Robust spin-orbit torque and spin-galvanic effect at the Fe/GaAs (001) interface at room temperature

Interfacial spin-orbit torques (SOTs) enable the manipulation of the magnetization through in-plane charge currents, which has drawn increasing attention for spintronic applications. The search for material systems providing efficient SOTs, has been focused on polycrystalline ferromagnetic metal/non-magnetic metal bilayers. In these systems, currents flowing in the non-magnetic layer generate—due to strong spin–orbit interaction—spin currents via the spin Hall effect and induce a torque at the interface to the ferromagnet. Here we report the observation of robust SOT occuring at a single crystalline Fe/GaAs (001) interface at room temperature. We find that the magnitude of the interfacial SOT, caused by the reduced symmetry at the interface, is comparably strong as in ferromagnetic metal/non-magnetic metal systems. The large spin-orbit fields at the interface also enable spin-to-charge current conversion at the interface, known as spin-galvanic effect. The results suggest that single crystalline Fe/GaAs interfaces may enable efficient electrical magnetization manipulation.


Oersted field
To demonstrate that the observed dc voltage is indeed induced by the effective spin-orbit fields (eSOFs) at the interface of single crystalline Fe/GaAs, we measure a polycrystalline 5.6-nm thick Fe film deposited on an amorphous SiOx substrate. The ferromagnetic resonance (FMR) absorption of the bare Fe/SiOx film, measured at 12 GHz, is presented in Supplementary Fig. 1a. One can see that a well-defined FMR line appears at a resonance field of HR ~ 110 mT. However, no characteristic dc voltage V is observed for the spin-orbit FMR (SO-FMR) device with the same device dimensions and the same microwave excitation as the single crystalline Fe/GaAs devices ( Supplementary Fig. 1b). This indicates that the observed dc voltages cannot arise from the current induced Oersted field, but stem from the eSOFs at the single crystalline Fe/GaAs interface.

Supplementary Note 2: dc voltages along different crystallographic orientations
As shown in Supplementary Fig. 2, we use two coordinate systems to derive the angular dependence of the dc voltages. In the measurement coordinate system (x', y', z'), the microwave current density j flows along the [100]-orientation and the dc voltage is also detected along this direction. In the coordinate system labelled (x, y, z), the magnetization M and the microwave current j can be described as M = (Mx, mye it , mze it ), and j = je i  t (cosM, sinM, 0), where Mx is the static magnetization along the x-direction, my (mz) the dynamic magnetization along the y (z)direction,  the angular frequency of the magnetization precession, the angular frequency of the driving current, j the magnitude of the microwave current density, and M the magnetization angle.
The phenomenological relationship between the electric field E and j can be expressed as 1 where  is the resistivity,  magnitude of the anisotropic magneto-resistance AMR, n (n = M/M) the unit vector of the dynamic magnetization, and H the anomalous Hall resistivity. The first term on the right-hand side of Eq. 1 corresponds to Ohm's law, the second term to the AMR effect, and the third term to the anomalous Hall effect. Note that the crystalline AMR effect 2 , which only contributes significantly when the Fe thickness is reduced to a few monolayers, is not taken into account here because of the thick Fe film used in this study (~35 monolayers  (2) where the overline denotes the time average, Re(my) the real part of the dynamic magnetization my,

Supplementary Note 3: Ferromagnetic resonance results and dynamic magnetic susceptibility
To obtain the magnitude and lineshape of Re(my), we solve the Landau-Lifshitz-Gilbert (LLG) equation analytically, which has been widely used to describe magnetization dynamics 3 : where 0 is the permeability in vacuum,  the gyromagnetic ratio,  the damping constant, Heff the effective magnetic field, which is the sum of the external magnetic field H, anisotropy fields, their dynamic components, as well as current induced effective spin-orbit fields. The first term on the  If we solve Eq. 4, by assuming  << 1, Mx = |M| and ignore second order terms 3 , one can obtain the complex dynamic magnetic susceptibility , which is related to the dynamic magnetization M and the eSOFs hso where H1 R and H2 R are H1 and H2 at resonance HR. Note the slight difference between diagonal components  I and O . Since the FMR absorption is determined by Im(), the magnetic-field angle H dependence of 0H can be well reproduced by calculating the linewidth of Im() by adopting  as a fitting parameter 3 ( Supplementary Fig. 3c). In the calculation, the H dependence of 0HR and M is used. The obtained value of  ~ 0.0036, is smaller than for most metallic ferromagnets. This low damping is an indication of the high quality of our samples. The reasonable fit as well as the small linewidth (~2.5 mT for the 100 orientations) make us believe that the main contribution to the linewidth is intrinsic, and the extrinsic contributions 4-6 , such as inhomogeneous broadening and/or two-magnon scattering, are small for the present sample.
It is known that Re( I ) has an anti-symmetric lineshape while Im(a O ) has a symmetric lineshape.

Supplementary Note 4: Magnetization angle dependence of the dc voltages
Supplementary Figure 5 shows the magnetization angle M dependence of the dc voltages for stripes prepared along different orientations. Since the magnitude of Re( I ) and Im(a O ) is angle dependent (Supplementary Fig. 4), Va-sym is normalized by Re( I ) and Vsym is normalized by Im(a O ).
The results can be well fitted by combining Eqs. 2, 3 and 11, and the resulting eSOFs are listed in Supplementary This large difference in amplitude also excludes the possibility that the Oersted field is the main contributor to the driving field.

Supplementary Note 5: Calibration of the magnitude of the microwave current
We use the resonance field shift induced by Joule heating to determine the magnitude of the microwave current. First, we measure the voltage spectrum at low microwave power by introducing a dc current Idc through the Bias Tee. The application of Idc results in a large background signal of ~ 1 V. To increase the sensitivity, we modulate the magnetic field by ~ 1 mT at a frequency of 86 Hz, and measure dV/dH using a lock-in amplifier. Supplementary Figure 6a shows the dV/dH curves as a function of Idc measured at Pin = 10 dBm. It can be seen that HR shifts to higher field values as Idc Figure 6b). The dependence of HR on Pin and Idc is summarized in Supplementary Figure 6c. The quadratic dependence indicates that the origin of the shift of HR is due to Joule heating. By comparing these two quantities, we obtain the microwave current Idc at each Pin. It should be noted that for this method the total current still contains a small ac current Iac due to the applied low microwave power (10 dBm). We determine Iac (10 dBm) in the following way: First, we measure the angular dependence of the dc voltage for each Pin.  . 3 and 11). From the fitting, C1 as a function of Pin is obtained.

increases. A similar behaviour is observed when increasing the microwave power (Supplementary
Second, due to the Pin ~ Idc calibration shown in Supplementary Figure 6c, the C1 ~ Pin relationship can be transferred to a C1 ~ Idc relationship (Supplementary Fig. 7). Third, the total current Itot is sum To do so, we set the external magnetic field along the stripe direction, and measure the dV/dH spectrum by tuning the magnitude and direction of the dc current Idc. Two points should be noted for this configuration: i) the dc voltage is still sizeable because of the deviation between magnetization and magnetic-field angle (H = 90 o but M = 76 o as shown in Supplementary Fig. 3b). ii) Only the Dresselhaus field can be detected since the external field points along the stripe. Supplementary Figure 9 shows the dc current Idc dependence of resonance field HR. We observe that the magnitude of HR(Idc) is slightly larger than that of HR(Idc). This asymmetric behaviour is expected when the Dresselhaus field is parallel to H for Idc, and anti-parallel to H for Idc. The inset shows the Idc dependence of the difference of the measured resonance fields HR(Idc)  HR(Idc). As can be seen, the relation is linear for currents larger than 3 mA. The magnitude of the Dresselhaus field, [HR(Idc)  HR(Idc)]/2, is determined to be 0.16 mT for Idc = 5 mA (corresponding to j = 1.6×10 11 Am -2 ), which is in good agreement with the value determined by SO-FMR (Supplementary Table 1

Supplementary Note 8: Theoretical estimation of the out-of-plane effective spinorbit fields
The out-of-plane component of the spin accumulation s [001] can arise from virtual transitions between the exchange-split bands, induced by the electric field E established in the sample by the current flow. The virtual transitions are due to the spin-orbit velocity coming from the spin-orbit fields [8][9][10]  presented. One can see that (/)out-of-plane and (/)in-plane are of the same sign and that the value of both is larger than 1 for all the devices, which indicates that the Rashba SOI dominates at the Fe/GaAs interface. The value of (/)out-of-plane is larger than that of (/)in-plane. This discrepancy is due to the different origins between in-plane and out-of-plane induced spin polarization. The inplane spin polarization is created only at the Fermi level, while the out-of-plane spin polarization is due to the electrical polarization (intrinsic effect) of the whole bands 11 .

Supplementary Note 9: Magnitude of spin-orbit torque in ferromagnetic metal/non-magnetic metal bi-layers
The spin-orbit torque (SOT) is defined by 0hSO×where  is the magnetic moment which increases with film thickness tFM while |hSO| decreases 14 . To compare with previous experiments we assume the magnetization of all the different ferromagnets to be the same and consider only the thickness of the ferromagnetic films.Furthermore, we normalize the spin-orbit field by the current density j. The resulting values of this rough estimate are listed in Supplementary Table 3.

Supplementary Note 10: Determination of the magnitude of the dc voltage induced by spin pumping
In ferromagnetic/non-magnetic bilayers, a long-standing issue has been the accurate determination of the magnitude of the voltage VSP induced by spin pumping. The difficulty lies in excluding parasitic effects, e.g., spin rectification effects (Refs. 4 and 19, and references therein), thermo-magnetic effects in the ferromagnetic layers 20 and the effect of spin memory loss at the ferromagnetic/nonmagnetic interface 21 . For the device we use here with the stripe integrated between the signal line and ground line (Fig. 5a in the main text), it has already been well established by using Py/Pt that VSP and rectification voltages show different angular dependencies.
Especially, VSP has a maximum value when the external in-plane magnetic field H is perpendicular to the stripe while rectification voltage vanishes for this geometry 22,23 .
Since spin pumping generates a pure spin current with symmetric line shape with respect to the external magnetic field H, usually, only Vsym is taken into account in the analysis. It has been shown, however, that there are three possible origins for Vsym: Rectification effects in the ferromagnetic layer due to AMR 19 , thermo-magnetic effects in the ferromagnetic layer 20 , as well as VSP.
Supplementary Figures 10a and b show the magnetic field dependence of Vsym for positive and negative fields. For Vsym induced by AMR or thermo-magnetic effects, the expected symmetry is In this case, voltages Vsym originating from AMR and thermo-magnetic effects ares cancelled. We have verified this method by using a Py/Pt bilayer where the measured magnetic-field angle dependence of VSP can be well fitted by the inverse spin Hall effect (ISHE) (not shown here), indicating the validity of the analysing method.
Supplementary Figure 10d shows the magnetic-field angle dependence of Vsym for a [110]oriented Fe/GaAs spin pumping device. One can see that Vsym(H) > Vsym(H) for H = 135 o (perpendicular to the stripe, the perpendicular orientation of external field and stripe is confirmed by the H dependence of HR, which is shown in Supplementary Figure 10c), Vsym(H) = Vsym(H) for H = 225 o (parallel to the stripe) and Vsym(H) < Vsym(H) H = 315 o (perpendicular to the stripe again). This is the expected symmetry of VSP with superimposed signals stemming from AMR and/or thermo-electric effects. The magnitude of VSP is then obtained by Eq. 12, which can be well fitted by the theoretical model of spin pumping (Fig. 5c in the main text). Since the non-magnetic layer is missing and Rashba dominated SOI has been demonstrated, this effect is called spingalvanic effect, SGE. We have also measured devices with other orientations and from different wafers; all of them show consistent results.
To eliminate the possibility that the observation of SGE is an artefact, we measure a polycrystalline Fe film deposited on an amorphous SiOx substrate. As shown in Supplementary Figure 11, the voltage goes to zero and no difference between Vsym(+H) and Vsym(-H) when magnetic field is perpendicular to the stripe. This indicates no existence of SGE for Fe/SiOx. The angular dependence can be well fitted by AMR effect of Fe. We have also measured Py/SiOx, and no SGE is observed (Supplementary Figure 12).