Introduction

When an in-plane charge current traverses a metal with strong spin-orbit coupling, it undergoes conversion into a spin current via the spin Hall effect (SHE) and the Rashba-Edelstein effect1,2,3,4,5,6. This spin current can then be used to exert torque on the magnetic moment of a ferromagnet. The phenomenon commonly known as spin-orbit torque (SOT)7,8. From an application standpoint, SOT has attracted significant attention for its high efficiency and simple device structure, making it a promising candidate in magnetization switching3,5,9, magnetic moment precession, and the rapid motion of domain walls10,11. From a physics perspective, SOT provides an effective means of detecting the spin current, which is essential for characterizing and understanding spin-charge conversion and spin scattering phenomena. In the case of a spin current with a spin direction σ, the induced SOT on the magnetization moment m can be decomposed into two components: a field-like torque τFL ∝ m × σ, and a damping-like torque τDL ∝ m × (m × σ). By measuring τFL and/or τDL through the second harmonic voltage12,13,14,15,16,17 or the ferromagnetic resonance voltage induced by SOT18,19,20,21, the spin direction and amplitude of the spin current can be determined.

Beyond charge current manipulation, spin current has also played a key role in the electrical manipulation of magnons due to the conversion between spin current and magnons22,23,24,25,26. By utilizing two Pt strips attached to a magnetic insulator as spin current injectors and detectors, a non-local voltage dependent on the magnetization of the magnetic insulator is observed. This observation suggests that magnons in the magnetic insulator are generated by the absorption of spin current in one Pt strip. As the generated magnons traverse the magnetic insulator and reach the other Pt strip, they convert back into spin current, inducing a charge voltage via the inverse SHE27. The conversion between magnon and spin current requires the conservation of angular momentum28. Specifically, a magnon (Smag=1) can be created by the spin flip from spin angular moment Sspin=1/2 to Sspin=−1/2,and is annihilated by the spin flip from Sspin=−1/2 to Sspin=1/2.This unidirectional feature is odd under either magnetization or current reversal, and has been further experimentally identified as one of possible sources of the unidirectional magnetoresistance (UMR), where the generated magnons influence the resistance of ferromagnet29,30,31,32,33. Since magnons are significantly suppressed by the magnetic field H, it limits magnon-induced UMR to low magnetic fields and causes it to scale with ∝ H−p.

Although the intimate relationships between spin current and SOT, as well as between spin current and magnons, are well-established, the use of magnons to manipulate SOT in devices mediated by spin current remains elusive. Recent theoretical work by Freimuth et al.34 has proposed the influence of magnons on SOT, referred to as magnonic SOT, to explain a significant increase of field-like torque with rising temperature in various SOT devices16,35,36. Very recently, the works of Noël et al. evidence the role of magnon creation-annihilation on the SOT and give a complete picture of the observed nonlinear Hall and longitudinal effects associated with magnonic contributions37,38. However, direct experimental evidence of magnonic SOT is still lacking.

In this study, we present our findings on magnonic SOT in an epitaxial Co film. To prevent decay of magnons caused by magnetic fields, we extended the second harmonic theory model within the low magnetic field range based on macro-spin model. Our experimental results demonstrate that the angular dependency of the second harmonic voltage in both longitudinal and transverse directions can be accurately described by our low-field SOT model. Due to their unique angular dependencies, the second harmonic voltages within low magnetic field range can be clearly separated into three distinct effects: the SOT effect, the anomalous Nernst effect (ANE), and the UMR effect. Interestingly, unlike the conventional field-like SOT, which is independent of the magnetic field, we observe an anomalous SOT at low magnetic fields that sharply decays with increasing magnitude of the external magnetic field. Furthermore, we reveal that the field decay behavior of the anomalous SOT exhibits a perfect linear correlation with the magnon-induced UMR effect, which provides strong evidence that the observed SOT is caused by magnons. Finally, we further confirm the existence of magnonic SOT, when the current flows along different crystal orientations. These findings highlight the potential of utilizing magnons to manipulate spin current in spintronics.

Results

Low field harmonic voltage model

To quantitatively analyze magnonic SOTs, we extended the macro-spin model to describe both the longitudinal and transverse harmonic voltages within the low magnetic field range. Considering an epitaxial FM layer with in-plane cubic anisotropy, the free energy density can be expressed as

$$E=\frac{1}{8}{H}_{{\rm{A}}}{M}_{{\rm{S}}}{\sin }^{2}2({\varphi }_{m}-{\varphi }_{{\rm{EA}}})+\frac{1}{2}{H}_{{\rm{K}}}{M}_{{\rm{S}}}{\cos }^{2}{\theta }_{m}-{\bf{m}}\cdot {\bf{H}},$$
(1)

where HA is the in-plane cubic anisotropic field, HK is the out-of-plane anisotropic field, φEA is the angle between the easy magnetization axis and the x axis, m represented as θm and φm in spherical coordinates is the magnetic moment of the FM film with saturation magnetization MS, and H is the external magnetic field. As shown in Fig. 1a, we in-plane rotated H by an angle φ. In the low magnetic field range, m lies in in-plane (θm = π/2), but it is not aligned with H due to the competition between the Zeeman energy and in-plane cubic anisotropic energy. We define the angle difference between m and H as α=φ−φm. By considering the minimum of the free energy density for a certain magnetic field amplitude H, we can obtain39.

$$\alpha =\frac{\sin 4(\varphi -{\varphi }_{{\rm{EA}}})}{4(H/{H}_{{\rm{A}}}+\cos 4(\varphi -{\varphi }_{{\rm{EA}}}))}.$$
(2)
Fig. 1: Crystal structure and static magnetic properties of Al/Co/SrTiO3 film.
figure 1

a Schematic representation of the measurement geometry. Longitudinal and transverse harmonic voltage are measured simultaneously. b High-resolution X-ray diffraction pattern of the epitaxial Co film. c In-plane hysteresis loops for easy magnetization axes (EA) along Co [110] and hard magnetization axes (HA) along Co [100] at room temperature. Inset displays in-plane remanence induced by in-plane cubic anisotropy energy.

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By injecting a charge current Jc into the SOT device, it induces an additional effective magnetic field ΔH. In our case, ΔH mainly arises from the current-induced SOT effective field and Oersted field, which can be expressed as:

$$\Delta {\bf{H}}=(0,{H}_{{\rm{FL}}}+{H}_{{\rm{Oe}}},{H}_{{\rm{DL}}}\cos (\varphi -\alpha )),$$
(3)

where HFL and HDL represent effective SOT magnetic fields induced by field-like torque and damping-like torque, respectively, and HOe is the current-induced Oersted field.

As ΔH exerts a torque on magnetization, it alters its equilibrium state from (θm,φm) to (θm+Δθ,φm+Δφ). Utilizing the first-order Taylor expansion of the minimum of the free energy density, a linear response is obtained between (Δθ,Δφ) and ΔH:

$$\Delta \theta =\frac{-\Delta {H}_{z}}{{H}_{{\rm{K}}}+H\cos \alpha -\frac{1}{4}{H}_{{\rm{A}}}{\sin }^{2}2(\varphi -\alpha -{\varphi }_{{\rm{EA}}})},$$
(4)
$$\Delta \varphi =\frac{\Delta {H}_{y}\cos (\varphi -\alpha )-\Delta {H}_{x}\sin (\varphi -\alpha )}{H\cos \alpha +{H}_{{\rm{A}}}\cos 4(\varphi -\alpha -{\varphi }_{{\rm{EA}}})}.$$
(5)

It should be noted that for α = 0 at large magnetic fields, Eqs. (4) and (5) reduce to the expressions for the saturated magnetic field as previously reported13. In this case, within the low magnetic field range, α significantly alters the behavior of Δθ and Δφ, causing the harmonic voltage to completely differentiate the curves at the saturated magnetic field, which we will show later. When an alternating charge current is applied, i.e., \({I}^{1\omega }={I}_{0}\sin \omega t\), ΔH oscillates, leading to magnetization oscillation around its equilibrium position at the same frequency ω. This magnetization oscillation is directly reflected in corresponding resistance changes R1ω due to the magnetoresistance effect, and further generating second harmonic voltages V2ω, according to V2ω=I1ωR1ω.

The angular dependence of the first harmonic longitudinal and transverse voltage changes can be expressed as follows:

$$\Delta {V}_{xx}^{1\omega }={I}_{0}{R}_{{\rm{AMR}}}\cos 2(\varphi -\alpha ),$$
(6)
$$\Delta {V}_{xy}^{1\omega }={I}_{0}{R}_{{\rm{PHE}}}\sin 2(\varphi -\alpha ),$$
(7)

where RAMR is the coefficient of anisotropic magnetoresistance (AMR) or spin Hall magnetoresistance (SMR) that exhibits the same angular dependence in x − y plane, and RPHE is the coefficient of planar Hall effect (PHE).

The corresponding second harmonic of longitudinal and transverse voltages at the low magnetic fields can be respectively expressed as following:

$${V}_{xx}^{2\omega }={I}_{0}{R}_{{\rm{AMR}}}\sin 2(\varphi -\alpha )\Delta \varphi ,$$
(8)
$${V}_{xy}^{2\omega }=\frac{1}{2}{I}_{0}({R}_{{\rm{AHE}}}\Delta \theta -2{R}_{{\rm{PHE}}}\cos 2(\varphi -\alpha )\Delta \varphi ),$$
(9)

where RAHE is the coefficient of anomalous Hall effect (AHE). By inserting the Δθ and Δφ from Eqs. (4) and (5) into Eqs. (8) and (9), the angular dependencies of the second harmonic voltages induced by ΔH in Eq. (3) are respectively given as:

$${V}_{xx}^{2\omega ,\Delta {\bf{H}}}={I}_{0}\frac{{R}_{{\rm{AMR}}}({H}_{{\rm{FL}}}+{H}_{{\rm{Oe}}})\sin 2(\varphi -\alpha )\cos (\varphi -\alpha )}{H\cos \alpha +{H}_{{\rm{A}}}\cos 4(\varphi -\alpha -{\varphi }_{{\rm{EA}}})},$$
(10)
$$\begin{array}{ll}{V}_{xy}^{2\omega ,\Delta {\bf{H}}}=-{I}_{0}\displaystyle\frac{{R}_{{\rm{PHE}}}({H}_{{\rm{FL}}}+{H}_{{\rm{Oe}}})\cos 2(\varphi -\alpha )\cos (\varphi -\alpha )}{H\cos \alpha +{H}_{{\rm{A}}}\cos 4(\varphi -\alpha -{\varphi }_{{\rm{EA}}})}\\\qquad\qquad\;\;-\displaystyle\frac{{I}_{0}}{2}\frac{{R}_{{\rm{AHE}}}{H}_{{\rm{DL}}}\cos (\varphi -\alpha )}{{H}_{{\rm{K}}}+H\cos \alpha -\frac{1}{4}{H}_{{\rm{A}}}{\sin }^{2}2(\varphi -\alpha -{\varphi }_{{\rm{EA}}})}.\end{array}$$
(11)

In addition to the SOT effective field, Joule heating is another significant factor influencing the second harmonic voltages. When a high-density charge current flows through the thin films, it induces a substantial thermal gradient perpendicular to the film, expressed as \(\nabla T\propto {I}_{0}^{2}\). This thermal gradient gives rise to thermoelectric effects in the ferromagnetic (FM) layer, such as the ANE effect14, which also contributes to the second harmonic voltage signal. Considering that the ANE-induced electric field is given by EANE ∝ ∇T × m, the longitudinal and transverse second harmonic voltages at low magnetic fields due to ANE can be expressed as:

$${V}_{xx}^{2\omega ,{\rm{ANE}}}=-\frac{1}{2}\beta l\nabla T\sin (\varphi -\alpha ),$$
(12)
$${V}_{xy}^{2\omega ,{\rm{ANE}}}=\frac{1}{2}\beta w\nabla T\cos (\varphi -\alpha ),$$
(13)

where β is the ANE coefficient, and l and w represent the length and width of the Hall bar, respectively.

Finally, we shift our focus to the UMR in FM/heavy metal (HM) bilayers, which also contributes to the second harmonic voltage. UMR arises from two primary mechanisms29,40. The first mechanism involves spin-dependent scattering (SD) at both the interface and within the bulk of the bilayer, similar to the giant magnetoresistance effect41,42. Here, the resistance of the bilayer changes depending on whether the spin polarization of the spin current generated in the HM is parallel or antiparallel to the magnetization of the ferromagnet (FM). This variation in resistance induced by an alternating charge current leads to the generation of the second harmonic voltage.

The second mechanism involves electron-magnon scattering43. When the spin current generated in the HM induces magnons in the FM layer, electron-magnon scattering alters the resistance of the FM, giving rise to UMR. Unlike SD-induced UMR, which remains constant with increasing magnetic field, magnon-induced UMR decays sharply as the magnetic field increases 31. Since the UMR is governed by the term Jc × m ⋅ z26,44, where z is the unit vector normal to the film plane, the second harmonic voltage due to UMR at low magnetic fields can be expressed as:

$${V}_{xx}^{2\omega ,{\rm{UMR}}}={I}_{0}{R}_{{\rm{UMR}}}\sin (\varphi -\alpha ),$$
(14)
$${V}_{xy}^{2\omega ,{\rm{UMR}}}=0,$$
(15)

where RUMR is the UMR coefficient.

Structure and static magnetism

High-resolution X-ray diffraction data in Fig. 1b confirms the Al/Co/SrTiO3 film exhibits a face-centered cubic (FCC) structure. The in-plane magnetization versus magnetic field curves (up to ±100 mT) is shown in Fig. 1c. When the external magnetic field exceeds 50.0 mT, no hysteresis is observed, and the system behavior can be accurately described by a macroscopic spin model. The hysteresis loop along the Co [100] direction reveals an in-plane magnetocrystalline anisotropy field of 37.3 mT45, as shown in Fig. 1c. Additionally, the in-plane remanence (inset of Fig. 1c) exhibits a fourfold symmetry, confirming that the sample possesses cubic magnetocrystalline anisotropy.

Harmonic measurement

Figure 2a, b present the angular dependence of the first harmonic voltages along the longitudinal and transverse directions, respectively, with the patterned Hall bar along the hard magnetization axis (HA) of Co. Consistent with our previous work 46, the first harmonic voltages in the low magnetic field regime slight deviations from the expected \(\cos 2\varphi\) and \(\sin 2\varphi\) angular dependencies until μ0H = 670 mT. However, across the entire measured magnetic field range, these angular dependencies are well described by Eqs. (6) and (7) of our low-field model.

Fig. 2: First and second harmonic voltages.
figure 2

Angular dependencies of the first harmonic voltages: a longitudinal and b transverse, measured for a current amplitude of I0 = 16 mA under 59.1 mT and 670.0 mT with φEA = 45°. Angular dependencies of the second harmonic voltages measured for a current amplitude of I0 = 16 mA under different external fields with φEA = 45°: c longitudinal and d transverse. The solid curves represent fitting results of the low magnetic field model. The variation of vertical axes are determined by the scale bars in the diagrams.

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The corresponding angular dependence of the second harmonic voltages under different magnetic fields are shown in Fig. 2c, d. In contrast to the first harmonic signals, both the longitudinal and transverse second harmonic voltage amplitudes increase significantly as the magnetic field decreases. Due to the noncollinearity between the magnetic field and magnetization, the angular behavior at low magnetic fields deviates sharply from that observed at saturated fields. However, the second harmonic signals across the entire magnetic field range can still be accurately fitted by our low-field model. Notably, the same fitting coefficients for SOT, ANE, and UMR were applied to both longitudinal and transverse results, further validating the accuracy of the model.

According to the fitting results, the contributions from SOT, ANE, and UMR are clearly distinguishable based on their different angular dependencies. The current-induced μ0HOe was carefully subtracted by applying a simplified Ampere’s law, assuming uniform current distribution within the Al/Co bilayer and using a parallel resistor model47,48. The details are reported in the supplemental information. The fitted angular dependencies of SOT, ANE, and UMR under low magnetic field range are further illustrated in Fig. 3. The increase in the second harmonic amplitude with decreasing magnetic field is mainly driven by the enhancement of SOT and UMR, while the amplitude of ANE remains nearly constant.

Fig. 3: Separation of SOT, ANE, and UMR contributions.
figure 3

Extracted angular dependencies of SOT, ANE, and UMR for a longitudinal and b transverse second harmonic voltages from Fig. 2 by fitting with the low magnetic field model. The variation of vertical axes are determined by the scale bars in the diagrams.

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Discussion

To confirm that the increase in UMR with decreasing magnetic field results from the suppression of magnons, we further investigated RUMR as a function of μ0H under different current conditions. As shown in Fig. 4a, the field dependence of RUMR follows a power law of the form \({({\mu }_{0}H)}^{-p}+{R}_{{\rm{0}}}\), where R0 arises from SD UMR that is independent of the magnetic field, and p ≈ 1, consistent with previous observations of magnon-induced UMR in weak magnetic fields 29. Additionally, we also investigated the current dependence of RUMR, and the results, shown in Fig. 4b, can be described by aI + bI3, where a and b are fitting parameters. This behavior is again consistent with previous findings on magnon-induced UMR29.

Fig. 4: UMR results.
figure 4

a RUMR as a function of μ0H for different current amplitudes with φEA = 45°. b RUMR as a function of I under different magnetic fields with φEA = 45°.

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The simultaneous increase of UMR and SOT at low magnetic fields suggests that the enhancement of SOT may also be linked to magnons. To explore this hypothesis, we systematically investigated the longitudinal and transverse μ0HFL as functions of both current I and magnetic field μ0H. As shown in Fig. 5a and b, the μ0HFL values, even in the low magnetic field range, are proportional to the current, consistent with the model that SOT is linearly driven by the spin current generated from the charge current. Figure 5c and d present μ0HFL as a function of magnetic field μ0H for different current amplitudes. Unlike the behavior at saturated magnetic fields, where μ0HFL remains relatively constant13,14,49, we observe a strong dependence of μ0HFL on the magnetic field in the low-field regime. Specifically, μ0HFL decreases significantly as the external magnetic field increases and tends to saturate at higher magnetic fields. More importantly, although the angular dependencies of SOT in the longitudinal and transverse directions differ significantly, the fitted values of μ0HFL are nearly identical and exhibit the same magnetic field-dependent behavior. This consistency reinforces the accuracy of our experimental results and further suggests a potential role of magnons in influencing SOT behavior.

Fig. 5: linear correlation between μ0HFL and RUMR.
figure 5

μ0HFL fitted from a longitudinal and b transverse measurements as a function of current I under different magnetic fields with φEA = 45°. Additionally, μ0HFL fitted from c longitudinal and d transverse measurements as a function of magnetic field μ0H for different current amplitudes with φEA = 45°. The solid curves represent fits to the data using Eq. (16). Panels e and f illustrate the linear correlation between μ0HFL from longitudinal and transverse measurements and RUMR for different current amplitudes with φEA = 45°.

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To further explore the potential correlation between the behavior of μ0HFL and the suppression of magnons, and to provide direct evidence supporting the proposed magnonic SOT, we analyzed the relationship between μ0HFL and RUMR at the same magnetic field. This analysis is akin to scaling laws that establish correlations between ρxx and ρxy under identical magnetic field and temperature conditions50,51. As shown in Fig. 5e, f, both longitudinal and transverse μ0HFL exhibit a clear linear relationship with RUMR. This provides strong evidence that magnons also play a crucial role in enhancing SOT at low magnetic fields. As the magnetic field increases, magnons are gradually suppressed, weakening the magnonic contribution to SOT. The observed linear correlation further demonstrates that the field dependence of μ0HFL follows a power law similar to that established for UMR, which can be expressed as:

$${\mu }_{0}{H}_{{\rm{FL}}}(H)={\mu }_{0}\Delta {H}_{{\rm{magnon}}}{H}^{-p}+{\mu }_{0}{H}_{0},$$
(16)

where μ0ΔHmagnon represents the amplitude of magnonic μ0HFL changes with p ≈ 1 and μ0H0 corresponds to the saturated value of μ0HFL. This strong evidence provides additional support for the notion that the magnon-induced UMR and magnonic SOT share a common underlying mechanism, reinforcing the critical role of magnons in enhancing spin-orbit torque at low magnetic fields.

Considering the nonlinear magnetoresistance effects arising from current-induced magnon creation-annihilation processes, Noël etal. reformulated the theoretical model for second harmonic voltages. We extended the theoretical framework developed by Noël et al. to account for epitaxial systems with magnetic anisotropy, resulting in the following expressions:

$$\begin{array}{ll}{V}_{xx}^{2\omega ,{\rm{magnon}}}/{I}_{0}={R}_{{\rm{MMR}}}^{2\omega }\sin (\varphi -\alpha )\\\qquad\qquad\qquad+{R}_{{\rm{AMR}}}^{2\omega }\sin 2(\varphi -\alpha )\cos (\varphi -\alpha )+{R}_{{\rm{SMR}}}^{2\omega }{\sin }^{3}(\varphi -\alpha ),\end{array}$$
(17)
$${V}_{xy}^{2\omega ,{\rm{magnon}}}/{I}_{0}={R}_{{\rm{PHE}}}^{2\omega }{\sin }^{2}(\varphi -\alpha )\cos (\varphi -\alpha ),$$
(18)

where \({R}_{{\rm{MMR}}}^{2\omega }{,}_{{\rm{AMR}}}^{2\omega }\), \({R}_{{\rm{SMR}}}^{2\omega }\), and \({R}_{{\rm{PHE}}}^{2\omega }\) are respectively the m†mPHE, m†mMMR, m†mAMR, and m†mSMR coefficients defined in refs. 37,38.

Notably, the \({R}_{{\rm{AMR}}}^{2\omega }\) term exhibits a lineshape nearly identical to that of the longitudinal FL torque contribution described in Eq. (10), In contrast, the \({R}_{{\rm{PHE}}}^{2\omega }\) term shows a distinctly different angular dependence compared to the transverse FL torque term in Eq. (11). In practice, due to the angular dependence introduced by both the damping-like torque and the anomalous Nernst effect, it becomes challenging to unambiguously separate these contributions through angular fitting alone37,38.

In our system, the low-field angular dependence of the second harmonic signal (as shown in Fig. 2) more closely resembles the FL torque signature. This behavior arises because the strong magnetocrystalline anisotropy enhances the amplitude of the FL torque contribution at low fields, allowing it to dominate. Our experimental data also reveal that the longitudinal effective field μ0HFL is generally slightly larger than the transverse component. This difference can be attributed to the difficulty in distinguishing the longitudinal magnon-induced m†mAMR signal from the μ0HFL contribution at low fields—a phenomenon previously discussed by Noël et al. In contrast, the transverse magnon-induced m†mPHE signal does not significantly mix with μ0HFL under similar conditions. If we consider only the magnon-related contributions proposed by Noël et al. and ignore possible variations in SOT, we are unable to achieve a satisfactory fit to the low-field experimental data, as shown in Supplemental Information. Therefore, we conclude that magnon excitation at low fields not only gives rise to the magnon-related nonlinear magnetoresistance terms proposed by Noël et al., but also enhances the SOT contributions we describe. Finally, due to the pronounced magnetocrystalline anisotropy in our epitaxial system, the theoretical model cannot be simplified into the forms \(A\sin \varphi +B{\sin }^{3}\varphi\) for \({V}_{xx}^{2\omega }\) and \(C\cos \varphi +D{\cos }^{3}\varphi\) for \({V}_{xy}^{2\omega }\) as were done in Noël’s isotropic model. Consequently, we cannot extract \({V}_{xx}^{2\omega ,{\rm{magnon}}}={V}_{\sin }^{2\omega }+{V}_{{\sin }^{3}}^{2\omega }\) by fitting \({V}_{xx}^{2\omega }\) alone, and the corrected SOT values cannot be determined using their fitting approach.

Finally, we demonstrate that the linear correlation between μ0HFL and RUMR represents a universal law, which might be independent of the form of the magnetocrystalline anisotropy energy. To validate this, we fabricated a Hall bar along the easy axis (EA). When I is aligned parallel to the EA (φEA = 0°), the magnetocrystalline anisotropy energy changes, leading to noticeable differences in both longitudinal and transverse second harmonic experimental data. However, as shown in Fig. 6a, b, the data can still be accurately fitted using our model by adjusting φEA = 45° to φEA = 0°. More importantly, the linear scaling law between longitudinal and transverse μ0HFL and RUMR, under different external fields, is still observed, as presented in Fig. 6c, d. The variations in the fitted values of RUMR and μ0HFL may be attributed to the anisotropy of spin-orbit coupling in the single crystal sample 52.

Fig. 6: Results of [110] crystal orientations with φEA = 0°.
figure 6

Angular dependencies of the second harmonic voltages: a longitudinal and b transverse, measured for a current amplitude of I0 = 8 mA under different magnetic fields with φEA = 0°. The solid curves represent fitting results of the low magnetic field model. Panels c, d illustrate the linear correlation between μ0HFL from longitudinal and transverse measurements and RUMR for different current amplitudes with φEA = 0°. The variation of vertical axes are determined by the scale bars in the diagrams.

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In summary, we have performed longitudinal and transverse second harmonic voltage measurements on epitaxial Co films to simultaneously probe SOT and magnons. To analyze the magnonic effects, we extended the second harmonic theoretical model to the low magnetic field regime. It was observed that μ0HFL decreases sharply with increasing external magnetic field, consistent with the power law of magnon-induced UMR. The emergence of magnonic SOT was demonstrated by identifying a linear scaling law between μ0HFL and UMR. The field dependence of magnonic μ0HFL was subsequently established based on this scaling law. Our results provide direct experimental evidence of magnonic SOT and offer a pathway for utilizing magnons to manipulate SOT.

Methods

Sample and its characterization

A 10 nm Co single-crystal film was grown on a SrTiO3(001) substrate using molecular beam epitaxy (MBE). The crystal structure of the sample was monitored in situ during the growth process via reflection high-energy electron diffraction (RHEED). To prevent oxidation, a 15 nm Al capping layer was deposited on top of the Co film. The samples were patterned into Hall bar structures with a length of 130 μm and a width of 10 μm using a combination of lithography and ion beam etching. Hall bars shown in Fig. 1a were aligned along the Co[100] (hard magnetization axis, φEA = 45° or Co[110] (easy magnetization axis, φEA = 0°) directions. The crystal structure of the samples was characterized using high-resolution X-ray diffraction, while the magnetic properties, including hysteresis loops and remanence, were measured using vibrating sample magnetometry (VSM).

Harmonic voltage measurement

To quantify the current-induced spin-orbit effective fields, a sinusoidal current with an amplitude of 16 mA and a frequency of 133 Hz was applied to the Hall bar device, and the time-dependent voltages in both the longitudinal and transverse directions were recorded simultaneously using a NI PXI-4461 card. These signals were subsequently decomposed into first and second harmonic components via fast Fourier transformation (FFT). This method, which allows for the simultaneous measurement of harmonic voltages in both directions, was well-suited for investigating the relationship between SOT effects and UMR in the low magnetic field regime.