Disentangling relativistic spin torques in a ferromagnet/semiconductor bilayer

Recently discovered relativistic spin torques induced by a lateral current at a ferromagnet/paramagnet interface are a candidate spintronic technology for a new generation of electrically-controlled magnetic memory devices. Phenomenologically, the torques have field-like and antidamping-like components with distinct symmetries. Microscopically, they are considered to have two possible origins. In one picture, a spin-current generated in the paramagnet via the relativistic spin Hall effect (SHE) is absorbed in the ferromagnet and induces the spin transfer torque (STT). In the other picture, a non-equilibrium spin-density is generated via the relativistic inverse spin galvanic effect (ISGE) and induces the spin-orbit torque (SOT) in the ferromagnet. From the early observations in paramagnetic semiconductors, SHE and ISGE are known as companion phenomena that can both allow for electrically aligning spins in the same structure. It is essential for our basic physical understanding of the spin torques at the ferromagnet/paramagnet interface to experimentally disentangle the SHE and ISGE contributions. To achieve this we prepared an epitaxial transition-metal-ferromagnet/semiconductor-paramagnet single-crystal structure and performed a room-temperature vector analysis of the relativistic spin torques by means of the all-electrical ferromagnetic resonance (FMR) technique. By design, the field-like torque is governed by the ISGE-based mechanism in our structure while the antidamping-like torque is due to the SHE-based mechanism

In the commonly studied polycrystalline transition-metal-ferromagnet/heavy-metalparamagnet samples, the dependence of the torques on the angle of the driving in-plane current also does not provide the direct means to disentangle the two microscopic origins.
The lowest order inversion-asymmetry spin-orbit terms in the Hamiltonian have the Rashba form for which the vectorζ is in the plane parallel to the interface and perpendicular to the current, independent of the current direction. The same applies to the spin-polarization of the SHE spin-current propagating from the paramagnet to the ferromagnet. TheM and ζ functional form of the field-like and antidamping-like SHE-STTs is the same as of the corresponding SOT components. In the observed lowest order torque terms in Pt/Co and Ta/CoFeB structures 28 the ISGE-based and the SHE-based mechanism remained, therefore, indistinguishable. The simultaneous observation of higher order torque terms in these samples pointed to SOTs due to structural inversion-asymmetry terms beyond the basic Rashba model. From the Ta thickness dependence measurements in the Ta/CoFeB structure it was concluded that in these samples both the ISGE-based and the SHE-based mechanisms contributed to both the field-like and the antidamping-like torques 29 .
The SHE and ISGE were originally discovered in III-V semiconductors [9][10][11]15,16  We have measured the relativistic spin torques using the electrically induced and detected FMR technique 1,18 (see Fig. 2a). In this method, a microwave current flowing in the device induces FMR when the externally applied magnetic field matches the resonant condition.
The resonance of the Fe magnetisation can be detected in the dc voltage induced across the bar, V dc . This is due to the homodyne mixing of the microwave current with the oscillating component of magnetoresistance caused by the magnetisation precession. In these measurements we increase the microwave current coupled into the sample, with a typical resistance of 8 kΩ, by using an impedance matching network 1 . The coefficients depend on the angle, θ, of the magnetisation vector relative to the current and are given by

Current
Here, V mix is the sensitivity of the mixing detection and is given by V mix = − 1 2 I 0 ∆R, where I 0 (e jωt , 0, 0) is the microwave current in the device and ∆R is the coefficient of the anisotropic magnetoresistance of the sample. A yy and A yz are the diagonal and off-diagonal components of the ac magnetic susceptibility, which depend on the magnetic anisotropies and Gilbert damping of the sample. In our devices, ∆R is typically 17 Ω which, assuming Fe carries the majority of the current in the bilayer, is consistent in sign and magnitude with literature values of 0.2% anisotropic magnetoresistance in Fe. 33 We estimate the proportion of total bilayer current in the Fe layer to be 79% by resistance measurements of Hall bars before and after removing the Fe and capping Al (see supplementary information).
FMR measurements were made using devices patterned in four crystal directions. The microwave power for all devices, incident on the impedance matching network, was 24 dBm.
For each angle, the resonances were fitted by symmetric and antisymmetric Lorentzian functions. A typical curve is shown in Fig. 2b. The in-plane uniaxial magnetic anisotropy of Fe implies that, in general, the magnetisation does not lie along the external field. The actual magnetisation angles and uniaxial anisotropy are self-consistently calculated from the dependence of the resonant field on the external magnetic field angle 34 . This also allows the susceptibilities, A yz and A yy , to be calculated. The magnetisation angle dependence of V sym /A yz and V asy /A yy is plotted in Fig. 2c for a bar patterned in the [010] crystal direction.
Our analysis of the current-induced torques using equation (1) is not necessarily valid if the torques do not act in phase with the microwave current. In comparison to electrically detected FMR measurements where the microwave current is capacitively or inductively coupled into the sample 35 , we do not expect a phase-shift between the microwave current and induced fields as the current is conducted ohmically. Nevertheless, we might worry that some part of our microwave resonator circuit leads to a phase shift. To test this, we repeated our measurements with a [100] device over a frequency range (11.8 to 14.4 GHz) using a microstrip resonator with a fundamental frequency close to 13 GHz (Fig. 3a). If there were some frequency dependent phase shift, we would expect the lineshape to oscillate between an antisymmetric and symmetric Lorentzian over this frequency range. However, the ratio of V sym to V asy remains constant in this frequency range to within experimental error ( Fig.   3b), confirming that our analysis is correct.
As shown in Fig. 4a, the in-plane current-induced field depends strongly on the crystal direction of the current and can be well fitted by the Dresselhaus-symmetry ISGE field, h ISGE ∼ cos 2φ [100] , − sin 2φ [100] , 0 , where φ [100] is the angle between the current and the [100] crystal direction. This is the expected symmetry of the current-induced non-equilibrium spin-polarisation of carriers in the semiconductor due to the inversionasymmetric crystal structure of the strained zinc-blende lattice of (Ga,Mn)As. The interface exchange coupling of these polarized carriers with the adjacent Fe moments induces the fieldlike SOT in Fe with the Dresselhaus symmetry. We note that other torque terms with the symmetry common to the Rashba ISGE, the field-like component of the SHE-STT, or the torque due to an Oersted field have only a minor contribution to the total measured field-like torque.
To highlight that carriers in the semiconductor layer are responsible for the Dresselhaussymmetry ISGE field, we compare Fig. 4a with previous measurements in which the in-plane current induced fields were measured in a bare (Ga,Mn)As epilayer 25 without the Fe film.
To observe the corresponding SOT in this sample, a larger concentration of magnetic Mnmoments was used, and the measurements were performed at low temperatures where the Mn moments are ferromagnetic in equilibrium. Instead of the interfacial exchange coupling to Fe, the current induced non-equilibrium spin-polarisation of carriers in the semiconduc-tor due to the Dresselhaus-symmetry ISGE is exchange-coupled directly to the ferromagnetic moments on which it exerts the field-like SOT. In both the Fe/(Ga,Mn)As and the (Ga,Mn)As samples the same crystal-symmetry field-like SOT is observed which confirms their common Dresselhaus ISGE origin.
In contrast to the in-plane field, the out-of-plane current induced field is independent of the crystal direction of the current but depends on the magnetisation angle. It is dominated by a term h SHE ∼M ×ŷ (ŷ is the direction perpendicular to the current) which generates the antidamping-like torque. As shown in Fig. 4b, the amplitudes of the field-like and antidamping-like torques are comparable in our Fe/(Ga,Mn)As structure. The underlying microscopic mechanism of the antidamping-component can only be of the SHE-STT origin.
In previous measurements in the bare (Ga,Mn)As epilayer, 25 the antidamping-like SOT was dominated by the counterpart microscopic mechanism to the Dresselhaus ISGE. This Dresselhaus-symmetry antidamping-like SOT is clearly missing in our measured data. It is suppressed in our Fe/(Ga,Mn)As structure by design because carriers in the semiconductor are not sufficiently magnetized at equilibrium due to the low Mn moment density and high temperature of the experiment.
A Rashba-symmetry antidamping-like SOT due to the carriers in the Fe experiencing the inversion-asymmetry of the interface could in principle also explain our measured data.
This antidamping SOT would have the same symmetry as the antidamping SHE-STT. This possibility is, however, ruled out by our control experiment in which we perform electrically detected FMR in a similar MBE-grown Fe (1 nm)/insulating GaAs structure at room temperature. In this case, we do not observe the anti-damping torque in the rectification effect, despite the sample possessing a similar magnetoresistance ratio (∼0.2%) to our Fe/(Ga,Mn)As. This is consistent with the carriers being removed from the semiconductor which eliminates the SHE source of the spin-current. We note that also consistently with the absence of carriers in the semiconductor in the Fe/insulating-GaAs structure, we do not observe the Dresselhaus-symmetry field-like SOT in this control sample.
To calibrate the microwave current in the sample we used a bolometric technique (see supplementary information). Using this calibration, we estimate amplitudes of |µ 0 h ISGE /J GaAs | = 16 ± 10 µT/10 6 Acm −2 and |µ 0 h SHE /J GaAs | = 20 ± 9 µT/10 6 Acm −2 . The error is found from the statistical variation from all of the devices measured. To verify the bolometric calibration, we also perform an additional check with a single device by measuring the change in Q-factor of the microstrip resonator loaded with and without a sample (see supplementary information). This calibration yields values of |µ 0 h ISGE /J GaAs | = 37 µT/10 6 Acm −2 and |µ 0 h SHE /J GaAs | = 47 µT/10 6 Acm −2 , close to the values of the bolometric technique.
From the measured h SHE in our Fe/(Ga,Mn)As structure we can infer the room-temperature spin Hall angle, θ SH , in the paramagnetic (Ga,Mn)As using the expression based on the antidamping-like STT 2 , Here it is assumed that the thickness of the semiconductor is much larger than its spin To conclude, we have experimentally disentangled the two archetype microscopic mechanisms that can drive relativistic current-induced torques in ferromagnet/paramagnet structures. In our epitaxial Fe/(Ga,Mn)As bilayer we simultaneously observed ISGE-based and SHE-based torques of comparable amplitudes. Designed magnetization-angle and currentangle symmetries of our single-crystal structure allowed us to split the two microscopic origins between the field-like and the antidamping-like torque components. Experimentally establishing the microscopic physics of the relativistic spin torques should stimulate both the fundamental and applied research of these intriguing and practical spintronic phenomena.

Methods
The semiconductor (Ga,Mn)As layer of thickness 20 nm was deposited on a GaAs (001) substrate at a temperature of 260 • C. The substrate temperature was then reduced to 0 • C, before depositing a 2 nm Fe layer, plus a 2 nm Al capping layer. In-situ reflection high energy electron diffraction and ex-situ x-ray reflectivity and diffraction measurements confirmed that the layers are single-crystalline with sub-nm interface roughness.
To improve the sensitivity of the FMR measurement, the sample is embedded in a mi-

S1. IMPEDANCE MATCHING NETWORK
We use a gap-coupled microstrip resonator, similar to one previously reported S1 , to impedance match the ∼ 8 kΩ sample to the 50 Ω transmission line. In contrast to the network previously reported, the sample is attached to the opposite end of the resonator with a wirebond. At the resonant frequency of the resonator, the impedance of the network becomes real and is given by (S1) C k is the capacitance of the interdigitated gap capacitor and R is the resistance of the sample. By choosing a suitable value of C k , at the resonant frequency Z ≈ 50Ω and most of the microwave power is transmitted from the transmission line to the matched network.
For the FMR measurements, we use the resonator at twice its fundamental frequency, ∼ 16 GHz, where the power is also matched to the sample. To measure V dc , a wirebond is attached at 1 4 of the length of the resonator. This causes little perturbation of the resonator mode, as at this point a node of electric field exists.

S2. Q FACTOR CALIBRATION OF MICROWAVE CURRENT
Close to the resonance, the microstrip resonator can be approximated as a series resonator.
The quality factor, Q, of the resonator can then be described by where Q sample represents Q due to power dissipated in the sample and Q board represents the Q due to power dissipated through losses in the PCB. When connected to an external network, the effect of dissipation externally can be described by a total Q factor where g is a coupling constant that describes the impedance matching of the resonator, given by g = Z 0 /Z and Z 0 = 50 Ω is the the characteristic impedance of the transmission line. Both Q and g can be found from measuring the reflected power from the resonator network around the resonant frequency. At resonance, the reflection coefficient is given by We can also determine the dissipation of the resonant circuit through the width of the absorption peak, with Q total given by where ∆ω FWHM is the full width at half maximum of the absorption peak.
To perform a calibration of the microwave current in a sample, we use a directional coupler and calibrated microwave diode (Fig. S1a) to measure the reflected signal with and without the resonator loaded by the sample. If we replace the diode with a microwave mixer ( Fig. S1b), we can measure the phase change of the reflected signal around the resonance of the microstrip resonator. We use the result that at resonance, the gradient of phase is given by ∂φ which allows us to determine the value of Q when the resonator is not loaded by the sample and the resonance is hard to observe in reflected power. We calibrate a microstrip resonator at its fundamental frequency, as this has previously been studied. To achieve a high enough frequency for our FMR measurements, we reduce the length of the resonator so that the fundamental frequency is around 13 GHz.
The measured magnitude and phase of the reflected signal is shown in Fig. S2a and b respectively. From the peak in absorption we estimate that for the loaded resonator, g = 4.3.
Although we cannot make out a clear peak from the noise for the unloaded case, we can put a lower limit on g > 20. From the gradient of the phase and the calculated values for g, we find that Q sample << Q board and so assume all of the power transmitted to the microstrip resonator is dissipated in the sample. On resonance, using the reflection measurements we then estimate, for a 10 dBm source power, that the microwave current in the device is 0.74 mA. We perform FMR measurements at 10 dBm in a [100] bar as discussed in the main text and, using the value of the calibrated microwave current, find values of |µ 0 h ISGE /J GaAs | = 37 µT/10 6 Acm −2 and |µ 0 h SHE /J GaAs | = 47 µT/10 6 Acm −2 .

S3. JOULE HEATING CALIBRATION OF MICROWAVE CURRENT
The microwave current can also be calibrated by a simple Joule heating method as the resistance of the semiconductor is sensitive to temperature. This calibration was performed for each device measured. To determine the microwave current in the device for a given source power, the change in resistance due to the heating of the microwave current is compared to the change due to heating of dc current.
The dc current is applied from the bias-T of the resonator, through the sample, to ground. An applied dc voltage is held for 10 seconds at increasing values before the current is recorded. The resistance is then calculated as a function of dc current. Microwave power is then applied at increasing increments and held for 10 seconds at each value as before, before the resistance is measured from a small dc bias applied concurrently. We then compare the gradients of resistance with microwave and dc power (Fig. S3). From the ratio of these

S5. DETERMINING THE RESISTIVITY OF EACH LAYER
To find the resistivity of the individual layers, the resistivity of a Fe (2 nm)/(Ga,Mn)As indicates that only 12% of the microwave current flows through the (Ga,Mn)As layer. The equivalent proportion for the 20 nm layers is 21%.