /Delta1H at perpendicular configuration (12 Oe, βM=0◦)i sa l m o s t three times larger than the minimum (4.2 Oe, βM=90◦). Figure 7(c) presents two typical V(H) curves measured at βM=10◦and 90◦to show the broadening of resonance peaks. Moreover, g↑↓are also found to be anisotropic in our Pt/YIG samples. Figures 7(d) and7(c) show the cavity FMR spectra of Pt/YIG and YIG at βM=10◦and 90◦. Although the linewidths of both samples become larger from in-plane to 134421-5ZHOU, FAN, MA, ZHANG, CUI, ZHOU, GUI, HU, AND XUE PHYSICAL REVIEW B 94, 134421 (2016) -100-50050100Vsp (μV) 100 0 -100 Vsp (μV) 360 270 180 90 0 βM (degrees)7 6 5 4 3 2Hr (kOe) 360 270 180 90 0 β(degrees)360 180 0βM (deg.) 360 180 0 β (deg.) xyzH MβMβ hz yz-plane hy yz-plane(a) (b) (c)-1000100V (μV) 360 270 180 90 0 γM (degrees) -1000100V (μV) 360 270 180 90 0 γM (degrees) hzxz-plane hyxz-plane FIG. 6. (a) Resonance positions as functions of βin the out- of-plane case; the inset shows the relation between βMandβ.( b ) Angular-dependent resonance amplitude for the hzdevice, accompa- nied by a fitting curve of the sin3βMfunction. (c) Angular-dependent resonance amplitude for the hydevice, accompanied by a fitting curve of the cos2βMsinβMfunction. The insets in (b) and (c) are angular-dependent resonance amplitudes measured in the xzplane in which the ISHE voltage vanishes. out-of-plane, the spin pumping-induced additional broadening is also enhanced. Figure 7(b) shows the angular-dependent g↑↓calculated using Eq. ( 9), which presents a cos 2 βM-like dependence. In Ref. [ 47], Chen and Zhang predicted an out-of- plane anisotropic g↑↓due to interfacial spin-orbit interaction (SOI), and a −cos 2βMdependence was expected. However, our results are inconsistent with their theoretical prediction: (1)the maximum g ↑↓is expected in-plane, but our experimental result is out-of-plane; (2) the relative variation between themaximum and minimum is 270% based on Fig. 7(b), which is far larger than the few percent estimated according toSOI intensity. In Ref. [ 38], Jungfleisch et al. , suggested that the incoherent short wavelength spin waves could make asubstantial contribution to spin pumping at the insulator/metalinterface. Since the coupling between FMR and spin waveshas strong angular dependence, this coupling could resultin the anisotropy in spin mixing conductance. We leave thisissue for a forthcoming paper and concentrate on the angulardependence of spin pumping. By taking the anisotropy of /Delta1H andg ↑↓into consid- eration, the combination of Vsp/Delta1H2/g↑↓would provide a pure trigonometric dependence according to Eq. ( 8). This14 12 10 8 6 4ΔH (Oe)(a) 16 12 8 4g (1018m-2) 270 180 90 βM (degrees)(b)1.2 0.8 0.4 0.0V/Vmax 1.0 0.5 0.0 -0.5 -1.0Int. (a.u.) -40-20 02040 H-Hr (Oe)-1.0-0.50.00.51.0Int. (a.u.) βΜ =90°βΜ =10°βM= 10o90o Pt/YIG YIG (c) (d) (e) FIG. 7. (a) Out-of-plane angular-dependent linewidth determined from ISHE voltage. (b) Angular-dependent g↑↓evaluated from the linewidth difference between YIG and Pt/YIG by using cavity FMR. (c) Two typical ISHE voltage spectra measured at βM=10◦and 90◦. (d) and (e) Cavity FMR spectra of the YIG film before and after the deposition of Pt measured at βM=10◦and 90◦. prediction was confirmed by the data shown in Fig. 8with the right-hand-side vertical axis, accompanied by a sin3βM (cos2βMsinβM) fitting curve for the hz(hy) device. In Fig. 8(a), the trigonometric fitting perfectly matches the experimental results; in Fig. 8(b), the fitting and data are essentially consistent, and the small discrepancy around 90◦ 8 4 0 -4VspΔH2 (mVOe2) 360 270 180 90 0 βM (degrees)-8-404VspΔH2/g (10-19mVOe2m2)8 4 0 -4VspΔH2 (mVOe2) 360 270 180 90 0 βM (degrees)-2-101VspΔH2/g (10-18mVOe2m2) hz yz-plane hy yz-plane(a) (b) FIG. 8. βM-dependent Vsp/Delta1H2/g↑↓(blue symbols) by using (a) thehzdevice and (b) the hydevice, which were fitted by sin3βMand cos2βMsinβM, respectively. The data of Vsp/Delta1H2are also plotted for comparison (black symbols), which cannot be well fitted using thetrigonometric functions. 134421-6SPATIAL SYMMETRY OF SPIN PUMPING AND INVERSE . . . PHYSICAL REVIEW B 94, 134421 (2016) and 270◦is due to the same reason as for the in-plane case [Fig. 5(d)]. To highlight the influence of g↑↓on the angular dependency, the data of Vsp/Delta1H2are also plotted in Fig. 8with a left vertical axis. Comparing the two types of angular-dependent data in each figure, it is clear that theanisotropy of g ↑↓has a significant contribution to the angular dependence of spin pumping. Moreover, the consistencybetween the experimental results and theoretical predictionsin both the in-plane and out-of-plane cases indicates that thespin Hall angle is an intrinsic constant that is independent ofthe polarization direction of the spin current. V . SUMMARY We have presented a systematic study on the spatial symmetry of the spin pumping effect associated with ISHE.The main symmetry of the voltage as a function of themagnetization direction is dominated by the ISHE and theeffective dynamic field acting on the magnetization, whichhave been summarized in Fig. 2; the anisotropy in the resonance linewidth, Pfactors, and spin mixing conductance, however, would distort the main symmetry. For the in-planecase, because these parameters are isotropic, our experimentalresults are consistent with simple trigonometric functions. Forthe out-of-plane case, although the basic symmetry of theexperimental results is similar to the theoretical predictions, asignificant discrepancy was found. This discrepancy has beendemonstrated to arise not only from the resonance linewidth, but also from the anisotropy in g ↑↓which is an open question in the spin pumping effect. ACKNOWLEDGMENTS The project is supported by the National Basic Re- search Program of China (No. 2012CB933101), NSFC(Nos. 11429401 and 51471081), the Program for ChangjiangScholars and Innovative Research Team in University (No.IRT1251), and the Fundamental Research Funds for theCentral Universities (Grant No. lzujbky-2016-k05). APPENDIX The purpose of this appendix is to deduce the spatial symmetry of spin pumping. Our theory starts from Eq. ( 5), where we need to calculate the Im( m∗ y/primemz/prime) term based on the Landau-Lifshitz-Gilbert (LLG) equation, which providesa phenomenological description of ferromagnetic dynamicsbased on the torque provided by the effective magnetic fieldH effwith a damping term: dm dt=−γm×Heff+αeffm×dm dt. (A1) Here, m=M/Msis the unit vector of dynamic magnetization, γ=μ0e/m eis the effective gyromagnetic ratio of electron with charge eand mass me,μ0is the permeability of vacuum, andαeffis the effective Gilbert damping parameter. Heff includes the applied dcmagnetic field H, dynamic magnetic field h, and demagnetization field HD=−NzMzˆz, where Nz=1 is the demagnetization factor of thin films and Mz is the out-of-plane component of magnetization.1. The dynamic magnetization in the xyplane When His applied in the xy plane, we assume thatmis parallel to Hbecause the in-plane anisotropy field is far smaller than the applied magnetic field, seeFig. 1(b). We solve the LLG equation in the ( x /prime,y/prime,z/prime) coordinate system with ˆx/primeparallel to mand ˆz/primeparal- lel to ˆz,m=(1,my/primee−iωt,mz/primee−iωt),H=(H,0,0),HD= −Ms(0,0,mz/primee−iωt), and h=(hx/prime,hy/prime,hz/prime)e−iωt. The solution is given by ⎛ ⎝0 my/prime mz/prime⎞ ⎠=⎛ ⎝00 0 0χxy y/primey/prime−iχxy y/primez/prime 0iχxy z/primey/primeχxy z/primez/prime⎞ ⎠⎛ ⎝0 hy/prime hz/prime⎞ ⎠, (A2) where χis the dynamic susceptibility, which is a function of the magnetic field at a certain frequency. To highlight the resonantfeature of the susceptibility tensor elements, we define thesymmetric Lorentz lineshape L(H) and the antisymmetric dispersive lineshape D(H)[41], L(H)=/Delta1H 2 (H−Hr)2+/Delta1H2 D(H)=/Delta1H(H−Hr) (H−Hr)2+/Delta1H2;( A 3 ) then the magnetic field-dependent χis given by χxy y/primey/prime(H)=1 /Delta1Haxy y/primey/prime[D(H)+iL(H)], χxy z/primez/prime(H)=1 /Delta1Haxy z/primez/prime[D(H)+iL(H)], χxy z/primey/prime(H)=χxy y/primez/prime(H)=1 /Delta1Haxy y/primez/prime[D(H)+iL(H)],(A4) with axy y/primey/prime=Ms+Hr Ms+2Hr, axy z/primez/prime=Hr Ms+2Hr, axy y/primez/prime=ω γ(Ms+2Hr). Therefore, the Im( m∗ y/primemz/prime) term in Eq. ( 5)i sg i v e nb y Im(˜m∗ y/prime˜mz/prime)=/parenleftBigg h2 y/prime /Delta1H2axy y/primey/primeaxy y/primez/prime+h2 z/prime /Delta1H2axy z/primez/primeaxy y/primez/prime/parenrightBigg L(H) =/parenleftBigg h2 y/prime /Delta1H2Pxy y+h2 z/prime /Delta1H2Pxy z/parenrightBigg L(H). (A5) Note that Eq. ( A5) is correct under the conditions that either considering only one dynamic magnetic field for each time(which is the case in our experiments) or h y/primeandhz/primeare in- phase. Otherwise, a coupling term of hyandhzwould emerge in Eq. ( A5) depending on the phase lag in between. Pis defined as the correction factor due to the ellipticity of dynamic 134421-7ZHOU, FAN, MA, ZHANG, CUI, ZHOU, GUI, HU, AND XUE PHYSICAL REVIEW B 94, 134421 (2016) magnetization, Pxy z=axy z/primez/primeaxy y/primez/prime=ωHr γ(Ms+2Hr)2, Pxy y=axy y/primey/primeaxy y/primez/prime=ω(Ms+Hr) γ(Ms+2Hr)2. (A6) It is clear that the Pfactor for the in-plane case is isotropic and only depends on the microwave frequency. In our experiments, because the dynamic magnetic fields were generated by the CPW in the device coordinate systemxyz, we need to project the microwave magnetic field h x,hy andhzto the x/primey/primez/primesystem, i.e., hy/prime=hxsinαM+hycosαM andhz/prime=hz. If we consider only one dynamic magnetic field for each time, Im( ˜m∗ y/prime˜mz/prime) for the in-plane case can be obtained Im(˜m∗ y/prime˜mz/prime)=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩h2 x /Delta1H2sin2αMPxy yL(H)f o r hx, h2 y /Delta1H2cos2αMPxy yL(H)f o r hy, h2 z/prime /Delta1H2Pxy zL(H)f o r hz.(A7) Placing Eq. ( A7) into Eq. ( 5), the spin pumping voltage measured along the xaxis is given as Vxy x,hx=Cxg↑↓Pxy yh2 x /Delta1H2sin2αMsinαML(H), (A8) Vxy x,hy=Cxg↑↓Pxy yh2 y /Delta1H2cos2αMsinαML(H), (A9) Vxy x,hz=Cxg↑↓Pxy zh2 z /Delta1H2sinαML(H). (A10) However, if the spin pumping signal was measured along ˆy, i.e., the Hall geometry, Vy=WNEy=WNjc·ˆy/σN, where WNis the width of the stripe (or the distance between Hall contacts). Together with Eqs. ( 1) and ( 4), the voltage along ˆy is given by Vy=Cyg↑↓Im(m∗ y/primemz/prime)(ˆz׈x/prime·ˆy), (A11) where Cy=−θSHeλsdWNω 2πσNtNtanh(tN/2λsd) is an angular- independent parameter similar to Cx. Combined with Eq. ( A7), the angular-dependent spin pumping voltage with differentdynamic magnetic fields can be described as Vxy y,hx=Cyg↑↓Pxy yh2 x /Delta1H2sin2αMcosαML(H), (A12) Vxy y,hy=Cyg↑↓Pxy yh2 y /Delta1H2cos3αML(H), (A13) Vxy y,hz=Cyg↑↓Pxy zh2 z /Delta1H2cosαML(H). (A14) 2. The dynamic magnetization in the yzplane When His applied in the yzplane, the LLG equation was solved in the ( x/prime,y/prime,z/prime) coordinate system in which ˆx/prime is parallel to mand ˆz/primeis parallel to ˆx, as shown in Fig. 1(c). The applied magnetic field H, demagnetization field HD, anddynamic magnetic field hare rewritten as H=H[cos(βM−β),sin(βM−β),0], HD=−Ms(cosβM+my/primee−iωtsinβM)⎛ ⎝cosβM −sinβM 0⎞ ⎠, h=⎛ ⎝hx/prime hy/prime hz/prime⎞ ⎠e−iωt, (A15) where βis the polar angle of Hdefined by experiments and βMis the equilibrium angle of M, which is determined by the static equilibrium condition Eq. ( 11). The solution is given by ⎛ ⎝0 my/prime mz/prime⎞ ⎠=⎛ ⎝00 0 0χyz y/primey/prime−iχyz y/primez/prime 0iχyz y/primez/primeχyz z/primez/prime⎞ ⎠⎛ ⎝0 hy/prime hz/prime⎞ ⎠. (A16) Because of the misalignment between MandH, we define /Delta1H/prime=/Delta1H/ cos(βM−β) as the linewidth in lineshape fit- ting, and the magnetic field-dependent χis given by χyz y/primey/prime(H)=1 /Delta1Hayz y/primey/prime[D/prime(H)+iL/prime(H)], χyz z/primez/prime(H)=1 /Delta1Hayz z/primez/prime[D/prime(H)+iL/prime(H)], χyz z/primey/prime(H)=χyz y/primez/prime(H)=1 /Delta1Hayz y/primez/prime[D/prime(H)+iL/prime(H)], (A17) with L/prime(H)=/Delta1H/prime2 (H−Hr)2+/Delta1H/prime2, D/prime(H)=/Delta1H/prime(H−Hr) (H−Hr)2+/Delta1H/prime2,(A18) and ayz y/primey/prime=Hrcos(β−βM)−Mscos2βM 2Hrcos(β−βM)−Ms(cos 2βM+cos2βM), ayz z/primez/prime=Hrcos(β−βM)−Mscos 2βM 2Hrcos(β−βM)−Ms(cos 2βM+cos2βM), ayz y/primez/prime=ω γ[2Hrcos(β−βM)−Ms(cos 2βM+cos2βM)]. (A19) The Im( m∗ y/primemz/prime) term in Eq. ( 5) is given by Im(m∗ y/primemz/prime)=/parenleftBigg h2 y/prime /Delta1H2ayz y/primey/primeayz y/primez/prime+h2 z/prime /Delta1H2ayz z/primez/primeayz y/primez/prime/parenrightBigg L/prime(H) =/parenleftBigg h2 y/prime /Delta1H2Pyz y+h2 z/prime /Delta1H2Pyz z/parenrightBigg L/prime(H). (A20) The ellipticity correction factors for the hzandhydevices in theyzcase are defined as Pyz z=ayz z/primez/primeayz y/primez/primeandPyz y=ayz y/primey/primeayz y/primez/prime, respectively. Clearly, the Pfactors for the out-of-plane case are anisotropic according to the results of Eqs. ( A19) and ( A20), which will be discussed later. After projecting the microwave magnetic field in the x/primey/primez/prime system via hy/prime=−hycosβM+hzsinβMandhz/prime=hxand 134421-8SPATIAL SYMMETRY OF SPIN PUMPING AND INVERSE . . . PHYSICAL REVIEW B 94, 134421 (2016) considering only one dynamic field for each time, Im( ˜m∗ y/prime˜mz/prime) is given by Im(˜m∗ y/prime˜mz/prime)=⎧ ⎪⎪⎨ ⎪⎪⎩h2 x /Delta1H2Pyz zL/prime(H)f o r hx, h2 y /Delta1H2cos2αMPyz yL/prime(H)f o r hy, h2 z /Delta1H2sin2αMPyz yL/prime(H)f o r hz.(A21) Combined with Eq. ( 5), the angular-dependent spin pumping voltages along ˆxwith different microwave magnetic fields are given by Vyz x,hx=Cxg↑↓Pyz zh2 x /Delta1H2sinβML/prime(H), (A22) Vyz x,hy=Cxg↑↓Pyz yh2 y /Delta1H2cos2βMsinβML/prime(H), (A23) Vyz x,hz=Cxg↑↓Pyz yh2 z /Delta1H2sin3βML/prime(H). (A24) The voltage along ˆywould vanish for the yzplane case. Because the polarization direction of the pumped spin current(σ=− ˆx /prime) lies in the yzplane, the charge accumulation due to ISHE ( ˆz׈x/prime) is along the xaxis; the detected voltage signal along the yaxis is zero ˆz׈x/prime·ˆy=0, i.e., Vyz y,hz=Vyz y,hy=Vyz y,hx=0. (A25) 3. The dynamic magnetization in the xzplane When His applied in the xzplane, the LLG equation was solved in the ( x/prime,y/prime,z/prime) coordinate system in which ˆx/primeis parallel tomand ˆz/primeis parallel to ˆy, as shown in Fig. 1(d). Because the in-plane anisotropy of YIG was ignored in our theoreticalcalculations, the dynamic magnetization in the xzplane is very similar to that in the yzplane, which can be obtained by simply replacing βandβ MwithγandγMin Eqs. ( A15)–(A19). After using hy/prime=−hxcosγM−hzsinγMandhz/prime=−hy,w e obtain Im(˜m∗ y/prime˜mz/prime)=⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩h2 x /Delta1H2cos2γMPxz yL/prime(H)f o r hx, h2 y /Delta1H2Pxz zL/prime(H)f o r hy, h2 z /Delta1H2sin2γMPxz yL/prime(H)f o r hz,(A26) where Pxz z=Pyz zandPxz y=Pyz y. In the xzplane configuration, no signal can be detected along the xaxis because ˆz׈x/prime·ˆx=0; only the voltage along theyaxis can be detected. Placing Eq. ( A26)i n t o( A11), Vxz y,hx=Cyg↑↓Pxz yh2 x /Delta1H2cos2γMsinγML/prime(H), (A27) Vxz y,hy=Cyg↑↓Pxz zh2 y /Delta1H2sinγML/prime(H), (A28) Vxz y,hz=Cyg↑↓Pxz yh2 z /Delta1H2sin3γML/prime(H). (A29) 4. Anisotropy of the Pfactor The correction factors due to the ellipticity of dynamic magnetization Pfor the out-of-plane case show strong anisotropy based on Eqs. ( A19) and ( A20), which wouldinfluence the spatial symmetry of the spin pumping voltage. We calculated the values of Pyz yas a function of βMat 7.5 GHz and 10 GHz as the data shown in Fig. 9(a). The maximum appears at 0◦and 180◦(Malong the film normal), with a value of 0.25 corresponding to a perfect circular precession [ 40], which is independent of the frequency. The minimum appearsfor the in-plane magnetization, and the values increase withincreasing frequency. Therefore, the anisotropy of the Pfactor becomes weak at high microwave frequencies. To visualize the influence of the Pfactors on the angular- dependent spin pumping signal, the calculated P yz ysinβ3 M andPyz ysinβMcos2βMwere plotted as the symbols shown in Fig. 9(b), corresponding to the cases shown in Figs. 8(a) and 8(b), respectively. The curves shown in Figs. 9(b) and 9(c) are the sin β3 Mand sin βMcos2βMfitting results. The symbols and curves are essentially consistent, which meansthat the influence of the Pfactors on the symmetry of spin pumping can be ignored in our cases. The reason comes fromthe fact that the variations in the Pfactors are mainly around 0 ◦and 180◦, i.e., magnetization along the film normal; at these 0.3 0.2 0.1 0.0 -0.1Pyyz sinβM cos2βM 360 270 180 90 0 βM (degrees)-0.2-0.10.00.10.2Pyyz sinβM cos2βM0.4 0.2 0.0 -0.2Pyyz sin3βM -0.3-0.2-0.10.00.10.2Pyyz sin3βM hzyz-plane hyyz-planeCalculated Pyyzsin3βM for 10 GHz 7.5 GHz Calculated Pyyz sinβM cos2βM for 10 GHz 7.5 GHz fit with sinβM cos2βM fit with sin3βM0.3 0.2 0.1Pyyz(a) (b) (c) 10 GHz 7.5 GHz FIG. 9. (a) Angular-dependent Pyz zcalculated using Eqs. ( A19) and ( A20). (b) Calculated Pyz zsin3βM(symbols) as functions ofβM, which were fitted with the sin3βMfunction. (c) Calcu- lated Pyz ycos2βMsinβM(symbols), which were fitted with the cos2βMsinβMfunction. 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PhysRevB.90.214436.pdf

PHYSICAL REVIEW B 90, 214436 (2014) Origin of the magnetostructural coupling in FeMnP 0.75Si0.25 E. K. Delczeg-Czirjak,1,*M. Pereiro,1L. Bergqvist,2Y . O. Kvashnin,1I. Di Marco,1Guijiang Li,3 L. Vitos,1,3,4and O. Eriksson1 1Division of Materials Theory, Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden 2Department of Materials and Nano Physics and Swedish e-Science Research Centre, Royal Institute of Technology (KTH), Electrum 229, SE-164 40 Kista, Sweden 3Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden 4Research Institute for Solid State Physics and Optics, Wigner Research Center for Physics, P .O. Box 49, HU-1525 Budapest, Hungary (Received 28 May 2014; revised manuscript received 28 October 2014; published 23 December 2014) The strong coupling between the crystal structure and magnetic state (ferromagnetic or helical antiferro- magnetic) of FeMnP 0.75Si0.25is investigated using density functional theory in combination with atomistic spin dynamics. We find many competing energy minima for drastically different ferromagnetic and noncollinearmagnetic configurations. We also find that the appearance of a helical spin-spiral magnetic structure at finitetemperature is strongly related to one of the crystal structures reported for this material. Shorter Fe-Fe distancesare found to lead to a destabilized ferromagnetic coupling, while out-of-plane Mn-Mn exchange interactionsbecome negative with the shortening of the interatomic distances along the caxis, implying an antiferromagnetic coupling for the nearest-neighbor Mn-Mn interactions. The impact of the local dynamical correlations is alsodiscussed. DOI: 10.1103/PhysRevB.90.214436 PACS number(s): 75 .30.Sg,75.50.Ee,75.40.Gb I. INTRODUCTION Magnetocaloric (MC) research has been primarily focused on the ferromagnetic (FM) to paramagnetic (PM) phasetransition [ 1–10]. Promising new directions toward finding well-performing MC materials are those focusing on materialswith an antiferromagnetic (AFM) to FM metamagnetic phasetransition [ 11–16]. Widely investigated representatives of this class of materials are the doped MnP-related Pnma com- pounds [ 13,14]. The strong connection between the magnetic properties and the crystal structure was observed recently forCo- and Ge-doped MnP compounds [ 17,18]. The magnetic state of these alloys is found to be strongly dependent onthe nearest-neighbor (NN) Mn-Mn separation. CoMnP andCoMnGe are collinear ferromagnets, whereas CoMnP 1−xGex (x≈0.5) has an AFM ground state. The shortest Mn-Mn distance has an intermediate value for CoMnP 1−xGex, while CoMnP possesses the shortest NN distance and CoMnGehas the largest Mn-Mn NN distance. The coupling betweenthe type of magnetic ordering (FM or AFM) and Mn-Mndistance was also discussed for other Mn-based compoundsin the past (e.g., in Ref. [ 19]). The shorter Mn-Mn distances result in an AFM coupling in the Mn 3Ni20P6compound, while the isostructural compound Mn 3Pd20P6shows ferromagnetism due to the larger Mn-Mn separation. Spichkin and Tishin showed that the major contribution to the MC effect comes from the change in the exchangeenergy, which can be related to the change in the exchangeparameters through the first-order magnetic phase transitionfor rare-earth metals (Tb, Dy, Er) and compounds (Tb 0.5Dy0.5, Gd5Si1.7Ge2.3)a sw e l la sF e 0.49Rh0.51[20,21]. Fe2P and its alloys, e.g., Fe 1−yMnyP1−xSix, were widely investigated experimentally from a magnetocaloric point of *erna.delczeg@physics.uu.seview [ 2,5–7,9,10,22] and are considered to be collinear ferro- magnetic materials [ 23–26]. These materials show a positive magnetocaloric effect when they go from the FM state to a PMstate at a Curie temperature T Ctypically ranging from 260 to 350 K [ 10]. Early on, the appearance of antiferromagnetism at ambient conditions was reported only for FeMnP 1−xAsx[27]. Recently, however, a state with very low net magnetization hasbeen observed for FeMnP 0.75Si0.25[28] at low temperatures. Further investigations [ 29] have shown the coexistence of two phases in the FeMnP 0.75Si0.25sample: phases A and B with a very distinguishable difference in the crystal structure. Afterthe first heat treatment of the sample the phase ratio is estimatedto be A/B ≈70/30 [29]. The collected magnetization data indicate a ferromagnetic ordering with T C=250 K due to the majority of phase A. After the second heat treatment of thesample the phase ratio changes to A/B ≈10/90. Magnetization data collected after the second heat treatment indicate theexistence of an incommensurate antiferromagnetic orderingdue to phase B with a N ´eel temperature T N≈150 K and propagation vector qx=0.363(1) measured at 16 K [ 29]. Neutron diffraction measurements reveal very small deviationsfrom the nominal composition [ 29]. The strong coupling between crystal structure and magnetic state (FM or incommensurate AFM) is interesting on itsown. However, since it is not explored if such strong localexchange striction has relevance for the performance of amagnetocaloric material, more investigations are necessary.The present study aims to identify the mechanism behindthe strong local exchange striction in terms of the electronicstructure of the material. Earlier electronic structure studies onthe relation between the structure and magnetic configurationwere based on total energy calculations [ 17,18]. Here we adopt a different approach to investigate the magnetostructuralcoupling. Namely, the magnetic exchange interactions areextracted from first-principles calculations, after which a spin-dynamics simulation is applied to obtain the final magnetic 1098-0121/2014/90(21)/214436(10) 214436-1 ©2014 American Physical SocietyE. K. DELCZEG-CZIRJAK et al. PHYSICAL REVIEW B 90, 214436 (2014) ab c FIG. 1. (Color online) Rhombohedral building blocks of FeMnP 0.75Si0.25in the ( a,b) plane. Fe-Fe (yellow balls) and Mn-Mn (purple balls) shortest-in-plane bonds are represented by gray rods. P/Si atoms (small black balls) are randomly distributed on the 2 c and 1bpositions. configuration at finite temperatures. Using these theoretical predictions, we discuss the available experimental data [ 29]. The rest of this paper is organized as follows. Section II gives an overview of the crystal structures and the appliedcomputational methods. Results and discussion are given inSec. III. Finally, the paper concludes with a brief summary of the reported results in Sec. IV. II. CRYSTAL STRUCTURE AND NUMERICAL DETAILS The experimentally reported A and B phases of FeMnP 0.75Si0.25alloys both crystallize in the hexagonal Fe 2P crystal structure (space group P62m, 189) [ 29]. The unit cell of Fe2P contains six Fe atoms situated in the threefold-degenerate 3f(tetrahedral) and 3 g(pyramidal) Wyckoff positions, and the three P atoms are accommodated on the 2 cand 1b positions. The basal ( a,b) plane of the rhombohedral subcell is presented in Fig. 1. Experimentally, it is known that Fe mainly occupies the 3 ftetrahedral position with the ( x1,0 , 0) coordinates, while Mn favors the 3 gpyramidal position with the ( x2,0 ,1/2) coordinates. Deviations from the site occupancy are less than 6% [ 29]. Fe atoms are surrounded by four P/Si atoms, while Mn has five P /Si atoms as neighbors. Alternating Fe and Mn layers along the ccrystallographic axis are shown in Fig. 2. There is no experimentally reported Si site preference in FeMnP 0.75Si0.25. Significant differences between the lattice parameters of phases A and B are given inTable I. The ground-state electronic structure was modeled on the level of density functional theory (DFT) using the Green’s-function-based exact muffin-tin orbitals (EMTO) method TABLE I. Experimental lattice parameters aandc(˚A), volume V(˚A3), hexagonal axial ratio c/a, and in-plane metal-metal distances dFe−Fe,dMn−Mn,a n ddFe−Mn(˚A) for phases A and B of FeMnP 0.75Si0.25. Phase aexpcexpVexp(c/a)expdFe−FedMn−MndFe−Mn A 6.163 3.304 108.66 0.536 2.753 3.241 2.709 B 5.976 3.488 107.86 0.584 2.634 3.126 2.728bc a FIG. 2. (Color online) Fe-Fe (yellow balls) and Mn-Mn (purple balls) out-of-plane [along the c(z) axis] NNs of FeMnP 0.75Si0.25.P/Si atoms (small black balls) are randomly distributed on the 2 cand 1b positions. [30–33]. EMTO theory formulates an efficient way to solve the Kohn-Sham equation [ 34]. A full description of the EMTO theory and the corresponding method may be found inRefs. [ 30–33]. Within this approach the compositional disorder for the P-Si atoms was treated using the coherent potentialapproximation [ 35,36]. Here we take a perfect ordering of the Fe and Mn atoms. The effect of Fe /Mn disorder is discussed elsewhere [ 37]. The ground-state information is then used to extract the intersite exchange integrals J ij(iandjdenote Fe and Mn positions). Jijwere calculated for collinear spins within the magnetic force theorem [ 38]: Jij=1 4πIm/integraldisplayEF −∞dETr L(/Delta1iTij,↑/Delta1jTji,↓), (1) where /Delta1i=t−1 i,↑−t−1 i,↓andti,sandTij,s(s=↑,↓) are the spin- projected single-site scattering matrices and the matrices of thescattering path operator, respectively. Note that the trace wastaken in angular momentum space, L=(/lscript,m), and we omitted labeling explicitly the energy dependence of the occurringscattering matrices. Recently, a noncollinear generalization ofEq. ( 1) was proposed [ 39], but that generalization was not pursued in this work. The one-electron equations were solved within the scalar- relativistic and soft-core approximations. The Green’s functionwas calculated for 20 complex energy points distributedexponentially on an ellipsoidal contour including states within1.5 Ry below the Fermi level. The s,p,d, andforbitals were included in the basis set, and an l h max=10 cutoff was used in the one-center expansion of the full charge density. Theelectrostatic correction to the single-site coherent potentialapproximation was described using the screened impuritymodel [ 40] with a screening parameter of 0.902. The Fe /Mn 3d and 4sand the P /Si 2sand 2pelectrons were treated as valence electrons, respectively. To obtain the accuracy needed for thecalculations, a 21 ×21×27k-point mesh was used within the Monkhorst-Pack scheme [ 41]. This gives more than 3900 kpoints in the irreducible wedge of the Brillouin zone. The self-consistent EMTO calculations were performed within thelocal-spin-density approximation (LSDA) [ 42], which gave a 214436-2ORIGIN OF THE MAGNETOSTRUCTURAL COUPLING IN . . . PHYSICAL REVIEW B 90, 214436 (2014) good description of the magnetic properties of Fe 2P and related systems [ 23,25,26,43–46]. In order to investigate the importance of the many-body effects in FeMnP 0.75Si0.25, we have performed a set of calcula- tions with special treatment of the localized 3 delectrons of Fe and Mn. For these electronic states an effective single-impurityproblem is solved by means of the dynamical mean-field theory(DMFT) equations [ 47]. Such treatment enables us to capture all the local correlation effects that the defined electrons areexposed to. The present technique, called DFT +DMFT, is the state-of-the-art method for modeling the electronic struc-ture from first principles [ 48]. We have used a charge self- consistent (CSC) version of this method, as incorporated inthe full-potential LMTO code RSPT [49]. As a DMFT solver we adapted the spin-polarized T-matrix and fluctuation exchange (SPTF) method [ 50]. It operates in a weak-correlation limit and was successfully applied to the series of bulk transitionmetals, their surfaces, and alloys [ 51,52]. Since transition- metal-transition-metal distances in FeMnP 0.75Si0.25are rather close to the bulk values of the corresponding 3 dmetals, we anticipate a similar degree of electron localization here. Inour study we have used a version of the solver in which theFeynman diagrams are calculated using partially renormalized(i.e., “dressed”) Green’s functions G HF, where HF stands for the Hartree-Fock treatment. Such a computational setup showsa systematic improvement in the description of the magnetic properties of various systems [ 53]. The localized 3 dorbitals to be treated in DMFT were defined by performing a projection onto a muffin-tin (MT)sphere around the chosen atom (so-called MT-head projec-tion). The parameter of the screened Coulomb interaction U was set to 2.3 and 3.0 eV for Fe and Mn, respectively. Intrasiteexchange was chosen to be 0.9 eV for both atomic species.These values are adapted from prior studies of bulk transitionmetals, which provided a fair agreement with existing experi-mental data [ 51,54,55]. The static part of the self-energy was used as a double-counting correction of the single-electronstates. The collinear EMTO calculations were augmented by noncollinear spin-spiral total-energy calculations using twodifferent implementations. Most of the noncollinear calcu-lations were performed using the linear muffin-tin orbitalmethod in the atomic sphere approximation, but the resultswere carefully validated against the more accurate but slowerfull potential linear augmented-plane-wave method (LAPW)as implemented in the ELK software package [ 56]. We found only minor differences between the two methods regardingmagnetic moments and ordering. In both softwares we employthe virtual crystal approximation (VCA) for the treatmentof the compositional disorder of P and Si. These softwaresallow for ground-state magnetic structure determination byminimizing the torque on each atom, i.e., zero-temperaturespin dynamics, as well as superimposing of a spin-spiralpropagation vector employing the generalized Bloch theoremsimulating helical magnetic structures [ 57]. The magnetic dynamical properties of the FeMnP 0.75Si0.25 system were investigated using an atomistic spin dynamics approach recently implemented in Ref. [ 58] at a temperature T=16 K. The equation of motion of the classical atomistic spins siat finite temperature is governed by Langevin dynamicsvia a stochastic differential equation, normally referred to as the atomistic Landau-Lifshitz-Gilbert (LLG) equations, whichcan be written in the form ∂s i ∂t=−γ 1+α2 isi×[Bi+bi(t)] −γαi s/parenleftbig 1+α2 i/parenrightbigsi×si×[Bi+bi(t)], (2) where γis the gyromagnetic ratio and αidenotes a dimen- sionless site-resolved damping parameter which accounts forthe energy dissipation that eventually brings the system intoa thermal equilibrium. The effective field in this equation iscalculated as B i=−∂H/∂si, where Hrepresents an effective Heisenberg Hamiltonian given by H=−/summationdisplay /angbracketleftij/angbracketrightJijsi·sj. (3) The exchange parameters Jijhave been taken from the EMTO method as given by Eq. ( 1). The temperature fluctuations T are considered through a random Gaussian-shaped field bi(t) with the following stochastic properties: /angbracketleftbi,μ(t)/angbracketright=0, /angbracketleftbi,μ(t)bj,ν(t/prime)/angbracketright=2αikBTδijδμνδ(t−t/prime) s(1+αi)2γ, (4) where iandjare atomic sites and μandνrepresent the Cartesian coordinates of the stochastic field. After solving theLLG equations, we have direct access to the dynamics of theatomic magnetic moments s i(t). III. RESULTS A. Electronic structure and magnetic moments of ferromagnetic FeMnP 0.75Si0.25 The electronic structure and magnetic properties discussed here were derived for the ferromagnetic FeMnP 0.75Si0.25in phase A. Spin-, site-, and orbital-projected densities of states(DOSs) presented in Fig. 3have been obtained using the EMTO method in combination with the coherent potentialapproximation for the P sites. Due to the fact that in theGreen’s function formalism the DOS is calculated on theenergy contour, the DOS plots do not show sharp peaks butcontain all the features typical for Fe 2P-type systems [ 23,45]. FeMnP 0.75Si0.25shows metallic behavior in both spin-up and spin-down channels. The low-energy region of the totalDOS (from −1t o−0.6 Ry) originates from the P and Si s states. The observed small shift between the spin-downand spin-up channels of P /Si is a result of a small exchange splitting induced by Fe and Mn and is consistent with thesmall magnetic moment of P and Si (see Table II). The P /Si pstates contribute to the middle-energy DOS from −0.6t o −0.2 Ry. Fe and Mn 3 dstates give the main contribution to the DOS between −0.2 Ry and the Fermi level (0 Ry). A clear hybridization can be observed between the Fe /Mn 3dstates and P/Sipstates. Phosphorus hybridizes more strongly than Si. More 3 dspin-down states are occupied for Fe than for Mn, in agreement with the larger number of valence electrons forFe. The spin-up channels for Fe and Mn look very similar. Theforbitals do not contribute to the DOS. 214436-3E. K. DELCZEG-CZIRJAK et al. PHYSICAL REVIEW B 90, 214436 (2014) -20-1001020Fe pDOS s p -20-1001020Mn pDOS d f -20-10010P pDOS - 1 btot -20-10010P pDOS - 2 c-20-10010Si pDOS - 1 b -1 -0.8 -0.6 -0.4 -0.2 0 E-EF (Ry)-20-10010Si pDOS - 2 c FIG. 3. (Color online) Site-, spin-, and orbital-projected DOSs (arbitrary units per atom) of ferromagnetic FeMnP 0.75Si0.25.T h e dashed black line stands for the total DOS, the orbital projected s,p,d,a n dfDOSs are represented by black, red, blue, and green solid lines, respectively.-75075150tDOS -20-1001020Fe pDOS -0.50 -0.25 0.00 0.25 E - EF (Ry)-20-1001020Mn pDOS FIG. 4. (Color online) Total and site-projected DOSs (arbitrary units per unit cell) of ferromagnetic FeMnP 0.75Si0.25calculated with RSPT . Solid lines represent the LSDA +DMFT results; dashed lines correspond to the LSDA. Disorder in P and Si is modeled using VCA.Site-projected DOS is computed by performing an integration within the muffin-tin spheres with radii of 2.1 a.u. The Fermi level is set to zero energy. In order to quantify the importance of correlation effects, we performed the DMFT-augmented calculations for the 3 dstates of Fe and Mn ions in FeMnP 0.75Si0.25. The computed DOSs of ferromagnetic FeMnP 0.75Si0.25obtained from LSDA and LSDA +DMFT calculations are depicted in Fig. 4. In general, it is seen that the many-body effects are more significant forthe majority-spin electrons. This is in line with the previousresults [ 51], and the explanation can be found in Ref. [ 59]. In a strong ferromagnet (i.e., a system where the majority-spin TABLE II. Experimental and theoretical site-projected magnetic moments in units of μB. Theoretical magnetic moments are calculated within DFT and DMFT framework. Method Phase MFe MMn MP/MSion 1bM P/MSion 2cM total/f.u. Experiment A 2.1 2.8 4.9 B 2.2 2.0 ≈0 EMTO-LSDA A 1.62 2.81 −0.14/−0.13 −0.10/−0.09 4.30 B 1.33 2.70 −0.15/−0.14 −0.10/−0.09 cAFM 0 2.55 0 /00 /00 RSPT -LSDA A 1.46 2.80 −0.09 −0.05 4.30 B 1.18 2.75 −0.08 −0.05 RSPT -LSDA +DMFT A 1.42 2.78 −0.09 −0.05 4.25 B 1.12 2.64 −0.08 −0.05 214436-4ORIGIN OF THE MAGNETOSTRUCTURAL COUPLING IN . . . PHYSICAL REVIEW B 90, 214436 (2014) band is filled), an excitation of a spin-up electron ubiquitously involves spin-flip transitions, which cost more energy thanelectron-hole excitations in the same spin channel (which ispermitted for the spin-down electrons). As a result of thecorrelation effects, the DOS in the spin-up channel becomesmore spread in energy, and its features become less pronouncedcompared to the LSDA results. The magnetic moments obtained through DFT and DFT+DMFT schemes for collinear ferromagnetic FeMnP 0.75Si0.25in both phases A and B are listed in Table II. The magnetic moments calculated within the DFT +DMFT framework are projected onto MT spheres, and the totalvalue takes into account the interstitial contribution. Thetheoretically obtained magnetic moments for Mn occupyingthe 3gposition are in good agreement with the experimental data for phase A. The discrepancy observed for the other localmoments is discussed later in Sec. III C . We find very small induced magnetic moments on P and Si. They couple antipar-allel to the Fe and Mn moments. The magnetization densitypresented in Fig. 5is taken in different cutting planes of the crystal structure for ferromagnetic FeMnP 0.75Si0.25and shows a strong localization of the magnetization around Fe and Mnatoms. Similar localization was found earlier for Fe 2P[60]. A comparison between DFT and DFT +DMFT results reveals that correlation has no significant effect on the elec-tronic structure and magnetic properties of FeMnP 0.75Si0.25. The magnetic moments and number of electrons ob- tained by the two theoretical approaches are very similar;therefore further analysis of the magnetic exchange cou-plings and noncollinearity is performed within the DFTframework. B. Collinear configurations Two sets of exchange interactions Jijhave been cal- culated in a collinear ferromagnetic configuration at 0 Kfor FeMnP 0.75Si0.25. The first set is generated using the experimental crystal structure for phase A, and the other oneis generated using the experimental crystallographic data forphase B. Both crystal structures contain six magnetic atoms(three Fe and three Mn atoms) per unit cell, which represent sixdifferent sublattices. Fe atoms located at the 3 fpositions will be labeled 1–3, and Mn atoms located at the 3 gpositions will be labeled 4–6. Due to symmetry relations, we can reduce thetotal number of J ijto six independent exchange parameters as follows: J11=J22=J33≡J(Fe−Fe) 1, J12=J13=J23≡J(Fe−Fe) 2, J14=J25=J36≡J(Fe−Mn) 1, J15=J16=J24=J26=J34=J35≡J(Fe−Mn) 2, J44=J55=J66≡J(Mn−Mn) 1, J45=J46=J56≡J(Mn−Mn) 2, where we label the interactions for the groups by Jij. Note that, by definition, Jij=Jji. For each sublattice it is, of course, possible to have the exchange interaction as a function ofthe distance between atoms. This is shown in Fig. 6for both phases A and B of FeMnP 0.75Si0.25calculated in a collinear ferromagnetic configuration. J(Fe−Fe) 1,J(Fe−Mn) 1, and J(Mn−Mn) 1interactions are domi- nated by the nearest-neighbor ferromagnetic exchange, whichdecays for longer range. The other three interactions, J (Fe−Fe) 2, J(Fe−Mn) 2, and J(Mn−Mn) 2, show oscillatory behavior, which is emphasized in the insets of Fig. 6. The first- and third-NN interactions are positive, while the second-NN interaction isnegative. Mixed interactions, J (Fe−Mn) 1and J(Fe−Mn) 2,h a v e the largest values; Jijbetween Mn atoms have intermediate values, while interactions between the Fe atoms are smaller.The differences between the various groups of interactions aresmaller in the case of the Mn- and Si-doped system comparedto pure Fe 2P[46]. The effect of the crystal structure on the magnetic interac- tions of FeMnP 0.75Si0.25is clear. The first-NN interactions forJ(Fe−Fe) 1,J(Fe−Mn) 1, and J(Mn−Mn) 1obtained for phase B are smaller than those for phase A. The rest of theseinteractions are small. This implies that the FM coupling inphase B is weaker than that in phase A. The weakening ofthe ferromagnetic coupling for phase B can be related to theshortening of the Fe-Fe separation. The Fe-Fe distance in phaseB is 4.5% smaller than that in the phase A, allowing for a largeroverlap between the wave functions that belong to differentsites. Similar results have been found for FeMnP 0.7As0.3[27]. The stabilization of the FM magnetic structure is due tothe localization of the Fe moments induced by the largerFe-Fe distance in the ferromagnetic phase. This is, in general, -2 -1 0 1 2 3 4 5012345 MnMn MnMnMny (Å) x (Å) -0.0620.0640.190.320.440.570.690.78 0123450.00.51.01.52.02.53.0MnMnz (Å) x (Å) -0.0650.0730.210.350.490.630.760.85Fe -2 -1 0 1 2 3 4 5 6012345Fe FeFe P/Si 1P/Si 1 P/Si 2 MnMnMn Mn(y,z) (Å) (x,z) (Å) -0.02-0.0040.0090.020.030.050.060.07 FIG. 5. (Color online) Magnetization density of ferromagnetic FeMnP 0.75Si0.25calculated at 0 K. The magnetization density in the left, middle and right panels is taken in the (0 ,0,1), (0,−1,0), and ( −1,−1,2) planes, respectively. The cutting planes are shown in the top right corner of each panel. 214436-5E. K. DELCZEG-CZIRJAK et al. PHYSICAL REVIEW B 90, 214436 (2014) 012J(Fe-Fe)1 phase B phase A 012J(Fe-Fe)2 012J(Fe-Mn)1 012J(Fe-Mn)2 012J(Mn-Mn)1 2468 1 0 distance (Å)012J(Mn-Mn)26 8 10 12-0.0500.05 6 8 10 12-0.0500.05 6 8 10 12-0.0500.05 FIG. 6. Calculated Fe-Fe, Fe-Mn, and Mn-Mn exchange interac- tions (mRy) for phase A (open squares) and phase B (solid circles) of FeMnP 0.75Si0.25as a function of distance ( ˚A). The insets show the oscillations at higher distances. accompanied by a narrowing of the Fe partial DOS, and according to Ref. [ 61], this leads to little overlap between spin- up and spin-down bands, which strengthens ferromagnetismover noncollinear or antiferromagnetic states. The other three types of interactions of phase B have a smaller magnitude, in both the positive and negative regions.J (Fe−Fe) 2is small compared to the other two interactions, i.e., J(Fe−Mn) 2andJ(Mn−Mn) 2.A l s o , J(Fe−Mn) 2andJ(Mn−Mn) 2reach larger absolute values in the negative region for phase B.Besides the weaker FM coupling, a strengthening of the AFMcoupling appears for phase B. Here the change in the magneticbehavior may be related to the large change in the Mn-Mnseparation. The mean Mn-Mn distance is 3.7% smaller forphase B than for phase A, which is expected to lead to anincreased tendency for AFM interactions. A closer analysis of the NN interactions was done using the c/a ratio as the most relevant materials parameter. Different-1012 J(Fe-Fe)1FM-B cAFM-B -1012 J(Fe-Fe)2 -101 J(Fe-Mn)1 -101 J(Fe-Mn)2 0.48 0.52 0.56 0.6 0.64-101 J(Mn-Mn)1 0.48 0.52 0.56 0.6 0.64 0.68 c/a-101 J(Mn-Mn)2 FIG. 7. First-nearest-neighbor interactions of different groups of Jijof phase B as a function of the c/aratio for FM-B (open circles) and cAFM-B (open triangles) configurations. The interactions Jij are sorted into symmetry-related groups as explained in the text. See the main text for the description of the FM-B and cAFM-B configurations. Positive Jijcorresponds to the FM ground state. groups of NN Jijare calculated for different c/aratios while keeping the value of the internal parameters and volumefixed to the experimental values of phase B for two magneticconfigurations: a collinear FM ordering (labeled FM-B) anda simple collinear commensurate ( q z=0.5) AFM structure (labeled cAFM-B). The results are presented in Fig. 7, where positive Jijindicates the stability of the FM state. The NN J(Fe−Fe) 1andJ(Mn−Mn) 1interactions correspond to the closest out-of-plane interactions along the ccrystallographic axis as illustrated in Fig. 2. The NN in-plane J(Fe−Fe) 2andJ(Mn−Mn) 2 interactions are presented in Fig. 1. Due to the crystal structure, the NN J(Fe−Mn) 1andJ(Fe−Mn) 2interactions have both in-plane and out-of-plane contributions. One can see from Fig. 7 that most of the interactions calculated for the collinearferromagnetic case FM-B are positive for the whole range ofc/aratios, implying a FM coupling. Only J (Mn−Mn) 1becomes negative for c/a values higher than that of the experimental value of phase B. However, the situation changes for thesimple collinear commensurate antiferromagnetic cAFM-Bcase. Fe moments then disappear for all values of the c/aratio, similar to what was discussed in Refs. [ 46,62,63]; therefore J (Fe−Fe) 1,J(Fe−Fe) 2,J(Fe−Mn) 1, and J(Fe−Mn) 2become zero. The Mn moments do not disappear, however, and have a value abit smaller (2.5 μ Bat the experimental c/a ratio of phase B) than that of Mn moments for a ferromagnetic configuration ofphase B at the experimental c/aratio (see Table II).J (Mn−Mn) 1 is positive for lower c/a ratios and changes sign at high c/a ratios above and close to the experimental c/avalue of phase 214436-6ORIGIN OF THE MAGNETOSTRUCTURAL COUPLING IN . . . PHYSICAL REVIEW B 90, 214436 (2014) B.J(Mn−Mn) 2is negative in the cAFM-B configuration for the whole c/arange. To conclude this section, Figs. 6and 7show that both crystal structure and magnetic structure critically influencethe exchange interactions between all atoms of this materialand are decisive if finite Fe moments are found in this material. C. Noncollinear configurations The existence of a helical magnetic structure for phase B has been further explored with noncollinear spin-spiral total-energy calculations. Without allowing for spin spirals theseLMTO-based total energies show that at 0 K the ground-statemagnetic structure has a ferromagnetic alignment of the Feand Mn moments for the structural parameters of phase B.The Fe moments are around 1.5 μ Band 2.6 μBfor Mn, in good agreement with the EMTO and RSPT results (see Table II), but with a marked deviation of the reported experimentalmagnetic moment of the noncollinear configuration of phaseB (Table II). We note, however, that the experimental analysis of phase B was not perfect in regard to the size of the magneticmoments, and hence the disagreement between experimentaland theoretical moments for the noncollinear phase B may notto be too alarming. By constraining the magnetic structurein different noncollinear orientations among the momentsin the calculations, we find that there are many magnetic configurations that are close in energy compared to the ground- state ferromagnetic configuration. This indicates that thesystem might form more complicated magnetic structures for aslightly modified composition and/or at elevated temperatures.Since the Fe moments are induced from the local magneticfield from the Mn moments and therefore are very soft, the Femoments become small or vanish when the internal orientationof the Mn atoms form configurations other than ferromagnetic.For instance, the Fe local moment decreases by a factor of 2(from 1.5 μ Bto around 0.8 μB) when the angle between the Mn moments is 180◦and vanishes completely when the Fe and Mn moments are perpendicular to each other. Allowing for a spin spiral propagating along the baxis sim- ulates a noncollinear helical magnetic configuration. However,for any finite wave vector of the spin spiral the total energyis higher compared to ferromagnetic alignment. At the zoneboundary, which corresponds to antiferromagnetic alignment,the total energy is around 4 mRy /f.u. higher than that of the FM structure. D. Magnetodynamical properties The results from all the calculations that we have performed for FeMnP 0.75Si0.25(ferromagnetic, antiferromagnetic, and noncollinear) show overall that the size of the Fe momentdepends on the type of magnetic configuration, similar toFe 2P[62,63] and Si-doped Fe 2P[62]. This becomes rather ex- treme in certain situations, as was pointed out in Refs. [ 62,63], where the Fe1 moment on the 3 fsite can disappear when their coupling to the Fe2 moments on the 3 gsites is purely antiferromagnetic. However, for smaller angles between Feand Mn moments, they maintain the size they have in aferromagnetic configuration, which indicates that a Heisenbergdescription is applicable, at least within some limits. We haveFIG. 8. (Color online) Ground-state helical configuration pre- dicted by ASD simulations for phase B. The Fourier transform ofthe static correlation function is plotted vs the q xvector. investigated how far such a picture can be used to explain the magnetic properties of FeMnP 0.75Si0.25and have therefore combined a Heisenberg Hamiltonian with an approach ofatomistic spin dynamics. We have hence evolved in timethe atomic magnetic moments of both phase A and phase Binitial magnetic configurations with the exchange parametersgiven by the EMTO method at T=16 K, the temperature at which measurements were conducted. The predicted magneticground state is FM for the exchange constants of phase A,while the system evolves into a helical spin structure for theexchange couplings of phase B. We calculated the Fouriertransform of the static correlation function with the aim ofdetermining the qvector of the spin spiral. The ASD method predicts a value for the qvector of (0 .25,0,0) along the propagation direction, as shown in Fig. 8. The angle between the Fe and Mn moments is about 14 ◦, and the moments lie almost exclusively in the ( a,b) plane. The predicted qvector is in rather good agreement with the reported experimentalresults (0 .363,0,0) [29]. Unfortunately, the complete magnetic structure of FeMnP 0.75Si0.25was not resolved in Ref. [ 29], so a more detailed comparison between experiment and theory isnot possible for phase B, e.g., with regard to the size of themagnetic moments. E. Discussion The total energies of the FM-B and cAFM-B configurations calculated using the internal parameters and volume of phase Bof FeMnP 0.75Si0.25are plotted in Fig. 9as a function of the c/a ratio and aaxis. The collinear ferromagnetic configuration FM-B has its total energy minimum at c/a=0.534 and a=6.166 ˚A, values close to the experimental value of phase A given in Table I. The 0.9 mRy energy difference between the energy minima of ferromagnetic phase B and phase A isdue to the difference in internal parameters between phases Aand B. The energy minimum for the cAFM-B configurationis atc/a=0.589 and a=5.963 ˚A, close to the experimental value for phase B (see Table I), indicating that even a simple antiferromagnetic configuration prefers higher c/a ratios. 214436-7E. K. DELCZEG-CZIRJAK et al. PHYSICAL REVIEW B 90, 214436 (2014) FIG. 9. Total energies E(mRy/atom) as a function of the c/a ratio and aaxis for FM-B (open circles) and cAFM-B (open triangles) configurations of phase B of FeMnP 0.75Si0.25. Total energies are expressed with respect to the total energy of phase A (open squares) of FeMnP 0.75Si0.25. The total-energy minimum for the cAFM-B configuration is 4.3 mRy higher than that of the FM-B configuration. However,for higher c/a ratios the cAFM-B configuration becomes lower in energy than the FM-B configuration. The commontangent of the two Evsc/a ratio curves indicates the coexistence of cAFM-B and FM-B phases between c/aratios of approximately 0.58 and 0.63. The magnetic phase changeseems to be accessible, applying a strain of ≈15 GPa to the FM-B phase along the aandbdirections. DOS analysis can give deeper insight into the relative stability of different phases. Figure 10shows DOS for different crystallographic and magnetic phases of FeMnP 0.75Si0.25. The spin-down channel is not significantly affected by the -0.2 -0.1 0 0.1 E-EF (Ry)-15-10-50510152025tot DOS (Ry/states)phase A phase B FM-B cAFM-B FIG. 10. (Color online) Total DOS for phase A (solid red line) and phase B (solid black line) of FeMnP 0.75Si0.25at the experimental c/aratio and for the FM-B (red dashed line) and cAFM (black dashed line) configurations of FeMnP 0.75Si0.25atc/a=0.62. See the main text for the description of the FM-B and cAFM-B configurations.different crystal structures and magnetic configurations of FeMnP 0.75Si0.25. However, there are major changes in the spin-up channel. Phase A possesses the lowest DOS at theFermi level in the spin-up channel N(E F)↑, indicating that this is the stable structure and magnetic configuration at0 K, in agreement with the total-energy results. Increasingthec/a ratio for phase A, one can get phase B in a ferromagnetic configuration, which has a higher N(E F)↑, indicating the relative instability of the structure and magneticconfiguration. As the c/a ratio is further increased up to 0.62 (FM-B), a stronger destabilization is observed. Thecrystal structure with a high c/aratio of 0.62 in the collinear commensurate antiferromagnetic configuration cAFM-B haslowerN(E F)↑than FM-B, indicating that this material prefers an antiparallel alignment for the magnetic moments with ahighc/aratio. IV . CONCLUSION We show that a detailed analysis of the magnetic exchange interactions as a function of structure and magnetic configu-ration of FeMnP 0.75Si0.25in combination with spin-dynamic simulation leads to agreement with experimental results inthat ferromagnetic and noncollinear configurations competeand depend crucially on crystalline structure and also mustpossess a strong temperature dependence. We also find many competing energy minima at drastically different magnetic configurations. We also find, for phase B, the appearance ofa helical spin-spiral magnetic structure at finite temperature.Theqvector of the helical spin-spiral magnetic structure is in satisfactory agreement with the experimental findings. In ourcalculations we find Mn moments that agree with observationsof the ferromagnetic phase, whereas the Fe moment is smallerthan that in experiments. Possibly, this can be explained by themany competing magnetic configurations of this material. Asmentioned, the lack of firm experimental information aboutthe magnetic moments of the noncollinear phase excludesa relevant comparison between experimental and magneticmoments for this phase. We find that the decrease in the Mn-Mn and Fe-Fe shortest interatomic distances leads to a destabilization of the FMcoupling. Out-of-plane Mn-Mn exchange interactions becomenegative with the shortening of the interatomic distancesalong the caxis, implying an AFM coupling for the NN Mn-Mn interactions. The appearance of the helical magneticstructure of FeMnP 0.75Si0.25results from two factors: the Mn-Mn separation plays an important role like in the caseofPnma compounds [ 17,18], and the shortening of the Fe-Fe interatomic distance is also important in the destabilization ofthe FM coupling. We also find that the size of the Fe moment depends criti- cally on the magnetic configuration. In a rather extreme way, itdisappears if it becomes perpendicular to the Mn moments orwhen Mn moments are antiferromagetically coupled to eachother along the zaxes. The sensitivity of the Fe moments in FeMnP 0.75Si0.25is similar to what has been suggested for Fe2P[46,62,63], and direct confirmation by, e.g., neutron scattering experiments would be very interesting. Finally, ourcalculations suggest that different magnetic configurations forFeMnP 0.75Si0.25can be accessed at low temperatures when 214436-8ORIGIN OF THE MAGNETOSTRUCTURAL COUPLING IN . . . PHYSICAL REVIEW B 90, 214436 (2014) strain is applied. A similar magnetic phase transition, e.g., from the ferromagnetic to antiferromagnetic configuration,was observed for Fe 2P under pressure [ 64–66]. ACKNOWLEDGMENTS Valuable discussions with V . H ¨oglin, M. Sahlberg, Y . Andersson, and P. Nordblad are acknowledged. The Euro-pean Research Council, Swedish Research Council, SwedishEnergy Agency, and the Swedish Foundation for InternationalCooperation in Research and Higher Education (STINT) are acknowledged for financial support. O.E. also acknowledgessupport from eSSCENCE, STANDUP, ERC (247062-ASD),and the KAW foundation. L.B. also acknowledges supportfrom the G ¨oran Gustafssons Foundation. The Hungarian Scientific Research Fund (research projects OTKA 84078 and109570) is also acknowledged for financial support (L.V .).Calculations were performed on UPPMAX and NSC-Matterresources. E.K.D.-Cz. acknowledges discussions with A. V .Ruban. Y .O.K. expresses his gratitude to D. Ius ¸an for numerous valuable discussions and assistance with the calculations. [1] V . K. Pecharsky and K. A. Gschneidner, Jr., Phys. Rev. Lett. 78, 4494 (1997 ). [ 2 ]O .T e g u s ,E .B r ¨uck, K. H. J. Buschow, and F. R. de Boer, Nature (London) 415,150 (2002 ) ;O .T e g u s ,E .B r ¨uck, L. Zhang, Dagula, K. H. J. 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PhysRevX.11.021063.pdf

Fibonacci Turbulence Natalia Vladimirova , Michal Shavit , and Gregory Falkovich Weizmann Institute of Science, Rehovot 76100 Israel Brown University, Providence, Rhode Island 02912, USA (Received 27 January 2021; revised 31 March 2021; accepted 29 April 2021; published 24 June 2021) Never is the difference between thermal equilibrium and turbulence so dramatic, as when a quadratic invariant makes the equilibrium statistics exactly Gaussian with independently fluctuating modes. That happens in two very different yet deeply connected classes of systems: incompressible hydrodynamics and resonantly interacting waves. This work presents the first detailed information-theoretic analysis of turbulencein such strongly interacting systems. The analysis involves both energy and entropy and elucidates the fundamental roles of space and time in setting the cascade direction and the changes of the statistics along it. We introduce a beautifully simple yet rich family of discrete models with triplet interactions of neighboringmodes and show that it has quadratic conservation laws defined by the Fibonacci numbers. Depending on how the interaction time changes with the mode number, three types of turbulence were found: single direct cascade, double cascade, and the first-ever case of a single inverse cascade. We describe quantitativelyhow deviation from thermal equilibrium all the way to turbulent cascades makes statistics increasingly non-Gaussian and find the self-similar form of the one-mode probability distribution. We reveal where the information (entropy deficit) is encoded and disentangle the communication channels between modes, asquantified by the mutual information in pairs and the interaction information inside triplets. DOI: 10.1103/PhysRevX.11.021063 Subject Areas: Fluid Dynamics, Nonlinear Dynamics, Statistical Physics I. INTRODUCTION The existence of quadratic invariants and Gaussianity of equilibrium in a strongly interacting system may seem exceptional. Indeed, generic systems have no invariantsexcept Hamiltonian. Strongly interacting systems have nonquadratic Hamiltonians, so that equilibrium Gibbs distribution (the exponent of the Hamiltonian) is generallynon-Gaussian. And yet two very distinct wide classes of physical systems have quadratic invariants and Gaussian statistics at thermal equilibrium. The first class is the familyof hydrodynamic models, starting from the celebrated hydrodynamic Euler equation and including many equa- tions for geophysical, astrophysical, and magnetohydrody-namic flows. The second class, as will be described in this paper, contains systems of resonantly interacting waves. We show that the discretized models of the first classexactly correspond to the second one. We shall considerone particular (arguably the simplest) family of such models and describe far-from equilibrium (turbulent) states of such systems.One calls turbulence a state of any system, where many degrees of freedom are deviated far from thermal equilib-rium. Therefore, studies of turbulence encompass a widevariety of phenomena in nature and industry, from pipeflows to ripples on a puddle. It can be studied from theviewpoint of a mathematician, engineer, or a physicist. Here, we employ the perspective of statistical physics, which is interested in fundamental principles that determinestatistical distributions in turbulence and thermal equilib-rium. We shall use both the traditional viewpoint ofcascades and the relatively recent viewpoint of informationtheory; that is, we address both energy and entropy of turbulence. So far, the statistical physics approach to turbulence was to a large extent devoted to two quitedistinct classes: systems of interacting waves like those onthe surface of the ocean or a puddle and incompressiblevortical flows where no waves are possible. Here, we builda bridge between these two classes and show that discretemodels of a certain kind can describe both. On the one hand, the vorticity, ω¼∇×v,o fa n isentropic flow of incompressible fluid satisfies the Eulerequation: ∂ω=∂t¼∇×ðv×ωÞ. Quite similar are two- dimensional hydrodynamic models, where a scalar field a (vorticity, temperature, potential) is linearly related tothe stream function ψof the velocity carrying the field:∂a=∂t¼−ðv·∇Þa,v¼ð∂ψ=∂y;−∂ψ=∂xÞ,ψðrÞ¼Rdr 0jr−r0jm−2aðr0Þ. For the 2D Euler equation, m¼2.Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article ’s title, journal citation, and DOI.PHYSICAL REVIEW X 11,021063 (2021) 2160-3308 =21=11(2) =021063(13) 021063-1 Published by the American Physical SocietyOther cases include surface geostrophic ( m¼1), rotating shallow fluid, or magnetized plasma ( m¼−2), etc. After Fourier transform, _ak¼X q½k×q/C138q−maqak−q: ð1Þ All such equations have quadratic nonlinearity and quad- ratic invariants. Then, it was suggested [1]to model different cases of fluid turbulence by the chains of ordinary differential equations having quadratic invariant gijuiujand these properties: _ui¼Γi jlujul;Γi il¼0¼gikΓk jlþglkΓk jiþgjkΓk li:ð2Þ On the other hand, consider resonantly interacting waves with the general Hamiltonian, Hw¼X iωijbij2þX ijlðVl;ijb/C3 ib/C3jblþV/C3 l;ijbibjb/C3lÞ;ð3Þ where Vl;ij≠0only if ωiþωj¼ωl. By the gauge trans- formation, ai¼biexpð{ωitÞ, we can turn the equations of motion, {_bi¼∂Hw=∂b/C3 iinto a system of the type [Eqs. (1)and(2)]: {_ai¼X jlðV/C3 i;jlajalþ2Vl;ija/C3 jalÞ: ð4Þ This means that quadratic and cubic parts of the Hamiltonian are conserved separately. If such a system is brought into contact with thermostat, it is straight-forward to show that the statistics is Gaussian: lnPfa ig∝−P iωijaij2. Our interest in resonances is connected to that in non- equilibrium. Thermal equilibrium does not distinguish between resonant and nonresonant interactions becauseof the detailed balance: Whatever correlations can be built over time between resonantly interacting modes, the reverse process destroying these correlations is equallyprobable. This is not true far away from thermal equilib- rium, especially in turbulence. Neglecting nonresonant and accounting only resonant interactions is the standard approach to weakly interacting systems, even though the weak nonlinearity assumptionbreaks for resonant modes. Weak turbulence theory gets around this by considering continuous distribution and integrating over resonances to get the kinetic wave equa-tion, which describes nonlinear evolution that is slowcompared to linear oscillations with wave frequencies [2–5]. There is a tendency in theoretical statistical physics to restrict consideration to two opposite limits: Either treatfew modes or infinitely many. That preference is even stronger in the studies of nonequilibrium. And yet, not only most of the real-world phenomena fall in between theselimits, but, as we show here, one learns some fundamental lessons comparing equilibrium and nonequilibrium states of systems with a finite number of degrees of freedom, where phase coherence can play a prominent role. Condensed matter physics taught us a similar lesson by discovering the world of mesoscopic phenomena, where the system size was made smaller than the phase coherence length. The previous treatment of mode discreteness was focused on the sparseness of resonances for the particular cases when resonant surfaces ω kþωq¼ωjkþqjdid not pass through integer lattice determined by a box [5,6]. Yet in many cases resonance surfaces lay in the lattice. For example, in a quite generic case of quadratic dispersion relation, ωk∝k2, Pythagorean theorem makes the resonance surface for three-wave interactions just perpendicular to any wave vector, so that in any rectangular box resonantly interacting triads fill the lattice of the box eigen modes. The class of model Eqs. (1),(2), and (4)is ideally suited for the comparative analysis of thermal equilibrium and turbulence. We show here that such analysis sheds light on the most fundamental aspects of turbulence, particularly the roles of spatial and temporal scales in determining cascade directions and buildup of intermittency. We consider the particular subclass of models that allow only neighboring interactions, and find it the most versatile tool to date to study turbulence as an ultimate far-from-equilibrium state. We carry here such detailed study of the known types of direct-only and double cascades with unprecedented numeri- cal resolution. Even more important, our models allow for an inverse-only cascade never encountered before. II. FIBONACCI TURBULENCE We consider a subclass of the model Eqs. (1),(2), and (4) which is Hamiltonian with a local interaction: H¼X iViða/C3 ia/C3 iþ1aiþ2þaiaiþ1a/C3 iþ2Þ: ð5Þ The equations of motion {_ai¼∂H=∂a/C3 iare as follows: {_ai¼Vi−2ai−1ai−2þVi−1a/C3 i−1aiþ1þVia/C3 iþ1aiþ2:ð6Þ This family of models (each characterized by Vi) can have numerous classical and quantum applications, since ican be denoting real-space sites, spectral modes, masses of particles, number of monomers in a polymers, etc. The Hamiltonian describes, in particular, decay and coalescence of waves or quantum particles, breakdown and coagulation of particles or polymerization of polymers, etc., when interactions of comparable entities are dominant. In par- ticular, the model describes the resonant interaction of waves whose frequencies are the Fibonacci numbers Fi¼f1;1;2;3;5;…gdefined by the identity FiþFiþ1¼ Fiþ2withF0¼0. Indeed, such waves are described by the HamiltonianVLADIMIROVA, SHAVIT, and FALKOVICH PHYS. REV . X 11,021063 (2021) 021063-2H0¼X i½Fijaij2þViða/C3 ia/C3 iþ1aiþ2þaiaiþ1a/C3 iþ2Þ/C138: ð7Þ The first term corresponds to the linear terms in the equations of motion, while the second term represents the only possible resonant interactions, since no noncon- secutive Fibonacci numbers sum into another Fibonaccinumber (Zeckendorf theorem). For any real t, the Hamiltonian (7)is invariant under the U ð1Þ×Uð1Þtrans- formation a i→aie{Fitdue to FiþFiþ1¼Fiþ2. The trans- formation (to the wave envelopes) reduces the equation of motion _ai¼∂H0=∂a/C3 ito Eq. (6). Ifiare spectral parameters, they are usually understood as shell numbers. That means that one can define wave numbers as k¼Fi¼½ϕi−ð−ϕÞ−i/C138=ffiffiffi 5p , where ϕ¼ ð1þffiffiffi 5p Þ=2is the golden mean. It plays here the role of an intershell ratio, since asymptotically at jij≫1, the wave number depends exponentially on the mode number: Fi∝ϕjij. The model Eq. (6)thus belongs to the class of the so-called shell models [7], that is, model Eq. (2)with neighboring interactions. Coefficients of shell models arechosen to have one or two quadratic integrals of motion. In particular, the Sabra shell model [8,9] for a particular choice of coefficients (not surprisingly, connected by thegolden ratio) coincides with model Eq. (6), which is Hamiltonian and has the cubic integral of motion, model Eq. (5). It is straightforward to show that for arbitrary V i, the dynamical equations (6)conserve a one-parameter family of quadratic invariants (generalizations of the Manley-Rowe invariants for three-wave interactions): F k¼X i¼1Fiþk−1jaij2; ð8Þ where kcould be of either sign if we define negative Fibonacci numbers: F−j¼ð−1Þjþ1Fj. All invariants can be obtained as linear combinations of any two of them. For example, the first two integrals are positive, independent, and in involution: F1¼X i¼1Fijaij2;F2¼X i¼1Fiþ1jaij2: ð9Þ In a closed system, the microcanonical equilibrium is P¼δðH−CÞδðF1−C1ÞδðF2−C2Þ. We now add dis- sipation and white-in-time pumping: _ai¼−{∂H=∂a/C3 iþξi−γiai: ð10Þ Here, hξia/C3 ji¼δijPi=2. It is straightforward to show, also in a general case [Eqs. (3)and (4)], that such forcing on average does not change the cubic Hamiltonian, since hξiaiþ1a/C3 iþ2i¼Pih∂ðaiþ1a/C3 iþ2Þ=∂a/C3 ii¼0for any i. Denoting Hi¼2ReðVia/C3 iai−1ai−2Þ, we then obtainP idhHii=dt¼−P iðγiþγi−1þγi−2ÞhHii, which must be zero in a steady state. At least when all sums γiþγi−1þγi−2are the same,P ihHii¼ hHi¼0(one can probably imagine exotic cases where separatehH ii≠0but we shall not consider them). If pumping and damping are in a detailed balance, so thatP kαkFiþk−1¼γi=Pifor every i, the thermal equilibrium distribution is Gaussian: P¼expð−P kαkFkÞ—is a steady solution of the Fokker-Planck equation: ∂tP¼fP;HgþX i½Pi∂ai∂a/C3 iþγið∂aiaiþ∂a/C3 ia/C3 iÞ/C138P ∝X i/C18 2γi−PiX kαkFiþk/C19 ¼0: That solution realizes maximum entropy for given values of the invariants. The distribution is exactly Gaussian despite the system being described by a cubic Hamiltonian and thus strongly interacting. The only restriction on thenumbers α kis normalization. In particular, when only α1¼ 1=2Tis nonzero, we get the equilibrium equipartition with the occupation numbers ni≡hjaij2i¼Pi=2γi¼T=F i. In a turbulent cascade, the fluxes of the quadratic invariants can be expressed via the third cumulant.Gauge invariance and Zeckendorf theorem ensure that the triple cumulants are nonzero only for consecutive modes in the inertial range: J i≡Imha/C3 iai−1ai−2i; ð11Þ Fiþk−1dhjaij2i dt¼2Fiþk−1ðVi−2Ji−Vi−1Jiþ1−ViJiþ2Þ ¼Πkði−1Þ−ΠkðiÞ¼−∂iΠkðiÞ: ð12Þ The right-hand side is the discrete divergence of the flux ΠkðmÞ≡−Xm iFiþk−1dhjaij2i dt ¼2FmþkVm−1Jmþ1þ2Fmþk−1VmJmþ2 ð13Þ defined up to a constant, independent of m. The third order cumulants are zero in equilibrium, but in turbulence they are nonzero to carry the flux. In the inertialinterval, the flux must be constant and its divergence zero.For our class of models, we are able to find analytically the form of the third cumulant (the analog of Kolmogorov ’s 4=5law for fluid turbulence): J m¼CFM−mþ1=Vm−2; ð14Þ where real constant Cand integer Mcan be of either sign. Let us substitute Eq. (14)into Eq. (13)and show that all the fluxes are nonzero constants independent of m:FIBONACCI TURBULENCE PHYS. REV . X 11,021063 (2021) 021063-3ΠkðmÞ¼2FmþkVm−1CFM−m=Vm−1 þ2Fmþk−1VmCFM−m−1=Vm¼CFMþk−1:ð15Þ The last equality follows from the Cassini identity: FmFnþFm−1Fn−1¼Fmþn−1. All the fluxes have the same sign for any k, that is, all the integrals Fkflow in the same direction for such solutions. We shall show in the nextsection what kind of fine-tuning is needed to get a doublecascade when both cascades carry the same integrals.In Ref. [8], the (quadric) spectral flux of the (cubic) Hamiltonian was also defined, but pumping does notproduce it, so that hHi¼0in a steady turbulent state, as well as in thermal equilibrium. Every model of our family is completely characterized by specifying the dependence of V ioni. While thermal equilibrium does not depend on Viand is universal for the whole family, turbulence depends on Vi, as clear from Eq.(14). In what follows, we shall consider the power-law dependence Vi¼Fα i, which turns into exponential depend- ence Vi≈ϕiαfori≫1. Therefore, the single real param- eterαdetermines the model. Our choice of particular values forαbelow will make the connection between wave and hydrodynamical turbulence through the Fibonacci modelmore explicit. III. CASCADE DIRECTION To get an analytic insight into our turbulence, particu- larly, to understand the flux direction, consider an invariantsubspace of solutions with purely imaginary a k¼iρk for all k: ∂ρi ∂t¼Vi−2ρi−1ρi−2−Vi−1ρi−1ρiþ1−Viρiþ1ρiþ2:ð16Þ In this case, H≡0. The invariant subspace owes its existence to the invariance of Eq. (6)with respect to the symmetry a→−a/C3. Consider the chain running between some integers Mand N, either positive or negative, and assume Vi=Vi−1¼ϕα. Then for ρi¼AϕiβandMþ1<i<N −1 we obtain ∂ρi ∂t¼A2Vi−2ϕ2iβðϕ−3β−ϕα−ϕ2αþ3βÞ: ð17Þ The right-hand side of Eq. (17) turns into zero for β¼−ð1þαÞ=3, which defines a steady solution ρi¼ ϕ−ið1þαÞ=3(also with the replacement ϕ→−1=ϕ). This solution can describe either direct or inverse cascade, sincethe symmetry ρ→−ρ,t→−tmeans that one reverses the flux by changing the sign of ρin this case. Indeed, consider the evolution from the initial state where all amplitudes arezero except the first two ρ M;ρMþ1. The first term in Eq. (16) then will produce ρMþ2of the same sign as VMρMρMþ1,which makes the flux positive, as it should be for a direct cascade. Alternately, by pumping the last two modes, the last term of Eq. (16) produces a negative flux. Which cascade can be realized in reality: direct, inverse, or both? Physically, it is clear that the sign of the flux must be determined by the only parameter α, that is, by how mode interaction depends on the mode number. Indeed,forα¼1=2, the scaling of the flux steady solution coincides with that of the thermal equilibrium: hρ ii¼0, hρiρji¼niδij¼δijT=F i∝ϕ−i, fori≫1. Such a state can be excited, for instance, by an imaginary pumping acting on every mode in detailed balance with dissipation. Physical common sense suggests that the cascade must carry the conserved quantityP iFiρ2 ifrom excess to scarcity [3,10] . Forα>1=2the steady solution ρ2 i¼ϕ−2ð1þαÞi=3decays withifaster than the equipartition ρ2 i∝1=Fi∝ϕ−i, so that it must correspond to a direct cascade. By the same token, we must have an inverse cascade for α<1=2. Of course, such consideration is a plausible argument, not a rigorousproof of the cascade sign. Getting a little ahead of ourselves, mention here that we observe a double-cascade turbulence exactly at α¼1=2. In a general complex case, arguing that the cascade changes direction when αcrosses 1=2is even less straight- forward. The flux constancy determines the third moment, which only bounds the product of the second and fourth moments (the claim that it bounds the square root of theproducts of three second moments made in Ref. [11] is incorrect). Yet a plausible argument can be made as follows. The input rate of F kis equal to Π¼PFpþk−1 where pis the position of the pumping. The input rate must be equal to the dissipation rate Π¼2γdFdþk−1ndfor any choice of γdtaken at the dissipation position d. In order for ndto smoothly match the cascade, one must choose γd comparable to the nonlinear interaction time: γd≃VdJ1=3 d≃ VdðΠ=VdFdÞ1=3. This gives an order-of-magnitude esti- mate nd≃ðΠ=VdFdÞ2=3. Such reasoning can be applied to every i, which in turn gives the estimate for the spectrum of occupation numbers: ni≃ðΠ=ViFiÞ2=3: ð18Þ Since the direction of the flux is toward the occupation numbers that are lower than thermal equilibrium, ni∝F−1 i, then again we see that the flux changes direction when Vi∝F1=2 i. The dimensionless degree of non-Gaussianity on such a spectrum, ξ≡Ji n3=2 i≃Π ViFin3=2 i; ð19Þ must be independent of i. For the spectrum close to equilibrium, ξ∝F3=2 i=ViFi¼F1=2 i=Vi.VLADIMIROVA, SHAVIT, and FALKOVICH PHYS. REV . X 11,021063 (2021) 021063-4Figures 1and2confirm these predictions. We place the pumping at a single mode, i¼p, between two dissipation regions on the ends, letting the system to choose thecascade direction. The system Eq. (10) with pumping and damping has been evolved numerically using LSODE solver [12]. At each step, random Gaussian noise of power Pis applied to the pumping-connected mode injecting fluxΠp¼PFp. Damping with γLandγRis applied to the two leftmost and two rightmost modes, respectively.Forα¼1=2(Vi¼ffiffiffiffiffiFip), the system is weakly distorted from equilibrium, with a constant flux on each side of the pumping. For α≠1=2we find that the invariants are absorbed only on one end of the spectrum. For α>1=2 (Vi¼Fi), we have a thermal equilibrium to the left of pumping and the direct cascade Eq. (18) with a constant ξ to the right. In the opposite case ( α<1=2,Vi¼const), we find an inverse cascade Eq. (18) with constant ξto the left and equilibrium equipartition to the right of pumping. In both cases, the damping on the flux side is carefully selected to avoid buildup in the spectrum (the damping on the equilibrium side can be then set to zero to establish cleaner scaling). We have chosen Vi¼FiandVi¼const because they qualitatively correspond to the Kolmogorov scaling of the direct energy cascade in incompressible turbulence and to the inverse wave action cascade in deep water turbulence, respectively. Thermal equilibrium at the scales exceeding the pump- ing scale together with a direct cascade at smaller scales have been predicted and observed [13]. To the best of our knowledge, nobody has seen before an inverse-only cas- cade together with a thermal equilibrium on the other side of the pumping, neither in hydrodynamic-type systems nor in wave turbulence or shell models. Inverse cascades play a prominent role in geophysics and astrophysics, from creation of planetary jets to Jupiter Great Red Spot and stormy seas. In all known cases inverse cascades appear in systems with at least two conserved quantities that scale differently. All our conserved quantities (8)scale the same in the limit i≫1. Probably closest to our findings are the results of Tom and Ray [14] who observed an inverse cascade in the limiting case of a shell model with two invariants having the same scaling. Their inverse cascade had normal scaling and run from fast to slow modes; the direct cascade was not resolved, but was likely present. Our observation poses the question: can one find another class of systems with a single conservation law and the 0 1 2 3 4 0 10 20 30 40ni FiΠp-2/3 i-1-0.5 0 0.5 1 0 10 20 30 40Π / Πp ip = 5 p = 10 p = 20 p = 30 p = 36 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40| ξ | i 0.05 0.1 0.2 0.5 1 2 1 2 4 10 20 40 i-1, m-i~ 1 / i FIG. 1. Compensated spectra, fluxes, and skewness for α¼1=2with different pumping locations: p¼5, 10, 20, 30, and 36 on the 40 mode interval. Pumping rate is selected to provide the same flux in all cases, Πp¼67.65. In all cases damping rates are γL¼γR¼1. The inset reproduces the longer arm of the cascades in log-log scale. 101102103104105106 0 20 40 60ni Fi i~ Fi1/3 p = 20 of 40 p = 30 of 40 p = 50 of 600.010.1110100 0 20 40 60ni Fi i~ Fi-1/3 p = 20 of 40 p = 10 of 40 p = 10 of 60 00.511.5 0 20 40 60| ξ | i00.10.20.3 0 20 40 60| ξ | i 0 20 40 60-1-0.5 0 Π / Πp 00.51 0 20 40 60Π / Πp FIG. 2. Compensated spectra, fluxes, and the dimensionless skewness for α¼0(left) and α¼1(right) for systems with different location of pumping. In all cases, Πp¼67.65. For α¼0damping rates are γL¼1.5andγR¼0; forα¼1damping rates are γL¼0andγR¼140 at mode 40 and γR¼3500 at mode 60.FIBONACCI TURBULENCE PHYS. REV . X 11,021063 (2021) 021063-5turbulent spectrum less steep than equilibrium. In weak wave turbulence, this requires the sum of the space dimensionality and the scaling exponent of the three-wave interaction to be less than the frequency scaling exponent [3]. We do not know such a physical system, nor are we aware of any fundamental law that forbids its existence. Remark that the connection between the cascade direction, its stability and steepness relative to equipartition, has been firmly established in the weak turbulence theory [3,10] .I n all known examples, the formal turbulent solution with awrong flux sign is not realized; the system chooses instead to stay close to equipartition with a slight deviation that provides for the flux in the right direction [3,15] . Similarly, when we place pumping and damping at the “wrong ”ends of a finite chain, our system heats up, staying close to thermal equilibrium. It is important that our system is a one-dimensional chain, as well as shell models, so that there is no space and consequently no distinction in the phase volume (number of modes) between infrared and ultraviolet parts of the spec- trum. The directions along the chain are only distinguishedtemporally, i.e., in terms of growth or decay of the typical interaction time. The same combination V 2 i=Fi∝ϕ2α−1 determines the idependence of the inverse interaction time both for the equilibrium, Vib1=2 i¼ViF−1=2 iT1=2,a n df o ra cascade, ViðΠ=ViFiÞ1=3¼ðV2 i=FiÞ1=3Π1=3. As the above consideration shows, the cascade proceeds from slow modes to fast modes in Fibonacci turbulence. Similarly, in shell models [11,16,17] (albeit with parameters and conservation laws distinct from our model), a cascade proceeding fromfast modes to slow modes was never observed. It was argued that this is because the fast modes act like thermal noise on the slow ones, which must lead to equilibrium [16]. That this cannot be generally true follows from the existence of the inverse energy cascade in 2D incompressible turbulence and from numerous examples in weak wave turbulence where nonlinear interaction time either grows or decays along thecascade. Moreover, the formation of the cascade spectrum proceeds from fast to slow modes (and not necessarily from pumping to damping), according to the information-theory argument [18]. Why is the flux direction unambiguously related to the cascade acceleration in shell models in general and in our model in particular, in distinction from other cases? The argument can be made by considering capacity, a measure that tells at which end the conserved quantity is stored — perturbations are known to run toward that end [3].F o r example, the power-law energy density spectrum ϵ k∝k−s inddimensions has the total energyR ϵkddk—at which end it diverges is determined by the sign of d−s. This is generally unrelated to the direction of the energy cascade,determined by the sign of s, which tells whether the spectrum is more or less steep than the equipartition. However, in shell models the exponential character of i dependencies makes the total energyP iFijaij2determinedby either the last or the first term of the sum, which solely depends on whether Fijaij2is steeper than equipartition or not, that is, by the sign of the flux. Which direction, then, does the cascade go in the symmetric case Vi¼ffiffiffiffiffiFip? Now the naive cascade solution (18)coincides with thermal equipartition Fini¼const, and the interaction time is independent of the mode numberfor such n i. If we start from thermal equilibrium and apply pumping to some intermediate mode, the system develops cascades in both directions. The left panel of Fig.1shows that the pumping at site pinside the interval ð1;NÞgenerates left and right fluxes in the proportion ΠL=ΠR≃ðN−pÞ=p. This seems natural as in the shorter interval the steeper spectrum falls away from the pumping,which must correspond to a larger flux. This means that if we want to keep the flux constant while increasing por N−p, we need to keep constant the ratio ðN−pÞ=p. We end this section with a general remark. Fibonacci Hamiltonian is not symmetric with respect to reversing the order of modes, it sets the preferred direction, which isphysically meaningful since the frequencies of two lowermodes sum into the frequency of a high one. Yet, as we see in the case V iF−1=2 i ¼const, direct and inverse cascades are pretty symmetric. So, it is natural to conclude that indeed the idependence of ViF−1=2 idetermines which way cascade goes. IV. ALONG THE CASCADES AND AWAY FROM EQUILIBRIUM As we have seen, thermal equilibrium statistics is exactly Gaussian with no correlation between modes, despitestrong interaction (which actually establishes equiparti- tion). The reason for the absence of correlation is appa- rently the detailed balance that cancels them. We do notexpect such cancellations in nonequilibrium states. In allcases of strong turbulence known before, the degree of non-Gaussianity increases along a direct cascade and staysconstant along an inverse cascade [19,20] . As we shall show now, non-Gaussianity always increases along the cascades in our one-dimensional chains. We present first the symmetric case, where the system is close to the equilibrium equipartition with the temperature set by pumping and slowly changing with the mode number: n iFi≈ðPFpÞ2=3fðiÞ. The slow function fðiÞ can be suggested by the analogy with the 2D enstrophy cascade [21,22] asfðiÞ∝ln2=3Fi∝i2=3, counting from the damping region. This gives the dimensionless cumulantξ∝1=i. This hypothesis is supported by the right panel of the Fig. 1, which shows that ξgrows along both cascades by a power law in irather than exponentially. Let us stress that count always starts from the dissipation region, where we have the balance condition Π¼γ dFdþk−1ndand where γd≃VdJ1=3 d≃VdðΠ=VdFdÞ1=3according to the dynamical estimate. This sets the nonlinearity parameter of order unityVLADIMIROVA, SHAVIT, and FALKOVICH PHYS. REV . X 11,021063 (2021) 021063-6at the damping region and decaying toward pumping; the longer the interval, the smaller is ξat any fixed distance from the pumping region. The limit of long intervals maythen be amenable to an analytical treatment. Indeed, Fig. 3 demonstrates that as the interval increases, the highercumulants remain small over longer and longer intervalsstarting from pumping. Despite the model having ultralocalinteractions (every mode participates in only three adjacentinteracting triplets), the cascade formation is very nonlocal.It is somewhat similar to thermal conduction: if we keep theflux but increase the distance, the distribution gets closer tothe thermal equilibrium at every point. Turning to asymmetric (one-cascade) cases, we see the cumulants higher than third growing with F iby a power law instead of logarithmic. Rather than look for scaling inthe mode number i, we find it more natural to use F i (playing the role of frequency); at large ione has Fi≈ϕi, where ϕis the golden mean. Traditional study of turbulence in general and shell models in particular was focused onthe single-mode moments (analog of structure functions), hja ijqi∝F−ζq i, whose anomalous scaling exponents ΔðqÞ¼qζ3=3−ζqgive particular measures of how non- Gaussianity grows along the cascade. For Vi¼Fα i, the flux law gives Ji∝Π=ViFi, that is, ζ3¼αþ1. The anomalous scaling is observable in numerics for the single-cascadecases α¼0andα¼1, as shown in the right panel of Fig. 6. This seems to be the first case of an anomalous scaling in an inverse cascade, with the anomalous dimen-sions having the opposite signs to those in direct cascades.The exponents start fairly small but grow fast with q. The anomalous exponents ΔðqÞcan be related to the statistical Lagrangian conservation laws [23,24] in fluid turbulence; no comparable physical picture was developed for shellmodels. Without physical guiding, the set of the anomalousexponents is not very informative, all the more that theycharacterize only one-mode distribution. Here, we suggest a complementary set of three infor- mation-theoretic measures, which shed a new light on theturbulent statistics emerging along the cascade. The maindistinction of any nonequilibrium state is that it has lowerentropy than the thermal equilibrium at the same energy. Turbulence has the entropy that is much lower, whichmeans that a lot of information is processed to excite the turbulence state. We pose the following question: Where is the information that distinguishes turbulence from equilib-rium encoded? V. WHERE IS THE INFORMATION ENCODED? First, the information is encoded in a single-mode statistics, which is getting more non-Gaussian deeper inthe cascade. This must be reflected in the decay of the one- mode entropy, S i¼SðxiÞ¼Sðjaij=ffiffiffiffinipÞ, with the growth ofji−pj. This can be computed using the multifractal formalism: the moments hxq ii∝F−ζqþqζ2=2 i in the limit of large ji−pjcorrespond to the multifractal distribution, PðxiÞ∝Z gðxi=Fh iÞx−1 iexp½fðhÞlnFi/C138dh; ð20Þ where gis the probability distribution of xion the subset with the scaling exponent hand fðhÞ¼ min qðζq−qζ2=2−qhÞ, that is, fðhÞis the Legendre transform of ζðqÞ. The entropy is then Si¼−Z dxPlnP∝½Δ0ð0Þ−Δð2Þ=2/C138lnFi: This decay is logarithmic in frequency Fi, that is, linear in i, as indeed can be seen in Fig. 6, where iis counted from pumping. Noticing that Δð1Þ≈Δð2Þand assuming quad- ratic dependence for q≤3, we estimate Δ0ð0Þ≈3Δð1Þ=2 and observe that the dashed lines in the right panel ofFig.6with the slopes Δð1Þlnϕby the order of magnitude represent the entropy decay in the inertial interval in both direct and inverse cascades. Second, the information is encoded in the correlations of different modes. It is natural to assume that correlations are strongest for modes in interacting triplets, a i;aiþ1;aiþ2. Disentangling of information encoded can be done by usingstructured groupings [25–27]: X n i¼1SðaiÞ−X ijSðai;ajÞþX ijkSðai;aj;akÞ −X ijklSðai;aj;ak;alÞþ/C1/C1/C1þð −1Þnþ1Sða1;…;anÞ; ð21Þ where Sða1;…;anÞis the entropy of the joint n-mode distribution. For n¼1, this gives the one-mode entropy Si which measures the total amount of information one can obtain by measuring or computing one-mode statistics. While the entropy itself depends on the units or para-metrization, all the quantities (21) forn> 1are 1 1.1 1.2 1.3 1.4 1.5 -30 -20 -10 0 10 20 30| ai|4 / 2ni2 i - p40 modes 60 modes 1 1.1 1.2 1.3 1.4 1.5 -30 -20 -10 0 10 20 30| ai|6 / 6ni4 i - p FIG. 3. Fourth and sixth moments for α¼1=2and center pumping in 40-mode system, with γL¼γR¼3,P¼0.1, and in the 60-mode system with γL¼γR¼30,P¼1.FIBONACCI TURBULENCE PHYS. REV . X 11,021063 (2021) 021063-7independent of units and invariant with respect to simulta- neous reparametrization of every single variable. For n¼2, we have the widely used mutual information, Iij¼SðaiÞþSðajÞ−Sðai;ajÞ; which measures the amount of information one can learn about one mode by measuring another, that is, characterizes the correlation between two modes. It is interesting that all pairs in the triplet have comparable mutual information inthe direct cascade ( V i¼Fi), while Ii;iþ1exceeds notice- ably Ii;iþ2in the inverse cascade ( Vi¼1), see the upper right panel in Fig. 8. One can also define the total (multimode) mutual information as the relative entropy between the true joint distribution and the product distri- bution: Iða1;…;akÞ¼Pk i¼1SðaiÞ−Sða1;…;akÞ.I ti s positive and monotonically decreases upon averaging over any of its arguments. As we see from Fig. 8, the changes along the cascade in one-mode entropy and in two-modeand three-mode mutual information are comparable, that is, one obtains a comparable amount of information about turbulence from these quantities. To see how much more information one gets by measuring or computing the three modes simultaneouslycompared to separately by pairs, one needs to use the measure of the irreducible information encoded in triplets, as given by the third member of the hierarchy (21).I ti s called interaction information in the classical statistics and topological entanglement entropy in the quantum statistics [25,28] : II i¼SðaiÞþSðaiþ1ÞþSðaiþ2ÞþSðai;aiþ1;aiþ2Þ −Sðai;aiþ1Þ−Sðai;aiþ2Þ−Sðaiþ1;aiþ2Þ ¼Ii;iþ1þIi;iþ2þIiþ1;iþ2−Ii;iþ1;iþ2 ¼Iði; iþ1Þ−Iði; iþ1jiþ2Þ: ð22Þ Here, the last term is computed using the probability distribution of the two modes i,iþ1, conditioned on a fixed amplitude of the third mode, iþ1. Interaction information measures the influence of the third variableon the amount of information shared between the other twoand could be of either sign. Positive IIðX;Y;Z Þmeasures the redundancy in the information about Yobtained by measuring XandZseparately, while the negative one measures synergy which is the extra information about Y received by knowing XandZtogether. While we cannot prove it mathematically, it seems physically plausible thatsystems with three-mode interaction must demonstrate synergy. Indeed, one finds a strong synergy for the cascades close to thermal equilibrium at V i¼ffiffiffiffiffiFipas seen in Fig. 7. The two-mode mutual information is much smaller than both the one-mode entropy and the absolute value of the interaction information, which is negative, that is, muchmore information is encoded in three modes than in the pairs separately. Let us stress that both the mutual information and the interaction information are symmetric, that is, they measure the degree of correlation rather than causal relationship orcascade direction. We compute the entropies and mutual information as follows. First, we obtain the probability distribution in 4D space ( x 2 i−2;x2 i−1;x2 i;θiÞand integrate it to get correspond- ing 1D and 2D distributions. Here, θi¼φi−φi−1−φi−2, where φiis the phase of mode i, and xi¼jaij=ffiffiffiffinip, while ni¼h jaij2iis the direct average. Mutual information and information interaction are computed directly from entro-pies, S¼−ΣPlog 2P, obtained for these distributions, since all normalization factors cancel out in subtraction. The entropy for an individual mode, however, is presentedrelative to the Gaussian entropy based on the average occupation number obtained for the binned, staircase distribution for x 2 i. We use the bin sizes Δx2 i¼1for α¼0andα¼1, andΔx2 i¼1=2forα¼1=2. In all cases Δθ¼2π=32. Far from equilibrium, we find synergy for the modes close to the pumping and redundancy for damping, see the last panel of Fig. 8. That means that the interaction information passes through zero in the inertial interval.There even seems to be a tendency to stick to zero in the inertial interval but this requires further studies with the number of modes exceeding our present abilities. (Ourcomputations are done with a record number of modes, upto 80, while previous studies were mostly done for 20 –30. The interaction times decrease exponentially with the mode number, which imposes heavy requirements on the com-putational time step. On top of that one needs very long runs to collect enough statistics to reliably represent the three-mode probability distribution in four-dimensionalspace.) With the present set of data we can suggest thatmost of the information about the three-mode correlation is in the sum of the pair correlations in the triplet. This is more pronounced in the direct cascade than in the inversecascade. Since the requirements on statistics grow expo- nentially with the dimensionality, the suggestion that one can get most of the information (or at least a large part of it)from lower-dimensional probability distributions is greatnews for turbulence measurements and modeling. To put it simply, comparable amounts of information can be brought from one-mode and from three-mode measurements indirect and inverse cascades; most of that information canbe inferred from two-mode measurements. It remains to be seen to what degree this property of small (asymptotically zero?) interaction information is a universal feature ofstrong turbulence. Insets in Figs. 4and5show the probability distribution of the relative phase θ i, which is closely related to the flux (skewness), proportional to hjaiai−1ai−2jsinθii. The prob- ability maxima are then at positive and negative angles forVLADIMIROVA, SHAVIT, and FALKOVICH PHYS. REV . X 11,021063 (2021) 021063-8direct and inverse cascades, respectively. Also, the i dependence of the phase distributions is in accordance with the changes in skewness along i. In the two-cascade symmetric case, the distribution is flat (the phases arerandom) near the pumping, and the phase correlations appear along the cascades, as can be seen comparing the last panel of Fig. 1with the inset in the right panel of Fig. 4. In the one-cascade cases, both skewness and the form of the spectrum are practically independent of the mode number, as seen from Figs. 2and5. The fact that the deviations from Gaussianity grow along our inverse cascade, in distinction from all the inverse cascades known before, calls for reflection. We used tothink about the anomalous scaling and intermittency in spatial terms: Direct cascades proceed inside the force correlation radius, which imposes nonlocality, while ininverse cascades one effectively averages over many small- scale fluctuations, which bring scale invariance [19,20] . The emphasis on the spatial features was reinforced by thesuccess of the Kraichnan ’s model of passive tracer turbu- lence, where it has been shown that the spatial (rather than temporal) structure of the velocity field is responsible for ananomalous scaling and intermittency of the tracer. There is no space in our case, so apparently it is all about time. Indeed, as we have seen, all our cascades propagate fromslow to fast modes, which leads to the buildup of non-Gaussianity and correlations. As a result, the entropy of every mode decreases and the intermode information grows along the cascade. This diminishes the overall entropycompared to the entropy of the same number of modes in thermal equilibrium with the same total energy. Despite qualitative similarity, there is a quantitative differ- ence between our direct and inverse cascades. Figures 5 and6show that the one-mode statistics and its moments faster deviate from Gaussian as one proceeds along the inverse cascade than the direct one. And yet one can see from Figs. 6and7that the one-mode entropy is essentially the same in both cascades, as well as the mutual information between two neighboring modes and the three-mode mutual information. The mutual information between non-neighboring modes I 13is about twice smaller, as seen in10-510-410-310-210-1100 0 4 8 12Probability |ai|2 / nii - p = –24 –14 –4 6 16 26 12345 0 4 8 12Compensated probability |ai|2 / ni0.40.50.6 -1 0 1 θi/ π FIG. 4. Probability (left) and deviation of probability from equilibrium (right) for α¼1=2. Main panels show probabilities of occupation numbers rescaled to their averages, the inset showsthe probability of phase difference, θ i¼φi−φi−1−φi−2. Refer to the first panel for the line color for different modes. Dataare shown for 60-mode system with center pumping andγ L¼γR¼30,P¼1. 10-610-410-2100 0 10 20 30α = 0 p - iProbability | ai|2 / ni44 36 28 20 12 4 10-610-410-2100 0 10 20 30α = 1 i - p | ai|2 / ni4 12 20 28 32 44 1248 -10 0 10 20 30 40 50α = 0| ai|4 / 2 ni2,| ai|6 / 6 ni3 (p + 1) - ip = 30 of 40 p = 50 of 601248 -10 0 10 20 30 40 50α = 1 i - (p + 1)p = 10 of 40 p = 10 of 600.40.50.6 -1 0 1 θi/ π0.40.50.6 -1 0 1 θi/ π FIG. 5. Probabilities (top) and fourth and sixth moments (bottom) for the inverse cascade, α¼0(left), and the direct cascade, α¼1(right). Probabilities for the rescaled occupation numbers are shown in the main panels, while probabilities for thephase difference, θ i¼φi−φi−1−φi−2, are shown in the insets. The variation between PðθiÞfor different iis minor. In all cases Πp¼67.65. For α¼0, the damping rates are γL¼1.5and γR¼0;f o r α¼1the damping rates are γL¼0andγR¼140at i¼40andγR¼3500 ati¼60. In the top panels the dashed lines indicate the Gaussian probability; in the bottom panels thedashed lines show linear fits to the data.-1-0.5 0 0.5 0 4 8 12Δ qα = 0 α = 1 -0.25-0.2-0.15-0.1-0.05 0 0 20 40 60Si iα = 0 α = 1 ~ 0.005 i ~ (-0.005) i-0.03-0.02-0.01 0 0.01 0.02 0.03 0 1 2 3 4 FIG. 6. Left panel: Anomalous exponents computed as ΔðqÞ¼qζ3=3−ζq. Right panel: Decay of entropy down the cascade for the one-mode complex amplitude normalized byffiffiffiffinip. The dashed lines Si−Sp≈−0.005ji−pjhave the slopes equal toΔ1lnðϕÞwithΔ1shown in the left panel. Direct cascade, blue; inverse cascade, red.FIBONACCI TURBULENCE PHYS. REV . X 11,021063 (2021) 021063-9Fig. 8. This difference can probably be related to the dynamics, which in our system is the coalescence of two neighboring modes into the next one and the inverse process of decay of one into two. In the dynamical equation (16), only one (first) term is responsible for the direct process (and the direct cascade), while two terms are responsible for the inverse process (and the inverse cascade). An important distinction between double-cascade and single-cascade turbulence in our system is the dependenceon the system size. The degree of non-Gaussianity of thecomplex amplitudes is fixed in the dissipation regions of the double cascade, so that in the thermodynamic limitthe statistics is Gaussian in the inertial intervals. On the contrary, the statistics of the amplitudes is fixed at the forcing scale for a single cascade, and it deviates more andmore from Gaussianity as one goes along the cascade. We end this section by a short remark on the production balance of the total entropy S¼−hlnρða 1;…;aNÞi. Here, ρða1;…;aNÞis the full N-mode probability distribution function. Since wave interaction does not change the totalentropy, then the entropy absorption by the dissipation must be equal to the entropy production by the pumping [18,29] : PZY idaida/C3 i 2ρ/C12/C12/C12/C12∂ρ ∂ap/C12/C12/C12/C122 ¼2X kγk;: ð23Þ For a single-cascade cases ( Vi¼1andVi¼Fi), the energy balance PFp¼2γFdndmeans that the left-hand side of Eq. (23) must be much larger than the Gaussian estimate P=n p[18]. It may seem to contradict our numeri- cal finding that the pumping-connected mode aphas its one-mode statistics close to Gaussian. Of course, there are nonzero triple correlation and the mutual information withtwo neighboring modes in the direction of the cascade. Yet, since ξ≃1, then the triple moment J p≃n3=2 pboth in direct and inverse cascades, so that the contribution to the left- hand side of Eq. (23) is comparable with P=n p.W e conclude then that even the pumping-connected mode must have strong correlations with many other modes. Since the triple correlation function of nonadjacent modesare zero, such correlations must be encoded in highercumulants. That deserves further study. VI. KOLMOGOROV MULTIPLIERS AND SELF-SIMILARITY An unbounded decrease of entropy along a single cascade prompts one to ask whether the total entropy ofturbulence is extensive (that is, proportional to the numberof modes) or grows slower than linear with the number of modes, so there could be some “area law of turbulence ” (like for the entropy of black holes). This question canbe answered with the help of the so-called Kolmogorov multipliers σ i¼lnjai=ai−1j[30]. Figure 9shows that in our cascades the multipliers have universal statistics inde-pendent of i, similar to shell models [31–34]. One conse- quence of the scale invariance of the statistics of the multipliers is that the entropy of the system is extensive, that is, proportional to the number of modes. Of course, theentropy depends on the representation. From the informa-tion theory viewpoint, the Kolmogorov multipliers realize representation by (almost) independent component, that is, allow for maximal entropy. In other words, computing ormeasuring turbulence in terms of multipliers gives maximal-0.02-0.01 0 0.01 -30 -20 -10 0 10 20 30S, I i - pS3 I23 I12 I31 -0.01-0.005 0 0.005 0.01 -30 -20 -10 0 10 20 30I, II i - pI23 I123 II FIG. 7. Deviation of entropies from equilibrium, mutual in- formation, and interaction information for α¼1=2and center pumping for a set of a 5×107data point. The same values of entropy were obtained for a set of 2×107data point, that is, Siis saturated. Both IandIIshow a slight decrease in absolute values with the increase of the ensemble size from 2×107to5×107. -0.4-0.3-0.2-0.1 0 0 20 40 60S | i - p |α = 0, 80 modes α = 1, 60 modes 0 0.05 0.1 0.15 0 20 40 60I | i - p |α = 0, I23 α = 0, I31 α = 1, I23 α = 1, I31 0 0.1 0.2 0.3 0.4 0 20 40 60I123 | i - p |α = 0 α = 1 -0.6-0.4-0.2 0 0.2 0 20 40 60II / I123 | i - p |α = 0 α = 1 FIG. 8. Deviation of the entropy from equilibrium, the mutual information, and the interaction information (all in bits) for α¼0 andα¼1and center pumping. Number of data points 2×108 forα¼0, 80 modes, 6×107forα¼0, 60 modes, and 108for α¼1. For the bin size selected, all quantities agree with those obtained in a half-reduced dataset.VLADIMIROVA, SHAVIT, and FALKOVICH PHYS. REV . X 11,021063 (2021) 021063-10information per measurement [the absolute maximum is achieved by using the flat distribution, that is, the variableuðσÞdefined by du¼PðσÞdσ]. The amplitudes are expressed via the multipliers: X k¼lnxk¼lnjakjffiffiffiffiffinkp ¼lnxpþXpþk i¼pþ1σiþ1 2lognp nk: The first term is due to the pumping-connected mode, which correlates weakly with σiin the inertial interval. As shown below, the correlation between multipliers decaysfast with the distance between them. That suggests that the statistics of the amplitude logarithm at large kmust have asymptotically a large-deviation form: lnPðX kÞ¼−kHðXk=kÞ: ð24Þ Indeed, the three upper curves in the top row of Fig. 5 collapse in these variables, as shown in the bottom row of Fig. 9. The self-similar distribution of the logarithm of amplitude, Eq. (24), is a dramatic simplification in com- parison with the general multifractal form (20). Technically, it means that gðxk=Fh kÞ¼gðeXk−khlnϕÞis such a sharp function that the integral in Eq. (20) is determined by thesingle Xk-dependent value, hðXkÞ¼Xk=klnϕ. We then identify f¼−H=lnϕ. The self-similarity of the amplitude distribution (plus the independence of the phase distribution on the modenumber) is great news, since it allows one to predict thestatistics of long cascades (at higher Reynolds number) from the study of shorter ones. In our case, Fig. 9shows that 28th mode already has the form close to asymptotic.Self-similarity and finite correlation radius of theKolmogorov multipliers have been also established exper-imentally for Navier-Stokes turbulence [35]. To avoid misunderstanding, let us stress that the self-similarity isfound for the probability distribution of the logarithm of the amplitude, which does not contradict the anomalous scal- ing of the amplitude moments with the exponents ζ q determined by the Legendre transform of forH. If the multipliers were statistically independent, one would compute ln PðXÞ¼−kHðX=kÞorζqproceeding from PðσÞby a standard large-deviation formalism: HðyÞ¼min z½zy−GðzÞ/C138, where GðzÞ¼lnRdσezσPðσÞ. Such derivation would express hjakjqiviaheqσki, which is impossible since the former moments exist for all q, while the latter do not because of the exponential tails of PðσÞ, see also Refs. [35,36] . Therefore, to describe properly the scaling of the amplitudes one needs to study correlations between multi- pliers. Physically, it is quite natural that the law of the distribution change along the cascade must be encoded incorrelations between the steps of the cascade. Indeed, wefind that the neighboring multipliers are dependent, albeitweakly, as expressed in their mutual information (tradi-tionally used pair correlation function [32,33,35] is not a proper measure of correlation for non-Gaussian statistics).We find that for the inverse cascade, Iðσ i;σiþ1Þ≃0.23, IIðσi;σiþ1;σiþ2Þ≃−0.1. For the direct cascade, Iðσi;σiþ1Þ≃ 0.3,IIðσi;σiþ1;σiþ2Þ≃−0.08. No discernible Iðσi;σiþkÞ were found for k> 1. While σiandσiþ2are practically uncorrelated, there is some small synergy in a triplet. To appreciate these numbers, let us present for com- parison the statistics of the Kolmogorov multipliers in thermal equilibrium. Normalized for zero mean and unitvariance, we have PðσÞ¼ZZ ∞ 0dxdye−x−yδ/C18 σ−1 2lnx y/C19 ¼1 2cosh2σ; Pðσi;σiþ1Þ¼8e4σiþ2σiþ1 ½1þe2σið1þe2σiþ1Þ/C1383: ð25Þ That gives Iðσi;σiþ1Þ¼ln2−1=2≈0.19. Figure 9shows that the equilibrium Gaussian statistics of independent amplitudes perfectly represents the statistics of a single multiplier. The joint probability distribution function Pðσi;σiþ1Þare shown in Fig. 10for thermal10-610-410-2100 -8 -4 0 4 8α = 0Probability σip - i = 42 30 6 10-610-410-2100 -8 -4 0 4 8α = 1Probability σii - p = 6 30 42 -0.4-0.20 0 0.05 0.1 0.15α = 0ln (P(X)) / k ln (P(X)) / k X / kp - i = 42 34 26 -0.4-0.20 0 0.05 0.1 0.15α = 1 X / ki - p = 46 38 30 FIG. 9. Top: probability distributions of the Kolmogorov multipliers σi¼lnjai=ai−1jfor different positions in the inverse (left) and direct (right) turbulent cascades. Solid lines correspond to the thermal equilibrium PðσÞ¼1=2cosh2ðσ−¯σÞ, where ¯σ¼ −ð1=3Þlnϕfor the inverse cascade and ¯σ¼−ð2=3Þlnϕfor the direct one. Bottom: probability distributions of X¼lnjakj2 collapse to the large-deviation form far away from the pumping, that is, for large k¼ji−pj.FIBONACCI TURBULENCE PHYS. REV . X 11,021063 (2021) 021063-11equilibrium and for two cascades. Again, the Gaussian statistics represents turbulence remarkably well. Thedifferences between the three cases are most pronouncedaround the peak at the origin, while the distant contours arehardly distinguishable. In plain words, the probabilities ofstrong fluctuations of the multipliers are the same inthermal equilibrium as in turbulence cascades. This is remarkably different from the statistics of the complex amplitudes, which demonstrate most difference betweenthe three cases for strong fluctuations and for highmoments. There seems to be a certain duality betweenfluctuations of the amplitudes and multipliers: strongfluctuations of the multipliers correspond to weakly corre-lated amplitudes, while strong fluctuations of the ampli-tudes may require their strong correlations and thuscorrespond to multipliers close to their mean values. Whether this duality can be exploited for an analytic treatment remains to be seen. The information about theanomalous scaling exponents of the amplitudes in turbu-lence must be encoded in the correlations between multi-pliers. Note that the mutual information Iðσ i;σiþ1Þ for both cascades ( I¼0.23andI¼0.30) is not that much higher than in thermal equilibrium ( I¼0.19bits). Physicists tend to be much excited about any broken symmetry; it is refreshing to notice that relatively little information is needed to encode the broken scale invari-ance in turbulence. How to decode this information fromthe joint statistics of multipliers remains the task for thefuture.VII. DISCUSSION The most surprising finding of our work is the existence of an inverse-only cascade and its anomalous scaling. In all cases known before, an inverse cascade appears only as an outlet for an extra invariant that cannot be transferred along the direct cascade with other invariant(s). In a truly weak turbulence, when the whole statistics is close to Gaussian, an inverse-only cascade is indeed impossible, since it would require an environment that provides rather than extracts entropy, which contradicts the second law of thermodynamics [18,29] . Here, we have shown that an inverse-only cascade is possible in a strong turbulence. As far as an anomalous scaling is concerned, we relate it to the change of the interaction time along the cascade. All the inverse cascades known before run from fast to slow modes and have a normal scaling. In our case, as in all shell models, cascades always proceed from slow to fast modes. Apparently, this is the reason that non-Gaussianity increases along all our cascades, and an anomalous scaling takes place in both single inverse and single direct cascades. Indeed, proceeding from fast to slow modes (in inverse cascades known before) involves an effective averaging over fast degrees of freedom, which diminishes intermittency. On the contrary, our cascades build up intermittency as they proceed. Another unexpected conclusion follows from the entropy production balance in a steady turbulent state: Even though the marginal statistics of the pumping-connected mode (averaged over all other modes) can be close to Gaussian, the correlations of that mode with other modes cannot be weak. Most of the present work is devoted to disentangling of the information encoded in strong turbulence. It is pre- dicted that in weak turbulence most of the information is encoded in the three-mode statistics [18], and Fig. 7 confirms this prediction. Yet in strong turbulence, we find that as much information is encoded in one-mode as in two- mode statistics, while three-mode statistics does not add much. This could be of practical importance for turbulence studies since it is much more difficult to collect, store, and analyze statistics for three-mode and multimode dis- tributions. Another important lesson is that measuring or computing mode amplitudes (or velocity structure func- tions) brings diminishing returns, that is, less and less information, as one goes deep into the cascade. The maximal information is encoded in the statistics of the Kolmogorov multipliers. Most of that information is encoded in the statistics of a single multiplier; less than 10% is encoded in the correlation of neighbors. How to decode it is the task for the future. ACKNOWLEDGMENTS We wish to thank Yotam Shapira for helpful discussions. The work was supported by the Scientific Excellence Center and Ariane de Rothschild Women DoctoralFIG. 10. Joint probability distributions of two neighboring Kolmogorov multipliers shifted to zero means. The contoursare at log 10ðPÞ¼−0.55;−1;−2;−3;−4;−5. Inverse cascade (α¼0) is red, direct cascade ( α¼1) is blue, black is the equilibrium distribution (25).VLADIMIROVA, SHAVIT, and FALKOVICH PHYS. REV . X 11,021063 (2021) 021063-12Program at WIS, Grant No. 662962 of the Simons Foundation, Grant No. 075-15-2019-1893 by the Russian Ministry of Science, Grant No. 873028 of theEU Horizon 2020 programme, and grants of ISF, BSF and Minerva. N. V. was in part supported by NSF Grant No. DMS-1814619. This work used the ExtremeScience and Engineering Discovery Environment(XSEDE), which is supported by NSF Grant No. ACI- 1548562, allocation DMS-140028. [1] A. M. Obukhov, Integral Invariants in Hydrodynamic Systems , Dokl. Akad. Nauk SSSR 184, 2 (1969), http:// www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid= dan&paperid=34380&option_lang=eng . [2] R. 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PhysRevB.100.104419.pdf

PHYSICAL REVIEW B 100, 104419 (2019) Competing cubic and uniaxial anisotropies on the energy barrier distribution of interacting magnetic nanoparticles Marcelo Salvador*and Lucas Nicolao† Departamento de Física, Universidade Federal de Santa Catarina, Campus Universitário, Trindade, 88040-900 Florianópolis, Santa Catarina, Brasil W. Figueiredo‡ Departamento de Física, Universidade Federal de Santa Catarina, Campus Universitário, Trindade, 88040-900 Florianópolis, Santa Catarina, Brasil and Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, 05508-090 São Paulo, São Paulo, Brasil (Received 6 July 2019; revised manuscript received 22 August 2019; published 16 September 2019) We study the magnetic behavior of a two-dimensional set of interacting magnetic nanoparticles. The single- domain nanoparticles exhibit competing cubic and uniaxial anisotropies, and they interact themselves throughlong-range dipolar interactions. We employ the stochastic Landau-Lifshitz-Gilbert equation to describe thetime evolution of the magnetic moments of the system. We determine the magnetic relaxation of the systemas a function of the ratio between cubic and uniaxial anisotropies, and from the strength of dipolar coupling.From the relaxation curves we calculate the effective energy barrier distribution by considering both situationswhere the uniaxial axes are completely aligned or randomly oriented relative to an external magnetic field.When the axes are randomly oriented, two peaks are observed in the distribution of energy barriers dependingon the ratio of cubic and uniaxial anisotropies, as well as on the intensity of dipolar coupling. Through thezero-field-cooled curves we also determine the blocking temperature of the system and we show that it increasesboth with the ratio between the cubic and uniaxial anisotropies, as well as with the magnitude of dipolarinteractions. DOI: 10.1103/PhysRevB.100.104419 I. INTRODUCTION The study of single-domain magnetic nanoparticles started around the middle of the last century with the pioneeringstudies of Stoner and Wohlfarth [ 1], Nèel [ 2], and Brown [3] concerning the magnetization reversal of the individual magnetic nanoparticles across energy barriers due to thermalfluctuations. After the synthesis of the first single-domainmagnetic nanoparticles [ 4–6] the theoretical predictions were finally confirmed. Since then, magnetic nanoparticles havebeen synthesized with specific properties that can be appliedin many different areas of science, ranging from engineeringto medicine [ 7–11]. Single-domain magnetic nanoparticles [12] are formed by hundreds or thousands of individual magnetic moments coupled by exchange interactions. Themagnetic energy of an isolated nanoparticle can generallybe described by uniaxial and cubic anisotropy contributions[13,14]. However, when they are put together, the long-range dipolar coupling appears among them, the strength of whichincreases with the concentration of the system. The roleplayed by these dipolar interactions in the magnetic propertiesof magnetic nanoparticles is still not completely understood.For instance, we find in the literature some conflicting trendsconcerning the effects of dipolar interactions on the blocking *celofsco@gmail.com †lucas.nicolao@gmail.com ‡wagner.figueiredo@ufsc.brtemperature of a system of magnetic nanoparticles [ 15]. In some studies the blocking temperature increases with thestrength of the dipolar interactions [ 16–18], while in other studies it decreases [ 19,20]. In this work we investigate the role played by dipolar interactions in the magnetic properties of a set of interactingmagnetic nanoparticles in a square lattice. The single-domainmagnetic nanoparticles exhibit uniaxial and cubic anisotropiesand the particle sizes are selected from a log-normal distri-bution. By employing the stochastic Landau-Lifshitz-Gilbertequation we determine the time evolution of the magneti- zation of the system, from which we are able to build the zero-field-cooled (ZFC) magnetization curve as a function oftemperature for different values of the ratio between uniaxialand cubic anisotropies, as well as the magnitude of the dipo-lar couplings. We show that the blocking temperature is anincreasing function of the strength of the dipolar coupling andalso of the ratio between the cubic and uniaxial anisotropies.We also determine the thermal magnetic relaxation curveof the system as a function of the time and temperature.From these curves we find the corresponding effective energybarrier distribution, which is determined in the cases bothwhere the uniaxial axes of the anisotropy are parallel, as wellas randomly oriented relative to an external magnetic field. Asin the previous studies of noninteracting systems [ 21] we find two peaks in the energy barrier distribution only when the uni- axial axes are randomly distributed in space. For the uniaxial axes parallel to the magnetic field we find only a single broadpeak. For the case of interacting magnetic nanoparticles our 2469-9950/2019/100(10)/104419(7) 104419-1 ©2019 American Physical SocietySALV ADOR, NICOLAO, AND FIGUEIREDO PHYSICAL REVIEW B 100, 104419 (2019) results are in agreement with recent Monte Carlo simulations [18,22] when both uniaxial and cubic anisotropies are also present in the magnetic energy of the system. The text is organized as follows: in Sec. IIwe present the model and details concerning the calculations based on thestochastic Landau-Lifshitz-Gilbert equation; in Sec. IIIwe present the results for the ZFC curves and determination ofthe blocking temperature as a function of the ratio betweenthe cubic and uniaxial anisotropies, and strength of dipolarcoupling. We also find the energy barrier distribution of thesystem of the magnetic nanoparticles for the cases where the uniaxial axes are parallel and random distributed relative to the field direction; finally, in Sec. IV, we summarize our conclusions. II. MODEL AND SIMULATION We consider a set of N=100 spherical single-domain magnetic nanoparticles with magnetocrystalline anisotropyand magnetostatic energy, for which we assume the followingenergy model: U=−μ 0/vectorH·N/summationdisplay i=1/vectorMi−1 M2sN/summationdisplay i=1Ku,i(/vectorMi·/vectorn1i)2 −1 2M2sN/summationdisplay i=13/summationdisplay j=1Kc,i(/vectorMi·/vectornji)4 −1 2gN/summationdisplay i=1N/summationdisplay j/negationslash=i/bracketleftBigg 3(/vectorMi·/vectoreij)(/vectorMj·/vectoreij)−(/vectorMi·/vectorMj) d3 ij/bracketrightBigg ,(1) where the first term represents the Zeeman contribution to the energy, the second and third terms are related to the uniaxialand cubic anisotropies, respectively, and the last one is themagnetic dipole-dipole energy. /vectorHis the external magnetic field, K u,iand Kc,iare the uniaxial and cubic anisotropy constants of the ith particle, and /vectornj,i(j=1,2,3) are unit vectors parallel to the cubic axes of the ith particle. The uniaxial easy axes are chosen in such a way that they arealways pointing along one of the cubic axes, that is, /vectorn 1,i, and /vectoreijis the unit vector along the line connecting the centers of theith and jth particles. The /vectorMivector is the magnetization at site iwhose magnitude is the saturation magnetization, Ms. The uniaxial and cubic anisotropies are written in the form Ku,i=kuViandKc,i=kcVi, respectively, where Viis the particle volume. The anisotropy density constants, kuandkc, are both assumed to be positive and the volume Viof the ith particle is selected from a log-normal size distribution. Themagnetic nanoparticles are centered on the sites of a squarelattice whose lattice parameter is a. In order to avoid the super- position of nearest-neighbor nanoparticles, we assume a max-imum diameter afor each nanoparticle; that is, they can eventually touch each other. The distance d ijbetween ith and jth particles is measured in units of the lattice parameter a. In order to determine the temporal evolution of the mag- netization vector /vectorMi(i=1,2,..., N) of the set of magnetic nanoparticles, we apply the micromagnetics formalism, in-troduced by Brown [ 23], so that the dynamic of the system is governed by the stochastic Landau-Lifshitz-Gilbert (sLLG)equation. In the Landau formulation [ 24,25], it reads d/vectorM i dt=−γ0 (1+α2)/vectorMi/bracketleftBig /vectorHef,i+/vectorWi+α Ms/vectorMi×(/vectorHef,i+/vectorWi)/bracketrightBig , (2) where α=γ0ηis the dimensionless damping constant with ηbeing the phenomenological damping parameter and γ0= γμ 0=2.2128×105m As, with γandμobeing the gyromag- netic ratio and magnetic permeability, respectively. The effec-tive field is given by /vectorH ef,i=−1 μ0∂U ∂/vectorMi, (3) which yields the following expression: /vectorHef,i=/vectorH+2Ku,i μ0M2s(/vectorMi·/vectorn1,i)/vectorn1,i +2Kc,i μ0M2s3/summationdisplay j=1(/vectorMi·/vectornji)3/vectornji +1 2gN/summationdisplay j/negationslash=i/bracketleftBigg 3(/vectorMj·/vectoreij)−/vectorMj d3 ij/bracketrightBigg . (4) In the micromagnetic approach, the thermal effects are introduced through suitable random fields, which are repre-sented by the /vectorW ivectors. The random field /vectorWiis assumed to be a Gaussian stochastic process with average and correlationsgiven by /angbracketleft/vectorW i,k(t)/angbracketright=0, (5) /angbracketleft/vectorWi,k(t)/vectorWi,j(t/prime)/angbracketright=2Diδkjδ(t−t/prime), (6) where kand jstand for the x,y, and zCartesian components. The properties, given by Eqs. ( 5) and ( 6), are related to the large number of microscopic degrees of freedom with equiva-lent statistical properties [ 24]. The parameter D iprovides the strength of the thermal fluctuations and was chosen in sucha way that the sLLG equation takes the magnetization to theequilibrium, as shown in Ref. [ 26]. It can be written in the following form: D i=αkBT μ0ViMs, (7) where Tis the absolute temperature and kBis Boltzmann’s constant. Because of the vector product occurring in Eq. ( 2) be- tween /vectorMiand/vectorWivectors, the stochastic field enters in a multiplicative way, so that to solve the sLLG equation onemust resort to the Stratonovich interpretation to ensure thephysical meaning of the system [ 24,27]. The stochastic Heun method has been applied to solve the sLLG equation, since themethod provides the solution of the general Langevin equationwith multiplicative noise when interpreted via Stratonovichprescription. For the sake of simplicity, we used all theequations in a dimensionless form. Defining τ=γ 0Mstas the dimensionless time and multiplying all the other fields by 1/Msto get /vectormi=/vectorMi/MS,/vectorhef,i=/vectorHef,i/Ms, and/vectorwi=/vectorWi/Ms, 104419-2COMPETING CUBIC AND UNIAXIAL ANISOTROPIES ON … PHYSICAL REVIEW B 100, 104419 (2019) we arrive at the following expressions for the sLLG equation and the corresponding effective field: d/vectormi dτ=−1 (1+α2)[/vectormi×/vectorhef,i+α/vectormi×/vectormi×/vectorhef,i +/vectormi×/vectorwi+α/vectormi×/vectormi×/vectorwi], (8) /vectorhef,i=/vectorh+2Ku,i μ0M2s(/vectormi·/vectorn1,i)/vectorn1,i +2Kc,i μ0M2s3/summationdisplay j=1(/vectormi·/vectornji)3/vectornji +g 2N/summationdisplay j/negationslash=i/bracketleftBigg 3(/vectormj·/vectoreij)−/vectormj d3 ij/bracketrightBigg , (9) /angbracketleft/vectorwi,k(τ)/angbracketright=0, (10) /angbracketleft/vectorwi,k(τ)/vectorwi,j(τ/prime)/angbracketright=2D/prime iδkjδ(τ−τ/prime), (11) and D/prime i=αkBT μ0ViM2s. (12) For the simulations of the ZFC curves we take a time step /Delta1τ=0.01, and for the simulations concerning the thermal magnetic relaxation we choose /Delta1τ=0.02. After each time step, the magnetization must be explicitly normalized, sincewe used Cartesian coordinates [ 28]. All the simulations are carried out in the limit of high damping, so we have takenα=1.0. III. RESULTS AND DISCUSSIONS In this section we present the results regarding the behavior of the blocking temperature and energy barrier distribution asa function of the uniaxial and cubic anisotropies, as well astheir dependence on the strength of the dipolar interaction.We use reduced variables to represent the physical quantities.The temperature ( T) and the external magnetic field ( H)a r e given by k BT/kuVmandHM s/ku, respectively, where Vmis the mean value of the volume distribution of the nanoparticles.In general, the effective anisotropy density constants, k uand kc, depend on some properties of the nanoparticle, such as its lattice structure, shape and size, the ferromagnetic exchangecoupling between spins inside the nanoparticle, and on-sitecore and surface anisotropies [ 14]. It is also interesting to define K uc=kc/ku, which measures the ratio between the cubic and uniaxial anisotropy densities. Another quantity ofinterest in this work is the ratio between the dipolar cou-pling gand the uniaxial anisotropy. For a spherical magnetic nanoparticle, we define α d=g/kuVm, which can be written as αd=πM2 s/6ku. We have determined the blocking temperature TBof the system from the maximum observed in the ZFC magnetiza-tion curve as a function of temperature. In order to generatethe ZFC curves we always start from an initial configurationwhere the total magnetization of the system is close to zero, atvery low temperatures. We get this by taking each reducedmagnetization vector /vectorm i(i=1,2,..., N), with magnitude equal to 1, and directing it randomly to one of the cubic(a) (b) (c) (d) FIG. 1. Zero-field-cooled curves for (a) noninteracting and (b–d) interacting magnetic nanoparticles, for four different values of the ratio Kuc=kc/kubetween cubic and uniaxial anisotropies, indicated in the figures. We plot the cases (a) αd=0, (b)αd=0.05, (c) αd= 0.1, and (d) αd=0.2. axes of the ith particle, /vectornj,i(j=1,2,3). Next, we apply a small magnetic field HM s/ku=0.1 to the system along a fixed direction in space. Then, increasing the tempera-ture, some magnetic moments become unblocked and a netmagnetization is observed in the direction of the field. Thisnet magnetization increases up to a maximum value, whosetemperature is defined as the blocking temperature T Bof the system. For temperatures larger than TBthe magnetization decreases and the nanoparticle exhibits a superparamagneticbehavior. For each value of temperature we need to take10 4time steps to allow the system to reach a stationary state. After this transient time, we record the component ofthe magnetic moment of each particle along the magneticfield direction over 10 5more time steps. We have considered 50 values of temperature, separated by T=0.01, up to a maximum temperature T=0.50 for each ZFC curve. For the case of noninteracting nanoparticles, the average value of themagnetization is calculated from 2 ×10 3samples, whereas for the interacting case, we used 1 .5×103samples. We show in Fig. 1some curves of the total magnetization as a function of temperature. We plot only the xcomponent of the total magnetization, which is the direction of the smallmagnetic field applied to generate the ZFC curves. In Fig. 1(a) we report the results for four different values of the ratioK uc=kc/kuandαd=0, that is, the case of noninteracting magnetic nanoparticles. Particularly if Kuc=0, we have only the contribution of the uniaxial anisotropy. The temperatureat which we observe the maximum in the ZFC curve de-fines the so-called blocking temperature. We note that theblocking temperature increases with the cubic contributionto the anisotropy. This behavior was also seen previously 104419-3SALV ADOR, NICOLAO, AND FIGUEIREDO PHYSICAL REVIEW B 100, 104419 (2019) FIG. 2. Dependence of the blocking temperature on the ratio between cubic and uniaxial anisotropies, Kuc, and some selected values of the dipolar coupling strength, αd, as indicated in the figure. through Monte Carlo simulations [ 18,21]. Figures 1(b)–1(d) represent the case of interacting magnetic nanoparticles. Ascan be seen the effect of dipolar interactions is to move to theright the peaks of the ZFC curves compared with the case ofnoninteracting nanoparticles, seen in Fig. 1(a).I nF i g . 2,w e show the behavior of the blocking temperature T Bas a function of the ratio between cubic and uniaxial anisotropies, Kuc,f o r different values of the strength of the dipolar coupling. Thisplot summarizes the results already seen in Fig. 1. For each value of the ratio K uc, the blocking temperature increases both with the magnitude of the dipolar coupling, as well as with Kuc for a given value of the dipolar interaction. The same trends were observed earlier in the case of interacting magneticnanoparticles with both cubic and uniaxial symmetries [ 18]. We now turn to the discussion of the role played by the uniaxial and cubic anisotropies, as well as the dipolar couplingon the relaxation processes observed just after an externalsaturating magnetic field applied to the system is turned off.As in the work of Correia et al. [21] we look at two different scenarios: first, where the uniaxial axis of each nanoparticleis initially aligned with the field, and another, where theuniaxial axes are randomly distributed relative to the fieldorientation. After the field is turned off, we start to recordthe magnetization as a function of time for different valuesof temperature. From these measurements we can extractinformation about the energy barrier distribution of the systemof magnetic nanoparticles as a function of the uniaxial andcubic anisotropies, the magnitude of dipolar coupling, andfrom the initial orientation of the easy axes relative to theexternal magnetic field. The basic principles concerning thisrelaxation process were established by Street and Woolley[29] and applied to noninteracting systems by Labarta et al.[30] and by Iglesias and Labarta [ 31] to a one-dimensional chain of spins interacting through dipolar forces. The energy barriers of a system consisting of single- domain magnetic nanoparticles depend on the volume sizedistribution of the nanoparticles and on the correspond-ing magnetic anisotropies. The distribution of energy bar-riers is represented by a function f(E) that is normalized,/integraltext f(E)dE=1. If at the beginning of the relaxation process the magnetization of the system is m 0, it can be shown [30] that it decreases with time following the law m(t)= m0/integraltext Ec(t)f(E)dE, where Ec(t)=kBTln(t/τ0). In this expres- sionτ0is the mean time spent by a nanoparticle to overcome the energy barrier, Tis the absolute temperature, and kBis Boltzmann’s constant. The interesting point is that the scalingfunction E c(t) being a function of the time and temperature allows one to measure the decay of the magnetization at verysmall temperatures as a function of ln( t), which is equivalent to measuring it at a fixed time as a function of temperature, arelatively simpler procedure, when compared with the verylong times required at low temperatures. The validity ofthis scaling relation is restricted to the cases for which thethermal energy k BTis very small compared with the width of the energy barrier distribution function f(E). In this way, from collapsing the measurements of the magnetization asa function of time for each value of temperature, we canobtain a master curve for the magnetization as a functionof the scaling variable E c(t), and from a simple derivative we obtain the energy barrier distribution function, f(E)= −d(m/m0)/d(Ec(t)). Although in this work we assume from the beginning a distribution for the size of the magnetic nanoparticles, thesame conclusions could be drawn from a different point ofview. Prozorov et al. [32] derived a phenomenological model to describe the magnetic relaxation of an assembly of inter-acting magnetic nanoparticles considering that the effectiveenergy barrier depends only on the instantaneous value of thetotal magnetic moment of the system. Therefore, adoptingthe idea of a time-varying energy barrier, they establisheda phenomenological equation that describes the logarithmicmagnetic relaxation of a set of nanoparticles. They appliedwith success this formalism to account for the measurementsof the magnetization decay of three different samples ofFe 2O3. The same formalism was also applied to explain the magnetic relaxation observed in frozen magnetic colloidsdepending on the corresponding volume fraction [ 33]. All the samples in our simulations are prepared so that att=0 they exhibit the maximum magnetization. When the external field is turned off, the magnetization relaxes, andwe compute the magnetization as a function of time for afixed value of temperature. For each value of temperature,we record the evolution of the magnetization during 2 ×10 5 time steps. We used 35 values of temperature, from the initial temperature T=0.05, and employed 1 .5×103independent samples. In Figs. 3(a) and3(b) we show the energy barrier distri- bution f(E) for the cases of uniaxial axes aligned and non- aligned to the direction of the initial magnetic field, respec-tively. We report the cases where the magnetic nanoparticlesare noninteracting and for some selected values of the ratiobetween the cubic and uniaxial anisotropies. 104419-4COMPETING CUBIC AND UNIAXIAL ANISOTROPIES ON … PHYSICAL REVIEW B 100, 104419 (2019) (a) (b) FIG. 3. Energy barrier distribution as a function of the scaling variable Ec(τ)=kBTln(τ/τ 0), for selected values of the ratio Kuc shown in the figures. Uniaxial axes are (a) parallel to the magnetic field and (b) randomly oriented to the direction of the applied field. Figure 3(a) shows the situation where the axes are aligned to the magnetic field and we note the presence ofa single broad peak that moves to the high energies aswe increase the ratio K ucbetween the cubic and uniaxial anisotropies. On the other hand, Fig. 3(b)accounts for the case where the uniaxial axes are randomly distributed in space.Now we have two peaks in the energy barrier distribution: oneat very low energies, and another one at high energies. Again,the peaks move to the high energies, increasing the ratio K uc. However, we note that for values of Kuclarger than 3 only one peak remains. These results based on the time integrationof the sLLG equation agree with those obtained previouslybased on the Monte Carlo simulations [ 21]. When the uniaxial axes of the magnetic nanoparticles are initially aligned withthe field the energy barrier distribution presents only a singlebroad peak. Figures 4(a) and 4(b) show the effects of the dipolar interactions on the behavior of the energy barrier distribu-tion when the uniaxial axes are randomly oriented in space.Figure 4(a) exhibits the component of total magnetization along the field direction as a function of the scaling variableE c(t)=kBTln(τ/τ 0), and its derivative f(E) for noninteract- ing magnetic nanoparticles. Two peaks appear for this particu-lar value of the ratio between cubic and uniaxial anisotropies. For this same value of the ratio ( K uc=2.5), we include in Fig. 4(b) the dipolar interactions, with αd=0.2. While the maximum at low energy is only slightly modified by the pres-ence of dipolar interactions, the second peak at higher energyclearly moves to the right. In this way, the net effect of thedipolar interactions is to increase the mean value of the energybarrier distribution. Finally, we exhibit in Figs. 5(a)–5(d) the behavior of the energy barrier distribution as a function of theratio between the cubic and uniaxial anisotropies and for someFIG. 4. Thermal relaxation of the magnetization as a function of the scaling variable Ec(τ)=kBTln(τ/τ 0) for uniaxial axes ran- domly oriented in space. Magnetization (left scale) and energybarrier distribution (right scale). (a) Free nanoparticles, α d=0, and (b) interacting nanoparticles, αd=0.20. selected values of the dipolar interaction. The uniaxial axes are randomly oriented relative to the direction of the mag-netic field. Except for the case K uc=0, where we have only uniaxial anisotropy, two peaks appear, one at low energies,due to the cubic component of the magnetic anisotropy, andanother broad one at large values of the energy. The peaksmove to higher energy values as a function of both the ratiobetween the cubic and uniaxial anisotropies, and the strengthof the dipolar interactions. These results are in line with ther-mal relaxation measurements performed in magnetoferritinnanoparticles as well as in magnetoferritin doped with Coatoms [ 34,35]. Magnetic relaxation experiments performed at zero field on a 2.5% cobalt-doped magnetoferritin sample inthe range of temperatures between 2 and 140 K have showntwo peaks in the magnetic energy barrier distribution only forthose samples with uniaxial axes randomly oriented in space.On the other hand, samples whose axes were aligned to thefield presented only a single peak in the corresponding energybarrier distribution. 104419-5SALV ADOR, NICOLAO, AND FIGUEIREDO PHYSICAL REVIEW B 100, 104419 (2019) (a) (b) (c) (d) FIG. 5. Energy barrier distribution as a function of the scaling variable Ec(τ)=kBTln(τ/τ 0), for selected values of the ratio Kuc shown in the figures: (a) free nanoparticles, αd=0, (b) αd=0.05, (c)αd=0.20, and (d) αd=0.30. IV . CONCLUSIONS In this work we have studied a system of magnetic nanoparticles through the integration of the stochasticLandau-Lifshitz-Gilbert differential equation. The magnetic nanoparticles, disposed in a two-dimensional array, interactvia long-range magnetic dipolar interactions and present bothcubic and uniaxial anisotropies. We have shown that theblocking temperature, which is determined from the maxi-mum of the ZFC curves, increases both with the ratio be-tween the cubic and uniaxial anisotropies, as well as withthe strength of the dipolar interactions. We have also inves-tigated the thermal behavior of the magnetization during therelaxation process from a saturated state. We have seen thatthe energy barrier distribution, which is obtained from thederivative of the master equation relating the magnetizationto the scaling variable Tln(t), presents a single peak when the uniaxial anisotropy axes are parallel to the magnetic field,and two peaks, one at low and the other at high energy, whenthe uniaxial anisotropy axes are randomly oriented in space.These energies increase with the ratio between the cubic anduniaxial anisotropies, and for large values of this ratio, onlyone peak remains in the energy barrier distribution. The maineffect of the dipolar coupling is to move the broad peakobserved at high energies towards still higher values of theenergy. ACKNOWLEDGMENTS M.S. was financed in part by the Coordenação de Aper- feiçoamento de Pessoal de Nível Superior - Brasil (CAPES)- Finance Code 001. 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PhysRevB.93.224421.pdf

PHYSICAL REVIEW B 93, 224421 (2016) Transformation of spin current by antiferromagnetic insulators Roman Khymyn,1,*Ivan Lisenkov,1,2Vasil S. Tiberkevich,1Andrei N. Slavin,1and Boris A. Ivanov3,4 1Department of Physics, Oakland University, Rochester, Michigan 48309, USA 2Institute of Radio-engineering and Electronics of RAS, Moscow 125009, Russia 3Institute of Magnetism, NASU and MESYSU, Kiev 03142, Ukraine 4Taras Shevchenko National University of Kiev, 01601 Kiev, Ukraine (Received 29 November 2015; revised manuscript received 19 May 2016; published 22 June 2016) It is demonstrated theoretically that a thin layer of an anisotropic antiferromagnetic (AFM) insulator can effectively conduct spin current through the excitation of a pair of evanescent AFM spin wave modes. The spincurrent flowing through the AFM is not conserved due to the interaction between the excited AFM modes andthe AFM lattice and, depending on the excitation conditions, can be either attenuated or enhanced. When thephase difference between the excited evanescent modes is close to π/2, there is an optimum AFM thickness for which the output spin current reaches a maximum, which can significantly exceed the magnitude of the inputspin current. The spin current transfer through the AFM depends on the ambient temperature and increasessubstantially when temperature approaches the N ´eel temperature of the AFM layer. DOI: 10.1103/PhysRevB.93.224421 I. INTRODUCTION Progress in modern spintronics critically depends on finding novel media that can serve as effective conduits of spin angularmomentum over large distances with minimum losses [ 1–3]. The mechanism of spin transfer is reasonably well understoodin ferromagnetic (FM) metals [ 4,5] and insulators [ 3,4,6–9], but there are very few theoretical papers describing spin currentin antiferromagnets (AFMs) (see, e.g., [ 10]). Recent experiments [ 11–13] have demonstrated that a thin layer of a dielectric AFM (NiO, CoO) could effectivelyconduct spin current. The transfer of spin current was studiedin the FM/AFM/Pt trilayer structure (see Fig. 1). The FM layer driven in ferromagnetic resonance (FMR) excited spin currentin a thin layer of AFM, which was detected in the adjacentPt film using the inverse spin Hall effect (ISHE). It was alsofound in [ 13] that the spin current through the AFM depends on the ambient temperature and goes through a maximum nearthe N ´eel temperature T N. The most intriguing feature of the experiments was the fact that for a certain optimum thicknessof the AFM layer ( ∼5 nm) the detected spin current had a maximum [ 11,12], which could be even higher than in the absence of the AFM spacer [ 12]. The spin current transfer in the reversed geometry, when the spin current flows from thePt layer driven by dc current through the AFM spacer intoa microwave-driven FM material, has been reported recentlyin [14]. The experiments [ 11–14] posed a fundamental question of the mechanism of the apparently rather effective spin currenttransfer through an AFM dielectric. A possible mechanism ofthe spin transfer through an easy-axis AFM has been recently proposed in [ 10]. However, this uniaxial model cannot explain the nonmonotonic dependence of the transmitted spin currenton the AFM layer thickness and the apparent “amplification”of the spin current seen in the experiments [ 11,12] performed with the biaxial NiO AFM layer [ 15]. *khiminr@gmail.comIn our current work, we propose a possible mechanism of spin current transfer through anisotropic AFM dielectrics,which may explain all the peculiarities of the experiments[11,12,14]. Namely, we show that the spin current can be effectively carried by the driven evanescent spin wave excitations, having frequencies that are much lower than thefrequency of the AFM resonance. We demonstrate that theangular momentum exchange between the spin subsystem andthe AFM lattice plays a crucial role in the process of spincurrent transfer, and may lead to the enhancement of the spin current by the angular momentum influx from the crystal latticeof the AFM. II. SPIN DYNAMICS IN THE AFM DIELECTRICS We consider a model of a simple AFM having two magnetic sublattices with the partial saturation magnetization Ms.T h e distribution of the magnetizations of each sublattice can bedescribed by the vectors M 1andM2,|M1|=|M2|=Ms. We use a conventional approach for describing the AFMdynamics by introducing the vectors of antiferromagnetism(l) and magnetism ( m)[16–19]: l=(M 1−M2)/(2Ms),m=(M1+M2)/(2Ms). (1) Assuming that all the magnetic fields are smaller then the exchange field Hexand neglecting the bias magnetic field, which is used to saturate the FM layer, the effective AFMLagrangian can be written as [ 16,18,19] L=μ[(∂l/∂t) 2−c2(∂l/∂y)2]−Wa−Wsδ(y). (2) Hereμ=Ms/(γ2Hex),γis the gyromagnetic ratio, cis the speed of the AFM spin waves ( c/similarequal33 km/s in NiO), and Wa= Msl·(ˆHa·l) is the energy of the anisotropy, defined by the matrix of the anisotropy fields ˆHa=diag(Ha 1,Ha 2,0) with the diagonal ( j,j) components Ha j=2Msβj(βjis the anisotropy constant along the jth axis). The equilibrium direction of the AFM vector l0=e3lies along the e3axis. The exchange coupling between the FM and AFM lay- ers is modeled in Eq. ( 2) by the surface energy term Ws=Es[(mFM·m)+α(mFM·l)], where Esis the surface 2469-9950/2016/93(22)/224421(7) 224421-1 ©2016 American Physical SocietyKHYMYN, LISENKOV , TIBERKEVICH, SLA VIN, AND IV ANOV PHYSICAL REVIEW B 93, 224421 (2016) FIG. 1. Sketch of the model of spin current transfer through an AFM insulator based on the experiment [ 11]. The FM layer excites spin wave excitations in the AFM layer. The output spin current (at the AFM/Pt interface) is detected by the Pt layer through the inverse spin Hall effect (ISHE). energy density and mFMis the unit vector of FM layer magnetization. We assumed that the net magnetization ofthe AFM at the FM/AFM interface layer could be partiallynoncompensated, and this “noncompensation” is characterizedby a dimensionless parameter α(0<α< 1); see also the Appendix. The dynamical equation for the AFM vector lfollows from the Lagrangian Eq. ( 2) and can be written as ∂ 2l/∂t2+/Delta1ω ∂ l/∂t−c2∂2l/∂y2+ˆ/Omega1·l=f(t)δ(y),(3) where /Delta1ωis the phenomenological damping parameter equal to the AFM resonance linewidth ( /Delta1ω/ 2π≈69 GHz for NiO [20]). Note that the damping-related decay length λG= 2c//Delta1ω ≈150 nm is much larger than the typical AFM thickness. Therefore, below we shall neglect damping exceptin Fig. 4, where the comparison of AFM spin currents in conservative and damped cases is presented. The matrix ˆ/Omega1=diag(ω 2 1,ω2 2,0),andωj=γ√ HexHa j,j=1,2, are the frequencies of the AFM resonance. In the case of NiO thetwo AFM resonance frequencies are substantially different:ω 1/2π/similarequal240 GHz and ω2/2π/similarequal1.1T H z[ 15]. We shall show below that the difference between the AFM resonancefrequencies is crucially important for the spin current transferthrough the AFM. The driving force in Eq. ( 3)f(t)=− (δW s/δl)/(2μ), local- ized at the FM/AFM interface, describes AFM excitation bythe precessing FM magnetization. In the absence of this termEq. ( 3) describes two branches of the eigenexcitations of the AFM with dispersion relations ω j(k)=√ ω2 j+c2k2. These propagating AFM spin waves have minimum frequenciesω jwhich are much higher than the excitation frequency (9.65 GHz in Ref. [ 11]) and, therefore, cannot be responsible for the spin current transfer. The presence of the FM layer, however, qualitatively changes the situation, as the driving force f(t) excites evanes- cent AFM spin wave modes at the frequency of the FM layer resonance (FMR), which is well below any of the AFMRfrequencies ω j. The profiles of the evanescent AFM modes can be easily found from Eq. ( 3): lj(t,y)=ej[Aje−y/λj+Bjey/λj]e−iωt+c.c., j=1,2, (4)where ωis the excitation frequency, λj=c/slashbig/radicalBig ω2 j−ω2 (5) is the penetration depth for the jth evanescent mode, and complex coefficients Aj,Bjare determined by the boundary conditions at the FM/AFM and AFM/Pt interfaces. Theinterfacial driving force f(t)δ(y) excites the AFM vector l(t,y=0) at the FM/AFM interface: l(t,y=0)=e 3+[(a1e1+a2e2)e−iωt+c.c.]. (6) The complex amplitudes a1anda2depend on the vector structure of the magnetization precession in the FM layer (seeAppendix for details), which opens a way to experimentallycontrol the input spin current in the AFM, and to directlyverify our theoretical predictions. Thus, if the FM layer ismagnetized along one of the AFM anisotropy axes e 1,2,t h e microwave magnetization component along that axis will be zero and the corresponding complex amplitude a1,2in Eq. ( 6) will vanish. On the other hand, if the FM layer is magnetizedalong the AFM equilibrium axis e 3, both amplitudes a1and a2will be nonzero with the phase shift φ=arg(a1/a2)≈π/2 between them. III. SPIN CURRENT THROUGH THE AFM LAYER At the AFM/Pt interface ( y=d) we adopt a simple form of the boundary conditions that were used previously for thedescription of spin current at the AFM/Pt [ 21] and FM/Pt [ 22] interfaces: P(y=d)=βcL (y=d), (7) where Pis the current of the e 3component of the spin angular momentum and Lis the corresponding angular momentum density inside the AFM: P=2μc2e3·[∂l/∂y×l],L=− 2Msγ−1e3·m,(8) andβis a dimensionless constant having magnitude in the range from 0 to 1 and being physically determined by thespin mixing conductance at the AFM/Pt interface [ 22]. The caseβ=0 corresponds to the conservative situation of a complete absence of the angular momentum flux, while thecaseβ=1 describes a “transparent” boundary, when the angular momentum freely moves across the AFM/Pt boundarywithout any reflection. Using Eqs. ( 8), the boundary conditions Eq. ( 7) can be rewritten as explicit conditions on the vector of antiferromag-netism lasβ∂l/∂t=−c∂l/∂y. This equation and Eq. ( 6) allow one to find all four coefficients A j,Bjin Eq. ( 4), and one can find the explicit expression for the spin current P(y) inside the AFM layer: P(y)=4μc2|a1a2|Re[Q(y)e−iφ], (9) where Q(y)=(e−y/λ 1+q1ey/λ 1)(e−y/λ 2−q∗ 2ey/λ 2) (1+q1)(1+q∗ 2)λ2 −(e−y/λ 1−q1ey/λ 1)(e−y/λ 2+q∗ 2ey/λ 2) (1+q1)(1+q∗ 2)λ1.(10) 224421-2TRANSFORMATION OF SPIN CURRENT BY . . . PHYSICAL REVIEW B 93, 224421 (2016) Hereqj=e2iψj−2d/λjandψj=arctan( βωλ j/c)≈βω/ω j. Equation ( 9) is the central result of this paper that allows one to find the spin current carried by the evanescent spin wavemodes in an AFM layer. Now we shall analyze the main features of the spin current transfer through an AFM dielectric that are described byEq. ( 9). First, one can see that the spin current Pis proportional to the product |a 1a2|of the amplitudes of both excited evanescent spin wave modes, and this current is completelyabsent if only one of the modes is excited. This is explainedby the fact that each of the modes Eq. ( 4) is linearly polarized, and, therefore, cannot alone carry any angular momentum. Second, the spin current in the AFM layer depends on the position yinside the AFM layer; i.e., it is not conserved .T h i s is a direct consequence of the assumed biaxial anisotropy ofthe AFM material, which allows for the transfer of the angularmomentum between the spin subsystem and the crystal latticeof the AFM layer. This effect is a magnetic analog of the opticaleffect of birefringence [ 23], where the spin angular momentum of light is dynamically changed during its propagation in abirefringent medium. In the case of a uniaxial anisotropy [ 10](λ 1=λ2=λ) Eq. ( 9) can be simplified to P=16μc2 λIm(q) |1+q|2|a1a2|sinφ, (11) and the spin current is conserved across the whole AFM layer. Equation ( 9) can also be simplified in the case of a semi- infinite AFM layer, in which case Bj=0 andq1=q2=0: P=4μc2(λ1−λ2) λ1λ2|a1a2|cosφe−y/λ eff. (12) In such a case the spin current decays monotonically inside the AFM layer with the effective penetration depth λeff= λ1λ2/(λ1+λ2)/similarequal5 nm for NiO. Another peculiarity of Eq. ( 9) and Eq. ( 11) is that the spin current Pdepends on the phase shift φbetween the two excited evanescent AFM spin wave modes l1andl2: P∝cos[φ−/Phi1(y)], (13) where /Phi1(y)=arg[Q(y)]. The maximum spin current at a given position yinside the AFM layer is achieved at φ=/Phi1(y). Since the AFM phase shift /Phi1(y), in general, depends on the position yinside the AFM layer, for any particular thickness d of the AFM layer it is possible to choose the excitation phaseshiftφthat would maximize the output spin current P(y=d), while the input spin current P(y→0) could be quite low. In such a case the additional angular momentum is taken fromthe crystal lattice of the AFM. This shows that, in principle,the AFM dielectrics can serve as “amplifiers” of a spin current. Figure 2shows the spatial profiles of the spin current density in a relatively thick AFM layer (thickness d=20 nm). This dependence is drastically different for different phase shiftsφbetween the excited evanescent spin wave modes. While forφ<π / 2 the spin current exponentially and monotonically decays inside the AFM layer (dashed blue line in Fig. 2), for φ>π / 2 (solid black line in Fig. 2) it initially increases at relatively small ydue to the angular momentum flow from the AFM crystal lattice to its spin subsystem. At larger values ofFIG. 2. Spatial distribution of the spin current P(y) inside the AFM layer for different phase shifts φbetween the two evanescent AFM spin wave modes calculated from Eq. ( 9). y, the spin current decays exponentially due to the decay of the excited evanescent spin wave modes. Figure 3demonstrates the dependencies of the spin current on the phase shift φat both interfaces FM/AFM (input spin current) and AFM/Pt (output spin current). It is clear fromFig. 3that the output spin current is shifted by ∼π/2 relative to the input spin current, and, for the phase shift φ≈π/2, the output spin current could have a maximum magnitude when theinput spin current is almost completely absent. This means thatat such a value of the phase shift between the evanescent spinwave modes practically all the output spin current is generatedas a result of interaction between the magnetic subsystem ofthe AFM layer and its crystal lattice. Thus, the AFM layer actsas a source of the spin current. On the other hand, at the phase shift of φ≈0o rφ≈π, the situation is opposite, as the input spin current is practically lost inside the AFM, and the AFMlayer acts as a spin current sink. Thus, we showed that a thin layer of AFM, driven by a constant flow of microwave energy from the FM layer, is ableto transform the angular momentum of a crystal lattice intothe spin current and vice versa. The described transfer of theangular momentum from the lattice to the spin system has asimple analog not only as a birefringence in optics, but also inmechanics: a mechanical oscillator which consists of a mass FIG. 3. Dependence of the input (dashed blue line) and output (solid red line) spin currents through the AFM layer on the phaseshiftφbetween the modes. 224421-3KHYMYN, LISENKOV , TIBERKEVICH, SLA VIN, AND IV ANOV PHYSICAL REVIEW B 93, 224421 (2016) FIG. 4. Spin current transfer factor of the AFM layer as a function of the AFM layer thickness for different values of the spin mixing conductance parameter β. The lines show the case of zero damping (/Delta1ω=0), while the circles correspond to the AFMR linewidth of /Delta1ω/ 2π=69 GHz [ 20]. suspended on two perpendicular springs with different stiffness attached to a fixed rectangular frame. The displacement of themass from its equilibrium position in the frame center along thedirection of one of the orthogonal springs results in the linearlypolarized oscillations along this direction, without any transferof the angular momentum from the frame to the oscillatingmass. In contrast, the linear displacement of the mass in adiagonal direction results in the rotation of the mass around its equilibrium position, and the angular momentum necessaryfor this rotation is taken from the frame (see animations inSupplemental Material [ 27]). The ratio of the output spin current to the input one (the spin current transfer factor) is shown in Fig. 4for different values of the constant β, i.e., for the different values of the spin mixing conductance at the AFM/Pt interface. This dependence has asharp maximum at the thickness of a few nanometers, wherethe input current is rather low, and the AFM layer acts as asource of a spin current. With the further increase of the AFMlayer thickness the transfer ratio is exponentially decreasing,while the position of the maximum shifts to the right withthe increase of the spin mixing conductance at the AFM/Ptinterface. As one can see from Fig. 4, the presence of the damping has little influence on the spin current, because thespatial decay of the amplitudes due to the evanescent characterof the modes l 1,2is dominant. IV . ENERGY EFFICIENCY OF THE SPIN TRANSFER It is obvious that, besides the spin transfer through the AFM dielectric, there is also a flux of energy through the AFM layer. This flux of energy /Pi1can be found from the Lagrangian Eq. ( 2) by applying the Noether theorem, and has the following form: /Pi1=8μc3ω2 β/summationdisplay i=1,2a2 i/parenleftbig c2/β2−ω2λ2 i/parenrightbig (1+cosh 2 d/λi).(14) As one can see from Eq. ( 14), the flux of energy does not depend on either the spatial coordinate yinside the AFM and the phase shift φbetween the excited evanescent AFM modes.Therefore, the FM layer is a source and the Pt layer is a receiver of the energy coming from the FM layer, and this flux of energyis not transformed inside the AFM layer (besides negligibleGilbert damping; see Fig. 4). At the same time, the situation with the spin current is quite different. Due to the anisotropy of the AFM layer the angularmomentum is not conserved inside the magnetic subsystem ofthe AFM layer, and, therefore, there appears a flux of angularmomentum between the spin subsystem and the lattice ofthe AFM. Therefore, as was discussed above, at certain parameters of the spin dynamics in the AFM it is possible to create a flux ofangular momentum from the lattice into the spin subsystem.In this case it is possible to get the output spin current that islarger than the input one, but, obviously, the flux of energy atthe output will never be larger than at the input. The efficiency of the spin transfer through the AFM layer can be characterized by the ratio of the spin current at theoutput of the AFM layer to the energy losses of the FM layer.Thus, we can introduce the value S eff=ωP|y=d//Pi1, which is defined as the ratio of the transferred angular momentum tothe energy flux, and, therefore, can be interpreted as “effectivespin” of the spin transfer. As one can see, the energy flux is the sum of the energies of both evanescent AFM modes, and has the form /Pi1= A 1a2 1+A2a2 2, while the spin current depends on the product of the modes’ amplitudes P=C|a1a2|. Thus, the effective spin has a maximum, and the value of this maximum isS eff=C/2√A1A2. Maximizing this value with respect to the phase shift φbetween the excited evanescent AFM modes one can obtain the maximum efficiency of the spin transferS max eff=1, which is the same as for propagating spin waves in ferromagnetic materials. Physically, the difference in the behavior of the energy and the spin flux originates from the symmetries of the Lagrangian(2). The flux of energy is defined by the infinitesimal shifts ofthe Lagrangian in time, which are symmetric in the absence of damping, resulting in the energy conservation. In contrast,the flux of the angular momentum is determined by theinfinitesimal rotations of the Lagrangian (2). Obviously, the operation of rotation does not transform the system to itselfin the case of a biaxial anisotropy, and, therefore, the angularmomentum in the spin system of an anisotropic AFM is notconserved. V . ISHE VOLTAGE IN PT LAYER Using Eq. ( 9), we estimated the ISHE voltage for an FM/AFM/Pt structure (and, in particular, for the NiFe/ NiO/Pt structure) with the AFM layer having thickness d, which is smaller than the penetration depths λjof both evanescent AFM modes. To find the amplitudes a1anda2of the evanescent modes, we adopt a simple model, where we assume that theprecession of the magnetization in the FM layer excites spindynamics of the AFM; see the Appendix. Following [ 24], the ISHE voltage can be written as V ISHE=ρ/Theta1SHw/parenleftbigg2e /planckover2pi1/parenrightbiggλPt dPttanh/parenleftbiggdPt 2λPt/parenrightbigg Pd, (15) 224421-4TRANSFORMATION OF SPIN CURRENT BY . . . PHYSICAL REVIEW B 93, 224421 (2016) where ρis the resistivity of the Pt, w=5 mm is the distance between the probe electrodes attached to the Pt layer, dPt= 10 nm is the thickness of the Pt layer, /Theta1SH=0.05 is the spin Hall angle in Pt, λPt=7.7 nm is the spin diffusion length in Pt,eis the electron charge, and /planckover2pi1is the Planck constant. It is important to note that the value of the interface exchange integral Jsand, correspondingly, the value of the surface energy density Esstrongly depend on the method of fabrication of the sample, and, therefore, can be measured onlyin an experiment performed for a particular sample. To givea reasonable numerical example, below we take the value ofE sto beEs=3.3×10−3J/m2, which was measured for the NiFe/NiO interface in [ 25]. For the given parameters, taking the angle of magnetization precession in the FM layer, sin θ=0.01 [see Eq. ( A5)], we obtain VISHE=40 mV for the uncompensated AFM boundary (α=1) and VISHE=4n Vf o rt h e compensated one (α=0). The first value is close to the ISHE voltage measured inRef. [ 11]. A partial interfacial magnetization of the antifer- romagnetic NiO in that case was confirmed by the XRDscan performed in [ 11]. The calculated ISHE voltage for the compensated AFM is closer to the experimental value obtainedin Ref. [ 12]. The reason for such a small magnitude of the ISHE voltage in the case of a compensated AFM interface is obvious. Since the dynamic magnetization min the “compensated” case is γH ex/ωtimes smaller than the magnitude of the AFM vector l, the energy of the exchange coupling Wsat the FM/AFM interface in Eq. ( 2) is rather small. VI. DISCUSSION The above presented results were obtained for the parame- ters of a bulk NiO sample at low temperature. However, it iswell known that such important parameters of AFM substancesas the anisotropy constants and N ´eel temperature in thin AFM films could be substantially smaller than in bulk crystals (see,e.g., [ 26]). Thus, the penetration depths of the evanescent spin wave modes Eq. ( 5), determined at a given driving frequency ωby the AFM anisotropy constants, would significantly depend on the thickness and the temperature of the AFMlayer. Particularly, with the increase of temperature the AFMRfrequencies ω jwould decrease, and would approach zero at the N ´eel temperature [ 20]. In accordance with Eq. ( 5), this means that the penetration depth of the evanescent AFM spinwave modes will increase substantially when the temperatureapproaches the N ´eel temperature of the AFM layer. This increase of the spin current transferred through the AFM layeris clearly seen in the experiments [ 13]. In conclusion, we demonstrated that the spin current can be effectively transmitted through thin dielectric AFM layersby a pair of externally excited evanescent AFM spin wavemodes. In the case of AFM materials with biaxial anisotropythe transfer of angular momentum between the spin subsystemand the crystal lattice of the AFM can lead to the enhancementor decrease of the transmitted spin current, depending onthe phase relation between the excited evanescent spin wavemodes. Our results explain all the qualitative features of therecent experiments [ 11–14], in particular, the existence of an optimum thickness of the AFM layer, for which the outputcurrent could reach a maximum value which is higher than the spin current magnitude in the absence of the AFM spacer, andthe increase of the transmitted spin current at the temperaturesclose to the N ´eel temperature of the AFM layer. ACKNOWLEDGMENTS This work was supported in part by Grant No. ECCS- 1305586 from the National Science Foundation of the USA,by the contract from the US Army TARDEC, RDECOM,by DARPA MTO/MESO Grant No. N66001-11-4114, andby the Center for NanoFerroic Devices (CNFD) and theNanoelectronics Research Initiative (NRI). APPENDIX: AFM DYNAMICS DRIVEN BY THE MAGNETIZATION PRECESSION IN AND ADJACENT TO THE FM LAYER Let us consider the excitation of dynamics in an AFM layer by the processes happening at the FM/AFM interface.Let us assume that the magnetic coupling between the FMand AFM is of the exchange origin and, therefore, is stronglylocalized at the AFM/FM interface. The spins existing at theAFM boundary can belong either to two different sublattices ofthe AFM, as shown in Fig. 5(a), or to the same sublattice [see Fig.5(b)]. In the first case the AFM has no static magnetization at the interface and will be called below a compensated AFM, while in the second case, the boundary of the AFMis magnetized, and such AFM will be called uncompensated . The exchange coupling between the FM and AFM layers creates an additional term in the energy of the AFM. Inthe case of a compensated AFM the additional energy is expressed as /Delta1E=/summationtextJ s(˜S·S1+˜S·S2), where Jsis the interface exchange integral, ˜Sis the FM spin at the interface, and the summation is taken over the whole FM/AFM interface. After the transition to a continuum limit and taking into account the relation Eq. ( 1) in the main text of the paper, one can write the additional term in the Lagrangian Eq. ( 2)a s Es(mFM·m)δ(y), where mFMis the unit vector defining the magnetization direction in the FM layer, Esis the density of the surface exchange energy describing FM/AFM coupling,andE sis proportional to the exchange integral Js:Es∝Js. FIG. 5. Two types of FM/AFM interface: (a) totally compensated AFM boundary with zero magnetization, (b) totally uncompensatedAFM boundary. 224421-5KHYMYN, LISENKOV , TIBERKEVICH, SLA VIN, AND IV ANOV PHYSICAL REVIEW B 93, 224421 (2016) Considering the case of an uncompensated boundary of the AFM, one can find the additional coupling energy as /Delta1E=/summationtext2Js˜S·S1, which leads to the term Es[mFM·(m+l)]δ(y). Usually, the AFM boundary is partially uncompensated , and we introduce the phenomenological parameter α∈[0...1], which describes the degree of the AFM noncompensation at the FM/AFM interface.Using the well-known expression for the AFM magnetiza- tion [ 16,18,19], m=1 γHex/bracketleftbigg l×∂l ∂t/bracketrightbigg , (A1) it is easy to obtain the Lagrange equations describing the spin dynamics inside the AFM: 2μ/braceleftbigg/bracketleftbigg l×∂2l ∂t2/bracketrightbigg −c2/bracketleftbigg l×∂2l ∂x2/bracketrightbigg/bracerightbigg −/bracketleftbigg l×∂Wa ∂l/bracketrightbigg =Esδ(y)/braceleftbigg α[l×mFM]+/bracketleftbigg l×1 γHex/parenleftbigg 2/bracketleftbigg∂l ∂t×mFM/bracketrightbigg +/bracketleftbigg l×∂mFM ∂t/bracketrightbigg/parenrightbigg/bracketrightbigg/bracerightbigg . (A2) Since vector lin the ground state is directed along the vector e3, we can, in the case of a negligibly small dissipation, write the dynamic equations for only two components l1andl2of the vector l. These equations have the form analogous to the form of the dynamic equation (3) in the main text of the paper: ∂2l1 ∂t2−c2∂2l1 ∂y2+ω2 1l1=Esδ(y) 2μ/braceleftbigg α(mFM·e1)+1 γHex/bracketleftbigg 2∂l2 ∂t(mFM·l)+l2/parenleftbigg∂mFM ∂t·l/parenrightbigg/bracketrightbigg −/parenleftbigg∂mFM ∂t·e2/parenrightbigg/bracerightbigg , (A3) ∂2l2 ∂t2−c2∂2l2 ∂y2+ω2 2l2=Esδ(y) 2μ/braceleftbigg α(mFM·e2)−1 γHex/bracketleftbigg 2∂l1 ∂t(mFM·l)+l1/parenleftbigg∂mFM ∂t·l/parenrightbigg/bracketrightbigg −/parenleftbigg∂mFM ∂t·e1/parenrightbigg/bracerightbigg . (A4) The above equations are the equations describing dynamics of an oscillatory system driven by an external force f(t)δ(y), where f(t) are the right-hand-side parts of the above equations. We consider the harmonic driving force and, therefore, mFM∝e−iωt. In this case, when ω<ω 1,ω2the solutions of these equations are the evanescent modes that exponentially decay with the increase of the coordinate yinside the AFM. 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PhysRevB.89.134405.pdf

PHYSICAL REVIEW B 89, 134405 (2014) Topological changes of two-dimensional magnetic textures Ricardo Gabriel El ´ıas and Alberto D. Verga* Universit ´e d’Aix-Marseille, IM2NP-CNRS, Campus St. J ´erˆome, Case 142, 13397 Marseille, France (Received 28 January 2013; revised manuscript received 28 March 2014; published 9 April 2014) We investigate the interaction of magnetic vortices and skyrmions with a spin-polarized current. In a square lattice, fixed classical spins and quantum itinerant electrons evolve according to the coupled Landau-Lifshitz andSchr ¨odinger equations. Changes in the topology occur at microscopic time and length scales and are shown to be triggered by the nucleation of a nontrivial electron-spin structure at the vortex core. DOI: 10.1103/PhysRevB.89.134405 PACS number(s): 75 .76.+j,75.70.Kw,75.78.−n I. INTRODUCTION Itinerant magnetism is a fascinating state of matter where the interplay of short-range coupling (exchange, spin-orbit,crystal anisotropy) and long-range dipolar interactions lead toa variety of spatial orders. Experiments show, in particular,that magnetic structures with nontrivial topology naturallyarise in nanosize ferromagnets and chiral metals. V orticesare present in confined geometries with closed magnetic fluxlines, such as permalloy nanodots [ 1–3]; the Dzyaloshinskii- Moriya spin-orbit coupling in magnetic metals with inversionasymmetry like bulk MnSi or Fe atomic films favors helicalordering in the form of a skyrmion lattice [ 4–6]. The exis- tence of inhomogeneous metastable states in two-dimensionalisotropic ferromagnets, distinct from the usual domains, wastheoretically predicted by Belavin and Polyakov [ 7], who exhibited an asymptotically uniform solution with a reversedmagnetization in the central region. Lattices of skyrmions wereshown to be thermodynamically allowed in chiral magnets,within a range of applied magnetic fields [ 8]. The topology of these magnetization fields can be characterized by theirdegree, or topological charge [ 9]; the skyrmion configuration realizes a map between the plane (the ferromagnetic film)and the sphere (the directions of the magnetization vector);it has therefore an integer topological charge [ 7,10]. From a topological point of view, the isolated vortices observed innanomagnets are more exotic, since their topological chargeis a half integer [ 11]. V ortices with an out-of-plane core magnetization can be viewed as half skyrmions, sometimescalled merons [ 12,13], because only a half sphere is mapped. In disk magnets, their stability is ensured by the constraint ofa tangent magnetization at the boundary that minimizes thedipolar magnetic energy [ 14]. Interestingly, experiments reveal that these topological con- figurations can be manipulated not only by external magneticfields but also using purely electric means, by a spin-polarizedcurrent through the spin-transfer torque mechanism [ 15,16]. The polarity of a vortex core can be reversed by applyinga short pulse of an in-plane magnetic field [ 17], or by a current [ 18]. More recently, ultrafast switching of a uniform magnetization, with the temporary formation of a magneticsingularity, was achieved in experiments using laser pulsesof circularly polarized light [ 19,20], a technique that can, in *Alberto.Verga@univ-amu.frprinciple, also be effective in vortex switching via a topological inverse Faraday effect [ 21]. The dynamics of the magnetization at the microscopic scale is governed by the Landau-Lifshitz equation [ 22], /planckover2pi1∂ ∂tS=S×f−αS×(S×f), (1) where Sis the dimensionless spin (treated here as a classical variable proportional to the magnetization in an atomic volumea 3) and fis the effective field derived from the free-energy functional ( /planckover2pi1is the Planck constant, αis the damping constant, andfhas the dimensions of energy). This equation takes into account the exact conservation of the magnetization norm toits saturation value (for convenience we define |S|=S=1). In addition, the special mathematical form of ( 1), where the right-hand side is perpendicular to the spin vector, ensures theconservation of the topological charge [ 23], defined by Q=/integraldisplay R2dx 4πq(x,t),q=S·∂xS×∂yS, (2) in a two-dimensional system, where x=(x,y) is a point in the plane perpendicular to z. It is worth noting that the conservation of the magnetization topology is independentof the effective field-specific form, and it holds even in thepresence of norm-preserving dissipation and time-dependentexternal fields. However, the topology conservation is violatedby stochastic perturbations, related, for instance, to thermal or quantum fluctuations. The micromagnetic approach was extensively used in recent years to investigate the dynamics of magnetic texturesinvolving monopoles and vortices. In spite of the fact thatthe Landau-Lifshitz equation conserves the topology of themagnetization field, micromagnetic simulations on discretelattices proved to be useful in describing complex topologicalchanges [ 24]. In particular, these simulations revealed the importance of the excitation of gyration modes and vortex- antivortex annihilation in vortex core reversal [ 25–29]. In this paper we investigate the topological changes of magnetic textures induced by a spin-polarized current, usinga semiclassical two-dimensional lattice model, in which theitinerant electrons are quantum and the fixed spins are classical.The interaction of the itinerant electrons with the fixed ones gives rise to a spin-transfer torque, generally described in the quasiadiabatic limit by adding terms in the gradients of themagnetization, v s·∇S, (3) 1098-0121/2014/89(13)/134405(9) 134405-1 ©2014 American Physical SocietyRICARDO GABRIEL EL ´IAS AND ALBERTO D. VERGA PHYSICAL REVIEW B 89, 134405 (2014) where vsis related to the electron-spin-polarized cur- rent [ 30,31]. This approximation is not well suited in the presence of strong magnetization gradients, as it is preciselythe case near vortex cores. In the present model we keep the fullquantum electron dynamics to resolve nonlocal effects that arefundamental, as we demonstrate, in the mechanisms involving the change of topology through the formation of magnetic singularities. As underlined by Miltat and Thiaville [ 32], the nucleation of Bloch points and vortex cores are at the edge ofquantum magnetism. II. LATTICE MODEL OF COUPLED ITINERANT AND FIXED SPINS We consider a periodic square lattice of fixed spins S (classical, |S|=1) and a single electron that can jump between neighboring sites i=(x,y)=xˆx+yˆyandj(the hopping energy is /epsilon1, the electron charge −e, the lattice parameter a, and sizeL). Periodic boundary conditions ensure a well-defined topology of the total system. A constant electric field Eˆxis applied to create an electron current; this current is polarizedby a fictive magnetic field acting only on the electrons B p. Electrons and fixed spins are coupled through an exchangeinteraction J s, which in ferromagnets can be much larger than the Curie energy J[33]. The electron Hamiltonian is He=−/epsilon1/summationdisplay /angbracketlefti,j/angbracketrighteiφi,j(t)c† icj−Js/summationdisplay iSi·(c† iσci)+Hp,(4) where ci=(c↑i,c↓i) is the annihilation operator at site iand spin-up ↑or spin-down ↓. To preserve the lattice periodicity we used a gauge transformation that introduces a time-dependentvector potential A=Etˆx, instead of a static electric potential. This allows us to take into account the constant electricfield through the phase φ i,j(t)=(i−j)·ˆxeaEt/ /planckover2pi1, which is zero if the neighboring sites i,jare not in the xdirection. The second term accounts for the interaction energy withthe fixed spins ( σare the Pauli matrices). The last term, H p=−μeBp·/summationtext ic† iσci, allows the current to polarize in the direction Bp(μeis the electron magnetic moment). The magnetic energy is the sum of exchange J> 0, anisotropy K(positive or negative for easy-plane or easy-axis cases, respectively), and Dzyaloshinskii-Moriya Dterms, HS=J 2/summationdisplay i(∇Si)2+K 2/summationdisplay iS2 zi−D 2/summationdisplay iSi·(∇× Si), (5) where ∇is here the discrete gradient operator (note that it is dimensionless). In the following we use units such thata=/epsilon1=/planckover2pi1=e=1. The typical microscopic scales are a∼0.3 nm and /epsilon1∼1 eV for a ferromagnet, or a∼0.5n m and/epsilon1∼0.1 eV for MnSi, given a time unit t0∼1–10 fs, depending on the energy scale; the unit of electric field is aboutE 0∼/epsilon1/(ea)≈0.1–1×109Vm−1and the unit of current I0∼e/epsilon1//planckover2pi1≈10–100 μA. These small time and length scales, related to the electron kinetic energy and lattice spacing, arenecessary to track the changes in topology. The system evolution is governed by the Schr ¨odinger equation (or, equivalently, the Heisenberg equation for theoperators) for the electrons, i˙c i(t)=He(t,Si)ci(t), (6) and ( 1) for the fixed spins, with fi=−∂HS/∂Si, fi=J∇2Si−KSzi+D∇× Si+Jsnesi, (7) where si=/angbracketleftc† iσci/angbracketrightis the itinerant electron spin and /angbracketleftc† ici/angbracketright= neis the number of electrons per lattice site. The last term gives the spin-transfer torque due to the moving electrons. At variance to the linear response or quasiadiabatic ap- proximations, leading to a modified Landau-Lifshitz equa-tion [ 31,33,34], we keep the full electron dynamics ( 6)t o compute the spin-transfer torque in ( 7)( s e eR e f .[ 35]f o ra related model). Indeed, the usual approach starts from thespin-current continuity equation (straightforwardly obtainedfrom the Schr ¨odinger equation): ∂s ∂t+∇· J=Jss×S−/Gamma1, (8) where Jis the spin-current density tensor, and the last term /Gamma1, absent in our model, takes into account the dissipation mechanisms. Equation ( 8) is then solved by approximating the spin-transfer torque by a series in the gradients ( 3),s≈ neS+δs, and J≈vs⊗S, to obtain [ 31], δs≈S×vs·∇S+βvs·∇S. (9) Here the first term [ 30] gives the adiabatic contribution to the spin-transfer torque, and the second term, referred to as the “ β” term [ 31,36], is the nonadiabatic contribution coming from the/Gamma1spin relaxation in ( 8). However, in the presence of large gradients and strongly nonstationary processes, characteristicof the topological transitions, the quasiadiabatic approxima-tion breaks down. In addition, the scattering of the itinerantelectrons by the fixed spin inhomogeneities, might naturallylead to inhomogeneities in the electron current, also neglectedin this approximation ( 9), which considers v sas a constant. In our model, the spin-current density Jis a dynamical variable computed from the electron wave function. We see that takinginto account the itinerant electron quantum dynamics revealsmechanisms such as localization, scattering, and angular andlinear momentum transfers that have an important role in thechanges of the magnetization distribution (fixed spins). III. CURRENT INDUCED TOPOLOGICAL CHANGES IN MAGNETIC TEXTURES We solved numerically the system ( 1),(6) to compute the evolution of the magnetization texture S(x,t) on the discrete lattice and its coupling with the itinerant spin density s(x,t) and to monitor changes in the topological charge Q(t). The Schr ¨odinger equation is solved using an unitary time-splitting method, and the Landau-Lifshitz equation, a fourth-orderRunge-Kutta time stepping; difference operators are exactlycomputed on the lattice using a pseudospectral algorithm. Thetime step is tuned to reach machine precision conservation ofthe magnetization norm. The initial magnetization distributionis carefully determined to satisfy the stationary Landau-Lifshitz equation for vanishing electron current. In practice,we start with an exact solution of the isotropic (continuous)magnetization equation ( K=D=J s=0), in the form of 134405-2TOPOLOGICAL CHANGES OF TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 89, 134405 (2014) FIG. 1. (Color online) Evolution of the topological charge Q(black) and Q+(red), of a skyrmion lattice ( Q=4 per cell) with x-polarized electrons (a), a single Q=−1 skyrmion (b), and a vortex array (c), with z-polarized electrons. The insets show the initial magnetization field, arrows are for the ( Sx,Sy) components, and Szis in color, from black Sz=−1t ow h i t e Sz=1 (bottom color bar). The lattice size is L=128a and the applied electric field E/E 0=10−4; electron densities are ne=0.02 for the skyrmion lattice (a) and ne=0.1 for (b) and (c). localized vortices, and let this distribution evolve in time with the full lattice Landau-Lifshitz equation (in zero externalelectric field E=0); the asymptotic stationary state is then used as the initial condition of the coupled system ( 1),(6). We verified that eventual changes of an initial metastable state(triggered by numerical noise, for example) actually occurat times much longer than the ones observed after injectionof the electron current; therefore, we may conclude that theobserved changes (notably in the topology of the spin texture)result from the interaction with the spin-polarized current. Wepresent results of initial arrays of vortices and skyrmions, andof isolated structures like the Belavin-Polyakov skyrmion. The typical parameters used in the simulations are J s= 1,J=0.1,0.4,K=0,±0.01,D=0,0.01, for the cou- pling energies in units of /epsilon1, and α=0.1 for the Gilbert constant. Finite values of Dare relevant for the skyrmion lattice [Figs. 1(a) and 2(top row)]; K< 0 (easy-axis) is used for the Belavin-Polyakov skyrmion [Figs. 1(b) and3]. For vortex arrays we have K> 0 (easy-plane) and D=0 [Figs. 1(c),2(bottom row), and 4]. The electron density and electric field are chosen in order to get effective spin-currentdensities of the order of the experimental ones (10 12Am−2 in three-dimensional ferromagnets) [ 18]:ne≈0.1 electrons per site, E/E 0≈10−5–5×10−4, and a spin polarization μeBp=0.1/epsilon1. We remark, however, that in our model we neglect dissipation effects other than αrelaxation, and as a consequence, the current density may increase over time (wefix the electric field); we see below examples of its behavior(see Figs. 5and6). As an illustration of the rich phenomenology exhibited by the Schr ¨odinger-Landau-Lifshitz system, we show in Figs. 1 and2the topological evolution of skyrmions and vortices in the presence of initial free electrons subject to an external electricfield [ 37]. The topological charge ( 2) decreases or increases by integer steps /Delta1Q=1. Variations with /Delta1Q > 1 result from the superposition of simultaneous and separated-in-space /Delta1Q= 1 events. In Fig. 1we also plot Q +(t) computed from the integral of |q|,a si n( 2), Q+=/integraldisplay R2dx 4π|q(x,t)|,which is a measure of the number of vortices present in the system at time t(at variance with Q, it is not a conserved quantity of the Landau-Lifshitz dynamics). In these exampleswe used a strong electric field ( E/E 0=10−4, equivalent to 105Vm−1) in order to clearly display the current-vortex interactions. We may distinguish two ways leading to a topological change, according to the value of /Delta1Q: the nucleation and annihilation of same polarity vortex-antivortex pairs that do notchange the total topological charge, /Delta1Q=0; and the reversal of a vortex core, the suppression of a skyrmion, or other vortexinteractions involving a change /Delta1Q=1. Figure 2shows the magnetization at selected times for an initial skyrmion latticeand an array of vortices, displaying a variety of topologicalchange events. The skyrmion lattice may be considered as a superposition of bounded meron-antimeron pairs [ 12], double-periodically distributed in the plane, and having a charge Q=8×(1/2) per cell. Under the action of a strong +x-spin electron current, they wander around as almost independent Q=1/2 structures (Fig. 2,t=2000), and when equal-charge pairs come close together, they annihilate emitting a burst of spinwaves (Fig. 2,t=4680,4760 and t=9004,9016). Lately, a uniform magnetization state is reached (Fig. 2,t=9600). The annihilation events are clearly identified by a discontinuity/Delta1Q=−2, as can be seen in Fig. 1(a). The vortex array is a superposition of vortex-antivortex opposite-sign pairs; the total topological charge is then Q=0. Even if the total charge vanishes, the dynamics and interactionsof individual structures are highly nontrivial. Subject to aspin-up current, the vortex array evolution (Fig. 2) in addition to the vortex annihilation event at times t=8990–9030, which is similar to the one observed in the skyrmion case,shows other interesting processes, such as the nucleation ofvortex-antivortex Q=0 pairs ( t=4500–7000) from magne- tization structures created by the polarized electrons (white-redpatches). Each of these events is easily correlated to a suddenchange in Q +[Fig. 1(c)]. In general, the mechanism of a /Delta1Q=1 topological change entails the formation of a virtual structure (a singularity inthe continuum limit) with a net unit charge opposite to the 134405-3RICARDO GABRIEL EL ´IAS AND ALBERTO D. VERGA PHYSICAL REVIEW B 89, 134405 (2014) FIG. 2. (Color online) Magnetization texture at selected times, for an initial skyrmion lattice (top row) and vortex array (bottom row), corresponding to Figs. 1(a) and1(c), respectively. V ortex interaction results in the change of the topological charge and the emission of spin waves. initial charge. For instance, in the case of a Q=1/2 vortex, a bump of opposite polarity forms near the core, even whenthe vortex is almost at rest, as a result of the torque exertedby the electrons. At the time when the two peaks approachclose enough, at most a few lattice steps away, a virtualantivortex should appear to provide the necessary charge toannihilate the original vortex, then allowing the growing bumpto become a new vortex. This pathway through the /Delta1Q=1 change is, with respect to the topology of the magnetizationfield, similar to the core reversal phenomenology observedin micromagnetic simulations for moving [ 27,38] or static structures [ 39,40]. However, at variance to these models where the driven mechanism is an external time-dependent magneticfield, here we take into account the self-consistent interactionwith the electron current (maintained by an external, constant,electric field). In addition to the precession impressed aroundthe local direction of the spin-polarized current, the actionof the moving electrons son the fixed spins texture is twofold: First they reduce the vortex core size through anonlocal interaction with the surrounding spin currents andwaves; and second, they are able, by a local spin-transfertorque, to reverse the orientation of individual spins (strongnon-adiabatic effect). The annihilation of a skyrmion core,presented in Fig. 3, is significant of the role played by the itinerant spins in the /Delta1Q=1 topological change. Figure 3presents the configuration of the Belavin-Polyakov skyrmion in the initial stage of the topological change,corresponding to t=1152 in Fig. 1(a). The skyrmion core was previously deformed by the spin-up polarized current,increasing the gradient of S zin the xdirection [Fig. 3(a)]; as in the case of a meron core switching, the core of the skyrmionis ultimately reversed, leaving a Q=0 final state (at time t=1168). One of the main characteristics of the free spins is its quasistochastic distribution, as one observes in Fig. 3(b). Remembering that the differentiability of the effective field f is a necessary condition for the conservation of the topologicalcharge, the spatiotemporal intermittency of the s=/angbracketleftc †σc/angbracketright field is a crucial ingredient in the microscopic mechanism ofthe topological change. The origin of this complex behavioris the multiple quantum scattering of the electron waves onthe magnetization inhomogeneities, as can be verified byfollowing the evolution of their wave function (compare thefixed and itinerant spin distributions of Fig. 3). We also show in Fig. 3the effective internal magnetic field created by the fluctuating spin texture of the itinerant electrons, b=n·∂ xn×∂yn,n=s/|s|. (10) FIG. 3. (Color online) Initial stage of the skyrmion annihilation process [ t=1152t0;s e eF i g . 1(b)]:S(a),s(b), spin torque |S×s|(c), and electron internal magnetic field bEq. ( 10) at different times during the topological transition (d); Szcontour lines (b)–(d); ( sx,sy) arrows (b)–(d); box size is 64 ×64a2for (a)–(c), and 24 ×24a2for the (d) panel. 134405-4TOPOLOGICAL CHANGES OF TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 89, 134405 (2014) FIG. 4. (Color online) Magnetization and electron bfield for the vortex array case (Fig. 2). The annihilation of equal-charge vortex- antivortex is accompanied by the nucleation of a strong electron vortex (black spot at t=8990t0). The change /Delta1Q+=4 corresponds to the simultaneous destruction of four Q=1/2 vortex pairs. It arises when imposing the electron spin direction as the natural quantization axis, leading to an effective gauge vectorpotential a=(n×∇n)·σ. This is opposite to the usual gauge transformation that takes the magnetization as the reference frame to locally rotate the quantization axis (used to eliminate the electron degrees of freedom from the action) [ 33,41]. The remarkable fact about this quantity is that it concentratesat the vortex core, presenting a strong gradient precisely inthe region where the core is reversing. The relation betweenthe internal bfield and the topological charge density of the electron spin field allows us to interpret the rotatingopposite-polarity peaks at the center of the vortex core asbeing the signature of a nontrivial topological structure thatwill trigger the formation of the Q=1 extra charge necessary to the transition. Therefore, this process is in some sense theFIG. 5. (Color online) Electron current for skyrmion lattice (a) and Bleavin-Polyakov (b) and vortex array (c) cases also shown inFig.2.(I 0, the unit of current, is in the range 10−4to 10−5A.) opposite of the quasiadiabatic mechanism: The magnetization vector, at a microscopic spatiotemporal scale, follows thedynamics imposed by the itinerant electron spins, which arethe source of the topological change. The formation of anelectron vortex, revealed by the presence of localized structureswith a strong bfield, is the main result of this paper. This is further evidenced by the observation of the topological changeunder rather different conditions, as in the annihilation of avortex-antivortex pair. In the case of the vortex array, the z-spin-polarized current drives the formation of vortex-antivortex pairs, which detachfrom the large polarized current aligned patches, as presentedin Fig. 2. (The current density is strongly inhomogeneous, even for very weak electric fields.) These vortices then interact withthe original vortex array and eventually annihilate with itsequal-sign pair. The detail of the magnetization field aroundthe annihilation time ( t≈8990t 0) and the corresponding b-electron field are presented in Fig. 4. The original negative polarity vortex [dashed contour lines at time 7000; Fig. 4(b)] and the nucleated positive polarity vortex (both having Q= 1/2) turn around each other; the negative polarity one is associated with a positive bfield. At time 8990 the two vortices rotated about π, and a spot of negative bfield appears. From the color map we note that this electron vortex possessesa−1 charge, opposite to the Q=1 charge of the former vortex-antivortex pair. After the annihilation, a strong emissionof spin waves is observed, and the b-field concentrations disappear. From this rich phenomenology of topological changes we may distinguish (see Table I) (i) the destruction of Belavin- Polyakov skyrmion subject to a spin current with polarizationantiparallel to its core magnetization, (ii) the equal-sign vortexannihilation, and (iii) the opposite-sign vortex annihilation.In cases (i) and (ii), for which the topological chargeschange, in spite of their differences the /Delta1Q=−1 driving microscopic mechanism is related to the formation of alocalized electron spin structure (compare Figs. 3and 4). Indeed, the superposition of the equal-sign vortex-antivortex 134405-5RICARDO GABRIEL EL ´IAS AND ALBERTO D. VERGA PHYSICAL REVIEW B 89, 134405 (2014) TABLE I. Elementary topology transformations. No. Configuration /Delta1Q /Delta1Q + Description (i) Skyrmion annihilation 1 −1b-field structure (ii) V-A Vaannihilation −1 −1 Collision, bfield (iii) V-A V nucleation 0 1 Current gradients (iv) V-A V annihilation 0 −1 Collision, smooth aV ortex-antivortex. of Fig. 4is topologically similar to the Belavin-Polyakov skyrmion, both having Q=1( o rQ=−1). The electric current in units of I0is computed from the formula I(t)=−/summationdisplay y/angbracketleftbigg∂He ∂A/angbracketrightbigg , (11) where the mean is taken over the quantum state ci(t) for all i=(x0,y) lattice sites at a fixed x=x0position ( A=Et is the xcomponent of the vector potential). It is plotted in Fig.5for the three cases presented in Fig. 1. We remark that in spite of the dissipationless dynamics of the electrons, thecurrent is not simply linear in time; it may even be almostsuppressed as in the skyrmion lattice case. In the case of theBelavin-Polyakov skyrmion, the current increases drasticallyonly when the vortex core disappears and an almost uniformmagnetization background remains. The actual evolution ofthe electric current drastically depends on the magnetizationdistribution: The three initial textures lead to very differentcurrent behavior and characteristic orders of magnitude. We also observed that the electronic density has a minimum inside the vortex cores, and tends to concentrate between thevortex structures, in regions with a smooth varying magnetiza-tion. From this observation one may think that under a certainthreshold, the polarized current would be unable to drive thetopological changes because of its weak interaction with thevortices. To investigate this point, we performed a series ofsimulations of the vortex array for different values of theelectric field and the anisotropy constant. Some representativeresults are shown in Fig. 6, where we plot the topological charge and the electric current as a function of time. Forweak easy-plane anisotropy [ K=0.01, Figs. 6(a) and6(b)] and weak electric field ( E/lessmuch10 −4), the nucleation of vortices by the polarized current is suppressed; however, it is able toslowly drive the opposite-sign original vortices close together,allowing their /Delta1Q=0 annihilation (times near 50 000 for E=10 −5and 20 000 for E=5×10−5). The increase of the easy-plane anisotropy [ K=0.05, Figs. 6(c) and 6(d)] contributes to the stabilization of the vortex array, and strongerelectric fields are needed to change the topology. A stationarystate naturally arises in the case K=0.05 and E=2×10 −4 of Fig. 6(c). For a stronger electric field, E=5×10−4,a rich dynamics develops, with nucleation and annihilation ofvortices [Fig. 6(d)]. IV . DISCUSSION AND CONCLUSION This paper focused on the influence of the itinerant electrons dynamics on the spin-transfer torque and the mechanisms oftopological changes in two-dimensional magnetic textures. Weproposed a simple model with a single coupling parameterbetween electrons and fixed magnetic moments, J s. Other parameters take into account the exchange, anisotropy, andspin-orbit effects. The unit of energy, which we took as/epsilon1∼J s, and the unit of length acan be considered as effective parameters whose values depend on the actual physical system.TheJ scoupling is typically of the order of the 1 eV in magnetic metals and one order of magnitude smaller in chiral magnetsor diluted magnetic semiconductors; the exchange constant isaboutJ/J s=0.1[42–44]. The elementary processes involving a topological change as found in our simulations, are summarized in Table I. Configuration (i) corresponds to the annihilation of a skyrmioncore; as shown in Fig. 3, this process involves an intermediate state characterized by an electron-spin structure that can berevealed by the topological bfield ( 10). Ab-field structure is also present in the annihilation of a nontrivial configuration (ii)of a vortex-antivortex pair. V ortices of equal-charge annihilateduring collisions and are followed by a burst of spin waves,as illustrated by the simulations of the skyrmion and vortexarrays (see Fig. 1and in the Supplemental Material [ 37]t h e corresponding movies). In contrast, topological trivial config-urations evolve smoothly, like (iii) and (iv), corresponding, re-spectively, to nucleation and annihilation of vortex-antivortexpairs of opposite charges. V ortices are generated in pairs bytraveling blobs of spin-polarized electrons. Indeed, strongcurrent inhomogeneities naturally appear by interaction of aninitially homogeneous current with existent vortices; furtherevolution of these inhomogeneities leads to the nucleation ofV-A V pairs. These elementary processes combine to display a FIG. 6. (Color online) Topological charge (top panels, in red Q+,Q=0) and current (bottom panels) for the vortex array case, as a function of the applied electric field [(a) E=10−5,( b )E=5×10−5,( c )E=2×10−4,a n d( d ) E=5×10−4, withne=0.1] and easy-plane anisotropy [ K=0.01 in (a) and (b) and K=0.05 in (c) and (d)]. In (a),(b), at variance to the case of Fig. 4, the change /Delta1Q+=2 corresponds to the trivial annihilation of two pairs of opposite-charge vortex-antivortex. 134405-6TOPOLOGICAL CHANGES OF TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 89, 134405 (2014) complex dynamics. For example, in the case of vortex arrays, nucleation of pairs at the edges of electron blobs is followedby their interaction with preexisting vortices; or isolatedcurrent inhomogeneities in an essentially uniform backgroundcreate a cascade of trivial and nontrivial pairs of vortices thatsubsequently annihilate [see Supplemental Material [ 37], the movie corresponding to the vortex array of Fig. 6(d)]. If one compares these results with previous micromagnetic simulations including the spin torque term [ 45–47], one finds that the phenomenology observed in the self-consistent caseis much richer, even if micromagnetism can capture some ofthe elementary mechanisms of nucleation and annihilation.In order to drive topological changes in the micromagneticframework, a current pulse [ 45] or an imposed stationary cur- rent inhomogeneity [ 47] was used. The injection of a current pulse in a ferromagnet induces the switching of a vortex corethrough the nucleation of a vortex-antivortex pair subsequentlyannihilated by the formation of a Bloch point [ 45]. This process is similar to the case (ii), where the electron spintexture plays the role of intermediate structure (a Bloch point isintrinsically three-dimensional, and then it cannot be realizedin our system; however a virtual superposition of a vortexand an antivortex having the same charge, is topologicallyequivalent to a Bloch point). Nucleation events as in case (iii)were also observed when the system is driven by a rotatinginhomogeneous current density [ 47]. In this case, the nontrivial topology of the current density is transferred by spin torqueto the magnetization texture. In our simulations we find thatthe current-magnetic texture interaction is accompanied by astrong modification of the current distribution itself, leadingsometimes to quasistationary states [as in the cases shown inFigs. 6(a)–6(c)]. Consideration of the electron dynamics makes it possible to show first that the hypothesis of uniform currentbreaks down in the vicinity of vortex cores, that quantumtransmission effects limit the current flow even without explicitdissipation effects, and, finally, that strongly nonadiabaticprocesses (notably through the electron bfield) are essential for the occurrence of topological changes of skyrmions andequal-charge vortex-antivortex pairs. The scattering of electrons off magnetization inhomo- geneities, notably vortex cores, is at the origin of the currentspatial variations and the formation of polarized electron blobs.We observed that vortex arrays are stabilized by easy-axisanisotropy, leading to a state characterized by a quasiperiodicemission of vortices [as in Fig. 6(d)]. One of the reasons of this stabilization is that the propagation of z-polarized electrons is limited in an easy-plane medium: the stronger theanisotropy, the stronger is the applied electric field neededfor generating the same current. Decreasing the electric field suppresses topological changes and an inhomogeneous stationary state settles in. The electric field threshold for thevortex array is found in the interval E/E 0=(2,5)×10−4 [compare Figs. 6(c) and6(d)]. However, the actual threshold values depend strongly on the anisotropy and other materialand geometric parameters. In the case of the Belavin-Polyakovskyrmion, our simulations show stability below electric fieldsof the order of E/E 0≈10−5, in the isotropic case K=0. Qualitatively, we note that topological changes produced forcurrents I/greaterorsimilar2I 0. These large currents are present in the case of the annihilation of the Belavin-Polyakov skyrmion or in thenucleation of vortices by current blobs. In fact, lower currents can also induce topological changes, but indirectly, because oftheir effect on the motion of vortices that eventually collide, asin the case of the skyrmion lattice (note that the currents are ofthe order I≈0.1I 0). Suppression of nucleation of new vortices from current inhomogeneities appears to occur below I≈2I0. Using a typical value for the current unit in ferromagnet,I 0≈5×10−5A, and a rough estimation for the current density as j∼I/(La), one obtains an order of magnitude j∼1012Am−2. At variance, in the case of a skyrmion lattice we did not find a minimum current below which vorticesremain static; even currents of the order of I/I 0∼10−2(with E/E 0=10−5) are able to induce a skyrmion motion. In summary, we investigated the topological changes in a two-dimensional ferromagnet driven by a self-consistentelectron current. The Landau-Lifshitz equation is coupledthrough the spin-transfer torque term with the Schr ¨odinger equation for the itinerant spins. At variance to the continuousmicromagnetic models, the system discreteness and, moreimportantly, the stochastic behavior of the driven term brokethe conservation of the topological charge. We observed thatboth local and nonlocal interactions play a role in the transitionbetween different topological configurations. In particular,the electron current tends to concentrate in channels thatavoid the vortex cores: Strong gradients of the magnetizationact as potential barriers, scattering off the electron waves.The phenomenology of a /Delta1Q=1 change of an initial Q=±1,±1/2 vortex, although rich, reduces to a single topological mechanism, the nucleation of a Q=±1 charge that annihilates the old structure, producing the new structurewith the opposite charge or Q=0. The interesting point is that this mechanism does not arise spontaneously but istriggered, above a threshold, by the spin-polarized current. Theelectron spin and its associated polarized current are stronglyfluctuating, up to the lattice and time unit scales, whichappear to be the relevant scales for the topological changesin the dissipationless limit. The spatial inhomogeneity andlocalization of the electrons is a general feature, systematicallyobserved, showing that the systems is relatively far fromthe quasiadiabatic regime. The spontaneous nucleation andannihilation of vortices are, in fact, driven by the stronginhomogeneity of the electron spin distribution. At the heart of the topological change is the formation of an electron nontrivial structure that induces the switchingmechanism of the magnetization in a strongly nonadiabaticprocess. This electron structure has a nontrivial topologycharacterized by an internal magnetic field. Localized spotsof this field appear during the transition and are observed inapparently different processes such as the annihilation of the Belavin-Polyakov skyrmion or the interaction of equal-sign vortices. Our model is limited to a simple two-dimensional geometry and boundary conditions; although it would be interestingto explore more complex situations, the main open questionis how to obtain a continuous limit (necessary to modellarger systems at longer time scales) that takes into accountthe effective dissipative coupling with the electrons (see therecent papers [ 41,48], where dissipation and dissipationless mechanisms are analyzed). In spite of these limitations, theobservation of topological changes driven by spin currents 134405-7RICARDO GABRIEL EL ´IAS AND ALBERTO D. VERGA PHYSICAL REVIEW B 89, 134405 (2014) should be accessible to experiments with chiral magnets like MnSi, where the effectiveness of spin-transfer torqueon skyrmion lattices was already demonstrated [ 49]. More recently [ 50], it was proved that individual skyrmions can be created and annihilated by means of an electron currentinjected by a local probe (using the tip of a spin-polarizedscanning tunneling microscope). The system undergoes atransition from a skyrmion to a ferromagnetic state, as theone shown in Fig. 1(b). The mechanism of this switching is attributed to a combination of nonthermal excitations ofthe injected electrons and the spin-transfer torque. Although a detailed comparison would require further investigation,these experimental results are in qualitative agreement withthe scenario presented in Fig. 3. ACKNOWLEDGMENTS We thank Riccardo Hertel and Jean-Christophe Toussaint for useful discussions. [ 1 ]R .P .C o w b u r n ,D .K .K o l t s o v ,A .O .A d e y e y e ,M .E .W e l l a n d , and D. M. Tricker, P h y s .R e v .L e t t . 83,1042 (1999 ). [2] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289,930(2000 ). [3] A. Wachowiak, J. Wiebe, M. Bode, O. 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PhysRevLett.125.077203.pdf

Unidirectional Pumping of Phonons by Magnetization Dynamics Xiang Zhang ,1Gerrit E. W. Bauer ,2,1and Tao Yu3,1,* 1Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, Netherlands 2WPI-AIMR & Institute for Materials Research & CSRN, Tohoku University, Sendai 980-8577, Japan 3Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany (Received 5 March 2020; revised 23 July 2020; accepted 24 July 2020; published 14 August 2020) We propose a method to control surface phonon transport by weak magnetic fields based on the pumping of surface acoustic waves (SAWs) by magnetostriction. We predict that the magnetization dynamics of ananowire on top of a dielectric films injects SAWs with opposite angular momenta into opposite directions. Two parallel nanowires form a phononic cavity that at magnetic resonances pump a unidirectional SAW current into half of the substrate. DOI: 10.1103/PhysRevLett.125.077203 Introduction. —Surface acoustic waves (SAWs) on the surface of high-quality piezoelectric crystals are frequently employed for traditional signal processing [1,2],b u t are also excellent mediators for coherent informationexchange between distant quantum systems such as super- conducting qubits and/or nitrogen-vacancy centers [3–6]. Piezoelectrically excited coherent SAWs drive the ferro-magnetic resonance (FMR) by magnetostriction [7–13], excite spin waves parametrically [14], and generate elec- tron spin currents by the rotation-spin coupling [15,16] . Conventional insulators often have good acoustic quality but only small piezoelectric effects, rendering the direct excitation, manipulation, and detection of the coherentSAWs challenging. The phonon pumping [17], i.e., the excitation of bulk sound waves in a high-quality acoustic insulator by the dynamics of a proximity magnetic layer via the magnetoelastic coupling [18,19] , may be useful here. Bulk phonons in the insulator gadolinium gallium garnet (GGG) can couple two yttrium iron garnet (YIG) magnetic layers over millimeters [20,21] . Here we address the coherent excitation and manipula- tion of Rayleigh SAWs by magnetization dynamics, whichis possible in a lateral planar configuration with ferromag- netic nanowires on top of a high-quality nonmagnetic insulator, as illustrated in Fig. 1. Similar configurations on magnetic substrates led to the electrical detection of diffuse magnon transport [22,23] and discovery of nonre- ciprocal magnon propagation [24], i.e., the generation of a unidirectional spin current in half space [25,26] . Magneticstray fields of the magnetization dynamics also generate chiral electron [27] and waveguide photon [28] transport. The unidirectional excitation of SAWs is important for acoustic device applications [29], which conventionally is achieved by metal electrodes on a piezoelectric crystal suchthat reflected SAWs constructively interfere with the source. This is a pure geometrical effect that is efficient at sub-GHz frequencies and sample dimensions that matchthe SAW wavelength [29,30] . We focus on the unidirectional [31]excitation of SAWs via magnetic nanostructures on top of a dielectric substrate that are brought into FMR by external microwaves. We predict effects that are very different from the reportednonreciprocity, i.e., a sound velocity that depends ondirection [11,32] , which is enhanced in magnetic multi- layers on top of a piezoelectric substrate [33–35]. The magnetic order of, e.g., a wire on top of a dielectric, doesnot couple nonreciprocally to the surface phonons in the configuration in Fig. 1, but excites both left- and right- propagating phonons, even though the angular momentum FIG. 1. Surface-phonon pumping by one magnetic nanowire (brown) on top of the acoustic insulator (blue). A static magneticfieldH 0applied in the ⃗xdirection saturates the magnetization. The pumped Rayleigh SAWs by the nanowire FMR propagateand rotate in opposite directions at the two sides of the nanowireas indicated by the green and black arrows, respectively.Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article ’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.PHYSICAL REVIEW LETTERS 125, 077203 (2020) 0031-9007 =20=125(7) =077203(6) 077203-1 Published by the American Physical Societycurrent has a direction because of the momentum-rotation coupling of Rayleigh SAWs. However, we predict robust unidirectional excitation of SAW phonons in a phononiccavity formed by two parallel wires. The SAWs actuated by the first wire interact with the second one (which does not see the microwaves) and excite its magnetization, whichin turn emits phonons. The phonons from both sourcesinterfere destructively on half of the surface and the net phonon pumping becomes unidirectional. Constructive interference between the two nanowires induces standingSAWs as in a Fabry-P´ erot cavity. Conventional unidirec- tional electric transducer [29,30] operate by a pure geomet- rical interference effect that works only for a fixed sub-GHzfrequency. Our magnetic unidirectional transducer operates by a dynamical phase shift and provides new functionalities, such as robust high frequency tunability and switchability. Model. —We consider a rectangular magnetic nanowire (YIG) on top of the surface of a thick dielectric (GGG) that spans the x,yplane. It extends along the ydirection with z∈½0;d/C138andx∈x iþ½−w=2;w =2/C138, as shown in Fig. 1.F o r an analytical treatment, dis assumed to be much smaller than the skin depth of the SAWs, such that the displacement field in the wire is nearly uniform in the zdependence. The lattice and elastic parameters at the YIG jGGG interface match well [17,20,21,36] and are assumed equal. A uniform and sufficiently large static magnetic field H0along ⃗xsaturates the equilibrium magnetization M0¼Ms ⃗xnormal to the wire. We can modulate the magnon-phonon coupling simply by rotating H0. The system Hamiltonian consists of the elastic energy ˆHe, the magnetoelastic coupling ˆHc, and the magnetic energy of the Kittel mode [37], ˆHm¼Z dr/C18 −MxH0þ1 2NxxM2xþ1 2NzzM2z/C19 ;ð1Þ where M¼ðMx;My;MzÞTis the magnetization vector and the demagnetization constants are taken as Nxx≃d=ðdþwÞ andNzz≃w=ðdþwÞ[25]. Although the predicted effects are classical, we use a quantum description for convenience and future applications in quantum phononics [3–6,38] ;w e can always recover the classical picture by replacing operators by amplitudes. The transverse magnetization is quantized by the Kittel-magnon operator ˆβðtÞwith nor- malized wave function my;z[39,40] , see Supplemental Material [41]: ˆMy;z¼−ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2γℏMsp ðmy;zˆβðtÞþm/C3y;zˆβ†ðtÞÞ; ð2Þ leading to ˆHm¼ℏωFˆβ†ˆβwith frequency ωF¼μ0γffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH0−NxxMsÞðH0−NxxMsþNzzMsÞp :Here,−γandμ0are the gyromagnetic ratio and vacuum permeability. In our configuration, only the Rayleigh SAWs couple efficiently with the magnet which by their surface nature and long decay length are well suited to exchange infor-mation with spatially remote magnets (see SupplementalMaterial [41]). Sufficiently thin and narrow wires do not affect the substrate strongly, so we may treat them pertur- batively. The surface eigenmodes of an isotropic elastic halfspace read [42] ψ x¼ikφk/C18 eqz−2qs k2þs2esz/C19 eikx; ψz¼qφk/C18 eqz−2k2 k2þs2esz/C19 eikx; ð3Þ where q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2−k2 lq and s¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2−k2tp with kl¼ ωkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ=ðλþ2μÞp andkt¼ωkffiffiffiffiffiffiffiffi ρ=μp are the wave vectors for longitudinal and transverse bulk waves, respectively. Here, ρis the material density, μand λare the elastic Lam´e constants, and φkis a normalization constant [41]. ωk¼jkjηffiffiffiffiffiffiffiffi μ=ρp ¼crjkjrepresents the eigenfrequency of Rayleigh SAWs with velocity crandη<1is the root of the SAW characteristic equation [42]that depends only on the Lam´e constants. The relative phase of the displacement field Arg ðψz=ψxÞjz¼0¼/C6iis opposite for right- and left- propagating waves, which reflects the rotation-momentum locking [42]. The quantized displacement field ðˆux;ˆuzÞcan be expanded into the eigenmodes ψðkÞand phonon operators ˆbkðtÞ[38] ˆuðx; z; t Þ¼X k½ψðx; z; k ÞˆbkðtÞþψ/C3ðx; z; k Þˆb† kðtÞ/C138: ð4Þ We normalize the mode amplitudes ψto recover the elastic Hamiltonian for Rayleigh SAWs [41] such that ˆHe¼ρZ dr_ˆu2ðx; z; t Þ¼X kℏωkˆb† kˆbk: ð5Þ In YIG films the magnetocrystalline anisotropy, which is important in CoFeB [12,43] , is relatively weak [17,21,36] and the (linearized) magnon-phonon coupling energy is dominated by ˆHm c¼ðB⊥=MsÞRðˆMy∂xuyþ ˆMz∂zuxþ ˆMz∂xuzÞdrwith magnetoelastic coupling con- stant B⊥[37,44] . The magnetic wire and nonmagnetic substrate are coupled by the dynamics of the surface strain. We require the interaction between a given SAW and the Kittel mode. By the translational symmetryalong the nanowire ydirection, the displacement field excited by the Kittel magnon does not depend on y, and SAWs propagating along xdo not contribute to u y.PHYSICAL REVIEW LETTERS 125, 077203 (2020) 077203-2The magnetoelastic energy contributed by the magnetic wire with length Lthen becomes ˆHm c¼B⊥L MsZd 0ˆMzðuzjx¼w=2þxi−uzjx¼−w=2þxiÞdz þB⊥L MsZw=2þxi −w=2þxiˆMzðuxjz¼d−uxjz¼0Þdx: ð6Þ We limit attention to the realistic situation in which the wire thickness dis much smaller than the decay length of the SAWs into the bulk. The strain in the magnet then mirrorsthat of the SAW at z¼0of the dielectric and the second term in Eq. (6)vanishes [12], ˆH m c→B⊥Ld MsˆMzðuzjx¼w=2þxi−uzjx¼−w=2þxiÞ;ð7Þ that corresponds to an oscillating surface force Fjx¼/C6w=2þxi¼ ∓B⊥Ld ˆMz=Ms ⃗zin the zdirection that excites SAWs traveling outwards in both directions [41]. Substituting Eqs. (2)and(4)into Eq. (7), we arrive at the interaction Hamiltonian ˆHc¼ℏX kgkˆβ†ˆbkþH:c:; ð8Þ in which the coupling constant ( qd≪1,sd≪1) gk≃−B⊥ffiffiffiffiffiffiffiffiffiffiffiffiffiγ Msρcrr ffiffiffiffi d wr sin/C18kw 2/C19 ξMξPeikxi; ð9Þ with factors ξMandξPgoverned by the magnetic and acoustic material parameters [41]. The form factor oscillates and decreases algebraically as a function of nanowire widthand phonon wavelength and vanishes when d; w→0.T h e coupling is reciprocal since jg kj¼jg−kj. SAW pumping. —We now calculate the phonon pumping by a single magnetic nanowire transducer centered at x0and excited by microwave photons represented by the (annihi- lation) operator ˆpin. The Hamiltonian ˆH¼ ˆHmþ ˆHeþ ˆHc leads to the Heisenberg equation of motion [45,46] dˆβ=dt¼−iωFˆβ−iX kjgkjeikx 0ˆbk−ðκmþκωÞˆβ=2−ffiffiffiffiffiκωpˆpin; dˆbk=dt¼−iωkˆbk−ijgkje−ikx0ˆβ−δkˆbk=2; ð10Þ where κmand δkare the intrinsic damping rates for the nanowire magnon and surface phonon, while κωis the radiative damping induced by the microwave field. In the frequency domain, ˆOðωÞ¼RdtˆOðtÞeiωt,ˆβðωÞ¼−iffiffiffiffiffiκωpˆpinðωÞ ω−ωFþiðκmþκωÞ=2−P kjgkj2GkðωÞ; ˆbkðωÞ¼GkðωÞjgkje−ikx 0ˆβðωÞ; ð11Þ where GkðωÞ¼1=ðω−ωkþiδk=2Þis the phonon Green function. The additional magnetic damping by the phononpumping at the FMR [17,47] is given by the imaginary part of the magnon self-energy σ kðωÞ¼−Im/C18X kjgkj2GkðωÞ/C19 ¼jgkrj2 cr; ð12Þ where we use the on-shell approximation [48,49] with ω→ωFandkr¼ωF=cr. The real part of the self-energy causes a small frequency shift that is absorbed into ωFin the following. The displacement field given by Eq. (4)is a super- position of coherent phonons hˆbkithat are excited by the microwave input hˆpinðωÞi. At resonance ω→ωF, the contour of the kintegral must be closed in the upper (lower) half of the complex plane for x>x 0(x<x 0), selecting the poles krþiϵð−kr−iϵÞin the denominator, where ϵis the inverse of the phonon propagation length. The low ultra-sonic attenuation in GGG at room temperature correspondsto characteristic SAW decay lengths of up to 6 mm [50].W e can therefore safely disregard the phonon damping(ϵ→0 þ), which leads to displacement fields uðz; tÞ¼−2 crRe/C26iψðkr;zÞg/C3 krhˆβðtÞi;x > x 0 iψð−kr;zÞg/C3 −krhˆβðtÞi;x < x 0:ð13Þ On the right (left) side of the nanowire x>x 0(x<x 0), the right- and left-propagating waves with opposite rotations,whose directions depend on z, are pumped as illustrated in Fig.1. A classical treatment leads to the same result [41]. These phonons carry a constant mechanical angular momentum density l DCðx; zÞ¼ρhu×_uit, where the sub- script tindicates time average, which is often referred to as phonon spin [18,51,52] : lDCðzÞ¼ð 4ρωF=c2rÞjhˆβij2jgkrj2⃗y ×I m/C26ψxðkr;zÞψ/C3zðkr;zÞ;x > x 0 ψxð−kr;zÞψ/C3zð−kr;zÞ;x < x 0:ð14Þ lDCis proportional to the excited magnon population, parallel to the wire, and opposite on both sides of thenanowire since ψ xð−kÞψ/C3zð−kÞ¼−ψxðkÞψ/C3zðkÞ. Into the substrate ( zdirection), the SAW eigenmodes have a node at which lDCchanges sign [41] as sketched in Fig. 1. The phonon pumping does not remove angular momentumfrom the ferromagnet, since only the xcomponent of the magnetic precession is damped. The force on theinterface is a superposition of opposite angular momentaPHYSICAL REVIEW LETTERS 125, 077203 (2020) 077203-32z¼ðzþixÞþðz−ixÞthat by the spin-momentum locking couple to phonons moving in opposite directions. The efficiency of phonon spin pumping depends on the nanowire and substrate. For GGG at room temperature, ρ¼7080 kg=m3,cl¼6545 m=s, and ct¼3531 m=s [53], leading to [42] η¼0.927,cr¼ηct¼3271 .8m=s, and ξP¼0.537. For YIG [54],γ¼1.82×1011s−1T−1, μ0Ms¼0.177T[55],B⊥¼6.96×105J=m3[17], and ξM≈1when H0is comparable to Ms. We plot the pumped phonon spin density at different zin Fig. 2(a) with ωF¼3GHz, d¼200nm and w¼2.5μm. We use a small precession cone angle 10−3degrees and phonon diffusion length ∼6mm. The spin density is opposite at the two sides of the nanowire and changes sign at larger z. Figure 2(b) is a plot of the additional magnon damping coefficient α¼σk=ωFin the dependence of FMR fre- quency ωFand nanowire width w. We observe geometric resonances ∼1=wwith α≲1×10−4, which is of the order of the intrinsic Gilbert damping of YIG single crystals α0∼4×10−5[56]and films 8×10−5[57]. In the thin YIG film, the additional damping α∼d. Unidirectional phonon pumping. —The single wire emits spin-momentum locked SAWs into two directions. We propose a truly unidirectional phonon source in the form oftwo parallel and identical nanowires located at r 1¼R1 ⃗xandr2¼R2 ⃗x, of which only the left one is addressed by a local microwave stripline [58]. The excited phonons below propagate to and are absorbed by the second nanowire. Its dynamics reemits phonons that subsequently interfere withthe original ones [20]. Denoting the magnon operators in the left and right nanowires as ˆβ Land ˆβR[41], ˆβRðωÞ¼P kjgkj2GkðωÞeikðR2−R1Þ ω−ωFþiκm=2−P kjgkj2GkðωÞˆβLðωÞ; ˆbkðωÞ¼jgkjGkðωÞðe−ikR 1ˆβLðωÞþe−ikR 2ˆβRðωÞÞ: ð15Þ At the FMR ω→ωF, ˆβRðωFÞ¼χðkrÞeiπþikrðR2−R1ÞˆβLðωFÞ; ð16Þ where χðkrÞ¼σðkrÞ=½κm=2þσðkrÞ/C138modulates the mag- netization amplitude in the second wire and krðR2−R1Þis the phase delay by the phonon transmission. The phaseshift πreflects the dynamical phase relation between magnons and phonons that is the key for the unidirection- ality. This relation can be observed inductively in micro-wave transmission spectra [41]. By substituting Eq. (16) into(15) at the FMR: ˆb −kr¼jgkrjGkreikrR1ˆβLðωFÞ½1−χðkrÞe2ikrðR2−R1Þ/C138; ˆbkr¼jgkrjGkre−ikrR1ˆβLðωFÞ½1−χðkrÞ/C138: ð17Þ In the strong magnon-phonon coupling limit σðkrÞ≫ κm=2,χðkrÞ→1, thus the right-going phonon kr>0is not excited by the double-wire configuration. Finite hˆb−kri but vanishing hˆbkriimplies a unidirectional phonon current. Such unidirectionality vanishes when the second wire is weakly coupled to the SAW, i.e., σðkrÞ≪κm=2, i.e., phonons transmit without interacting with the magnet. By Eqs. (4)and(15), the displacement fields of fre- quency ωFread uðx; tÞ¼2jgkrj crIm8 >>< >>:ψð−k rÞeikrR1hˆβLðtÞið1−χðkrÞe2ikrðR2−R1ÞÞ x<R 1 eikrðR2−R1ÞhˆβLðtÞiðψðkrÞe−ikrR2−χðkrÞψð−krÞeikrR2ÞforR1<x<R 2 ψðkrÞe−ikrR1hˆβLðtÞið1−χðkrÞÞ x>R 2: ð18Þ When χðkrÞ→1, the displacement field vanishes in the region x>R 2, but is a traveling wave for x<R 1.B e t w e e n the two nanowires with R1<x<R 2, the SAWs form standing waves with uz∼sinkrðx−R2Þandux∼coskrðx−R2Þ. The pumping is unidirectional apart from special cases: with frequency or distance krðR2−R1Þ¼nπwith n∈Z0,t h e SAWs on the left-hand side vanishes as well and the phononis fully trapped between the two wires to form a cavity. The phonon emission is not perfectly unidirectional when χðkrÞ<1,h o w e v e r .F i g u r e 3(a)is a plot of the magnitude of the displacement field at the GGG surface juðx; z¼0Þj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2xþu2zp jz¼0with an intrinsic Gilbert damping α0¼ 8×10−5[57]. We choose the YIG wires of d¼200nm andw¼2.5μm to center at R1¼0andR2¼30μm, and -2-1.5-1-0.5 0 0.5 1 1.5 2 -300 -200 -100 0 100 200 300 /g80/g68(a) z=0 /g80m -1.5 /g80m(b) FIG. 2. Pumped phonon spin density [(a)] and additional magnon damping coefficient α[(b)] for a YIG wire on a GGG substrate as a function of FMR frequencies and wire widths.PHYSICAL REVIEW LETTERS 125, 077203 (2020) 077203-4the Kittel frequency ωF¼3GHz such that the additional damping coefficient is 1.2×10−4.I nF i g . 3(b) we plot the phonon (DC) spin density at the GGG surface for a precession cone angle 10−3degrees in the left wire. The asymmetry of the pumped phonon spin at the two sides of YIG cavity is not perfect, but clearly survives a largerdamping. In YIG jGGG systems, phonon pumping should be measurable even in less than perfect samples. Devices with more than two wires or made from magnetic materials withlarger magnetoelasticity may achieve full unidirectionality. The pumped phonon (spin) can propagate coherently over millimeters on the substrate surface, which is verypromising for classical and quantum transport of spininformation. It can be measured by Brillouin light scatter-ing[18], the spin-rotation coupling by fabricating a con- ductor on top of the acoustic medium [15,16] , and other techniques [52]. The generation of unidirectionality by interference does not require a nonreciprocal couplingmechanism [25–28]but only an out-of-phase relation of the two fields at resonance. The phenomenon shouldbe universal for many field propagation phenomena,such as exchange coupled magnetic nanowires and films[25] and reciprocally coupled magnons and waveguide photons [28]. Discussion. —In conclusion, we developed a theory for pumping SAWs and proposed a phonon cavity device that realizes unidirectional phonon current in a reciprocalsystem. When exciting a single magnetic nanowire bymicrowaves, we predict a passive wire induces theunidirectional phonon current and formation of standingwaves in the region between two magnetic nanowires. Thismechanism should also lead to a unidirectional spinSeebeck effect [26] generated by a temperature gradient between the magnetic and acoustic insulators [59]. Unidirectionality emerges here from a dynamical phaseshift, rather than the purely geometrical interferenceemployed by electrical unidirectional SAW generators[29,30] . In the strong coupling regime [20] two magnetic wires on a simple dielectric form a fully unidirectionalSAW phonon source or transducer that opens intriguingperspectives for magnonics and spintronics, but alsoplasmonics [60,61] , nano-optics [62], and quantum communication. This work is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) aswell as JSPS KAKENHI Grant No. 19H006450. *tao.yu@mpsd.mpg.de [1] E. A. Ash, A. A. 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PhysRevB.101.020403.pdf

PHYSICAL REVIEW B 101, 020403(R) (2020) Rapid Communications Magnon decay theory of Gilbert damping in metallic antiferromagnets Haakon T. Simensen , Akashdeep Kamra, Roberto E. Troncoso, and Arne Brataas Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Received 26 June 2019; revised manuscript received 25 November 2019; published 9 January 2020) Gilbert damping is a key property governing magnetization dynamics in ordered magnets. We present a theoretical study of intrinsic Gilbert damping induced by magnon decay in antiferromagnetic metals throughs-dexchange interaction. Our theory delineates the qualitative features of damping in metallic antiferromagnets owing to their bipartite nature. It provides analytic expressions for the damping parameters yielding valuesconsistent with recent first-principles calculations. Magnon-induced intraband electron scattering is found topredominantly cause magnetization damping, whereas the Néel field is found to be damped via disorder.Depending on the conduction electron band structure, we predict that magnon-induced interband electronscattering around band crossings may be exploited to engineer a strong Néel field damping. DOI: 10.1103/PhysRevB.101.020403 Introduction. The dynamical properties of a harmonic mode are captured by its frequency and lifetime [ 1,2]. While the eigenfrequency is typically determined by the linearizedequations of motion, or equivalently by a noninteracting de-scription of the corresponding quantum excitation, the life-time embodies rich physics stemming from its interactionwith one or more dissipative baths [ 1,3]. Dissipation plays a central role in the system response time. In the context ofmagnetic systems employed as memories, the switching timesdecrease with increasing damping thereby requiring a strongerdissipation for fast operation [ 4–6]. The dissipative properties of the system also result in rich phenomena such as quantumphase transitions [ 7–10]. Furthermore, the formation of hybrid excitations, such as magnon-polarons [ 11–18] and magnon- polaritons [ 19–24], requires the dissipation to be weak with respect to the coupling strengths between the two participatingexcitations [ 25]. Therefore, in several physical phenomena that have emerged in recent years [ 12,16,26–30], damping not only determines the system response but also the very natureof the eigenmodes themselves. Understanding, exploiting, andcontrolling the damping in magnets is thus a foundationalpillar of the field. The success of Landau-Lifshitz-Gilbert (LLG) phe- nomenology [ 31,32] in describing ferromagnetic dynamics has inspired vigorous efforts towards obtaining the Gilbertdamping parameter using a wide range of microscopic the-ories. The quantum particles corresponding to magnetizationdynamics—magnons—provide one such avenue for micro-scopic theories and form the central theme in the field ofmagnonics [ 33,34]. While a vast amount of fruitful research has provided a good understanding of ferromagnets (FMs)[35–54], analogous studies on antiferromagnets (AFMs) are relatively scarce and have just started appearing [ 55,56] due to the recently invigorated field of antiferromagnetic spintronics[57–62]. Among the ongoing discoveries of niches borne by AFMs, from electrically and rapidly switchable memo-ries [ 63], topological spintronics [ 60], long-range magnonic transport [ 64], to quantum fluctuations [ 65], an unexpectedsurprise has been encountered in the first-principles evaluation of damping in metallic AFMs. Liu and co-workers [ 56] and another more recent first-principles study [ 66] both found the magnetization dissipation parameter to be much larger thanthe corresponding Néel damping constant, in stark contrastwith previous assumptions, exhibiting richer features than inFMs. An understanding of this qualitative difference as well asthe general AFM dissipation is crucial for the rapidly growingapplications and fundamental novel phenomena based onAFMs. Here, we accomplish an intuitive and general understand- ing of the Gilbert damping in metallic AFMs based on themagnon picture of AFM dynamics. Employing the s-d,t w o - sublattice model for a metallic AFM, in which the dands electrons constitute the magnetic and conduction subsystems,we derive analytic expressions for the Gilbert damping pa-rameters as a function of the conduction electron density ofstates at the Fermi energy and s -dexchange strength. Our analytic results yield values consistent with available numer-ical studies [ 56,66] and experiments [ 67,68]. The presence of spin-degenerate conduction bands in AFMs is found tobe the key in their qualitatively different damping propertiesas compared to FMs. This allows for absorption of AFMmagnons via s-dexchange-mediated intraband conduction electron spin-flip processes leading to strong damping of themagnetization as compared to the Néel field [ 69]. We also show that interband spin-flip processes, which are forbiddenin our simple AFM model but possible in AFMs with bandcrossings in the conduction electron dispersion, result in astrong Néel field damping. Thus, the general qualitative fea-tures of damping in metallic AFMs demonstrated herein allowus to understand the Gilbert damping given the conductionelectron band structure. These insights provide guidance forengineering AFMs with desired damping properties, whichdepend on the exact role of the AFM in a device. Model. We consider two-sublattice metallic AFMs within thes-dmodel [ 35,36,44]. The delectrons localized at lat- tice sites constitute the magnetic subsystem responsible for 2469-9950/2020/101(2)/020403(7) 020403-1 ©2020 American Physical SocietySIMENSEN, KAMRA, TRONCOSO, AND BRATAAS PHYSICAL REVIEW B 101, 020403(R) (2020) z yx FIG. 1. Schematic depiction of our model for a metallic AFM. The red and blue arrows represent the localized delectrons with spin up and down, and constitute the Néel ordered magnetic subsystems. The green cloud illustrates the delocalized, itinerant selectrons that forms the conduction subsystem. antiferromagnetism, while the itinerant selectrons form the conduction subsystem that accounts for the metallic traits.The two subsystems interact via s-dexchange [Eq. ( 3)]. For ease of depiction and enabling an understanding of qualitativetrends, we here consider a one-dimensional AFM (Fig. 1). The results within this simple model are generalized to AFMs withany dimensionality in a straightforward manner. Furthermore,we primarily focus on the uniform magnetization dynamicsmodes. At each lattice site i, there is a localized delectron with spin S i. The ensuing magnetic subsystem is antifer- romagnetically ordered (Fig. 1), and the quantized excita- tions are magnons [ 70,71]. Disregarding applied fields for simplicity and assuming an easy-axis anisotropy along thezaxis, the magnetic Hamiltonian, H m=˜J/summationtext /angbracketlefti,j/angbracketrightSi·Sj− K/summationtext i(Sz i)2, where /angbracketlefti,j/angbracketrightdenotes summation over nearest- neighbor lattice sites, is quantized and mapped to thesublattice-magnon basis [ 71] H m=/summationdisplay q[Aq(a† qaq+b† qbq)+B† qa† qb† q+Bqaqbq],(1) where we substitute ¯ h=1,Aq=(2˜J+2K)S, and Bq= ˜JSe−iq·a/summationtext /angbracketleftδ/angbracketrighteiq·δ, where S=|Si|,ais the displacement be- tween the two atoms in the basis, and /angbracketleftδ/angbracketrightdenotes summing over nearest-neighbor displacement vectors. aqand bqare bosonic annihilation operators for plane-wave magnons onthe A and B sublattices, respectively. We diagonalize theHamiltonian [Eq. ( 1)] through a Bogoliubov transformation [71]t oH m=/summationtext qωq(α† qαq+β† qβq), with eigenenergies ωq=√ A2 q−|Bq|2. In the absence of an applied field, the magnon modes are degenerate. The selectron conduction subsystem is described by a tight-binding Hamiltonian that includes the “static” contri-bution from the s-dexchange interaction [Eq. ( 3)] discussed below: H e=−t/summationdisplay /angbracketlefti,j/angbracketright/summationdisplay σc† iσcjσ−J/summationdisplay i(−1)i(c† i↑ci↑−c† i↓ci↓).(2) Here ciσis the annihilation operator for an selectron at site iwith spin σ.t(>0) is the hopping parameter, and J(>0) accounts for s-dexchange interaction [Eq. ( 3)]. The ( −1)i factor in the exchange term reflects the two-sublattice nature of the AFM. The conduction subsystem unit cell consists-/2 - /4 0 /2 /4ee kF,1a=μ1mee m=μ2 kF,2a FIG. 2. The selectron dispersion in the metallic AFM model with illustrations of intraband electron-magnon scattering at two different Fermi levels, μ1andμ2. The depicted momentum transfer is exaggerated for clarity. of two basis atoms, similar to the magnetic subsystem. As a result, there are four distinct electron bands: two due tothere being two basis atoms per unit cell, and twice this dueto the two possible spin polarizations. Disregarding appliedfields, these constitute two spin-degenerate bands. We labelthese bands 1 and 2, where the latter is higher in energy.The itinerant electron Hamiltonian [Eq. ( 2)] is diagonalized into an eigenbasis ( c 1kσ,c2kσ) with eigenenergies /epsilon11k=−/epsilon1k and/epsilon12k=+/epsilon1k, where /epsilon1k=/radicalbig J2S2+t2|γk|2, where γk=/summationtext /angbracketleftδ/angbracketrighte−ik·δ. The itinerant electron dispersion is depicted in Fig.2. The magnetic and conduction subsystems interact through s-dexchange interaction, parametrized by J: HI=−J/summationdisplay iSi·si, (3) where si=/summationtext σσ/primec† iσσσσ/primeciσ/primeis the spin of the itinerant elec- trons at site i, where σis the vector of Pauli matrices. The term which is zeroth order in the magnon operators, and thusaccounts for the static magnetic texture, is already includedinH e[Eq. ( 2)]. To first order in magnon operators, the interaction Hamiltonian can be compactly written as He-m=/summationdisplay λρ/summationdisplay kk/primeqc† λk↑cρk/prime↓/parenleftbig WA,λρ kk/primeqa† −q+WB,λρ kk/primeqbq/parenrightbig +H.c., (4) where λandρare summed over the electron band indices. As detailed in the Supplemental Material [ 72],WA,λρ kk/primeqand WB,λρ kk/primeq, both linear in J, are coefficients determining the am- plitudes for scattering between the itinerant electrons and the aqandbqmagnons, respectively. Specifically, when consid- ering plane-wave states, WA/B,λρ kk/primeqbecomes a delta function, thereby enforcing the conservation of crystal momentum in a translationally invariant lattice. Inclusion of disorder orother many-body effects results in deviation of the eigenstates 020403-2MAGNON DECAY THEORY OF GILBERT DAMPING IN … PHYSICAL REVIEW B 101, 020403(R) (2020) from ideal plane waves causing a wave vector spread around its mean value [ 2]. The delta function, associated with an exact crystal momentum conservation, is thus transformed to apeaked function with finite width ( /Delta1k). The λρcombinations 11 and 22 describe intraband electron scattering, while 12 and 21 describe interband scattering. Intraband scattering is illustrated in Fig. 2. The scattering described by H e-m[Eq. ( 4)] transfers spin angular momentum between the magnetic and conductionsubsystems. The itinerant electrons are assumed to maintaina thermal distribution thereby acting as a perfect spin sink.This is consistent with a strong conduction electron spinrelaxation observed in metallic AFMs [ 73,74]. As a result, the magnetic subsystem spin is effectively damped through thes-dexchange interaction. Gilbert damping. In the LLG phenomenology for two- sublattice AFMs, dissipation is accounted via a 2 ×2 Gilbert damping matrix [ 75,76]. Our goal here is to determine the elements of this matrix in terms of the parameters and physicalobservables within our microscopic model. To this end, weevaluate the spin current “pumped” by the magnetic sub-system into the sconduction electrons, which dissipate the spins immediately within our model. The angular momen-tum thus lost by the magnetic subsystem appears as Gilbertdamping in its dynamical equations [ 75,77]. The second es- sential ingredient in identifying the Gilbert damping matrixfrom our microscopic theory is the idea of coherent states[78,79]. The classical LLG description of the magnetization is necessarily equivalent to our quantum formalism, when themagnetic eigenmode is in a coherent state [ 78–80]. Driving the magnetization dynamics via a microwave field, such asin the case of ferromagnetic resonance experiments, achievessuch a coherent magnetization dynamics [ 77,81]. The spin current pumped by a two-sublattice magnetic system into an electronic bath may be expressed as [ 82] I z=Gmm(m×˙m)z+Gnn(n×˙n)z +Gmn[(m×˙n)z+(n×˙m)z],(5) where mandnare the magnetization and Néel field nor- malized by the sublattice magnetization, respectively. Here,G ij=αij×(M/|γ|), where αijare the Gilbert damping coef- ficients, γis the gyromagnetic ratio of the delectrons, and M is the sublattice magnetization. Considering the uniform mag-netization mode, I zis the spin current operator Iz=i[He-m,Sz] [83], where Sz=/summationtext iSz i. We get Iz=i/summationdisplay λρ/summationdisplay kk/primeq/braceleftbig c† λk↑cρk/prime↓/parenleftbig WA,λρ kk/primeqa† −q+WB,λρ kk/primeqbq/parenrightbig −H.c./bracerightbig . (6) The expectation value of this operator assuming the uniformmagnetization mode to be in a coherent state corresponds tothe spin pumping current [Eq. ( 5)]. In order to evaluate the spin pumping current from Eq. ( 6), we follow the method employed to calculate interfacial spinpumping current into normal metals in Refs. [ 77,81,82], and the procedure is described in detail therein. Briefly, thismethod entails assuming the magnetic and conduction subsys-tems to be independent and in equilibrium at t=− ∞ , when the mutual interaction [Eq. ( 4)] is turned on. The subsequenttime evolution of the coupled system allows evaluating its physical observables in steady state. The resulting coherentspin current corresponds to the classical spin current I zthat can be related to the motion of the magnetization and theNéel field [Eq. ( 5)]. As a last step, we identify expressions for (m×˙m) z,(m×˙n)z, and ( n×˙n)zin terms of coherent magnon states, which enables us to identify the Gilbert damp-ing coefficients α mm,αnn, andαmn. Results. Relegating the detailed evaluation to Supplemental Material [ 72], we now present the analytic expression ob- tained for the various coefficients [Eq. ( 5)]. A key assumption that allows these simple expressions is that the electronicdensity of states in the conduction subsystem does not varysignificantly over the magnon energy scale. Furthermore, weaccount for a weak disorder phenomenologically via a finitescattering length lassociated with the conduction electrons. This results in an effective broadening of the electron wavevectors determined by the inverse electron scattering length, (/Delta1k)=2π/l. As a result, the crystal momentum conservation in the system is enforced only within the wave vector broad-ening. By weak disorder we mean that the electron scatteringlength is much larger than the lattice parameter a.I fkandk /prime are the wave vectors of the incoming and outgoing electrons, respectively, we then have ( k−k/prime)a=(/Delta1k)a/lessmuch1. This jus- tifies an expansion in the wave vector broadening ( /Delta1k)a. The Gilbert damping coefficients stemming from intrabandelectron scattering are found to be α mm=α0(ξJ) −α0(ξJ) 4/parenleftBigg 1+ξ2 J/bracketleftbig ξ2 J+8−4 cos2(kFa)/bracketrightbig /bracketleftbig ξ2 J+4 cos2(kFa)/bracketrightbig2/parenrightBigg [(/Delta1k)a]2, αnn=α0(ξJ) 4/parenleftBigg 1+sin2(kFa) cos2(kFa)ξ2 J/bracketleftbig ξ2 J+4 cos2(kFa)/bracketrightbig/parenrightBigg [(/Delta1k)a]2, (7) where ξJ=JS/t,kFis the Fermi momentum, and ais the lattice parameter, and where α0(ξJ)=πv2J2 2g2(μ)|˜V|24 cos2(kFa) ξ2 J+4 cos2(kFa). (8) αmn=αnm=0 due to the equivalence between the two sub- lattices [ 75] in an AFM. Here, vis the unit cell volume, g(/epsilon1) is the density of states per unit volume, μis the Fermi level, andω0is the energy of the q=0magnon mode. ˜Vis a dimen- sionless and generally complex function introduced to accountfor the momentum broadening dependency of the scatteringamplitudes. It satisfies ˜V(0)=1 and 0 /lessorequalslant|˜V(/Delta1k)|/lessorequalslant1 within our model. These analytic expressions for the Gilbert dampingparameters constitute one of the main results of this RapidCommunication. Discussion. The Gilbert damping in metallic AFMs [Eq. ( 7)] bears dependencies similar to the analogous case of spin pumping in AFM|NM bilayers with interfacial exchangecoupling [ 82]. There are, however, two key differences: The s-dexchange coupling exists in the bulk of metallic AFMs, whereas it is localized at the interface in the bilayer structures.Additionally, the itinerant electron wave functions follow 020403-3SIMENSEN, KAMRA, TRONCOSO, AND BRATAAS PHYSICAL REVIEW B 101, 020403(R) (2020) kFaμ ee m Δω0 ka FIG. 3. A schematic depiction of magnon-induced interband scattering in a band (anti-)crossing at the Fermi level. distinct periodicities in metallic AFMs [ 72], FMs and NMs, amounting to qualitative differences. In the limit of weak momentum broadening ( /Delta1k)a/lessmuch1 and weak s-dexchange ξJ/lessmuch1, we arrive at αnn≈0 and αmm=πv2J2g2(μ)/2 for intraband scattering. In this exper- imentally relevant limit we need only two material inputparameters, Jand g(μ), to compute the Gilbert damping. Using a typical value of J∼0.1e V[ 36], and density of states from a selection of experiments and density functional theory(DFT) calculations [ 84–88], we obtain α mmcomparable to the first-principles calculation results [ 56,66] (see Supple- mental Material [ 72] for details). For instance, we find that αmm=0.35 in FeMn, while the DFT calculation in Ref. [ 56] findsαmm=0.38. Moreover, experiments in Mn 90Cu10imply αmm=0.3[67,68]. Since the relevant material parameters are not available in Mn 90Cu10, we assume a g(μ) similar to other manganese-based AFMs, e.g., IrMn, and obtain αmm=0.14. The uniform mode magnon energy is much smaller than the electron band gap within our simple model, prohibitinginterband scattering. However, in real AFM metals with morecomplex band structures, band crossings with gaps smallerthan the magnon energy may exist [ 89–91]. In materials with band crossings at the Fermi level, magnon-induced interbandscattering should also contribute to Gilbert damping. Moti-vated by this, we now consider Gilbert damping stemmingfrom interband scattering within the minimal model, whiledisregarding the energy conservation for the moment, labelingthe coefficients α I mmandαI nn. We then find the same expres- sions as in Eq. ( 7) with the roles of αI mm,nninterchanged with respect to αmm,nn, giving a non-negligible αI nn. Although arriving at this result required disregarding energy conser-vation, the qualitative effect in itself is not an artifact ofthis. For Gilbert damping resulting from band (anti-)crossings(with band gap /Delta1/epsilon1 < ω 0) as depicted in Fig. 3,αI nn/αI mm/greaterorsimilar αnn/αmm. This generic principle derived within our simple model provides valuable guidance for designing materialswith an engineered Gilbert damping matrix. We now provide a rough intuitive picture for the damp- ing dependencies obtained above followed by a moremathematical discussion. Consider a conventional diffractionexperiment where an incident probing wave is able to resolvethe two slits only when the wavelength is comparable to the physical separation between the two slits. In the case at hand,the wave functions of electrons and magnons participating ina scattering process combine in a way that the wave numberby which the conservation of crystal momentum is violatedbecomes the probing wave number within a diffraction pic-ture. Therefore, the processes conserving crystal momentumhave vanishing probing wave number and are not able toresolve the opposite spins localized at adjacent lattice sites.Therefore, only the average magnetization is damped leavingthe Néel field unaffected. With disorder, the probing wavenumber becomes nonzero and thus also couples to the Néelfield. The interband scattering, on the other hand, is rem-iniscent of umklapp scattering in a single-sublattice modeland the probing wave number matches with the inverse lat-tice spacing. Therefore, the coupling with the Néel field isstrong. The Gilbert damping in metallic AFMs here considered is caused by spin pumping from the magnetic subsysteminto the sband, and depends thus on transition amplitudes proportional to products of itinerant electron wave functionssuch as ψ † λk↑(x)ψρk/prime↓(x). The damping on sublattices A and B is therefore a function of/summationtext jcos2(πxj 2a)ψ† λk↑(xj)ψρk/prime↓(xj) and/summationtext jsin2(πxj 2a)ψ† λk↑(xj)ψρk/prime↓(xj), respectively. Equiva- lently, αmmis a function of/summationtext jψ† λk↑(xj)ψρk/prime↓(x), and αnnis af u n c t i o no f/summationtext jcos(πxj a)ψ† λk↑(xj)ψρk/prime↓(x). Assuming plane- wave solutions of the electron wave functions, considering intraband scattering only, we find that αmmis a function of [1−i(/Delta1k)a], where iis the imaginary unit, whereas αnn is a function of ( /Delta1k)a. This coincides well with Eq. ( 7). In the limit of a negligible band gap in the simple modelpresented previously, the upper electron band is a continuationof the lower band with a ±π/amomentum shift. Therefore, interband scattering at momentum kis equivalent to intraband scattering between kandk±π/a. This is indeed the exact phase shift which results in a large α nn. The magnon decay picture developed herein goes be- yond metals, where electron-magnon scattering constitutesthe dominant mechanism for magnon decay and thus Gilbertdamping. In insulators, magnon-phonon and magnon-magnonscattering provide comparable competing damping channels.The present methodology can thus be generalized to insula-tors. From an experimental perspective, magnetic resonancelinewidth measurement has been a standard approach toextracting Gilbert damping. Recent studies [ 55]h a v ee m - ployed terahertz spectroscopy as a means of studying damp-ing in AFMs, while time-resolved magnetization dynamicsoffers further possibilities [ 92,93]. As regards corroborat- ing and exploiting our finding of interband scattering ef-fects, a way to control the degree of interband scatteringat the Fermi level will be helpful in isolating its contribu- tion. This may be achieved by doping an antiferromagnet in which a band crossing is close to the Fermi level [ 89–91], and investigating the Gilbert damping as a function ofdoping. Conclusion. We have provided a microscopic derivation of Gilbert damping resulting from magnon decay through s-d exchange interaction in metallic antiferromagnets. Analyticexpressions for Gilbert damping coefficients resulting from 020403-4MAGNON DECAY THEORY OF GILBERT DAMPING IN … PHYSICAL REVIEW B 101, 020403(R) (2020) intraband electron scattering are presented, while Gilbert damping resulting from interband electron scattering is dis-cussed on a conceptual level. We find that intraband electronscattering gives rise to a large magnetization damping anda negligible Néel field damping. 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PhysRevLett.122.147201.pdf

Anisotropic and Controllable Gilbert-Bloch Dissipation in Spin Valves Akashdeep Kamra,1,*Dmytro M. Polishchuk,2Vladislav Korenivski,2and Arne Brataas1 1Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, Trondheim, Norway 2Nanostructure Physics, Royal Institute of Technology, Stockholm, Sweden (Received 31 October 2018; published 9 April 2019) Spin valves form a key building block in a wide range of spintronic concepts and devices from magnetoresistive read heads to spin-transfer-torque oscillators. We elucidate the dependence of themagnetic damping in the free layer on the angle its equilibrium magnetization makes with that in the fixed layer. The spin pumping-mediated damping is anisotropic and tensorial, with Gilbert- and Bloch-like terms. Our investigation reveals a mechanism for tuning the free layer damping in situ from negligible to a large value via the orientation of fixed layer magnetization, especially when the magnets are electricallyinsulating. Furthermore, we expect the Bloch contribution that emerges from the longitudinal spinaccumulation in the nonmagnetic spacer to play an important role in a wide range of other phenomena inspin valves. DOI: 10.1103/PhysRevLett.122.147201 Introduction. —The phenomenon of magnetoresistance is at the heart of contemporary data storage technologies [1,2]. The dependence of the resistance of a multilayered heterostructure comprising two or more magnets on theangles between their respective magnetizations has beenexploited to read magnetic bits with a high spatial reso- lution [3]. Furthermore, spin valves comprising two mag- netic layers separated by a nonmagnetic conductor havebeen exploited in magnetoresistive random access memo-ries[2,4,5] . Typically, in such structures, one “free layer ” is much thinner than the other “fixed layer ”allowing for magnetization dynamics and switching in the former. The latter serves to spin polarize the charge currents flowing across the device and thus exert spin torques on the former[6–9]. Such structures exhibit a wide range of phenomena from magnetic switching [5]to oscillations [10,11] driven by applied electrical currents. With the rapid progress in taming pure spin currents [12–20], magnetoresistive phenomena have found a new platform in hybrids involving magnetic insulators (MIs).The electrical resistance of a nonmagnetic metal ( N)w a s found to depend upon the magnetic configuration of an adjacent insulating magnet [21–24]. This phenomenon, dubbed spin Hall magnetoresistance (SMR), relies on thepure spin current generated via spin Hall effect (SHE) in N [25,26] . The SHE spin current accumulates spin at the MI=Ninterface, which is absorbed by the MI depending on the angle between its magnetization and the accumulated spin polarization. The net spin current absorbed by the MImanifests as additional magnetization-dependent contribu-tion to resistance in Nvia the inverse SHE. The same principle of magnetization-dependent spin absorption byMI has also been exploited in demonstrating spin Nernsteffect [27], i.e., thermally generated pure spin current, in platinum. Although the ideas presented above have largely been exploited in sensing magnetic fields and magnetizations,tunability of the system dissipation is a valuable, under- exploited consequence of magnetoresistance. Such an electrically controllable resistance of a magnetic wire hosting a domain wall [28] has been suggested as a basic circuit element [29] in a neuromorphic computing [30] architecture. In addition to the electrical resistance or dissipation, the spin valves should allow for controllingthe magnetic damping in the constituent magnets [31]. Such an in situ control can be valuable in, for example, architectures where a magnet is desired to have a large damping to attain low switching times and a low dissipation for spin dynamics and transport [13,16] . Furthermore, a detailed understanding of magnetic damping in spin valves is crucial for their operation as spin-transfer-torque oscil- lators [10] and memory cells [5]. Inspired by these new discoveries [21,27] and previous related ideas [31–34], we suggest new ways of tuning the magnetic damping of the free layer F1in a spin valve (Fig. 1) via controllable absorption by the fixed layer F2of the spin accumulated in the spacer Ndue to spin pumping [31,35] . The principle for this control over spin absorption is akin to the SMR effect discussed above and capitalizeson altering the F 2magnetization direction. When spin relaxation in Nis negligible, the spin lost by F1is equal to the spin absorbed by F2. This lost spin appears as tensorial Gilbert [36] and Bloch [37] damping in F1magnetization dynamics. In its isotropic form, the Gilbert contributionarises due to spin pumping and is well established [31–33, 35,38 –40]. We reveal that the Bloch term results from backflow due to a finite dc longitudinal spin accumulationPHYSICAL REVIEW LETTERS 122, 147201 (2019) 0031-9007 =19=122(14) =147201(6) 147201-1 © 2019 American Physical SocietyinN. Our results for the angular and tensorial dependence of the Gilbert damping are also, to best of our knowledge,new. We show that the dissipation in F 1, expressed in terms of ferromagnetic resonance (FMR) linewidth, varies withthe angle θbetween the two magnetizations (Fig. 3). The maximum dissipation is achieved in collinear or orthogonal configurations depending on the relative size of the spin-mixing g 0rand longitudinal spin glconductances of the NjF2subsystem. For very low gl, which can be achieved employing insulating magnets, the spin pumping mediatedcontribution to the linewidth vanishes for collinear con- figurations and attains a θ-independent value for a small noncollinearity. This can be used to strongly modulate themagnetic dissipation in F 1electrically via, for example, an F2comprised by a magnetoelectric material [41]. FMR linewidth. —Disregarding intrinsic damping for convenience, the magnetization dynamics of F1including a dissipative spin transfer torque arising from the spin current lost Is1may be expressed as _ˆm¼−jγjðˆm×μ0HeffÞþjγj MsVIs1: ð1Þ Here, ˆmis the unit vector along the F1magnetization M treated within the macrospin approximation, γð<0Þis the gyromagnetic ratio, Msis the saturation magnetization, Vis the volume of F1, and Heffis the effective magnetic field. Under certain assumptions of linearity as will be detailedlater, Eq. (1)reduces to the Landau-Lifshitz equation with Gilbert-Bloch damping [36,37] _ˆm¼−jγjðˆm×μ 0HeffÞþð ˆm×GÞ−B: ð2Þ Considering the equilibrium orientation ˆmeq¼ˆz, Eq. (2)is restricted to the small transverse dynamics described by mx;y≪1, while the zcomponent is fully determined by the constraint ˆm·ˆm¼1. Parametrized by a diagonal dimen- sionless tensor ˇα, the Gilbert damping has been incorporatedviaG¼αxx_mxˆxþαyy_myˆyin Eq. (2). The Bloch damping is parametrized via a diagonal frequency tensor ˇΩas B¼ΩxxmxˆxþΩyymyˆy. A more familiar, although insuffi- cient for the present considerations, form of Bloch damping can be obtained by assuming isotropy in the transverse plane:B¼Ω 0ðˆm−ˆmeqÞ. This form, restricted to transverse dynamics, makes its effect as a relaxation mechanism with characteristic time 1=Ω0evident. The Bloch damping, in general, captures the so-called inhomogeneous broadening and other, frequency independent contributions to the mag-netic damping. Considering uniaxial easy-axis and easy-plane anisotro- pies, parametrized, respectively, by K zand Kx[42], the magnetic free energy density Fmis expressed as Fm¼−μ0M·Hext−KzM2zþKxM2x, with Hext¼H0ˆzþ hrfas the applied static plus microwave field. Employing the effective field μ0Heff¼−∂Fm=∂Min Eq. (2)and switching to Fourier space [ ∼expðiωtÞ], we obtain the resonance frequency ωr¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω0ðω0þωaxÞp . Here, ω0≡ jγjðμ0H0þ2KzMsÞandωax≡jγj2KxMs. The FMR line- width is evaluated as jγjμ0ΔH¼αxxþαyy 2ωþtΩxxþΩyy 2 þtωax 4ðαyy−αxxÞ; ð3Þ where ωis the frequency of the applied microwave field hrfand is approximately ωrclose to resonance, and t≡ ω=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2þω2ax=4p ≈1for a weak easy-plane anisotropy. Thus, in addition to the anisotropic Gilbert contributions, the Bloch damping provides a nearly frequency-independent offset in the linewidth. Spin flow. —We now examine spin transport in the device with the aim of obtaining the damping parameters that determine the linewidth [Eq. (3)]. The Nlayer is considered thick enough to eliminate static exchange interactionbetween the two magnetic layers [31,40] . Furthermore, we neglect the imaginary part of the spin-mixing conduct- ance, which is small in metallic systems and does notaffect dissipation in any case. Disregarding longitudinal spin transport and relaxation in the thin free layer, the net spin current I s1lost by F1is the difference between the spin pumping and backflow currents [31] Is1¼gr 4πðℏˆm×_ˆm−ˆm×μs׈mÞ; ð4Þ where gris the real part of the F1jNinterfacial spin-mixing conductance, and μsis the spatially homogeneous spin accumulation in the thin Nlayer. The spin current absorbed byF2may be expressed as [31]FIG. 1. Schematic depiction of the device under investigation. The blue arrows denote the magnetizations. The fixed layer F2 magnetization remains static. The free layer F1magnetization precesses about the zaxis with an average cone angle Θ≪1. The two layers interact dynamically via spin pumping and backflowcurrents.PHYSICAL REVIEW LETTERS 122, 147201 (2019) 147201-2Is2¼g0r 4πˆm2×μs׈m2þgl 4πðˆm2·μsÞˆm2; ≡X i;j¼fx;y;z ggij 4πμsjˆi; ð5Þ where glandg0rare, respectively, the longitudinal spin conductance and the real part of the interfacial spin-mixingconductance of the NjF 2subsystem, ˆm2denotes the unit vector along F2magnetization, and gij¼gjiare the components of the resulting total spin conductance tensor. glquantifies the absorption of the spin current along the direction of ˆm2, the so-called longitudinal spin current. For metallic magnets, it is dominated by the diffusive spincurrent carried by the itinerant electrons, which is dissi- pated over the spin relaxation length [31]. On the other hand, for insulating magnets, the longitudinal spin absorp-tion is dominated by magnons [43,44] and is typically much smaller than for the metallic case, especially at low temperatures. Considering ˆm 2¼sinθˆyþcosθˆz(Fig. 1), Eq.(5)yields gxx¼g0r,gyy¼g0rcos2θþglsin2θ,gzz¼ g0rsin2θþglcos2θ,gxy¼gyx¼gxz¼gzx¼0, and gyz¼ gzy¼ðgl−g0rÞsinθcosθ. Relegating the consideration of a small but finite spin relaxation in the thin Nlayer to the Supplemental Material [45], we assume here that the spin current lost by F1is absorbed by F2, i.e., Is1¼Is2. Imposing this spin current conservation condition, the spin accumulation in Nalong with the currents themselves can be determined. We are primarily interested in the transverse ( xandy) components of the spin current because these fully determine themagnetization dynamics ( ˆm·ˆm¼1) I s1x¼1 4πgrgxx grþgxxð−ℏ_myþmxμszÞ; Is1y¼1 4π/C18grgyy grþgyyðℏ_mxþmyμszÞþgyzμszð1−lyÞ/C19 ; μsz¼ℏgrðlxmx_my−lymy_mx−p_mxÞ gzz−pgyzþgrðlxm2xþlym2yþ2pm yÞ; ð6Þ where lx;y≡gxx;yy=ðgrþgxx;yyÞand p≡gyz=ðgrþgyyÞ. The spin lost by F1appears as damping in the magneti- zation dynamics [Eqs. (1)and(2)][31,35] . We pause to comment on the behavior of μszthus obtained [Eq. (6)]. Typically, μszis considered to be first or second order in the cone angle, and thus negligibly small. However, as discussed below, an essential new finding is that it becomes independent of the cone angleand large under certain conditions. For a collinear con- figuration and vanishing g l,gzz¼gyz¼0results in ˜μsz≡ μsz=ℏω→1[38]. Its finite dc value contributes to the Bloch damping [Eq. (6)][38]. For a noncollinear configu- ration, μsz≈−ℏgrp_mx=ðgzz−pgyzÞand contributes toGilbert damping via Is1y[Eq. (6)]. Thus, in general, we may express the spin accumulation as μsz¼μsz0þμsz1 [46], where μsz0is the dc value and μsz1∝_mxis the linear oscillating component. μsz0andμsz1contribute, respec- tively, to Bloch and Gilbert damping. Gilbert-Bloch dissipation. —Equations (1)and(6)com- pletely determine the magnetic damping in F1. However, these equations are nonlinear and cannot be captured within our linearized framework [Eqs. (2)and(3)]. The leading order effects, however, are linear in all but a narrow range of parameters. Evaluating these leading order terms within reasonable approximations detailed in the SupplementalMaterial [45], we are able to obtain the Gilbert and Bloch damping tensors ˇαand ˇΩ. Obtaining the general result numerically [45], we present the analytic expressions for two cases covering a large range of the parameter space below. First, we consider the collinear configurations in the limit of ˜g l≡gl=gr→0. As discussed above, we obtain ˜μsz0≡ μsz0=ℏω→1and ˜μsz1≡μsz1=ℏω→0[Eq. (6)]. Thus, the components of the damping tensors can be directly read from Eq. (6)as ˜αxx;yy≡αxx;yy=αss¼ly;x¼g0r= ðgrþg0rÞ¼ ˜g0r=ð1þ˜g0rÞ, and ˜Ωxx;yy≡Ωxx;yy=ðαssωÞ¼ −lx;yμsz0=ðℏωÞ¼−g0r=ðgrþg0rÞ¼−˜g0r=ð1þ˜g0rÞ. Here, we defined ˜g0r≡g0r=grandαss≡ℏgrjγj=ð4πMsVÞis the Gilbert constant for the case of spin pumping into an idealspin sink [31,35] . Substituting these values in Eq. (3),w e find that the linewidth, or equivalently damping, vanishes. This is understandable because the system we haveconsidered is not able to relax the zcomponent of the spin at all. There can, thus, be no net contribution to magnetic damping. μ sz0accumulated in Nopposes the Gilbert relaxation via a negative Bloch contribution [38]. The latter may also be understood as an antidamping spin transfer torque due to the accumulated spin [6].FIG. 2. Normalized damping parameters for F1magnetization dynamics vs spin-valve configuration angle θ(Fig. 1).˜αxx≠˜αyy signifies the tensorial nature of the Gilbert damping. The Bloch parameters ˜Ωxx≈˜Ωyyare largest for the collinear configuration. The curves are mirror symmetric about θ¼90°. ˜g0r¼1, ˜gl¼0.01,Θ¼0.1,ω0¼10×2πGHz, and ωax¼1×2πGHz.PHYSICAL REVIEW LETTERS 122, 147201 (2019) 147201-3Next, we assume the system to be in a noncollinear configuration such that ˜μsz0→0and may be disregarded, while ˜μsz1simplifies to ˜μsz1¼−_mx ωð˜gl−˜g0rÞsinθcosθ ˜g0r˜glþ˜glcos2θþ˜g0rsin2θ; ð7Þ where ˜gl≡gl=grand ˜g0r≡g0r=gras above. This in turn yields the following Gilbert parameters via Eq. (6), with the Bloch tensor vanishing on account of ˜μsz0→0 ˜αxx¼˜g0r˜gl ˜g0r˜glþ˜glcos2θþ˜g0rsin2θ; ˜αyy¼˜g0r 1þ˜g0r;ð8Þ where ˜αxx;yy≡αxx;yy=αssas above. Thus, ˜αyyisθinde- pendent because ˆm2lies in the y-zplane and the x component of spin, the absorption of which is captured by ˜αyy, is always orthogonal to ˆm2.˜αxx, on the other hand, strongly varies with θand is generally not equal to ˜αyy highlighting the tensorial nature of the Gilbert damping. Figure 2depicts the configurational dependence of normalized damping parameters. The Bloch parameters are appreciable only close to the collinear configurations on account of their proportionality to μsz0. The θrange over which they decrease to zero is proportional to the cone angleΘ[Eq. (6)]. The Gilbert parameters are described sufficiently accurately by Eq. (8). The linewidth [Eq. (3)] normalized to its value for the case of spin pumping into a perfect spin sink has been plotted in Fig. 3. For low ˜gl, the Bloch contribution partially cancels the Gilbert dissipation, which results in a smaller linewidth close to the collinear configurations [38].A s ˜glincreases, the relevance of Bloch contribution and μsz0diminishes, and the results approachthe limiting condition described analytically by Eq. (8). In this regime, the linewidth dependence exhibits a maxi- mum for either collinear or orthogonal configurationdepending on whether ˜g l=˜g0ris smaller or larger than unity. Physically, this change in the angle with maximum line-width is understood to reflect whether transverse orlongitudinal spin absorption is stronger. We focus now on the case of very low ˜g lwhich can be realized in structures with electrically insulating magnets.Figure 4depicts the linewidth dependence close to the collinear configurations. The evaluated points are markedwith stars and squares, whereas the lines smoothly connectthe calculated points. The gap in data for very small anglesreflects the limited validity of our linear theory, as dis-cussed in the Supplemental Material [45]. As per the limiting case ˜g l→0discussed above, the linewidth should vanish in perfectly collinear states. A more precise state-ment for the validity of this limit is reflected in Fig. 4and Eq.(6)as˜g l=Θ2→0. For sufficiently low ˜gl, the linewidth changes sharply from a negligible value to a large valueover a θrange approximately equal to the cone angle Θ. This shows that systems comprising magnetic insulatorsbearing a very low ˜g lare highly tunable in regard to magnetic or spin damping by relatively small deviation from the collinear configuration. The latter may be accom-plished electrically by employing magnetoelectric material[41] forF 2or via current driven spin transfer torques [6,9,47] . Discussion. —Our identification of damping contribu- tions as Gilbert-like and Bloch-like [Eq. (6)] treats μszas an independent variable that may result from SHE, forexample. When it is caused by spin pumping current and μ sz∝ω, this Gilbert-Bloch distinction is less clear and becomes a matter of preference. Our results demonstratethe possibility of tuning the magnetic damping in an activeFIG. 4. Normalized FMR linewidth of F1for very small ˜gl. The squares and circles denote the evaluated points while thelines are guides to the eye. The linewidth increases from beingnegligible to its saturation value as θbecomes comparable to the average cone angle Θ.˜g 0r¼1,ω0¼10×2πGHz, and ωax¼1×2πGHz.FIG. 3. Normalized ferromagnetic resonance (FMR) linewidth ofF1for different values of the longitudinal spin conductance ˜gl≡gl=grofNjF2bilayer. The various parameters employed are ˜g0r≡g0r=gr¼1,Θ¼0.1rad,ω0¼10×2πGHz, and ωax¼ 1×2πGHz. grandg0rare the spin-mixing conductances of F1jN and NjF2interfaces, respectively. Only the spin pumping- mediated contribution to the linewidth has been consideredand is normalized to its value for the case of spin pumping intoa perfect spin sink [31].PHYSICAL REVIEW LETTERS 122, 147201 (2019) 147201-4magnet via the magnetization of a passive magnetic layer, especially for insulating magnets. In addition to controllingthe dynamics of the uniform mode, this magnetic “gate ” concept [48] can further be employed for modulating the magnon-mediated spin transport in a magnetic insulator [43,44] . The anisotropy in the resulting Gilbert damping may also offer a pathway toward dissipative squeezing [49] of magnetic modes, complementary to the internalanisotropy-mediated “reactive ”squeezing [50,51] . We also found the longitudinal accumulated spin, which is oftendisregarded, to significantly affect the dynamics. This con-tribution is expected to play an important role in a wide range of other phenomena such as spin-valve oscillators. Conclusion. —We have investigated the angular modu- lation of the magnetic damping in a free layer via control ofthe static magnetization in the fixed layer of a spin-valve device. The damping can be engineered to become larger for either collinear or orthogonal configuration by choosingthe longitudinal spin conductance of the fixed layer smalleror larger than its spin-mixing conductance, respectively.The control over damping is predicted to be sharp for spinvalves made from insulating magnets. Our results pave the way for exploiting magneto-damping effects in spin valves. We acknowledge financial support from the Research Council of Norway through its Centers of Excellence funding scheme, Project No. 262633, “QuSpin, ”and from the Swedish Research Council, Project No. 2018-03526,and Stiftelse Olle Engkvist Byggmästare. *akashdeep.kamra@ntnu.no [1] A. Fert, Nobel lecture: Origin, development, and future of spintronics, Rev. Mod. Phys. 80, 1517 (2008) . [2] S. Parkin, X. Jiang, C. Kaiser, A. Panchula, K. Roche, and M. Samant, Magnetically engineered spintronic sensors andmemory, Proc. IEEE 91, 661 (2003) . [3] K. Nagasaka, CPP-GMR technology for magnetic read heads of future high-density recording systems, J. Magn. Magn. Mater. 321, 508 (2009) . [4] J. Åkerman, Toward a universal memory, Science 308, 508 (2005) . [5] S. Bhatti, R. Sbiaa, A. Hirohata, H. Ohno, S. Fukami, and S. N. Piramanayagam, Spintronics based random accessmemory: A review, Mater. Today 20, 530 (2017) . 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[42] The easy plane may stem from the shape anisotropy in thin films, in which case K x¼μ0=2while the easy axis may be magnetocrystalline in nature [52].[43] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J. van Wees, Long-distance transport of magnon spininformation in a magnetic insulator at room temperature,Nat. Phys. 11, 1022 (2015) . [44] S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn, M. Althammer, R. Gross, and H. Huebl, Non-local magnetoresistance in YIG/Pt nanostructures, Appl. Phys. Lett. 107, 172405 (2015) . [45] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.122.147201 for a dis- cussion on the case of collinear configurations with small ˜g l, details on the numerical analysis with related approxima-tions, dependence of the FMR linewidth on the easy-planeanisotropy and spin-mixing conductance g 0r, and the effect of finite spin relaxation in the spacer layer. [46] As detailed in the Supplemental Material [45], we have disregarded the term in μszwhich oscillates with a frequency 2ω. Strictly speaking, this term needs to be included even in our linear analysis, because it produces terms oscillatingwith ωwhen multiplied with another term at ω.H o w e v e r , such a contribution is only relevant in a narrow parameterrange which may be hard to resolve in an experiment.Furthermore, it requires a nonlinear solution to the equationsand is beyond the scope of the present work. [47] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015) . [48] L. J. Cornelissen, J. Liu, B. J. van Wees, and R. A. Duine, Spin-current-controlled modulation of the magnon spin conductance in a three-terminal magnon transistor, Phys. Rev. Lett. 120, 097702 (2018) . [49] A. Kronwald, F. Marquardt, and A. A. Clerk, Dissipative optomechanical squeezing of light, New J. Phys. 16, 063058 (2014) . [50] A. Kamra and W. Belzig, Super-poissonian shot noise of squeezed-magnon mediated spin transport, Phys. Rev. Lett. 116, 146601 (2016) . [51] A. Kamra, U. Agrawal, and W. Belzig, Noninteger-spin magnonic excitations in untextured magnets, Phys. Rev. B 96, 020411(R) (2017) . [52] A. I. Akhiezer, V. G. Bar ’iakhtar, and S. V. Peletminski, Spin Waves (North-Holland Publishing Company, Amsterdam, 1968).PHYSICAL REVIEW LETTERS 122, 147201 (2019) 147201-6

PhysRevA.3.1939.pdf

I'HYSICA LREVIEW A VOLUME 3,NUMBER 6 JUNE1971 Broadening oftheSodiumDLinesbyAtomicHydrogen. AnAnalysis inTerms oftheNaHMolecular Potentials E.L.Lewis,*L.F.McNamara, fandH.H.Michelsf- &ointInstituteforLaboratory Astrophysics,~University ofColorado, Boulder, Colorado 80302 (Received 17December 1970) Theinteratomic forcesbetween sodium andhydrogen atomswhichareresponsible forthe broadening ofthesodiumDlinesarediscussed intermsofcalculated interatomic potential curvesfortheNaHmolecule. Theimportance ofincluding overlapinteractions andofcon- sidering bothupperandlowerstatesofthetrans;tions isemphasized. Acalculation ofthe linebroadening usingamolecular modelforthepairofcolliding atomswiththecorrect adia- baticlimitsyieldsdamping parameters of8.3&&10radsec withanestimated accuracy of 5-10%undertheconditions ofthesolarphotosphere andfitswellwiththeobserved wingsof theabsorption lineprofiles. Thetemperature dependence ofthebroadening isfoundtobe approximately T',andthewidth/shift ratiosareoftheorderof30:1forbothrsonance lines.Crosssections fortransitions induced between thefine-structure states P3~2andP&~22 2 areestimated, andthevalue 0&F23~2=(70+10)7tao atavelocity of1.28&10cmsec'isin agreement withprevious calculations. Ground-state spin-exchange crosssections areesti- matedforhydrogen-sodium collisions as22~aointhesamevelocity range. I.INTRODUCTION Thebroadening ofspectral linesduetocollisions withneutralperturbers isusuallyrelatedtopoten- tialsexpressed intheformofaninverse power seriesintheinteratomic distanceR.'3Suchan approach isconvenient whenthecrosssectionfor broadening collisions ismuchgreater thanthe extentoftheexcited-state wavefunctions, andthe leadingtermintheinteraction seriesissufficient. Thecrosssectionisessentially determined bythe largest valuebooftheimpactparameter 5atwhich thephaseshiftisunity.Sincethephaseshiftcan beexpressed approximately asP=V7,where V istheaverage energy ofinteraction duringthecol- lisionandv=b/vistheduration ofthecoilision, weseethatthefasterthecollision, thegreateris theenergyofinteraction required atboandthe moreimportant istherangeofinteratomic separa- tionsatwhichseveraltermsintheseriesexpan- sionarenecessary andatwhichoverlapcontribu- tionsareimportant. Inmanycasesasingleterminthepowerseries hasbeenusedtodiscuss experimental results, buttheinadequacy ofsuchatreatment andthe importance ofrepulsive contributions evenatlabo- ratorytemperatures hasbeenemphasized bysev- eralauthors. Inclusion ofrepulsion intheform ofaLennard- Jonespotential ofthe6-12typewas investigated byHindmarsh, Petford, andSmith, butsuchananalysis doesnotgivethecorrect temperature dependence andshift/width ratiosin caseswhererepulsive andattractive forcesare approximately ofequalimportance. 'Also,the usualapproximation ofasinglesetofparameters todescribe theinteratomic potential forallrela-tiveorientations ofthecolliding atomsisclearly invalidforallbutSstates,sincetherealsituation involves manyangle-dependent termsandthe variation ofthecoupling ofthespinandorbital angular momenta withinteratomic separation. TheLennard- Jonespotential cantherefore bebest considered asaconvenient parametrization in theabsence ofmorerealistic potentials. Theinclusion ofrepulsive forcesisimportant fortheanalysis ofthebroadening ofstrongabsorp- tionlinesinthesolarspectrum. Itiswellknown thatthewingregions oftheselinesaredetermined bytheconditions ofthephotosphere (-5000'K) andcannotbepredicted usingelemental abundances derived fromequivalent widthanalysis ofweak lines.Particular casesarediscussed by O'Mara,'byLambert andWarner,'byChame- reaux,'andbyMuller, Bascheck, andHolweger.' Inthepresent paperweareconcerned withthe particular caseofthesodium Dlinesforwhich thedominant broadening mechanism iscollisions withhydrogen atoms, thevelocity anddensitycon- ditionsbeingsuchastofallwithintheregionof validity oftheimpactapproximations. Thesystem Na-Hissufficiently simpletoallow reliable calculation ofthemolecular energies using modern high-speed computers. Wehavetherefore proceeded fromsuchmolecular potentials and treated thecolliding atomsasamolecule during thecollisions. Thisapproach avoidsthedifficulties ofseparately computing manytermsintheseries expansion forthepotential inaddition totheover- lapcontributions. II.MOLECULAR POTENTIAL CURVES Ourscattering analysis, intermsofmolecular 1939 1940 LEWIS, McNAMARA, ANDMICHELS potential curvesratherthaninaslowlyconvergent multipole expansion, requires adetailed knowledge ofthemolecular eigenfunctions andenergies associ- atedwiththepossible molecular statesconnecting the'S(3s)andP,&„, &a(3P)statesofsodium and thegroundS(ls)stateofhydrogen. Themolecular statesarisingfromtheinteraction ofground-state sodium andhydrogen are'Z'and'Z'.Thestates arisingfromground-state hydrogen andtheexcited Pq/23/pstateofsodiumare'Z','Z','Il,and Itisimperative, forthepurposes ofthis study,tochooseaprocedure forconstructing these molecular eigenfunctions thatassures proper limiting behavior attheseparated atomlimits. An analysis ofthemolecular symmetry typeindicates thatthelowest'Z'and'IIstateshavemolecular orbital (MO)assignments (lo)(2o)(3o)(1v)(3o) (4o)and(lo)(2o)(3o)(lx)(4o)(2v),respectively. Thesetwostatesare,therefore, correctly repre- sentable byHartree-Fock (HF)wavefunctions in thelimitoflargeinternuclear separations. All othermolecular statesneededforthescattering analysis eitherdonothavetheproperlimiting behavior withintheHFframework orarenotrep- resentable atallwithinthismodel. Thesecon- siderations dictatethatamoregeneral methodof analysis isrequired toadequately describe theNaH molecular system. Thecalculation procedure chosenforthese studiesisthevalence-configuration-interaction (VCI)method.''~Aspin-free nonrelativistic elec- trostatic Hamiltonian isemployed withintheBorn- Oppenheimer approximation. Foradiatomic mol- eculethisapproximation leadstoanelectronic Hamiltonian depending parametrically ontheinter- nuclear separation A.Electronic wavefunctions P(R)aremadetobeoptimum approximations to solutions oftheSchrodinger equation H(R)q(R)=E(R)q(R) byinvoking thevariational principle. Thespecific formfor((R)maybewritten a.s g(R)=Z„c„g„(R), (2) whereeachg(R)isreferred toasaconfiguration andhasthegeneralstructure $(R)=A6,IIP„;(r(,R)8~ (3) i=1 whereP„;isaspatialorbital, Aistheantisym- metrizing operator, p,isthespin-projection opera- torforspinquantum number S,andtI)„isaproduct ofnandPone-electron spinfunctions ofmagnetic quantum number M,.Ifnorestriction isimposed astothedoubleoccupancy ofthespatialorbitals, Eqs.(2)and(3)candescribe acompletely general wavefunction. InHFcalculations, $(R)isrestricted toasingle |)~(R)whichisassumed toconsistasnearlyaspos-sibleofdoublyoccupied orbitals. Theseorbitals arethenselected tobethebestlinearcombination ofbasisorbitals (LCAO) thatsatisfy thevariational principle. TheHFcomputational method hasbeen thoroughly discussed intheliterature.''"Incon- figuration- interaction calculations, thesummation inEq.(2)hasmorethanoneterm,andthec,.are determined fromthesecular equation g„(H„—ES„„)c„=0, where H„„=fg,*(R)H(R)tt'„(R)d7., S„„=fy*,(R)p„(R)dr.(4) Equation (4)issolved bymatrixdiagonalization. Thematrixelements H„„andS„„appearing inEq. (4)mayfinallybereduced byappropriate operator algebra tosummations ofproducts ofone-andtwo- electron integrals overthebasisfunctions. Thebasisfunctions chosenfordescribing the molecular system NaHwereoptimized Slater-type atomicorbitals(STO's). Aminimum basiswas chosenfortheKandLinnershellsofNaandan extended basiswasusedtodescribe thevalence electrons. Configuration interaction wastakenonly overthevalence-shell electrons (frozenK,L,shell approximation). Suchawavefunction givesanac- curaterepresentation oftheinteratomic potentials atlargeinternuclear separations butmustbecome poorer atshortseparations whereL-shellpolariza- tioneffectsareimportant. Both0-andm-coupled valence configurations wereincluded aswellas ionicconfigurations corresponding toNa'Hand NaO'.Theresults ofthesecalculations aregiven inTableIandFig.1. Thecalculated potential curvesfortheexcited stateshavebeenuniformly adjusted bythesmall calculated error(0.031eV)inthePStermvalue- inordertoensureproperatomiclimiting behavior. Anaverage termvalueisemployed sincetheII molecular statesaredegenerate withinourapproxi- mateHamiltonian. Todetermine theregionof validity oftheCIwavefunctions, aparallel HF analysis wascarried outforthelowest'Z'andII states. TheHFwavefunction wasoptimized asa function oftheinternuclear separation andthus includes inner-shell distortion andpolarization effects. Theresultsofthesecalculations arealso showninTableIandFig.1.Theyindicate that theregionofvalidity ofthepotential curvesderived fromtheCIwavefunctions extends downtoabout 3A,whichisadequate forouruse. Themostcharacteristic feature ofthesecalcu- latedpotentials isthestrongattractive behavior of thefirstexcited'Z'state.Examination oftheCI expansion forthisstateindicates thatthewave function canbewritten toafirstapproximation as BROADENING 0FSODIUM DLINES ~.. 1941 -2 C9 CL LIJ LIJ I LIJ O CL0—2,I03 Na+H Na+H 4,384eV AT R=oo 0.0 Na+H 4 I I 289 I I II0 j I I 3 4 5 INTERNUCLEAR SEPARATION00QO FIG.1.Sodiumhydride molecular potentials. p=tI„«[(ls„+3pz, }—X(lsH—3pw,}'), where~-1forlargeRandX(1atshortersepara- tions.Thusthebindingischaracterized bya doublyoccupied oMOformedfromthe1sorbital ofHandthe3pporbitalofNa.Theground'Z' stateissimilarly characterized byadoublyoc- cupied0MOformedfrom1sofHand3sofNa,butheretheionicconfiguration Na'Happears to havegreater weight. The'Z'statesarecharac- teristically repulsive. TheIIstatesappeartohave nostronglong-range interaction although the'II stateindicates aslightattraction forshortersep- arations. Theseresultsareallsimilar tothose foundbyBender andDavidson' inaparallel study oftheexcitedstatesofLiH. Animportant characteristic oftheseNaHmolec- ularwavefunctions iscommon toallofthediatomic alkalihydrides. Forallsuchsystems itisneces- sarytoincludeconfigurations ofanioniccharacter. Inthiscasetheimportant configuration isNa'H whichliesat4.383eVabovetheNagroundstate. Thevariation oftheenergyofsuchaconfiguration withRisessentially aCoulomb curveindicated by thebrokenlineinFig.1andisof'Z'symmetry. Itisimportant tonotethatthe'Z'ionicstatecan causeperturbation oftheA'Z'andX'Z'statesat comparatively largeinternuclear distances. The perturbation oftheX'Z'stateisparticularly im- portant andattheequilibrium separation ithasan ioniccharacter. Fromthepointofthebroadening oftheDlinestheground-state perturbation issig- nificant sincethewholeoftheground-state pertur- bationisintheZ'stateswhiletheZ'statesderived fromtheresonance levelsarestatistically only 3 oftheinteraction intheexcitedstates ~Itisthere- foreexpected thatthegroundstatewillcontribute significantly tothebroadening oftheresonance lines. Theuseofthesepotentials isvalidonlywhenthe Born-Oppenheimer approximation isvalidinits fullestsense. Theneglected termscanbethought ofasintroducing correction to,andmixingbetween, TABLEI.Calculated potential energies forNaH.Thenumbers ofconfigurations foreachsymmetry typeareZ'(12), Z(8),II(8),andII(8).Thecalculated energyatinfiniteseparations forthegroundstateis—161.6800hartree. R (bohr) x'z' CI3g+I HF CIa3II HF 15.0 14.0 13.0 12.0 11.0 10.5 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.0 5.0 4.00.0 0.0 0.0—0.00001 —0.00004 —0.00013 —P.00022—0.00037—0.00063 000105—0.00175—0.00291 —0.00477 PP0775—0.01877—P.03735—0.055580.07731 0.07731 0.07728 0.07726 0.07722 0.07646 0.07588 0.07492 0.07340 0.07109 0.06777 0.06330 0.05777 0.05154 0.03973 0.03318 0.035990.0 0.0 0~0 0.0 0.00001 0.00003 0.00005 0.00010 0.00018 0.00033 0.00057 0~00096 000156 0.00246 0.00548 0.01097 0.020010.0 0.0 0.0 —0.00002 0.00035 0.00238 0.00405 0.00925 0.017660.07731 0.07731 0.07731 0.07730 0.07730 0.07733 0.07740 0.07754 0.07781 0.07829 0.07911 0.08048 0.08266 0.08604 0.09366 0.12450 0.166750.07731 0.07731 0.07731 0.07730 ~~~ 0.07728 0.07732 P.07759 0.07871 0.083880.07731 0.07731 P.07731 P.07728 ~~~ 0.07701 0.07651 0.07535 0.07337 0.073520.07731 0.07731 0.07735 0.07718 ~d~ 0.07636 0.07537 0.07385 0.07198 0.07107 1942 LEWIS, McNAMARA, ANDMICHELS x&j(m(ls, lj~ma&&jtmi Is.'lgfmf& (9a} FIG.2.Collision axes. theBorn-Oppenheimer statesPt.Hirschfelder andMeath' havediscussed thesetermswhichare oftheform andEfp,"(r,p)v',p(,))(er Atlowdensities andreasonably highvelocities linebroadening isdetermined bytwo-body colli- sionsofshortduration between theexcited atom andperturbers. Thepressure broadened compo- nentofthelineshapeisthenoftheLorentzian form f((v)~1/[((u, -(u-d)'+u'I, (7} whereuanddaretherealandimaginary partsof thedamping parameter &Na(v)v&„=Nff(v)dv2vfb[1-ll(b, v)jdb (9) Weworkinaproductspace IAmi;j2.mm&=IA. ,Ami&IH, jzm2&, whereonlyground-state termsofhydrogen are considered sothattheexpression forfl(b,v)may bewrittenZ )t),(r,R)(p'(~(p',P,(r,K)dr,a+ ()frk wherepisthereduced massoftheatomsofmass M,andM~.Acorrection isalsointroduced dueto thecoupling between theelectronic angular momen- tumandthemol.ecularrotation whichleadstoA doubling inrealmolecules. Wheretheyhavebeen considered indetailtheseeffectsintroduce only smallcorrections totheBorn-Oppenheimer ener- giesandaretherefore neglected inourtreatment. Thepresent work,therefore, represents anat- tempttouserealistic molecular potentials ina linebroadening calculation foracaseinwhichthe useofanexpansion inaconventional seriesin powers of8'wouldentailmanyterms. III.COLLISION THEORYItisunderstood thatatraceovertheperturber statesweighted bythedensitymatrixhasbeen taken.TheS,denotethescattering matrices for theatomiccollision calculated withreference framedefined bythecollision forboththeinitial andfinalstates. Theaverage overtheorientation ofthecollision frames withrespecttothelabora- toryreference framehasbeenmade,foraniso- tropicenvironment, inderiving(9).Onesuitable collision framedefined inFig.2isthatforwhich z,liesalongtheimpactparameter b.Inourcal- culation wemaketheapproximation, usuallyre- ferredtoasthe"classical path,"thatthepertur- berwithvelocity vtravels alongarectilinear path perpendicular tob. Wedenotethesodium groundstateS~~zasthe statefandtheexcitedstatesP,q„,I,asthestate i,thensincetheSmatrixforanySstateisneces- sarilydiagonal, Eq,(Qa)reduces to =...(',.'",)' x&j,m,Is,Ij,m,&&.'m,Is-I,'m,)-,(9b) andobtainasolution s,=(a(t=+~)Ia(t=-~)& Theadiabatic solution of(11)is t e'(t)=exp (-~p E(t)er'}p(t),' where [~(t)-E(t)j(t (t)=0. Usingamolecular basis Im(t))withinternuclear axisaszaxis,(11)becomessinceallj'sandm'sareoddhalf-integers. Tocalculate theSmatrices weproceedasfol- lows.Firstwewritethecoefficients oftheatomic functionsIj,m„j,m,&asacolumnvector la&and proceed inasimilar wayforthecoefficients ofthe molecular stateswhichdescribe NaH,theninthe notation described &=Ai where Aisthematrixforthecoupling oftheatomic functions intomolecular functions. WehavetosolvetheSchrodinger equation itf—„,Ie(t)&=@I' (t))d ff(h,v)= fft)75j(yftlI%)~N(1)2J(em(erie)itf—Im(t))=v,Im(t)&,d BROADENING 0FSODIUM DLINES... 1943 andinthebasisIm'(t)&referred tobaszaxis,we have fn„—,/~'«)&=[DV.D-'-fh(DD ')](~'(f)&,(»") whereDistheappropriate rotation matrixforthe angle8.Neglecting thesecondterminsquare brackets weobtain fI— ~m'(t)&=V,' ~m'(f)& (»'") andV,inEq.(11')isalwaysdiagonal. Thusthe neglect oftermsinDisequivalent tothemodelin whichthepairofatomsisdescribed asarotating molecule throughout thecollision. Theapproxima- tionisvalidiftermsofD,whichareoftheorder 8=v/5=1/7,aresmallcompared totheinter- atomicperturbation V.Clearlyforclosecollisions suchatreatment isvalidbutcannotbeused,in general, fortheregion b&bo,where V7&1. Ourderivation oftheSmatrixisbestunderstood intermsofFig.2.Ifwedefineaninteraction spherewithinwhichthemajorpartofthephase shiftoccurs andwithinwhichthemolecular ap- proximation isvalid,then(13)becomes (.-)&=s'&~ (14a) whereS''isthephase-shift matrix +OO S«=exp ——V«tdt=e'« m40(14b) The ft)«calculated foravelocity of10cmsec' areplottedagainst binFig.3.Foracollision in whichthemolecule rotates through2awehavein thecollision frame gim)D()g(~)D(+)-1 andintheatomicbasiswithrespecttothecollision frame, D(/)St~D(Q) (16) Forclosecollisions e-&mandfordistantcollisions a-0,i.e.,D(n)=I.Thesearethetwoapproxi- mations weusefortheregion b&boandb&bo,re- spectively. Whiletheapproximation atlargebhas, ingeneral, littleformalvalidity, thefactthatan averageiseventually takenoverthemlevelsof theatomicsystemleadstocloseagreement with theresults ofnumerical solutions of(11)inthe atomicbasis.~'Clearly, thefirsttermintheex- pansion oftheSmatrix, whichcontributed tothe shift,isthesameintherigorous andtheapproxi- matetreatments whichthusgivethesamelimiting behavior atlargeb. TheformofD(zv)wasderived withtheassump- tionthattheelectron spinandtheorbitalangular momentum ofthesodium atomweredecoupledduringthecollision andthatonlytheorbitalangular momentum followed therotation ofthemolecular axis.Thisisjustifiable fortheexcitedconfigura- tionsincethespin-orbit parameters insodium and magnesium (theunitedatomlimitofNaH)areonly 11.5and40.5cm',respectively. Thereisthus nottimeforthespintoprecess abouttheorbital angular momentum andfollowthemolecular axis duringaclosecollision. Thegroundstatesand theionicstatesleadtostatesof"Z'symmetry onlyfc,rwhichD(a)=—I.Inthepresentcase,as willbeshownbelow,themajorcontribution tothe crosssections atlargebcomesfromthelowest"Z'statessothattheuseof(16)withD-=Ijsa goodapproximation forb&bp. Themolecular basisstatesusedwereofthetype A,Mi.„,Mz,D&,ratherthanthe+and- stateswhicharethecorrectbasisfortherotational statesofamolecule. TheAforcombinations of ground-state sodium andhydrogen atomshasasim- pleform,theiX'Z',1,1;2&andiX'Z';—1,—1;—2& beingidentical toatomicproducts ofatomicstates andthe iX~Z',0,1,;0)and iX'Z',0,0;0)being symmetric andantisymmetric combinations of atomicstates.Forthestatesarisingfromthe sodiumP»„, ~~resonance levelsandthehydrogen state,theAmatrixismorecomplicated, butmay bederived simplyfromthestarting pointofdeter- minantal productstatesid).Since ij,m„jzm2& =Pld)and i'"A,Mr„Mz, 0&=Qid&then 1jgmg,j2ll1g&=PQ'I'''A,M~,Mz;ft&.Here wehavemadethereasonable assumption, justified below,thatthespin-orbit interaction inthesodium atomislessthantheelectrostatic termsforallthe interatomic separations Rofinterest. Thuswepro- ceeddirectly toHund'scase(b)whereas thereis inrealityatransition regionfromcase(c)tocase (b)atabout14ao. Inthisregionthephaseshifts areoftheorderof0.1sothatneglect ofthecor- recttransition between thetwocoupling conditions willnotintroduce anappreciable error. Theexis- tenceofthe'Z'ionictermwhichstrongly perturbs themolecular levelsatabout13aonarrows the transition regionandreduces theimportance ofthe effect. Theresulting Smatrixfortheex"itedstateis 12-dimensional, andforthegroundstate4-dimen- sional. Sinceweareconsidering thesimultaneous effectofacollision ontheupperandlowerstates, andtherearenodipolematrixelements between thehydrogen groundstateswithm,=&andm,=—&, weinclude inouraverage onlytermswheretheS matrices fortheupperandlowerlevelshavesimul- taneously hydrogen statesm,=&orm,=——,.The resultsforthetwocasesareobviously identical, sothatwerequire onlya6-dimensional excited- stateanda2-dimensional ground-state Smatrix. FromEq.(15)onethenobtains, withD(o)=I, 1944 LEWIS, McNAMARA, ANDMICHELS cmsec' 2 LL I CAf) 0 LL)(ff&-I CL2FIG.3.Phaseshiftscalculated forrelative velocity ofcollision 106cmsec 1. T4 789 IO II IMPACT PARAMETER, (BOHRS)l2 l4 (b)~Afe&El& 1gie&x L& 3/2&=Se +Ss (b)Igsic&E/&IDie&x E&1/2 &~18 +18 -rerr)-rc()2-reE)~-re+ 7-re(rr)g-re(rr)-re(E)2-re(E)-2 +2e =Veie(rr)2e-re('rr)+ Se'~''~e'~+e+~e +4e -fe(rr)-re(rrr) 1-ry(E)+ye +4(1Va) (1vb) fl(bE)3efo&x E&&efo&x&L&,v—,e+,e7 (18) andifonlytheexcited-state interactions areused weagainobtainequalshiftsandwidthsforthetwo lineswith Il(b,I&)=[(2L+1)Z(2S+1)]'Ifonlytheground-state interactions areimportant thelineswouldshowequalbroadening andshift withD'1(-re(3H) 'e(1D)-18r'(E)—24~ —1«'"").(20) Inthiscasetheaverage overperturber states requires theuseofthefull12-and4-dimensional Smatrices fortheupperandlowerstates,re-spectivelyy. Qurformulation alsoyieldsanexpression forthe crosssectionfortransitions between thefine-struc- turelevels P1/2and'Ps/2ofthesodium atomin- ducedbycollisions withatomichydrogen. Thisex- pression isvalidwhenthefine-structure energy separation isneglected andisthusthehigh-velocity limit.Wefindthat o3/31/32&f J&&bdb—,3[(2e -«e&'n& -e&3n&)(c.c) -ie&3c& -ie&&c&2-ie&311&)( (2S+1) (2S1)e-foI &I/L&I S,Ng Usingtheapproximation D(&I)=D(3'I/)andEq.(16) weobtaintheresult,validforsmallb,intheform of(17a)butwiththequantities A,etc.,replaced by+2(e-ie &L&s-ie&E&)(c.c) -ie&III& -ie&n&)()] andnecessarily obtain +1/23/22~3/2 1/2' (21b) &(e-io& II&e-ie& II&14e-ie&E&4e-ie&E&) e-io&11&+e-fe&'11&4-ie&3c&2-fo&I)E&I) C'=—,',(ve'"'"'-Ve"""'—176e"'"' -ie&E&)Fromtheground-state collision matrix wefind thatthecrosssectionforspinexchange between thesodium andhydrogen atomsis cr=2&fJ-',(1—cos[p(X'Z) —p(X'Z)])bdb (22) intheformgivenbyPurcell andFieldwhenhy- perfinestructure isneglected. BROADENING OFSOSODIUM DLINES l61945 OADENING BYBOTHSTATES '—BROADENING BYLOWER STATE —----ROTATI ONSTATE ONLY l4— INCLUDED I fl ll IIIII Il„i,g)I'l2— VELOCITY = l.28IIflocm.sec Xl6— I I I 0 2468 IMPACTPARAMETER, b(BOHRS)68 IO 12 FIG.4.I(v,b)=[l-—ReII(b,v)]bforthesodiumD lingatvelocjtJJ ofl.28x10cmsec~.l4 IV.RESULTS ANDDISCUSSION A.LineBroadening TheSmatrices fortheuerppandlowerstates cuatedusingboth(17)and'20 oFig —ReeeviorofP(b,v)= 2S.(.))-..:)- II(b,v)fortheP «2lineforvelocity1.28x10cmse locitydistribution tth atco pproximateIy thepeakoftheve- pe opoo uionattheternrat ecorresponding uanti ~ ~ 2 2oiePIi2—Sg/2line. Itisinteresting toconsider theco thecrosssetfecionfromtheuerierecontributions to separatel aspperandlowerstates eyasgivenbyEqs.(18)and~19' brokenlinesinFi45rethoseobta'ned ings.4and5a eowerstateinteractions onlyandcl givethemajorcontributcearly nriuiontothewidthsatla atomicaxisthrouhthe,erotation oftheinter- contribution oftheseugeco11ision doesnotaffectthe oesestatestothecrosss 1,3+rpeaof~b,v&isdu wicstatistically constit1 th ttWh levelaloes. entheeffectsoftheuper loneareconsidered P(b,v)doesn'tliti 1ofP'btt1oot'btonriutesignificantl . y goog aaictheoryincludi miingbehavior atsma115asth* zaxisapproximation Inasefixed crosssectionsion.Incalculatin gthebroadening ,various valuesofbweretakenforSHIFTS BYBOTHLEVELS—.—SHIFTS BYLOWER LEVEL ONLY R0TAT I0NINCLUDED IIIii&IIIIIII IIIII~siIIIIItii 2—il ''i) VEIOCITY =I l.28Ixl06cm.s ~I.sec. 6— iI I I I I 68 IO I2 IMPACT PARI2I4 ARAMETER, b(BQHRS) 5.q(b,v)=rmrr(b v,v)bforthesodiumD lo'tof1.28&110cmsec'.-8 2 l6 FIG.thechanechangeoverinEq.(16)froD—'tD0 eregionof8aoneitherofth imations isent1eseapprox- uncertaintireysatisfactor y,andtheexpected ofthiyinourresultsisd ssection.discussed attheend Theshiftsobtainedarevereareverysmallandtothered broadau30oftheful1widthofthoepressure adenedcomponent ofthli 11hift f1tthenes.These ecefactthatthelevelswhich e ~oem,the'Z'levels inpairswhosecet e nersofgravitare portantregionofRye,intheim- free-atom eneno,displaced verlitt energy.ylefromthe Thebroadening constants calculat oupperandlowerstateinterac areessentially equalfineractions quaorthetwoDlines. velocity dependenc fthThe ningandshiftcross eoebroadenin vR,and0,aregiveninFi.6an regionofourcalculatcuaions,correspond a matelytoavariati f inbt th dWionoTwhichisint vanerWaalsresultofT th1ott'o1 rsaeinteractions leadstoaim sameexponent forthoalmostthe acalculation whichioretemperature deependence, but wcincludes onlytheuer interactions leadstliupperstate totheincreasinsoaslightlylareregrexponent due singimportance ofthe' increasing terntemperature.e'llstateswith TableIIsumsummarizes thecontr'but 11tdd havebeenaveragedowandd,aftertheerosssections tribution for5000'K.WesrageoveraMaxwellian velocity dis— K.Weseethattheinclusion of i946 LEWIS, McNAMARA, ANDMICHELS l40 l20— l00— a080— 60—FIG.6.Temperature dependence ofcrosssections. 40— 20— 0 )05IIitII l 5 l06 2 VELOCITY, crnsec'l07 onlythelowerstateleadstowidthsapproximately 10ksmaller thanthecomplete result, whilethein- clusionofonlytheupperstateinteractions reduces thewidthsto-,'oftheirfullvalue. Inallcasesthe w,d,andwidth/shift ratioarethesameforthetwo resonance lines.Thefine-structure transition rate isincluded inTableIIinthesameunitsas~andd forcomparison. Inastrophysical calculations itisusualtowork withasingle-parameter interaction V(R)=—hC6R whereC~isthedifference ofthequantities Cs(i) =(e/h)o(r,),oisthepolarizability oftheground stateoftheperturber, and(r,)isobtained from theapproximate expression;'a,n*'[5n*'+ 1 —3l(f+I)],forthetwolevelsinvolved inthetran- sition. Withtheseexpressions hC,=90.3hartree, whereRisinbohrs. Withtheaidofconventional phase-shift theory oneobtains thevaluegivenin thefourthlineofTableIIwithawidth-to-shift ratioof2.76andatemperature dependence ofT'. Theresultofusingourcalculated molecular poten- tialsratherthantheestimated vanderWaalscoef- ficientsistoincrease thebroadening ofthereso- nancelinesbyafactorof1.4,whichimproves the fittothesolarabsorption profiles considerably, whereas theexisting discrepancy wouldimplyun- realistically anorder-of-magnitude errorintheCs constant. Asyettherearenolaboratory measurements of thebroadening ofthesodium Dlinesbyatomichy- drogen, butsuchexperiments maywellbepossible inthenearfutureusingshocktubetechniques. We therefore givethevelocity dependence ofshiftsandTABLEII.Broadening andshiftconstants forthe sodium Dlines. Commentf'ine-strueturc m/N d/N transition rate (109radsec)(10radsec)—»&/~ Bothlevels Upperlevelonly Lov'erlevelonly vander6'aals8.3 6.08 7.62 5.95—0.,)9 —0.72 —1.'~0 —'&.163.03 3.()3widthsinfullinFig.6. Theresultsofcorrectly including therotation of theintermolecular axisatlargeRhavebeenfound tobeoftheorderof5-10%inthecaseofresonance broadening'inwhichthisregionisofimportance duetotheslowlyvarying (R)natureoftheforces. Forthereasonance interaction thecontribution of collisions withb&b,isabouthalfthecrosssection, andforvanderWaals(R)forcesthisregioncon- tributes only20%ofthecrosssectionsothatthe effectsofneglecting rotation willbeonlyabout 2-5/o.Thishasbeenconfirmed byarecentstudy byHerman andLamb. Forsmall5wehavecor- rectlyincluded theeffectsofrotation withanerror ofabout57'introduced bytheuncertainty inthe transition region. Sincethelong-range partofthe interactions inthepresentcasevariesratherfaster thanRwewouldexpectcorrections fortherota- tionatlargebtobenotgreater than5%.Compu- tationalerrorsareestimated tobelessthan2% forthewidthsandabout10%fortheshifts. In drawingFig.6smalloscillations ofabout5a0have beensuppressed. Although thesearetosomeex- tentreal,theytendtocanceloutinavelocityaver- BROADENING OFSODIUM DLINES... 1947 ageforagiventemperature andaretherefore not ofpractical interest. Afterthevelocity average hasbeenperformed thewidthandshiftconstants shouldtherefore beaccurate toabout10%and20/c, respectively, B.Fine-Structure andSpin-Exchange Transitions Ourresultof63&aoforthecrosssection cr,&23/p atabout10cmsec'isingoodagreement withthe result(70+20)vaocalculated byBenderetal.and shouldberegarded asalowerlimittothecross section. Thevelocity dependence ofthecrosssec- tionisgiveninFig.6.Asdiscussed above,the molecular description willhavegreatest validity atsmallvaluesofRandshouldbeagooddescrip- tionwithintheopticalbroadening crosssection. Thequestion stillremains astowhether thereis asignificant contribution tothecrosssectionfrom theregionR&11ao. Inthisregionadescription of thecolliding atomsintermsofanatomicbasisset ismoreappropriate although themagnitude ofthe interatomic interactions willbeindicated bythe molecular energies. IntheregionR13aothein- teratomic interactions arecomparable withthefine- structure splitting ofsodium(7.8&10'a.u.).We mayestimate thecontribution ofthisregiontothe fine-structure transition crosssection usingtheap- proximate theoryofCallaway andBauer.' Ifwewritetheinteraction Hamiltonian inthe form Vq(R)=U(R)oTo+U(R)2TO, wheretheT,"aretensoroperators intheproduct spaceoftheatomsandweneglect therotation of thesystem duringthecollision, wefindthatonly stateswithm=&orm=—,'-aremixedandtheprob- abilityofatransition resulting fromthecollision is 9sin2(0.3q),whereg=(1/g)J'"U(R)2dt. Theupper limitmaybeestimated byputting p=@,theoptical phaseshiftwhichhasamaximum value0.2inthe regionofinterest. Thecontribution tothecross sectionfromtheregion1la,to15aoisthusatmost 5~ao.Theinclusion oftherotation ofthesystem duringthecollision alsocouples thestates IP,&2,+-,')and lP,&2,+-,'),andthismightbeex- pectedtodoublethecontribution ofthisregion. Assuming thatwemayaddtheeffectsofthetwo regions incoherently wesuggestavalueof(70+10) xmaoforthecrosssection 0$/g3/2 Fromtheordersofmagnitude discussed abovewe seethatthechangeinthediagonal elements ofthe Smatrix, whichcontribute tothelinebroadeningandshifts,willhavenegligible effectontheline broadening crosssection butmaywellbemoreim- portantfortheshifts. Theground-state spin-exchange crosssection thatwehaveobtained without considering theeffects ofhyperfine structure isabout20%smaller than thefine-structure transition crosssection andthe velocity dependence isshowninFig.5.Thecompu- tationalerrorinthesevaluesisapproximately 5%. V.SUMMARY Wehaveshown howthemolecular potentials for NaHcanbeusedtocalculate linebroadening and shiftparameters. Thecontributions oftheground— stateperturbations areconsiderably largerthan expected onthebasisofvanderWaalsapproxima- tionsandcontribute significantly tothebroadening. Inseveralcaseswhereneutralatombroadening has beenobserved athightemperatures eitherinshock tubes'orinastrophysical sources''thebroaden- inghasbeenfoundtobeapproximately 50/&larger thanthatcalculated usingestimated vanderWaals constants. Ourcalculation indicates thatthisisthe sizeofthecorrection tobeexpected whenrealistic interatomic potentials areused. Inthepresentcase observations ofthewingsoftheabsorption profiles ofthesodium Dlinesfromthesolarspectrum are insatisfactory agreement withthecalculation pre- sentedaboveusingourpotentials, whereas thevan derWaalsapproximation required anadjustment of thepolarizabilities byanorderofmagnitude. The smallredshiftscalculated inourapproximation aresensitive todistantcollisions whereourmethod haslessvaliditysothattheshiftconstants willbe lessaccurate thanthoseforthebroadening. Exten- sionofthemethods wehaveuseddepends onthein- creasing availability ofsuchcalculated molecular potentials andthedetermination oitheirlong-range limiting behavior fromexperimental data. ACKNOWLEDGMENTS Theauthors wishtoacknowledge manyfruitful discussions withcolleagues attheJointInstitute for Laboratory Astrophysics, inparticular discussions withDr.R.M.Hedges,Dr.A.Gallagher, Dr.J. Cooper, andDr.R.Garstang. Wealsogratefully acknowledge thatsupportforoneofus(L.F.M.) wasprovided throughagrantfromtheNational Aeronautics andSpaceAdministration totheUni- versity ofColorado (Contract No.NGR-06-003- 057). *Visiting Fellow,1969-1970; onleavefromClaren- donLaboratory, University ofOxford, Oxford, England. Presentaddress: Department ofAtomicPhysics, The University, Newcastle uponTyne,NE1VRU,England. tPresentaddress: Ionospheric Prediction ServiceDivision, Commonwealth Center, Sydney N.S.W.,2000, Australia. ~Visiting Fellow, 1969—1970;onleavefromUnited Aircraft Research Laboratories, EastHartford, Conn. ~OftheNational Bureau ofStandards and. 1g48 LEWIS, McNAMARA, ANDMICHELS University ofColorado, Boulder, Colo. H.Marganau, Rev.Mod.Phys. ~111(1939). H.Marganau andN.R.Kestner, TheoryofInter- molecular forces(Pergamon, NewYork,1969). 3W.R.Hindmarsh, Monthly NoticesRoy.Astron. Soc.119,11(1959);121,48(1960). G.Smith,Proc.Roy.Soc.(London) A297,288(1967). H.G.KuhnandE.L.Lewis,Proc.Roy.Soc.(Lon- don)A299,423(1967). W.R.Hindmarsh, A.D.Petford, andG.Smith, Proc.Roy.Soc.(London) A297,296(1967).J.M.Vaughan, Phys.Rev.166,13(1968). J.M.Vaughan andG.Smith,Phys.Rev.166,17 (1968). 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J.124, 542(1956). 23L.H.Aller,astrophysics, Theatmospheres ofthe ~unand&tars(Ronald, NewYork,1963). 4H.MFoley,Phys.Rev.69,616(1946). 25J.Weddell, Astrophys. J.136,223(1962). A.Omont andJ.Meunier, Phys.Rev.169,92 (1968). D.N.Stacey andJ.Cooper, Phys.Letters 30A,49 (1969). P.R.Berman andW.E.LambJr.,Phys.Rev. 187,221(1969). 9P.L.Bender, D.R.Crosley, D.R.Palmer and R.N.Zare,inFifthInternational Conference onthe physicsof&lectronic andAtomicCollisions: Abstracts of&apers, editedbyI.P.Flaks(Nauka, Leningrad, 1967),p.510. J.Callaway andE.Bauer,Phys.Rev.140,A1072 (1965). ~J.Evans,thesis(University ofColorado, 1970) (unpublished) .

PhysRevB.86.054445.pdf

PHYSICAL REVIEW B 86, 054445 (2012) Domain wall motion induced by the magnonic spin current Xi-guang Wang, Guang-hua Guo,*Yao-zhuang Nie, Guang-fu Zhang, and Zhi-xiong Li School of Physics Science and Technology, Central South University, Changsha, 410083, China (Received 28 April 2012; revised manuscript received 17 August 2012; published 31 August 2012) The spin-wave induced domain wall motion in a nanostrip with perpendicular magnetic anisotropy is studied. It is found that the domain wall can move either in the same direction or in the opposite direction to that ofspin-wave propagation depending on whether the spin wave is reflected by the wall or transmitted through thewall. A magnonic momentum transfer mechanism is proposed and, together with the magnonic spin-transfertorque, a one-dimensional phenomenal model is constructed. The wall motion calculated based on this model isin qualitative agreement with micromagnetic simulations, showing that the model can describe the characteristicsof spin-wave-induced wall motion and, especially, the wall motion direction. DOI: 10.1103/PhysRevB.86.054445 PACS number(s): 75 .60.Ch, 75 .30.Ds, 75 .40.Gb, 85 .70.Kh The magnetic domain wall motion induced by spin-transfer torque (STT) from a spin-polarized electrical current hasattracted growing interest because of its fundamental relevanceand potential applications in spintronic devices such as Race-track memory and logic devices. 1–3It is widely recognized that there are two types of STT acting on the wall when aspin-polarized current flows through it, namely the adiabaticSTT and nonadiabatic STT. 4,5The adiabatic STT comes from the adiabatic reversal of conduction electron spins, whichinduces a reaction torque on the wall as required by theconservation of angular momentum. 6However, the origin of nonadiabatic STT still remains controversial. There are manycontributions to this torque, such as spin-orbit interactions,spin-flip scattering, etc. 7–9When the domain wall is narrow, the momentum transfer due to the electron reflection by thewall also contributes to the nonadiabatic STT. 10The influence of the adiabatic and nonadiabatic STT on the wall motion isdifferent. The adiabatic STT plays a more important role atthe initial motion of the wall. It provides an initial velocityand causes the wall to move, but finally, it is balanced byan internal restoring torque and the wall motion ceases. 6In contrast, the nonadiabatic STT behaves like a magnetic fieldand can sustain a steady-state wall motion. 5 Recently, the noncharge-based spin current, magnonic spin current, is proposed11and experimentally demonstrated.12 Similar to the spin-polarized electrical current, the magnonicspin current also leads to the STT in the magnet andcan be exploited to control the spin structures, includingthe displacement of the domain wall. Hinzke et al. 13first demonstrated theoretically that a single domain wall in ananowire can be displaced by a magnonic spin current due tothe temperature gradient. Yan et al. 14showed that a spin wave excited by a microwave field can also drive a wall motion.They suggested that the mechanism of wall motion is the spintransfer torque resulting from the angular momentum transferbetween the magnons and the local magnetization in the wall.As a magnon moves across the wall, its angular moment ischanged by 2¯ hwhich is absorbed by the wall, making the wall propagate in the opposite direction to that of the magnon.It is worth noting that the spin-wave-induced domain wallmotion was studied earlier in Refs. 15–17. By micromagnetic simulations, the authors noticed that, contrary to the result inRefs. 13and14, the spin wave causes the wall to move in thesame direction to that of spin-wave propagation, and the wall velocity is strongly dependent on the transmission coefficientof spin wave. So far, there is no theory to explain the wallmotion induced by spin waves. In this letter, by using micromagnetic simulations, we study a single 180 ◦Bloch domain wall motion induced by spin waves in a nanostrip with perpendicular magnetic anisotropy (PMA).It is found that the direction and velocity of wall motion arestrongly dependent on the spin-wave frequency. In the high-frequency region, where the transmission coefficient of spinwaves passing through the wall is very close to a unit, the wallmoves in the opposite direction to that of magnons. While inthe low-frequency region, an obvious reflection of spin wavesoccurs. In this case, the wall propagates in the same directionto that of incident magnons, and its velocity increases with thedecrease in the spin-wave frequency. In order to explain thecharacteristics of spin-wave-induced wall motion, we proposea magnonic momentum transfer mechanism and construct aone-dimensional phenomenological model to account for thesimulation results. The PMA nanostrip studied here is 6 μm long in the x direction, 50 nm wide in the ydirection, and 10 nm thick in the zdirection, as shown in Fig. 1. For micromagnetic simulation, the following material parameters are used: satu-ration magnetization M s=8.6×105A/m, exchange stiffness Aex=1.3×10−11J/m, perpendicular anisotropy constant K1=5.8×105J/m3and Gilbert damping constant α=0.01. Micromagnetic simulations presented here are performed withthe micromagnetic code of OOMMF .18The simulation cell size is chosen to be 2 ×2×10 nm3. No thermal effects are considered. A 180◦Bloch wall is placed at the center position ( X=0) and relaxed to stable. The wall width is δ= π/Delta1=19.1 nm. An external harmonic sinusoidal field H= H0sin(2πvHt)ˆyalong the yaxis is applied locally in an area (2×50×10 nm3) in the left side of the strip to excite spin waves, which propagates along the nanostrip and induces thewall to move. The displacements of the wall driven by the spin wave with frequencies of 70 and 22 GHz are shown in Fig. 2.F o r the spin wave with frequency f=70 GHz [Fig. 2(b)], the wall moves in the opposite direction to that of the magnon,and its speed is almost constant. This result is in agreementwith Ref. 14. However, for the spin wave of frequency 054445-1 1098-0121/2012/86(5)/054445(5) ©2012 American Physical SocietyWANG, GUO, NIE, ZHANG, AND LI PHYSICAL REVIEW B 86, 054445 (2012) FIG. 1. (Color online) Illustration of the Model PMA nanostrip with the geometry and dimensions. A 180◦Bloch domain wall is positioned at the center X=0. The magnetization direction is represented by arrows. The gray area with H(t) represents the region where spin waves are excited. The Cartesian coordinate system is shown on the higher left. The lower right inset shows the magnetization components of the wall profiles. f=22 GHz [Fig. 2(a)], the situation appears to be quite different. The domain wall moves in the same direction tothat of the spin-wave propagation and displays a complicatedbehavior. At first, the wall moves with an acceleration stage,and then it enters into a relatively steady-state motion withan almost constant velocity (denoted as v s). Finally, the wall velocity decreases as it moves gradually away from the excitingregion. Figure 3(a) shows the frequency dependence of wall velocity. In the case that the wall moves in the same directionto that of the spin-wave propagation, this velocity correspondsto the steady-state motion v s. Considering that the efficiency of spin-wave excitation is strongly dependent on the frequencyof the microwave field as indicated by Fig. 4, the velocity is normalized by ( ρ/M s)2for comparison, where ρis the amplitude of spin waves. It shows that the wall velocitydecreases rapidly with the increase in frequency. In the (a) (b) FIG. 2. (Color online) The wall displacement Xas a function of timet, induced by propagating spin waves of frequency (a) 22 GHz and (b) 70 GHz. The black squares are the micromagnetic simulation results. The red solid line and green dashed line represent the modelcalculation with α=0.01 and 0, respectively. The blue circle is the simulation results of wall motion driven by the equivalent spin- polarized electrical current.(a) (b) FIG. 3. (Color online) (a) Simulated (black squares) and model calculated (red circles) wall velocity vand (b) initial wall velocity vi as a function of the frequency. The amplified scale of the wall velocity induced by the spin wave with frequencies larger than 55 GHz is shown as an inset of (a). All the velocities are normalized by ( ρ/M s)2. The dashed lines are guides for the eye. low-frequency region, the wall moves in the same direction to that of the spin wave ( v> 0). At a frequency of about 50 GHz, there is a sudden increase in the velocity comparedwith neighboring frequencies. For frequencies larger than55 GHz, the velocity becomes negative ( v< 0) as indicated by the inset of Fig. 3(a), meaning that the spin wave drags the wall to the opposite direction. It should be mentioned that theactual maximum value of the speed is approximately 40 m /s, and Walker breakdown is not observed in the range of oursimulation. FIG. 4. (Color online) Transmission coefficient Tof spin wave passing through the domain wall (black solid line) and amplitude of spin wave ρ(red/dark gray solid line) as a function of the frequency. 054445-2DOMAIN WALL MOTION INDUCED BY THE MAGNONIC ... PHYSICAL REVIEW B 86, 054445 (2012) In order to understand the above results, we calculated the frequency dependences of the spin-wave amplitude ρand the transmission coefficients Tpassing through the domain wall, w h i c ha r es h o w ni nF i g . 4. For calculations, a sinc function fieldH0sin(2πvHt)/(2πvHt)ˆywithH0=3 mT and vH= 150 GHz is used. Thus, spin waves of frequencies from 0 to150 GHz can be excited. Spin waves having frequencies lowerthan 18 GHz are prohibited to propagate in the nanostrip. Itcan be seen from Fig. 4that the amplitude ρdecreases with the increase in frequency. The transmission coefficients Tare also sensitively dependent on the frequency. In the region offrequencies larger than 55 GHz, the spin waves pass throughthe domain wall without reflection. For frequencies lowerthan 55 GHz, the spin waves are partially reflected, and thetransmission coefficients decrease with frequency. It has beenestablished that, for a one-dimensional domain wall, a spinwave can be transmitted without reflection, 19,20but for the wall in nanostrip, partial or complete reflection of a spin wavemay occur when the wavelength is larger than the wall width. 21 The reflective behavior results from the stray field due to thetransversely confined dimension. Also, the reflectivity is alsorelated to the inherent oscillation modes of the wall. 15This is displayed on our T(f) curve as well, where at frequencies of about 50 GHz, there is a clearly drop of transmission. Thisdrop corresponds to a normal oscillation mode of the wall. Therefore, it is clearly shown that, for the 180 ◦Bloch domain wall in PMA nanostrips, the spin-wave-induced wallmotion exhibits different behaviors depending on whether thespin wave passes through the wall or it is reflected by thewall. Very recently, Kim et al. studied the spin-wave-induced N´eel-wall motion in a nanostrip by using micromagnetic simulations. 22They also found that the wall velocity is strongly dependent on the spin-wave frequency. At certain frequencies,the wall velocity is negative, while at other frequencies wherestrong spin-wave reflection occurs, the wall has a positivevelocity. Moreover, the internal normal oscillation modesof the wall also influence the wall motion. According toour studies, the wall motion direction is determined by thetransmission or reflection characteristics of a spin wave. Whenthe spin-wave frequency is the same as that of an internal modeof the wall, the resonant reflection occurs and acts on the wallmotion. Hinzke et al. 13and Yan et al.14have proposed a magnonic spin-transfer torque mechanism to explain the spin-wave-induced wall motion. When a magnon passes through thewall, its spin is adiabatically reversed, and this induces areaction torque on the wall as required by the conservationof angular momentum, which makes the wall propagate inthe opposite direction to that of the magnon. Obviously, thismechanism cannot explain the feature of wall motion when aspin wave is reflected by the wall. In the case that a magnonis reflected, its spin keeps constant, but its momentum ischanged by 2¯ hk. Here, kis the wave vector of the spin wave. The transfer of momentum between the magnon and the wallalso induces torque acting on the wall and pushes it movingin the same direction to that of the incident magnon. Thistype of momentum transfer torque was proposed by Tataraet al. 10regarding the spin-polarized electrical-current-induced nonadiabatic STT.In the following, we will establish a one-dimensional phenomenological model to account for the spin-wave-induced wall motion by introducing the above-mentionedmagnonic spin transfer torque and momentum transfer torqueof magnonic currents. When a magnonic current passesthrough a domain wall, the magnon spin, which is oppositeto local magnetization, is adiabatically reversed. 13,14Similar to the spin-polarized current, an magnonic STT is brought out,which can be expressed in a one-dimensional situation as: 23 /parenleftbigg∂/vectorM ∂t/parenrightbigg ST=−∂/vectorJm ∂x. (1) Here, /vectorJm=−u/vectorM. The negative sign means that the magnon spin is antiparallel to the direction of the local magnetization.Here, u=(γ¯hnv k)/μoMs,nis the number of magnons per unit area, vkis the propagation velocity of magnons, and γ=μ0ge/2merepresents the gyromagnetic ratio. In the case that the magnonic current is reflected completely bythe wall, the momentum transfer between the magnons andthe wall gives rise to a force F=dP magnon/dt=2nvk¯hk, which pushes the wall to move forward. We can introducean effective field H magnon =(nvk¯hk)/(μ0Ms)=uk/γ along the direction perpendicular to the nanostrip to describe thismomentum transfer mechanism. By assuming a constant domain wall profile, the wall dy- namics can be described by two collective coordinates, the wallposition Xand the tilt angle ϕof the wall magnetization. 24,25 For a one-dimensional Bloch wall profile, the equations of wall motion including the spin-wave induced torques become (1+α2)˙X=γ/Delta1K d μ0Mssin(2ϕ)−Tu+(1−T)α/Delta1uk, (2) (1+α2)˙ϕ=−γαK d μ0Mssin(2ϕ)+(1−T)uk+Tαu /Delta1.(3) Here, Kd=1 2μ0M2 s(Nx−Ny) is the effective anisotropy, representing the magnetostatic energy difference betweenthe Bloch wall and the N ´eel wall, 26where NxandNy are the demagnetizing factors along the xandydirections, respectively. Also, Nx−Nyis calculated to be 0.05, and Tis the transmission coefficient of spin waves passing through thewall. The initial wall velocity corresponding to the conditionϕ=π/2 can be derived from Eq. (2)as: v i=−T 1+α2u+(1−T)α/Delta1k 1+α2u. (4) Equation (4)clearly indicates that transmitted magnons drag the wall to move backward, and the reflected magnonsdrive the wall to propagate forward with respect to thepropagation direction of spin waves. As the damping constantα/lessmuch1, the initial wall velocity is mainly determined by the first term. After the initial acceleration motion, the wall enters intoa steady-state motion corresponding to the condition ∂ϕ/∂t = 0. The steady-state motion velocity can be expressed as thefollowing: v s=(1−T)/Delta1k αu. (5) 054445-3WANG, GUO, NIE, ZHANG, AND LI PHYSICAL REVIEW B 86, 054445 (2012) The above formula shows that vsis determined by the reflected magnons, which transfer their linear momentum tothe wall and push it to move. For numerical calculations of wall position X(t), initial velocity v ias well as steady-state motion velocity vs,t h e magnon density nmust be decided at first. Here, nis directly proportional to the square of the spin-wave amplitude, whichcan be approximated as n=ρ 2/(2MsgμB),27where ρis the spin-wave amplitude. The transmission coefficient T, spin-wave amplitude ρ, wave vector k, and velocity vk, which are all functions of the spin-wave frequency, are taken fromthe micromagnetic simulation results. Also, ρis a function of wall position because of the gradual attenuation of the spinwave as it propagates away from exciting source. The wall displacements calculated from the above one-dimensional model are depicted in Fig. 2together with the data obtained by micromagnetic simulations. Fora spin wave with f=22 GHz [see Fig. 2(a)], the X(t) curve calculated based on the one-dimensional model isin qualitative agreement with the simulation results. Thewall motion can be classified into three phases: initialacceleration motion, a relatively steady-state motion, andfinally, the gradual deceleration motion, but for a spin wavewith frequency f=70 GHz [see Fig. 2(b)], there is a large difference between the theoretically calculated results andmicromagnetic simulation data. In the case of the theoreticalcalculation, the wall moves a certain distance and then ceasesto move. To shed more light on this difference, we simulatethe wall motion driven by the adiabatic STT of spin-polarizedelectrical currents. The strength of this STT is the same value asthe magnon’s. As the adiabatic STT of spin-polarized electrical currents is τ e=−ue∂/vectorM ∂x, this gives the equivalent value ue= u=(γ¯hnvk)/μoMs. Here, we get a similar result as that of the simulated spin-wave-induced wall motion; the wall moves amuch longer distance compared to the one-dimensional modelcalculation. This illuminates that the difference between thesimulated and model-calculated data mainly results from theone-dimensional approximation of wall motion. In addition,the wall motion induced by the magnonic STT is sensitive tothe damping constant α, as indicated by Fig. 2.The initial wall velocity v ias well as velocity vsin the steady-state motion vs the frequency of spin waves calculatedfrom Eqs. (4)and (5)are shown in Figs. 3(b) and 3(a), respectively, together with the simulated data. It can be seenthat the analytical results are in qualitative agreement withthe simulation. The initial v iincreases with the frequency. As indicated by Eq. (4),viis mainly determined by the magnons passing through the wall. The transmission coefficient Tand the spin velocity vkall lead vito increase with the frequency. It is worth noting that viis much lower than vs, meaning the reflected magnons can give rise to a larger torque on the wallthan the transmitted magnons. This can be understood fromEqs. (4)and(5)thatv s/vi≈(1+α2)/Delta1k/α when the same number of magnons is reflected or passing through the wall.Taking k=1×10 8m−1givesvs/vi≈61. In conclusion, we have studied a Bloch domain wall motion in a PMA nanostrip induced by spin waves (or magnoniccurrents). It is demonstrated that the behavior of the wallmotion is strongly dependent on whether the magnons passthrough the wall or they are reflected by the wall. When themagnons pass through the wall, the spin transfer torque, whichoriginates from the spin transfer between the magnons and thewall, makes the wall propagate in the opposite direction to thatof the magnons. When the magnons are reflected by the wall,the momentum transfer between the magnons and the wallalso gives rise to torque, which drives the wall to move in thesame direction to that of the incident magnons. By controllingthe transmission coefficient of the spin wave, the wall motionvelocity as well as direction can be manipulated. The resultsobtained in this work may find their use in designing magnonicspin devices. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China under Grant No. 60571043, DoctoralFund of Ministry of Education of China, and the ScientificPlane Project of Hunan Province of China under GrantNo. 2011FJ3193. *Corresponding author: guogh@mail.csu.edu.cn 1D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 (2005). 2M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin, Science 320, 209 (2008). 3S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 4S. Zhang and Z. Li, P h y s .R e v .L e t t . 93, 127204 (2004). 5A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys. Lett. 69, 990 (2005). 6Z. Li and S. Zhang, P h y s .R e v .B 70, 024417 (2004). 7I. Garate, K. Gilmore, M. D. Stiles, and A. H. MacDonald, Phys. Rev. B 79, 104416 (2009). 8R. A. Duine, A. S. N ´u˜nez, J. Sinova, and A. H. MacDonald, Phys. Rev. B 75, 214420 (2007).9P. Bal ´aˇz, V . K. Dugaev, and J. Barna ´s,Phys. Rev. B 85, 024416 (2012). 10G. Tatara and H. Kohno, P h y s .R e v .L e t t . 92, 086601 (2004). 11Y . Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi,S. Maekawa, and E. Saitoh, Nature (London) 464, 262 (2010). 12A. V . Chumak, A. A. Serga, M. B. Jungfleisch, R. Neb, D. A.Bozhko, V . S. Tiberkevich, and B. Hillebrands, Appl. Phys. Lett. 100, 082405 (2012). 13D. Hinzke and U. Nowak, P h y s .R e v .L e t t . 107, 027205 (2011). 14P. Yan, X. S. Wang, and X. R. Wang, P h y s .R e v .L e t t . 107, 177207 (2011). 15D. S. Han, S. K. Kim, J. Y . Lee, S. J. Hermsdoerfer, H. Schultheiss,B. Leven, and B. Hillebrands, Appl. Phys. Lett. 94, 112502 (2009). 054445-4DOMAIN WALL MOTION INDUCED BY THE MAGNONIC ... PHYSICAL REVIEW B 86, 054445 (2012) 16M. Jamali, H. Yang, and K. J. Lee, Appl. Phys. Lett. 96, 242501 (2010). 17S. M. Seo, H. W. Lee, H. Kohno, and K. J. Lee, Appl. Phys. Lett. 98, 012514 (2011). 18M. J. Donahue and D. G. Porter, The Object Oriented MicroMagnetic Framework ( OOMMF ) project at ITL/NIST, http://math.nist.gov/oommf/ . 19A. A. Thiele, Phys. Rev. B 7, 391 (1973). 20C. Bayer, H. Schultheiss, B. Hillebrands, and R. L. Stamps, IEEE Trans. Magn. 41, 3094 (2005). 21S. Macke and D. Goll, J. Phys.: Conf. Ser. 200, 042015 (2010).22J. S. Kim, M. St ¨ark, M. Kl ¨aui, J. Yoon, C. Y . You, L. Lopez-Diaz, and E. Martinez, P h y s .R e v .B 85, 174428 (2012). 23J. Fern ´andez-Rossier, M. Braun, A. S. N ´u˜nez, and A. H. MacDonald, P h y s .R e v .B 69, 174412 (2004). 24N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). 25L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. S. P. Parkin, Nature (London) 443, 197 (2006). 26S. W. Jung, W. Kim, T. D. Lee, K. J. Lee, and H. W. Lee, Appl. Phys. Lett. 92, 202508 (2008). 27A. G. Gurevich and G. A. 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PhysRevApplied.14.014003.pdf

PHYSICAL REVIEW APPLIED 14,014003 (2020) Analytical Criteria for Magnetization Reversal in a ϕ0Josephson Junction A.A. Mazanik,1,2I.R. Rahmonov,1,3A.E. Botha ,4and Yu.M. Shukrinov1,4,5, * 1BLTP , JINR, Dubna, Moscow Region 141980, Russia 2MIPT, Dolgoprudny, Moscow Region 141700, Russia 3Umarov Physical Technical Institute, TAS, Dushanbe 734063, Tajikistan 4Department of Physics, University of South Africa, Florida, Johannesburg 1710, South Africa 5Dubna State University, Dubna, Moscow Region 141980, Russia (Received 6 March 2020; revised 17 May 2020; accepted 20 May 2020; published 1 July 2020) The switching of magnetization by electric current pulse in the ϕ0Josephson junction formed by ordi- nary superconductors and a magnetic noncentrosymmetric interlayer is studied. The ground state of this junction is characterized by the finite phase difference ϕ0, which is proportional to the strength of the spin-orbit interaction and the exchange field in the normal metal. Based on the Landau-Lifshits-Gilbert and resistively shunted junction model equations we build an analytical description of the magnetization dynamics induced by an arbitrary current pulse. We formulate the criteria for magnetization reversal and, using the obtained results, the form and duration of the current pulse are optimized. The analytical and numerical results are in excellent agreement at GrI p/greatermuch1, where Gis a Josephson-to-magnetic energy ratio, ris a strength of spin-orbit interaction, and Ipis a value of the current pulse. The analytical result allows one to predict magnetization reversal at the chosen system parameters and explains the features of magnetization reversal in the G-rand G-αdiagrams, where αis the Gilbert damping. We propose to use such a ϕ0Josephson junction as a memory element, with the information encoded in the magnetization direction of the ferromagnetic layer. DOI: 10.1103/PhysRevApplied.14.014003 I. INTRODUCTION The ability to manipulate magnetic properties by the Josephson current and its opposite, i.e., to influence the Josephson current by a magnetic moment, has attracted much recent attention [ 1–6]. In the superconductor- ferromagnet-superconductor ( S-F-S) Josephson junctions, the spin-orbit interaction in a ferromagnet without inver- sion symmetry provides a mechanism for a direct (linear) coupling between the magnetic moment and the super- conducting current. In such junctions, with noncentrosym- metric ferromagnetic interlayer and broken time-reversal symmetry, called ϕ0junctions, the current-phase relation (CPR) is given by I=Icsin(ϕ−ϕ0), where the phase shift ϕ0is proportional to the magnetic moment perpendicular to the gradient of the asymmetric spin-orbit potential [ 7]. Theϕ0junctions are possible due to the anomalous Josephson effect in different hybrid heterostructures, which reflect the simultaneous interplay of superconductivity, spin-orbit interaction and magnetism [ 8–22], gives insight into the problem of the mutual influence of superconduc- tivity and ferromagnetism, allows a realization of exotic *shukrinv@theor.jinr.rusuperconducting states such as the Larkin-Ovchinnikov-Fulde-Ferrell state and triplet ordering. The anomalous Josephson effect promises applications that utilize the spin degree of freedom [ 13] and demonstrates a number of unique features, important for superconducting spintron- ics and modern informational technologies. These features allow one to manipulate the internal magnetic moment by using the Josephson current [ 7,23]. Thus, once the magnetization rotates, a reverse phenomenon should be expected. Namely, magnetization rotation might pump cur- rent through the ϕ 0phase shift, which is fueled by the term proportional to magnetization and spin-orbit cou- pling. Such pumping of current leads to the appearance of a dc component in the superconducting current, and plays an important role in the transformation of the I-V characteristics in the resonance region [ 24]. The application of a dc voltage to the ϕ0junction produces current oscillations, and consequently, magnetic precession. As shown in Ref. [ 23], this precession may be monitored by the appearance of higher harmonics in the CPR, as well as by the presence of a dc component in the superconducting current. The latter increases substantially near the ferromagnetic resonance (FMR). Konschelle and Buzdin [ 23] stressed that the magnetic dynamics of the S- F-Sϕ0junction may be quite complicated and strongly 2331-7019/20/14(1)/014003(12) 014003-1 © 2020 American Physical SocietyMAZANIK, RAHMONOV, BOTHA, and SHUKRINOV PHYS. REV. APPLIED 14,014003 (2020) anharmonic. In contrast to these results, it was demon- s t r a t e di nR e f .[ 24] that precession of the magnetic moment in some current intervals along I-Vcharacteristics may be very simple and harmonic. It is expected that external radiation would lead to a series of interesting phenomena. Among these, there is the possibility of the appearance of half-integer Shapiro steps (in addition to the conventional integer steps) and the generation of an additional magnetic precession at the frequency of the external radiation [ 23]. Heterostructures that demonstrate the anomalous Josep- hson effect are being intensively developed. The anoma- lous Josephson effect was predicted in two types of sys- tems: one with both, spin-orbit interaction and exchange field and another with a noncoplanar magnetization. In Ref. [ 25] a full microscopic theory to describe the Joseph- son current through an extended superconductor–normal- metal–superconductor ( S-N-S) diffusive junction with an intrinsic spin-orbit coupling (SOC) in the presence of a spin-splitting field hwas presented. It was demonstrated that the ground state of the junction corresponds to a finite intrinsic phase difference 0 <ϕ< 2πbetween the superconductor electrodes provided that both hand the SOC-induced SU(2)Lorentz force are finite. The authors found the ϕ0as a function of the strengths of the spin fields, the length of the junction, the temperature, and the properties of S-Ninterfaces. The proper geometry of the system was discussed in Ref. [ 26], where a quasiclassi- cal transport theory to deal with magnetoelectric effects in superconducting structures was developed. For Joseph- son junctions the authors establish a direct connection between the inverse Edelstein effect and the appearance of an anomalous phase shift ϕ0in the current-phase rela- tion. In particular they show that ϕ0is proportional to the equilibrium spin current in the weak link. Predictions of anomalous Josephson junctions with a noncoplanar magnetization have been done in Ref. [ 27], for ballistic systems, where the anomalous Josephson cur- rent appears at zero phase difference in junctions coupled with a ferromagnetic trilayer having noncoplanar mag- netizations. The anomalous current was calculated using the Bogoliubov-de Gennes equation, and a clear physical explanation was given of the anomalous Josephson effect in this structure [ 27]. The authors of Ref. [ 27] also showed that the triplet proximity correlation and the phase shift in the anomalous current-phase relation all stem from the spin precession in the first and third ferromagnetic lay- ers. In Ref. [ 28] it was demonstrated that the conditions for the observation of the anomalous Josephson current in diffusive S-F-Sjunctions are a noncoplanar magneti- zation distribution and a broken magnetization inversion symmetry of the superconducting current. The authorsshow that this symmetry can be removed by introducing spin-dependent boundary conditions for the quasiclassical equations at the superconducting-ferromagnet interfaces in diffusive systems [ 28]. Using this recipe, they thendetermine the ideal experimental conditions in order to maximize the anomalous current [ 28]. The Josephson junctions composed of two semicon- ducting nanowires with Rashba spin-orbit coupling and induced superconductivity from the proximity effect dis- play a geometrically induced anomalous Josephson effect, the flow of a supercurrent in the absence of external phase bias [ 22]. A generic nonaligned Josephson junction in the presence of an external magnetic field reveals an unusual flux-dependent current-phase relation [ 20]. Such nonaligned Josephson junctions can be utilized to obtain a ground state other than 0 and π, corresponding to the ϕ junction, which is tunable via the external magnetic flux. A tunable ±ϕand hybrid system between ϕandϕ 0junctions were investigated in Refs. [ 17–19]. Recently, an anomalous phase shift was experimentally observed in different systems, particularly, in the ϕ0junc- tion based on a nanowire quantum dot [ 29]. A quantum interferometer device was used in order to investigate phase offsets and demonstrate that ϕ0can be controlled by electrostatic gating. The presence of an anomalous phase shift of ϕ0was also experimentally observed directly through CPR measurement in a hybrid S-N-SJoseph- son junction fabricated using Bi 2Se3(which is a topo- logical insulator with strong spin-orbit coupling) in the presence of an in-plane magnetic field [ 30]. This consti- tutes a direct experimental measurement of the spin-orbit coupling strength and opens up different possibilities for phase-controlled Josephson devices made from materials with strong spin-orbit coupling. In Refs. [ 31]a n d[ 32], the authors argued that the ϕ0Josephson junction is ideally suited for studying of quantum tunneling of the magnetic moment. They proposed that magnetic tunneling would show up in the ac voltage across the junction and it could be controlled by the bias current applied to the junction. Though the static properties of the S-F-Sstructures are well studied both theoretically and experimentally, much less is known about the magnetic dynamics of these sys- tems [ 33–35]. The observation of a tunable anomalous Josephson effect in InAs/Al Josephson junctions mea- sured via a superconducting quantum interference device (SQUID) reported in Ref. [ 36]. By gate controlling the density of InAs the authors were able to tune the spin- orbit coupling of the Josephson junction by more than one order of magnitude. This gives the ability to tune ϕ0,a n d opens several opportunities for superconducting spintron- ics [1], and the possibilities for realizing and characterizing topological superconductivity [ 37–39]. One of the milestones for superconducting electronics, which stands out by ultralow energy dissipation is the creation of cryogenic memory [ 40–42]. Different realiza- tions for such devices were proposed including devices based on the ϕ0Josephson junctions [ 41,43–45]. The dc superconducting current applied to a S-F-Sϕ0junc- tion might produce a strong orientation effect on the 014003-2ANALYTICAL CRITERIA FOR MAGNETIZATION REVERSAL... PHYS. REV. APPLIED 14,014003 (2020) ferromagnetic layered magnetic moment [ 46]. The full magnetization reversal can be realized by applying an elec- tric current pulse [ 46]. Detailed pictures representing the intervals of the damping parameter α, Josephson to mag- netic energy relation G, and the spin-orbit coupling param- eter rwere obtained with the full magnetization reversal [47]. It was demonstrated that the appearance of the rever- sal was sensitive to changing the system parameters and showed some periodic structure. Guarcello and Bergeret [45] suggested to use a ϕ0S-F-Sjunction as a cryogenic memory element, based on the current pulse switching of magnetization proposed in Ref. [ 46]. In this scheme, a bit of information is associated with the direction of the magnetic moment along or opposite the direction of the easy axis of the ferromagnetic layer. The writing is carried out as a reversal of the magnetic moment by a pulse of current and the readout is performed by detec- tion of the magnetic flux by SQUID inductively coupled to the ϕ0junction. They also explored the robustness of the current-induced magnetization reversal against thermal fluctuations and suggested a way of decoupling the Joseph- son phase and the magnetization dynamics by tuning the Rashba spin-orbit interaction strength via a gate voltage. A suitable nondestructive readout scheme based on a dc SQUID inductively coupled to the ϕ0junction was also discussed. We stress that in all the above mentioned works the magnetization reversal was studied numerically only. In our present work we derive an analytical solution for the magnetization dynamics induced by an arbitrary current pulse and formulate the criteria for magnetiza- tion reversal in the ϕ0Josephson junctions formed by ordinary superconductors and a magnetic, noncentrosym- metric interlayer. Using the obtained analytical results, we optimize the form and duration of the current pulse. The agreement between analytical and numerical investi- gations is reached in the case of a large product of the ratio of the Josephson energy to the magnetic energy, strength of spin-orbit interaction and a minimum value of the flowing current. The obtained results explain the periodicity in the appearance of the magnetization-reversal intervals observed in Ref. [ 47] and allow one to predict magnetization reversal at the chosen system parameters. The plan of the rest of this work is as follows. In Sec. IIwe introduce the model and methods, particu- larly, the derivation of effective field and current pulse. This is followed by Sec. III, where the relation between expressions for the temporal dependence of the current’s pulse and the superconducting current is obtained for dif- ferent Josephson-to-characteristic frequency ratios of the junction. In Sec. IVwe present the solution of the Landau- Lifshiz-Gilbert equation for the case when the product ofthe ratio of the Josephson-to-magnetic energy and spin- orbit coupling is much more that one. Section Vis devoted to the small damping regime. We discuss the periodicity in the appearance of the magnetization-reversal intervals inthe diagrams “Gilbert damping—Josephson-to-magnetic energy ratio.” The periodicity for the diagram “spin- orbit coupling—Josephson-to-magnetic energy ratio” is discussed in Sec. VI. Finally, in Sec. VIIwe summarize our main results and conclude. II. MODEL AND METHODS We study the anomalous Josephson effect in the sys- tem with both spin-orbit interaction and exchange field. Such a case was also demonstrated in a recent experi- ment by Strambini and collaborators [ 48] who showed that the combined actions of spin-orbit coupling and exchange interaction breaks the phase rigidity of the sys- tem, inducing a strong coupling between charge, spin, and superconducting phase. Here we investigate the system, which consists of a simple ϕ 0Josephson junction with planar geometry made up with ordinary superconductors and noncentrosymmet- ric magnetic interlayer with intrinsic Rashba spin-orbit coupling [ 23,45,46]. From the microscopic point of view such Rashba coupling is considered as vSO[/vectorσ×/vectorp]·/vectorn, where /vectornis the unit vector along the asymmetric poten- tial gradient and parameter vSOdescribes its strength. In the following text the microscopical details of the Rashba SOC and exchange field are included in one dimension- less parameter rand the current-phase relation written as Is=Icsin(ϕ−ϕ0), where ϕ0=rM y/M0,Mydenotes the component of magnetic moment in ˆydirection, M0 is the modulus of the magnetization. So, the physics of S-F-SJosephson structures is determined by the system of equations that consists of the Landau-Lifshits-Gilbert(LLG), resistively shunted junction (RSJ) model with I s=Icsin(ϕ−ϕ0), and Josephson relation between phase difference and voltage. The dynamics of the magnetic moment in the ϕ0Josephson junction is described by [ 49] dM dt=γmHeff×M+α M0/parenleftbigg M×dM dt/parenrightbigg ,( 1 ) where Mis the magnetization vector, γmis the gyromag- netic relation, Heffis the effective magnetic field, αis the Gilbert damping parameter, M0=|M|. In order to find the expression for the effective magnetic field we use the model developed in Ref. [ 23], where it is assumed that the gradient of the spin-orbit potential is along the easy axis of magnetization taken to be along ˆz. In this case the total energy of the system can be written as Etot=−/Phi10 2πϕI+Es(ϕ,ϕ0)+EM(ϕ0),( 2 ) where ϕis the phase difference between the supercon- ductors across the junction, Iis the external current, Es(ϕ,ϕ0)=EJ[1−cos(ϕ−ϕ0)],a n d EJ=/Phi10Ic/2πis the Josephson energy. Here /Phi10is the flux quantum, Icis 014003-3MAZANIK, RAHMONOV, BOTHA, and SHUKRINOV PHYS. REV. APPLIED 14,014003 (2020) the critical current, ϕ0=lυSOMy/(υ FM0),l=4hL//planckover2pi1υF,L is the length of Flayer, his the exchange field of the F layer, EM=− KVM2 z/(2M2 0), the parameter υSO/υ Fchar- acterizes a relative strength of spin-orbit interaction, K is the anisotropic constant, and Vis the volume of the ferromagnetic ( F) layer. As mentioned in the Introduction, the physics of the S-F-SJosephson junctions is mostly described by the qua- siclassical Green-function method. Here we reduce it to the RSJ model assuming the current-phase relation Is= Icsin(ϕ−ϕ0), calculated by the Green-function method in Refs. [ 7], [25], and [ 26]. We consider a low-frequency regime /planckover2piωJ/lessmuch Tc(ωJ=2eV//planckover2pibeing the Josephson angu- lar frequency [ 50]), which allows us to use the qua- sistatic approach to treat the superconducting subsystem. If /planckover2piωJ/lessmuch Tc, one can use the static value for the total energy of the junction, Eq. (2), considering ϕ(t)as an external potential. Derivative of total energy, Eq. (2), on phase dif- ference gives the current-phase relation, which is used in the RSJ model. The effective field for the LLG equation is determined by Heff=−1 V∂Etot ∂M =K M0/bracketleftbigg Grsin/parenleftbigg ϕ−rMy M0/parenrightbigg /hatwidey+Mz M0/hatwidez/bracketrightbigg ,( 3 ) where r=lυSO/υ F,a n d G=EJ/(KV). Using Eqs. (1)and(3), we obtain the system of equa- tions, which describes the dynamics of the magnetization ofFlayer in S-F-Sstructure ˙mx=1 1+α2{−mymz+Grm zsin(ϕ−rm y) −α[mxm2 z+Grm xmysin(ϕ−rm y)]}, ˙my=1 1+α2{mxmz −α[mym2 z−Gr(m2 z+m2 x)sin(ϕ−rm y)]}, ˙mz=1 1+α2{−Grm xsin(ϕ−rm y) −α[Grm ymzsin(ϕ−rm y)−mz(m2 x+m2 y)]},(4) where mx,y,z=Mx,y,z/M0satisfy the constraint/summationtext i=x,y,z m2 i(t)=1. In this system of equations, time is normalized to the inverse ferromagnetic resonance frequency ωF= γK/M0:(t→ tωF). In order to describe the full dynamics of S-F-Sstruc- ture the LLG equations should be supplemented by the equation for phase difference ϕ, i.e., the equation of the RSJ model. According to the extended RSJ model [ 51], which takes into account the derivative of the ϕ0phaseshift, the current flowing through the system in the over- damped case is determined by I=/planckover2pi1 2eR/bracketleftbiggdϕ dt−r M0dM y dt/bracketrightbigg +Icsin/parenleftbigg ϕ−r M0My/parenrightbigg ,( 5 ) or in the normalized variables, I=wd/Phi1 dt+sin/Phi1,( 6 ) where the bias current Iis normalized to the critical one Ic and/Phi1=ϕ−rm y. We note that in order to use the same time scale in the LLG and RSJ equations we normalize time to the ω−1 F.I n this case an additional parameter given by the w=ωF/ω c appears, where ωc=2eIcR//planckover2pi1is a characteristic frequency of the Josephson junction. As we see, the behavior of the system depends on the value of this parameter and it characterizes its different regimes. The current pulse is I=Ipand it has rectangular form Ip(t)=/braceleftbigg As, t∈[t0,t0+δt]; 0, otherwise,(7) where Asandδtare the pulse amplitude and width, respec- tively. We note that if any other form is not specified below, the rectangular form is considered. The initial conditions for the LLG equation are the mx(t=0)=0, my(t=0)=0, mz(t=0)=1a n df o r RSJ-model equation ϕ(t=0)=0. Via the numerical solu- tion [ 46,47]o fE q . (4), taking into account Eqs. (6)and(7), we obtain the time dependence of magnetization mx,y,z(t), phase difference ϕ(t), and normalized superconducting current Is(t)=sin(ϕ−rm y). In this paper we also compare the analytical and numer- ical results concerning the periodicity of magnetization reversal in the α-Gand r-Gplanes. In order to demonstrate the realization of the magnetization-reversal intervals, we solve numerically the system of differential Eqs. (4)and(6) for the fixed values of Gandα(or Gand r). Then we check the value of mzat the end of each time domain, and if the reversal is realized, the values of Gandα(or r)a r e recorded to the files. Repeating this procedure for the dif- ferent values of our parameters we build the figures, which demonstrate the magnetization-reversal appearance in α-G and r-Gplanes. III. RELATION BETWEEN Ip(t)AND sin /Phi1( t)AT DIFFERENT w As mentioned above, the physics of switching in the ϕ0is determined by the LLG equation, Eq. (1), and the RSJ-model equation, Eq. (6). An interesting feature of this system of equations in the overdamped case is a decou- pling, i.e., Eq. (6)for/Phi1is decoupled from the LLG 014003-4ANALYTICAL CRITERIA FOR MAGNETIZATION REVERSAL... PHYS. REV. APPLIED 14,014003 (2020) equation. Eq. (4). This allows us to find the analytical solu- tion for /Phi1and build the theory for magnetization reversal at some values of model and pulse parameters. To investigate the magnetization dynamics, we solve Eq.(6)for the pulse Ip(t)and calculate sin /Phi1, which is needed to determine the effective magnetic field, Eq. (3). We find that the sin /Phi1profile consists of two regions: the first one is the pumping of the sin /Phi1during the pulse and the second one is the dropping of sin /Phi1to zero when the pulse has been switched off. We investigate these processes in the case of the rectangular pulse, Eq. (7). During the pulse t0≤t≤t0+δt, the equation for /Phi1has a form As=wd/Phi1 dt+sin/Phi1,( 8 ) with the initial condition /Phi1(t=t0)=0. For As<1i t gives tan/Phi1(t)/2=Astanh/bracketleftBig t−t0 τ0/bracketrightBig tanh/bracketleftBig t−t0 τ0/bracketrightBig +/radicalbig 1−A2s.( 9 ) Hereτ−1 0=/radicalbig 1−A2s/2w, which determines the time scale for approaching a constant value of /Phi1. So, the formula (9) allows calculation of the sin /Phi1during the pulse t0≤t≤ t0+δt. In the second region t≥t0+δt, when the pulse has been switched off [ Ip(t)=0], we have sin/Phi1(t)=2t a n/bracketleftBig /Phi1(t0+δt) 2/bracketrightBig exp/bracketleftBig t−t0−δt w/bracketrightBig +tan2/bracketleftBig /Phi1(t0+δt) 2/bracketrightBig exp/bracketleftBig −t−t0−δt w/bracketrightBig, (10) which exponentially drops to zero with a time scale τ1∼ w. Here tan {[/Phi1(t0+δt)]/2}is determined by Eq. (9). We see that there are two physically distinguishable cases. The first case of small wis realized when the conditions τ0=2w//radicalbig 1−A2s/lessmuchδtandτ1∼w/lessmuchδtare fulfilled, i.e., when the pumping time τ0and the time τ1 of dropping to zero, are small in comparison with the pulse duration: τ0/lessmuchδtandτ1/lessmuchδt. These conditions mean that sin/Phi1approaches Asand drops to 0 for short periods of timeτ0,τ1, correspondingly, in comparison with the pulse duration δt, thus sin /Phi1shows nearly a rectangular form, which coincides with the pulse Ip(t). This case is demon- strated in Fig. 1(a). Here the magnetic moment feels an approximately constant field sin /Phi1during the pulse. In the opposite case of w/greaterorsimilarτ0,τ1the profile of sin /Phi1 becomes more complicated. First, the pumping process ofsin/Phi1to A sbecomes broader. Second, a significant tail emerges where Ip(t)=0, but sin /Phi1/negationslash=0, which influences the magnetization dynamics. This situation is shown in Fig.1(b). (a) (b) FIG. 1. (a) Dynamics of sin /Phi1(green line), based on the ana- lytical formulas (9)and(10), and numerical solution of Eq. (4) (blue dots). The red line corresponds to the current pulse Ip(t) atw=0.1. The other parameters of calculations are As=0.5, t0=1, and δt=3; (b) the same at w=1. So, one can notice that parameter wmeasures time of /Phi1reaction to the external current. It could be concluded that for small win comparison with the characteristic time scale of Ip(t)and for arbitrary current pulse Ip(t), where the values are not very close to 1 (to Ic), time derivative w˙/Phi1 in Eq. (6)can be neglected and then, the relation sin /Phi1= Ip(t)works well. The following question appears: is magnetization rever- sal determined by the value of mzat the end of the current’s pulse? The answer is positive for small wonly, i.e., when ωF/lessmuchωc. In this case, when the pulse is switched off, the y component of effective field, Eq. (3), in the LLG equation can be neglected, because Ip≈sin/Phi1. Then, the dynamics ofmzis determined by parameters of the LLG equation: ifmz<0, we observe the magnetization reversal, and it does not happen in the opposite case ( mz>0). At a large value of wthe magnetization reversal is determined by the tails in sin /Phi1after switching the pulse off. This feature is demonstrated in Fig. 2, where the influence of sin /Phi1tails on the dynamics of magnetization component mzis shown at different w. In the case w=0.01 [see Fig. 2(a)] and the chosen set of parameters we observe the fastest reversal, i.e., the magnetization reversal happens in the current pulse time interval and after switching the pulse off, the mz=−1. In the case of w=0.1, [Fig. 2(b)] the magnetization rever- sal is also realized, but the value of mzafter switching the pulse off is mz=−0.82 and it reaches the mz=−1 att=40. At w=1 [Fig. 2(c)], even the value of mzis positive and has a large enough value at the end of the current pulse ( mz=0.5), we nevertheless observe the mag- netization reversal, but it reaches the mz=−1a t t=60 only. The magnetization reversal is not realized at w=10 [Fig. 2(d)] for the chosen values of the system parameters. So, the value of w, i.e., the relation of ωFtoωcplays an important role for the magnetization reversal. 014003-5MAZANIK, RAHMONOV, BOTHA, and SHUKRINOV PHYS. REV. APPLIED 14,014003 (2020) (a) (c)(b) (d) FIG. 2. Time dependence of mzatG=60,α=0.1, r=0.1, As=0.5, and δt=1 for different values of parameter windicated in the figures. Insets demonstrate the enlarged parts showing them zdynamics in the time interval around current pulse. IV. SOLUTION OF THE LLG EQUATION FOR THE Gr/greatermuch1 CASE Our theory is based on a few key observations. The first one is that for small wand current pulse Ip(t), where the value Asis not close to 1 (to Ic), as discussed in the previ- ous section, we can neglect the term w×d/Phi1/ dtin Eq. (6), which implies the relation Ip(t)=sin/Phi1. (11) The second observation is that the condition w/lessmuch1 can be rewritten as w=(1/G)(/Phi1 0/2πV)(γ/planckover2pi1/2eRM 0)∼const× 1/G/lessmuch1. It means that w/lessmuch1 does not imply the case of small G/lessmuch1, therefore we can use the LLG equation in the Gr/greatermuch1 limit as carried out in Ref. [ 23]. It was also estimated there, that it is plausible for Gto vary in a wide range, starting from G/lessmuch1 till G∼100/greatermuch1. The third observation is that the Gilbert damping can be relatively small α/lessmuch1[52–54]. So, if the duration of the current pulse is not long, the damping cannot influence the magnetization significantly, and the system may be consid- ered as it is at α=0. Estimations for this case is given in the next section. According to the previous remarks, using Eq. (11),w e can write the LLG equation during the pulse as ⎧ ⎨ ⎩˙mx=Grm zsin/Phi1=GrI p(t)mz, ˙my=mxmz, ˙mz=− Grm xsin/Phi1=− GrI p(t)mx.(12)The limit of the strong coupling Gr/greatermuch1 (but r/lessmuch1) can be treated analytically [ 23]. In this case my(t)≈0a n d for applicability of this method we also need GrI p(t)/greatermuch1 during the pulse. In the opposite case, zeroes of current pulse Ip(t)destroy the predominance of the used terms and more careful consideration should be carried out. Because my(t)≈0, then mx=ρsinφ,mz=ρcosφand we find directly ˙φ=GrI p.S o , φ(t)=Gr/integraldisplayt t0dt1Ip(t1). (13) As we see from Eq. (9), after the pulse has been switched off, the sin /Phi1has a fast drop to 0 due to condition w/lessmuch1. In this time region the dynamics of the magnetization is deter- mined only by the interplay of the magnetic anisotropy and the Gilbert damping, which makes the magnetization line up along the easy axis [ 55]. So, as follows from Eq. (13), the magnetization reversal occurs when cos/bracketleftbigg Gr/integraldisplayt0+δt t0dt1Ip(t1)/bracketrightbigg <0, (14) where δtis the pulse duration. We illustrate this idea in Figs. 3(a) and3(b) for a rect- angular pulse Ip(t)=As[θ(t−t0)−θ(t−t0−δt)]with As=0.5 for two pulse durations δt1=1a n dδt2=3. The parameters G=100, r=0.1,α=0.005, w= 0.01 are used. In the first case our criteria, Eq. (14), gives cos (GrA sδt1)=0.28>0, so the reversal is absent, whereas for δt2=3 we get cos (GrA sδt1)=−0.76<0 and the reversal occurs. We see that the solution, Eq. (13), represented by the blue dashed curve, coincides with the numerical one, represented by the green solid curve, using the complete Eqs. (6)and (1)with Eq. (3)during the pulse. When the pulse has been switched off, the damp- ing destroys any deviations from the easy axis mz=±1. It is demonstrated in the insets to Fig. 3. It should be noted, that the magnetization reversal is not affected by the form of the current pulse, but by its inte- gral over the pulse duration only. This is demonstrated in Fig. 4(c) for the pulse Ip(t)=0.75−| t−t0−t/2|/3, δt=3. The integral/integraltext dt1Ip(t1)for such a pulse is the same as for the pulse in Fig. 3(b), so we see that dynamics of mz and the magnetization reversal appearance are not different from the case presented in Fig. 3(b). V. SMALL DAMPING REGIME It was demonstrated in Ref. [ 47] by numerical simu- lations that there was a periodicity in the appearance of intervals of the magnetization reversal under the variation of the spin-orbit coupling, Gilbert damping parameter, and Josephson-to-magnetic energy ratio. Now we can see that 014003-6ANALYTICAL CRITERIA FOR MAGNETIZATION REVERSAL... PHYS. REV. APPLIED 14,014003 (2020) (a) (b) FIG. 3. Dynamics of mzbased on numerical solution of Eqs. (4)and(6)(blue dots) and analytical solutions of Eq. (13) (green line) for different pulse amplitudes and widths. The cur-rent pulse is shown by red. The parameters of calculations are G=100, r=0.1,α=0.005, w=0.01, t 0=1. (a) As=0.5, δt=1; (b) As=0.5,δt=3. the origin of this feature follows from Eq. (14), which leads to such a periodicity by changing parameters of the system and current pulse. As a result, we observe the intervals of parameters with the magnetization reversal and its absence. Based on this equation we can repro- duce results of the numerical simulations of Ref. [ 47]a n d show the way to optimize magnetization reversal at differ- ent conditions. Under current pulse the magnetic moment has complex oscillations determined by system and pulse parameters. Due to the Gilbert damping, the deviated mag- netic moment returns back to the stable states with mz= 1o r mz=−1. To describe its dynamics, we write the equation for myincluding the first non-neglecting term in damping parameter α ˙my=mxmz+αGr(1−m2 y)sin/Phi1. (15) At the beginning of the pulse my(t)≈0,mxmzmakes fast oscillations due to Gr/greatermuch1, so rising of myis determined FIG. 4. Dynamics of mz(green line, analytical; blue dots, numerical) with the current pulse Ip(t)=0.75−| t−t0− δt/2|/3,δt=3,δt=3 (red line). Insets show dynamics after switching current pulse off. only by the term αGrsin/Phi1. For applicability of Eq. (13) we need to keep my(t)≈0, which imposes the condition for the small damping regime /integraldisplayt0+δt t0dt1αGrsin/Phi1(t1)=αGr/integraldisplayt0+δt t0dt1Ip(t1)/lessmuch1. (16) For example, at G=100, r=0.1, As/Ic=0.5,δt=3 we have α/lessmuch0.07, which corresponds to the experimental value of the Gilbert damping parameter [ 52–54]. According to Eq. (14), the magnetization reversal in the r-Gplane under pulse Ip(t)=As[θ(t−t0)−θ(t−t0−δt)] occurs in the hyperbolic areas at π 2+2πn≤GnrA sδt≤3π 2+2πn (17) for n=0,±1,..., whereas the most efficient reversal appears when the condition cos(GrA sδt)=−1 (18) is fulfilled, i.e., GrA sδt=π+2πn. Equation (17) does not depend on α, but it indicates the intervals of MRatα=0. These intervals are shown in Fig. 5by dashed lines. We see that analytical intervals coincide with the numerical ones, calculated at small α.I t allows us to make a conclusion that magnetization reversal does not depend on αat its small values. In order to test the effect of the small damping regime determined by Eq. (16), we calculate numerically the areas in theα-Gdiagram where the reversal appears. Results are demonstrated in Fig. 5(a). As we see, in the small damping regime the magneti- zation reversal does not depend on α. The areas, where it 014003-7MAZANIK, RAHMONOV, BOTHA, and SHUKRINOV PHYS. REV. APPLIED 14,014003 (2020) (a) (b) FIG. 5. (a) Comparison of analytical and numerical results demonstrating the periodicity MR realization in the α-Gplane at small Gilbert damping. The orange stripes reflect the areaswhere magnetization reversal is realized. The dashed blue lines correspond to the areas obtained analytically by Eq. (17).T h e calculation is performed for the Gwith the step /Delta1G=0.5 and for αwith the step /Delta1α=0.0001. Other parameters are r=0.1, w= 0.01, A s/Ic=0.5,δt=3; (b) results of numerical calculations at large Gilbert damping produced with the same parameters. occurs, are periodic in parameter G, which is determined by Eq. (17). As we see, the realization of the magnetiza- tion reversal intervals in α-Gplane obtained by numerical simulations at small damping, is in agreement with the analytical results. Results of numerical calculations of the magnetization reversal intervals in α-Gplane at large Gilbert damping produced with the same parameters are demonstrated in Fig.5(b). We see essential variations of the stripes at large Gandα. The estimations made for different S-F-SJoseph- son junctions show small values of Gilbert damping whenour theory works [ 29,30,36]. Our theory works also in the limit of small Gilbert damping only. Nevertheless, we con- sider that the presented results are a challenge for future theoretical considerations.VI. PERIODICITY OF THE MAGNETIZATION REVERSAL IN r-GPLANE For simplicity we again consider a rectangular pulse in the low damping regime and small w. Equation (17) gives the hyperbolic curves for different n. From a physical point of view they are the curves of a constant ampli- tude for the driving force in the LLG equation, Eq. (1). In this situation the magnetic moment becomes aligned in the m z=−1 direction exactly after the pulse has been switched off, and the relevant time scale is determined only by the pulse duration, not by the Gilbert damping. It helps us to optimize the pulse duration in order to make the fastest reversal. We see from Eq. (18) that the shortest time is realized for n=0, i.e., δteff=π GrA s. (19) This situation is demonstrated in Fig. 6for G=100, r= 0.1,α=0.005, w=0.01, As=0.5, and δteff=0.628. It leads to the reversal time δtrev≈0.6×10−10s for typical ωF∼10 GHz. This time is 2 orders of magnitude smaller than the estimated one in Ref. [ 46]. Similar hyperbolic profiles of 1 /δteffon Ip(t)were obtained theoretically in Ref. [ 56] and experimentally in Refs. [ 44]a n d[ 57] for a spin-transfer-induced magneti- zation reversal setup in current-perpendicular spin-valve nanomagnetic junctions. In contrast to our case, this type of setup needs some critical spin-polarized current for magnetization reversal. In order to test our analytical results, we calculate numerically the areas of the magnetization reversal in the r-Gplane, using the complete Eq. (1)with Eqs. (3)and(6). In Fig. 7we compare them with the analytical results (shown by dashed lines) based on Eqs. (17) and(18). FIG. 6. The same as in Fig. 3for the current-pulse duration δteff=0.628 together with analytic according to Eqs. (19).T h e green line corresponds to the analytical solution and blue dots to the numerical one. The current pulse is shown by the red line. 014003-8ANALYTICAL CRITERIA FOR MAGNETIZATION REVERSAL... PHYS. REV. APPLIED 14,014003 (2020) FIG. 7. Periodicity of the magnetization reversal in the r-G plane. Realization of magnetization reversal is shown by the green stripes, the borders of the areas, Eq. (17), by the blue dashed lines and the curves for the most efficient reversal, Eq.(18), by the red lines. The calculation is performed with the step /Delta1G=0.5 and step /Delta1r=0.001. Other parameters are α=0.005, w=0.01, As/Ic=0.7,δt=3. The solid lines cor- respond to the analytical expression (17) with n=0, 1, 2, ...,9 . Black circles indicate the point where the dynamics of mz(t)is shown in Fig. 8. The almost perfect agreement between numerical and analytical calculations stresses the validity of our theory at the chosen system’s parameters. It should be noticed that the periodicity of the magnetization reversal in the r- Gplane was observed in Ref. [ 47] numerically only and for a nongauge-invariant scheme. Comparing both results we may conclude that, actually, the term ˙ϕ0in Eq. (6), which makes the equations gauge invariant, only slightly shifts these areas of magnetization reversal. But, from the other point of view, the gauge-invariant form of equa- tions gives a possibility for analytical consideration of Eq.(6). Finally, we discuss the magnetization reversal at the parameters corresponding to the different points in the stripes, indicated in Fig. 7. The results of numerical sim- ulations of mztemporal dependence in the first, second, and third stripes at different values of spin-orbit coupling are shown in Fig. 8. First we compare the magnetization dynamics for three points in the lowest stripe shown in Figs. 8(a)–8(c).A t the boundaries of this stripe we observe a slow reversal, while at the point corresponded to the center of stripe (r=0.021) the magnetization reversal is the fastest one [Fig. 8(b)]. The similar behavior we observe at the points corresponded to the second and the third stripes shown in Fig.8. The main important difference between dynamics of the mzin the centers of the different stripes is the following: for the second stripe the mzmakes an additional rotation in compare with a case of the first stripe [see Fig. 8(f)], also for the third stripe mzmakes one more additional rota- tion [see Fig. 8(i)]. So, it could be directly concluded, thatthe stripes in Fig. 7differ from each other by the number of oscillations made by the mz(t)component during the current pulse. It should be noted that the obtained periodicity of the magnetization reversal in the S-F-Sϕ0junction is similar to the well-known effect following from the Bloch equa- tions in quantum optics and nuclear magnetic resonance [47,58]. Generally speaking, here we find the limits of parameters, where the famous πpulse is realized in our system. As seen from Figs. 7,8,a n d (13), the number of oscillations nmade by mzduring the reversal pro- cess is proportional to the integral of the pulse function/integraltext dtI p(t)over time, multiplied by Gr/π. This property takes place for Gr/greatermuch1,w/lessmuch1 and small damping regime, Eq.(16), otherwise the process of the reversal becomes more complicated, as discussed in Ref. [ 47]. To implement the magnetization-switching element we propose using a memory element that is based on a fer- romagnetic anomalous Josephson junction [ 7]. The latter consists of a S-F-SJosephson junction with a Rashba-like spin-orbit coupling. Its ground state corresponds to a finite phase shift in its current-phase-relation 0 <ϕ 0<π. In our proposal, the magnetization-switching element is a Joseph- son junction with a ferromagnetic link, with time-reversal broken intrinsically by the exchange field. As demon- strated theoretically, in these junctions the magnetization of the Flayer can be controlled by passing an electric cur- rent through the device [ 23,33,35,46,59,60]. We propose to use such a junction as a memory element with the informa- tion encoded in the magnetization direction of the Flayer. An important issue for the memory element is the effects stemming from the unavoidable thermal fluctuations on both phase and magnetization dynamics. An exhaustive analysis of the noisy dynamics of a current-biased S-F-S Josephson junction, considering the influence of stochas- tic thermal fluctuations were presented in Ref. [ 45] and the robustness of such memory against noise-induced effects was investigated. Finally, we specify the set of ferromagnetic layer param- eters and junction geometry for the possible experimental observation of the predicted effect. In a ferromagnet with weak magnetic anisotropy, as in the permalloy with [ 61] K∼4×10−5KA−3, a junction with a relatively high crit- ical current ( Ic∼3×105–5×106A/cm2), as reported for the Nb/NiGdNi/Nb junction [ 62], and a ferromagnetic film volume V=10−15cm3, we find that the ratio between the Josephson and the magnetic anisotropy energy to be G∼20–200 [ 55]. Another suitable candidate may be the permalloy doped with Pt[63]. As estimated in Ref. [ 23], if the length of the ferromagnetic layer is of the order of the magnetic decaying length we have r∼0.1, the value of the parameter Gris in the range 2–20. Note that ferro- magnets without inversion symmetry, like MnSi or FeGe, may also be interesting candidates for the observation of the discussed phenomena. 014003-9MAZANIK, RAHMONOV, BOTHA, and SHUKRINOV PHYS. REV. APPLIED 14,014003 (2020) (a) (b) (c)(d) (e) (f)(g) (h) (i) FIG. 8. Results of numerical simulations of mztemporal dependence in the first stripe (a)–(c), the second stripe (d)–(f), and the third stripe (g)–(i) for the set of parameters G=100,α=0.005 w=0.01, As=0.5,δt=3, and different value of the SO coupling parameters indicated in the figures. VII. CONCLUSIONS Theϕ0Josephson junction is an interesting and important object for superconducting electronics. Its experimental realization opened the way for its use in different applications, particularly, as a cryogenic mem- ory element. In our paper we study the reversal of the magnetic moment in the superconductor-ferromagnet-superconductor ϕ 0Josephson junction and developed a theory, which allows us to understand the phenomena of magnetization reversal in this system, and also to predict its occurrence at the chosen system parameters. The ana- lytical criteria for the magnetization reversal are derivedin Eqs. (14) and (18), which clarify the dependence of the reversal on the Josephson-to-magnetic energy ratio, spin-orbit coupling, and pulse amplitude, as well as its duration. We compare our analytical results with numer- ical simulations, explain the observed diagrams G-rand G-α, and demonstrate their almost perfect agreement. We demonstrate the magnetization reversal for different forms of the current pulse. In particular, we find the conditions for faster reversal, which is important for the creation of cryogenic memory, based on this system. We consider that the obtained analytical criteria will be useful in the development of a memory element based on the S-F-Sϕ0 Josephson junction. 014003-10ANALYTICAL CRITERIA FOR MAGNETIZATION REVERSAL... PHYS. REV. APPLIED 14,014003 (2020) ACKNOWLEDGMENTS The reported study is partially funded by the RFBR, Projects 18-02-00318 and 18-52-45011-IND. Sections V and VIof the reported study is funded by the RFBR, Project 20-37-70056. 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PhysRevB.91.214424.pdf

PHYSICAL REVIEW B 91, 214424 (2015) Phase diagram and optimal switching induced by spin Hall effect in a perpendicular magnetic layer Shu Yan*and Ya. B. Bazaliy† Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA (Received 26 January 2015; revised manuscript received 31 May 2015; published 17 June 2015) In a ferromagnet/heavy-metal bilayer device with strong spin Hall effect an in-plane current excites magnetic dynamics through spin torque. We analyze bilayers with perpendicular magnetization and calculatethree-dimensional phase diagrams describing switching by external magnetic field at a fixed current. We thenconcentrate on the case of a field applied in the plane formed by the film normal and the current direction.Here we analytically study the evolution of both the conventional “up”/“down” magnetic equilibria and theadditional equilibria created by the spin torque. Expressions for the stability regions of all equilibria arederived, and the nature of switching at each critical boundary is discussed. The qualitative picture obtainedthis way predicts complex hysteresis patterns that should occur in bilayers. Analyzing the phase portraits of thesystem we find regimes where switching between the up and down states proceeds through the current-inducedstate as an intermediate. The first step of such two-step process is fast and resembles ballistic switching forthe reasons discussed in the paper. Using numeric simulations we analyze the switching time and compareit to that of a conventional spin torque device with collinear magnetizations of the polarizer and the freelayer. DOI: 10.1103/PhysRevB.91.214424 PACS number(s): 75 .78.−n,85.75.−d I. INTRODUCTION Recently a number of investigations focused on bilayer structures consisting of a ferromagnetic (F) layer and anonmagnetic (N) layer with strong spin-orbit interaction madeof heavy metals such as Pt, Ta, or W [ 1–10]. It was theoretically predicted and experimentally observed that when an in-planeelectric current is being applied, the itinerant electrons insidethe nonmagnetic layers become spin polarized due to thestrong spin-orbit coupling and exert a spin torque on the fer-romagnetic layers. This additional torque contributes to themagnetization dynamics described by the Landau-Lifshitz-Gilbert (LLG) equation. Up to now two models have beenproposed to account for the effects. One of them [ 2,5] treats the bilayer structure as a two-dimensional system withstrong interfacial Rashba spin-orbital coupling due to thestructural inversion symmetry breaking in the direction normalto the interface [ 11]. This model leads to a fieldlike torque directed along ˆm×(j e׈z)[12–16], where ˆm=M/Msis the magnetization unit vector ( Msrepresents the constant absolute value of the magnetization M),jeis the in-plane electric current density, and ˆzis a unit vector perpendicular to the plane of the layers. The other model [ 1,4,6–9] is based on the interfacial diffusion of the pure spin current that originatesin the heavy-metal layers due to the bulk spin Hall effect(SHE) [ 17–20] and leads to spin-transfer torque dynamics [21,22] in the magnetic layers. In the SHE model the torque is directed along ˆm×[ˆm×(j e׈z)] [23]. This type of torque is called a Slonczewski, or dampinglike, or adiabatic torque inthe literature. Several experiments showed that an in-plane electric current flowing through the structure is able to switch themagnetization of the ferromagnetic layer [ 5,7–9]. In those *syan@physics.sc.edu †yar@physics.sc.eduexperiments the F layers were magnetized perpendicular to the film plane. It is believed that the observed magneticreversal can only be induced by the dampinglike torque. Thereasons for this are (a) experimentally measured switchingphase diagrams are in accord with the macrospin modelcalculations [ 7,24], and (b) due to its symmetry, a fieldlike torque, if there is any, does not favor either the “up”or “down” state of the perpendicular magnetization, andtherefore should not contribute to switching. These argumentsseem to favor the SHE-based model; however, subsequentcalculations [ 25–27] suggested that the model based on Rashba coupling generates both fieldlike and dampingliketorques, and thus is also capable of describing the switching(Fig. 1). Despite the fact that the underlying torque mechanism is still not fully understood, a thorough study of the switchingbehavior is of importance for analysis and prediction. Inthis paper we perform such a study describing the magneticlayer within the framework of a macrospin model. This isa reasonably good approximation when the sample size issmall enough for the magnetization to be close to uniform.In larger samples the situation may be different and recentexperiments have shown that domain nucleation [ 28] and propagation [ 24,29,30] need to be taken into account in those cases. With macrospin approximation in place, our goal willbe to describe magnetic switching induced by an externallyapplied field Hat a fixed in-plane electric current. It will be further assumed that the magnetic anisotropy energy of the Flayer corresponds to an easy axis directed normal to the layers(perpendicular anisotropy) E(m)=−K(ˆm·ˆz) 2. (1) Magnetization is switched between the up and down states at critical fields Hcthat form a surface in three-dimensional H space. Without electric current and for the magnetic anisotropygiven by ( 1) this surface is an axially symmetric figure of 1098-0121/2015/91(21)/214424(12) 214424-1 ©2015 American Physical SocietySHU Y AN AND Y A. B. BAZALIY PHYSICAL REVIEW B 91, 214424 (2015) FIG. 1. Schematic diagram of the bilayer spin Hall effect device. Electric current jflows along the yaxis. revolution with a cross section given by the astroid curve [31]. The presence of the in-plane current jebreaks the axial symmetry through the damping and the fieldlike spin torques.However, the fieldlike torque can be compensated by anexternal field in the j e׈zdirection, thus, its presence simply shifts the entire critical surface without changing its shape.For the purpose of finding the shape of the current-modifiedastroid we may consider the damping torque alone. In this paper we calculate the three-dimensional (3D) critical surface in Hspace using the method of Refs. [ 32,33]. Astroid modification in the same setting was previouslystudied in Refs. [ 7,34–36]. Earlier analytic studies dealt with the properties of equilibria that exist at zero current and aremodified when the spin torque is turned on. This paper providesresults on the novel equilibria that are produced by the current.The evidence for their existence was previously obtainednumerically [ 37] but an in-depth study of their properties, including their role in magnetic switching, was lacking. The outline of the paper is as follows. In Sec. II,w er e v i e w the theoretical approach of Refs. [ 32,33]. In Sec. III, we find the three-dimensional H cat a given current magnitude and give analytic formulas for this multisheet surface. However,the surface itself does not provide all the details of switching.To understand them one has to determine which equilibriaare destabilized on each of its sheets. In Sec. IV, we perform such an analysis for a particular cross section of the criticalsurface, the one with Hconfined to the plane formed by the electric current j eand the film normal ˆz. After resolving the implicit 3D analytic expressions, we find the correspondingtwo-dimensional (2D) cross section of H c. We further observe, in accord with the previous numeric investigations [ 37]. that sufficiently large currents produce an extra equilibriumin addition to the existing up and down ones. Analyticalexpressions are derived for the position and stability regionof this equilibrium. Knowing the equilibrium states of thesystem, we provide the description of all possible switchingevents and discuss the usage of the modified astroid invarious experimental situations. In Sec. V, we analyze the phase portraits of the system and study its evolution after thedestabilization of a given equilibrium. Here we show thatthe extra equilibrium plays an important role in the switchingprocess, influencing its speed and fashion.II. GENERAL DESCRIPTION OF THE THEORETICAL APPROACH Magnetization dynamics of the ferromagnetic layer in the macrospin approximation is governed by the LLG equation: dM dt=−γμ 0(M×Heff)+α Ms/parenleftbigg M×dM dt/parenrightbigg , (2) where αis the Gilbert damping factor, γis the gyromagnetic ratio, and Heffis the total effective field. The standard LLG equation can be transformed into dˆm dt=− ˆm×heff−αˆm׈m×heff, (3) where the field is rescaled as heff=Heff/Hkusing the char- acteristic anisotropy field Hk=2K/μ 0Ms, and the time is rescaled as t→t/prime=γμ 0Hkt/(1+α2). Hereafter, all the field-related terms that are written in lowercase letters aredimensionless (normalized by H k). The method used to find the critical surfaces [ 32,33] can be summarized as follows. A stationary solution ˆm0of Eq. ( 3) satisfies the equilibrium condition ˆm×heff=0, which indi- cates that the magnetization at equilibrium should be parallel tothe total effective field, i.e., h eff=λˆm0with arbitrary λ. Total effective field is given by heff=h−∇ε+hsp, where his the external field, ε=E/(μ0HkMs) is the rescaled anisotropy energy, and hspis the spin-torque effective field hsp=αj[m×(ej׈z)], (4) where ejis a unit vector in the electric current direction and αj is a spin-torque strength parameter, proportional to the electric current density. Equation heff=λˆm0can be solved for the external field as h=λm0+∇ε(m0)−hsp(m0). The meaning of this formula is that for any given magnetization directionthere is a whole line of external fields, parametrized by λ, which make it an equilibrium—stable or unstable. In sphericalcoordinates with three orthogonal unit vectors defined as ˆm=sinθcosφˆx+sinθsinφˆy+cosθˆz,ˆθ=∂ˆm/∂θ, and ˆφ=(1/sinθ)∂ˆm/∂φ, we get h=λˆm 0+/parenleftbig ∂θε−hθ sp/parenrightbig 0ˆθ0+/parenleftbigg1 sinθ∂φε−hφ sp/parenrightbigg 0ˆφ0,(5) where ∂θstands for ∂/∂θ and the superscript θindicates theˆθcomponent of a vector field (e.g., hθ eff=heff·ˆθ), etc. Equation ( 5) maps the 3D space {λ,θ 0,φ0}to the 3D field space{hx,hy,hz}. Next, stability of equilibrium states is analyzed. This is done by expanding Eq. ( 3) in small deviations ˆm=ˆm0+δˆm up to the linear terms. Such an expansion produces two coupledlinear differential equations /parenleftbigg˙δθ sinθ 0˙δφ/parenrightbigg =A(θ0,φ0)/parenleftbigg δθ sinθ0δφ/parenrightbigg , (6) with matrix A(θ0,φ0) given by A=/bracketleftBigg ∂θ/parenleftbig αhθ eff+hφ eff/parenrightbig1 sinθ∂φ/parenleftbig αhθ eff+hφ eff/parenrightbig ∂θ/parenleftbig αhφ eff−hθ eff/parenrightbig1 sinθ∂φ/parenleftbig αhφ eff−hθ eff/parenrightbig/bracketrightBigg . (7) Stationary solutions can be classified as stable or unstable using the eigenvalues of A.F o ra2 ×2 matrix the two 214424-2PHASE DIAGRAM AND OPTIMAL SWITCHING INDUCED . . . PHYSICAL REVIEW B 91, 214424 (2015) eigenvalues μ±are uniquely determined by its determinant, detA, and trace, tr A[38]. An equilibrium is stable when both eigenvalues μ±are either complex numbers with negative real parts, or negative real numbers, which leads to the stabilitycriterion detA> 0 and tr A< 0. (8) For a given direction ( θ 0,φ0) this criterion selects the parts of the field line h(λ,θ 0,φ0)(5)f o rw h i c h( θ0,φ0)i sn o tj u s t an equilibrium but specifically a stable equilibrium. Thosestable parts are specified by the intervals of λwhere conditions (8) are satisfied. By evaluating expression ( 7) at external field specified by Eq. ( 5) one obtains A(λ). We find that for the arbitrary form of spin torque and arbitrary anisotropyenergy, tr A(λ) is a linear function λwith a negative linear coefficient, and det A(λ) is a quadratic function of λwith a positive quadratic coefficient. To simplify the expressions, weintroduce a vector field f=− ∇ ε+h spand its matrix gradient ∇f≡/bracketleftbigg ∂θfθ∂θfφ ∂φfθ∂φfφ/bracketrightbigg (9) (see Appendix Afor the explicit expressions). The roots of equations tr A(λ)=0 and det A(λ)=0 can be respectively calculated as λT(θ0,φ0)=1 2/bracketleftbigg ∂θfθ+∂φfφ+1 α(∂θfφ−∂φfθ)/bracketrightbigg ,(10) λ±(θ0,φ0)=∂θfθ+∂φfφ 2 ±/radicalBigg/parenleftbigg∂θfθ−∂φfφ 2/parenrightbigg2 +∂θfφ∂φfθ.(11) In terms of the critical values λTandλ±, the stability criteria become λ> Max(λT,λ+)i f λT/greaterorequalslantλ− λT<λ<λ −orλ>λ +ifλT<λ−.(12) When λ±are complex, det Ais always positive and criteria ( 12) can be further simplified as λ>λ T. The full classification is given in Appendix B. Substituting functions λ+,λ−,o rλTforλin Eq. ( 5) one generates three surfaces in the field space, which are denotedS +,S−, andST, respectively. Their physical meaning is that at least one equilibrium changes its stability when external fieldcrosses such a surface. It is either locally destabilized whentheS Tsurface is crossed, or merges with a saddle point when theS±surfaces are crossed [ 39]. The entire critical surface S is constructed from the parts of S+,S−, andSTas explained in Ref. [ 33]. III. 3D PHASE DIAGRAM In this section we construct the three-dimensional critical surface using the method of Sec. II. The dimensionless perpendicular anisotropy energy has the form ε=− cos2θ/2. We set the in-plane current to be along the +ˆydirection, and the current-induced field is then hsp=αjˆm׈xwithαj given by αj=θSHje/j0, where θSHis the spin Hall angle, j0=2eM sdFHk//planckover2pi1is the characteristic current density, and dF FIG. 2. (Color online) Illustrative example of critical values (a) λT(red) and (b) λ+/−(blue/green) as functions of ( θ,φ)f o rα=0.1 andαj=0.1. Zero current critical values (independent of φ)a r e plotted for reference as additional lines at φ=−π/2u s i n gt h es a m e color convention. For consistency, the same parameters and color convention are used in the remaining figures in this section. is the thickness of the F layer. For brevity, we drop index “0” for the equilibrium direction. The critical values of λcalculated according to ( 10) and ( 11) specialize to λT(θ,φ)=sin2θ 2−cos2θ−αj αsinθcosφ, (13) λ±(θ,φ)=sin2θ 2−cos2θ±sinθ/radicalbigg sin2θ 4−α2 jcos2φ. (14) Figure 2shows three critical values as functions of equilibrium angles at nonzero current with αj=0.1 andα=0.1. For practical calculations we decompose Eq. ( 5)i n t o Cartesian coordinates hx=(λ+cos2θ)s i nθcosφ, (15a) hy=(λ+cos2θ)s i nθsinφ−αjcosθ, (15b) hz=(λ−sin2θ) cosθ+αjsinθsinφ. (15c) For an arbitrary λthese equations represent the mapping of the ( λ,θ,φ ) space to the ( hx,hy,hz) space for the case of uniaxial anisotropy and chosen electric current direction.When functions λ T(θ,φ)(13)o rλ±(θ,φ)(14) are substituted forλ, one obtains parametric expressions for the critical surfaces STandS±with parameters ( θ,φ) running through all possible values, 0 /lessorequalslantθ/lessorequalslantπand 0/lessorequalslantφ/lessorequalslant2π. For each ( θ,φ) one has to choose the relevant Ssurfaces [33] that correspond to critical λ’s bracketing the stability intervals ( 12). For example, for αj=0 one finds λTto be in the midpoint of the interval ( λ−,λ+) for any direction ( θ,φ), and therefore only S+is relevant and constitutes the entire critical surface. For nonzero current λTmay leave the interval ( λ−, λ+) for some values of ( θ,φ). In those cases destabilization boundaries should be selected for every direction individually.The selection of relevant critical λsurfaces is illustrated in Fig. 3. By substituting the relevant critical λvalues into Eq. ( 15), we plot the critical surface Sas shown in Fig. 4. Three types of modifications due to spin torque can be observed. First, theoriginal astroid is distorted forming the blue region bounded bytheS +surface. Second, part of the S+surface, where λT>λ+ is satisfied, becomes irrelevant. A red region in the figure 214424-3SHU Y AN AND Y A. B. BAZALIY PHYSICAL REVIEW B 91, 214424 (2015) FIG. 3. (Color online) Selection of relevant λsurfaces: (a) 3D view of the λTandλ±surfaces and (b) the cross section at θ=π/2. The relevant surfaces separate stable and unstable values of λ,a s illustrated by solid lines in (b). is formed this way. Third, extra equilibrium points appear in the field space where λT<λ−is satisfied, forming the green region. While the task of constructing Sis achieved, its interpretation requires more work. Critical surface can bea complicated self-crossing manifold [ 33]. To understand it one has to specify which equilibrium is destabilized at eachsurface, and on which side of the surface is this equilibriumstable. A corresponding study for one cross section of Sis given in the next section. FIG. 4. (Color online) Critical surface S: (a) 3D view; (b) side view along the xaxis; (c) side view along the yaxis. For reference purposes the conventional Stoner-Wohlfarth (SW) astroid is plotted in (b) and (c) using a dashed black line. The same color conventionand parameters as in Fig. 2are adopted.IV . PHASE DIAGRAM IN THE y-zPLANE The 3D phase diagram is quite difficult to use due to the complicated shape of the critical surface S. Moreover, experiments are often performed with the field being confinedwithin the y-zplane [ 7,24]. Here we study in detail a section of Scorresponding to the external field hconfined to such a plane. This section is a line ¯Sin the 2D space ( h y,hz). Note that if a fieldlike component spin torque is present in the system, it hasto be compensated by an appropriate constant h xcomponent of external field in order for our results to be applicable. A field in the y-zplane satisfies a constraint hx=0. According to Eq. ( 15a) this implies a relationship between θ,φ, andλ. On the one hand, this relationship allows one to express the equilibrium angles ( θ,φ) as functions of ( hy,hz) and study how the equilibria evolve as a function of appliedfield. On the other hand, Eq. ( 15a) can be used to find the section ¯S. While the surface Sis given by a parametric formula with independently varying θandφas explained in Sec. III, the line ¯Sis found from the same formula with θandφbeing related to each other by a requirement that Eq. ( 15a) holds withλ=λ T,+,−(θ,φ). A. Evolution of equilibrium states Equation ( 15a) has three types of solutions: (I) φ=±π/2, (II)λ=− cos2θ, and (III) θ=0,π. Since the value of φat θ=0,πis immaterial, type III can be considered as a special case of type I. Thus we focus on the first two cases. Fordefiniteness, assume α j>0. Solutions of type I have sin φ=± 1. They are located on the meridian of the unit sphere lying in the y-zplane. We will call them on-meridian states. Eliminating λfrom Eqs. ( 15b) and ( 15c) one finds a system of equations for their polar angles φ=±π/2, (16) hycosθ∓hzsinθ=± sinθcosθ−αj. Depending on hy,hz, andαjthere can be four, two, or zero equilibrium states of type I. Solutions of type II have λ=− cos2θ. Equations ( 15b) and (15c) read hy=−αjcosθ, hx=− cosθ+αjsinθsinφ. Solving them one finds cosθ=−hy αj, (17) sinφ=hz−hy/αj/radicalBig α2 j−h2y with associated requirements |hy|/lessorequalslantαjand |(hz− hy/αj)/√ α2 j−h2 y|/lessorequalslant1 that define their domain of existence. Having φ/negationslash=±π/2, type II solutions are away from the y-z plane and will be called off-meridian states. They exist as apair with the same polar angle θand azimuthal angles φand π−φ. Equilibrium states can be visualized as points on the unit sphere that change their positions when the experimental 214424-4PHASE DIAGRAM AND OPTIMAL SWITCHING INDUCED . . . PHYSICAL REVIEW B 91, 214424 (2015) xyz h S2S1 OM 1OM 2Mup MdownX FIG. 5. Equilibrium states shown as points on the unit sphere for small current αj>0a n dfi e l d h. Solid arrows show equilibria displacements as his increased at fixed αj. Dashed empty arrows show equilibria displacements as αjis increased at fixed h.F o r example, increasing field at fixed current causes an eventual merging of points OM 1,OM 2,a n dS2that produces point X. parameters handαjare varied (Fig. 5). In the absence of current and external field the uniaxial magnet exhibits twostable equilibria θ=0,π: the up and down states. Due to the axial symmetry of the system the entire equator of the unitsphere forms a circle of unstable equilibrium states. At nonzero current spin torque breaks the axial symmetry of the problem even in the absence of magnetic field. Forα j/negationslash=0,h=0 the continuous set of unstable equilibria along the sphere’s equator is reduced to four isolated equilibriumpoints. Two of them are off-meridian states ( OM 1,2in Fig. 5) for which Eq. ( 17)g i v e s θ=π/2 and φ=0,π, i.e., the ±ˆx directions. The other two are on-meridian states S1,2given by Eq. ( 16). It will be shown below that they are saddle points. For small αjthe system has six equilibria: slightly displaced up (Mup) and down ( Mdown) on-meridian states, on-meridian states S1,2that are slightly displaced from the equator of the sphere, and the ±ˆxstatesOM 1,2(Fig. 5). The following useful rules apply to the on-meridian equilib- ria described by Eq. ( 16): (1) Increasing current shifts points Mup/down clockwise and points S1,2counterclockwise along the meridian (solid arrows in Fig. 5); asαjis increased, the statesS1andS2approach the up and down states, respectively. At a critical current they merge pairwise and disappear. (2)Increasing magnetic field shifts points M up/down along the meridian towards the field directions and points S1,2away from the field direction (dashed, empty arrows in Fig. 5). Consider now the situation with a small fixed current and a variable external field. For the discussion we will assume afixed direction of hbetween +ˆyand+ˆzdirections (see Fig. 5). Equations ( 17) show that as the field magnitude his increased, the off-meridian states approach the meridian and reach it at acritical field magnitude. Since the two off-meridian states aremirror symmetric with respect to the y-zplane, they reach the meridian simultaneously and merge. Moreover, using Eq. ( 17)one can show that the merging point also satisfies Eq. ( 16), so actually a merging of two off-meridian and one on-meridianequilibrium takes place. This triequilibrium merging is notaccidental. As discussed in Ref. [ 39], merging of equilibria has to conserve the winding number and it would be impossible forthe two off-meridian equilibria with equal winding numbers tomerge without the participation of a third equilibrium with theopposite winding number. Ashis increased further, the new equilibrium X, resulting from the merging of S 2,OM 1, andOM 2, remains on-meridian. Analysis in the next section shows that it is an unstable focus,analogous to the maximum energy point equilibrium of aconventional (no spin torque) uniaxial magnet subjected to theexternal field. In general, above the critical field the evolutionof the four on-meridian equilibria M up,Mdown,S1, andXis qualitatively similar to that found at αj=0. We may conclude that our system has two regimes: one at low magnetic fieldwhere spin torque dominates, and another one at high fieldwhere magnetic torque dominates. The spin-torque-dominatedregime is characterized by the presence of two OM equilibria produced by current. In the field-dominated regime the current-induced equilibria are gone. These results are quite natural. The SHE system is equiva- lent to a conventional spin-transfer device with spin polarizer p directed along +ˆx[40]. Spin torque attracts the magnetization topand repels from −p. At very large currents spin torque dominates all other torques, so, only two equilibrium points— one close to pand another close to −pshould remain. In our system we find that it is enough for the spin torque to dominatethe magnetic field torque in order to produce these equilibria.This happens because pis directed into a point on the equator that is already an equilibrium, albeit unstable, of the system atzero current. B. Stability of equilibria analysis and switching phase diagram In this section we are going to find the critical line ¯Sof equilibrium destabilization. It will be composed from partsproduced by type I and type II solutions. 1. On-meridian equilibria Equations ( 13) and ( 14) show that for the on-meridian statesλTis the midpoint of λ±interval for any current value. Therefore only λ+is needed to calculate the critical surface. By substituting λ=λ+(θ,φ) and sin φ=± 1i n t oE q s .( 15b) and ( 15c), we get an exact parametric form of ¯SM, the line of on-meridian equilibria destabilization. It turns out to bethe same as the one found before [ 34] using an approximate method. h y=± sin3θ−αjcosθ, (18) hz=− cos3θ±αjsinθ. By evaluating det Aand trAfor each on-meridian equilibrium it is possible to conclude that the up and down equilibria arestable foci, while the S 1,2equilibria are unstable saddle points. The ¯SMcurve for various spin-torque strengths are shown in Fig. 6. When magnetic field hcrosses ¯SM, one of the stable equilibria Mup/down merges with one of the saddles S1,2and 214424-5SHU Y AN AND Y A. B. BAZALIY PHYSICAL REVIEW B 91, 214424 (2015) FIG. 6. (Color online) Modified astroid composed of ¯SM(black) and ¯SOM[red (gray)] lines. Spin-torque strengths are (a) αj=0, (b) αj=0.3, (c)αj=0.5, and (d) αj=0.7. disappears. In fact, ¯SMrepresents the SW astroid boundary modified by the current [ 34]: The original astroid shape is squeezed along one of the diagonal directions. The segments of ¯SMconnecting the corner points of the astroid become unequal: two of them grow with increasing αj, and the other two shrink. The details of merging depend on whether the long or theshort segment is crossed by the field. On the short segment thesign of the m zcomponent of the disappearing equilibrium is always opposite to the sign of the field component hz. This property was satisfied everywhere on the conventionalSW astroid boundary, and we denote the short segment of ¯S Mas¯SMcwith index “ c” meaning “conventional.” On the long segment the sign of mzis not determined by the sign of hz. Indeed, points MupandS1merging on this segment have mz>0, and at the same time it can be crossed by a field with hz>0, if his directed at a small enough angle to the yaxis. We denote the long segment as ¯SMuwith index “ u” meaning “unconventional.” 2. Off-meridian equilibria Next, we analyze the stability of the off-meridian equilibria. Recall that for them λ=− cos2θand according to Eq. ( 14) λ/lessorequalslantλ−is automatically satisfied when λ±are real. Thus, according to criteria ( 12), only the ¯STcritical line is relevant whether λ±are real or complex, and the stability condition for these states is given by λ−λT=/parenleftbiggαj αcosφ−1 2sinθ/parenrightbigg sinθ> 0. (19) This inequality can be satisfied only for cos φ> 0, which means that the OM 2equilibrium characterized by π/2/lessorequalslant φ/lessorequalslant3π/2 is always unstable. The off-meridian equilibrium with−π/2<φ<π / 2 can be stable. The critical line λ=λT gives a destabilization boundary ¯SOMfor this equilibrium. Its0.5 0 0.51010.5 0 0.5 101 hyhz FIG. 7. (Color online) ¯SOMboundaries for various damping fac- tors with spin-torque strength set to αj=0.5. To make lines distinguishable, we adopt large damping parameters, with α/αjset to be 0.8, 1.5, 1.9, and 2.1 (going from the outermost to the innermost curves). At low damping regime ¯SOMis very close to the boundary of existence of the OM points [red (gray) dashed line]. analytic form is obtained from Eqs. ( 15b) and ( 15c)a s hz=hy αj±/radicalBig αj2−hy2/radicalBigg 1−α2 4αj2/parenleftbigg 1−hy2 αj2/parenrightbigg . (20) The ¯SOMcurve for various damping parameters at a fixed spin- torque strength is shown in Fig. 7. From det Aand trAanalysis it is possible to extract more detailed information about thenature of the OM 1,2equilibria. We find that inside the domain bounded by the ¯SOMlineOM 2is always an unstable node (two real positive eigenvalues), and the OM 1equilibrium is a stable focus (complex conjugate eigenvalues with negative realparts)—see Appendix Cfor a complete analysis. As the field increases and moves out of this domain, OM 1is destabilized but not destroyed. It continues to exist, first as an unstablefocus, and then as an unstable node, until it finally mergeswith the points OM 2andS2, as discussed in Sec. IV A .M o r e details of the OM 1state evolution are provided in Appendix C. C. Current-field diagrams Switching phase diagrams Figs. 6and 7describe exper- iments performed at fixed current with magnetic field of afixed direction increased until switching happens at a criticalvalueh c. In a different type of experiment one can measure how the hcthreshold depends on the current magnitude. Such experiments were indeed recently performed by Yu et al. [24] (also numerically modeled earlier [ 35]). The hc(αj) dependencies were measured for different field directions and,quite surprisingly, it was found that for fields making a finiteangle with the yaxis multiple switchings may occur. This fact finds a natural explanation in the framework of our theory.I nt e r m so fF i g . 6the critical fields are determined by the intersections of a straight line representing the field directionwith the lines ¯S Mand ¯SOM. If the direction of the field is defined by the angle θhwith respect to the zaxis, the former 214424-6PHASE DIAGRAM AND OPTIMAL SWITCHING INDUCED . . . PHYSICAL REVIEW B 91, 214424 (2015) 1.0 0.5 0.0 0.5 1.01.00.50.00.51.0 Αjh FIG. 8. (Color online) Tilted current-field phase diagram at α= 0. The black and red (gray) lines correspond to hcMandhcOM, respectively. The solid lines represent the diagrams with a tiltingangle of the field set to be 15 ◦with respect to the ˆydirection. The dashed lines represent the diagram with h||ˆy. Opposite tilting happens when the angle is negative. intersection point hcM(αj) can be found by solving Eq. ( 18) withhy=hcsinθh,hz=hccosθh. For the latter intersection pointhcOM(αj)E q .( 20) should be used. The results are shown in Fig. 8. One can see that for the field directed along the yaxis (θh=π/2) the dependence hcM(αj) exhibits a sharp peak at αj=0. As the field is tilted away from the yaxis, the position of the peak moves and its initially symmetricshape deforms. Eventually the deformation grows so big thatthe function h cM(αj) becomes multivalued, in accord with experimental findings. Comparing the current-field diagramwith the experimental diagram (Fig. 3of Ref. [ 24]) one can see a good qualitative correspondence. Here we show how the shape of h cM(αj) can be understood from the evolution of the modified astroid ¯SM. As the current is increased with αj>0, the astroid is squeezed in the (1 ,−1) direction and expanded in the (1 ,1) direction. The ¯SMulines approach the origin in the hspace and the ¯SMclines move away from it. When the field is directed along the yaxis, its line intersects the ¯SMuboundary. Since this boundary moves towards the origin with increasing αj, the function hcM(αj)i s decreasing. However, when the field is directed at an angle totheyaxis, its line may initially cross the ¯S Mcboundary. Since ¯SMcmoves away from the origin, the function hcM(αj) would increase. At a threshold value of current the field line goesexactly through the corner point between the ¯S Mcand ¯SMu. At this point hcM(αj) exhibits a cusp. For currents above the threshold, the field line crosses ¯SMuand, just like in the h||ˆy case,hcM(αj) becomes a decreasing function. For some angles θhthere may be situations when the field line crosses both ¯SMc and ¯SMulines. This is when hcM(αj) becomes multivalued and complicated hysteresis patterns are realized. The form of the other critical field line, hcOM(αj)( r e d curves in Fig. 8), is related to the evolution of the ¯SOMline. Since this line moves away from the origin in all directions,h cOM(αj) turns out to be an increasing function.D. Discussion of the phase diagram The ¯SMand ¯SOMlines together give the complete switching phase diagram in the y-zfield plane. For small values of αjthe critical line ¯SMis qualitatively equivalent to the conventional SW astroid, and the equilibrium merging process is similar:There are four on-meridian equilibria for hinside the astroid, and as the field crosses its boundary two of them merge anddisappear. Above the critical current α j=1/2, the ¯SMcritical line becomes self-crossing (Fig. 6). At the critical current the ¯SMulines touch each other at h=0, so, the threshold can be found from Eq. ( 18) with hy=hz=0. Inside the region of self-crossing there are no on-meridian equilibria, asalready observed in Ref. [ 34] and the Supplemental Material of Ref. [ 7]. However, the Poincar ´e-Hopf theorem is not violated due to the presence of the off-meridian equilibria. In the absence of current the system in constant external field hresides in one of the two stable Mequilibria. As α jis increased, the oval-shaped region of stability of the OM 1state grows, and the area inside the modified astroid ¯SMshrinks. Moreover, the self-crossing region of ¯SM, where no on-meridian equilibria exist, also grows. Thus both Mup andMdown states eventually become unstable at some critical currents αMup j(αMdown j ) and mswitches to the OM 1state. What happens if the current is subsequently decreased? The answerto this question can be read from the h c(αj) dependence shown in Fig. 8.A tag i v e n hthe off-meridian state remains stable down to the current αOM jobtained from the equation h=hcOM(αj). IfαOM j<αM j, one would observe hysteretic behavior of the system in the current interval between αOM jand αM j. At the higher end of this interval the system switches from anMs t a t et ot h e OM 1state. At the lower end it switches back to anMstate. As seen from Fig. 8, the length of the bistable interval becomes larger for smaller h.A th=0u s i n gE q .( 20) one finds αOM j|h=0=α/2. The value of αM jath=0w a s already discussed—it corresponds to the first self-crossing of ¯SM, thusαM j|h=0=1/2. For typical values of Gilbert damping α∼0.01 the hysteresis range is very large. It requires an initial pulse of current of the order αj∼1 to get to the OM state, but after that the current can be reduced to αj∼α, and the OM state can be comfortably studied at low currents. Experimentswith SHE devices [ 5,7] are already performed in the regime α j∼1, so the discussed hysteresis should be observable. When magnetic field is set inside the domain of existence of OM states but outside of their domain of stability, the system has two unstable OM equilibria. It is possible to arrange parameters so that there no Mequilibria either (this happens in the high damping, high current regime). In this case thesystem has no choice but to follow some precession cycle, theanalysis of which is beyond the scope of the present paper. V . DYNAMIC PROPERTIES In this section we discuss what happens after the stability boundaries are crossed and equilibria are destabilized. A. Switching to the off-meridian state Existence of a stable OM state within the area given by Eq. ( 20) raises a question: When an Mstate is destabilized at 214424-7SHU Y AN AND Y A. B. BAZALIY PHYSICAL REVIEW B 91, 214424 (2015) S XMdown(a) Mup S OM 1 OM2S1 2 MdownMup(b) S OM 1 OM2 2 Mdown(c) FIG. 9. Sketches of the phase portraits on the stereographic plane describing the evolution of the system in a constant field h=+hˆywith increasing current. The north and south poles are projected to the circle center and to infinity, respectively; the dashed circle is the projection of the equator. (a) Field-dominated regime. Two basins of attraction exist for the MupandMdownequilibria. (b) Current-dominated regime with the additional OM 1,2equilibria. The basins of attraction of MupandMdown are separated by the basin of attraction of OM 1. Spin torques had shifted Mdownthrough the south pole and it shows up in a different place on the stereographic plane. (c) Further increase of current results in the merging of MupandS1. The magnetization finds itself in the basin of attraction of OM 1and switches to the off-meridian state. A set of phase portraits that represents the three cases above can be obtained with parameters (a) hy<1,αj=0; (b)hy=0.05,αj=0.3; and (c) hy=0.15, αj=0.45. the¯SMboundary, will the system switch to the other Mstate, or to a stable OM state? To answer this question we plot the flow diagrams (phase portraits) of the system. The results ofsimulations are presented in the form of qualitative sketchesthat emphasize the structure of the flow (Fig. 9). In the field-dominated regime the flow is qualitatively similar to that in the absence of the current. There are twobasins of attraction of stable points M upandMdown [white and gray areas in Fig.9(a)]. The separatrix between the twobasins winds around the unstable focus Xmaking an infinite number of turns. As a result, near Xthe basins are finely intermixed and a small change in initial conditions may changethe equilibrium where the system ends up. When the modifiedastroid boundary is crossed, one of Mpoints is destroyed. A system initially residing in this point will switch to the otherMpoint. Simulations in the current-dominated regime show three basins of attraction. The one of the OM 1point [darkest area in Fig. 9(b)] separates those of Mup(white area) and Mdown (gray area). The white and gray areas touch at the point of unstable equilibrium OM 2. The important difference from the field-dominated regime is that OM 2is an unstable node, and not a focus. Thus, there is no winding of the separatingline around it and, consequently, no area of fine intermixing.Figure 9(c) shows what happens when the current is increased further so that M upandS1merge at the modified astroid boundary. The phase portrait in the upper part of the unitsphere qualitatively changes: The basins of attraction of M up andOM 1merge, forming a larger basin of attraction of OM 1. This transformation of the phase portrait does not affect thequalitative picture in the lower part of the unit sphere andthe boundary between the basin of OM 1andMdown. The end result is that a system initially residing in Mupwill switch to the OM 1state with certainty. The latter statement, of course, only applies to the case of slow, quasistatic change of parameters,in which case one can be sure that mfollows the stable point with great accuracy. If parameters are changed at a finite speed,there will be a lag between mand the equilibrium point, and a more careful investigation is necessary. B. Two-stage switching through the off-meridian equilibrium Magnetization reversal is one of the most important processes in magnetism that is linked to the magnetic datastorage process, such as in hard disk drives. Switching speedand reliability are two crucial factors to the design of suchsystems. In conventional spin-transfer torque switching spinpolarizer is directed along the easy axis of the free layer. Thenthe magnetization moves towards the new equilibrium along aspiral trajectory in a reliable but fairly slow manner [ 22,41–45]. Much faster reversals, which are often called precessionalswitchings, have been designed. Some have magnetic fieldapplied orthogonally to the easy axis. Others use spin polarizerperpendicular to the easy plane of the free layer (“magneticfan” geometry) [ 46–48]. In both cases the reversal process is fast but requires the current or field pulse length to becarefully adjusted. This is experimentally hard to achieve, andconsequently such methods may lead to greater error rates. It was numerically found [ 35,37] that switching from an Ms t a t et ot h e OM 1state happens fast, without precession or “ringing.” A recent micromagnetic simulation [ 49] confirmed that this result is not an artifact of the macrospin approxi-mation. Figure 10shows the process of switching from the M upto the OM 1state. It is seen that the switching time is of the order of ferromagnetic resonance period T(T=2π in dimensionless unit used in Fig. 10). In the framework of our theory the absence of precession is explained as follows.Switching is initiated by the destruction of an Mequilibrium due to its merging with an Spoint. Since in the current- dominated regime there is no fine intermixing of the basinsof attractions, the flow lines originating from the mergingpoint do not exhibit a winding pattern, and consequently thereare no oscillations in the beginning of the switching process.Oscillations at the end of the switching process, when m 214424-8PHASE DIAGRAM AND OPTIMAL SWITCHING INDUCED . . . PHYSICAL REVIEW B 91, 214424 (2015) FIG. 10. (Color online) Evolution of the three components of the magnetization in a two-stage switching process by direct numericalintegration of the LLG equation. The parameters are h y=0.15,αj= 0.45, and α=0.05. The field-current pulse is turned on at t=0a n d off att=20. approaches the focus OM 1, are suppressed for another reason. In order to cause a merging of MandSspin torque has to be large with αj∼1. But a large spin torque strongly increases the effective damping, especially since the damping actionachieves its maximum at the position of OM 1[39]. This is why, while strictly speaking OM 1is a focus, oscillations are almost absent in practice. Is it possible to utilize the fast nature of the Mup→OM 1 switching to achieve a useful and fast procedure for transitions between the zero-current states MupandMdown? We will consider one possible Mup→Mdown switching scenario with an intermediate stop in the OM 1state. Consider a system that is initially in the Mupstate. Magnetic field his set in the negative zdirection during the whole switching procedure with its magnitude satisfying h< 1, so that the Mup=+ ˆz state remains stable. First, we apply a short pulse of strongcurrent α j∼1. The rise and fall times of the pulse are assumed to be negligible. During the pulse time Mupdoes not exist and mswitches to OM 1. The pulse length tpis selected to be large enough for the switching process to beaccomplished. Importantly, this requirement sets only a lowbound for t p—there will be no harm in keeping the current switched on for a longer time. According to Eq. ( 20)f o rhy=0 the state OM 1hasθ=π/2 and sits on the equator of the unit sphere. After the end of the pulse the current is switched offand the second stage of switching begins. Now the states M up toMdown are stable again and mshould go to one of them. With field pointing down and αj=0, the boundary between the basins of attraction of MupandMdown is a parallel circle, located above the equator of the unit sphere. Thus the secondstage starts with mresiding in the basin of attraction of M down, to which meventually relaxes in a precessional manner. The whole process is characterized by a fast first stage with strongcurrent and a slow second stage, during which the system isnot driven externally. While the total switching time is of thesame order of magnitude as in the conventional switching, the“active” stage requires much shorter time, comparable to thatof precessional switching, making the procedure potentiallyuseful for special applications. An important drawback of thisswitching scenario is that for a given direction of hit can be performed only in one direction, e.g., in the discussion abovefromM uptoMdown. To switch back one would have to reverse the direction of h. It is interesting to note that the SHE device switching between MupandMdownin a two-stage manner described above can be alternatively viewed as a realization of a controlled- NOT gate with hzbeing the control parameter. Finally, we want to compare the duration of the fast stage of SHE switching with the switching time of a conventional spin-torque device, where the magnetic polarizer and the externalfield are both pointing along the easy axis of the free layer.Assuming the conventional spin torque to be described by aconstant spin-transfer efficiency factor g(θ)=¯g, we get h sp= αj[m׈z] with αj=¯gj/j 0for its effective field. In this fully axially symmetric case the switching time can be computedanalytically [ 50]a s t s=1 2α(1−h/prime)ln/parenleftbigg1−mz0 1−mz0/h/prime/parenrightbigg , (21) withh/prime=h+αj/αandmz0being the initial value of the magnetization component along the easy axis. In the small-damping ( α/lessmuch1), large-current ( α j/α/greatermuchh∼1) regime this simplifies to ts≈− ln(1−mz0)/(2αj). Spin-torque switching in collinear geometry requires some initial deviation of mfrom equilibrium. This deviation is usually thought to come fromthermal fluctuations and can be evaluated by using Maxwellequilibrium distribution for m z0, ρ(mz0)=κ/radicalBig 1−m2 z0e−E(m)/kbT, (22) where κis the normalizing constant. To compare the switching times we adopt a typical expected value at room temperature ofm z0≈0.99 (θ0≈0.5◦)[51]. For the purpose of switching time comparison it is important to remember that conventional andSHE devices differ in two aspects. On the theoretical level,in conventional devices switching occurs at α j∼α, while αj∼1 is required for SHE switching. On a practical level, conventional devices can bear smaller currents due to heatingproblems. Thus achieving α j∼1 in them is problematic. In view of that, we perform two comparisons. First, we comparethe SHE and conventional switching times for α j=0.5 and h=0. Here we get tp≈14 and ts≈5. Given the same normalized spin-torque strength, a conventional device is fasterthan the SHE one. Second, we compare the two devicesoperating at their critical switching current with a small field,sayh=0.02, pointing toward the −zdirection. For the SHE device we again use α j=0.5 and the resulting switching time does not change much, tp≈13.5. For a conventional device we use αj=α, then ts≈29/α. In this sense the SHE switching turns out to be much faster. In addition, since theinitial condition is a statistical average, the switching timeestimated in this fashion may cause a non-negligible error inexperiments. VI. SUMMARY We considered magnetic switching in a bilayer F/N struc- ture with strong spin-orbit interaction using a macrospin 214424-9SHU Y AN AND Y A. B. BAZALIY PHYSICAL REVIEW B 91, 214424 (2015) TABLE I. Classification of stability. Stability type Equilibrium type Eigenvalue equivalent tr A-detAequivalent λequivalent Stable focus Complex Re[ μ±]<0 λ> Max(λT,λ+)i f λT/greaterorequalslantλ−Sink tr A< 0, detA> 0Stable node Real μ−<μ +<0 λT<λ<λ − ifλT<λ− Unstable focus Complex Re[ μ±]>0 λ< Min(λT,λ−)i fλT/lessorequalslantλ+Source tr A> 0, detA> 0Unstable node Real 0 <μ −<μ + λ+<λ<λ T ifλT>λ+ Saddle Saddle point Real μ−<0<μ + detA< 0 λ−<λ<λ + approximation that is applicable to sufficiently small, single domain devices. The method of Refs. [ 32,33] provides a framework that can be applied to find the critical switchingsurfaces for any magnetic single domain system with arbitraryanisotropy and spin torque in an exact fashion. In this paperwe calculated the three-dimensional critical surface for a SHEbilayer system with perpendicular anisotropy and in-planecurrent using single domain approximation. For external fieldsin the y-zplane, the SHE-induced spin torque not only shifts the existing equilibria, but also generates two new,current-induced equilibria. First , we derive an analytic formula for the stability boundary for the current-induced equilibria—in previousnumeric research this boundary was not distinguished fromthe boundary of its existence. Second , in contrast to the other authors we discuss the switching phase diagrams in the fieldspace at a constant current, and plot the modified astroid andthe oval-shaped stability region of the off-meridian state. Wethen show how our qualitative description of the evolutionof the constant current switching boundaries can explain theresults of the other authors obtained for variable currents.Third , we discuss in detail the evolution of equilibrium points and the character of their destabilization on the switchingboundaries. This allows us to put forward a qualitativeunderstanding of the complicated hysteresis processes thatare found in SHE devices. Fourth , we point out that while a large current is required to set magnetization into the current-induced state, it remains in this state when current is decreasedto values that are αtimes smaller. Thus the spin-torque-induced state can be studied at low currents. Fifth , we show that in the current-dominated regime switching between up anddown equilibria can deterministically proceed through thecurrent-induced state. The first stage of this switching is veryfast, without ringing effects in the beginning or at the end,and an explanation to this fact is provided. An example oftwo-stage switching is considered and the switching time iscompared to that of a conventional collinear spin-torque devicewith magnetic polarizer. Here it is found that, depending onthe limitations on current magnitude imposed by the factorsnot related to spin-torque physics, either of the two devicescan operate faster.ACKNOWLEDGMENTS This research was supported by National Science Founda- tion Grant No. DMR-0847159. APPENDIX A The explicit expression of each component in Eq. ( 9) can be derived as ∂θfθ=−∂θθε+∂θhθ sp, (A1a) ∂θfφ=−∂θ/parenleftbigg1 sinθ∂φε/parenrightbigg +∂θhφ sp, (A1b) ∂φfθ=cosθ sinθ/parenleftbigg1 sinθ∂φε−hφ sp/parenrightbigg −1 sinθ/parenleftbig ∂2 θφε−∂φhθ sp/parenrightbig , (A1c) ∂φfφ=−cosθ sinθ/parenleftbig ∂θε−hθ sp/parenrightbig −1 sin2θ∂2 φφε+1 sinθ∂φhφ sp. (A1d) APPENDIX B For a planar linear system of the form ˙X=AX,t h e eigenvalues of the 2 ×2 coefficient matrix Acan be calculated in terms of its trace and determinant as [ 38] μ±=1 2(trA±/radicalbig (trA)2−4 detA). (B1) Therefore knowing tr Aand det Atells us virtually everything about the geometry of its solutions. Besides stability, the types of an equilibrium is also of importance in understanding the switching process. Anequilibrium of the same stability (except saddle point) canbe a node or a focus, depending on whether the eigenvalues(B1) are real or complex. Therefore the differentiation of the focus set and the node set requires another pair of criticalvalues λ c±which satisfies (trA)2−4 detA=0. (B2) It can be demonstrated that the relationships λc+/greaterorequalslantλ+and λc−/lessorequalslantλ−are always satisfied. TABLE II. Classification of focus and node. Equilibrium type Eigenvalues tr A-detAequivalent λequivalent Node Real (tr A)2−4detA> 0, detA> 0aλc−<λ<λ −orλ+<λ<λ c+ Focus Complex (tr A)2−4detA< 0aλ<λ c−orλ>λ c+ aNote that (tr A)2−4detA< 0 guarantees det A> 0. 214424-10PHASE DIAGRAM AND OPTIMAL SWITCHING INDUCED . . . PHYSICAL REVIEW B 91, 214424 (2015) FIG. 11. The ¯SOM,¯Sc, and the equilibrium merging boundary are plotted with solid, dashed, and dotted lines, respectively. The unstablenode, unstable focus, and stable focus regions are marked in different gray scales. For illustrative purposes, we adopt α j=0.6a n d α=1 to make the unstable regions large enough to be seen. The classification of stability and equilibrium type in terms of the eigenvalues and λare summarized in Tables Iand II, respectively. The complete dynamic analysis of an equilibriumneeds to take into account the two factors together. APPENDIX C We mentioned the equilibrium types of the two off-meridian equilibrium states and their evolution in Secs. IV A andIVB2 .To quantitatively understand the evolution of these two states, we also need to find the critical λwhich separates nodes and foci, i.e., to solve Eq. ( B2). Its solution in the off-meridian case is given by λc±=sin2θ 2−cos2θ±1 2sin2θ/radicalbig 1+α2 +αjαsinθcosφ. (C1) The two critical values λc±give another two surfaces in the parameter space: one above λ+and the other below λ−. 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PhysRevB.90.184401.pdf

PHYSICAL REVIEW B 90, 184401 (2014) Spin transport parameters in Ni 80Fe20/Ru and Ni 80Fe20/Ta bilayers J. E. G ´omez,1,*B. Zerai Tedlla,2N. R. ´Alvarez,1G. Alejandro,1E. Goovaerts,2and A. Butera3 1Centro At ´omico Bariloche, Instituto de Nanociencia y Nanotecnolog ´ıa (CNEA) and Conicet, 8400 Bariloche, R ´ıo Negro, Argentina 2Experimental Condensed Matter Physics, Physics Department, University of Antwerp, Universiteitsplein 1, BE-2610 Antwerpen, Belgium 3Centro At ´omico Bariloche, Instituto de Nanociencia y Nanotecnolog ´ıa (CNEA), Instituto Balseiro (U. N. Cuyo), and Conicet, 8400 Bariloche, R ´ıo Negro, Argentina (Received 29 May 2014; revised manuscript received 16 October 2014; published 3 November 2014) We present a systematic study of the spin transport properties in two different bilayer systems, Ni 80Fe20/Ru and Ni 80Fe20/Ta, combining ferromagnetic resonance (FMR) and inverse spin Hall effect (ISHE) voltage measurements. We have estimated the effective spin mixing conductance g↑↓by analyzing the permalloy (Py) thickness dependence of the FMR linewidth obtaining g↑↓=(3.8±0.7)×1015cm−2andg↑↓=(1.3±0.4)× 1015cm−2for Py/Ru and Py/Ta, respectively. Analyzing the Ta thickness dependence of the ISHE voltage, we have been able to extract the spin diffusion length, λSD=1.5±0.5 nm, and spin Hall angle, /Theta1SH=−0.03±0.01, of Ta. From the two series of Py/Ta bilayers—with thickness variation of ferromagnetic and nonmagnetic layers,respectively—we demonstrate a path to estimate the spin diffusion length from the experimental data, independentof the spin Hall angle and the microwave field amplitude. DOI: 10.1103/PhysRevB.90.184401 PACS number(s): 81 .15.Cd,76.50.+g,75.70.Tj,75.40.Gb I. INTRODUCTION The manipulation of the charge and spin degrees of freedom in electron transport has attracted continuous interest in thescientific community for many decades, with a remarkable andpersistent revival since the early 1990s [ 1]. In recent years, the feasibility of spin pumping from a ferromagnetic (FM)layer towards an adjacent nonmagnetic metallic film (NM)was demonstrated in the form of a pure spin current [ 2,3], which opened paths for the research in several new and novelphenomena in this area [ 4]. The ability of some metals, like Pt, Pd, or Ta, among others, to convert a spin current into a chargecurrent via the inverse spin Hall effect (ISHE) opens the wayto not only qualitative but also quantitative investigation andunderstanding of the spin transport that takes place in this kindof system. Earlier studies on inverse spin Hall voltage induced by spin pumping in FM/NM bilayers were carried out using preferablyPt as the NM layer because of its large spin orbit interaction.But several NM metals, like Pd, Mo, Au, and Ta, among others,have been used to demonstrate the spin Hall effect in suchmetals [ 5,6]. Applying the theory developed to describe this phenomenon, it is possible to estimate different parameters likethe spin diffusion length, the spin Hall angle, or the effectivespin mixing conductance. Some of these parameters have abroad range of reported values that depends mainly on themethod used for their estimation. For example, spin diffusionlength values found for Pt range roughly from 0.5 nm [ 7]t o 10 nm [ 8]. This broad range can be also explained by spin memory loss effects occurring at the interface [ 9]. In the following sections we present a detailed study of two different systems of bilayers of permalloy (Py) with Ru and Ta,characterized as a function of the thickness of the FM and/orthe NM layer. We show that Ru can effectively be used todetect a spin current by measuring the voltage generated by theinverse spin Hall effect. For both systems we have estimated *gomezj@cab.cnea.gov.arthe effective spin mixing conductance parameter, g↑↓.I nt h e case of the Py/Ta bilayer we present a set of spin transportparameters estimated from the Py thickness dependence of theFMR spectra and the inverse spin Hall voltage as a functionof the thickness of both the NM and the FM layer. We finallydescribe a simple method for determining the spin diffusionlength independent of the spin Hall angle. II. FILM FABRICATION AND STRUCTURAL CHARACTERIZATION Films have been fabricated by dc magnetron sputtering on naturally oxidized Si (100) substrates. The chamber waspumped down to a base pressure of 1 ×10 −6Torr and the films were sputtered at 3 mTorr of Ar pressure, a power of20 W, and a target-substrate distance of 10 cm. With theseparameters we obtained a sputtering rate of 0.06, 0.11, and0.15 nm /s for Ru, Py, and Ta, respectively. Two different series were grown keeping fixed the NM layer: t Ru=10 nm andtTa=5 nm and varying the thickness of the Py layer, tPy=5, 7, 9, and 11 nm. Complementarily, another series was grown with different Ta layer thickness, tTa=2, 5, 8, and 11 nm, and with Py thickness tPy=10 nm (see Table I). Our sputtering machine can host several substrates at the sametime, allowing one to deposit a complete series in exactlythe same sputtering conditions. In the case of Ta films thesputtering variables were optimized in order to promote thegrowth of the crystallographic βphase, as confirmed by x-ray diffraction patterns performed on 133 nm reference films, witha high electrical resistivity ( ρ∼2000/Omega1nm). Although there is a tendency in very thin Ta films to grow in an amorphousphase, there are several reports in which it is shown thatby choosing the correct sputtering conditions the βphase is dominant down to thicknesses of a few nanometers [ 10–13]. However, possible mixing of both phases should not be totallydiscarded, especially in the thinnest samples studied in thiswork ( t Ta=2n m ) . FMR spectra were acquired at room temperature with a commercial Bruker ESP 300 spectrometer at frequencies of 1098-0121/2014/90(18)/184401(8) 184401-1 ©2014 American Physical SocietyJ. E. G ´OMEZ et al. PHYSICAL REVIEW B 90, 184401 (2014) TABLE I. Nominal layer thickness in the three different series of bilayers. Series 1 Series 2 Series 3 tPy(nm) tRu(nm) tPy(nm) tTa(nm) tPy(nm) tTa(nm) 51 0 5 5 1 0 2 71 0 7 5 1 0 5 91 0 9 5 1 0 811 10 11 5 10 11 9.4 GHz (X band), 24 GHz (K band), and 34 GHz (Q band). The samples were placed at the center of a resonant cavitywhere the derivative of the absorbed power was measuredusing a standard field modulation and lock-in detectiontechnique, with a modulation frequency of 100 kHz and amodulation amplitude of 10 Oe. The film plane was in all casesparallel to the microwave excitation field. Angular variationswith respect to the external dc field were made around a verticalaxis parallel to the microwave field (see Fig. 1). The inverse spin Hall voltage ( V ISHE) has been acquired by measuring the voltage between the edges of the sample[see sketch of Fig. 1(a)], while the external magnetic field was swept in the vicinity of the resonance field of the ferromagnet.Electrical contacts were made with two 25 μm diameter aluminum wires fixed to the nonmagnetic layer of the samplewith silver paste. The signal was amplified by a high-gaincalibrated differential amplifier located close to the cavity toreduce electronic noise. The amplified signal was connectedto the digitizer input of the FMR spectrometer in order tosynchronize this signal with the magnetic field sweep. TheISHE measurements were all performed inside of the X-bandTE102 rectangular cavity (microwave frequency of 9.4 GHz)and with the field modulation turned off. III. EXPERIMENTAL RESULTS AND DISCUSSION A. Effective spin mixing conductance in Ru and Ta series We have measured the FMR spectra and the inverse spin Hall voltage in the whole set of samples. In Fig. 2we show (a) (b) FIG. 1. (Color online) (a) Sketch of the geometrical setup of the sample, the microwave field ( hRF), and the external dc field, H. (b) Coordinate reference system chosen to analyze the FMR andthe inverse spin Hall voltage measurements.(a) (b) FIG. 2. (Color online) Typical voltage signal ( VISHE)a n dF M R absorption spectra (inset) in two bilayers of Py in contact withdifferent metals: (a) Ru; (b) Ta. The distance ( L) between the electrical contacts is 2.2 and 2.3 mm, respectively. typical FMR spectra and the corresponding VISHE measured on the bilayers Py(5 nm)/Ru(10 nm) and Py(5 nm)/Ta(5 nm).In the two samples the FMR absorption derivative has, asexpected, near perfect odd symmetry with respect to theresonance field and the ISHE signal is close to symmetricwhich indicates that the contribution of anomalous Hall effectis small compared to the ISHE part [ 5]. In addition, one finds that the sign of the induced voltage is positive for Ru andnegative for Ta. The theory describing the spin injection from a FM layer fulfilling the resonance condition into an adjacent NM layeris well established [ 2,14,15] and numerous experimental results supporting these models can be found in the litera-ture [ 5,6,16,17]. More precisely, a pure spin current appears as a consequence of an additional interfacial relaxation process,mediated by the conduction electrons of the normal metal, thatcontributes to relax the magnetization of the FM layer to theequilibrium configuration. This additional relaxation throughthe interface is added to the intrinsic volume relaxation ofthe FM and then an increment in the linewidth ( /Delta1H)o ft h e bilayer system is expected if compared with a single FM layerwhich is not in contact with a NM. Note that if the thicknessof the FM layer is reduced, an increment of the linewidthis also expected because the interfacial relaxation processbecomes more important when compared to volume processes,which are reduced by the decreasing thickness. The intrinsiclinewidth is governed by the total Gilbert damping parameterα, and includes a thickness independent volume term ( α 0) and an interfacial term that increases with decreasing thickness.The thickness dependence can be described by the following 184401-2SPIN TRANSPORT PARAMETERS IN Ni 80Fe . . . PHYSICAL REVIEW B 90, 184401 (2014) expression [ 2]: α=α0+γ 4πMs/planckover2pi1 tFMg↑↓, (1) where γ=gμB//planckover2pi1is the gyromagnetic ratio, gis thegfactor, /planckover2pi1is the reduced Planck constant, Msis the saturation mag- netization, and g↑↓is the effective spin mixing conductance. The quantity g↑↓is a property of the FM/NM interface and is phenomenologically interpreted as the net efficiency of theFM layer to transfer angular momentum across the FM/NMinterface [ 18]. The parameter α 0is the volume Gilbert damping parameter and is expected to be close to that of a single FMlayer without any NM layer in contact. In order to properly estimate the effective spin mixing conductance from the linewidth it is important to considerthat/Delta1H is often enhanced by extrinsic contributions [ 19–21]. For example, the distribution of anisotropy axis (both inmagnitude and orientation), inhomogeneities within the films,among other effects, can result in a wrong estimation of theparameter g ↑↓if they are not properly considered. To separate the extrinsic contributions (which may be assumed to beindependent of the excitation frequency) from the intrinsicrelaxation, a linear frequency dependence of /Delta1H is often assumed and is phenomenologically described [ 22]b yE q .( 2), /Delta1H=/Delta1H 0+2√ 3ω γα. (2) In this equation the total peak-to-peak linewidth /Delta1H is the sum of the extrinsic contribution /Delta1H 0and the intrinsic term proportional to the microwave excitation frequencyf=ω/2π. We show in Fig. 3the frequency dependence of /Delta1H for the set of samples (a) Py( t Py)/Ta(5 nm) and (b) Py( tPy)/ Ru(10 nm) measured at X-, K-, and Q-band frequencies withthe external field applied parallel to the film. We found that,within experimental error, the extrapolation to zero frequencyyields /Delta1H 0≈0, indicating that the extrinsic contribution /Delta1H 0 is negligibly small in all cases. Note that we have neglected any extrinsic contribution coming from the two magnon scatteringprocess based on the fact that similar linewidths were observedwhen the external field was applied parallel or perpendicularto the film plane. Also, the typical square root frequencydependence for this mechanism is not observable in theexperimental data of Fig. 3. This figure shows an increment of the slope in both series of bilayers when the thickness of the Pylayer is decreased. Analyzing these data by using Eq. ( 2)w e can obtain the total Gilbert damping parameter for each sampleand then, from the dependence of this parameter as a functionof the inverse of the Py thickness [Eq. ( 1)], it is possible to separate the volume and interfacial relaxation terms. InFig. 3(c) we plot αvs 1/t Pyand, by performing a linear fit, we estimated α0and the effective spin mixing conductance. We list in Table IIthe values of g↑↓andα0obtained for the two sets of samples. For this estimation we used a saturationmagnetization M s=770 emu /cm3. The procedure used to determine Msand the surface anisotropy Ksis discussed in Sec. III B. As already mentioned, α0corresponds to the volume Gilbert damping parameter of the Py layer withoutthe interfacial contribution to the relaxation. We obtained thesame value for both bilayers, α 0=0.0083±0.0022, that is(a) (b) (c) FIG. 3. (Color online) Dependence of the peak to peak linewidth of (a) Py( tPy)/Ta(5 nm) and (b) Py( tPy)/Ru(10 nm) series as a function of the microwave excitation frequency. (c) Total Gilbert damping parameter as a function of the inverse of the Py layer thickness forboth sets of samples. comparable to previous reports in Py which range between α0∼0.006 [ 23] andα0∼0.013 [ 24]. The value for the effective spin mixing conductance in the Py/Ru interface is about three times larger than g↑↓for Py/Ta. The spin current injected in the NM layer is proportional tog ↑↓and can be estimated [ 14,16]f r o m Jsˆs=/planckover2pi1 4πg↑↓m×dm dt, (3) where m=M Msis the reduced magnetization. Thus, from the obtained values of g↑↓, we can conclude that Ru mediates better than Ta the relaxation of the magnetization of the Pylayer by creating a larger pure spin current. In this work we have used a model similar to that developed in Ref. [ 16], by considering that the spin Hall voltage is originated by the injected spin current averaged over a time TABLE II. Effective spin mixing conductance g↑↓and volume Gilbert damping parameter α0for the two series of samples Py(tPy)/Ta(5 nm) and Py( tPy)/Ru(10 nm) obtained from a linear fit of the data shown in Fig. 3(c). Interface g↑↓(1015cm−2) α0 Py/Ta 1 .3±0.40 .0083±0.0012 Py/Ru 3 .8±0.70 .0083±0.0022 184401-3J. E. G ´OMEZ et al. PHYSICAL REVIEW B 90, 184401 (2014) period. This results in a pure dc spin current propagating in the ˆxdirection (perpendicular to the interface) and polarized along the magnetization direction ˆs[the reference system is sketched in Fig. 1(b)]. The dc pure spin current is converted into a charge current via the inverse spin Hall effect that deflects theelectrons preferentially to one end of the sample. The edge ofthe sample where the charge accumulation takes place dependson the sign of the deflection of the carriers, given by the signof the spin Hall angle, /Theta1 SH, Jc=/Theta1SH/parenleftbigg−2e /planckover2pi1/parenrightbigg Jx s(ˆx׈s), (4) where Jcis the charge current, Jx sis the dc spin current that takes different values inside of the NM layer along the ˆx direction, and eis the electron charge. The unit vectors ˆx and ˆsare the propagation and polarization directions of the spin current, respectively. We mentioned that Py/Ru and Py/Ta samples have opposite signs of the measured VISHE signal (as can be seen in Fig. 2), from which we conclude that Ru and Ta have opposite spin Hallangle signs. This result was already pointed out in Ref. [ 25], and it is related to the fact that the sign of the spin Hallconductivity depends on the sign of the spin-orbit interaction,being positive for more than half-filled dorbitals (as happens in Ru, which has a 4 d 7configuration, according to Hund’s rules) and negative in the other case (Ta is a 5 d3). The negative sign of the Ta spin Hall angle was experimentally observedin Ref. [ 6]. We have also observed that the ISHE voltage measured on these two kinds of bilayers (with the same Pythickness and similar electrodes separation) is always larger(about one order of magnitude) in the series with Ta, althoughthe FMR linewidth and the effective spin mixing conductancewere always larger in the Ru series. B. Angular variation of VISHEin the Ru series As far as we know the inverse spin Hall voltage generated in a Py/Ru bilayer has not been yet reported, which led usto perform careful measurements to characterize the angulardependence of the V ISHE signal on φH, the angle between the external magnetic field, H,and the film normal [see Fig. 1(b)]. We show in Fig. 4(a)the angular dependence of the voltage measured in the neighborhood of the FMR resonance field, inthe sample Py(5 nm)/Ru(10 nm) at different angles, startingfrom+90 ◦and ending in −90◦. We have centered the VISHE peak at the resonance field position to better appreciate the evolution of the spectra for different angles. We have fittedall measured spectra by an absorption (assumed of Lorentzianshape) and a dispersion curve (Lorentzian-like derivative) assuggested in Ref. [ 5]. With this procedure we could separate the ISHE signal from the contribution originating from theanomalous Hall effect in the FM layer, which are even and oddfunctions, respectively. From the fitting it is possible to extractthe intensity of the symmetric component ( V ISHE), its field position ( Hr), and linewidth ( /Delta1H). As expected, these two last quantities coincide with the corresponding values measured inFMR experiments. The antisymmetric contribution in each ofthese spectra is very small as is readily observed in Fig. 4(a). Also, the intensity of the spectra at 90 ◦is equal, within(a) (b) FIG. 4. (Color online) (a) ISHE voltage signal measured in an X-Band FMR experiment performed in the Py(5 nm)/Ru(10 nm) bilayer for different orientations of the magnetic field with respect to the film normal ( φH). (b) Peak value of the symmetric contribution as a function of the angle of the equilibrium magnetization ( φ) normalized by the VISHEacquired at φH=90◦. The solid line corresponds to the values obtained from Eq. ( 9) using the FMR data. Explored angles areφH=±90◦,±60◦,±40◦,±20◦,±10◦,a n d0◦. experimental error, but opposite in sign to that acquired at −90◦, allowing us to discard any contributions coming from anisotropic magnetoresistance effects [ 5]. Figure 5shows the angular dependence of the resonance field position and the linewidth of the bilayer Py(5 nm)/Ru(10nm). A typical behavior for thin films is found, in whichH ris maximum when His applied perpendicular to the film plane and /Delta1H peaks approximately where ∂Hr/∂φH is maximum [ 26–28]. The Smit and Beljers formalism is generally used [ 22] to obtain the dispersion relation for the uniform mode of precession [Eq. ( 5)] and the equilibrium angle of the magnetization [Eq. ( 6)]. From the measurements of the resonance field with the applied field parallel andperpendicular to the film plane it is then possible to obtainthe effective anisotropy field ( H eff) and the gvalue. Once the effective anisotropy field is calculated, from Eq. ( 6) it is possible to estimate the angle of equilibrium of themagnetization, φ. These quantities, H effandφ, are necessary for fitting the angular dependence of VISHE: /parenleftbigω γ/parenrightbig2=[Hrcos(φH−φ)−Heffcos2(φ)] ×[Hrcos(φH−φ)−Heffcos(2φ)], (5) 2Hrsin(φH−φ)+Heffsin(2φ)=0. (6) 184401-4SPIN TRANSPORT PARAMETERS IN Ni 80Fe . . . PHYSICAL REVIEW B 90, 184401 (2014) FIG. 5. (Color online) Angular variation of the resonance field (full symbols) and the linewidth (open symbols) in the bilayer Py(5 nm) /Ru(10 nm) obtained from X-Band FMR experiments. The solid line corresponds to the fitting using Eqs. ( 5)a n d( 6), from which we obtained Heff=5920 Oe. The dotted line is a guide to the eye. FMR experiments performed at parallel and perpendicular geometries for different Py thicknesses show a relativelyconstant g-factor value of g=2.095±0.009 and a decreasing value of H effwhen the thickness of Py is reduced. The quantity Heffis related to the shape anisotropy (4 πMs)o f the sample, but in thin films the surface anisotropy must bealso considered, especially for thicknesses in the nm range.The usual expression to consider these two anisotropies isgiven [ 29]b y H eff=4πMs−4Ks MstPy, (7) where the last term is the surface anisotropy field written as a function of the surface anisotropy constant ( Ks) and the Py thickness. To estimate Ms, we used a reference thick film of approximately 313 nm, which is thick enough to discardsurface effects. From the FMR data and Eq. ( 5) we deduced M s=770 emu /cm3andg=2.091, which is coincident with thegvalue of Py obtained in the studied bilayers. Note that in very thin films a magnetically “dead” layer is oftenconsidered in order to account for the reduced magnetizationof the surface layers which are in contact with the substrate orthe nonmagnetic metal. In this situation the nominal thicknessof the ferromagnetic layer should be corrected to an effectiveone. In the present work we used the nominal Py thickness andconsidered an uncertainty of 0.5 nm in the Py thicknesses totake into account the possible presence of a dead layer. Smalldepartures from the proposed model could be due to this effect. In Fig. 6we show the dependence of the gfactor and H eff of the Py( t)/Ta and Py( t)/Ru series as a function of the inverse of the Py thickness. It is important to mention that values ofH effwere obtained in most cases using only the resonance fieldFIG. 6. (Color online) Thickness dependence of the effective anisotropy field ( Heff) and the gfactor of the Py(t)/Ta and Py(t)/Ru series. values parallel and perpendicular to the film plane. In Fig. 5 we fit the angular variation of the resonance field using allexperimental points and although this procedure is expectedto be more precise for the estimation of H effthe differences between both methods are always smaller than a few percent. By solving the Landau-Lifshitz-Gilbert equation of motion, and following the same reasoning of Ref. [ 16] it is possible to derive the angular dependence of the injected dc spin currentat the resonance condition [i.e., when Eq. ( 5) is satisfied] J 0 s(φ)=γ/planckover2pi1 4πg↑↓/parenleftbigghRF α/parenrightbigg2 ×/bracketleftbigg[Hrcos(φH−φ)−Heffcos(2φ)] [2Hrcos(φH−φ)−Heff(3 cos2φ−1)]2/bracketrightbigg , (8) where hRFis the amplitude of the microwave magnetic field inside of the resonant cavity. This expression is totallyequivalent to Eq. ( 12)o fR e f .[ 16], but has been rearranged to emphasize the dependence on the effective anisotropy fieldand the relative orientation between the magnetization and theexternal field. The spin current shown in Eq. ( 8) corresponds to the current at the interface of the FM/NM bilayer that propagatesdiffusively inside the NM layer. Using the coordinate systemshown in Fig. 1(b) and from Eq. ( 4) one can figure out that the pure spin current propagates in xdirection and is polarized along the magnetization direction. As V ISHE∝J0 s(φ)s i n (φ), it is convenient to define a normalized spin Hall effect voltage 184401-5J. E. G ´OMEZ et al. PHYSICAL REVIEW B 90, 184401 (2014) as VN=J0 s(φ) J0s(φ=90◦)sin(φ) =[Hrcos(φH−φ)−Heffcos(2φ)] [2Hrcos(φH−φ)−Heff(3 cos2φ−1)]2 ×[2H||+Heff]2 H||+Heffsin(φ), (9) where we defined H||as the resonance field measured at φ=φH=90◦. The normalized experimental data of Fig. 4(b) has been superimposed with the curve obtained from Eq. ( 9), which depends exclusively on parameters obtained from theFMR spectra. A very close agreement was found, supportingthe proposed model. C. Microwave power dependence We also studied the microwave power ( P) dependence of the ISHE voltage in the sample Py(5 nm)/Ru(10 nm). Allmeasurements were performed with the external field parallelto the plane of the sample along the ydirection. We show in Fig. 7the obtained linear behavior of the maximum of the measured voltage as a function of P. As explained in the literature [ 16] this is a consequence of the dependence of Eq. ( 8) with h 2 RF. To understand the linear dependence of VISHE as a function of Pwe can inspect the time averaged y component of Eq. ( 3),J0 s∝/integraltext2π ω 0[mxdmz dt−mzdmx dt]dt.A sw e are in the linear response regime ( m=χhRF) each transversal component mi=x,zcontributes with a hRFfactor, making J0 s∝ FIG. 7. (Color online) Power dependence of the maximum in- verse spin Hall voltage measured at H=Hr. In the inset we show the Lorentzian shape of VISHEcurves in the neighborhood of Hr. Data correspond to the bilayer Py(5 nm)/Ru(10 nm) varying Pfrom 4 mW to 200 mW.h2 RFand hence VISHE∝h2 RF.A sPis proportional to h2 RF[30] we obtain the linear behavior of the maximum of the measuredvoltage as a function of the microwave power, as shown inFig.7. D. Estimation of the transport parameters in the Py/Ta series To estimate the transport parameters involved in the ISHE effect, we have studied two different series of Py/Ta bilayers.The first set, Py( t Py)/Ta(5 nm), has variable Py thickness with tPy=5, 7, 9, and 11 nm and a fixed thickness of 5 nm for the Ta layer. The second series, Py(10 nm)/Ta( tTa), has the same Py thickness in all samples and the Ta layer thicknesstakes the values t Ta=2, 5, 8, and 11 nm. In all cases we have acquired the voltage signal by applying the magnetic field inthe plane of the sample with a microwave power P=200 mW. To perform the analysis we have extracted the V ISHEsymmetric contribution of the signal, as already explained. The dc spin current injected by the spin pumping phe- nomena diffusively propagates in the NM layer and, via theinverse spin Hall effect, it is converted into a charge current.Because we work in an open circuit condition, this chargecurrent creates an electric field that cancels its propagation. Thetheory and expression for the resulting voltage, see Eq. ( 10), associated to this electric field is well established and can befound in several articles [ 9,16], V ISHE=L/parenleftbigg2e /planckover2pi1/parenrightbigg/Theta1SHλSDtanh(tTa/2λSD) tPyσPy+tTaσTaJ0 s. (10) In this equation Lis the separation between the electrodes where the voltage is measured, λSDis the spin diffusion length, andσPyandσTaare the electrical conductivities of Py and Ta, respectively. All FMR effects are contained in J0 sthat takes the form of Eq. ( 11) when the external field is applied parallel to the film plane ( φH=φ=90◦) J0 s=γ/planckover2pi1 4πg↑↓/parenleftbigghRF α/parenrightbigg2H||+Heff [2H||+Heff]2. (11) Combining Eq. ( 10) with Eq. ( 11) we can define the quantity ρ ρ=VISHE L/parenleftbigg2π e/parenrightbiggα2 γg↑↓/bracketleftbigg[2H||+Heff]2 H||+Heff/bracketrightbigg =h2 RF/Theta1SHλSDtanh(tTa/2λSD) tPyσPy+tTaσTa(12) that, apart from a factor h2 RF, has units of resistivity ( /Omega1nm). In Fig.8we show the ρcurves obtained as a function of magnetic field for each series. As expected from Eq. ( 12),ρdecreases when tPyincreases and has a more complex behavior as a function of tTa. This suggests that the marked different behavior ofρwithtPyandtTamay be used to extract separately the values ofλSDand/Theta1SH. We obtained the maximum of the symmetric contribution in each of the curves, normalized this value inorder to calculate ρ, and plotted this variable as a function of tPyandtTa,i nF i g s . 9(a) and9(b), respectively. In Fig. 9(a) we can observe that the experimental ρas a function of tPy follows approximately the hyperbolic functional dependence 184401-6SPIN TRANSPORT PARAMETERS IN Ni 80Fe . . . PHYSICAL REVIEW B 90, 184401 (2014) (a) (b) FIG. 8. (Color online) ρsignal as a function of magnetic field for (a) Py( tPy)/Ta(5 nm) and (b) Py(10 nm)/Ta( tTa)s e r i e s . of Eq. ( 12),ρ(tPy)=K/(tPyσPy+tTaσTa). Using the bulk con- ductivities of Ta and Py [ 10,31],σTa=0.0005/Omega1−1nm−1and σPy=0.0022/Omega1−1nm−1, and fitting the experimental data we have obtained K=0.091(7) (Oe2nm). The constant Kcan then be used to fit the experimental values of ρas a function of tTafor a fixed value of tPyusing ρ(tTa)=1 tPyσPy+tTaσTaK tanh5 2λSDtanhtTa 2λSD. (13) The best fit of the experimental data is shown in Fig. 9(b) and was obtained for λSD=1.5±0.5n m . It is important to note here that this procedure allowed us to experimentally determine the spin diffusion length in Tawithout knowing the spin Hall angle /Theta1 SHor the microwave field amplitude hRF. With this estimation of λSDand knowing the value of hRF, it is possible to calculate /Theta1SH.T h eT E 102resonant cavity used in our experiment has a quality factor Q∼2400 (defined asQ=ω//Delta1ω ) that was not significantly changed when the sample was placed inside. With this value of Qand following Ref. [ 32] we estimated hRF∼1.5(3) Oe, obtaining /Theta1SH∼ −0.03±0.01. It is worth mentioning that the experimental determination of hRFis not straightforward and usually comes with a large associated error. As a consequence, the spin Hallangle is also expected to be affected by a large uncertainty, andwas estimated to be around 50% in our case. We also note that our fittings of the experimental data were made using the bulk electrical conductivities of Ta and Py. Itis well known that the conductivity depends on film thickness,and the correction becomes important when the thickness isreduced to 1 nm or less [ 33]. If we consider this correction(a) (b) FIG. 9. (Color online) Dependence of ρas function of the thick- ness of (a) Py layer and (b) Ta layer. The continuous lines are the fitsobtained with Eq. ( 12) and the parameters indicated in the figure. only a minor improvement is obtained in our data. However, there are reports in the literature in which an experimentalthickness dependence of the conductivity of Ta [ 6] and Py [ 15] is observed well above 1 nm. If we take into account thesevalues a slightly better fit of the experimental data can beobtained but the spin diffusion length is almost unaffected. IV . CONCLUSIONS In summary, we have studied two bilayer systems, Py/Ru and Py/Ta, as a function of the permalloy thickness tPy. In both systems we estimated the effective spin mixing conductanceby analyzing the dependence of the FMR linewidth as afunction of t Py. We obtained a larger value of g↑↓in the Py/Ru system indicating that Ru mediates better the relaxationof the magnetization of the Py layer during the resonance,creating a larger pure spin current. In the same analysis,extrapolating the tendency of the Gilbert damping parameteras a function of the inverse of the film thickness, we have foundα 0=0.0083±0.0022 for the polycrystalline Py layer in both systems. Because the ability of Ru to convert a pure spin current into a charge current has not been yet reported, we exploredthe angular and microwave power dependence of the inversespin Hall voltage in the Py/Ru system obtaining an excellentagreement with the predictions of the inverse spin Hall effecttheory. On the other hand, we estimated the spin diffusionlength and the spin Hall angle of Ta by analyzing thedependence of the inverse spin Hall voltage in this system as afunction of both Py and Ta layer thicknesses obtaining λ SD= 184401-7J. E. G ´OMEZ et al. PHYSICAL REVIEW B 90, 184401 (2014) 1.5±0.5 nm and /Theta1SH=−0.03±0.01 in agreement with recently published results [ 6]. We have also experimentally verified that Ru and Ta have opposite spin Hall angle signs. Comparing both systems we have found that Ru produces a larger spin current but, because of its higher conductivityand probably a smaller spin Hall angle, it produces a weakerinverse spin Hall voltage. On the contrary, despite its relativelysmall g ↑↓value, Ta generates a larger VISHE because of its higher resistivity and a larger spin Hall angle. The latterresult makes Ta a good candidate to be used as a spin currentdetector without affecting considerably the Py magnetizationrelaxation.ACKNOWLEDGMENTS This work was supported by Conicet (Grant No. PIP 112-201101-00482), ANPCyT (Grant No. PICT-2010-0773),and U.N. Cuyo (Grant No. 06/C352), all from Argentina.Investment funds for advanced electron-paramagnetic reso-nance instrumentation from the Flemish Hercules foundationin Project No. AUHA013 and support from the MINCyT–FWO international cooperation project (No. FW/11/04) arekindly acknowledged. We would also like to acknowledgethe very fruitful discussions with Dr. Juan Carlos RojasS´anchez. [1] F. Mott, Proc. R. Soc. A 153,699(1936 ); M. Johnson and R. H. Silsbee, P h y s .R e v .L e t t . 55,1790 (1985 ); T. Valet and A. Fert, Phys. Rev. B 48,7099 (1993 ). [2] Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett.88,117601 (2002 ). [3] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. 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PhysRevB.93.224427.pdf

PHYSICAL REVIEW B 93, 224427 (2016) Twist in the bias dependence of spin torques in magnetic tunnel junctions S¨oren Boyn,1Jo˜ao Sampaio,1,*Vincent Cros,1Julie Grollier,1,†Akio Fukushima,2 Hitoshi Kubota,2Kay Yakushiji,2and Shinji Yuasa2 1Unit´e Mixte de Physique, Centre National de la Recherche Scientifique, Thales, Universit ´e Paris-Sud, Universit ´e Paris-Saclay, 91767 Palaiseau, France 2National Institute of Advanced Industrial Science and Technology, Spintronics Research Center, Tsukuba, Ibaraki 305-8568, Japan (Received 15 April 2016; published 27 June 2016) The spin torque in magnetic tunnel junctions possesses two components that both depend on the applied voltage. Here, we develop a method for the accurate extraction of this bias dependence from experiments overlarge voltage ranges. We study several junctions with different magnetic layer structures of the top electrode. Ourresults obtained on junctions with symmetric CoFeB electrodes agree well with theoretical calculations. The biasdependences of asymmetric samples, with top electrodes containing NiFe, however, are twisted compared to thequadratic form generally assumed. Our measurements reveal the complexity of spin-torque mechanisms at largebias. DOI: 10.1103/PhysRevB.93.224427 I. INTRODUCTION The effect of spin torque [ 1,2] is widely used to control the magnetization of the free layer in spin valves and magnetictunnel junctions [ 3,4]. It can induce both stable switching and dynamic oscillations [ 5]. Owing to their high tunneling magnetoresistance (TMR) [ 6], MgO-based magnetic tunnel junctions are promising candidates for applications to memorydevices [ 7], oscillators [ 8], logic [ 9], and neural networks [ 10]. The mechanism underlying spin torque is an essential part ofthese techniques. However, the dependence of spin torque onthe applied bias shows a wide influence of material parametersthat have not allowed a full picture of this effect to be drawnyet [11]. In general, spin torque in magnetic tunnel junctions has two components: one acting as a damping or antidampingdepending on the direction of the current, the other oneacting similarly to a magnetic field along the polarizermagnetization axis [ 12]. The dampinglike torque (T DL) has first been predicted and observed in spin valve structures whereit increases linearly with current [ 1,13]. In tunnel junctions, however, the presence of the tunneling barrier leads to astrong filtering of the electron wave vectors so that onlyelectrons from a small fraction of the Fermi surface contributeto the tunneling current [ 14]. As a result, the electron-spin dephasing length is strongly increased. This gives rise to thefieldlike torque (T FL). Measurements have revealed that its amplitude can be of the same order as that of the dampingliketorque [ 2,15,16]. For the case of symmetric magnetic tunnel junctions with identical top and bottom electrodes, theoretical calculationsbased on different methods unanimously yield a quasilinearbias dependence of the dampinglike torque and a quadratic biasdependence of the fieldlike torque [ 12,17–23]. These depen- dences have also been measured experimentally [ 15,24,25]. *Present address: Laboratoire de Physique des Solides, CNRS, Universit ´e Paris-Sud, Universit ´e Paris-Saclay, 91405 Orsay, France. †julie.grollier@thalesgroup.comFor the case of asymmetric magnetic tunnel junctions with different compositions of the electrodes, however, theoreticalresults that take into account the different electronic band struc-tures predict a large variety of bias dependences [ 26–30]. In general, the dampinglike torque is assumed to gain a quadraticterm whereas the fieldlike torque becomes linear at small bias.However, there are few experimental measurements [ 31,32] testing these predictions in junctions with different electrodes,and many questions remain unanswered. In particular thebias dependence of spin torques in junctions with compositemagnetic layers has never been studied. In order to probe the bias dependence experimentally several techniques have been developed [ 15,32–35]. How- ever, they either are limited to low voltages and demandhigh-frequency measurements or rely on critical externalparameters. Here, we present a technique to determine thebias dependences of spin torques. It is based on simple dcmeasurements as a function of applied magnetic field andbias. Thanks to the TMR effect, the relative magnetizationof the two electrodes is derived from the resistance. Thisallows us to create the phase diagram of each junctionfrom which we determine the spin-torque evolutions withvoltage [ 32,36]. For some special cases of bias dependence, the critical fields and voltages for magnetization reversal canbe calculated analytically [ 32,37,38]. However, no analytical solution exists in the general case. We therefore calculate thephase diagram numerically, which leaves us free from anyrestrictions on the bias dependence of spin torques and enablesus to reveal dependencies beyond usual assumptions. Thecomparison of these numerical simulations to the experimentalphase diagrams finally allows for the determination of the biasdependent spin torques. We studied tunnel junctions with different top electrode compositions while keeping the same bottom electrode. Thesamples cover the range from a fully symmetric structurewith electrodes of the same materials, over a sample withidentical interfaces but different materials in the bulk, toentirely different electrode materials. The results of the biasdependent torques in these samples reveal the need for a morecomplex description of spin torques than commonly assumed. 2469-9950/2016/93(22)/224427(6) 224427-1 ©2016 American Physical SocietyS¨OREN BOYN et al. PHYSICAL REVIEW B 93, 224427 (2016) II. SIMULATIONS The simulations of the phase diagrams are based on a macrospin model of the normalized free-layer magnetiza- tion/vectorm=/vectorM/M s. We describe its time-dependent dynamics using the Landau-Lifshitz-Gilbert (LLG) equation with spintorques [ 1,39]: d/vectorm dt=−γ0/vectorm×μ0/vectorHeff+α/vectorm×d/vectorm dt +γ0TIP/vectorm×(/vectorm×/vectorp)−γ0TOOP/vectorm×/vectorp.( 1 ) Here, the effective magnetic field /vectorHeffincludes the externally applied field /vectorHextas well as the contribution from the shape anisotropy of our films: approximating the thin free layer asan ellipsoid, the latter is taken into account by a dampinglike uniaxial anisotropy field /vectorH anand a demagnetizing field Hd= μ0MsNzwith demagnetization factor Nzperpendicular to the free layer [ 40]. We simulate the influence of finite temperature by a three-dimensional thermal fluctuation field μ0/vectorHTat each integration step. This field’s Cartesian component amplitudesare chosen independently from a Gaussian distribution ofvariance σ T=√2αkBT/(γ0MsV/Delta1 t ) with Gilbert damping constant α, Boltzmann constant k B, temperature T, absolute electron gyromagnetic ratio γ0, saturation magnetization Ms, volume V, and time step /Delta1t[41]. The fixed-layer magne- tization /vectorpthat determines the polarization of the tunneling electrons is kept constant along the easy axis of the free layer.Its amplitude is set to |/vectorp|=1, normalizing the resulting torque amplitudes. The amplitudes of dampinglike and fieldlike torque are functions of applied bias Vthat are expressed as Taylor polynomials: T IP(V)=a1V+a2V2+..., (2a) TOOP(V)=b1V+b2V2+.... (2b) The fixed parameters used in all simulations are α=0.01, an elliptical free layer with a minor axis of 70 nm and a majoraxis of 170 nm, /Delta1t=5 ps, total integration time t end=1m s ,a small initial angle between /vectormand/vectorpof 1 mrad, and temperature T=300 K. The effect of Joule heating can be neglected in our samples [ 31,42]. The LLG equation ( 1) is then solved using the classical fourth-order Runge-Kutta method. For eachnumerical integration, the final state reported in the switchingphase diagrams corresponds to time-averaged values at theend of each simulation. In this way, oscillatory states resultin intermediate, nonsaturated values as it is the case in dcmeasurements. As shown in Fig. 1, different bias dependences of damping- like and fieldlike torques result in unique features in the switch-ing phase diagrams. First, the influence of the two types of spintorques is fundamentally distinct. As the applied bias increases,the dampinglike component eventually closes the bistable areain which parallel (P) and antiparallel (AP) configurations ofthe fixed and free-layer magnetizations coexist. In contrast,the fieldlike torque directly adds up to the external magneticfield and results in a shift of the hysteresis curve as a functionof applied bias. Therefore, the bias dependence of the fieldliketorque directly shows up in the switching phase diagram[Figs. 1(c) and 1(d)]. At higher voltages, the dampinglike(a) (b) (c) (d) 1 0 -1simulationHext / HanT / HanAP / PP U (V)AP -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-0.50.00.5 TOOP TIP simulation AP / PP AP TOOPTIP 1 0 -1Hext / HanT / HanAP / PP APsimulation -0.50.00.5 TOOPTIP simulation AP / PP U (V)AP -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6TOOPTIPTDL TFLTDL TFL TDLTFL TDL TFL FIG. 1. Simulated switching phase diagrams and corresponding spin torques as a function of voltage. Colors indicate magnetizationstate where dark (blue) regions correspond to parallel (P) alignment and bright (yellow) regions correspond to antiparallel (AP) alignment of the fixed and free layer magnetizations. The results shown here areobtained for increasing field values; the white line marks the switching border in the case of decreasing field. Each panel shows the effect of one bias dependence of spin torque at a time. (a) Linear dampingliketorque. (b) Quadratic dampinglike torque. (c) Linear fieldlike torque. (d) Quadratic fieldlike torque. torque leads to sustained magnetization oscillations, resulting in reduced averaged magnetization amplitudes [Figs. 1(a) and 1(b)]. This allows for an unambiguous determination of the respective torque amplitudes for which we adapt thecoefficients of the torques in Eq. ( 2) until the simulated phase diagrams match the experimental ones. In order toestimate the uncertainty of our results on the bias dependentspin torques, we vary each coefficient [ a i, respectively, biin Eq. ( 2)] to the point where the agreement between simulation and measurement is palpably lost. The errors on the finalresults that are given in Figs. 2(m) –2(o) are calculated from the combination yielding the highest error for each bias.As we are only interested in the bias dependences of thetorques and not in their absolute amplitudes, the assumptionof a fully spin-polarized current does not represent anyrestriction. III. EXPERIMENTS The samples have a common layer stack of CoFe(2.5)/Ru(0.85)/Co 60Fe20B20(3)/MgO(1.1)/free layer (thicknesses in nanometers). In order to elucidate theinfluence of asymmetry in electrode composition, sampleswith three different compositions of the top electrode, thatforms the free layer, have been fabricated [see Figs. 2(a)–2(c)]. The films were deposited by sputtering. Then the junctionswere patterned into an elliptical shape (170 ×70 nm) by 224427-2TWIST IN THE BIAS DEPENDENCE OF SPIN TORQUES . . . PHYSICAL REVIEW B 93, 224427 (2016) 15 10 5 0 -5 -10-15 -0.4 -0.2 0.0 0.2 0.4Pexperiment AP / Pµ0(Hext - Hsh) (mT) Voltage (V)AP 10 8 6 4 2 0 -2 -4 -6 -8 -10 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6Pexperimentµ0(Hext - Hsh) (mT) Voltage (V)AP-0.4 -0.2 0.0 0.2 0.4300400500600700800900Resistance (Ohm) Voltage (V)8090100110120130140150TMR (%)AP (µH = -18 mT) P(µH = 25 mT) -0.4 -0.2 0.0 0.2 0.4300350400450500550600650700 P(µH= 23 mT)TMR (%)Resistance (Ohm) Voltage (V)AP (µH= -20 mT) 406080100 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6400450500550600650Resistance (Ohm) Voltage (V)10203040TMR (%)AP (µH= -6 mT) P(µH= 17 mT) (i) (l)(h) (k) (n) (o)(g) (j) (m)MgOCoFeB (2.5 nm) CoFeB (3.0 nm)MgO CoFeB (3.0 nm)NiFe (3.0 nm)symmetric asymmetric symmetric interfaces, asymmetric bulk MgO CoFeB (3.0 nm)NiFe (1.0 nm) CoFeB (1.0 nm)(a) (d)(b) (e)(c) (f) 8 6 4 2 0 -2 -4 -6-8 -0.4 -0.2 0.0 0.2 0.4 Voltage (V)experimentµ0(Hext-Hsh) (mT)AP / PP AP 8 6 4 2 0 -2 -4 -6 -8 -0.4 -0.2 0.0 0.2 0.4simulation AP / PPµ0Hext (mT) Voltage (V)AP -0.4 -0.2 0.0 0.2 0.4-1.0-0.50.00.51.0T / Han Voltage (V) 10 8 6 4 2 0 -2 -4 -6 -8 -10 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Voltage (V)simulation Pµ0Hext (mT) AP -0.4 -0.2 0.0 0.2 0.4-1.0-0.50.00.51.0T / Han Voltage (V)-0.1 0.0 0.1-0.10-0.050.000.05 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-1.0-0.50.00.51.0T / Han Voltage (V) 15 10 5 0 -5 -10 -15 -0.4 -0.2 0.0 0.2 0.4 Voltage (V)simulationµ0Hext (mT)AP / PP AP TDL TFLTDL TFLTDL TFLTFL FIG. 2. (a–c) Simplified layer stacks of the samples used in this study. (d–f) V oltage dependent resistances of the saturated parallel (P) and antiparallel (AP) states and corresponding TMR values. Experimental (g–i) and simulated (j–l) switching phase diagrams. The whitelines indicate the border for AP to P state switching for increasing applied fields. (m–o) Resulting bias dependent spin torques as used in the simulations. ion milling in order to introduce an easy magnetization axis thanks to shape anisotropy. A more detailed description of thefabrication process is given in Ref. [ 43].The experiments consist of dc resistance measurements while sweeping an external magnetic field parallel to theeasy axis at a certain fixed applied voltage bias. Under 224427-3S¨OREN BOYN et al. PHYSICAL REVIEW B 93, 224427 (2016) positive voltage, the electrons flow from the free to the fixed layer and a positive magnetic field favors parallelalignment of fixed and free layer magnetizations. In orderto remove the strong voltage dependence of the resistance,we use normalized values in the diagrams: R rel(V,H ext)= [R(V,H ext)−RP(V)]/[RAP(V)−RP(V)], where RP(V) and RAP(V) are fully saturated P and AP states, respectively, stabilized by high external magnetic fields [see Figs. 2(d)– 2(f)]. As the measurements are taken on dc time scales, the free layer magnetization is expected to be in equilibrium state. Theresults for a range of bias values are represented in switchingphase diagrams [see Figs. 2(g)–2(i)]. In these diagrams, intermediate resistance values between the minimum andmaximum values of the static P and AP states correspondto dynamic oscillatory states. A small field shift H shin the low-bias hysteresis curves has been subtracted from thepresented data. It most certainly results from the dipolarcoupling to the reference layer due to an incomplete screening.This subtraction does not influence our bias dependent results,but implies that we ignore any zero-bias interlayer exchangecoupling [ 44,45]. IV . RESULTS AND DISCUSSIONS A. Symmetric junction We first performed measurements on samples with symmet- ric electrode interfaces, both composed of the same material(Co 60Fe20B20) giving rise to TMR values of about 130% at low bias [see Fig. 2(d)]. The symmetry of the junction is also translated into highly symmetric bias dependences ofthe saturated P and AP states resistances [see Fig. 2(d)]. The experimentally obtained switching phase diagram [Fig. 2(g)] shows a fully closed hysteretic zone at the center and ismainly symmetric. This can also be observed in the biasdependent resistance for the saturated states in Fig. 2(d). These measurements could be reproduced by our simulation to a verygood agreement [Fig. 2(j), using a saturation magnetization of M s=1.38 T, Ref. [ 46]]. The resulting spin torques [Fig. 2(m) ] show the ex- pected bias dependences of a symmetric magnetic tunneljunction [ 21]: the dampinglike torque is a strongly linear function of voltage with a small quadratic contribution. Incontrast, the fieldlike torque is purely quadratic and thereforea symmetric function of the applied bias. These resultsagree very well with other results obtained with symmetricmagnetic tunnel junctions by both experimental [ 15,24,25] and theoretical [ 12,17–19,22,23] techniques. They also val- idate our method for extracting the spin-torque bias de-pendences by matching experimental and simulated phasediagrams. B. Symmetric interfaces, asymmetric bulk The second sample with a free layer of Co 60Fe20B20 (1 nm)/Ni 81Fe19(1 nm) [see Fig. 2(b)] possesses symmetric interfaces at the MgO barrier but different bulk materialsin the electrodes resulting in a slight asymmetry in thebias dependence of the TMR [Fig. 2(e)]. The experimental switching phase diagram resembles that obtained on the fullysymmetric sample [Fig. 2(h)]. However, the hysteretic zone shows a stronger, linear shift with voltage which indicates anonvanishing linear component in the bias dependence of thefieldlike torque. We were able to reproduce the experimentalresult by simulation with only a small deviation at highnegative voltages [Fig. 2(k), using the mean of the saturation magnetizations of CoFeB [ 46] and NiFe [ 47]M s=1.11 T]. The strong oscillations observed in the simulations are mostprobably suppressed by micromagnetic effects in our samples.The best agreement to measurements is achieved under theassumption of an increased quadratic dampinglike torque incombination with a clearly linear bias dependence of thefieldlike torque at small voltages [Fig. 2(n) and inset therein]. At higher bias it finally recovers its quadratic form as observedin the symmetric junction. C. Asymmetric junction In the case of a free layer of Ni 81Fe19(3 nm) [see Fig. 2(c)], the bias dependent P and AP state resistances reflect thestrong asymmetry in the junction’s structure [see Fig. 2(f)]. The switching phase diagram features strong differences fromthose obtained for symmetric interfaces [Fig. 2(i)]. Addition- ally, at room temperature, the sample is superparamagnetic,not displaying any hysteresis, which can be ascribed atleast partially to its lower saturation magnetization of M s= 0.81 T [ 47]. In order to reproduce the phase diagram [Fig. 2(l)], it is necessary to include a third-order voltage term into the de-scription of the spin torques’ bias dependences. The switchingborder is mainly influenced by the fieldlike torque which isa nearly linear function of voltage at small bias. A similardependence has been measured in other (slightly) asymmetricmagnetic tunnel junctions [ 31,32,48]. Theoretical calculations also yield this linear component [ 26–28]. At higher voltages, however, the evolution deviates from the previous results.The bias dependence is twisted and reigned by a third-orderterm. The dampinglike component also exhibits a mainlylinear bias dependence at small voltage amplitude as it ismeasured in symmetric junctions. However the apparition andsubsequent suppression of oscillations, observable through theintermediate relative resistance values at voltages amplitudesof±0.4 V , can only result from a reduction of the dampinglike torque amplitude at high absolute voltage values. Theseassumptions are in good agreement with the results of thebias dependent torques [Fig. 2(o)]. Higher-order voltage terms in the bias dependence have also resulted from the calculations by Tang et al. [27], modeling the asymmetry of the junction by a parameter describing a shift inthe spin-dependent on-site energies of the two ferromagneticelectrodes. Kalitsov et al. [30] have shown similar results by introducing different electrode interfaces in a differenttunneling regime. However, these theoretical results are linkedto negative TMR values appearing in the low-bias regionwhich we did not observe in our sample [see Fig. 2(f)]. In summary, although many models are in a good partialagreement with our experimental results, a full understandingand theoretical description of the bias dependence of spintorques in asymmetric junctions is still missing. 224427-4TWIST IN THE BIAS DEPENDENCE OF SPIN TORQUES . . . PHYSICAL REVIEW B 93, 224427 (2016) V . CONCLUSIONS We have developed a method to extract the bias depen- dences of the dampinglike and fieldlike spin torques in MgO-based magnetic tunnel junctions without any assumption ontheir variations. A symmetric junction with identical electrodesof CoFeB exhibits a linear dampinglike and a quadraticfieldlike torque, as expected from theory. By introducing anasymmetry in the bulk of an CoFeB(1 nm)/NiFe(1 nm) freelayer, both torques are modified: the dampinglike componentgains a small quadratic dependence and the fieldlike torquebecomes linear at small bias. For the case of a completelyasymmetric junction with a NiFe free layer we found thatthe bias dependences are twisted. Both spin torques acquirehigher-order terms and the amplitude of the fieldlike torquebecomes similar to that of the dampinglike torque. Althoughthis has been partially predicted by theoretical calculations, more experimental results are required to fully understandthe parameters that influence spin torques in magnetic tunneljunctions. The technique we developed in this paper allows aneasy-to-implement and fast way to extract the bias dependenceof torques in any kind of sample. This will eventually allowthe systematic design of magnetic tunnel junction structuresfor specific applications. ACKNOWLEDGMENTS The authors would like to thank Aur ´elien Manchon and Mairbek Chshiev for useful discussions, and Canon ANELV Afor preparation of the magnetic films. 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PhysRevB.90.054414.pdf

PHYSICAL REVIEW B 90, 054414 (2014) Spin-wave-mediated mutual synchronization of spin-torque nano-oscillators: A micromagnetic study of multistable phase locking T. Kendziorczyk,1,*S. O. Demokritov,1,2and T. Kuhn1 1Department of Physics and Center for Nonlinear Science, University of Muenster, 48149 Muenster, Germany 2Institute of Metal Physics, Ural Division of RAS, Yekaterinburg 620041, Russia (Received 29 May 2014; revised manuscript received 31 July 2014; published 19 August 2014) We present a detailed micromagnetic study of the synchronization dynamics of two spin-torque nano-oscillators which are dynamically coupled by the emission of spin waves through an extended magnetic layer. We showthat the intercontact distance and the wavelength of the emitted spin waves have a large impact on the possiblesynchronization states. Depending on the intercontact distance, there exist three different synchronization regionswhich can be characterized as in-phase and antiphase synchronization as well as a bistability region with bothin-phase and antiphase synchronization. We demonstrate under which conditions bistability can be achieved andhow the system can be switched from one to the other synchronization state. Finally, we develop a reducedsemianalytical model which reproduces the simulation results and allows us to understand the physical origin ofthe bistability. It turns out that the bistability is caused by time-delay effects. DOI: 10.1103/PhysRevB.90.054414 PACS number(s): 75 .78.Cd,75.75.−c,85.75.−d,05.45.Xt I. INTRODUCTION In various studies it has been shown theoretically [ 1] and experimentally [ 2–4] that transfer of angular momentum from a spin-polarized current to a thin ferromagnetic layer canexert a sufficiently large spin-transfer torque to compensatethe natural damping and lead to magnetic auto-oscillations.Subsequent theoretical work [ 5,6] demonstrated that devices based on this principle, referred to as spin-torque nano-oscillators (STNOs), can also be used to excite spin wavesin the surrounding ferromagnetic material. This predictionhas been only recently confirmed experimentally by directmapping of the emitted spin waves [ 7,8]. STNOs have attracted much interest in the past years especially because oftheir potential use as tunable nanoscale microwave emitters,which are compatible with the present semiconductor planartechnology [ 9]. Another large field for potential applications is the emerging field of magnonics [ 10], which deals with the transmission and processing of information based on spinwaves rather than electric currents. In this field, STNOs couldbe used as effective spin-wave injectors, which have clearbenefits compared with the use of conventional inductiveantennas [ 11]. For both routes of application it is desirable to increase the low output power of a STNO, which is typicallybelow 100 nW [ 9], by using arrays of STNOs. Theoretically, STNOs can be described as auto-oscillators with negative damping and a nonlinear frequency shift[12–16]. For systems of nonlinear oscillators, the phenomenon of synchronization is a general concept in nature [ 17–19], which is not only interesting from the fundamental pointof view, but can also pave the way for the coherent powerenhancement in STNO arrays. Furthermore, coherently gen-erating STNOs can create stationary spin-wave interferencepatterns, which might have potential use in on-chip commu-nication and information processing [ 20]. Experimentally, the synchronization of STNOs has been demonstrated both for thephase locking to external signals [ 21–23] and the mutual phase *t.kend@uni-muenster.delocking of STNO pairs, which is mediated mainly by spin waves [ 24–26]. During the past years enormous progress has been achieved in the understanding of the synchronization dy-namics for STNOs from the theoretical point of view in termsof analytical models based on the auto-oscillator theory [ 15,27] and on the basis of micromagnetic simulations [ 28,29]. Here, we present a detailed micromagnetic study of the synchronization dynamics for a pair of STNOs in the point-contact geometry. The geometrical and material parametersof the single STNO are adapted to the experiments inRef. [ 30]. We extend previous research by investigating systematically the influence of the system’s initial conditionson the synchronization. We show that the phase locking ofSTNOs has a bistability in the stationary phase differencecorresponding to in-phase and antiphase synchronization forone and the same set of parameters. We investigate in detailunder which conditions this bistability is achieved and howthe system can be switched from the in-phase to the antiphasestate and vice versa. Furthermore, we will show how thereported effect can be incorporated in the auto-oscillator theoryof synchronization, which allows us to identify the physicalorigin of the bistability. The paper is organized as follows:In Sec. II, we describe details of the studied system of two STNOs and summarize the underlying theory and simulationmethods of our study. Section IIIcontains an overview of the current induced dynamics for a single STNO and acomparison with experiments and previous theoretical studies.We also discuss the choice of parameters for the study of thesynchronization dynamics of a STNO pair which then followsin Sec. IV. The impact of the intercontact distance on the synchronization dynamics is addressed in Sec. V. Finally, in Sec. VIwe demonstrate how the synchronization dynamics can be modeled by means of a reduced model based on theauto-oscillator theory. The paper concludes with a summaryand final remarks in Sec. VII. II. SYSTEM AND THEORETICAL FRAMEWORK The device under investigation is shown in Fig. 1.T h e system consists of two identical spin-torque nano-oscillators, 1098-0121/2014/90(5)/054414(9) 054414-1 ©2014 American Physical SocietyT. KENDZIORCZYK, S. O. DEMOKRITOV , AND T. KUHN PHYSICAL REVIEW B 90, 054414 (2014) FIG. 1. (Color online) Pictorial representation of the system. Two STNOs are placed on a common free magnetic layer of permalloy. The free layer is separated by a Cu spacer from the CoFepolarizer. The emitted spin waves in the permalloy layer can mediate synchronization between the STNOs. which are separated by a center-to-center intercontact distance ofd12. The properties of the single contacts are adapted from the experimental work by Bonetti et al. [30]. Consequently, the STNOs are defined through the direct currents I1andI2which traverse a structure of Co 50Fe50(20 nm)/Cu(6 nm)/Py(4 .5n m ) perpendicularly through circular nanocontacts with respectiveradiir 1=r2=20 nm. The CoFe layer acts as a spin polarizer for the reflected electrons, which have a velocity with thedirection −e zforI1,I2>0. For this current polarity, the spin- transfer torque in the permalloy layer compensates the Gilbertdamping and leads to auto-oscillations and emission of spinwaves. The free layer (thin permalloy) is extended betweenthe point contacts to promote mutual interaction through theemission of spin waves similar to Refs. [ 24,25]. An external magnetic field with magnitude μ 0H0=1.1 T is applied at an angle of θwith respect to the film plane in the xzplane. The theoretical framework for the description of the dynam- ics of the magnetization M(r,t) is given by the micromagnetic model. We have implemented our own finite differencescode to numerically integrate the Landau-Lifschitz-Gilbertequation with a supplementary Slonczewski term to take intoaccount the spin-transfer torque. Thus, the system dynamicsis described by dM dt=−γμ 0M×Heff+αG M0/bracketleftbigg M×dM dt/bracketrightbigg +β M2 0[M×[M×p]] (1) with Heff=−1 μ0δW δM, (2) where γis the gyromagnetic ratio, αGthe Gilbert damping con- stant, and M0the saturation magnetization. The Slonczewski term contains the magnetization direction pof the polarizing CoFe layer and the prefactor β=gμBεj(r,t)/(2edM2 0), which depends on the spin-polarization efficiency ε, the thickness of the free permalloy layer d, and the current density j(r,t). The free energy W, from which the effective field Heffis derived, contains the standard micromagnetic contributions resultingfrom the external field, the exchange interaction, and theOersted field, which is produced by the current flowing throughthe point contacts. The demagnetization field is calculated in the thin-film approximation H demag=− (M·ez)ez[14]. This approximation is good in the case kd/lessmuch1, where kis the in-plane wave vector and dthe thickness of the layer. This condition is fulfilled when the characteristic length scalesof the magnetic system (spin-wave wavelength, size of thenanocontact) are much larger than the film thickness. The ob-tained results using this approximation show good quantitativeagreement with simulations including the numerically exactdemagnetization field as shown below (c.f. Fig. 2). We use the standard parameters for permalloy, corre- sponding to a saturation magnetization μ 0M0=0.8T , t h e dimensionless Gilbert damping constant αG=0.01, and an exchange stiffness of A=1.4×10−11J/m. The CoFe po- larizer is assumed to be fixed, therefore, pis determined by initially calculating the equilibrium of the CoFe layer using thevalueμ 0M0=1.8 T for the saturation magnetization. Based on the analysis in Ref. [ 30], the point contacts are modeled by a constant out-of-plane current density below the pointcontacts with effective radii r 1=r2=32 nm and zero current density outside the contacts. To confirm this simplified model,we performed additional simulations of the current densityin a realistic point-contact geometry based on the data givenin Refs. [ 30,31]. The simulations show that the current flows mainly perpendicular to the layer structure, however, it is notentirely localized below the point contacts, which is consistentwith the increased effective radius of the point contact. Itshould be noted that for certain contact geometries, the currentcan also have substantial lateral components [ 32], which could lead to an additional spin-transfer torque [ 33]. The spin-polarization efficiency is ε=0.3, which is a typical value for the considered trilayer structure. The Oerstedfields are calculated analytically for infinitely long, straight,cylindrical wires. III. SINGLE STNO IN THE FREE-RUNNING REGIME Before considering the synchronization dynamics in the system of two STNOs, we analyze a single STNO to validatethe model and the chosen set of parameters. As discussedin Sec. II, the system parameters are chosen to match the setup which was used in the experiment by Bonetti et al. in Ref. [ 30]. To compare with those results, we perform micromagnetic simulations of a single STNO and calculatethe auto-oscillation frequency as a function of the currentfor different angles θof the external magnetic field. The simulation is performed for a computational region of thesize 1200 nm ×1200 nm ×4.5 nm with a cell size of 4 nm × 4n m×4.5 nm. Because of the relatively low damping of the permalloy we employ a spatially increasing damping towardsthe boundary of the computational region to avoid artificialspin-wave reflections [ 34]. In order to calculate the auto- oscillation spectrum, we spatially average the M ycomponent in the contact region and perform a spectral analysis of thetime-dependent signal /angbracketleftM y/angbracketrightSTNO . We take into account only the signal which belongs to auto-oscillation after the systemhas reached a stationary oscillation amplitude. The resultingfrequency spectrum (amplitude spectral density) for a constantcurrent of I=13 mA as a function of the angle θis shown in Fig. 2. The black dots show the experimental data extracted 054414-2SPIN-WA VE-MEDIATED MUTUAL SYNCHRONIZATION OF . . . PHYSICAL REVIEW B 90, 054414 (2014) 10 20 30 40 50 60 70 80 90 field angle (deg)1520253035frequency (GHz) ASD (arb. units) I=1 3m A f=2 6.0G H zθ=6 5◦ f=3 4.3G H zθ=3 5◦ f=3 1.9G H zθ=3 5◦ FIG. 2. (Color online) Auto-oscillation spectrum (amplitude spectral density) for a single STNO at a current of I=13 mA as a function of the angle θ. The black dots show the experimental data extracted from Ref. [ 30]. The red dots show the results from micromagnetic simulations using the exact formula for the calculation of the demagnetization field (without thin-film approximation). Theinsets show the spatial dependence of the M ysignal power for the different modes. The red circles in the insets indicate the size of the point contacts. The arrows indicate the in-plane component H/bardbl 0 of the external field. from Ref. [ 30], while the red dots show the results from micromagnetic simulations using the exact formula [ 35]f o rt h e calculation of the demagnetization field. The insets show thespatial profile of the different excited modes. This informationis obtained by computing the temporal Fourier transform of theM ytime traces for each computational cell and evaluating the square modulus of this spectrum at frequencies correspondingto the main peaks in the auto-oscillation spectrum. The auto-oscillation spectrum clearly shows a single auto-oscillationmode for angles θ> 50 ◦. According to the spatial distribution of the intensity, which is shown in the inset for θ=65◦, this mode represents a propagating Slonczewski mode. Theasymmetry of the spin-wave intensity perpendicular to thedirection of the in-plane component of the external magneticfield (ydirection) can be explained by the inhomogeneity of the internal field, which is caused by the Oersted field producedby the nanocontact [ 36,37]. For smaller angles of the external fieldθ< 50 ◦, there exist two modes. The spatial distribution of the power according to the inset for θ=34◦reveals that those modes are both localized. All those results for a single STNOare in good quantitative agreement with the experiment [ 30]a s well as with the simulation results from Ref. [ 37]. In this paper, we restrict the angle of the external field to θ> 50 ◦because only in this case the STNO always excites spin waves, whichprovide a sufficiently large coupling between two STNOs inorder to achieve synchronization. In Fig. 3, we show the frequency of the auto-oscillation forθ=65 ◦,75◦,85◦as a function of the current. The data forθ=65◦show good agreement with the experiment from Ref. [ 30] for the onset current of auto-oscillations as well as for the nonlinear frequency shift. This angle will be used inthe following simulations. For larger angles of the externalfield the onset current and the nonlinear frequency shiftincrease in agreement with previous simulations and analyticalmodels [ 38,39]. 6 8 10 12 14 16 18 20 current (mA)1015202530frequency (GHz) ASD (arb. units) θ=8 5◦θ=7 5◦θ=6 5◦ FIG. 3. (Color online) Auto-oscillation spectrum (amplitude spectral density) for a single STNO at the angles θ=65◦,75◦,85◦ as a function of the current through the point contact. The dots show the experimental data extracted from Ref. [ 30]. The nonlinear frequency shift is also an important quantity for an auto-oscillator in the context of thermal fluctuationswhich can be generally important due to the finite operationtemperature of the device [ 40]. The nonlinear frequency shift gives rise to a renormalization of the phase fluctuations in theSTNO because power fluctuations are directly translated to ad-ditional phase fluctuations [ 41]. Therefore, the role of thermal fluctuations strongly depends on the angle θof the external magnetic field and is expected to be less important for smallerangles θ. This has been confirmed experimentally in Ref. [ 42]. For angles in the vicinity of θ=60 ◦, a substantially reduced linewidth of the auto-oscillation has been reported. Therefore,in the following we will analyze the synchronization dynamicswithout the influence of thermal fluctuations. IV . SYNCHRONIZATION OF A STNO PAIR In this section, we consider a system which consists of two identical STNOs. We begin the analysis for an intercontactcenter-to-center distance of d 12=460 nm, which is in the order of the typical distance for which synchronization ofSTNOs was observed experimentally [ 24,25]. In order to achieve a symmetric interaction by spin waves, the STNOsare positioned at y=0.6μm along a line which is parallel to the in-plane component of an external field with θ=65 ◦. Otherwise, according to the spatial profile of the spin-waveintensity, shown in the inset of Fig. 2, a substantial asymmetry would be introduced. In the simulation, the current through the first contact is kept fixed at a value of I 1=13 mA, while the current I2is varied. We initialize the simulation at /Delta1I=I2−I1=0, where both STNOs are identical, except for minor asymmetries due tothe Oersted field. To simulate a situation which is similar toan experimental measurement, /Delta1Iis changed linearly with a relatively slow rate of roughly 1 .5m A/μs to its next value, where it is halted for a time period of 100 ns in order tocalculate the frequencies and the instantaneous relative phasedifference /Psi1=φ 2−φ1of the auto-oscillators. The ramping speed has been chosen so that a further reduction of the speeddid not affect the simulation results. 054414-3T. KENDZIORCZYK, S. O. DEMOKRITOV , AND T. KUHN PHYSICAL REVIEW B 90, 054414 (2014) π 0 -1.5 -1 -0.5 0 0.5 1 1.5Ψ0,( r a d ) ΔI(mA) 2525.52626.5frequency (GHz)ωf/2πanti-phase in-phase -1.5 -1 -0.5 0 0.5 1 1.5 ΔI(mA) 00.20.40.60.81 ASD (arb. units)ωf/2πanti-phase in-phase 0.4 0.6 0.8 1 1.2 x(µm)0.40.60.81y(µm) 123 45 0.4 0.6 0.8 1 1.2 x(µm)0.40.60.81y(µm) 123456 t1t2t3t4 FIG. 4. (Color online) (a), (b) Combined frequency spectra (amplitude spectral density) and stationary phase difference /Psi10for two coupled STNOs for two different initial conditions of the simulation (a) /Psi10(t=0)=0, (b)/Psi10(t=0)=π; (c) dynamics of the synchronization during a current cycle of I2withI1fixed, top panel: current ramping protocol, center panel: total signal s(t)=/angbracketleftMy/angbracketrightSTNO1+/angbracketleftMy/angbracketrightSTNO2 , bottom panel: instantaneous phase difference /Psi1(t) of the oscillators; (d), (e) spin-wave intensity showing the interference patterns for /Delta1I=0 and in-phase synchronization (d)/antiphase synchronization (e). The arrows indicate the in-plane component H/bardbl 0of the external field. The ramping procedure is repeated until the desired maximum value (here, /Delta1I=1.5 mA) is reached. Then, the same procedure is followed for decreasing /Delta1Ifrom/Delta1I=0 to/Delta1I=−1.5 mA. The frequency spectra are determined analogously to Sec. II, while the instantaneous phases φ1,φ2 are determined from the time-dependent spatially averaged components /angbracketleftMy/angbracketrightSTNO1 and/angbracketleftMy/angbracketrightSTNO2 . This is done by applying the analytical signal method [ 19] based on the Hilbert transform [ 43]. Figure 4(a)shows the resulting sum of the frequency spectra of STNO 1 and STNO 2 as a function of /Delta1I. At large values of|/Delta1I|, the frequencies ω1andω2of the two STNOs can be clearly distinguished in the spectrum. The frequency ω1is approximately equal to the free-running frequency ωf(dashed line). The frequency ω2experiences a nonlinear blue-shift with increasing /Delta1Iin agreement with Fig. 3. We can also identify weaker signals at the frequencies 2 ω1−ω2and 2ω2−ω1. These signals belong to a third-order intermodulation whichis expected due to the intrinsic nonlinearity of the magneticsystem [ 37]. The frequency spectrum shows clearly a single frequency for both auto-oscillators at −0.25 mA </Delta1 I< 0.75 mA. In this region of /Delta1Ithe time derivative of the instantaneous phase difference is given by d/Psi1/dt =0, which means that the STNOs are phase locked and oscillate coherently withone common frequency ω 1=ω2=ωs. The bottom panel of Fig. 4(a) shows the value of the stationary phase difference /Psi1(t)=/Psi10=const inside of the phase-locking region. The most striking feature in the frequency spectrum and in thebehavior of /Psi1 0(/Delta1I) is the discontinuity of the frequency and the phase difference at /Delta1I=0.29 mA, where the phase- locking region is separated in an in-phase and antiphase region.In order to investigate these features, we calculate the timedependence of the total STNO signal s(t)=/angbracketleftM y/angbracketrightSTNO1+/angbracketleftMy/angbracketrightSTNO2 and the instantaneous phase difference /Psi1(t) during a current ramping cycle from /Delta1I=0t o/Delta1I=1 mA and vice versa during a total time period of 250 ns. Figure 4(c) shows the current ramping protocol, the total signal s(t), and the instantaneous phase difference /Psi1(t). At the beginning of the simulation the system clearly evolves into the in-phase synchronized state with /Delta1I=0. Deviations from /Psi1≈0 can be attributed to an Oersted field- induced asymmetry in the system. The internal field, which isproduced by the Oersted field created by STNO 1, changes theinternal field at the position at STNO 2 and vice versa. That iswhy both oscillators are not strictly identical. At the time t=t 1we start to increase /Delta1I.F o rt1<t<t 2 the system remains in the state with in-phase synchronization. Att=t2a sudden transition into antiphase synchronization can be observed, becoming also apparent in the amplitude ofthe total signal s(t), which drops to approximately 0 .15. If /Delta1Iis increased even further, the system leaves the phase- locking region for antiphase synchronization at t=t 3which becomes visible by the low-frequency beating of s(t) with the frequency ω2−ω1fort3<t<t 4. When the current difference /Delta1Iis decreased again, the system returns to the antiphase synchronization state at t=t4, which means that the phase-locking bandwidth at /Delta1I > 0 is larger for antiphase synchronization. The most interesting feature of Fig. 4(c) is that the system stays in the antiphase synchronizationstate, if we decrease /Delta1Ibelow the value of /Delta1I=0.29 mA, where the discontinuity in Fig. 4(a) appeared. Therefore, the presented current ramping protocol represents a completehysteresis loop, which switches the system from in-phaseto antiphase synchronization. Consequently, there exists abistability in the phase difference with stable in-phase andantiphase synchronization for the same set of parametersbut different initial condition of the system similar to the 054414-4SPIN-WA VE-MEDIATED MUTUAL SYNCHRONIZATION OF . . . PHYSICAL REVIEW B 90, 054414 (2014) results in Ref. [ 29]. Figure 4(b)shows the frequency spectrum, which is determined analogously to 4(a), but with the initial condition of antiphase synchronization at /Delta1I=0. Comparing the two frequency spectra we find two phase-locking regionscorresponding to in-phase and antiphase synchronization. Forthe given set of parameters, the antiphase synchronizationregion is significantly larger than the in-phase synchronizationregion, but the phase-locking region is more extended to the leftside (/Delta1I < 0) for in-phase synchronization. Thus, a second discontinuity of the frequency and the phase difference is foundat/Delta1I=−0.17 mA. It is interesting to note that the frequency ω sof the syn- chronized oscillators, even for identical oscillators at /Delta1I=0, is different from the free-running frequency ωf, which can be observed in Figs. 4(a)and4(b). For in-phase synchronization, the synchronization frequency experiences a down-shift ωs< ωf, while for antiphase synchronization the frequency is shifted upwards, i.e., ωs>ωf. This behavior is due to a general property of asynchronous auto-oscillators which arecoupled through a signal with a finite velocity, which causesa time delay in the coupling [ 15]. A direct consequence of the observed frequency shifts is the change of the spin-wave wave-length λ SW. In order to analyze this effect, we calculate the spatial profile of the spin-wave intensity for /Delta1I=0. Figure 4 shows the spatial intensity profile for in-phase synchronizationin Fig. 4(d) and for antiphase synchronization in Fig. 4(e) for the same set of parameters, respectively. We can observe thespin-wave interference patterns, which are periodic in λ SW/2. Due to the symmetry of the system with /Delta1I=0w ec a n observe a maximum in the interference pattern in the center be-tween the contacts for in-phase synchronization. For antiphasesynchronization, there is a minimum in the center. Therefore,the interference patterns are shifted by λ SW/4 between the two cases and one additional maximum appears for antiphasesynchronization. Additionally to this effect a more detailedanalysis of the interference pattern reveals that the wavelengthsfor in-phase synchronization are increased by up to 10 nm incomparison with the case of antiphase synchronization. This isconsistent with the found frequency differences. However, forboth cases the wavelengths are not constant in the region be-tween the contacts, which can be attributed to inhomogeneousstatic dipolar fields created by the large-angle auto-oscillationbelow the contacts. Although the wavelength shift betweenin-phase and antiphase synchronization seems to be small, thecoupling phase β, which is defined as the phase difference of the spin waves accumulated while propagating from theemitting STNO to the other one, is changed significantly. Thecoupling phase can be estimated to be approximately β≈ 2πd/λ SW. This yields a difference between the coupling phase for in-phase and antiphase synchronization of roughly π/2. It has to be noted that the difference in βhas not to be necessarily πin order to induce a modification of the phase difference /Psi10 byπ. This follows from the auto-oscillator theory, which will be shown in Sec. VI. There, we will show how the possible sta- ble solutions with a stationary phase difference /Psi10depend on the value of βand that the interplay between the dependence of the oscillation frequency on the coupling and the alterationof the coupling signal through the modification of the spin-wave wavelength is closely related to the effect of bistabilityin the studied system.V . IMPACT OF THE INTERCONTACT DISTANCE In the following, we analyze the influence of the intercon- tact distance d12on the in-phase and antiphase synchroniza- tion. Figure 5shows the stationary phase difference /Psi10, which is determined after the system reached synchronization, fordifferent intercontact distances as a function of the currentdifference /Delta1I. The calculation is performed for different initial conditions. The initial conditions were set to in-phaseor antiphase synchronization by applying specific currentramping protocols in order to switch the system to therespective synchronization state. The details of these switchingprotocols will be discussed below (cf. Fig. 6). After the in- phase or antiphase synchronization for /Delta1I=0 is established, /Delta1Iis ramped to the final value which is indicated in Fig. 5. The top panel shows the case where the initial phase difference is set to /Psi1(t=0)=0, whereas in the bottom panel it is set to /Psi1(t=0)=π. In the black regions of Fig. 5,n o synchronization is observed, meaning d/Psi1/dt /negationslash=0. For both initial conditions, the phase-locking bandwidth is a quasiperiodic function of the intercontact distance with theperiodicity λ 0=124 nm, which is indicated by the arrow in the bottom panel. This value is consistent with the value which canbe determined from the analytical spin-wave dispersion [ 44] substituting the free-running frequency ω f/2π=25.73 GHz. The maximum values of the phase-locking bandwidth decreasefor larger values of d 12due to the attenuation of the spin-wave amplitude, which can be described by [ 7] A(r)=A01√re−r/λr,λ r=vgr/(αGωSW), (3) I/A I/A I/A A A A/II I/A I/A I/A A A A/II FIG. 5. (Color online) Stationary phase difference /Psi10as a func- tion of the intercontact distance and the current difference /Delta1I;t o p panel: initial condition /Psi1(t=0)=0; bottom panel: initial condition /Psi1(t=0)=π. I and A denote regions of in-phase and antiphase synchronization, respectively. The dashed lines are guides for the eyeand indicate the borders of the regions of multistability (I/A). 054414-5T. KENDZIORCZYK, S. O. DEMOKRITOV , AND T. KUHN PHYSICAL REVIEW B 90, 054414 (2014) ramping Indownramping Inup 13 14 15 16 17 18 19 20 current (mA)(˜A) ( ˜I/˜A) ( ˜I) ( ˜I/˜A)1 2 3 5 413.016.520.0In(t)I1=I2ΔI=0 -1.00.01.0s(t) 0πΨ(t) 3.54.55.5 0 50 100 150 200 250d12/λ time (ns)0.61 2 3 4 5 FIG. 6. (Color online) (a) Dynamics of the synchronization dur- ing a current cycle with I1=I2, first panel: current ramping protocol; second panel: total signal s(t)=/angbracketleftMy/angbracketrightSTNO1+/angbracketleftMy/angbracketrightSTNO2 ; third panel: instantaneous phase difference /Psi1(t) of the oscillators; fourth panel: dynamic change of the ratio between the intercontact distance and the wavelength d12/λ. (b) Illustration of the occurring switching events between in-phase and antiphase synchronizationduring the current ramping process. where the factor 1 /√ris due to the propagation of the wavefront in the two-dimensional (2D) system and λris the relaxation length due to the Gilbert damping, which dependson the frequency ω SWof the emitted spin waves and their group velocity vgr. We show in the top panel of Fig. 5that the maximum phase-locking bandwidth is proportional toA(d 12), which is consistent with the analytical results from Ref. [ 27]. If we analyze the dependence on the intercontact distance for/Delta1I=0, we can clearly identify three different regions, which are labeled as (I/A), (I), and (A). Comparison of the topand bottom panels of Fig. 5reveals that (I/A) corresponds to a bistability region for in-phase and antiphase synchronization.However, towards the borders of the (I/A) region, the phase-locking bandwidths of in-phase and antiphase synchronization,respectively, are continuously shrinking to zero. According tothe vanishing phase-locking bandwidth, the (I) and (A) regionspossess only one stable synchronization state. Therefore, the bistability can be avoided by adjusting the intercontact distance d 12. This fact leads naturally to a switching scheme which is based on driving the system fromthe bistability region (I/A) to a region (I) or (A) where onlyone phase-locked state is possible and back to (I/A) again.However, it is not possible to change the distance betweenthe contacts dynamically. On the other hand, due to the factthat the phase-locking bandwidths and regions of stability arequasiperiodic functions of d 12with the periodicity λ0,i ti s sufficient to alter the ratio d12/λ0. Thus, alternatively it is also possible to modify the wavelength for example by changingthe currents through both contacts simultaneously. We present a switching scheme based on this idea in Fig. 6. The top panel of Fig. 6(a) illustrates the currentramping protocol, which corresponds to a linear simultaneous increase of I 1andI2to a maximum value of I1=I2=20 mA during a time interval of roughly 100 ns. After reachingthe maximum, the current is decreased again with the samerate to the initial value I 1=I2=13 mA. We can clearly observe discontinuities in the total signal amplitude (secondpanel) and the instantaneous phase difference (third panel).These correspond to the three switching events 1/circlecopyrt,3/circlecopyrtand 5/circlecopyrtfrom in-phase to antiphase and the two switching events 2/circlecopyrtand 4/circlecopyrtfrom antiphase to in-phase synchronization. To understand the observed switching events, we calculate thevalues of d 12/λ. The wavelength is dynamically calculated from the analytical dispersion relation [ 44] substituting the frequency ωs=dφ1/dt=dφ2/dt, where φiare the instanta- neous phases which are obtained from the signals /angbracketleftMy/angbracketrightSTNO1 and/angbracketleftMy/angbracketrightSTNO2 . In the bottom panel of Fig. 6(a), we can observe that the value of d12/λincreases with increasing current and vice versa. This can be explained by the nonlinearblue-shift for increasing currents, which can be observed inFig. 3for a single STNO. The red arrow in the bottom panel of Fig. 6(a) indicates that the difference of d 12/λbetween two subsequent switching events is approximately 0 .6. This is consistent with the distance of λ/2 between the beginning of the (I) and (A) regions in Fig. 5. Thus, for example, the switching event 1/circlecopyrtoccurs because the system is driven by the change of d12/λfrom the bistability region into a region where only antiphase synchronization is possible and theswitching event 2/circlecopyrtoccurs at the border between the bistability region and the in-phase synchronization region. This becomesclearer when we systematically show the current values ofall switching events 1/circlecopyrt–5/circlecopyrt, which are plotted in Fig. 6(b). Similar to Fig. 5we can define different synchronization regions as a function of the current: ( ˜I/˜A) represents a bistability region for in-phase and antiphase synchronization,(˜I) corresponds to only in-phase synchronization, and in the region ( ˜A) only antiphase synchronization is possible. The switching events occur only when the system enters the regions(˜I) or ( ˜A) from ( ˜I/˜A). This also explains why the switching events 4/circlecopyrtand 5/circlecopyrtduring the down-ramping phase take place for other currents than the respective switching events 1/circlecopyrt–3/circlecopyrtduring the up-ramping phase of the current cycling protocol. In summary, the proposed method represents a robust switching scheme for transitions between in-phase and an-tiphase synchronization. Furthermore, the adjustment of λ by changing both currents simultaneously could be a goodmethod to demonstrate the dependence of the bistability on theratiod 12/λexperimentally because in this case a qualitatively similar behavior to Fig. 5is observed without the need of varying the intercontact distance. VI. PHASE BISTABILITY IN THE AUTO-OSCILLATOR THEORY In order to clarify the origin of the reported synchronization bistability, we consider a simplified model based on the auto-oscillator theory [ 15,27]. According to Ref. [ 15], the dynamics of a STNO pair can be effectively described by the two coupled 054414-6SPIN-WA VE-MEDIATED MUTUAL SYNCHRONIZATION OF . . . PHYSICAL REVIEW B 90, 054414 (2014) nonlinear ordinary differential equations (ODEs) dc1 dt+iω1(|c1|2)c1+/Gamma11(|c1|2,I1)c1=/Omega112exp(iβ12)c2, (4) dc2 dt+iω2(|c2|2)c2+/Gamma12(|c2|2,I2)c2=/Omega121exp (iβ21)c1, (5) where cj(t),j=1,2 are complex amplitudes for the oscillation in the jth STNO. The functions ωj(|cj|2) describe the nonlinear dependence of the STNOs’ frequency on the os-cillation power p j=|cj|2./Gamma1j(|cj|2,Ij) are effective damping functions, which contain information about the positive Gilbertdamping, the negative damping introduced by the spin-transfer torque, and an effective positive damping which is caused by the spin waves, which radiate out of the contact region [ 13]. The mutual coupling through the emission of spin waves isdescribed by the external forces F ij=/Omega1ijexp (iβij)cjwhich act on the oscillator i, where /Omega1ijis the coupling strength and βijis a time-delay-induced coupling phase [ 15]. We developed an approach in order to extract the functions ωj(|cj|2),/Gamma1i(|cj|2,Ij) from the micromagnetic simulation of a single STNO for an out-of-plane external field with θ=90◦. With the ansatz cj=√pjexp [iφj(t)], the dynamics of the oscillator power is described by dpj/dt=−2/Gamma1j(|cj|2,Ij)pj. Forθ=90◦, the instantaneous power pjcan be easily related to the normalized magnetization m=M/M 0bypj=(1− mz)/2[15,45]. In this, case we can extract the function /Gamma1j(|cj|2,Ij) by performing micromagnetic simulations and calculate the value of −(dpj/dt)/2pjfor different currents and oscillation powers. It has to be noted that during the auto-oscillation with the stationary power pj(t)=p0,j=const, it is not possible to get any information about the damping function, because bydefinition the various damping terms compensate each other inthis case, resulting in /Gamma1 j(p0,j,Ij)=0. Therefore, the damping function can only be extracted during the transient to thelimit cycle. Thus, we initialize the micromagnetic simulationatp j=0 with Ij=0 and apply instantaneously a current Ij. The magnetization trajectory for Ij=14 mA following from this simulation is shown in the inset of Fig. 7.W e extract the function /Gamma11(|c1|2,I1) for powers close to the limit cycle for simulations with different currents in the rangeI j=13 mA ,..., 15 mA as shown in Fig. 7. In this case the simulations show that the damping function can be expressedas a linear function in p j(red lines in Fig. 7): /Gamma1j(pj,Ij)=˜/Gamma1(pj−p0,j),p 0,j=˜p+˜σIj. (6) For the constants in the damping functions we find ˜/Gamma1/2π= 16.59 GHz, ˜p=−0.139, and ˜ σ=0.02078 /mA. The fre- quency of the auto-oscillation ωj(p0,j) has been calculated analogously to Sec. III. It can be expressed as ωj(pj)=ωon+Np, (7) where ωon/2π=10.26 GHz is the oscillation frequency at the onset current and N/2π=55.59 GHz is the nonlinear frequency shift. With these fitted parameters for the single oscillators we analyze the synchronization of two identical oscillators with-3-2-101 0.05 0.1 0.15 0.2 0.25 0.3−(dpj/dt)/(2pj)/2π(GHz) pjΓj(pj,Ij) limit cycledpj/dt=0 Ij=1 3m Ap0,j Ij=1 5m A Ij=1 4m A-0.500.5 -1 -0.5 0 0.5 1my mxIj=1 4m A pj=p0,jlimit cycle pj=0 FIG. 7. (Color online) Procedure to extract the damping function /Gamma1j(|cj|2,Ij) from the micromagnetic simulation. The dots show the values of −(dpj/dt)/2pj, which have been calculated from the magnetization trajectories obtained by micromagnetic simulations. The red lines are the fitted linear functions /Gamma1j(pj,Ij) as described in the main text. The inset shows the magnetization trajectory for asimulation with I j=14 mA. different currents. We consider two different models for the coupling phases βin the differential equations ( 4) and ( 5). In model (I) we set the coupling phase to β12=β21=β= 2πd12/λ0, where λ0is the wavelength of the spin waves which are excited in a single free-running oscillator [ 15]. For the chosen parameters with an out-of-plane external field, thevalue of the wavelength is given according to the analyticaldispersion relation [ 44]b yλ 0=83.5 nm. In model (II) we take into account the dynamical change of the oscillation frequencyduring the synchronization, which was observed in Sec. IV. Therefore, the coupling phases are set to β ij=2πdij λj, (8) where λjis the dynamical wavelength for the spin waves which are generated by the jth oscillator. In order to obtain a simple analytical formula for the dynamical wavelength, we expandthe dispersion relation to second order in kcorresponding to ω=ω 0+Dk2. The coefficient Dcan be expressed [ 27]a s D=λ2 exωM ω0/parenleftbigg ωH+1 2cos2(θ)ωM/parenrightbigg , (9) with the exchange length λ2 ex=2A/(μ0M2 0),ωH=γμ 0Hint, andωM=γμ 0M0, where ω0is the ferromagnetic reso- nance frequency and Hint=H0−Mzthe internal field. With this dispersion formula the dynamic wavelength can beexpressed as λ j=2π/radicalBigg D |˙φj|−ω0. (10) Rewriting the system of ODEs ( 4) and ( 5) in terms of phases φ1,φ2and powers p1,p2, the dynamical coupling phase can be directly included into the coupling force, leading to asystem of four coupled implicit ODEs, which can be solvednumerically. Figure 8shows the stationary phase difference /Psi1 0 as a function of the intercontact distance. We choose the initial condition /Psi1(t=0)=0 and a fixed phenomenological value for the symmetrical coupling strength /Omega112/2π=/Omega121/2π= 054414-7T. KENDZIORCZYK, S. O. DEMOKRITOV , AND T. KUHN PHYSICAL REVIEW B 90, 054414 (2014) 400 450 500 distance (nm)-2-1012ΔI(mA) −π/20π/2π3π/2no sync Ψ0(rad)model (II) 2 n o syn c I/A I/A A AI I I/A FIG. 8. (Color online) Stationary phase difference /Psi10as a func- tion of the intercontact distance and the current difference /Delta1Ifor the models (I) and (II), as described in the main text. I and A denote regions of in-phase and antiphase synchronization, respectively. The dashed lines are guides for the eye and indicate the borders of the regions of multistability (I/A). 200 MHz. The results for model (II) are in good qualitative agreement with the results from the micromagnetic simulation,despite the fact that in our simplified model /Omega1 ijhas no dependence of d12. We observe three different regions with in-phase, antiphase, and in-phase/antiphase bistability. For thesimpler model (I) with a fixed wavelength we do not find anyregions with bistability. According to the analytical theory for mutual synchro- nization of auto-oscillators [ 15], the stability of the in-phase and antiphase locked solution depends on the time delay,respectively, the coupling phase βaccording to /Psi1 0=/braceleftbigg arcsin (δ//Delta1), cos [β−arctan ( ν)]>0, π−arcsin ( δ//Delta1),cos [β−arctan ( ν)]<0,(11) where δis the difference of the free-running frequencies for STNO 1 and STNO 2, /Delta1is the phase-locking bandwidth, i.e., the maximum δwhere phase locking is possible, and ν= dω(p) dp/d/Gamma1(p,I) dpis the normalized frequency nonlinearity, which only gives rise to a constant offset in the coupling phase β. Our numerical results for model (I) are consistent with Eq. (11). Thus, for /Delta1I=0 the in-phase and antiphase synchro-nized states alternatively exhibit a transcritical bifurcation, i.e., they interchange their stability while always one state remainsstable. The distance between two such bifurcations is given byλ 0/2. The presented semianalytical model gives insight in the physical mechanism underlying the bistability for the phaselocking of two STNO. Due to the fact that the bistabilityis only present in model (II), we can conclude that themechanism relies on two effects which are necessary toprovide the dynamical change of the coupling phase: (i) thesynchronization must contain a time delay to allow dynamialchange of the synchronization frequency on the coupling forceand (ii) the explicit dynamical change of the wavelength dueto (i) in accordance with the spin-wave dispersion is crucial. VII. CONCLUSION In conclusion, we have performed a detailed micromag- netic study of two synchronized spin-torque nano-oscillators(STNOs) which are coupled by the emission of spin waves.The simulation results show a bistability in the stationaryphase difference of the oscillators corresponding to in-phaseand antiphase synchronization. We showed how this bistabilitycan be avoided and we presented switching schemes to inducetransitions between the in-phase and antiphase synchroniza-tion of two STNOs. Finally, by applying a modified couplingterm to the auto-oscillator theory, we reproduced these resultsin this simplified model and identified the physical origin ofthe bistability, which is the time delay due to the finite groupvelocity of spin waves. On the one hand, the time delay causesa change of the oscillation frequency for a phase-locked STNOpair, which on the other hand influences the wavelength of thespin waves in accordance to the dispersion relation and leads toa modified coupling phase of the spin waves. We observe onlymultistable synchronization, if these two effects are explicitlyincluded in the auto-oscillator model. Due to the large qualitative impact of the multistability on the synchronization dynamics, we think that this effect has tobe considered in the design of larger STNO array. 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