Field-like spin–orbit torque induced by bulk Rashba channels in GeTe/NiFe bilayers

Most studies of the Rashba effect have focused on interfacial Rashba spin–orbit coupling. Recently, bulk Rashba materials have attracted considerable interest owing to their potential to enhance the Rashba spin–orbit torque. By employing a bulk Rashba material, GeTe, as a spin–orbit channel in GeTe/NiFe bilayers, a large field-like spin–orbit torque up to 15.8 mT/(107 A cm−2) is measured. This value is one of the largest reported field-like torques and is attributed to the interfacial spin–orbit coupling being enhanced by the bulk Rashba effect in the GeTe channel. Furthermore, the large field-like torque is maintained even for a 20-nm-thick NiFe layer. This unconventional dependence on the thickness of both the GeTe and NiFe layers cannot be described by conventional theory, but it is believed to stem from the additional bulk Rashba effect-induced term. The large field-like torque over a wide range of ferromagnet thicknesses results in scalable in-plane spin–orbit torque devices. This result calls for a further theoretical study on spin transport in heterostructures, including bulk Rashba materials. We find a very large field-like spin-orbit torque in a bulk Rashba channel/ferromagnet structure and this large value of spin-orbit torque is attributed to the interfacial spin-orbit coupling enhanced by the well-grown bulk Rashba material.


Introduction
Rashba spin-orbit coupling (RSOC) is a crucial element for spin-based information devices. The control of spin precession concomitant with the modulation of RSOC by an electric field has been theoretically suggested and experimentally demonstrated using a two-dimensional quantum well [1][2][3][4][5][6] . In addition to gate-controlled spin precession, the RSOC also contributes to spin-orbit torques (SOTs) in a normal metal (NM)/ferromagnet (FM) bilayer, the basic structure for SOT-active devices, where inversion symmetry breaks at the NM/FM interface gives rise to RSOC [6][7][8][9][10] . As the SOT serves as a writing scheme for energy-efficient spintronic devices [11][12][13] , enhancing the RSOC effect in NM/ FM bilayers is an important task for practical applications.
Most RSOC studies have focused on interfacial effects in two-dimensional systems or bilayers. Considering spin transport, it is expected that the RSOC effect becomes stronger when the RSOC is present in the bulk of a system rather than being limited to just the interface. Recent studies have revealed that several polar crystals, such as BiTeCl 14 , BiTeI 15 , and GeTe 16 , have strong bulk RSOC, where the bulk inversion asymmetry is related to the ferroelectricity. These materials are classified as ferroelectric-Rashba semiconductors 17,18 . Among these materials, α-GeTe is remarkable in that it has a relatively high ferroelectric Curie temperature of 670 K 19 , and its ferroelectricity survives at a thickness of~5 nm 20 . In addition, a spin-and angle-resolved photoemission spectroscopy (SARPES) study reported that GeTe has a large Rashba constant of~4.2 × 10 −10 eV m 21 .
The SOT strength in NM/FM bilayers is related to the spin Hall conductivity of the spin-orbit channel and NM layer. The aforementioned attractive features of GeTe have led to recent density functional theory calculations of the spin Hall conductivity of GeTe and its controllability through the modulation of electric polarization 22,23 . Even considering various values of electric polarizations, however, these theoretical studies predicted that the spin Hall effect (or spin Hall angle) of bulk Rashba GeTe at the Fermi level is not superior to that of Pt, a widely used spin-orbit channel for SOT devices. This result is attributed to the fact that the SOC effect of GeTe is mainly observed in the valence bands (Te 5p-orbitals) below the Fermi level due to Ge vacancies.
Despite the above prediction, it is worthwhile to experimentally investigate the SOT in GeTe/FM bilayers because the above theories 24

Results
Polar crystal structure of α-GeTe with a strong bulk Rashba effect The X-ray diffraction (XRD) 2θ scan of a single-layer GeTe film [ Fig. 1a] shows that the film is oriented with its hexagonal (000h) axis normal to the Si(111) substrate. The full width at half maximum (FWHM) of the GeTe (0003) reflection is estimated to be~0.075°[Supplementary Fig. S1a], indicating the high crystalline quality of the film. In Fig. 1b, the XRD ϕ-scan of the GeTe(0-114) shows that it has threefold rotational symmetry mixed with a small amount of twinned grains, which is consistent with the rhombohedral crystal structure of α-GeTe. The XRD reciprocal lattice map of the GeTe film indicates that the film is fully relaxed from the substrate or free of stress [ Supplementary Fig. S1b]. Figure 1c shows a fast Fourier transform (FFT)-filtered atomic-resolution scanning transmission electron microscopy (STEM) image, which clearly indicates Te (brighter balls) and Ge (dimmer balls). Note that the Ge atoms are not positioned at the center between two adjacent Te atoms but rather shifted to the right. This implies an uncompensated polarization, which is consistent with the known ferroelectric nature of α-GeTe 21,25 . For quantitative analysis, a line profile is also displayed along the red-dashed line in Fig. 1c, which is composed of two kinds of peaks. From the peak positions, the distances of Te(left)-Ge(right) and Ge(left)-Te(right) are estimated to be 2.08 and 1.51 Å, respectively [ Fig. 1d]. These values are very close to the known values of 2.14 and 1.49 Å 26,27 . Figure 1e shows the selected-area electron diffraction (SAED) pattern of a 5 μm spot size, indicating that the GeTe film is highly crystalline with the R3m space group mixed with a small amount of twin grains, as indicated by the yellow arrow. The ferroelectric nature of GeTe films is also confirmed by a piezoresponse force microscopy (PFM) study ( Supplementary Fig. S2), where clear spontaneous polarization is uniformly observed. Figure 1f shows the magnetoresistance (MR) of a 50-nmthick GeTe film as a function of the external magnetic field (B) applied perpendicular to the film plane. The MR curves exhibit weak antilocalization (WAL)-like features (sharp dip near B = 0; for detailed analysis, see Supplementary Note 1), indicating a strong SOC in the film. Using the Fukuyama-Hoshino model for three-dimensional WAL 28 , we fit the MR curves [black dashed lines in Fig. 1f] and obtain a Rashba constant (α R ) of~0.3 × 10 −10 eV m at 20 K, which is approximately three times larger than that of a strong semiconducting Rashba system 2 .

Harmonic Hall measurements
An in-plane charge current flowing through an SOC channel induces a perpendicular spin current, which exerts an SOT on the nearby ferromagnet. In this experiment, we extract the magnitude of the SOT in a GeTe (130 nm)/NiFe (20 nm) structure via harmonic Hall measurements (see the Experimental Section). We investigate a field-like torque (T FL ∝ m × y) and a damping-like torque [T DL ∝ m × (y × m)], where m and y are unit vectors along the magnetization and perpendicular to both directions of charge-current flow (x) and spin-current flow (z), respectively. Regarding in-plane magnetization, the field-like and damping-like torques correspond to the in-plane (B FL ) and out-of-plane (B DL ) effective spin-orbit fields, respectively, as shown in Fig. 2a. We note that according to our coordinate system and geometry, a current-induced Oersted field is in the −y direction for a positive current [ Fig. 2a]. The sign of the current-induced Oersted field is also confirmed by the harmonic Hall measurement of a Cu/NiFe bilayer where the Cu layer creates a negligible SOT ( Supplementary Fig. S3). Figure 2b shows the optical microscopy image and measurement geometry of the Hall bar structure. An external in-plane magnetic field B ext (= 25 mT − 4 T) is applied while varying the azimuthal angle (φ) between B ext and the AC current I [ Fig. 2a]. Because of the easy-plane anisotropy of the NiFe layer, the magnetization is aligned along B ext . In this field geometry for in-plane magnetization, the first ðR ω xy Þ and second ðR 2ω xy Þ harmonic Hall resistances are expressed as 24,29 where R PHE is the planar Hall resistance; R AHE is the anomalous Hall resistance; B eff is the sum of B ext and the effective out-of-plane demagnetizing field B d ; A is the coefficient of the thermomagnetic effects, including the anomalous Nernst effect and spin-Seebeck effect; I 0 is the amplitude of the AC current; α is the geometric factor; ∇T is the vertical temperature gradient; N is the ordinary Nernst coefficient, and B Oe is the current-induced Oersted field. The vertical temperature gradient may originate from current-induced Joule heating and the asymmetric heat flow between layers. We note that in comparison to ref. 24 , the ordinary Nernst term, which is linear in B ext , is included in Eq. (2) because this term is found to be large in our GeTe/NiFe structure, which we show below. A recent experiment 29 reported that the ordinary Nernst term was also large in a Bi-Sb/Co heterostructure. Figure 2c, d shows the measured firstand secondharmonic Hall resistances, respectively. The first harmonic Hall resistance R ω xy follows the angular dependence of sin 2φ as expected from the planar Hall effect [Eq. (1)]. On the other hand, the second-harmonic Hall resistance R 2ω xy shows the angular dependences of cosφ and cos2φcosφ, which is consistent with Eq. (2) and will be analyzed in detail in the next section.
Field-like torque of the GeTe/NiFe structure The field-like torque is described by the in-plane effective spin-orbit field (B FL ) inherently separated from thermal contributions, as shown in the second-harmonic Hall resistance (R 2ω xy ) of Eq. (2). Thus, this term can be directly extracted from the first term of Eq. (2). Figure 3a shows the amplitude of the cos2φcosφ term, C FL , in the GeTe (130 nm)/NiFe (20 nm) bilayer. The C FL is inversely proportional to the external field B ext , which is consistent with Eq. (2). From the first harmonic results of Fig. 2c and Eq. (1), we find that R PHE is 14.95 mΩ. From the channel dimension of the Hall bar structure and resistivities (ρ GeTe = 176.7 μΩ cm, ρ NiFe = 41.7 μΩ cm) along with Ampere's law 30 , we estimate that the current-induced Oersted field (B Oe /J) is −8.1 mT/(10 7 A cm −2 ), where J is the current density flowing through the GeTe layer.
Combining these values with C FL ¼ 2R PHE ðB FL þ B Oe Þ=B ext , we find a field-like torque (B FL /J) of +12.0 mT/ (10 7 A cm −2 ), which is larger in magnitude than B Oe /J, and, more importantly, its sign is opposite to B Oe /J. This sign difference unambiguously demonstrates that a spin-orbit-originated field-like torque is present in GeTe/ NiFe bilayers. Previous studies reported B FL = 2 mT/(10 7 A cm −2 ) and 4 mT/(10 7 A cm −2 ) for Pt/CoFe 31 and Pt/ Co 32 channels, respectively. In a Ta/CoFeB channel 33 , the observed spin torques were B FL = −2.1 mT/(10 7 Acm −2 ). Thus, the B FL in our system is as large of a value as that obtained in the heavy metal system. The field-like torque is found to be linear with the AC current [ Fig. 3b; see  Supplementary Fig. S4 for the detailed harmonic curves], validating the estimation of B FL /J. We also confirm that the cos2φcosφ term is absent in a GeTe/Cu structure ( Supplementary Fig. S5), showing that B FL is not an artifact originating from GeTe itself.
Assuming that the bulk SOC of GeTe is the only source of spin current, one can analyze the magnitude of B FL /J based on the spin drift-diffusion model with the quantum mechanical boundary condition 8 . This model predicts that B FL /J increases and then saturates when the NM thickness exceeds its spin diffusion length. It also predicts that B FL /J is inversely proportional to the FM thickness, as reported in a recent experiment 34 . To check whether this diffusion model is applicable to our study, we performed harmonic Hall measurements on GeTe/NiFe bilayers with varying GeTe thicknesses [ Fig. 3c] and NiFe thicknesses [ Fig. 3(d)]. Regarding the GeTe thickness dependence [ Fig. 3c], B FL /J shows a slightly nonlinear dependence on the GeTe thickness but does not exhibit a clear saturation behavior. One possible explanation is that the spin diffusion length of GeTe is comparable to or larger than the largest tested GeTe thickness (170 nm). However, this explanation is unlikely because the spin diffusion length of most spin-orbit channels is on the nanometer scale 35 . Moreover, for the NiFe-thickness dependence [ Fig. 3d], we find that B FL /J is almost constant even with a threefold change in the NiFe thickness. This behavior is evidently inconsistent with the prediction of the spin drift-diffusion model. Haney et al. 8 reported Boltzmann solutions for the SOT but did not consider the bulk Rashba SOC in the normal metal layer. Another missing part is that it does not consider the SOC in ferromagnets. A recent theory 36 based on quantum Boltzmann equations reported that the SOC in a ferromagnet also generates an SOT. In this respect, we need a theory based on the Boltzmann equations, which takes into account the SOC in ferromagnets and the bulk Rashba SOC in the nonmagnetic layer, which demands further theoretical progress. Recently, nonreciprocal charge transport of the GeTe channel was observed 37 , and this term is possibly added to our observed SOT signal. Moreover, this unique property of the field-like torque, which maintains a large magnitude even for a thick FM layer, is applicable for scalable in-plane SOT memories [Fig. 3f, see the "Discussion" section].
We check several possibilities for the unconventional thickness dependence of field-like torque. We first check whether a thick magnetic dead layer is formed at the GeTe/NiFe interface. We find that the magnetic dead layer is only 0.36 nm thick [ Fig. 3e], so we exclude this possibility for the unconventional dependence on NiFe thickness of field-like torque. Recent studies have reported that oxidation significantly affects the SOT 38-40 , we also checked whether an oxide layer is formed at the GeTe/NiFe interface with energy dispersive spectroscopy (EDS) but found no noticeable oxidation signature (Supplementary Fig. S6).
However, another possibility is the interface SOC at the GeTe/NiFe interface. The spin drift-diffusion model is obtained by integrating the Boltzmann equation; thus, the Boltzmann equation is better suited to describe in-plane transport 8 . The Boltzmann transport calculation 8 for the interfacial SOC shows that the scaled field-like torque [= (field-like torque) × (FM thickness)] increases and then saturates with increasing FM thickness. In other words, it predicts that the field-like torque is not scaled with 1/(FM thickness) but is more or less constant in some ranges of FM thickness. Our observation [ Fig. 3d] is qualitatively consistent with this prediction that considers the interface SOC. Therefore, we attribute the unconventional NiFethickness dependence of field-like torque to the interface effect, which may be enhanced due to the bulk Rashba SOC of GeTe.
Damping-like torque of the GeTe/NiFe structure Based on the second-harmonic Hall resistance (R 2ω xy ), we attempted to extract the magnitude of damping-like torque in the GeTe/NiFe bilayers. The damping-like torque is expressed as the out-of-plane effective spin-orbit field (B DL ), which can be extracted from the amplitude of the cosφ term in Eq. (2). However, several thermomagnetic effects 24,29 , such as the ordinary Nernst effect, the anomalous Nernst effect, and the spin-Seebeck effect, are also present in the second-harmonic signals. The damping-like term amplitudeC DL ¼ R AHE B DL B ext þB d þ AI 0 α∇T þ NI 0 αB ext ∇T includes three terms that depend on B ext in different ways. From the independent experiments, we find an R AHE of 3.41 mΩ and a B d of 1 T (Supplementary Fig. S7). As shown in Fig. 4a, C DL fits well to Eq. (2). However, the third term of C DL , NI 0 αB ext ∇T , which is linear in B ext, dominates the other two terms [ Fig.  4b]. Even though the first (R AHE B DL =B eff ) and second ( AI 0 α∇T ) terms have different dependences on B ext , the combined magnitude of these terms is too small to reliably separate. As a result, the damping-like torque of the GeTe/NiFe bilayer is believed to be much smaller than the field-like torque. From the spin-torque ferromagnetic resonance measurement, we also confirm the weak damping-like torque (Supplementary Note 2 and Supplementary Fig. S8).

Discussion
Now, we focus on the field-like torque and its possible utilization. An interesting observation in our work is that the field-like torque is weakly dependent on the FM thickness, which is in stark contrast to the typical inverse proportionality of the SOT magnitude to the FM thickness. This suggests that the bulk SOC of GeTe is not the only source of spin current in the GeTe/NiFe bilayers. Rather, the spin current is noticeably attributed to the interfacial SOC at the GeTe/NiFe interface. The large field-like torque indicates a large interfacial SOC effect. The bulk Rashba effect of GeTe itself may have a role in the large interfacial SOC effect, which demands further theoretical studies.
Another interesting observation is that the field-like torque is large even with a thick FM layer. When considering the FM thickness, the field-like torque of the GeTe/NiFe bilayer in this study is noticeably large, and this large field-like torque suggests that the GeTe/FM bilayers can be used for scalable in-plane SOT magnetic random access memories (MRAMs). Even though the SOT is able to switch the in-plane magnetization 12,41 , inplane MRAMs suffer from inferior scalability to perpendicular MRAMs. The thermal energy barrier Δ of in-plane MRAM is given by K s V/k B T, where K s is the shape anisotropy, V is the FM-layer volume, and k B T is the thermal energy at room temperature. Figure 3f shows a contour map of the calculated Δ as a function of the in-plane length L and thickness t of the FM layer, assuming an inplane aspect ratio of 2 and a saturation magnetization of 800 kA/m. It clearly shows that Δ >50 for 10-year retention cannot be reached for a thin FM layer. One has to increase the FM thickness to meet the criteria of Δ >50. However, when the SOT is inversely proportional to the FM thickness, it costs an increased write current, resulting in high power consumption. In this respect, the large field-like torque of the GeTe/NiFe bilayers for a thick NiFe layer and its weak dependence on the NiFe thickness are attractive for scalable in-plane MRAMs.
Finally, we checked the possibility of electrically controlling the SOT of GeTe/NiFe bilayers. This is one of our original motivations of this study because GeTe is a wellknown ferroelectric material and is expected to exhibit gate-controlled electric polarization and subsequent Rashba effect modulation in a nonvolatile manner. However, we find no noticeable modulation of the fieldlike torque by gating ( Supplementary Fig. S9) due to the relatively high conductivity of GeTe and/or the inefficiency of back-gating. Therefore, we need to engineer the material itself or to develop an alternative method for electrically controlling the polarization, which is beyond the scope of this work.

Materials and methods
Material growth and characterization α-GeTe thin films have been grown on Si(111) substrates by the thermal evaporation of single elemental sources of Ge and Te. Just before loading into the growth chamber, the substrate was carefully prepared via three cleaning steps: (1) (3) in diluted HF solution for a few minutes to remove the native oxide. After loading the substrate, the growth chamber was evacuated below 5 × 10 −8 Torr, and the substrate was quickly heated and held at 270°C for 1 h. The fluxes of Ge and Te gas were separately controlled by using a thickness monitor and optimized with respect to the composition, crystallinity, and surface morphology. After the optimized flux of Ge and Te gas was stabilized, the growth started with the substrate temperature at 270°C, and the thickness of GeTe was controlled by the deposition time with an average deposition rate of 0.44 Å s −1 . The structural characteristics of the grown GeTe films were investigated by using X-ray diffraction (ATX-G, Rigaku), atomic force microscopy (AFM, XE-70, Park systems), and transmission electron microscopy (TEM, TitanTM 80-300, FEI). The composition of the GeTe films was analyzed by using Auger electron spectroscopy (AES, PHI-700, ULVAC-PHI).

Device fabrication
The device consists of a GeTe/Ni 81 Fe 19 /MgO/Ta multilayer, where a MgO (2 nm)/Ta (2 nm) layer is deposited to protect the ferromagnet layer. After the grown GeTe films underwent an ion-milling process, the ferromagnetic layer and the capping layers were deposited by DC magnetron sputtering without vacuum breaking (see Supplementary Fig. S10 for the FM characteristics). The transport channels were patterned by photolithography and Ar-ion milling. Then, contact electrodes made of Ti (5 nm)/Au (100 nm) were formed by photolithography and lift-off. The lateral dimension of the channel was 100 μm × 6 μm.

Harmonic measurement
All measurements were performed in a physical property measurement system (PPMS) at room temperature. The Hall voltage was measured by supplying AC current from a Keithley 6221 device and simultaneously monitoring the firstand second-harmonic Hall voltages by SR850 and SR860 lock-in amplifiers, respectively. All harmonic transport measurements were performed at room temperature with an alternating current at a frequency of 17 Hz.