Terahertz control of linear and nonlinear Magno-phononics

Abstract

Coherent manipulation of magnetism through the lattice provides opportunities for controlling spintronic functionalities on the ultrafast timescale. Such nonthermal control typically involves nonlinear excitation of Raman-active phonons which are coupled to the magnetic order. Linear excitation, in contrast, holds potential for more efficient and selective modulation of magnetic properties. However, since the linear excitation of Raman-active phonons is conventionally considered forbidden in inversion symmetric quantum materials, the simultaneous linear and nonlinear excitation of a collective mode involving lattice component has remained elusive. Here, we harness strong coupling between magnons and Raman-active phonons to achieve both linear and quadratic excitation regimes of magnon-polarons, magnon-phonon hybrid quasiparticles. We demonstrate this by driving magnon-polarons with an intense terahertz pulse in the van der Waals antiferromagnet FePS₃. Such excitation behavior enables a unique way to coherently control the amplitude of magnon-polaron oscillations by tuning the terahertz field strength and its polarization. The polarimetry of the resulting coherent oscillation amplitude breaks the crystallographic C₂ symmetry due to strong interference between different excitation channels. Our findings unlock a wide range of possibilities to manipulate material properties, including modulation of exchange interactions by phonon-Floquet engineering.

It is generally assumed that modulating magnetic properties via linear excitations of Raman-active phonons is forbidden in inversion symmetric magnets. Here, Luo, Ning, Ilyas, von Hoegen, and coauthors demonstrate a linear excitation of Raman-active lattice vibrations, via magnon-polaron excitation.

Introduction

Coupling between various degrees of freedom in quantum materials gives rise to exotic phases and powerful controls over functional properties. Exploiting the lattice degree of freedom in magnetic materials offers unique perspectives into emerging ultrafast spintronic and straintronic applications^1,2. By coherently driving phonons in the terahertz (THz) spectral range, an ultrafast and nonthermal control of magnetism circumventing laser-induced heating can be realized. It has been experimentally demonstrated that coherent phonons can impact the magnetic degree of freedom through a large variety of pathways, including facilitating ultrafast demagnetization^3–5, driving spin-wave precession^6,7, modulating exchange interactions and magnetic anisotropy^8–11, inducing dynamical magnetization^12–16, mediating angular momentum transfer^17,18, flipping magnetic order parameter^19,20, stabilizing fluctuating magnetization²¹, and inducing exotic magnetic states inaccessible via equilibrium approaches^22–24. Conventionally, such control has been achieved through the displacive or impulsive excitation of Raman-active phonons or rectifications to the microscopic parameters with Raman symmetry, whose amplitudes both scale quadratically with the electric field component of light (Fig. 1a, see also Supplementary Note 1). Modulating the magnetic properties via linear excitation of a collective mode that involves lattice vibrations in inversion-symmetric systems has not been widely studied. While in inversion-asymmetric systems, Raman-phonons can be simultaneously infrared-active (IR-active) and linearly excitable by resonant electric field, in systems with inversion symmetry, such excitation channel is traditionally considered forbidden due to the absence of an electric-dipole moment of Raman-active phonons. Confining excitation exclusively to nonlinear pathways may constrain the attainable amplitude of lattice vibrations and hinders a detailed control over lattice oscillation patterns, thereby limiting the scope of phononic control of magnetic properties.

Fig. 1. Nonlinear and linear excitation pathways.

a–c Schematics of the excitation mechanisms for a Raman-active phonon, magnon, and magnon-polaron (MP). d–f Driving field E_THz-dependence of the amplitudes of the modes shown in panels (a–c). The Raman-active phonon shows a quadratic dependence (d). The magnon shows a linear dependence (e). The MP shows a dependence described by Eq. (1) (violet line in f), a combination of quadratic (dotted red line in f) and linear (dotted blue line in f) dependence. g–i Driving field polarization ϕ-dependence of the amplitudes of different modes. Filled and white lobes denote opposite phases. g The Raman-active phonon exhibits a four-petal pattern described by a second-order homogeneous polynomial of $\sin \varphi$ and $\cos \varphi$. h The magnon exhibits a two-petal pattern described by a linear homogeneous polynomial of $\sin \varphi$ and $\cos \varphi$. i The MP exhibits a pattern breaking the C₂ symmetry as a result of the combination of nonlinear and linear excitation pathways. j Schematic of the experimental setup. A strong THz pump (yellow) and a near-IR probe (red) are focused on the (001) surface of FePS₃ with a tunable time delay. Both E_THz and ϕ can be controlled (Methods and Supplementary Note 2). Different modes in FePS₃ (bottom right) are detected through the change in probe ellipticity (Δη) of the transmitted probe. k Δη transient obtained at 10 K at maximal E_THz and ϕ = 12°. l FFT spectrum of the time trace in (k). The shaded regions are fits to the FFT peaks by damped oscillators. MPs with dominant phonon weight (MP1, MP1', and MP2), MPs with dominant magnon weight (M' and M), and Raman-active phonons (P) are shaded in violet, blue, and red, respectively. Inset shows the FFT spectrum at 130 K, where only the P mode is observed.

A promising avenue to overcome this obstacle involves leveraging Raman-active phonon modes that are linearly coupled to magnons. Since magnons can be linearly excited by magnetic field through the magnetic dipolar process^25,26 (Fig. 1b), its linear coupling to a Raman-active phonon imparts the phonon with a magnetic dipole, thereby offering a linear pathway for exciting the lattice oscillations using an oscillating magnetic field with a frequency matching that of the Raman-active phonon, referred to as a resonant magnetic field. The strength of the magnetic-dipolar pathway can be comparable to or even stronger than that of the quadratic Raman process. The hybrid magnon-phonon mode, denoted as magnon-polaron (MP), can arise when the frequencies of the two quanta are in vicinity to each other in a system with strong spin-lattice coupling. MPs exhibit the characteristics of both a Raman-active phonon and a magnon (Fig. 1c), holding the potential of both nonlinear and linear manipulations. As the hybridization strength increases, the MPs acquire a greater magnon component, leading to enhanced linear excitation efficiency. Notably, manipulating MPs is physically equivalent to simultaneously controlling both the magnon and phonon components. Therefore, linear excitation of MPs enables the linear excitation of the constituent lattice vibrations. However, despite its promise, a coherent control of the two pathways, including tuning their interference and switching their relative weights, has remained an elusive goal.

Previous investigations on MPs largely focused on the acoustic phonon-magnon hybrid in ferromagnetic materials, where the coupling is prominent at finite momentum and in gigahertz range^27–35. Here, we demonstrate both linear and nonlinear coherent excitation of MPs in a quasi-two-dimensional van der Waals antiferromagnet FePS₃, which exhibits inversion symmetry^36–38 and strong spin-lattice coupling^39–43. FePS₃ respects 2/m point group symmetry and orders into a zigzag antiferromagnetic pattern with an easy axis along the out-of-plane direction below 118 K (Fig. 1j inset)³⁷. Importantly, the magnon frequency of FePS₃ lies in proximity to the Raman-active phonon frequencies, enabling strong hybridization as evidenced by the avoided crossing between them in the THz range, as reported in previous Raman and Fourier-transform infrared spectroscopy (FTIR) studies^44–47. Here, we use high-field, ultrashort THz electromagnetic pulses that possess significant spectral weight at the MP frequencies to resonantly excite the MPs and enable their linear excitation. Meanwhile, the THz pulses are tailored to cover a broad bandwidth to facilitate the nonlinear Raman excitation pathway (Supplementary Note 2). Although our studies are conducted under zero magnetic field and far from the avoided-crossing point, the effects of magnon-phonon hybridization persist, as they are governed by the same magnon-phonon hybridization Hamiltonian (Supplementary Note 10). These resulting effects can be experimentally detected as demonstrated below.

To distinguish between the linear and nonlinear pathways, we analyze their distinct dependencies on the THz electric field strength E_THz and THz electric field polarization angle ϕ (Fig. 1j). The amplitude (A) of a mode driven by the Raman excitation channel follows a quadratic dependence on E_THz (Fig. 1d), whereas it exhibits a linear dependence on E_THz for the resonant pathway (Fig. 1e). The mode amplitude is then determined by the sum of their contributions (Fig. 1f):

$$A=∣a{E}_{\mathrm{THz}}^2+b{E}_{\mathrm{THz}}∣,$$ 1

where a and b represent the excitation efficiencies of the nonlinear and linear pathways, respectively.

Additionally, the amplitude exhibits a distinct behavior as a function of the THz polarization angle ϕ (see Fig. 1g–i). In the linear pathway, the magnetic component of the THz driving field directly interacts with the magnetic moment induced by magnons and can thus linearly excite them. The MP amplitude is maximal when the magnetic field is aligned with the magnetization generated by the magnon. Therefore, the ϕ-dependence of the MP amplitude follows a two-petal pattern represented by a linear polynomial of $\cos \left(\varphi \right)$ or $\sin \left(\varphi \right)$ (Fig. 1h). On the other hand, the nonlinear Raman excitation pathway follows a homogeneous quadratic polynomial of $\cos \left(\varphi \right)$ and $\sin \left(\varphi \right)$ (Fig. 1g), with the specific form determined by the symmetry of the mode^26,48. The phase of the mode driven by the two channels exhibit different ϕ dependencies (Fig. 1g,h shading of petals). Therefore, when the two channels are simultaneously activated, they superpose in a phase-sensitive way, leading to constructive or destructive interference. The interference between these two routes will lead to a polar pattern with unequal amplitudes when the direction of the driving field is flipped (ϕ → ϕ + 180°), breaking the C₂ symmetry along the b-axis of FePS₃ (Fig. 1i). Therefore, by scanning E_THz and ϕ, we can differentiate the two excitation methods. More importantly, we gain control of their relative strengths and can therefore coherently shape the symmetry of the MP dynamics.

Results

In our experiment, we measure the MPs by tracking the polarization state of an ultrashort, 800 nm probe pulse that passes through the sample. By controlling the relative delay Δt between the probe pulse and the THz excitation, the MPs appear as coherent oscillations in the probe pulse ellipticity (Δη). Figure 1k shows a typical Δη transient measured in FePS₃ at 10 K as a function of Δt. The initial positive signal close to Δt = 0 arises from the THz Kerr effect when pump and probe temporally overlap. A beating pattern emerges at later delays, which indicates the coexistence of multiple coherent oscillations. A fast Fourier transform (FFT) of the time trace in Fig. 1k reveals a series of peaks corresponding to various low-energy excitations of the system (Fig. 1l). Here, the three lowest energy modes (violet shadings) and the two modes around 3.7 THz (blue shadings) disappear above the magnetic transition temperature (inset of Fig. 1l). They have been shown to result from the hybridization between phonons and magnons^44–47. Therefore, they should be referred to as MPs rather than pure phonons or magnons. We label the bare phonons involved in the hybridization as P1, P1’, and P2, and the bare magnons as bare-M’, and bare-M. Due to hybridization, these modes (P1, P1’, P2, bare-M’, and bare-M) no longer represent the elementary excitations of the system. Instead, we only detect the hybridized modes MP1 (2.7 THz), MP1’ (2.8 THz), MP2 (3.3 THz), M’ (3.7 THz), and M (3.7 THz). Since the coupling strength under zero magnetic field is significantly smaller than the frequency differences between the magnons (bare-M’ and bare-M) and the phonons (P1, P1’, and P2)⁴⁴, the hybridization has negligible effects on the mode frequencies. Therefore we have $f_{{\mathrm{P}}_{\mathrm{i}}}\approx f_{{\mathrm{MP}}_{\mathrm{i}}}$ for ${\mathrm{P}}_{\mathrm{i}}=\mathrm{P1},{\mathrm{P1}}^{\prime },\mathrm{P2}$ and $f_{\mathrm{bare}-{\mathrm{M}}^{\prime }}\approx f_{{\mathrm{M}}^{\prime }}$, $f_{\mathrm{bare}-\mathrm{M}}\approx f_{\mathrm{M}}$. The modes MP1, MP1’, and MP2 predominantly exhibit phonon characteristics, while M’ and M are primarily magnon-like (Supplementary Note 10).

To gain insight into the symmetry and the excitation mechanisms of these modes, we investigate their amplitudes as a function of THz polarization ϕ^49,50. We rotate the THz polarization ϕ while keeping E_THz, the probe polarization, and the sample orientation unchanged. Mode amplitudes are determined by integrating the area under their respective FFT peaks or by fitting to damped harmonic oscillators (Supplementary Note 3). The excitation spectrum of the modes at different ϕ are shown by the color plot in Fig. 2. Here we focus on four modes: the 2.7 THz mode (MP1), 3.3 THz mode (MP2), the mode with higher frequency around 3.7 THz (M), and the 7.5 THz mode (P). Further details regarding the remaining modes are provided in Supplementary Notes 4 and 5. We first measure the polarimetry of different modes at a low E_THz (~50 kV/cm peak electric field). Both MP1 and MP2 show distorted two-petal patterns breaking the C₂ symmetry along the b-axis (Fig. 2a, b). For MP1, the two lobes are rotated towards ϕ = ± 45° (Fig. 2e), while for MP2, the left lobe is noticeably larger than the right lobe (Fig. 2f). In comparison, M shows two identical lobes (Fig. 2c, g), and P exhibits a symmetric nodeless pattern oriented along the a-axis (Fig. 2d, h). Both of M and P preserve the C₂ symmetry, in contrast with the observed asymmetry in MP1 and MP2. We then repeat the polarimetry measurements at a high E_THz (~150 kV/cm peak electric field), where M and P retain their equivalent polarimetry compared to their low E_THz cases (Fig. 2k, l, o, p). On the other hand, remarkable change in the polar pattern of MPs can be observed: MP1 develops two additional lobes along ϕ = ± 135° (Fig. 2i, m), while MP2 maintains two lobes but with increased asymmetry (Fig. 2j, n).

Fig. 2. ϕ dependence of the mode amplitudes.

a–d FFT spectra of MP1, MP2, M, and P as a function of THz polarization ϕ at low E_THz. The amplitudes are normalized to the maximal value for each mode (see colorbar at the bottom). The horizontal dashed line at 3.68 THz marks the frequency of M. The MP1, MP2, and M modes are measured at 30% of the maximal E_THz and the P mode is measured at 50% of the maximal E_THz. E_THz = 150 kV/cm. e–h Amplitudes of MP1 (e), MP2 (f), M (g), and P (h) as a function of ϕ extracted from (a–d). Solid lines are fits. MP1 is fit by the sum of Eq. (2) and Eq. (3) for a B_g mode. MP2 is fit by the sum of Eq. (2) and Eq. (3) for a A_g mode. M is fit by Eq. (3) for a A_g mode. P is fit by Eq. (2) for a A_g mode. The white and gray shadings in the polar plots indicate opposite phases. The numbers on the polar plots represent the maximal amplitudes normalized to the maximal amplitude measured at 30% of the maximal E_THz of different modes. The color bars next to the polar plots indicate the relative strengths of the nonlinear (red) and linear (blue) pathways obtained by the fitting. i–p are identical to panels (a–h) but acquired at 100% of the maximal E_THz. The error bars show experimental standard error. q–u Schematics of the interference. q shows a typical fitting (violet) of the ϕ dependence of MP1 at highest E_THz (m), along with the linear (blue) and nonlinear (red) components of the fitting. r–u shows schematics of the MP oscillations excited by the linear channel alone (L, blue curve), the nonlinear channel alone (NL, red curve), and both linear and nonlinear channels (L+NL, violet curves). The blue shaded regions highlight different phases of the oscillations. The curves are offset for clarity.

To quantitatively understand the distinct symmetries of the polar patterns, we fit the nonlinear and linear excitation channels based on the symmetry of the the Raman phonons and the magnon, respectively. The point group 2/m allows Raman-active phonons with A_g and B_g irreducible representations (irreps). Based on the symmetry constraints, both the nonlinear and linear driving forces must share the same irrep as the corresponding phonon mode (see Supplementary Note 7). Therefore, their ϕ-dependences can be derived from the basis functions of A_g and B_g irreps in the 2/m point group (Supplementary Note 7). To this end, we expect the following ϕ-dependence of the driven amplitudes of A_g and B_g phonons from nonlinear channels:

$$A_{\mathrm{NL},A_g}=a_1{E}_a^2+a_2{E}_b^2=\left(a_1{\cos }^2\varphi +a_2{\sin }^2\varphi \right)E_{\mathrm{THz}}^2,A_{\mathrm{NL},B_g}=2a_3{E}_a{E}_b=2a_3\cos \varphi \sin \varphi E_{\mathrm{THz}}^2,$$ 2

where E_a and E_b are the THz electric fields along the crystallographic a- and b-axes, respectively, and a₁, a₂, a₃ are fitting coefficients (Supplementary Note 7). Importantly, Eq. (2) is valid for any potential quadratic nonlinear pathway, making our description universally applicable. For the linear channels, since the magnetization along a- and b-axes (M_a and M_b) respects B_g and A_g symmetry, respectively, the ϕ-dependence of the linear excitation channel of MP can be described as:

$$\begin{array}{c}{A}_{\mathrm{L},A_g}\propto M_b\propto B_{\mathrm{THz}}\cos \varphi =b_1{E}_{\mathrm{THz}}\cos \varphi ,\hfill \\ A_{\mathrm{L},B_g}\propto M_a\propto B_{\mathrm{THz}}\sin \varphi =b_2{E}_{\mathrm{THz}}\sin \varphi ,\hfill \end{array}$$ 3

where B_THz is the magnetic field of the driving THz pulse, which is perpendicular to the electric field, and b₁, b₂ are fitting coefficients (Supplementary Note 7). The absolute value and sign of Eq. (2) and Eq. (3) represent the amplitude and phase of the oscillations induced by each pathway. The distinct angular dependence of the linear and nonlinear pathways allow for the control of their relative phase and therefore the interference. The C₂ symmetry breaking behavior of MP1 and MP2 strongly suggests that they are simultaneously driven by both channels, consistent with their MP nature. We can quantitatively fit the polar patterns of MP1 and MP2 by the sum of Eq. (2) and Eq. (3) (violet lines in Fig. 2e, f, m, and n) of B_g and A_g symmetries, respectively. The mode symmetries determined here agree with assignments from polarization-dependent Raman studies⁵¹. Moreover, the pattern change of MPs at different E_THz can be explained by the alteration in the relative strengths of the nonlinear and linear pathways, which can be extracted by the ratio between the maximum fit values of the nonlinear and linear polar patterns (color bars next to each polar pattern in Fig. 2). We find that the relative strengths not only depend on E_THz but also differ between the two MPs. For MP1, the dominant excitation pathway switches from linear at low E_THz to quadratic at high E_THz. On the other hand, for MP2, the linear pathway dominates within our attainable E_THz range. While quantitative comparison between the linear excitation efficiency b_1,2 of the MPs is difficult due to their different detection sensitivity, qualitatively, the large linear to nonlinear ratio of MP2 can be attributed to the small frequency difference between MP2 and M, with additional influence from the linear magnon-phonon coupling strength and nonlinear excitation coefficients (Supplementary Note 10). In comparison, M and P can be fit by considering only the linear A_g (Eq. (3)) and the quadratic A_g pathways (Eq. (2)) at all E_THz values, respectively (blue and red lines in Fig. 2g, h, o, and p). This indicates that the hybridization has negligible effects on the excitation of M, allowing it to be treated as almost a pure A_g magnon, and P is an A_g phonon that does not significantly hybridize with the magnons. Therefore, our method not only reveals and controls the nonlinear and linear excitation pathways, but also identifies the nature of the modes and determine their symmetries, without requiring an external magnetic field.

The agreement between Eqs. (2), (3) and the experiments unambiguously points out that the polar patterns of MP1 and MP2 arise from interference between the linear and nonlinear pathway. A schematics of this interference for MP1 at high E_THz (Fig. 2i and m) is provided in Fig. 2q–u for demonstration. At ϕ = 45° (Fig. 2t), the linear (blue curve) and nonlinear pathways (red curve) are in phase, resulting in constructive interference (violet curve). Rotating the THz polarization by 180° to ϕ = 225° causes the linear channel to experience a π phase shift (Eq. (3)), while the phase of the nonlinear channel remains unaffected (Eq. (2)). Consequently, destructive interference occurs, exhibiting smaller amplitude compared to the constructive interference at ϕ = 45°.

The phase of the interference pattern of MP1 (violet curves) remains unchanged for ϕ = 45° and ϕ = 225° due to the dominance of the nonlinear channel. However, when the THz polarization moves to around ϕ = 135° and ϕ = 315°, the nonlinear channel undergoes a π-phase shift, causing the interference pattern to flip phase (violet curves). In this case, constructive interference occurs around ϕ = 315°, while destructive interference is observed at ϕ = 135°.

To corroborate our interpretations and directly demonstrate the existence of both linear and nonlinear pathways, we measure the E_THz dependencies of mode amplitudes at characteristic ϕ values where the linear and nonlinear responses should be nearly maximal based on the polarimetry analysis. The amplitudes of MP1 and MP2 exhibit both linear and nonlinear behaviors depending on ϕ. Specifically, at ϕ = 92°, which is around the maximum of the linear channel (Eq. (3)) and the node of the nonlinear channel for B_g (Eq. (2)), the amplitude of MP1 shows nearly linear dependence on E_THz (Fig. 3a dotted blue line). The E_THz dependence becomes almost quadratic when we rotate the THz polarization to ϕ = 52°, close to the maxima of the nonlinear channel (Fig. 3b dotted red line). For MP2, the linear and quadratic dependence on E_THz appear at ϕ = 12° (Fig. 3c) and ϕ = 92° (Fig. 3d), respectively, distinct from MP1. The E_THz-dependencies of the MP amplitudes can all be quantitatively captured by Eq. (1). This combined linear and quadratic E_THz-dependencies at distinct ϕ values confirm the coexistence of the two pathways. In contrast, such strong ϕ dependence is not observed for M and P. At all selected ϕ values, the amplitude of M is linear in E_THz (Fig. 3e), validating the magnetic-dipole excitation pathway. Similarly, the amplitude of P consistently shows a quadratic E_THz-dependence (Fig. 3f), as expected for the sum-frequency excitation of a Raman-active phonon mode.

Fig. 3. E_THz dependence of the mode amplitudes.

a, b MP1 amplitude as a function of E_THz obtained at ϕ = 92° and ϕ = 52°, respectively. c, d, MP2 amplitude as a function of E_THz obtained at ϕ = 12° and ϕ = 92°, respectively. e M amplitude as a function of E_THz at different ϕ values. Data are normalized for a better comparison. f P amplitude as a function of E_THz at different ϕ values. Data are normalized for better comparison. In all the panels, red and blue lines are quadratic and linear fits, while violet lines are the fits to Eq. (1). The insets show the THz polarization with respect to the crystallographic axes. E_THz is normalized by 180 kV/cm. The chosen ϕ values are not perfectly aligned with the crystallographic axes due to difficulties in sample alignment.

The combined measurements of ϕ and E_THz dependencies of the MP1 and MP2 amplitudes substantiate the coexistence of linear and nonlinear pathways and their interference. To further corroborate the symmetry of different MPs, we fit the E_THz-dependence of MP amplitudes using Eq. (1) systematically at various ϕ values. We thereby obtain the ϕ-dependence of parameters a and b of Eq. (1), which directly reflect the symmetry of the nonlinear and linear excitation channels. Notably, a(ϕ) of MP1 exhibits four equal petals, each offset by 45° from the crystallographic axes (Fig. 4a), suggesting the B_g-symmetry as we previously assigned. Conversely, a(ϕ) of MP2 shows four petals aligned with the crystallographic axes (Fig. 4b), in agreement with the expected A_g-symmetry. Similarly, b(ϕ) of MP1 and MP2 exhibit two equal lobes aligned with the b- and a-axes (Fig. 4c and d), respectively, consistent with the assignment of B_g- and A_g-symmetry as predicted by Eq. (3). Hence, the coexistence of both nonlinear and linear channels yields either constructive (a(ϕ)b(ϕ) > 0, see the shadings of Fig. 4a–d) or destructive (a(ϕ)b(ϕ) < 0) interference depending on the mode symmetry and ϕ, giving rise to the ostensible C₂-symmetry breaking.

Fig. 4. Nonlinear and linear pathways for MP excitation and their applications.

a, b Nonlinear pathway efficiency a for MP1 (a) and MP2 (b) as a function of ϕ. Red lines are fits for the B_g and A_g modes to Eq. (2). c, d Linear pathway efficiency b for MP1 (c) and MP2 (d) as a function of ϕ. Blue lines are fits for the B_g and A_g modes to Eq. (3). The white and gray shaded regions indicate different signs of the coefficients a and b, which will lead to opposite phases of MP oscillations. The error bars show the standard error of fittings. e Schematics of the dynamical phase diagram induced by coherent MPs. Red region: M_z ≠ 0 state induced by MP2 net displacement Q₀. The red and blue spheres are Fe atoms with equilibrium spin up and spin down. The bonds with exchange interaction enhanced (thicker bonds) and weakened (thinner bonds) by the phonon displacement are highlighted in yellow. The red and blue arrows indicate the average spin of different Fe atoms upon THz excitation. Top inset shows the schematic of the potential energy landscape change upon driving the MP through the nonlinear displacive pathway. Blue region: spin-canting state induced by MP2 oscillation Q₁. The solid yellow lines depict the new spin interaction terms (S_i × S_j) ⋅ (S_k × S_l) induced by hopping between Floquet bands. Left inset shows schematic of the potential energy landscape upon driving the MP through the linear pathway, where the phonon oscillates symmetrically around the equilibrium position. White region: more magnetic phases induced by different relative strength between Q₀ and Q₁.

To substantiate our phenomenological descriptions and quantitatively understand the experimental observations, we perform microscopic dynamical simulations (Methods). We start by constructing an equilibrium Hamiltonian describing the spin degree of freedom which includes magnetic exchange interactions and anisotropy with all the microscopic parameters determined by first-principles calculations. We then derive the spin dynamics by solving the Landau-Lifshitz-Gilbert (LLG) equation based on the Hamiltonian with a time-dependent magnetic driving field. This allows us to obtain the dynamics of the magnon-induced magnetization M(t), which determines the nonlinear effective force T_Q(t) (see Supplementary Note 6 and 8 for more discussions) and the linear effective force from the magnetic-dipole process gM(t), where g is the magnon-phonon coupling strength. The dynamics of the MP can then be simulated by its equations of motion:

$$\ddot{Q}+2{\gamma }_Q\stackrel{°}{Q}+{\omega }_Q^{2}Q=T_Q\left(t\right)+gM\left(t\right),$$ 4

where Q, γ_Q, and ω_Q are the displacement amplitude, damping rate, and frequency of the MP. By comparing the theoretical simulations with the experimental observations, we achieve a remarkable agreement in terms of the dependence on E_THz and ϕ, confirming the validity of our proposed excitation mechanisms (Supplementary Note 8).

Discussion

Therefore, we have comprehensively demonstrated that by selecting the appropriate ϕ and E_THz values, we can achieve a nonlinear-to-linear excitation crossover. Interestingly, for a specific phonon mode that exhibits strong coupling to the spin, such tunability allows different mechanisms for phonon-mediated control and potentially a range of different magnetic phases inaccessible in equilibrium. As an outlook, we theoretically explore different scenarios by coherently exciting MP2. In the nonlinear pathway, the quadratic driving generates a unidirectional force that results in a net displacement along the phonon coordinate⁵²(Fig. 4e top inset), which lasts at least within the laser pulse duration. Such displacement can effectively and selectively modify exchange interactions, giving rise to a net magnetization along the spin direction (Fig. 4 red region). Our frozen phonon calculations show that the phonon displacement Q₀ of MP2 can bilinearly couple to the magnetization along z (M_z) and AFM order parameter (L): $\mathcal{H}\propto Q_{0}L{M}_z$⁵³. The presence of displacement will thus generate a finite magnetization in the AFM motif ^22,53. On the other hand, the linear excitation induces a symmetric oscillation Q₁(t) of the Raman-active phonon mode without net displacement (Fig. 4e left inset). This coherent periodic modulation can produce phonon-dressed side bands⁵⁴, establishing a fertile playground for manipulating electronic and magnetic properties via the phonon Floquet mechanism^55,56 (Fig. 4e blue region). Our microscopic calculations demonstrate that to the lowest order, the phonon Floquet Hamiltonian produce interactions between four nearest neighbour spins: $\mathcal{H}\propto Q_1{\left(t\right)}^2\left({\mathbf{S}}_i\times {\mathbf{S}}_j\right)\cdot \left({\mathbf{S}}_k\times {\mathbf{S}}_l\right)$, where S are spins at nearest neighbour sites i, j, k, l. These terms are only nonzero when the spins are neither parallel nor anti-parallel to each other, thereby favoring a state with canted magnetic order (Fig. 4e blue region and Supplementary Note 9). Compared to the conventional photon-based Floquet engineering of magnetism^57–60, the extremely long lifetime (Supplementary Note 6) of the MPs in our sample allows for a sustained Floquet engineering effect that persists for over 100 ps. Finally, the crossover between nonlinear and linear excitation controlled by THz polarization ϕ and field strength E_THz tips the intricate balance between different ground states with distinguished magnetic structures, allowing for the access to a rich landscape of magnetic phases without equilibrium analogues.

Altogether, we have demonstrated not only linear and nonlinear coherent excitation of MPs that are dominated by their phononic components but also the ability to manipulate their ratio by adjusting the strength and polarization of the driving field, irrespective of the type of nonlinear excitation route involved. Our high-field, broadband THz pulses are crucial for the simultaneous activation of the linear and nonlinear excitation pathways (Supplementary Note 6 and ref. ³⁹). Such capability enables the control over the dynamical symmetry of MPs and significantly expands the horizon of magnon- and phonon-mediated control over material properties on demand, a burgeoning field known as magno-phononics. Furthermore, applying the methodology developed in this work to the recently discovered chiral MPs in a similar compound FePSe₃ and other systems with strong magnon-optical phonon interactions^61,62 affords an exceptional avenue for the manipulation of phonons and spins^63,64. We also note that while combining high-field THz pulses with an external magnetic field is technically challenging, it offers the potential for more volatile control over the ratio between the magnon and phonon excitation pathways for each MPs. By tuning the magnon eigenvectors and frequencies through magnetic fields — for instance, bringing magnon frequencies closer to phonon frequencies — the MP1 and MP2 modes could become magnon-dominated, enhancing the linear pathway while reducing the nonlinear pathway (Supplementary Note 10). We also envision that further nonlinear THz spectroscopy measurements on FePS₃ will provide conclusive insights into the microscopic nature of the two-particle excitation pathways for different modes⁶⁵ (Supplementary Note 6).

Methods

Sample preparation

FePS₃ single crystals were synthesized from iron (Sigma-Aldrich, 99.99% purity), phosphorus (Sigma-Aldrich, 99.99%), and sulfur (Sigma-Aldrich, 99.998%) using the chemical vapor transport method. The powdered elements were prepared inside an argon-filled glove box. We weigh the starting materials in the correct stoichiometric ratio and added an additional 5 wt of sulfur to compensate for its high vapor pressure. We carried out the chemical analysis of the synthesized single-crystal using a COXEM-EM30 scanning electron microscope equipped with a Bruker QUANTAX 70 energy-dispersive X-ray system and confirmed the correct stoichiometry. The crystal structure of the sample is checked with a commercial X-ray diffractometer (Rigaku Miniflex II). Single crystals were cleaved along the [001] direction immediately before the experiment. The sample thickness after exfoliation is 20 μm as measured by a Bruker Dektak DXT-A stylus profilometer.

THz experiment

A detailed description can be found in Supplementary Note 2. The broadband THz pump is generated by pumping N-benzyl-2-methyl-4-nitroaniline (BNA) crystal with the 1300 nm output from an optical parametric amplifier (OPA) seeded by a Ti:Sapphire amplifier at a repetition rate of 1 kHz. A weak 800 nm pulse from the Ti:Sapphire amplifier is focused on the sample to probe the phonon and magnon dynamics through the light-induced change in linear birefringence. The THz and 800 nm pulses are incident normally on the sample. The ellipticity of the 800 nm pulse after transmitting through the sample is measured by the balanced detection scheme with a pair of photodiodes. The output of the photodiodes is collected by a data acquisition (DAQ) card.

Theoretical simulations

The microscopic Hamiltonian describing the spin degree of freedom of the system can be expressed as:

$$\mathcal{H}=J_1{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j+J_2{\sum }_{⟨⟨ij⟩⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j+J_3{\sum }_{⟨⟨⟨ij⟩⟩⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j-\mathrm{\Delta }{\sum }_i{S}_{iz}^2-\gamma \mathbf{B}\cdot {\sum }_i{\mathbf{S}}_i,$$ 5

where S_i is the spin of Fe at site i. The first three terms represent Heisenberg interactions between nearest, next-nearest, and next-next-nearest in-plane neighbors, the fourth term corresponds to an Ising type easy-axis anisotropy, and the last term is the Zeeman energy due to an in-plane magnetic field which represents the THz driving field. Interactions between layers can be neglected. γ = gμ_B/ℏ is the gyromagnetic ratio. The values of all exchange interactions and anisotropies have been obtained through first-principles calculations^44,53. Then we solve the Landau-Lifshitz-Gilbert (LLG) equation, which describes the dynamics of the motion of spins under an effective magnetic field ${\mathbf{H}}_{\mathrm{eff}}=-\frac{\partial \mathcal{H}}{\gamma \partial {\mathbf{S}}_i}$. It can be expressed as:

$$\frac{d{\mathbf{S}}_i}{dt}=\frac{\gamma }{1+{\alpha }^2}\left[{\mathbf{S}}_i\times {\mathbf{H}}_{\mathbf{eff}}+\frac{\alpha }{\mid {\mathbf{S}}_i\mid }{\mathbf{S}}_i\times \left({\mathbf{S}}_i\times {\mathbf{H}}_{\mathrm{eff}}\right)\right],$$ 6

where γ = gμ_B/ℏ is the gyromagnetic ratio and α is a phenomenological Gilbert damping constant that accounts for energy dissipation. We can then solve the dynamical equation for the magnetization M, summing over the four spins within one unit cell: M = S₁ + S₂ + S₃ + S₄. As we solve the dynamics of M, we can study their impact on phonons through magnon-phonon coupling based on the symmetry constraints. In the linear coupling channel, A_g phonon exclusively couples to A_g magnon while B_g phonon exclusively couples to B_g magnon, which induce magnetization along y (M_y) and x (M_x), respectively. Thus, we have:

$$\begin{array}{c}{\ddot{Q}}_{Ag}+2{\stackrel{°}{Q}}_{Ag}/{\tau }_{Ag}+{\omega }_{Ag}^2{Q}_{Ag}=g_{Ag}{M}_y,\hfill \\ {\ddot{Q}}_{Bg}+2{\stackrel{°}{Q}}_{Bg}/{\tau }_{Bg}+{\omega }_{Bg}^2{Q}_{Bg}=g_{Bg}{M}_x,\hfill \end{array}$$ 7

where Q_Ag/Bg, τ_Ag/Bg, ω_Ag/Bg, and g_Ag/Bg are the phonon coordinate, phonon lifetime, phonon frequency, and phonon-magnon coupling constant of A_g and B_g symmetry phonons, respectively. For the nonlinear driving, we consider the one-magnon-one-photon excitation pathway. As discussed in Supplementary Note 7, other quadratic driving pathways should produce qualitatively similar E_THz and ϕ dependence. Based on the symmetry, we have:

$$\begin{array}{c}{\ddot{Q}}_{Ag}+2{\stackrel{°}{Q}}_{Ag}/{\tau }_{Ag}+{\omega }_{Ag}^2{Q}_{Ag}=g_{Ag1}{M}_y{B}_y+g_{Ag2}{M}_x{B}_x,\hfill \\ {\ddot{Q}}_{Bg}+2{\stackrel{°}{Q}}_{Bg}/{\tau }_{Bg}+{\omega }_{Bg}^2{Q}_{Bg}=g_{Bg1}{M}_y{B}_x+g_{Bg2}{M}_x{B}_y,\hfill \end{array}$$ 8

where g_Ag/Bg are the nonlinear phonon-magnon coupling constants of A_g and B_g-symmetry phonons, respectively. The simulation results of LLG simulations are elaborated in Supplementary Note 8.

Author contributions

T.L., H.N., and B.I. performed the measurements. H.N. performed dynamical simulations with help from E.V.B. E.V.B. and A.R. developed the theoretical model of phonon Floquet effect. J.P. and J.K. synthesized and characterized FePS₃ single crystals under the supervision of J.-G.P. T.L., H.N., A.v.H. and B.I. performed the data analysis. T.L., H.N., B.I., A.v.H., D.M.J., E.V.B. and N.G. interpret the data and wrote the manuscript with critical inputs from all other authors. The project was supervised by N.G.

Peer review

Peer review information

Nature Communications thanks the anonymous, reviewers for their contribution to the peer review of this work. A peer review file is available.

Data availability

The experimental data associated with this paper are available on the Harvard Dataverse at 10.7910/DVN/LTGABG⁶⁶.

Code availability

The code used for the current study are available from the corresponding author on request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-62091-4.