Coupling of terahertz light with nanometre-wavelength magnon modes via spin–orbit torque

Spin-based technologies can operate at terahertz frequencies but require manipulation techniques that work at ultrafast timescales to become practical. For instance, devices based on spin waves, also known as magnons, require efficient generation of high-energy exchange spin waves at nanometre wavelengths. To achieve this, a substantial coupling is needed between the magnon modes and an electro-magnetic stimulus such as a coherent terahertz field pulse. However, it has been difficult to excite non-uniform spin waves efficiently using terahertz light because of the large momentum mismatch between the submillimetre-wave radiation and the nanometre-sized spin waves. Here we improve the light–matter interaction by engineering thin films to exploit relativistic spin–orbit torques that are confined to the interfaces of heavy metal/ferromagnet heterostructures. We are able to excite spin-wave modes with frequencies of up to 0.6 THz and wavelengths as short as 6 nm using broadband terahertz radiation. Numerical simulations demonstrate that the coupling of terahertz light to exchange-dominated magnons originates solely from interfacial spin–orbit torques. Our results are of general applicability to other magnetic multilayered structures, and offer the prospect of nanoscale control of high-frequency signals. Engineering of the spin–orbit interactions in a magnetic multilayered structure makes it possible to coherently generate coherent spin waves using terahertz radiation, which could benefit the development of spintronic devices.

Spin-based technologies can operate at terahertz frequencies but require manipulation techniques that work at ultrafast timescales to become practical. For instance, devices based on spin waves, also known as magnons, require efficient generation of high-energy exchange spin waves at nanometre wavelengths. To achieve this, a substantial coupling is needed between the magnon modes and an electro-magnetic stimulus such as a coherent terahertz field pulse. However, it has been difficult to excite non-uniform spin waves efficiently using terahertz light because of the large momentum mismatch between the submillimetre-wave radiation and the nanometre-sized spin waves. Here we improve the light-matter interaction by engineering thin films to exploit relativistic spin-orbit torques that are confined to the interfaces of heavy metal/ferromagnet heterostructures. We are able to excite spin-wave modes with frequencies of up to 0.6 THz and wavelengths as short as 6 nm using broadband terahertz radiation. Numerical simulations demonstrate that the coupling of terahertz light to exchange-dominated magnons originates solely from interfacial spinorbit torques. Our results are of general applicability to other magnetic multilayered structures, and offer the prospect of nanoscale control of high-frequency signals.
The increasing demand for fast and energy-efficient communication and data processing stimulates new concepts for terahertz spintronics and magnonics. These concepts require ultrafast (that is, at timescales of picoseconds and below) manipulation of the magnetic spin degree of freedom for the generation of spin currents and short-wavelength magnons at terahertz frequencies [1][2][3] . Owing to their terahertz spin dynamics, antiferromagnetic materials have been the particular focus of research within the past decade [4][5][6] . Terahertz spin pumping via antiferromagnetic magnon excitations can also be realized, as demonstrated recently 7,8 . To generate resonant excitations at terahertz frequencies in technologically relevant metallic ferromagnets, high-energy spin waves have been proposed [9][10][11][12][13][14] . In these studies, nanoscale-wavelength spin waves are launched by making use of laser-pulse-driven spin currents-that is, optically induced spin-polarized electrons that move Article https://doi.org/10.1038/s41567-022-01908-1 Different substrates are used to account for the back-reflected terahertz signal in transmission geometry. We employed a 50 mT external magnetic field (H Ext ) to keep the samples magnetically saturated with the magnetization (M) in the film plane, with the option to make it parallel or orthogonal to the magnetic field component of the incident terahertz pulse (H THz ). Figure 1b presents a comparison of the time-resolved magnetization dynamics in asymmetric Ta/Py/Pt versus symmetric Ta/Py/Ta samples (the symmetric Pt/Py/Pt sample shows an identical response to the Ta/Py/Ta; see Supplementary Note 1 and Supplementary Fig. 1). A typical coherent response driven by the Zeeman interaction between the Py magnetization and H THz (seen as a single-cycle transient of about 1 ps duration) is evident for all samples 17,[20][21][22][23] . The Zeeman torque-driven magnetization dynamics in the asymmetric and symmetric samples result in a similar signal, which we will refer to as the Zeeman torque signal (ZTS). In contrast to the samples with symmetric interfaces, once the terahertz pulse has passed through the sample, the signal of the Ta/Py/Pt sample clearly exhibits additional features in the form of oscillations at longer delay times (see the black curve in Fig. 1b). The amplitude of the magnetization's precessional signal, A SWR , is about 7.5% of the ZTS amplitude A ZTS for this particular sample, as indicated by the dashed lines in Fig. 1b. The frequency of the oscillations is calculated to be at 0.29 ± 0.01 THz, which we attribute to the SWR mode with n = 2 in equation (1). Both the ZTS and SWR show a linear response with respect to the amplitude of the terahertz field (Supplementary Note 2 and Supplementary Fig. 2), indicating that they are not due to the heat deposited by the radiation. Furthermore, the absence of the SWR in the Ta/Py/Ta (and Pt/Py/Pt) sample suggests that H THz alone cannot be the driving force for the detected spin-wave excitations. In the following, we show instead that the SOT is the mechanism responsible for the observed magnon modes, employing the characteristic symmetry of SOTs induced by the terahertz field.
To confirm that both ZTS and SWR signals from the Ta/Py/Pt sample are magnetic in origin, we present in Fig. 1c the response of the Ta(3 nm)/Py(8.6 nm)/Pt(2 nm) sample with the magnetization aligned perpendicular (black curve) or parallel (red curve) to H THz . The vanishing ZTS in the parallel configuration indicates its magnetic origin, since the Zeeman torque is zero for a magnetization oriented parallel to H THz (ref. 23 ). Similarly, the SOT is inefficient for electrical fields orthogonal to the in-plane magnetization [26][27][28][29][30] , hence the SWR signal was not detected. We attribute the small ZTS visible in Fig. 1 to a slight misalignment, possibly also due to a non-ideal polarization of the terahertz field. We also checked the effect of reversing the external magnetic field H Ext on the Faraday signal (Fig. 1d). Both the coherent and the oscillatory responses reversed their sign on reversal of the sign of H Ext . The sign inversion of the ZTS was expected, as the Zeeman torque is asymmetric under magnetization reversal 23 . The asymmetric behaviour of the SWR signal is a further indication that the SOT is the driving mechanism, as we reason below.
To explain the asymmetric behaviour of the SWR signal in Fig. 1d, we start with the Landau-Lifshitz-Gilbert equation: (2) In equation (2) m = M/M s is the magnetization unit vector normalized to the magnetization at saturation M s , γ is the gyromagnetic ratio, t is time and α is the Gilbert damping parameter. The effective field H eff is given by: where we consider the SOT-induced field-like H FL and damping-like H DL equivalent fields with H FL ∝ (z × j) and H DL ∝ (j × z) × m (refs. 26,29,30 ).
Here j is the electrical current density induced by the terahertz field into the interface region of an adjacent ferromagnetic layer. There, the net angular momentum flow creates a spin-transfer torque that acts on the magnetization, consequently exciting high-frequency spin dynamics. As a result, an interface-localized broadband wave packet of nanometre-wavelength magnons launches from the top surface into the depth of the ferromagnetic layer 11 , followed by its reflection at the layer's bottom surface. This finally results in perpendicular standing spin waves (PSSW), which are detected by using magneto-optical techniques [9][10][11][12][13][14] .
Besides optical light, broadband terahertz radiation has been suggested as a means to access ultrafast magnetization dynamics [15][16][17][18][19][20][21][22][23][24][25] . In this case, the magnetic field component of the terahertz pulse is coherently coupled to the spin system via the Zeeman interaction, and accordingly results in a torque that drives magnetization dynamics at picosecond timescales [15][16][17][18][19][20][21][22][23][24] . However, the excitation of non-uniform and short-wavelength coherent spin waves using a terahertz light source has not been reported yet. This is due to the very inefficient coupling involved, as the momentum of the terahertz light and that of short-wavelength magnons are different by orders of magnitude. Here we overcome this obstacle and demonstrate an efficient and coherent excitation of nanometre-wavelength spin-wave modes in Ta/Py/Pt (where Py is Ni 81 Fe 19 ) trilayers using a broadband terahertz pulse with about 1 ps duration. Unlike the above-mentioned excitation schemes, the coupling of the terahertz radiation to the spin waves is realized via the spin-orbit torque (SOT) 26 , which has been predicted to be efficient even at terahertz frequencies 27,28 . We used a single-cycle terahertz pulse to induce a current in the Ta and Pt metallic layers. This electrical current was then converted into a spin current via the spin Hall effect, which in turn generated a torque acting on the magnetization of the Py layer. Owing to the strong localization of the spin polarization, short-wavelength coherent magnons were excited at the two Py interfaces, and propagated towards each other, resulting in PSSW modes. We demonstrate that the terahertz SOT induces PSSW modes in Py films of 6-12 nm thickness with mode dispersion in agreement with the exchange-dominated analytical relation of the spin-wave-resonance (SWR) frequency f SWR (k) (Kittel formula) given by: (1) In equation (1) f 0 is the uniform mode frequency of the ferromagnetic layer, D ex is the exchange stiffness parameter, ħ is the reduced Planck constant and k = πn/d Py is the wavenumber, which is indexed by the number of nodes, n, in the Py layer of thickness d Py . As we show below, no PSSW modes were detected in Ta/Py/Ta and Pt/Py/Pt trilayers with symmetric interfaces, confirming that the direct coupling of the magnetic field of the terahertz radiation to the Py layer cannot be the source of the detected spin waves.

Experiment
To study spin dynamics, we employed time-resolved magneto-optics in a Faraday configuration, as schematically shown in Fig. 1a. An intense single-cycle terahertz pulse is focused onto a trilayer sample and the sample's magnetization state is then probed by polarization rotation of a delayed 100 fs probe pulse with a central wavelength of 800 nm. The terahertz and probe pulses are collinear and have normal incidence with respect to the sample surface. In this configuration, the polarization of the probe pulse is sensitive to the out-of-plane (z axis) magnetization component, which is driven by the terahertz radiation. A more detailed description of the pump-probe experiment is provided in the Methods. We studied Ta(3 nm)/Py(9 nm)/Pt(2 nm) samples, as shown in Fig. 1a (the thickness of each individual layer is given in parentheses). As reference samples, we also used Ta(3 nm)/Py(9 nm)/Ta(3 nm) and Pt(2 nm)/Py(9 nm)/Pt(2 nm) samples with symmetric interfaces. All samples were deposited on transparent double-sided polished quartz glass, as well as on Al 2 O 3 substrates with a substrate thickness of 3 mm. Article https://doi.org/10.1038/s41567-022-01908-1 and z is the unit vector normal to the sample plane. According to equations (2) and (3), the field-like torque is antisymmetric on inversion of the magnetization, whereas the damping-like torque is symmetric. This points to the fact that the field-like torque is responsible for the terahertz-generated SWR signals, explaining the 180° phase shift in Fig. 1d. The reason for the field-like torque prevailing over the damping-like one in our experiment requires further study. We note, however, that when the magnetization lies in the film plane, H FL could be much larger than H DL , in contrast to a canted magnetization state 31 . Finally, we exploited the asymmetry of the field-like torque when the sample was flipped with respect to the incident beam direction. This is shown schematically in Fig. 2a.
According to equations (2) and (3), inverting the order of the Ta/ Py/Pt layers with respect to the beam direction should not change the sign of the signals induced by H THz , whereas H FL is expected to reverse its direction due to the opposite sign of the spin Hall angle in Ta and Pt, as illustrated in Fig. 2a. Accordingly, opposite directions of the torque would lead to a 180° phase shift in the dynamical components of the magnetization. This scenario is revealed in our experiment in Fig. 2b: when the sample is mounted with the Ta layer side towards the incoming beam, the ZTS induced by H THz is identical to that observed with the flipped sample (that is, with the Pt layer towards the incoming beam). The SWR signal, however, displays a 180° phase shift, confirming the SOT scenario illustrated in Fig. 2a. Exploiting the opposite symmetry of the two main effects, we could isolate the dynamics of the SOT-induced effect by determining the difference of the signals recorded with the sample mounted in opposite configurations. The difference signal, shown in the inset of Fig. 2b, reveals clear damped oscillations at a frequency of 0.29 ± 0.01 THz.
The time delay scans of the difference signal for different Py thicknesses are presented in Fig. 3a. All samples displayed damped oscillations, and we fitted our data using a decaying cosine function. The thickest 12 nm Py sample exhibited a more complex dynamical behaviour, therefore the best fit to the curve was achieved using a superposition of three damped cosine functions 32 : where ψ (t) is the time-dependent difference signal, a i and φ i are the initial amplitudes and phases of the ith mode and f i and τ i are the frequency modes and characteristic damping times. The inset in Fig. 3b shows the mode frequencies as a function of k 2 for the 12 nm Py sample. The dependence on the wavenumber squared is evident, which we fitted using equation (1). The SWR frequency as a function of Py layer thickness for all Ta/Py (d Py )/Pt samples is plotted in Fig. 3b. As expected, the frequency increased with decreasing Py thickness and reached 0.6 ± 0.01 THz for d Py = 6 nm. The fit using equation (1) revealed that the SWR mode in all samples corresponded to the n = 2 mode, and the extracted exchange stiffness was D ex = 240 ± 20 meV Å 2 (A ex = 8 ± 0.8 pJ m −1 , where A ex is the exchange constant). This value is similar to that reported previously for thin Py films 33,34 ; furthermore, the fit to the multiple modes of the 12 nm Py sample (inset in Fig. 3b) revealed identical D ex , indicating that the exchange stiffness did not change much when the Py thickness was decreased down to 6 nm. Modes larger than n = 2 became visible in 12 nm Py as the thickness of the metallic sample was comparable to the probe light penetration depth, which was typically 10-15 nm (refs. [35][36][37]. The details of the mode-selective detection in our experiment are discussed below. The relaxation time of the SWR increased with Py thickness. The effective damping parameter of the n = 2 mode in the 9 nm Py sample was calculated to be α eff = 0.062 ± 0.002, which is comparable to the values reported for the PSSW modes at 0.3 THz (refs. 12,13 ). This value, however, is larger than α eff = 0.014 of the n = 2 mode in 50-nm-thick Py 32 , suggesting a substantial wave number-dependent damping contribution at terahertz frequencies 12,38,39 .

Micromagnetic modelling
To gain further insight into the origin of the SWR signal in the Ta/Py/Pt asymmetric system, we modelled the magnetization dynamics microscopically using the MuMax3 code 40 (see the Methods for details). The magnetic properties of the samples, as well as the asymmetric pinning due to different surface anisotropies at Ta/Py and Py/Pt interfaces, were extracted from the ferromagnetic resonance characterization discussed in the Supplementary Note 3 and Supplementary Fig. 3. We note that no SWR was obtained in the simulations when a homogeneous or linearly decaying profile of H THz (ref. 22 ) was assumed within the Py thickness. This also confirmed that the direct action of H THz on the magnetization (Zeeman torque itself) cannot be the origin of the SWR signal. Article https://doi.org/10.1038/s41567-022-01908-1 Having verified that the terahertz magnetic field does not contribute to the spin-wave excitation, we estimated the magnitude of H FL . To do so, we first considered the large refractive index of the metallic films, and calculated the absorbed terahertz power using the transfer matrix method [41][42][43] . Next, we estimated the electric current density, which causes the terahertz power absorption (see Supplementary Note 4 and Supplementary Fig. 4 for details). The current density was calculated to be about j = 10 12 A m −2 . We assumed an SOT efficiency µ 0 H FL /j = 10 −14 T A −1 m 2 (ref. 26 ) and, accordingly, arrived at µ 0 H FL = 10 mT. We further localized the field b SOT = µ 0 H FL = 10 mT at the two interfaces of the 9-nm-thick Py layer within a depth comparable to the spin coherence length, as schematically shown in Fig. 4a. The spin coherence length was assumed to be 2 nm, which is a reasonable value for 3d metals and for Py in particular 39,44,45 . Furthermore, our simulations revealed that the chosen spin coherence length in the interval between 1 nm and 3 nm did not notably influence the SWR amplitudes in the 9 nm Py. Figure 4b plots the simulated spectra for the parallel (as shown in the top panel in Fig. 4a) and antiparallel (bottom panel in Fig. 4a) alignment of b SOT at the two interfaces of the Py layer. The parallel alignment corresponds to the asymmetric Ta/Py/Pt sample, as the opposite sign of the spin Hall angle in Ta and Pt resulted in the same torque direction at both Py interfaces, as schematically shown in Fig. 2a. Accordingly, the opposite direction of the field-like torque in the Ta/Py/Ta sample corresponded to the antiparallel alignment of b SOT . The n = 2 mode in the asymmetric sample is evident (blue line in Fig. 4b). Furthermore, as expected, the symmetric sample did not display even-n modes (red line in Fig. 4b), instead n = 1 and n = 3 modes have substantial amplitudes. These modes were not detected in our experiment as counter-propagating coherent spin waves (which are antiphase and have identical amplitudes) within the transient time of about 2 ps develop standing spin waves with a net-zero dynamical component projected along the z axis. This did not hold for the asymmetric Ta/Py/Pt sample, as the precession amplitudes at the Ta and Pt interfaces were different due to different SOTs acting on the magnetization. This resulted in an in-depth asymmetrical profile of the dynamical components during the transient time, which was longer than the studied time-delay interval of 25 ps in Fig. 3a. The asymmetrical magnetization profile in turn allowed the detection of the magnetization dynamics in a Faraday geometry. Different magnetization excitation amplitudes at the two interfaces are also expected to result in the odd-n modes. This scenario was verified using micromagnetic modelling by adjusting b SOT at the two interfaces to be 10 mT at the Pt side and 2 mT at the Ta side. The corresponding Fourier spectra are presented in Fig. 4b (dashed black line). Besides the even-n modes, we obtained the n = 3 mode with a substantial amplitude. This mode, however, was not detected in the Ta/Py/Pt with 9 nm Py, presumably due to the fact that at 0.6 THz the mode was strongly damped. Additionally, at 0.6 THz, the pump pulse amplitude in the experimental spectral density was about 30% of the peak amplitude at 0.3 THz (Extended Data Fig. 1). The signal from the n = 3 mode is therefore expected to be threefold smaller than the signal from the n = 2 mode. Furthermore, the asymmetry ratio in b SOT (10 mT/2 mT) at the two interfaces could be smaller, resulting in a smaller amplitude of the n = 3 mode. Nevertheless, the higher-order modes (n = 3 and n = 4) were resolved when the thickness of the Py layer became comparable to the laser-probe light penetration depth in the 12 nm Py sample (Fig. 3).

Discussion
Using time-resolved magneto-optics, we unambiguously demonstrate the efficient excitation of nanometre-wavelength coherent spin waves, driven by terahertz radiation, in a ferromagnetic layer sandwiched by different heavy metals with spin Hall angles of opposite sign. Our experimental and theoretical studies of asymmetric Ta/Py/Pt versus symmetric Ta/Py/Ta systems rule out the possibility that the terahertz magnetic field alone is the source for the SWR, detected in the asymmetric sample. Exploiting the opposite sign of the spin Hall angle in Ta and Pt, and the antisymmetric nature of the field-like torque under Article https://doi.org/10.1038/s41567-022-01908-1 magnetization reversal and sample flipping, we demonstrate that the SOT induced by the terahertz field pulse in the heavy metal layer is the driving force for the terahertz magnons. The efficient excitation of the n = 2 mode in the asymmetric Ta/Py/Pt samples is explained by the fact that the magnetization at the two interfaces of the Py layer experiences a torque in the same direction (Fig. 2a). Such in-phase precession at the two interfaces meets the criteria for the observation of the n = 2 mode, as schematically shown in the top panel in Fig. 4a. The mode frequency increases quadratically with decreasing Py thickness, according to equation (1), and reaches a maximum of 0.6 THz in 6-nm-thick Py. The lifetime of the 0.6 THz mode is, however, two times shorter than the 0.29 THz mode in 9 nm Py. The decrease of the mode lifetime with frequency is presumably one of the main reasons why the higher harmonic signals (above 0.6 THz) are absent in our experiment.
As soon as the laser-probe light penetration depth becomes comparable to the Py thickness in the 12 nm Py, we were able to resolve the n = 3 mode. This indicates that the SOT-induced magnetization dynamics amplitudes are different at the Ta and Pt interfaces. The asymmetric torques at two different interfaces are expected, as the SOT efficiency at the Ta interface is usually smaller than at the Pt one 26 . In the case of symmetric interfaces (that is, in the Ta/Py/Ta sample), the SOT generates counter-propagating coherent magnon packets, which are in antiphase. Consequently, they cancel out each other in the Py layer, and standing spin-wave modes with an even number of n are not excited. The odd-number PSSW modes in the Ta/Py/Ta sample cannot be detected in our experimental geometry for the reasons discussed above. The assumed SOT efficiency µ 0 H FL /j = 10 −14 T A −1 m 2 is equivalent to about a µ 0 H FL = 10 mT field and is assumed to be similar to that in magnetotransport measurements 26,46 according to the theoretical predictions 27,28 . We note, however, that the exact verification of the SOT efficiency at terahertz frequencies requires further studies with calibrated Faraday angles using high magnetic fields. These studies are of great interest, as they offer contactless and lithography-free characterization of SOT efficiency in complex heterostructures. Our work demonstrates the coupling of terahertz light with magnon modes of corresponding energy. Unlike optically induced spin waves [9][10][11][12][13][14] , this terahertz-frequency magnon excitation via SOT is linear with the amplitude of the terahertz field, which allows us to achieve coherent and efficient control over the excitation, which is of much fundamental and applied interest. Furthermore, the layered metallic stacks based on an encapsulated ferromagnetic thin film act as an intrinsic terahertz magnon source at zero magnetic field, and offer low-cost fabrication and easy integration into existing technologies [47][48][49] , thus opening an attractive pathway towards low-power terahertz-frequency magnonics and spintronics at the nanoscale.

Online content
Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41567-022-01908-1.

Sample fabrication
Samples of Ta(3 nm)/Py(9 nm)/Ta(3 nm) and Ta(3 nm)/Py(d Py )/ Pt(2 nm) where d Py = 6 nm, 7 nm, 8.1 nm, 8.6 nm, 9 nm, 9.6 nm and 12 nm were fabricated at room temperature by d.c.-magnetron sputter deposition in a 0.4 Pa Ar atmosphere in an ultrahigh-vacuum BESTEC system. We used 3-mm-thick double-side polished quartz (SiO 2 ) and sapphire (Al 2 O 3 ) substrates. The thicknesses of all layers were controlled via the deposition time. Before the sample fabrication, the sputter rate of each individual material was calibrated using X-ray reflectivity characterization of the corresponding film. For the magnetic characterization we used a 0.1-50 GHz vector network analyser ferromagnetic resonance set-up.

Pump-probe measurement
In the experiment a laser system with a 1 kHz repetition rate, 1.5 mJ pulse energy, 800 nm central wavelength and 100 fs pulse duration was used. The laser pulses were split into two branches: one for terahertz pulse generation and the other for magneto-optical probing. The first pulse, with a pulse energy of about 1.2 mJ, generated a single-cycle terahertz pulse via a tilted pulse front scheme in a lithium niobate (LiNbO 3 ) crystal. The terahertz pulse was then refocused onto the sample surface with full-width at half-maximum spot diameter of 0.8 mm and a pulse energy of 2 µJ. The probe pulse was transmitted through the sample and was spatially overlapped with the pump pulse. The temporal delay between the two pulses was controlled using a linear delay stage. The terahertz-induced rotation of the probe pulse polarization was measured with a balanced pair of photodiodes using a half-wave plate and Wollaston prism placed behind the sample.
The peak electric field of (50 ± 5 MV m −1 of the linearly polarized terahertz pulse was measured using electro-optical sampling. The H-field amplitude was estimated to be 160 mT using B = E/c, where B is the terahertz induction and c is the velocity of light. All experiments were performed in a dry nitrogen gas atmosphere to suppress absorption of the terahertz radiation by water vapour in air.

Micromagnetic modelling
Micromagnetic simulations were performed using a custom version of the open-source GPU-accelerated micromagnetic solver MuMax3 40 , which relies on the time integration of the Landau-Lifshitz-Gilbert equation of motion of the magnetization on a rectangular grid. The ferromagnetic layers of different thicknesses were modelled using a segment of 3 × 3 cells in the layer plane and 64 cells along the layer's thickness. The lateral edge length of the cells was set to 5 nm, whereas the cell size along the thickness direction varied between 78 and 203 pm, depending on the chosen layer thickness. To simulate an infinitely extended layer, we applied true periodic boundary conditions in the lateral directions. For the material parameters, we set a saturation magnetization of µ 0 M s = 1 T, an exchange stiffness constant of A ex = 8 pJ m −1 , an out-of-plane anisotropy of K u = 25 kJ m −3 and a Gilbert damping constant of α = 0.01. Asymmetric surface anisotropy was implemented using additional out-of-plane anisotropies with constants K u,i = lK u,i in only the surface cells of the layer, where l is the cell size along the thickness direction and the surface anisotropy constants K s,1 = 0.1 mJ m −2 and K s,2 = 0.2 mJ m −2 . The in-plane-saturated equilibrium state was found by applying an external magnetic field of B DC = 50 mT. Spin-wave dynamics were excited using a SOT field with a pulse profile: where the pulse parameters ω p = 4.4 THz and t p 2 = 1.6 ps 2 are empirical, fitting the terahertz pump pulse spectrum. The field b SOT (b p = 10 mT with a skin depth λ p = 1-3 nm, according to the spin coherence length in Py) were applied in the film plane and perpendicular to the direction of magnetization (that is, perpendicular to B DC ). The magnetization dynamics was simulated for 250 ps and sampled every 100 fs. Subsequently, a fast Fourier transform of the time evolution of the magnetization was performed for each discretization cell to obtain the spectral response as a sum of all cell contributions.

Data availability
Source data are provided with this paper. All other data that support the finding of this work are available from the corresponding authors upon request.

Code availability
The mathematical code that supports the findings of the study is available via GitHub at https://github.com/mumax/3. The MuMax3 scripts are provided in Supplementary Code 1.