# Controlling 4 f antiferromagnetic dynamics via itinerant electronic susceptibility

Sang-Eun Lee, Yoav William Windsor, Daniela Zahn, Alexej Kraiker, Kurt Kummer, Kristin Kliemt, Cornelius Krellner, Christian Schüßler-Langeheine, Niko Pontius, Urs Staub, et al.

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Sang-Eun Lee, Yoav William Windsor, Daniela Zahn, Alexej Kraiker, Kurt Kummer, et al., Controlling 4 f antiferromagnetic dynamics via itinerant electronic susceptibility. Physical Review Research, 2024, 6 (4), pp.043019. 10.1103/PhysRevResearch.6.043019. hal-05098764

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Distributed under a Creative Commons Attribution 4.0 International License
Controlling 4f antiferromagnetic dynamics via itinerant electronic susceptibility

Sang-Eun Lee,¹,²,* Yoav William Windsor,¹ Daniela Zahn,¹ Alexej Kraiker,³ Kurt Kummer,⁴ Kristin Kliemt,³

Cornelius Krellner,³ Christian Schüßler-Langeheine,⁵ Niko Pontius,⁵ Urs Staub,⁶ Denis V. Vyalikh,⁷,⁸

Arthur Ernst,⁹,¹⁰,† and Laurenz Rettig,¹,‡

¹Department of Physical Chemistry, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

²Center of Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

³Physikalisches Institut, Goethe-Universität Frankfurt, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany

⁴European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex, France

⁵Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Albert-Einstein-Strasse 15, 12489 Berlin, Germany

⁶Swiss Light Source, Paul Scherrer Institut, Forschungsstrasse 111, 5232 Villigen PSI, Switzerland

⁷Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia/San Sebastián, Basque Country, Spain

⁸IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain

⁹Institute for Theoretical Physics, Johannes Kepler University, Altenberger Strasse 69, 4040 Linz, Austria

¹⁰Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle (Saale), Germany

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(Received 15 April 2024; accepted 23 August 2024; published 4 October 2024)

Optical manipulation of magnetism holds promise for future ultrafast spintronics, especially with lanthanides and their huge, localized 4f magnetic moments. These 4f moments interact indirectly by spin polarizing the conduction electrons (the Ruderman-Kittel-Kasuya-Yosida exchange interaction), influenced by interatomic orbital overlap, and the conduction electron's susceptibility around the Fermi level. Here, we study this influence in a series of 4f antiferromagnets, $GdT_2Si_2$ ($T$ = Co, Rh, Ir), using ultrafast resonant x-ray diffraction. We observe a twofold increase in the ultrafast intersublattice angular momentum transfer rate between the materials, originating from modifications in the conduction electron susceptibility, as confirmed by first-principles calculations.

DOI: 10.1103/PhysRevResearch.6.043019

## I. INTRODUCTION

Lanthanides are becoming increasingly important in technology due to their exceptionally large magnetic moments. Since the majority of the magnetic moments of lanthanides resides in spatially localized 4f electron shells [1,2], magnetic moments from different lanthanide atoms can only interact indirectly by spin-polarizing itinerant conduction electrons that surround the 4f moments. This indirect interaction is called the Rudermann-Kittel-Kasuya-Yosida (RKKY) exchange interaction [3]. As its mechanism implies, not only the localized 4f moments but also the itinerant conduction electrons play an important role in determining the strength of the RKKY interaction $J_{RKKY}$ in lanthanide-based magnetic materials. $J_{RKKY}$ is proportional to the squared overlap integral between the 4f and conduction electrons, $|I|^2$, and the susceptibility $\chi$ of the

conduction electrons' spin polarization around the Fermi level to an effective magnetic field formed by 4f magnetic moments ($J_{RKKY} \propto |I|^2\chi$) [3].

The ultrafast dynamics of magnetic devices after femtosecond laser excitation is governed by the transfer speed of angular momentum between its microscopic subsystems. While the investigation of such ultrafast spin dynamics has been employed to study these interactions for several decades [4,5], and also numerous studies of lanthanide magnetism have been reported [6–14], the influence of the individual contributions of $J_{RKKY}$ to ultrafast spin dynamics has remained mostly elusive. In particular, whereas the role of the localized 4f magnetic moments has been extensively studied mostly in the heavy trivalent lanthanides (Gd-Tm) [6,12,13], the role of the itinerant conduction electrons in lanthanide magnetism has not been systematically investigated so far since many elemental lanthanides share very similar conduction electron structures, making it difficult to isolate their specific role in ultrafast magnetization dynamics. However, due to their central role in the RKKY interaction, a systematic investigation of the influence of the conduction electron properties on the magnetization dynamics in lanthanide-based compounds is of strong interest.

For this purpose, we investigated the ultrafast magnetization dynamics in a series of 4f antiferromagnetic (AF) compounds $LnT_2Si_2$ [Ln: lanthanides; T: transition metals; see Fig. 1(a)], which share almost identical magnetic and

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. Open access publication funded by Max Planck Society.

<footnote>*Contact author: sang-eun@lanl.gov</footnote>

<footnote>†Contact author: arthur.ernst@jku.at</footnote>

<footnote>‡Contact author: rettig@fhi-berlin.mpg.de</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 1. (a) Crystal structure of GdT₂Si₂ (T = Co, Rh, and Ir). J₁, J₂ are exchange coupling between the nearest and the next-nearest in-plane 4f moments, and J₃ is the exchange coupling between the nearest out-of-plane 4f moments. (b) Sketch of the experimental setup. Top: Pump-induced suppression of the (001) magnetic diffraction peak of GdRh₂Si₂.</figcaption></figure>

lattice structures [15,16]. Using this similarity, we recently investigated the role of the 4f moments on the ultrafast spin dynamics of LnRh₂Si₂ by varying the lanthanide Ln [14], which demonstrated that the direct spin transfer between antiferromagnetically coupled 4f moments is an essential demagnetization channel during ultrafast spin dynamics scaling with the strength of the RKKY interaction. In this study, in a similar approach we systematically vary the nonmagnetic transition metal T occupation within GdT₂Si₂ from 3d to 5d (Co, Rh, Ir), and single out the influence on the RKKY interaction and the ultrafast spin dynamics in these 4f antiferromagnets. Surprisingly, we find a nonmonotonous variation of angular momentum transfer rates with d-shell occupation, with GdRh₂Si₂ showing larger transfer rates than the other two compounds. Using *ab initio* calculations, we explain this behavior by the variation of the conduction electron susceptibility χ due to a competition of the T-ion d-orbital extension and the d-level energy splitting.

## II. METHODS

### A. Experiments

The family of intermetallics GdT₂Si₂ (T = Co 3d, Rh 4d, and Ir 5d) crystallizes in the tetragonal ThCr₂Si₂ structure (a = b ~ 4 Å, c ~ 10 Å) and are A-type antiferromagnets, where antiferromagnetically ordered Gd ions are separated by T₂Si₂ blocks along the c axis [Fig. 1(a)] [16,17]. The sample growth condition and characterization are elaborated in Appendix A. The Néel temperatures T_N of the three samples are 45 K (Co), 107 K (Rh), and 85 K (Ir) [11,18]. In GdT₂Si₂, the magnetic moments are carried by the seven unpaired Gd 4f electrons, while the conduction electrons including the T states are only indirectly spin polarized [15,19].

Employing resonant soft x-ray diffraction (RXD), we measured the resonantly enhanced [0 0 L] magnetic diffraction intensity, sensitive to long-range AF ordering of Gd 4f moments along the c axis [Fig. 1(b)]. The samples were characterized at the RESOXS end station of the SIM beam line of the Swiss Light Source at the Paul Scherrer Institute, Switzerland, and the RIXS end station of the ID32 beam line

of the European Synchrotron Radiation Facility in Grenoble, France [20,21]. Time-resolved resonant soft x-ray diffraction (trRXD) experiments were performed at the FemtoSpeX beam line UE56/1-ZPM of BESSY II of the Helmholtz-Zentrum Berlin, Germany, which uses femtosecond slicing to provide ultrashort soft x-ray pulses [22]. We used 50-fs-long laser pulses centered at 1.55 eV, at a repetition rate of 3 kHz, to excite the sample and measured the transient diffraction intensity of 100-fs-long sliced soft x-ray pulses centered at the Gd M₅ absorption edge at ~1188 eV with an avalanche photodiode (APD), at a repetition rate of 6 kHz [Fig. 1(b)]. All dynamical experiments were conducted at a temperature of 20 K.

### B. First-principles calculations

First-principles calculations were carried out using a self-consistent Green's function method [23,24] within the density functional theory in a generalized gradient approximation (GGA) [25]. Strongly localized Gd 4f electronic state were treated within a GGA+U method applying U = 6.0 eV [26]. Exchange constants were obtained utilizing the magnetic force theorem, implemented within the multiple scattering theory [27]. Critical temperatures were estimated within a random phase approximation [28].

## III. RESULTS AND DISCUSSION

### A. Characterization of antiferromagnetic GdT₂Si₂

While GdIr₂Si₂ shows a commensurate magnetic diffraction peak at constant L = 1 at all temperatures [Fig. 2(a)] similar to GdRh₂Si₂ [11], GdCo₂Si₂ displays incommensurate magnetic diffraction peaks at L=q and 2-q with q ~ 0.966 at 20 K, which shift with temperature [Fig. 2(b)]. Since the two incommensurate peaks exhibit an almost identical temperature and photon-energy dependence, we concentrate on the (0 0 q) peak in this study.

The magnetic diffraction intensity of the three samples exhibits almost identical temperature dependencies following a mean-field-like behavior (Gd, S = 7/2) [Fig. 2(c)]. The similar photon-energy dependence of resonant enhancement at the Gd M₅ absorption edge (hv ~ 1188 eV; 3d → 4f) demonstrates their similar orbital and magnetic configuration [Fig. 2(d)], and leads to very similar penetration depths of ~4 nm at resonance, corresponding to the x-ray light probing ~4 unit cells of GdT₂Si₂ [11,14,29]. These similarities in both temperature and photon-energy dependence demonstrate their similar long-range 4f antiferromagnetism, justifying the following comparative analysis of ultrafast spin dynamics.

### B. Ultrafast dynamics of long-range 4f antiferromagnetic order in GdT₂Si₂

The femtosecond dynamics of the AF order parameter, given by the square-root of the (0 0 L) magnetic diffraction peak amplitude are shown for selected fluences in Fig. 3(a). The dynamics at all measured pump fluences discussed in this study are presented in Appendix B. In the case of GdRh₂Si₂, the peak amplitudes have been separated from a transient reorientation of the magnetic in-plane
<figure><img src="image_4.png" /><figcaption>FIG. 2. Temperature-dependent behavior of the magnetic diffraction intensity of (a) GdIr$_{2}$Si$_{2}$ and (b) GdCo$_{2}$Si$_{2}$. (c) Equilibrium temperature-dependent behavior of the magnetic diffraction amplitude of GdT$_{2}$Si$_{2}$ (symbols). The dashed line is a mean-field curve for $S = 7/2$ corresponding to Gd [1]. Bold solid symbols indicate the estimated staggered magnetization of each sample at 20 K, the base temperature of the dynamic measurements. (d) Photon-energy-dependent resonant amplification of the magnetic diffraction intensity of GdT$_{2}$Si$_{2}$ (symbols). The vertical dashed line indicates the photon energy that we chose for the rest of this study (1187.5 eV).</figcaption></figure>

easy axis using a procedure that combines several azimuthal orientations [11]. To account for pump-induced transient peak shifts, the peak amplitudes for GdCo$_{2}$Si$_{2}$ have been corrected by considering the transient peak position evolution of the (0 0 L) diffraction peak, as detailed in Appendix C.

For a quantitative comparison of the demagnetization dynamics of the three materials, we modeled the demagnetization curves using a phenomenological exponential decaying functions describing different timescales,

$$A(t) = 1 - \Theta(t, t_0) \sum_{i}^{f,s} A_i (1 - e^{-(t-t_0)/\tau_i}), \quad (1)$$

where $A_f$, $A_s$, and $\tau_f$, $\tau_s$ are the amplitude and the time constant of the fast (~1 ps) and slow (~30–100 ps) processes, respectively. $t_0$ corresponds to the temporal overlap of pump and probe pulses, and $\Theta(t, t_0)$ is the Heaviside function. The observed demagnetization timescales of the three compounds are comparable with the dynamics in other antiferro- and ferromagnetic 4f compounds [6,10,14].

Figure 3(b) presents the demagnetization amplitude $\Delta m = A_f + A_s$ as functions of fluence normalized to the critical fluence $F_C$, defined for each material as the fluence necessary to induce 50% demagnetization [14] (Details of the absorbed fluence estimation are provided in Appendix D.) We find values for $F_C$ of 0.60 (Co), 1.74 (Rh), and 1.36 (Ir) mJ/cm$^2$, respectively, which surprisingly do not follow the d-shell occupation of the T ions, but instead show a linear relationship to $T_N$ [inset of Fig. 3(b)]. A similar relation with a comparable slope has also been observed in LnRh$_{2}$Si$_{2}$ (Ln=Pr-Ho) [14]. This scaling relation between the critical fluence $F_C$ and the Néel

<figure><img src="image_5.png" /><figcaption>FIG. 3. (a) Ultrafast dynamics of the normalized (0 0 L) magnetic diffraction amplitude of GdT$_{2}$Si$_{2}$ ($T = \text{Co}$, Rh, and Ir) at a selected pump fluence acquired at a constant momentum transfer $Q$. Error bars denote uncertainties from Poisson statistics. Solid lines are exponential decaying functions for phenomenological description (see text). (b) Demagnetization amplitude of the three samples plotted along the fluence $F$ normalized by the critical fluence $F_C$ (see text) of each sample. Inset: The relation between the critical fluence and the Néel temperature of each sample. (c) Normalized slow demagnetization time constants $\tau_f/\tau_c$ of the three samples plotted along the normalized fluence $F/F_C$. Inset: Time constants $\tau_c$ at the critical fluence used for normalization.</figcaption></figure>

temperature $T_N$ implies that the relevant magnetic interactions follow a classical mean-field-like behavior [14], supporting the validity of our comparative analysis of the ultrafast spin dynamics of GdT$_{2}$Si$_{2}$.

The normalized time constants of the slow demagnetization process $\tau_s$, which is present in all materials, are shown in Fig. 3(c). They exhibit a qualitatively similar square-root-like behavior with $F/F_C$, albeit the absolute timescales differ substantially [see the inset of Fig. 3(c)], similar to our previous study [14]. Thus, while all of the studied materials exhibit qualitatively similar demagnetization behavior, there are also notable differences. Whereas GdRh$_{2}$Si$_{2}$ and GdIr$_{2}$Si$_{2}$ display a two-step decay in their demagnetization dynamics ($\tau_f \sim 1$ ps, $\tau_s > 10$ ps), followed by a slow recovery after ~100 ps, $\tau_f$ is almost absent in GdCo$_{2}$Si$_{2}$. Furthermore, similar to the critical fluences, we find substantial differences in the demagnetization rate, following the same sequence (GdRh$_{2}$Si$_{2}$ and GdCo$_{2}$Si$_{2}$ exhibiting the fastest and the slowest dynamics, respectively).

For accurate comparison of the demagnetization rates, the ultrafast angular momentum transfer rate $\alpha = m_{\text{stag}}\mu_{4f}A_s/\tau_s$ is calculated for each material [14], where $\mu_{4f}=7\mu_B$ is the size of the Gd 4f moments [3]. The staggered magnetization $m_{\text{stag}}$ of each sample at 20 K is indicated in Fig. 1(c) with bold solid markers. The fluence-dependent behavior of the
<figure><img src="image_6.png" /></figure>

FIG. 4. (a) Energy distribution of calculated spin polarization of Gd 5d states of GdT<sub>2</sub>Si<sub>2</sub> in the vicinity of the Fermi level E<sub>F</sub>. (b) Calculated indirect RKKY exchange interaction between the nearest in-plane 4f moments J<sub>1</sub> (closed markers, dashed line, right axis) and the nearest out-of-plane antiferromagnetically coupled 4f moments J<sub>3</sub> (open markers, solid line, left axis) plotted along the calculated Gd 5d spin polarization. (c) Angular momentum transfer rate at F<sub>C</sub> interpolated from fluence dependence (see Appendix B) plotted along the Gd 5d spin polarization of the three samples. The gray line is a guide to the eyes. Error bars are derived from error propagation of the demagnetization time constant and amplitude of each compound.

angular momentum transfer rates is shown in Appendix B. Similarly to the inverse time constants and critical fluences, GdRh<sub>2</sub>Si<sub>2</sub> (4d) has the largest α, followed by GdIr<sub>2</sub>Si<sub>2</sub> (5d) and GdCo<sub>2</sub>Si<sub>2</sub> (3d). Although they share the same Gd 4f moments, the angular momentum transfer rate of the GdT<sub>2</sub>Si<sub>2</sub> series varies by ~100% (Fig 4(c)).

### C. Scaling of the angular momentum transfer rate with Gd 5d spin polarization around the Fermi level

In order to understand the reason for the varying ultrafast angular momentum transfer rate of the GdT<sub>2</sub>Si<sub>2</sub> series, we calculated exchange coupling constants and electronic densities of states (eDOS) of GdT<sub>2</sub>Si<sub>2</sub> employing density functional theory (DFT) (see Sec. II B for details). According to the DFT calculations, in GdT<sub>2</sub>Si<sub>2</sub>, the conduction electrons are mostly composed of 5d electrons. Thus, the interaction between the local magnetic moments is mediated predominantly via spin-polarized Gd 5d states. As discussed in Sec. I, the strength of the RKKY interaction J<sub>RKKY</sub> is determined by the overlap integral between 4f and conduction electrons and the nonlocal susceptibility of the conduction electrons' spin polarization around the Fermi level E<sub>F</sub>. Since all the studied compounds share the same local magnetic moments and Gd 5d electrons, the overlap integral factor does not vary much among the GdT<sub>2</sub>Si<sub>2</sub> series.

In contrast, the calculations show that the spin polarization of Gd 5d electrons around E<sub>F</sub>, i.e., the difference in eDOS of majority- and minority-spin states, varies substantially between the different compounds [Fig. 4(a)]. At the Fermi level, GdRh<sub>2</sub>Si<sub>2</sub> has the largest spin polarization, followed by GdIr<sub>2</sub>Si<sub>2</sub> and GdCo<sub>2</sub>Si<sub>2</sub>. As shown in Fig. 4(b), the spin polarization at E<sub>F</sub> also directly correlates with the strength of the calculated RKKY interaction between the nearest in-plane and out-of-plane Gd 4f moments [J<sub>1</sub>, J<sub>3</sub>, respectively, in Fig. 1(a)]. This varying spin polarization implies that the transition metal ions modify the eDOS of both majority- and minority-spin states and thus the nonlocal susceptibility of the conduction electrons' spin polarization in the vicinity of the Fermi level. At the same time, we also find a clear scaling relation of the experimental angular momentum transfer rate with the Gd 5d spin polarization at E<sub>F</sub> [Fig. 4(c)]. As shown for the LnRh<sub>2</sub>Si<sub>2</sub> series [14], the ultrafast angular momentum transfer rate in this series of compounds scales with the strength of the RKKY interaction. Therefore, the observed scaling relation with the Gd 5d spin polarization [Fig. 4(c)] reflects the influence of the conduction electrons' susceptibility on the ultrafast spin dynamics of the GdT<sub>2</sub>Si<sub>2</sub> series due to varying nonmagnetic T ions.

Besides the antiferromagnetic interplane coupling J<sub>3</sub>, the in-plane coupling constants J<sub>1</sub>, which are responsible for establishing in-plane ferromagnetic order, also vary among the series [Fig. 4(b)]. Among the antiferromagnetic LnT<sub>2</sub>Si<sub>2</sub> compounds, GdT<sub>2</sub>Si<sub>2</sub> exhibit the strongest intersublattice angular momentum transfer between the antiparallel Gd 4f spins [14]. Thus, phonon-mediated Elliott-Yafet spin-flip scattering processes within the planes due to J<sub>1</sub> are expected to contribute only weakly. Therefore, even though the in-plane exchange couplings vary with the T ions, the general conclusion that we yield from the antiferromagnetic dynamics does not change substantially.

### D. Two competing factors affecting the Gd 5d spin polarization around E<sub>F</sub>

The behavior of the 5d electrons in GdT<sub>2</sub>Si<sub>2</sub> explored in our calculations can be explained by two important factors. The first important factor is the extension of the transition metal wave functions. These orbitals show an increasing degree of delocalization when going from the 3d to the 5d shell, with GdCo<sub>2</sub>Si<sub>2</sub> showing the strongest localization (see the regions highlighted in magenta in Fig. 5). Therefore, the hybridization between the Si and Co states is much weaker than the hybridization between the Si and Ir/Rh states in GdIr<sub>2</sub>Si<sub>2</sub> and GdRh<sub>2</sub>Si<sub>2</sub>. Consequently, with increasing delocalization of T d orbitals along the series, the vacant Si valence electrons hybridize less with the Gd 5d states, increasing the 5d eDOS and hence the spin polarization at the Fermi level [Fig. 6(a)]. The second factor influencing the 5d DOS is the bonding/antibonding splitting of T d states, and their distance to E<sub>F</sub>. Here, the distance of the antibonding states from E<sub>F</sub> increases from Co to Ir (see the regions highlighted in yellow in Fig. 5), leading to a reduction of the eDOS near E<sub>F</sub> along the series [Fig. 6(b)], which in consequence decreases the spin polarization near E<sub>F</sub> as well. Combined with the first factor,
<figure><img src="image_7.png" /><figcaption>FIG. 5. Total and element-resolved densities of states (DOS) of GdT<sub>2</sub>Si<sub>2</sub>: T= Co (left panel), Rh (middle), Ir (right panel). In the element-resolved DOS, only the most important orbital contributions are shown. The shaded area in the total and transition metal (T) d-orbital DOS highlights localization of T d-wave functions (magenta) and splitting of the bonding and antibonding states around the Fermi level (yellow), which are discussed in the main text.</figcaption></figure>

this explains the observed behavior of the largest eDOS and angular momentum transfer in GdRh<sub>2</sub>Si<sub>2</sub> [Fig. 6(c)].

In addition, the particular crystalline structure of GdT<sub>2</sub>Si<sub>2</sub> supports this trend. As shown in Ref. [30], the magnetic properties of the lanthanide compounds are highly sensitive

<figure><img src="image_8.png" /><figcaption>FIG. 6. Cartoon summarizing the factors contributing to the scaling of the ultrafast angular momentum transfer rate of GdT<sub>2</sub>Si<sub>2</sub> with respect to the transition metal (T) ions, 3d Co, 4d Rh, and 5d Ir. (a) The extension of T d orbitals is reflected by their calculated bandwidth (Fig. 5, purple areas). (b) The distance between the bonding/antibonding orbitals with respect to the Fermi level is reflected by the energy splitting of the calculated unoccupied antibonding T d and occupied T d states (Fig. 5, yellow areas). (c) The combined effect of the trends in (a) and (b) leads to the observed behavior of spin susceptibility and angular momentum transfer rates.</figcaption></figure>

to changes in unit cell volumes: a reduction of the unit cell volume leads to a reduction of 5d eDOS at the Fermi level, thereby modifying the magnetic interaction in the system. In our case, the variation of the unit cell volume (GdCo<sub>2</sub>Si<sub>2</sub>: 150.0 Å<sup>3</sup> [17] < GdIr<sub>2</sub>Si<sub>2</sub>: 156.0 Å<sup>3</sup> [16] < GdRh<sub>2</sub>Si<sub>2</sub>: 162.9 Å<sup>3</sup> [15]) reflects the changes in eDOS and spin polarization at E<sub>F</sub>.

Finally, a comparison to semimetallic ferromagnets appears interesting. While, in these systems, enhanced spin polarization accompanied by reduced band overlap and DOS results in decreased demagnetization rates by blocking either phonon-mediated spin-flip scattering or diffusive spin transport [31–33], our study found the opposite behavior for antiferromagnetic intersublattice angular momentum transfer due to enhanced RKKY coupling for larger DOS and spin polarization. This observation underlines the importance of RKKY-driven intersublattice angular momentum transfer for the ultrafast spin dynamics in these 4f antiferromagnetic systems.

## IV. CONCLUSIONS

In summary, we investigated the role of the itinerant conduction electrons in ultrafast spin dynamics of 4f antiferromagnets. By substituting the T ions in GdT<sub>2</sub>Si<sub>2</sub> (T= Co, Rh, Ir), we selectively modified their conduction electron susceptibility and measured the femtosecond dynamics of magnetic diffraction intensity at various pump fluences employing time-resolved resonant magnetic soft x-ray diffraction. While we found qualitatively similar demagnetization behavior upon optical excitation at 1.55 eV, the observed critical fluences and ultrafast angular momentum transfer rates α vary drastically, and nonmonotonously with T orbital shell (α: GdRh<sub>2</sub>Si<sub>2</sub> > GdIr<sub>2</sub>Si<sub>2</sub> > GdCo<sub>2</sub>Si<sub>2</sub>). First-principles calculations of the electronic density of states and exchange coupling constants of GdT<sub>2</sub>Si<sub>2</sub> employing density functional theory show that the spin polarization of Gd 5d electrons scales with the in-plane and out-of-plane nearest-neighbor exchange coupling constants, and with the experimental angular momentum transfer rate. This implies that varying the T ions modifies the non-local susceptibility of conduction electrons’ spin polarization and, hence, the strength of the RKKY interaction. We explain this effect by a combination of d-orbital wave-function localization and bonding/antibonding splitting of T d states, modifying the electronic density of states around the Fermi level and their nonlocal susceptibility. Our findings provide important insights for designing lanthanide-based magnetic devices, showing how a modification of the itinerant conduction electrons, which could, e.g., be implemented by chemical or electrostatic doping, impacts ultrafast angular momentum transfer processes.

The data that support the findings of this article are openly available [34].

## ACKNOWLEDGMENTS

We thank the Helmholtz-Zentrum Berlin für Materialien und Energie for the allocation of synchrotron radiation beam time. The experimental support of the staff at beam lines UE56/1 (HZB), X11MA (SLS), and PM3 (HZB) is gratefully
acknowledged. This work received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the Emmy Noether program (Grant No. RE 3977/1), within the Transregio TRR 227 - 328545488 Ultrafast Spin Dynamics (Projects No. A03, No. B07 and No. A10), within TRR 288 - 422213477 (Project No. A03), SFB1143 (Grant No. 247310070), and within Grant GU 2228/2-1 (Project No. 512344413). Funding was also received from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant Agreement No. ERC-2015-CoG-682843). We also are thankful for support from the Spanish Ministry of Science and Innovation, Project No. PID2020-116093RB-C44, funded by MCIN/AEI/10.13039/501100011033. We acknowledge funding by Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Grant No. I 5384. We acknowledge support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Triad National Security, LLC ("Triad") contract Grant No. 89233218CNA000001 FWP: LANLE1NT and the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy Office of Science. Los Alamos National Laboratory, an affirmative action equal opportunity employer, is managed by Triad National Security, LLC for the U.S. DOE NNSA, under Contract No. 89233218CNA000001. Calculations were carried out at the Rechenzentrum Garching of the Max-Planck Society. We acknowledge the European Synchrotron Radiation Facility (ESRF) for provision of synchrotron radiation facilities under Proposal No. IH-HC-3815.

## APPENDIX A: SAMPLE GROWTH AND CHARACTERIZATION

The GdRh₂Si₂ and GdIr₂Si₂ single crystals were obtained according to the procedure described earlier [15,16]. For the growth of GdCo₂Si₂ single crystals, high-purity starting materials were weighed in a graphite crucible and sealed in a niobium crucible under argon atmosphere. The stoichiometric mixture of the elements was used with the ratio 1 : 2 : 2 : 24 (Gd : Co : Si : In) with indium as flux. The growth was performed as described in Ref. [16] with a maximum temperature of the furnace of 1550 °C.

Powder x-ray diffraction yielded lattice parameters of a = b = 3.911 Å and c = 9.803 Å, which are in good agreement with the literature [35]. Energy dispersive x-ray spectroscopy (EDX) revealed the stoichiometry of Gd : Co : Si = 20 ± 1 : 39 ± 2 : 41 ± 2, which is in good agreement with the 122 target stoichiometry within the experimental error. The orientation of the single crystals was determined using the Laue method.

## APPENDIX B: ULTRAFAST ANTIFERROMAGNETIC ORDER DYNAMICS OF GdT₂Si₂

Ultrafast long-range 4f sublattice magnetization dynamics of GdT₂Si₂ (T = Co, Rh, Ir) at various pump fluences are presented in Figs. 7(a)–7(c). Curves presented in Fig. 2 of the main text are selected from this set of demagnetization curves in Fig. 7. As discussed in the main text, note that the demagnetization dynamics of GdCo₂Si₂ are corrected for a

<figure><img src="image_9.png" /><figcaption>FIG. 7. (a)–(c) Ultrafast antiferromagnetic order dynamics of GdT₂Si₂ (T = Co, Rh, Ir) at various pump fluences. Solid lines are a phenomenological description of the decaying part of the curves using exponentially decaying functions (see main text). Numbers along with each curve indicate the absorbed fluence. (d) Ultrafast angular momentum transfer of GdT₂Si₂ plotted along the normalized fluence (fluence divided by the critical fluence of each material; see main text).</figcaption></figure>

possible transient magnetic diffraction peak shift, which is detailed in Appendix C.

The fluence-dependent behavior of the ultrafast angular momentum transfer rate of GdT₂Si₂ upon optical excitation is plotted in Fig. 7(d). While the angular momentum transfer rate increases with the normalized fluence, the systematic difference between the investigated compounds with GdRh₂Si₂ having the largest rate and GdCo₂Si₂ the smallest transfer rate remains consistent.

## APPENDIX C: CORRECTION OF THE DEMAGNETIZATION AMPLITUDE CONSIDERING THE TRANSIENT PEAK SHIFT IN GdCo₂Si₂

Since the delay scans were acquired at constant momentum transfer Q, they may not reflect the correct antiferromagnetic order dynamics due to a transient peak shift in GdCo₂Si₂. In order to correct possible intensity modulation from the peak shift, we measured the transient evolution of the diffraction peak position of GdCo₂Si₂ at selected fluences and at selected delays [Fig. 8(a)]. The delay scans were acquired at the "shoulder" (the vertical dashed line) of the equilibrium peak to minimize such intensity variation. The diffraction peak intensity of GdCo₂Si₂ is modeled with a phenomenological Doniach-Sunjic function for precise description of the asymmetric peak shape,

$$DS(L; \alpha, \Gamma, Q, A) = A \left( \frac{\cos \frac{\pi \alpha}{2} + (1 - \alpha) \tan \frac{L-Q-1}{\Gamma}}{[\Gamma^2 + (L-Q)^2]^{(1-\alpha)/2}} \right), \quad (C1)$$
<figure><img src="image_10.png" /><figcaption>FIG. 8. (a) Transient diffraction intensity evolution of the (0 0 q) reflection of GdCo₂Si₂ upon ultrafast optical excitation at selected delays. The vertical dashed line indicates the diffraction geometry for acquiring delay scans (see text). (b) The ratio between the intensity for probing the delay scan *I*<sub>probe</sub> and the actual maximum intensity of the (0 0 q) diffraction intensity *I*<sub>max</sub> at various pump fluences. Dashed lines indicate the linearly interpolated ratio based on the experimental data points. (c) The raw diffraction intensity dynamics (dashed lines) and the corrected diffraction intensity dynamics (solid lines) at various pump fluences.</figcaption></figure>

where *L* is the position in reciprocal space [0 0 *L*]. *A* is the amplitude of the peak, *Q* is the effective peak position, Γ is the effective peak width, and *α* determines the degree of asymmetry of the peak. For *α* = 0, the equation becomes a symmetric Lorentzian profile, and asymmetry increases as *α* increases towards 1. The modeled intensity at the delay scan acquisition point *I*<sub>probe</sub> (vertical dashed line) is clearly different from the modeled maximum intensity of the peak *I*<sub>max</sub> for most pump-probe delays. The ratio between the two (*I*<sub>probe</sub>/*I*<sub>max</sub>) is plotted in Fig. 8(b). As we acquired the diffraction peak at selected delays, delay points in between the θ − 2θ scans were linearly interpolated. Similarly, the fluence dependence in between measured fluences was linearly interpolated using the delay-interpolated ratio that was constructed in the previous step. The interpolated correction factors shown in Fig. 8(b) were applied to the raw delay scan intensity.

Figure 8(c) presents the corrected delay scans (solid lines) along with the raw delay scans (dashed lines) at selected fluences. As we see, the corrected intensity compensates the intensity loss from the transient peak shift at early and later phases of the dynamics.

APPENDIX D: ESTIMATION OF ABSORBED FLUENCE

All the reported fluences are total absorbed fluences, which
were calculated using the measured incident fluences cor-
rected for reflection and refraction effects. Calculation of
absorbed fluence of GdRh₂Si₂ is reported in Ref. [11]. We

conducted reflectivity measurements of the other two samples
using 1.55 eV light. We estimate the complex index of refrac-
tion at this photon energy as n = n₀ + ik; GdIr₂Si₂: (2.97
± 0.18) + (2.52 ± 0.25)i; GdCo₂Si₂: (4.64 ± 0.46) + (3.77
± 0.68)i. Using the indices of refraction, the total absorbed
fluences were estimated. We estimate the penetration depth of
1.55 eV light to be 15.3 and 26.3 nm at the Bragg angle for
GdCo₂Si₂ and GdIr₂Si₂, respectively.

APPENDIX E: ELECTRONIC STRUCTURE OF GdT$_{2}$Si$_{2}$

To elucidate the nature of the exchange coupling and ob-
served magnetic properties in GdT<sub>2</sub>Si<sub>2</sub> (T= Co 3d, Rh 4d,
Ir 5d), the densities of states were calculated and analyzed
(see Fig. 5). Gd 4f states are localized and located mostly
8 eV below the Fermi level. These states form localized
magnetic moments which interact with each other via the
itinerant conduction electrons (RKKY interaction). Since the
spin polarization of the free electrons is crucial for the RKKY
interaction, mainly Gd 5d and Si 3p states participate in the
magnetic interaction. Si has an induced magnetic moment:
0.08 µ<sub>B</sub>, 0.11 µ<sub>B</sub>, 0.09 µ<sub>B</sub>, for the cases with Co, Rh and Ir,
respectively. Gd 5d electrons carry moments almost two times
larger than Si 3p: 0.18 µ<sub>B</sub>, 0.21 µ<sub>B</sub>, 0.19 µ<sub>B</sub>, respectively.
It is remarkable that the conduction electrons are more spin
polarized in GdRh<sub>2</sub>Si<sub>2</sub> than in the other two compounds and
this is in line with the strength of the calculated J's presented
in the main text and observed Néel temperatures.
The *d* electrons of the transition metals are not spin polar-
ized and therefore do not directly participate in the magnetic
coupling. However, they are crucial for the covalent bonding
in the compounds and are responsible for the formation of
structural and electronic properties. Co 3d states are strongly
localized (mainly between 1 and 4 eV below the Fermi level).
The localized nature of Co 3d electrons in GdCo₂Si₂ is
responsible for the significantly smaller unit cell volume than
in GdRh₂Si₂ and GdIr₂Si₂: both *a* and *c* lattice parameters are
about 5% smaller than in the other two compounds. Despite
the smaller volume, Co 3d states remain to be localized and
hybridize less with Gd 5d and Si 3p states compared to Rh 4d
and Ir 5d states. The latter two are more delocalized: The Rh
4d and Ir 5d bandwidth extends to the bottom of the valence
zone at 6.7 and 7.6 eV below the Fermi level, respectively.
From Fig. 5, one recognizes that Si 3p and Gd 5d electrons
hybridize with the transition metal *d* orbitals only in the lower
part of the valence zone: States below −2 eV belong mainly to
*d*<sup>xz</sup>, *d*<sup>3z<sup>2</sup>−r</sup>, and *d*<sup>yz</sup> symmetries, while close to the Fermi level,
the in-plane *d*<sup>xy</sup>, *d*<sup>x<sup>2</sup>−y<sup>2</sup></sup> symmetries dominate. In the case

of GdCo₂Si₂, Gd 5d states with *d*xy and *dx²-y²* symmetries
are more dispersive since the in-plane lattice constant *a* is
small and this leads to a stronger hybridization within the Gd
layer. In the other two compounds, out-of-plane and in-plane
Gd 5d orbitals are clearly separated. This separation and a
stronger localization of the orbitals result in a larger DOS at
the Fermi level in GdRh₂Si₂ and GdIr₂Si₂, which increases
the strength of the RKKY interaction. However, despite the
more extended Ir 5d orbitals, the DOS at the Fermi level
is larger in GdRh₂Si₂ (and, thereby, the RKKY interaction
is stronger). The main reason for this is the position of the
transition metal antibonding *d* states above the Fermi level: in
GdRh₂Si₂, they are located at 0.5 eV above the Fermi level,
which is about 0.4 eV lower in energy than in GdIr₂Si₂. This
leads to a stronger accumulation of the DOS at the Fermi
level. In the case of GdCo₂Si₂, this argument does not hold
since the strong in-plane hybridization of Gd 5d electrons
makes the states broader and reduces the DOS at the Fermi
level, as discussed above.

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