Sample preparation
Samples with various Fe thicknesses tFe are grown by molecular-beam epitaxy (MBE). First, a GaAs buffer layer of 100 nm is grown in a III–V MBE. After that the substrate (semi-insulating wafer, which has a resistivity ρ between 1.72 × 108 Ω cm and 2.16 × 108 Ω cm) is transferred to a metal MBE without breaking the vacuum for the growth of the metal layers. For a better comparison of the physical properties of different samples, various Fe thicknesses are grown on a single two-inch wafer by stepping the main shadow shutter of the metal MBE. After the growth of the step-wedged Fe film, 1.5-nm Al/6-nm Pt layers are deposited on the whole wafer. Sharp reflection high-energy electron diffraction patterns have been observed after the growth of each layer (Supplementary Note 1), which indicate the epitaxial growth mode as well as good surface (interface) flatness. High-resolution transmission electron microscopy measurements (Supplementary Note 1) show that (1) all the layers are crystalline and (2) there is diffusion of Al into Pt but no significant Al–Fe and Pt–Fe interdiffusion. Therefore, the magnetic proximity effect between Fe and Pt is reduced. The intermixed Pt–Al alloy can be a good spin current generator. Previous work49 has shown that alloying Pt with Al enhances the spin-torque efficiency.
Device fabrication
First, Pt/Al/Fe stripes with a dimension of 4 μm × 20 μm and with the long side along the [110] and [100] orientations are defined by a mask-free writer and Ar-etching. After that, contact pads for the application of the d.c. current, which are made from 3-nm Ti and 50 nm Au, are prepared by evaporation and lift-off. Then, a 70-nm Al2O3 layer is deposited by atomic layer deposition to electrically isolate the d.c. contacts and the coplanar waveguide (CPW). Finally, the CPW consisting of 5 nm Ti and 150 nm Au is fabricated by evaporation, and the Fe/Al/Pt stripes are located in the gap between the signal line and ground line of the CPW (Fig. 2a). During the fabrication, the highest baking temperature is 110 °C. The CPW is designed to match the radiofrequency network that has an impedance of 50 Ω. The width of the signal line and the gap are 50 μm and 30 μm, respectively. Magnetization dynamics of Fe are excited by out-of-plane Oersted field induced by the radiofrequency microwave currents flowing in the signal and ground lines.
FMR measurements
The FMR method is used in this study for several reasons: (1) FMR has a higher sensitivity than static magnetization measurements. (2) The FMR method, together with angle and frequency-dependent measurements, is a standard way to quantify the effective magnetization, magnetic anisotropies and Gilbert damping. (3) Damping-like and field-like torques can be determined simultaneously in a single experiment, and thus we can establish a connection between damping-like torque and the modification of magnetic anisotropies. (4) The Joule heating effect, which also alters the magnetic properties of Fe, can be easily excluded from the I dependence of HR.
The FMR spectra are measured optically by time-resolved magneto-optical Kerr microscopy; a pulse train of a Ti:sapphire laser (repetition rate of 80 MHz and pulse width of 150 fs) with a wavelength of 800 nm is phase-locked to a microwave current. A phase shifter is used to adjust the phase between the laser pulse train and microwave, and the phase is kept constant during the measurement. The polar Kerr signal at a certain phase, VKerr, is detected by a lock-in amplifier by phase modulating the microwave current at a frequency of 6.6 kHz. The VKerr signal is measured by sweeping the external magnetic field, and the magnetic field can be rotated in-plane by 360°. A Keithley 2400 device is used as the d.c. current source for linewidth and resonance field modifications. All measurements are performed at room temperature.
The FMR spectra are well fitted by combining a symmetric (Lsym = ΔH2/[4(H − HR)2 + ΔH2]) and an anti-symmetric Lorentzian (La-sym = −4ΔH(H − HR)/[4(H − HR)2 + ΔH2]), VKerr = VsymLsym + Va-symLa-sym + Voffset, where HR is the resonance field, ΔH is the full width at half maximum, Voffset is the offset voltage, and Vsym (Va-sym) is the magnitude of the symmetric (anti-symmetric) component of VKerr. It is worth mentioning that, by analysing the position of HR, we have also confirmed that the application of the charge currents does not have a detrimental effect on the magnetic properties of the Fe films (Supplementary Note 2).
Magnetic anisotropies in Pt/Al/Fe/GaAs multilayers
A typical in-plane magnetic field angle φH dependence of the resonance field HR for tFe = 1.2 nm measured at f = 13 GHz is shown in Extended Data Fig. 2a. The sample shows typical in-plane uniaxial anisotropy with two-fold symmetry, that is, a magnetically HA for φH = −45° and 135° (\(\langle \bar{1}10\rangle \) orientations) and a magnetically EA for φH = 45° and 225° (⟨110⟩ orientations), which originates from the anisotropic bonding at the Fe/GaAs interface33. To quantify the magnitude of the anisotropies, we further measure the f dependence of HR both along the EA and the HA (Extended Data Fig. 2b). Both the angle and frequency dependence of HR are fitted according to34,50
$${\left(\frac{2{\rm{\pi }}f}{\gamma }\right)}^{2}={\mu }_{0}^{2}{H}_{1}^{\text{R}}{H}_{2}^{\text{R}},$$
(5)
with \({H}_{1}^{\text{R}}\) = HR cos(φ − φH) + HK + HB(3 + cos 4φ)/4 − HU sin2(φ − 45°) and \({H}_{2}^{\text{R}}\) = HR cos(φ − φH) + HB cos 4φ − HU sin 2φ. Here γ (= gμB/ħ) is the gyromagnetic ratio, g is the Landé g-factor, μB is the Bohr magneton, ħ is the reduced Planck constant, HK (= M − H⊥) is the effective demagnetization magnetic anisotropy field, including the perpendicular magnetic anisotropy field H⊥, HB is the biaxial magnetic anisotropy field along the ⟨100⟩ orientations, HU is the in-plane UMA field along ⟨110⟩ orientations and φ is the in-plane angle of magnetization as defined in Extended Data Fig. 1. The magnitude of φ is obtained by the equilibrium condition
$${H}_{{\rm{R}}}\,\sin (\varphi -{\varphi }_{H})+({H}_{{\rm{B}}}/4)\sin 4\varphi +({H}_{{\rm{U}}}/2)\cos 2\varphi =0.$$
(6)
It can be checked that φ = φH holds when H is along ⟨110⟩ and \(\langle \bar{1}10\rangle \) orientations. From the fits of HR, the magnitude of the magnetic anisotropy fields HA (HA = HK, HB, HU) for each tFe is obtained, and their dependences on inverse Fe thickness \({t}_{\text{Fe}}^{{-}1}\), together with the results obtained from the AlOx/Fe/GaAs samples, are shown in Extended Data Fig. 2c. The results show that the Pt/Al/Fe/GaAs samples have virtually identical magnetic anisotropies as the AlOx/Fe/GaAs samples, and introducing the Pt/Al layer neither enhances the magnetization leading to an increase in HK nor generates a perpendicular anisotropy leading to a decrease in HK. By comparing the values of HK and M, we confirm that the main contribution to HK stems from the magnetization due to the demagnetization field. For both sample series, HK and HB decrease as tFe decreases because of the reduction of the magnetization as tFe decreases, and both of them scale linearly with \({t}_{\text{Fe}}^{{-}1}\). The intercept (about 2,220 mT) of the \({H}_{{\rm{K}}}-{t}_{\text{Fe}}^{-1}\) trace corresponds to the saturation magnetization of bulk Fe, and the intercept (around 45 mT) of the \({H}_{{\rm{B}}}-{t}_{\text{Fe}}^{-1}\) trace corresponds to the biaxial anisotropy of bulk Fe. In contrast to HK and HB, HU shows a linear dependence on \({t}_{\text{Fe}}^{{-}1}\) with a zero intercept, indicative of the interfacial origin of HU.
Effective mixing conductance in Pt/Al/Fe/GaAs multilayers
Extended Data Fig. 3a,b shows the φH dependence and f dependence, respectively, of linewidth ΔH for tFe = 1.2 nm. The magnitude of ΔH varies strongly with φH because of the presence of in-plane anisotropy, and the dependencies of ΔH on f along both EA and HA show linear behaviour. Both the angular and frequency dependence of ΔH can be well fitted by51
$$\Delta H=\Delta [\text{Im}(\chi )]+\Delta {H}_{0}=\Delta \left[\frac{\alpha \sqrt{{H}_{1}^{\text{R}}{H}_{2}^{\text{R}}}({H}_{1}{H}_{1}+{H}_{1}^{\text{R}}{H}_{2}^{\text{R}})M}{{({H}_{1}{H}_{2}-{H}_{1}^{\text{R}}{H}_{2}^{\text{R}})}^{2}+{\alpha }^{2}{H}_{1}^{\text{R}}{H}_{2}^{\text{R}}{({H}_{1}+{H}_{2})}^{2}}\right]+\Delta {H}_{0},$$
(7)
where Δ[Im(χ)] is the linewidth of the imaginary part of the dynamic magnetic susceptibility Im(χ), H1 and H2 are defined in equation (5) for arbitrary H values, and ΔH0 is the residual linewidth (zero-frequency intercept). As the angular trace can be well fitted by using a damping value of 0.0078, there is no need to consider other extrinsic effects (that is, inhomogeneity and/or two-magnon scattering) contributing to ΔH. It is worth mentioning that the angular trace gives a slightly higher α value because ΔH0, which also depends on φH, is not considered in the fit. In this case, the frequency dependence of linewidth gives more reliable damping values (Extended Data Fig. 3b). Extended Data Fig. 3c compares the magnitude of damping for Pt/Al/Fe/GaAs and AlOx/Fe/GaAs samples. For both sample series, the Gilbert damping increases as tFe decreases and a linear dependence of α on \({t}_{\text{Fe}}^{{-}1}\) is observed. The enhancement of α is because of the spin pumping effect, which is given by52,53
$$\alpha ={\alpha }_{0}\,+\,{g}_{{\rm{eff}}}^{\uparrow \downarrow }\frac{\gamma \hbar }{4{\rm{\pi }}M}{t}_{{\rm{Fe}}}^{-1},$$
(8)
where α0 is the intrinsic damping of pure bulk Fe and \({g}_{{\rm{eff}}}^{\uparrow \downarrow }\) is the effective spin mixing conductance quantifying the spin pumping efficiency. By using μ0M = 2.2 T and γ = 1.80 × 1011 rad s−1 T−1, the magnitude of \({g}_{{\rm{eff}}}^{\uparrow \downarrow }\) for Pt/Al/Fe/GaAs is determined to be 4.6 × 1018 m−2, and \({g}_{{\rm{eff}}}^{\uparrow \downarrow }\) at the Fe/GaAs interface is determined to be 1.9 × 1018 m−2. Therefore, by subtracting these two values, the magnitude of \({g}_{{\rm{eff}}}^{\uparrow \downarrow }\) at Pt/Al/Fe interface is determined to be 2.7 × 1018 m−2. The spin transparency Tint of the Pt/Al/Fe interface is given by ref. 53
$${T}_{{\rm{int}}}=\frac{2{e}^{2}}{h}\frac{{g}_{{\rm{eff}}}^{\uparrow \downarrow }}{{G}_{{\rm{Pt}}}}$$
(9)
where 2e2/h is the conductance quantum, GPt [= 1/(ρxxλs)] is the spin conductance of Pt, ρxx is the resistivity and λs is the spin diffusion length. By using λs = 4 nm and an averaged ρxx = 40 μΩ cm, Tint = 0.21 is determined. We note that the magnitude of \({g}_{{\rm{eff}}}^{\uparrow \downarrow }\) at the Pt/Al/Fe interface is about one order of magnitude smaller than the experimental values found at heavy metal/ultrathin ferromagnet interfaces54, but very close to the value obtained by the first-principles calculations55. The previously overestimated \({g}_{{\rm{eff}}}^{\uparrow \downarrow }\) and thus Tint at heavy metal/ultrathin ferromagnet interfaces is probably because the enhancement of α by two-magnon scattering56 as well as by the magnetic proximity effect (see Supplementary Note 3) is not properly excluded. Moreover, the obtained α0 values for Pt/Al/Fe/GaAs (α0 = 0.0039) and AlOx/Fe/GaAs (α0 = 0.0033) slightly differ; the reason is unclear to us, but might be because of a small error in the Fe thickness, which is hard to be determined accurately in the ultrathin regime.
Theory of the modulation of the linewidth
To model the modulation of the FMR linewidth by the application of d.c. current, the Landau–Lifshitz–Gilbert equation with damping-like spin-torque term is considered18,35,
$$\frac{{\rm{d}}{\bf{M}}}{{\rm{d}}t}=-\gamma {\bf{M}}\times {\mu }_{0}{{\bf{H}}}_{{\rm{e}}{\rm{f}}{\rm{f}}}+\frac{\alpha }{M}{\bf{M}}\times \frac{{\rm{d}}{\bf{M}}}{{\rm{d}}t}-\frac{\gamma {\mu }_{0}{h}_{{\rm{D}}{\rm{L}}}}{M}{\bf{M}}\times {\bf{M}}\times {\boldsymbol{\sigma }}.$$
(10)
The terms on the right side of equation (10) correspond to the precession torque, the damping torque and the damping-like spin torque induced by the spin current. Here σ is the spin polarization unit vector, and hDL is the effective anti-damping-like magnetic field. The effective magnetic field Heff, containing both external and internal fields, is expressed in terms of the free energy density F, which can be obtained as
$${{\bf{H}}}_{\text{eff}}=-\frac{1}{{\mu }_{0}}\frac{\partial F}{\partial {\bf{M}}}.$$
(11)
For single-crystalline Fe films grown on GaAs(001) substrates with in-plane magnetic anisotropies, F is given by34,58
$$\,F=\frac{{\mu }_{0}M}{2}\left\{-2H[\cos \theta \cos {\theta }_{H}+\sin \theta \sin {\theta }_{\text{H}}\cos (\varphi -{\varphi }_{H})]+{H}_{\text{K}}{\cos }^{2}\theta -\frac{{H}_{\text{B}}}{2}{\sin }^{4}\theta \frac{3+\cos 4\varphi }{4}-{H}_{\text{U}}{\sin }^{2}\theta {\sin }^{2}\left(\varphi -\frac{{\rm{\pi }}}{4}\right)\right\}.$$
(12)
Bringing equations (11) and (12) into equation (10), the time-resolved magnetization dynamics for current flowing along the [110] orientation (that is, σ ∥ \([\bar{1}10]\)) is obtained as
$$\left\{\begin{array}{l}\frac{\partial \varphi }{\partial t}=\frac{\gamma {\mu }_{0}}{\left(1+{\alpha }^{2}\right)M\sin \theta }\left(\frac{\partial F}{\partial \theta }-\frac{\alpha }{\sin \theta }\frac{\partial F}{\partial \varphi }\right)+\frac{\gamma {\mu }_{0}{h}_{{DL}}}{\left(1+{\alpha }^{2}\right)\sin \theta }\frac{\sqrt{2}}{2}\left[\alpha \cos \theta \left(\sin \varphi -\cos \varphi \right)+\cos \varphi +\sin \varphi \right]\\ \frac{\partial \theta }{\partial t}=\frac{\gamma {\mu }_{0}}{M\sin \theta }\left(\frac{{\alpha }^{2}}{1+{\alpha }^{2}}-1\right)\frac{\partial F}{\partial \varphi }-\frac{\alpha }{1+{\alpha }^{2}}\frac{\gamma {\mu }_{0}}{M}\frac{\partial F}{\partial \theta }+\left(1+\frac{{\alpha }^{2}}{1+{\alpha }^{2}}\right)\gamma {\mu }_{0}{h}_{{DL}}\frac{\sqrt{2}}{2}\cos \theta \left(\sin \varphi -\cos \varphi \right)+\frac{\alpha }{1+{\alpha }^{2}}\gamma {\mu }_{0}{h}_{{DL}}\frac{\sqrt{2}}{2}\left(\cos \varphi +\sin \varphi \right)\end{array}\right.$$
(13)
Similarly, for the current flowing along the [100]-orientation (that is, σ ∥ [010]), we have
$$\left\{\begin{array}{l}\frac{\partial \varphi }{\partial t}=\frac{\gamma {\mu }_{0}}{\left(1+{\alpha }^{2}\right)M\sin \theta }\left(\frac{\partial F}{\partial \theta }-\frac{\alpha }{\sin \theta }\frac{\partial F}{\partial \varphi }\right)+\frac{\gamma {\mu }_{0}{h}_{{DL}}}{\left(1+{\alpha }^{2}\right)\sin \theta }\left(\alpha \cos \theta \sin \varphi +\cos \varphi \right)\\ \frac{\partial \theta }{\partial t}=\frac{\gamma {\mu }_{0}}{M\sin \theta }\left(\frac{{\alpha }^{2}}{1+{\alpha }^{2}}-1\right)\frac{\partial F}{\partial \varphi }-\frac{\alpha }{1+{\alpha }^{2}}\frac{\gamma {\mu }_{0}}{M}\frac{\partial F}{\partial \theta }-\gamma {\mu }_{0}{h}_{{DL}}\left[\frac{{\alpha }^{2}}{1+{\alpha }^{2}}\left(\alpha \cos \theta \sin \varphi +\cos \varphi \right)-\cos \theta \sin \varphi \right]\end{array}.\right.$$
(14)
The time dependence of φ(t), θ(t) and then m(t) can be readily obtained from equations (13) and (14), and Extended Data Fig. 4a shows an example of the time-dependent mz by using μ0H = 101 mT, μ0HK = 1,350 mT, μ0HU = 128 mT, μ0HB = 10 mT, α = 0.0063 and μ0HDL = 0. The damped oscillating dynamic magnetization can be well fitted by
$${m}_{z}(t)=A{\text{e}}^{-t/\tau }\cos (2{\rm{\pi }}ft+\phi )$$
(15)
where A is the amplitude, τ is the magnetization relaxation time and ϕ is the phase shift. The connection between τ and ΔH is given by
$$\Delta H=\frac{1}{2{\rm{\pi }}}\left|\frac{{\rm{d}}{H}_{\text{R}}}{{\rm{d}}f}\right|\frac{1}{\tau }$$
(16)
where dHR/df can be readily obtained from equation (5). We confirm the validity of the above method in Extended Data Fig. 4b by showing that the angle dependence of ΔH obtained from the time domain (equation (16)) at hDL = 0 is identical to the linewidth obtained by the dynamic susceptibility in the magnetic field domain (equation (7)).
Having obtained the linewidth for I = 0, the next step is to calculate the influence of the linewidth by spin–orbit torque. The magnitude of hDL is given by
$${\mu }_{0}{h}_{\text{DL}}=\frac{\hbar }{2e}\frac{\xi }{M{t}_{\text{Fe}}}{j}_{\text{Pt}}$$
(17)
where ξ is the effective damping-like torque efficiency and jPt is the current density in Pt. For the Pt/Al/Fe multilayer, jPt is determined by the parallel resistor model
$${j}_{\text{Pt}}=\frac{{t}_{\text{Pt}}\,{\rho }_{\text{Al}}{\rho }_{\text{Fe}}}{{t}_{\text{Pt}}\,{\rho }_{\text{Al}}{\rho }_{\text{Fe}}+{t}_{\text{Al}}{\rho }_{\text{Pt}}{\rho }_{\text{Fe}}+{t}_{\text{Fe}}{\rho }_{\text{Pt}}{\rho }_{\text{Al}}}\frac{I}{w{t}_{\text{Pt}}}$$
(18)
where ρPt (= 40 μΩ cm), ρAl (= 10 μΩ cm) and ρFe (= 50 μΩ cm) are the resistivities of the Pt, Al and Fe layers, respectively; tPt, tAl and tFe are the thicknesses of the Pt, Al and Fe layers, respectively; I is the d.c. current; and w is the width of the device. Plugging equations (17) and (18) into equations (13) and (14), the I dependence of ΔH can be obtained. An example is shown in Extended Data Fig. 4c, which shows a linear ΔH−I relationship. From the linear fit (equation (1) in the main text), we obtain the modulation amplitude of ΔH, that is, d(ΔH)/dI. Extended Data Fig. 4d presents the calculated d(ΔH)/dI as a function of the magnetic field angle, which shows a strong variation around the HA.
To reproduce the experimental data as shown in Fig. 1f in the main text, the magnitude of the magnetic anisotropies and the damping parameter obtained in Extended Data Fig. 3 as well as ξ = 0.06 are used. Note that the distinctive presence of robust UMA at the Fe/GaAs interface significantly alters the angular dependence of d(ΔH)/dI. This deviation is remarkable when compared with the sinφI–H dependence of d(ΔH)/dI as observed in polycrystalline samples, such as Pt/Py (refs. 57,58).
To understand the strong deviation of d(ΔH)/dI around the HA, we plot the in-plane angular dependence of F in Extended Data Fig. 5 for θ = θH = 90°, that is,
$$F=\frac{{\mu }_{0}M}{2}\left[-2{H}_{\text{R}}\cos (\varphi -{\varphi }_{H})-\frac{{H}_{\text{B}}}{2}\frac{3+\cos 4\varphi }{4}-{H}_{\text{U}}{\sin }^{2}\left(\varphi -\frac{{\rm{\pi }}}{4}\right)\right].$$
(19)
It shows that, around the HA (approximately ±15°), the magnetic potential barrier completely vanishes and \(\frac{\partial F}{\partial \varphi }=0\) and \(\frac{{\partial }^{2}F}{\partial \varphi } < 0\) hold. This indicates that the net static torques induced by internal and external magnetic fields acting on the magnetization cancel and the magnetization has a large cone angle for precession59. Consequently, the magnetization behaves freely with no constraints in the vicinity of the HA, and the low stiffness allows larger d(ΔH)/dI values induced by spin current60. If there are no in-plane magnetic anisotropies, the free energy is constant and is independent of the angle, the magnetization always follows the direction of the applied magnetic field and has the same stiffness at each position. Therefore, the modulation shows no deviation around the HA.
Frequency dependence of the linewidth modulation
Extended Data Fig. 6a shows the frequency dependence of the modulation of linewidth d(ΔH)/dI for tFe = 2.8 nm and 1.2 nm, in which the current flows along the [100] orientation. For both samples, the modulation changes polarity as the direction of M is changed by 180°. The modulation amplitude increases quasi-linearly with frequency, and the experimental results can be also reproduced by equation (14) using ξ = 0.06, consistent with the angular modulation shown in Fig. 2f. For H along the ⟨110⟩ and \(\langle \bar{1}10\rangle \) orientations, the frequency and the Fe thickness dependence of linewidth modulation is approximately given by24
$$\frac{\text{d}({\mu }_{0}\Delta H)}{\text{d}(I)}=2\frac{2{\rm{\pi }}f}{\gamma }\frac{\sin {\varphi }_{I-H}}{{H}_{\text{R}}+{H}_{\text{K}}/2}\frac{\hbar }{2e}\frac{\xi }{{Mt}_{\text{Fe}}}\frac{1}{{t}_{\text{Pt}}w},$$
(20)
where φI–H = 45°, 135°, 225° and 315° as shown by the inset of each panel in Extended Data Fig. 6. The damping-like torque efficiency can be further quantified by the slope s of f-dependence modulation, that is, \(s=\frac{\text{d}[\text{d}(\Delta H)\,/\,\text{d}I]}{\text{d}f}\). Extended Data Fig. 7 shows the absolute value of s values as a function of \({t}_{\text{Fe}}^{{-}1}\). A linear dependence of |s| on \({t}_{\text{Fe}}^{{-}1}\) is observed, which indicates that the damping-like torque is an interfacial effect, originating from the absorption of spin current generated in Pt (ref. 61).
Quantifying the modification of the magnetic anisotropies
In this section, we show our procedure to quantify the modulation of magnetic anisotropies by spin currents. According to equation (5), the f dependencies of HR along the EA (φH = φ = 45° and 225°) and the HA (φH = φ = 135° and 315°) are given by equation (3). From the angle and frequency dependencies of HR as shown in Extended Data Fig. 2, μ0HK = 1,350 mT, μ0HU = 128 mT, μ0HB = 10 mT and g = 2.05 are determined for tFe = 1.2 nm. Extended Data Fig. 8a shows the HR dependence of f for μ0HK = 1,350 mT (blue solid line) and μ0HK + Δμ0HK = 1,400 mT (red solid line) along the HA calculated by equation (3). To exaggerate the difference, μ0ΔHK of 50 mT is assumed. The shift of the resonance field ΔHR is obtained as ΔHR = HR(HK) − HR(HK + ΔHK), and the frequency dependence of ΔHR is plotted in Extended Data Fig. 8b, which shows a linear behaviour with respect to f between 10 GHz and 20 GHz (in the experimental range), that is, ΔHR = kKf. Note that, to simplify the analysis, the zero-frequency intercept is ignored because the magnitude is much smaller than the intercept induced by ΔHU and ΔHB. The sign of the slope kK is the same as that of ΔHK and its magnitude is proportional to ΔHK, that is, kK ∝ ΔHK. For the EA as shown in Extended Data Fig. 8c,d, the ΔHR–f relationship induced by ΔHK remains the same as for the HA, that is, ΔHR = kKf still holds.
Extended Data Fig. 8e shows the HRdependence of f for μ0HU = 128 mT (blue solid line) and μ0HU + μ0ΔHU = 178 mT (red solid line) along the HA. As shown in Extended Data Fig. 8f, the shift of the resonance field along the HA is independent of f with a negative intercept, that is, ΔHR = −ΔHU. However, for the EA, as shown in Extended Data Fig. 8g,h, the f-dependent ΔHR can be expressed as ΔHR = ΔHU − kUf, which has an opposite slope compared with the ΔHR–f relationships induced by ∆HK (Extended Data Fig. 8d), that is, kU ∝ −ΔHU.
If the modulation is induced by a change in the biaxial anisotropy as shown in Extended Data Fig. 8i–l, ΔHR along both the HA and EA shows a linear dependence on f, which is expressed as ΔHR = −ΔHB + kBf, and kB ∝ ΔHB holds.
Extended Data Table 1 summarizes the ΔHR–f relationships both along the EA and HA induced by ΔHK, ΔHU and ΔHB.
As hOe/FL generated by the d.c. current also shifts the resonance field along the EA and HA axes by \(\pm \frac{\sqrt{2}}{2}{h}_{\text{Oe}/\text{FL}}\), where plus corresponds to the [110] (EA) and the \([\bar{1}10]\) (HA) directions, and minus corresponds to the \([\bar{1}\bar{1}0]\) (EA) and the \([1\bar{1}0]\) (HA) directions, the total ΔHR induced by ΔHK, ΔHU and ΔHB along the EA and HA is, respectively, given by equation (4).
Based on equations (4) and (5), the values of ΔHK, ΔHU, ΔHB and hOe/FL for tFe ≤ 2.2 nm are extracted as follows:
-
1.
We consider the results obtained for H ∥ M ∥ [110] (EA) and H ∥ M/\([1\bar{1}0]\) (HA) as shown in Extended Data Fig. 9a (the same results as shown in Fig. 4 in the main text for I = 1 mA), where the net magnetization is parallel to I. At f = 0, equation (4) is reduced to
$$\Delta {H}_{\text{R}}^{\text{EA}}(0)=\Delta {H}_{\text{U}}-\Delta {H}_{\text{B}}+\frac{\sqrt{2}}{2}{h}_{\text{Oe}/\text{FL}}=-0.20\,{\rm{mT}}$$
(21)
$$\Delta {H}_{\text{R}}^{\text{HA}}(0)=-(\Delta {H}_{\text{U}}+\Delta {H}_{\text{B}})-\frac{\sqrt{2}}{2}{h}_{\text{Oe}/\text{FL}}=-0.32\,{\rm{mT}}.$$
(22)
By adding equations (21) and (22), the magnitude of ΔHB is determined to be 0.26 mT, which corresponds to kB of 4 × 10−3 mT GHz−1 according to equation (3).
-
2.
From Extended Data Fig. 9a, the slope along the HA is determined to be kK + kB = 0.025 mT GHz−1. Thus, the magnitude of kK is determined by kK = 0.025 mT GHz−1 − kB = 0.021 mT GHz−1, which corresponds to ΔHK = 2.0 mT according to equation (3).
-
3.
As \(\Delta {H}_{\text{R}}^{\text{EA}}\) is frequency independent, this requires that kU = kK + kB = 0.025 mT GHz−1, which corresponds ΔHU = 2.5 mT.
-
4.
As the magnetization along EA and HA is, respectively, rotated by 180° to the \([\bar{1}\bar{1}0]\) and \([\bar{1}10]\) directions, and the net magnetization is antiparallel to I (Extended Data Fig. 9b), we obtain ΔHB = −0.26 mT, ΔHK = −2.0 mT and ΔHU = −2.5 mT, which are of opposite sign as the results obtained from Extended Data Fig. 9a.
-
5.
Finally, bringing the magnitude of ΔHB and ΔHU back into equations (21) and (22), \(\frac{\sqrt{2}}{2}{h}_{\text{Oe}/\text{FL}}\) is determined to be −2.24 mT. The negative sign of hOe/FL indicates that it is along the \([0\bar{1}0]\) orientation.
Similarly, the corresponding ΔHB, ΔHK and ΔHU values can be determined for tFe = 2.2 nm (Extended Data Fig. 10). Extended Data Table 2 summarizes the magnitudes of the magnetic anisotropy modifications as well as the hOe/FL values for all the devices. The enhancement of the field-like torque in thinner samples has been observed in other systems and is probably because of the enhanced Bychkov–Rashba spin–orbit interaction61,62 and/or the orbital angular momentum (orbital Hall effect and orbital Rashba effect) at the ferromagnetic metal/heavy metal interface62.
It is worth mentioning that, once the magnetization direction is fixed, ΔHB, ΔHK and ΔHU obtained either from Extended Data Fig. 9a (Extended Data Fig. 10a) or from Extended Data Fig. 9b (Extended Data Fig. 10b) have the same sign (either positive or negative depending on the direction of M). This is consistent with the change in magnetic anisotropies by temperature (Supplementary Fig. 7), which shows that the magnitude of ΔHB, HK and ΔHU increases as the temperature decreases and decreases as the temperature increases. This indicates that the increase in the magnetic anisotropies is dominated by the increase in M as temperature decreases and the decrease in the magnetic anisotropies is dominated by the decrease in M as temperature increases. For the spin current modification demonstrated here, the temperature is not changed but the change in M is induced by populating the electronic bands by the spin current. More interestingly, the new modification method can control the increase or decrease in M simply by the direction of current and/or the direction of magnetization, which is not accessible by other controls.
Alternative interpretation of the experimental results
It is known that the starting point of the FMR analysis is the static magnetic energy landscape, which is related to the magnetic anisotropies. Therefore, it is natural to consider that the modification of magnetic anisotropy accounts for the f-linear dHR/dI curves as observed in the experiment. Although the data analysis discussed in the previous section is self-consistent, there could be alternative interpretations of the data. One possibility could be the current-induced modification of the Landé g-factor of Fe. In magnetic materials, it is known that g is related to the orbital moment μL and the spin moment μS:
$$g=\frac{2{\mu }_{\text{L}}}{{\mu }_{\text{S}}}+2.$$
(23)
A flow of spin and orbital angular momentum induced by charge current could, respectively, modify the orbital and spin moment of Fe by ΔμS and ΔμL, and then a change in the gyromagnetic ratio of Fe is expected. This could, in turn, lead to a shift of FMR resonance fields linearly depending on the frequency. However, if this were the case, an anisotropic modification of g is needed to interpret the data as observed in Extended Data Figs. 9 and 10 (that is, there is sizeable modification along the HA, but no modification along the EA). As we cannot figure out why the modification of g could be anisotropic, we ignore the discussion of the g-factor modification in the main text. We are also open to other possible explanations for the experimental observations.
Estimation of the magnitude of spin transfer electrons
The change in magnetization is attributed to the additional filling of the electronic d-band. The induced filling of the bands in Fe occurs mainly close to the interface and is not homogeneously distributed, as it depends on the spin diffusion length of the spin current in Fe. In other words, the measured modulated magnetic anisotropies are averaged over the whole ferromagnetic film. For simplicity, we neglect the spin current distribution in Fe and assume that it is homogeneously distributed. The spin chemical potential at the interface63 is given by \({u}_{\text{s}}^{0}=2e\lambda \xi E\tanh \left(\frac{{t}_{\text{Pt}}}{2\lambda }\right)\), where e is the elementary charge, λ is the spin diffusion length, E (= j/σ) is the electric field, j is the current density and σ is the conductivity of Pt. The areal spin density ns transferred into Fe is obtained as \({n}_{\text{s}}={u}_{\text{s}}^{0}\lambda N\) (ref. 18), where N is the density of states at the Fermi level. Using N = 6 × 1048 J−1 m−3, λ = 4 nm, ξ = 0.06, σ = 2.0 × 106 Ω−1 m−1, ns = 4.2 × 1012 μB cm−2 is obtained for I = 1 mA. As Fe has a bcc structure (lattice constant a = 2.8 Å) with a moment of about 1.0 μB for tFe = 1.2 nm at room temperature64, the areal density of the magnetic moment of Fe nFe is determined to be 2.6 × 1014 μB cm−2. In this case, the filling of the d-band by spin current leads to a change in the magnetic moment of the order of ns/nFe ≈ 0.16%, which agrees with the ratio between ΔHK and HK, that is, ΔHK/HK ≈ 2.0 mT/ 1 T ≈ 0.2%.
