Spin–Phonon Interactions and Anharmonic Lattice Dynamics in Fe3GeTe2

Raman scattering is performed on Fe3GeTe2 (FGT) at temperatures from 8 to 300 K and under pressures from the ambient pressure to 9.43 GPa. Temperature‐dependent and pressure‐dependent Raman spectra are reported. The results reveal respective anomalous softening and moderate stiffening of the two Raman active modes as a result of the increase of pressure. The anomalous softening suggests anharmonic phonon dynamics and strong spin–phonon coupling. Pressure‐dependent density functional theory and phonon calculations are conducted and used to study the magnetic properties of FGT and assign the observed Raman modes, E2g2$E_{2{\rm{g}}}^2$ and A1g1$A_{1{\rm{g}}}^1$ . The calculations proved the strong spin–phonon coupling for the E2g2$E_{2{\rm{g}}}^2$ mode. In addition, a synergistic interplay of pressure‐induced reduction of spin exchange interactions and spin–orbit coupling effect accounts for the softening of the E2g2$E_{2{\rm{g}}}^2$ mode as pressure increases.


Introduction
The intensive research on magnetic thin films has been driven by the rapid development of nanoelectronic and spintronic devices. [1][2][3] In two-dimensional (2D) materials, according to the conventional Mermin-Wagner theorem, [4] thermal fluctuations could strongly suppress the magnetic order of materials. However, the discovery of long-range ferromagnetic order in Cr 2 Ge 2 Te 6 [5] and CrI 3 [6] monolayers breaks the conventional theorem. [4] This breakthrough promotes tremendous effort in exploring the potential applications of 2D magnetic materials in magnetoelectrics, electrical control of magnetism, and magnetic tunnel junction. [7][8][9] Fe 3 GeTe 2 (FGT), a valuable member of 2D layered magnetic materials, has attracted special interests recently due to its rare metallic itinerant ferromagnetism with high Curie temperature (≈230 K) [10,11] and novel physical properties, including anomalous hall effect, Kondo effect, and giant tunneling magnetoresistance. [12][13][14] FGT crystallizes in a hexagonal structure with space group P6 3 /mmc (No. 194), as shown in Figure 1. The unit cell has two layers that are bonded by interlayer van der Waals (vdW) interactions. Each layer comprises five covalently bonded atomic planes. The planar Fe II Ge is sandwiched by two planes of Fe I atoms, and the triple planes are then sandwiched by two layers of Te atoms. The Fe I and Fe II atoms in each layer contribute to both the itinerant electrons and local ferromagnetic moments, which play significant roles in the magnetic spin-order transition with pressure and temperature dependence. [15,16] The first-principles calculations reported an increase of the frequencies of the A 1 1g and E 2g Raman modes when the spin ordering in FGT changes from FM to AFM, indicating notable spin-phonon coupling for these Raman modes in FGT. [17] Though, there is a lack of study on how strong/weak the spin-phonon coupling is for each of the modes. Raman spectroscopy is a powerful technique for probing the lattice vibrations in a crystal. Previous studies have reported an anomalous pressure-induced phonon softening behavior [18] and a temperature-driven strong spin-phonon coupling in FGT [19] through Raman studies. However, simultaneous pressure-and temperature-dependent lattice vibrations and spin-phonon interactions have not been reported experimentally. The combined effects of temperature and pressure on lattice dynamics may provide information that is not available through only one of them. In addition, what causes anomalous pressure-induced phonon softening is still not very clear.
In this work, we conducted the pressure-dependent (PD) Raman measurement on FGT at room temperature. In addition, we also performed the combined pressure-and temperaturedependent (PTD) measurements. Two Raman active modes (E 2 2g and A 1 1g ) were observed. The two types of measurements show that increasing pressure softens the E 2 2g mode anomalously and the decreasing temperature suppresses the softening. Temperature-dependent (TD) results were extracted from the two. The extracted TD phonon frequency is much higher than the reported direct measurement. Our density functional theroy (DFT) calculations examine the pressure-dependent magnetic properties of FGT. With phonon calculations, we also predict the two Raman active modes, investigate their spin-phonon couplings, and analyze how their frequencies are affected by the changes of pressure and temperature. Our work provides insightful information for studying the strain effect and thermal properties of FGT for its applications in nanoelectronic and spintronic devices.

Figure 2a
shows the Raman spectra of FGT in the frequency range of 80-200 cm −1 from ambient pressure to 9.52 GPa at 300 K. Bulk FGT has 12 atoms per unit cell and 36 phonon modes. [17] The two observed Raman active modes E 2 2g and A 1 1g are at 122.7 and 139.8 cm −1 at room temperature under ambient pressure, similar to the reported results. [18,19,20] Our spinpolarized DFT calculation for the antiferromagnetic state predicts the phonon frequencies of E 2 2g and A 1 1g modes, respectively, at 115.7 and 126.6 cm −1 , showing decent agreement with the measurement. As shown in Figure 2b, A 1 1g mode stiffens slightly with increasing pressure, but E 2 2g mode softens by 21 cm −1 mono- tonically from ambient pressure to 9.52 GPa at room temperature. As for the pressure dependence of the phonon linewidth of these two modes, Figure 2c shows an obvious broadening with increasing pressure. Figure 3 shows that E 2 2g mode involves the in-plane atomic vibrations and A 1 1g mode is related to the out-of-plane vibrations of Te, Fe, and/or Ge atoms. Increasing pressure usually decreases the bond lengths and unit cell volume [21] leading to larger interatomic force constants and higher vibrational frequency. However, our results reveal a pressure-induced softening of E 2 2g mode.  . Pressure and temperature-dependent (PTD) Raman spectra. a) Raman spectra of FGT at temperatures from 8 to 300 K and under pressures from 9.43 GPa to ambient pressure. b) Comparison between PD data and PTD data. The solid red squares and blue circles represent the PTD E 2 2g and A 1 1g , respectively. The hollow red squares and blue circles represent the PD E 2 2g and A 1 1g , calculated from the fitted pressure dependence. c) Extracted temperature-dependent (TD) phonon frequency. The solid red squares and blue circles represent the PTD E 2 2g and A 1 1g , respectively. The hollow red squares and blue circles represent the extracted TD E 2 2g and A 1 1g , respectively. The green triangles represent the TD E 2 2g obtained from the literature. [19] To better understand the abnormal softening, quasi-harmonic approximation (QHA) and DFT calculations were employed, and the results are analyzed in Section 4.

Pressure-and Temperature-Dependent (PTD) Raman Spectra
The PTD phonon frequencies and linewidths of E 2 2g and A 1 1g modes are shown in Figures 4 and 5. Figure 4a shows Raman spectra of FGT at temperatures from 8 to 300 K and pressures  [15] The red circle dashed line represents the phase boundary from Jie-Min et al. [18] The black triangle solid line represents our measured data points. b) Phonon linewidth of the two measured modes. from 9.43 GPa to ambient pressure. Our PTD data points cross the phase transition boundary near 6.3 GPa (Figure 5a). The PTD phonon frequencies of the two modes are higher than those of the PD data, as shown in Figure 4b, suggesting that the decrease in temperature suppresses the E 2 2g softening and enhances the A 1 1g stiffening. The PTD phonon linewidths are shown in Figure 5b. The phonon linewidth of A 1 1g mode increases as pressure increases (temperature decreases), but for E 2 2g mode, its phonon linewidth decreases cross the phase transition (near 160 K and 6.3 GPa).
To quantify the temperature effects on the phonon frequencies, we assume that the PTD phonon shifts are a linear combination of the temperature contribution and the pressure contribution. In this way, the temperature induced shift Δ AP,T could be calculated based on the following equations: where P,T represents the PTD phonon frequency. AP,300K is our measured phonon frequency at 300 K and ambient pressure. AP,T represents the TD phonon frequency at ambient pressure. Δ AP,T represents the temperature induced frequency shift under ambient pressure. P,300K represents our measured PD phonon frequency at 300 K. Δ P,300K represents the pressureinduced frequency shift at 300 K. After subtracting the pressure-induced frequency shift Δ P,300K from Equation (1), AP,T of the two modes are obtained and shown in Figure 4c. Like the behavior of pressure dependence, the effect of temperature on A 1 1g mode is much weaker than that on E 2 2g . A 1 1g mode stiffens moderately by 4.4 cm −1 from 300 to 8 K, while E 2 2g mode stiffens by 10.6 cm −1 in the same temperature range. For A 1 1g mode, both the increasing pressure and the descending temperature could lead to the phonon stiffening. Therefore, the PTD phonon frequency is higher than the PD one. For E 2 2g mode, the increasing pressure softens the phonon frequency, but decreasing temperature results in stiffening. The reported TD results under ambient pressure show a slight stiffening from 300 to 8 K, [19] which is much weaker than our extracted results. This discrepancy indicates that the effect of decreasing temperature will be enhanced and could compensate for pressure-induced phonon softening. The discrepancy also suggests that the linear subtraction in Equation (1) could not quantify the effect of temperature and pressure on the phonon frequency of E 2 2g mode. Considering higher-order effects by simultaneous pressure and temperature is needed to explain the PTD phonon frequency.

Discussion
Quasi-harmonic approximation (QHA) assumes that the phonon frequency is volume-dependent and that the phonon mode remains harmonic at each volume. The Grüneisen parameter provides the connections between phonon frequency and volume change. The isothermal mode Grüneisen parameter can be expressed as: iT . [22] i is the phonon frequency of ith phonon mode, B is the bulk modulus, which is calculated according to the power X-ray diffraction measurement. [21] Based on our PD data, the T of E 2 2g and A 1 1g are −0.88 and 0.11 respectively. The negative value of E 2 2g mode suggests significant phonon anharmonicity. Furthermore, with increasing pressure, the broadening phonon linewidth in Figure 2c reveals a shorter phonon lifetime and higher phonon scattering rates, which also suggest the anharmonicity of this mode. On the other hand, our PTD results indicate that the decreasing temperature could sup-press the E 2 2g softening with increasing pressure. The deep of phonon linewidth near phase transition in Figure 5b also suggests that the phonon scattering rates are suppressed with increasing pressure and decreasing temperature. In Figure 2c, we find that increasing pressure only slightly broadens the phonon linewidth of the E 2 2g mode. Therefore, we can infer that the temperature effect is opposite and plays a more important role in suppressing the phonon scattering rates below the phase transition.
To understand pressure-induced abnormal softening and anharmonic lattice dynamics in FGT, DFT and phonon calculations were applied to study the magnetic properties and Raman frequencies of the experimental structure at ambient pressure and the relaxed structures at external pressure of 1, 3, 5, and 7 GPa.
Although most studies proposed that Fe 3 FeTe 2 has a ferromagnetic interlayer spin order, [17] it was reported that the ferromagnetic layers of Fe 3 GeTe 2 order antiferromagnetically along the c axis below 152 K. [23] To resolve this inconsistency, we examined the interlayer magnetic interactions in FGT by comparing the total energy of two magnetic states, the ferromagnetic (FM) and antiferromagnetic (AFM) states. Both states contain ferromagnetic intralayer interactions, but ferromagnetic and antiferromagnetic interlayer interactions for the FM and AFM states, respectively. Based on our results in Figure 6b, the AFM state is lower in energy for all pressures used, indicating that the AFM state is the ground state. However, the FM states are only slightly higher in energy (1-10 meV Fe −1 ) than the AFM states. Therefore, there might be a competition between the FM and AFM ordering between the FGT layers. Additionally, as pressure www.advancedsciencenews.com www.advphysicsres.com increases, the lattice parameters ( Figure 6a) and Fe-Fe distances ( Table 1) decrease, leading to more Fe─Fe orbital overlaps, and less localized magnetic moments on Fe atoms. As a result, spin exchange interactions become weaker in FGT as the pressure increases. This is supported by the spin exchange parameters calculated from the SPRKKR calculations. Table 1 lists the values of eight spin exchange parameters in FGT at ambient pressure and at 7 GPa. The spin exchange interaction between adjacent Fe I atoms, J 1 , decreases from 86.9 to 45.2 meV. J 2 -J 6 and J 8 also decrease dramatically from ambient pressure to 7 GPa. Therefore, external pressure can reduce spin exchange interactions in FGT and decrease its ordering temperature, which is consistent with the lower Curie temperature at higher pressure demonstrated in Figure 5a. Furthermore, magnetic anisotropy was investigated using spin-orbit coupling calculations. We evaluated the magnetocrystalline anisotropy energy for FGT at ambient pressure and external pressures of 1, 3, 5, 7 GPa by comparing the energy of each FGT with spins in the ab-plane (E SOC (∥a)) and parallel to the c-axis (E SOC (∥c)), i.e., ∆E SOC = E SOC (∥a) − E SOC (∥c). The results, listed in Figure 6c, demonstrate that spin parallel to the c-axis is lower in energy at all pressures, indicating easy axis anisotropy. As pressure increases, the magnetocrystalline anisotropy energy becomes smaller, suggesting that the magnetic anisotropy in FGT becomes weaker under pressure. In our phonon calculation, we examined the Raman active modes, E 2 2g and A 1 1g , for nonmagnetic, FM state, AFM state without and with spin-orbit coupling. Results are shown in Tables 2 and 3. The renormalized phonon frequency due to a spinphonon coupling ( R ) can be described by the spin-correlation function S i · S j , R = 0 + S i · S j , where 0 is the phonon frequency in the absence of spin correlations and is the spinphonon coupling constant. [24] From the spin-orbit coupling calculations along the c-axis, we can easier estimate S i · S j . The calculated magnetic moments of Fe I and Fe II are about 2 B and 1 B . If we only consider the nearest neighbor Fe I -Fe I and Fe I -Fe II couplings in a FGT unit cell, S i · S j = 8. Therefore, we can obtain the spin-phonon coupling constant using the equation R = 0 + S i · S j . The calculated values of are listed in Tables 2 and 3, which demonstrate that the spin-phonon coupling is substantial for the E 2 2g mode but very weak for the A 1 1g mode. Therefore,  the phonon frequency of the E 2 2g mode could be highly affected by spin ordering, which can be controlled by changing pressure and temperature (see Figure 5a).
Let's first focus on the E 2 2g mode. Under perfect spin ordering states, the calculated Raman frequency increases as pressure increases (see results in Table 2). This can be easily explained by the decrease in lattice parameters shown in Figure 6a. However, FGT does not have perfect spin ordering when performing Raman spectroscopy measurements under various pressures and temperatures. The spin ordering and magnetic domain alignment in FGT are highly dependent on the strength of spin exchange interactions and magnetic anisotropy controlled by pressure. As discussed above, increased pressure could reduce spin exchange interactions and spin-orbit coupling effect in FGT, therefore the system becomes more disordered and the spin-correlation function S i · S j gets smaller. Based on R = 0 + S i · S j , the overall renormalized frequency R decreases as pressure increases, leading to a remarkable softening of the E 2 2g mode. Therefore, the pressure-induced decline of spin exchange interactions and the spin orbit coupling effect cause an overall softening of the E 2 2g Raman mode. As the temperature decreases, the FGT becomes more ordered, resulting in a larger spin-correction function S i · S j then higher renormalized frequency R and a stiffening of the E 2 2g mode. While for the A 1 1g mode, much smaller values indicate a very weak spin-phonon coupling. The Raman frequency is mainly controlled by changes in the lattice parameters. Consequently, the A 1 1g mode is slightly stiffened under pressure.

Conclusion
We conducted the pressure-dependent (PD) and pressure-and temperature-dependent (PTD) Raman scattering measurements and performed first-principles pressure-dependent phonon calculations on FGT. We observed the anomalous E 2 2g softening in PD data. Using the linear pressure dependence of E 2 2g and A 1 1g modes, we extracted the temperature-induced phonon frequency shift from PTD data. The obtained temperature effect was stronger than the reported direct results, indicating that the higher-order contribution from simultaneous pressure and temperature effect is non-negligible. DFT and phonon calculations demonstrate that the spin-phonon couplings are significant for the E 2 2g mode but very moderate for the A 1 1g mode. Our results suggest that the anomalous phonon softening of the E 2 2g mode is caused by the decrease of spin ordering due to the decline of the spin-orbit coupling effect and spin exchange interactions under pressure.

Experimental Section
Experimental Method: The FGT samples were prepared by solid-state reaction of elements at 800°C for 5 days. After mixing the elements Fe, Ge, and Te in their stoichiometric molar ratio, the mixture was pressed into a pellet, sealed in a quartz glass ampule under vacuum, and loaded into the furnace for reaction (see details in ref. [25]). The phase purity and crystallinity of the sample were determined by powder X-ray diffraction using a Rigaku Miniflex diffractometer. The excitation source for the Raman spectrometer was 532 nm. The laser power was set at 30 mW to minimize sample damage. An ultrasteep long-pass edge filter (OD abs > 6) was used to block the laser line, and a spectrometer (PI Acton Series 500 mm) was used for spectral imaging on a thermoelectrically cooled 2D CCD camera (PI PIXIS 400B). An Almax plate diamond anvil cell (DAC) with tungsten carbide seats was used for high-pressure environments inside a closedcycle cryostat. The culet size of the diamonds was 250 m. A stainless-steel gasket with a 100 m hole was used. Silicone oil was used as the pressure medium in the PD experiment and sodium chloride was used for the PTD experiment. At lower temperatures, thermal contraction of the DAC and gasket shrinks the gasket hole and increases pressure. Therefore, temperature and pressure changes were correlated in the PTD experiments. The PD measurement was conducted at room temperature.
Computational Method: To investigate the phonon dynamics of Fe 3 GeTe 2 , non-spin and spin-polarized density functional theory (DFT) calculations and phonon calculations were performed. Electronic structure calculations were performed using the projector augmented wave method of Blöchl [26,27] coded in the Vienna ab initio simulation package (VASP). [28] All VASP calculations employed the generalized gradient approximation (GGA) with exchange and correlation treated by the Perdew-Burke-Enzerhoff functional. [29] A Plane wave energy cutoff of 500 eV (550 eV for structure relaxation) and an energy convergence criterion of 10 −7 eV were used. The Brillouin zone integrations were carried out using Г-centered 11 × 11 × 3 and 4 × 4 × 2 k-point mesh for a unit cell and 2 × 2 × 1 supercell, respectively. To study the pressure dependence of the phonon vibrations, structure optimization was performed under external pressures of 1, 3, 5, and 7 GPa and followed by phonon calculations using the Phonopy software. [30] The convergence threshold for structural relaxation was set to be 0.01 eV Å −1 in force. GGA + SOC calculations were employed to examine the spin-orbit coupling (SOC) effect. VASP total energies of spin in the ab-plane (E SOC (∥a)) and parallel to the c-axis (E SOC (∥c)) for each compound were calculated. The magnetocrystalline anisotropy energy was then obtained using the relation ∆E SOC = E SOC (∥a) − E SOC (∥c). For the experimental structure and four fully relaxed structures under external pressure, phonon calculations were carried out using the finite displacement (frozen phonon) method implemented in the Phonopy software [30] to obtain the Raman peak energies at the Gamma point. Phonon calculations were applied to non-spin polarized state, the ferromagnetic state (FM interlayer interaction), the antiferromagnetic state (AFM interlayer interaction), and the antiferromagnetic state with spin-orbit coupling for each structure using 2 × 2 × 1 supercell.
To study the effect of pressure on spin interactions in Fe 3 GeTe 2 , the effective Fe-Fe exchange parameters in the experimentally observed structure and the relaxed structure under 7 GPa external pressure were evaluated using the spin polarized, relativistic Korringa−Kohn−Rostoker (SPRKKR) package [31] with GGA-PBE as the exchange and correlation corrections, an energy convergence criterion of 10 -6 Rd, and 500 k-points in the Brillouin zone.