Atomically Resolved Acoustic Dynamics Coupled with Magnetic Order in a Van der Waals Antiferromagnet

ABSTRACT

Magnetoelastic coupling in van der Waals (vdW) magnetic materials enables a unique interplay between the spin and lattice degrees of freedom. Characterizing the elastic responses with atomic and femtosecond resolution across the magnetic transition is essential for guiding the design of magnetically tunable actuators and strain‐mediated spintronic devices. Here, ultrafast X‐ray diffraction employed at a free‐electron laser reveals that the atomic displacements, wave vectors, and dispersion relations of acoustic phonon modes in a vdW antiferromagnet FePS₃ are coupled with the magnetic order, by tracking both in‐plane and out‐of‐plane Bragg peaks upon optical excitation across the Néel temperature (T _N). One transverse mode shows that a quasi‐out‐of‐plane atomic displacement undergoes a significant directional change across T _N. Its quasi‐in‐plane wave vector is derived by comparing the measured sound velocity and the first‐principles calculations. The other transverse mode is an interlayer shear acoustic mode whose amplitude is strongly enhanced in the antiferromagnetic phase, exhibiting eight times stronger amplitude than the longitudinal acoustic mode below T _N. The atomically resolved characterization of acoustic phonon dynamics that couple with magnetic ordering opens opportunities for harnessing unique magnetoelastic coupling in vdW magnets on ultrafast timescales.

Magnetically coupled acoustic phonon modes (f ₁, f ₂, f ₃) are captured by ultrafast X‐ray diffraction in a prototypical van der Waals antiferromagnet FePS₃, including their atomic displacement vectors (u) and the acoustic propagation wave vectors (k). Notably, the atomic displacement vector and amplitude of these modes, represented by the direction and size of double arrows, significantly change across the magnetic ordering temperature T _N. The time‐resolved and atomistic insights revealed by this work deepen the understanding of the unique magnetoelastic coupling in van der Waals magnets and pave the way for interfacing spins and mechanical degrees of freedom for applications.

1. Introduction

Magnetoelastic coupling in magnetic materials holds great promise for sensing, actuation, and energy harvesting applications [1, 2, 3, 4]. Van der Waals (vdW) magnets are particularly compelling materials for these applications due to their diverse magnetic states, remarkable mechanical flexibility, and strong magnetoelastic coupling [5, 6, 7, 8, 9]. Coherent acoustic phonons play critical roles in governing nanoscale energy and momentum transport. However, the characteristics of acoustic phonons, such as their atomic displacements, wave vectors, and dispersion, are challenging to study using conventional optical techniques [10] that have negligible momentum transfer. By contrast, ultrafast X‐ray diffraction enables momentum‐resolved tracking of acoustic wave propagation with atomic sensitivity [11]. Quantitative and direct measurements of coherent acoustic phonons and their relations with magnetic order offer insights into the unique magnetoelastic coupling in vdW magnets.

FePS₃ is featured in this study as an exemplary vdW magnet because it exhibits strong spin‐lattice coupling. In this material, both the magnetic and structural phase transitions occur at the Néel temperature (T _N = 117 K) [12, 13, 14]. Below T _N, FePS₃ hosts a zigzag in‐plane antiferromagnetic order within each honeycomb layer, as well as out‐of‐plane antiferromagnetic order between layers that are stacked in a monoclinic crystal structure [15]. The strong spin‐lattice coupling is evidenced in several observations, including the mechanical resonance sensitive to spin orders [4], demagnetization‐coupled optical phonons [16], magnon‐phonon hybridization [17], coupled critical slowing down [5], spin‐mediated shear oscillations [18], and exotic THz field‐induced metastable ferromagnetic states [19]. Previous ultrafast electron scattering and imaging studies [18, 20] have focused on in‐plane Bragg peaks, thus mainly sensitive to in‐plane atomic motions, leaving the coherent acoustic phonons with out‐of‐plane atomic displacements largely unexplored.

Here, we report atomically resolved characterization of three coherent acoustic modes in the vdW antiferromagnet FePS₃ using ultrafast X‐ray diffraction. By tracking the dynamics of both in‐plane and out‐of‐plane Bragg peaks following photoexcitation above the bandgap, we directly measure the atomic displacements and momentum‐energy dispersions of these three modes. The first mode with the lowest frequency exhibits atomic displacements along the out‐of‐plane direction while its wave vector is aligned with a quasi‐in‐plane direction. This mode is a quasi‐transverse mode that propagates nearly perpendicular to the transient temperature gradient. Notably, its atomic displacement vector rotates by ∼20° and exhibits a ∼5% change in sound velocity across T _N, indicating strong coupling to the magnetic order. The second mode with an intermediate frequency is identified as an interlayer shear acoustic mode with in‐plane atomic displacements and an out‐of‐plane propagation direction. Its amplitude is significantly suppressed above T _N, which is consistent with earlier reports [18, 20]. The third mode with the highest frequency is a longitudinal interlayer breathing mode [21], characterized by both the atomic displacement and wave vector oriented along the out‐of‐plane direction. Below T _N, the oscillation amplitude of the interlayer shear mode is eight times larger than that of the longitudinal interlayer breathing mode, a ratio that exceeds the largest previously reported value of six in the multiferroic perovskite system BiFeO₃ [22].

2. Results and Discussions

We performed 400‐nm‐optical‐pump, hard‐X‐ray‐diffraction‐probe measurements of nanometer‐thick FePS₃ flakes at the Femtosecond X‐ray Scattering (FXS) endstation of the Pohang Accelerator Laboratory X‐ray Free‐Electron Laser (PAL‐XFEL), as illustrated in Figure 1a. Multiple flakes with uniform thickness ranging from 40 to 200 nm were studied. The 100‐fs optical pump pulses were focused to a spot size with a spatial extent of ∼300 µm × 1500 µm in full width at half maximum (FWHM) on the sample, which was larger than the probed region to ensure homogeneous excitation. The pump photon energy of 3.1 eV is above the bandgap (1.7 eV) of FePS₃ [23]. The 12.0‐keV X‐ray pulses, operating at a repetition rate of 60 Hz, were focused to a ∼15 µm × 75 µm (FWHM) spot on the sample surface. At an incident optical fluence of 2.6 mJ cm⁻², the top surface of the sample absorbed 0.16 photons per unit cell, leading to a transient lattice temperature rise of ∼120 K (see Section S1) after the electron‐phonon coupling. The transiently established temperature gradient along the sample depth, together with the released structural instabilities associated with ultrafast demagnetization [18], leads to the coherent generation of acoustic phonons that are captured in real time using ultrafast X‐ray diffraction.

FIGURE 1.

Schematic experimental setup and time‐dependent Bragg peak shifts. (a) Schematic of the ultrafast X‐ray diffraction experimental setup at an XFEL (X‐ray free‐electron laser). (b) Real‐space structure viewed from the b‐axis. Only Fe atoms are shown for clarity. The double blue arrows indicate the atomic vibration and polarization angle P relative to the a‐axis. (c) Time‐dependent center‐of‐mass shift (∆q _z ) of the 002 Bragg peak along the c ^* direction at various temperatures. (d) Fast Fourier transform (FFT) of the time‐resolved data in (c). Inset: schematic of the reciprocal lattice of FePS₃. Green and red symbols represent the peak positions before and after laser excitation, respectively. (e) Time‐dependent center‐of‐mass shift (∆q _z ) of the $\overline{2}02$ Bragg peak along the c ^* direction at various temperatures. (f) FFT of the time‐resolved data in (e). The vertical dashed lines indicate the frequencies of the three phonon modes, labeled as f ₁, f ₂, and f ₃.

We focused our measurements on the shifts of 002 and $\overline{2}02$ Bragg peaks, which are sensitive to the atomic displacements in the ac‐plane (Figure 1b), as the laser‐induced structural changes in FePS₃ primarily occur within the ac‐plane [5]. Upon optical excitation, the 002 Bragg peak was observed to shift toward the ‐c ^* direction, as captured by a 2D X‐ray area detector, indicating a laser‐induced interlayer expansion (Figure 1c). This observation, which was quantified by analyzing the peak shift in the 2D reciprocal space cut accessible to the detector, is consistent with the 3D reciprocal space mapping of the peak shift extracted from the time‐resolved rocking curve measurements (Figure S1). In addition to the step‐like peak shift, oscillatory modulations were superimposed on the 002 peak dynamics. A fast Fourier transform (FFT) of the time‐dependent 002 peak shift revealed a strong frequency component at 3.5 GHz and a weak one at 11.1 GHz (Figure 1d), indicating the excitation of at least two coherent phonon modes. Temperature‐dependent FFT amplitudes showed that both modes persist across T _N. The mode at 3.5 GHz dominates the oscillation, with an FFT amplitude more than four times larger than that of the 11.1‐GHz mode. On the other hand, for the $\overline{2}02$ Bragg peak dynamics (Figure 1e), the most prominent temperature‐dependent feature was a reversal in the sign of the peak shift (Δ q _z): negative above T _N and positive below T _N. This behavior is consistent with an interlayer shear motion, attributed to a sudden decrease in the monoclinic angle β from its equilibrium value of 107.2°, triggered by the melting of the antiferromagnetic order as reported previously [5, 18]. In addition, the oscillatory signal of $\overline{2}02$ peak exhibited different periods and the oscillation phase underwent a π shift across T _N. The oscillation amplitude was shown by the FFT spectra (Figure 1f), which revealed the emergence of a new mode with a large FFT amplitude at 4.6 GHz below T _N. These coherent acoustic modes were consistently observed across multiple samples, despite the frequency variations due to differences in sample thickness. The linear dependence of the oscillation periods on the sample thickness (Figure S2) confirmed that all three modes we observed were neither optical phonons nor magnons, but acoustic phonons whose wave vectors k have an out‐of‐plane component. For clarity, we labeled the three acoustic phonon modes as f ₁, f ₂, and f ₃, corresponding to the 3.5, 4.6, and 11.1 GHz modes, respectively.

When multiple oscillatory signals have similar frequencies and finite lifetimes, as in the case of the f ₁ and f ₂ modes, it is difficult to determine their amplitudes and frequencies precisely based on the FFT analysis alone. To complement the FFT analysis, we fitted the data by a function that is composed of three sinusoidal functions with exponentially decaying amplitudes: $F(t)={\sum }_{i=1,2,3}(A_i{e}^{-\frac{t}{{\tau }_i}}\cos (2\pi f_{i}t+{\phi }_i))+cc$, where A_i , τ _i , f_i , and φ _i are the amplitude, decay constant, frequency, and phase of the i‐th (i = 1, 2, 3) mode, respectively, and cc is a constant offset (See Section S2for details). Figure 2a–d show the fittings for the 002 and $\overline{2}02$ peak, respectively, with data from below T _N in the top panels and above T _N in the bottom panels. The temperature‐dependent fitting results were summarized in Figures S3 and S4. For the 002 peak at all temperatures, the average amplitude of the f ₁ mode was four times larger than the f ₃ mode and ten times larger than that of f ₂ mode (Figure S3a), indicating a dominant f ₁ component and a negligible f ₂ component measured in the 002 peak. For the $\overline{2}02$ peak above T _N, the fitting results were similar to those of the 002 peak. By contrast, for the $\overline{2}02$ peak below T _N, the f ₂ mode dominates the oscillation with the largest amplitude (Figure S3a) and exhibits π out of phase (Figure S4a) compared to the other two modes.

FIGURE 2.

Time‐domain fitting of the oscillatory dynamics. (a,b) Oscillatory dynamics of the 002 Bragg peak measured at 99 K (a) and 140 K (b). To emphasize the oscillatory behavior, the data points at negative time delays are omitted. Red solid lines are fits using a three‐component damped sinusoidal model. (c,d) Same as (a,b), but for the $\overline{2}02$ Bragg peak. (e) Temperature‐dependent atomic polarization angle P ₁ relative to the a‐axis direction and oscillation period T ₁ for the f ₁ mode. The definition of polarization angle is illustrated in Figure 1b. (f) Temperature‐dependent atomic polarization angle P ₂ relative to the a‐axis direction and oscillation amplitudes A ₂ of two peaks for the f ₂ mode. The polarization angle above T _N is omitted due to the disappearance of the f ₂ mode. The error bars in (e) and (f) are errors propagated from the standard deviation of the time‐domain fitting. (g) Schematic drawing of the atomic displacement vector u and the wave vector k direction for the three modes. The displacement vector u is based on the time‐domain fittings, while the wave vector k will be discussed in later sections. The parallogram represents the sample flake that establishes a temperature gradient after optical excitation.

Based on the reciprocal space geometry (inset of Figure 1d) and the oscillation amplitudes, we were able to extract the polarization angles (See Section S3), i.e., the direction of atomic displacement vectors u, for all three modes. The shift of the 002 peak along c ^* (Δq_z ) is a result of the out‐of‐plane atomic displacements, while the shift of $\overline{2}02$ peak is determined by both the in‐plane (along the a‐axis) and the out‐of‐plane atomic displacements. Since both peaks share the same L‐index, any difference in their Δq_z shift (Figure S3b) reflects the changes in the monoclinic tilt angle β and in‐plane atomic displacements. As shown in Figure 2e, the polarization of the f ₁ mode was primarily along the c ^* direction below T _N and rotated by about 20° above T _N, which likely originates from the strong magnetoelastic coupling. Crossing T _N, the change in spin exchange interactions leads to a renormalization of the inter‐atomic force constants and elastic tensor [4, 24], which subsequently leads to a rotation of the displacement polarization. In addition, the oscillation period (Figure 2e) of the f ₁ mode changed by 5% across T _N, further showing the effect of magnetic ordering on the elastic properties associated with the f ₁ mode. For the f ₂ mode (Figure 2f), its amplitude is the largest among all three modes below T _N, but is suppressed to nearly zero above T _N. In contrast to the out‐of‐plane atomic displacement for the f ₁ and f ₃ modes (Figure S3c), the polarization of the f ₂ mode is mostly in the in‐plane direction. We estimate the systematic error is 5° for the absolute value of the measured polarization angles. The atomic displacements and wave vectors of all three modes are summarized in Figure 2g, which is further supported by the dispersion analysis and first‐principles calculations detailed below.

Besides the atomic displacements, ultrafast X‐ray diffraction offers momentum‐resolved dynamics that enable the direct characterization of the acoustic phonon dispersions. We analyzed the intensity oscillations of the Kiessig fringes on the 2D X‐ray area detector (Figure 3a) that arise from the finite‐thickness effect of the crystal and cover a span of momentum space. To extend the accessible momentum range afforded by these fringes, we decreased the sample θ angle by 0.1° so that the fringes were stronger on the lower‐q_z side. We note that the frequency we measured here in the dispersion relation is not the second harmonic of the acoustic waves as measured by the time‐domain inelastic X‐ray scattering technique [25], because the dominant dynamical signal in our measurement is still near the Bragg peak rather than from the diffuse scattering. After optical excitation, different Kiessig fringes exhibited distinct oscillation periods, as highlighted by the red dashed curves in Figure 3b. With the correction that removed the contribution from the strain pulse propagation (Figure S5), we performed FFT on the time evolution of each momentum point on the shift‐corrected image to obtain the dispersion map shown in Figure 3c. A prominent linear dispersion in Figure 3c confirms the acoustic nature of this mode. By fitting the slope of the dispersion (Figure S6a), we extract a sound velocity of 3560 ± 20 m s⁻¹. This value is lower than the calculated value of 4200 ± 20 m s⁻¹ by $v=\frac{2d}{T}$, where d  =  195.8 nm is the flake thickness as measured by the Kiessig fringe and T  =  93.3 ps is the fitted oscillation period. This discrepancy might be due to the difference in the effective propagation distance of the acoustic wave in the two measurements, as detailed in Section S4. Nevertheless, the measured sound velocity of the f ₃ mode is highest among all the measured modes and agrees with the calculated longitudinal acoustic velocity (the value of the purple curve at 90° in Figure 4a). Therefore, the mode f ₃ is ascribed as a quasi‐longitudinal acoustic mode, i.e., the interlayer breathing mode.

FIGURE 3.

Momentum‐resolved dynamics and phonon dispersions. (a) Diffraction image of the 002 Bragg peak displayed in a logarithmic scale. The central peak at q _z ‐ q _z (002) = 0 is the 002 peak, and the side peaks along q _z are the Kiessig fringes. (b) Momentum‐resolved dynamics of the 002 peak at room temperature. Red dashed lines are visual guides indicating the q _z‐dependent oscillation, the same as in (e). (c) FFT spectra of the data in (b). The white dashed line highlights the dispersion of the longitudinal acoustic mode (f ₃), and the red rectangle indicates the expected dispersion for the f ₁ mode. (d) Diffraction image of the $\overline{2}02$ Bragg peak displayed on a logarithmic scale. (e) Momentum‐resolved dynamics of the $\overline{2}02$ peak at 99 K. (f) FFT spectra of the data in (e). The white dashed line traces the dispersions of the f ₂ branch. (g–i) Same as (d–f), but measured at 140 K. The red rectangle indicates suppression of the f ₂ mode above T _N.

FIGURE 4.

Sound velocity and thermal transport calculations. (a) Polar plot of the sound velocities for three acoustic branches as a function of the wave vector directions (θ _k ) in the ac‐plane. θ _k =  0° in the graph represents the a‐axis, while θ _k =  90° corresponds to the plane‐normal c ^* direction. Calculation details are in the Experimental Section. LA: longitudinal acoustic. TA: transverse acoustic. The horizontal dashed line represents the experimentally measured f ₁ sound velocity, which is a projected velocity along c ^* label v _1p. The crossing point between the dashed line and the red line TA1 branch represents the f ₁ mode with wave vector label k ₁. (b) Depth‐dependent sample temperature rise (ΔT) calculated by a microscopic three‐temperature model, detailed in Section S5. (c) Depth‐dependent demagnetization dynamics normalized to the initial magnetization M ₀.

The dispersion of the $\overline{2}02$ peak below T _N is shown in Figure 3d–f. The sound velocity from linear dispersion fitting is 1670 ± 20 m s⁻¹ (Figure S6b), in agreement with the sound velocity of the f ₂ mode (1710 ± 20 m s⁻¹) estimated by the oscillation of the $\overline{2}02$ peak shift. Furthermore, the dispersion of this mode was prominent below T _N but strongly suppressed above T _N (Figure 3g–i), in agreement with the temperature dependence of the mode amplitude measured by the dynamics of the peak shift (Figure 2c,d) as well as by the previous ultrafast electron diffraction and microscopy measurements [18, 20]. Therefore, the f ₂ mode is an interlayer shear acoustic mode that propagates along the out‐of‐plane direction with an in‐plane atomic displacement. We note that the sound velocity measured by this work is higher than the value measured by ultrafast electron diffraction [18], which may be related to the different sample environment as detailed in Section S4.

For the lowest‐frequency f ₁ mode, its Δ q _z oscillation (Figures 1c and 2a) dominates the 002 peak dynamics, with an amplitude four times larger than the interlayer breathing f ₃ mode. However, in the dispersion map shown in Figure 3c, only the f ₃ mode is prominent, while the f ₁ mode is not resolvable along the expected dispersion trajectory indicated by the red rectangle. This absence is consistent with the interpretation that the wave vector of the f ₁ mode is mainly in‐plane with little projection along the c ^* direction, so that its dispersion is not discernible along the c ^* direction. The wave vector of the f ₁ mode cannot be purely in‐plane either, because the oscillation period of the f ₁ mode scales linearly with the sample thickness (Figure S2), implying the presence of an out‐of‐plane component in its wave vector. Therefore, the wave vector of the f ₁ mode is tilted significantly away from the sample surface normal, with a dominant in‐plane and a small out‐of‐ plane component, as schematically shown in Figure 2g.

To further assess the wave vector of the f ₁ mode, we compare the measured sound velocity with the theoretical values. We performed first‐principles calculations of FePS₃ and extracted sound velocities in all three dimensions. The sound velocities of three acoustic branches were plotted in Figure 4a along each wave vector direction in the ac‐plane. The theory calculations were first verified by comparing with the experimental sound velocities of the f ₂ and f ₃ modes. The theory values for the two modes (marked in the vertical axis in Figure 4a) were 1715 and 3332 m s⁻¹, respectively, in agreement with the measured sound velocities. For the f ₁ mode, since its wave vector is neither purely in‐plane nor out of plane as discussed earlier, we need to search for a point on the TA1 curve between 0 and 90 degrees that matches the measured sound velocity. TA1 branch is chosen instead of TA2 branch because the phonon polarization of TA1 branch matches that of the f ₁ mode. A horizontal dashed line with a height of the measured velocity V _1p along the c^* direction is drawn to find its crossing point with the TA1 branch. The origin to this crossing point defines the wave vector of the f ₁ mode, which is oriented at 30° with respect to the in‐plane direction. The length of the vector represents the velocity along this direction, whose value can be calculated as V _1p / sin 30° = 2600±120 m s⁻¹.

3. Discussion

It is unusual that the wave vector of the f ₁ mode is mostly in‐plane. In previous studies of photoexcited coherent acoustic phonons, the phonon propagation direction, regardless of a longitudinal or transverse mode, has always been expected to be along the surface normal direction [11, 26, 27, 28]. This is because photoexcitation of the sample's top surface creates a transient temperature gradient and thus thermal stress along the sample depth direction. While it is common to generate in‐plane or surface acoustic waves  from the laser‐pumped region propagating toward unpumped areas due to the in‐plane thermal stress [29], our measurements focused on the homogeneously excited region within the pump laser spot. In this case, the pump laser spot size is much larger than the probe spot size, ensuring homogeneous excitation on the sample's top surface of the probed region. In the previous reports of in‐plane travelling acoustic phonons, analogous to the f ₁ mode, the coherent phonons have always been launched from the sample edges or defects [20, 30, 31, 32]. In our study, the f ₁ mode cannot originate from edges because the probed region is at least 10 µm away from the sample edges. The acoustic wave launched at the edges would result in a delayed onset time on nanosecond timescale, which we did not observe. On the other hand, it is possible that the f ₁ mode is launched from nanoscale structural defects such as step edges, wrinkles, or surface roughness [33], which are commonly present in vdW materials. These structural defects in FePS₃ were confirmed by the previous ultrafast electron microscopy investigations [18, 20].

Our results show that the wave vector of the f ₁ mode is directed in the quasi‐in‐plane direction, indicating a transverse thermal flow perpendicular to the temperature gradient. Strong magnetoelastic coupling in FePS₃ has been proposed to induce hybridization between magnon and phonon bands, which, combined with time‐reversal symmetry breaking, gives rise to a nonzero Berry curvature and the thermal Hall effect [34]. The predicted band hybridization between magnons and optical phonons occurs near 14 meV (∼3.4 THz). At the energy range of acoustic phonons observed in this study, the acoustic phonon and magnon hybridization are not expected in antiferromagnetic materials due to their energy and momentum mismatch. Nevertheless, our temperature‐dependent characterization of acoustic phonons clearly indicates that the magnetic order plays an important role in governing the dynamics of f ₁ mode. The efficient transverse thermal flow carried by f ₁ mode presents a promising avenue for future phononic and elastic actuation applications. Conversely, the strong magnetoelastic coupling demonstrated here could be potentially harnessed for coherent control of magnetism via acoustic phonon excitation.

The interlayer shear acoustic mode f ₂ has been reported in the previous electron scattering studies [18, 20]. Here, with ultrafast X‐ray diffraction measurements, we provided direct evidence for both the atomic displacements and phonon dispersions across T _N. Apart from the f ₁ and f ₂ modes, the interlayer breathing mode f ₃ in FePS₃ was reported and comprehensively characterized for the first time. These results allow for quantitative comparisons between these modes. Specifically, the ratio of oscillation amplitudes A ₂/A ₃ reveals that the interlayer shear mode exhibits an amplitude eight times larger than that of the interlayer breathing mode (Figure S3a). This striking contrast underscores the presence of giant shear instabilities strongly coupled to the zigzag antiferromagnetic order, which has been demonstrated to be effective for ultrafast demagnetization‐driven shear oscillations [18, 20]. Below T _N, ultrafast demagnetization alters the spin exchange interactions, which subsequently modifies the inter‐atomic potential and thus shifts the minimum‐energy atomic positions. Above T _N, photo excitation does not change the spin exchange interactions, resulting in diminishing amplitude for the f ₂ mode.

To corroborate the atomic dynamics measured by ultrafast X‐ray diffraction, we performed thermal and spin transport simulations using a microscopic three‐temperature model (M3TM, see Section S5 for details). The simulation captures the evolution of the lattice and spin temperatures (Figure 4b,c) across the sample, in which the thermal gradient persists beyond 1 ns. Since the sample is in a mixed state of antiferromagnet and paramagnet after excitation in thick samples but mostly a paramagnetic state in thinner samples, the speed of sound for the interlayer shear mode is expected to be higher in the antiferromagnetic state. As the magnetization state of the sample still evolves in the thick sample during the measured time window (Figure 4c), the resulting acoustic transport is thus more complicated to model. This complication may partially explain why the sound velocity of the interlayer shear (f ₂) mode we measured in this work is 33% larger than that measured in significantly thinner samples by ultrafast electron diffraction and microscopy [18]. Other factors may also contribute to this discrepancy (Section S4).

4. Conclusion

We have investigated the coherent acoustic phonon dynamics in the exemplary vdW antiferromagnet FePS₃ across its magnetic ordering temperature using ultrafast time‐resolved X‐ray diffraction. In addition to the previous electron scattering studies [18, 20], which primarily probed in‐plane atomic displacements, our measurements revealed previously unobserved coherent phonon modes (f ₁ and f ₃) exhibiting out‐of‐plane atomic displacements and unconventional propagation characteristics. Notably, direct characterization of the interlayer shear mode (f ₂) and the interlayer breathing mode (f ₃) yields a giant shear‐to‐longitudinal oscillation amplitude ratio of up to eight, which highlights the shear degrees of freedom that couple to magnetic order. Further, the f ₁ mode exhibits a transverse propagation direction perpendicular to the transient temperature gradient and a 20° rotation in its atomic polarization vector at T _N. This observation suggests strong magnetoelastic coupling and may have connections to the theoretically predicted nonzero Berry curvature‐induced topological transverse transport phenomenon in FePS₃ [34]. Together, the two transverse modes observed here thus provide a unique platform for exploring magnetoelastic coupling and engineering unconventional phonon transport in future phononic and spintronic devices.

5. Experimental Section

5.1. Sample Growth and Sample Preparation

Single crystals of FePS₃ were synthesized by the chemical vapour transport method using iodine as the transport agent. Stoichiometric amounts of iron powder (99.998%), phosphorus powder (98.9%), and sulfur pieces (99.9995%) were mixed with iodine (1 mg cm⁻³) and sealed in quartz tubes (10 cm in length) under high vacuum. The tubes were placed in a horizontal one‐zone tube furnace with the charge near the center of the furnace. Sizeable crystals (10 × 10 × 0.5 mm³) were obtained after gradually heating the precursor up to 750°C, dwelling for a week and cooling down to room temperature. Ultrathin crystals were obtained by mechanical exfoliation, and thin flakes were transferred to sapphire substrates. The layer thickness was measured by atomic force microscopy as well as by measuring the Kiessig fringes in situ.

5.2. Ultrafast X‐ray Diffraction Experiment

The ultrafast X‐ray diffraction experiments were performed at the Femtosecond X‐ray Scattering (FXS) endstation of the Pohang Accelerator Laboratory X‐ray Free‐Electron Laser (PAL‐XFEL). The 400 nm pump laser pulse was derived by doubling the fundamental wavelength of an amplified Ti: Sapphire laser. The pump beam has a 10°‐crossing angle relative to the X‐ray probe beam. At the scattering geometry of the 002 Bragg peak, the pump laser footprint on the sample was 300 µm × 1500 µm (FWHM), with an incident fluence up to 4 mJ cm⁻² for the experiment. The repetition rates of the pump laser and the probe X‐ray were 30 and 60 Hz, respectively. The X‐ray pulse with a duration of 40 fs was monochromatized to an energy of 12 keV and focused to a size of 15 µm × 75 µm (FWHM) on the sample by a pair of Kirkpatrick–Baez mirrors. The X‐ray footprint is aligned at the center of the sample (∼100 µm in lateral size, see Figure S1a), which makes the probe beam more than 10 µm from the sample edge as quoted in the Discussion section. The sample was mounted on a four‐circle diffractometer (Huber GmbH.) and cooled down by a helium cryoblower (Oxford Instruments). A temperature sensor was mounted right underneath the sample for monitoring the sample temperature. An area detector (Jungfrau 0.5 m) was used to record the shot‐by‐shot diffraction pattern. Regarding the absence of delayed onset dynamics in FePS₃, the time‐zero was independently calibrated by a standard bithmuth sample with femtosecond precision.

5.3. First‐Principles Calculations of Sound Velocity

First‐principles density functional theory (DFT) calculations were carried out for FePS₃ in the zigzag antiferromagnetic configuration using the VASP package [35]. The local density approximation (LDA) functional was employed, with a Hubbard U correction applied to the Fe 3d orbitals. The correction was treated using the Dudarev approach with an effective U–J value of 3.5 eV [36]. A Γ‐centered 6 × 3 × 3 k‐point mesh was used for Brillouin zone sampling, and the plane‐wave basis set was truncated at a kinetic energy cut‐off of 500 eV. Structural relaxation was performed until both the lattice parameters and atomic positions were fully optimized. The elastic moduli tensor was then calculated using the finite difference method. Finally, sound velocities were derived from the computed elastic constants using the Christoffel package [37].

5.4. Microscopic Three‐Temperature Model (M3TM)

Our theoretical approach to understanding ultrafast magnetization dynamics is based on the microscopic three‐temperature model (M3TM) [38, 39, 40, 41, 42]. In this framework, the ultrafast laser pulse S(z,t) interacts with the electronic subsystem in the electric dipole approximation, where electrons near the Fermi energy get excited above the bandgap. Upon fast thermalization of the electron system through Coulomb scattering, the laser pulse causes a substantial elevation in the electronic temperature T _e. Electron‐phonon scattering processes allow for the exchange of energy and equilibration of T _e and the phonon bath, characterized by its temperature T _p. Depth‐dependent laser absorption leads to temperature gradients within the sample that cause diffusion processes in both baths. The energetic cost of demagnetization get compensated by the electronic system. Additional details are in Section S5.

Author Contributions

F.Z., S.S.H., S.H.C., J.P., I.E., M.Z., S.K., S.C., Z.C., K.H.O., Y.S., A.Z., S.O.H., N.G., H.K., X.X., and H.W. performed ultrafast X‐ray diffraction experiments. F. Z. and H.W. performed data analysis and interpreted the data. Q.J. grew the single crystals under the supervision of J.‐H.C., K.H. prepared the samples supervised by X.X., X.Z., T.C, and D.X. did first‐principles calculations. Z.Y. and E.J.G.S. simulated the time‐dependent temperature profile. F.Z. and H.W. wrote the manuscript with input from all authors. H.W. conceived and supervised the project.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supporting File: adma72325‐sup‐0001‐SuppMat.pdf.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.