To study spin dynamics, we employ time-resolved magneto-optics in Faraday configuration, as schematically shown in Fig. 1a. An intense single-cycle THz pulse is focused onto a trilayer sample, and then the sample magnetization state is probed by polarization rotation of a delayed 100-fs-probe pulse with 800 nm central wavelength. The THz 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 THz radiation. A more detailed description of the pump-probe experiment is provided in the Methods section.
We study Ta(3 nm)/Py(9 nm)/Pt(2 nm) samples, as shown in Fig. 1a. The thickness of each individual layer is given in the parentheses. As reference samples, we also use 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 are deposited on transparent double-side polished quartz glass as well as on Al2O3 substrates with a substrate thickness of 3 mm. Different substrates are used in order to account for the back-reflected THz signal in transmission geometry. We employ a 50 mT external magnetic field (HExt) to keep the samples magnetically saturated with the magnetization (M) lying in the film plane, with the option to make it parallel or orthogonal to the magnetic field component of the incident THz pulse (HTHz).
Figure 1b presents the comparison of 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 Fig. S1). A typical coherent response driven by the Zeeman interaction between the Py magnetization and HTHz (seen as a single-cycle transient of about 1 ps duration) is evident for all samples [17, 20–23]. 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 THz pulse has passed through the sample, the signal of the Ta/Py/Pt clearly exhibits additional features in the form of oscillations at longer delay times (see black curve in Fig. 1b). The amplitude of the magnetization’s precessional signal, ASWR, is about 7.5% of the ZTS amplitude AZTS for this particular sample, as indicated by the dashed lines in Fig. 1b. The frequency of the oscillations is calculated to be at about 0.29 THz, which we attribute to the spin-wave resonance (SWR) mode with n = 2 in Eq. (1). Both, the ZTS and SWR show a linear response with respect to the THz field amplitude (Supplementary Note 2 and Fig. S2), 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) suggests that HTHz 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 THz 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 HTHz. The vanishing ZTS in the parallel configuration indicates its magnetic origin, since the Zeeman torque is zero for a magnetization being oriented parallel to HTHz [23]. Similarly, the SOT is inefficient for electrical fields orthogonal to the in-plane magnetization [26–30], hence the SWR signal is not detected. We attribute the small ZTS visible in the figure to a slight misalignment, possibly also due to a non-ideal polarization of the THz field. We additionally check the effect of reversing the external magnetic field HExt on the Faraday signal, Fig. 1d. Both the coherent and the oscillatory responses reverse their sign upon reversal of the sign of HExt. The sign inversion of the ZTS is expected, since the Zeeman torque is asymmetric under magnetization reversal [23]. The asymmetric behavior of the SWR signal is a further indication that the SOT is the driving mechanism, as we reason below. We note that the asymmetric behavior of the SWR is just opposite to the behavior of thermally excited uniform modes, which are symmetric under field inversion [23].
In order to explain the asymmetric behavior of the SWR signal in Fig. 1d, we start with the Landau-Lifshitz-Gilbert equation
∂ t m = - γ(m × Heff) + αm × ∂tm. (2)
In Eq. (2) m = M/Ms is the magnetization unit vector normalized to the magnetization at saturation Ms, γ is the gyromagnetic ratio, and α is the Gilbert damping parameter. The effective field Heff is given by
H eff = HExt + HTHz + HFL + HDL, (3)
where we consider the SOT-induced field-like HFL and damping-like HDL equivalent fields with HFL ∝ (z×j) and HDL ∝ (j×z)×m [26, 29, 30]. Here j is the electrical current density induced by the THz and z is the unit vector normal to the sample plane. According to Eqs. (2) and (3), the field-like torque is antisymmetric upon 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 THz-generated SWR signals, explaining the 180-degree phase shift in Fig. 1d. Finally, we exploit the asymmetry of the field-like torque under a flip of the sample with respect to the incident beam direction. This is shown schematically in Fig. 2a.
According to Eqs. (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 HTHz, whereas HFL is expected to inverse 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 the HTHz is identical to the one, observed with the flipped sample, i.e., 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 can isolate the dynamics of the SOT-induced effect by taking the difference of the signals recorded with the sample mounted in the opposite configuration. The difference signal, shown in the inset of Fig. 2b, reveals clear damped oscillations at a frequency of 0.29 THz.
The time delay scans of the difference signal for different Py thicknesses are presented in Fig. 3a. All samples display damped oscillations, and we fit our data using a decaying cosine function. The thickest 12-nm Py sample exhibits a more complex dynamical behavior, therefore the best fit to the curve is achieved using a superposition of three damped cosine functions [31]:
$$\psi \left(t\right)={\sum }_{i=1}^{3}\left({a}_{i}{e}^{\frac{-t}{{\tau }_{i}}}\text{cos}(2{\pi }{f}_{i}t+{\phi }_{i})\right)$$
4
,
where \(\psi \left(t\right)\) is the time-dependent difference signal, \({a}_{i}\) and \({\phi }_{i}\) are the initial amplitudes and phases of the i-th mode, fi and τi are the frequency modes and characteristic damping times. The inset in Fig. 3b shows the mode frequencies as a function of k2 for the 12-nm-Py sample. The dependence on the wave-vector squared is evident, which we fit using equation (1). The SWR frequency as a function of Py layer thickness for all Ta/Py (dPy)/Pt samples is plotted in Fig. 3b. As expected, the frequency increases with decreasing Py thickness and reaches 0.6 THz for dPy = 6 nm. The fit using equation (1) reveals that the SWR mode in all samples corresponds to the n = 2 mode, and the extracted exchange stiffness is Dex = 240 meV·Å2 (Aex = 8 pJ/m, where Aex is the exchange constant). This value is similar to the one reported previously for thin Py films [32, 33]; furthermore, the fit to the multiple modes of the 12-nm Py sample (inset in Fig. 2b) reveals identical Dex, indicating that the exchange stiffness does not change significantly with decreasing the Py thickness down to 6 nm. The larger than n = 2 modes in 12-nm Py become visible, since the thickness of the metallic sample is comparable to the probe-light penetration depth, which is typically 10 - 15 nm [34–36]. The details for the mode-selective detection in our experiment are discussed below. The relaxation time of the SWR increases with Py thickness. The effective damping parameter of the n = 2 mode in 9-nm Py sample is calculated to be αeff = 0.06, which is comparable to the values reported for the PSSW modes at 0.3 THz [12, 13]. This value, however, is larger than αeff = 0.014 of the n = 2 mode in 50-nm thick Py [31], suggesting a significant wave-vector-dependent damping contribution at THz frequencies [12, 37, 38].
Micromagnetic modeling. In order to gain further insight into the origin of the SWR signal in the Ta/Py/Pt asymmetric system, we model the magnetization dynamics microscopically using the MuMax3 code [39] (see the Methods section 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 are extracted from ferromagnetic resonance (FMR) characterization discussed in the Supplementary Note 3 and Fig. S3. We note that no SWR is obtained in the simulations, when assuming a homogeneous or linearly decaying profile of the HTHz [22] within the Py thickness. This additionally confirms that the direct action of the HTHz on the magnetization (Zeeman torque itself) cannot be the origin of the SWR signal.
Having verified that the THz magnetic field contributes to the spin-wave excitation insignificantly, we estimate the magnitude of HFL. For that, we first consider the large refractive index of the metallic films, and calculate the absorbed THz power using the transfer matrix method [40–42]. Next, we estimate the electric current density, which causes the THz power absorption (see Supplementary Note 4 and Fig. S4 for details). The current density is calculated to be of about j = 1012 A/m2. We assume the SOT efficiency µ0HFL/j = 1 mT/(1011A/m2) [26] and, accordingly, arrived at µ0HFL = 10 mT. We further localize the field bSOT = µ0HFL = 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 is assumed to be 2 nm, which is a reasonable value for 3d metals and for Py in particular [38, 43, 44]. Furthermore, our simulations reveal that the chosen spin coherence length in the interval between 1 nm and 3 nm does not significantly 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 the bSOT at the two interfaces of the Py layer. The parallel alignment corresponds to the asymmetric Ta/Py/Pt sample, since the opposite sign of the spin Hall angle in Ta and Pt results 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 corresponds to the antiparallel alignment of the bSOT. The n = 2 mode in the asymmetric sample is evident (blue line in Fig. 4b). Furthermore, as expected, the symmetric sample does not display even-n modes (red line in Fig. 4b), instead n = 1 and n = 3 modes show substantial amplitudes. These modes were not detected in our experiment, since 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 zero net z-axis projected dynamical component. This does not hold for the asymmetric Ta/Py/Pt sample, since the precession amplitudes at the Ta and Pt interfaces are different due to different SOTs acting on the magnetization. This results in an in-depth asymmetrical profile of the dynamical components during the transient time, which is longer than the studied time-delay interval of 25 ps in Fig. 3a. The asymmetrical magnetization profile in turn allows the detection of the magnetization dynamics in Faraday geometry. Additionally, different magnetization excitation amplitudes at the two interfaces are expected to result in the odd-n modes as well. This scenario is verified using micromagnetic modeling by adjusting the bSOT at the two interfaces to be 10 mT at Pt side and 2 mT at the Ta one. 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 the 0.6 THz frequency the mode is strongly damped. Furthermore, the asymmetry ratio in bSOT (10 mT/2 mT) at the two interfaces could be smaller, resulting in a smaller n = 3 mode amplitude. Nevertheless, the higher-order modes (n = 3 and n = 4) are resolved when the thickness of the Py layer becomes comparable to the laser-probe light penetration depth in the 12-nm Py sample (Fig. 3).