Direct observation of current-induced nonlinear spin torque in Pt-Py bilayers

Toshiyuki Kodama tkodama@tohoku.ac.jp Institute for Excellence in Higher Education, Tohoku University, Sendai, 980-8576, Japan Nobuaki Kikuchi Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan Takahiro Chiba Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan Satoshi Okamoto Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan Center for Science and Innovation in Spintronics, Tohoku University, Sendai 980-8577, Japan Seigo Ohno Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan Satoshi Tomita tomita@tohoku.ac.jp Institute for Excellence in Higher Education, Tohoku University, Sendai, 980-8576, Japan Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan

Abstract

We experimentally observe nonlinear spin torque in metallic bilayers of platinum and permalloy by means of spin-torque ferromagnetic-resonance (ST-FMR) under massive dc current injection. The observed nonlinear spin torque exerted to permalloy magnetization is attributed primarily to nonlinear spin polarization. Additional origin of the nonlinear spin torque is magnon generation (annihilation) followed by shrinkage (expansion) of effective magnetization, which is reveled by ST-FMR and unidirectional spin Hall magnetoresistance measurements. The present study paves a way to spin-Hall effect based nonlinear spintronic devices as well as time-varying nonlinear magnetic metamaterials with tailor-made permeability.

^†^†preprint: APS/123-QED

I Introduction

Nonlinear phenomena are common and intriguing topics in condensed matter physics Robert2008 . Among the fascinating achievements in spintronics and magnonics are nonlinear spin torque oscillator Yamaguchi2019-xt ; Iwakiri2020-ac , nonlinear spin-wave interference toward memory or neural network systems Adhikari2020-aa ; Hula2022-jh , and nonlinear magnon polariton for quantum information technologies Lee2023-ar . Recently, several theoretical studies predicted nonlinear spin polarization in non-centrosymmetric system Hamamoto2017 , $\mathcal{P}\mathcal{T}$ -symmetric collinear magnets Hayami2022-kt , and time-reversal centrosymmetric materials Xiao2022PRL ; Xiao2023PRL . The nonlinear spin polarization is of great interest because it probes novel band geometric quantities and offers new tools to characterize and control material properties. However, lacking is experimental studies of the nonlinear spin polarization. Therefore, in this work, we investigate experimentally the nonlinear spin polarization using spin-torque ferromagnetic-resonance (ST-FMR).

When electric current flows in a bilayer system consisting of heavy-metal, for example platinum (Pt), and ferromagnetic-metal, for example, permalloy (Py), the spin-Hall effect due to strong spin-orbit interaction in the Pt layer gives rise to spin polarization. The spin polarization causes spin current injected to the Py layer, bringing about spin torque exerted to precessing Py magnetization on resonance under magnetic fields; this is referred to as ST-FMR Liu2011-vd . In this paper, we carry out ST-FMR measurements under large dc current injection up to 20 mA ( $\sim 6.5\times 10^{11}$ A/m²) to directly observe current-induced nonlinear spin torque in Pt-Py bilayers. An undoped silicon (Si) substrate with excellent thermal conductivity enables us to inject such a large current without sample degradation due to the Joule heating. The ST-FMR signals demonstrate that the massive dc current affects the resonance field and Gilbert damping parameter nonlinearly. The nonlinear changes are traced back to the nonlinear spin torque caused by the nonlinear spin polarization. Furthermore, ST-FMR study reveals that the nonlinear spin torque is attributed also to magnon generation (annihilation) followed by effective magnetization shrinkage (expansion), which is confirmed by the observation of unidirectional spin Hall magnetoresistance (USMR) Avci2015 .

Eventually we evaluate the origins of the nonlinear spin torque, i.e., the nonlinear spin polarization and magnon generation/annihilation, by introducing indices of nonlinearity, $\eta$ and $\xi$ , obtained from ST-FMR and USMR measurements. The experimentally evaluated $\eta$ is larger than $\xi$ , indicating that the nonlinear spin polarization is dominant rather than the magnon generation/annihilation in the nonlinear spin torque. The $\eta$ and $\xi$ correspond respectively to the 2nd- and 3rd-order nonlinear susceptibilities, $\chi^{(2)}$ and $\chi^{(3)}$ , in nonlinear photonics Robert2008 . In analogy between photonics and electronics, $\eta$ and $\xi$ can be used to evaluate spintronic nonlinearity, elucidate the origins of nonlinear phenomena, and realize nonlinear spintronic effects, for example, second harmonic generation and rectification. Furthermore, the nonlinear spin torque leads to time-varying nonlinear magnetic metamaterials for 6th-generation mobile communication light sources of millimeter waves and THz light Kodama2023 .

II Experimental setup

We study metallic bilayers composed of 5 nm thick Pt top layer and 2 nm thick Py bottom layer. The Pt-Py bilayer is deposited after a 3 nm thick tantalum buffer layer on an undoped Si substrate having electrical resistivity at least 1 k $\Omega\cdot$ cm (Crystal Base, Inc.) Kodama2023 . An inset of Fig. 1(a) shows an optical microscopic image of the specimen consisting of lithographically-prepared Pt-Py strip attached to gold electrodes. The width of the strip is 5 $\mu$ m and the length is 24 $\mu$ m. In ST-FMR measurements, an in-plane external dc magnetic field $H_{\rm ext}$ is applied with a relative angle $\theta=45^{\circ}$ to the $y$ -axis as shown in Figs. 1(a) and 1(b). An ac current $I_{\rm ac}$ with microwave frequencies is applied between the signal (S) and ground (G) lines by a signal generator. The $I_{\rm ac}$ in the Pt layer generates an oscillating Oersted magnetic field, which primarily drives ST-FMR of the Py magnetization $M$ . Additionally, the spin-Hall effect in the Pt layer gives rise to ac spin current, which is injected into the Py layer. The spin angular momentum is transferred to the in-plane Py magnetization, exerting a field-like torque (FLT) that secondary drives ST-FMR and a damping-like torque (DLT) that enhances or reduces magnetic relaxation chiba2014prappl ; chiba2015jap ; schreier2015prb . Mixing of $I_{\rm ac}$ and oscillating anisotropic magnetoresistance (AMR) in Py gives rise to a time-independent longitudinal dc voltage $V_{\rm AMR}$ . We measure $V_{\rm AMR}$ as a function of $\mu_{0}H_{\rm ext}$ using a bias tee to obtain ST-FMR signals. All measurements are carried out at room temperature.

Refer to caption — Figure 1: (a) Measured ST-FMR signal (blue circles), as a function of external dc magnetic field, $\mu_{0}H_{\rm ext}$ , with $I_{\rm ac}$ at 9 GHz. Green and red solid lines represent fitting curves with symmetric ( $V_{\rm S}$ ) and anti-symmetric coefficients ( $V_{\rm A}$ ), respectively. Blue solid line corresponds to the sum of $V_{\rm S}$ and $V_{\rm A}$ . Inset: optical microscopic image of Py-Pt bilayer strip with Au electrodes. (b) Schematic of Py magnetization ( $M$ ), $H_{\rm ext}$ , and Oersted field ( $H_{\rm Oe}$ ). Effective field $H_{\rm FLT}$ and $H_{\rm DLT}$ correspond to field-like torque and damping-like torque, respectively. In-plane effective field $H_{\rm in}$ is a component of $H_{\rm FLT}$ and $H_{\rm Oe}$ parallel to $H_{\rm ext}$ .

III Results

III.1 Spin-torque ferromagnetic resonance measurement

Figure 1(a) shows a typical ST-FMR signal probed by $V_{\rm AMR}$ with $I_{\rm ac}$ at 9 GHz. The blue circles correspond to measured $V_{\rm AMR}$ . The $V_{\rm AMR}$ in a thin film is expressed as $V_{\rm AMR}=V_{\rm S}+V_{\rm A}$ , where $V_{\rm S}$ and $V_{\rm A}$ are symmetric and anti-symmetric components, respectively Liu2011-vd . Both $V_{\rm S}$ and $V_{\rm A}$ are described using $\mu_{0}H_{\rm ext}$ , the resonance field ( $\mu_{0}H_{\rm FMR}$ ), and the half width at half maximum ( $\mu_{0}\Delta_{\rm FMR}$ ) of the FMR signal. The fitting in Fig. 1(a) gives $\mu_{0}H_{\rm FMR}$ = 119.8 mT and $\mu_{0}\Delta_{\rm{FMR}}$ = 14.8 mT. The sum of $V_{\rm S}$ (green line) and $V_{\rm A}$ (red line) after the fitting is represented by the blue line, which reproduces well the measured $V_{\rm AMR}$ .

Together with $I_{\rm ac}$ , a dc current $I_{\rm dc}$ is applied to the bilayer to modify the FMR condition Kasai2014-ia ; Nan2015-yg . The $I_{\rm dc}>0$ ( $I_{\rm dc}<0$ ) corresponds to the current in the $+y$ ( $-y$ ) direction. The $I_{\rm dc}$ causes a time-independent dc Oersted field $\bm{H}_{\rm Oe}$ along $\pm x$ axis as shown in Fig. 1(b). Additionally, $I_{\rm dc}$ generates a time-independent FLT and DLT on $\bm{M}$ . As in Fig. 1(b), FLT and DLT are regarded as effective fields $\bm{H}_{\rm FLT}\propto{\delta}{\bm{s}}$ and $\bm{H}_{\rm DLT}\propto{\bm{m}}\times{\delta}{\bm{s}}$ , respectively. The ${\delta}{\bm{s}}$ is spin polarization and ${\bm{m}}$ is a unit vector of magnetization Fan2013-tq ; Karube2020-df . The $\bm{H}_{\rm FLT}$ and $\bm{H}_{\rm DLT}$ affect the FMR condition, resulting in a shift of $\mu_{0}H_{\rm FMR}$ and a change in $\mu_{0}\Delta_{\rm FMR}$ .

To study the shift of $\mu_{0}H_{\rm FMR}$ and change in $\mu_{0}\Delta_{\rm FMR}$ by $\bm{H}_{\rm FLT}$ and $\bm{H}_{\rm DLT}$ , the ST-FMR signals with $I_{\rm dc}$ between $-$ 20 and $+$ 20 mA are measured at various $I_{\rm ac}$ frequency ( $f_{\rm ac}$ ) from 3 to 9 GHz. Thanks to the undoped Si substrate with a better thermal conductivity of 150 W/mK Slack1964-pl compared to quartz (1.4 W/mK) Zhu2018-ib and magnesium oxide (56 W/mK) Stackhouse2010-ph substrates, a large $I_{\rm dc}$ up to $\pm$ 20 mA can be applied (See Supplemental Material SM1 SM ). After the fitting of the ST-FMR signals, we evaluate $\mu_{0}H_{\rm FMR}$ and $\mu_{0}\Delta_{\rm FMR}$ at a specific $I_{\rm dc}$ value. The resonance field shift $\mu_{0}H^{\rm shift}_{\rm FMR}$ by $I_{\rm dc}$ injection is derived from $\mu_{0}H^{\rm shift}_{\rm FMR}(I_{\rm dc})=\mu_{0}H_{\rm FMR}(I_{\rm dc})-\mu_{0}H_{\rm FMR}(0)$ , where $\mu_{0}H_{\rm FMR}(I_{\rm dc})$ corresponds to $\mu_{0}H_{\rm FMR}$ at non-zero $I_{\rm dc}$ and $\mu_{0}H_{\rm FMR}(0)$ corresponds to $\mu_{0}H_{\rm FMR}$ at zero $I_{\rm dc}$ . Moreover, $f_{\rm ac}$ -dependence of $\mu_{0}\Delta_{\rm FMR}$ gives Gilbert damping parameter $\alpha$ at a specific $I_{\rm dc}$ value.

Figure 2 shows $I_{\rm dc}$ versus $\mu_{0}H^{\rm shift}_{\rm FMR}$ at $f_{\rm ac}$ = 3 GHz (red asterisks), 6 GHz (red squares), and 9 GHz (red circles) as indicated from the left vertical axis. In addition, the variation in $\alpha$ by $I_{\rm dc}$ injection, $\Delta\alpha(I_{\rm dc})=\alpha(I_{\rm dc})-\alpha(0)$ , reproduced from our previous report Kodama2023 , is plotted as black triangles indicated from the right vertical axis. The $\mu_{0}H^{\rm shift}_{\rm FMR}$ and $\Delta\alpha$ are odd functions of $I_{\rm dc}$ , because $\delta{\bm{s}}$ is odd function of the dc current. Figure 2 highlights two striking features: i) $\mu_{0}H^{\rm shift}_{\rm FMR}$ and $\Delta\alpha$ are dependent nonlinearly on $I_{\rm dc}$ , and ii) a higher $f_{\rm ac}$ results in a larger $|\mu_{0}H^{\rm shift}_{\rm FMR}|$ at the same $I_{\rm dc}$ . Note that these features are observed in another specimen with a longer Pt-Py strip of 45 $\mu$ m length (see Supplemental Material SM2 SM ).

III.2 Evaluation of variation in effective magnetization

The $\mu_{0}H_{\rm FMR}$ with various $I_{\rm dc}$ of $-$ 20, 0, and 20 mA are plotted as a function of $f_{\rm ac}$ from 3 to 9 GHz in Fig. 3. Blue triangles, black circles, and red triangles correspond respectively to $\mu_{0}H_{\rm FMR}$ by $I_{\rm dc}$ = $-$ 20, 0, and 20 mA. Figures 3(b) and 3(c) show enlarged plots of Fig. 3(a), in which the horizontal axis variations are identical to be 25 mT for direct comparison. At $f_{\rm ac}$ = 3 GHz as in Fig. 3(c), the resonance field shifts upward by 5.7 mT when $I_{\rm dc}$ increases from $-$ 20 mA to 20 mA. At a higher $f_{\rm ac}$ of 9 GHz as in Fig. 3(b), the shift amount is larger to be 16.2 mT.

As shown in Fig. 1(b), a component of $H_{\rm FLT}$ and $H_{\rm Oe}$ parallel to the $H_{\rm ext}$ corresponds to the in-plane effective field $H_{\rm in}$ . The $H_{\rm in}$ expressed as $\mu_{0}H_{\rm in}=(\mu_{0}H_{\rm FLT}+\mu_{0}H_{\rm Oe})\times\sin\theta$ , where $\theta=\pi/4$ in the present ST-FMR study, is small, but affects the FMR condition. (see Supplemental Material SM3 SM ). The Kittel equation for FMR is described as

	$\displaystyle 2\pi f_{\rm ac}=\gamma\sqrt{\mu_{0}H_{\rm FMR}(I_{\rm dc}=0)+\mu_{0}H_{\rm in}}$
	$\displaystyle\times\sqrt{\mu_{0}H_{\rm FMR}(I_{\rm dc}=0)+\mu_{0}H_{\rm in}+\mu_{0}M_{\rm eff}},$		(1)

where $\gamma$ is gyromagnetic ratio, $\mu_{0}H_{\rm FMR}(I_{\rm dc}=0)$ is resonance field without $I_{\rm dc}$ injection, and $\mu_{0}M_{\rm eff}$ is effective magnetization. When $I_{\rm dc}$ = 0 mA, the black circles in Fig. 3 are fitted by Eq. (III.2) with $\mu_{0}H_{\rm in}=0$ . The fitting gives $\mu_{0}M_{\rm eff}(I_{\rm dc}=0)=\mu_{0}M_{\rm{s}}=$ 658 mT, where $\mu_{0}M_{\rm{s}}$ is the saturation magnetization. The $\mu_{0}M_{\rm{s}}$ of 658 mT is smaller than a typical value of Py saturation magnetization, probably due to magnetic dead layers at the Pt-Py interface Kodama2023 ; Hirayama2017-kz .

Equation (III.2) is fitted to $\mu_{0}H_{\rm FMR}$ at $I_{\rm dc}=$ $\pm$ 8, $\pm$ 12, $\pm$ 16, and $\pm$ 20 mA to evaluate $\mu_{0}M_{\rm eff}$ and $\mu_{0}H_{\rm in}$ . The fitting curves with $I_{\rm dc}$ = $-$ 20 and +20 mA are drawn by blue and red solid lines in Fig. 3, respectively. In Fig. 4, evaluated $\mu_{0}H_{\rm in}$ (red circles) and $\mu_{0}\Delta M_{\rm eff}=\mu_{0}M_{\rm eff}-\mu_{0}M_{\rm s}$ (blue square) are plotted as a function of $I_{\rm dc}$ . The upper horizontal axis indicates the $J_{\rm Pt}$ , electrical current density in the Pt layer (see Supplemental Material SM4 SM ). The inset shows an enlarged view of $\mu_{0}H_{\rm in}$ versus $I_{\rm dc}$ . The $\mu_{0}H_{\rm in}$ is very small, slightly decreases with increasing $I_{\rm dc}$ , and reaches at $-3.3$ mT when $I_{\rm dc}$ is 20 mA.

Contrastingly, $\mu_{0}\Delta M_{\rm eff}$ is affected significantly by $I_{\rm dc}$ . The maximum value of $\mu_{0}\Delta M_{\rm eff}$ at $I_{\rm dc}=$ 20 mA is $-60$ mT, which includes $\mu_{0}H_{\rm DLT}$ as shown in Fig. 1(b). However, $\mu_{0}\Delta M_{\rm eff}$ = $-60$ mT is nonetheless larger than the $\mu_{0}H_{\rm DLT}$ evaluated from previous reports, for example, $-2.4$ mT at approximately $I_{\rm dc}=$ 20 mA ( $\sim 6.5\times 10^{11}\rm A/m^{2})$ in Refs. Karube2020-df . Note here that $\mu_{0}H_{\rm FMR}$ with non-zero $I_{\rm dc}$ in Fig. 3 cannot be reproduced using Eq. (III.2) without $\mu_{0}\Delta M_{\rm eff}$ . This is clearly indicated in Eq. (III.2), i.e., $\mu_{0}H_{\rm in}$ shifts the curves whereas $\mu_{0}\Delta M_{\rm eff}$ changes the gradient of the curves. The large $\mu_{0}\Delta M_{\rm eff}$ is indispensable in explaining a larger resonance field shift at a higher $f_{\rm ac}$ . The negative $\mu_{0}\Delta M_{\rm eff}$ corresponds to shrinkage of $\mu_{0}M_{\rm eff}$ , whereas the positive $\mu_{0}\Delta M_{\rm eff}$ corresponds to expansion. Therefore, we consider shrinkage/expansion of effective magnetization by nonlinear magnon generation/annihilation.

III.3 Unidirectional spin-Hall magnetoresistance measurement

The nonlinear magnon generation/annihilation due to massive spin current injection likely brings about USMR Borisenko2018APL ; Kim2019APEX . We thus conduct USMR measurements with the large $I_{\rm dc}$ injection using the same specimen. Figures 5(a) and 5(b) show schematics of the sample cross section in the $x$ - $z$ plane viewed to the $+y$ direction. The $\theta$ is $+90^{\circ}$ or $-90^{\circ}$ in the USMR measurements, i.e.,the $\mu_{0}H_{\rm ext}$ = $+100$ mT is applied to the sample in the $+x$ direction (Fig. 5(a)), while $\mu_{0}H_{\rm ext}$ = $-100$ mT in the $-x$ direction (Fig. 5(b)). The dc current $I_{\rm dc}$ between $-$ 30 mA and $+$ 30 mA is chopped to be 0.2 ms width pulses. The longitudinal dc voltage $V_{\rm dc}$ are measured 100 times to obtain an averaged value of $V_{\rm dc}$ .

The measured $V_{\rm dc}-I_{\rm dc}$ curve is converted to the dc electric resistance $R$ - $I_{\rm dc}$ curve as shown in Fig. 5(c). Red crosses ( $+$ ) correspond to $R$ under $\mu_{0}H_{\rm ext}$ = $+100$ mT, referred to as $R_{+\rm H}$ , while blue crosses ( $\times$ ) corresponds to $R$ under $\mu_{0}H_{\rm ext}$ = $-100$ mT, referred to as $R_{-\rm H}$ . Because the current is applied beyond the ohmic region, Fig. 5(c) shows a parabolic increase in $R_{+\rm H}$ and $R_{-\rm H}$ due to the Joule heating Chiang2019-ju . Note here that a very similar increase in $V_{\rm dc}$ is confirmed in a measurement using un-chopped continuous $I_{\rm dc}$ as same as in the ST-FMR study (see Supplemental Material SM1 SM ). Figure 5(c) shows that $R_{\rm\pm H}$ increases from 245 $\Omega$ at $|I_{\rm dc}|\sim$ 0 mA to 255 $\Omega$ at $|I_{\rm dc}|\sim$ 20 mA. Given that a temperature coefficient of Pt resistance is 0.002 ${\rm K}^{-1}$ , the increase in $R_{\rm\pm H}$ from 245 to 255 $\Omega$ corresponds to the sample temperature elevation of approximately 20 K Belser1959-pc . This evaluation clearly indicates that the Joule heating component is small and not dominant in the present ST-FMR experiments with $|I_{\rm dc}|$ up to 20 mA.

As highlighted in the inset of Fig. 5(c), an enlarged view at approximately $I_{\rm dc}=$ 30 mA, $R_{\rm+H}$ (red crosses) is slightly larger than $R_{\rm-H}$ (blue crosses). The difference between $R_{\rm+H}$ and $R_{\rm-H}$ is plotted as a function of $I_{\rm dc}$ in Fig. 5(d). Note that the small Joule heating contribution, which is independent of the Py magnetization reversal, is already removed in the $[R_{\rm+H}]-[R_{\rm-H}]$ plot. As $I_{\rm dc}$ increases from 0 mA to 30 mA, $[R_{\rm+H}]-[R_{\rm-H}]$ increases slowly and then rapidly at above 20 mA. More strikingly, $[R_{\rm+H}]-[R_{\rm-H}]$ is odd under the $I_{\rm dc}$ direction reversal; this is the hallmark of USMR.

The USMR is caused by the electron scattering by magnons. When the spins are injected into the Py layer, parallel spin injection to the Py magnetization annihilates the magnons as in Fig. 5(a) whereas anti-parallel spin injection generates the magnons as in Fig. 5(b). The magnon generation/annihilation influences the electron-magnon scattering, resulting in a resistance change of the Py layer as USMR. The excited magnon number is increased nonlinearly when the inherent damping of Py is compensated by the anti-damping DLT Borisenko2018APL . This is consistent with nonlinear decrease in $\Delta\alpha$ observed in Fig. 2. The magnon excitation depending on the current is expressed as $a_{\rm USMR}I_{\rm dc}+c_{\rm USMR}(I_{\rm dc})^{3}$ , where $a_{\rm USMR}$ and $c_{\rm USMR}$ are linear and 3rd-order nonlinear coefficients, respectively Borisenko2018APL ; Avci2018 . The fitting gives parameters of $(a_{\rm USMR},c_{\rm USMR})=(1.08\times 10^{-3},5.67\times 10^{-6})$ , resulting in the ratio $\xi_{\rm USMR}=c_{\rm USMR}/a_{\rm USMR}\sim 5.25\times 10^{-3}$ $\rm(mA)^{-2}$ as summarized in Table 1. The fitting curve represented by a black solid line in Fig. 5(d) reproduces well the experimental results. This indicates that the USMR in Fig. 5(d) is traced back to the magnon generation/annihilation.

III.4 Magnetization measurement

Magnetization of the Pt-Py bilayer are measured using vibrating sample magnetometer (VSM). Figure 6(a) shows magnetization curves at 296 (room temperature, black) and 346 K (red). The magnetization is normalized by the saturation magnetization $\mu_{0}M_{\rm S}$ at 296 K. By elevating the temperature from 296 K to 346 K, $\mu_{0}M_{\rm S}$ decreases slightly. In Fig. 6(b), normalized $\mu_{0}M_{\rm S}$ is plotted as a function of temperature (red circles). $\mu_{0}M_{\rm S}$ decrease monotonically as temperature increases. The solid black line is a linear function obtained by the fitting of red circles. The gradient of the linear function is $-1.5\times 10^{-3}$ $K^{-1}$ . When sample temperature is 316 K, corresponding to 20 K elevation from room temperature, the normalized $\mu_{0}M_{\rm S}$ is 0.97.

IV Discussion

The 20 K temperature elevation due to the Joule heating confirmed in the USMR study causes a saturation magnetization decrease by 3 $\%$ evaluated by magnetization measurements. This value is smaller than the decrease in $|\mu_{0}\Delta M_{\rm eff}|$ of 60 mT evaluated in Fig. 4 corresponding to 9 $\%$ of $\mu_{0}M_{\rm{s}}=$ 658 mT. Moreover, $|\mu_{0}\Delta M_{\rm eff}|$ variation is asymmetric for the sign of $I_{\rm dc}$ . Therefore, the Joule heating is not dominant, indicating that the nonlinear spin polarization Xiao2022PRL ; Xiao2023PRL causes the nonlinear spin torque observed in the present ST-FMR study.

The spin polarization ${\delta}s_{i}$ is described by an electronic response to an applied electric field $E_{j}$ as ${\delta}s_{i}=\chi_{ij}^{s(1)}E_{j}+\chi_{ijk}^{s(2)}E_{j}E_{k}$ , where $i$ , $j$ , and $k$ are Cartesian indices and the Einstein summation convention is adopted Xiao2022PRL ; Xiao2023PRL . The $\chi_{ijk}^{s(2)}$ corresponds to the 2nd-order nonlinear response tensor, which is relevant to the Berry connection polarizability determined by the electronic band structures. In Fig. 2, we focus on the nonlinear variation of $\Delta\alpha$ because $\Delta\alpha$ is relevant to the magnitude of spin torque. The nonlinear spin polarization in Pt is partially absorbed by the Py magnetization, resulting in a nonlinear torque ( $\propto\delta{\bm{s}}\times{\bf M}$ ). In this way, the nonlinear torque caused by the nonlinear spin polarization gives rise to $\Delta\alpha$ depending on $(I_{\rm dc})^{2}$ . Indeed, the variation of $\Delta\alpha$ in Fig. 2 is reproduced well by the fitting curve $a_{\rm damp}I_{\rm dc}+b_{\rm damp}(I_{\rm dc})^{2}$ as represented by the black solid lines with $(a_{\rm damp},b_{\rm damp})=(1.35\times 10^{-3},2.10\times 10^{-5})$ . An index of nonlinearity is evaluated to be $\eta_{\rm damp}=b_{\rm damp}/a_{\rm damp}\sim 1.54\times 10^{-2}$ $\rm(mA)^{-1}$ as shown in Table 1.

Based on the Holstein-Primakoff picture Holstein1940-vb , the magnon number $\langle\hat{a}^{{\dagger}}\hat{a}\rangle$ is related to the precession angle $\varphi$ as $\langle\hat{a}^{{\dagger}}\hat{a}\rangle=S(1-\cos\varphi)=2S\sin(\varphi/2)$ , where $\hat{a}^{{\dagger}}(\hat{a})$ is the creation (annihilation) operators for magnons and the $S$ is the magnitude squared of the spin angular momentum Nakata2017-ps . Hence, the magnon generation (annihilation), $\delta\langle\hat{a}^{{\dagger}}\hat{a}\rangle$ , by the spin injection gives rise to the increase (decrease) of the precession angle $\delta\varphi$ , which corresponds to shrinkage (expansion) of the effective magnetization in the $x$ -axis direction as in Figs. 5(a) and 5(b) Nakata2017-ps . Because the spin torque is expressed as $\propto{\delta\bm{s}}\times{\bf M}_{\rm eff}$ , the nonlinear magnon generation/annihilation confirmed by the USMR study and followed by the magnetization shrinkage/expansion can thus be another origin of the nonlinear spin torque.

Given that the magnetization shrinkage/expansion contains both the 2nd- and 3rd-order nonlinearity Borisenko2018APL , $\mu_{0}\Delta M_{\rm eff}$ is expressed as $a_{\rm mag}I_{\rm dc}+b_{\rm mag}(I_{\rm dc})^{2}+c_{\rm mag}(I_{\rm dc})^{3}$ , where $a_{\rm mag}$ , $b_{\rm mag}$ and $c_{\rm mag}$ correspond respectively to the linear coefficient, and the 2nd- and 3rd-order nonlinear coefficients. A fitting curve with $(a_{\rm mag},b_{\rm mag},c_{\rm mag})=(9.51\times 10^{-4},4.34\times 10^{-5},2.35\times 10^{-6})$ represented by a solid black line in Fig. 4 reproduces experimentally obtained values of $\mu_{0}\Delta M_{\rm eff}$ . The $\xi_{\rm mag}=c_{\rm mag}/a_{\rm mag}$ is $2.47\times 10^{-3}$ ( $\rm mA)^{-2}$ , which is similar to $\xi_{\rm USMR}\sim 5.25\times 10^{-3}$ ( $\rm mA)^{-2}$ obtained from USMR measurements. In addition, the $\eta_{\rm mag}=b_{\rm mag}/a_{\rm mag}$ is $\sim 4.56\times 10^{-2}$ $\rm(mA)^{-1}$ , which is similar to $\eta_{\rm damp}\sim 1.54\times 10^{-2}$ $\rm(mA)^{-1}$ as summarized in Table 1.

Table 1: Nonlinearity indices

\eta

and

\xi

evaluated from ST-FMR and USMR measurements.

	$\eta$ $(\rm mA)^{-1}$	$\xi$ $(\rm mA)^{-2}$
USMR	-	$5.25\times 10^{-3}$
ST-FMR magnetization	$4.56\times 10^{-2}$	$2.47\times 10^{-3}$
ST-FMR damping	$1.54\times 10^{-2}$	-

The present paper reveals that massive $I_{\rm dc}$ causes the nonlinear spin torque. The nonlinear spin torque has two origins at least. The first origin is electronic one, i.e., the nonlinear spin polarization in the Pt layer. The nonlinear spin polarization results in nonlinear spin torque directly observed by nonlinear $\Delta\alpha$ variation and represented by $\eta_{\rm damp}$ and $\eta_{\rm mag}$ . The second origin is magnonic one, i.e., the nonlinear magnon generation/annihilation confirmed by USMR and effective magnetization shrinkage/expansion, and represented by $\xi_{\rm USMR}$ and $\xi_{\rm mag}$ . In comparison between $\eta$ and $\xi$ , Table 1 indicates that nonlinear spin polarization is dominant rather than nonlinear magnon generation/annihilation. However, the nonlinear magnon generation/annihilation is caused by the linear spin polarization as well as nonlinear spin polarization Sandweg2011 . Therefore, the magnonic origin could be comparable to the electronic origin. This is a future issue for the theoretical consideration.

The indices $\eta$ and $\xi$ obtained in ST-FMR and USMR measurements are utilized in nonlinear spintronics of magnetic insulators chiba2014prappl . The nonlinear spin polarization is anticipated to bring about spin-torque oscillation in the Pt-Py bilayer Divinskiy2019 ; Fulara2020-ka . Furthermore, the nonlinear variation of the $\alpha$ and $\mu_{0}M_{\rm eff}$ due to the nonlinear spin torque enables us to vary significantly the magnetic permeability of the magnetic bilayer system. Figures 7(a) shows nonlinear variation of $\alpha$ calculated using $\eta=1.54\times 10^{-2}$ . Nonlinear variation of $\mu_{0}M_{\rm eff}$ calculated using $\eta=4.56\times 10^{-2}$ and $\xi=2.47\times 10^{-3}$ is shown in Fig. 7(b). The relationship between $\mu_{0}M_{\rm eff}$ and the FMR resonance frequency $f_{\rm FMR}$ is expressed by the Kittel equation as

\displaystyle 2\pi f_{\rm FMR}=\omega_{\rm FMR}=\gamma\sqrt{\mu_{0}H_{\rm ext}(\mu_{0}H_{\rm ext}+\mu_{0}M_{\rm eff})},

(2)

where $\omega_{\rm FMR}$ is the resonance angular frequency. Nonlinear variation of $\omega_{\rm FMR}$ is thus obtained as plotted in Fig. 7(c). Using $\alpha$ , $\mu_{0}M_{\rm eff}$ and $\omega_{\rm FMR}$ , we evaluate the magnetic permeability variation under a dc external magnetic field $\mu_{0}H_{\rm FMR}=58.4$ mT. The relative permeability is written by real $\mu_{\rm r}^{\prime}$ and imaginary $\mu_{\rm r}^{\prime\prime}$ parts Kodama2023 as

\displaystyle\mu_{\rm r}(\omega)=\mu_{\rm r}^{\prime}(\omega)-j\mu_{\rm r}^{\prime\prime}(\omega),

(3)

where


$\displaystyle\mu_{\rm r}^{\prime}(\omega)$	$\displaystyle=1+\gamma\mu_{0}M_{\rm eff}\frac{\omega_{\rm FMR}(\omega_{\rm FMR}^{2}-\omega^{2})+\omega_{\rm FMR}\omega^{2}\alpha^{2}}{[\omega_{\rm FMR}^{2}-\omega^{2}(1+\alpha^{2})]^{2}+4\omega_{\rm FMR}^{2}\omega^{2}\alpha^{2}},$	(4a)
$\displaystyle\mu_{\rm r}^{\prime\prime}(\omega)$	$\displaystyle=\gamma\mu_{0}M_{\rm eff}\frac{\alpha\omega[\omega_{\rm FMR}^{2}-\omega^{2}(1+\alpha^{2})]}{[\omega_{\rm FMR}^{2}-\omega^{2}(1+\alpha^{2})]^{2}+4\omega_{\rm FMR}^{2}\omega^{2}\alpha^{2}}.$	(4b)

By substituting $\alpha$ , $\mu_{0}M_{\rm eff}$ and $\omega_{\rm FMR}$ into the Eq. (4), $\mu_{\rm r}(\omega)$ at each $I_{\rm dc}$ is evaluated.

Figures 7(d) and 7(e) show dispersion curves of $\mu_{\rm r}^{\prime}(\omega)$ and $\mu_{\rm r}^{\prime\prime}(\omega)$ , respectively, at various $I_{\rm dc}$ from $-$ 20 (blue) to $+$ 20 mA (red) at 1 mA intervals. In Fig. 7(f), $\mu_{\rm r}^{\prime}$ and $\mu_{\rm r}^{\prime\prime}$ obtained at 5.4 GHz are plotted as a function of $I_{\rm dc}$ . When $I_{\rm dc}$ is varied from $-$ 20 to 10 mA, only the $\mu_{\rm r}^{\prime}$ can be modified. Contrastingly, at 5.9 GHz, only $\mu_{\rm r}^{\prime\prime}$ can be modified as in Fig. 7(g). This is an advantageous in realizing time-varying permeability metamaterials for the microwave frequency conversion towards 6th-generation mobile communication light source.

V Conclusion

We directly observe the nonlinear spin torque in the Pt-Py bilayer by means of ST-FMR with a large dc current. The nonlinear spin torque observed by nonlinear $\Delta\alpha$ variation is attributed primarily to nonlinear spin polarization represented by $\eta$ . Moreover, ST-FMR and USMR measurements demonstrate that nonlinear magnon generation/annihilation followed by shrinkage/expansion of effective magnetization, represented by $\xi$ , is another origin of the nonlinear spin torque. Comparison between $\eta$ and $\xi$ indicates that nonlinear spin polarization is dominant rather than nonlinear magnon generation/annihilation. The real and imaginary parts of permeability can be varied independently using nonlinear spin torque. The present paper paves a way to spin-Hall effect based nonlinear spintronic devices as well as time-varying nonlinear magnetic metamaterials with tailor-made permeability.

Acknowledgements

The authors acknowledge Masatoshi Hatayama, Takashi Komine, Saburo Takahashi and Yoshiaki Kanamori for their valuable contributions in this work. N. K., S. O., and S. T. also thank Network Joint Research Center for Materials and Devices (NJRC). This work is financially supported by JST- CREST (JPMJCR2102).

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