Sample preparation
Samples with various Fe thicknesses t_{Fe} are grown by molecularbeam epitaxy (MBE). First, a GaAs buffer layer of 100 nm is grown in a III–V MBE. After that the substrate (semiinsulating wafer, which has a resistivity ρ between 1.72 × 10^{8} Ω cm and 2.16 × 10^{8} 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 twoinch wafer by stepping the main shadow shutter of the metal MBE. After the growth of the stepwedged Fe film, 1.5nm Al/6nm Pt layers are deposited on the whole wafer. Sharp reflection highenergy 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. Highresolution 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 work^{49} has shown that alloying Pt with Al enhances the spintorque 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 maskfree writer and Aretching. After that, contact pads for the application of the d.c. current, which are made from 3nm Ti and 50 nm Au, are prepared by evaporation and liftoff. Then, a 70nm Al_{2}O_{3} 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 outofplane 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 frequencydependent measurements, is a standard way to quantify the effective magnetization, magnetic anisotropies and Gilbert damping. (3) Dampinglike and fieldlike torques can be determined simultaneously in a single experiment, and thus we can establish a connection between dampinglike 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 H_{R}.
The FMR spectra are measured optically by timeresolved magnetooptical 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 phaselocked 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, V_{Kerr}, is detected by a lockin amplifier by phase modulating the microwave current at a frequency of 6.6 kHz. The V_{Kerr} signal is measured by sweeping the external magnetic field, and the magnetic field can be rotated inplane 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 (L_{sym} = ΔH^{2}/[4(H − H_{R})^{2} + ΔH^{2}]) and an antisymmetric Lorentzian (L_{asym} = −4ΔH(H − H_{R})/[4(H − H_{R})^{2} + ΔH^{2}]), V_{Kerr} = V_{sym}L_{sym} + V_{asym}L_{asym} + V_{offset}, where H_{R} is the resonance field, ΔH is the full width at half maximum, V_{offset} is the offset voltage, and V_{sym} (V_{asym}) is the magnitude of the symmetric (antisymmetric) component of V_{Kerr}. It is worth mentioning that, by analysing the position of H_{R}, 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 inplane magnetic field angle φ_{H} dependence of the resonance field H_{R} for t_{Fe} = 1.2 nm measured at f = 13 GHz is shown in Extended Data Fig. 2a. The sample shows typical inplane uniaxial anisotropy with twofold symmetry, that is, a magnetically HA for φ_{H} = −45° and 135° (\(\langle \bar{1}10\rangle \) orientations) and a magnetically EA for φ_{H} = 45° and 225° (⟨110⟩ orientations), which originates from the anisotropic bonding at the Fe/GaAs interface^{33}. To quantify the magnitude of the anisotropies, we further measure the f dependence of H_{R} both along the EA and the HA (Extended Data Fig. 2b). Both the angle and frequency dependence of H_{R} are fitted according to^{34,50}
$${\left(\frac{2{\rm{\pi }}f}{\gamma }\right)}^{2}={\mu }_{0}^{2}{H}_{1}^{\text{R}}{H}_{2}^{\text{R}},$$
(5)
with \({H}_{1}^{\text{R}}\) = H^{R} cos(φ − φ_{H}) + H_{K} + H_{B}(3 + cos 4φ)/4 − H_{U} sin^{2}(φ − 45°) and \({H}_{2}^{\text{R}}\) = H^{R} cos(φ − φ_{H}) + H_{B} cos 4φ − H_{U} sin 2φ. Here γ (= gμ_{B}/ħ) is the gyromagnetic ratio, g is the Landé gfactor, μ_{B} is the Bohr magneton, ħ is the reduced Planck constant, H_{K} (= M − H_{⊥}) is the effective demagnetization magnetic anisotropy field, including the perpendicular magnetic anisotropy field H_{⊥}, H_{B} is the biaxial magnetic anisotropy field along the ⟨100⟩ orientations, H_{U} is the inplane UMA field along ⟨110⟩ orientations and φ is the inplane 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 = φ_{H} holds when H is along ⟨110⟩ and \(\langle \bar{1}10\rangle \) orientations. From the fits of H_{R}, the magnitude of the magnetic anisotropy fields H_{A} (H_{A} = H_{K}, H_{B}, H_{U}) for each t_{Fe} is obtained, and their dependences on inverse Fe thickness \({t}_{\text{Fe}}^{{}1}\), together with the results obtained from the AlO_{x}/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 AlO_{x}/Fe/GaAs samples, and introducing the Pt/Al layer neither enhances the magnetization leading to an increase in H_{K} nor generates a perpendicular anisotropy leading to a decrease in H_{K}. By comparing the values of H_{K} and M, we confirm that the main contribution to H_{K} stems from the magnetization due to the demagnetization field. For both sample series, H_{K} and H_{B} decrease as t_{Fe} decreases because of the reduction of the magnetization as t_{Fe} 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 H_{K} and H_{B}, H_{U} shows a linear dependence on \({t}_{\text{Fe}}^{{}1}\) with a zero intercept, indicative of the interfacial origin of H_{U}.
Effective mixing conductance in Pt/Al/Fe/GaAs multilayers
Extended Data Fig. 3a,b shows the φ_{H} dependence and f dependence, respectively, of linewidth ΔH for t_{Fe} = 1.2 nm. The magnitude of ΔH varies strongly with φ_{H} because of the presence of inplane 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 by^{51}
$$\Delta H=\Delta [\text{Im}(\chi )]+\Delta {H}_{0}=\Delta \left[\frac{\alpha \sqrt{{H}_{1}^{\text{R}}{H}_{2}^{\text{R}}}({H}_{1}{H}_{1}+{H}_{1}^{\text{R}}{H}_{2}^{\text{R}})M}{{({H}_{1}{H}_{2}{H}_{1}^{\text{R}}{H}_{2}^{\text{R}})}^{2}+{\alpha }^{2}{H}_{1}^{\text{R}}{H}_{2}^{\text{R}}{({H}_{1}+{H}_{2})}^{2}}\right]+\Delta {H}_{0},$$
(7)
where Δ[Im(χ)] is the linewidth of the imaginary part of the dynamic magnetic susceptibility Im(χ), H_{1} and H_{2} are defined in equation (5) for arbitrary H values, and ΔH_{0} is the residual linewidth (zerofrequency 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 twomagnon scattering) contributing to ΔH. It is worth mentioning that the angular trace gives a slightly higher α value because ΔH_{0}, which also depends on φ_{H}, 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 AlO_{x}/Fe/GaAs samples. For both sample series, the Gilbert damping increases as t_{Fe} 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 by^{52,53}
$$\alpha ={\alpha }_{0}\,+\,{g}_{{\rm{eff}}}^{\uparrow \downarrow }\frac{\gamma \hbar }{4{\rm{\pi }}M}{t}_{{\rm{Fe}}}^{1},$$
(8)
where α_{0} 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 μ_{0}M = 2.2 T and γ = 1.80 × 10^{11} rad s^{−1} T^{−1}, the magnitude of \({g}_{{\rm{eff}}}^{\uparrow \downarrow }\) for Pt/Al/Fe/GaAs is determined to be 4.6 × 10^{18} m^{−2}, and \({g}_{{\rm{eff}}}^{\uparrow \downarrow }\) at the Fe/GaAs interface is determined to be 1.9 × 10^{18} 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 × 10^{18} m^{−2}. The spin transparency T_{int} 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 2e^{2}/h is the conductance quantum, G_{Pt} [= 1/(ρ_{xx}λ_{s})] is the spin conductance of Pt, ρ_{xx} is the resistivity and λ_{s} is the spin diffusion length. By using λ_{s} = 4 nm and an averaged ρ_{xx} = 40 μΩ cm, T_{int} = 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 interfaces^{54}, but very close to the value obtained by the firstprinciples calculations^{55}. The previously overestimated \({g}_{{\rm{eff}}}^{\uparrow \downarrow }\) and thus T_{int} at heavy metal/ultrathin ferromagnet interfaces is probably because the enhancement of α by twomagnon scattering^{56} as well as by the magnetic proximity effect (see Supplementary Note 3) is not properly excluded. Moreover, the obtained α_{0} values for Pt/Al/Fe/GaAs (α_{0} = 0.0039) and AlO_{x}/Fe/GaAs (α_{0} = 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 dampinglike spintorque term is considered^{18,35},
$$\frac{{\rm{d}}{\bf{M}}}{{\rm{d}}t}=\gamma {\bf{M}}\times {\mu }_{0}{{\bf{H}}}_{{\rm{e}}{\rm{f}}{\rm{f}}}+\frac{\alpha }{M}{\bf{M}}\times \frac{{\rm{d}}{\bf{M}}}{{\rm{d}}t}\frac{\gamma {\mu }_{0}{h}_{{\rm{D}}{\rm{L}}}}{M}{\bf{M}}\times {\bf{M}}\times {\boldsymbol{\sigma }}.$$
(10)
The terms on the right side of equation (10) correspond to the precession torque, the damping torque and the dampinglike spin torque induced by the spin current. Here σ is the spin polarization unit vector, and h_{DL} is the effective antidampinglike magnetic field. The effective magnetic field H_{eff}, 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 singlecrystalline Fe films grown on GaAs(001) substrates with inplane magnetic anisotropies, F is given by^{34,58}
$$\,F=\frac{{\mu }_{0}M}{2}\left\{2H[\cos \theta \cos {\theta }_{H}+\sin \theta \sin {\theta }_{\text{H}}\cos (\varphi {\varphi }_{H})]+{H}_{\text{K}}{\cos }^{2}\theta \frac{{H}_{\text{B}}}{2}{\sin }^{4}\theta \frac{3+\cos 4\varphi }{4}{H}_{\text{U}}{\sin }^{2}\theta {\sin }^{2}\left(\varphi \frac{{\rm{\pi }}}{4}\right)\right\}.$$
(12)
Bringing equations (11) and (12) into equation (10), the timeresolved 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 timedependent m_{z} by using μ_{0}H = 101 mT, μ_{0}H_{K} = 1,350 mT, μ_{0}H_{U} = 128 mT, μ_{0}H_{B} = 10 mT, α = 0.0063 and μ_{0}H_{DL} = 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 dH_{R}/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 h_{DL} = 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 h_{DL} 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 dampinglike torque efficiency and j_{Pt} is the current density in Pt. For the Pt/Al/Fe multilayer, j_{Pt} 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 ρ_{Pt} (= 40 μΩ cm), ρ_{Al} (= 10 μΩ cm) and ρ_{Fe} (= 50 μΩ cm) are the resistivities of the Pt, Al and Fe layers, respectively; t_{Pt}, t_{Al} and t_{Fe} 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 sinφ_{I–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 inplane angular dependence of F in Extended Data Fig. 5 for θ = θ_{H} = 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 precession^{59}. 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 current^{60}. If there are no inplane 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 t_{Fe} = 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 quasilinearly 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 by^{24}
$$\frac{\text{d}({\mu }_{0}\Delta H)}{\text{d}(I)}=2\frac{2{\rm{\pi }}f}{\gamma }\frac{\sin {\varphi }_{IH}}{{H}_{\text{R}}+{H}_{\text{K}}/2}\frac{\hbar }{2e}\frac{\xi }{{Mt}_{\text{Fe}}}\frac{1}{{t}_{\text{Pt}}w},$$
(20)
where φ_{I–H} = 45°, 135°, 225° and 315° as shown by the inset of each panel in Extended Data Fig. 6. The dampinglike torque efficiency can be further quantified by the slope s of fdependence 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 dampinglike 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 H_{R} along the EA (φ_{H} = φ = 45° and 225°) and the HA (φ_{H} = φ = 135° and 315°) are given by equation (3). From the angle and frequency dependencies of H_{R} as shown in Extended Data Fig. 2, μ_{0}H_{K} = 1,350 mT, μ_{0}H_{U} = 128 mT, μ_{0}H_{B} = 10 mT and g = 2.05 are determined for t_{Fe} = 1.2 nm. Extended Data Fig. 8a shows the H_{R} dependence of f for μ_{0}H_{K} = 1,350 mT (blue solid line) and μ_{0}H_{K} + Δμ_{0}H_{K} = 1,400 mT (red solid line) along the HA calculated by equation (3). To exaggerate the difference, μ_{0}ΔH_{K} of 50 mT is assumed. The shift of the resonance field ΔH_{R} is obtained as ΔH_{R} = H_{R}(H_{K}) − H_{R}(H_{K} + ΔH_{K}), and the frequency dependence of ΔH_{R} 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, ΔH_{R} = k_{K}f. Note that, to simplify the analysis, the zerofrequency intercept is ignored because the magnitude is much smaller than the intercept induced by ΔH_{U} and ΔH_{B}. The sign of the slope k_{K} is the same as that of ΔH_{K} and its magnitude is proportional to ΔH_{K}, that is, k_{K} ∝ ΔH_{K}. For the EA as shown in Extended Data Fig. 8c,d, the ΔH_{R}–f relationship induced by ΔH_{K} remains the same as for the HA, that is, ΔH_{R} = k_{K}f still holds.
Extended Data Fig. 8e shows the H_{R}dependence of f for μ_{0}H_{U} = 128 mT (blue solid line) and μ_{0}H_{U} + μ_{0}ΔH_{U} = 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, ΔH_{R} = −ΔH_{U}. However, for the EA, as shown in Extended Data Fig. 8g,h, the fdependent ΔH_{R} can be expressed as ΔH_{R} = ΔH_{U} − k_{U}f, which has an opposite slope compared with the ΔH_{R}–f relationships induced by ∆H_{K} (Extended Data Fig. 8d), that is, k_{U} ∝ −ΔH_{U}.
If the modulation is induced by a change in the biaxial anisotropy as shown in Extended Data Fig. 8i–l, ΔH_{R} along both the HA and EA shows a linear dependence on f, which is expressed as ΔH_{R} = −ΔH_{B} + k_{B}f, and k_{B} ∝ ΔH_{B} holds.
Extended Data Table 1 summarizes the ΔH_{R}–f relationships both along the EA and HA induced by ΔH_{K}, ΔH_{U} and ΔH_{B}.
As h_{Oe/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 ΔH_{R} induced by ΔH_{K}, ΔH_{U} and ΔH_{B} along the EA and HA is, respectively, given by equation (4).
Based on equations (4) and (5), the values of ΔH_{K,} ΔH_{U}, ΔH_{B} and h_{Oe/FL} for t_{Fe} ≤ 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 ΔH_{B} is determined to be 0.26 mT, which corresponds to k_{B} 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 k_{K} + k_{B} = 0.025 mT GHz^{−1}. Thus, the magnitude of k_{K} is determined by k_{K} = 0.025 mT GHz^{−1} − k_{B} = 0.021 mT GHz^{−1}, which corresponds to ΔH_{K} = 2.0 mT according to equation (3).

3.
As \(\Delta {H}_{\text{R}}^{\text{EA}}\) is frequency independent, this requires that k_{U} = k_{K} + k_{B} = 0.025 mT GHz^{−1}, which corresponds ΔH_{U} = 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 ΔH_{B} = −0.26 mT, ΔH_{K} = −2.0 mT and ΔH_{U} = −2.5 mT, which are of opposite sign as the results obtained from Extended Data Fig. 9a.

5.
Finally, bringing the magnitude of ΔH_{B} and ΔH_{U} 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 h_{Oe/FL} indicates that it is along the \([0\bar{1}0]\) orientation.
Similarly, the corresponding ΔH_{B}, ΔH_{K} and ΔH_{U} values can be determined for t_{Fe} = 2.2 nm (Extended Data Fig. 10). Extended Data Table 2 summarizes the magnitudes of the magnetic anisotropy modifications as well as the h_{Oe/FL} values for all the devices. The enhancement of the fieldlike torque in thinner samples has been observed in other systems and is probably because of the enhanced Bychkov–Rashba spin–orbit interaction^{61,62} and/or the orbital angular momentum (orbital Hall effect and orbital Rashba effect) at the ferromagnetic metal/heavy metal interface^{62}.
It is worth mentioning that, once the magnetization direction is fixed, ΔH_{B}, ΔH_{K} and ΔH_{U} 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 ΔH_{B}, H_{K} and ΔH_{U} 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 flinear dH_{R}/dI curves as observed in the experiment. Although the data analysis discussed in the previous section is selfconsistent, there could be alternative interpretations of the data. One possibility could be the currentinduced modification of the Landé gfactor of Fe. In magnetic materials, it is known that g is related to the orbital moment μ_{L} and the spin moment μ_{S}:
$$g=\frac{2{\mu }_{\text{L}}}{{\mu }_{\text{S}}}+2.$$
(23)
A flow of spin and orbital angular momentum induced by charge current could, respectively, modify the orbital and spin moment of Fe by Δμ_{S} and Δμ_{L}, 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 gfactor 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 dband. 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 interface^{63} 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 n_{s} 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 × 10^{48} J^{−1} m^{−3}, λ = 4 nm, ξ = 0.06, σ = 2.0 × 10^{6} Ω^{−1} m^{−1}, n_{s} = 4.2 × 10^{12} μ_{B} 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 μ_{B} for t_{Fe} = 1.2 nm at room temperature^{64}, the areal density of the magnetic moment of Fe n_{Fe} is determined to be 2.6 × 10^{14} μ_{B} cm^{−2}. In this case, the filling of the dband by spin current leads to a change in the magnetic moment of the order of n_{s}/n_{Fe} ≈ 0.16%, which agrees with the ratio between ΔH_{K} and H_{K}, that is, ΔH_{K}/H_{K} ≈ 2.0 mT/ 1 T ≈ 0.2%.